text
stringlengths 0
3.34M
|
|---|
------------------------------------------------------------------------
-- The Agda standard library
--
-- Properties of rose trees
------------------------------------------------------------------------
{-# OPTIONS --without-K --safe --sized-types #-}
module Data.Tree.Rose.Properties where
open import Level using (Level)
open import Size
open import Data.List.Base as List using (List)
open import Data.List.Extrema.Nat
import Data.List.Properties as Listₚ
open import Data.Nat.Base using (ℕ; zero; suc)
open import Data.Tree.Rose
open import Function.Base
open import Relation.Binary.PropositionalEquality
open ≡-Reasoning
private
variable
a b c : Level
A : Set a
B : Set b
C : Set c
i : Size
------------------------------------------------------------------------
-- map properties
map-compose : (f : A → B) (g : B → C) →
map {i = i} (g ∘′ f) ≗ map {i = i} g ∘′ map {i = i} f
map-compose f g (node a ts) = cong (node (g (f a))) $ begin
List.map (map (g ∘′ f)) ts ≡⟨ Listₚ.map-cong (map-compose f g) ts ⟩
List.map (map g ∘ map f) ts ≡⟨ Listₚ.map-compose ts ⟩
List.map (map g) (List.map (map f) ts) ∎
depth-map : (f : A → B) (t : Rose A i) → depth {i = i} (map {i = i} f t) ≡ depth {i = i} t
depth-map f (node a ts) = cong (suc ∘′ max 0) $ begin
List.map depth (List.map (map f) ts) ≡˘⟨ Listₚ.map-compose ts ⟩
List.map (depth ∘′ map f) ts ≡⟨ Listₚ.map-cong (depth-map f) ts ⟩
List.map depth ts ∎
------------------------------------------------------------------------
-- foldr properties
foldr-map : (f : A → B) (n : B → List C → C) (ts : Rose A i) →
foldr {i = i} n (map {i = i} f ts) ≡ foldr {i = i} (n ∘′ f) ts
foldr-map f n (node a ts) = cong (n (f a)) $ begin
List.map (foldr n) (List.map (map f) ts) ≡˘⟨ Listₚ.map-compose ts ⟩
List.map (foldr n ∘′ map f) ts ≡⟨ Listₚ.map-cong (foldr-map f n) ts ⟩
List.map (foldr (n ∘′ f)) ts ∎
|
Formal statement is: lemma compact_insert [simp]: assumes "compact s" shows "compact (insert x s)" Informal statement is: If $s$ is a compact set, then $s \cup \{x\}$ is compact. |
## Configurações
## Função para conferir se um pacote já está instalado; se estiver, carregá-lo, se não, instalar e depois carregar
lock.and.load <- function(list.of.packages){
new.packages <- list.of.packages[!(list.of.packages %in% installed.packages()[,"Package"])]
if(length(new.packages)) {
install.packages(new.packages)
}
for(i in 1:length(list.of.packages)){
library(list.of.packages[1], character.only=T)
}
}
## Instalar e carregar pacotes necessários abaixo
lock.and.load('dplyr')
lock.and.load('tidyr')
lock.and.load('plumber')
#* Geolocalizar endereço
#* @param address endereço
#* @param code_ibge código ibge onde o endereço tem que estar localizado
#* @post /geo
function(address="", code_ibge="", libpostal="False"){
# testar que address é string
# testar que libpostal é booleano
# se code_ibge for inválido, não aplicar teste
data <- list(
address = address
, code_ibge = code_ibge
) %>%
createInput()
result <-
data %>%
{if(as.logical(libpostal)){generateAlternatives(.)} else {.}} %>%
geocode() %>%
checkIbge()
return(result)
}
|
[STATEMENT]
lemma prod_mset_prime_factorization_weak:
assumes "x \<noteq> 0"
shows "normalize (prod_mset (prime_factorization x)) = normalize x"
[PROOF STATE]
proof (prove)
goal (1 subgoal):
1. normalize (\<Prod>\<^sub># (prime_factorization x)) = normalize x
[PROOF STEP]
using assms
[PROOF STATE]
proof (prove)
using this:
x \<noteq> (0::'a)
goal (1 subgoal):
1. normalize (\<Prod>\<^sub># (prime_factorization x)) = normalize x
[PROOF STEP]
proof (induction x rule: prime_divisors_induct)
[PROOF STATE]
proof (state)
goal (3 subgoals):
1. (0::'a) \<noteq> (0::'a) \<Longrightarrow> normalize (\<Prod>\<^sub># (prime_factorization (0::'a))) = normalize (0::'a)
2. \<And>x. \<lbrakk>is_unit x; x \<noteq> (0::'a)\<rbrakk> \<Longrightarrow> normalize (\<Prod>\<^sub># (prime_factorization x)) = normalize x
3. \<And>p x. \<lbrakk>prime p; x \<noteq> (0::'a) \<Longrightarrow> normalize (\<Prod>\<^sub># (prime_factorization x)) = normalize x; p * x \<noteq> (0::'a)\<rbrakk> \<Longrightarrow> normalize (\<Prod>\<^sub># (prime_factorization (p * x))) = normalize (p * x)
[PROOF STEP]
case (factor p x)
[PROOF STATE]
proof (state)
this:
prime p
x \<noteq> (0::'a) \<Longrightarrow> normalize (\<Prod>\<^sub># (prime_factorization x)) = normalize x
p * x \<noteq> (0::'a)
goal (3 subgoals):
1. (0::'a) \<noteq> (0::'a) \<Longrightarrow> normalize (\<Prod>\<^sub># (prime_factorization (0::'a))) = normalize (0::'a)
2. \<And>x. \<lbrakk>is_unit x; x \<noteq> (0::'a)\<rbrakk> \<Longrightarrow> normalize (\<Prod>\<^sub># (prime_factorization x)) = normalize x
3. \<And>p x. \<lbrakk>prime p; x \<noteq> (0::'a) \<Longrightarrow> normalize (\<Prod>\<^sub># (prime_factorization x)) = normalize x; p * x \<noteq> (0::'a)\<rbrakk> \<Longrightarrow> normalize (\<Prod>\<^sub># (prime_factorization (p * x))) = normalize (p * x)
[PROOF STEP]
have "normalize (prod_mset (prime_factorization (p * x))) =
normalize (p * normalize (prod_mset (prime_factorization x)))"
[PROOF STATE]
proof (prove)
goal (1 subgoal):
1. normalize (\<Prod>\<^sub># (prime_factorization (p * x))) = normalize (p * normalize (\<Prod>\<^sub># (prime_factorization x)))
[PROOF STEP]
using factor.prems factor.hyps
[PROOF STATE]
proof (prove)
using this:
p * x \<noteq> (0::'a)
prime p
goal (1 subgoal):
1. normalize (\<Prod>\<^sub># (prime_factorization (p * x))) = normalize (p * normalize (\<Prod>\<^sub># (prime_factorization x)))
[PROOF STEP]
by (simp add: prime_factorization_times_prime)
[PROOF STATE]
proof (state)
this:
normalize (\<Prod>\<^sub># (prime_factorization (p * x))) = normalize (p * normalize (\<Prod>\<^sub># (prime_factorization x)))
goal (3 subgoals):
1. (0::'a) \<noteq> (0::'a) \<Longrightarrow> normalize (\<Prod>\<^sub># (prime_factorization (0::'a))) = normalize (0::'a)
2. \<And>x. \<lbrakk>is_unit x; x \<noteq> (0::'a)\<rbrakk> \<Longrightarrow> normalize (\<Prod>\<^sub># (prime_factorization x)) = normalize x
3. \<And>p x. \<lbrakk>prime p; x \<noteq> (0::'a) \<Longrightarrow> normalize (\<Prod>\<^sub># (prime_factorization x)) = normalize x; p * x \<noteq> (0::'a)\<rbrakk> \<Longrightarrow> normalize (\<Prod>\<^sub># (prime_factorization (p * x))) = normalize (p * x)
[PROOF STEP]
also
[PROOF STATE]
proof (state)
this:
normalize (\<Prod>\<^sub># (prime_factorization (p * x))) = normalize (p * normalize (\<Prod>\<^sub># (prime_factorization x)))
goal (3 subgoals):
1. (0::'a) \<noteq> (0::'a) \<Longrightarrow> normalize (\<Prod>\<^sub># (prime_factorization (0::'a))) = normalize (0::'a)
2. \<And>x. \<lbrakk>is_unit x; x \<noteq> (0::'a)\<rbrakk> \<Longrightarrow> normalize (\<Prod>\<^sub># (prime_factorization x)) = normalize x
3. \<And>p x. \<lbrakk>prime p; x \<noteq> (0::'a) \<Longrightarrow> normalize (\<Prod>\<^sub># (prime_factorization x)) = normalize x; p * x \<noteq> (0::'a)\<rbrakk> \<Longrightarrow> normalize (\<Prod>\<^sub># (prime_factorization (p * x))) = normalize (p * x)
[PROOF STEP]
have "normalize (prod_mset (prime_factorization x)) = normalize x"
[PROOF STATE]
proof (prove)
goal (1 subgoal):
1. normalize (\<Prod>\<^sub># (prime_factorization x)) = normalize x
[PROOF STEP]
by (rule factor.IH) (use factor in auto)
[PROOF STATE]
proof (state)
this:
normalize (\<Prod>\<^sub># (prime_factorization x)) = normalize x
goal (3 subgoals):
1. (0::'a) \<noteq> (0::'a) \<Longrightarrow> normalize (\<Prod>\<^sub># (prime_factorization (0::'a))) = normalize (0::'a)
2. \<And>x. \<lbrakk>is_unit x; x \<noteq> (0::'a)\<rbrakk> \<Longrightarrow> normalize (\<Prod>\<^sub># (prime_factorization x)) = normalize x
3. \<And>p x. \<lbrakk>prime p; x \<noteq> (0::'a) \<Longrightarrow> normalize (\<Prod>\<^sub># (prime_factorization x)) = normalize x; p * x \<noteq> (0::'a)\<rbrakk> \<Longrightarrow> normalize (\<Prod>\<^sub># (prime_factorization (p * x))) = normalize (p * x)
[PROOF STEP]
finally
[PROOF STATE]
proof (chain)
picking this:
normalize (\<Prod>\<^sub># (prime_factorization (p * x))) = normalize (p * normalize x)
[PROOF STEP]
show ?case
[PROOF STATE]
proof (prove)
using this:
normalize (\<Prod>\<^sub># (prime_factorization (p * x))) = normalize (p * normalize x)
goal (1 subgoal):
1. normalize (\<Prod>\<^sub># (prime_factorization (p * x))) = normalize (p * x)
[PROOF STEP]
by simp
[PROOF STATE]
proof (state)
this:
normalize (\<Prod>\<^sub># (prime_factorization (p * x))) = normalize (p * x)
goal (2 subgoals):
1. (0::'a) \<noteq> (0::'a) \<Longrightarrow> normalize (\<Prod>\<^sub># (prime_factorization (0::'a))) = normalize (0::'a)
2. \<And>x. \<lbrakk>is_unit x; x \<noteq> (0::'a)\<rbrakk> \<Longrightarrow> normalize (\<Prod>\<^sub># (prime_factorization x)) = normalize x
[PROOF STEP]
qed (auto simp: prime_factorization_unit is_unit_normalize) |
module Server.InigoServer.KVOps
import Inigo.Async.Promise
import Data.List
import Data.Maybe
import Data.SortedSet as Set
import Extra.String
import Fmt
import Inigo.Account.Account as Account
import Inigo.Async.Base
import Inigo.Async.CloudFlare.KV
import Inigo.Async.SubtleCrypto.SubtleCrypto
import Inigo.Package.Package
import Inigo.Package.PackageDeps
import Inigo.Package.PackageIndex
import SemVar.Data
import Server.InigoServer.Util
||| TODO: Should package technically be a list of strings?
||| TODO: We should be able to rebuild if any of the redundant stores becomes corrupted
sessionTTL : Int
sessionTTL = 60 * 60 * 24 * 10 -- 10 days
packageIndexKey : (String, String)
packageIndexKey = ("index", "index")
packageKey : String -> String -> Version -> (String, String)
packageKey ns package version = ("packages", fmt "%s.%s@%s" ns package (show version))
readmeKey : String -> String -> Version -> (String, String)
readmeKey ns package version = ("readme", fmt "%s.%s@%s" ns package (show version))
archiveKey : String -> String -> Version -> (String, String)
archiveKey ns package version = ("archives", fmt "%s.%s@%s" ns package (show version))
packageDepKey : String -> String -> (String, String)
packageDepKey ns package = ("deps", fmt "%s.%s" ns package)
accountKey : String -> (String, String)
accountKey ns = ("accounts", ns)
sessionKey : String -> (String, String)
sessionKey session = ("sessions", session)
pkgVerKey : Package -> (String -> String -> Version -> a) -> a
pkgVerKey pkg f =
f (ns pkg) (package pkg) (version pkg)
pkgKey : Package -> (String -> String -> a) -> a
pkgKey pkg f =
f (ns pkg) (package pkg)
read : (String, String) -> Promise (Maybe String)
read key =
do
res <- (uncurry KV.read key)
if res == ""
then pure $ Nothing
else pure $ Just res
write : (String, String) -> String -> Promise ()
write key val = (uncurry KV.write key) val
writeTTL : (String, String) -> String -> Int -> Promise ()
writeTTL key val ttl = (uncurry KV.writeTTL key) val ttl
export
readIndex : Promise PackageIndex
readIndex =
do
indexRes <- read packageIndexKey
log ("Package Index: " ++ show indexRes)
expectResult $ fromMaybe (Right []) $ map parsePackageIndex indexRes
export
writeIndex : PackageIndex -> Promise ()
writeIndex index =
write packageIndexKey (encodePackageIndex index)
export
readArchive : String -> String -> Version -> Promise (Maybe String)
readArchive packageNS packageName version =
read (archiveKey packageNS packageName version)
export
writeArchive : String -> String -> Version -> String -> Promise ()
writeArchive packageNS packageName version body =
write (archiveKey packageNS packageName version) body
export
readReadme : String -> String -> Version -> Promise (Maybe String)
readReadme packageNS packageName version =
read (readmeKey packageNS packageName version)
export
writeReadme : String -> String -> Version -> String -> Promise ()
writeReadme packageNS packageName version body =
write (readmeKey packageNS packageName version) body
-- TODO: This logic is a little complex to be here
export
readAllDeps : String -> String -> Promise (List (List String, PackageDeps))
readAllDeps packageNS packageName =
readAllDeps_ Set.empty [packageNS, packageName]
where
readDep : List String -> Promise PackageDeps
readDep [packageNS, packageName] =
do
depsRes <- read (packageDepKey packageNS packageName)
log ("Package Deps: " ++ show depsRes)
expectResult $ fromMaybe (Right []) $ map parsePackageDeps depsRes
readDep els =
reject ("Invalid package name: " ++ show els)
readAllDeps_ : SortedSet (List String) -> List String -> Promise (List (List String, PackageDeps))
readAllDeps_ curr q =
do
packageDeps <- readDep q
let allDeps = concat $ map (\(_, packageDep) => deps packageDep ++ dev packageDep) packageDeps
let subDeps = foldl (\acc, (pkgName, _) => Set.insert pkgName acc) (the (SortedSet (List String)) Set.empty) allDeps
let newDeps = difference subDeps curr
let nextSet = Set.insert q curr
res <- all $ map (readAllDeps_ (union nextSet newDeps)) (Set.toList newDeps)
pure $ (q, packageDeps) :: (concat res)
export
readDeps : Package -> Promise PackageDeps
readDeps pkg =
do
depsRes <- read (pkgKey pkg packageDepKey)
log ("Package Deps: " ++ show depsRes)
expectResult $ fromMaybe (Right []) $ map parsePackageDeps depsRes
export
writeDeps : Package -> PackageDeps -> Promise ()
writeDeps pkg packageDeps =
write (pkgKey pkg packageDepKey) (encodePackageDeps packageDeps)
export
writePackage : Package -> Promise ()
writePackage pkg =
write (pkgVerKey pkg packageKey) (encodePackage pkg)
export
readPackage : String -> String -> Version -> Promise (Either String Package)
readPackage packageNS packageName version =
do
Just contents <- read (packageKey packageNS packageName version)
| Nothing => pure $ Left "package not found"
pure $ parsePackage contents
export
readVersions : String -> String -> Promise (List Version)
readVersions packageNS packageName =
do
depsRes <- read (packageDepKey packageNS packageName)
deps <- expectResult $ fromMaybe (Right []) $ map parsePackageDeps depsRes
pure $ map fst deps
export
latestVersion : String -> String -> Promise (Maybe Version)
latestVersion packageNS packageName =
do
versions <- readVersions packageNS packageName
pure $ head' $ reverse $ sort $ versions --'
-- Note: there could be race conditions, and since this
-- is EV, it would be difficult to prevent overall
-- TODO: Validate account details on server-side
export
newAccount : Account -> Promise (Maybe (Int, String))
newAccount account =
do
Nothing <- read (accountKey (ns account))
| Just _ => pure $ Just (400, "account already exists")
write (accountKey (ns account)) (encode account)
pure Nothing
export
readAccountHash : String -> Promise (Maybe (Algorithm, String))
readAccountHash ns =
do
Just accountToml <- read (accountKey ns)
| Nothing => pure Nothing
let Just account = Account.decode accountToml
| Nothing => pure Nothing
pure (Just $ (kdf account, hash account))
export
readSession : String -> Promise (Maybe String)
readSession session =
read (sessionKey session)
||| TODO: write session should have an expiration
export
writeSession : String -> String -> Promise ()
writeSession session ns =
writeTTL (sessionKey session) ns sessionTTL
|
= = = Manifestations and combinations = = =
|
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% Medium Length Graduate Curriculum Vitae
% LaTeX Template
% Version 1.1 (9/12/12)
%
% This template has been downloaded from:
% http://www.LaTeXTemplates.com
%
% Original author:
% Rensselaer Polytechnic Institute (http://www.rpi.edu/dept/arc/training/latex/resumes/)
%
% Important note:
% This template requires the res.cls file to be in the same directory as the
% .tex file. The res.cls file provides the resume style used for structuring the
% document.
%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%----------------------------------------------------------------------------------------
% PACKAGES AND OTHER DOCUMENT CONFIGURATIONS
%----------------------------------------------------------------------------------------
\documentclass[margin, 10pt]{res} % Use the res.cls style, the font size can be changed to 11pt or 12pt here
\usepackage{fontawesome}
\usepackage{helvet} % Default font is the helvetica postscript font
\usepackage[shortlabels]{enumitem}
%\usepackage{newcent} % To change the default font to the new century schoolbook postscript font uncomment this line and comment the one above
\usepackage{url}
\usepackage{color,hyperref}
\definecolor{darkblue}{rgb}{0.0,0.0,0.55}
\hypersetup{colorlinks,breaklinks,
linkcolor=darkblue,urlcolor=darkblue,
anchorcolor=darkblue,citecolor=darkblue}
\setlength{\textwidth}{6in} % Text width of the document
\newlist{innerlist}{itemize}{3}
\setlist[innerlist]{label=\enskip\textbullet,leftmargin=*,parsep=0pt,itemsep=0pt,topsep=0pt,partopsep=0pt}
\begin{document}
%----------------------------------------------------------------------------------------
% NAME AND ADDRESS SECTION
%----------------------------------------------------------------------------------------
\moveleft0.7\hoffset\centerline{\Huge\bf Ali Gholami} % Your name at the top
\moveleft0.7\hoffset\vbox{\hrule width 6in height 1.5pt}\smallskip % Horizontal line after name; adjust line thickness by changing the '1pt'
\moveleft0.7\hoffset\centerline{\textsc{Computer Engineering \& Information Technology Department}} % Your address
\moveleft0.7\hoffset\centerline{\textsc{Amirkabir University of Technology}}
\moveleft0.7\hoffset\centerline{\url{https://aligholamee.github.io}}
\moveleft0.7\hoffset\centerline{\url{[email protected]}}
\vspace{0.1cm}
\moveleft0.7\hoffset\centerline{\href{https://stackoverflow.com/users/4723938/aligholamee}{\UrlFont\color{darkblue}\small{[\ \faCodeFork}\ ]}\href{https://www.linkedin.com/in/ali-gholami-987136b5/}{\UrlFont\color{darkblue}\small{[\ \faLinkedin}\ ]}\href{https://github.com/aligholamee}{\UrlFont\color{darkblue}\small{[\ \faGithubAlt}\ ]}}
\moveleft0.7\hoffset\centerline{\tiny\textsc{\underline{Updated on July 4, 2018}}}
%----------------------------------------------------------------------------------------
\begin{resume}
%----------------------------------------------------------------------------------------
% EDUCATION SECTION
%----------------------------------------------------------------------------------------
\section{EDUCATION}
\textbf{B.S. Computer Engineering} \textsc{@}
\href{http://aut.ac.ir/aut/}{\textsc{Amirkabir University of Technology}}\\
{\UrlFont[ 1'st Industrial University in Iran ]}\hfill {\UrlFont{\underline{GPA: 3.6/4}}}\\
\vspace{0.1cm}
\textbf{Mathematics \& Physics Diploma} \textsc{@}
\href{http://www.kamal.sch.ir/}{\textsc{Kamal Highschool}}\hfill {\UrlFont{\underline{GPA: 19/20}}}
\section{RESEARCH \\ INTERESTS}
\vspace{0.6cm}
\begin{innerlist}
\begin{minipage}{0.4\linewidth}
\item Visual Question Answering
\item Image Captioning
\item Image Segmentation
\end{minipage}
\begin{minipage}{0.4\linewidth}
\item High Performance Computing
\item Heterogeneous Programming
\item Applied Deep Learning
\end{minipage}
\end{innerlist}
\vspace{0.2cm}
\section{RELATED \\ COURSES}
\textbf{Machine Learning} \textsc{@} \textsc{Amirkabir University of Technology}\\
\textbf{Computer Vision} \textsc{@} \textsc{Udacity}\\
\textbf{Deep Learning} \textsc{@} \textsc{Udacity}\\
\textbf{cs231n} \textsc{@} \textsc{Stanford University}\\
\textbf{OpenMP} \textsc{@} \textsc{Intel}\\
%----------------------------------------------------------------------------------------
% Technology SKILLS SECTION
%----------------------------------------------------------------------------------------
\section{RESEARCH \\ EXPERIENCE}
\textbf{Machine Learning Lab} \textsc{@} \textsc{Amirkabir University of Technology}
\hfill {July 2018 -- Present}\\
\textit{Computer Vision | Pattern Recognition}
\vspace{0.15cm}
\begin{innerlist}
\item Generative Adversarial Models for Style Transfer in Images
\item Analysis of Formal Representation of Deep Learning Models
\item One-shot Imitation Learning
\end{innerlist}
\textbf{CEIT} \textsc{@} \textsc{Amirkabir University of Technology}
\hfill {Dec 2017 -- Present}\\
\textit{Computer Vision | Pattern Recognition}
\vspace{0.15cm}
\begin{innerlist}
\item Implementation of \textit{AlexNet CNN} architecture using \textit{Tensorflow}
\href{https://github.com/aligholamee/AlexNet-Tensorflow}{\hfill{\UrlFont[code]}}
\item Implementation of a \textit{DCGAN} to draw \textit{MNIST} using \textit{Tensorflow}
\href{https://github.com/aligholamee/MNIST-Drawer-DCGAN}{\hfill{\UrlFont[code]}}
\item Implementation of a \textit{Variational Autoencoder} using \textit{Tensorflow}
\href{https://github.com/aligholamee/MNIST-Drawer-VAE}{\hfill{\UrlFont[code]}}
\item Implementation of various \textit{Deep Learning} techniques using \textit{Tensorflow}
\href{https://github.com/aligholamee/notMNIST}{\hfill{\UrlFont[code]}}
\end{innerlist}
%----------------------------------------------------------------------------------------
% PAPERS AND TECHNICAL REPORTS SECTION
%----------------------------------------------------------------------------------------
\section{TECHNICAL REPORTS}
\textbf{Statistical Pattern Recognition}\\
\textit{Advisor: Prof. Mohammad Rahmati}
\begin{innerlist}
\item \textit{Introduction to Linear Algebra -- Statistics -- Probabilities} \href{https://github.com/aligholamee/Patterns/raw/master/docs/assignment-1/SPR_9531504_HW1.pdf}{\hfill\UrlFont[docs]}
\item \textit{Bayesian Decision Boundaries -- Bayes Error Bounds -- Risk Minimization} \href{https://github.com/aligholamee/Patterns/raw/master/docs/assignment-2/SPR_9531504_HW2.pdf}{\hfill\UrlFont[docs]}
\item \textit{Maximum Likelihood \& Bayesian Parameter Estimation} \href{https://github.com/aligholamee/Patterns/raw/master/docs/assignment-3/SPR_9531504_HW3.pdf}{\hfill\UrlFont[docs]}
\item \textit{Kernel Density Estimation -- KNN Density Estimation} \href{https://github.com/aligholamee/Patterns/raw/master/docs/assignment-4/SPR_9531504_HW4.pdf}{\hfill\UrlFont[docs]}
\item \textit{PCA -- FLDA -- Feature Subset Selection} \href{https://github.com/aligholamee/Patterns/raw/master/docs/assignment-5/SPR_9531504_HW5.pdf}{\hfill\UrlFont[docs]}
\item \textit{LDF -- SVM -- Clustering} \href{https://github.com/aligholamee/Patterns/raw/master/docs/assignment-6/SPR_9531504_HW6.pdf}{\hfill\UrlFont[docs]}
\end{innerlist}
\textbf{Foundations of Data Mining}\\
\textit{Advisor: Prof. Ehsan Nazerfard}
\begin{innerlist}
\item \textit{Association Rule Mining -- Feature Engineering} \href{https://github.com/aligholamee/Datadigger/raw/master/docs/assignment-1/report/DM_9531504_HW1.pdf}{\hfill\UrlFont[docs]}
\item \textit{Decision Tree Classifier -- Data Cleaning} \href{https://github.com/aligholamee/Datadigger/raw/master/docs/assignment-2/report/DM_9531504_HW2.pdf}{\hfill\UrlFont[docs]}
\item \textit{Naive Bayes Classifier -- Spam Filtering -- Text Processing} \href{https://github.com/aligholamee/Datadigger/raw/master/docs/assignment-3/report/DM_9531504_HW3.pdf}{\hfill\UrlFont[docs]}
\item \textit{Data Preprocessing \& Model Building with \textit{RapidMiner}} \href{https://github.com/aligholamee/Datadigger/raw/master/docs/assignment-4/report/DM_9531504_HW4.pdf}{\hfill\UrlFont[docs]}
\end{innerlist}
\textbf{Design \& Implementation of Programming Languages}\\
\textit{Advisor: Prof. Mehran S. Fallah}
\begin{innerlist}
\item \textit{Induction \& Denotational Semantics -- Lambda Calculus} \href{https://github.com/aligholamee/HALFLIFE/raw/master/reports/PL29531504.pdf}{\hfill\UrlFont[docs]}
\item \textit{Lisp \& Garbage Collection -- Higher-order Functions} \href{https://github.com/aligholamee/HALFLIFE/raw/master/reports/PL39531504.pdf}{\hfill\UrlFont[docs]}
\item \textit{Algol \& Meta Language -- ML Data Types \& Patterns} \href{https://github.com/aligholamee/HALFLIFE/raw/master/reports/PL49531504.pdf}{\hfill\UrlFont[docs]}
\item \textit{Type Safety \& Type Inference -- Polymorphism} \href{https://github.com/aligholamee/HALFLIFE/raw/master/reports/PL49531504.pdf}{\hfill\UrlFont[docs]}
\end{innerlist}
\textbf{Computer Networks}\\
\textit{Advisor: Prof. Siavash Khorsandi}
\begin{innerlist}
\item \textit{Introduction to Computer Networks} \href{https://github.com/aligholamee/Compnets/raw/master/docs/assignment-1/compnet_assignment_1_9531504.pdf}{\hfill\UrlFont[docs]}
\item \textit{Packet Transmission Approaches -- Congestion Control} \href{https://github.com/aligholamee/Compnets/raw/master/docs/assignment-2/compnet_assignment_2_9531504.pdf}{\hfill\UrlFont[docs]}
\item \textit{Queuing \& Delay Analysis \& Network Protocol Stack 1} \href{https://github.com/aligholamee/Compnets/raw/master/docs/assignment-3/compnet_assignment_3_9531504.pdf}{\hfill\UrlFont[docs]}
\item \textit{Queuing \& Delay Analysis \& Network Protocol Stack 2} \href{https://github.com/aligholamee/Compnets/raw/master/docs/assignment-4/compnet_assignment_4_9531504.pdf}{\hfill\UrlFont[docs]}
\item \textit{Multiplexing \& Multiple Access Medium -- IETF} \href{https://github.com/aligholamee/Compnets/raw/master/docs/assignment-5/compnet_assignment_5_9531504.pdf}{\hfill\UrlFont[docs]}
\item \textit{Application Layer Protocols; HTTP -- FTP -- SMTP -- CDN -- DNS} \href{https://github.com/aligholamee/Compnets/raw/master/docs/assignment-6/compnet_assignment_6_9531504.pdf}{\hfill\UrlFont[docs]}
\item \textit{Reliable Data Transfer; Stop \& Wait Protocol} \href{https://github.com/aligholamee/Compnets/raw/master/docs/assignment-8/compnet_assignment_8_9531504.pdf}{\hfill\UrlFont[docs]}
\item \textit{Reliable Data Transfer; Go-Back-N \& Selective Retransmit Protocols} \href{https://github.com/aligholamee/Compnets/raw/master/docs/assignment-9/compnet_assignment_9_9531504.pdf}{\hfill\UrlFont[docs]}
\end{innerlist}
\textbf{Multi-core Programming}\\
\textit{Advisor: Prof. Mahmoud Momtazpour}
\begin{innerlist}
\item \textit{Parallel Architectures -- Speedup Metrics} \href{https://github.com/aligholamee/Parallax/raw/master/docs/assignment-1/MCP_9531504_HW1.pdf}{\hfill\UrlFont[docs]}
\item \textit{OpenMP -- Parallelization of Matrix Computations} \href{https://github.com/aligholamee/Parallax/raw/master/docs/assignment-2/MCP_9531504_HW2.pdf}{\hfill\UrlFont[docs]}
\item \textit{OpenMP -- Parallelization of Sort Algorithms} \href{https://github.com/aligholamee/Parallax/raw/master/docs/assignment-3/MCP_9531504_HW3.pdf}{\hfill\UrlFont[docs]}
\item \textit{Nvidia GPUs Architecture -- Memory Types -- GPU Characteristics} \href{https://github.com/aligholamee/Parallax/raw/master/docs/assignment-4/MCP_9531504_HW4.pdf}{\hfill\UrlFont[docs]}
\item \textit{Parallel Scan -- Reduction Strategies on CUDA} \href{https://github.com/aligholamee/Parallax/raw/master/docs/assignment-5/MCP_9531504_HW5.pdf}{\hfill\UrlFont[docs]}
\item \textit{Reduction Tuning -- Bank Conflicts -- Loop Unrolling} \href{https://github.com/aligholamee/Parallax/raw/master/docs/assignment-6/MCP_9531504_HW6.pdf}{\hfill\UrlFont[docs]}
\end{innerlist}
\textbf{Engineering Ethics}\\
\textit{Advisor: Prof. Ali Dizani}
\begin{innerlist}
\item \textit{A Deep Analysis of Ethical Dilemmas in Computer Engineering.} \href{https://www.dropbox.com/s/hysshvi811nqwm4/Release_1_0_0.pdf?dl=0}{\hfill\UrlFont[docs]}
\end{innerlist}
%----------------------------------------------------------------------------------------
% WORK EXPERIENCE
%----------------------------------------------------------------------------------------
\section{WORK \\ EXPERIENCE}
\textbf{Internship} \textsc{@}
\href{https://www.arvancloud.com/}{\textsc{Arvan Cloud}}
\hfill {Jun -- Sep 2017}\\
\textit{Web Application Development}
\begin{innerlist}
\item \textit{HTML, CSS, PHP, Laravel, Javascript, ECMAScript, Node.js, Vue.js, React.js}
\end{innerlist}
\textbf{Internship} \textsc{@}
\href{http://www.fandogh.org/}{\textsc{Fandogh}}
\hfill {Jun -- Aug 2017}\\
\textit{Mobile Application Development}
\begin{innerlist}
\item \textit{Java, React Native}
\end{innerlist}
%----------------------------------------------------------------------------------------
% TEACHING EXPERIENCE
%----------------------------------------------------------------------------------------
\section{TEACHING \\ EXPERIENCE}
\textbf{T.A.} \textsc{@}
\textsc{CEIT} \textsc{@} \textsc{Amirkabir University of Technology}
\hfill {Sep -- Dec 2017}\\
\textit{Microprocessors \& Assembly Programming}\\
\textit{Advisor: Prof. Mahdi Homayounpour}
\href{https://github.com/aligholamee/Microprocessors}{\hfill\UrlFont[Resources]}
\textbf{T.A.} \textsc{@}
\textsc{ENG} \textsc{@} \textsc{Kharazmi University of Tehran}
\hfill {Sep -- Dec 2015}\\
\textit{Foundations of Programming in C++}\\
\textit{Advisor: Dr. Azadeh Mansouri}
\href{https://github.com/aligholamee/Foundations-of-Programming}{\hfill\UrlFont[Resources]}
%----------------------------------------------------------------------------------------
% HONORS
%----------------------------------------------------------------------------------------
\section{HONORS}
\textbf{Ranked top 3} among most active \textbf{GitHub} developers in Iran. \hfill {June 2018}
\textbf{Admitted} to
\textbf{Amirkabir University of Technology}
among all\hfill {Aug 2018}\\ bachelor students at Computer Engineering
Department,\\ Kharazmi University of Tehran.
\textbf{Member of Executive Team}
at the 17'th \href{https://icpc.baylor.edu/regionals/finder/tehran-2017}{International Collegiate} \hfill {Nov 2017}\\ \href{https://icpc.baylor.edu/regionals/finder/tehran-2017}{Programming Contest} held at the \href{http://ceit.aut.ac.ir}{Amirkabir University of Technology}.
\textbf{Participated}
in the 4'th national programming contest contest held \hfill {Sep 2017}\\at the \href{http://sharif.ir}{Sharif University of Technology} as a member of \textit{Morph} team.
\textbf{Ranked top 3}
among all bachelor students at Computer Engieering \hfill {July 2016}\\ Department, \href{http://khu.ac.ir}{Kharazmi University of Tehran}.
\textbf{Participated}
in the \href{http://avatech.ir}{Avatech}'s Educational Startup Weekend held \hfill {Jun 2015}\\at the \href{http://ut.ac.ir}{University of Tehran}, as a member of \textit{3-mim} team.
\textbf{Ranked top 0.006}
in the Nationwide University Entrance Exam \hfill {July 2014}\\ among all students in
Mathemathics and physics (approximately 250,000).
\textbf{ٍElected}
as the \textbf{tidiest} student at the campus of international \hfill {Aug 2011}\\ summer school, \textit{\href{https://www.monterosa.ch/}{Institute Monte Rosa}}, Montreux, Switzerland.
\end{resume}
%----------------------------------------------------------------------------------------
% TALKS
%----------------------------------------------------------------------------------------
\section{TALKS}
\textbf{Automatic Image Captioning with Attention Mechanism}
\hfill {June 2018}
\vspace{0.15cm}
\begin{innerlist}
\item Based on the paper \textit{Show and tell: A neural image caption generator}\\ by \href{http://www.jmlr.org/proceedings/papers/v37/xuc15.pdf}{Vinyals, O., Toshev, A., Bengio, S. and Erhan, D}.
\end{innerlist}
\textbf{Visual Question Answering with CNNs and RNNs}
\hfill {May 2018}
\vspace{0.15cm}
\begin{innerlist}
\item Based on the paper \textit{Visual question answering: Datasets, algorithms,\\ and future challenges} by \href{https://www.sciencedirect.com/science/article/pii/S1077314217301170}{K. Kafle and C. Kanan}.
\end{innerlist}
\textbf{Machine Learning at Scale}
\hfill {Oct 2017}
\vspace{0.15cm}
\begin{innerlist}
\item Based on the paper \textit{Rules of Machine Learning} by \href{http://martin.zinkevich.org/rules_of_ml/rules_of_ml.pdf}{Dr. Martin Zinkevich}.
\end{innerlist}
\textbf{Energy Awareness}
\hfill {July 2017}
\vspace{0.15cm}
\begin{innerlist}
\item Based on the paper \textit{Energy-aware adaptation for mobile applications} by \href{http://www-cgi.cs.cmu.edu/afs/cs.cmu.edu/Web/People/odyssey/docdir/s17.pdf5}{Dr. Jason Flinn}.
\end{innerlist}
\textbf{Metasploit Framework}
\hfill {May 2017}
\vspace{0.15cm}
\begin{innerlist}
\item Introduction to \textit{Metasploit Framework} \& \textit{Social Engineering} techniques.
\end{innerlist}
%----------------------------------------------------------------------------------------
% COMPETENCES
%----------------------------------------------------------------------------------------
\section{SKILLS}
\textbf{Languages }
Persian (\emph{native}), English (\emph{advanced working proficiency})
\textbf{Data Science}
\textit{Python, Scikit-learn, Numpy, Pandas, Matplotlib, RapidMiner, Weka.}
\textbf{Computer Vision \& Deep Learning}
\textit{Tensorflow, OpenCV.}
\textbf{Parallel Processing Frameworks}
\textit{C/C++ @ OpenMP, Intel VTune Amplifier, Intel Inspector, C/C++ @ CUDA, Nsight Monitor.}
\textbf{Functional Programming}
\textit{Racket, ML, Scheme.}
\textbf{Hardware Design}
\textit{VHDL, Verilog, HLS, AVR, ARM, Arduino, Xillinx Vivado, ModelSim, Atmel Studio, Proteus, Cadence PSpice, Keil.}
\textbf{Networking}
\textit{GNS3, Wireshark, Packet Tracer, Boson NetSim.}
\textbf{Mobile Application Development}
\textit{Java, React Native, Android Studio.}
\textbf{Web Application Development}
\textit{HTML/CSS, Javascript, Node.js, React, PHP, Laravel.}
\textbf{Databases}
\textit{PostgreSQL, MySQL, MongoDB.}
%----------------------------------------------------------------------------------------
% PROJECTS
%----------------------------------------------------------------------------------------
\section{NOTABLE \\ PROJECTS}
\textbf{KDEPlot},
\textit{Density Estimation}
\begin{innerlist}
\item Implementation of 1-D and 2-D Kernel Density Estimation Methods in Python using Numpy and Matplotlib Only.
\hfill{\UrlFont\href{https://github.com/aligholamee/KDEPlot}{[code]}}
\end{innerlist}
\textbf{KNNCC},
\textit{Image Classification}
\begin{innerlist}
\item Implementation of a K-Nearest Neighbour \textit{CIFAR-10} Classifier in Python.
\hfill{\UrlFont\href{https://github.com/aligholamee/KNNCC}{[code]}}
\end{innerlist}
\textbf{Paralab},
\textit{Parallel Processing}
\begin{innerlist}
\item Implementation of frequent parallel problems using OpenMP and Intel Parallel Studio in C.\\ \phantom{km}
\hfill{\UrlFont\href{https://github.com/aligholamee/ParaLab/tree/master}{[code]}}
\end{innerlist}
\textbf{Annealing},
\textit{Data Cleaning \& Preprocessing}
\begin{innerlist}
\item Preprocessing and training the dataset of annealing. Reached 98\% accuracy with a decision tree classifier.\\ \phantom{km}
\hfill{\UrlFont{\href{https://github.com/aligholamee/Datadigger/raw/master/docs/assignment-2/report/DM_9531504_HW2.pdf}{[docs]}\href{https://github.com/aligholamee/Datadigger/tree/master/docs/assignment-2/src}{[code]}}}
\end{innerlist}
\textbf{Titanic},
\textit{Data Science \& Feature Engineering}
\begin{innerlist}
\item Prediction of Titanic survivals as a part of Kaggle competition. Reached an\\ Accuracy of 83\% and Recall of 76\%. \href{https://github.com/aligholamee/Titanic}{\hfill{\UrlFont[code]}}
\end{innerlist}
\textbf{MNIST-Drawer},
\textit{Variational Autoencoder}
\begin{innerlist}
\item Implementation of a \textit{Variational Autoencoder} to draw \textit{MNIST} dataset characters\\ using \textit{Tensorflow}. \href{https://github.com/aligholamee/MNIST-Drawer}{\hfill{\UrlFont[code]}}
\end{innerlist}
\textbf{notMNIST},
{\textit{Convolutional Neural Network}
\begin{innerlist}
\item Implementation of multiple machine learning classifiers and regularization techniques\\ on the \textit{notMNIST} dataset using \textit{Tensorflow}. \href{https://github.com/aligholamee/notMNIST}{\hfill{\UrlFont[code]}}
\end{innerlist}
\textbf{Freeman},
{\textit{Hardware Programming \& Co-design}
\begin{innerlist}
\item Implementation of a \textit{Parking Controller} \& \textit{Security Controller} using \textit{VHDL}.\\ \ \href{https://github.com/aligholamee/Freeman}{\hfill{\UrlFont[code]}}
\end{innerlist}
\textbf{Numex},
\textit{Functional Programming}
\begin{innerlist}
\item Implementation of an \textit{Advanced Functional Interpreter} using \textit{Racket}. \href{https://github.com/aligholamee/NUMEX}{\hfill{\UrlFont[code]}}
\end{innerlist}
\textbf{Hornburg},
\textit{Deep Learning Basics}
\begin{innerlist}
\item Implementation of \textit{Principal Machine Learning Algorithms} using \textit{Python}. \href{https://github.com/aligholamee/Hornburg}{\hfill{\UrlFont[code]}}
\end{innerlist}
\textbf{Iris},
\textit{Multi-nomial classification}
\begin{innerlist}
\item Multi-nomial classification of \textit{Iris} dataset using \textit{scikit-learn}. \href{https://github.com/aligholamee/IRIS}{\hfill{\UrlFont[code]}}
\end{innerlist}
\textbf{ARMHE},
\textit{Advanced RISC Machine Programming}
\begin{innerlist}
\item Implementation of the \textit{Histogram Equalization} algorithm on the \textit{STMF32F407VGT6}\\ with \textit{ARMv4T} architecture using \textit{ARM Assembly}. \href{https://github.com/aligholamee/ARMHE}{\hfill{\UrlFont[code]}}
\end{innerlist}
\textbf{Cinder},
{\textit{Low Level Programming}
\begin{innerlist}
\item Implementation of a basic \textit{Operating System} with \textit{C}. \href{https://github.com/aligholamee/CinderOS}{\hfill{\UrlFont[code]}}
\end{innerlist}
\textbf{Chronicle},
{\textit{Compiler Design}
\begin{innerlist}
\item Implementation of a \textit{lexical analyzer}, \textit{syntax analyzer} and a \textit{partial code generator} using \textit{Lex \& Yacc} in \textit{Java}. \href{https://github.com/aligholamee/Chronicle}{\hfill{\UrlFont[code]}}
\end{innerlist}
\textbf{Sockets},
\textit{Socket Programming}
\begin{innerlist}
\item Implementation of various types of \textit{Sockets} in \textit{Interprocess Communication} \& \textit{TCP/IP\\ Protocol} with \textit{C}. \href{https://github.com/aligholamee/Socket-Programming-Package}{\hfill{\UrlFont[code]}}
\end{innerlist}
\textbf{Toofan},
\textit{Android Application Development}
\begin{innerlist}
\item Implementation of a \textit{Weather Forecast Application} on the \textit{Android} platform using\\ \textit{Java} \& \textit{Android Studio}. \href{https://github.com/aligholamee/Toofan}{\hfill{\UrlFont[code]}}
\end{innerlist}
\textbf{Huffman},
\textit{Huffman Coding}
\begin{innerlist}
\item Implementation of the \textit{Huffman Text Compression Algorithm} using \textit{Java}. \href{https://github.com/aligholamee/Java-Huffman-Zipper}{\hfill{\UrlFont[code]}}
\end{innerlist}
\textbf{2048},
\textit{C++ Programming}
\begin{innerlist}
\item Implementation of the \textit{2048 Puzzle Game} with various gameplay tweaks using \textit{C++}. \href{https://github.com/aligholamee/2048-Console-Game}{\hfill{\UrlFont[code]}}
\end{innerlist}
\textbf{Manobase},
\textit{VHDL Programming}
\begin{innerlist}
\item Implementation of the \textit{Morris Mano's Base Computer} using \textit{VHDL}. \href{https://github.com/aligholamee/Mano-Basic-Computer-Design}{\hfill{\UrlFont[code]}}
\end{innerlist}
\end{document} |
Formal statement is: lemma supp_sum_empty[simp]: "supp_sum f {} = 0" Informal statement is: The support of the sum of the empty set is zero. |
function read_bv
% read_bv - read data from an eeg-File
%
% SYNOPSIS
% data = read_bv(file, HDR, OPT);
%
% ARGUMENTS
% file - Name of EEG file (.eeg) is appended)
% HDR - Information about the file (read from the *.vhdr header file)
% .fs - Sampling rate
% .nChans - Number of channels
% .nPoints - Number of data points in the file (optional)
% .scale - Scaling factors for each channel
% .endian - Byte ordering: 'l' little or 'b' big
% OPT - Struct with following fields
% .chanidx - Indices of the channels that are to be read
% .fs - Down sample to this sampling rate
% .filt_b - Filter coefficients of IIR filter
% applied to raw data (b part)
% (optional)
% .filt_a - Filter coefficients of IIR filter
% applied to raw data (a part)
% (optional)
% .filt_subsample - Filter coefficients of FIR filter
% used for sub sampling (optional)
% .data - A matrix where the data is stored
% (optional)
% .dataPos - The position in the matrix
% [dataStart dataEnd fileStart
% fileEnd](optional)
%
% The filter parts of the OPT structure are optional fields.
% The default for the filt_subsample is a filter which takes the last
% value of filtered block e.g. [0 ... 0 1]
%
% With opt.data and opt.dataPos read_bv can write directly to a
% matrix. dataPos is an optional value for opt.data where you can set
% the position of the read data. dataStart is the position in data
% where the first read datasample is stored.
%
% Please note, that the fields chanidx and dataPos used as c indices
% starting at 0.
% RETURNS
% data: [nChans, len] the actual data
%
% DESCRIPTION
% Open a file and read the eeg data. The data is filtered with an IIR
% filter and an FIR filter.
%
% For the eeg file we assume that it was written with the following
% settings: DataFormat = BINARY
% DataOrientation = MULTIPLEXED
% BinaryFormat = INT_16
%
% COMPILE WITH
% mex read_bv.c
%
% AUTHOR
% Max Sagebaum
%
% 2008/04/15 - Max Sagebaum
% - file created
% (c) 2005 Fraunhofer FIRST |
Formal statement is: lemma pdivmod_via_pseudo_divmod: "(f div g, f mod g) = (if g = 0 then (0, f) else let ilc = inverse (coeff g (degree g)); h = smult ilc g; (q,r) = pseudo_divmod f h in (smult ilc q, r))" (is "?l = ?r") Informal statement is: The quotient and remainder of the division of two polynomials $f$ and $g$ can be computed by dividing $f$ by $g$ using the pseudo-division algorithm. |
module Limit where
-- Statement that the limit of the function f at point l exists (and its value is L)
-- This is expressed by converting the standard (ε,δ)-limit definition to Skolem normal form (TODO: ...I think? Is this correct?
record Lim (f : ℝ → ℝ) (p : ℝ) (L : ℝ) : Stmt where
field
δ : ℝ₊ → ℝ₊ -- The delta function that is able to depend on epsilon
satisfaction : ∀{ε : ℝ} → ⦃ ε > 𝟎 ⦄ → ∀{x : ℝ} → (𝟎 < ‖ x − p ‖ < δ(ε)) → (‖ f(x) − L ‖ < ε)
-- Limit value function f (if the limit exists)
lim : (f : ℝ → ℝ) → (p : ℝ) → ⦃ _ : ∃(Lim f(p)) ⦄ → ℝ
lim _ _ ⦃ l ⦄ = Lim.L(l)
module Proofs where
postulate [+]-limit : ∀{f g p} → ⦃ _ : ∃(Lim f(p)) ⦄ → ⦃ _ : ∃(Lim g(p)) ⦄ → Lim(x ↦ f(x) + g(x))(p)
postulate [−]-limit : ∀{f g p} → ⦃ _ : ∃(Lim f(p)) ⦄ → ⦃ _ : ∃(Lim g(p)) ⦄ → Lim(x ↦ f(x) − g(x))(p)
postulate [⋅]-limit : ∀{f g p} → ⦃ _ : ∃(Lim f(p)) ⦄ → ⦃ _ : ∃(Lim g(p)) ⦄ → Lim(x ↦ f(x) ⋅ g(x))(p)
postulate [/]-limit : ∀{f g p} → ⦃ _ : ∃(Lim f(p)) ⦄ → ⦃ _ : ∃(Lim g(p)) ⦄ → Lim(x ↦ f(x) / g(x))(p)
postulate [+]-lim : ∀{f g p} → ⦃ _ : ∃(Lim f(p)) ⦄ → ⦃ _ : ∃(Lim g(p)) ⦄ → (lim(x ↦ f(x) + g(x))(p) ≡ lim f(p) + lim g(p))
postulate [−]-lim : ∀{f g p} → ⦃ _ : ∃(Lim f(p)) ⦄ → ⦃ _ : ∃(Lim g(p)) ⦄ → (lim(x ↦ f(x) − g(x))(p) ≡ lim f(p) − lim g(p))
postulate [⋅]-lim : ∀{f g p} → ⦃ _ : ∃(Lim f(p)) ⦄ → ⦃ _ : ∃(Lim g(p)) ⦄ → (lim(x ↦ f(x) ⋅ g(x))(p) ≡ lim f(p) ⋅ lim g(p))
postulate [/]-lim : ∀{f g p} → ⦃ _ : ∃(Lim f(p)) ⦄ → ⦃ _ : ∃(Lim g(p)) ⦄ → (lim(x ↦ f(x) / g(x))(p) ≡ lim f(p) / lim g(p))
|
\PassOptionsToPackage{unicode=true}{hyperref} % options for packages loaded elsewhere
\PassOptionsToPackage{hyphens}{url}
\PassOptionsToPackage{dvipsnames,svgnames*,x11names*}{xcolor}
%
\documentclass[ignorenonframetext,aspectratio=169]{beamer}
\usepackage{pgfpages}
\setbeamertemplate{caption}[numbered]
\setbeamertemplate{caption label separator}{: }
\setbeamercolor{caption name}{fg=normal text.fg}
\beamertemplatenavigationsymbolsempty
% Prevent slide breaks in the middle of a paragraph:
\widowpenalties 1 10000
\raggedbottom
\setbeamertemplate{part page}{
\centering
\begin{beamercolorbox}[sep=16pt,center]{part title}
\usebeamerfont{part title}\insertpart\par
\end{beamercolorbox}
}
\setbeamertemplate{section page}{
\centering
\begin{beamercolorbox}[sep=12pt,center]{part title}
\usebeamerfont{section title}\insertsection\par
\end{beamercolorbox}
}
\setbeamertemplate{subsection page}{
\centering
\begin{beamercolorbox}[sep=8pt,center]{part title}
\usebeamerfont{subsection title}\insertsubsection\par
\end{beamercolorbox}
}
\AtBeginPart{
\frame{\partpage}
}
\AtBeginSection{
\ifbibliography
\else
\frame{\sectionpage}
\fi
}
\AtBeginSubsection{
\frame{\subsectionpage}
}
\usepackage{lmodern}
\usepackage{amssymb,amsmath}
\usepackage{ifxetex,ifluatex}
\usepackage{fixltx2e} % provides \textsubscript
\ifnum 0\ifxetex 1\fi\ifluatex 1\fi=0 % if pdftex
\usepackage[T1]{fontenc}
\usepackage[utf8]{inputenc}
\usepackage{textcomp} % provides euro and other symbols
\else % if luatex or xelatex
\usepackage{unicode-math}
\defaultfontfeatures{Ligatures=TeX,Scale=MatchLowercase}
\fi
\usetheme[]{Frankfurt}
\usecolortheme{beaver}
\usefonttheme{structuresmallcapsserif}
% use upquote if available, for straight quotes in verbatim environments
\IfFileExists{upquote.sty}{\usepackage{upquote}}{}
% use microtype if available
\IfFileExists{microtype.sty}{%
\usepackage[]{microtype}
\UseMicrotypeSet[protrusion]{basicmath} % disable protrusion for tt fonts
}{}
\IfFileExists{parskip.sty}{%
\usepackage{parskip}
}{% else
\setlength{\parindent}{0pt}
\setlength{\parskip}{6pt plus 2pt minus 1pt}
}
\usepackage{xcolor}
\usepackage{hyperref}
\hypersetup{
pdftitle={Overview of Biotechnology},
pdfauthor={Deependra Dhakal},
colorlinks=true,
linkcolor=red,
filecolor=Maroon,
citecolor=blue,
urlcolor=red,
breaklinks=true}
\urlstyle{same} % don't use monospace font for urls
\newif\ifbibliography
\setlength{\emergencystretch}{3em} % prevent overfull lines
\providecommand{\tightlist}{%
\setlength{\itemsep}{0pt}\setlength{\parskip}{0pt}}
\setcounter{secnumdepth}{0}
% set default figure placement to htbp
\makeatletter
\def\fps@figure{htbp}
\makeatother
\usepackage{booktabs}
\usepackage{longtable}
\usepackage{array}
\usepackage{multirow}
\usepackage{wrapfig}
\usepackage{float}
\usepackage{colortbl}
\usepackage{pdflscape}
\usepackage{tabu}
\usepackage{threeparttable}
\usepackage{threeparttablex}
\usepackage[normalem]{ulem}
\usepackage{makecell}
\usepackage{xcolor}
% % set background image if you will
% \usebackgroundtemplate%
% {%
% \includegraphics[width=\paperwidth,height=\paperheight]{02-dna_modification_background_dna_helix.jpg}%
% }
% % set background in a TikZ node for modifications
% % set background in a TikZ node for modifications
\usepackage{tikz}
\usepackage[absolute,overlay]{textpos}
\setbeamertemplate{title page}{
\tikz\node[opacity=0.8] {\includegraphics[width=0.8\paperwidth]{./01-overview_mutation_cover.png}};
% \tikz\node[opacity=0.8] {\includegraphics[width=5cm,ext=.1-overview_mutation_cover.png,type=png,read=.1-overview_mutation_cover.png]{1}} % if name shall contain dots, suppose filename = 01.1-overview_mutation_cover.png
\begin{textblock}{15}(1.5,2.8)\usebeamerfont{title} % 1.5 is x position in page
{\color{white}\raggedright\par\inserttitle}
\end{textblock}
\begin{textblock}{7.5}(1.5,7) % 1.5 is x position in page
{\color{white}\raggedright{\insertauthor}\mbox{}\\[0.2cm]
\insertdate}
\end{textblock}} % dd_rookie modified, figure is top aligned
% % set caption font size
% % note that beamer presentation native captions have their own configs
% \usepackage{caption}
% \captionsetup{font=footnotesize}
% this font option is amenable for beamer
\setbeamerfont{caption}{size=\tiny}
% some beamer themes naturally might not support navigation symbols
% \setbeamertemplate{navigation symbols}{} % remove navigation symbols
\setbeamertemplate{footline}[page number] % insert page number in footline
% \setbeamertemplate{navigation symbols}{slide} % insert slide indication in navigation
% \setbeamertemplate{navigation symbols}{frame} % insert frame indication in navigation
% \setbeamertemplate{navigation symbols}{section} % insert section indication in navigation
% \setbeamertemplate{navigation symbols}{subsection} % insert subsection indication in navigation
% \AtBeginSubsection{} % supress subsection display
\title{Overview of Biotechnology}
\providecommand{\subtitle}[1]{}
\subtitle{History process and product}
\author{Deependra Dhakal}
\providecommand{\institute}[1]{}
\institute{GAASC, Baitadi \and Tribhuwan University}
\date{Academic year 2019-2020}
\begin{document}
\frame{\titlepage}
\begin{frame}
\tableofcontents[hideallsubsections]
\end{frame}
\hypertarget{background}{%
\section{Background}\label{background}}
\begin{frame}{Overview}
\protect\hypertarget{overview}{}
\begin{block}{The U.S. Department of Agriculture (USDA)'s classical interpretation}
Agricultural biotechnology is a collection of scientific techniques, including genetic engineering, that are used to create, improve, or modify plants, animals, and microorganisms...
\end{block}
\begin{itemize}
\tightlist
\item
Agricultural Research Service (ARS), the in-house research agency of
USDA, classifies biotechnology research into six components:
\begin{itemize}
\tightlist
\item
basic engineering of recombinant DNA;
\item
DNA sequencing;
\item
genomic mapping with molecular markers;
\item
monoclonal antibodies;
\item
cell fusion and chromosome transfer;
\item
biologically-based processing
\end{itemize}
\end{itemize}
\end{frame}
\begin{frame}{Overview}
\protect\hypertarget{overview-1}{}
\begin{itemize}
\tightlist
\item
Humans have continually improved crop plants and animals by selective
breeding, mostly by trial and error.
\item
However, the field of genetics is quickly transitting into a highly
advanced scientic practice, wherein there is now not much place for
trial and error studies.
\item
The modern biotechnology of interest centers upon the newfound ability
to remove DNA from cells of an organism, modify it, and reinsert it
into cells where it will be functional.
\item
This process is sometimes called ``genetic engineering,'' and products
therefrom have often been ingenuously termed ``genetically modified
organisms'' (GMOs), even though all crop plants are genetically
modified in one way or another.
\item
The traditional breeder's available gene pool is predominantly limited
to those genes in sexually-compatible organisms, whereas modern
biotechnology enables some new, wider-ranging, choices.
\end{itemize}
\end{frame}
\begin{frame}{Classical plant breeding}
\protect\hypertarget{classical-plant-breeding}{}
\begin{itemize}
\tightlist
\item
Practicing classical plant breeding means many thousands of plants
must be cross-pollinated to find the one offspring with higher yield.
\item
In crossing plants,
\begin{itemize}
\tightlist
\item
Pollen must be taken from one plant and manually placed on another.
\item
The possibility of finding improved traits is limited by the amount
of genetic diversity already present in the plants.
\item
Consequently, if the two plants that are crossed share many of the
same genes, the amount of possible improvement is limited.
\end{itemize}
\item
Therefore, scientists have searched for better ways to improve plants.
\end{itemize}
\end{frame}
\begin{frame}{Mutation breeding}
\protect\hypertarget{mutation-breeding}{}
\begin{itemize}
\tightlist
\item
In the 1920s, scientists realized that \emph{mutations} could be
induced in seeds by using chemical mutagens or by exposure to X-rays
or gamma rays.
\item
Outcome of such treatments is even less predictable than traditional
breeding methods.
\item
Successful in world of flowers; new colors and more petals have been
expressed in flowers such as tulips, snapdragons, roses,
chrysanthemums, and many others.
\item
Mutation breeding has also been tried on vegetables, fruits, and
crops. For instance, peppermint plants that are resistant to fungus
were generated this way.
\end{itemize}
\end{frame}
\begin{frame}{Modern breeding}
\protect\hypertarget{modern-breeding}{}
\begin{itemize}
\tightlist
\item
Introduction of molecular breeding-- more predictable way to enhance
crops.
\item
Movement of genes from foreign sources into a specific plant,
resulting in a \emph{transgenic} plant.
\item
The foreign gene, or transgene, may confer specific resistance to an
insect, protect the plant against a specific herbicide, or enhance the
vitamin content of the crop.
\item
With something so powerful as genetic engineering, one mistake could
have profound and wide-ranging effects. We must impose tough controls
on the genetics supply industry and work to make sure that the new
techniques are in the service of the global community
\end{itemize}
\end{frame}
\hypertarget{history}{%
\section{History}\label{history}}
\begin{itemize}
\tightlist
\item
The science of genetics was transformed by the discovery of DNA
(deoxyribonucleic acid)
\item
Francis Crick and James Watson along with Rosalind Franklin, in 1953,
discovered that the DNA structure was a double helix: two strands
twisted around each other like a spiral staircase with bars across
like rings
\item
The structure, function, and composition of DNA are virtually
identical in all living organisms---from a blade of grass to an
elephant
\item
Difference only in precise ordering of chemical base that make up DNA
\item
This formed idea that changing this ordering will lead to modification
of lifeforms
\item
Marshall Nirenberg and H. Gobind Khorana carried out the deciphering
of the genetic code in 1961
\end{itemize}
\begin{frame}{History}
\protect\hypertarget{history-1}{}
\begin{itemize}
\tightlist
\item
While some were seeking to alter the genetic make-up of living things
by transferring specific genes from one organism to another, they now
had tools to alter exactly the hereditary material at the molecular
level.
\item
Walter Gilbert carried out the first recombinant DNA experiments in
1973
\item
First hybridomas created in 1975
\item
The production of monoclonal antibodies for diagnostics was carried
out in 1982,
\item
The first recombinant human therapeutic protein, insulin (humulin),
was produced in 1982.
\item
In 1976, the U.S. company Genentech became the first biotech company
to develop technologies to rearrange DNA.
\item
1980 ruling of the U.S. Supreme Court allowed genetically-engineered
microorganisms to be patented. This means that virtually any lifeform
on this planet can theoretically become the private property of the
company or person who ``creates'' it.
\end{itemize}
\end{frame}
\begin{frame}{History}
\protect\hypertarget{history-2}{}
\begin{itemize}
\tightlist
\item
Clues to understanding fermentation emerged in the seventeenth century
when Dutch experimentalist Anton Van Leeuwenhoek discovered
microorganisms using his microscope.
\item
He unraveled the chemical basis of the process of fermentation using
analytical techniques for the estimation of carbon dioxide.
\item
Two centuries later, in 1857, a French scientist Louis Pasteur
published his first report on lactic acid formation from sugar by
fermentation. He proved that fermentation is the consequence of
anaerobic life and identified three of its types:
\begin{itemize}
\tightlist
\item
Fermentation, which generates gas;
\item
Fermentation that results in alcohol; and
\item
Fermentation, which results in acids.
\end{itemize}
\end{itemize}
\end{frame}
\begin{frame}{History}
\protect\hypertarget{history-3}{}
\begin{itemize}
\tightlist
\item
At the end of the nineteenth century, Eduard Buchner observed the
formation of ethanol and carbon dioxide when cell-free extract of
yeast was added to an aqueous solution of sugars. Thus, he proved that
cells are not essential for the fermentation process and the
components responsible for the process are dissolved in the extract.
He named that substance ``Zymase''.
\item
During World war I,
\begin{itemize}
\tightlist
\item
Germany produced glycerine for making the explosive nitroglycerine
\item
Bacteria that converts corn or molasses into acetone for making the
explosive cordite.
\item
Sir Alexander Fleming's discovery of penicillin, the first
antibiotic, proved highly successful in treating wounded soldiers.
\end{itemize}
\end{itemize}
\end{frame}
\begin{frame}{Transgenic technology vs traditional breeding}
\protect\hypertarget{transgenic-technology-vs-traditional-breeding}{}
\begin{itemize}
\tightlist
\item
A plant can be transformed with a gene from any source, including
animals, bacteria, or viruses as well as other plants, whereas
traditional cross-breeding methods move genes only between members of
a particular genus of plants.
\item
Furthermore, transgenes can be placed in precise locations within the
genome and have known functions that have been evaluated extensively
before being inserted into the plant.
\item
In traditional breeding, on the other hand, the identity of genes
responsible for improving the crop is rarely known.
\end{itemize}
\end{frame}
\begin{frame}[allowframebreaks]{Timeline}
\protect\hypertarget{timeline}{}
\begin{table}[t]
\caption{\label{tab:biotechnology-history1}History of biotechnology}
\centering
\fontsize{5}{7}\selectfont
\begin{tabular}{>{\bfseries}l>{\raggedright\arraybackslash}p{40em}}
\toprule
Date & Event\\
\midrule
\rowcolor{gray!6} 5000 BC & Indus and Indo-Aryan civilizations practiced biotechnology to produce fermented foods and medicines and to keep the environment clean.\\
4000 BC & Egyptians used yeasts to make wine and bread.\\
\rowcolor{gray!6} 1750 BC & The Sumerians brewed beer.\\
250 BC & The Greeks used crop rotation to maximize crop fertility.\\
\rowcolor{gray!6} 1500 AD & The Aztecs made cake from spirulina.\\
\addlinespace
1663 AD & Robert Hook first described cells.\\
\rowcolor{gray!6} 1675 AD & Microbes were first described by Anton Van Leeuwenhock.\\
1859 AD & Darwin published his theory of evolution in ‘The Origin of Species.’\\
\rowcolor{gray!6} 1866 AD & Gregor John Mendel published the basic laws of genetics.\\
1869 AD & DNA was isolated by Friederich Miescher.\\
\addlinespace
\rowcolor{gray!6} 1910 AD & Genes were discovered to be present in chromosomes.\\
1917 AD & The term ‘biotechnology’ was used to describe fermentation technology.\\
\rowcolor{gray!6} 1928 AD & The first antibiotic, penicillin, was discovered by Alexander Flemming.\\
1941 AD & The term ‘genetic engineering’ was first used.\\
\rowcolor{gray!6} 1944 AD & Hereditary material was identified as DNA.\\
\addlinespace
1953 AD & Watson and Crick proposed the double helix structure of DNA.\\
\bottomrule
\end{tabular}
\end{table}
\end{frame}
\begin{frame}{Timeline}
\protect\hypertarget{timeline-1}{}
\begin{table}[t]
\caption{\label{tab:biotechnology-history2}History of biotechnology (...continued)}
\centering
\fontsize{5}{7}\selectfont
\begin{tabular}{>{\bfseries}l>{\raggedright\arraybackslash}p{40em}}
\toprule
Date & Event\\
\midrule
\rowcolor{gray!6} 1961 AD & Deciphering of genetic code by M.Nirenberg and H.G. Khorana.\\
1969 AD & The first gene was isolated.\\
\rowcolor{gray!6} 1973 AD & The first genetic engineering experiment was carried out by Walter Gilbert.\\
1975 AD & Creation of the first hybridomas.\\
\rowcolor{gray!6} 1976 AD & The first biotech company.\\
\addlinespace
1978 AD & World’s first ‘test-tube baby,’ Louise Brown, was born through in vitro fertilization.\\
\rowcolor{gray!6} 1981 AD & The first gene was synthesized. The first DNA synthesizer was developed.\\
1982 AD & The first genetically engineered drug, human insulin, produced by bacteria, was manufactured and marketed by a U.S. company. Production of the first monoclonal antibodies for diagnostics.\\
\rowcolor{gray!6} 1983 AD & The first transgenic plant was created—a petunia plant was genetically engineered to be resistant to kanamycin, an antibiotic.\\
1983 AD & The chromosomal location of the gene responsible for the genetic disorder, Huntington’s disease, was discovered leading to the development of genetic screening test.\\
\addlinespace
\rowcolor{gray!6} 1985 AD & DNA fingerprinting was first used in a criminal investigation.\\
1986 AD & The first field tests of genetically-engineered plants (tobacco) were conducted.\\
\rowcolor{gray!6} 1990 AD & Chymosin, an enzyme used in cheese making, became the first product of genetic engineering to be introduced into the food supply\\
1990 AD & Human genome project was launched.\\
\rowcolor{gray!6} 1990 AD & The first human gene therapy trial was performed on a four-year-old girl with an immune disorder.\\
\addlinespace
1991 AD & The gene implicated in the inherited form of breast cancer was discovered\\
\bottomrule
\end{tabular}
\end{table}
\end{frame}
\begin{frame}{Timeline}
\protect\hypertarget{timeline-2}{}
\begin{table}[t]
\caption{\label{tab:biotechnology-history3}History of biotechnology (...continued)}
\centering
\fontsize{5}{7}\selectfont
\begin{tabular}{>{\bfseries}l>{\raggedright\arraybackslash}p{40em}}
\toprule
Date & Event\\
\midrule
\rowcolor{gray!6} 1992 AD & Techniques for testing embryos for inherited diseases were developed\\
1994 AD & First commercial approval for transgenic plant by the U.S. government.\\
\rowcolor{gray!6} 1995 AD & First successful xenotransplantation trial was conducted, transplanting a heart from a genetically-engineered pig into a baboon.\\
1996 AD & First commercial introduction of a ‘gene chip’ designed to rapidly detect variances in the HIV virus and select the best drug treatment for patients.\\
\rowcolor{gray!6} 1996 AD & Dolly, the sheep was cloned from a cell of an adult sheep.\\
\addlinespace
1998 AD & Embryonic stem cells were grown successfully, opening new doors to cell- or tissue-based therapies.\\
\rowcolor{gray!6} 1999 AD & A U.S. company announced the successful cloning of human embryonic cells from an adult skin cell.\\
1999 AD & Chinese scientists cloned a giant panda embryo.\\
\rowcolor{gray!6} 1999 AD & Indian scientists and companies started producing recombinant vaccines, hormones, and other drugs.\\
1999 AD & The draft of human genome sequence was published.\\
\bottomrule
\end{tabular}
\end{table}
\end{frame}
\begin{frame}{Major cellular techniques in biology}
\protect\hypertarget{major-cellular-techniques-in-biology}{}
\begin{itemize}
\tightlist
\item
Microscopy
\item
Cell sorting
\item
Cell fractionation
\item
Cell-growth determination
\end{itemize}
\end{frame}
\begin{frame}{Major genetic techniques}
\protect\hypertarget{major-genetic-techniques}{}
\begin{itemize}
\tightlist
\item
Chromosomal techniques
\item
Mutagenic technique
\item
Recombination in bacteria (Recombination DNA technology)
\begin{itemize}
\tightlist
\item
Tools
\item
Making Recombinant DNA
\item
DNA library
\item
Transgenics (Introduction of Recombinant DNA into host cells)
\item
Identification of recombinants
\item
Polymerase chain reaction
\item
DNA probes
\item
Hybridization techniques
\item
DNA sequencing
\item
Site-directed mutagenesis
\end{itemize}
\item
Pedigree analysis in humans
\item
DNA isolation and purification techniques
\item
Molecular markers, TILLING and ZFN technology in plants
\end{itemize}
\end{frame}
\begin{frame}{Major applications of biotechnology}
\protect\hypertarget{major-applications-of-biotechnology}{}
\begin{itemize}
\tightlist
\item
Biological fuel generation
\item
Single-cell protein
\item
Sewage treatment
\item
Environmental biotechnology
\item
Medical biotechnology
\item
Agriculture and forest biotechnology
\item
Food and beverage biotechnology
\item
Safety in biotechnology
\end{itemize}
\end{frame}
\begin{frame}{Biotechnology process}
\protect\hypertarget{biotechnology-process}{}
\begin{itemize}
\tightlist
\item
In previous century, industries linked to the fermentation technology
had grown tremendously because of the high demand for various
chemicals such as ethanol, butanol, glycerine, acetone, etc.
\item
The advancement in fermentation process by its interaction with
chemical engineering has given rise to a new area -- the bioprocess
technology.
\item
Large-scale production of proteins and enzymes can be carried out by
applying bioprocess technology in fermentation.
\item
Processes to create large quantities of chemicals, antibiotics,
proteins, and enzymes in an economical manner.
\item
Bioprocess technology includes media and buffer preparation, upstream
processing and downstream processing.
\end{itemize}
\end{frame}
\begin{frame}{Biotechnology process: Upstream processing}
\protect\hypertarget{biotechnology-process-upstream-processing}{}
\begin{itemize}
\tightlist
\item
Includes:
\begin{itemize}
\tightlist
\item
microorganism media (development of processes for aseptic treatment
of substrates or raw materials with the microorganism or the
biocatalyst)
\item
substrate, and the correct chemical environment to carry out the
required biochemical reactions to produce the product.
\end{itemize}
\item
Unit operations involved in upstream processing are: Milling, Mixing,
Media preparation, Sterilization, Cooling, Heating.
\end{itemize}
\end{frame}
\begin{frame}{Biotechnology process: Bioreactor}
\protect\hypertarget{biotechnology-process-bioreactor}{}
\begin{itemize}
\tightlist
\item
Initial step: designing of the appropriate type of bioreactor or
fermentor.
\item
Bioreactors are vessels in which raw materials are biologically
converted into specific products, using microorganisms, plants,
animals, or human cells or individual enzymes.
\item
Bioreactor supports the natural process of cells by trying to maintain
their environment to provide optimum growth conditions by providing
appropriate temperature, pH, substrates, salts, vitamins, and oxygen.
\item
In most of the bioreaction processes the substrate of the
biotransformation and the carbon source of the organisms will be the
same.
\end{itemize}
\end{frame}
\begin{frame}{Biotechnology process: Bioreactor}
\protect\hypertarget{biotechnology-process-bioreactor-1}{}
\begin{itemize}
\tightlist
\item
Unit operations involved in bioreactions are: Mixing, Handling of
microorganisms, plant or animal cells, Inoculation of the cells,
Heating and cooling.
\item
Bioreactors can be classified according to the type of biocatalysts
and the type of bioreaction.
\item
The first classification is based on the type of biological agent
used:
\begin{itemize}
\tightlist
\item
microbial fermentors or
\item
enzyme (cell-free) reactors
\end{itemize}
\item
Further classification is possible based on biochemical reactions and
process requirements.
\end{itemize}
\end{frame}
\begin{frame}{Biotechnology process: Downstream processing}
\protect\hypertarget{biotechnology-process-downstream-processing}{}
\begin{itemize}
\tightlist
\item
recovery and purification of the required product from the growth
medium through a set of separation and purification techniques.
\item
each stage in the overall separation procedure is strongly dependent
on the history and quality of the biological production process.
\item
maximization of production can lead to great difficulties in
downstreaming and recently more attention is being paid to overall
process optimization.
\item
includes techniques such as filtration, centrifugation, sedimentation,
various types of chromatographic techniques, electrophoresis, etc.
\end{itemize}
\end{frame}
\begin{frame}{Biotechnology process}
\protect\hypertarget{biotechnology-process-1}{}
\begin{figure}
\includegraphics[width=0.4\linewidth]{../images/bioprocess_flow_chart} \caption{A typical biotechnology flow process}\label{fig:biotechnology-process}
\end{figure}
\end{frame}
\begin{frame}{Industrial biotechnology}
\protect\hypertarget{industrial-biotechnology}{}
\begin{itemize}
\tightlist
\item
Recombinant microorganisms, plant cells, and animal cells can be
cultivated and used for large-scale production of
industrially-important enzymes and chemicals. A list of such enzymes
is given in Table \ref{tab:industrial-chemicals}
\end{itemize}
\begin{table}[t]
\caption{\label{tab:industrial-chemicals}Some major industrial enzymes and their sources and uses.}
\centering
\fontsize{5}{7}\selectfont
\begin{tabular}{>{\raggedright\arraybackslash}p{8em}>{\raggedright\arraybackslash}p{26em}>{\raggedright\arraybackslash}p{30em}}
\toprule
Enzymes & Sources & Uses\\
\midrule
\rowcolor{gray!6} Amylases & Aspergillus niger, A. oryzae, B. licheniformis, B. subtilis, germinating cereals germinating barley & Hydrolyze starch to glucose, detergents, baked goods, milk cheese, fruit juice, digestive medicines, dental care\\
Invertases & Saccharomyces cerevisiae & Production of invert sugar, confectionery\\
\rowcolor{gray!6} Glucose isomerase & Arthrobacter globiformis, Actomoplanes missouriensis, Streptomyces solivaceus and E. coli & Conversion of glucose to fructose production of high fructose syrup, other beverages, and food\\
$\alpha$ D-Galactosidase & Mortierella vinacease & Raffinose hydrolysis\\
\rowcolor{gray!6} $\beta$ D-Galactosidase & Aspergillus niger & Lactose hydrolysis\\
\addlinespace
Papain & Papaya & Meat, beer, leather, textiles, pharmaceuticals, meat industry, digestive aid, dental hygiene, etc.\\
\rowcolor{gray!6} Proteases & Bacillus subtilis, B. licheniformis & Detergents, meat tenderizers, beer, cheese, flavor production\\
Pepsin & Hog (pig) stomachs & Cereals, pharmaceuticals\\
\rowcolor{gray!6} Trypsin & Hog and calf pancreases & Meat, pharmaceuticals\\
11-$\beta$-Hydroxylase & Curvularia lunata & Steroid conversion, bioconversion of organic chemicals\\
\addlinespace
\rowcolor{gray!6} Ficin & Figs & Leather, meat, pharmaceuticals\\
Bromelain & Pineapple & Meat, beer, pharmaceuticals\\
\bottomrule
\end{tabular}
\end{table}
\end{frame}
\begin{frame}{Biotechnology product}
\protect\hypertarget{biotechnology-product}{}
\begin{itemize}
\tightlist
\item
Traditional biotechnology products also include biologically-processed
items like bread, cheese, and wine.
\item
Other modern biotechnologies, such as monoclonal antibodies or
molecular markers as aids for traditional breeding and selection, are
not relevant to genetic engineering but they constitute biotechnology
products.
\end{itemize}
\end{frame}
\hypertarget{fermentation}{%
\section{Fermentation}\label{fermentation}}
\begin{frame}{Introduction}
\protect\hypertarget{introduction}{}
\begin{itemize}
\tightlist
\item
Decomposition of foodstuffs generally accompanied by the evolution of
gas.
\item
The best-known example is alcoholic fermentation, in which sugar is
converted into alcohol and carbon dioxide.
\item
This conversion, described by the equation below, was established by
J. L. Gay-Lussac in 1815.
\end{itemize}
\[
\begin{aligned}
&C_6 H_{12} O_6 &\rightarrow &2CO_2 + &2C_2H_5OH \\
&\textrm{Sugar} &&\textrm{Carbon dioxide} &\textrm{Alcohol}
\end{aligned}
\]
\end{frame}
\begin{frame}{History}
\protect\hypertarget{history-4}{}
\begin{itemize}
\tightlist
\item
Before 1800 the association of yeast or leaven with fermentation had
been noted, but the nature of these agents was not understood.
\item
Experiments of C. Cagniard-Latour, of F. T. Kutzing, and of T Schwann
in 1837 indicated that yeast is a living organism and is the cause of
fermentation.
\item
This view was opposed by such leading chemists as J. von Liebig and F.
Wohler, who sought a chemical rather than a biological explanation of
the process.
\item
The biological concept became generally accepted following the work of
Louis Pasteur, who concluded that fermentation is a physiological
counterpart of oxidation, and permits organisms to live and grow in
the absence of air (anaerobically).
\end{itemize}
\end{frame}
\begin{frame}{History}
\protect\hypertarget{history-5}{}
\begin{itemize}
\tightlist
\item
This linked fermentation and putrefaction as comparable processes;
both represent decompositions of organic matter brought about by
microorganisms in the absence of air.
\item
The difference is determined by the nature of the decomposable
material;
\begin{itemize}
\tightlist
\item
sugary substances generally yield products with pleasant odor and
taste (fermentation),
\item
proteins give rise to evil-smelling products (putrefaction).
\end{itemize}
\item
Pasteur also discovered the lactic acid and butyric acid
fermentations, and from his experiments concluded that each kind of
fermentation was caused by a specific microbe.
\item
Later work supported this idea to a large extent, and considerably
increased the number of specific fermentations.
\end{itemize}
\end{frame}
\begin{frame}{Process}
\protect\hypertarget{process}{}
During fermentation organic matter is decomposed in the absence of air
(oxygen); hence, there is always an accumulation of reduction products,
or incomplete oxidation products. Some of these products (for example,
alcohol and lactic acid) are of importance to society, and fermentation
has therefore been used for their manufacture on an industrial scale.
With regard to historic roots of the process, Converting dry grains and
other seeds into something more appetizing than a gruel must have made
agriculture more attractive and valuable. Alcohol, despite its dangers,
provided (and still provides), in reasonable moderation, a basis for
social interaction. There are also many microbiological processes that
go on in the presence of air while yielding incomplete oxidation
products. Good examples are the formation of acetic acid (vinegar) from
alcohol by vinegar bacteria, and of citric acid from ugar by certain
molds (for example, \emph{Aspergillus niger}). These microbial
processes, too, have gained industrial importance, and are often
referred to as fermentations, even though they do not conform to
Pasteur's concept of fermentation as a decomposition in the absence of
air.
\end{frame}
\begin{frame}{Fermentation technology: Milk fermentation}
\protect\hypertarget{fermentation-technology-milk-fermentation}{}
\begin{figure}
\includegraphics[width=0.55\linewidth]{../images/milk_fermentation} \caption{Basic steps in manufacture of fermented milks. From International Dairy Federation (1988) Fermented Milks-Science and Technology. International Dairy Federation Bulletin No. 227.}\label{fig:milk-fermentation-steps}
\end{figure}
\end{frame}
\hypertarget{bibliography}{%
\section{Bibliography}\label{bibliography}}
\begin{frame}{Further study}
\protect\hypertarget{further-study}{}
Also see: Nair (\protect\hyperlink{ref-nair2008introduction}{2008})
\end{frame}
\begin{frame}{References}
\protect\hypertarget{references}{}
\hypertarget{refs}{}
\leavevmode\hypertarget{ref-nair2008introduction}{}%
Nair, A Jayakumaran. 2008. \emph{Introduction to Biotechnology and
Genetic Engineering}. Laxmi Publications, Ltd.
\end{frame}
\end{document}
|
[GOAL]
J✝ : Type
inst✝⁴ : SmallCategory J✝
inst✝³ : FinCategory J✝
k : Type v
inst✝² : Field k
J : Type
inst✝¹ : Fintype J
Z : J → ModuleCat k
inst✝ : ∀ (j : J), FiniteDimensional k ↑(Z j)
⊢ FiniteDimensional k ↑(ModuleCat.of k ((j : J) → ↑(Z j)))
[PROOFSTEP]
unfold ModuleCat.of
[GOAL]
J✝ : Type
inst✝⁴ : SmallCategory J✝
inst✝³ : FinCategory J✝
k : Type v
inst✝² : Field k
J : Type
inst✝¹ : Fintype J
Z : J → ModuleCat k
inst✝ : ∀ (j : J), FiniteDimensional k ↑(Z j)
⊢ FiniteDimensional k ↑(ModuleCat.mk ((j : J) → ↑(Z j)))
[PROOFSTEP]
infer_instance
[GOAL]
J✝ : Type
inst✝⁴ : SmallCategory J✝
inst✝³ : FinCategory J✝
k : Type v
inst✝² : Field k
J : Type
inst✝¹ : Fintype J
Z : J → ModuleCat k
inst✝ : ∀ (j : J), FiniteDimensional k ↑(Z j)
this : FiniteDimensional k ↑(ModuleCat.of k ((j : J) → ↑(Z j)))
⊢ Mono (ModuleCat.piIsoPi fun j => Z j).hom
[PROOFSTEP]
infer_instance
[GOAL]
J : Type
inst✝² : SmallCategory J
inst✝¹ : FinCategory J
k : Type v
inst✝ : Field k
F : J ⥤ FGModuleCat k
⊢ ∀ (j : J), FiniteDimensional k ↑((F ⋙ forget₂ (FGModuleCat k) (ModuleCat k)).obj j)
[PROOFSTEP]
intro j
[GOAL]
J : Type
inst✝² : SmallCategory J
inst✝¹ : FinCategory J
k : Type v
inst✝ : Field k
F : J ⥤ FGModuleCat k
j : J
⊢ FiniteDimensional k ↑((F ⋙ forget₂ (FGModuleCat k) (ModuleCat k)).obj j)
[PROOFSTEP]
change FiniteDimensional k (F.obj j)
[GOAL]
J : Type
inst✝² : SmallCategory J
inst✝¹ : FinCategory J
k : Type v
inst✝ : Field k
F : J ⥤ FGModuleCat k
j : J
⊢ FiniteDimensional k ↑(F.obj j)
[PROOFSTEP]
infer_instance
|
Summer is the time that many of us have a lot more free time than we usually do, and this is a great thing! This means that you have time to do all of the fun things that you can’t seem to make room for when you’re always working or studying.
Perhaps you already have a bucket list in your mind, or you have no idea where to start. It doesn’t matter because after reading this you will know exactly what you still want to do this summer. So let’s get to the fun part of this blog!
If you always buy the same kind of clothes and you’ve been rocking the same old hairstyle for three years then it is time to try something different.
Not only will all of this be a lot of fun, you may also develop a new style that you are totally in love with.
Festival season is happening right now, and why wouldn’t you want to participate? It doesn’t matter if you visit three festivals all-weekend or one free festival for a day, as long as you have fun. Festivals are a great way to enjoy music, the good weather and playing dress-up with all that fringe and glitter. I know that I always love visiting a festival.
Yes, this is a generic tip, which is why you can choose to do what this tip what you want. You can take some cooking lessons and impress your upcoming dates, you can learn how to write an amazing novel or you could even follow a workshop in Burlesque!
Fulfil your craziest dreams and learn something that you always wanted to be able to do, or know. Now is your time to develop yourself.
Of course, it’s important that you’re safe when doing this, so don’t drink too much and make sure someone knows where you are at all times.
If you’ve got that figured out then it can be incredibly exciting, but also a little scary, to go out by yourself. If you’re the kind of girl who likes a hook-up now and then, then you’ll love all the attention that you get as a girl in a club or bar all by yourself. Or perhaps you don’t like it at all, you can always go home, it’s definitely something to experience.
Nothing screams summer like a good ol’ BBQ, so I had to include it in this list. You can keep it simple, or you can go all out for once. Get the best meat from an actual butcher and search the internet for some delicious salad recipes.
If you want to upgrade your BBQ party even further then you can make sure that you have some kind of pool in your garden, whether that is a deep built-in pool or small inflatable pool.
I don’t know about you, but I usually feel guilty when I watch too much Netflix while I should actually be doing work, so a marathon is just not going to work. Summer is a completely different time though, a lot more time mostly. This is the perfect period for that all-day Netflix marathon where you don’t even get dressed and eat a lot of take-out food. You are allowed to treat yourself now and then.
Even if you stayed in all summer watching movies and eating ice cream, your followers are allowed to think that you went on all these crazy adventures. So take one day, put together a lot of outfits, gather your girlfriends, and travel all across town to take some of your best pictures. You will have a lot of fun and enough Instagram photo’s for the coming months.
Don’t forget to share this post if you liked it and follow me on my social media to always be updated and see much more of my work! |
{- Copyright © 2015 Benjamin Barenblat
Licensed under the Apache License, Version 2.0 (the ‘License’); you may not use
this file except in compliance with the License. You may obtain a copy of the
License at
http://www.apache.org/licenses/LICENSE-2.0
Unless required by applicable law or agreed to in writing, software distributed
under the License is distributed on an ‘AS IS’ BASIS, WITHOUT WARRANTIES OR
CONDITIONS OF ANY KIND, either express or implied. See the License for the
specific language governing permissions and limitations under the License. -}
module B.Prelude.Product where
import Data.Product
open Data.Product
using (_×_; proj₁; proj₂; _,_; _,′_; curry; uncurry; uncurry′)
public
module Product where
|
\<^marker>\<open>creator "Kevin Kappelmann"\<close>
subsubsection \<open>Order Equivalence\<close>
theory Transport_Natural_Functors_Order_Equivalence
imports
Transport_Natural_Functors_Base
begin
lemma inflationary_on_in_dom_FrelI:
assumes "inflationary_on (in_dom R1) R1 f1"
and "inflationary_on (in_dom R2) R2 f2"
and "inflationary_on (in_dom R3) R3 f3"
defines "R \<equiv> Frel R1 R2 R3"
shows "inflationary_on (in_dom R) R (Fmap f1 f2 f3)"
apply (unfold R_def)
apply (rule inflationary_onI)
apply (subst (asm) in_dom_Frel_eq_Fpred_in_dom)
apply (erule FpredE)
apply (subst Frel_Fmap_eq2)
apply (rule Frel_refl_strong)
apply (rule inflationary_onD[where ?R=R1] inflationary_onD[where ?R=R2]
inflationary_onD[where ?R=R3],
rule assms,
assumption+)+
done
lemma inflationary_on_in_codom_FrelI:
assumes "inflationary_on (in_codom R1) R1 f1"
and "inflationary_on (in_codom R2) R2 f2"
and "inflationary_on (in_codom R3) R3 f3"
defines "R \<equiv> Frel R1 R2 R3"
shows "inflationary_on (in_codom R) R (Fmap f1 f2 f3)"
apply (unfold R_def)
apply (rule inflationary_onI)
apply (subst (asm) in_codom_Frel_eq_Fpred_in_codom)
apply (erule FpredE)
apply (subst Frel_Fmap_eq2)
apply (rule Frel_refl_strong)
apply (rule inflationary_onD[where ?R=R1] inflationary_onD[where ?R=R2]
inflationary_onD[where ?R=R3],
rule assms,
assumption+)+
done
lemma inflationary_on_in_field_FrelI:
assumes "inflationary_on (in_field R1) R1 f1"
and "inflationary_on (in_field R2) R2 f2"
and "inflationary_on (in_field R3) R3 f3"
defines "R \<equiv> Frel R1 R2 R3"
shows "inflationary_on (in_field R) R (Fmap f1 f2 f3)"
apply (unfold R_def)
apply (subst in_field_eq_in_dom_sup_in_codom)
apply (subst inflationary_on_sup_eq)
apply (unfold inf_apply)
apply (subst inf_bool_def)
apply (rule conjI;
rule inflationary_on_in_dom_FrelI inflationary_on_in_codom_FrelI;
rule inflationary_on_if_le_pred_if_inflationary_on,
rule assms,
rule le_predI,
rule in_field_if_in_dom in_field_if_in_codom,
assumption)
done
lemma deflationary_on_in_dom_FrelI:
assumes "deflationary_on (in_dom R1) R1 f1"
and "deflationary_on (in_dom R2) R2 f2"
and "deflationary_on (in_dom R3) R3 f3"
defines "R \<equiv> Frel R1 R2 R3"
shows "deflationary_on (in_dom R) R (Fmap f1 f2 f3)"
apply (unfold R_def)
apply (subst deflationary_on_eq_inflationary_on_rel_inv)
apply (subst in_codom_rel_inv_eq_in_dom[symmetric])
apply (unfold Frel_rel_inv_eq_rel_inv_Frel[symmetric])
apply (rule inflationary_on_in_codom_FrelI;
subst deflationary_on_eq_inflationary_on_rel_inv[symmetric],
subst in_codom_rel_inv_eq_in_dom,
rule assms)
done
lemma deflationary_on_in_codom_FrelI:
assumes "deflationary_on (in_codom R1) R1 f1"
and "deflationary_on (in_codom R2) R2 f2"
and "deflationary_on (in_codom R3) R3 f3"
defines "R \<equiv> Frel R1 R2 R3"
shows "deflationary_on (in_codom R) R (Fmap f1 f2 f3)"
apply (unfold R_def)
apply (subst deflationary_on_eq_inflationary_on_rel_inv)
apply (subst in_dom_rel_inv_eq_in_codom[symmetric])
apply (unfold Frel_rel_inv_eq_rel_inv_Frel[symmetric])
apply (rule inflationary_on_in_dom_FrelI;
subst deflationary_on_eq_inflationary_on_rel_inv[symmetric],
subst in_dom_rel_inv_eq_in_codom,
rule assms)
done
lemma deflationary_on_in_field_FrelI:
assumes "deflationary_on (in_field R1) R1 f1"
and "deflationary_on (in_field R2) R2 f2"
and "deflationary_on (in_field R3) R3 f3"
defines "R \<equiv> Frel R1 R2 R3"
shows "deflationary_on (in_field R) R (Fmap f1 f2 f3)"
apply (unfold R_def)
apply (subst deflationary_on_eq_inflationary_on_rel_inv)
apply (subst in_field_rel_inv_eq[symmetric])
apply (unfold Frel_rel_inv_eq_rel_inv_Frel[symmetric])
apply (rule inflationary_on_in_field_FrelI;
subst deflationary_on_eq_inflationary_on_rel_inv[symmetric],
subst in_field_rel_inv_eq,
rule assms)
done
lemma rel_equivalence_on_in_field_FrelI:
assumes "rel_equivalence_on (in_field R1) R1 f1"
and "rel_equivalence_on (in_field R2) R2 f2"
and "rel_equivalence_on (in_field R3) R3 f3"
defines "R \<equiv> Frel R1 R2 R3"
shows "rel_equivalence_on (in_field R) R (Fmap f1 f2 f3)"
apply (unfold R_def)
apply (subst rel_equivalence_on_eq)
apply (unfold inf_apply)
apply (subst inf_bool_def)
apply (insert assms)
apply (elim rel_equivalence_onE)
apply (rule conjI;
rule inflationary_on_in_field_FrelI deflationary_on_in_field_FrelI;
assumption)
done
context transport_natural_functor
begin
lemmas inflationary_on_in_field_unitI = inflationary_on_in_field_FrelI
[of L1 "\<eta>\<^sub>1" L2 "\<eta>\<^sub>2" L3 "\<eta>\<^sub>3", folded transport_defs unit_eq_Fmap]
lemmas deflationary_on_in_field_unitI = deflationary_on_in_field_FrelI
[of L1 "\<eta>\<^sub>1" L2 "\<eta>\<^sub>2" L3 "\<eta>\<^sub>3", folded transport_defs unit_eq_Fmap]
lemmas rel_equivalence_on_in_field_unitI = rel_equivalence_on_in_field_FrelI
[of L1 "\<eta>\<^sub>1" L2 "\<eta>\<^sub>2" L3 "\<eta>\<^sub>3", folded transport_defs unit_eq_Fmap]
interpretation flip :
transport_natural_functor R1 L1 r1 l1 R2 L2 r2 l2 R3 L3 r3 l3
rewrites "flip.unit \<equiv> \<epsilon>" and "flip.t1.unit \<equiv> \<epsilon>\<^sub>1"
and "flip.t2.unit \<equiv> \<epsilon>\<^sub>2" and "flip.t3.unit \<equiv> \<epsilon>\<^sub>3"
by (simp_all only: order_functors.flip_counit_eq_unit)
lemma order_equivalenceI:
assumes "((\<le>\<^bsub>L1\<^esub>) \<equiv>\<^sub>o (\<le>\<^bsub>R1\<^esub>)) l1 r1"
and "((\<le>\<^bsub>L2\<^esub>) \<equiv>\<^sub>o (\<le>\<^bsub>R2\<^esub>)) l2 r2"
and "((\<le>\<^bsub>L3\<^esub>) \<equiv>\<^sub>o (\<le>\<^bsub>R3\<^esub>)) l3 r3"
shows "((\<le>\<^bsub>L\<^esub>) \<equiv>\<^sub>o (\<le>\<^bsub>R\<^esub>)) l r"
apply (insert assms)
apply (elim order_functors.order_equivalenceE)
apply (rule order_equivalenceI;
rule mono_wrt_rel_leftI
flip.mono_wrt_rel_leftI
rel_equivalence_on_in_field_unitI
flip.rel_equivalence_on_in_field_unitI;
assumption)
done
end
end |
import evaluation
import backends.bfs.baseline
import all
open baseline
example {p q : Prop} (h₁ : p) (h₂ : q) : p ∧ q :=
begin
tidy
end
universe u
example {α : Type u} (p : α → Prop) [decidable_pred p] (l : list α) :
list.partition p l = (list.filter p l, list.filter (not ∘ p) l) :=
begin
simp at *
end
-- example : ∀ (b : bool) (n : ℕ), (nat.bit b n).bodd = b :=
-- begin
-- -- simp at *
-- tidy_bfs_proof_search 25 tt 1
-- end
|
# This file is part of IntegerSequences.
# Copyright Peter Luschny. License is MIT.
(@__DIR__) ∉ LOAD_PATH && push!(LOAD_PATH, (@__DIR__))
module Kolakoski
using Nemo
export ModuleKolakoski
export KolakoskiList, C000002, I000002, L000002
"""
* KolakoskiList, C000002, I000002, L000002
"""
const ModuleKolakoski = ""
"""
Generate the Kolakoski sequence which is the unique sequence over the alphabet ``{1, 2}`` starting with ``1`` and having the sequence of run lengths identical with itself.
"""
C000002() =
Channel(csize = 10) do c
x = y = Int(-1)
while true
put!(c, [2, 1][(x&1)+1])
f = y & ~(y + 1)
x = xor(x, f)
y = (y + 1) | (f & (x >> 1))
end
end
struct KolakoskiSeq
count::Int
ch::Channel
KolakoskiSeq(count) = new(count, C000002())
end
function Base.iterate(I::KolakoskiSeq)
if I.count == 0
close(I.ch)
return nothing
end
(take!(I.ch), (0))
end
function Base.iterate(I::KolakoskiSeq, S)
j = S[1] + 1
if I.count == j
close(I.ch)
return nothing
end
(take!(I.ch), (j))
end
Base.length(I::KolakoskiSeq) = I.count
Base.eltype(I::KolakoskiSeq) = Int
"""
Iterate over the first ``n`` Kolakoski numbers.
"""
I000002(n::Int) = KolakoskiSeq(n)
"""
Return the list of the first ``n`` terms of the Kolakoski sequence.
"""
function KolakoskiList(len::Int)
len ≤ 0 && return []
generator = C000002()
L = [take!(generator) for _ ∈ 1:len]
close(generator)
return L
end
"""
Return the list of the first ``n`` terms of the Kolakoski sequence.
"""
L000002(n::Int) = KolakoskiList(n)
#START-TEST-########################################################
using Test
function test()
@testset "Kolakoski" begin
K = KolakoskiList(100)
@test K[1] == 1
@test K[33] == 2
@test K[72] == 2
generator = C000002()
for n ∈ [1, 33, 72]
k = take!(generator)
@test K[n] == k
end
close(generator)
end
end
function demo()
println(KolakoskiList(20))
generator = C000002()
o = e = 0
for n ∈ 1:80
take!(generator) == 1 ? o += 1 : e += 1
print(o - e, " ")
end
println()
close(generator)
for f ∈ I000002(20)
print(f, ", ")
end
println()
print(L000002(20))
println()
end
"""
I000002(100000) ::
0.000035 seconds (31 allocations: 2.969 KiB)
KolakoskiList(10000) ::
0.086202 seconds (120.03 k allocations: 10.226 MiB, 17.61% gc time)
"""
function perf()
@time I000002(100000)
@time KolakoskiList(100000)
end
function main()
test()
demo()
perf()
end
main()
end # module
#=
[1, 2, 2, 1, 1, 2, 1, 2, 2, 1, 2, 2, 1, 1, 2, 1, 1, 2, 2, 1]
1 0 -1 0 1 0 1 0 -1 0 -1 -2 -1 0 -1 0 1 0 -1 0 -1 0 1 0 1 0 -1 0
1 0 1 2 1 2 1 0 1 0-1 0 1 0 1 0 -1 0 -1 0 1 0 1 2 1 0 1 0 -1 0 1
0 1 0 -1 0 -1 -2 -1 0 -1 0 1 0 1 0 -1 0 -1 0 1 0
1, 2, 2, 1, 1, 2, 1, 2, 2, 1, 2, 2, 1, 1, 2, 1, 1, 2, 2, 1,
[1, 2, 2, 1, 1, 2, 1, 2, 2, 1, 2, 2, 1, 1, 2, 1, 1, 2, 2, 1]
0.000035 seconds (31 allocations: 2.969 KiB)
0.086202 seconds (120.03 k allocations: 10.226 MiB, 17.61% gc time)
=#
|
module accumulate_flow_mod
use flow_accumulation_algorithm_mod
use coords_mod
implicit none
contains
subroutine accumulate_flow_icon_single_index(cell_neighbors, &
input_river_directions, &
output_cumulative_flow)
integer, dimension(:), pointer, intent(inout) :: input_river_directions
integer, dimension(:), pointer, intent(out) :: output_cumulative_flow
class(*), dimension(:), pointer :: input_river_directions_ptr
class(*), dimension(:), pointer :: output_cumulative_flow_ptr
integer, dimension(:,:), pointer, intent(in) :: cell_neighbors
integer, dimension(:,:), pointer :: secondary_neighbors
type(icon_icosohedral_grid), pointer :: coarse_grid
type(generic_1d_section_coords) :: coarse_grid_shape
type(icon_single_index_flow_accumulation_algorithm) :: flow_acc_alg
where (input_river_directions == 0 .or. &
input_river_directions == -1 .or. &
input_river_directions == -2 .or. &
input_river_directions == -5)
input_river_directions = -3
end where
input_river_directions_ptr => input_river_directions
output_cumulative_flow_ptr => output_cumulative_flow
coarse_grid => icon_icosohedral_grid(cell_neighbors)
call coarse_grid%calculate_secondary_neighbors()
secondary_neighbors => coarse_grid%get_cell_secondary_neighbors()
coarse_grid_shape = generic_1d_section_coords(cell_neighbors, &
secondary_neighbors)
flow_acc_alg = &
icon_single_index_flow_accumulation_algorithm(coarse_grid_shape, &
input_river_directions_ptr, &
output_cumulative_flow_ptr)
call flow_acc_alg%generate_cumulative_flow(.false.)
call flow_acc_alg%icon_single_index_destructor()
call coarse_grid_shape%generic_1d_section_coords_destructor()
deallocate(coarse_grid)
deallocate(secondary_neighbors)
end subroutine accumulate_flow_icon_single_index
end module accumulate_flow_mod
|
/-
Copyright (c) 2022 Michael Stoll. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Michael Stoll
-/
import number_theory.legendre_symbol.jacobi_symbol
/-!
# A `norm_num` extension for Jacobi and Legendre symbols
We extend the `tactic.interactive.norm_num` tactic so that it can be used to provably compute
the value of the Jacobi symbol `J(a | b)` or the Legendre symbol `legendre_sym p a` when
the arguments are numerals.
## Implementation notes
We use the Law of Quadratic Reciprocity for the Jacobi symbol to compute the value of `J(a | b)`
efficiently, roughly comparable in effort with the euclidean algorithm for the computation
of the gcd of `a` and `b`. More precisely, the computation is done in the following steps.
* Use `J(a | 0) = 1` (an artifact of the definition) and `J(a | 1) = 1` to deal
with corner cases.
* Use `J(a | b) = J(a % b | b)` to reduce to the case that `a` is a natural number.
We define a version of the Jacobi symbol restricted to natural numbers for use in
the following steps; see `norm_num.jacobi_sym_nat`. (But we'll continue to write `J(a | b)`
in this description.)
* Remove powers of two from `b`. This is done via `J(2a | 2b) = 0` and
`J(2a+1 | 2b) = J(2a+1 | b)` (another artifact of the definition).
* Now `0 ≤ a < b` and `b` is odd. If `b = 1`, then the value is `1`.
If `a = 0` (and `b > 1`), then the value is `0`. Otherwise, we remove powers of two from `a`
via `J(4a | b) = J(a | b)` and `J(2a | b) = ±J(a | b)`, where the sign is determined
by the residue class of `b` mod 8, to reduce to `a` odd.
* Once `a` is odd, we use Quadratic Reciprocity (QR) in the form
`J(a | b) = ±J(b % a | a)`, where the sign is determined by the residue classes
of `a` and `b` mod 4. We are then back in the previous case.
We provide customized versions of these results for the various reduction steps,
where we encode the residue classes mod 2, mod 4, or mod 8 by using terms like
`bit1 (bit0 a)`. In this way, the only divisions we have to compute and prove
are the ones occurring in the use of QR above.
-/
section lemmas
namespace norm_num
/-- The Jacobi symbol restricted to natural numbers in both arguments. -/
def jacobi_sym_nat (a b : ℕ) : ℤ := jacobi_sym a b
/-!
### API Lemmas
We repeat part of the API for `jacobi_sym` with `norm_num.jacobi_sym_nat` and without implicit
arguments, in a form that is suitable for constructing proofs in `norm_num`.
-/
/-- Base cases: `b = 0`, `b = 1`, `a = 0`, `a = 1`. -/
lemma jacobi_sym_nat.zero_right (a : ℕ) : jacobi_sym_nat a 0 = 1 :=
by rwa [jacobi_sym_nat, jacobi_sym.zero_right]
lemma jacobi_sym_nat.one_right (a : ℕ) : jacobi_sym_nat a 1 = 1 :=
by rwa [jacobi_sym_nat, jacobi_sym.one_right]
lemma jacobi_sym_nat.zero_left_even (b : ℕ) (hb : b ≠ 0) : jacobi_sym_nat 0 (bit0 b) = 0 :=
by rw [jacobi_sym_nat, nat.cast_zero, jacobi_sym.zero_left (nat.one_lt_bit0 hb)]
lemma jacobi_sym_nat.zero_left_odd (b : ℕ) (hb : b ≠ 0) : jacobi_sym_nat 0 (bit1 b) = 0 :=
by rw [jacobi_sym_nat, nat.cast_zero, jacobi_sym.zero_left (nat.one_lt_bit1 hb)]
lemma jacobi_sym_nat.one_left_even (b : ℕ) : jacobi_sym_nat 1 (bit0 b) = 1 :=
by rw [jacobi_sym_nat, nat.cast_one, jacobi_sym.one_left]
lemma jacobi_sym_nat.one_left_odd (b : ℕ) : jacobi_sym_nat 1 (bit1 b) = 1 :=
by rw [jacobi_sym_nat, nat.cast_one, jacobi_sym.one_left]
/-- Turn a Legendre symbol into a Jacobi symbol. -/
lemma legendre_sym.to_jacobi_sym (p : ℕ) (pp : fact (p.prime)) (a r : ℤ) (hr : jacobi_sym a p = r) :
legendre_sym p a = r :=
by rwa [@legendre_sym.to_jacobi_sym p pp a]
/-- The value depends only on the residue class of `a` mod `b`. -/
lemma jacobi_sym.mod_left (a : ℤ) (b ab' : ℕ) (ab r b' : ℤ) (hb' : (b : ℤ) = b')
(hab : a % b' = ab) (h : (ab' : ℤ) = ab) (hr : jacobi_sym_nat ab' b = r) :
jacobi_sym a b = r :=
by rw [← hr, jacobi_sym_nat, jacobi_sym.mod_left, hb', hab, ← h]
lemma jacobi_sym_nat.mod_left (a b ab : ℕ) (r : ℤ) (hab : a % b = ab)
(hr : jacobi_sym_nat ab b = r) :
jacobi_sym_nat a b = r :=
by { rw [← hr, jacobi_sym_nat, jacobi_sym_nat, _root_.jacobi_sym.mod_left a b, ← hab], refl, }
/-- The symbol vanishes when both entries are even (and `b ≠ 0`). -/
lemma jacobi_sym_nat.even_even (a b : ℕ) (hb₀ : b ≠ 0) :
jacobi_sym_nat (bit0 a) (bit0 b) = 0 :=
begin
refine jacobi_sym.eq_zero_iff.mpr ⟨nat.bit0_ne_zero hb₀, λ hf, _⟩,
have h : 2 ∣ (bit0 a).gcd (bit0 b) := nat.dvd_gcd two_dvd_bit0 two_dvd_bit0,
change 2 ∣ (bit0 a : ℤ).gcd (bit0 b) at h,
rw [← nat.cast_bit0, ← nat.cast_bit0, hf, ← even_iff_two_dvd] at h,
exact nat.not_even_one h,
end
/-- When `a` is odd and `b` is even, we can replace `b` by `b / 2`. -/
lemma jacobi_sym_nat.odd_even (a b : ℕ) (r : ℤ) (hr : jacobi_sym_nat (bit1 a) b = r) :
jacobi_sym_nat (bit1 a) (bit0 b) = r :=
begin
have ha : legendre_sym 2 (bit1 a) = 1 :=
by simp only [legendre_sym, quadratic_char_apply, quadratic_char_fun_one, int.cast_bit1,
char_two.bit1_eq_one, pi.one_apply],
cases eq_or_ne b 0 with hb hb,
{ rw [← hr, hb, jacobi_sym_nat.zero_right], },
{ haveI : ne_zero b := ⟨hb⟩, -- for `jacobi_sym.mul_right`
rwa [bit0_eq_two_mul b, jacobi_sym_nat, jacobi_sym.mul_right,
← _root_.legendre_sym.to_jacobi_sym, nat.cast_bit1, ha, one_mul], }
end
/-- If `a` is divisible by `4` and `b` is odd, then we can remove the factor `4` from `a`. -/
lemma jacobi_sym_nat.double_even (a b : ℕ) (r : ℤ) (hr : jacobi_sym_nat a (bit1 b) = r) :
jacobi_sym_nat (bit0 (bit0 a)) (bit1 b) = r :=
begin
have : ((2 : ℕ) : ℤ).gcd ((bit1 b) : ℕ) = 1,
{ rw [int.coe_nat_gcd, nat.bit1_eq_succ_bit0, bit0_eq_two_mul b, nat.succ_eq_add_one,
nat.gcd_mul_left_add_right, nat.gcd_one_right], },
rwa [bit0_eq_two_mul a, bit0_eq_two_mul (2 * a), ← mul_assoc, ← pow_two, jacobi_sym_nat,
nat.cast_mul, nat.cast_pow, jacobi_sym.mul_left, jacobi_sym.sq_one' this, one_mul],
end
/-- If `a` is even and `b` is odd, then we can remove a factor `2` from `a`,
but we may have to change the sign, depending on `b % 8`.
We give one version for each of the four odd residue classes mod `8`. -/
lemma jacobi_sym_nat.even_odd₁ (a b : ℕ) (r : ℤ)
(hr : jacobi_sym_nat a (bit1 (bit0 (bit0 b))) = r) :
jacobi_sym_nat (bit0 a) (bit1 (bit0 (bit0 b))) = r :=
begin
have hb : (bit1 (bit0 (bit0 b))) % 8 = 1,
{ rw [nat.bit1_mod_bit0, nat.bit0_mod_bit0, nat.bit0_mod_two], },
rw [jacobi_sym_nat, bit0_eq_two_mul a, nat.cast_mul, jacobi_sym.mul_left,
nat.cast_two, jacobi_sym.at_two (odd_bit1 _), zmod.χ₈_nat_mod_eight, hb],
norm_num,
exact hr,
end
lemma jacobi_sym_nat.even_odd₇ (a b : ℕ) (r : ℤ)
(hr : jacobi_sym_nat a (bit1 (bit1 (bit1 b))) = r) :
jacobi_sym_nat (bit0 a) (bit1 (bit1 (bit1 b))) = r :=
begin
have hb : (bit1 (bit1 (bit1 b))) % 8 = 7,
{ rw [nat.bit1_mod_bit0, nat.bit1_mod_bit0, nat.bit1_mod_two], },
rw [jacobi_sym_nat, bit0_eq_two_mul a, nat.cast_mul, jacobi_sym.mul_left,
nat.cast_two, jacobi_sym.at_two (odd_bit1 _), zmod.χ₈_nat_mod_eight, hb],
norm_num,
exact hr,
end
lemma jacobi_sym_nat.even_odd₃ (a b : ℕ) (r : ℤ)
(hr : jacobi_sym_nat a (bit1 (bit1 (bit0 b))) = r) :
jacobi_sym_nat (bit0 a) (bit1 (bit1 (bit0 b))) = -r :=
begin
have hb : (bit1 (bit1 (bit0 b))) % 8 = 3,
{ rw [nat.bit1_mod_bit0, nat.bit1_mod_bit0, nat.bit0_mod_two], },
rw [jacobi_sym_nat, bit0_eq_two_mul a, nat.cast_mul, jacobi_sym.mul_left,
nat.cast_two, jacobi_sym.at_two (odd_bit1 _), zmod.χ₈_nat_mod_eight, hb],
norm_num,
exact hr,
end
lemma jacobi_sym_nat.even_odd₅ (a b : ℕ) (r : ℤ)
(hr : jacobi_sym_nat a (bit1 (bit0 (bit1 b))) = r) :
jacobi_sym_nat (bit0 a) (bit1 (bit0 (bit1 b))) = -r :=
begin
have hb : (bit1 (bit0 (bit1 b))) % 8 = 5,
{ rw [nat.bit1_mod_bit0, nat.bit0_mod_bit0, nat.bit1_mod_two], },
rw [jacobi_sym_nat, bit0_eq_two_mul a, nat.cast_mul, jacobi_sym.mul_left,
nat.cast_two, jacobi_sym.at_two (odd_bit1 _), zmod.χ₈_nat_mod_eight, hb],
norm_num,
exact hr,
end
/-- Use quadratic reciproity to reduce to smaller `b`. -/
lemma jacobi_sym_nat.qr₁ (a b : ℕ) (r : ℤ) (hr : jacobi_sym_nat (bit1 b) (bit1 (bit0 a)) = r) :
jacobi_sym_nat (bit1 (bit0 a)) (bit1 b) = r :=
begin
have ha : (bit1 (bit0 a)) % 4 = 1,
{ rw [nat.bit1_mod_bit0, nat.bit0_mod_two], },
have hb := nat.bit1_mod_two,
rwa [jacobi_sym_nat, jacobi_sym.quadratic_reciprocity_one_mod_four ha (nat.odd_iff.mpr hb)],
end
lemma jacobi_sym_nat.qr₁_mod (a b ab : ℕ) (r : ℤ) (hab : (bit1 b) % (bit1 (bit0 a)) = ab)
(hr : jacobi_sym_nat ab (bit1 (bit0 a)) = r) :
jacobi_sym_nat (bit1 (bit0 a)) (bit1 b) = r :=
jacobi_sym_nat.qr₁ _ _ _ $ jacobi_sym_nat.mod_left _ _ ab r hab hr
lemma jacobi_sym_nat.qr₁' (a b : ℕ) (r : ℤ) (hr : jacobi_sym_nat (bit1 (bit0 b)) (bit1 a) = r) :
jacobi_sym_nat (bit1 a) (bit1 (bit0 b)) = r :=
begin
have hb : (bit1 (bit0 b)) % 4 = 1,
{ rw [nat.bit1_mod_bit0, nat.bit0_mod_two], },
have ha := nat.bit1_mod_two,
rwa [jacobi_sym_nat, ← jacobi_sym.quadratic_reciprocity_one_mod_four hb (nat.odd_iff.mpr ha)]
end
lemma jacobi_sym_nat.qr₁'_mod (a b ab : ℕ) (r : ℤ) (hab : (bit1 (bit0 b)) % (bit1 a) = ab)
(hr : jacobi_sym_nat ab (bit1 a) = r) :
jacobi_sym_nat (bit1 a) (bit1 (bit0 b)) = r :=
jacobi_sym_nat.qr₁' _ _ _ $ jacobi_sym_nat.mod_left _ _ ab r hab hr
lemma jacobi_sym_nat.qr₃ (a b : ℕ) (r : ℤ)
(hr : jacobi_sym_nat (bit1 (bit1 b)) (bit1 (bit1 a)) = r) :
jacobi_sym_nat (bit1 (bit1 a)) (bit1 (bit1 b)) = -r :=
begin
have hb : (bit1 (bit1 b)) % 4 = 3,
{ rw [nat.bit1_mod_bit0, nat.bit1_mod_two], },
have ha : (bit1 (bit1 a)) % 4 = 3,
{ rw [nat.bit1_mod_bit0, nat.bit1_mod_two], },
rwa [jacobi_sym_nat, jacobi_sym.quadratic_reciprocity_three_mod_four ha hb, neg_inj]
end
lemma jacobi_sym_nat.qr₃_mod (a b ab : ℕ) (r : ℤ) (hab : (bit1 (bit1 b)) % (bit1 (bit1 a)) = ab)
(hr : jacobi_sym_nat ab (bit1 (bit1 a)) = r) :
jacobi_sym_nat (bit1 (bit1 a)) (bit1 (bit1 b)) = -r :=
jacobi_sym_nat.qr₃ _ _ _ $ jacobi_sym_nat.mod_left _ _ ab r hab hr
end norm_num
end lemmas
section evaluation
/-!
### Certified evaluation of the Jacobi symbol
The following functions recursively evaluate a Jacobi symbol and construct the
corresponding proof term.
-/
namespace norm_num
open tactic
/-- This evaluates `r := jacobi_sym_nat a b` recursively using quadratic reciprocity
and produces a proof term for the equality, assuming that `a < b` and `b` is odd. -/
meta def prove_jacobi_sym_odd : instance_cache → instance_cache → expr → expr →
tactic (instance_cache × instance_cache × expr × expr)
| zc nc ea eb := do
match match_numeral eb with
| match_numeral_result.one := -- `b = 1`, result is `1`
pure (zc, nc, `(1 : ℤ), `(jacobi_sym_nat.one_right).mk_app [ea])
| match_numeral_result.bit1 eb₁ := do -- `b > 1` (recall that `b` is odd)
match match_numeral ea with
| match_numeral_result.zero := do -- `a = 0`, result is `0`
b ← eb₁.to_nat,
(nc, phb₀) ← prove_ne nc eb₁ `(0 : ℕ) b 0, -- proof of `b ≠ 0`
pure (zc, nc, `(0 : ℤ), `(jacobi_sym_nat.zero_left_odd).mk_app [eb₁, phb₀])
| match_numeral_result.one := do -- `a = 1`, result is `1`
pure (zc, nc, `(1 : ℤ), `(jacobi_sym_nat.one_left_odd).mk_app [eb₁])
| match_numeral_result.bit0 ea₁ := do -- `a` is even; check if divisible by `4`
match match_numeral ea₁ with
| match_numeral_result.bit0 ea₂ := do
(zc, nc, er, p) ← prove_jacobi_sym_odd zc nc ea₂ eb, -- compute `jacobi_sym_nat (a / 4) b`
pure (zc, nc, er, `(jacobi_sym_nat.double_even).mk_app [ea₂, eb₁, er, p])
| _ := do -- reduce to `a / 2`; need to consider `b % 8`
(zc, nc, er, p) ← prove_jacobi_sym_odd zc nc ea₁ eb, -- compute `jacobi_sym_nat (a / 2) b`
match match_numeral eb₁ with
-- | match_numeral_result.zero := -- `b = 1`, not reached
| match_numeral_result.one := do -- `b = 3`
r ← er.to_int,
(zc, er') ← zc.of_int (- r),
pure (zc, nc, er', `(jacobi_sym_nat.even_odd₃).mk_app [ea₁, `(0 : ℕ), er, p])
| match_numeral_result.bit0 eb₂ := do -- `b % 4 = 1`
match match_numeral eb₂ with
-- | match_numeral_result.zero := -- not reached
| match_numeral_result.one := do -- `b = 5`
r ← er.to_int,
(zc, er') ← zc.of_int (- r),
pure (zc, nc, er', `(jacobi_sym_nat.even_odd₅).mk_app [ea₁, `(0 : ℕ), er, p])
| match_numeral_result.bit0 eb₃ := do -- `b % 8 = 1`
pure (zc, nc, er, `(jacobi_sym_nat.even_odd₁).mk_app [ea₁, eb₃, er, p])
| match_numeral_result.bit1 eb₃ := do -- `b % 8 = 5`
r ← er.to_int,
(zc, er') ← zc.of_int (- r),
pure (zc, nc, er', `(jacobi_sym_nat.even_odd₅).mk_app [ea₁, eb₃, er, p])
| _ := failed
end
| match_numeral_result.bit1 eb₂ := do -- `b % 4 = 3`
match match_numeral eb₂ with
-- | match_numeral_result.zero := -- not reached
| match_numeral_result.one := do -- `b = 7`
pure (zc, nc, er, `(jacobi_sym_nat.even_odd₇).mk_app [ea₁, `(0 : ℕ), er, p])
| match_numeral_result.bit0 eb₃ := do -- `b % 8 = 3`
r ← er.to_int,
(zc, er') ← zc.of_int (- r),
pure (zc, nc, er', `(jacobi_sym_nat.even_odd₃).mk_app [ea₁, eb₃, er, p])
| match_numeral_result.bit1 eb₃ := do -- `b % 8 = 7`
pure (zc, nc, er, `(jacobi_sym_nat.even_odd₇).mk_app [ea₁, eb₃, er, p])
| _ := failed
end
| _ := failed
end
end
| match_numeral_result.bit1 ea₁ := do -- `a` is odd
-- use Quadratic Reciprocity; look at `a` and `b` mod `4`
(nc, bma, phab) ← prove_div_mod nc eb ea tt, -- compute `b % a`
(zc, nc, er, p) ← prove_jacobi_sym_odd zc nc bma ea, -- compute `jacobi_sym_nat (b % a) a`
match match_numeral ea₁ with
-- | match_numeral_result.zero := -- `a = 1`, not reached
| match_numeral_result.one := do -- `a = 3`; need to consider `b`
match match_numeral eb₁ with
-- | match_numeral_result.zero := -- `b = 1`, not reached
-- | match_numeral_result.one := -- `b = 3`, not reached, since `a < b`
| match_numeral_result.bit0 eb₂ := do -- `b % 4 = 1`
pure (zc, nc, er, `(jacobi_sym_nat.qr₁'_mod).mk_app [ea₁, eb₂, bma, er, phab, p])
| match_numeral_result.bit1 eb₂ := do -- `b % 4 = 3`
r ← er.to_int,
(zc, er') ← zc.of_int (- r),
pure (zc, nc, er', `(jacobi_sym_nat.qr₃_mod).mk_app [`(0 : ℕ), eb₂, bma, er, phab, p])
| _ := failed
end
| match_numeral_result.bit0 ea₂ := do -- `a % 4 = 1`
pure (zc, nc, er, `(jacobi_sym_nat.qr₁_mod).mk_app [ea₂, eb₁, bma, er, phab, p])
| match_numeral_result.bit1 ea₂ := do -- `a % 4 = 3`; need to consider `b`
match match_numeral eb₁ with
-- | match_numeral_result.zero := do -- `b = 1`, not reached
-- | match_numeral_result.one := do -- `b = 3`, not reached, since `a < b`
| match_numeral_result.bit0 eb₂ := do -- `b % 4 = 1`
pure (zc, nc, er, `(jacobi_sym_nat.qr₁'_mod).mk_app [ea₁, eb₂, bma, er, phab, p])
| match_numeral_result.bit1 eb₂ := do -- `b % 4 = 3`
r ← er.to_int,
(zc, er') ← zc.of_int (- r),
pure (zc, nc, er', `(jacobi_sym_nat.qr₃_mod).mk_app [ea₂, eb₂, bma, er, phab, p])
| _ := failed
end
| _ := failed
end
| _ := failed
end
| _ := failed
end
/-- This evaluates `r := jacobi_sym_nat a b` and produces a proof term for the equality
by removing powers of `2` from `b` and then calling `prove_jacobi_sym_odd`. -/
meta def prove_jacobi_sym_nat : instance_cache → instance_cache → expr → expr →
tactic (instance_cache × instance_cache × expr × expr)
| zc nc ea eb := do
match match_numeral eb with
| match_numeral_result.zero := -- `b = 0`, result is `1`
pure (zc, nc, `(1 : ℤ), `(jacobi_sym_nat.zero_right).mk_app [ea])
| match_numeral_result.one := -- `b = 1`, result is `1`
pure (zc, nc, `(1 : ℤ), `(jacobi_sym_nat.one_right).mk_app [ea])
| match_numeral_result.bit0 eb₁ := -- `b` is even and nonzero
match match_numeral ea with
| match_numeral_result.zero := do -- `a = 0`, result is `0`
b ← eb₁.to_nat,
(nc, phb₀) ← prove_ne nc eb₁ `(0 : ℕ) b 0, -- proof of `b ≠ 0`
pure (zc, nc, `(0 : ℤ), `(jacobi_sym_nat.zero_left_even).mk_app [eb₁, phb₀])
| match_numeral_result.one := do -- `a = 1`, result is `1`
pure (zc, nc, `(1 : ℤ), `(jacobi_sym_nat.one_left_even).mk_app [eb₁])
| match_numeral_result.bit0 ea₁ := do -- `a` is even, result is `0`
b ← eb₁.to_nat,
(nc, phb₀) ← prove_ne nc eb₁ `(0 : ℕ) b 0, -- proof of `b ≠ 0`
let er : expr := `(0 : ℤ),
pure (zc, nc, er, `(jacobi_sym_nat.even_even).mk_app [ea₁, eb₁, phb₀])
| match_numeral_result.bit1 ea₁ := do -- `a` is odd, reduce to `b / 2`
(zc, nc, er, p) ← prove_jacobi_sym_nat zc nc ea eb₁,
pure (zc, nc, er, `(jacobi_sym_nat.odd_even).mk_app [ea₁, eb₁, er, p])
| _ := failed
end
| match_numeral_result.bit1 eb₁ := do -- `b` is odd
a ← ea.to_nat,
b ← eb.to_nat,
if b ≤ a then do -- reduce to `jacobi_sym_nat (a % b) b`
(nc, amb, phab) ← prove_div_mod nc ea eb tt, -- compute `a % b`
(zc, nc, er, p) ← prove_jacobi_sym_odd zc nc amb eb, -- compute `jacobi_sym_nat (a % b) b`
pure (zc, nc, er, `(jacobi_sym_nat.mod_left).mk_app [ea, eb, amb, er, phab, p])
else
prove_jacobi_sym_odd zc nc ea eb
| _ := failed
end
/-- This evaluates `r := jacobi_sym a b` and produces a proof term for the equality.
This is done by reducing to `r := jacobi_sym_nat (a % b) b`. -/
meta def prove_jacobi_sym : instance_cache → instance_cache → expr → expr
→ tactic (instance_cache × instance_cache × expr × expr)
| zc nc ea eb := do
match match_numeral eb with -- deal with simple cases right away
| match_numeral_result.zero := pure (zc, nc, `(1 : ℤ), `(jacobi_sym.zero_right).mk_app [ea])
| match_numeral_result.one := pure (zc, nc, `(1 : ℤ), `(jacobi_sym.one_right).mk_app [ea])
| _ := do -- Now `1 < b`. Compute `jacobi_sym_nat (a % b) b` instead.
b ← eb.to_nat,
(zc, eb') ← zc.of_int (b : ℤ),
-- Get the proof that `(b : ℤ) = b'` (where `eb'` is the numeral representing `b'`).
-- This is important to avoid inefficient matching between the two.
(zc, nc, eb₁, pb') ← prove_nat_uncast zc nc eb',
(zc, amb, phab) ← prove_div_mod zc ea eb' tt, -- compute `a % b`
(zc, nc, amb', phab') ← prove_nat_uncast zc nc amb, -- `a % b` as a natural number
(zc, nc, er, p) ← prove_jacobi_sym_nat zc nc amb' eb₁, -- compute `jacobi_sym_nat (a % b) b`
pure (zc, nc, er,
`(jacobi_sym.mod_left).mk_app [ea, eb₁, amb', amb, er, eb', pb', phab, phab', p])
end
end norm_num
end evaluation
section tactic
/-!
### The `norm_num` plug-in
-/
namespace tactic
namespace norm_num
/-- This is the `norm_num` plug-in that evaluates Jacobi and Legendre symbols. -/
@[norm_num] meta def eval_jacobi_sym : expr → tactic (expr × expr)
| `(jacobi_sym %%ea %%eb) := do -- Jacobi symbol
zc ← mk_instance_cache `(ℤ),
nc ← mk_instance_cache `(ℕ),
(prod.snd ∘ prod.snd) <$> norm_num.prove_jacobi_sym zc nc ea eb
| `(norm_num.jacobi_sym_nat %%ea %%eb) := do -- Jacobi symbol on natural numbers
zc ← mk_instance_cache `(ℤ),
nc ← mk_instance_cache `(ℕ),
(prod.snd ∘ prod.snd) <$> norm_num.prove_jacobi_sym_nat zc nc ea eb
| `(@legendre_sym %%ep %%inst %%ea) := do -- Legendre symbol
zc ← mk_instance_cache `(ℤ),
nc ← mk_instance_cache `(ℕ),
(zc, nc, er, pf) ← norm_num.prove_jacobi_sym zc nc ea ep,
pure (er, `(norm_num.legendre_sym.to_jacobi_sym).mk_app [ep, inst, ea, er, pf])
| _ := failed
end norm_num
end tactic
end tactic
|
#
# This file is part of the Actors.jl Julia package,
# MIT license, part of https://github.com/JuliaActors
#
#
# remote node failure detection (RNFD) actor
# ----------------------------------------------
# if an actor on a (relative) remote node is added to a
# supervisor an actor is started scanning the remote link
# in regular intervals. If a ProcessExitedException is
# detected, it notifies the supervisor about it.
#
const scan_interval = 1.0
struct RNFD{L,S,T}
sv::L # supervisor link
lks::S # links to remote actors
pids::T # supervised pids
end
struct Add{L}
lk::L
end
struct Remove{L}
lk::L
end
struct Scan end
#
# RNFD behavior
#
function (rfd::RNFD)(msg::Add)
if msg.lk ∉ rfd.lks
push!(rfd.lks, msg.lk)
msg.lk.pid ∈ rfd.pids || push!(rfd.pids, msg.lk.pid)
end
end
function (rfd::RNFD)(msg::Remove)
if msg.lk ∈ rfd.lks
filter!(≠(msg.lk), rfd.lks)
pids = (lk.pid for lk ∈ rfd.lks)
filter!(∈(pids), rfd.pids)
end
end
function (rfd::RNFD)(::Scan)
isempty(rfd.lks) && return nothing
excs = Int[]
for lk in rfd.lks
try
isready(lk.chn)
catch exc
filter!(≠(lk), rfd.lks)
exc isa ProcessExitedException && push!(excs, exc.worker_id)
end
end
if !isempty(excs)
filter!(lk->lk.pid ∉ excs, rfd.lks)
filter!(p->p ∉ excs, rfd.pids)
send(rfd.sv, NodeFailure(unique(excs)))
end
end
#
# Supervisor behavior for NodeFailure
#
#
# remove temporary childs from both s and fchilds
#
function remove_temporary!(s, fchilds)
act = task_local_storage("_ACT")
filter!(fchilds) do child
if child.info.restart == :temporary
log_warn("temporary actor $(isnothing(child.name) ? :noname : child.name) failed, $(ProcessExitedException(child.lk.pid))")
filter!(c->c.lk!=child.lk, act.conn)
filter!(c->c.lk!=child.lk, s.childs)
return false
else
return true
end
end
end
function restart_child!(c::Child, pid::Int)
log_warn("supervisor: restarting child $(isnothing(c.name) ? :noname : c.name) on pid $pid")
if c.lk isa Link
lk = !isnothing(c.start) ? c.start(pid) : spawn(c.init; pid)
c.lk.chn = lk.chn
c.lk.pid = lk.pid
isnothing(c.name) || update!(lk, c.name, s=:name)
end
end
function restart!(s::Supervisor, cs::Vector{Child}, pids::Vector{Int})
if s.option[:strategy] == :one_for_one
for (i, c) in enumerate(cs)
restart_child!(c, pids[i])
rnfd_add(s, c.lk)
end
elseif s.option[:strategy] == :one_for_all
log_warn("supervisor: restarting all")
for child in s.childs
child ∈ cs ?
begin
restart_child!(child, pids[findfirst(==(child),cs)])
rnfd_add(s, child.lk)
end :
child.lk.mode ≠ :rnfd && shutdown_restart_child!(child)
end
else
log_warn("supervisor: restarting rest")
ix = findfirst(c->c ∈ cs, s.childs)
for child in s.childs[ix:end]
child ∈ cs ?
begin
restart_child!(child, pids[findfirst(==(child),cs)])
rnfd_add(s, child.lk)
end :
child.lk.mode ≠ :rnfd && shutdown_restart_child!(child)
end
end
end
#
# return spare pids for failed childs cs and delete them from
# the s.option[:spares] dict entry
#
function spare_pids!(s::Supervisor, cs)
spares = if haskey(s.option, :spares) && !isempty(s.option[:spares])
s.option[:spares]
else
used = unique(map(c->c.lk.pid, s.childs))
filter(p->p ∉ used, reverse(procs()))
end
filter!(p->p ∈ procs(), spares)
pids = map(c->c.lk.pid, cs)
p_old = sort(unique(pids))
if length(p_old) ≤ length(spares)
p_new = spares[1:length(p_old)]
rp = [p_old[i]=>p_new[i] for i ∈ 1:length(p_old)]
replace!(pids, rp...)
elseif !isempty(spares)
pids = rand(spares, length(pids))
else
pids = rand(procs(), length(pids))
end
haskey(s.option, :spares) && filter!(p->p ∉ pids, s.option[:spares])
return pids
end
function (s::Supervisor)(msg::NodeFailure)
foreach(msg.pids) do pid
log_warn("supervisor: Process $pid exited!")
end
failed_childs = filter(c->c.lk.pid ∈ msg.pids, s.childs)
remove_temporary!(s, failed_childs)
if !isempty(failed_childs)
if restart_limit!(s)
log_warn("supervisor: restart limit $(s.option[:max_restarts]) exceeded!")
send(self(), Exit(:shutdown, fill(nothing, 3)...))
else
restart!(s, failed_childs, spare_pids!(s, failed_childs))
end
end
end
#
# RNFD API
#
"""
rnfd_start(sv::Link;; interval=1, kwargs...)
Start a RNFD actor and return a link to it.
# Arguments
- `sv::Link`: supervisor to the actor,
- `interval=1`: interval in seconds for checking remote nodes.
"""
function rnfd_start(sv::Link; interval=1, kwargs...)
lk = spawn(RNFD(sv, Link[], Int[]); mode = :rnfd, kwargs...)
exec(lk, supervise, sv)
timer = Timer(interval; interval) do t
send(lk, Scan())
end
term!(lk, (exp)->close(timer))
return lk
end
#
# get a link to the RNFD actor, create it if it doesn't exist.
#
function rnfd(s::Supervisor)
i = findfirst(c->c.lk.mode == :rnfd, s.childs)
return !isnothing(i) ? s.childs[i].lk : rnfd_start(self())
end
rnfd_exists(s::Supervisor) = !isnothing(findfirst(c->c.lk.mode == :rnfd, s.childs))
#
# add a remote child to an RNFD actor,
# create it first if it doesn't exist.
#
rnfd_add(s::Supervisor, child::Link) = send(rnfd(s), Add(child))
|
/-
Copyright (c) 2022 Oliver Nash. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Oliver Nash
! This file was ported from Lean 3 source module algebra.module.bimodule
! leanprover-community/mathlib commit 58cef51f7a819e7227224461e392dee423302f2d
! Please do not edit these lines, except to modify the commit id
! if you have ported upstream changes.
-/
import Mathbin.RingTheory.TensorProduct
/-!
# Bimodules
One frequently encounters situations in which several sets of scalars act on a single space, subject
to compatibility condition(s). A distinguished instance of this is the theory of bimodules: one has
two rings `R`, `S` acting on an additive group `M`, with `R` acting covariantly ("on the left")
and `S` acting contravariantly ("on the right"). The compatibility condition is just:
`(r • m) • s = r • (m • s)` for all `r : R`, `s : S`, `m : M`.
This situation can be set up in Mathlib as:
```lean
variables (R S M : Type*) [ring R] [ring S]
variables [add_comm_group M] [module R M] [module Sᵐᵒᵖ M] [smul_comm_class R Sᵐᵒᵖ M]
```
The key fact is:
```lean
example : module (R ⊗[ℕ] Sᵐᵒᵖ) M := tensor_product.algebra.module
```
Note that the corresponding result holds for the canonically isomorphic ring `R ⊗[ℤ] Sᵐᵒᵖ` but it is
preferable to use the `R ⊗[ℕ] Sᵐᵒᵖ` instance since it works without additive inverses.
Bimodules are thus just a special case of `module`s and most of their properties follow from the
theory of `module`s`. In particular a two-sided submodule of a bimodule is simply a term of type
`submodule (R ⊗[ℕ] Sᵐᵒᵖ) M`.
This file is a place to collect results which are specific to bimodules.
## Main definitions
* `subbimodule.mk`
* `subbimodule.smul_mem`
* `subbimodule.smul_mem'`
* `subbimodule.to_submodule`
* `subbimodule.to_submodule'`
## Implementation details
For many definitions and lemmas it is preferable to set things up without opposites, i.e., as:
`[module S M] [smul_comm_class R S M]` rather than `[module Sᵐᵒᵖ M] [smul_comm_class R Sᵐᵒᵖ M]`.
The corresponding results for opposites then follow automatically and do not require taking
advantage of the fact that `(Sᵐᵒᵖ)ᵐᵒᵖ` is defeq to `S`.
## TODO
Develop the theory of two-sided ideals, which have type `submodule (R ⊗[ℕ] Rᵐᵒᵖ) R`.
-/
open TensorProduct
attribute [local instance] TensorProduct.Algebra.module
namespace Subbimodule
section Algebra
variable {R A B M : Type _}
variable [CommSemiring R] [AddCommMonoid M] [Module R M]
variable [Semiring A] [Semiring B] [Module A M] [Module B M]
variable [Algebra R A] [Algebra R B]
variable [IsScalarTower R A M] [IsScalarTower R B M]
variable [SMulCommClass A B M]
/-- A constructor for a subbimodule which demands closure under the two sets of scalars
individually, rather than jointly via their tensor product.
Note that `R` plays no role but it is convenient to make this generalisation to support the cases
`R = ℕ` and `R = ℤ` which both show up naturally. See also `base_change`. -/
@[simps]
def mk (p : AddSubmonoid M) (hA : ∀ (a : A) {m : M}, m ∈ p → a • m ∈ p)
(hB : ∀ (b : B) {m : M}, m ∈ p → b • m ∈ p) : Submodule (A ⊗[R] B) M :=
{ p with
carrier := p
smul_mem' := fun ab m =>
TensorProduct.induction_on ab (fun hm => by simpa only [zero_smul] using p.zero_mem)
(fun a b hm => by simpa only [TensorProduct.Algebra.smul_def] using hA a (hB b hm))
fun z w hz hw hm => by simpa only [add_smul] using p.add_mem (hz hm) (hw hm) }
#align subbimodule.mk Subbimodule.mk
theorem smul_mem (p : Submodule (A ⊗[R] B) M) (a : A) {m : M} (hm : m ∈ p) : a • m ∈ p :=
by
suffices a • m = a ⊗ₜ[R] (1 : B) • m by exact this.symm ▸ p.smul_mem _ hm
simp [TensorProduct.Algebra.smul_def]
#align subbimodule.smul_mem Subbimodule.smul_mem
theorem smul_mem' (p : Submodule (A ⊗[R] B) M) (b : B) {m : M} (hm : m ∈ p) : b • m ∈ p :=
by
suffices b • m = (1 : A) ⊗ₜ[R] b • m by exact this.symm ▸ p.smul_mem _ hm
simp [TensorProduct.Algebra.smul_def]
#align subbimodule.smul_mem' Subbimodule.smul_mem'
/-- If `A` and `B` are also `algebra`s over yet another set of scalars `S` then we may "base change"
from `R` to `S`. -/
@[simps]
def baseChange (S : Type _) [CommSemiring S] [Module S M] [Algebra S A] [Algebra S B]
[IsScalarTower S A M] [IsScalarTower S B M] (p : Submodule (A ⊗[R] B) M) :
Submodule (A ⊗[S] B) M :=
mk p.toAddSubmonoid (smul_mem p) (smul_mem' p)
#align subbimodule.base_change Subbimodule.baseChange
/-- Forgetting the `B` action, a `submodule` over `A ⊗[R] B` is just a `submodule` over `A`. -/
@[simps]
def toSubmodule (p : Submodule (A ⊗[R] B) M) : Submodule A M :=
{ p with
carrier := p
smul_mem' := smul_mem p }
#align subbimodule.to_submodule Subbimodule.toSubmodule
/-- Forgetting the `A` action, a `submodule` over `A ⊗[R] B` is just a `submodule` over `B`. -/
@[simps]
def toSubmodule' (p : Submodule (A ⊗[R] B) M) : Submodule B M :=
{ p with
carrier := p
smul_mem' := smul_mem' p }
#align subbimodule.to_submodule' Subbimodule.toSubmodule'
end Algebra
section Ring
variable (R S M : Type _) [Ring R] [Ring S]
variable [AddCommGroup M] [Module R M] [Module S M] [SMulCommClass R S M]
/-- A `submodule` over `R ⊗[ℕ] S` is naturally also a `submodule` over the canonically-isomorphic
ring `R ⊗[ℤ] S`. -/
@[simps]
def toSubbimoduleInt (p : Submodule (R ⊗[ℕ] S) M) : Submodule (R ⊗[ℤ] S) M :=
baseChange ℤ p
#align subbimodule.to_subbimodule_int Subbimodule.toSubbimoduleInt
/-- A `submodule` over `R ⊗[ℤ] S` is naturally also a `submodule` over the canonically-isomorphic
ring `R ⊗[ℕ] S`. -/
@[simps]
def toSubbimoduleNat (p : Submodule (R ⊗[ℤ] S) M) : Submodule (R ⊗[ℕ] S) M :=
baseChange ℕ p
#align subbimodule.to_subbimodule_nat Subbimodule.toSubbimoduleNat
end Ring
end Subbimodule
|
include("../benchmark/bench_van_der_pol.jl")
@time lc_solver = pre_solve!(make_van_der_pol_lc_solver())
@time sweep!(lc_solver)
@time lc_solver = pre_solve!(make_van_der_pol_lc_solver())
Profile.clear()
@profile sweep!(lc_solver)
|
You may remember, dear reader, that I have called a bear market for equities. How am I doing so far? Well, the S&P 500 is up 13% from its February 11th intra-day low (1812). That sure doesn’t look good, does it? But I’m still convinced I’m right. I haven’t shorted the market yet, and won’t until the S&Ps break that February low. So this rally hasn’t hurt me, it’s just annoying to have to wait.Maybe Wall Street still has inventory to unload, as David Stockman says.
The signs of an impending flush are now everywhere. The March 15 release of business sales for January, for example, showed another down month. The critical inventory-to-sales ratio for the entire economy is now at 1.40—–a ratio last recorded in May 2009.
Once upon a time, when I was a young lad, investors used to watch the real economy, not the latest Fed head jabbering on CNBC. One of the things we paid keen attention to was the inventory-to-sales ratio because we knew that’s where recessions came from, and recessions were not good for the stock market.
When sales slow down, inventories build up. The build in inventories actually continues to boost GDP because the supply chain hasn’t got the message yet. When they do, they stop producing as much, inventories fall faster than sales and the GDP falls, giving us an official recession…ALWAYS after the fact. Those who watch GDP, like most economists, totally miss the boat. They never call the recession before it starts because they follow their models and not the real economy. Only highly intelligent and educated people can be this stupid, as I am fond of saying.
Business sales as reported on March 15 were down by 5.1% from their July 2014 peak. Declines of this magnitude have occurred only twice since 1992 and both times they signalled a recession.
Here is the chart of total business sales for all levels of the economy. Is this hard to understand?
The shaded areas mark recessions.
If you haven’t sold your stocks yet, the gods of the markets are giving you another chance, just an eyelash below the all-time highs. Accept the gift, dear reader, accept the gift.
Next Next post: Is China Dodging Its Debt Bomb? |
theory prop_10
imports Main
"$HIPSTER_HOME/IsaHipster"
begin
datatype 'a list = Nil2 | Cons2 "'a" "'a list"
fun append :: "'a list => 'a list => 'a list" where
"append (Nil2) y = y"
| "append (Cons2 z xs) y = Cons2 z (append xs y)"
fun rev :: "'a list => 'a list" where
"rev (Nil2) = Nil2"
| "rev (Cons2 y xs) = append (rev xs) (Cons2 y (Nil2))"
(*hipster append rev *)
(*hipster rev*)
lemma lemma_a [thy_expl]: "append x2 Nil2 = x2"
by (hipster_induct_schemes rev.simps)
lemma lemma_aa [thy_expl]: "append (append x2 y2) z2 = append x2 (append y2 z2)"
by (hipster_induct_schemes rev.simps)
lemma lemma_ab [thy_expl]: "append (rev x5) (rev y5) = rev (append y5 x5)"
by (hipster_induct_schemes rev.simps)
lemma unknown [thy_expl]: "rev (rev x) = x"
oops
theorem x0 :
"(rev (rev x)) = x"
by(hipster_induct_schemes rev.simps append.simps)
(* by (tactic {* Subgoal.FOCUS_PARAMS (K (Tactic_Data.hard_tac @{context})) @{context} 1 *})*)
end
|
/*
This file is part of solidity.
solidity is free software: you can redistribute it and/or modify
it under the terms of the GNU General Public License as published by
the Free Software Foundation, either version 3 of the License, or
(at your option) any later version.
solidity is distributed in the hope that it will be useful,
but WITHOUT ANY WARRANTY; without even the implied warranty of
MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the
GNU General Public License for more details.
You should have received a copy of the GNU General Public License
along with solidity. If not, see <http://www.gnu.org/licenses/>.
*/
#include <test/libyul/YulInterpreterTest.h>
#include <test/tools/yulInterpreter/Interpreter.h>
#include <test/Common.h>
#include <libyul/backends/evm/EVMDialect.h>
#include <libyul/AsmParser.h>
#include <libyul/AssemblyStack.h>
#include <libyul/AsmAnalysisInfo.h>
#include <liblangutil/ErrorReporter.h>
#include <liblangutil/SourceReferenceFormatter.h>
#include <libsolutil/AnsiColorized.h>
#include <boost/test/unit_test.hpp>
#include <boost/algorithm/string.hpp>
#include <fstream>
using namespace solidity;
using namespace solidity::util;
using namespace solidity::langutil;
using namespace solidity::yul;
using namespace solidity::yul::test;
using namespace solidity::frontend;
using namespace solidity::frontend::test;
using namespace std;
YulInterpreterTest::YulInterpreterTest(string const& _filename):
EVMVersionRestrictedTestCase(_filename)
{
m_source = m_reader.source();
m_expectation = m_reader.simpleExpectations();
}
TestCase::TestResult YulInterpreterTest::run(ostream& _stream, string const& _linePrefix, bool const _formatted)
{
if (!parse(_stream, _linePrefix, _formatted))
return TestResult::FatalError;
m_obtainedResult = interpret();
if (m_expectation != m_obtainedResult)
{
string nextIndentLevel = _linePrefix + " ";
AnsiColorized(_stream, _formatted, {formatting::BOLD, formatting::CYAN}) << _linePrefix << "Expected result:" << endl;
// TODO could compute a simple diff with highlighted lines
printIndented(_stream, m_expectation, nextIndentLevel);
AnsiColorized(_stream, _formatted, {formatting::BOLD, formatting::CYAN}) << _linePrefix << "Obtained result:" << endl;
printIndented(_stream, m_obtainedResult, nextIndentLevel);
return TestResult::Failure;
}
return TestResult::Success;
}
void YulInterpreterTest::printSource(ostream& _stream, string const& _linePrefix, bool const) const
{
printIndented(_stream, m_source, _linePrefix);
}
void YulInterpreterTest::printUpdatedExpectations(ostream& _stream, string const& _linePrefix) const
{
printIndented(_stream, m_obtainedResult, _linePrefix);
}
void YulInterpreterTest::printIndented(ostream& _stream, string const& _output, string const& _linePrefix) const
{
stringstream output(_output);
string line;
while (getline(output, line))
_stream << _linePrefix << line << endl;
}
bool YulInterpreterTest::parse(ostream& _stream, string const& _linePrefix, bool const _formatted)
{
AssemblyStack stack(
solidity::test::CommonOptions::get().evmVersion(),
AssemblyStack::Language::StrictAssembly,
solidity::frontend::OptimiserSettings::none()
);
if (stack.parseAndAnalyze("", m_source))
{
m_ast = stack.parserResult()->code;
m_analysisInfo = stack.parserResult()->analysisInfo;
return true;
}
else
{
AnsiColorized(_stream, _formatted, {formatting::BOLD, formatting::RED}) << _linePrefix << "Error parsing source." << endl;
printErrors(_stream, stack.errors());
return false;
}
}
string YulInterpreterTest::interpret()
{
InterpreterState state;
state.maxTraceSize = 10000;
state.maxSteps = 10000;
Interpreter interpreter(state, EVMDialect::strictAssemblyForEVMObjects(langutil::EVMVersion{}));
try
{
interpreter(*m_ast);
}
catch (InterpreterTerminatedGeneric const&)
{
}
stringstream result;
state.dumpTraceAndState(result);
return result.str();
}
void YulInterpreterTest::printErrors(ostream& _stream, ErrorList const& _errors)
{
SourceReferenceFormatter formatter(_stream);
for (auto const& error: _errors)
formatter.printErrorInformation(*error);
}
|
/*
Copyright 2014 Glen Joseph Fernandes
([email protected])
Distributed under the Boost Software License, Version 1.0.
(http://www.boost.org/LICENSE_1_0.txt)
*/
#ifndef BOOST_ALIGN_IS_ALIGNED_HPP
#define BOOST_ALIGN_IS_ALIGNED_HPP
#include <boost/align/detail/is_aligned.hpp>
#include <boost/align/detail/not_pointer.hpp>
namespace boost {
namespace alignment {
template<class T>
BOOST_CONSTEXPR inline typename detail::not_pointer<T, bool>::type
is_aligned(T value, std::size_t alignment) BOOST_NOEXCEPT
{
return (value & (T(alignment) - 1)) == 0;
}
} /* alignment */
} /* boost */
#endif
|
#include <Python.h>
#include <boost/python.hpp>
#include "FabMapVocabluary.h"
#include "ChowLiuTree.h"
#include "openFABMAPPython.h"
BOOST_PYTHON_MODULE(openFABMAP)
{
boost::python::class_<pyof2::FabMapVocabluary, std::shared_ptr<pyof2::FabMapVocabluary>>(
"Vocabluary", boost::python::no_init);
boost::python::class_<pyof2::FabMapVocabluaryBuilder, std::shared_ptr<pyof2::FabMapVocabluaryBuilder>>(
"VocabluaryBuilder", boost::python::init<boost::python::dict>())
.def("add_training_image", &pyof2::FabMapVocabluaryBuilder::addTrainingImage)
.def("build_vocabluary", &pyof2::FabMapVocabluaryBuilder::buildVocabluary);
boost::python::class_<pyof2::ChowLiuTree, std::shared_ptr<pyof2::ChowLiuTree>>(
"ChowLiuTree", boost::python::init<std::shared_ptr<pyof2::FabMapVocabluary>, boost::python::dict>())
.def("add_training_image", &pyof2::ChowLiuTree::addTrainingImage)
.def("build_chow_liu_tree", &pyof2::ChowLiuTree::buildChowLiuTree)
.def("save", &pyof2::ChowLiuTree::save)
.def("load", &pyof2::ChowLiuTree::load)
.staticmethod("load");
boost::python::class_<pyof2::OpenFABMAPPython, std::shared_ptr<pyof2::OpenFABMAPPython>>(
"OpenFABMAP", boost::python::init<
std::shared_ptr<pyof2::ChowLiuTree>,
boost::python::dict>())
.def("load_and_process_image", &pyof2::OpenFABMAPPython::loadAndProcessImage)
.def("get_last_match", &pyof2::OpenFABMAPPython::getLastMatch)
.def("get_all_loop_closures", &pyof2::OpenFABMAPPython::getAllLoopClosures);
}
|
function showmesh3(node,elem,expr,varargin)
%% SHOWMESH3 displays a tetrahedron mesh in 3-D.
%
% showmesh3(node,elem) displays a 3-dimensional tetrahderon mesh given
% by node and elem matrices; see <a href="matlab:ifem('meshdoc')">meshdoc</a> for the data structure:
% node and elem.
%
% showmesh3(node,elem,expr) displays parts of the mesh specificed by the
% expression. For example, showmesh3(node,elem,'~(x>=0 & y>=0)') only
% shows the tetrahedron not in the first quadrant.
%
% showmesh3(node,elem,expr,'param','value','param','value'...) allows
% additional patch param/value pairs to be used when displaying the
% mesh. For example, the default transparency parameter is set to 0.5.
% You can overwrite this value by using the param pair ('FaceAlpha',
% value). The value has to be a number between 0 and 1. Other parameters
% include: 'Facecolor', 'Edgecolor' etc.
%
% For meshes with large data, the 3-D graphics is very slow. You may use
% <a href="matlab:help showboundary3">showboundary3</a> to display the boundary surface mesh only.
%
% Example:
% % A mesh for a cube
% node = [-1,-1,-1; 1,-1,-1; 1,1,-1; -1,1,-1; -1,-1,1; 1,-1,1; 1,1,1; -1,1,1];
% elem = [1,2,3,7; 1,6,2,7; 1,5,6,7; 1,8,5,7; 1,4,8,7; 1,3,4,7];
% [node,elem] = uniformbisect3(node,elem);
% subplot(1,2,1);
% showmesh3(node,elem); pause(1)
% subplot(1,2,2);
% showmesh3(node,elem,'~(x>=0 & y>=0 & z>=0)','FaceAlpha',0.25);
% axis on; view([59,20])
%
% See also showboundary3, showsolution3, showmesh.
%
% Copyright (C) Long Chen. See COPYRIGHT.txt for details.
if (nargin >= 3) && (any(expr))
x = node(:,1); y = node(:,2); z = node(:,3); %#ok<NASGU>
incl = find(eval(expr));
elem = elem(any(ismember(elem,incl),2),:);
end
h = tetramesh(elem(:,1:4),node,ones(size(elem,1),1));
set(h,'facecolor',[0.5 0.9 0.45],'edgecolor','k');
if nargin > 3
set(h,varargin{1:end})
else % default display properties
set(h,'FaceAlpha',0.4);
end
view(3);
axis off; axis equal; axis tight |
## Exercise 1 (4 points): Idea of Maximal Margin (SVM) [Pen and Paper]
With the help of support vector machines, we can find a separation line which maximize the margin between two classes. In this exercise, we use a two-class example and approach to such a line in a geometric way. Figure 1 shows the two-dimensional data we are using throughout this exercise. Each point 𝑥 ∈ ℝ<sup>2</sup> belongs either to the positive 𝜔<sub>1</sub> (blue, circles) or negative class 𝜔<sub>−1</sub> (orange, squares).
1. To find the separation line, the convex hull of a dataset is helpful. Start by drawing the convex hull for each class in Figure 1.
2. The line with the maximal margin is defined by the two points from the convex hulls of the two classes with minimal distance to each other. Find these two points on the convex hulls you have just drawn and label them with 𝑐<sub>1</sub> and 𝑐<sub>-1</sub> (for the class 𝜔<sub>1</sub> and 𝜔<sub>-1</sub>, respectively). Note that every point on a convex hull is a possible candidate and they do not necessarily need to correspond with the data points.
3. Connect the points 𝑐<sub>1</sub> and 𝑐<sub>-1</sub> with a line.
4. The separation line goes through the centre of the line you have just drawn and is orthogonal to it, i.e. the two lines enclose a 90° angle. Draw the separation line 𝑠, the line 𝑙<sub>1</sub> which passes through the support vectors of 𝜔<sub>1</sub> and the line 𝑙<sub>-1</sub> which passes through the support vectors of 𝜔<sub>-1</sub>.
5. Add two arbitrary points from each class, i.e. x<sub>1</sub>, 𝑥<sub>2</sub> ∈ $\omega$𝜔<sub>1</sub> and 𝑥<sub>3</sub>, 𝑥<sub>4</sub> ∈ 𝜔<sub>-1</sub>, to the feature space so that the separation line 𝑠 found previously does not change.
6. Start fresh with Figure 2 and add a new data point 𝑥<sub>5</sub> which belongs to a class of your choice to the feature space so that the new margin between the two classes equals 1. As before, sketch the three lines $\tilde{l}$<sub>1</sub> , $\tilde{l}$<sub>-1</sub> and $\tilde{s}$ based on the points $\tilde{c}$<sub>1</sub> and $\tilde{l}$<sub>-1</sub> on the convex hull.
7. In general, what is the difference between a line obtained via SVM and a line obtained via the perceptron learning algorithm?
## Exercise 2 (6 points): SVM [Pen and Paper]
From the previous exercise, we know which kind of separation line we can expect when using support vector machines and how to find it graphically. Determining the separation line analytically, however, is very hard; even for the simple problem of Exercise 1. The number of variables and conditions make this task impractical for a “Pen and Paper” exercise. But, to still get a basic idea of the SVM algorithm, we consider here an even simpler problem where we have only two points $x_i \in \mathbb{R}^{2}$ which belong to their own class each. The first point is from the set of positive samples $\omega_1 = \{x_i | T_i = 1\}$ and the second from the set of negative samples $\omega_{-1} = \{x_i | T_i = −1\}$. The two points are defined as
$$ x_1 =
\begin{pmatrix}
1\\
1\\
\end{pmatrix} \in \omega_1
\hspace{0.5cm}and\hspace{0.5cm}
\begin{pmatrix}
2\\
3\\
\end{pmatrix} \in \omega_{-1}
$$
Obviously, both points are also support vectors since they are the only representatives from their classes and hence must lie on the margin. Our goal is to find the parameters of a separation line with maximal margin, i.e. we search for 𝑤 and 𝑤<sub>0</sub>. In order to find our parameters, we need to optimize the Lagrange function
\begin{equation}
L(w, w_0, \alpha_1, \alpha_2) = L(\alpha_1, \alpha_2) = \sum_{i=1}^2 \alpha_i - \frac{1}{2}\sum_{i=1}^2\sum_{j=1}^2\alpha_i\alpha_jT_iT_jx_i^Tx_j
\end{equation}
subject to the constraint
$$\sum_{i=1}^2\alpha_iT_i=0$$
Technically, we also need the constraints $\alpha_i \geq 0$. But to keep things simple, we are assuming them satisfied here.
1. Set up the Lagrange function (Equation 1) as well as the constraint (Equation 2) of our problem set, i.e. plug in the values and simplify.
2. To optimize Equation 1 under the constraint of Equation 2, we need an additional Lagrange function with the Lagrange multiplier 𝜆
\begin{equation} \Lambda(\alpha_1, \alpha_2, \lambda) = L(\alpha_1, \alpha_2) + \lambda\left(\sum_{i=1}^2\alpha_iT_i\right) \end{equation}
Optimize this function and show that the optimal values are given as
$$\alpha_1^* = \frac{2}{5} \hspace{0.5cm} and \hspace{0.5cm} \alpha_2^* = \frac{2}{5} $$
It is sufficient to calculate only the extrema values. You do not need to show that they belong to a minimum/maximum (and not a saddle point) of $L(\alpha_1, \alpha_2)$. You can use Figure 3 to convince yourself that this is indeed the case here.<br/>
3. Based on the previous results, calculate the line parameters 𝑤 and 𝑤<sub>0</sub>.<br/>
4. Draw the resulting separation line into Figure 4.<br/>
5. Suppose you want to classify the new point 𝑥3 = (2, 2). Assign this point the correct label
a) graphically based on the line you have just drawn into Figure 4.
b) analytically by using the decision function 𝑓(𝑥) (script page 102).<br/>
<br/>
6. Some general questions to conclude this exercise (no calculations required):
a) How would Equation 1 change if we had more than three points?
b) How can we infer from the coefficient 𝛼∗ 𝑖 whether the corresponding point 𝑥𝑖 contributes to the separation line (i.e. whether 𝑥𝑖 is a support vector)?
|
A set $S$ is connected if and only if it is not the union of two nonempty open sets that are disjoint. |
(*<*)
theory SPRView
imports
KBPsAuto
begin
(*>*)
subsection{* The Synchronous Perfect-Recall View *}
text{*
\label{sec:kbps-theory-pr-view}
The synchronous perfect-recall (SPR) view records all observations the
agent has made on a given trace. This is the canonical
full-information synchronous view; all others are functions of this
one.
Intuitively it maintains a list of all observations made on the trace,
with the length of the list recording the time:
*}
definition (in Environment) spr_jview :: "('a, 's, 'obs Trace) JointView" where
"spr_jview a = tMap (envObs a)"
(*<*)
context Environment
begin
lemma spr_jview_length_eq:
"tLength (spr_jview a t) = tLength t"
by (simp add: spr_jview_def)
lemma spr_jview_tInit_inv[simp]:
"spr_jview a t = tInit obs \<longleftrightarrow> (\<exists>s. t = tInit s \<and> envObs a s = obs)"
by (cases t) (simp_all add: spr_jview_def)
lemma spr_jview_tStep_eq_inv:
"spr_jview a t' = spr_jview a (t \<leadsto> s) \<Longrightarrow> \<exists>t'' s'. t' = t'' \<leadsto> s'"
by (cases t') (simp_all add: spr_jview_def)
lemma spr_jview_prefix_closed[dest]:
"spr_jview a (t \<leadsto> s) = spr_jview a (t' \<leadsto> s') \<Longrightarrow> spr_jview a t = spr_jview a t'"
by (simp add: spr_jview_def)
end
(*>*)
text{*
The corresponding incremental view appends a new observation to the
existing ones:
*}
definition (in Environment) spr_jviewInit :: "'a \<Rightarrow> 'obs \<Rightarrow> 'obs Trace" where
"spr_jviewInit \<equiv> \<lambda>a obs. tInit obs"
definition (in Environment)
spr_jviewIncr :: "'a \<Rightarrow> 'obs \<Rightarrow> 'obs Trace \<Rightarrow> 'obs Trace"
where
"spr_jviewIncr \<equiv> \<lambda>a obs' tobs. tobs \<leadsto> obs'"
sublocale Environment
< SPR!: IncrEnvironment jkbp envInit envAction envTrans envVal
spr_jview envObs spr_jviewInit spr_jviewIncr
(*<*)
proof
{ fix a t t' assume "spr_jview a t = spr_jview a t'"
hence "tLength t = tLength t'"
using spr_jview_length_eq[where a=a, symmetric] by simp }
thus "\<forall>a t t'. spr_jview a t = spr_jview a t' \<longrightarrow> tLength t = tLength t'"
by blast
next
show "\<forall>a s. spr_jviewInit a (envObs a s) = spr_jview a (tInit s)"
unfolding spr_jviewInit_def by (simp add: spr_jview_def)
next
show "\<forall>a t s. spr_jview a (t \<leadsto> s) = spr_jviewIncr a (envObs a s) (spr_jview a t)"
unfolding spr_jviewIncr_def by (simp add: spr_jview_def)
qed
(* These need to follow the locale instantiation as we appeal to
sync. *)
lemma (in Environment) spr_tFirst[dest]:
assumes v: "spr_jview a t = spr_jview a t'"
shows "envObs a (tFirst t) = envObs a (tFirst t')"
using SPR.sync[rule_format, OF v] v
apply (induct rule: trace_induct2)
apply (simp_all add: spr_jview_def)
done
lemma (in Environment) spr_tLast[dest]:
assumes v: "spr_jview a t = spr_jview a t'"
shows "envObs a (tLast t) = envObs a (tLast t')"
using SPR.sync[rule_format, OF v] v
apply (induct rule: trace_induct2)
apply (simp_all add: spr_jview_def)
done
(*>*)
text{*
\citet[Theorem~5]{Ron:1996} showed that it is not the case that
finite-state implementations always exist with respect to the SPR
view, and so we consider three special cases:
\begin{itemize}
\item[\S\ref{sec:kbps-spr-single-agent}] where there is a single
agent;
\item[\S\ref{sec:kbps-theory-spr-deterministic-protocols}] when the
protocols of the agents are deterministic and communication is by
broadcast; and
\item[\S\ref{sec:kbps-theory-spr-non-deterministic-protocols}] when
the agents use non-deterministic protocols and again use broadcast to
communicate.
\end{itemize}
Note that these cases do overlap but none is wholly
contained in another.
*}
(*<*)
end
(*>*)
|
theory prop_24
imports Main
"$HIPSTER_HOME/IsaHipster"
begin
datatype Nat = Z | S "Nat"
fun plus :: "Nat => Nat => Nat" where
"plus (Z) y = y"
| "plus (S z) y = S (plus z y)"
fun even :: "Nat => bool" where
"even (Z) = True"
| "even (S (Z)) = False"
| "even (S (S z)) = even z"
(*hipster plus even *)
lemma lemma_a [thy_expl]: "plus x2 Z = x2"
by (hipster_induct_schemes plus.simps)
lemma lemma_aa [thy_expl]: "plus (plus x2 y2) z2 = plus x2 (plus y2 z2)"
by (hipster_induct_schemes plus.simps)
lemma lemma_ab [thy_expl]: "plus x2 (S y2) = S (plus x2 y2)"
by (hipster_induct_schemes plus.simps)
lemma lemma_ac [thy_expl]: "plus x1 (plus y1 x1) = plus y1 (plus x1 x1)"
by (hipster_induct_schemes plus.simps)
lemma lemma_ad [thy_expl]: "plus x2 (plus y2 y2) = plus y2 (plus y2 x2)"
by (hipster_induct_schemes plus.simps)
lemma lemma_ae [thy_expl]: "plus x2 (S y2) = S (plus y2 x2)"
by (hipster_induct_schemes plus.simps)
lemma lemma_af [thy_expl]: "plus (S x2) y2 = S (plus y2 x2)"
by (hipster_induct_schemes plus.simps)
lemma lemma_ag [thy_expl]: "plus (plus x2 y2) (plus x2 z2) =
plus (plus x2 z2) (plus x2 y2)"
by (hipster_induct_schemes plus.simps)
lemma lemma_ah [thy_expl]: "plus (plus x2 y2) (plus z2 x2) = plus (plus z2 x2) (plus x2 y2)"
by (hipster_induct_schemes plus.simps)
lemma lemma_ai [thy_expl]: "plus x2 (plus y2 z2) = plus y2 (plus z2 x2)"
by (hipster_induct_schemes plus.simps)
(*hipster even plus*)
lemma lemma_aj []: "even (plus x1 x1) = True"
by (hipster_induct_schemes even.simps plus.simps)
theorem x0 :
"(even (plus x y)) = (even (plus y x))"
by (tactic \<open>Subgoal.FOCUS_PARAMS (K (Tactic_Data.hard_tac @{context})) @{context} 1\<close>)
end
|
(* * Correctness proof of the lowering pass *)
(* ** Imports and settings *)
From mathcomp Require Import all_ssreflect all_algebra.
From mathcomp Require Import word_ssrZ.
Require Import ZArith psem compiler_util lea_proof x86_instr_decl x86_extra.
Require Import
lowering
lowering_lemmas.
Require Import
arch_extra
sem_params_of_arch_extra.
Require Export x86_lowering.
Import Utf8.
Import Psatz.
Import Order.POrderTheory Order.TotalTheory.
Import ssrring.
Set Implicit Arguments.
Unset Strict Implicit.
Unset Printing Implicit Defensive.
Local Open Scope vmap_scope.
Local Open Scope seq_scope.
Section PROOF.
Context {syscall_state : Type} {sc_sem : syscall_sem syscall_state} {T:eqType} {pT:progT T} {sCP: semCallParams}.
Variable p : prog.
Variable ev : extra_val_t.
Notation gd := (p_globs p).
Context (is_regx: var -> bool) (options: lowering_options).
Context (warning: instr_info -> warning_msg -> instr_info).
Variable fv : fresh_vars.
Context (is_var_in_memory: var_i → bool).
Notation lower_prog :=
(lower_prog (asmop := _asmop) (lower_i is_regx) options warning fv is_var_in_memory).
Notation lower_cmd :=
(lower_cmd (asmop := _asmop) (lower_i is_regx) options warning fv is_var_in_memory).
Hypothesis fvars_correct: fvars_correct fv (p_funcs p).
Definition disj_fvars := disj_fvars fv.
Definition vars_p := vars_p (p_funcs p).
Definition fvars := fvars fv.
Lemma fvars_fresh: disj_fvars vars_p.
Proof. by move: fvars_correct => /andP []. Qed.
Lemma of_neq_cf : fv.(fresh_OF) != fv.(fresh_CF).
Proof. by move: fvars_correct=> /and5P [] ???? /and3P []. Qed.
Lemma of_neq_sf : fv.(fresh_OF) != fv.(fresh_SF).
Proof. by move: fvars_correct=> /and5P [] ???? /and3P []. Qed.
Lemma of_neq_zf : fv.(fresh_OF) != fv.(fresh_ZF).
Proof. by move: fvars_correct=> /and5P [] ???? /and3P []. Qed.
Lemma cf_neq_sf : fv.(fresh_CF) != fv.(fresh_SF).
Proof. by move: fvars_correct=> /and5P [] ???? /and3P []. Qed.
Lemma cf_neq_zf : fv.(fresh_CF) != fv.(fresh_ZF).
Proof. by move: fvars_correct=> /and5P [] ???? /and3P []. Qed.
Lemma sf_neq_zf : fv.(fresh_SF) != fv.(fresh_ZF).
Proof. by move: fvars_correct=> /and5P [] ???? /and3P []. Qed.
Lemma of_in_fv: Sv.In (vbool fv.(fresh_OF)) fvars.
Proof. by rewrite /fvars /x86_lowering.fvars /= /fv_of; SvD.fsetdec. Qed.
Lemma cf_in_fv: Sv.In (vbool fv.(fresh_CF)) fvars.
Proof. by rewrite /fvars /x86_lowering.fvars /= /fv_cf; SvD.fsetdec. Qed.
Lemma sf_in_fv: Sv.In (vbool fv.(fresh_SF)) fvars.
Proof. by rewrite /fvars /x86_lowering.fvars /= /fv_sf; SvD.fsetdec. Qed.
Lemma pf_in_fv: Sv.In (vbool fv.(fresh_PF)) fvars.
Proof. by rewrite /fvars /x86_lowering.fvars /= /fv_pf; SvD.fsetdec. Qed.
Lemma zf_in_fv: Sv.In (vbool fv.(fresh_ZF)) fvars.
Proof. by rewrite /fvars /x86_lowering.fvars /= /fv_zf; SvD.fsetdec. Qed.
Lemma multiplicand_in_fv sz : Sv.In (vword sz (fv.(fresh_multiplicand) sz)) fvars.
Proof. by rewrite /fvars /x86_lowering.fvars /=; case: sz; SvD.fsetdec. Qed.
Local Hint Resolve of_neq_cf of_neq_sf of_neq_zf cf_neq_sf cf_neq_zf sf_neq_zf : core.
Local Hint Resolve of_in_fv cf_in_fv sf_in_fv pf_in_fv zf_in_fv multiplicand_in_fv : core.
Local
Definition p' := lower_prog p.
Definition eq_exc_fresh s1 s2 := estate_eq_except fvars s1 s2.
Lemma disj_fvars_subset s1 s2 :
Sv.Subset s1 s2 →
disj_fvars s2 →
disj_fvars s1.
Proof.
move => Hle h1; rewrite /disj_fvars /x86_lowering.disj_fvars.
by apply: disjoint_w; eauto.
Qed.
Global Instance disj_fvars_m : Proper (Sv.Equal ==> iff) disj_fvars.
Proof.
by move=> s1 s2 Heq; split; rewrite /disj_fvars /x86_lowering.disj_fvars Heq.
Qed.
Lemma fvars_fun fn f:
get_fundef (p_funcs p) fn = Some f ->
disj_fvars (vars_fd f).
Proof.
have := fvars_fresh; rewrite /vars_p.
move: (p_funcs p) fn f.
elim=> // [[fn0 fd0]] l Hl fn f.
rewrite get_fundef_cons /=.
move=> /disjoint_union [Hq Hh].
case: ifP=> Hfn.
+ by move=> []<-.
+ move=> Hf.
exact: (Hl _ _ Hq Hf).
Qed.
Let Pi s (i:instr) s' :=
disj_fvars (vars_I i) ->
forall s1, eq_exc_fresh s1 s ->
exists s1', sem p' ev s1 (lower_i is_regx options warning fv is_var_in_memory i) s1' /\ eq_exc_fresh s1' s'.
Let Pi_r s (i:instr_r) s' :=
forall ii, Pi s (MkI ii i) s'.
Let Pc s (c:cmd) s' :=
disj_fvars (vars_c c) ->
forall s1, eq_exc_fresh s1 s ->
exists s1', sem p' ev s1 (lower_cmd c) s1' /\ eq_exc_fresh s1' s'.
Let Pfor (i:var_i) vs s c s' :=
disj_fvars (Sv.union (vars_c c) (Sv.singleton i)) ->
forall s1, eq_exc_fresh s1 s ->
exists s1', sem_for p' ev i vs s1 (lower_cmd c) s1' /\ eq_exc_fresh s1' s'.
Let Pfun scs1 m1 fn vargs scs2 m2 vres :=
sem_call p' ev scs1 m1 fn vargs scs2 m2 vres.
Local Lemma Hskip : sem_Ind_nil Pc.
Proof. move=> s ? s1 [H1 H2]; exists s1; repeat split=> //; exact: Eskip. Qed.
Local Lemma Hcons : sem_Ind_cons p ev Pc Pi.
Proof.
move=> s1 s2 s3 i c Hsi Hi Hsc Hc Hdisj s1' Hs1'.
move: Hdisj.
rewrite
/disj_fvars
/x86_lowering.disj_fvars vars_c_cons
=> /disjoint_union [Hdisji Hdisjc].
have [s2' [Hs2'1 Hs2'2]] := Hi Hdisji _ Hs1'.
have [s3' [Hs3'1 Hs3'2]] := Hc Hdisjc _ Hs2'2.
exists s3'; repeat split=> //.
exact: (sem_app Hs2'1 Hs3'1).
Qed.
Local Lemma HmkI : sem_Ind_mkI p ev Pi_r Pi.
Proof. move=> ii i s1 s2 _ Hi; exact: Hi. Qed.
Lemma type_of_get_gvar vm sz vn vi vs v:
get_gvar gd vm {| gv := {| v_var := {| vtype := sword sz; vname := vn |} ; v_info := vi |} ; gs := vs |} = ok v ->
∃ sz', type_of_val v = sword sz' ∧ (sz' ≤ sz)%CMP.
Proof.
rewrite /get_gvar; case: vs => /=; last first.
- by case/get_globalI => gv [] _ -> ->; exists sz.
rewrite /get_var /on_vu.
case Heq: (vm.[_])=> [a|[]] // [<-] /=; eauto.
case: a {Heq} => /= sz' _; eauto.
Qed.
Lemma add_inc_dec_classifyP' sz a b:
match add_inc_dec_classify sz a b with
| AddInc y => (a = y ∧ b = Papp1 (Oword_of_int sz) (Pconst 1)) ∨ (a = Papp1 (Oword_of_int sz) (Pconst 1) ∧ b = y)
| AddDec y => (a = y ∧ b = Papp1 (Oword_of_int sz) (Pconst (-1))) ∨ (a = Papp1 (Oword_of_int sz) (Pconst (-1)) ∧ b = y)
| AddNone => True
end.
Proof.
rewrite /add_inc_dec_classify.
repeat match goal with
| |- True => exact: I
| |- _ ∨ _ => (left; split; reflexivity) || (right; split; reflexivity)
| |- match (if _ == _ then _ else _) with _ => _ end => case: eqP => // ?; subst
| |- match match ?x with _ => _ end with _ => _ end => destruct x
end.
Qed.
Lemma add_inc_dec_classifyP s sz (a b : pexpr) w1 (z1: word w1) w2 (z2 : word w2) :
sem_pexprs gd s [:: a; b] = ok [:: Vword z1; Vword z2] ->
match add_inc_dec_classify sz a b with
| AddInc y => exists sz' (w: word sz'), (sz' = w1 ∨ sz' = w2) ∧ sem_pexpr gd s y = ok (Vword w) /\ zero_extend sz w + 1 = zero_extend sz z1 + zero_extend sz z2
| AddDec y => exists sz' (w: word sz'), (sz' = w1 ∨ sz' = w2) ∧ sem_pexpr gd s y = ok (Vword w) /\ zero_extend sz w - 1 = zero_extend sz z1 + zero_extend sz z2
| AddNone => True
end%R.
Proof.
have := add_inc_dec_classifyP' sz a b.
case: (add_inc_dec_classify sz a b)=> [y|y|//].
+ case=> [[??]|[??]]; subst; rewrite /sem_pexprs /=; t_xrbindP.
+ by move => z -> -> -> [<-]; exists w1, z1; do 2 (split; first by eauto); rewrite zero_extend_u /wrepr mathcomp.word.word.mkword1E.
by move => ? z -> <- -> [<-] [->]; exists w2, z2; do 2 (split; first by eauto); rewrite zero_extend_u /wrepr mathcomp.word.word.mkword1E GRing.addrC.
+ case=> [[??]|[??]]; subst; rewrite /sem_pexprs /=; t_xrbindP.
+ by move => z -> -> -> [<-]; exists w1, z1; do 2 (split; first by eauto); rewrite zero_extend_u /wrepr mathcomp.word.word.mkwordN1E.
by move => ? z -> <- -> [<-] [->]; exists w2, z2; do 2 (split; first by eauto); rewrite zero_extend_u /wrepr mathcomp.word.word.mkwordN1E GRing.addrC.
Qed.
Lemma sub_inc_dec_classifyP sz e:
match sub_inc_dec_classify sz e with
| SubInc => e = Papp1 (Oword_of_int sz) (Pconst (-1))
| SubDec => e = Papp1 (Oword_of_int sz) (Pconst 1)
| SubNone => True
end.
Proof.
by case: e => // -[] // ? [] // [] // [] //=; case: eqP => // ->.
Qed.
Lemma write_lval_word l sz v s s':
stype_of_lval l = sword sz →
write_lval gd l v s = ok s' →
∃ sz', type_of_val v = sword sz'.
Proof.
case: l => /= [ _ [] // sz' | [[vt vn] vi] | sz' [[vt vn] vi] e | aa sz' [[vt vn] vi] e | aa sz' len [[vt vn] vi] e ] /=.
- case => ->; case: v => //=; eauto => -[] //=; eauto.
- move => ->; case: v => //=; eauto => -[] //=; eauto.
- move => ->; t_xrbindP => w1 v1 _ h1 w n _ hn w' /to_wordI [ws [? [??]]]; subst => /=; eauto.
- by move => ->; apply: on_arr_varP.
by move => ->; apply: on_arr_varP.
Qed.
Lemma between_ZR (a b c: Z) :
(a <= b < c)%R →
(a <= b < c)%Z.
Proof. by case/andP => /word_ssrZ.lezP ? /word_ssrZ.ltzP. Qed.
Lemma wleuE' sz (α β: word sz) :
wle Unsigned β α = (wunsigned (β - α) != (wunsigned β - wunsigned α)%Z) || (β == α).
Proof.
case: (β =P α).
+ by move => <-; rewrite orbT /= lexx.
rewrite orbF /wunsigned /=.
case: α β => α hα [] β hβ ne'.
Transparent word.
repeat rewrite /mathcomp.word.word.urepr /=.
Opaque word.
have ne : α ≠ β.
- move => ?; subst; apply: ne'.
by rewrite (Eqdep_dec.UIP_dec Bool.bool_dec hα).
case/between_ZR: hα hβ {ne'} => hα hα' /between_ZR [hβ hβ'].
elim_div => z a [] //.
elim_div => z1 b [] //.
set m := (wsize_size_minus_1 sz).+1.
have /word_ssrZ.ltzP := mathcomp.word.word.modulus_gt0 m.
match goal with |- (?x < _)%Z → _ => have hz : x = 0%Z by [] end.
rewrite hz in hα, hβ |- * => {hz}.
move => hm /Z.eq_opp_r ?; subst α => - []; last Psatz.lia.
case => ??? []; last Psatz.lia.
case => ??.
symmetry; case: word_ssrZ.lezP => h; apply/eqP; first Psatz.nia.
fold m in hα', hβ'.
suff: z = (- z1)%Z; Psatz.nia.
Qed.
Lemma lower_condition_corr ii ii' i e e' s1 cond:
(i, e') = lower_condition fv ii' e ->
forall s1', eq_exc_fresh s1' s1 ->
sem_pexpr gd s1' e = ok cond ->
exists s2',
sem p' ev s1' (map (MkI ii) i) s2' /\ eq_exc_fresh s2' s1 /\ sem_pexpr gd s2' e' = ok cond.
Proof.
move=> Hcond s1' Hs1' He.
move: Hcond.
rewrite /lower_condition.
case heq : lower_cond_classify => [ [[[[lv ws] c] x] y]| ]; last first.
+ by move=> [ -> ->]; exists s1' => /=; split => //; constructor.
case: ifP; last first.
+ by move=> _ [ -> ->]; exists s1' => /=; split => //; constructor.
move=> hws [??]; subst i e'.
case: e He heq => // o e1 e2 /=; t_xrbindP => v1 hv1 v2 hv2.
set Of := {| v_var := {| vname := fresh_OF _ |} |}.
set Cf := {| v_var := {| vname := fresh_CF _ |} |}.
set Sf := {| v_var := {| vname := fresh_SF _ |} |}.
set Zf := {| v_var := {| vname := fresh_ZF _ |} |}.
have hw : forall (bof bcf bsf bpf bzf: bool),
exists s2',
[/\
write_lvals gd s1' [:: Lvar Of; Lvar Cf; Lvar Sf; Lnone ii' sbool; Lvar Zf]
[:: Vbool bof; Vbool bcf; Vbool bsf; Vbool bpf; Vbool bzf] = ok s2',
eq_exc_fresh s1' s2' &
[/\ sem_pexpr gd s2' (Plvar Of) = ok (Vbool bof),
sem_pexpr gd s2' (Plvar Cf) = ok (Vbool bcf),
sem_pexpr gd s2' (Plvar Sf) = ok (Vbool bsf) &
sem_pexpr gd s2' (Plvar Zf) = ok (Vbool bzf) ]].
+ eexists; split => /=; first reflexivity.
+ split.
+ by rewrite !escs_with_vm. + by rewrite !emem_with_vm.
rewrite evm_with_vm => z hz.
by rewrite !Fv.setP_neq //; apply/eqP => heq; subst z; elim hz;
auto using of_in_fv, cf_in_fv, sf_in_fv, pf_in_fv.
split=> /=.
+ rewrite get_gvar_neq; last by move=> _ [] h; have := of_neq_zf; rewrite h eqxx.
rewrite get_gvar_neq; last by move=> _ [] h; have := of_neq_sf; rewrite h eqxx.
rewrite get_gvar_neq; last by move=> _ [] h; have := of_neq_cf; rewrite h eqxx.
by rewrite (@get_gvar_eq gd (mk_lvar Of)).
+ rewrite get_gvar_neq; last by move=> _ [] h; have := cf_neq_zf; rewrite h eqxx.
rewrite get_gvar_neq; last by move=> _ [] h; have := cf_neq_sf; rewrite h eqxx.
by rewrite (@get_gvar_eq gd (mk_lvar Cf)).
+ rewrite get_gvar_neq; last by move=> _ [] h; have := sf_neq_zf; rewrite h eqxx.
by rewrite (@get_gvar_eq gd (mk_lvar Sf)).
by rewrite (@get_gvar_eq gd (mk_lvar Zf)).
have {hw}hw : forall wx wy,
to_word ws v2 = ok wy ->
to_word ws v1 = ok wx ->
∃ s2' : estate,
[/\ sem p' ev s1' [:: MkI ii (Copn [:: Lvar Of; Lvar Cf; Lvar Sf; Lnone ii' sbool; Lvar Zf] AT_none
(Ox86 (CMP ws)) [:: e1; e2])] s2',
eq_exc_fresh s2' s1 &
[/\ sem_pexpr gd s2' (Plvar Of) = ok (Vbool ((wsigned (wx - wy) != (wsigned wx - wsigned wy)%Z))),
sem_pexpr gd s2' (Plvar Cf) = ok (Vbool (wunsigned (wx - wy) != (wunsigned wx - wunsigned wy)%Z)),
sem_pexpr gd s2' (Plvar Sf) = ok (Vbool (SF_of_word (wx - wy))) &
sem_pexpr gd s2' (Plvar Zf) = ok (Vbool (ZF_of_word (wx - wy)))]].
+ move=> wx wy hx hy;
have [s2' [{hw}hw heq he]] := hw (wsigned (wx - wy) != (wsigned wx - wsigned wy)%Z)
(wunsigned (wx - wy) != (wunsigned wx - wunsigned wy)%Z)
(SF_of_word (wx - wy))
(PF_of_word (wx - wy)) (ZF_of_word (wx - wy)).
exists s2'; split => //.
+ apply: sem_seq1; econstructor; econstructor.
rewrite /sem_sopn /= hv1 hv2 /= /exec_sopn /= hx hy /= /sopn_sem /= /x86_CMP.
rewrite /check_size_8_64 hws //.
by apply: eeq_excT Hs1'; apply eeq_excS.
case: o => //.
+ case=> // ws' /sem_sop2I /= [wx [wy [b [hw2 hw1]]]] hs ? [] ?????; subst cond e1 e2 ws' c lv.
have [s2' [hsem heqe [hof hcf hsf hzf]]]:= hw _ _ hw1 hw2.
exists s2'; split => //; split => //.
by case: hs => <-; rewrite hzf /ZF_of_word GRing.subr_eq0.
+ case=> // ws' /sem_sop2I /= [wx [wy [b [hw2 hw1]]]] hs ? [] ?????; subst cond e1 e2 ws' c lv.
have [s2' [hsem heqe [hof hcf hsf hzf]]]:= hw _ _ hw1 hw2.
exists s2'; split => //; split => //.
move: hzf; rewrite /enot /= => -> /=.
rewrite /sem_sop1 /=.
by case: hs => <-; do 3! f_equal; rewrite /ZF_of_word GRing.subr_eq0.
+ case => // -[] ws' /sem_sop2I /= [wx [wy [b [hw2 hw1]]]] hs ? [] ?????; subst cond e1 e2 ws' c lv;
have [s2' [hsem heqe [hof hcf hsf hzf]]]:= hw _ _ hw1 hw2;
exists s2'; split => //; split => //; case: hs => <-.
+ move: hof hsf => /= -> -> /=; rewrite /sem_sop1 /= /SF_of_word.
by rewrite eq_sym -wltsE.
by move: hcf => /= -> /=; rewrite -wleuE /= ltNge.
+ case => // -[] ws' /sem_sop2I /= [wx [wy [b [hw2 hw1]]]] hs ? [] ?????; subst cond e1 e2 ws' c lv;
have [s2' [hsem heqe [hof hcf hsf hzf]]]:= hw _ _ hw1 hw2;
exists s2'; split => //; split => //; case: hs => <-.
+ move: hof hsf hzf => /= -> -> -> /=; rewrite /sem_sop2 /= /SF_of_word /ZF_of_word.
rewrite eq_sym -wltsE GRing.subr_eq0 le_eqVlt orbC eqtype.inj_eq //.
by apply word.srepr_inj.
move: hcf hzf => /= -> -> /=; rewrite /sem_sop2 /= /ZF_of_word.
by rewrite GRing.subr_eq0 -wleuE'.
+ case => // -[] ws' /sem_sop2I /= [wx [wy [b [hw2 hw1]]]] hs ? [] ?????; subst cond e1 e2 ws' c lv;
have [s2' [hsem heqe [hof hcf hsf hzf]]]:= hw _ _ hw1 hw2;
exists s2'; split => //; split => //; case: hs => <-.
+ move: hof hsf hzf => /= -> -> -> /=; rewrite /sem_sop2 /= /SF_of_word /ZF_of_word.
rewrite ltNge -(negbK (_ == msb _)).
rewrite -negb_or (eq_sym _ (msb _)) -wltsE GRing.subr_eq0 orbC /= le_eqVlt.
by rewrite eqtype.inj_eq //; apply word.srepr_inj.
move: hcf hzf => /= -> -> /=; rewrite /sem_sop2 /= /ZF_of_word.
by rewrite -negb_or GRing.subr_eq0 ltNge -wleuE'.
case => // -[] ws' /sem_sop2I /= [wx [wy [b [hw2 hw1]]]] hs ? [] ?????; subst cond e1 e2 ws' c lv;
have [s2' [hsem heqe [hof hcf hsf hzf]]]:= hw _ _ hw1 hw2;
exists s2'; split => //; split => //; case: hs => <-.
+ move: hof hsf => /= -> -> /=; rewrite /sem_sop2 /= /SF_of_word.
by rewrite eq_sym -(negbK (_ == _)) -wltsE /= leNgt.
by move: hcf => /= -> /=; rewrite /sem_sop1 /= -wleuE negbK.
Qed.
Lemma read_es_swap x y : Sv.Equal (read_es [:: x ; y ]) (read_es [:: y ; x ]).
Proof. by rewrite ! read_es_cons; SvD.fsetdec. Qed.
(* ---------------------------------------------------------- *)
Lemma is_leaP f sz x e l :
is_lea f sz x e = Some l ->
[/\ (U16 ≤ sz)%CMP && (sz ≤ U64)%CMP,
Sv.Subset (read_lea l) (read_e e),
mk_lea sz e = Some l & check_scale l.(lea_scale)].
Proof.
rewrite /is_lea; case: ifP => // /andP [-> _].
case h: mk_lea => [[d b sc o]|] //.
move /mk_lea_read in h.
by case: ifP => // /andP [] /andP [] heq _ _ [<-].
Qed.
Lemma zquot_bound m x y :
(y ≠ 0 → x ≠ -m ∨ y ≠ -1 → -m <= x <= m - 1 → -m <= y <= m - 1 → -m <= x ÷ y <= m - 1)%Z.
Proof.
move => hnz hn1 hx hy.
move: (x ÷ y)%Z (Z.quot_div x y hnz) => z.
elim_div => ? ? []; first lia.
move => h []; last lia.
nia.
Qed.
Lemma wsigned_quot_bound sz (w1 w2:word sz) :
w2 ≠ 0%R →
(wsigned w1 == wmin_signed sz) && (w2 == (-1)%R) = false →
[|| wsigned w2 == 0%Z, (wsigned w1 ÷ wsigned w2 <? wmin_signed sz)%Z
| (wsigned w1 ÷ wsigned w2 >? wmax_signed sz)%Z] = false.
Proof.
move => hnz hn1.
case: eqP.
+ by rewrite -(@wsigned0 sz) => /(can_inj (@word.sreprK _)).
move => hnz' /=.
apply: negbTE; rewrite negb_or; apply/andP.
rewrite Z.gtb_ltb -!Z.leb_antisym -!(rwP lezP).
apply: zquot_bound => //; try exact: wsigned_range.
case /Bool.andb_false_elim: hn1 => /eqP h; [ left | right ] => //.
by rewrite -(@wsignedN1 sz) => /(can_inj (@word.sreprK _)).
Qed.
Lemma wunsigned_div_bound sz (w1 w2: word sz) :
wunsigned w2 != 0%Z ->
~~(wunsigned w1 / wunsigned w2 >? wmax_unsigned sz)%Z.
Proof.
have ? := wunsigned_range w2.
move/eqP => hnz.
rewrite Z.gtb_ltb -Z.leb_antisym; apply/leZP.
rewrite /wmax_unsigned.
have := wunsigned_range w1.
elim_div; nia.
Qed.
Lemma check_size_16_64_ve (ve:velem) : (U16 ≤ ve)%CMP -> check_size_16_64 ve = ok tt.
Proof. by rewrite /check_size_16_64 => ->; case:ve. Qed.
Lemma check_size_32_64_ve (ve:velem) : (U32 ≤ ve)%CMP -> check_size_32_64 ve = ok tt.
Proof. by rewrite /check_size_32_64 => ->; case:ve. Qed.
Lemma check_size_128_256_ge sz : (U128 <= sz)%CMP -> check_size_128_256 sz = ok tt.
Proof. by move=> h; rewrite /check_size_128_256 h wsize_ge_U256. Qed.
Lemma mulr_ok l sz w1 w2 (z1 : word w1) (z2:word w2) e1 e2 o e' s s':
sem_pexpr gd s e1 = ok (Vword z1) ->
sem_pexpr gd s e2 = ok (Vword z2) ->
(sz ≤ w1)%CMP ->
(sz ≤ w2)%CMP ->
(U16 ≤ sz)%CMP && (sz ≤ U64)%CMP ->
write_lval gd l (Vword (zero_extend sz z1 * zero_extend sz z2)) s = ok s'->
mulr sz e1 e2 = (o, e') ->
Sv.Subset (read_es e') (read_e (Papp2 (Omul (Op_w sz )) e1 e2))
∧ Let x := Let x := sem_pexprs gd s e' in exec_sopn (Ox86 o) x
in write_lvals gd s
[:: Lnone (var_info_of_lval l) sbool; Lnone (var_info_of_lval l) sbool;
Lnone (var_info_of_lval l) sbool; Lnone (var_info_of_lval l) sbool;
Lnone (var_info_of_lval l) sbool; l] x = ok s'.
Proof.
rewrite /mulr => ok_v1 ok_v2 hle1 hle2 hsz64 Hw.
case Heq: (is_wconst _ _) => [z | ].
* have! := (is_wconstP gd s Heq); t_xrbindP => v1 h1 hz [<- <-].
split; first done.
rewrite /= ok_v1 ok_v2 /= /exec_sopn /sopn_sem /= /truncate_word hle1 hle2.
by rewrite /x86_IMULt /check_size_16_64 hsz64 /= GRing.mulrC Hw.
case Heq2: (is_wconst _ _) => [z | ].
* have! := (is_wconstP gd s Heq2); t_xrbindP => v2 h2 hz [<- <-].
split; first by rewrite read_es_swap.
rewrite /= ok_v1 ok_v2 /= /exec_sopn /sopn_sem /= /truncate_word hle1 hle2 /=.
by rewrite /x86_IMULt /check_size_16_64 hsz64 /= Hw.
move=> [<- <-];split; first by rewrite read_es_swap.
rewrite /= ok_v1 ok_v2 /= /exec_sopn /sopn_sem /= /truncate_word hle1 hle2 /=.
by rewrite /x86_IMULt /check_size_16_64 hsz64 /= Hw.
Qed.
Lemma lower_cassgn_classifyP e l s s' v ty v' (Hs: sem_pexpr gd s e = ok v)
(Hv': truncate_val ty v = ok v')
(Hw: write_lval gd l v' s = ok s'):
match lower_cassgn_classify is_var_in_memory ty e l with
| LowerMov _ =>
exists2 sz, ty = sword sz & (sz ≤ U64)%CMP ∧
∃ sz' (w : word sz'), (sz ≤ sz')%CMP ∧ v = Vword w
| LowerCopn o a =>
sem_pexprs gd s a >>= exec_sopn o = ok [:: v' ]
| LowerInc o a =>
∃ b1 b2 b3 b4, sem_pexprs gd s [:: a] >>= exec_sopn o = ok [:: Vbool b1; Vbool b2; Vbool b3; Vbool b4; v']
| LowerFopn _ o e' _ =>
let vi := var_info_of_lval l in
let f := Lnone vi sbool in
Sv.Subset (read_es e') (read_e e) ∧
sem_pexprs gd s e' >>= exec_sopn o >>=
write_lvals gd s [:: f; f; f; f; f; l] = ok s'
| LowerDiscardFlags n op e' =>
let f := Lnone (var_info_of_lval l) sbool in
Sv.Subset (read_es e') (read_e e)
/\ sem_pexprs gd s e'
>>= exec_sopn op
>>= write_lvals gd s (nseq n f ++ [:: l ]) = ok s'
| LowerDivMod p u sz o a b =>
let vi := var_info_of_lval l in
let f := Lnone vi sbool in
let lv :=
match p with
| DM_Fst => [:: f; f; f; f; f; l; Lnone vi (sword sz)]
| DM_Snd => [:: f; f; f; f; f; Lnone vi (sword sz); l]
end in
[/\ (exists (va:value)(wa:word sz),
[/\ (sem_pexpr gd s a) = ok va,
to_word sz va = ok wa &
(forall s1,
eq_exc_fresh s1 s ->
disj_fvars (vars_lval l) ->
disj_fvars (read_e e) ->
[/\ (sem_pexpr gd s1 a) = ok va &
exists s1',
(Let vb := (sem_pexpr gd s1 b) in
let v0 : word sz :=
if u is Unsigned then 0%R
else if msb wa then (-1)%R else 0%R in
exec_sopn o [::Vword v0; va; vb] >>=
write_lvals gd s1 lv) = ok s1' /\
eq_exc_fresh s1' s'])]),
ty = sword sz , (U16 ≤ sz)%CMP & (sz ≤ U64)%CMP]
| LowerCond => True
| LowerIf t a e1 e2 =>
check_size_16_64 (wsize_of_lval l) = ok tt ∧ e = Pif t a e1 e2 ∧ wsize_of_lval l = wsize_of_stype ty ∧ ∃ sz', stype_of_lval l = sword sz'
| LowerLea sz l =>
((U16 ≤ sz)%CMP && (sz ≤ U64)%CMP ∧ check_scale (lea_scale l) ∧
Sv.Subset (read_lea l) (read_e e) ∧
exists w: word sz,
v' = Vword w /\ sem_lea sz (evm s) l = ok w)
| LowerConcat hi lo =>
sem_pexprs gd s [:: hi ; lo ] >>= exec_sopn (Oasm (ExtOp Oconcat128)) = ok [:: v' ]
| LowerAssgn => True
end.
Proof.
rewrite /lower_cassgn_classify.
move: e Hs=> [z|b|n|x|aa ws x e | aa ws len x e |sz x e| o e|o e1 e2| op es |e e1 e2] //.
+ case: x => - [] [] [] // sz vn vi vs //= /dup[] ok_v.
case/type_of_get_gvar => sz' [Hs Hs'].
have := truncate_val_subtype Hv'. rewrite Hs -(truncate_val_has_type Hv').
case hty: (type_of_val v') => [ | | | sz'' ] //= hle.
case: (write_lval_undef Hw hty) => w ? {hty}; subst v'.
case/truncate_valI: Hv' => s'' [] w'' [] ? ok_w ?; subst.
case: Hs => ?; subst s''.
case: ifP.
* move => h; eexists; first reflexivity.
split; first exact: (cmp_le_trans hle (cmp_le_trans Hs' h)).
by eexists _, _; split; last reflexivity.
rewrite eqxx andbT => _.
case: ifP => // hsz''.
by rewrite /= ok_v /exec_sopn /sopn_sem /= /x86_MOVX /check_size_32_64 hsz'' ok_w.
+ rewrite /=; apply: rbindP => - [] // len a /= ok_a; t_xrbindP => i j ok_j ok_i w ok_w ?; subst v.
case: x ok_a => x xs ok_a.
case/truncate_valE: Hv' => sz' [] w' [] -> {ty} ok_w' ?; subst v'.
case: ifP.
* move => h.
eexists; first reflexivity.
case/truncate_wordP: ok_w' => hle _.
split; first exact: (cmp_le_trans hle).
by eauto.
rewrite eqxx andbT => _.
case: ifP => // hsz''.
by rewrite /= ok_a ok_j /= ok_i /= ok_w /exec_sopn /sopn_sem /= /x86_MOVX /check_size_32_64 hsz'' ok_w'.
+ rewrite /=; t_xrbindP => ???????? w _ ?; subst v; case: ifP => // ?.
have {Hv'} [sz' [? [? /truncate_wordP [hle _] ?]]] := truncate_valE Hv'.
subst v' ty => /=.
eexists; first reflexivity.
split; first exact: (cmp_le_trans hle).
by eauto.
+ case: o => //.
(* Oword_of_int *)
- move => sz; case: e => // z [?]; subst v.
have {Hv'} [sz' [? [? /truncate_wordP [hle _] ?]]] := truncate_valE Hv'.
subst v' ty => /=.
by case: ifP => // hle'; eauto 6.
(* Osignext *)
+ rewrite /= /sem_sop1 /=; t_xrbindP => sz sz' x ok_x x' /to_wordI' [szx [wx [hle ??]]] ?.
subst x x' v.
case: sz' Hv' hle => // /truncate_valE [sz' [? [-> /truncate_wordP [_ ->] ->]]] hle.
- case: andP => // - [] hs /eqP[] ?; subst sz.
by rewrite /= ok_x /= zero_extend_sign_extend /exec_sopn //= /truncate_word hle /=
/sopn_sem /= /x86_MOVSX /check_size_16_64 hs.
- case: andP => // - [] hs /eqP[] ?; subst sz.
by rewrite /= ok_x /= zero_extend_sign_extend /exec_sopn //= /truncate_word hle /=
/sopn_sem /= /x86_MOVSX /check_size_16_64 hs.
case: andP => // - [] hs /eqP[] /= ?; subst sz'.
by rewrite ok_x /= zero_extend_sign_extend // /exec_sopn /= /truncate_word hle
/sopn_sem /= /x86_MOVSX /check_size_32_64 hs.
(* Ozeroext *)
+ rewrite /= /sem_sop1 /=; t_xrbindP => sz sz' x ok_x x' /to_wordI' [szx [wx [hle ??]]] ?.
subst x x' v.
case: sz' Hv' hle => // /truncate_valE [sz' [? [? /truncate_wordP[hle' ->] ?]]] hle; subst ty v'.
- case: andP => // - [] hs /eqP[] ?; subst sz.
by rewrite /= ok_x /= zero_extend_u /exec_sopn /= /truncate_word hle /sopn_sem /= /x86_MOVZX /check_size_16_64 hs.
- case: andP => // - [] hs /eqP[] ?; subst sz.
by rewrite /= ok_x /= zero_extend_u /exec_sopn /= /truncate_word hle /sopn_sem /= /x86_MOVZX /check_size_32_64 hs.
- case: sz Hw hle' => // Hw hle'; case: eqP => // - [] ?; subst sz'.
1-3: rewrite /= ok_x /exec_sopn /= /truncate_word hle /= zero_extend_u //.
do 3 f_equal.
exact: zero_extend_cut.
case: sz Hw hle' => // Hw hle'; case: eqP => // - [] ?; subst sz'.
1-2: rewrite /= ok_x /exec_sopn /= /truncate_word hle /= zero_extend_u //.
do 3 f_equal.
exact: zero_extend_cut.
(* Olnot *)
+ rewrite /= /sem_sop1 => sz; t_xrbindP => w Hz z' /to_wordI' [sz' [z [Hsz ? ->]]] ?; subst.
case: andP => // - [hsz] /eqP ?; subst ty.
rewrite /truncate_val /= /truncate_word cmp_le_refl zero_extend_u in Hv'.
case: Hv' => ?; subst v'.
by rewrite /sem_pexprs /= Hz /exec_sopn /= /truncate_word Hsz /= /sopn_sem /=
/x86_NOT /check_size_8_64 hsz.
(* Oneg *)
+ rewrite /= /sem_sop1 => - [] // sz; t_xrbindP => w Hv z' /to_wordI' [sz' [z [Hsz ? ->]]] ?; subst.
case: andP => // - [hsz] /eqP ?; subst ty.
split. reflexivity.
rewrite /truncate_val /= /truncate_word /= cmp_le_refl /= zero_extend_u in Hv'.
case: Hv' => ?; subst v'.
by rewrite /sem_pexprs /= Hv /exec_sopn /= /truncate_word Hsz /sopn_sem /= /x86_NEG /check_size_8_64 hsz /= Hw.
+ case: o => // [[] sz |[] sz|[] sz|[]// u sz| []// u sz|sz|sz|sz|sz|sz|sz|sz|sz| ve sz | ve sz | ve sz | ve sz | ve sz | ve sz] //.
case: andP => // - [hsz64] /eqP ?; subst ty.
(* Oadd Op_w *)
+ rewrite /= /sem_sop2 /=; t_xrbindP => v1 ok_v1 v2 ok_v2.
move => ? /to_wordI' [w1] [z1] [hle1 ??]; subst.
move => ? /to_wordI' [w2] [z2] [hle2 ??]; subst.
move => ?; subst v.
rewrite /truncate_val /= /truncate_word /= cmp_le_refl /= zero_extend_u in Hv'.
case: Hv' => ?; subst v'.
case Heq: is_lea => [lea|].
+ (* LEA *)
case/is_leaP: Heq => /andP [hsz1 hsz2] hsub hlea hsc.
split; first by rewrite hsz1 hsz2.
split => //; split => //=.
eexists; split; first reflexivity.
rewrite -(zero_extend_u (_ + _)).
apply: (mk_leaP (gd := gd) _ (cmp_le_refl _) hlea) => //.
by rewrite /= ok_v1 ok_v2 /= /sem_sop2 /= /truncate_word hle1 hle2.
move => {Heq}.
have /= := @add_inc_dec_classifyP s sz e1 e2.
rewrite ok_v1 ok_v2 => /(_ _ _ _ _ erefl).
case: (add_inc_dec_classify _ _ _) => [y|y|//].
(* AddInc *)
* case => sz' [w'] [hsz] []; rewrite /sem_pexprs /= => -> /= <-.
have hsz' : (sz ≤ sz')%CMP by case: hsz => ->.
by rewrite /exec_sopn /sopn_sem /= /x86_INC /rflags_of_aluop_nocf_w /flags_w /truncate_word hsz'
/= /check_size_8_64 hsz64 /=; eauto.
(* AddDec *)
* case => sz' [w'] [hsz] []; rewrite /sem_pexprs /= => -> /= <-.
have hsz' : (sz ≤ sz')%CMP by case: hsz => ->.
by rewrite /exec_sopn /sopn_sem /= /x86_DEC /rflags_of_aluop_nocf_w /flags_w /truncate_word hsz' /= /check_size_8_64 hsz64 /=; eauto.
(* AddNone *)
move=> _;split.
rewrite read_es_cons {2}/read_e /= !read_eE. SvD.fsetdec.
by rewrite /= ok_v1 ok_v2 /= /exec_sopn /= /sem_sopn /= /truncate_word hle1 hle2 /= /sopn_sem /=
/x86_ADD /= /check_size_8_64 hsz64 /= Hw.
(* Omul Op_w *)
+ rewrite /= /sem_sop2 /=; t_xrbindP => v1 ok_v1 v2 ok_v2.
move => ? /to_wordI' [w1] [z1] [hle1 ??]; subst.
move => ? /to_wordI' [w2] [z2] [hle2 ??]; subst.
move => ?; subst v.
case: andP => // - [hsz64] /eqP ?; subst ty.
rewrite /truncate_val /= /truncate_word cmp_le_refl zero_extend_u in Hv'.
case: Hv' => ?; subst v'.
case Heq: is_lea => [lea|].
(* LEA *)
+ case/is_leaP: Heq => /andP [hsz1 hsz2] hsub hlea hsc.
split; first by rewrite hsz1 hsz2.
split => //; split => //=.
eexists; split; first reflexivity.
rewrite -(zero_extend_u (_ * _)).
apply: (mk_leaP (gd := gd) _ (cmp_le_refl _) hlea) => //.
by rewrite /= ok_v1 ok_v2 /= /sem_sop2 /= /truncate_word hle1 hle2.
move => {Heq}.
case Heq : mulr => [o e'].
by apply: mulr_ok ok_v1 ok_v2 hle1 hle2 hsz64 Hw Heq.
(* Osub Op_w *)
+ rewrite /= /sem_sop2 /=; t_xrbindP => v1 ok_v1 v2 ok_v2.
move => ? /to_wordI' [w1] [z1] [hle1 ??]; subst.
move => ? /to_wordI' [w2] [z2] [hle2 ??]; subst.
move => ?; subst v.
case: andP => // - [hsz64] /eqP ?; subst ty.
rewrite /truncate_val /= /truncate_word cmp_le_refl zero_extend_u in Hv'.
case: Hv' => ?; subst v'.
case Heq: is_lea => [lea|].
(* LEA *)
* case/is_leaP: Heq => /andP [hsz1 hsz2] hsub hlea hsc.
split; first by rewrite hsz1 hsz2.
split => //; split => //=.
eexists; split; first reflexivity.
rewrite -(zero_extend_u (_ - _)).
apply: (mk_leaP (gd := gd) _ (cmp_le_refl _) hlea) => //.
by rewrite /= ok_v1 ok_v2 /= /sem_sop2 /= /truncate_word hle1 hle2.
have := sub_inc_dec_classifyP sz e2.
case: (sub_inc_dec_classify _ _)=> [He2|He2|//]; try subst e2.
(* SubInc *)
* move: ok_v2 => /ok_word_inj [??]; subst.
rewrite ok_v1 /= /exec_sopn /sopn_sem /= /truncate_word hle1 /=.
rewrite /x86_INC /check_size_8_64 hsz64 /rflags_of_aluop_nocf_w /flags_w /=.
eexists _, _, _, _. repeat f_equal.
rewrite zero_extend_u /wrepr mathcomp.word.word.mkwordN1E.
ssring.
(* SubDec *)
* move: ok_v2 => /ok_word_inj [??]; subst.
rewrite ok_v1 /= /exec_sopn /sopn_sem /= /truncate_word hle1 /=.
rewrite /x86_DEC /check_size_8_64 hsz64 /rflags_of_aluop_nocf_w /flags_w /=.
by eexists _, _, _, _; repeat f_equal; rewrite zero_extend_u /wrepr mathcomp.word.word.mkword1E.
(* SubNone *)
+ split. by rewrite read_es_swap.
by rewrite /= ok_v1 ok_v2 /= /exec_sopn /sopn_sem /= /truncate_word hle1 hle2 /x86_SUB /check_size_8_64 hsz64 /= Hw.
(* Odiv (Cmp_w u sz) *)
+ case: ifP => // /andP [] /andP [] hsz1 hsz2 /eqP ?;subst ty.
rewrite /sem_pexprs /=; t_xrbindP => v1 hv1 v2 hv2.
rewrite /sem_sop2 /= /mk_sem_divmod;t_xrbindP => /= w1 hw1 w2 hw2 w3 hw3 ?; subst v.
move: Hv'; rewrite /truncate_val /= /truncate_word cmp_le_refl /= => /ok_inj ?; subst v'.
split => //.
exists v1, w1;split => //.
move=> s1 hs1 hl he.
have -> /= := eeq_exc_sem_pexpr _ hs1 hv1; last first.
+ move: he; rewrite /read_e /= /disj_fvars /x86_lowering.disj_fvars !read_eE /disjoint.
by rewrite /is_true !Sv.is_empty_spec;SvD.fsetdec.
split => //.
have -> /= := eeq_exc_sem_pexpr _ hs1 hv2; last first.
+ move: he; rewrite /read_e /= /disj_fvars /x86_lowering.disj_fvars !read_eE /disjoint.
by rewrite /is_true !Sv.is_empty_spec;SvD.fsetdec.
case: ifP hw3 => // hdiv []; simpl in * => {he}.
case/Bool.orb_false_elim: hdiv => /eqP neq hdiv.
case: u => /= ?; subst w3;
rewrite /= /exec_sopn /sopn_sem /= /x86_IDIV /x86_DIV !truncate_word_u
/check_size_16_64 /= hsz1 hsz2 /= hw2 /=.
+ rewrite hw1 /= wdwords0 (wsigned_quot_bound neq hdiv) /=.
move: Hw;rewrite /wdivi zero_extend_u => /(eeq_exc_write_lval hl hs1) [s1' -> ?].
by exists s1'.
have hw2' : (wunsigned w2 == 0%Z) = false.
+ by apply /negbTE; apply /eqP => h; apply neq, wunsigned_inj.
rewrite hw2' hw1 /= wdwordu0.
move: hw2' => /negbT -/(wunsigned_div_bound w1) -/negbTE -> /=.
move: Hw;rewrite /wdivi zero_extend_u => /(eeq_exc_write_lval hl hs1) [s1' -> ?].
by exists s1'.
(* Omod (Cmp_w u sz) *)
+ case: ifP => // /andP [] /andP [] hsz1 hsz2 /eqP ?; subst ty.
rewrite /sem_pexprs /=; t_xrbindP => v1 hv1 v2 hv2.
rewrite /sem_sop2 /= /mk_sem_divmod;t_xrbindP => /= w1 hw1 w2 hw2 w3 hw3 ?; subst v.
move: Hv'; rewrite /truncate_val /= /truncate_word cmp_le_refl /= => /ok_inj ?; subst v'.
split => //.
exists v1, w1;split => //.
move=> s1 hs1 hl he.
have -> /= := eeq_exc_sem_pexpr _ hs1 hv1; last first.
+ move: he; rewrite /read_e /= /disj_fvars /x86_lowering.disj_fvars !read_eE /disjoint.
by rewrite /is_true !Sv.is_empty_spec;SvD.fsetdec.
split => //.
have -> /= := eeq_exc_sem_pexpr _ hs1 hv2; last first.
+ move: he; rewrite /read_e /= /disj_fvars /x86_lowering.disj_fvars !read_eE /disjoint.
by rewrite /is_true !Sv.is_empty_spec;SvD.fsetdec.
case: ifP hw3 => // hdiv []; simpl in * => {he}.
case/Bool.orb_false_elim: hdiv => /eqP neq hdiv.
case: u => /= ?; subst w3;
rewrite /= /exec_sopn /sopn_sem /= /x86_IDIV /x86_DIV !truncate_word_u
/check_size_16_64 /= hsz1 hsz2 /= hw2 /=.
+ rewrite hw1 /= wdwords0 (wsigned_quot_bound neq hdiv) /=.
move: Hw;rewrite /wdivi zero_extend_u => /(eeq_exc_write_lval hl hs1) [s1' -> ?].
by exists s1'.
have hw2' : (wunsigned w2 == 0%Z) = false.
+ by apply /negbTE; apply /eqP => h; apply neq, wunsigned_inj.
rewrite hw2' hw1 /= wdwordu0.
move: hw2' => /negbT -/(wunsigned_div_bound w1) -/negbTE -> /=.
move: Hw;rewrite /wdivi zero_extend_u => /(eeq_exc_write_lval hl hs1) [s1' -> ?].
by exists s1'.
(* Oland Op_w *)
+ case handn : is_andn => [[a1 a2] | ].
+ move=> he.
have : sem_pexpr gd s (Papp2 (Oland sz) (Papp1 (Olnot sz) a1) a2) = ok v /\
Sv.Subset (read_es [:: a1; a2]) (read_e (Papp2 (Oland sz) e1 e2)).
+ have hlnot : forall e, match is_lnot e with
| Some a => exists sz, e = Papp1 (Olnot sz) a
| _ => True
end.
+ by case => //= -[] // ??;eexists;eauto.
move: handn (hlnot e1) (hlnot e2); rewrite /is_andn.
case: is_lnot.
+ move=> a1' [] ?? [sz1 ?] ?; subst e1 a1' a2.
move: he;rewrite /= /sem_sop2 /= /sem_sop1 /=.
t_xrbindP => y h ha1 h' /to_wordI' [sz' [w' [hsz]]] ???;subst y h h'.
move=> w2 -> wn /to_wordI' [sz1' [wn1 [hsz1]]].
move=> /Vword_inj [heq ]; subst sz1' => /= ??; subst wn1 wn.
move=> w3 /to_wordI' [sz2 [wn2 [hsz2]]] ???; subst w2 w3 v.
have hle := cmp_le_trans hsz1 hsz.
rewrite ha1 /= /truncate_word hle /= truncate_word_u /= hsz2 /=.
rewrite !wnot_zero_extend // zero_extend_idem //; split => //.
by rewrite /read_e /read_es /= !read_eE; SvD.fsetdec.
case: is_lnot => //.
move=> a1' [] ?? _ [sz1 ?]; subst e1 a1' e2.
move: he;rewrite /= /sem_sop2 /= /sem_sop1 /=.
t_xrbindP => y -> w wa -> h3 /to_wordI' [sz' [w' [hsz]]] ???; subst wa h3 w.
move=> w2 /to_wordI' [sz1' [wn1 [hsz1]]] ??; subst y w2.
move=> w3 /to_wordI' [sz2 [wn2 [hsz2]]].
move=> /Vword_inj [heq ]; subst sz1 => /= ???; subst wn2 w3 v.
have hle := cmp_le_trans hsz2 hsz.
rewrite /truncate_word hle hsz1 /= truncate_word_u /=.
by rewrite !wnot_zero_extend // zero_extend_idem // (@wandC sz); split.
move=> []; rewrite /= /sem_sop1 /sem_sop2 /=.
t_xrbindP => v1 va1 ha1 wa1 hva1 hv1 va2 ha2 wa2 hwa2 twa2 hva2 ? hread.
subst v v1.
case hty: (_ ≤ _)%CMP => /=.
+ case hty32: (_ ≤ _)%CMP => //=.
case : eqP => //= ?; subst ty.
split;first by apply hread.
rewrite /exec_sopn /sopn_sem /= ha1 /= ha2 /= hva1 /= hva2 /=.
rewrite /x86_ANDN /check_size_32_64 hty32 hty /=.
move: Hv' hwa2; rewrite /truncate_val /= /truncate_word cmp_le_refl /=.
rewrite !zero_extend_u => /ok_inj ? /ok_inj ?; subst wa2 v'.
by rewrite /wandn Hw.
case : eqP => //= ?; subst ty.
rewrite /exec_sopn /sopn_sem /= ha1 /= ha2 /= hva1 /= hva2 /=.
rewrite /x86_VPANDN /x86_u128_binop (wsize_nle_u64_check_128_256 hty) /=.
move: Hv' hwa2; rewrite /truncate_val /= /truncate_word cmp_le_refl /=.
by rewrite !zero_extend_u => /ok_inj <- /ok_inj <-.
case: eqP; last by rewrite andbF => _ _ /=; case: ifP.
move => ?; subst ty; rewrite /= /sem_sop2 /=; t_xrbindP => v1 ok_v1 v2 ok_v2.
move => ? /to_wordI' [sz1] [w1] [hw1 ??]; subst.
move => ? /to_wordI' [sz2] [w2] [hw2 ??]; subst.
move => ?; subst v.
move: Hv'; rewrite /truncate_val /= /truncate_word cmp_le_refl zero_extend_u => /ok_inj ?; subst v'.
case hty: (_ ≤ _)%CMP; rewrite /exec_sopn /sopn_sem /= ok_v1 ok_v2 /= /truncate_word hw1 hw2 /=.
* (* AND *)
split. by rewrite read_es_swap.
by rewrite /x86_AND /check_size_8_64 hty /= Hw.
(* VPAND *)
rewrite /x86_VPAND /x86_u128_binop /=.
by rewrite (wsize_nle_u64_check_128_256 hty) /=.
(* Olor Op_w *)
+ case: eqP; last by rewrite andbF => _ _ /=; case: ifP.
move => ?; subst ty; rewrite /= /sem_sop2 /=; t_xrbindP => v1 ok_v1 v2 ok_v2.
move => ? /to_wordI' [sz1] [w1] [hw1 ??]; subst.
move => ? /to_wordI' [sz2] [w2] [hw2 ??]; subst.
move => ?; subst v.
move: Hv'; rewrite /truncate_val /= /truncate_word cmp_le_refl zero_extend_u => /ok_inj ?; subst v'.
case hty: (_ ≤ _)%CMP; rewrite /exec_sopn /sopn_sem /= ok_v1 ok_v2 /= /truncate_word hw1 hw2 /=.
* (* OR *)
split. by rewrite read_es_swap.
by rewrite /x86_OR /check_size_8_64 hty /= Hw.
(* VPOR *)
rewrite /x86_VPOR /x86_u128_binop /=.
by rewrite (wsize_nle_u64_check_128_256 hty).
(* Olxor Op_w *)
+ case: eqP; last by rewrite andbF => _ _ /=; case: ifP.
move => ?; subst ty; rewrite /= /sem_sop2 /=; t_xrbindP => v1 ok_v1 v2 ok_v2.
move => ? /to_wordI' [sz1] [w1] [hw1 ??]; subst.
move => ? /to_wordI' [sz2] [w2] [hw2 ??]; subst.
move => ?; subst v.
move: Hv'; rewrite /truncate_val /= /truncate_word cmp_le_refl zero_extend_u => /ok_inj ?; subst v'.
case hty: (_ ≤ _)%CMP; rewrite /exec_sopn /sopn_sem /= ok_v1 ok_v2 /= /truncate_word hw1 hw2 /=.
* (* XOR *)
split. by rewrite read_es_swap.
by rewrite /x86_XOR /check_size_8_64 hty /= Hw.
(* VPXOR *)
rewrite /x86_VPXOR /x86_u128_binop /=.
by rewrite (wsize_nle_u64_check_128_256 hty).
(* Olsr *)
+ case: andP => // - [hsz64] /eqP ?; subst ty.
rewrite /sem_pexprs /=; t_xrbindP => v1 -> v2 ->.
rewrite /sem_sop2 /exec_sopn /sopn_sem /=.
t_xrbindP => w1 -> w2 -> /= ?; subst v.
move: Hv'; rewrite /truncate_val /= /truncate_word cmp_le_refl zero_extend_u => /ok_inj ?; subst v'.
split. by rewrite read_es_swap.
move: Hw; rewrite /sem_shr /sem_shift /x86_SHR /check_size_8_64 hsz64 /=.
case: eqP.
* by move => ->; rewrite /= wshr0 => ->.
move => _ /=.
by case: ifP => /= _ ->.
(* Olsl *)
+ case: sz => // sz.
case: andP => // - [hsz64] /eqP ?; subst ty.
rewrite /sem_pexprs /=; t_xrbindP => v1 -> v2 ->.
rewrite /sem_sop2 /exec_sopn /sopn_sem /=; t_xrbindP => w1 -> w2 -> /= ?; subst v.
move: Hv'; rewrite /truncate_val /= /truncate_word cmp_le_refl zero_extend_u => /ok_inj ?; subst v'.
split. by rewrite read_es_swap.
move: Hw; rewrite /sem_shl /sem_shift /x86_SHL /check_size_8_64 hsz64 /=.
case: eqP.
* by move => ->; rewrite /= wshl0 => ->.
move => _ /=.
by case: ifP => /= _ ->.
(* Oasr *)
+ case: sz => // sz.
case: andP => // - [hsz64] /eqP ?; subst ty.
rewrite /sem_pexprs /=; t_xrbindP => v1 -> v2 ->.
rewrite /sem_sop2 /exec_sopn /sopn_sem /=; t_xrbindP => w1 -> w2 -> /= ?; subst v.
move: Hv'; rewrite /truncate_val /= /truncate_word cmp_le_refl zero_extend_u => /ok_inj ?; subst v'.
split. by rewrite read_es_swap.
move: Hw; rewrite /sem_sar /sem_shift /x86_SAR /check_size_8_64 hsz64 /=.
case: eqP.
* by move => ->; rewrite /= wsar0 => ->.
move => _ /=.
by case: ifP => /= _ ->.
(* Oror *)
+ case: andP => // - [hsz64] /eqP ?; subst ty.
rewrite /=; t_xrbindP => v1 -> v2 ->.
rewrite /sem_sop2 /exec_sopn /sopn_sem /=.
t_xrbindP => w1 -> w2 -> /= ?; subst v.
move: Hv'; rewrite /truncate_val /= /truncate_word cmp_le_refl zero_extend_u => /ok_inj ?; subst v'.
split. by rewrite read_es_swap.
move: Hw; rewrite /sem_shr /sem_shift /x86_ROR /check_size_8_64 hsz64 /=.
case: eqP.
* rewrite /sem_ror /sem_shift.
move=> -> /=.
rewrite wunsigned0 wror0.
by move=> ->.
move=> _ /=.
by case: ifP => /= _ ->.
(* Orol *)
+ case: andP => // - [hsz64] /eqP ?; subst ty.
rewrite /=; t_xrbindP => v1 -> v2 ->.
rewrite /sem_sop2 /exec_sopn /sopn_sem /=.
t_xrbindP => w1 -> w2 -> /= ?; subst v.
move: Hv'; rewrite /truncate_val /= /truncate_word cmp_le_refl zero_extend_u => /ok_inj ?; subst v'.
split. by rewrite read_es_swap.
move: Hw; rewrite /sem_shr /sem_shift /x86_ROL /check_size_8_64 hsz64 /=.
case: eqP.
* rewrite /sem_rol /sem_shift.
move=> -> /=.
rewrite wunsigned0 wrol0.
by move=> ->.
move=> _ /=.
by case: ifP => /= _ ->.
(* Ovadd ve sz *)
+ case: ifP => // /andP [hle /eqP ?]; subst ty.
rewrite /= /sem_sop2 /exec_sopn /sopn_sem /=;t_xrbindP => v1 ok_v1 v2 ok_v2.
move => ? /to_wordI' [sz1] [w1] [hw1 ??]; subst.
move => ? /to_wordI' [sz2] [w2] [hw2 ??]; subst.
move => ?; subst v.
move: Hv'; rewrite /truncate_val /= /truncate_word cmp_le_refl zero_extend_u => /ok_inj ?; subst v'.
rewrite ok_v1 /= ok_v2 /= /x86_VPADD /x86_u128_binop /=.
by rewrite (check_size_128_256_ge hle) /= /truncate_word hw1 hw2.
(* Ovsub ve sz *)
+ case: ifP => // /andP [hle /eqP ?]; subst ty.
rewrite /= /sem_sop2 /exec_sopn /sopn_sem /=;t_xrbindP => v1 ok_v1 v2 ok_v2.
move => ? /to_wordI' [sz1] [w1] [hw1 ??]; subst.
move => ? /to_wordI' [sz2] [w2] [hw2 ??]; subst.
move => ?; subst v.
move: Hv'; rewrite /truncate_val /= /truncate_word cmp_le_refl zero_extend_u => /ok_inj ?; subst v'.
rewrite ok_v1 /= ok_v2 /= /x86_VPSUB /x86_u128_binop /=.
by rewrite (check_size_128_256_ge hle) /= /truncate_word hw1 hw2.
(* Ovmul ve sz *)
+ case: ifP => // /andP [/andP[hle1 hle2] /eqP ?]; subst ty.
rewrite /= /sem_sop2 /exec_sopn /sopn_sem /=;t_xrbindP => v1 ok_v1 v2 ok_v2.
move => ? /to_wordI' [sz1] [w1] [hw1 ??]; subst.
move => ? /to_wordI' [sz2] [w2] [hw2 ??]; subst.
move => ?; subst v.
move: Hv'; rewrite /truncate_val /= /truncate_word cmp_le_refl zero_extend_u => /ok_inj ?; subst v'.
rewrite ok_v1 /= ok_v2 /= /x86_VPMULL /x86_u128_binop /=.
rewrite /check_size_16_32 hle1 (check_size_128_256_ge hle2).
by rewrite /truncate_word hw1 hw2.
(* Ovlsr ve sz *)
+ case: ifP => // /andP [/andP [hle1 hle2] /eqP ?]; subst ty.
rewrite /= /sem_sop2 /exec_sopn /sopn_sem /=;t_xrbindP => v1 ok_v1 v2 ok_v2.
move => ? /to_wordI' [sz1] [w1] [hw1 ??]; subst.
move => ? /to_wordI' [sz2] [w2] [hw2 ??]; subst.
move => ?; subst v.
move: Hv'; rewrite /truncate_val /= /truncate_word cmp_le_refl zero_extend_u => /ok_inj ?; subst v'.
rewrite ok_v1 /= ok_v2 /= /x86_VPSRL /x86_u128_shift /=.
rewrite (check_size_128_256_ge hle2) (check_size_16_64_ve hle1) /=.
by rewrite /truncate_word hw1 hw2.
(* Ovlsl ve sz *)
+ case: ifP => // /andP [/andP [hle1 hle2] /eqP ?]; subst ty.
rewrite /= /sem_sop2 /exec_sopn /sopn_sem /=;t_xrbindP => v1 ok_v1 v2 ok_v2.
move => ? /to_wordI' [sz1] [w1] [hw1 ??]; subst.
move => ? /to_wordI' [sz2] [w2] [hw2 ??]; subst.
move => ?; subst v.
move: Hv'; rewrite /truncate_val /= /truncate_word cmp_le_refl zero_extend_u => /ok_inj ?; subst v'.
rewrite ok_v1 /= ok_v2 /= /x86_VPSLL /x86_u128_shift /=.
rewrite (check_size_128_256_ge hle2) (check_size_16_64_ve hle1) /=.
by rewrite /truncate_word hw1 hw2.
(* Ovasr ve sz *)
+ case: ifP => // /andP [/andP [hle1 hle2] /eqP ?]; subst ty.
rewrite /= /sem_sop2 /exec_sopn /sopn_sem /=;t_xrbindP => v1 ok_v1 v2 ok_v2.
move => ? /to_wordI' [sz1] [w1] [hw1 ??]; subst.
move => ? /to_wordI' [sz2] [w2] [hw2 ??]; subst.
move => ?; subst v.
move: Hv'; rewrite /truncate_val /= /truncate_word cmp_le_refl zero_extend_u => /ok_inj ?; subst v'.
rewrite ok_v1 /= ok_v2 /= /x86_VPSRA /x86_u128_shift /=.
rewrite (check_size_128_256_ge hle2) (check_size_16_64_ve hle1) /=.
by rewrite /truncate_word hw1 hw2.
(* PappN *)
+ case: op => // - [] // - [] //.
case: es => // - [] // [] // [] // hi.
case => // [] // [] // [] // [] // [] // lo [] //.
case: ty Hv' => // - [] //= ok_v'.
rewrite /= /sem_opN /exec_sopn /sem_sop1 /=.
t_xrbindP => ??? -> _ /to_wordI'[] szhi [] whi [] szhi_ge -> -> <- ??? ->.
move => ? /to_wordI'[] szlo [] wlo [] szlo_ge -> -> <- <- <- ?.
t_xrbindP => _ /to_intI[] <- _ /to_intI[] <- [] <- ?; subst => /=.
case: ok_v' => <-{Hw v'}.
rewrite /truncate_word zero_extend_u szlo_ge /=.
rewrite szhi_ge /=.
congr (ok [:: (Vword (wrepr _ (word.wcat_r _))) ]).
by rewrite /= -!/(wrepr U128 _) !wrepr_unsigned.
(* Pif *)
rewrite /check_size_16_64.
by case: stype_of_lval => // w hv; case: andP => // - [] /andP[] -> -> /eqP <-; eauto.
Qed.
Lemma vmap_eq_except_set q s x v:
Sv.In x q → s.[ x <- v] = s [\q].
Proof.
move=> h a ha. apply: Fv.setP_neq.
by case: eqP => // ?; subst.
Qed.
Definition pwrepr64 n :=
{| pw_size := U64 ; pw_word := wrepr _ n ; pw_proof := erefl (U64 ≤ U64)%CMP |}.
Lemma opn_no_immP (P: sopn → sopn → Prop) :
(∀ ws sz, P (Oasm (BaseOp (ws, IMULri sz))) (Oasm (BaseOp (ws, IMULr sz)))) →
(∀ op, (∀ ws sz, op ≠ Oasm (BaseOp (ws, IMULri sz))) → P op op) →
∀ op, P op (opn_no_imm op).
Proof.
clear => A B.
case.
1-5: move => >; exact: B.
case; last by move => >; exact: B.
case => ws.
case; try by move => >; exact: B.
move => sz; exact: A.
Qed.
Lemma opn_5flags_casesP a m sz x y z :
opn_5flags_cases a m sz = Opn5f_large_immed x y z ->
exists2 n : Z, a = x :: y :: z & y = Papp1 (Oword_of_int U64) n.
Proof.
rewrite /opn_5flags_cases.
case: a => [//|] x' [//|] y' z'.
case: is_wconst_of_sizeP => [n|//].
case: check_signed_range => //.
move=> [] ???; subst x y z.
by eexists.
Qed.
Lemma opn_5flags_correct vi ii s a t o cf r xs ys m sz s' :
disj_fvars (read_es a) →
disj_fvars (vars_lvals [:: cf ; r ]) →
sem_pexprs gd s a = ok xs →
exec_sopn o xs = ok ys →
write_lvals gd s [:: Lnone_b vi ; cf ; Lnone_b vi ; Lnone_b vi ; Lnone_b vi ; r] ys = ok s' →
∃ s'',
sem p' ev s [seq MkI ii i | i <- opn_5flags fv m sz vi cf r t o a] s''
∧ eq_exc_fresh s'' s'.
Proof.
move=> da dr hx hr hs; rewrite/opn_5flags.
case hopn: opn_5flags_cases => [x y z|] /=.
+ move: hopn => /opn_5flags_casesP [n ??]; subst a y.
set ℓ :=
with_vm s
(evm s).[{| vtype := sword64; vname := fresh_multiplicand fv U64 |} <- ok (pwrepr64 n)].
assert (eq_exc_fresh ℓ s) as e.
+ subst ℓ; case:(s) => ?? /=;split => //.
by apply vmap_eq_except_set, multiplicand_in_fv.
assert (disj_fvars (read_e x) ∧ disj_fvars (read_es z)) as dxz.
{ eapply disj_fvars_m in da.
2: apply SvP.MP.equal_sym; eapply read_es_cons.
apply disjoint_union in da;intuition. }
case: dxz => dx dz.
case:(eeq_exc_write_lvals _ e hs). exact dr.
move=> s'' hs' e'.
exists s''. refine (conj _ e'). repeat econstructor.
rewrite /sem_sopn /= !zero_extend_u -/(pwrepr64 _) -/ℓ.
move: hx; rewrite /sem_pexprs /=; t_xrbindP => y hy z' z1 hz1 ? ?; subst z' xs.
rewrite (eeq_exc_sem_pexpr dx e hy) /=.
fold (sem_pexprs gd s) in hz1.
rewrite /get_gvar /get_var /on_vu Fv.setP_eq /= -/(sem_pexprs gd ℓ).
rewrite (eeq_exc_sem_pexprs dz e hz1) /= /exec_sopn /sopn_sem /=.
move: hr.
apply opn_no_immP.
- rewrite /exec_sopn /sopn_sem; case.
+ by move => ws ? /=; case: eqP => /= ? ->.
by move => _ /= ->.
by rewrite /exec_sopn => op _ ->.
+ exists s'. repeat econstructor. by rewrite /sem_sopn hx /= hr.
Qed.
(* This situation comes up several times below. *)
Lemma aux_eq_exc_trans P r r' :
eq_exc_fresh r r'
-> forall s,
P s /\ eq_exc_fresh s r
-> exists s', P s' /\ eq_exc_fresh s' r'.
Proof.
move=> Hr' s [Ps Hs].
exists s.
split; first exact Ps.
exact: (eeq_excT Hs Hr').
Qed.
Lemma reduce_wconstP s e sz sz' (v: word sz') :
sem_pexpr gd s e = ok (Vword v) →
∃ sw (w: word sw),
sem_pexpr gd s (reduce_wconst sz e) = ok (Vword w) ∧
(cmp_min sz sz' ≤ sw)%CMP ∧
zero_extend sz v = zero_extend sz w.
Proof.
rewrite /reduce_wconst.
case: e; eauto using cmp_min_leR.
case; eauto using cmp_min_leR.
move => sw []; eauto using cmp_min_leR.
move => z /ok_word_inj [?]; subst => /= <- {v}.
eexists _, _; split; first reflexivity.
split => //.
refine (cmp_minP (P := λ x, zero_extend _ _ = zero_extend sz (wrepr x z)) _ _) => //.
by move => hle; rewrite !zero_extend_wrepr.
Qed.
Lemma mov_wsP (p1: prog) s1 e ws tag i x w s2 :
(ws <= U64)%CMP ->
(Let i' := sem_pexpr (p_globs p1) s1 e in to_word ws i') = ok i
-> write_lval (p_globs p1) x (Vword i) s1 = ok s2
-> sem_i p1 w s1 (mov_ws is_regx ws x e tag) s2.
Proof.
by move=> hws he hx; rewrite /mov_ws; case: ifP => [ /andP [] _ h | _];
constructor; rewrite /sem_sopn /= /exec_sopn /=;
move: he; t_xrbindP => _ -> /= -> /=;
rewrite /sopn_sem /= /x86_MOVX /x86_MOV /check_size_32_64 /check_size_8_64 hws ?h /= hx.
Qed.
Local Lemma Hassgn : sem_Ind_assgn p Pi_r.
Proof.
move => s1 s2 l tag ty e v v' Hv hty Hw ii /= Hdisj s1' Hs1'.
move: Hdisj; rewrite /disj_fvars /x86_lowering.disj_fvars vars_I_assgn=> /disjoint_union [Hdisjl Hdisje].
have Hv' := eeq_exc_sem_pexpr Hdisje Hs1' Hv.
have [s2' Hw' Hs2'] := eeq_exc_write_lval Hdisjl Hs1' Hw.
rewrite /= /lower_cassgn.
have := lower_cassgn_classifyP Hv' hty Hw'.
case: (lower_cassgn_classify is_var_in_memory _ e l).
(* LowerMov *)
+ move=> b [tw ?] [hle'] [sz'] [w] [hsz' ?]; subst ty v.
move: hty; rewrite /truncate_val; apply: rbindP => w' /truncate_wordP [] hle -> {w'} [?]; subst v'.
have [sz [vw [h [hsz hw]]]] := reduce_wconstP tw Hv'.
rewrite (cmp_le_min hle) in hsz.
case: b.
* set ℓ :=
with_vm s1'
(evm s1').[{| vtype := sword tw; vname := fresh_multiplicand fv tw |} <- ok (pword_of_word (zero_extend tw vw)) ].
assert (eq_exc_fresh ℓ s1') as dℓ.
+ subst ℓ; case:(s1') => ?? /=; split => //.
by apply vmap_eq_except_set, multiplicand_in_fv.
case: (eeq_exc_write_lval Hdisjl dℓ Hw') => ℓ' hℓ' dℓ'.
eexists; split.
repeat econstructor.
by rewrite /sem_sopn /sem_pexprs /= h /= /exec_sopn /sopn_sem /= /truncate_word hsz
/x86_MOV /check_size_8_64 hle' /= /write_var /set_var /= sumbool_of_boolET.
by rewrite /sem_sopn /sem_pexprs/= /get_gvar /get_var Fv.setP_eq /= /exec_sopn /sopn_sem /= /truncate_word cmp_le_refl /x86_MOV /check_size_8_64 hle' /= zero_extend_u /= -/ℓ -hw hℓ'.
exact: (eeq_excT dℓ' Hs2').
* exists s2'; split=> //=.
case: ifP => [/andP [] /andP [] /is_zeroP he ??| _ ];first last.
- apply/sem_seq1/EmkI/mov_wsP => //.
+ by rewrite h /= /truncate_word hsz.
by rewrite -hw.
move: h; rewrite he => /ok_word_inj [?]; subst => /= ?; subst vw.
rewrite hw zero_extend_u wrepr0 in Hw' => {hw}.
by case: ifP => hsz64; apply: sem_seq1; apply: EmkI; apply: Eopn;
rewrite /sem_sopn /sem_pexprs /exec_sopn /sopn_sem /= /Oset0_instr hsz64 /= Hw'.
(* LowerCopn *)
+ move=> o e' H.
exists s2'; split=> //.
apply: sem_seq1; apply: EmkI; apply: Eopn.
by rewrite /sem_sopn H /= Hw'.
(* LowerInc *)
+ move=> o e' [b1 [b2 [b3 [b4 H]]]].
exists s2'; split=> //; apply: sem_seq1; apply: EmkI; apply: Eopn.
by rewrite /sem_sopn H /= Hw'.
(* LowerLea *)
+ move => sz [d b sc o] /= [hsz] [Hsc] [hrl] [w [? Hslea]]; subst v'; set f := Lnone_b _.
set ob := oapp Plvar _ b; set oo := oapp Plvar _ o.
have [wb [wo [Hwb Hwo Ew ]]]:
exists (wb wo: word sz),
[/\ sem_pexpr gd s1' ob >>= to_word sz = ok wb,
sem_pexpr gd s1' oo >>= to_word sz = ok wo &
w = (wrepr sz d + (wb + (wrepr sz sc * wo)))%R].
+ move: Hslea; rewrite /sem_lea /=; t_xrbindP => wb Hwb wo Hwo H.
exists wb, wo; split.
- subst ob; case: b Hwb {hrl} => [ b | ] /=; t_xrbindP.
* by rewrite /get_gvar => vb -> /to_wordI' [sz'] [w'] [h -> ->]; rewrite /= /truncate_word h.
by move => <-; rewrite truncate_word_u; f_equal; apply: word_ext.
- subst oo; case: o Hwo {hrl} => [ o | ] /=; t_xrbindP.
* by rewrite /get_gvar => vb -> /to_wordI' [sz'] [w'] [h -> ->]; rewrite /= /truncate_word h.
by move => <-; rewrite truncate_word_u; f_equal; apply: word_ext.
by subst.
move: Hwb; apply: rbindP => vb Hvb Hwb.
move: Hwo; apply: rbindP => vo Hvo Hwo.
set elea := Papp2 (Oadd (Op_w sz)) (wconst (wrepr Uptr d)) (Papp2 (Oadd (Op_w sz)) ob (Papp2 (Omul (Op_w sz)) (wconst (wrepr Uptr sc)) oo)).
case /andP: hsz => hsz1 hsz2.
have Hlea :
Let vs := sem_pexprs gd s1' [:: elea ] in
exec_sopn (Ox86 (LEA sz)) vs = ok [:: Vword w ].
+ rewrite /sem_pexprs /= Hvb Hvo /= /exec_sopn /sopn_sem /sem_sop2 /= /truncate_word hsz2 /=.
rewrite Hwb Hwo /= truncate_word_u /= truncate_word_u /= truncate_word_u /= /x86_LEA /check_size_16_64 hsz1 hsz2 /=.
by rewrite Ew -!/(zero_extend _ _) !zero_extend_wrepr.
have Hlea' : sem p' ev s1'
[:: MkI (warning ii Use_lea) (Copn [:: l] tag (Ox86 (LEA sz)) [:: elea])] s2'.
+ by apply: sem_seq1; apply: EmkI; apply: Eopn; rewrite /sem_sopn Hlea /= Hw'.
case: use_lea; first by exists s2'.
subst w.
case: eqP => [ ? | _ ].
+ subst d; case: eqP => [ ? | _].
+ subst sc; exists s2'; split => //; apply: sem_seq1; constructor; constructor.
move: Hw'; rewrite /sem_sopn /sem_pexprs /exec_sopn /sopn_sem /= Hvb Hvo /= Hwb Hwo /= /x86_ADD /=.
by rewrite /check_size_8_64 hsz2 /= wrepr0 wrepr1 GRing.add0r GRing.mul1r => ->.
case: is_zeroP => [ Eob | _ ]; last by exists s2'.
case Heq : mulr => [o1 e'].
move: Hvb; rewrite Eob /= /sem_sop1 /= => -[?]; subst vb.
have [sz1 [w1 [hle1 ??]]]:= to_wordI' Hwo;subst vo wo.
have Hsc1 : sem_pexpr gd s1' (wconst (wrepr Uptr sc)) = ok (Vword (wrepr Uptr sc)).
+ by rewrite /wconst /= /sem_sop1 /= wrepr_unsigned.
move: Hwb; rewrite /= truncate_word_u wrepr_unsigned => -[?];subst wb.
rewrite wrepr0 !GRing.add0r GRing.mulrC in Hw'.
rewrite -(zero_extend_wrepr sc hsz2) in Hw'.
have [] := mulr_ok Hvo Hsc1 hle1 hsz2 _ Hw' Heq; first by rewrite hsz1.
move=> hsub; t_xrbindP => vo vs hvs hvo hw.
case: (opn_5flags_correct ii tag (Some U32) sz _ _ hvs hvo hw).
+ apply: disjoint_w Hdisje .
apply: SvP.MP.subset_trans hrl.
apply: (SvP.MP.subset_trans hsub).
rewrite /read_e /= /read_lea /= /oo read_eE.
by case: (o) => [ ?|]; rewrite /= /read_e /=;SvD.fsetdec.
+ by apply Hdisjl.
exact: (aux_eq_exc_trans Hs2').
case: is_zeroP => [ Eoo | _]; last by exists s2'.
move: Hvo Hwo Hw'; rewrite Eoo => - [<-] {Eoo oo elea Hlea Hlea'}.
rewrite wrepr_unsigned /= truncate_word_u => - [?]; subst wo.
rewrite GRing.mulr0 GRing.addr0 GRing.addrC => Hw'.
case: eqP => [ Ed | _ ].
+ subst d; exists s2'; split => //; apply: sem_seq1; constructor; constructor.
by rewrite /sem_sopn /sem_pexprs /exec_sopn /sopn_sem /= Hvb /= Hwb /= /x86_INC
/check_size_8_64 hsz2 /= -(zero_extend1 sz sz) Hw'.
case: ifP => [ hrange | _ ].
+ exists s2'; split => //; apply: sem_seq1; constructor; constructor.
by rewrite /sem_sopn /sem_pexprs /exec_sopn /sopn_sem /= Hvb /= Hwb /=
/truncate_word hsz2 zero_extend_wrepr //= /x86_ADD /check_size_8_64 hsz2 /=
-/(zero_extend _ _) zero_extend_wrepr // Hw'.
case: eqP => [ Ed | _ ].
+ exists s2'; split => //; apply: sem_seq1; constructor; constructor.
rewrite /sem_sopn /sem_pexprs /exec_sopn /sopn_sem /= Hvb /= Hwb /=.
rewrite truncate_word_u /x86_SUB /check_size_8_64 hsz2 /=.
by rewrite wrepr_unsigned wrepr_opp GRing.opprK Hw'.
set si :=
with_vm s1'
(evm s1').[{| vtype := sword64; vname := fresh_multiplicand fv U64 |} <- ok {| pw_size := U64 ; pw_word := wrepr U64 d ; pw_proof := erefl (U64 ≤ U64)%CMP |}].
have hsi : eq_exc_fresh si s1'.
+ by rewrite /si; case: (s1') => ?? /=; split => //= k hk; rewrite Fv.setP_neq //; apply/eqP => ?; subst k; apply: hk; exact: multiplicand_in_fv.
have [si' Hwi hsi'] := eeq_exc_write_lval Hdisjl hsi Hw'.
eexists; split.
+ apply: Eseq; first by repeat constructor.
apply: sem_seq1. repeat constructor.
rewrite /sem_sopn /exec_sopn /sopn_sem /=.
rewrite zero_extend_u wrepr_unsigned /get_gvar /get_var Fv.setP_eq /=.
rewrite (eeq_exc_sem_pexpr (xs := fvars) _ _ Hvb) //=.
- by rewrite Hwb /= /truncate_word /= /x86_ADD /check_size_8_64 hsz2 /= zero_extend_wrepr // Hwi.
apply: (disj_fvars_subset _ Hdisje).
apply: (SvD.F.Subset_trans _ hrl).
rewrite /read_lea /=; subst ob; case: (b) => [ x | ] /=.
- SvD.fsetdec.
exact: SvP.MP.subset_empty.
exact: (eeq_excT hsi' Hs2').
(* LowerFopn *)
+ set vi := var_info_of_lval _.
move=> sz o a m [] LE. t_xrbindP => ys xs hxs hys hs2.
case: (opn_5flags_correct ii tag m sz _ _ hxs hys hs2).
move: LE Hdisje. apply disjoint_w.
exact Hdisjl.
exact: (aux_eq_exc_trans Hs2').
(* LowerDiscardFlags *)
+ set vi := var_info_of_lval _.
move=> n o es [] hreades.
t_xrbindP=> ys xs hxs hys hs2.
exists s2'.
split; last exact: Hs2'.
apply: sem_seq1. constructor. constructor.
rewrite /sem_sopn hxs {hxs} /=.
rewrite hys {hys} /=.
exact: hs2.
(* LowerCond *)
+ move=> _.
case heq: lower_condition => [i e'].
have [s2'' [hs2'' [ heqex he']]]:= lower_condition_corr ii (sym_eq heq) Hs1' Hv'.
have [s3 hw3 heqex3] := eeq_exc_write_lval Hdisjl heqex Hw.
exists s3; split => //.
rewrite map_cat; apply: (sem_app hs2'') => /=.
apply: sem_seq1; constructor; econstructor; eauto.
(* LowerIf *)
+ move=> t cond e1 e2 [Hsz64] [He] [Hsz] [sz' Ht]; subst e.
set x := lower_condition _ _ _.
have Hcond: x = lower_condition fv (var_info_of_lval l) cond by [].
move: x Hcond=> [i e'] Hcond.
clear s2' Hw' Hs2'.
move: Hv' => /=; t_xrbindP=> b bv Hbv Hb trv1 v1 Hv1 Htr1 trv2 v2 Hv2 Htr2 ?;subst v.
have [s2' [Hs2'1 [Hs2'2 Hs2'3]]] := lower_condition_corr ii Hcond Hs1' Hbv.
have [s3' Hw' Hs3'] := eeq_exc_write_lval Hdisjl Hs2'2 Hw.
exists s3'; split=> //.
rewrite map_cat.
apply: sem_app.
+ exact: Hs2'1.
apply: sem_seq1; apply: EmkI; apply: Eopn.
move: bv Hbv Hb Hs2'3=> [] //=; last by case.
move=> b0 Hb [?] Hb'; subst b0.
rewrite /sem_sopn /sem_pexprs /= Hb' /=.
have Heq' := eeq_excT Hs2'2 (eeq_excS Hs1').
rewrite /read_e /= /disj_fvars /x86_lowering.disj_fvars in Hdisje; move: Hdisje.
rewrite read_eE read_eE -/(read_e _).
move=> /disjoint_union [He /disjoint_union [He1 He2]].
rewrite (eeq_exc_sem_pexpr He1 Heq' Hv1) (eeq_exc_sem_pexpr He2 Heq' Hv2) /=.
have [sz Hvt] := write_lval_word Ht Hw'.
have [w Hvw] := write_lval_undef Hw' Hvt; subst.
rewrite /exec_sopn /sopn_sem /= /x86_CMOVcc.
have /=? := truncate_val_has_type hty; subst ty.
rewrite Hsz64 Hsz /=.
have [sz'' [w' [_ /truncate_wordP[hle ?] hw']]]:= truncate_valI hty; subst.
have : exists w1 w2, to_word sz v1 = ok w1 /\ to_word sz v2 = ok w2 /\
(if b then w1 else w2) = zero_extend sz w'.
+ case: (b) hw' => ?; subst.
+ have [sz3 [w1 [? /truncate_wordP[hle3 ->] ?]]] /= := truncate_valI Htr1; subst.
rewrite /= zero_extend_idem // /truncate_word (cmp_le_trans hle hle3).
move: Htr2 => /= /truncate_val_typeE[? [? [? [/truncate_wordP[hle' ?] ??]]]];subst.
by rewrite /= /truncate_word (cmp_le_trans hle hle');eauto.
have [sz3 [w1 [? /truncate_wordP[hle3 ?] ->]]] /= := truncate_valI Htr2; subst.
rewrite zero_extend_idem // /truncate_word (cmp_le_trans hle hle3).
move: Htr1 => /=; rewrite /truncate_val; t_xrbindP => /= ? /to_wordI' [? [?[hle'??]]] ?;subst.
by rewrite /= /truncate_word (cmp_le_trans hle hle');eauto.
move=> [w1 [w2 [ -> [->]]]] /=.
by case: (b) => ?;subst => /=;rewrite Hw'.
(* LowerDivMod *)
+ move=> d u w s p0 p1 /= [[va [wa [hva hwa hdiv]]] ? hle1 hle2];subst ty.
set vf := {| v_var := _ |}.
set i1 := match u with Signed => _ | _ => _ end.
move: hdiv; set va0 := Vword (match u with Signed => _ | _ => _ end) => hdiv.
have [s1'1 [hsem1 hget heq1]]: exists s1'1,
[/\ sem_I p' ev s1' (MkI ii i1) s1'1,
get_var (evm s1'1) (v_var vf) = ok va0 &
eq_exc_fresh s1'1 s1'].
+ rewrite /i1 /va0; case: (u); eexists; split.
+ by apply: EmkI; rewrite /i1; apply: Eopn; rewrite /sem_sopn /exec_sopn /sopn_sem /= hva /= hwa /x86_CQO /=
/check_size_16_64 hle1 hle2 /= sumbool_of_boolET;eauto.
+ by rewrite /get_var Fv.setP_eq.
+ by split => //; apply vmap_eq_except_set; apply multiplicand_in_fv.
+ by apply: EmkI; apply: Eopn; rewrite /sem_sopn /exec_sopn /sopn_sem /= truncate_word_u /=
/x86_MOV /check_size_8_64 hle2 /=;eauto.
+ by rewrite /= sumbool_of_boolET /get_var /= Fv.setP_eq /= wrepr0.
rewrite sumbool_of_boolET; split => //.
by apply vmap_eq_except_set; apply multiplicand_in_fv.
have [hwa1 [s3 [hsem heqe] {hdiv}]]:= hdiv _ heq1 Hdisjl Hdisje.
exists s3;split.
+ econstructor;first by eassumption.
by case: d hsem => hsem;apply: sem_seq1;apply: EmkI; apply: Eopn;
move: hsem; rewrite /sem_sopn /= /get_gvar hget /= hwa1 /=; t_xrbindP => ? -> ? /= ->.
apply: eeq_excT heqe Hs2'.
(* LowerConcat *)
+ t_xrbindP => hi lo vs ok_vs ok_v'.
exists s2'; split; last exact: Hs2'.
apply: sem_seq1; apply: EmkI; apply: Eopn.
by rewrite /sem_sopn ok_vs /= ok_v' /= Hw'.
(* LowerAssgn *)
move=> _.
exists s2'; split=> //.
apply: sem_seq1; apply: EmkI; apply: Eassgn.
* by rewrite Hv'.
* exact: hty.
exact: Hw'.
Qed.
Lemma app_wwb_dec T' sz (f:sem_prod [::sword sz; sword sz; sbool] (exec T')) x v :
app_sopn _ f x = ok v ->
∃ sz1 (w1: word sz1) sz2 (w2: word sz2) b,
(sz ≤ sz1)%CMP ∧ (sz ≤ sz2)%CMP ∧
x = [:: Vword w1; Vword w2; Vbool b] ∧
f (zero_extend _ w1) (zero_extend _ w2) b = ok v.
Proof.
case: x => // -[] //; last by case => //= ? ?; case: ifP.
move => sz1 w1 [ | x y ] //=; rewrite /truncate_word; case: ifP => //= hle.
t_xrbindP => wx /to_wordI' [sz'] [wx'] [hle' -> ->] {x wx}.
case: y => // y z; t_xrbindP => b /to_boolI -> {y}; case: z => // h.
by eexists _, w1, _, wx', b.
Qed.
Lemma app_ww_dec T' sz (f:sem_prod [::sword sz; sword sz] (exec T')) x v :
app_sopn _ f x = ok v ->
exists sz1 (w1: word sz1) sz2 (w2: word sz2),
(sz ≤ sz1)%CMP ∧ (sz ≤ sz2)%CMP ∧
x = [:: Vword w1; Vword w2] ∧
f (zero_extend _ w1) (zero_extend _ w2) = ok v.
Proof.
case: x => // -[] //; last by case => //= ? ?; case: ifP.
move => sz1 w1 [ | x y ] //=; rewrite /truncate_word; case: ifP => //= hle.
t_xrbindP => wx /to_wordI' [sz'] [wx'] [hle' -> ->] {x wx}.
case: y => // h.
by eexists _, w1, _, wx'.
Qed.
Lemma add_carry_overflow sz (w1 w2: word sz) (b: bool) :
(wbase sz <=? wunsigned w1 + wunsigned w2 + Z.b2z b)%Z =
(wunsigned (add_carry sz (wunsigned w1) (wunsigned w2) (Z.b2z b)) != (wunsigned w1 + wunsigned w2 + Z.b2z b))%Z.
Proof.
rewrite unsigned_overflow //.
have := wunsigned_range w1; have := wunsigned_range w2.
case: b => /=; lia.
Qed.
Lemma sub_underflow sz (w1 w2: word sz) :
(wunsigned w1 - wunsigned w2 <? 0)%Z = (wunsigned (w1 - w2) != (wunsigned w1 - wunsigned w2))%Z.
Proof.
have hn: forall b, ~~b = true <-> ~b.
+ by case;split.
have -> : (wunsigned w1 - wunsigned w2 <? 0)%Z =
~~(wunsigned w2 <=? wunsigned w1)%Z.
+ apply Bool.eq_true_iff_eq.
rewrite hn /is_true Z.ltb_lt Z.leb_le; lia.
by f_equal; rewrite -wleuE.
Qed.
Lemma sub_borrow_underflow sz (w1 w2: word sz) (b:bool) :
(wunsigned w1 - wunsigned w2 - Z.b2z b <? 0)%Z =
(wunsigned (sub_borrow sz (wunsigned w1) (wunsigned w2) (Z.b2z b)) != (wunsigned w1 - (wunsigned w2 + Z.b2z b)))%Z.
Proof.
rewrite /sub_borrow.
case: b => /=;last first.
+ by rewrite Z.sub_0_r Z.add_0_r wrepr_sub !wrepr_unsigned sub_underflow.
have -> : (wunsigned w1 - wunsigned w2 - 1 =
wunsigned w1 - (wunsigned w2 + 1))%Z by ring.
case : (wunsigned w2 =P wbase sz - 1)%Z => hw2.
+ have -> : (wunsigned w1 - (wunsigned w2 + 1) <? 0)%Z.
+ by rewrite /is_true Z.ltb_lt; have := wunsigned_range w1;lia.
symmetry;apply /eqP.
have ->: (wunsigned w2 + 1)%Z = wbase sz by rewrite hw2;ring.
rewrite wrepr_sub wreprB GRing.subr0 wrepr_unsigned.
by have := @wbase_n0 sz;lia.
have -> : (wunsigned w2 + 1 = wunsigned (w2 + 1))%Z.
+ rewrite -wunsigned_add ?wrepr1 //.
by have := wunsigned_range w2;lia.
by rewrite wrepr_sub !wrepr_unsigned sub_underflow.
Qed.
Lemma sem_pexprs_dec2 s e1 e2 v1 v2:
sem_pexprs gd s [:: e1; e2] = ok [:: v1; v2] ->
sem_pexpr gd s e1 = ok v1 /\
sem_pexpr gd s e2 = ok v2.
Proof.
rewrite /sem_pexprs /=.
t_xrbindP=> v1' -> [] // v1'' [] // v2' -> []<- <- []<-.
by split.
Qed.
Lemma sem_pexprs_dec3 s e1 e2 e3 v1 v2 v3:
sem_pexprs gd s [:: e1; e2; e3] = ok [:: v1; v2; v3] ->
sem_pexpr gd s e1 = ok v1 /\
sem_pexpr gd s e2 = ok v2 /\
sem_pexpr gd s e3 = ok v3.
Proof.
rewrite /sem_pexprs /=.
t_xrbindP=> v1' -> [] // v2' [] // v3' [] // v4' Hv4' [] // v5' [] // v6' Hv6' []<- []<- <- <- []<- <-.
by split.
Qed.
Lemma write_lvals_dec2_s s1 s2 v1 v2 xs:
write_lvals gd s1 xs [:: v1; v2] = ok s2 ->
exists x1 x2, xs = [:: x1; x2].
Proof.
move: xs=> [] // x1 [] //=.
+ by apply: rbindP.
move=> x2 [] //; last first.
+ by move=> x3 ? /=; t_xrbindP.
t_xrbindP=> s1' Hs1' s2' Hs2' /= []Hs2; subst s2'.
by eauto.
Qed.
Lemma sem_pexprs_dec2_s s es v1 v2:
sem_pexprs gd s es = ok [:: v1; v2] ->
exists e1 e2, es = [:: e1; e2].
Proof.
move: es=> [] // e1 [] //.
+ by rewrite /sem_pexprs /=; apply: rbindP.
move=> e2 []; last first.
+ move=> a l; rewrite /sem_pexprs /=; t_xrbindP=> ??????????.
by move=> <- <-.
rewrite /sem_pexprs /=.
t_xrbindP=> v1' Hv1' [] // v1'' [] // v2' Hv2' []??[]?; subst v1'' v1' v2'.
by eauto.
Qed.
Lemma lower_addcarry_classifyP sub xs es :
if lower_addcarry_classify sub xs es
is Some (vi, op, es', cf, r)
then
xs = [:: cf; r ] ∧
∃ x y b,
es = [:: x ; y ; b ] ∧
((b = Pbool false ∧ vi = var_info_of_lval r ∧ op = (if sub then SUB else ADD) ∧ es' = [:: x ; y ])
∨
(∃ cfi, b = Plvar cfi ∧ vi = v_info cfi ∧ op = (if sub then SBB else ADC) ∧ es' = es))
else True.
Proof. clear.
case xs => // cf [] // r [] //.
case es => // x [] // y [] // [] //.
+ by move => [] // [] //=; eauto 10.
by rewrite /Plvar /mk_lvar => -[cfi []] // [] //=; eauto 11.
Qed.
Lemma lower_addcarry_correct ii si so si' sub sz xs t es x v :
eq_exc_fresh si' si →
disj_fvars (vars_lvals xs) →
disj_fvars (read_es es) →
sem_pexprs gd si' es = ok x →
exec_sopn ((if sub then Osubcarry else Oaddcarry) sz) x = ok v →
write_lvals gd si' xs v = ok so →
∃ so',
sem p' ev si' (map (MkI ii) (lower_addcarry fv sz sub xs t es)) so' ∧
eq_exc_fresh so' so.
Proof.
move=> hi dxs des hx hv ho.
rewrite/lower_addcarry /=.
set default := [:: Copn _ _ _ _ ].
have hdefault : ∃ so', sem p' ev si' [seq MkI ii i | i <- default] so' ∧ eq_exc_fresh so' so.
+ by repeat econstructor; rewrite /sem_sopn hx /= hv.
case: ifP => // hsz64.
generalize (lower_addcarry_classifyP sub xs es); case: lower_addcarry_classify => //.
move => [[[[vi op] es'] cf] r] [? [x' [y' [b [?]]]]] C; subst.
assert (
disj_fvars (read_es es') ∧
∃ x',
sem_pexprs gd si' es' = ok x' ∧
∃ v',
exec_sopn (Ox86 (op sz)) x' = ok v' ∧
let f := Lnone_b vi in
write_lvals gd si' [:: f ; cf ; f ; f ; f ; r ] v' = ok so) as D.
{
clear - hsz64 des hx hv C ho.
case: C => [ [? [? [? ?]]] | [cfi [?[?[? ?]]]]]; subst; apply (conj des).
+ move: hv hx; rewrite /exec_sopn; t_xrbindP; case: sub => y hy;
have {hy} := app_wwb_dec hy=> -[sz1] [w1] [sz2] [w2] [b] [hsz1] [hsz2] [?] [?] ?;subst x y v =>
/sem_pexprs_dec3 [hx] [hy] [?]; subst b;
(exists [:: Vword w1; Vword w2]; split; [by rewrite /sem_pexprs /= hx /= hy|]);
rewrite /= /sopn_sem /= /truncate_word hsz1 hsz2 /x86_SUB /x86_ADD /check_size_8_64 hsz64; eexists; split; first reflexivity.
+ by rewrite /= Z.sub_0_r sub_underflow wrepr_sub !wrepr_unsigned in ho.
+ by [].
by rewrite /= Z.add_0_r add_overflow wrepr_add !wrepr_unsigned in ho.
exists x; split; [ exact hx |]; clear hx.
move: hv;rewrite /exec_sopn; t_xrbindP; case: sub => y hy;
have {hy} := app_wwb_dec hy=> -[sz1] [w1] [sz2] [w2] [b] [hsz1] [hsz2] [?] [?] ?;
subst x y v; rewrite /= /sopn_sem /= /truncate_word hsz1 hsz2 /x86_SBB /x86_ADC /check_size_8_64 hsz64;
eexists; split; first reflexivity;
rewrite //=.
+ by rewrite /= sub_borrow_underflow in ho.
by rewrite /= add_carry_overflow in ho.
}
clear C.
case: D => des' [ xs' [ hxs' [ v' [hv' ho'] ] ] ].
case: (opn_5flags_correct ii t (Some U32) sz des' dxs hxs' hv' ho') => {hv' ho'} so'.
intuition eauto using eeq_excT.
Qed.
Opaque lower_addcarry.
Local Lemma Hopn : sem_Ind_opn p Pi_r.
Proof.
move => s1 s2 t o xs es.
apply: rbindP=> v; apply: rbindP=> x Hx Hv Hw ii Hdisj s1' Hs1'.
move: Hdisj; rewrite /disj_fvars /x86_lowering.disj_fvars vars_I_opn=> /disjoint_union [Hdisjl Hdisje].
have Hx' := eeq_exc_sem_pexprs Hdisje Hs1' Hx; have [s2' Hw' Hs2'] := eeq_exc_write_lvals Hdisjl Hs1' Hw.
have default : ∃ s2', sem p' ev s1' [:: MkI ii (Copn xs t o es)] s2' ∧ eq_exc_fresh s2' s2.
+ by exists s2'; split=> //; apply: sem_seq1; apply: EmkI; apply: Eopn; rewrite /sem_sopn Hx' /=; rewrite /= in Hv; by rewrite Hv.
case: o Hv default => //; (move => sz Hv default || move => Hv default).
(* Omulu *)
+ move: Hv; rewrite /exec_sopn; t_xrbindP => y hy.
have := app_ww_dec hy => -[sz1] [w1 [sz2 [w2 [hsz1 [hsz2 [? [?]]]]]]] ?; subst x y v.
move=> {Hx Hw}.
have [x1 [x2 ?]] := write_lvals_dec2_s Hw'; subst xs.
have [e1 [e2 ?]] := sem_pexprs_dec2_s Hx'; subst es.
rewrite /=.
have [He1 He2] := sem_pexprs_dec2 Hx'.
have hdefault : ∃ s1'0,
sem p' ev s1'
[seq MkI ii i | i <- [:: Copn [:: x1; x2] t (Omulu sz) [:: e1; e2]]] s1'0
∧ eq_exc_fresh s1'0 s2.
+ exists s2'; split=> //; apply: sem_seq1; apply: EmkI; apply: Eopn.
by rewrite /sem_sopn /= /exec_sopn /sopn_sem /= He1 He2 /= /truncate_word hsz1 hsz2.
rewrite /lower_mulu; case hsz: check_size_16_64 => //.
have /andP [hsz16 hsz64] := assertP hsz.
have! := (is_wconstP gd s1' (sz := sz) (e := e1)).
case: is_wconst => [ n1 | _ ].
+ move => /(_ _ erefl) /=; rewrite He1 /= /truncate_word hsz1 => - [?]; subst n1.
set s2'' := with_vm s1'
(evm s1').[vword sz (fv.(fresh_multiplicand) sz) <- ok (pword_of_word (zero_extend _ w1)) ].
have Heq: eq_exc_fresh s2'' s1'.
split=> //.
rewrite /s2'' /= => x Hx.
rewrite Fv.setP_neq //.
apply/eqP=> Habs; apply: Hx; rewrite -Habs //.
have [s3'' Hw'' Hs3''] := eeq_exc_write_lvals Hdisjl Heq Hw'.
have Hd2 : disj_fvars (read_e e2).
- move: Hdisje.
rewrite (disj_fvars_m (read_es_cons _ _)) => /disjoint_union [_].
rewrite (disj_fvars_m (read_es_cons _ _)) => /disjoint_union [//].
have He2' := eeq_exc_sem_pexpr Hd2 Heq He2.
eexists; split.
+ apply: Eseq.
+ apply: EmkI; apply: Eopn; eauto.
rewrite /sem_sopn /sem_pexprs /= /exec_sopn /sopn_sem /= He1 /= /truncate_word hsz1 /= /x86_MOV /check_size_8_64 hsz64 /=.
by rewrite sumbool_of_boolET.
+ apply: sem_seq1; apply: EmkI; apply: Eopn=> /=.
rewrite /= /read_es /= in Hdisje.
rewrite /sem_sopn /sem_pexprs /= He2' /=.
rewrite /get_gvar /get_var /on_vu /= Fv.setP_eq /= /exec_sopn /sopn_sem /= /truncate_word hsz2 cmp_le_refl /x86_MUL hsz /= zero_extend_u wmulhuE Z.mul_comm GRing.mulrC wmulE.
exact Hw''.
+ exact: (eeq_excT Hs3'' Hs2').
have! := (is_wconstP gd s1' (sz := sz) (e := e2)).
case: is_wconst => [ n2 | _ ].
+ move => /(_ _ erefl) /=; rewrite He2 /= /truncate_word hsz2 => - [?]; subst n2.
set s2'' := with_vm s1' (evm s1').[vword sz (fv.(fresh_multiplicand) sz) <- ok (pword_of_word (zero_extend _ w2)) ].
have Heq: eq_exc_fresh s2'' s1'.
* split=> //.
rewrite /s2'' /= => x Hx.
rewrite Fv.setP_neq //.
apply/eqP=> Habs; apply: Hx; rewrite -Habs //.
have [s3'' Hw'' Hs3''] := eeq_exc_write_lvals Hdisjl Heq Hw'.
have Hd1 : disj_fvars (read_e e1).
* by move: Hdisje; rewrite (disj_fvars_m (read_es_cons _ _)) => /disjoint_union [].
have He1' := eeq_exc_sem_pexpr Hd1 Heq He1.
eexists; split.
+ apply: Eseq.
+ apply: EmkI; apply: Eopn; eauto.
rewrite /sem_sopn /sem_pexprs /= He2 /= /exec_sopn /sopn_sem /= /truncate_word hsz2 /= /x86_MOV /check_size_8_64 hsz64 /=.
by rewrite /write_var /set_var /= sumbool_of_boolET.
+ apply: sem_seq1; apply: EmkI; apply: Eopn=> /=.
rewrite /= /read_es /= in Hdisje.
rewrite /sem_sopn /sem_pexprs /= He1' /=.
rewrite /get_gvar /get_var /on_vu /= Fv.setP_eq /= /exec_sopn /sopn_sem /= /truncate_word hsz1 cmp_le_refl /x86_MUL hsz /= zero_extend_u wmulhuE wmulE.
exact: Hw''.
+ exact: (eeq_excT Hs3'' Hs2').
exists s2'; split=> //; apply: sem_seq1; apply: EmkI; apply: Eopn.
rewrite /sem_sopn Hx' /= /exec_sopn /sopn_sem /= /truncate_word hsz1 hsz2 /x86_MUL hsz /=.
by rewrite /wumul -wmulhuE in Hw'.
(* Oaddcarry *)
+ case: (lower_addcarry_correct ii t (sub:= false) Hs1' Hdisjl Hdisje Hx' Hv Hw').
exact: (aux_eq_exc_trans Hs2').
(* Osubcarry *)
+ case: (lower_addcarry_correct ii t (sub:= true) Hs1' Hdisjl Hdisje Hx' Hv Hw').
exact: (aux_eq_exc_trans Hs2').
Qed.
Local Lemma Hsyscall : sem_Ind_syscall p Pi_r.
Proof.
move=> s1 scs m s2 o xs es ves vs hes ho hw ii hdisj s1' hs1' /=.
move: hdisj; rewrite /disj_fvars /x86_lowering.disj_fvars vars_I_syscall => /disjoint_union [hdisjx hdisje].
have hes' := eeq_exc_sem_pexprs hdisje hs1' hes.
have hs1'w: eq_exc_fresh (with_scs (with_mem s1' m) scs) (with_scs (with_mem s1 m) scs).
+ by rewrite /eq_exc_fresh /estate_eq_except /=; case: hs1' => ?? ->.
have [s2' hw' hs2'] := eeq_exc_write_lvals hdisjx hs1'w hw.
exists s2'; split => //.
apply: sem_seq1; constructor; econstructor; eauto.
by case: hs1' => -> ->.
Qed.
Local Lemma Hif_true : sem_Ind_if_true p ev Pc Pi_r.
Proof.
move=> s1 s2 e c1 c2 Hz _ Hc ii /= Hdisj s1' Hs1' /=.
move: Hdisj; rewrite /disj_fvars /x86_lowering.disj_fvars vars_I_if=> /disjoint_union [Hdisje /disjoint_union [Hc1 Hc2]].
set x := lower_condition _ _ _.
have Hcond: x = lower_condition fv dummy_var_info e by [].
move: x Hcond=> [i e'] Hcond.
have [s2' [Hs2'1 [Hs2'2 Hs2'3]]] := lower_condition_corr ii Hcond Hs1' (eeq_exc_sem_pexpr Hdisje Hs1' Hz).
have [s3' [Hs3'1 Hs3'2]] := Hc Hc1 _ Hs2'2.
exists s3'; split=> //.
rewrite -cats1.
rewrite map_cat.
apply: (sem_app Hs2'1).
apply: sem_seq1; apply: EmkI; apply: Eif_true.
+ by rewrite Hs2'3.
exact: Hs3'1.
Qed.
Local Lemma Hif_false : sem_Ind_if_false p ev Pc Pi_r.
Proof.
move=> s1 s2 e c1 c2 Hz _ Hc ii /= Hdisj s1' Hs1' /=.
move: Hdisj; rewrite /disj_fvars /x86_lowering.disj_fvars vars_I_if=> /disjoint_union [Hdisje /disjoint_union [Hc1 Hc2]].
set x := lower_condition _ _ _.
have Hcond: x = lower_condition fv dummy_var_info e by [].
move: x Hcond=> [i e'] Hcond.
have [s2' [Hs2'1 [Hs2'2 Hs2'3]]] := lower_condition_corr ii Hcond Hs1' (eeq_exc_sem_pexpr Hdisje Hs1' Hz).
have [s3' [Hs3'1 Hs3'2]] := Hc Hc2 _ Hs2'2.
exists s3'; split=> //.
rewrite -cats1.
rewrite map_cat.
apply: (sem_app Hs2'1).
apply: sem_seq1; apply: EmkI; apply: Eif_false.
+ by rewrite Hs2'3.
exact: Hs3'1.
Qed.
Local Lemma Hwhile_true : sem_Ind_while_true p ev Pc Pi_r.
Proof.
move=> s1 s2 s3 s4 a c e c' _ Hc Hz _ Hc' _ Hwhile ii Hdisj s1' Hs1' /=.
have := Hdisj; rewrite /disj_fvars /x86_lowering.disj_fvars vars_I_while=> /disjoint_union [Hdisje /disjoint_union [Hc1 Hc2]].
set x := lower_condition _ _ _.
have Hcond: x = lower_condition fv dummy_var_info e by [].
move: x Hcond=> [i e'] Hcond.
have [s2' [Hs2'1 Hs2'2]] := Hc Hc1 _ Hs1'.
have [s3' [Hs3'1 [Hs3'2 Hs3'3]]] :=
lower_condition_corr
dummy_instr_info
Hcond
Hs2'2
(eeq_exc_sem_pexpr Hdisje Hs2'2 Hz).
have [s4' [Hs4'1 Hs4'2]] := Hc' Hc2 _ Hs3'2.
have [s5' [Hs5'1 Hs5'2]] := Hwhile ii Hdisj _ Hs4'2.
exists s5'; split=> //.
apply: sem_seq1; apply: EmkI; apply: Ewhile_true.
apply: (sem_app Hs2'1 Hs3'1).
by rewrite Hs3'3.
exact: Hs4'1.
rewrite /= -Hcond in Hs5'1.
rewrite {1}/map /= in Hs5'1.
by case/semE: Hs5'1 => ? [/sem_IE H] /semE ->.
Qed.
Local Lemma Hwhile_false : sem_Ind_while_false p ev Pc Pi_r.
Proof.
move=> s1 s2 a c e c' _ Hc Hz ii Hdisj s1' Hs1' /=.
move: Hdisj; rewrite /disj_fvars /x86_lowering.disj_fvars vars_I_while=> /disjoint_union [Hdisje /disjoint_union [Hc1 Hc2]].
set x := lower_condition _ _ _.
have Hcond: x = lower_condition fv dummy_var_info e by [].
move: x Hcond=> [i e'] Hcond.
have [s2' [Hs2'1 Hs2'2]] := Hc Hc1 _ Hs1'.
have [s3' [Hs3'1 [Hs3'2 Hs3'3]]] :=
lower_condition_corr
dummy_instr_info
Hcond
Hs2'2
(eeq_exc_sem_pexpr Hdisje Hs2'2 Hz).
exists s3'; split=> //.
apply: sem_seq1; apply: EmkI; apply: Ewhile_false.
exact: (sem_app Hs2'1 Hs3'1).
by rewrite Hs3'3.
Qed.
Local Lemma Hfor : sem_Ind_for p ev Pi_r Pfor.
Proof.
move=> s1 s2 i d lo hi c vlo vhi Hlo Hhi _ Hfor ii Hdisj s1' Hs1' /=.
move: Hdisj; rewrite /disj_fvars /x86_lowering.disj_fvars vars_I_for=> /disjoint_union [Hdisjc /disjoint_union [Hdisjlo Hdisjhi]].
have [s2' [Hs2'1 Hs2'2]] := Hfor Hdisjc _ Hs1'.
exists s2'; split=> //.
apply: sem_seq1; apply: EmkI; apply: Efor; eauto.
+ by rewrite (eeq_exc_sem_pexpr Hdisjlo Hs1' Hlo).
by rewrite (eeq_exc_sem_pexpr Hdisjhi Hs1' Hhi).
Qed.
Local Lemma Hfor_nil : sem_Ind_for_nil Pfor.
Proof. move=> s i c _ s' Hs'; exists s'; split=> //; exact: EForDone. Qed.
Local Lemma Hfor_cons : sem_Ind_for_cons p ev Pc Pfor.
Proof.
move=> s1 s1' s2 s3 i w ws c Hw _ Hc _ Hfor Hdisj s1'' Hs1''.
have := Hdisj=> /disjoint_union [Hdisjc Hdisji].
have Hw1: write_lval gd (Lvar i) w s1 = ok s1' by exact: Hw.
have [|s2'' Hs2''1 Hs2''2] := eeq_exc_write_lval _ Hs1'' Hw1.
rewrite /=; have H: Sv.Equal (Sv.union Sv.empty (Sv.add i Sv.empty)) (Sv.singleton i).
by SvD.fsetdec.
rewrite /vars_lval /= /disj_fvars.
by move: Hdisji; rewrite /disj_fvars /x86_lowering.disj_fvars /vars_lval H.
have [s3'' [Hs3''1 Hs3''2]] := Hc Hdisjc _ Hs2''2.
have [s4'' [Hs4''1 Hs4''2]] := Hfor Hdisj _ Hs3''2.
exists s4''; split=> //.
by apply: EForOne; eauto.
Qed.
Local Lemma Hcall : sem_Ind_call p ev Pi_r Pfun.
Proof.
move=> s1 scs m2 s2 ii xs fn args vargs vs Harg _ Hfun Hret ii' Hdisj s1' Hs1'; move: Hdisj.
rewrite /disj_fvars /x86_lowering.disj_fvars vars_I_call=> /disjoint_union [Hxs Hargs].
have Heq: eq_exc_fresh (with_scs (with_mem s1' m2) scs) (with_scs (with_mem s1 m2) scs).
+ by case: Hs1' => * /=.
have [s2' Hs2'1 Hs2'2] := eeq_exc_write_lvals Hxs Heq Hret.
exists s2'; split=> //.
apply: sem_seq1; apply: EmkI; apply: Ecall; eauto.
rewrite (eeq_exc_sem_pexprs Hargs Hs1' Harg) //.
move: Hs1'=> [-> -> _]; exact: Hfun.
Qed.
Local Lemma Hproc : sem_Ind_proc p ev Pc Pfun.
Proof.
move=> scs1 m1 scs2 m2 fn f vargs vargs' s0 s1 s2 vres vres' Hget Htya Hi Harg _ Hc Hres Htyr Hsys Hfi.
have H: eq_exc_fresh s1 s1 by [].
have Hdisj := fvars_fun Hget.
rewrite /vars_fd in Hdisj.
move: Hdisj=> /disjoint_union [Hdisjp /disjoint_union [Hdisjr Hdisjc]].
have [[scs1' m1' vm1'] [Hs1'1 [/= ? Hs1'2 Hs1'3]]] := Hc Hdisjc _ H; subst scs1' m1'.
apply: EcallRun=> //.
+ by rewrite get_map_prog Hget.
+ exact: Htya.
+ exact: Hi.
+ exact: Harg.
+ exact: Hs1'1.
+ rewrite /=.
have ->: vm1' = evm (with_vm s2 vm1') by rewrite evm_with_vm.
rewrite -(sem_pexprs_get_var gd).
rewrite -(sem_pexprs_get_var gd) in Hres.
have H': forall l, Sv.Equal (read_es (map Plvar l)) (vars_l l).
+ elim=> // a l /= Hl.
rewrite read_es_cons Hl /read_e /= /mk_lvar /read_gvar /=.
by SvD.fsetdec.
apply: (eeq_exc_sem_pexprs _ _ Hres).
* rewrite /disj_fvars /x86_lowering.disj_fvars H'. exact: Hdisjr.
done.
+ exact: Htyr.
done. done.
Qed.
Lemma lower_callP f scs mem scs' mem' va vr:
sem_call p ev scs mem f va scs' mem' vr ->
sem_call p' ev scs mem f va scs' mem' vr.
Proof.
exact:
(sem_call_Ind
Hskip
Hcons
HmkI
Hassgn
Hopn
Hsyscall
Hif_true
Hif_false
Hwhile_true
Hwhile_false
Hfor
Hfor_nil
Hfor_cons
Hcall
Hproc).
Qed.
End PROOF.
|
lemma closed_halfspace_component_le: "closed {x::'a::euclidean_space. x\<bullet>i \<le> a}" |
module Tests.SpecFunctions where
import Test.QuickCheck hiding (choose,within)
--import Test.Framework
--import Test.Framework.Providers.QuickCheck2
--import Numeric.SpecFunctions
|
(* Title: Multisets_Extras
Author: Chelsea Edmonds
*)
section \<open>Micellanious Helper Functions on Sets and Multisets\<close>
theory Multisets_Extras imports
"HOL-Library.Multiset"
Card_Partitions.Set_Partition
Nested_Multisets_Ordinals.Multiset_More
Nested_Multisets_Ordinals.Duplicate_Free_Multiset
"HOL-Library.Disjoint_Sets"
begin
subsection \<open>Set Theory Extras\<close>
text \<open>A number of extra helper lemmas for reasoning on sets (finite) required for Design Theory
proofs\<close>
lemma card_Pow_filter_one:
assumes "finite A"
shows "card {x \<in> Pow A . card x = 1} = card (A)"
using assms
proof (induct rule: finite_induct)
case empty
then show ?case by auto
next
case (insert x F)
have "Pow (insert x F) = Pow F \<union> insert x ` Pow F"
by (simp add: Pow_insert)
then have split: "{y \<in> Pow (insert x F) . card y = 1} =
{y \<in> (Pow F) . card y = 1} \<union> {y \<in> (insert x ` Pow F) . card y = 1}"
by blast
have "\<And> y . y \<in> (insert x ` Pow F) \<Longrightarrow> finite y"
using finite_subset insert.hyps(1) by fastforce
then have single: "\<And> y . y \<in> (insert x ` Pow F) \<Longrightarrow> card y = 1 \<Longrightarrow> y = {x}"
by (metis card_1_singletonE empty_iff image_iff insertCI insertE)
then have "card {y \<in> (insert x ` Pow F) . card y = 1} = 1"
using empty_iff imageI is_singletonI is_singletonI' is_singleton_altdef (* LONG *)
by (metis (full_types, lifting) Collect_empty_eq_bot Pow_bottom bot_empty_eq mem_Collect_eq)
then have " {y \<in> (insert x ` Pow F) . card y = 1} = {{x}}"
using single card_1_singletonE card_eq_0_iff
by (smt empty_Collect_eq mem_Collect_eq singletonD zero_neq_one)
then have split2:"{y \<in> Pow (insert x F) . card y = 1} = {y \<in> (Pow F) . card y = 1} \<union> {{x}}"
using split by simp
then show ?case
proof (cases "x \<in> F")
case True
then show ?thesis using insert.hyps(2) by auto
next
case False
then have "{y \<in> (Pow F) . card y = 1} \<inter> {{x}} = {}" by blast
then have fact:"card {y \<in> Pow (insert x F) . card y = 1} =
card {y \<in> (Pow F) . card y = 1} + card {{x}}"
using split2 card_Un_disjoint insert.hyps(1) by auto
have "card (insert x F) = card F + 1"
using False card_insert_disjoint by (metis Suc_eq_plus1 insert.hyps(1))
then show ?thesis using fact insert.hyps(3) by auto
qed
qed
lemma elem_exists_non_empty_set:
assumes "card A > 0"
obtains x where "x \<in> A"
using assms card_gt_0_iff by fastforce
lemma set_self_img_compr: "{a | a . a \<in> A} = A"
by blast
lemma card_subset_not_gt_card: "finite A \<Longrightarrow> card ps > card A \<Longrightarrow> \<not> (ps \<subseteq> A)"
using card_mono leD by auto
lemma card_inter_lt_single: "finite A \<Longrightarrow> finite B \<Longrightarrow> card (A \<inter> B) \<le> card A"
by (simp add: card_mono)
lemma set_diff_non_empty_not_subset:
assumes "A \<subseteq> (B - C)"
assumes "C \<noteq> {}"
assumes "A \<noteq> {}"
assumes "B \<noteq> {}"
shows " \<not> (A \<subseteq> C)"
proof (rule ccontr)
assume " \<not> \<not> (A \<subseteq> C)"
then have a: "\<And> x . x \<in> A \<Longrightarrow> x \<in> C" by blast
thus False using a assms by blast
qed
lemma set_card_diff_ge_zero: "finite A \<Longrightarrow> finite B \<Longrightarrow> A \<noteq> B \<Longrightarrow> card A = card B \<Longrightarrow>
card (A - B) > 0"
by (meson Diff_eq_empty_iff card_0_eq card_subset_eq finite_Diff neq0_conv)
lemma set_filter_diff: "{a \<in> A . P a } - {x} = {a \<in> (A - {x}) . (P a )}"
by (auto)
lemma set_filter_diff_card: "card ({a \<in> A . P a } - {x}) = card {a \<in> (A - {x}) . (P a )}"
by (simp add: set_filter_diff)
lemma obtain_subset_with_card_int_n:
assumes "(n ::int) \<le> of_nat (card S)"
assumes "(n ::int) \<ge> 0"
obtains T where "T \<subseteq> S" "of_nat (card T) = (n ::int)" "finite T"
using obtain_subset_with_card_n assms
by (metis nonneg_int_cases of_nat_le_iff)
lemma transform_filter_img_empty_rm:
assumes "\<And> g . g \<in> G \<Longrightarrow> g \<noteq> {}"
shows "{g - {x} | g. g \<in> G \<and> g \<noteq> {x}} = {g - {x} | g. g \<in> G } - {{}}"
proof -
let ?f = "\<lambda> g . g - {x}"
have "\<And> g . g \<in> G \<Longrightarrow> g \<noteq> {x} \<longleftrightarrow> ?f g \<noteq> {}" using assms
by (metis Diff_cancel Diff_empty Diff_insert0 insert_Diff)
thus ?thesis by auto
qed
lemma bij_betw_inter_subsets: "bij_betw f A B \<Longrightarrow> a1 \<subseteq> A \<Longrightarrow> a2 \<subseteq> A
\<Longrightarrow> f ` (a1 \<inter> a2) = (f ` a1) \<inter> (f ` a2)"
by (meson bij_betw_imp_inj_on inj_on_image_Int)
text\<open>Partition related set theory lemmas\<close>
lemma partition_on_remove_pt:
assumes "partition_on A G"
shows "partition_on (A - {x}) {g - {x} | g. g \<in> G \<and> g \<noteq> {x}}"
proof (intro partition_onI)
show "\<And>p. p \<in> {g - {x} |g. g \<in> G \<and> g \<noteq> {x}} \<Longrightarrow> p \<noteq> {}"
using assms partition_onD3 subset_singletonD by force
let ?f = "(\<lambda> g . g - {x})"
have un_img: "\<Union>({?f g | g. g \<in> G }) = ?f (\<Union> G)" by blast
have empty: "\<Union> {g - {x} |g. g \<in> G \<and> g \<noteq> {x}} = \<Union>({g - {x} | g. g \<in> G } - {{}})"
by blast
then have "\<Union>({g - {x} | g. g \<in> G } - {{}}) = \<Union>({g - {x} | g. g \<in> G })" by blast
then show " \<Union> {g - {x} |g. g \<in> G \<and> g \<noteq> {x}} = A - {x}" using partition_onD1 assms un_img
by (metis empty)
then show "\<And>p p'.
p \<in> {g - {x} |g. g \<in> G \<and> g \<noteq> {x}} \<Longrightarrow>
p' \<in> {g - {x} |g. g \<in> G \<and> g \<noteq> {x}} \<Longrightarrow> p \<noteq> p' \<Longrightarrow> p \<inter> p' = {}"
proof -
fix p1 p2
assume p1: "p1 \<in> {g - {x} |g. g \<in> G \<and> g \<noteq> {x}}"
and p2: "p2 \<in> {g - {x} |g. g \<in> G \<and> g \<noteq> {x}}"
and ne: "p1 \<noteq> p2"
obtain p1' p2' where orig1: "p1 = p1' - {x}" and orig2: "p2 = p2' - {x}"
and origne: "p1' \<noteq> p2'" and ne1: "p1' \<noteq> {x}" and ne2:"p2' \<noteq> {x}" and ing1: "p1' \<in> G"
and ing2: "p2' \<in> G"
using p1 p2 using mem_Collect_eq ne by blast
then have "p1' \<inter> p2' = {}" using assms partition_onD2 ing1 ing2 origne disjointD by blast
thus "p1 \<inter> p2 = {}" using orig1 orig2 by blast
qed
qed
lemma partition_on_cart_prod:
assumes "card I > 0"
assumes "A \<noteq> {}"
assumes "G \<noteq> {}"
assumes "partition_on A G"
shows "partition_on (A \<times> I) {g \<times> I |g. g \<in> G}"
proof (intro partition_onI)
show "\<And>p. p \<in> {g \<times> I |g. g \<in> G} \<Longrightarrow> p \<noteq> {}"
using assms(1) assms(4) partition_onD3 by fastforce
show "\<Union> {g \<times> I |g. g \<in> G} = A \<times> I"
by (metis Setcompr_eq_image Sigma_Union assms(4) partition_onD1)
show "\<And>p p'. p \<in> {g \<times> I |g. g \<in> G} \<Longrightarrow> p' \<in> {g \<times> I |g. g \<in> G} \<Longrightarrow> p \<noteq> p' \<Longrightarrow> p \<inter> p' = {}"
by (smt (verit, best) Sigma_Int_distrib1 Sigma_empty1 assms(4) mem_Collect_eq partition_onE)
qed
subsection \<open>Multiset Helpers\<close>
text \<open>Generic Size, count and card helpers\<close>
lemma count_size_set_repr: "size {# x \<in># A . x = g#} = count A g"
by (simp add: filter_eq_replicate_mset)
lemma mset_nempty_set_nempty: "A \<noteq> {#} \<longleftrightarrow> (set_mset A) \<noteq> {}"
by simp
lemma mset_size_ne0_set_card: "size A > 0 \<Longrightarrow> card (set_mset A) > 0"
using mset_nempty_set_nempty by fastforce
lemma set_count_size_min: "count A a \<ge> n \<Longrightarrow> size A \<ge> n"
by (metis (full_types) count_le_replicate_mset_subset_eq size_mset_mono size_replicate_mset)
lemma card_size_filter_eq: "finite A \<Longrightarrow> card {a \<in> A . P a} = size {#a \<in># mset_set A . P a#}"
by simp
lemma size_multiset_set_mset_const_count:
assumes "card (set_mset A) = ca"
assumes "\<And>p. p \<in># A \<Longrightarrow> count A p = ca2"
shows "size A = (ca * ca2)"
proof -
have "size A = (\<Sum> p \<in> (set_mset A) . count A p)" using size_multiset_overloaded_eq by auto
then have "size A = (\<Sum> p \<in> (set_mset A) . ca2)" using assms by simp
thus ?thesis using assms(1) by auto
qed
lemma size_multiset_int_count:
assumes "of_nat (card (set_mset A)) = (ca :: int)"
assumes "\<And>p. p \<in># A \<Longrightarrow> of_nat (count A p) = (ca2 :: int)"
shows "of_nat (size A) = ((ca :: int) * ca2)"
proof -
have "size A = (\<Sum> p \<in> (set_mset A) . count A p)" using size_multiset_overloaded_eq by auto
then have "of_nat (size A) = (\<Sum> p \<in> (set_mset A) . ca2)" using assms by simp
thus ?thesis using assms(1) by auto
qed
lemma mset_union_size: "size (A \<union># B) = size (A) + size (B - A)"
by (simp add: union_mset_def)
lemma mset_union_size_inter: "size (A \<union># B) = size (A) + size B - size (A \<inter># B)"
by (metis diff_add_inverse2 size_Un_Int)
text \<open>Lemmas for repeat\_mset\<close>
lemma repeat_mset_size [simp]: "size (repeat_mset n A) = n * size A"
by (induction n) auto
lemma repeat_mset_subset_in:
assumes "\<And> a . a \<in># A \<Longrightarrow> a \<subseteq> B"
assumes "X \<in># repeat_mset n A"
assumes "x \<in> X"
shows " x \<in> B"
using assms by (induction n) auto
lemma repeat_mset_not_empty: "n > 0 \<Longrightarrow> A \<noteq> {#} \<Longrightarrow> repeat_mset n A \<noteq> {#}"
by (induction n) auto
lemma elem_in_repeat_in_original: "a \<in># repeat_mset n A \<Longrightarrow> a \<in># A"
by (metis count_inI count_repeat_mset in_countE mult.commute mult_zero_left nat.distinct(1))
lemma elem_in_original_in_repeat: "n > 0 \<Longrightarrow> a \<in># A \<Longrightarrow> a \<in># repeat_mset n A"
by (metis count_greater_zero_iff count_repeat_mset nat_0_less_mult_iff)
text \<open>Lemmas on image and filter for multisets\<close>
lemma multiset_add_filter_size: "size {# a \<in># (A1 + A2) . P a #} = size {# a \<in># A1 . P a #} +
size {# a \<in># A2 . P a #}"
by simp
lemma size_filter_neg: "size {#a \<in># A . P a #} = size A - size {# a \<in># A . \<not> P a #}"
using size_filter_mset_lesseq size_union union_filter_mset_complement
by (metis ordered_cancel_comm_monoid_diff_class.le_imp_diff_is_add)
lemma filter_filter_mset_cond_simp:
assumes "\<And> a . P a \<Longrightarrow> Q a"
shows "filter_mset P A = filter_mset P (filter_mset Q A)"
proof -
have "filter_mset P (filter_mset Q A) = filter_mset (\<lambda> a. Q a \<and> P a) A"
by (simp add: filter_filter_mset)
thus ?thesis using assms
by (metis (mono_tags, lifting) filter_mset_cong)
qed
lemma filter_filter_mset_ss_member: "filter_mset (\<lambda> a . {x, y} \<subseteq> a) A =
filter_mset (\<lambda> a . {x, y} \<subseteq> a) (filter_mset (\<lambda> a . x \<in> a) A)"
proof -
have filter: "filter_mset (\<lambda> a . {x, y} \<subseteq> a) (filter_mset (\<lambda> a . x \<in> a) A) =
filter_mset (\<lambda> a . x \<in> a \<and> {x, y} \<subseteq> a) A" by (simp add: filter_filter_mset)
have "\<And> a. {x, y} \<subseteq> a \<Longrightarrow> x \<in> a" by simp
thus ?thesis using filter by auto
qed
lemma multiset_image_do_nothing: "(\<And> x .x \<in># A \<Longrightarrow> f x = x) \<Longrightarrow> image_mset f A = A"
by (induct A) auto
lemma set_mset_filter: "set_mset {# f a . a \<in># A #} = {f a | a. a \<in># A}"
by (simp add: Setcompr_eq_image)
lemma mset_exists_imply: "x \<in># {# f a . a \<in># A #} \<Longrightarrow> \<exists> y \<in># A . x = f y"
by auto
lemma filter_mset_image_mset:
"filter_mset P (image_mset f A) = image_mset f (filter_mset (\<lambda>x. P (f x)) A)"
by (induction A) auto
lemma mset_bunion_filter: "{# a \<in># A . P a \<or> Q a #} = {# a \<in># A . P a #} \<union># {# a \<in># A . Q a #}"
by (rule multiset_eqI) simp
lemma mset_inter_filter: "{# a \<in># A . P a \<and> Q a #} = {# a \<in># A . P a #} \<inter># {# a \<in># A . Q a #}"
by (rule multiset_eqI) simp
lemma image_image_mset: "image_mset (\<lambda> x . f x) (image_mset (\<lambda> y . g y) A) =
image_mset (\<lambda> x. f (g x)) A"
by simp
text \<open>Big Union over multiset helpers\<close>
lemma mset_big_union_obtain:
assumes "x \<in># \<Sum>\<^sub># A"
obtains a where "a \<in># A" and "x \<in># a"
using assms by blast
lemma size_big_union_sum: "size (\<Sum>\<^sub># (M :: 'a multiset multiset)) = (\<Sum>x \<in>#M . size x)"
by (induct M) auto
text \<open>Cartesian Product on Multisets\<close>
lemma size_cartesian_product_singleton [simp]: "size ({#a#} \<times># B) = size B"
by (simp add: Times_mset_single_left)
lemma size_cartesian_product_singleton_right [simp]: "size (A \<times># {#b#}) = size A"
by (simp add: Times_mset_single_right)
lemma size_cartesian_product_empty [simp]: "size (A \<times># {#}) = 0"
by simp
lemma size_add_elem_step_eq:
assumes "size (A \<times># B) = size A * size B"
shows "size (add_mset x A \<times># B) = size (add_mset x A) * size B"
proof -
have "(add_mset x A \<times># B) = A \<times># B + {#x#} \<times># B"
by (metis Sigma_mset_plus_distrib1 add_mset_add_single)
then have "size (add_mset x A \<times># B) = size (A \<times># B) + size B" by auto
also have "... = size A * size B + size B"
by (simp add: assms)
finally have "size (add_mset x A \<times># B) = (size A + 1) * size B"
by auto
thus ?thesis by simp
qed
lemma size_cartesian_product: "size (A \<times># B) = size A * size B"
by (induct A) (simp_all add: size_add_elem_step_eq)
lemma cart_prod_distinct_mset:
assumes assm1: "distinct_mset A"
assumes assm2: "distinct_mset B"
shows "distinct_mset (A \<times># B)"
unfolding distinct_mset_count_less_1
proof (rule allI)
fix x
have count_mult: "count (A \<times># B) x = count A (fst x) * count B (snd x)"
using count_Sigma_mset by (metis prod.exhaust_sel)
then have "count A (fst x) * count B (snd x) \<le> 1" using assm1 assm2
unfolding distinct_mset_count_less_1 using mult_le_one by blast
thus "count (A \<times># B) x \<le> 1" using count_mult by simp
qed
lemma cart_product_single_intersect: "x1 \<noteq> x2 \<Longrightarrow> ({#x1#} \<times># A) \<inter># ({#x2#} \<times># B) = {#}"
using multiset_inter_single by fastforce
lemma size_union_distinct_cart_prod: "x1 \<noteq> x2 \<Longrightarrow> size (({#x1#} \<times># A) \<union># ({#x2#} \<times># B)) =
size ({#x1#} \<times># A) + size ({#x2#} \<times># B)"
by (simp add: cart_product_single_intersect size_Un_disjoint)
lemma size_Union_distinct_cart_prod: "distinct_mset M \<Longrightarrow>
size (\<Sum>p\<in>#M. ({#p#} \<times># B)) = size (M) * size (B)"
by (induction M) auto
lemma size_Union_distinct_cart_prod_filter: "distinct_mset M \<Longrightarrow>
(\<And> p . p \<in># M \<Longrightarrow> size ({# b \<in># B . P p b #}) = c) \<Longrightarrow>
size (\<Sum>p\<in>#M. ({#p#} \<times># {# b \<in># B . P p b #})) = size (M) * c"
by (induction M) auto
lemma size_Union_distinct_cart_prod_filter2: "distinct_mset V \<Longrightarrow>
(\<And> b . b \<in># B \<Longrightarrow> size ({# v \<in># V . P v b #}) = c) \<Longrightarrow>
size (\<Sum>b\<in>#B. ( {# v \<in># V . P v b #} \<times># {#b#})) = size (B) * c"
by (induction B) auto
lemma cart_product_add_1: "(add_mset a A) \<times># B = ({#a#} \<times># B) + (A \<times># B)"
by (metis Sigma_mset_plus_distrib1 add_mset_add_single union_commute)
lemma cart_product_add_1_filter: "{#m \<in># ((add_mset a M) \<times># N) . P m #} =
{#m \<in># (M \<times># N) . P m #} + {#m \<in># ({#a#} \<times># N) . P m #}"
unfolding add_mset_add_single [of a M] Sigma_mset_plus_distrib1
by (simp add: Times_mset_single_left)
lemma cart_product_add_1_filter2: "{#m \<in># (M \<times># (add_mset b N)) . P m #} =
{#m \<in># (M \<times># N) . P m #} + {#m \<in># (M \<times># {#b#}) . P m #}"
unfolding add_mset_add_single [of b N] Sigma_mset_plus_distrib1
by (metis Times_insert_left Times_mset_single_right add_mset_add_single filter_union_mset)
lemma cart_prod_singleton_right_gen:
assumes "\<And> x . x \<in># (A \<times># {#b#}) \<Longrightarrow> P x \<longleftrightarrow> Q (fst x)"
shows "{#x \<in># (A \<times># {#b#}). P x#} = {# a \<in># A . Q a#} \<times># {#b#}"
using assms
proof (induction A)
case empty
then show ?case by simp
next
case (add x A)
have "add_mset x A \<times># {#b#} = add_mset (x, b) (A \<times># {#b#})"
by (simp add: Times_mset_single_right)
then have lhs: "filter_mset P (add_mset x A \<times># {#b#}) = filter_mset P (A \<times># {#b#}) +
filter_mset P {#(x, b)#}" by simp
have rhs: "filter_mset Q (add_mset x A) \<times># {#b#} = filter_mset Q A \<times># {#b#} +
filter_mset Q {#x#} \<times># {#b#}"
by (metis Sigma_mset_plus_distrib1 add_mset_add_single filter_union_mset)
have "filter_mset P {#(x, b)#} = filter_mset Q {#x#} \<times># {#b#}"
using add.prems by fastforce
then show ?case using lhs rhs add.IH add.prems by force
qed
lemma cart_prod_singleton_left_gen:
assumes "\<And> x . x \<in># ({#a#} \<times># B) \<Longrightarrow> P x \<longleftrightarrow> Q (snd x)"
shows "{#x \<in># ({#a#} \<times># B). P x#} = {#a#} \<times># {#b \<in># B . Q b#}"
using assms
proof (induction B)
case empty
then show ?case by simp
next
case (add x B)
have lhs: "filter_mset P ({#a#} \<times># add_mset x B) = filter_mset P ({#a#} \<times># B) +
filter_mset P {#(a, x)#}"
by (simp add: cart_product_add_1_filter2)
have rhs: "{#a#} \<times># filter_mset Q (add_mset x B) = {#a#} \<times># filter_mset Q B +
{#a#} \<times># filter_mset Q {#x#}"
using add_mset_add_single filter_union_mset by (metis Times_mset_single_left image_mset_union)
have "filter_mset P {#(a, x)#} = {#a#} \<times># filter_mset Q {#x#}"
using add.prems by fastforce
then show ?case using lhs rhs add.IH add.prems by force
qed
lemma cart_product_singleton_left: "{#m \<in># ({#a#} \<times># N) . fst m \<in> snd m #} =
({#a#} \<times># {# n \<in># N . a \<in> n #})" (is "?A = ?B")
proof -
have stmt: "\<And>m. m \<in># ({#a#} \<times># N) \<Longrightarrow> fst m \<in> snd m \<longleftrightarrow> a \<in> snd m"
by (simp add: mem_Times_mset_iff)
thus ?thesis by (metis (no_types, lifting) Sigma_mset_cong stmt cart_prod_singleton_left_gen)
qed
lemma cart_product_singleton_right: "{#m \<in># (N \<times># {#b#}) . fst m \<in> snd m #} =
({# n \<in># N . n \<in> b #} \<times># {# b #})" (is "?A = ?B")
proof -
have stmt: "\<And>m. m \<in># (N \<times># {#b#}) \<Longrightarrow> fst m \<in> snd m \<longleftrightarrow> fst m \<in>b"
by (simp add: mem_Times_mset_iff)
thus ?thesis by (metis (no_types, lifting) Sigma_mset_cong stmt cart_prod_singleton_right_gen)
qed
lemma cart_product_add_1_filter_eq: "{#m \<in># ((add_mset a M) \<times># N) . (fst m \<in> snd m) #} =
{#m \<in># (M \<times># N) . (fst m \<in> snd m) #} + ({#a#} \<times># {# n \<in># N . a \<in> n #})"
unfolding add_mset_add_single [of a M] Sigma_mset_plus_distrib1
using cart_product_singleton_left cart_product_add_1_filter by fastforce
lemma cart_product_add_1_filter_eq_mirror: "{#m \<in># M \<times># (add_mset b N) . (fst m \<in> snd m) #} =
{#m \<in># (M \<times># N) . (fst m \<in> snd m) #} + ({# n \<in># M . n \<in> b #} \<times># {#b#})"
unfolding add_mset_add_single [of b N] Sigma_mset_plus_distrib1 (* longish *)
by (metis (no_types) add_mset_add_single cart_product_add_1_filter2 cart_product_singleton_right)
lemma set_break_down_left:
shows "{# m \<in># (M \<times># N) . (fst m) \<in> (snd m) #} = (\<Sum>m\<in>#M. ({#m#} \<times># {#n \<in># N. m \<in> n#}))"
by (induction M) (auto simp add: cart_product_add_1_filter_eq)
lemma set_break_down_right:
shows "{# x \<in># M \<times># N . (fst x) \<in> (snd x) #} = (\<Sum>n\<in>#N. ({#m \<in># M. m \<in> n#} \<times># {#n#}))"
by (induction N) (auto simp add: cart_product_add_1_filter_eq_mirror)
text \<open>Reasoning on sums of elements over multisets\<close>
lemma sum_over_fun_eq:
assumes "\<And> x . x \<in># A \<Longrightarrow> f x = g x"
shows "(\<Sum>x \<in># A . f(x)) = (\<Sum> x \<in># A . g (x))"
using assms by auto
lemma sum_mset_add_diff_nat:
fixes x:: 'a and f g :: "'a \<Rightarrow> nat"
assumes "\<And>x . x \<in># A \<Longrightarrow> f x \<ge> g x"
shows "(\<Sum> x \<in># A. f x - g x) = (\<Sum> x \<in># A . f x) - (\<Sum> x \<in># A . g x)"
using assms by (induction A) (simp_all add: sum_mset_mono)
lemma sum_mset_add_diff_int:
fixes x:: 'a and f g :: "'a \<Rightarrow> int"
shows "(\<Sum> x \<in># A. f x - g x) = (\<Sum> x \<in># A . f x) - (\<Sum> x \<in># A . g x)"
by (induction A) (simp_all add: sum_mset_mono)
context ring_1
begin
lemma sum_mset_add_diff: "(\<Sum> x \<in># A. f x - g x) = (\<Sum> x \<in># A . f x) - (\<Sum> x \<in># A . g x)"
by (induction A) (auto simp add: algebra_simps)
end
context ordered_semiring
begin
lemma sum_mset_ge0:"(\<And> x . f x \<ge> 0) \<Longrightarrow> (\<Sum> x \<in># A. f x ) \<ge> 0"
proof (induction A)
case empty
then show ?case by simp
next
case (add x A)
then have hyp2: "0 \<le> sum_mset (image_mset f A)" by blast
then have " sum_mset (image_mset f (add_mset x A)) = sum_mset (image_mset f A) + f x"
by (simp add: add_commute)
then show ?case
by (simp add: add.IH add.prems)
qed
lemma sum_order_add_mset: "(\<And> x . f x \<ge> 0) \<Longrightarrow> (\<Sum> x \<in># A. f x ) \<le> (\<Sum> x \<in># add_mset a A. f x )"
by (simp add: local.add_increasing)
lemma sum_mset_0_left: "(\<And> x . f x \<ge> 0) \<Longrightarrow> (\<Sum> x \<in># A. f x ) = 0 \<Longrightarrow> (\<forall> x \<in># A .f x = 0)"
apply (induction A)
apply auto
using local.add_nonneg_eq_0_iff sum_mset_ge0 apply blast
using local.add_nonneg_eq_0_iff sum_mset_ge0 by blast
lemma sum_mset_0_iff_ge_0:
assumes "(\<And> x . f x \<ge> 0)"
shows "(\<Sum> x \<in># A. f x ) = 0 \<longleftrightarrow> (\<forall> x \<in> set_mset A .f x = 0)"
using sum_mset_0_left assms by auto
end
lemma mset_set_size_card_count: "(\<Sum>x \<in># A. x) = (\<Sum>x \<in> set_mset A . x * (count A x))"
proof (induction A)
case empty
then show ?case by simp
next
case (add y A)
have lhs: "(\<Sum>x\<in>#add_mset y A. x) = (\<Sum>x\<in># A. x) + y" by simp
have rhs: "(\<Sum>x\<in>set_mset (add_mset y A). x * count (add_mset y A) x) =
(\<Sum>x\<in>(insert y (set_mset A)) . x * count (add_mset y A) x)"
by simp
then show ?case
proof (cases "y \<in># A")
case True
have x_val: "\<And> x . x \<in> (insert y (set_mset A)) \<Longrightarrow> x \<noteq> y \<Longrightarrow>
x* count (add_mset y A) x = x * (count A x)"
by auto
have y_count: "count (add_mset y A) y = 1 + count A y"
using True count_inI by fastforce
then have "(\<Sum>x\<in>set_mset (add_mset y A). x * count (add_mset y A) x) =
(y * (count (add_mset y A) y)) + (\<Sum>x\<in>(set_mset A) - {y}. x * count A x)"
using x_val finite_set_mset sum.cong sum.insert rhs
by (smt DiffD1 Diff_insert_absorb insert_absorb mk_disjoint_insert sum.insert_remove)
then have s1: "(\<Sum>x\<in>set_mset (add_mset y A). x * count (add_mset y A) x) =
y + y * (count A y) + (\<Sum>x\<in>(set_mset A) - {y}. x * count A x)"
using y_count by simp
then have "(\<Sum>x\<in>set_mset (add_mset y A). x * count (add_mset y A) x) =
y + (\<Sum>x\<in>insert y ((set_mset A) - {y} ) . x * count A x)"
by (simp add: sum.insert_remove)
then have "(\<Sum>x\<in>set_mset (add_mset y A). x * count (add_mset y A) x) =
y + (\<Sum>x\<in>(set_mset A) . x * count A x)"
by (simp add: True insert_absorb)
then show ?thesis using lhs add.IH
by linarith
next
case False
have x_val: "\<And> x . x \<in> set_mset A \<Longrightarrow> x* count (add_mset y A) x = x * (count A x)"
using False by auto
have y_count: "count (add_mset y A) y = 1" using False count_inI by fastforce
have lhs: "(\<Sum>x\<in>#add_mset y A. x) = (\<Sum>x\<in># A. x) + y" by simp
have "(\<Sum>x\<in>set_mset (add_mset y A). x * count (add_mset y A) x) =
(y * count (add_mset y A) y) + (\<Sum>x\<in>set_mset A. x * count A x)"
using x_val rhs by (metis (no_types, lifting) False finite_set_mset sum.cong sum.insert)
then have "(\<Sum>x\<in>set_mset (add_mset y A). x * count (add_mset y A) x) =
y + (\<Sum>x\<in>set_mset A. x * count A x)"
using y_count by simp
then show ?thesis using lhs add.IH by linarith
qed
qed
subsection \<open>Partitions on Multisets\<close>
text \<open>A partition on a multiset A is a multiset of multisets, where the sum over P equals A and the
empty multiset is not in the partition. Based off set partition definition.
We note that unlike set partitions, there is no requirement for elements in the multisets to be
distinct due to the definition of union on multisets \<^cite>\<open>"benderPartitionsMultisets1974"\<close>\<close>
lemma mset_size_partition_dep: "size {# a \<in># A . P a \<or> Q a #} =
size {# a \<in># A . P a #} + size {# a \<in># A . Q a #} - size {# a \<in># A . P a \<and> Q a #}"
by (simp add: mset_bunion_filter mset_inter_filter mset_union_size_inter)
definition partition_on_mset :: "'a multiset \<Rightarrow> 'a multiset multiset \<Rightarrow> bool" where
"partition_on_mset A P \<longleftrightarrow> \<Sum>\<^sub>#P = A \<and> {#} \<notin># P"
lemma partition_on_msetI [intro]: "\<Sum>\<^sub>#P = A \<Longrightarrow> {#} \<notin># P \<Longrightarrow> partition_on_mset A P"
by (simp add: partition_on_mset_def)
lemma partition_on_msetD1: "partition_on_mset A P \<Longrightarrow> \<Sum>\<^sub>#P = A"
by (simp add: partition_on_mset_def)
lemma partition_on_msetD2: "partition_on_mset A P \<Longrightarrow> {#} \<notin># P"
by (simp add: partition_on_mset_def)
lemma partition_on_mset_empty: "partition_on_mset {#} P \<longleftrightarrow> P = {#}"
unfolding partition_on_mset_def
using multiset_nonemptyE by fastforce
lemma partition_on_mset_all: "A \<noteq> {#} \<Longrightarrow> partition_on_mset A {#A #}"
by (simp add: partition_on_mset_def)
lemma partition_on_mset_singletons: "partition_on_mset A (image_mset (\<lambda> x . {#x#}) A)"
by (auto simp: partition_on_mset_def)
lemma partition_on_mset_not_empty: "A \<noteq> {#} \<Longrightarrow> partition_on_mset A P \<Longrightarrow> P \<noteq> {#}"
by (auto simp: partition_on_mset_def)
lemma partition_on_msetI2: "\<Sum>\<^sub>#P = A \<Longrightarrow> (\<And> p . p \<in># P \<Longrightarrow> p \<noteq> {#}) \<Longrightarrow> partition_on_mset A P"
by (auto simp: partition_on_mset_def)
lemma partition_on_mset_elems: "partition_on_mset A P \<Longrightarrow> p1 \<in># P \<Longrightarrow> x \<in># p1 \<Longrightarrow> x \<in># A"
by (auto simp: partition_on_mset_def)
lemma partition_on_mset_sum_size_eq: "partition_on_mset A P \<Longrightarrow> (\<Sum>x \<in># P. size x) = size A"
by (metis partition_on_msetD1 size_big_union_sum)
lemma partition_on_mset_card: assumes "partition_on_mset A P" shows " size P \<le> size A"
proof (rule ccontr)
assume "\<not> size P \<le> size A"
then have a: "size P > size A" by simp
have "\<And> x . x \<in># P \<Longrightarrow> size x > 0" using partition_on_msetD2
using assms nonempty_has_size by auto
then have " (\<Sum>x \<in># P. size x) \<ge> size P"
by (metis leI less_one not_less_zero size_eq_sum_mset sum_mset_mono)
thus False using a partition_on_mset_sum_size_eq
using assms by fastforce
qed
lemma partition_on_mset_count_eq: "partition_on_mset A P \<Longrightarrow> a \<in># A \<Longrightarrow>
(\<Sum>x \<in># P. count x a) = count A a"
by (metis count_sum_mset partition_on_msetD1)
lemma partition_on_mset_subsets: "partition_on_mset A P \<Longrightarrow> x \<in># P \<Longrightarrow> x \<subseteq># A"
by (auto simp add: partition_on_mset_def)
lemma partition_on_mset_distinct:
assumes "partition_on_mset A P"
assumes "distinct_mset A"
shows "distinct_mset P"
proof (rule ccontr)
assume "\<not> distinct_mset P"
then obtain p1 where count: "count P p1 \<ge> 2"
by (metis Suc_1 distinct_mset_count_less_1 less_Suc_eq_le not_less_eq)
then have cge: "\<And> x . x \<in># p1 \<Longrightarrow> (\<Sum>p \<in># P. count p x ) \<ge> 2"
by (smt count_greater_eq_one_iff count_sum_mset_if_1_0 dual_order.trans sum_mset_mono zero_le)
have elem_in: "\<And> x . x \<in># p1 \<Longrightarrow> x \<in># A" using partition_on_mset_elems
by (metis count assms(1) count_eq_zero_iff not_numeral_le_zero)
have "\<And> x . x \<in># A \<Longrightarrow> count A x = 1" using assms
by (simp add: distinct_mset_def)
thus False
using assms partition_on_mset_count_eq cge elem_in count_inI local.count multiset_nonemptyE
by (metis (mono_tags) not_numeral_le_zero numeral_One numeral_le_iff partition_on_mset_def semiring_norm(69))
qed
lemma partition_on_mset_distinct_disjoint:
assumes "partition_on_mset A P"
assumes "distinct_mset A"
assumes "p1 \<in># P"
assumes "p2 \<in># P - {#p1#}"
shows "p1 \<inter># p2 = {#}"
using Diff_eq_empty_iff_mset assms diff_add_zero distinct_mset_add multiset_inter_assoc sum_mset.remove
by (smt partition_on_msetD1 subset_mset.inf.absorb_iff2 subset_mset.le_add_same_cancel1 subset_mset.le_iff_inf)
lemma partition_on_mset_diff:
assumes "partition_on_mset A P"
assumes "Q \<subseteq>#P"
shows "partition_on_mset (A - \<Sum>\<^sub>#Q) (P - Q)"
using assms partition_on_mset_def
by (smt diff_union_cancelL subset_mset.add_diff_inverse sum_mset.union union_iff)
lemma sigma_over_set_partition_count:
assumes "finite A"
assumes "partition_on A P"
assumes "x \<in># \<Sum>\<^sub># (mset_set (mset_set ` P))"
shows "count (\<Sum>\<^sub># (mset_set (mset_set ` P))) x = 1"
proof -
have disj: "disjoint P" using assms partition_onD2 by auto
then obtain p where pin: "p \<in># mset_set (mset_set ` P)" and xin: "x \<in># p"
using assms by blast
then have "count (mset_set (mset_set ` P)) p = 1"
by (meson count_eq_zero_iff count_mset_set')
then have filter: "\<And> p' . p' \<in># ((mset_set (mset_set` P)) - {#p#}) \<Longrightarrow> p \<noteq> p'"
using count_eq_zero_iff count_single by fastforce
have zero: "\<And> p'. p' \<in># mset_set (mset_set ` P) \<Longrightarrow> p' \<noteq> p \<Longrightarrow> count p' x = 0"
proof (rule ccontr)
fix p'
assume assm: "p' \<in># mset_set (mset_set ` P)" and ne: "p' \<noteq> p" and n0: "count p' x \<noteq> 0"
then have xin2: "x \<in># p'" by auto
obtain p1 p2 where p1in: "p1 \<in> P" and p2in: "p2 \<in> P" and p1eq: "mset_set p1 = p"
and p2eq: "mset_set p2 = p'" using assm assms(1) assms(2) pin
by (metis (no_types, lifting) elem_mset_set finite_elements finite_imageI image_iff)
have origne: "p1 \<noteq> p2" using ne p1eq p2eq by auto
have "p1 = p2" using partition_onD4 xin xin2
by (metis assms(2) count_eq_zero_iff count_mset_set' p1eq p1in p2eq p2in)
then show False using origne by simp
qed
have one: "count p x = 1" using pin xin assms count_eq_zero_iff count_greater_eq_one_iff
by (metis count_mset_set(3) count_mset_set_le_one image_iff le_antisym)
then have "count (\<Sum>\<^sub># (mset_set (mset_set ` P))) x =
(\<Sum>p' \<in># (mset_set (mset_set ` P)) . count p' x)"
using count_sum_mset by auto
also have "... = (count p x) + (\<Sum>p' \<in># ((mset_set (mset_set ` P)) - {#p#}) . count p' x)"
by (metis (mono_tags, lifting) insert_DiffM pin sum_mset.insert)
also have "... = 1 + (\<Sum>p' \<in># ((mset_set (mset_set ` P)) - {#p#}) . count p' x)"
using one by presburger
finally have "count (\<Sum>\<^sub># (mset_set (mset_set ` P))) x =
1 + (\<Sum>p' \<in># ((mset_set (mset_set ` P)) - {#p#}) . 0)"
using zero filter by (metis (mono_tags, lifting) in_diffD sum_over_fun_eq)
then show "count (\<Sum>\<^sub># (mset_set (mset_set ` P))) x = 1" by simp
qed
lemma partition_on_mset_set:
assumes "finite A"
assumes "partition_on A P"
shows "partition_on_mset (mset_set A) (mset_set (image (\<lambda> x. mset_set x) P))"
proof (intro partition_on_msetI)
have partd1: "\<Union>P = A" using assms partition_onD1 by auto
have imp: "\<And>x. x \<in># \<Sum>\<^sub># (mset_set (mset_set ` P)) \<Longrightarrow> x \<in># mset_set A"
proof -
fix x
assume "x \<in># \<Sum>\<^sub># (mset_set (mset_set ` P))"
then obtain p where "p \<in> (mset_set ` P)" and xin: "x \<in># p"
by (metis elem_mset_set equals0D infinite_set_mset_mset_set mset_big_union_obtain)
then have "set_mset p \<in> P"
by (metis empty_iff finite_set_mset_mset_set image_iff infinite_set_mset_mset_set)
then show "x \<in># mset_set A"
using partd1 xin assms(1) by auto
qed
have imp2: "\<And>x . x \<in># mset_set A \<Longrightarrow> x \<in># \<Sum>\<^sub># (mset_set (mset_set ` P))"
proof -
fix x
assume "x \<in># mset_set A"
then have "x \<in> A" by (simp add: assms(1))
then obtain p where "p \<in> P" and "x \<in> p" using assms(2) using partd1 by blast
then obtain p' where "p' \<in> (mset_set ` P)" and "p' = mset_set p" by blast
thus "x \<in># \<Sum>\<^sub># (mset_set (mset_set ` P))" using assms \<open>p \<in> P\<close> \<open>x \<in> p\<close> finite_elements partd1
by (metis Sup_upper finite_imageI finite_set_mset_mset_set in_Union_mset_iff rev_finite_subset)
qed
have a1: "\<And> x . x \<in># mset_set A \<Longrightarrow> count (mset_set A) x = 1"
using assms(1) by fastforce
then show "\<Sum>\<^sub># (mset_set (mset_set ` P)) = mset_set A" using imp imp2 a1
by (metis assms(1) assms(2) count_eq_zero_iff multiset_eqI sigma_over_set_partition_count)
have "\<And> p. p \<in> P \<Longrightarrow> p \<noteq> {} " using assms partition_onD3 by auto
then have "\<And> p. p \<in> P \<Longrightarrow> mset_set p \<noteq> {#}" using mset_set_empty_iff
by (metis Union_upper assms(1) partd1 rev_finite_subset)
then show "{#} \<notin># mset_set (mset_set ` P)"
by (metis elem_mset_set equals0D image_iff infinite_set_mset_mset_set)
qed
lemma partition_on_mset_distinct_inter:
assumes "partition_on_mset A P"
assumes "distinct_mset A"
assumes "p1 \<in># P" and "p2 \<in># P" and "p1 \<noteq> p2"
shows "p1 \<inter># p2 = {#}"
by (metis assms in_remove1_mset_neq partition_on_mset_distinct_disjoint)
lemma partition_on_set_mset_distinct:
assumes "partition_on_mset A P"
assumes "distinct_mset A"
assumes "p \<in># image_mset set_mset P"
assumes "p' \<in># image_mset set_mset P"
assumes "p \<noteq> p'"
shows "p \<inter> p' = {}"
proof -
obtain p1 where p1in: "p1 \<in># P" and p1eq: "set_mset p1 = p" using assms(3)
by blast
obtain p2 where p2in: "p2 \<in># P" and p2eq: "set_mset p2 = p'" using assms(4) by blast
have "distinct_mset P" using assms partition_on_mset_distinct by blast
then have "p1 \<noteq> p2" using assms using p1eq p2eq by fastforce
then have "p1 \<inter># p2 = {#}" using partition_on_mset_distinct_inter
using assms(1) assms(2) p1in p2in by auto
thus ?thesis using p1eq p2eq
by (metis set_mset_empty set_mset_inter)
qed
lemma partition_on_set_mset:
assumes "partition_on_mset A P"
assumes "distinct_mset A"
shows "partition_on (set_mset A) (set_mset (image_mset set_mset P))"
proof (intro partition_onI)
show "\<And>p. p \<in># image_mset set_mset P \<Longrightarrow> p \<noteq> {}"
using assms(1) partition_on_msetD2 by fastforce
next
have "\<And> x . x \<in> set_mset A \<Longrightarrow> x \<in> \<Union> (set_mset (image_mset set_mset P))"
by (metis Union_iff assms(1) image_eqI mset_big_union_obtain partition_on_msetD1 set_image_mset)
then show "\<Union> (set_mset (image_mset set_mset P)) = set_mset A"
using set_eqI' partition_on_mset_elems assms by auto
show "\<And>p p'. p \<in># image_mset set_mset P \<Longrightarrow> p' \<in># image_mset set_mset P \<Longrightarrow>
p \<noteq> p' \<Longrightarrow> p \<inter> p' = {}"
using partition_on_set_mset_distinct assms by blast
qed
lemma partition_on_mset_eq_imp_eq_carrier:
assumes "partition_on_mset A P"
assumes "partition_on_mset B P"
shows "A = B"
using assms partition_on_msetD1 by auto
lemma partition_on_mset_add_single:
assumes "partition_on_mset A P"
shows "partition_on_mset (add_mset a A) (add_mset {#a#} P)"
using assms by (auto simp: partition_on_mset_def)
lemma partition_on_mset_add_part:
assumes "partition_on_mset A P"
assumes "X \<noteq> {#}"
assumes "A + X = A'"
shows "partition_on_mset A' (add_mset X P)"
using assms by (auto simp: partition_on_mset_def)
lemma partition_on_mset_add:
assumes "partition_on_mset A P"
assumes "X \<in># P"
assumes "add_mset a X = X'"
shows "partition_on_mset (add_mset a A) (add_mset X' (P - {#X#}))"
using add_mset_add_single assms empty_not_add_mset mset_subset_eq_single partition_on_mset_all
by (smt partition_on_mset_def subset_mset.add_diff_inverse sum_mset.add_mset sum_mset.remove union_iff union_mset_add_mset_left)
lemma partition_on_mset_elem_exists_part:
assumes "partition_on_mset A P"
assumes "x \<in># A"
obtains p where "p \<in># P" and "x \<in># p"
using assms in_Union_mset_iff partition_on_msetD2 partition_on_msetI
by (metis partition_on_mset_eq_imp_eq_carrier)
lemma partition_on_mset_combine:
assumes "partition_on_mset A P"
assumes "partition_on_mset B Q"
shows "partition_on_mset (A + B) (P + Q)"
unfolding partition_on_mset_def
using assms partition_on_msetD1 partition_on_msetD2 by auto
lemma partition_on_mset_split:
assumes "partition_on_mset A (P + Q)"
shows "partition_on_mset (\<Sum>\<^sub>#P) P"
using partition_on_mset_def partition_on_msetD2 assms by fastforce
end |
(* Author: Norbert Schirmer
Maintainer: Norbert Schirmer, norbert.schirmer at web de
License: LGPL
*)
(* Title: UserGuide.thy
Author: Norbert Schirmer, TU Muenchen
Copyright (C) 2004-2008 Norbert Schirmer
Some rights reserved, TU Muenchen
This library is free software; you can redistribute it and/or modify
it under the terms of the GNU Lesser General Public License as
published by the Free Software Foundation; either version 2.1 of the
License, or (at your option) any later version.
This library is distributed in the hope that it will be useful, but
WITHOUT ANY WARRANTY; without even the implied warranty of
MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU
Lesser General Public License for more details.
You should have received a copy of the GNU Lesser General Public
License along with this library; if not, write to the Free Software
Foundation, Inc., 59 Temple Place, Suite 330, Boston, MA 02111-1307
USA
*)
section \<open>User Guide \label{sec:UserGuide}\<close>
(*<*)
theory UserGuide
imports HeapList Vcg
"HOL-Statespace.StateSpaceSyntax" "HOL-Library.LaTeXsugar"
begin
(*>*)
(*<*)
syntax
"_statespace_updates" :: "('a \<Rightarrow> 'b) \<Rightarrow> updbinds \<Rightarrow> ('a \<Rightarrow> 'b)" ("_\<langle>_\<rangle>" [900,0] 900)
(*>*)
text \<open>
We introduce the verification environment with a couple
of examples that illustrate how to use the different
bits and pieces to verify programs.
\<close>
subsection \<open>Basics\<close>
text \<open>
First of all we have to decide how to represent the state space. There
are currently two implementations. One is based on records the other
one on the concept called `statespace' that was introduced with
Isabelle 2007 (see \texttt{HOL/Statespace}) . In contrast to records a
'satespace' does not define a new type, but provides a notion of state,
based on locales. Logically
the state is modelled as a function from (abstract) names to
(abstract) values and the statespace infrastructure organises
distinctness of names an projection/injection of concrete values into
the abstract one. Towards the user the interface of records and
statespaces is quite similar. However, statespaces offer more
flexibility, inherited from the locale infrastructure, in
particular multiple inheritance and renaming of components.
In this user guide we prefer statespaces, but give some comments on
the usage of records in Section \ref{sec:records}.
\<close>
hoarestate vars =
A :: nat
I :: nat
M :: nat
N :: nat
R :: nat
S :: nat
text (in vars) \<open>The command \isacommand{hoarestate} is a simple preprocessor
for the command \isacommand{statespaces} which decorates the state
components with the suffix \<open>_'\<close>, to avoid cluttering the
namespace. Also note that underscores are printed as hyphens in this
documentation. So what you see as @{term "A_'"} in this document is
actually \texttt{A\_'}. Every component name becomes a fixed variable in
the locale \<open>vars\<close> and can no longer be used for logical
variables.
Lookup of a component @{term "A_'"} in a state @{term "s"} is written as
@{term "s\<cdot>A_'"}, and update with a value @{term "term v"} as @{term "s\<langle>A_' := v\<rangle>"}.
To deal with local and global variables in the context of procedures the
program state is organised as a record containing the two componets @{const "locals"}
and @{const "globals"}. The variables defined in hoarestate \<open>vars\<close> reside
in the @{const "locals"} part.
\<close>
text \<open>
Here is a first example.
\<close>
lemma (in vars) "\<Gamma>\<turnstile> \<lbrace>\<acute>N = 5\<rbrace> \<acute>N :== 2 * \<acute>N \<lbrace>\<acute>N = 10\<rbrace>"
apply vcg
txt \<open>@{subgoals}\<close>
apply simp
txt \<open>@{subgoals}\<close>
done
text \<open>We enable the locale of statespace \<open>vars\<close> by the
\texttt{in vars} directive. The verification condition generator is
invoked via the \<open>vcg\<close> method and leaves us with the expected
subgoal that can be proved by simplification.\<close>
text (in vars) \<open>
If we refer to components (variables) of the state-space of the program
we always mark these with \<open>\<acute>\<close> (in assertions and also in the
program itself). It is the acute-symbol and is present on
most keyboards. The assertions of the Hoare tuple are
ordinary Isabelle sets. As we usually want to refer to the state space
in the assertions, we provide special brackets for them. They can be written
as {\verb+{| |}+} in ASCII or \<open>\<lbrace> \<rbrace>\<close> with symbols. Internally,
marking variables has two effects. First of all we refer to the implicit
state and secondary we get rid of the suffix \<open>_'\<close>.
So the assertion @{term "{|\<acute>N = 5|}"} internally gets expanded to
\<open>{s. locals s \<cdot>N_' = 5}\<close> written in ordinary set comprehension notation of
Isabelle. It describes the set of states where the \<open>N_'\<close> component
is equal to \<open>5\<close>.
An empty context and an empty postcondition for abrupt termination can be
omitted. The lemma above is a shorthand for
\<open>\<Gamma>,{}\<turnstile> \<lbrace>\<acute>N = 5\<rbrace> \<acute>N :== 2 * \<acute>N \<lbrace>\<acute>N = 10\<rbrace>,{}\<close>.
\<close>
text \<open>We can step through verification condition generation by the
method \<open>vcg_step\<close>.
\<close>
lemma (in vars) "\<Gamma>,{}\<turnstile> \<lbrace>\<acute>N = 5\<rbrace> \<acute>N :== 2 * \<acute>N \<lbrace>\<acute>N = 10\<rbrace>"
apply vcg_step
txt \<open>@{subgoals}\<close>
txt \<open>The last step of verification condition generation,
transforms the inclusion of state sets to the corresponding
predicate on components of the state space.
\<close>
apply vcg_step
txt \<open>@{subgoals}\<close>
by simp
text \<open>
Although our assertions work semantically on the state space, stepping
through verification condition generation ``feels'' like the expected
syntactic substitutions of traditional Hoare logic. This is achieved
by light simplification on the assertions calculated by the Hoare rules.
\<close>
lemma (in vars) "\<Gamma>\<turnstile> \<lbrace>\<acute>N = 5\<rbrace> \<acute>N :== 2 * \<acute>N \<lbrace>\<acute>N = 10\<rbrace>"
apply (rule HoarePartial.Basic)
txt \<open>@{subgoals}\<close>
apply (simp only: mem_Collect_eq)
txt \<open>@{subgoals}\<close>
apply (tactic
\<open>Hoare.BasicSimpTac @{context} Hoare.Function false
[] (K all_tac) 1\<close>)
txt \<open>@{subgoals}\<close>
by simp
text \<open>The next example shows how we deal with the while loop. Note the
invariant annotation.
\<close>
lemma (in vars)
"\<Gamma>,{}\<turnstile> \<lbrace>\<acute>M = 0 \<and> \<acute>S = 0\<rbrace>
WHILE \<acute>M \<noteq> a
INV \<lbrace>\<acute>S = \<acute>M * b\<rbrace>
DO \<acute>S :== \<acute>S + b;; \<acute>M :== \<acute>M + 1 OD
\<lbrace>\<acute>S = a * b\<rbrace>"
apply vcg
txt \<open>@{subgoals [display]}\<close>
txt \<open>The verification condition generator gives us three proof obligations,
stemming from the path from the precondition to the invariant,
from the invariant together with loop condition through the
loop body to the invariant, and finally from the invariant together
with the negated loop condition to the postcondition.\<close>
apply auto
done
subsection \<open>Procedures\<close>
subsubsection \<open>Declaration\<close>
text \<open>
Our first procedure is a simple square procedure. We provide the
command \isacommand{procedures}, to declare and define a
procedure.
\<close>
procedures
Square (N::nat|R::nat)
where I::nat in
"\<acute>R :== \<acute>N * \<acute>N"
text \<open>A procedure is given by the signature of the procedure
followed by the procedure body. The signature consists of the name of
the procedure and a list of parameters together with their types. The
parameters in front of the pipe \<open>|\<close> are value parameters and
behind the pipe are the result parameters. Value parameters model call
by value semantics. The value of a result parameter at the end of the
procedure is passed back to the caller. Local variables follow the
\<open>where\<close>. If there are no local variables the \<open>where \<dots>
in\<close> can be omitted. The variable @{term "I"} is actually unused in
the body, but is used in the examples below.\<close>
text \<open>
The procedures command provides convenient syntax
for procedure calls (that creates the proper @{term init}, @{term return} and
@{term result} functions on the fly) and creates locales and statespaces to
reason about the procedure. The purpose of locales is to set up logical contexts
to support modular reasoning. Locales can be seen as freeze-dried proof contexts that
get alive as you setup a new lemma or theorem (\cite{Ballarin-04-locales}).
The locale the user deals with is named \<open>Square_impl\<close>.
It defines the procedure name (internally @{term "Square_'proc"}), the procedure body
(named \<open>Square_body\<close>) and the statespaces for parameters and local and
global variables.
Moreover it contains the
assumption @{term "\<Gamma> Square_'proc = Some Square_body"}, which states
that the procedure is properly defined in the procedure context.
The purpose of the locale is to give us easy means to setup the context
in which we prove programs correct.
In this locale the procedure context @{term "\<Gamma>"} is fixed.
So we always use this letter for the procedure
specification. This is crucial, if we prove programs under the
assumption of some procedure specifications.
\<close>
(*<*)
context Square_impl
begin
(*>*)
text \<open>The procedures command generates syntax, so that we can
either write \<open>CALL Square(\<acute>I,\<acute>R)\<close> or @{term "\<acute>I :== CALL
Square(\<acute>R)"} for the procedure call. The internal term is the
following:
\<close>
(*<*) declare [[hoare_use_call_tr' = false]] (*>*)
text \<open>\small @{term [display] "CALL Square(\<acute>I,\<acute>R)"}\<close>
(*<*) declare [[hoare_use_call_tr' = true]] (*>*)
text \<open>Note the
additional decoration (with the procedure name) of the parameter and
local variable names.\<close>
(*<*)
end
(*>*)
text \<open>The abstract syntax for the
procedure call is @{term "call init p return result"}. The @{term
"init"} function copies the values of the actual parameters to the
formal parameters, the @{term return} function copies the global
variables back (in our case there are no global variables), and the
@{term "result"} function additionally copies the values of the formal
result parameters to the actual locations. Actual value parameters can
be all kind of expressions, since we only need their value. But result
parameters must be proper ``lvalues'': variables (including
dereferenced pointers) or array locations, since we have to assign
values to them.
\<close>
subsubsection \<open>Verification\<close>
text (in Square_impl) \<open>
A procedure specification is an ordinary Hoare tuple.
We use the parameterless
call for the specification; \<open>\<acute>R :== PROC Square(\<acute>N)\<close> is syntactic sugar
for \<open>Call Square_'proc\<close>. This emphasises that the specification
describes the internal behaviour of the procedure, whereas parameter passing
corresponds to the procedure call.
The following precondition fixes the current value \<open>\<acute>N\<close> to the logical
variable @{term n}.
Universal quantification of @{term "n"} enables us to adapt
the specification to an actual parameter. The specification is
used in the rule for procedure call when we come upon a call to @{term Square}.
Thus @{term "n"} plays the role of the auxiliary variable @{term "Z"}.
\<close>
text \<open>To verify the procedure we need to verify the body. We use
a derived variant of the general recursion rule, tailored for non recursive procedures:
@{thm [source] HoarePartial.ProcNoRec1}:
\begin{center}
@{thm [mode=Rule,mode=ParenStmt] HoarePartial.ProcNoRec1 [no_vars]}
\end{center}
The naming convention for the rule
is the following: The \<open>1\<close> expresses that we look at one
procedure, and \<open>NoRec\<close> that the procedure is non
recursive.
\<close>
lemma (in Square_impl)
shows "\<forall>n. \<Gamma>\<turnstile>\<lbrace>\<acute>N = n\<rbrace> \<acute>R :== PROC Square(\<acute>N) \<lbrace>\<acute>R = n * n\<rbrace>"
txt \<open>The directive \<open>in\<close> has the effect that
the context of the locale @{term "Square_impl"} is included to the current
lemma, and that the lemma is added as a fact to the locale, after it is proven. The
next time locale @{term "Square_impl"} is invoked this lemma is immediately available
as fact, which the verification condition generator can use.
\<close>
apply (hoare_rule HoarePartial.ProcNoRec1)
txt "@{subgoals[display]}"
txt \<open>The method \<open>hoare_rule\<close>, like \<open>rule\<close> applies a
single rule, but additionally does some ``obvious'' steps:
It solves the canonical side-conditions of various Hoare-rules and it
automatically expands the
procedure body: With @{thm [source] Square_impl}: @{thm [names_short] Square_impl [no_vars]} we
get the procedure body out of the procedure context @{term "\<Gamma>"};
with @{thm [source] Square_body_def}: @{thm [names_short] Square_body_def [no_vars]} we
can unfold the definition of the body.
The proof is finished by the vcg and simp.
\<close>
txt "@{subgoals[display]}"
by vcg simp
text \<open>If the procedure is non recursive and there is no specification given, the
verification condition generator automatically expands the body.\<close>
lemma (in Square_impl) Square_spec:
shows "\<forall>n. \<Gamma>\<turnstile>\<lbrace>\<acute>N = n\<rbrace> \<acute>R :== PROC Square(\<acute>N) \<lbrace>\<acute>R = n * n\<rbrace>"
by vcg simp
text \<open>An important naming convention is to name the specification as
\<open><procedure-name>_spec\<close>. The verification condition generator refers to
this name in order to search for a specification in the theorem database.
\<close>
subsubsection \<open>Usage\<close>
text\<open>Let us see how we can use procedure specifications.\<close>
(* FIXME: maybe don't show this at all *)
lemma (in Square_impl)
shows "\<Gamma>\<turnstile>\<lbrace>\<acute>I = 2\<rbrace> \<acute>R :== CALL Square(\<acute>I) \<lbrace>\<acute>R = 4\<rbrace>"
txt \<open>Remember that we have already proven @{thm [source] "Square_spec"} in the locale
\<open>Square_impl\<close>. This is crucial for
verification condition generation. When reaching a procedure call,
it looks for the specification (by its name) and applies the
rule @{thm [source,mode=ParenStmt] HoarePartial.ProcSpec}
instantiated with the specification
(as last premise).
Before we apply the verification condition generator, let us
take some time to think of what we can expect.
Let's look at the specification @{thm [source] Square_spec} again:
@{thm [display] Square_spec [no_vars]}
The specification talks about the formal parameters @{term "N"} and
@{term R}. The precondition @{term "\<lbrace>\<acute>N = n\<rbrace>"} just fixes the initial
value of \<open>N\<close>.
The actual parameters are @{term "I"} and @{term "R"}. We
have to adapt the specification to this calling context.
@{term "\<forall>n. \<Gamma>\<turnstile> \<lbrace>\<acute>I = n\<rbrace> \<acute>R :== CALL Square(\<acute>I) \<lbrace>\<acute>R = n * n\<rbrace>"}.
From the postcondition @{term "\<lbrace>\<acute>R = n * n\<rbrace>"} we
have to derive the actual postcondition @{term "\<lbrace>\<acute>R = 4\<rbrace>"}. So
we gain something like: @{term "\<lbrace>n * n = (4::nat)\<rbrace>"}.
The precondition is @{term "\<lbrace>\<acute>I = 2\<rbrace>"} and the specification
tells us @{term "\<lbrace>\<acute>I = n\<rbrace>"} for the pre-state. So the value of @{term n}
is the value of @{term I} in the pre-state. So we arrive at
@{term "\<lbrace>\<acute>I = 2\<rbrace> \<subseteq> \<lbrace>\<acute>I * \<acute>I = 4\<rbrace>"}.
\<close>
apply vcg_step
txt "@{subgoals[display]}"
txt \<open>
The second set looks slightly more involved:
@{term "\<lbrace>\<forall>t. \<^bsup>t\<^esup>R = \<acute>I * \<acute>I \<longrightarrow> \<acute>I * \<acute>I = 4\<rbrace>"}, this is an artefact from the
procedure call rule. Originally \<open>\<acute>I * \<acute>I = 4\<close> was \<open>\<^bsup>t\<^esup>R = 4\<close>. Where
@{term "t"} denotes the final state of the procedure and the superscript notation
allows to select a component from a particular state.
\<close>
apply vcg_step
txt "@{subgoals[display]}"
by simp
text \<open>
The adaption of the procedure specification to the actual calling
context is done due to the @{term init}, @{term return} and @{term result} functions
in the rule @{thm [source] HoarePartial.ProcSpec} (or in the variant
@{thm [source] HoarePartial.ProcSpecNoAbrupt} which already
incorporates the fact that the postcondition for abrupt termination
is the empty set). For the readers interested in the internals,
here a version without vcg.
\<close>
lemma (in Square_impl)
shows "\<Gamma>\<turnstile>\<lbrace>\<acute>I = 2\<rbrace> \<acute>R :== CALL Square(\<acute>I) \<lbrace>\<acute>R = 4\<rbrace>"
apply (rule HoarePartial.ProcSpecNoAbrupt [OF _ _ Square_spec])
txt "@{subgoals[display]}"
txt \<open>This is the raw verification condition,
It is interesting to see how the auxiliary variable @{term "Z"} is
actually used. It is unified with @{term n} of the specification and
fixes the state after parameter passing.
\<close>
apply simp
txt "@{subgoals[display]}"
prefer 2
apply vcg_step
txt "@{subgoals[display]}"
apply (auto intro: ext)
done
subsubsection \<open>Recursion\<close>
text \<open>We want to define a procedure for the factorial. We first
define a HOL function that calculates it, to specify the procedure later on.
\<close>
primrec fac:: "nat \<Rightarrow> nat"
where
"fac 0 = 1" |
"fac (Suc n) = (Suc n) * fac n"
(*<*)
lemma fac_simp [simp]: "0 < i \<Longrightarrow> fac i = i * fac (i - 1)"
by (cases i) simp_all
(*>*)
text \<open>Now we define the procedure.\<close>
procedures
Fac (N::nat | R::nat)
"IF \<acute>N = 0 THEN \<acute>R :== 1
ELSE \<acute>R :== CALL Fac(\<acute>N - 1);;
\<acute>R :== \<acute>N * \<acute>R
FI"
text \<open>
Now let us prove that our implementation of @{term "Fac"} meets its specification.
\<close>
lemma (in Fac_impl)
shows "\<forall>n. \<Gamma>\<turnstile> \<lbrace>\<acute>N = n\<rbrace> \<acute>R :== PROC Fac(\<acute>N) \<lbrace>\<acute>R = fac n\<rbrace>"
apply (hoare_rule HoarePartial.ProcRec1)
txt "@{subgoals[display]}"
apply vcg
txt "@{subgoals[display]}"
apply simp
done
text \<open>
Since the factorial is implemented recursively,
the main ingredient of this proof is, to assume that the specification holds for
the recursive call of @{term Fac} and prove the body correct.
The assumption for recursive calls is added to the context by
the rule @{thm [source] HoarePartial.ProcRec1}
(also derived from the general rule for mutually recursive procedures):
\begin{center}
@{thm [mode=Rule,mode=ParenStmt] HoarePartial.ProcRec1 [no_vars]}
\end{center}
The verification condition generator infers the specification out of the
context @{term "\<Theta>"} when it encounters a recursive call of the factorial.
\<close>
subsection \<open>Global Variables and Heap \label{sec:VcgHeap}\<close>
text \<open>
Now we define and verify some procedures on heap-lists. We consider
list structures consisting of two fields, a content element @{term "cont"} and
a reference to the next list element @{term "next"}. We model this by the
following state space where every field has its own heap.
\<close>
hoarestate globals_heap =
"next" :: "ref \<Rightarrow> ref"
cont :: "ref \<Rightarrow> nat"
text \<open>It is mandatory to start the state name with `globals'. This is exploited
by the syntax translations to store the components in the @{const globals} part
of the state.
\<close>
text \<open>Updates to global components inside a procedure are
always propagated to the caller. This is implicitly done by the
parameter passing syntax translations.
\<close>
text \<open>We first define an append function on lists. It takes two
references as parameters. It appends the list referred to by the first
parameter with the list referred to by the second parameter. The statespace
of the global variables has to be imported.
\<close>
procedures (imports globals_heap)
append(p :: ref, q::ref | p::ref)
"IF \<acute>p=Null THEN \<acute>p :== \<acute>q
ELSE \<acute>p\<rightarrow>\<acute>next :== CALL append(\<acute>p\<rightarrow>\<acute>next,\<acute>q) FI"
(*<*)
context append_impl
begin
(*>*)
text \<open>
The difference of a global and a local variable is that global
variables are automatically copied back to the procedure caller.
We can study this effect on the translation of @{term "\<acute>p :== CALL append(\<acute>p,\<acute>q)"}:
\<close>
(*<*)
declare [[hoare_use_call_tr' = false]]
(*>*)
text \<open>
@{term [display] "\<acute>p :== CALL append(\<acute>p,\<acute>q)"}
\<close>
(*<*)
declare [[hoare_use_call_tr' = true]]
end
(*>*)
text \<open>Below we give two specifications this time.
One captures the functional behaviour and focuses on the
entities that are potentially modified by the procedure, the second one
is a pure frame condition.
\<close>
text \<open>
The functional specification below introduces two logical variables besides the
state space variable @{term "\<sigma>"}, namely @{term "Ps"} and @{term "Qs"}.
They are universally quantified and range over both the pre-and the postcondition, so
that we are able to properly instantiate the specification
during the proofs. The syntax \<open>\<lbrace>\<sigma>. \<dots>\<rbrace>\<close> is a shorthand to fix the current
state: \<open>{s. \<sigma> = s \<dots>}\<close>. Moreover \<open>\<^bsup>\<sigma>\<^esup>x\<close> abbreviates
the lookup of variable \<open>x\<close> in the state
\<open>\<sigma>\<close>.
The approach to specify procedures on lists
basically follows \cite{MehtaN-CADE03}. From the pointer structure
in the heap we (relationally) abstract to HOL lists of references. Then
we can specify further properties on the level of HOL lists, rather then
on the heap. The basic abstractions are:
@{thm [display] Path.simps [no_vars]}
@{term [show_types] "Path x h y ps"}: @{term ps} is a list of references that we can obtain
out of the heap @{term h} by starting with the reference @{term x}, following
the references in @{term h} up to the reference @{term y}.
@{thm [display] List_def [no_vars]}
A list @{term "List p h ps"} is a path starting in @{term p} and ending up
in @{term Null}.
\<close>
lemma (in append_impl) append_spec1:
shows "\<forall>\<sigma> Ps Qs.
\<Gamma>\<turnstile> \<lbrace>\<sigma>. List \<acute>p \<acute>next Ps \<and> List \<acute>q \<acute>next Qs \<and> set Ps \<inter> set Qs = {}\<rbrace>
\<acute>p :== PROC append(\<acute>p,\<acute>q)
\<lbrace>List \<acute>p \<acute>next (Ps@Qs) \<and> (\<forall>x. x\<notin>set Ps \<longrightarrow> \<acute>next x = \<^bsup>\<sigma>\<^esup>next x)\<rbrace>"
apply (hoare_rule HoarePartial.ProcRec1)
txt \<open>@{subgoals [margin=80,display]}
Note that @{term "hoare_rule"} takes care of multiple auxiliary variables!
@{thm [source] HoarePartial.ProcRec1} has only one auxiliary variable, namely @{term Z}.
But the type of @{term Z} can be instantiated arbitrarily. So \<open>hoare_rule\<close>
instantiates @{term Z} with the tuple @{term "(\<sigma>,Ps,Qs)"} and derives a proper variant
of the rule. Therefore \<open>hoare_rule\<close> depends on the proper quantification of
auxiliary variables!
\<close>
apply vcg
txt \<open>@{subgoals [display]}
For each branch of the \<open>IF\<close> statement we have one conjunct to prove. The
\<open>THEN\<close> branch starts with \<open>p = Null \<longrightarrow> \<dots>\<close> and the \<open>ELSE\<close> branch
with \<open>p \<noteq> Null \<longrightarrow> \<dots>\<close>. Let us focus on the \<open>ELSE\<close> branch, were the
recursive call to append occurs. First of all we have to prove that the precondition for
the recursive call is fulfilled. That means we have to provide some witnesses for
the lists @{term Psa} and @{term Qsa} which are referenced by \<open>p\<rightarrow>next\<close> (now
written as @{term "next p"}) and @{term q}. Then we have to show that we can
derive the overall postcondition from the postcondition of the recursive call. The
state components that have changed by the recursive call are the ones with the suffix
\<open>a\<close>, like \<open>nexta\<close> and \<open>pa\<close>.
\<close>
apply fastforce
done
text \<open>If the verification condition generator works on a procedure
call it checks whether it can find a modifies clause in the
context. If one is present the procedure call is simplified before the
Hoare rule @{thm [source] HoarePartial.ProcSpec} is
applied. Simplification of the procedure call means that the ``copy
back'' of the global components is simplified. Only those components
that occur in the modifies clause are actually copied back. This
simplification is justified by the rule @{thm [source]
HoarePartial.ProcModifyReturn}.
So after this simplification all global
components that do not appear in the modifies clause are treated
as local variables.\<close>
text \<open>We study the effect of the modifies clause on the following
examples, where we want to prove that @{term "append"} does not change
the @{term "cont"} part of the heap.
\<close>
lemma (in append_impl)
shows "\<Gamma>\<turnstile> \<lbrace>\<acute>cont=c\<rbrace> \<acute>p :== CALL append(Null,Null) \<lbrace>\<acute>cont=c\<rbrace>"
proof -
note append_spec = append_spec1
show ?thesis
apply vcg
txt \<open>@{subgoals [display]}\<close>
txt \<open>Only focus on the very last line: @{term conta} is the heap component
after the procedure call,
and @{term cont} the heap component before the procedure call. Since
we have not added the modified clause we do not know that they have
to be equal.
\<close>
oops
text \<open>
We now add the frame condition.
The list in the modifies clause names all global state components that
may be changed by the procedure. Note that we know from the modifies clause
that the @{term cont} parts are not changed. Also a small
side note on the syntax. We use ordinary brackets in the postcondition
of the modifies clause, and also the state components do not carry the
acute, because we explicitly note the state @{term t} here.
\<close>
lemma (in append_impl) append_modifies:
shows "\<forall>\<sigma>. \<Gamma>\<turnstile>\<^bsub>/UNIV\<^esub> {\<sigma>} \<acute>p :== PROC append(\<acute>p,\<acute>q)
{t. t may_only_modify_globals \<sigma> in [next]}"
apply (hoare_rule HoarePartial.ProcRec1)
apply (vcg spec=modifies)
done
text \<open>We tell the verification condition generator to use only the
modifies clauses and not to search for functional specifications by
the parameter \<open>spec=modifies\<close>. It also tries to solve the
verification conditions automatically. Again it is crucial to name
the lemma with this naming scheme, since the verfication condition
generator searches for these names.
\<close>
text \<open>The modifies clause is equal to a state update specification
of the following form.
\<close>
lemma (in append_impl) shows "{t. t may_only_modify_globals Z in [next]}
=
{t. \<exists>next. globals t=update id id next_' (K_statefun next) (globals Z)}"
apply (unfold mex_def meq_def)
apply simp
done
text \<open>Now that we have proven the frame-condition, it is available within
the locale \<open>append_impl\<close> and the \<open>vcg\<close> exploits it.\<close>
lemma (in append_impl)
shows "\<Gamma>\<turnstile> \<lbrace>\<acute>cont=c\<rbrace> \<acute>p :== CALL append(Null,Null) \<lbrace>\<acute>cont=c\<rbrace>"
proof -
note append_spec = append_spec1
show ?thesis
apply vcg
txt \<open>@{subgoals [display]}\<close>
txt \<open>With a modifies clause present we know that no change to @{term cont}
has occurred.
\<close>
by simp
qed
text \<open>
Of course we could add the modifies clause to the functional specification as
well. But separating both has the advantage that we split up the verification
work. We can make use of the modifies clause before we apply the
functional specification in a fully automatic fashion.
\<close>
text \<open>
To prove that a procedure respects the modifies clause, we only need
the modifies clauses of the procedures called in the body. We do not need
the functional specifications. So we can always prove the modifies
clause without functional specifications, but we may need the modifies
clause to prove the functional specifications. So usually the modifies clause is
proved before the proof of the functional specification, so that it can already be used
by the verification condition generator.
\<close>
subsection \<open>Total Correctness\<close>
text \<open>When proving total correctness the additional proof burden to
the user is to come up with a well-founded relation and to prove that
certain states get smaller according to this relation. Proving that a
relation is well-founded can be quite hard. But fortunately there are
ways to construct and stick together relations so that they are
well-founded by construction. This infrastructure is already present
in Isabelle/HOL. For example, @{term "measure f"} is always well-founded;
the lexicographic product of two well-founded relations is again
well-founded and the inverse image construction @{term "inv_image"} of
a well-founded relation is again well-founded. The constructions are
best explained by some equations:
@{thm in_measure_iff [no_vars]}\\
@{thm in_lex_iff [no_vars]}\\
@{thm in_inv_image_iff [no_vars]}
Another useful construction is \<open><*mlex*>\<close> which is a combination
of a measure and a lexicographic product:
@{thm in_mlex_iff [no_vars]}\\
In contrast to the lexicographic product it does not construct a product type.
The state may either decrease according to the measure function @{term f} or the
measure stays the same and the state decreases because of the relation @{term r}.
Lets look at a loop:
\<close>
lemma (in vars)
"\<Gamma>\<turnstile>\<^sub>t \<lbrace>\<acute>M = 0 \<and> \<acute>S = 0\<rbrace>
WHILE \<acute>M \<noteq> a
INV \<lbrace>\<acute>S = \<acute>M * b \<and> \<acute>M \<le> a\<rbrace>
VAR MEASURE a - \<acute>M
DO \<acute>S :== \<acute>S + b;; \<acute>M :== \<acute>M + 1 OD
\<lbrace>\<acute>S = a * b\<rbrace>"
apply vcg
txt \<open>@{subgoals [display]}
The first conjunct of the second subgoal is the proof obligation that the
variant decreases in the loop body.
\<close>
by auto
text \<open>The variant annotation is preceded by \<open>VAR\<close>. The capital \<open>MEASURE\<close>
is a shorthand for \<open>measure (\<lambda>s. a - \<^bsup>s\<^esup>M)\<close>. Analogous there is a capital
\<open><*MLEX*>\<close>.
\<close>
lemma (in Fac_impl) Fac_spec':
shows "\<forall>\<sigma>. \<Gamma>\<turnstile>\<^sub>t {\<sigma>} \<acute>R :== PROC Fac(\<acute>N) \<lbrace>\<acute>R = fac \<^bsup>\<sigma>\<^esup>N\<rbrace>"
apply (hoare_rule HoareTotal.ProcRec1 [where r="measure (\<lambda>(s,p). \<^bsup>s\<^esup>N)"])
txt \<open>In case of the factorial the parameter @{term N} decreases in every call. This
is easily expressed by the measure function. Note that the well-founded relation for
recursive procedures is formally defined on tuples
containing the state space and the procedure name.
\<close>
txt \<open>@{subgoals [display]}
The initial call to the factorial is in state @{term "\<sigma>"}. Note that in the
precondition @{term "{\<sigma>} \<inter> {\<sigma>'}"}, @{term "\<sigma>'"} stems from the lemma we want to prove
and @{term "\<sigma>"} stems from the recursion rule for total correctness. Both are
synonym for the initial state. To use the assumption in the Hoare context we
have to show that the call to the factorial is invoked on a smaller @{term N} compared
to the initial \<open>\<^bsup>\<sigma>\<^esup>N\<close>.
\<close>
apply vcg
txt \<open>@{subgoals [display]}
The tribute to termination is that we have to show \<open>N - 1 < N\<close> in case of
the recursive call.
\<close>
by simp
lemma (in append_impl) append_spec2:
shows "\<forall>\<sigma> Ps Qs. \<Gamma>\<turnstile>\<^sub>t
\<lbrace>\<sigma>. List \<acute>p \<acute>next Ps \<and> List \<acute>q \<acute>next Qs \<and> set Ps \<inter> set Qs = {}\<rbrace>
\<acute>p :== PROC append(\<acute>p,\<acute>q)
\<lbrace>List \<acute>p \<acute>next (Ps@Qs) \<and> (\<forall>x. x\<notin>set Ps \<longrightarrow> \<acute>next x = \<^bsup>\<sigma>\<^esup>next x)\<rbrace>"
apply (hoare_rule HoareTotal.ProcRec1
[where r="measure (\<lambda>(s,p). length (list \<^bsup>s\<^esup>p \<^bsup>s\<^esup>next))"])
txt \<open>In case of the append function the length of the list referenced by @{term p}
decreases in every recursive call.
\<close>
txt \<open>@{subgoals [margin=80,display]}\<close>
apply vcg
apply (fastforce simp add: List_list)
done
text \<open>
In case of the lists above, we have used a relational list abstraction @{term List}
to construct the HOL lists @{term Ps} and @{term Qs} for the pre- and postcondition.
To supply a proper measure function we use a functional abstraction @{term list}.
The functional abstraction can be defined by means of the relational list abstraction,
since the lists are already uniquely determined by the relational abstraction:
@{thm islist_def [no_vars]}\\
@{thm list_def [no_vars]}
\isacommand{lemma} @{thm List_conv_islist_list [no_vars]}
\<close>
text \<open>
The next contrived example is taken from \cite{Homeier-95-vcg}, to illustrate
a more complex termination criterion for mutually recursive procedures. The procedures
do not calculate anything useful.
\<close>
procedures
pedal(N::nat,M::nat)
"IF 0 < \<acute>N THEN
IF 0 < \<acute>M THEN
CALL coast(\<acute>N- 1,\<acute>M- 1) FI;;
CALL pedal(\<acute>N- 1,\<acute>M)
FI"
and
coast(N::nat,M::nat)
"CALL pedal(\<acute>N,\<acute>M);;
IF 0 < \<acute>M THEN CALL coast(\<acute>N,\<acute>M- 1) FI"
text \<open>
In the recursive calls in procedure \<open>pedal\<close> the first argument always decreases.
In the body of \<open>coast\<close> in the recursive call of \<open>coast\<close> the second
argument decreases, but in the call to \<open>pedal\<close> no argument decreases.
Therefore an relation only on the state space is insufficient. We have to
take the procedure names into account, too.
We consider the procedure \<open>coast\<close> to be ``bigger'' than \<open>pedal\<close>
when we construct a well-founded relation on the product of state space and procedure
names.
\<close>
ML \<open>ML_Thms.bind_thm ("HoareTotal_ProcRec2", Hoare.gen_proc_rec @{context} Hoare.Total 2)\<close>
text \<open>
We provide the ML function {\tt gen\_proc\_rec} to
automatically derive a convenient rule for recursion for a given number of mutually
recursive procedures.
\<close>
lemma (in pedal_coast_clique)
shows "(\<forall>\<sigma>. \<Gamma>\<turnstile>\<^sub>t {\<sigma>} PROC pedal(\<acute>N,\<acute>M) UNIV) \<and>
(\<forall>\<sigma>. \<Gamma>\<turnstile>\<^sub>t {\<sigma>} PROC coast(\<acute>N,\<acute>M) UNIV)"
apply (hoare_rule HoareTotal_ProcRec2
[where r= "((\<lambda>(s,p). \<^bsup>s\<^esup>N) <*mlex*>
(\<lambda>(s,p). \<^bsup>s\<^esup>M) <*mlex*>
measure (\<lambda>(s,p). if p = coast_'proc then 1 else 0))"])
txt \<open>We can directly express the termination condition described above with
the \<open><*mlex*>\<close> construction. Either state component \<open>N\<close> decreases,
or it stays the same and \<open>M\<close> decreases or this also stays the same, but
then the procedure name has to decrease.\<close>
txt \<open>@{subgoals [margin=80,display]}\<close>
apply simp_all
txt \<open>@{subgoals [margin=75,display]}\<close>
by (vcg,simp)+
text \<open>We can achieve the same effect without \<open><*mlex*>\<close> by using
the ordinary lexicographic product \<open><*lex*>\<close>, \<open>inv_image\<close> and
\<open>measure\<close>
\<close>
lemma (in pedal_coast_clique)
shows "(\<forall>\<sigma>. \<Gamma>\<turnstile>\<^sub>t {\<sigma>} PROC pedal(\<acute>N,\<acute>M) UNIV) \<and>
(\<forall>\<sigma>. \<Gamma>\<turnstile>\<^sub>t {\<sigma>} PROC coast(\<acute>N,\<acute>M) UNIV)"
apply (hoare_rule HoareTotal_ProcRec2
[where r= "inv_image (measure (\<lambda>m. m) <*lex*>
measure (\<lambda>m. m) <*lex*>
measure (\<lambda>p. if p = coast_'proc then 1 else 0))
(\<lambda>(s,p). (\<^bsup>s\<^esup>N,\<^bsup>s\<^esup>M,p))"])
txt \<open>With the lexicographic product we construct a well-founded relation on
triples of type @{typ "(nat\<times>nat\<times>string)"}. With @{term inv_image} we project
the components out of the state-space and the procedure names to this
triple.
\<close>
txt \<open>@{subgoals [margin=75,display]}\<close>
apply simp_all
by (vcg,force)+
text \<open>By doing some arithmetic we can express the termination condition with a single
measure function.
\<close>
lemma (in pedal_coast_clique)
shows "(\<forall>\<sigma>. \<Gamma>\<turnstile>\<^sub>t {\<sigma>} PROC pedal(\<acute>N,\<acute>M) UNIV) \<and>
(\<forall>\<sigma>. \<Gamma>\<turnstile>\<^sub>t {\<sigma>} PROC coast(\<acute>N,\<acute>M) UNIV)"
apply(hoare_rule HoareTotal_ProcRec2
[where r= "measure (\<lambda>(s,p). \<^bsup>s\<^esup>N + \<^bsup>s\<^esup>M + (if p = coast_'proc then 1 else 0))"])
apply simp_all
txt \<open>@{subgoals [margin=75,display]}\<close>
by (vcg,simp,arith?)+
subsection \<open>Guards\<close>
text (in vars) \<open>The purpose of a guard is to guard the {\bf (sub-) expressions} of a
statement against runtime faults. Typical runtime faults are array bound violations,
dereferencing null pointers or arithmetical overflow. Guards make the potential
runtime faults explicit, since the expressions themselves never ``fail'' because
they are ordinary HOL expressions. To relieve the user from typing in lots of standard
guards for every subexpression, we supply some input syntax for the common
language constructs that automatically generate the guards.
For example the guarded assignment \<open>\<acute>M :==\<^sub>g (\<acute>M + 1) div \<acute>N\<close> gets expanded to
guarded command @{term "\<acute>M :==\<^sub>g (\<acute>M + 1) div \<acute>N"}. Here @{term "in_range"} is
uninterpreted by now.
\<close>
lemma (in vars) "\<Gamma>\<turnstile>\<lbrace>True\<rbrace> \<acute>M :==\<^sub>g (\<acute>M + 1) div \<acute>N \<lbrace>True\<rbrace>"
apply vcg
txt \<open>@{subgoals}\<close>
oops
text \<open>
The user can supply on (overloaded) definition of \<open>in_range\<close>
to fit to his needs.
Currently guards are generated for:
\begin{itemize}
\item overflow and underflow of numbers (\<open>in_range\<close>). For subtraction of
natural numbers \<open>a - b\<close> the guard \<open>b \<le> a\<close> is generated instead
of \<open>in_range\<close> to guard against underflows.
\item division by \<open>0\<close>
\item dereferencing of @{term Null} pointers
\item array bound violations
\end{itemize}
Following (input) variants of guarded statements are available:
\begin{itemize}
\item Assignment: \<open>\<dots> :==\<^sub>g \<dots>\<close>
\item If: \<open>IF\<^sub>g \<dots>\<close>
\item While: \<open>WHILE\<^sub>g \<dots>\<close>
\item Call: \<open>CALL\<^sub>g \<dots>\<close> or \<open>\<dots> :== CALL\<^sub>g \<dots>\<close>
\end{itemize}
\<close>
subsection \<open>Miscellaneous Techniques\<close>
subsubsection \<open>Modifies Clause\<close>
text \<open>We look at some issues regarding the modifies clause with the example
of insertion sort for heap lists.
\<close>
primrec sorted:: "('a \<Rightarrow> 'a \<Rightarrow> bool) \<Rightarrow> 'a list \<Rightarrow> bool"
where
"sorted le [] = True" |
"sorted le (x#xs) = ((\<forall>y\<in>set xs. le x y) \<and> sorted le xs)"
procedures (imports globals_heap)
insert(r::ref,p::ref | p::ref)
"IF \<acute>r=Null THEN SKIP
ELSE IF \<acute>p=Null THEN \<acute>p :== \<acute>r;; \<acute>p\<rightarrow>\<acute>next :== Null
ELSE IF \<acute>r\<rightarrow>\<acute>cont \<le> \<acute>p\<rightarrow>\<acute>cont
THEN \<acute>r\<rightarrow>\<acute>next :== \<acute>p;; \<acute>p:==\<acute>r
ELSE \<acute>p\<rightarrow>\<acute>next :== CALL insert(\<acute>r,\<acute>p\<rightarrow>\<acute>next)
FI
FI
FI"
lemma (in insert_impl) insert_modifies:
"\<forall>\<sigma>. \<Gamma>\<turnstile>\<^bsub>/UNIV\<^esub> {\<sigma>} \<acute>p :== PROC insert(\<acute>r,\<acute>p)
{t. t may_only_modify_globals \<sigma> in [next]}"
by (hoare_rule HoarePartial.ProcRec1) (vcg spec=modifies)
lemma (in insert_impl) insert_spec:
"\<forall>\<sigma> Ps . \<Gamma>\<turnstile>
\<lbrace>\<sigma>. List \<acute>p \<acute>next Ps \<and> sorted (\<le>) (map \<acute>cont Ps) \<and>
\<acute>r \<noteq> Null \<and> \<acute>r \<notin> set Ps\<rbrace>
\<acute>p :== PROC insert(\<acute>r,\<acute>p)
\<lbrace>\<exists>Qs. List \<acute>p \<acute>next Qs \<and> sorted (\<le>) (map \<^bsup>\<sigma>\<^esup>cont Qs) \<and>
set Qs = insert \<^bsup>\<sigma>\<^esup>r (set Ps) \<and>
(\<forall>x. x \<notin> set Qs \<longrightarrow> \<acute>next x = \<^bsup>\<sigma>\<^esup>next x)\<rbrace>"
(*<*)
apply (hoare_rule HoarePartial.ProcRec1)
apply vcg
apply (intro conjI impI)
apply fastforce
apply fastforce
apply fastforce
apply (clarsimp)
apply force
done
(*>*)
text \<open>
In the postcondition of the functional specification there is a small but
important subtlety. Whenever we talk about the @{term "cont"} part we refer to
the one of the pre-state.
The reason is that we have separated out the information that @{term "cont"} is not
modified by the procedure, to the modifies clause. So whenever we talk about unmodified
parts in the postcondition we have to use the pre-state part, or explicitly
state an equality in the postcondition.
The reason is simple. If the postcondition would talk about \<open>\<acute>cont\<close>
instead of \mbox{\<open>\<^bsup>\<sigma>\<^esup>cont\<close>}, we get a new instance of \<open>cont\<close> during
verification and the postcondition would only state something about this
new instance. But as the verification condition generator uses the
modifies clause the caller of @{term "insert"} instead still has the
old \<open>cont\<close> after the call. Thats the sense of the modifies clause.
So the caller and the specification simply talk about two different things,
without being able to relate them (unless an explicit equality is added to
the specification).
\<close>
subsubsection \<open>Annotations\<close>
text \<open>
Annotations (like loop invariants)
are mere syntactic sugar of statements that are used by the \<open>vcg\<close>.
Logically a statement with an annotation is
equal to the statement without it. Hence annotations can be introduced by the user
while building a proof:
@{thm [source] HoarePartial.annotateI}: @{thm [mode=Rule] HoarePartial.annotateI [no_vars]}
When introducing annotations it can easily happen that these mess around with the
nesting of sequential composition. Then after stripping the annotations the resulting statement
is no longer syntactically identical to original one, only equivalent modulo associativity of sequential composition. The following rule also deals with this case:
@{thm [source] HoarePartial.annotate_normI}: @{thm [mode=Rule] HoarePartial.annotate_normI [no_vars]}
\<close>
text_raw \<open>\paragraph{Loop Annotations}
\mbox{}
\medskip
\mbox{}
\<close>
procedures (imports globals_heap)
insertSort(p::ref| p::ref)
where r::ref q::ref in
"\<acute>r:==Null;;
WHILE (\<acute>p \<noteq> Null) DO
\<acute>q :== \<acute>p;;
\<acute>p :== \<acute>p\<rightarrow>\<acute>next;;
\<acute>r :== CALL insert(\<acute>q,\<acute>r)
OD;;
\<acute>p:==\<acute>r"
lemma (in insertSort_impl) insertSort_modifies:
shows
"\<forall>\<sigma>. \<Gamma>\<turnstile>\<^bsub>/UNIV\<^esub> {\<sigma>} \<acute>p :== PROC insertSort(\<acute>p)
{t. t may_only_modify_globals \<sigma> in [next]}"
apply (hoare_rule HoarePartial.ProcRec1)
apply (vcg spec=modifies)
done
text \<open>Insertion sort is not implemented recursively here, but with a
loop. Note that the while loop is not annotated with an invariant in the
procedure definition. The invariant only comes into play during verification.
Therefore we annotate the loop first, before we run the \<open>vcg\<close>.
\<close>
lemma (in insertSort_impl) insertSort_spec:
shows "\<forall>\<sigma> Ps.
\<Gamma>\<turnstile> \<lbrace>\<sigma>. List \<acute>p \<acute>next Ps \<rbrace>
\<acute>p :== PROC insertSort(\<acute>p)
\<lbrace>\<exists>Qs. List \<acute>p \<acute>next Qs \<and> sorted (\<le>) (map \<^bsup>\<sigma>\<^esup>cont Qs) \<and>
set Qs = set Ps\<rbrace>"
apply (hoare_rule HoarePartial.ProcRec1)
apply (hoare_rule anno=
"\<acute>r :== Null;;
WHILE \<acute>p \<noteq> Null
INV \<lbrace>\<exists>Qs Rs. List \<acute>p \<acute>next Qs \<and> List \<acute>r \<acute>next Rs \<and>
set Qs \<inter> set Rs = {} \<and>
sorted (\<le>) (map \<acute>cont Rs) \<and> set Qs \<union> set Rs = set Ps \<and>
\<acute>cont = \<^bsup>\<sigma>\<^esup>cont \<rbrace>
DO \<acute>q :== \<acute>p;; \<acute>p :== \<acute>p\<rightarrow>\<acute>next;; \<acute>r :== CALL insert(\<acute>q,\<acute>r) OD;;
\<acute>p :== \<acute>r" in HoarePartial.annotateI)
apply vcg
txt \<open>\<open>\<dots>\<close>\<close>
(*<*)
apply fastforce
prefer 2
apply fastforce
apply (clarsimp)
apply (rule_tac x=ps in exI)
apply (intro conjI)
apply (rule heap_eq_ListI1)
apply assumption
apply clarsimp
apply (subgoal_tac "x\<noteq>p \<and> x \<notin> set Rs")
apply auto
done
(*>*)
text \<open>The method \<open>hoare_rule\<close> automatically solves the side-condition
that the annotated
program is the same as the original one after stripping the annotations.\<close>
text_raw \<open>\paragraph{Specification Annotations}
\mbox{}
\medskip
\mbox{}
\<close>
text \<open>
When verifying a larger block of program text, it might be useful to split up
the block and to prove the parts in isolation. This is especially useful to
isolate loops. On the level of the Hoare calculus
the parts can then be combined with the consequence rule. To automate this
process we introduce the derived command @{term specAnno}, which allows to introduce
a Hoare tuple (inclusive auxiliary variables) in the program text:
@{thm specAnno_def [no_vars]}
The whole annotation reduces to the body @{term "c undefined"}. The
type of the assertions @{term "P"}, @{term "Q"} and @{term "A"} is
@{typ "'a \<Rightarrow> 's set"} and the type of command @{term c} is @{typ "'a \<Rightarrow> ('s,'p,'f) com"}.
All entities formally depend on an auxiliary (logical) variable of type @{typ "'a"}.
The body @{term "c"} formally also depends on this variable, since a nested annotation
or loop invariant may also depend on this logical variable. But the raw body without
annotations does not depend on the logical variable. The logical variable is only
used by the verification condition generator. We express this by defining the
whole @{term specAnno} to be equivalent with the body applied to an arbitrary
variable.
The Hoare rule for \<open>specAnno\<close> is mainly an instance of the consequence rule:
@{thm [mode=Rule,mode=ParenStmt] HoarePartial.SpecAnno [no_vars]}
The side-condition @{term "\<forall>Z. c Z = c undefined"} expresses the intention of body @{term c}
explained above: The raw body is independent of the auxiliary variable. This
side-condition is solved automatically by the \<open>vcg\<close>. The concrete syntax for
this specification annotation is shown in the following example:
\<close>
lemma (in vars) "\<Gamma>\<turnstile> {\<sigma>}
\<acute>I :== \<acute>M;;
ANNO \<tau>. \<lbrace>\<tau>. \<acute>I = \<^bsup>\<sigma>\<^esup>M\<rbrace>
\<acute>M :== \<acute>N;; \<acute>N :== \<acute>I
\<lbrace>\<acute>M = \<^bsup>\<tau>\<^esup>N \<and> \<acute>N = \<^bsup>\<tau>\<^esup>I\<rbrace>
\<lbrace>\<acute>M = \<^bsup>\<sigma>\<^esup>N \<and> \<acute>N = \<^bsup>\<sigma>\<^esup>M\<rbrace>"
txt \<open>With the annotation we can name an intermediate state @{term \<tau>}. Since the
postcondition refers to @{term "\<sigma>"} we have to link the information about
the equivalence of \<open>\<^bsup>\<tau>\<^esup>I\<close> and \<open>\<^bsup>\<sigma>\<^esup>M\<close> in the specification in order
to be able to derive the postcondition.
\<close>
apply vcg_step
apply vcg_step
txt \<open>@{subgoals [display]}\<close>
txt \<open>The first subgoal is the isolated Hoare tuple. The second one is the
side-condition of the consequence rule that allows us to derive the outermost
pre/post condition from our inserted specification.
\<open>\<acute>I = \<^bsup>\<sigma>\<^esup>M\<close> is the precondition of the specification,
The second conjunct is a simplified version of
\<open>\<forall>t. \<^bsup>t\<^esup>M = \<acute>N \<and> \<^bsup>t\<^esup>N = \<acute>I \<longrightarrow> \<^bsup>t\<^esup>M = \<^bsup>\<sigma>\<^esup>N \<and> \<^bsup>t\<^esup>N = \<^bsup>\<sigma>\<^esup>M\<close> expressing that the
postcondition of the specification implies the outermost postcondition.
\<close>
apply vcg
txt \<open>@{subgoals [display]}\<close>
apply simp
apply vcg
txt \<open>@{subgoals [display]}\<close>
by simp
lemma (in vars)
"\<Gamma>\<turnstile> {\<sigma>}
\<acute>I :== \<acute>M;;
ANNO \<tau>. \<lbrace>\<tau>. \<acute>I = \<^bsup>\<sigma>\<^esup>M\<rbrace>
\<acute>M :== \<acute>N;; \<acute>N :== \<acute>I
\<lbrace>\<acute>M = \<^bsup>\<tau>\<^esup>N \<and> \<acute>N = \<^bsup>\<tau>\<^esup>I\<rbrace>
\<lbrace>\<acute>M = \<^bsup>\<sigma>\<^esup>N \<and> \<acute>N = \<^bsup>\<sigma>\<^esup>M\<rbrace>"
apply vcg
txt \<open>@{subgoals [display]}\<close>
by simp_all
text \<open>Note that \<open>vcg_step\<close> changes the order of sequential composition, to
allow the user to decompose sequences by repeated calls to \<open>vcg_step\<close>, whereas
\<open>vcg\<close> preserves the order.
The above example illustrates how we can introduce a new logical state variable
@{term "\<tau>"}. You can introduce multiple variables by using a tuple:
\<close>
lemma (in vars)
"\<Gamma>\<turnstile> {\<sigma>}
\<acute>I :== \<acute>M;;
ANNO (n,i,m). \<lbrace>\<acute>I = \<^bsup>\<sigma>\<^esup>M \<and> \<acute>N=n \<and> \<acute>I=i \<and> \<acute>M=m\<rbrace>
\<acute>M :== \<acute>N;; \<acute>N :== \<acute>I
\<lbrace>\<acute>M = n \<and> \<acute>N = i\<rbrace>
\<lbrace>\<acute>M = \<^bsup>\<sigma>\<^esup>N \<and> \<acute>N = \<^bsup>\<sigma>\<^esup>M\<rbrace>"
apply vcg
txt \<open>@{subgoals [display]}\<close>
by simp_all
text_raw \<open>\paragraph{Lemma Annotations}
\mbox{}
\medskip
\mbox{}
\<close>
text \<open>
The specification annotations described before split the verification
into several Hoare triples which result in several subgoals. If we
instead want to proof the Hoare triples independently as
separate lemmas we can use the \<open>LEMMA\<close> annotation to plug together the
lemmas. It
inserts the lemma in the same fashion as the specification annotation.
\<close>
lemma (in vars) foo_lemma:
"\<forall>n m. \<Gamma>\<turnstile> \<lbrace>\<acute>N = n \<and> \<acute>M = m\<rbrace> \<acute>N :== \<acute>N + 1;; \<acute>M :== \<acute>M + 1
\<lbrace>\<acute>N = n + 1 \<and> \<acute>M = m + 1\<rbrace>"
apply vcg
apply simp
done
lemma (in vars)
"\<Gamma>\<turnstile> \<lbrace>\<acute>N = n \<and> \<acute>M = m\<rbrace>
LEMMA foo_lemma
\<acute>N :== \<acute>N + 1;; \<acute>M :== \<acute>M + 1
END;;
\<acute>N :== \<acute>N + 1
\<lbrace>\<acute>N = n + 2 \<and> \<acute>M = m + 1\<rbrace>"
apply vcg
apply simp
done
lemma (in vars)
"\<Gamma>\<turnstile> \<lbrace>\<acute>N = n \<and> \<acute>M = m\<rbrace>
LEMMA foo_lemma
\<acute>N :== \<acute>N + 1;; \<acute>M :== \<acute>M + 1
END;;
LEMMA foo_lemma
\<acute>N :== \<acute>N + 1;; \<acute>M :== \<acute>M + 1
END
\<lbrace>\<acute>N = n + 2 \<and> \<acute>M = m + 2\<rbrace>"
apply vcg
apply simp
done
lemma (in vars)
"\<Gamma>\<turnstile> \<lbrace>\<acute>N = n \<and> \<acute>M = m\<rbrace>
\<acute>N :== \<acute>N + 1;; \<acute>M :== \<acute>M + 1;;
\<acute>N :== \<acute>N + 1;; \<acute>M :== \<acute>M + 1
\<lbrace>\<acute>N = n + 2 \<and> \<acute>M = m + 2\<rbrace>"
apply (hoare_rule anno=
"LEMMA foo_lemma
\<acute>N :== \<acute>N + 1;; \<acute>M :== \<acute>M + 1
END;;
LEMMA foo_lemma
\<acute>N :== \<acute>N + 1;; \<acute>M :== \<acute>M + 1
END"
in HoarePartial.annotate_normI)
apply vcg
apply simp
done
subsubsection \<open>Total Correctness of Nested Loops\<close>
text \<open>
When proving termination of nested loops it is sometimes necessary to express that
the loop variable of the outer loop is not modified in the inner loop. To express this
one has to fix the value of the outer loop variable before the inner loop and use this value
in the invariant of the inner loop. This can be achieved by surrounding the inner while loop
with an \<open>ANNO\<close> specification as explained previously. However, this
leads to repeating the invariant of the inner loop three times: in the invariant itself and
in the the pre- and postcondition of the \<open>ANNO\<close> specification. Moreover one has
to deal with the additional subgoal introduced by \<open>ANNO\<close> that expresses how
the pre- and postcondition is connected to the invariant. To avoid this extra specification
and verification work, we introduce an variant of the annotated while-loop, where one can
introduce logical variables by \<open>FIX\<close>. As for the \<open>ANNO\<close> specification
multiple logical variables can be introduced via a tuple (\<open>FIX (a,b,c).\<close>).
The Hoare logic rule for the augmented while-loop is a mixture of the invariant rule for
loops and the consequence rule for \<open>ANNO\<close>:
\begin{center}
@{thm [mode=Rule,mode=ParenStmt] HoareTotal.WhileAnnoFix' [no_vars]}
\end{center}
The first premise expresses that the precondition implies the invariant and that
the invariant together with the negated loop condition implies the postcondition. Since
both implications may depend on the choice of the auxiliary variable @{term "Z"} these two
implications are expressed in a single premise and not in two of them as for the usual while
rule. The second premise is the preservation of the invariant by the loop body. And the third
premise is the side-condition that the computational part of the body does not depend on
the auxiliary variable. Finally the last premise is the well-foundedness of the variant.
The last two premises are usually discharged automatically by the verification condition
generator. Hence usually two subgoals remain for the user, stemming from the first two
premises.
The following example illustrates the usage of this rule. The outer loop increments the
loop variable @{term "M"} while the inner loop increments @{term "N"}. To discharge the
proof obligation for the termination of the outer loop, we need to know that the inner loop
does not mess around with @{term "M"}. This is expressed by introducing the logical variable
@{term "m"} and fixing the value of @{term "M"} to it.
\<close>
lemma (in vars)
"\<Gamma>\<turnstile>\<^sub>t \<lbrace>\<acute>M=0 \<and> \<acute>N=0\<rbrace>
WHILE (\<acute>M < i)
INV \<lbrace>\<acute>M \<le> i \<and> (\<acute>M \<noteq> 0 \<longrightarrow> \<acute>N = j) \<and> \<acute>N \<le> j\<rbrace>
VAR MEASURE (i - \<acute>M)
DO
\<acute>N :== 0;;
WHILE (\<acute>N < j)
FIX m.
INV \<lbrace>\<acute>M=m \<and> \<acute>N \<le> j\<rbrace>
VAR MEASURE (j - \<acute>N)
DO
\<acute>N :== \<acute>N + 1
OD;;
\<acute>M :== \<acute>M + 1
OD
\<lbrace>\<acute>M=i \<and> (\<acute>M\<noteq>0 \<longrightarrow> \<acute>N=j)\<rbrace>"
apply vcg
txt \<open>@{subgoals [display]}
The first subgoal is from the precondition to the invariant of the outer loop.
The fourth subgoal is from the invariant together with the negated loop condition
of the outer loop to the postcondition. The subgoals two and three are from the body
of the outer while loop which is mainly the inner while loop. Because we introduce the
logical variable @{term "m"} here, the while Rule described above is used instead of the
ordinary while Rule. That is why we end up with two subgoals for the inner loop. Subgoal
two is from the invariant and the loop condition of the outer loop to the invariant
of the inner loop. And at the same time from the invariant of the inner loop to the
invariant of the outer loop (together with the proof obligation that the measure of the
outer loop decreases). The universal quantified variables @{term "Ma"} and @{term "N"} are
the ``fresh'' state variables introduced for the final state of the inner loop.
The equality @{term "Ma=M"} is the result of the equality \<open>\<acute>M=m\<close> in the inner
invariant. Subgoal three is the preservation of the invariant by the
inner loop body (together with the proof obligation that the measure of
the inner loop decreases).
\<close>
(*<*)
apply (simp)
apply (simp,arith)
apply (simp,arith)
done
(*>*)
subsection \<open>Functional Correctness, Termination and Runtime Faults\<close>
text \<open>
Total correctness of a program with guards conceptually leads to three verification
tasks.
\begin{itemize}
\item functional (partial) correctness
\item absence of runtime faults
\item termination
\end{itemize}
In case of a modifies specification the functional correctness part
can be solved automatically. But the absence of runtime faults and
termination may be non trivial. Fortunately the modifies clause is
usually just a helpful companion of another specification that
expresses the ``real'' functional behaviour. Therefor the task to
prove the absence of runtime faults and termination can be dealt with
during the proof of this functional specification. In most cases the
absence of runtime faults and termination heavily build on the
functional specification parts. So after all there is no reason why
we should again prove the absence of runtime faults and termination
for the modifies clause. Therefor it suffices to have partial
correctness of the modifies clause for a program were all guards are
ignored. This leads to the following pattern:\<close>
procedures foo (N::nat|M::nat)
"\<acute>M :== \<acute>M
\<comment> \<open>think of body with guards instead\<close>"
foo_spec: "\<forall>\<sigma>. \<Gamma>\<turnstile>\<^sub>t (P \<sigma>) \<acute>M :== PROC foo(\<acute>N) (Q \<sigma>)"
foo_modifies: "\<forall>\<sigma>. \<Gamma>\<turnstile>\<^bsub>/UNIV\<^esub> {\<sigma>} \<acute>M :== PROC foo(\<acute>N)
{t. t may_only_modify_globals \<sigma> in []}"
text \<open>
The verification condition generator can solve those modifies clauses automatically
and can use them to simplify calls to \<open>foo\<close> even in the context of total
correctness.
\<close>
subsection \<open>Procedures and Locales \label{sec:Locales}\<close>
text \<open>
Verification of a larger program is organised on the granularity of procedures.
We proof the procedures in a bottom up fashion. Of course you can also always use Isabelle's
dummy proof \<open>sorry\<close> to prototype your formalisation. So you can write the
theory in a bottom up fashion but actually prove the lemmas in any other order.
Here are some explanations of handling of locales. In the examples below, consider
\<open>proc\<^sub>1\<close> and \<open>proc\<^sub>2\<close> to be ``leaf'' procedures, which do not call any
other procedure.
Procedure \<open>proc\<close> directly calls \<open>proc\<^sub>1\<close> and \<open>proc\<^sub>2\<close>.
\isacommand{lemma} (\isacommand{in} \<open>proc\<^sub>1_impl\<close>) \<open>proc\<^sub>1_modifies\<close>:\\
\isacommand{shows} \<open>\<dots>\<close>
After the proof of \<open>proc\<^sub>1_modifies\<close>, the \isacommand{in} directive
stores the lemma in the
locale \<open>proc\<^sub>1_impl\<close>. When we later on include \<open>proc\<^sub>1_impl\<close> or prove
another theorem in locale \<open>proc\<^sub>1_impl\<close> the lemma \<open>proc\<^sub>1_modifies\<close>
will already be available as fact.
\isacommand{lemma} (\isacommand{in} \<open>proc\<^sub>1_impl\<close>) \<open>proc\<^sub>1_spec\<close>:\\
\isacommand{shows} \<open>\<dots>\<close>
\isacommand{lemma} (\isacommand{in} \<open>proc\<^sub>2_impl\<close>) \<open>proc\<^sub>2_modifies\<close>:\\
\isacommand{shows} \<open>\<dots>\<close>
\isacommand{lemma} (\isacommand{in} \<open>proc\<^sub>2_impl\<close>) \<open>proc\<^sub>2_spec\<close>:\\
\isacommand{shows} \<open>\<dots>\<close>
\isacommand{lemma} (\isacommand{in} \<open>proc_impl\<close>) \<open>proc_modifies\<close>:\\
\isacommand{shows} \<open>\<dots>\<close>
Note that we do not explicitly include anything about \<open>proc\<^sub>1\<close> or
\<open>proc\<^sub>2\<close> here. This is handled automatically. When defining
an \<open>impl\<close>-locale it imports all \<open>impl\<close>-locales of procedures that are
called in the body. In case of \<open>proc_impl\<close> this means, that \<open>proc\<^sub>1_impl\<close>
and \<open>proc\<^sub>2_impl\<close> are imported. This has the neat effect that all theorems that
are proven in \<open>proc\<^sub>1_impl\<close> and \<open>proc\<^sub>2_impl\<close> are also present
in \<open>proc_impl\<close>.
\isacommand{lemma} (\isacommand{in} \<open>proc_impl\<close>) \<open>proc_spec\<close>:\\
\isacommand{shows} \<open>\<dots>\<close>
As we have seen in this example you only have to prove a procedure in its own
\<open>impl\<close> locale. You do not have to include any other locale.
\<close>
subsection \<open>Records \label{sec:records}\<close>
text \<open>
Before @{term "statespaces"} where introduced the state was represented as a @{term "record"}.
This is still supported. Compared to the flexibility of statespaces there are some drawbacks
in particular with respect to modularity. Even names of local variables and
parameters are globally visible and records can only be extended in a linear fashion, whereas
statespaces also allow multiple inheritance. The usage of records is quite similar to the usage of statespaces.
We repeat the example of an append function for heap lists.
First we define the global components.
Again the appearance of the prefix `globals' is mandatory. This is the way the syntax layer distinguishes local and global variables.
\<close>
record globals_list =
next_' :: "ref \<Rightarrow> ref"
cont_' :: "ref \<Rightarrow> nat"
text \<open>The local variables also have to be defined as a record before the actual definition
of the procedure. The parent record \<open>state\<close> defines a generic @{term "globals"}
field as a place-holder for the record of global components. In contrast to the
statespace approach there is no single @{term "locals"} slot. The local components are
just added to the record.
\<close>
record 'g list_vars = "'g state" +
p_' :: "ref"
q_' :: "ref"
r_' :: "ref"
root_' :: "ref"
tmp_' :: "ref"
text \<open>Since the parameters and local variables are determined by the record, there are
no type annotations or definitions of local variables while defining a procedure.
\<close>
procedures
append'(p,q|p) =
"IF \<acute>p=Null THEN \<acute>p :== \<acute>q
ELSE \<acute>p \<rightarrow>\<acute>next:== CALL append'(\<acute>p\<rightarrow>\<acute>next,\<acute>q) FI"
text \<open>As in the statespace approach, a locale called \<open>append'_impl\<close> is created.
Note that we do not give any explicit information which global or local state-record to use.
Since the records are already defined we rely on Isabelle's type inference.
Dealing with the locale is analogous to the case with statespaces.
\<close>
lemma (in append'_impl) append'_modifies:
shows
"\<forall>\<sigma>. \<Gamma>\<turnstile> {\<sigma>} \<acute>p :== PROC append'(\<acute>p,\<acute>q)
{t. t may_only_modify_globals \<sigma> in [next]}"
apply (hoare_rule HoarePartial.ProcRec1)
apply (vcg spec=modifies)
done
lemma (in append'_impl) append'_spec:
shows "\<forall>\<sigma> Ps Qs. \<Gamma>\<turnstile>
\<lbrace>\<sigma>. List \<acute>p \<acute>next Ps \<and> List \<acute>q \<acute>next Qs \<and> set Ps \<inter> set Qs = {}\<rbrace>
\<acute>p :== PROC append'(\<acute>p,\<acute>q)
\<lbrace>List \<acute>p \<acute>next (Ps@Qs) \<and> (\<forall>x. x\<notin>set Ps \<longrightarrow> \<acute>next x = \<^bsup>\<sigma>\<^esup>next x)\<rbrace>"
apply (hoare_rule HoarePartial.ProcRec1)
apply vcg
apply fastforce
done
text \<open>
However, in some corner cases the inferred state type in a procedure definition
can be too general which raises problems when attempting to proof a suitable
specifications in the locale.
Consider for example the simple procedure body @{term "\<acute>p :== NULL"} for a procedure
\<open>init\<close>.
\<close>
procedures init (|p) =
"\<acute>p:== Null"
text \<open>
Here Isabelle can only
infer the local variable record. Since no reference to any global variable is
made the type fixed for the global variables (in the locale \<open>init'_impl\<close>) is a
type variable say @{typ "'g"} and not a @{term "globals_list"} record. Any specification
mentioning @{term "next"} or @{term "cont"} restricts the state type and cannot be
added to the locale \<open>init_impl\<close>. Hence we have to restrict the body
@{term "\<acute>p :== NULL"} in the first place by adding a typing annotation:
\<close>
procedures init' (|p) =
"\<acute>p:== Null::(('a globals_list_scheme, 'b) list_vars_scheme, char list, 'c) com"
subsubsection \<open>Extending State Spaces\<close>
text \<open>
The records in Isabelle are
extensible \cite{Nipkow-02-hol,NaraschewskiW-TPHOLs98}. In principle this can be exploited
during verification. The state space can be extended while we we add procedures.
But there is one major drawback:
\begin{itemize}
\item records can only be extended in a linear fashion (there is no multiple inheritance)
\end{itemize}
You can extend both the main state record as well as the record for the global variables.
\<close>
subsubsection \<open>Mapping Variables to Record Fields\<close>
text \<open>
Generally the state space (global and local variables) is flat and all components
are accessible from everywhere. Locality or globality of variables is achieved by
the proper \<open>init\<close> and \<open>return\<close>/\<open>result\<close> functions in procedure
calls. What is the best way to map programming language variables to the state records?
One way is to disambiguate all names, by using the procedure names as prefix or the
structure names for heap components. This leads to long names and lots of
record components. But for local variables this is not necessary, since
variable @{term i} of procedure @{term A} and variable @{term "i"} of procedure @{term B}
can be mapped to the same record component, without any harm, provided they have the
same logical type. Therefor for local variables it is preferable to map them per type. You
only have to distinguish a variable with the same name if they have a different type.
Note that all pointers just have logical type \<open>ref\<close>. So you even do not
have to distinguish between a pointer \<open>p\<close> to a integer and a pointer \<open>p\<close> to
a list.
For global components (global variables and heap structures) you have to disambiguate the
name. But hopefully the field names of structures have different names anyway.
Also note that there is no notion of hiding of a global component by a local one in
the logic. You have to disambiguate global and local names!
As the names of the components show up in the specifications and the
proof obligations, names are even more important as for programming. Try to
find meaningful and short names, to avoid cluttering up your reasoning.
\<close>
(*<*)
text \<open>
in locales, includes, spec or impl?
Names: per type not per procedure\<dots>
downgrading total to partial\<dots>
\<close>
(*>*)
text \<open>\<close>
(*<*)
end
(*>*)
|
module New.Correctness where
open import Function hiding (const)
open import New.Lang
open import New.Changes
open import New.Derive
open import New.LangChanges
open import New.LangOps
open import New.FunctionLemmas
open import New.Unused
⟦Γ≼ΔΓ⟧ : ∀ {Γ} (ρ : ⟦ Γ ⟧Context) (dρ : ChΓ Γ) → validΓ ρ dρ →
ρ ≡ ⟦ Γ≼ΔΓ ⟧≼ dρ
⟦Γ≼ΔΓ⟧ ∅ ∅ tt = refl
⟦Γ≼ΔΓ⟧ (v • ρ) (dv • .v • dρ) (vdv , refl , ρdρ) = cong₂ _•_ refl (⟦Γ≼ΔΓ⟧ ρ dρ ρdρ)
fit-sound : ∀ {Γ τ} → (t : Term Γ τ) →
(ρ : ⟦ Γ ⟧Context) (dρ : ChΓ Γ) → validΓ ρ dρ →
⟦ t ⟧Term ρ ≡ ⟦ fit t ⟧Term dρ
fit-sound t ρ dρ ρdρ = trans
(cong ⟦ t ⟧Term (⟦Γ≼ΔΓ⟧ ρ dρ ρdρ))
(sym (weaken-sound t _))
correctDeriveConst : ∀ {τ} (c : Const τ) → ⟦ c ⟧Const ≡ ⟦ c ⟧Const ⊕ (⟦_⟧ΔConst c)
correctDeriveConst (lit n) = sym (right-id-int n)
correctDeriveConst plus = ext (λ m → ext (lemma m))
where
lemma : ∀ m n → m + n ≡ m + n + (m + - m + (n + - n))
lemma m n rewrite right-inv-int m | right-inv-int n | right-id-int (m + n) = refl
correctDeriveConst minus = ext (λ m → ext (λ n → lemma m n))
where
lemma : ∀ m n → m - n ≡ m - n + (m + - m - (n + - n))
lemma m n rewrite right-inv-int m | right-inv-int n | right-id-int (m - n) = refl
correctDeriveConst cons = ext (λ v1 → ext (λ v2 → sym (update-nil (v1 , v2))))
correctDeriveConst fst = ext (λ vp → sym (update-nil (proj₁ vp)))
correctDeriveConst snd = ext (λ vp → sym (update-nil (proj₂ vp)))
correctDeriveConst linj = ext (λ va → sym (cong inj₁ (update-nil va)))
correctDeriveConst rinj = ext (λ vb → sym (cong inj₂ (update-nil vb)))
correctDeriveConst (match {t1} {t2} {t3}) = ext³ lemma
where
lemma : ∀ s f g →
⟦ match {t1} {t2} {t3} ⟧Const s f g ≡
(⟦ match ⟧Const ⊕ ⟦ match ⟧ΔConst) s f g
lemma (inj₁ x) f g rewrite update-nil x | update-nil (f x) = refl
lemma (inj₂ y) f g rewrite update-nil y | update-nil (g y) = refl
validDeriveConst : ∀ {τ} (c : Const τ) → valid ⟦ c ⟧Const (⟦_⟧ΔConst c)
validDeriveConst (lit n) = tt
validDeriveConst {τ = t1 ⇒ t2 ⇒ pair .t1 .t2} cons = binary-valid (λ a da ada b db bdb → (ada , bdb)) dcons-eq
where
open BinaryValid ⟦ cons {t1} {t2} ⟧Const (⟦ cons ⟧ΔConst)
dcons-eq : binary-valid-eq-hp
dcons-eq a da ada b db bdb rewrite update-nil (a ⊕ da) | update-nil (b ⊕ db) = refl
validDeriveConst fst (a , b) (da , db) (ada , bdb) = ada , update-nil (a ⊕ da)
validDeriveConst snd (a , b) (da , db) (ada , bdb) = bdb , update-nil (b ⊕ db)
validDeriveConst plus = binary-valid (λ a da ada b db bdb → tt) dplus-eq
where
open BinaryValid ⟦ plus ⟧Const (⟦ plus ⟧ΔConst)
dplus-eq : binary-valid-eq-hp
dplus-eq a da ada b db bdb rewrite right-inv-int (a + da) | right-inv-int (b + db) | right-id-int (a + da + (b + db)) = mn·pq=mp·nq {a} {da} {b} {db}
validDeriveConst minus = binary-valid (λ a da ada b db bdb → tt) dminus-eq
where
open BinaryValid ⟦ minus ⟧Const (⟦ minus ⟧ΔConst)
dminus-eq : binary-valid-eq-hp
dminus-eq a da ada b db bdb rewrite right-inv-int (a + da) | right-inv-int (b + db) | right-id-int (a + da - (b + db)) | sym (-m·-n=-mn {b} {db}) = mn·pq=mp·nq {a} {da} { - b} { - db}
validDeriveConst linj a da ada = sv₁ a da ada , cong inj₁ (update-nil (a ⊕ da))
validDeriveConst rinj b db bdb = sv₂ b db bdb , cong inj₂ (update-nil (b ⊕ db))
validDeriveConst (match {t1} {t2} {t3}) =
ternary-valid dmatch-valid dmatch-eq
where
open TernaryValid {{chAlgt (sum t1 t2)}} {{chAlgt (t1 ⇒ t3)}} {{chAlgt (t2 ⇒ t3)}} {{chAlgt t3}} ⟦ match ⟧Const (⟦ match ⟧ΔConst)
dmatch-valid : ternary-valid-preserve-hp
dmatch-valid .(inj₁ a) .(inj₁ (inj₁ da)) (sv₁ a da ada) f df fdf g dg gdg = proj₁ (fdf a da ada)
dmatch-valid .(inj₂ b) .(inj₁ (inj₂ db)) (sv₂ b db bdb) f df fdf g dg gdg = proj₁ (gdg b db bdb)
dmatch-valid .(inj₁ a1) .(inj₂ (inj₂ b2)) (svrp₁ a1 b2) f df fdf g dg gdg
rewrite changeMatchSem-lem1 f df g dg a1 b2
= ⊝-valid (f a1) (g b2 ⊕ dg b2 (nil b2))
dmatch-valid .(inj₂ b1) .(inj₂ (inj₁ a2)) (svrp₂ b1 a2) f df fdf g dg gdg
rewrite changeMatchSem-lem2 f df g dg b1 a2
= ⊝-valid (g b1) (f a2 ⊕ df a2 (nil a2))
dmatch-eq : ternary-valid-eq-hp
dmatch-eq .(inj₁ a) .(inj₁ (inj₁ da)) (sv₁ a da ada) f df fdf g dg gdg
rewrite update-nil (a ⊕ da)
| update-nil (f (a ⊕ da) ⊕ df (a ⊕ da) (nil (a ⊕ da))) = proj₂ (fdf a da ada)
dmatch-eq .(inj₂ b) .(inj₁ (inj₂ db)) (sv₂ b db bdb) f df fdf g dg gdg
rewrite update-nil (b ⊕ db)
| update-nil (g (b ⊕ db) ⊕ dg (b ⊕ db) (nil (b ⊕ db))) = proj₂ (gdg b db bdb)
dmatch-eq .(inj₁ a1) .(inj₂ (inj₂ b2)) (svrp₁ a1 b2) f df fdf g dg gdg
rewrite changeMatchSem-lem1 f df g dg a1 b2
| update-nil b2
| update-diff (g b2 ⊕ dg b2 (nil b2)) (f a1)
| update-nil (g b2 ⊕ dg b2 (nil b2))
= refl
dmatch-eq .(inj₂ b1) .(inj₂ (inj₁ a2)) (svrp₂ b1 a2) f df fdf g dg gdg
rewrite changeMatchSem-lem2 f df g dg b1 a2
| update-nil a2
| update-diff (f a2 ⊕ df a2 (nil a2)) (g b1)
| update-nil (f a2 ⊕ df a2 (nil a2))
= refl
validDeriveVar : ∀ {Γ τ} → (x : Var Γ τ) →
(ρ : ⟦ Γ ⟧Context) (dρ : ChΓ Γ) →
validΓ ρ dρ → valid (⟦ x ⟧Var ρ) (⟦ x ⟧ΔVar ρ dρ)
validDeriveVar this (v • ρ) (dv • .v • dρ) (vdv , refl , ρdρ) = vdv
validDeriveVar (that x) (v • ρ) (dv • .v • dρ) (vdv , refl , ρdρ) = validDeriveVar x ρ dρ ρdρ
correctDeriveVar : ∀ {Γ τ} → (x : Var Γ τ) →
IsDerivative ⟦ x ⟧Var (⟦ x ⟧ΔVar)
correctDeriveVar this (v • ρ) (dv • v' • dρ) ρdρ = refl
correctDeriveVar (that x) (v • ρ) (dv • .v • dρ) (vdv , refl , ρdρ) = correctDeriveVar x ρ dρ ρdρ
validDerive : ∀ {Γ τ} → (t : Term Γ τ) →
(ρ : ⟦ Γ ⟧Context) (dρ : ChΓ Γ) → validΓ ρ dρ →
valid (⟦ t ⟧Term ρ) (⟦ t ⟧ΔTerm ρ dρ)
correctDerive : ∀ {Γ τ} → (t : Term Γ τ) →
IsDerivative ⟦ t ⟧Term ⟦ t ⟧ΔTerm
correctDerive (const c) ρ dρ ρdρ rewrite ⟦ c ⟧ΔConst-rewrite ρ dρ = correctDeriveConst c
correctDerive (var x) ρ dρ ρdρ = correctDeriveVar x ρ dρ ρdρ
correctDerive (app s t) ρ dρ ρdρ rewrite sym (fit-sound t ρ dρ ρdρ) =
let
open ≡-Reasoning
a0 = ⟦ t ⟧Term ρ
da0 = ⟦ derive t ⟧Term dρ
a0da0 = validDerive t ρ dρ ρdρ
in
begin
⟦ s ⟧Term (ρ ⊕ dρ) (⟦ t ⟧Term (ρ ⊕ dρ))
≡⟨ correctDerive s ρ dρ ρdρ ⟨$⟩ correctDerive t ρ dρ ρdρ ⟩
(⟦ s ⟧Term ρ ⊕ ⟦ s ⟧ΔTerm ρ dρ) (⟦ t ⟧Term ρ ⊕ ⟦ t ⟧ΔTerm ρ dρ)
≡⟨ proj₂ (validDerive s ρ dρ ρdρ a0 da0 a0da0) ⟩
⟦ s ⟧Term ρ (⟦ t ⟧Term ρ) ⊕ (⟦ s ⟧ΔTerm ρ dρ) (⟦ t ⟧Term ρ) (⟦ t ⟧ΔTerm ρ dρ)
∎
where
open import Theorem.CongApp
correctDerive (abs t) ρ dρ ρdρ = ext $ λ a →
let
open ≡-Reasoning
ρ1 = a • ρ
dρ1 = nil a • a • dρ
ρ1dρ1 = nil-valid a , refl , ρdρ
in
-- equal-future-expand-derivative ⟦ t ⟧Term ⟦ t ⟧ΔTerm (correctDerive t)
-- ρ1 dρ1 ρ1dρ1
-- (a • (ρ ⊕ dρ))
-- (cong (_• ρ ⊕ dρ) (sym (update-nil a)))
begin
⟦ t ⟧Term (a • ρ ⊕ dρ)
≡⟨ cong (λ a′ → ⟦ t ⟧Term (a′ • ρ ⊕ dρ)) (sym (update-nil a)) ⟩
⟦ t ⟧Term (ρ1 ⊕ dρ1)
≡⟨ correctDerive t ρ1 dρ1 ρ1dρ1 ⟩
⟦ t ⟧Term ρ1 ⊕ ⟦ t ⟧ΔTerm ρ1 dρ1
∎
validDerive (app s t) ρ dρ ρdρ =
let
f = ⟦ s ⟧Term ρ
df = ⟦ derive s ⟧Term dρ
v = ⟦ t ⟧Term ρ
dv = ⟦ derive t ⟧Term dρ
vdv = validDerive t ρ dρ ρdρ
fdf = validDerive s ρ dρ ρdρ
fvdfv = proj₁ (fdf v dv vdv)
in subst (λ v′ → valid (f v) (df v′ dv)) (fit-sound t ρ dρ ρdρ) fvdfv
validDerive (abs t) ρ dρ ρdρ =
λ a da ada →
let
ρ1 = a ⊕ da • ρ
dρ1 = nil (a ⊕ da) • (a ⊕ da) • dρ
ρ2 = a • ρ
dρ2 = da • a • dρ
ρ1dρ1 = nil-valid (a ⊕ da) , refl , ρdρ
ρ2dρ2 = ada , refl , ρdρ
rdr = validDerive t ρ2 dρ2 ρ2dρ2
open ≡-Reasoning
in
rdr ,
equal-future-derivative ⟦ t ⟧Term ⟦ t ⟧ΔTerm (correctDerive t)
ρ1 dρ1 ρ1dρ1
ρ2 dρ2 ρ2dρ2
(cong (λ a′ → (a′ • ρ ⊕ dρ)) (update-nil (a ⊕ da)))
validDerive (var x) ρ dρ ρdρ = validDeriveVar x ρ dρ ρdρ
validDerive (const c) ρ dρ ρdρ rewrite ⟦ c ⟧ΔConst-rewrite ρ dρ = validDeriveConst c
|
About existT.
Print existT.
Print Implicit existT.
Print eq_refl.
About eq_refl.
Print Implicit eq_refl.
Print Nat.add.
About Nat.add.
Print Implicit Nat.add.
About plus_n_O.
Arguments le_S {n} [m] _.
Print le_S.
About comparison.
Print comparison.
Definition foo := forall x, x = 0.
Parameter bar : foo.
Arguments bar [x].
About bar.
Print bar.
About Peano. (* Module *)
About sym_eq. (* Notation *)
Arguments eq_refl {A} {x}, {A} x.
Print eq_refl.
Definition newdef := fun x:nat => x.
Goal forall n:nat, n <> newdef n -> newdef n <> n -> False.
intros n h h'.
About n. (* search hypothesis *)
About h. (* search hypothesis *)
Abort.
Goal forall n:nat, let g := newdef in n <> newdef n -> newdef n <> n -> False.
intros n g h h'.
About g. (* search hypothesis *)
About h. (* search hypothesis *)
Abort.
|
main@global(128) {
#// printf ("Hello World\n");
: nop
: mov eax, 1
: push eax
: int 0x80
}
|
// $Id$
//
// Copyright (C) 2003-2013 Greg Landrum and Rational Discovery LLC
//
// @@ All Rights Reserved @@
// This file is part of the RDKit.
// The contents are covered by the terms of the BSD license
// which is included in the file license.txt, found at the root
// of the RDKit source tree.
//
#include <GraphMol/RDKitBase.h>
#include <GraphMol/SmilesParse/SmilesParse.h>
#include <GraphMol/SmilesParse/SmilesWrite.h>
#include <GraphMol/SmilesParse/SmartsWrite.h>
#include <GraphMol/Subgraphs/Subgraphs.h>
#include <GraphMol/Subgraphs/SubgraphUtils.h>
#include <boost/foreach.hpp>
#include <iostream>
using namespace std;
using namespace RDKit;
void test1() {
std::cout << "-----------------------\n Test1: pathToSubmol" << std::endl;
{
std::string smiles = "CC1CC1";
RWMol *mol = SmilesToMol(smiles);
TEST_ASSERT(mol);
PATH_LIST sgs;
sgs = findAllSubgraphsOfLengthN(*mol, 3, false, 0);
TEST_ASSERT(sgs.size() == 3);
BOOST_FOREACH (PATH_TYPE tmp, sgs) {
TEST_ASSERT(tmp[0] == 0);
TEST_ASSERT(tmp.size() == 3);
ROMol *frag = Subgraphs::pathToSubmol(*mol, tmp, false);
smiles = MolToSmiles(*frag, true, false, 0, false);
if (tmp[1] == 1) {
if (tmp[2] == 2) {
TEST_ASSERT(smiles == "CCCC");
} else if (tmp[2] == 3) {
TEST_ASSERT(smiles == "CC(C)C");
} else {
TEST_ASSERT(0);
}
} else if (tmp[1] == 3) {
if (tmp[2] == 2) {
TEST_ASSERT(smiles == "CCCC");
} else if (tmp[2] == 1) {
TEST_ASSERT(smiles == "CC(C)C");
} else {
TEST_ASSERT(0);
}
} else {
TEST_ASSERT(0);
}
delete frag;
}
delete mol;
}
std::cout << "Finished" << std::endl;
}
void test2() {
std::cout << "-----------------------\n Test2: Atom Environments"
<< std::endl;
{
std::string smiles = "CC1CC1";
RWMol *mol = SmilesToMol(smiles);
TEST_ASSERT(mol);
PATH_TYPE pth = findAtomEnvironmentOfRadiusN(*mol, 1, 0);
TEST_ASSERT(pth.size() == 1);
TEST_ASSERT(pth[0] == 0);
pth = findAtomEnvironmentOfRadiusN(*mol, 2, 0);
TEST_ASSERT(pth.size() == 3);
TEST_ASSERT(pth[0] == 0);
pth = findAtomEnvironmentOfRadiusN(*mol, 3, 0);
TEST_ASSERT(pth.size() == 4);
TEST_ASSERT(pth[0] == 0);
pth = findAtomEnvironmentOfRadiusN(*mol, 4, 0);
TEST_ASSERT(pth.size() == 0);
pth = findAtomEnvironmentOfRadiusN(*mol, 1, 1);
TEST_ASSERT(pth.size() == 3);
pth = findAtomEnvironmentOfRadiusN(*mol, 2, 1);
TEST_ASSERT(pth.size() == 4);
pth = findAtomEnvironmentOfRadiusN(*mol, 3, 1);
TEST_ASSERT(pth.size() == 0);
delete mol;
}
{
std::string smiles = "CC1CC1";
RWMol *mol = SmilesToMol(smiles);
TEST_ASSERT(mol);
ROMol *mH = MolOps::addHs(static_cast<const ROMol &>(*mol));
PATH_TYPE pth = findAtomEnvironmentOfRadiusN(*mH, 1, 0);
TEST_ASSERT(pth.size() == 1);
TEST_ASSERT(pth[0] == 0);
pth = findAtomEnvironmentOfRadiusN(*mH, 1, 0, true);
TEST_ASSERT(pth.size() == 4);
delete mol;
delete mH;
}
{
std::string smiles = "O=C(O)CCCC=CC(C1C(O)CC(O)C1(C=CC(O)CCCCC))";
RWMol *mol = SmilesToMol(smiles);
TEST_ASSERT(mol);
smiles = MolToSmiles(*mol);
PATH_TYPE pth = findAtomEnvironmentOfRadiusN(*mol, 2, 9);
TEST_ASSERT(pth.size() == 8);
ROMol *frag = Subgraphs::pathToSubmol(*mol, pth, false);
smiles = MolToSmiles(*frag, true, false, 0, false);
TEST_ASSERT(smiles == "C(C(C(O)C)C(C)C)C");
delete frag;
delete mol;
}
std::cout << "Finished" << std::endl;
}
void testGithubIssue103() {
std::cout << "-----------------------\n Testing github Issue103: "
"stereochemistry and pathToSubmol" << std::endl;
{
std::string smiles = "O=C(O)C(=O)C[C@@]1(C(=O)O)C=C[C@H](O)C=C1";
RWMol *mol = SmilesToMol(smiles);
TEST_ASSERT(mol);
PATH_TYPE pth = findAtomEnvironmentOfRadiusN(*mol, 2, 12);
TEST_ASSERT(pth.size() == 5);
ROMol *frag = Subgraphs::pathToSubmol(*mol, pth, false);
smiles = MolToSmiles(*frag, true);
TEST_ASSERT(smiles == "C=CC(O)C=C");
delete frag;
delete mol;
}
{
std::string smiles = "O=C(O)C(=O)C[C@@]1(C(=O)O)C=C[C@H](O)C=C1";
RWMol *mol = SmilesToMol(smiles);
TEST_ASSERT(mol);
PATH_TYPE pth = findAtomEnvironmentOfRadiusN(*mol, 2, 12);
TEST_ASSERT(pth.size() == 5);
ROMol *frag = Subgraphs::pathToSubmol(*mol, pth, false);
smiles = MolToSmarts(*frag);
TEST_ASSERT(smiles == "[#6](-[#6H](-[#8])-[#6]=[#6])=[#6]");
delete frag;
delete mol;
}
{
std::string smiles = "O=C(O)C(=O)C[C@@]1(C(=O)O)C=C[C@H](O)C=C1";
RWMol *mol = SmilesToMol(smiles);
TEST_ASSERT(mol);
PATH_TYPE pth = findAtomEnvironmentOfRadiusN(*mol, 2, 12);
TEST_ASSERT(pth.size() == 5);
ROMol *frag = Subgraphs::pathToSubmol(*mol, pth, true);
smiles = MolToSmarts(*frag);
TEST_ASSERT(smiles == "[#6](-[#6](-[#8])-[#6]=[#6])=[#6]");
delete frag;
delete mol;
}
std::cout << "Finished" << std::endl;
}
// -------------------------------------------------------------------
int main() {
test1();
test2();
testGithubIssue103();
return 0;
}
|
subroutine test1
character*8 c
character*2 d, f
dimension d(2), f(2)
character*4 e
equivalence (c(1:1), d(1)(2:)), (c(3:5), e(2:4))
equivalence (c(6:6), f(2)(:))
d(1)='AB'
c='abcdefgh'
if (c.ne.'abcdefgh'.or.d(1).ne.'Aa'.or.d(2).ne.'bc') call abort
if (e.ne.'bcde'.or.f(1).ne.'de'.or.f(2).ne.'fg') call abort
end subroutine test1
subroutine test2
equivalence (c(1:1), d(1)(2:2)), (c(3:5), e(2:4))
equivalence (c(6:6), f(2)(1:))
character*8 c
character*2 d, f
dimension d(2), f(2)
character*4 e
d(1)='AB'
c='abcdefgh'
if (c.ne.'abcdefgh'.or.d(1).ne.'Aa'.or.d(2).ne.'bc') call abort
if (e.ne.'bcde'.or.f(1).ne.'de'.or.f(2).ne.'fg') call abort
end subroutine test2
subroutine test3
character*8 c
character*2 d, f
character*4 e
equivalence (c(1:1), d(1)(2:)), (c(3:5), e(2:4))
equivalence (c(6:6), f(2)(:1))
dimension d(2), f(2)
d(1)='AB'
c='abcdefgh'
if (c.ne.'abcdefgh'.or.d(1).ne.'Aa'.or.d(2).ne.'bc') call abort
if (e.ne.'bcde'.or.f(1).ne.'de'.or.f(2).ne.'fg') call abort
end subroutine test3
subroutine test4
dimension d(2), f(2)
equivalence (c(1:1), d(1)(2:2)), (c(3:5), e(2:4))
equivalence (c(6:6), f(2)(1:2))
character*8 c
character*2 d, f
character*4 e
d(1)='AB'
c='abcdefgh'
if (c.ne.'abcdefgh'.or.d(1).ne.'Aa'.or.d(2).ne.'bc') call abort
if (e.ne.'bcde'.or.f(1).ne.'de'.or.f(2).ne.'fg') call abort
end subroutine test4
program main
call test1
call test2
call test3
call test4
end program main
|
module PackageCompiler
using Libdl, SnoopCompile
Sys.iswindows() && using WinRPM
include("compiler_flags.jl")
include("static_julia.jl")
include("api.jl")
include("snooping.jl")
include("system_image.jl")
const sysimage_binaries = ("sys.$(Libdl.dlext)",)
function copy_system_image(src, dest, ignore_missing = false)
for file in sysimage_binaries
# backup
srcfile = joinpath(src, file)
destfile = joinpath(dest, file)
if !isfile(srcfile)
ignore_missing && continue
error("No file: $srcfile")
end
if isfile(destfile)
if isfile(destfile * ".backup")
rm(destfile * ".backup", force = true)
end
mv(destfile, destfile * ".backup", force = true)
end
@info "Copying system image: $srcfile to $destfile"
cp(srcfile, destfile, force = true)
end
end
julia_cpu_target(x) = error("CPU target needs to be a string or `nothing`")
julia_cpu_target(x::String) = x # TODO: match against available targets
function julia_cpu_target(::Nothing)
replace(Base.julia_cmd().exec[2], "-C" => "")
end
"""
Reverts a forced compilation of the system image.
This will restore any previously backed up system image files, or
build a new, clean system image.
"""
function revert(debug = false)
syspath = default_sysimg_path(debug)
sysimg_backup = dirname(get_backup!(debug))
copy_system_image(sysimg_backup, syspath)
end
function get_root_dir(path)
path, name = splitdir(path)
if isempty(name)
return splitdir(path)[2]
else
name
end
end
function sysimg_folder(files...)
base_path = normpath(abspath(joinpath(@__DIR__, "..", "sysimg")))
isdir(base_path) || mkpath(base_path)
normpath(abspath(joinpath(base_path, files...)))
end
function sysimgbackup_folder(files...)
backup = sysimg_folder("backup")
isdir(backup) || mkpath(backup)
sysimg_folder("backup", files...)
end
function package_folder(package...)
packages = normpath(abspath(joinpath(@__DIR__, "..", "packages")))
isdir(packages) || mkpath(packages)
normpath(abspath(joinpath(packages, package...)))
end
"""
compile_package(packages...; kw_args...)
with packages being either a string naming a package, or a tuple `(package_name, precompile_file)`.
If no precompile file is given, it will use the packages `runtests.jl`, which is a good canditate
for figuring out what functions to compile!
"""
function compile_package(packages...; kw_args...)
args = map(packages) do package
# If no explicit path to a seperate precompile file, use runtests
isa(package, String) && return (package, "test/runtests.jl")
isa(package, Tuple{String, String}) && return package
error("Unrecognized package. Use `packagename::String`, or `(packagename::String, rel_path_to_testfile::String)`. Found: `$package`")
end
compile_package(args...; kw_args...)
end
"""
compile_package(packages::Tuple{String, String}...; force = false, reuse = false, debug = false, cpu_target = nothing)
Compile a list of packages. Each package comes as a tuple of `(package_name, precompile_file)`
where the precompile file should contain all function calls, that should get compiled into the system image.
Usually the `runtests.jl` file is a good candidate, since it should run all important functions of a package.
"""
function compile_package(packages::Tuple{String, String}...; force = false, reuse = false, debug = false, cpu_target = nothing)
userimg = sysimg_folder("precompile.jl")
if !reuse
snoop_userimg(userimg, packages...)
end
!isfile(userimg) && reuse && error("Nothing to reuse. Please run `compile_package(reuse = true)`")
image_path = sysimg_folder()
build_sysimg(image_path, userimg, cpu_target=cpu_target)
imgfile = joinpath(image_path, "sys.$(Libdl.dlext)")
syspath = joinpath(default_sysimg_path(debug), "sys.$(Libdl.dlext)")
if force
try
backup = syspath * ".packagecompiler_backup"
isfile(backup) || mv(syspath, backup)
cp(imgfile, syspath)
@info """Replaced system image successfully. Next start of julia will load the newly compiled system image.
If you encounter any errors with the new julia image, try `PackageCompiler.revert([debug = false])`."""
catch e
@warn "An error occured while replacing sysimg files:" error=e
@info "Recovering old system image from backup"
# if any file is missing in default system image, revert!
if !isfile(syspath)
@info "$syspath missing. Reverting!"
revert(debug)
end
end
else
@info """Not replacing system image.
You can start julia with $(`julia -J $imgfile`) at a posix shell to load the compiled files."""
end
imgfile
end
function __init__()
if Base.julia_cmd().exec[2] != "-Cnative"
@warn "Your Julia system image is not compiled natively for this CPU architecture.\n" *
"Please run `PackageCompiler.force_native_image!()` for optimal Julia performance."
end
end
export compile_package, revert, force_native_image!, executable_ext, build_executable, build_shared_lib, static_julia
end # module
|
module adios2_adios
use adios2_adios_init
implicit none
contains
subroutine adios2_declare_io(io, adios, io_name, ierr)
integer(kind=8), intent(out) :: io
integer(kind=8), intent(in) :: adios
character*(*), intent(in) :: io_name
integer, intent(out) :: ierr
call adios2_declare_io_f2c(io, adios, TRIM(ADJUSTL(io_name))//char(0), &
& ierr)
end
subroutine adios2_finalize(adios, ierr)
integer(kind=8), intent(in) :: adios
integer, intent(out) :: ierr
call adios2_finalize_f2c(adios, ierr)
end
end module
|
Personalize your space with cyber monday gym membership in a wide variety of style. We have all of the little details you need to turn your house into a home with style. The cyber monday gym membership with good price can always be found at Beddinginn.com. For you who love great deals and gorgeous cyber monday gym membership, Beddinginn sell lovely things for even lovelier prices. |
% $Id: WB08_Fig_2.m,v 1.1.1.1 2008/05/09 21:34:52 myself Exp $
%
% This prepares data for Figure 2 which is created with the corresponding
% shell script using GMT. It also makes a plot in Matlab/Octave.
%
% Wessel, P. and J. M. Becker, 2008, Interpolation using a
% generalized Green's function for a spherical surface spline
% in tension, Geophys. J. Int., doi:10.1111/j.1365-246X.2008.03829.x
%
% Replicate Parker and then find the tension that minimizes the misfit at
% his 8 extra validation stations.
load mag_obs_1990.d
loni = mag_obs_1990(:,1);
lati = mag_obs_1990(:,2);
zi = mag_obs_1990(:,3);
d=1;
% Set global 1x1 grid output coordinates
[X Y] = meshgrid (0:d:360, 0:d:90);
% First Parker's solution (p = 0)
Z = sphsplinet (loni, lati, zi, X, Y);
figure(1); clf
subplot (2,1,1)
contour (X, Y, Z)
drawnow
A = [X(:) Y(:) Z(:)];
save Fig_2_p0.d A -ascii -tabs
%Then used the wrong Oslo longitude to recreate Parker's figure
k = find (loni == 10.45)
loni(k) = 104.5;
Z = sphsplinet (loni, lati, zi, X, Y);
subplot (2,1,2)
contour (X, Y, Z)
A = [X(:) Y(:) Z(:)];
save Fig_2_orig.d A -ascii -tabs
|
import counit_ker_abelian
import pq_induction_principles
import group_theory.abelianization
universe u
section commutator_counit_kernel
variables {G : Type u} [group G]
theorem commutator_intersect_counit_kernel_trivial (x y : pq_group G) (hxy : counit (x * y * x⁻¹ * y⁻¹) = 1) : x * y * x⁻¹ * y⁻¹ = 1 :=
begin
simp only [counit_of, monoid_hom.map_mul, monoid_hom.map_mul_inv] at hxy,
rw ←rhd_def_group,
rw inner_aut_eq,
generalize ha : counit x = a,
rw ha at hxy,
clear ha x,
suffices : of a ▷ y = y,
{
rw this,
simp only [mul_right_inv],
},
have hya : a ▷ (counit y) = counit y,
{
rw ←rhd_def_group at hxy,
exact mul_inv_eq_one.mp hxy,
},
clear hxy,
suffices : (of a) ▷ (y * (of (counit y))⁻¹) = y * (of (counit y))⁻¹,
{
rw rhd_mul at this,
suffices this2 : (of a▷(of (counit y))⁻¹) = (of (counit y))⁻¹,
{
rw this2 at this,
simp only [mul_left_inj] at this,
exact this,
},
clear this,
rw of_inv,
rw rhd_of_eq_of_rhd,
apply congr_arg,
rw ←power_quandle.pow_rhd,
rw hya,
},
suffices : (y * (of (counit y))⁻¹) ▷ (of a) = of a,
{
generalize ha1 : of a = a1,
generalize ha2 : (y * (of (counit y))⁻¹) = a2,
rw ha1 at this,
rw ha2 at this,
clear ha1 ha2 hya a y,
rw rhd_def_group at *,
rw ←center_reformulate at *,
symmetry,
exact this,
},
rw inner_aut_eq,
simp only [counit_of, mul_right_inv, monoid_hom.map_mul_inv],
rw of_one,
rw power_quandle.one_rhd,
end
end commutator_counit_kernel
|
import Language.Reflection
%language ElabReflection
logPrims : Elab a
logPrims
= do ns <- getType `{ (++) }
traverse_ (\ (n, ty) =>
do logMsg "" 0 ("Name: " ++ show n)
logTerm "" 0 "Type" ty) ns
fail "Not really trying"
logDataCons : Elab a
logDataCons
= do [(n, _)] <- getType `{ Nat }
| _ => fail "Ambiguous name"
logMsg "" 0 ("Resolved name: " ++ show n)
logMsg "" 0 ("Constructors: " ++ show !(getCons n))
fail "Still not trying"
logBad : Elab a
logBad
= do [(n, _)] <- getType `{ DoesntExist }
| [] => fail "Undefined name"
| _ => fail "Ambiguous name"
logMsg "" 0 ("Resolved name: " ++ show n)
logMsg "" 0 ("Constructors: " ++ show !(getCons n))
fail "Still not trying"
-- because the exact sequence number in a gensym depends on
-- the library and compiler internals we need to censor it so the
-- output won't be overly dependent on the exact versions used.
censorDigits : String -> String
censorDigits str = pack $ map (\c => if isDigit c then 'X' else c) (unpack str)
tryGenSym : Elab a
tryGenSym
= do n <- genSym "plus"
ns <- inCurrentNS n
fail $ "failed after generating " ++ censorDigits (show ns)
dummy1 : a
dummy1 = %runElab logPrims
dummy2 : a
dummy2 = %runElab logDataCons
dummy3 : a
dummy3 = %runElab logBad
dummy4 : a
dummy4 = %runElab tryGenSym
|
Require Import ssreflect ssrbool ssrfun seq eqtype fintype.
Set Implicit Arguments.
Unset Strict Implicit.
Unset Printing Implicit Defensive.
Require Import sepcomp. Import SepComp.
Lemma reestablish_locBlocksSrc mu0 mu :
locBlocksSrc (reestablish mu0 mu) = locBlocksSrc mu0.
Proof. by case: mu0; case: mu. Qed.
Lemma reestablish_locBlocksTgt mu0 mu :
locBlocksTgt (reestablish mu0 mu) = locBlocksTgt mu0.
Proof. by case: mu0; case: mu. Qed.
Lemma reestablish_pubBlocksSrc mu0 mu :
pubBlocksSrc (reestablish mu0 mu) = pubBlocksSrc mu0.
Proof. by case: mu0; case: mu. Qed.
Lemma reestablish_pubBlocksTgt mu0 mu :
pubBlocksTgt (reestablish mu0 mu) = pubBlocksTgt mu0.
Proof. by case: mu0; case: mu. Qed.
Lemma reestablish_extBlocksSrc mu0 mu :
extBlocksSrc (reestablish mu0 mu)
= (fun b => if locBlocksSrc mu0 b then false else DomSrc mu b).
Proof. by case: mu0; case: mu. Qed.
Lemma reestablish_extBlocksTgt mu0 mu :
extBlocksTgt (reestablish mu0 mu)
= (fun b => if locBlocksTgt mu0 b then false else DomTgt mu b).
Proof. by case: mu0; case: mu. Qed.
Lemma reestablish_frgnBlocksSrc mu0 mu :
frgnBlocksSrc (reestablish mu0 mu)
= frgnBlocksSrc mu0.
Proof. by case: mu0; case: mu. Qed.
Lemma reestablish_frgnBlocksTgt mu0 mu :
frgnBlocksTgt (reestablish mu0 mu)
= frgnBlocksTgt mu0.
Proof. by case: mu0; case: mu. Qed.
Lemma reestablish_local_of mu0 mu :
local_of (reestablish mu0 mu) = local_of mu0.
Proof. by case: mu0; case: mu. Qed.
Lemma reestablish_extern_of mu0 mu :
extern_of (reestablish mu0 mu)
= (fun b => if locBlocksSrc mu0 b then None
else as_inj mu b).
Proof. by case: mu0; case: mu. Qed.
|
(*
Copyright 2018
Licensed under the Apache License, Version 2.0 (the "License");
you may not use this file except in compliance with the License.
You may obtain a copy of the License at
http://www.apache.org/licenses/LICENSE-2.0
Unless required by applicable law or agreed to in writing, software
distributed under the License is distributed on an "AS IS" BASIS,
WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied.
See the License for the specific language governing permissions and
limitations under the License.
*)
theory task_list_push_back_mem
imports tasks
begin
text \<open>Up to two locales per function in the binary.\<close>
locale task_list_push_back_function = tasks_context +
fixes rsp\<^sub>0 rbp\<^sub>0 a task_list_push_back_ret :: \<open>64 word\<close>
and v\<^sub>0 :: \<open>8 word\<close>
and blocks :: \<open>(nat \<times> 64 word \<times> nat) set\<close>
assumes seps: \<open>seps blocks\<close>
and masters:
\<open>master blocks (a, 1) 0\<close>
\<open>master blocks (rsp\<^sub>0, 8) 1\<close>
\<open>master blocks (rsp\<^sub>0-8, 8) 2\<close>
\<open>master blocks (rsp\<^sub>0-16, 8) 3\<close>
\<open>master blocks (rsp\<^sub>0-24, 8) 4\<close>
and ret_address: \<open>outside task_list_push_back_ret 139 317\<close> \<comment> \<open>Only works for non-recursive functions.\<close>
begin
text \<open>
The Floyd invariant expresses for some locations properties that are invariably true.
Simply expresses that a byte in the memory remains untouched.
\<close>
definition pp_\<Theta> :: \<open>_ \<Rightarrow> _ \<Rightarrow> _ \<Rightarrow> floyd_invar\<close> where
\<open>pp_\<Theta> list task last' \<equiv> [
\<comment> \<open>precondition\<close>
boffset+139 \<mapsto> \<lambda>\<sigma>. regs \<sigma> rsp = rsp\<^sub>0
\<and> regs \<sigma> rbp = rbp\<^sub>0
\<and> regs \<sigma> rdi = list
\<and> regs \<sigma> rsi = task
\<and> \<sigma> \<turnstile> *[list + 8,8] = last'
\<and> \<sigma> \<turnstile> *[rsp\<^sub>0,8] = boffset+task_list_push_back_ret
\<and> \<sigma> \<turnstile> *[a,1] = v\<^sub>0,
boffset+314 \<mapsto> \<lambda>\<sigma>. regs \<sigma> rsp = rsp\<^sub>0-8
\<and> regs \<sigma> rbp = rsp\<^sub>0-8
\<and> \<sigma> \<turnstile> *[list + 8,8] = last'
\<and> \<sigma> \<turnstile> *[rsp\<^sub>0-24,8] = task
\<and> \<sigma> \<turnstile> *[rsp\<^sub>0-16,8] = list
\<and> \<sigma> \<turnstile> *[rsp\<^sub>0-8,8] = rbp\<^sub>0
\<and> \<sigma> \<turnstile> *[rsp\<^sub>0,8] = boffset+task_list_push_back_ret
\<and> \<sigma> \<turnstile> *[a,1] = v\<^sub>0,
boffset+315 \<mapsto> \<lambda>\<sigma>. regs \<sigma> rsp = rsp\<^sub>0-8
\<and> regs \<sigma> rbp = rsp\<^sub>0-8
\<and> \<sigma> \<turnstile> *[rsp\<^sub>0-8,8] = rbp\<^sub>0
\<and> \<sigma> \<turnstile> *[rsp\<^sub>0,8] = boffset+task_list_push_back_ret
\<and> \<sigma> \<turnstile> *[a,1] = v\<^sub>0,
\<comment> \<open>postcondition\<close>
boffset+task_list_push_back_ret \<mapsto> \<lambda>\<sigma>. \<sigma> \<turnstile> *[a,1] = v\<^sub>0
\<and> regs \<sigma> rsp = rsp\<^sub>0+8
\<and> regs \<sigma> rbp = rbp\<^sub>0
]\<close>
text \<open>Adding some rules to the simplifier to simplify proofs.\<close>
schematic_goal pp_\<Theta>_zero[simp]:
\<open>pp_\<Theta> list task last' boffset = ?x\<close>
unfolding pp_\<Theta>_def
by simp
schematic_goal pp_\<Theta>_numeral_l[simp]:
\<open>pp_\<Theta> list task last' (n + boffset) = ?x\<close>
unfolding pp_\<Theta>_def
by simp
schematic_goal pp_\<Theta>_numeral_r[simp]:
\<open>pp_\<Theta> list task last' (boffset + n) = ?x\<close>
unfolding pp_\<Theta>_def
by simp
lemma rewrite_task_list_push_back_mem:
assumes
\<open>master blocks (list, 8) 5\<close>
\<open>master blocks (list + 8, 8) 6\<close>
\<open>master blocks (task + 0x58, 8) 7\<close>
\<open>master blocks (task + 0x60, 8) 8\<close>
\<open>master blocks (last' + 0x58, 8) 9\<close>
shows \<open>is_std_invar task_list_push_back_ret (floyd.invar task_list_push_back_ret (pp_\<Theta> list task last'))\<close>
proof -
note masters = masters assms
show ?thesis
text \<open>Boilerplate code to start the VCG\<close>
apply (rule floyd_invarI)
apply (rewrite at \<open>floyd_vcs task_list_push_back_ret \<hole> _\<close> pp_\<Theta>_def)
apply (intro floyd_vcsI)
text \<open>Subgoal for rip = boffset+139\<close>
subgoal premises prems for \<sigma>
text \<open>Insert relevant knowledge\<close>
apply (insert prems seps ret_address)
text \<open>Apply VCG/symb.\ execution\<close>
apply (restart_symbolic_execution?, (symbolic_execution masters: masters)+, (finish_symbolic_execution masters: masters)?)+
done
text \<open>Subgoal for rip = boffset+314\<close>
subgoal premises prems for \<sigma>
text \<open>Insert relevant knowledge\<close>
apply (insert prems seps ret_address)
text \<open>Apply VCG/symb.\ execution\<close>
apply (restart_symbolic_execution?, (symbolic_execution masters: masters)+, (finish_symbolic_execution masters: masters)?)+
done
text \<open>Subgoal for rip = boffset+315\<close>
subgoal premises prems for \<sigma>
text \<open>Insert relevant knowledge\<close>
apply (insert prems seps ret_address)
text \<open>Apply VCG/symb.\ execution\<close>
apply (restart_symbolic_execution?, (symbolic_execution masters: masters)+, (finish_symbolic_execution masters: masters)?)+
done
text \<open>Trivial ending subgoal.\<close>
subgoal
by simp
done
qed
end
end
|
import numpy as np
from scipy.signal import correlate
from scipy.linalg import eigh, blas
from alphacsc.utils import check_random_state
def learn_atoms(X, n_atoms, n_times_atom, n_iter=10, max_shift=11,
random_state=None):
"""Learn atoms using the MoTIF algorithm.
Parameters
----------
X : array, shape (n_trials, n_times)
The data on which to apply MoTIF.
n_atoms : int
The number of atoms.
n_times_atom : int
The support of the atoms
n_iter : int
The number of iterations
max_shift : int
The maximum allowable shift for the atoms.
random_state : int | None
The random initialization.
"""
rng = check_random_state(random_state)
n_trials, n_times = X.shape
atoms = rng.rand(n_atoms, n_times_atom)
corrs = np.zeros(n_trials)
match = np.zeros((n_atoms, n_trials), dtype=np.int)
# loop through atoms
for k in range(n_atoms):
aligned_data = np.zeros((n_times_atom, n_trials))
# compute Bk
B = np.zeros((n_times_atom, n_times_atom), order='F')
for l in range(k):
for p in np.arange(max_shift):
atom_shifted = np.roll(atoms[l], -p)[np.newaxis, :]
# B += np.dot(atom_shifted.T, atom_shifted)
B = blas.dger(1, atom_shifted, atom_shifted, a=B,
overwrite_a=1)
# make B invertible by adding a full-rank matrix
B += np.eye(B.shape[0]) * np.finfo(np.float32).eps
for i in range(n_iter):
print('[seed %s] Atom %d Iteration %d' % (random_state, k, i))
# loop through training data
for n in range(n_trials):
# ### STEP 1: Find out where the data and atom align ####
# which of these to use for template matching?
vec1 = (X[n] - np.mean(X[n])) / (np.std(X[n]) * len(X[n]))
vec2 = (atoms[k] - np.mean(atoms[k])) / np.std(atoms[k])
tmp = np.abs(correlate(vec1, vec2, 'same'))
offset = n_times_atom // 2
match[k, n] = tmp[offset:-offset].argmax() + offset
corrs[n] = tmp[match[k, n]]
# aligned_data[:, n] = np.roll(X[n], -shift[n])[:n_times_atom]
aligned_data[:, n] = X[n, match[k, n] - offset:
match[k, n] + offset].copy()
# ### STEP 2: Solve the generalized eigenvalue problem ####
A = np.dot(aligned_data, aligned_data.T).copy()
if k == 0:
B = None
e, U = eigh(A, B)
# e, U = eigh(A)
atoms[k, :] = U[:, -1] / np.linalg.norm(U[:, -1])
return atoms
|
import topology.basic
import topology.compact_open
import data.nat.prime
import data.real.basic
import data.real.irrational
import data.complex.basic
import data.fin.basic
import geometry.euclidean.basic
import analysis.inner_product_space.pi_L2
import algebra.group.defs
import algebra.field.basic
import combinatorics.configuration
import ring_theory.polynomial.basic
import group_theory.free_group
import combinatorics.simple_graph.basic
import ring_theory.integral_closure
import data.fintype.card
import category_theory.category.basic
import ring_theory.discrete_valuation_ring
import group_theory.torsion
import linear_algebra.matrix.charpoly.basic
import algebra.order.absolute_value
import analysis.convex.basic
import topology.uniform_space.uniform_convergence_topology
import topology.sequences
import analysis.normed.group.infinite_sum
import data.nat.choose.sum
import group_theory.specific_groups.cyclic
import group_theory.order_of_element
import analysis.mean_inequalities
import analysis.normed_space.banach
import topology.algebra.continuous_monoid_hom
import linear_algebra.matrix.symmetric
import analysis.inner_product_space.spectrum
import ring_theory.class_group
import ring_theory.dedekind_domain.basic
import ring_theory.principal_ideal_domain
import model_theory.satisfiability
import probability.integration
import ring_theory.simple_module
import category_theory.preadditive.schur
import representation_theory.maschke
import topology.paracompact
import combinatorics.simple_graph.coloring
--Prove that Overflow in Lean mathlib
--OUTPUT 1
theorem overflow_add_sub (x y : ℤ) : x + y = x - -y :=
begin
rw [int.add_sub_cancel_left, int.sub_neg_eq_add],
--OUTPUT 2
theorem overflow_thm (a b : ℕ) (h : a+b < a) : b = 0 :=
begin
cases h with d hd,
have h1 : a + b = a + d,
{ rw hd },
rw add_comm at h1,
rw add_right_cancel at h1,
assumption,
--OUTPUT 3
theorem overflow_thm (n : ℕ) : n + n = n ↔ n = 0 :=
begin
split,
{ intro h,
have h' : n + n = n + 0,
{ rw add_zero },
rw h at h',
apply add_right_cancel h', },
{ intro h,
rw h,
rw add_zero, }
|
This refined home in desirable Cow Hollow has a stately façade and shingled roofline that belie the tasteful contemporary style of its spacious multilevel interiors, all recently updated. Spanning four levels and offering five bedrooms, three and a half baths, the home blends San Francisco classicism with modern fixtures and amenities, creating a uniquely appealing home.
Upon entering the main level, one immediately appreciates the delicate balance between preserving historical architectural details and creating a contemporary atmosphere. A highlight of the home is the chic living/dining room, which boasts a regal arched entryway, handsome hardwood floors, classical moldings, a wall of windows, and marble fireplace flanked by plentiful built-in bookshelves. Beyond the central gallery landing is the elegant family room, with its hardwood floors and French doors that lead to a private deck, allowing for a great flow between indoor and outdoor living. The adjoining kitchen has been expertly outfitted with white Carrera marble counters, white subway tile backsplash, stainless steel appliances including a 6-burner gas range, white shaker cabinetry, and great storage. The adjacent bay-windowed breakfast area features built-in seating and glorious natural light with direct access through a series of french doors to a private deck, perfect for enjoying meals, coffee, or cocktails alfresco. A powder room is conveniently tucked below the stairs.
The third level of the home hosts three bedrooms and two baths, including the spacious master suite. The sophisticated and serene master suite has incredible scale and enjoys a tranquil sitting area nestled in a sun-dappled bay window. En suite is a luxurious bath with a soaking tub, glass-enclosed shower, and dual sinks – all appointed with timeless finishes, including white Carrera marble and subway tiles. A thoughtfully appointed walk-in closet and a series of built-ins complete the master suite.
The guest and master bedrooms are separated for privacy by the grand stairway hall and a convenient laundry room. Two restful guest bedrooms share a bath with timeless fixtures. The south bedroom is a stylish space with hardwood floors and a light-filled bay window. The north bedroom enjoys access to the enviable roof terrace, perfect for relaxation with its whimsical astroturf and views of the Golden Gate Bridge. Completing the top floor of the residence is a built-in window bench with serene outlooks.
On the entry level of the home, beyond the formal main entry and past the stairway, rests the office. The home office, which could also serve as a family room, is a cool, quiet, stylish space with a bay window and a series of built-ins. A peaceful bedroom with plentiful windows and a nearby bath are secluded on the rear of the home. A discreet rear stairway affords access from the entry level to the breakfast nook above.
An additional bedroom, an office/lounge space, and extra storage are located on the home’s lowest level, accessible via the rear stairway. The incredibly private rear garden, accessed from the lower level, is surrounded by mature landscaping, including a wall of hydrangeas, and enjoys a built-in spa and seating bench, creating a serene garden oasis.
The garage accommodates two vehicles and includes built-in shelving for storage and direct access into the home.
3041 Divisadero St is located on the western side of Cow Hollow and is perfectly positioned near the restaurants and shops along Chestnut St and the Presidio National Park. |
State Before: α : Type u
inst✝ : IsEmpty α
⊢ (#α →₀ ℕ) = 1 State After: no goals Tactic: simp |
[STATEMENT]
lemma parts_analz [simp]: "parts (analz H) = parts H"
[PROOF STATE]
proof (prove)
goal (1 subgoal):
1. parts (analz H) = parts H
[PROOF STEP]
apply (rule equalityI)
[PROOF STATE]
proof (prove)
goal (2 subgoals):
1. parts (analz H) \<subseteq> parts H
2. parts H \<subseteq> parts (analz H)
[PROOF STEP]
apply (rule analz_subset_parts [THEN parts_mono, THEN subset_trans], simp)
[PROOF STATE]
proof (prove)
goal (1 subgoal):
1. parts H \<subseteq> parts (analz H)
[PROOF STEP]
apply (blast intro: analz_increasing [THEN parts_mono, THEN subsetD])
[PROOF STATE]
proof (prove)
goal:
No subgoals!
[PROOF STEP]
done |
Require Import Coq.Reals.Rdefinitions.
Require Import ChargeTactics.Lemmas.
Require Import TLA.TLA.
Require Import TLA.EnabledLemmas.
Require Import TLA.ProofRules.
Require Import Examples.System2.
Require Import Examples.UpperLowerSecond.
Require Import Examples.UpperLowerFirst.
Local Open Scope string_scope.
Local Open Scope HP_scope.
Module Type UpperLowerParams.
Parameter ub : R.
Parameter d : R.
Parameter d_gt_0 : (d > 0)%R.
Parameter amin : R.
Parameter amin_lt_0 : (amin < 0)%R.
Parameter ubv : R.
Axiom ubv_gt_amin_d : (ubv >= -amin*d)%R.
Parameter ub_ubv :
(ubv*d + ubv*ubv*(0 - /2)*(/amin) <= ub)%R.
End UpperLowerParams.
Module UpperLower (Import P : UpperLowerParams).
Module Y <: UpperLowerSecondParams.
Definition ub := P.ub.
Definition d := P.d.
Definition d_gt_0 := P.d_gt_0.
Definition amin := P.amin.
Definition amin_lt_0 := P.amin_lt_0.
Definition ubv := P.ubv.
Definition ub_ubv := P.ub_ubv.
End Y.
Module V <: UpperLowerFirstParams.
Definition ub := P.ubv.
Definition d := P.d.
Definition d_gt_0 := P.d_gt_0.
End V.
Module Position := UpperLowerSecond Y.
Module Vel := UpperLowerFirst V.
Definition Next : ActionFormula :=
SysCompose Vel.Next Position.Next.
Definition IndInv : StateFormula :=
Vel.IndInv //\\ Position.IndInv.
Lemma Vel_IndInv_Position_Assumption :
Vel.IndInv //\\ TimeBound V.d |-- Position.Next_Assumption.
Proof.
rewrite Vel.IndInv_impl_Inv.
unfold Position.Next_Assumption, Vel.Safe, V.ub,
Position.Monitor.Next_Assumption, Position.Params.ubv, Y.ubv.
rewrite <- Rename_ok by eauto with rw_rename.
solve_linear.
Qed.
Lemma TimedPreserves_Next :
|-- TimedPreserves d IndInv Next.
Proof with (refine _).
unfold IndInv, Next. unfold SysCompose.
rewrite SysCompose_simpl.
rewrite <- TimedPreserves_And...
charge_split.
{ apply TimedPreserves_intro.
rewrite <- Vel.TimedPreserves_Next.
charge_tauto. }
{ apply TimedPreserves_intro.
rewrite <- Position.TimedPreserves_Next.
rewrite Vel_IndInv_Position_Assumption.
charge_tauto. }
Qed.
Definition Constraint :=
P.amin <= "a" <= --P.amin.
Lemma SysNeverStuck_Discr :
|-- Enabled ((0 <= "T"! <= d //\\ Sys_D Next) //\\
next Constraint).
Proof.
enable_ex_st.
pose proof P.amin_lt_0. pose proof P.d_gt_0.
pose proof P.ubv_gt_amin_d.
unfold Vel.V.ub, Vel.V.d, V.ub, V.d.
unfold Position.Params.amin.
unfold Y.amin.
destruct (RIneq.Rgt_dec (st "y") R0);
destruct (RIneq.Rgt_dec (st "v") R0).
{ exists amin; do 2 eexists; exists d; solve_linear. }
{ do 3 eexists; exists d; solve_linear. }
{ do 3 eexists; exists d; solve_linear. }
{ exists (-amin)%R; do 2 eexists; exists d; solve_linear. }
Qed.
Theorem SysNeverStuck_Next : |-- SysNeverStuck d IndInv Next.
Proof.
eapply SysNeverStuck_Sys';
[ refine _ | pose proof d_gt_0 ; solve_linear | | ].
{ pose proof SysNeverStuck_Discr.
etransitivity; [ apply H; clear H |
apply Proper_Enabled_lentails ].
charge_tauto. }
{ admit. (** Provable, but we won't worry about it *) }
Qed.
Definition Safe : StateFormula :=
Vel.Safe //\\ Position.Safe.
Lemma IndInv_impl_Safe : IndInv //\\ TimeBound d |-- Safe.
Proof with (eauto with rw_rename).
charge_split.
{ rewrite <- Vel.IndInv_impl_Inv.
unfold IndInv, TimeBound, V.d.
charge_tauto. }
{ rewrite <- Position.IndInv_impl_Safe.
unfold Y.d, IndInv.
rewrite <- Vel_IndInv_Position_Assumption.
unfold V.d. charge_tauto. }
Qed.
Lemma UpperLower_safe :
|-- (IndInv //\\ TimeBound d) //\\ []Next -->> []Safe.
Proof.
rewrite <- IndInv_impl_Safe.
eapply Inductively.Preserves_Inv.
3: apply TimedPreserves_Next.
- compute; tauto.
- apply always_tauto. charge_tauto.
Qed.
End UpperLower. |
module Types
mutual
public export
data Val = A String -- atom
| L (List Val) -- list
| D (List Val) Val -- Dotted List
| N Integer -- num
| S String -- string
| B Bool -- bool
| PrimitiveFunc (List Val -> ThrowsError Val)
| Func (List String) (Maybe String) (List Val) Env
public export
data Error = ParserE String
| BadSpecialForm String Val
| NotFunction String String
| NumArgs Integer (List Val)
| TypeMismatch String Val
| UnboundVar String String
public export
ThrowsError : Type -> Type
ThrowsError = Either Error
public export
Env : Type
Env = List (String, Val)
mutual
unwordsList : List Val -> String
unwordsList = unwords . map showVal
p : Maybe String -> String
p varargs = case varargs of
Nothing => ""
Just arg => " . " ++ arg
export
showVal : Val -> String
showVal (A x) = x
showVal (L xs) = "(" ++ unwordsList xs ++ ")"
showVal (D xs x) = "(" ++ unwordsList xs ++ " . " ++ showVal x ++ ")"
showVal (N x) = show x
showVal (S x) = "\"" ++ x ++ "\""
showVal (B True) = "#t"
showVal (B False) = "#f"
showVal (PrimitiveFunc _) = "<primitive>"
showVal (Func args varargs body closure) = "(lambda (" ++ (unwords (map show args)) ++ (p varargs) ++ ") ...)"
export
Show Val where
show = showVal
export
Show Error where
show (ParserE e) = "Parse error " ++ e
show (UnboundVar m v) = m ++ ": " ++ v
show (BadSpecialForm s v) = s ++ " : " ++ show v
show (NotFunction s f) = s ++ " : " ++ show f
show (NumArgs e f) = "Expected " ++ show e ++ " args; found " ++ (show $ length f)
show (TypeMismatch e f) = "Invalid type: expected " ++ e ++ ", found " ++ show f
show _ = "Error!!!"
|
lemma prime_int_iff': "prime (p :: int) \<longleftrightarrow> p > 1 \<and> (\<forall>n \<in> {2..<p}. \<not> n dvd p)" (is "?P \<longleftrightarrow> ?Q") |
#ミラーの指定
options(repos="http://cran.ism.ac.jp/")#東京の統計数理研究所
# パッケージのインストール
install.packages("partitions",dependencies=TRUE)
install.packages("DoE.base",dependencies=TRUE) |
#ifndef DIRICHLETSAMPLE_H
#define DIRICHLETSAMPLE_H
#include <stdio.h>
#include <iostream>
#include <vector>
#include <gsl/gsl_rng.h>
#include <gsl/gsl_randist.h>
#include <sys/time.h>
using namespace std;
std::vector<double> DirSample(std::vector<double> alpha, int k, int N);
#endif
|
Song lyrics by Crystal Pride. Check-out these awesome song lyrics by the artist, learn every word of your favourite song and sing it like Crystal Pride. Get one of the 8 lyrics and watch the video by artist Crystal Pride. |
"""
bresenhams_line_algorithm(maxx, maxy, x1, y1, x2, y2)
maxx, maxy: Integer max values
Assuming x1 and x2 are coordinates > 0.5, return integer coordinates between the two coordinates.
x1 to y2 may be floats, but this is intended to be 'floating point indexes' in a matrix.
Based on
https://stackoverflow.com/questions/40273880/draw-a-line-between-two-pixels-on-a-grayscale-image-in-julia
Adapted for float input arguments and limit checks.
"""
function bresenhams_line_algorithm(maxx::T, maxy::T, x1, y1, x2, y2) where {T<:Int}
@assert round(Int, x1) <= maxx
@assert round(Int, y1) <= maxy
@assert round(Int, x1) >= 1
@assert round(Int, y1) >= 1
@assert round(Int, x2) <= maxx
@assert round(Int, y2) <= maxy
@assert round(Int, x2) >= 1
@assert round(Int, y2) >= 1
# Calculate distances
dx = x2 - x1
dy = y2 - y1
# Determine how steep the line is
is_steep = abs(dy) > abs(dx)
# Rotate line
if is_steep
x1, y1 = y1, x1
x2, y2 = y2, x2
maxy, maxx = maxx, maxy
end
# Swap start and end points if necessary and store swap state
swapped = if x1 > x2
x1, x2 = x2, x1
y1, y2 = y2, y1
true
else
false
end
# Recalculate differentials
dx = x2 - x1
dy = y2 - y1
# Calculate error
error = round(Int, dx / 2.0)
ystep = if y1 < y2
1
else
-1
end
vy = Vector{Int}()
# Iterate while generating points between start and end
y = round(Int, y1)
xiter = round(Int, x1):(round(Int, x2))
for x in xiter
push!(vy, y)
error -= abs(dy)
if error < 0
y += ystep
# edge case, a subpixel line might cover two pixels due to rounding.
# However, if that happens to be on the edge, don't bleed over and don't cause an
# error throw.
if y > maxy || y < 1
y -= ystep
error -= dx
end
error += dx
end
end
# Reverse the list if the coordinates were swapped
if swapped
if is_steep
zip(reverse(vy), reverse(xiter))
else
zip(reverse(xiter), reverse(vy))
end
else
if is_steep
zip(vy, xiter)
else
zip(xiter, vy)
end
end
end
"""
crossing_line_algorithm(maxx, maxy, cx, cy)
maxx, maxy: Integer max values
Return indices for pixels in a 10x10 to 11x11 cross centered at cx, cy
cx, cy may be floats, but this is intended to be 'floating point indexes' in a matrix.
"""
function crossing_line_algorithm(maxx::T, maxy::T, cx, cy) where T<:Int
indices = Set{Tuple{Int64, Int64}}()
d = 5
xb = clamp(cx - d, 1, maxx)
xt = clamp(cx + d, 1, maxx)
yb = clamp(cy - d, 1, maxy)
yt = clamp(cy + d, 1, maxy)
lineindices = bresenhams_line_algorithm( maxx, maxy, xt, yb, xb, yt)
push!(indices, lineindices...)
lineindices = bresenhams_line_algorithm( maxx, maxy, xb, yb, xt, yt)
push!(indices, lineindices...)
indices
end
"""
box_line_algorithm(maxx, maxy, cx, cy)
maxx, maxy: Integer max values
Return indices for pixels in a 10x10 to 11x11 cross centered at cx, cy
cx, cy may be floats, but this is intended to be 'floating point indexes' in a matrix.
"""
function box_line_algorithm(maxx::T, maxy::T, cx, cy) where T<:Int
indices = Set{Tuple{Int64, Int64}}()
d = 5
xb = clamp(cx - d, 1, maxx)
xt = clamp(cx + d, 1, maxx)
yb = clamp(cy - d, 1, maxy)
yt = clamp(cy + d, 1, maxy)
lineindices = bresenhams_line_algorithm( maxx, maxy, xt, yb, xb, yb)
push!(indices, lineindices...)
lineindices = bresenhams_line_algorithm( maxx, maxy, xt, yt, xb, yt)
push!(indices, lineindices...)
lineindices = bresenhams_line_algorithm( maxx, maxy, xt, yb, xt, yt)
push!(indices, lineindices...)
lineindices = bresenhams_line_algorithm( maxx, maxy, xb, yb, xb, yt)
push!(indices, lineindices...)
indices
end
"""
circle_algorithm(maxx, maxy, cx, cy)
maxx, maxy: Integer max values
Return indices for pixels in a 10 pixel diameter circle centered at cx, cy
cx, cy may be floats, but this is intended to be 'floating point indexes' in a matrix.
"""
function circle_algorithm(maxx::T, maxy::T, cx, cy) where T<:Int
indices = Set{Tuple{Int64, Int64}}()
d = 10
pts = 2 * round(Int, π * d)
for ϕ in range(0.0, 2π, length = pts)
x = clamp(round(Int, cx + d * cos(ϕ) / 2), 1, maxx)
y = clamp(round(Int, cy + d * cos(ϕ) / 2), 1, maxy)
push!(indices, (x, y))
end
indices
end |
theory clique
imports Main
begin
text \<open>Formalise the polynomial-time reduction between vertex cover,
clique and independent set\<close>
section \<open>definitions\<close>
type_synonym 'a graph = "'a set \<times> ('a set set)"
definition invar :: "'a graph => bool" where
"invar g = (
let (V, E) = g in (\<forall>s \<in> E. (\<forall>x \<in> s. x \<in> V) \<and> card s = 2)
)"
fun vertex_cover :: "'a graph => 'a set => bool" where
"vertex_cover g s = (
let (_, E) = g in (\<forall>s1 \<in> E. \<exists>x \<in> s1. x \<in> s)
)"
fun clique :: "'a graph => 'a set => bool" where
"clique g s = (
let (_, E) = g in (\<forall>a \<in> s. \<forall> b \<in> s. a \<noteq> b \<longrightarrow> {a, b} \<in> E)
)"
fun vc_to_clique :: "'a graph => 'a graph" where
"vc_to_clique g = (
let (V, E) = g in (V, {s. \<exists>a \<in> V. \<exists>b \<in> V. s = {a, b} \<and> s \<notin> E \<and> a \<noteq> b})
)"
fun T_vc_to_clique :: "'a graph => nat" where
"T_vc_to_clique (V, E) = card {s. \<exists>a \<in> V. \<exists>b \<in> V. s = {a, b} \<and> s \<notin> E \<and> a \<noteq> b}"
section \<open>proofs of invariant, correctness and polynomial time\<close>
theorem invar_vc_to_clique : "invar (V, E) \<Longrightarrow> invar (vc_to_clique (V, E))"
by (auto simp add: invar_def)
theorem vc_clique_correct:
assumes "invar (V, E)"
shows "clique (vc_to_clique (V, E)) (V - s) = vertex_cover (V, E) s"
proof
have 1:"\<forall>a. {a} \<notin> E" using assms invar_def by force
from assms have prems: "\<forall>s \<in> E. (\<forall>x \<in> s. x \<in> V)" "\<forall>s \<in> E. \<exists>a \<in> V. \<exists> b \<in> V. s = {a, b}"
apply (auto simp: invar_def) by (metis card_2_iff insert_iff)
assume "clique (vc_to_clique (V, E)) (V - s)"
hence "\<forall>a \<in> V-s. \<forall>b \<in> V-s. a \<noteq> b \<longrightarrow> {a, b} \<in> {s. \<exists>a\<in>V. \<exists>b\<in>V. s = {a, b} \<and> s \<notin> E \<and> a \<noteq> b}" by simp
hence "\<forall>a \<in> V-s. \<forall>b \<in> V-s. a \<noteq> b \<longrightarrow> {a, b} \<notin> E" by auto
hence "\<forall>a \<in> V-s. \<forall>b \<in> V-s. {a, b} \<notin> E" using 1 by force
hence "\<forall>s1 \<in> E. \<exists>a b. s1 = {a, b} \<and> (a \<notin> V-s \<or> b \<notin> V-s)"
using prems(2) doubleton_eq_iff by fast
hence "\<forall>s1 \<in> E. \<exists>a \<in> s1. a \<notin> V-s"
by auto
hence "\<forall>s1 \<in> E. \<exists>a \<in> s1. a \<in> s"
using prems(1) by simp
thus "vertex_cover (V, E) s" by simp
next
assume "vertex_cover (V, E) s"
hence "\<forall>s1 \<in> E. \<exists>a \<in> s1. a \<in> s" by simp
hence "\<forall>s1 \<in> E. \<exists>a \<in> s1. a \<notin> V-s" by auto
hence "\<forall>a \<in>V-s. \<forall>b \<in>V-s. a \<noteq> b \<longrightarrow> {a, b} \<notin> E" by fast
thus "clique (vc_to_clique (V, E)) (V - s)" by auto
qed
lemma aux0 :
assumes "finite A" "x \<in> A"
shows "card {s. \<exists>a\<in>A. s={x, a} \<and> a \<noteq> x} = card A - 1"
using assms proof (induction A rule: remove_induct)
case empty
then show ?case by simp
next
case infinite
then show ?case by simp
next
case (remove A)
hence 0:"\<forall>y \<in> A - {x}. card {s. \<exists>a\<in>A - {y}. s = {x, a} \<and> a \<noteq> x} = card (A - {y}) - 1"
by auto
have "\<forall>y \<in> A - {x}. {s. \<exists>a\<in>A. s = {x, a} \<and> a \<noteq> x} = insert {x, y} {s. \<exists>a\<in>A - {y}. s = {x, a} \<and> a \<noteq> x}"
by auto
moreover have "\<forall>y \<in> A - {x}. {x, y} \<notin> {s. \<exists>a\<in>A - {y}. s = {x, a} \<and> a \<noteq> x}"
by auto
ultimately have 1:"\<forall>y \<in> A - {x}. card {s. \<exists>a\<in>A. s = {x, a} \<and> a \<noteq> x} = card {s. \<exists>a\<in>A - {y}. s = {x, a} \<and> a \<noteq> x} + 1"
using remove by simp
from 0 1 have "\<forall>y \<in> A - {x}. card {s. \<exists>a\<in>A. s = {x, a} \<and> a \<noteq> x} = card (A - {y}) - 1 + 1" by simp
hence 3: "\<forall>y \<in> A - {x}. card {s. \<exists>a\<in>A. s = {x, a} \<and> a \<noteq> x} = card (A) - 1"
by (metis (no_types, lifting) One_nat_def add.right_neutral add_Suc_right card_Diff_singleton
card_Suc_Diff1 finite_insert insert_Diff_single insert_iff remove.prems(1) remove.prems(2))
from 3 show ?case apply auto by (metis card_le_Suc0_iff_eq remove.prems(1))
qed
lemma aux:
assumes "finite V"
shows "card {s. \<exists>a \<in> V. \<exists>b \<in> V. s = {a, b} \<and> a \<noteq> b} = card V * (card V - 1) div 2"
using assms proof (induction V rule: finite_remove_induct)
case empty
then show ?case by auto
next
case (remove A)
have "\<forall>x \<in> A. {s. \<exists>a\<in>A - {x}. \<exists>b\<in>A - {x}. s = {a, b} \<and> a \<noteq> b} =
{s. \<exists>a\<in>A. \<exists>b\<in>A. s = {a, b} \<and> a \<noteq> b} - {s. \<exists>a\<in>A. s={x, a} \<and> a \<noteq> x}" by auto
hence "\<forall>x \<in> A. {s. \<exists>a\<in>A. \<exists>b\<in>A. s = {a, b} \<and> a \<noteq> b} =
{s. \<exists>a\<in>A - {x}. \<exists>b\<in>A - {x}. s = {a, b} \<and> a \<noteq> b} \<union> {s. \<exists>a\<in>A. s={x, a} \<and> a \<noteq> x}"
by auto
moreover have "\<forall>x \<in> A. finite {s. \<exists>a\<in>A - {x}. \<exists>b\<in>A - {x}. s = {a, b} \<and> a \<noteq> b}"
using remove by simp
moreover have "\<forall>x \<in> A. finite {s. \<exists>a\<in>A. s={x, a} \<and> a \<noteq> x}" using remove by simp
moreover have "\<forall>x \<in> A. {s. \<exists>a\<in>A - {x}. \<exists>b\<in>A - {x}. s = {a, b} \<and> a \<noteq> b}
\<inter> {s. \<exists>a\<in>A. s={x, a} \<and> a \<noteq> x} = {}" by auto
ultimately have "\<forall>x \<in> A. card {s. \<exists>a\<in>A. \<exists>b\<in>A. s = {a, b} \<and> a \<noteq> b}
= card {s. \<exists>a\<in>A - {x}. \<exists>b\<in>A - {x}. s = {a, b} \<and> a \<noteq> b} + card {s. \<exists>a\<in>A. s={x, a} \<and> a \<noteq> x}"
using card_Un_disjoint by fastforce
hence "\<forall>x \<in> A. card {s. \<exists>a\<in>A. \<exists>b\<in>A. s = {a, b} \<and> a \<noteq> b}
= card (A - {x}) * (card (A - {x}) - 1) div 2 + (card A - 1)"
using aux0 remove by fastforce
hence "\<forall>x \<in> A. card {s. \<exists>a\<in>A. \<exists>b\<in>A. s = {a, b} \<and> a \<noteq> b}
= (card A - 1) * (card A - 2) div 2 + (card A - 1)"
by (metis (no_types, lifting) card_Diff_singleton diff_diff_left nat_1_add_1)
hence "\<forall>x \<in> A. card {s. \<exists>a\<in>A. \<exists>b\<in>A. s = {a, b} \<and> a \<noteq> b}
= ((card A - 1) * (card A - 2) + (card A - 1) * 2) div 2"
by simp
hence "\<forall>x \<in> A. card {s. \<exists>a\<in>A. \<exists>b\<in>A. s = {a, b} \<and> a \<noteq> b}
= card A * (card A - 1) div 2"
by (metis (no_types, lifting) One_nat_def cancel_comm_monoid_add_class.diff_cancel
card_0_eq distrib_left le_add_diff_inverse2 less_Suc0 less_Suc_eq
linorder_not_less mult.commute mult_zero_right one_add_one plus_1_eq_Suc remove.hyps(1) remove.hyps(2))
then show ?case by auto
qed
theorem vc_to_clique_polynomial : "\<lbrakk>invar (V, E); finite E; finite V\<rbrakk>
\<Longrightarrow> T_vc_to_clique (V, E) = card V * (card V -1) div 2 - card E"
proof-
assume assms: "invar (V, E)" "finite E" "finite V"
hence "\<forall>s \<in> E. \<exists>a \<in> V. \<exists> b \<in> V. s = {a, b} \<and> a \<noteq> b"
apply (auto simp add: invar_def) by (metis card_2_iff insert_iff)
hence 1: "E \<subseteq> {s. \<exists>a \<in> V. \<exists>b \<in> V. s = {a, b} \<and> a \<noteq> b}" by auto
have "{s. \<exists>a \<in> V. \<exists>b \<in> V. s = {a, b} \<and> s \<notin> E \<and> a \<noteq> b}
= {s. \<exists>a \<in> V. \<exists>b \<in> V. s = {a, b} \<and> a \<noteq> b} - E" by auto
from card_Diff_subset[OF assms(2) 1] this
have "card {s. \<exists>a \<in> V. \<exists>b \<in> V. s = {a, b} \<and> s \<notin> E \<and> a \<noteq> b} =
card {s. \<exists>a \<in> V. \<exists>b \<in> V. s = {a, b} \<and> a \<noteq> b} - card E" by argo
also have "... = card V * (card V - 1) div 2 - card E" by (auto simp add: aux[OF assms(3)])
finally show ?thesis by simp
qed
section \<open>independent set\<close>
fun independent_set :: "'a graph => 'a set => bool" where
"independent_set g s = (
let (V, E) = g in
(\<forall>a \<in>s. \<forall>b \<in>s. a \<noteq> b \<longrightarrow> {a, b} \<notin> E)
)"
text \<open>constant reduction from independet set to vertex cover\<close>
fun is_to_vc :: "'a graph => 'a graph" where
"is_to_vc g = g"
fun T_is_to_vc :: "'a graph => nat" where
"T_is_to_vc _ = 1"
theorem is_to_vc_correct:
assumes "invar (V, E)"
shows "independent_set (V, E) s = vertex_cover (is_to_vc (V, E)) (V-s)"
proof
from assms have prems: "\<forall>s \<in> E. (\<forall>x \<in> s. x \<in> V)" "\<forall>s \<in> E. \<exists>a \<in> V. \<exists> b \<in> V. s = {a, b}"
apply (auto simp: invar_def) by (metis card_2_iff insert_iff)
assume "independent_set (V, E) s"
hence "\<forall>a \<in>s. \<forall>b \<in>s. a \<noteq> b \<longrightarrow> {a, b} \<notin> E" by simp
hence "(\<forall>a \<in>s. \<forall>b \<in>s. {a, b} \<notin> E)" using assms by (force simp add: invar_def)
hence "\<forall>s1 \<in>E. \<exists>a b. s1 = {a, b} \<and> (a \<notin> s \<or> b \<notin> s)"
using prems(2) by metis
hence "\<forall>s1 \<in>E. \<exists>a \<in>s1. a\<notin>s" by auto
hence "\<forall>s1 \<in>E. \<exists>a \<in>s1. a \<in> V-s" using prems(1) by simp
then show "vertex_cover (is_to_vc (V, E)) (V-s)" by simp
next
assume "vertex_cover (is_to_vc (V, E)) (V-s)"
hence "\<forall>s1 \<in>E. \<exists>a \<in>s1. a \<in>V-s" by simp
hence "\<forall>s1 \<in>E. \<exists>a \<in>s1. a \<in> V-s" by auto
hence "\<forall>a \<in>s. \<forall>b \<in>s. a \<noteq> b \<longrightarrow> {a, b} \<notin> E" by fastforce
then show "independent_set (V, E) s" by simp
qed
theorem is_to_vc_polynomial: "T_is_to_vc g = 1" by simp
text \<open>reduction from clique to independent set\<close>
fun clique_to_is :: "'a graph => 'a graph" where
"clique_to_is g = (
let (V, E) = g in (V, {s. \<exists>a \<in> V. \<exists>b \<in> V. s = {a, b} \<and> s \<notin> E \<and> a \<noteq> b})
)"
fun T_clique_to_is :: "'a graph => nat" where
"T_clique_to_is (V, E) = card {s. \<exists>a \<in> V. \<exists>b \<in> V. s = {a, b} \<and> s \<notin> E \<and> a \<noteq> b}"
theorem clique_to_is_correct :
assumes "invar (V, E)" "s \<subseteq> V"
shows "clique (V, E) s = independent_set (clique_to_is (V, E)) s"
using assms apply (auto simp add: invar_def) apply metis by blast
theorem clique_to_is_polynomial : "\<lbrakk>invar (V, E); finite E; finite v\<rbrakk>
\<Longrightarrow> T_clique_to_is (V, E) = card V * (card V -1) div 2 - card E"
using vc_to_clique_polynomial by auto
theorem threeway_reduction_correct:
assumes "invar (V, E)" "s \<subseteq> V"
shows "clique (V, E) s = vertex_cover (is_to_vc (clique_to_is (V, E))) (V - s)"
proof-
have "clique (V, E) s = independent_set (clique_to_is (V, E)) s"
using clique_to_is_correct assms by blast
also have "... = vertex_cover (is_to_vc (clique_to_is (V, E))) (V - s)"
using is_to_vc_correct assms
by (metis (mono_tags, lifting) clique_to_is.elims invar_vc_to_clique prod.simps(2) vc_to_clique.simps)
finally show ?thesis by simp
qed
end |
# Built-in
Binomial(5, 3);
# 10
|
import tactic -- hide
open function nat -- hide
/-
## More on `apply`
In the following example, `h` can be seen as a map
$$h : \mathbb{N} \rightarrow \\{\text{proofs}\\}$$
which gives,
for each natural number $x$, a proof of the fact that this particular $x$ satisfies $0 \leq x^2$.
This is why `apply` will work in the following example. *Lean* is smart enough to figure out which value of $x$
needs to be plugged in to match the conclusion of `h` with the goal.
-/
/- Symbol:
≤ : \leq
-/
/- Lemma : no-side-bar
If for all x we know $0 ≤ x^2$, then $0 ≤ 3^2$.
-/
lemma l6 (h : ∀ x, 0 ≤ x^2) : 0 ≤ 3^2:=
begin
apply h,
end |
(* Exercise 2: Definiți o funcție de egalitate peste acest tip de date. *)
(* Define the 'Day' data type. *)
Inductive Day :=
| Monday
| Tuesday
| Wednesday
| Thursday
| Friday
| Saturday
| Sunday.
(* Define the 'equality' function. *)
Definition equality (firstDay secondDay : Day) : bool :=
match firstDay, secondDay with
| Monday, Monday => true
| Tuesday, Tuesday => true
| Wednesday, Wednesday => true
| Thursday, Thursday => true
| Friday, Friday => true
| Saturday, Saturday => true
| Sunday, Sunday => true
| _, _ => false
end.
(* Test the 'equality' function. *)
Compute equality Monday Monday.
Compute equality Tuesday Tuesday.
Compute equality Wednesday Wednesday.
Compute equality Thursday Thursday.
Compute equality Friday Friday.
Compute equality Saturday Saturday.
Compute equality Sunday Sunday.
Compute equality Monday Friday.
Compute equality Thursday Sunday.
Compute equality Tuesday Sunday.
Compute equality Thursday Monday.
Compute equality Friday Saturday. |
-- Infinite series of analytic functions
import analysis.analytic.basic
import data.complex.basic
import data.real.basic
import data.real.ennreal
import data.real.nnreal
import data.real.pi.bounds
import data.set.basic
import topology.metric_space.basic
import topology.uniform_space.uniform_convergence
import topology.algebra.infinite_sum
import analytic
import bounds
import simple
import tactics
import uniform
open complex (abs)
open filter (at_top)
open metric (ball closed_ball sphere)
open_locale real nnreal ennreal topological_space
noncomputable theory
-- Summability restricted to sets
def summable_on (f : ℕ → ℂ → ℂ) (s : set ℂ) := ∀ z, z ∈ s → summable (λ n, f n z)
def has_sum_on (f : ℕ → ℂ → ℂ) (g : ℂ → ℂ) (s : set ℂ) := ∀ z, z ∈ s → has_sum (λ n, f n z) (g z)
noncomputable def tsum_on (f : ℕ → ℂ → ℂ) := λ z, tsum (λ n, f n z)
def has_uniform_sum (f : ℕ → ℂ → ℂ) (g : ℂ → ℂ) (s : set ℂ) :=
tendsto_uniformly_on (λ (N : finset ℕ) z, N.sum (λ n, f n z)) g at_top s
-- Uniform vanishing means late sums are uniformly small
def uniform_vanishing (f : ℕ → ℂ → ℂ) (s : set ℂ) :=
∀ e : ℝ, e > 0 → ∃ n : ℕ, ∀ (N : finset ℕ) z, late N n → z ∈ s → N.sum (λ n, abs (f n z)) < e
lemma uniform_vanishing_to_summable {f : ℕ → ℂ → ℂ} {s : set ℂ} {z : ℂ}
(zs : z ∈ s) (h : uniform_vanishing f s) : summable (λ n, f n z) := begin
rw [summable_iff_cauchy_seq_finset, metric.cauchy_seq_iff],
intros e ep,
rcases h e ep with ⟨m,hm⟩,
existsi finset.range m,
intros A HA B HB,
calc dist (A.sum (λ n, f n z)) (B.sum (λ n, f n z)) ≤ (A ∆ B).sum (λ n, abs (f n z)) : symm_diff_bound _ _ _
... < e : hm (A ∆ B) z (symm_diff_late HA HB) zs
end
lemma uniform_vanishing_to_uniform_cauchy_series {f : ℕ → ℂ → ℂ} {s : set ℂ}
(h : uniform_vanishing f s) : uniform_cauchy_seq_on (λ (N : finset ℕ) z, N.sum (λ n, f n z)) at_top s := begin
rw metric.uniform_cauchy_seq_on_iff,
intros e ep,
rcases h e ep with ⟨m,hm⟩,
existsi finset.range m,
intros A HA B HB z zs,
calc dist (A.sum (λ n, f n z)) (B.sum (λ n, f n z)) ≤ (A ∆ B).sum (λ n, abs (f n z)) : symm_diff_bound _ _ _
... < e : hm (A ∆ B) z (symm_diff_late HA HB) zs
end
lemma uniform_vanishing_to_tendsto_uniformly_on {f : ℕ → ℂ → ℂ} {s : set ℂ}
(h : uniform_vanishing f s) : has_uniform_sum f (tsum_on f) s := begin
rw [has_uniform_sum, metric.tendsto_uniformly_on_iff],
intros e ep,
rcases h (e/4) (by bound) with ⟨m,hm⟩,
rw filter.eventually_at_top,
existsi finset.range m,
intros N Nm z zs,
rw tsum_on, simp,
generalize G : tsum (λ n, f n z) = g,
have S : summable (λ n, f n z) := uniform_vanishing_to_summable zs h,
have GS : has_sum (λ n, f n z) g, { rw ←G, exact summable.has_sum S }, clear S,
rw has_sum at GS,
rw metric.tendsto_at_top at GS,
rcases GS (e/4) (by bound) with ⟨M,HM⟩, clear GS G h,
set A := N ∪ (M \ N),
have AM : M ⊆ A := simple.subset_union_sdiff _ _,
simp at HM,
specialize HM A AM,
rw dist_comm at HM,
calc dist g (N.sum (λ n, f n z)) ≤ dist g (A.sum (λ n, f n z)) + dist (A.sum (λ n, f n z)) (N.sum (λ n, f n z)) : by bound
... ≤ e/4 + dist (A.sum (λ n, f n z)) (N.sum (λ n, f n z)) : by bound
... = e/4 + dist (N.sum (λ n, f n z) + (M \ N).sum (λ n, f n z)) (N.sum (λ n, f n z))
: by rw finset.sum_union finset.disjoint_sdiff
... = e/4 + abs (N.sum (λ n, f n z) + (M \ N).sum (λ n, f n z) - N.sum (λ n, f n z)) : by rw complex.dist_eq
... = e/4 + abs ((M \ N).sum (λ n, f n z)) : by ring_nf
... ≤ e/4 + (M \ N).sum (λ n, abs (f n z)) : by bound [simple.finset_complex_abs_sum_le (M \ N) (λ n, f n z)]
... ≤ e/4 + e/4 : by bound [hm (M \ N) z (sdiff_late M Nm) zs]
... = e/2 : by ring
... < e : by bound
end
-- Geometric bounds with c ≤ 0 are degenerate
lemma c_nonpos.degenerate {f : ℕ → ℂ → ℂ} {s : set ℂ} {c a : ℝ}
(c0 : c ≤ 0) (a0 : 0 ≤ a) (hf : ∀ n z, z ∈ s → abs (f n z) ≤ c * a^n)
: ∀ n z, z ∈ s → f n z = 0 := begin
intros n z zs, specialize hf n z zs,
have ca : c * a^n ≤ 0 := mul_nonpos_iff.mpr (or.inr ⟨c0, by bound⟩),
exact complex.abs_eq_zero.mp (le_antisymm (trans hf ca) (complex.abs_nonneg _))
end
-- Uniformly exponentially converging series converge uniformly.
theorem fast_series_converge_uniformly_on {f : ℕ → ℂ → ℂ} {s : set ℂ} {c a : ℝ}
(a0 : 0 ≤ a) (a1 : a < 1) (hf : ∀ n z, z ∈ s → abs (f n z) ≤ c * a^n)
: has_uniform_sum f (tsum_on f) s := begin
by_cases c0 : c ≤ 0, {
have fz := c_nonpos.degenerate c0 a0 hf, simp at fz,
rw [has_uniform_sum, metric.tendsto_uniformly_on_iff],
intros e ep, apply filter.eventually_of_forall, intros n z zs,
rw tsum_on, simp,
simp_rw fz _ z zs, simp,
assumption
}, {
simp at c0,
apply uniform_vanishing_to_tendsto_uniformly_on,
-- ∀ e : ℝ, e > 0 → ∃ n : ℕ, ∀ (N : finset ℕ) z, late N n → z ∈ s → N.sum (λ n, abs (f n z)) < e
intros e ep,
set t := (1-↑a)/↑c*(e/2),
have tp : t > 0 := by bound,
rcases exists_pow_lt_of_lt_one tp a1 with ⟨n,nt⟩,
existsi n,
intros N z NL zs,
have a1p : 1 - (a : ℝ) > 0 := by bound,
calc N.sum (λ n, abs (f n z)) ≤ N.sum (λ n, c * a^n) : finset.sum_le_sum (λ n _, hf n z zs)
... = c * N.sum (λ n, a^n) : finset.mul_sum.symm
... ≤ c * (a^n * (1 - a)⁻¹) : by bound [late_geometric_bound _ (by bound) a1]
... = a^n * (c * (1 - a)⁻¹) : by ring
... ≤ t * (c * (1 - a)⁻¹) : by bound
... = (1 - a) / c * (e / 2) * (c * (1 - a)⁻¹) : rfl
... = (1 - a) * (1 - a)⁻¹ * (c / c) * (e / 2) : by ring
... = 1 * 1 * (e / 2) : by rw [field.mul_inv_cancel (ne_of_gt a1p), simple.div_self (ne_of_gt c0)]
... = e / 2 : by ring
... < e : by bound
}
end
-- Exponentially converging series converge.
theorem fast_series_converge_at {f : ℕ → ℂ} {c a : ℝ}
(a0 : 0 ≤ a) (a1 : a < 1) (hf : ∀ n, abs (f n) ≤ c * a^n) : summable f := begin
set s : set ℂ := {0},
set g : ℕ → ℂ → ℂ := λ n _, f n,
have hg : ∀ n z, z ∈ s → abs (g n z) ≤ c * a^n := λ n z zs, hf n,
have u := fast_series_converge_uniformly_on a0 a1 hg,
simp at u,
rw has_uniform_sum at u,
rw tendsto_uniformly_on_singleton_iff_tendsto at u,
apply has_sum.summable, assumption
end
-- Finite sums of analytic functions are analytic
lemma finite_sums_are_analytic {f : ℕ → ℂ → ℂ} {s : set ℂ}
(h : ∀ n, analytic_on ℂ (f n) s) (N : finset ℕ) : analytic_on ℂ (λ z, N.sum (λ n, f n z)) s := begin
induction N using finset.induction with a B aB hB, {
simp, intros z zs, exact entire.zero z
}, {
intros z zs,
simp_rw finset.sum_insert aB,
apply analytic_at.add,
exact h a z zs,
exact hB z zs
}
end
-- Analytic series that converge exponentially converge to analytic functions.
theorem fast_series_converge {f : ℕ → ℂ → ℂ} {s : set ℂ} {c a : ℝ}
(o : is_open s) (a0 : 0 ≤ a) (a1 : a < 1)
(h : ∀ n, analytic_on ℂ (f n) s) (hf : ∀ n z, z ∈ s → abs (f n z) ≤ c * a^n)
: ∃ (g : ℂ → ℂ), analytic_on ℂ g s ∧ has_sum_on f g s := begin
set g := tsum_on f,
existsi g,
have su : has_uniform_sum f g s := fast_series_converge_uniformly_on a0 a1 hf,
constructor, {
refine uniform_analytic_lim o _ su,
exact finite_sums_are_analytic h
}, {
intros z zs,
exact summable.has_sum (fast_series_converge_at a0 a1 (λ n, hf n z zs))
}
end |
theory Chapter12_4
imports "HOL-IMP.VCG" "HOL-IMP.Hoare_Examples"
begin
text \<open>
\exercise
Let @{term "asum i"} be the annotated command \texttt{y := 0; W}
where \texttt{W} is defined in Example~12.7. Prove
\<close>
definition asum :: "int \<Rightarrow> acom"where
"asum i = ''y'' ::= (N 0);;
{\<lambda>s. s ''y'' + sum (s ''x'') = sum i}
WHILE Less (N 0) (V ''x'') DO
(''y'' ::= Plus (V ''y'') (V ''x'');; ''x'' ::= Plus (V ''x'') (N (-1)))"
lemma "\<turnstile> {\<lambda>s. s ''x'' = i} strip(asum i) {\<lambda>s. s ''y'' = sum i}"
unfolding asum_def
by (rule vc_sound', auto)
text \<open>
with the help of corollary @{thm[source] vc_sound'}.
\endexercise
\exercise
Solve exercises \ref{exe:Hoare:sumeq} to \ref{exe:Hoare:sqrt} using the VCG:
for every Hoare triple @{prop"\<turnstile> {P} c {Q}"} from one of those exercises
define an annotated version @{text C} of @{text c} and prove
@{prop"\<turnstile> {P} strip C {Q}"} with the help of %Corollary~\ref{cor:vc_sound}
corollary @{thm[source] vc_sound'}.
\<close>
definition Eq :: "aexp \<Rightarrow> aexp \<Rightarrow> bexp" where
"Eq a1 a2 = (And (Not (Less a1 a2)) (Not (Less a2 a1)))"
lemma bval_Eq[simp]: "bval (Eq a1 a2) s = (aval a1 s = aval a2 s)"
unfolding Eq_def by auto
lemma "\<turnstile> {\<lambda>s. s ''x'' = i \<and> 0 \<le> i} strip (
''y'' ::= N 0;;
{\<lambda>s. s ''y'' = sum i - sum (s ''x'') \<and> 0 \<le> s ''x''}
WHILE Not(Eq (V ''x'') (N 0))
DO (''y'' ::= Plus (V ''y'') (V ''x'');;
''x'' ::= Plus (V ''x'') (N (-1))))
{\<lambda>s. s ''y'' = sum i}"
by (rule vc_sound', auto)
lemma "\<turnstile> {\<lambda>s. s ''x'' = x \<and> s ''y'' = y \<and> 0 \<le> x} strip (
{\<lambda>s. s ''y'' - s ''x'' = y - x \<and> 0 \<le> s ''x''}
WHILE Less (N 0) (V ''x'')
DO (''x'' ::= Plus (V ''x'') (N (-1));;
''y'' ::= Plus (V ''y'') (N (-1))))
{\<lambda>t. t ''y'' = y - x}"
by (rule vc_sound', auto)
abbreviation cmult :: com where
"cmult \<equiv> ''z'' ::= N 0;;
WHILE Less (N 0) (V ''y'') DO (''y'' ::= (Plus (V ''y'') (N (-1)));; ''z'' ::= (Plus (V ''z'') (V ''x'')))"
lemma
"\<turnstile> {\<lambda>s. s ''x'' = x \<and> s ''y'' = y \<and> 0 \<le> y} strip (
''z'' ::= N 0;;
{\<lambda>s. s ''x'' = x \<and> s ''z'' = s ''x'' * (y - s ''y'') \<and> 0 \<le> s ''y''}
WHILE Less (N 0) (V ''y'')
DO (''y'' ::= (Plus (V ''y'') (N (-1)));;
''z'' ::= (Plus (V ''z'') (V ''x''))))
{\<lambda>t. t ''z'' = x*y}"
by (rule vc_sound', auto simp add: algebra_simps)
lemma
"\<turnstile> { \<lambda>s. s ''x'' = i \<and> 0 \<le> i} strip (
''r'' ::= N 0;; ''r2'' ::= N 1;;
{\<lambda>s. s ''x'' = i \<and> (s ''r'')\<^sup>2 \<le> s ''x'' \<and> s ''r2'' = (s ''r'' + 1)\<^sup>2}
WHILE (Not (Less (V ''x'') (V ''r2'')))
DO (''r'' ::= Plus (V ''r'') (N 1);;
''r2'' ::= Plus (V ''r2'') (Plus (Plus (V ''r'') (V ''r'')) (N 1))))
{\<lambda>s. (s ''r'')^2 \<le> i \<and> i < (s ''r'' + 1)^2}"
proof (rule vc_sound', auto simp add: algebra_simps)
fix s :: state
have "\<And>x::int. 3 + (2 * x + (1 + x)\<^sup>2) = (2 + x)\<^sup>2"
proof -
fix x :: int
have "(1 + x)\<^sup>2 = x\<^sup>2 + 2 * x + 1" by (simp add: power2_sum)
then have "3 + (2 * x + (1 + x)\<^sup>2) = 3 + (2 * x + (x\<^sup>2 + 2 * x + 1))" by simp
also have "\<dots> = 4 + 4 * x + x\<^sup>2" by simp
also have "\<dots> = (2 + x)\<^sup>2" by (simp add: power2_sum)
finally show "3 + (2 * x + (1 + x)\<^sup>2) = (2 + x)\<^sup>2" .
qed
then show "3 + (2 * s ''r'' + (1 + s ''r'')\<^sup>2) = (2 + s ''r'')\<^sup>2" by simp
qed
text \<open>
\endexercise
\exercise
Having two separate functions @{const pre} and @{const vc} is inefficient.
When computing @{const vc} one often needs to compute @{const pre} too,
leading to multiple traversals of many subcommands. Define an optimised function
\<close>
fun prevc :: "acom \<Rightarrow> assn \<Rightarrow> assn \<times> bool" where
"prevc SKIP Q = (Q, True)" |
"prevc (x ::= a) Q = (\<lambda>s. Q(s(x := aval a s)), True)" |
"prevc (C\<^sub>1;; C\<^sub>2) Q = (\<lambda>(P', S'). (\<lambda>(P, S). (P, S \<and> S')) (prevc C\<^sub>1 P')) (prevc C\<^sub>2 Q)" |
"prevc (IF b THEN C\<^sub>1 ELSE C\<^sub>2) Q =
(\<lambda>(P\<^sub>1, S\<^sub>1). (\<lambda>(P\<^sub>2, S\<^sub>2). (\<lambda>s. if bval b s then P\<^sub>1 s else P\<^sub>2 s, S\<^sub>1 \<and> S\<^sub>2) ) (prevc C\<^sub>2 Q)) (prevc C\<^sub>1 Q)" |
"prevc ({I} WHILE b DO C) Q = (\<lambda>(P, S).
(I, (\<forall>s.
(I s \<and> bval b s \<longrightarrow> P s) \<and>
(I s \<and> \<not> bval b s \<longrightarrow> Q s)) \<and>
S)) (prevc C I)"
text \<open> that traverses the command only once. Prove \<close>
lemma "prevc C Q = (pre C Q, vc C Q)"
by (induct C arbitrary: Q) auto
text \<open>
\endexercise
\exercise
Design a VCG that computes post rather than preconditions.
Start by solving Exercise~\ref{exe:fwdassign}. Now modify theory
@{short_theory "VCG"} as follows. Instead of @{const pre} define a function
\<close>
fun post :: "acom \<Rightarrow> assn \<Rightarrow> assn" where
"post SKIP P = P" |
"post (x ::= a) P = (\<lambda>s. \<exists> x'. P (s(x := x')) \<and> s x = aval a (s(x := x')))" |
"post (C\<^sub>1;; C\<^sub>2) P = post C\<^sub>2 (post C\<^sub>1 P)" |
"post (IF b THEN C\<^sub>1 ELSE C\<^sub>2) P =
(\<lambda>t. post C\<^sub>1 (\<lambda>s. P s \<and> bval b s) t \<or> post C\<^sub>2 (\<lambda>s. P s \<and> \<not>bval b s) t)" |
"post ({I} WHILE b DO c) P = (\<lambda>s. I s \<and> \<not>bval b s)"
text \<open>
such that (with the execption of loops) @{term "post C P"} is the strongest
postcondition of @{text C} w.r.t.\ the precondition @{text P} (see also
Exercise~\ref{exe:sp}). Now modify @{const vc} such that is uses
@{const post} instead of @{const pre} and prove its soundness
and completeness.
\<close>
fun vc :: "acom \<Rightarrow> assn \<Rightarrow> bool" where
"vc SKIP P = True" |
"vc (x ::= a) P = True" |
"vc (C\<^sub>1;; C\<^sub>2) P \<longleftrightarrow> vc C\<^sub>1 P \<and> vc C\<^sub>2 (post C\<^sub>1 P)" |
"vc (IF b THEN C\<^sub>1 ELSE C\<^sub>2) P \<longleftrightarrow> vc C\<^sub>1 (\<lambda>s. P s \<and> bval b s) \<and> vc C\<^sub>2 (\<lambda>s. P s \<and> \<not>bval b s)" |
"vc ({I} WHILE b DO C) P =
((\<forall>s. (post C (\<lambda>s. I s \<and> bval b s) s \<longrightarrow> I s) \<and>
(P s \<longrightarrow> I s)) \<and>
vc C (\<lambda>s. I s \<and> bval b s))"
lemma vc_sound: "vc C P \<Longrightarrow> \<turnstile> {P} strip C {post C P}"
proof (induction C arbitrary: P)
case (Aassign x x2)
show ?case
proof (rule strengthen_pre; (simp, rule hoare.Assign)?; simp, intro allI impI)
fix s :: state
let ?H = "\<lambda>x'. P (s(x := x')) \<and> aval x2 s = aval x2 (s(x := x'))"
assume "P s"
then have "?H (s x)" by auto
from exI [of ?H, OF this]
show "\<exists>x'. ?H x'" by simp
qed
next
case (Aif b C1 C2)
show ?case by (simp, rule hoare.If; rule weaken_post; (rule Aif(1) | rule Aif(2))?; insert Aif(3)) auto
next
case (Awhile I b C)
let ?c = "strip C"
show ?case by (simp, rule strengthen_pre; (rule hoare.While, rule weaken_post; (rule Awhile(1))?)?; insert Awhile(2); auto)
qed auto
lemma post_mono: "\<forall>s. P s \<longrightarrow> P' s \<Longrightarrow> post C P s \<Longrightarrow> post C P' s"
proof (induction C arbitrary: P P' s)
case (Aassign x1 x2)
then show ?case by simp metis
next
case (Aseq C1 C2)
then show ?case by simp metis
next
case (Aif b C1 C2)
let ?PT = "\<lambda>P s. P s \<and> bval b s"
let ?PF = "\<lambda>P s. P s \<and> \<not>bval b s"
from Aif(3) have HT: "\<forall>s. ?PT P s \<longrightarrow> ?PT P' s" by simp
from Aif(3) have HF: "\<forall>s. ?PF P s \<longrightarrow> ?PF P' s" by simp
from Aif(4) consider (T) "post C1 (?PT P) s" | (F) "post C2 (?PF P) s" by auto
then show ?case
proof cases
case T
from Aif(1) [OF HT T] show ?thesis by simp
next
case F
from Aif(2) [OF HF F] show ?thesis by simp
qed
qed auto
lemma vc_antimono: "\<forall>s. P s \<longrightarrow> P' s \<Longrightarrow> vc C P' \<Longrightarrow> vc C P"
proof(induct C arbitrary: P P')
case (Aseq C1 C2) thus ?case by simp (metis post_mono)
next
case (Aif b C1 C2)
let ?PT = "\<lambda>P s. P s \<and> bval b s"
let ?PF = "\<lambda>P s. P s \<and> \<not>bval b s"
from Aif(3) have HT: "\<forall>s. ?PT P s \<longrightarrow> ?PT P' s" by simp
from Aif(3) have HF: "\<forall>s. ?PF P s \<longrightarrow> ?PF P' s" by simp
from Aif(1) [OF HT] Aif(2) [OF HF] Aif(4) show ?case by auto
qed simp_all
lemma vc_complete: "\<turnstile> {P} c {Q}
\<Longrightarrow> \<exists>C. strip C = c \<and> vc C P \<and> (\<forall>s. post C P s \<longrightarrow> Q s)"
(is "_ \<Longrightarrow> \<exists>C. ?G P c Q C")
proof (induct rule: hoare.induct)
case (Skip P)
show ?case (is "\<exists>C. ?C C")
by (rule exI [of ?C Askip], simp)
next
case (Assign P a x)
show ?case (is "\<exists>C. ?C C")
proof (rule exI [of ?C "Aassign x a"], auto)
fix s :: state and x'
assume "s x = aval a (s(x := x'))"
then have "s(x := aval a (s(x := x'))) = s" by auto
moreover assume "P (s(x := aval a (s(x := x'))))"
ultimately show "P s" by simp
qed
next
case (Seq P c\<^sub>1 Q c\<^sub>2 R)
from Seq(4) obtain C\<^sub>2 where IH2: "?G Q c\<^sub>2 R C\<^sub>2" by blast
from Seq(2) obtain C\<^sub>1 where IH1: "?G P c\<^sub>1 Q C\<^sub>1" by blast
have "?G P (c\<^sub>1;; c\<^sub>2) R (C\<^sub>1;; C\<^sub>2)"
proof (intro conjI)
from IH1 IH2 show "strip (C\<^sub>1;; C\<^sub>2) = c\<^sub>1;; c\<^sub>2" by auto
from IH1 IH2 show "vc (C\<^sub>1;; C\<^sub>2) P" by (fastforce elim!: post_mono vc_antimono)
show "\<forall>s. post (C\<^sub>1;; C\<^sub>2) P s \<longrightarrow> R s"
proof (intro allI)
fix s
from IH1 have "post C\<^sub>2 (post C\<^sub>1 P) s \<Longrightarrow> post C\<^sub>2 Q s" by (auto elim!: post_mono)
with IH2 show "post (C\<^sub>1;; C\<^sub>2) P s \<longrightarrow> R s" by auto
qed
qed
then show ?case by blast
next
case (If P b c\<^sub>1 Q c\<^sub>2)
from If(2) obtain C\<^sub>1 where IH1: "?G (\<lambda>s. P s \<and> bval b s) c\<^sub>1 Q C\<^sub>1" by blast
from If(4) obtain C\<^sub>2 where IH2: "?G (\<lambda>s. P s \<and> \<not>bval b s) c\<^sub>2 Q C\<^sub>2" by blast
from IH1 IH2 have "?G P (IF b THEN c\<^sub>1 ELSE c\<^sub>2) Q (IF b THEN C\<^sub>1 ELSE C\<^sub>2)" by simp
then show ?case by blast
next
case (While P b c)
from While(2) obtain C where IH: "?G (\<lambda>s. P s \<and> bval b s) c P C" by blast
then have "?G P (WHILE b DO c) (\<lambda>s. P s \<and> \<not>bval b s) ({P} WHILE b DO C)" by auto
then show ?case by blast
next
case (conseq P' P c Q Q')
from conseq(3) obtain C where HC: "strip C = c" "vc C P" "\<forall>s. post C P s \<longrightarrow> Q s" by blast+
from conseq(1) HC(2) have "vc C P'" by (simp add: vc_antimono)
moreover from conseq(1, 4) HC(3) have "\<forall>s. post C P' s \<longrightarrow> Q' s" by (simp add: post_mono vc_antimono)
ultimately show ?case using HC(1) by auto
qed
text \<open>
\endexercise
\<close>
end
|
State Before: α : Type u_1
β : Type ?u.122213
γ : Type ?u.122216
ι : Sort ?u.122219
ι' : Sort ?u.122222
ι₂ : Sort ?u.122225
κ : ι → Sort ?u.122230
κ₁ : ι → Sort ?u.122235
κ₂ : ι → Sort ?u.122240
κ' : ι' → Sort ?u.122245
S : Set (Set α)
⊢ Set.Nonempty (⋃₀ S) ↔ ∃ s, s ∈ S ∧ Set.Nonempty s State After: no goals Tactic: simp [nonempty_iff_ne_empty] |
/-
Copyright (c) 2020 Aaron Anderson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Aaron Anderson
-/
import data.polynomial.degree.definitions
/-!
# Cancel the leading terms of two polynomials
## Definition
* `cancel_leads p q`: the polynomial formed by multiplying `p` and `q` by monomials so that they
have the same leading term, and then subtracting.
## Main Results
The degree of `cancel_leads` is less than that of the larger of the two polynomials being cancelled.
Thus it is useful for induction or minimal-degree arguments.
-/
namespace polynomial
noncomputable theory
variables {R : Type*}
section comm_ring
variables [comm_ring R] (p q : polynomial R)
/-- `cancel_leads p q` is formed by multiplying `p` and `q` by monomials so that they
have the same leading term, and then subtracting. -/
def cancel_leads : polynomial R :=
C p.leading_coeff * X ^ (p.nat_degree - q.nat_degree) * q -
C q.leading_coeff * X ^ (q.nat_degree - p.nat_degree) * p
variables {p q}
@[simp] lemma neg_cancel_leads : - p.cancel_leads q = q.cancel_leads p := neg_sub _ _
lemma dvd_cancel_leads_of_dvd_of_dvd {r : polynomial R} (pq : p ∣ q) (pr : p ∣ r) :
p ∣ q.cancel_leads r :=
dvd_sub (pr.trans (dvd.intro_left _ rfl)) (pq.trans (dvd.intro_left _ rfl))
end comm_ring
lemma nat_degree_cancel_leads_lt_of_nat_degree_le_nat_degree [comm_ring R] [is_domain R]
{p q : polynomial R} (h : p.nat_degree ≤ q.nat_degree) (hq : 0 < q.nat_degree) :
(p.cancel_leads q).nat_degree < q.nat_degree :=
begin
by_cases hp : p = 0,
{ convert hq,
simp [hp, cancel_leads], },
rw [cancel_leads, sub_eq_add_neg, tsub_eq_zero_iff_le.mpr h, pow_zero, mul_one],
by_cases h0 :
C p.leading_coeff * q + -(C q.leading_coeff * X ^ (q.nat_degree - p.nat_degree) * p) = 0,
{ convert hq,
simp only [h0, nat_degree_zero], },
have hq0 : ¬ q = 0,
{ contrapose! hq,
simp [hq] },
apply lt_of_le_of_ne,
{ rw [← with_bot.coe_le_coe, ← degree_eq_nat_degree h0, ← degree_eq_nat_degree hq0],
apply le_trans (degree_add_le _ _),
rw ← leading_coeff_eq_zero at hp hq0,
simp only [max_le_iff, degree_C hp, degree_C hq0, le_refl q.degree, true_and, nat.cast_with_bot,
nsmul_one, degree_neg, degree_mul, zero_add, degree_X, degree_pow],
rw leading_coeff_eq_zero at hp hq0,
rw [degree_eq_nat_degree hp, degree_eq_nat_degree hq0, ← with_bot.coe_add, with_bot.coe_le_coe,
tsub_add_cancel_of_le h], },
{ contrapose! h0,
rw [← leading_coeff_eq_zero, leading_coeff, h0, mul_assoc, mul_comm _ p,
← tsub_add_cancel_of_le h, add_comm _ p.nat_degree],
simp only [coeff_mul_X_pow, coeff_neg, coeff_C_mul, add_tsub_cancel_left, coeff_add],
rw [add_comm p.nat_degree, tsub_add_cancel_of_le h, ← leading_coeff, ← leading_coeff,
mul_comm _ q.leading_coeff, ← sub_eq_add_neg, ← mul_sub, sub_self, mul_zero] }
end
end polynomial
|
module SizedIO.coIOIO where
open import Size
mutual
data coIO² (i : Size) (j : Size)
(Cin : Set ) (Rin : Cin → Set)
(Cext : Set) (Rext : Cext → Set)
(A : Set) : Set where
return : A → coIO² i j Cin Rin Cext Rext A
dof : (i' : Size< i) →
(c : Cin) → (Rin c → coIO² i' j Cin Rin Cext Rext A)
→ coIO² i j Cin Rin Cext Rext A
do∞ : (c : Cext)
→ (Rext c → coIO²∞ j Cin Rin Cext Rext A)
→ coIO² i j Cin Rin Cext Rext A
record coIO²∞ (j : Size) (Cin : Set ) (Rin : Cin → Set)
(Cext : Set) (Rext : Cext → Set)
(A : Set) : Set where
coinductive
field
force : (j' : Size< j)
→ coIO² ∞ j' Cin Rin Cext Rext A
open coIO²∞ public
|
theory Chapter12_3
imports "HOL-IMP.Hoare_Sound_Complete"
begin
text\<open>
\exercise
Prove
\<close>
lemma "\<Turnstile> {P} c {Q} \<longleftrightarrow> (\<forall>s. P s \<longrightarrow> wp c Q s)"
(* your definition/proof here *)
text\<open>
\endexercise
\begin{exercise}
Replace the assignment command with a new command \mbox{@{term"Do f"}} where
@{text "f ::"} @{typ "state \<Rightarrow> state"} can be an arbitrary state transformer.
Update the big-step semantics, Hoare logic and the soundness and completeness proofs.
\end{exercise}
\exercise
Which of the following rules are correct? Proof or counterexample!
\<close>
lemma "\<lbrakk>\<turnstile> {P} c {Q}; \<turnstile> {P'} c {Q'}\<rbrakk> \<Longrightarrow>
\<turnstile> {\<lambda>s. P s \<or> P' s} c {\<lambda>s. Q s \<or> Q' s}"
(* your definition/proof here *)
lemma "\<lbrakk>\<turnstile> {P} c {Q}; \<turnstile> {P'} c {Q'}\<rbrakk> \<Longrightarrow>
\<turnstile> {\<lambda>s. P s \<and> P' s} c {\<lambda>s. Q s \<and> Q' s}"
(* your definition/proof here *)
lemma "\<lbrakk>\<turnstile> {P} c {Q}; \<turnstile> {P'} c {Q'}\<rbrakk> \<Longrightarrow>
\<turnstile> {\<lambda>s. P s \<longrightarrow> P' s} c {\<lambda>s. Q s \<longrightarrow> Q' s}"
(* your definition/proof here *)
text\<open>
\endexercise
\begin{exercise}
Based on Exercise~\ref{exe:IMP:OR}, extend Hoare logic and the soundness and completeness proofs
with nondeterministic choice.
\end{exercise}
\begin{exercise}
Based on Exercise~\ref{exe:IMP:REPEAT}, extend Hoare logic and the soundness and completeness proofs
with a @{text REPEAT} loop. Hint: think of @{text"REPEAT c UNTIL b"} as
equivalent to \noquotes{@{term[source]"c;; WHILE Not b DO c"}}.
\end{exercise}
\exercise\label{exe:sp}
The dual of the weakest precondition is the \concept{strongest postcondition}
@{text sp}. Define @{text sp} in analogy with @{const wp} via the big-step semantics:
\<close>
definition sp :: "com \<Rightarrow> assn \<Rightarrow> assn" where
(* your definition/proof here *)
text\<open> Prove that @{const sp} really is the strongest postcondition: \<close>
lemma "(\<Turnstile> {P} c {Q}) \<longleftrightarrow> (\<forall>s. sp c P s \<longrightarrow> Q s)"
(* your definition/proof here *)
text\<open>
In analogy with the derived equations for @{const wp} given in the text,
give and prove equations for \<open>\<close>calculating'' @{const sp} for three constructs:
@{prop"sp (x ::= a) P = Q\<^sub>1"}, @{prop"sp (c\<^sub>1;;c\<^sub>2) P = Q\<^sub>2"}, and
@{prop"sp (IF b THEN c\<^sub>1 ELSE c\<^sub>2) P = Q\<^sub>3"}.
The @{text Q\<^sub>i} must not involve the semantics and may only call
@{const sp} recursively on the subcommands @{text c\<^sub>i}.
Hint: @{text Q\<^sub>1} requires an existential quantifier.
\<close>
text\<open>
\endexercise
\<close>
end
|
c-----------------------------------------------------------------------------
c
c Copyright (c) 2000 - 2014, Lawrence Livermore National Security, LLC
c Produced at the Lawrence Livermore National Laboratory
c LLNL-CODE-442911
c All rights reserved.
c
c This file is part of VisIt. For details, see https://visit.llnl.gov/. The
c full copyright notice is contained in the file COPYRIGHT located at the root
c of the VisIt distribution or at http://www.llnl.gov/visit/copyright.html.
c
c Redistribution and use in source and binary forms, with or without
c modification, are permitted provided that the following conditions are met:
c
c - Redistributions of source code must retain the above copyright notice,
c this list of conditions and the disclaimer below.
c - Redistributions in binary form must reproduce the above copyright notice,
c this list of conditions and the disclaimer (as noted below) in the
c documentation and/or other materials provided with the distribution.
c - Neither the name of the LLNS/LLNL nor the names of its contributors may
c be used to endorse or promote products derived from this software without
c specific prior written permission.
c
c THIS SOFTWARE IS PROVIDED BY THE COPYRIGHT HOLDERS AND CONTRIBUTORS "AS IS"
c AND ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED TO, THE
c IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR PURPOSE
c ARE DISCLAIMED. IN NO EVENT SHALL LAWRENCE LIVERMORE NATIONAL SECURITY,
c LLC, THE U.S. DEPARTMENT OF ENERGY OR CONTRIBUTORS BE LIABLE FOR ANY
c DIRECT, INDIRECT, INCIDENTAL, SPECIAL, EXEMPLARY, OR CONSEQUENTIAL
c DAMAGES (INCLUDING, BUT NOT LIMITED TO, PROCUREMENT OF SUBSTITUTE GOODS OR
c SERVICES; LOSS OF USE, DATA, OR PROFITS; OR BUSINESS INTERRUPTION) HOWEVER
c CAUSED AND ON ANY THEORY OF LIABILITY, WHETHER IN CONTRACT, STRICT
c LIABILITY, OR TORT (INCLUDING NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY
c OUT OF THE USE OF THIS SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF SUCH
c DAMAGE.
c
c-----------------------------------------------------------------------------
program main
implicit none
include "silo.inc"
integer dbfile, ierr
c Create the Silo file
ierr = dbcreate("fucdvar2d.silo", 14, DB_CLOBBER, DB_LOCAL,
. "Unstructured 2D mesh with variable", 34,
. DB_HDF5, dbfile)
if(dbfile.eq.-1) then
write (6,*) 'Could not create Silo file!\n'
goto 10000
endif
c Add other Silo calls here.
call write_ucd2d(dbfile)
call write_ucdvars(dbfile)
c Close the Silo file.
ierr = dbclose(dbfile)
10000 stop
end
c
c *---*---*--* nodes (5,6,7,8)
c | |\5/ 3 |
c | 1 |4*----* nodes (3,4)
c | |/ 2 |
c *---*------* nodes(0,1,2)
c
c cell 1 quad(0,1,6,5)
c cell 2 quad(1,2,4,3)
c cell 3 quad(3,4,8,7)
c cell 4 tri(1,3,6)
c cell 5 tri(3,7,6)
c
subroutine write_ucd2d(dbfile)
implicit none
integer dbfile
include "silo.inc"
integer err, ierr, ndims, nshapetypes, nnodes, nzones
c Node coordinates
real x(9) /0., 2., 5., 3., 5., 0., 2., 4., 5./
real y(9) /0., 0., 0., 3., 3., 5., 5., 5., 5./
c Connectivity
integer LNODELIST
parameter (LNODELIST = 18)
integer nodelist(LNODELIST) /2,4,7,
. 4,8,7,
. 1,2,7,6,
. 2,3,5,4,
. 4,5,9,8/
c Shape type 1 has 3 nodes (tri), shape type 2 is quad
integer shapesize(2) /3, 4/
c We have 2 tris and 3 quads
integer shapecounts(2) /2, 3/
nshapetypes = 2
nnodes = 9
nzones = 5
ndims = 2
c Write out connectivity information.
err = dbputzl(dbfile, "zonelist", 8, nzones, ndims, nodelist,
. LNODELIST, 1, shapesize, shapecounts, nshapetypes, ierr)
c Write an unstructured mesh
err = dbputum(dbfile, "mesh", 4, ndims, x, y, DB_F77NULL,
. "X", 1, "Y", 1, DB_F77NULL, 0, DB_FLOAT, nnodes, nzones,
. "zonelist", 8, DB_F77NULL, 0, DB_F77NULL, ierr)
end
subroutine write_ucdvars(dbfile)
implicit none
integer dbfile
include "silo.inc"
integer err, ierr, NNODES, NZONES, optlistid
parameter (NNODES = 9)
parameter (NZONES = 5)
real nodal(NNODES) /1.,2.,3.,4.,5.,6.,7.,8.,9./
real zonal(NZONES) /1.,2.,3.,4.,5./
err = dbmkoptlist(1, optlistid)
err = dbaddcopt(optlistid, DBOPT_UNITS, "g/cc", 4)
c Write a zone-centered variable.
err = dbputuv1(dbfile, "zonal", 5, "mesh", 4, zonal, NZONES,
. DB_F77NULL, 0, DB_FLOAT, DB_ZONECENT, optlistid, ierr)
err = dbfreeoptlist(optlistid)
c Write a node-centered variable.
err = dbputuv1(dbfile, "nodal", 5, "mesh", 4, nodal, NNODES,
. DB_F77NULL, 0, DB_FLOAT, DB_NODECENT, DB_F77NULL, ierr)
end
|
* hot_100: https://leetcode-cn.com/problemset/leetcode-hot-100/
leetcode提供的最热100道题，可以快速刷下
* picked_200 : https://leetcode-cn.com/problemset/leetcode-200/
leetcode精选的200道算法题目，大部分都是locked，T_T
* 网上找到的一个帖子积累的经典的题目：
https://blog.csdn.net/haolexiao/article/details/53487436
* curated list of top 100， 有分类，感觉还不错的
https://h1ros.github.io/posts/coding/leetcode-top-100-problem-selection/
* 网上看到的一些算法题的总结，可以用来思考下：
https://wizardforcel.gitbooks.io/the-art-of-programming-by-july/content/02.15.html
|
{-# LANGUAGE CPP, TypeOperators, TypeFamilies #-}
{-# LANGUAGE DeriveFunctor, DeriveDataTypeable #-}
{-# OPTIONS_GHC -Wall #-}
-- {-# OPTIONS_GHC -fno-warn-unused-imports #-} -- TEMP
-- {-# OPTIONS_GHC -fno-warn-unused-binds #-} -- TEMP
#define CustomComplex
#if !defined CustomComplex
{-# OPTIONS_GHC -fno-warn-orphans #-} -- for HasRep
#endif
----------------------------------------------------------------------
-- |
-- Module : ConCat.Complex
-- Copyright : (c) 2015 Conal Elliott and David Banas
-- License : BSD3
--
-- Maintainer : [email protected]
-- Stability : experimental
--
-- Simplified Complex type
----------------------------------------------------------------------
module ConCat.Complex (Complex(..),cis) where
#if defined CustomComplex
import Control.Applicative (liftA2)
import Data.Typeable
import Data.Data
import Test.QuickCheck (Arbitrary(..),CoArbitrary(..))
#else
import Data.Complex
#endif
#if defined CustomComplex
infixl 1 :+
data Complex a = a :+ a deriving (Functor,Eq,Show,Typeable,Data,Ord)
-- | The nonnegative magnitude of a complex number.
-- {-# SPECIALISE magnitude :: Complex Double -> Double #-}
magnitude :: (RealFloat a) => Complex a -> a
magnitude (x:+y) = sqrt (x*x + y*y)
-- From Data.Complex
-- magnitude (x:+y) = scaleFloat k
-- (sqrt (sqr (scaleFloat mk x) + sqr (scaleFloat mk y)))
-- where k = max (exponent x) (exponent y)
-- mk = - k
-- sqr z = z * z
-- | The phase of a complex number, in the range @(-'pi', 'pi']@.
-- If the magnitude is zero, then so is the phase.
-- {-# SPECIALISE phase :: Complex Double -> Double #-}
phase :: (RealFloat a) => Complex a -> a
phase (0 :+ 0) = 0 -- SLPJ July 97 from John Peterson
phase (x:+y) = atan2 y x
-- | Complex Identity Scaler
-- Returns a point on the unit circle, such that the angle between the
-- vector to that point and the positive x-axis is equal to the function
-- argument, given in radians.
cis :: (RealFloat a) => a -> Complex a
cis w = cos w :+ sin w
instance (RealFloat a) => Num (Complex a) where
-- {-# SPECIALISE instance Num (Complex Float) #-}
-- {-# SPECIALISE instance Num (Complex Double) #-}
(x:+y) + (x':+y') = (x+x') :+ (y+y')
(x:+y) - (x':+y') = (x-x') :+ (y-y')
(x:+y) * (x':+y') = (x*x'-y*y') :+ (x*y'+y*x')
negate (x:+y) = negate x :+ negate y
abs z = magnitude z :+ 0
signum (0:+0) = 0
signum z@(x:+y) = x/r :+ y/r where r = magnitude z
fromInteger n = fromInteger n :+ 0
{-# INLINE (+) #-}
{-# INLINE (-) #-}
{-# INLINE (*) #-}
{-# INLINE negate #-}
{-# INLINE abs #-}
{-# INLINE signum #-}
{-# INLINE fromInteger #-}
instance (RealFloat a) => Fractional (Complex a) where
-- {-# SPECIALISE instance Fractional (Complex Float) #-}
-- {-# SPECIALISE instance Fractional (Complex Double) #-}
recip z@(x:+y) = x/mag :+ (-y)/mag where mag = magnitude z
-- (x:+y) / (x':+y') = (x*x''+y*y'') / d :+ (y*x''-x*y'') / d
-- where x'' = scaleFloat k x'
-- y'' = scaleFloat k y'
-- k = - max (exponent x') (exponent y')
-- d = x'*x'' + y'*y''
fromRational a = fromRational a :+ 0
instance (RealFloat a) => Floating (Complex a) where
-- {-# SPECIALISE instance Floating (Complex Float) #-}
-- {-# SPECIALISE instance Floating (Complex Double) #-}
pi = pi :+ 0
exp (x:+y) = expx * cos y :+ expx * sin y
where expx = exp x
log z = log (magnitude z) :+ phase z
sqrt (0:+0) = 0
sqrt z@(x:+y) = u :+ (if y < 0 then -v else v)
where (u,v) = if x < 0 then (v',u') else (u',v')
v' = abs y / (u'*2)
u' = sqrt ((magnitude z + abs x) / 2)
sin (x:+y) = sin x * cosh y :+ cos x * sinh y
cos (x:+y) = cos x * cosh y :+ (- sin x * sinh y)
tan (x:+y) = (sinx*coshy:+cosx*sinhy)/(cosx*coshy:+(-sinx*sinhy))
where sinx = sin x
cosx = cos x
sinhy = sinh y
coshy = cosh y
sinh (x:+y) = cos y * sinh x :+ sin y * cosh x
cosh (x:+y) = cos y * cosh x :+ sin y * sinh x
tanh (x:+y) = (cosy*sinhx:+siny*coshx)/(cosy*coshx:+siny*sinhx)
where siny = sin y
cosy = cos y
sinhx = sinh x
coshx = cosh x
asin z@(x:+y) = y':+(-x')
where (x':+y') = log (((-y):+x) + sqrt (1 - z*z))
acos z = y'':+(-x'')
where (x'':+y'') = log (z + ((-y'):+x'))
(x':+y') = sqrt (1 - z*z)
atan z@(x:+y) = y':+(-x')
where (x':+y') = log (((1-y):+x) / sqrt (1+z*z))
asinh z = log (z + sqrt (1+z*z))
acosh z = log (z + (z+1) * sqrt ((z-1)/(z+1)))
atanh z = 0.5 * log ((1.0+z) / (1.0-z))
instance Arbitrary a => Arbitrary (Complex a) where
arbitrary = liftA2 (:+) arbitrary arbitrary
shrink (x :+ y) = map (uncurry (:+)) (shrink (x,y))
instance CoArbitrary a => CoArbitrary (Complex a) where
coarbitrary (x :+ y) = coarbitrary x . coarbitrary y
#endif
|
import Kenny.sites.basic order.complete_boolean_algebra
universes u v
namespace lattice
class Sup_lattice (X : Type u) extends lattice X, has_Sup X :=
(le_Sup : ∀ {s : set X} {a : X}, a ∈ s → a ≤ Sup s)
(Sup_le : ∀ {s : set X} {a : X}, (∀ (b : X), b ∈ s → b ≤ a) → Sup s ≤ a)
class Sup_distrib_lattice (X : Type u) extends Sup_lattice X :=
(inf_Sup_le {} : ∀ {x : X} {s : set X}, x ⊓ lattice.Sup s ≤ lattice.Sup ((⊓) x '' s))
section Sup_lattice
instance complete_lattice.to_Sup_lattice {X : Type u} [complete_lattice X] : Sup_lattice X :=
{ .. (infer_instance : complete_lattice X) }
variables {X : Type u} [Sup_lattice X]
theorem le_Sup' {s : set X} {a : X} : a ∈ s → a ≤ Sup s :=
Sup_lattice.le_Sup
theorem Sup_le' {s : set X} {a : X} : (∀ (b : X), b ∈ s → b ≤ a) → Sup s ≤ a :=
Sup_lattice.Sup_le
theorem Sup_singleton' (x : X) : Sup {x} = x :=
le_antisymm (Sup_le' $ λ b hb, set.eq_of_mem_singleton hb ▸ le_refl _) $
le_Sup' $ set.mem_singleton x
end Sup_lattice
section Sup_discrete_lattice
instance complete_distrib_lattice.to_Sup_distrib_lattice {X : Type u} [complete_distrib_lattice X] : Sup_distrib_lattice X :=
{ inf_Sup_le := λ x s, by rw [inf_Sup_eq, Sup_image],
.. (infer_instance : complete_distrib_lattice X) }
variables {X : Type u} [Sup_distrib_lattice X]
theorem inf_Sup {x : X} {s : set X} : x ⊓ lattice.Sup s = lattice.Sup ((⊓) x '' s) :=
le_antisymm Sup_distrib_lattice.inf_Sup_le $ Sup_le' $ λ b ⟨c, hcs, hxcb⟩, hxcb ▸ inf_le_inf (le_refl x) (le_Sup' hcs)
end Sup_discrete_lattice
end lattice
namespace category_theory
open lattice
variables {X : Type u}
class is_univalent (X : Type u) [category.{v} X] : Prop :=
(univalent : ∀ x y : X, ∀ e : x ≅ y, x = y)
theorem eq_of_iso [category.{v} X] [is_univalent X] {x y : X} (e : x ≅ y) : x = y :=
is_univalent.univalent x y e
instance is_univalent_partial_order [partial_order X] : is_univalent X :=
⟨λ x y e, le_antisymm e.1.1.1 e.2.1.1⟩
instance semilattice_inf.has_pullback [semilattice_inf X] : has_pullback X :=
⟨λ F,
{ cone :=
{ X := F.obj pullback_diagram.base_left ⊓ F.obj pullback_diagram.base_right,
π :=
{ app := λ p, pullback_diagram.rec_on p ⟨⟨inf_le_left⟩⟩ ⟨⟨inf_le_right⟩⟩
⟨⟨le_trans inf_le_left (F.map pullback_diagram.hom.to_target_left).down.down⟩⟩,
naturality' := by intros; ext } },
is_limit :=
{ lift := λ c, ⟨⟨le_inf (c.π.app pullback_diagram.base_left).down.down (c.π.app pullback_diagram.base_right).down.down⟩⟩,
fac' := by intros; ext,
uniq' := by intros; ext } }⟩
instance Sup_lattice.has_site [Sup_distrib_lattice X] : has_site X :=
{ cov := λ U, { c | U ≤ Sup (sigma.fst '' c) },
iso_mem := λ U V e, show U ≤ _, by rw [set.image_singleton, Sup_singleton']; exact e.2.1.1,
comp_mem := λ U S HS F HF, le_trans HS $ Sup_le' $ λ x hx, let ⟨m, hmS, hmx⟩ := hx in
hmx ▸ le_trans (HF m hmS) (Sup_le' $ λ y hy, let ⟨n, hnFS, hny⟩ := hy in
le_Sup' ⟨⟨n.1, ⟨⟨le_trans n.2.1.1 m.2.1.1⟩⟩⟩, ⟨m, hmS, n, hnFS, rfl⟩, hny⟩),
pullback_mem := λ U S HS V f,
calc V
≤ V ⊓ Sup (sigma.fst '' S) : le_inf (le_refl V) (le_trans f.1.1 HS)
... = Sup ((⊓) V '' (sigma.fst '' S)) : inf_Sup
... = Sup ((⊓) V ∘ sigma.fst '' S) : congr_arg Sup (set.image_comp _ _ S).symm
... ≤ Sup (sigma.fst '' {m | ∃ t ∈ S, (⟨_, pullback.fst f t.2⟩ : Σ W, W ⟶ V) = m}) :
Sup_le' (λ b ⟨c, hcs, hb⟩, le_Sup' ⟨⟨V ⊓ c.1, ⟨⟨inf_le_left⟩⟩⟩, ⟨c, hcs, rfl⟩, hb⟩) }
end category_theory
|
Formal statement is: lemma bigo_bigomega_trans: "f \<in> O[F](g) \<Longrightarrow> h \<in> \<Omega>[F](g) \<Longrightarrow> f \<in> O[F](h)" and bigo_smallomega_trans: "f \<in> O[F](g) \<Longrightarrow> h \<in> \<omega>[F](g) \<Longrightarrow> f \<in> o[F](h)" and smallo_bigomega_trans: "f \<in> o[F](g) \<Longrightarrow> h \<in> \<Omega>[F](g) \<Longrightarrow> f \<in> o[F](h)" and smallo_smallomega_trans: "f \<in> o[F](g) \<Longrightarrow> h \<in> \<omega>[F](g) \<Longrightarrow> f \<in> o[F](h)" and bigomega_bigo_trans: "f \<in> \<Omega>[F](g) \<Longrightarrow> h \<in> O[F](g) \<Longrightarrow> f \<in> \<Omega>[F](h)" and bigomega_smallo_trans: "f \<in> \<Omega>[F](g) \<Longrightarrow> h \<in> o[F](g) \<Longrightarrow> f \<in> \<omega>[F](h)" and smallomega_bigo_trans: "f \<in> \<omega>[F](g) \<Longrightarrow> h \<in> O[F](g) \<Longrightarrow> f \<in> \<omega>[F](h)" and smallomega_smallo_trans: "f \<in> \<omega>[F](g) \<Longrightarrow> h \<in> o[F](g) \<Longrightarrow> f \<in> \<omega>[F](h)" Informal statement is: If $f \in O(g)$ and $h \in \Omega(g)$, then $f \in O(h)$. |
Require Import Ascii.
Inductive Decidable (P : Prop) : Prop :=
| dec_true : P -> Decidable P
| dec_false : ~ P -> Decidable P.
Inductive pbool : Prop :=
| ptrue
| pfalse.
Definition compute_dec (P : Prop) {d : Decidable P} :=
match d with
| dec_true _ _ => ptrue
| dec_false _ _ => pfalse
end.
Lemma DecidableBoolEq (a b : bool) : (Decidable (a = b)).
Proof.
destruct a; destruct b; try (now left); now right.
Qed.
Lemma DecidableAsciiEq (a b : ascii) : (Decidable (a = b)).
Proof.
assert ({a = b} + {a <> b}) by repeat decide equality.
destruct H; [ now left | now right ].
Defined.
|
lemma (in algebra) is_sigma_algebra: assumes "finite M" shows "sigma_algebra \<Omega> M" |
module Simpson
using Base.Threads
function simpson(func, x1::BigFloat, x2::BigFloat, n)
dh = (x2 - x1) / BigFloat(n)
# Simpsonの公式によって数値積分する
m = div(n, 2)
results = Array{BigFloat}(undef, m)
@inbounds @threads for i = 1:m
fx1 = func(x1 + dh * BigFloat(i * 2 - 1))
fx2 = func(x1 + dh * BigFloat(i * 2))
results[i] = BigFloat(4) * fx1 + BigFloat(2) * fx2
end
sum = func(x1) + reduce(+, results) - func(x2)
return dh / BigFloat(3) * sum
end
end |
[GOAL]
X Y : Scheme
f : X ⟶ Y
x✝ : QuasiCompact f
h : ∀ (U : Set ↑↑Y.toPresheafedSpace), IsOpen U → IsCompact U → IsCompact (↑f.val.base ⁻¹' U)
⊢ Continuous ↑f.val.base
[PROOFSTEP]
continuity
[GOAL]
X✝ Y✝ : Scheme
f✝ : X✝ ⟶ Y✝
X Y : Scheme
f : X ⟶ Y
inst✝ : IsIso f
⊢ QuasiCompact f
[PROOFSTEP]
constructor
[GOAL]
case isCompact_preimage
X✝ Y✝ : Scheme
f✝ : X✝ ⟶ Y✝
X Y : Scheme
f : X ⟶ Y
inst✝ : IsIso f
⊢ ∀ (U : Set ↑↑Y.toPresheafedSpace), IsOpen U → IsCompact U → IsCompact (↑f.val.base ⁻¹' U)
[PROOFSTEP]
intro U _ hU'
[GOAL]
case isCompact_preimage
X✝ Y✝ : Scheme
f✝ : X✝ ⟶ Y✝
X Y : Scheme
f : X ⟶ Y
inst✝ : IsIso f
U : Set ↑↑Y.toPresheafedSpace
a✝ : IsOpen U
hU' : IsCompact U
⊢ IsCompact (↑f.val.base ⁻¹' U)
[PROOFSTEP]
convert hU'.image (inv f.1.base).continuous_toFun using 1
[GOAL]
case h.e'_3.h
X✝ Y✝ : Scheme
f✝ : X✝ ⟶ Y✝
X Y : Scheme
f : X ⟶ Y
inst✝ : IsIso f
U : Set ↑↑Y.toPresheafedSpace
a✝ : IsOpen U
hU' : IsCompact U
e_1✝ : (forget TopCat).obj ↑X.toPresheafedSpace = ↑↑X.toPresheafedSpace
⊢ ↑f.val.base ⁻¹' U = (inv f.val.base).toFun '' U
[PROOFSTEP]
rw [Set.image_eq_preimage_of_inverse]
[GOAL]
case h.e'_3.h.h₁
X✝ Y✝ : Scheme
f✝ : X✝ ⟶ Y✝
X Y : Scheme
f : X ⟶ Y
inst✝ : IsIso f
U : Set ↑↑Y.toPresheafedSpace
a✝ : IsOpen U
hU' : IsCompact U
e_1✝ : (forget TopCat).obj ↑X.toPresheafedSpace = ↑↑X.toPresheafedSpace
⊢ Function.LeftInverse (↑f.val.base) (inv f.val.base).toFun
case h.e'_3.h.h₂
X✝ Y✝ : Scheme
f✝ : X✝ ⟶ Y✝
X Y : Scheme
f : X ⟶ Y
inst✝ : IsIso f
U : Set ↑↑Y.toPresheafedSpace
a✝ : IsOpen U
hU' : IsCompact U
e_1✝ : (forget TopCat).obj ↑X.toPresheafedSpace = ↑↑X.toPresheafedSpace
⊢ Function.RightInverse (↑f.val.base) (inv f.val.base).toFun
[PROOFSTEP]
delta Function.LeftInverse
[GOAL]
case h.e'_3.h.h₁
X✝ Y✝ : Scheme
f✝ : X✝ ⟶ Y✝
X Y : Scheme
f : X ⟶ Y
inst✝ : IsIso f
U : Set ↑↑Y.toPresheafedSpace
a✝ : IsOpen U
hU' : IsCompact U
e_1✝ : (forget TopCat).obj ↑X.toPresheafedSpace = ↑↑X.toPresheafedSpace
⊢ ∀ (x : ↑↑Y.toPresheafedSpace), ↑f.val.base (ContinuousMap.toFun (inv f.val.base) x) = x
case h.e'_3.h.h₂
X✝ Y✝ : Scheme
f✝ : X✝ ⟶ Y✝
X Y : Scheme
f : X ⟶ Y
inst✝ : IsIso f
U : Set ↑↑Y.toPresheafedSpace
a✝ : IsOpen U
hU' : IsCompact U
e_1✝ : (forget TopCat).obj ↑X.toPresheafedSpace = ↑↑X.toPresheafedSpace
⊢ Function.RightInverse (↑f.val.base) (inv f.val.base).toFun
[PROOFSTEP]
exacts [IsIso.inv_hom_id_apply f.1.base, IsIso.hom_inv_id_apply f.1.base]
[GOAL]
X✝ Y✝ : Scheme
f✝ : X✝ ⟶ Y✝
X Y Z : Scheme
f : X ⟶ Y
g : Y ⟶ Z
inst✝¹ : QuasiCompact f
inst✝ : QuasiCompact g
⊢ QuasiCompact (f ≫ g)
[PROOFSTEP]
constructor
[GOAL]
case isCompact_preimage
X✝ Y✝ : Scheme
f✝ : X✝ ⟶ Y✝
X Y Z : Scheme
f : X ⟶ Y
g : Y ⟶ Z
inst✝¹ : QuasiCompact f
inst✝ : QuasiCompact g
⊢ ∀ (U : Set ↑↑Z.toPresheafedSpace), IsOpen U → IsCompact U → IsCompact (↑(f ≫ g).val.base ⁻¹' U)
[PROOFSTEP]
intro U hU hU'
[GOAL]
case isCompact_preimage
X✝ Y✝ : Scheme
f✝ : X✝ ⟶ Y✝
X Y Z : Scheme
f : X ⟶ Y
g : Y ⟶ Z
inst✝¹ : QuasiCompact f
inst✝ : QuasiCompact g
U : Set ↑↑Z.toPresheafedSpace
hU : IsOpen U
hU' : IsCompact U
⊢ IsCompact (↑(f ≫ g).val.base ⁻¹' U)
[PROOFSTEP]
rw [Scheme.comp_val_base, coe_comp, Set.preimage_comp]
[GOAL]
case isCompact_preimage
X✝ Y✝ : Scheme
f✝ : X✝ ⟶ Y✝
X Y Z : Scheme
f : X ⟶ Y
g : Y ⟶ Z
inst✝¹ : QuasiCompact f
inst✝ : QuasiCompact g
U : Set ↑↑Z.toPresheafedSpace
hU : IsOpen U
hU' : IsCompact U
⊢ IsCompact (↑f.val.base ⁻¹' (↑g.val.base ⁻¹' U))
[PROOFSTEP]
apply QuasiCompact.isCompact_preimage
[GOAL]
case isCompact_preimage.a
X✝ Y✝ : Scheme
f✝ : X✝ ⟶ Y✝
X Y Z : Scheme
f : X ⟶ Y
g : Y ⟶ Z
inst✝¹ : QuasiCompact f
inst✝ : QuasiCompact g
U : Set ↑↑Z.toPresheafedSpace
hU : IsOpen U
hU' : IsCompact U
⊢ IsOpen (↑g.val.base ⁻¹' U)
[PROOFSTEP]
exact
Continuous.isOpen_preimage
(by
-- porting note: `continuity` failed
-- see https://github.com/leanprover-community/mathlib4/issues/5030exact Scheme.Hom.continuous g)
_ hU
[GOAL]
X✝ Y✝ : Scheme
f✝ : X✝ ⟶ Y✝
X Y Z : Scheme
f : X ⟶ Y
g : Y ⟶ Z
inst✝¹ : QuasiCompact f
inst✝ : QuasiCompact g
U : Set ↑↑Z.toPresheafedSpace
hU : IsOpen U
hU' : IsCompact U
⊢ Continuous ↑g.val.base
[PROOFSTEP]
exact Scheme.Hom.continuous g
[GOAL]
case isCompact_preimage.a
X✝ Y✝ : Scheme
f✝ : X✝ ⟶ Y✝
X Y Z : Scheme
f : X ⟶ Y
g : Y ⟶ Z
inst✝¹ : QuasiCompact f
inst✝ : QuasiCompact g
U : Set ↑↑Z.toPresheafedSpace
hU : IsOpen U
hU' : IsCompact U
⊢ IsCompact (↑g.val.base ⁻¹' U)
[PROOFSTEP]
apply QuasiCompact.isCompact_preimage
[GOAL]
case isCompact_preimage.a.a
X✝ Y✝ : Scheme
f✝ : X✝ ⟶ Y✝
X Y Z : Scheme
f : X ⟶ Y
g : Y ⟶ Z
inst✝¹ : QuasiCompact f
inst✝ : QuasiCompact g
U : Set ↑↑Z.toPresheafedSpace
hU : IsOpen U
hU' : IsCompact U
⊢ IsOpen U
[PROOFSTEP]
assumption
[GOAL]
case isCompact_preimage.a.a
X✝ Y✝ : Scheme
f✝ : X✝ ⟶ Y✝
X Y Z : Scheme
f : X ⟶ Y
g : Y ⟶ Z
inst✝¹ : QuasiCompact f
inst✝ : QuasiCompact g
U : Set ↑↑Z.toPresheafedSpace
hU : IsOpen U
hU' : IsCompact U
⊢ IsCompact U
[PROOFSTEP]
assumption
[GOAL]
X✝ Y : Scheme
f : X✝ ⟶ Y
X : Scheme
U : Set ↑↑X.toPresheafedSpace
⊢ IsCompact U ∧ IsOpen U ↔ ∃ s, Set.Finite s ∧ U = ⋃ (i : ↑(Scheme.affineOpens X)) (_ : i ∈ s), ↑↑i
[PROOFSTEP]
apply Opens.IsBasis.isCompact_open_iff_eq_finite_iUnion (fun (U : X.affineOpens) => (U : Opens X.carrier))
[GOAL]
case hb
X✝ Y : Scheme
f : X✝ ⟶ Y
X : Scheme
U : Set ↑↑X.toPresheafedSpace
⊢ Opens.IsBasis (Set.range fun U => ↑U)
[PROOFSTEP]
rw [Subtype.range_coe]
[GOAL]
case hb
X✝ Y : Scheme
f : X✝ ⟶ Y
X : Scheme
U : Set ↑↑X.toPresheafedSpace
⊢ Opens.IsBasis (Scheme.affineOpens X)
[PROOFSTEP]
exact isBasis_affine_open X
[GOAL]
case hb'
X✝ Y : Scheme
f : X✝ ⟶ Y
X : Scheme
U : Set ↑↑X.toPresheafedSpace
⊢ ∀ (i : ↑(Scheme.affineOpens X)), IsCompact ↑↑i
[PROOFSTEP]
exact fun i => i.2.isCompact
[GOAL]
X Y : Scheme
f : X ⟶ Y
⊢ QuasiCompact f ↔ ∀ (U : Opens ↑↑Y.toPresheafedSpace), IsAffineOpen U → IsCompact (↑f.val.base ⁻¹' ↑U)
[PROOFSTEP]
rw [QuasiCompact_iff]
[GOAL]
X Y : Scheme
f : X ⟶ Y
⊢ (∀ (U : Set ↑↑Y.toPresheafedSpace), IsOpen U → IsCompact U → IsCompact (↑f.val.base ⁻¹' U)) ↔
∀ (U : Opens ↑↑Y.toPresheafedSpace), IsAffineOpen U → IsCompact (↑f.val.base ⁻¹' ↑U)
[PROOFSTEP]
refine' ⟨fun H U hU => H U U.isOpen hU.isCompact, _⟩
[GOAL]
X Y : Scheme
f : X ⟶ Y
⊢ (∀ (U : Opens ↑↑Y.toPresheafedSpace), IsAffineOpen U → IsCompact (↑f.val.base ⁻¹' ↑U)) →
∀ (U : Set ↑↑Y.toPresheafedSpace), IsOpen U → IsCompact U → IsCompact (↑f.val.base ⁻¹' U)
[PROOFSTEP]
intro H U hU hU'
[GOAL]
X Y : Scheme
f : X ⟶ Y
H : ∀ (U : Opens ↑↑Y.toPresheafedSpace), IsAffineOpen U → IsCompact (↑f.val.base ⁻¹' ↑U)
U : Set ↑↑Y.toPresheafedSpace
hU : IsOpen U
hU' : IsCompact U
⊢ IsCompact (↑f.val.base ⁻¹' U)
[PROOFSTEP]
obtain ⟨S, hS, rfl⟩ := (isCompact_open_iff_eq_finset_affine_union U).mp ⟨hU', hU⟩
[GOAL]
case intro.intro
X Y : Scheme
f : X ⟶ Y
H : ∀ (U : Opens ↑↑Y.toPresheafedSpace), IsAffineOpen U → IsCompact (↑f.val.base ⁻¹' ↑U)
S : Set ↑(Scheme.affineOpens Y)
hS : Set.Finite S
hU : IsOpen (⋃ (i : ↑(Scheme.affineOpens Y)) (_ : i ∈ S), ↑↑i)
hU' : IsCompact (⋃ (i : ↑(Scheme.affineOpens Y)) (_ : i ∈ S), ↑↑i)
⊢ IsCompact (↑f.val.base ⁻¹' ⋃ (i : ↑(Scheme.affineOpens Y)) (_ : i ∈ S), ↑↑i)
[PROOFSTEP]
simp only [Set.preimage_iUnion]
[GOAL]
case intro.intro
X Y : Scheme
f : X ⟶ Y
H : ∀ (U : Opens ↑↑Y.toPresheafedSpace), IsAffineOpen U → IsCompact (↑f.val.base ⁻¹' ↑U)
S : Set ↑(Scheme.affineOpens Y)
hS : Set.Finite S
hU : IsOpen (⋃ (i : ↑(Scheme.affineOpens Y)) (_ : i ∈ S), ↑↑i)
hU' : IsCompact (⋃ (i : ↑(Scheme.affineOpens Y)) (_ : i ∈ S), ↑↑i)
⊢ IsCompact (⋃ (i : ↑(Scheme.affineOpens Y)) (_ : i ∈ S), ↑f.val.base ⁻¹' ↑↑i)
[PROOFSTEP]
exact Set.Finite.isCompact_biUnion hS (fun i _ => H i i.prop)
[GOAL]
X✝ Y✝ : Scheme
f✝ : X✝ ⟶ Y✝
X Y : Scheme
f : X ⟶ Y
⊢ AffineTargetMorphismProperty.toProperty affineProperty f ↔ IsAffine Y ∧ CompactSpace ↑↑X.toPresheafedSpace
[PROOFSTEP]
delta AffineTargetMorphismProperty.toProperty QuasiCompact.affineProperty
[GOAL]
X✝ Y✝ : Scheme
f✝ : X✝ ⟶ Y✝
X Y : Scheme
f : X ⟶ Y
⊢ (∃ h, CompactSpace ↑↑X.toPresheafedSpace) ↔ IsAffine Y ∧ CompactSpace ↑↑X.toPresheafedSpace
[PROOFSTEP]
simp
[GOAL]
X Y : Scheme
f : X ⟶ Y
⊢ QuasiCompact f ↔ targetAffineLocally QuasiCompact.affineProperty f
[PROOFSTEP]
rw [quasiCompact_iff_forall_affine]
[GOAL]
X Y : Scheme
f : X ⟶ Y
⊢ (∀ (U : Opens ↑↑Y.toPresheafedSpace), IsAffineOpen U → IsCompact (↑f.val.base ⁻¹' ↑U)) ↔
targetAffineLocally QuasiCompact.affineProperty f
[PROOFSTEP]
trans ∀ U : Y.affineOpens, IsCompact (f.1.base ⁻¹' (U : Set Y.carrier))
[GOAL]
X Y : Scheme
f : X ⟶ Y
⊢ (∀ (U : Opens ↑↑Y.toPresheafedSpace), IsAffineOpen U → IsCompact (↑f.val.base ⁻¹' ↑U)) ↔
∀ (U : ↑(Scheme.affineOpens Y)), IsCompact (↑f.val.base ⁻¹' ↑↑U)
[PROOFSTEP]
exact ⟨fun h U => h U U.prop, fun h U hU => h ⟨U, hU⟩⟩
[GOAL]
X Y : Scheme
f : X ⟶ Y
⊢ (∀ (U : ↑(Scheme.affineOpens Y)), IsCompact (↑f.val.base ⁻¹' ↑↑U)) ↔ targetAffineLocally QuasiCompact.affineProperty f
[PROOFSTEP]
apply forall_congr'
[GOAL]
case h
X Y : Scheme
f : X ⟶ Y
⊢ ∀ (a : ↑(Scheme.affineOpens Y)), IsCompact (↑f.val.base ⁻¹' ↑↑a) ↔ QuasiCompact.affineProperty (f ∣_ ↑a)
[PROOFSTEP]
exact fun _ => isCompact_iff_compactSpace
[GOAL]
X Y : Scheme
f : X ⟶ Y
⊢ @QuasiCompact = targetAffineLocally QuasiCompact.affineProperty
[PROOFSTEP]
ext
[GOAL]
case h.h.h.a
X Y : Scheme
f : X ⟶ Y
x✝² x✝¹ : Scheme
x✝ : x✝² ⟶ x✝¹
⊢ QuasiCompact x✝ ↔ targetAffineLocally QuasiCompact.affineProperty x✝
[PROOFSTEP]
exact quasiCompact_iff_affineProperty _
[GOAL]
X✝ Y : Scheme
f✝ : X✝ ⟶ Y
X : Scheme
U : Opens ↑↑X.toPresheafedSpace
hU : IsCompact ↑U
f : ↑(X.presheaf.obj (op U))
⊢ IsCompact ↑(Scheme.basicOpen X f)
[PROOFSTEP]
classical
refine' ((isCompact_open_iff_eq_finset_affine_union _).mpr _).1
obtain ⟨s, hs, e⟩ := (isCompact_open_iff_eq_finset_affine_union _).mp ⟨hU, U.isOpen⟩
let g : s → X.affineOpens := by
intro V
use V.1 ⊓ X.basicOpen f
have : V.1.1 ⟶ U := by
apply homOfLE; change _ ⊆ (U : Set X.carrier); rw [e]
convert @Set.subset_iUnion₂ _ _ _ (fun (U : X.affineOpens) (_ : U ∈ s) => ↑U) V V.prop using 1
erw [← X.toLocallyRingedSpace.toRingedSpace.basicOpen_res this.op]
exact IsAffineOpen.basicOpenIsAffine V.1.prop _
haveI : Finite s := hs.to_subtype
refine' ⟨Set.range g, Set.finite_range g, _⟩
refine' (Set.inter_eq_right_iff_subset.mpr (SetLike.coe_subset_coe.2 <| RingedSpace.basicOpen_le _ _)).symm.trans _
rw [e, Set.iUnion₂_inter]
apply le_antisymm <;> apply Set.iUnion₂_subset
· intro i hi
exact
Set.Subset.trans (Set.Subset.rfl : _ ≤ g ⟨i, hi⟩)
(@Set.subset_iUnion₂ _ _ _ (fun (i : Scheme.affineOpens X) (_ : i ∈ Set.range g) => (i : Set X.toPresheafedSpace))
_ (Set.mem_range_self ⟨i, hi⟩))
· rintro ⟨i, hi⟩ ⟨⟨j, hj⟩, hj'⟩
rw [← hj']
refine' Set.Subset.trans _ (Set.subset_iUnion₂ j hj)
exact Set.Subset.rfl
[GOAL]
X✝ Y : Scheme
f✝ : X✝ ⟶ Y
X : Scheme
U : Opens ↑↑X.toPresheafedSpace
hU : IsCompact ↑U
f : ↑(X.presheaf.obj (op U))
⊢ IsCompact ↑(Scheme.basicOpen X f)
[PROOFSTEP]
refine' ((isCompact_open_iff_eq_finset_affine_union _).mpr _).1
[GOAL]
X✝ Y : Scheme
f✝ : X✝ ⟶ Y
X : Scheme
U : Opens ↑↑X.toPresheafedSpace
hU : IsCompact ↑U
f : ↑(X.presheaf.obj (op U))
⊢ ∃ s, Set.Finite s ∧ ↑(Scheme.basicOpen X f) = ⋃ (i : ↑(Scheme.affineOpens X)) (_ : i ∈ s), ↑↑i
[PROOFSTEP]
obtain ⟨s, hs, e⟩ := (isCompact_open_iff_eq_finset_affine_union _).mp ⟨hU, U.isOpen⟩
[GOAL]
case intro.intro
X✝ Y : Scheme
f✝ : X✝ ⟶ Y
X : Scheme
U : Opens ↑↑X.toPresheafedSpace
hU : IsCompact ↑U
f : ↑(X.presheaf.obj (op U))
s : Set ↑(Scheme.affineOpens X)
hs : Set.Finite s
e : ↑U = ⋃ (i : ↑(Scheme.affineOpens X)) (_ : i ∈ s), ↑↑i
⊢ ∃ s, Set.Finite s ∧ ↑(Scheme.basicOpen X f) = ⋃ (i : ↑(Scheme.affineOpens X)) (_ : i ∈ s), ↑↑i
[PROOFSTEP]
let g : s → X.affineOpens := by
intro V
use V.1 ⊓ X.basicOpen f
have : V.1.1 ⟶ U := by
apply homOfLE; change _ ⊆ (U : Set X.carrier); rw [e]
convert @Set.subset_iUnion₂ _ _ _ (fun (U : X.affineOpens) (_ : U ∈ s) => ↑U) V V.prop using 1
erw [← X.toLocallyRingedSpace.toRingedSpace.basicOpen_res this.op]
exact IsAffineOpen.basicOpenIsAffine V.1.prop _
[GOAL]
X✝ Y : Scheme
f✝ : X✝ ⟶ Y
X : Scheme
U : Opens ↑↑X.toPresheafedSpace
hU : IsCompact ↑U
f : ↑(X.presheaf.obj (op U))
s : Set ↑(Scheme.affineOpens X)
hs : Set.Finite s
e : ↑U = ⋃ (i : ↑(Scheme.affineOpens X)) (_ : i ∈ s), ↑↑i
⊢ ↑s → ↑(Scheme.affineOpens X)
[PROOFSTEP]
intro V
[GOAL]
X✝ Y : Scheme
f✝ : X✝ ⟶ Y
X : Scheme
U : Opens ↑↑X.toPresheafedSpace
hU : IsCompact ↑U
f : ↑(X.presheaf.obj (op U))
s : Set ↑(Scheme.affineOpens X)
hs : Set.Finite s
e : ↑U = ⋃ (i : ↑(Scheme.affineOpens X)) (_ : i ∈ s), ↑↑i
V : ↑s
⊢ ↑(Scheme.affineOpens X)
[PROOFSTEP]
use V.1 ⊓ X.basicOpen f
[GOAL]
case property
X✝ Y : Scheme
f✝ : X✝ ⟶ Y
X : Scheme
U : Opens ↑↑X.toPresheafedSpace
hU : IsCompact ↑U
f : ↑(X.presheaf.obj (op U))
s : Set ↑(Scheme.affineOpens X)
hs : Set.Finite s
e : ↑U = ⋃ (i : ↑(Scheme.affineOpens X)) (_ : i ∈ s), ↑↑i
V : ↑s
⊢ ↑↑V ⊓ Scheme.basicOpen X f ∈ Scheme.affineOpens X
[PROOFSTEP]
have : V.1.1 ⟶ U := by
apply homOfLE; change _ ⊆ (U : Set X.carrier); rw [e]
convert @Set.subset_iUnion₂ _ _ _ (fun (U : X.affineOpens) (_ : U ∈ s) => ↑U) V V.prop using 1
[GOAL]
X✝ Y : Scheme
f✝ : X✝ ⟶ Y
X : Scheme
U : Opens ↑↑X.toPresheafedSpace
hU : IsCompact ↑U
f : ↑(X.presheaf.obj (op U))
s : Set ↑(Scheme.affineOpens X)
hs : Set.Finite s
e : ↑U = ⋃ (i : ↑(Scheme.affineOpens X)) (_ : i ∈ s), ↑↑i
V : ↑s
⊢ ↑↑V ⟶ U
[PROOFSTEP]
apply homOfLE
[GOAL]
case h
X✝ Y : Scheme
f✝ : X✝ ⟶ Y
X : Scheme
U : Opens ↑↑X.toPresheafedSpace
hU : IsCompact ↑U
f : ↑(X.presheaf.obj (op U))
s : Set ↑(Scheme.affineOpens X)
hs : Set.Finite s
e : ↑U = ⋃ (i : ↑(Scheme.affineOpens X)) (_ : i ∈ s), ↑↑i
V : ↑s
⊢ ↑↑V ≤ U
[PROOFSTEP]
change _ ⊆ (U : Set X.carrier)
[GOAL]
case h
X✝ Y : Scheme
f✝ : X✝ ⟶ Y
X : Scheme
U : Opens ↑↑X.toPresheafedSpace
hU : IsCompact ↑U
f : ↑(X.presheaf.obj (op U))
s : Set ↑(Scheme.affineOpens X)
hs : Set.Finite s
e : ↑U = ⋃ (i : ↑(Scheme.affineOpens X)) (_ : i ∈ s), ↑↑i
V : ↑s
⊢ ↑↑↑V ⊆ ↑U
[PROOFSTEP]
rw [e]
[GOAL]
case h
X✝ Y : Scheme
f✝ : X✝ ⟶ Y
X : Scheme
U : Opens ↑↑X.toPresheafedSpace
hU : IsCompact ↑U
f : ↑(X.presheaf.obj (op U))
s : Set ↑(Scheme.affineOpens X)
hs : Set.Finite s
e : ↑U = ⋃ (i : ↑(Scheme.affineOpens X)) (_ : i ∈ s), ↑↑i
V : ↑s
⊢ ↑↑↑V ⊆ ⋃ (i : ↑(Scheme.affineOpens X)) (_ : i ∈ s), ↑↑i
[PROOFSTEP]
convert @Set.subset_iUnion₂ _ _ _ (fun (U : X.affineOpens) (_ : U ∈ s) => ↑U) V V.prop using 1
[GOAL]
case property
X✝ Y : Scheme
f✝ : X✝ ⟶ Y
X : Scheme
U : Opens ↑↑X.toPresheafedSpace
hU : IsCompact ↑U
f : ↑(X.presheaf.obj (op U))
s : Set ↑(Scheme.affineOpens X)
hs : Set.Finite s
e : ↑U = ⋃ (i : ↑(Scheme.affineOpens X)) (_ : i ∈ s), ↑↑i
V : ↑s
this : ↑↑V ⟶ U
⊢ ↑↑V ⊓ Scheme.basicOpen X f ∈ Scheme.affineOpens X
[PROOFSTEP]
erw [← X.toLocallyRingedSpace.toRingedSpace.basicOpen_res this.op]
[GOAL]
case property
X✝ Y : Scheme
f✝ : X✝ ⟶ Y
X : Scheme
U : Opens ↑↑X.toPresheafedSpace
hU : IsCompact ↑U
f : ↑(X.presheaf.obj (op U))
s : Set ↑(Scheme.affineOpens X)
hs : Set.Finite s
e : ↑U = ⋃ (i : ↑(Scheme.affineOpens X)) (_ : i ∈ s), ↑↑i
V : ↑s
this : ↑↑V ⟶ U
⊢ RingedSpace.basicOpen (LocallyRingedSpace.toRingedSpace X.toLocallyRingedSpace)
(↑((LocallyRingedSpace.toRingedSpace X.toLocallyRingedSpace).toPresheafedSpace.presheaf.map this.op) f) ∈
Scheme.affineOpens X
[PROOFSTEP]
exact IsAffineOpen.basicOpenIsAffine V.1.prop _
[GOAL]
case intro.intro
X✝ Y : Scheme
f✝ : X✝ ⟶ Y
X : Scheme
U : Opens ↑↑X.toPresheafedSpace
hU : IsCompact ↑U
f : ↑(X.presheaf.obj (op U))
s : Set ↑(Scheme.affineOpens X)
hs : Set.Finite s
e : ↑U = ⋃ (i : ↑(Scheme.affineOpens X)) (_ : i ∈ s), ↑↑i
g : ↑s → ↑(Scheme.affineOpens X) :=
fun V => { val := ↑↑V ⊓ Scheme.basicOpen X f, property := (_ : ↑↑V ⊓ Scheme.basicOpen X f ∈ Scheme.affineOpens X) }
⊢ ∃ s, Set.Finite s ∧ ↑(Scheme.basicOpen X f) = ⋃ (i : ↑(Scheme.affineOpens X)) (_ : i ∈ s), ↑↑i
[PROOFSTEP]
haveI : Finite s := hs.to_subtype
[GOAL]
case intro.intro
X✝ Y : Scheme
f✝ : X✝ ⟶ Y
X : Scheme
U : Opens ↑↑X.toPresheafedSpace
hU : IsCompact ↑U
f : ↑(X.presheaf.obj (op U))
s : Set ↑(Scheme.affineOpens X)
hs : Set.Finite s
e : ↑U = ⋃ (i : ↑(Scheme.affineOpens X)) (_ : i ∈ s), ↑↑i
g : ↑s → ↑(Scheme.affineOpens X) :=
fun V => { val := ↑↑V ⊓ Scheme.basicOpen X f, property := (_ : ↑↑V ⊓ Scheme.basicOpen X f ∈ Scheme.affineOpens X) }
this : Finite ↑s
⊢ ∃ s, Set.Finite s ∧ ↑(Scheme.basicOpen X f) = ⋃ (i : ↑(Scheme.affineOpens X)) (_ : i ∈ s), ↑↑i
[PROOFSTEP]
refine' ⟨Set.range g, Set.finite_range g, _⟩
[GOAL]
case intro.intro
X✝ Y : Scheme
f✝ : X✝ ⟶ Y
X : Scheme
U : Opens ↑↑X.toPresheafedSpace
hU : IsCompact ↑U
f : ↑(X.presheaf.obj (op U))
s : Set ↑(Scheme.affineOpens X)
hs : Set.Finite s
e : ↑U = ⋃ (i : ↑(Scheme.affineOpens X)) (_ : i ∈ s), ↑↑i
g : ↑s → ↑(Scheme.affineOpens X) :=
fun V => { val := ↑↑V ⊓ Scheme.basicOpen X f, property := (_ : ↑↑V ⊓ Scheme.basicOpen X f ∈ Scheme.affineOpens X) }
this : Finite ↑s
⊢ ↑(Scheme.basicOpen X f) = ⋃ (i : ↑(Scheme.affineOpens X)) (_ : i ∈ Set.range g), ↑↑i
[PROOFSTEP]
refine' (Set.inter_eq_right_iff_subset.mpr (SetLike.coe_subset_coe.2 <| RingedSpace.basicOpen_le _ _)).symm.trans _
[GOAL]
case intro.intro
X✝ Y : Scheme
f✝ : X✝ ⟶ Y
X : Scheme
U : Opens ↑↑X.toPresheafedSpace
hU : IsCompact ↑U
f : ↑(X.presheaf.obj (op U))
s : Set ↑(Scheme.affineOpens X)
hs : Set.Finite s
e : ↑U = ⋃ (i : ↑(Scheme.affineOpens X)) (_ : i ∈ s), ↑↑i
g : ↑s → ↑(Scheme.affineOpens X) :=
fun V => { val := ↑↑V ⊓ Scheme.basicOpen X f, property := (_ : ↑↑V ⊓ Scheme.basicOpen X f ∈ Scheme.affineOpens X) }
this : Finite ↑s
⊢ ↑U ∩ ↑(RingedSpace.basicOpen (LocallyRingedSpace.toRingedSpace X.toLocallyRingedSpace) f) =
⋃ (i : ↑(Scheme.affineOpens X)) (_ : i ∈ Set.range g), ↑↑i
[PROOFSTEP]
rw [e, Set.iUnion₂_inter]
[GOAL]
case intro.intro
X✝ Y : Scheme
f✝ : X✝ ⟶ Y
X : Scheme
U : Opens ↑↑X.toPresheafedSpace
hU : IsCompact ↑U
f : ↑(X.presheaf.obj (op U))
s : Set ↑(Scheme.affineOpens X)
hs : Set.Finite s
e : ↑U = ⋃ (i : ↑(Scheme.affineOpens X)) (_ : i ∈ s), ↑↑i
g : ↑s → ↑(Scheme.affineOpens X) :=
fun V => { val := ↑↑V ⊓ Scheme.basicOpen X f, property := (_ : ↑↑V ⊓ Scheme.basicOpen X f ∈ Scheme.affineOpens X) }
this : Finite ↑s
⊢ ⋃ (i : ↑(Scheme.affineOpens X)) (_ : i ∈ s),
↑↑i ∩ ↑(RingedSpace.basicOpen (LocallyRingedSpace.toRingedSpace X.toLocallyRingedSpace) f) =
⋃ (i : ↑(Scheme.affineOpens X)) (_ : i ∈ Set.range g), ↑↑i
[PROOFSTEP]
apply le_antisymm
[GOAL]
case intro.intro.a
X✝ Y : Scheme
f✝ : X✝ ⟶ Y
X : Scheme
U : Opens ↑↑X.toPresheafedSpace
hU : IsCompact ↑U
f : ↑(X.presheaf.obj (op U))
s : Set ↑(Scheme.affineOpens X)
hs : Set.Finite s
e : ↑U = ⋃ (i : ↑(Scheme.affineOpens X)) (_ : i ∈ s), ↑↑i
g : ↑s → ↑(Scheme.affineOpens X) :=
fun V => { val := ↑↑V ⊓ Scheme.basicOpen X f, property := (_ : ↑↑V ⊓ Scheme.basicOpen X f ∈ Scheme.affineOpens X) }
this : Finite ↑s
⊢ ⋃ (i : ↑(Scheme.affineOpens X)) (_ : i ∈ s),
↑↑i ∩ ↑(RingedSpace.basicOpen (LocallyRingedSpace.toRingedSpace X.toLocallyRingedSpace) f) ≤
⋃ (i : ↑(Scheme.affineOpens X)) (_ : i ∈ Set.range g), ↑↑i
[PROOFSTEP]
apply Set.iUnion₂_subset
[GOAL]
case intro.intro.a
X✝ Y : Scheme
f✝ : X✝ ⟶ Y
X : Scheme
U : Opens ↑↑X.toPresheafedSpace
hU : IsCompact ↑U
f : ↑(X.presheaf.obj (op U))
s : Set ↑(Scheme.affineOpens X)
hs : Set.Finite s
e : ↑U = ⋃ (i : ↑(Scheme.affineOpens X)) (_ : i ∈ s), ↑↑i
g : ↑s → ↑(Scheme.affineOpens X) :=
fun V => { val := ↑↑V ⊓ Scheme.basicOpen X f, property := (_ : ↑↑V ⊓ Scheme.basicOpen X f ∈ Scheme.affineOpens X) }
this : Finite ↑s
⊢ ⋃ (i : ↑(Scheme.affineOpens X)) (_ : i ∈ Set.range g), ↑↑i ≤
⋃ (i : ↑(Scheme.affineOpens X)) (_ : i ∈ s),
↑↑i ∩ ↑(RingedSpace.basicOpen (LocallyRingedSpace.toRingedSpace X.toLocallyRingedSpace) f)
[PROOFSTEP]
apply Set.iUnion₂_subset
[GOAL]
case intro.intro.a.h
X✝ Y : Scheme
f✝ : X✝ ⟶ Y
X : Scheme
U : Opens ↑↑X.toPresheafedSpace
hU : IsCompact ↑U
f : ↑(X.presheaf.obj (op U))
s : Set ↑(Scheme.affineOpens X)
hs : Set.Finite s
e : ↑U = ⋃ (i : ↑(Scheme.affineOpens X)) (_ : i ∈ s), ↑↑i
g : ↑s → ↑(Scheme.affineOpens X) :=
fun V => { val := ↑↑V ⊓ Scheme.basicOpen X f, property := (_ : ↑↑V ⊓ Scheme.basicOpen X f ∈ Scheme.affineOpens X) }
this : Finite ↑s
⊢ ∀ (i : ↑(Scheme.affineOpens X)),
i ∈ s →
↑↑i ∩ ↑(RingedSpace.basicOpen (LocallyRingedSpace.toRingedSpace X.toLocallyRingedSpace) f) ⊆
⋃ (i : ↑(Scheme.affineOpens X)) (_ : i ∈ Set.range g), ↑↑i
[PROOFSTEP]
intro i hi
[GOAL]
case intro.intro.a.h
X✝ Y : Scheme
f✝ : X✝ ⟶ Y
X : Scheme
U : Opens ↑↑X.toPresheafedSpace
hU : IsCompact ↑U
f : ↑(X.presheaf.obj (op U))
s : Set ↑(Scheme.affineOpens X)
hs : Set.Finite s
e : ↑U = ⋃ (i : ↑(Scheme.affineOpens X)) (_ : i ∈ s), ↑↑i
g : ↑s → ↑(Scheme.affineOpens X) :=
fun V => { val := ↑↑V ⊓ Scheme.basicOpen X f, property := (_ : ↑↑V ⊓ Scheme.basicOpen X f ∈ Scheme.affineOpens X) }
this : Finite ↑s
i : ↑(Scheme.affineOpens X)
hi : i ∈ s
⊢ ↑↑i ∩ ↑(RingedSpace.basicOpen (LocallyRingedSpace.toRingedSpace X.toLocallyRingedSpace) f) ⊆
⋃ (i : ↑(Scheme.affineOpens X)) (_ : i ∈ Set.range g), ↑↑i
[PROOFSTEP]
exact
Set.Subset.trans (Set.Subset.rfl : _ ≤ g ⟨i, hi⟩)
(@Set.subset_iUnion₂ _ _ _ (fun (i : Scheme.affineOpens X) (_ : i ∈ Set.range g) => (i : Set X.toPresheafedSpace)) _
(Set.mem_range_self ⟨i, hi⟩))
[GOAL]
case intro.intro.a.h
X✝ Y : Scheme
f✝ : X✝ ⟶ Y
X : Scheme
U : Opens ↑↑X.toPresheafedSpace
hU : IsCompact ↑U
f : ↑(X.presheaf.obj (op U))
s : Set ↑(Scheme.affineOpens X)
hs : Set.Finite s
e : ↑U = ⋃ (i : ↑(Scheme.affineOpens X)) (_ : i ∈ s), ↑↑i
g : ↑s → ↑(Scheme.affineOpens X) :=
fun V => { val := ↑↑V ⊓ Scheme.basicOpen X f, property := (_ : ↑↑V ⊓ Scheme.basicOpen X f ∈ Scheme.affineOpens X) }
this : Finite ↑s
⊢ ∀ (i : ↑(Scheme.affineOpens X)),
i ∈ Set.range g →
↑↑i ⊆
⋃ (i : ↑(Scheme.affineOpens X)) (_ : i ∈ s),
↑↑i ∩ ↑(RingedSpace.basicOpen (LocallyRingedSpace.toRingedSpace X.toLocallyRingedSpace) f)
[PROOFSTEP]
rintro ⟨i, hi⟩ ⟨⟨j, hj⟩, hj'⟩
[GOAL]
case intro.intro.a.h.mk.intro.mk
X✝ Y : Scheme
f✝ : X✝ ⟶ Y
X : Scheme
U : Opens ↑↑X.toPresheafedSpace
hU : IsCompact ↑U
f : ↑(X.presheaf.obj (op U))
s : Set ↑(Scheme.affineOpens X)
hs : Set.Finite s
e : ↑U = ⋃ (i : ↑(Scheme.affineOpens X)) (_ : i ∈ s), ↑↑i
g : ↑s → ↑(Scheme.affineOpens X) :=
fun V => { val := ↑↑V ⊓ Scheme.basicOpen X f, property := (_ : ↑↑V ⊓ Scheme.basicOpen X f ∈ Scheme.affineOpens X) }
this : Finite ↑s
i : Opens ↑↑X.toPresheafedSpace
hi : i ∈ Scheme.affineOpens X
j : ↑(Scheme.affineOpens X)
hj : j ∈ s
hj' : g { val := j, property := hj } = { val := i, property := hi }
⊢ ↑↑{ val := i, property := hi } ⊆
⋃ (i : ↑(Scheme.affineOpens X)) (_ : i ∈ s),
↑↑i ∩ ↑(RingedSpace.basicOpen (LocallyRingedSpace.toRingedSpace X.toLocallyRingedSpace) f)
[PROOFSTEP]
rw [← hj']
[GOAL]
case intro.intro.a.h.mk.intro.mk
X✝ Y : Scheme
f✝ : X✝ ⟶ Y
X : Scheme
U : Opens ↑↑X.toPresheafedSpace
hU : IsCompact ↑U
f : ↑(X.presheaf.obj (op U))
s : Set ↑(Scheme.affineOpens X)
hs : Set.Finite s
e : ↑U = ⋃ (i : ↑(Scheme.affineOpens X)) (_ : i ∈ s), ↑↑i
g : ↑s → ↑(Scheme.affineOpens X) :=
fun V => { val := ↑↑V ⊓ Scheme.basicOpen X f, property := (_ : ↑↑V ⊓ Scheme.basicOpen X f ∈ Scheme.affineOpens X) }
this : Finite ↑s
i : Opens ↑↑X.toPresheafedSpace
hi : i ∈ Scheme.affineOpens X
j : ↑(Scheme.affineOpens X)
hj : j ∈ s
hj' : g { val := j, property := hj } = { val := i, property := hi }
⊢ ↑↑(g { val := j, property := hj }) ⊆
⋃ (i : ↑(Scheme.affineOpens X)) (_ : i ∈ s),
↑↑i ∩ ↑(RingedSpace.basicOpen (LocallyRingedSpace.toRingedSpace X.toLocallyRingedSpace) f)
[PROOFSTEP]
refine' Set.Subset.trans _ (Set.subset_iUnion₂ j hj)
[GOAL]
case intro.intro.a.h.mk.intro.mk
X✝ Y : Scheme
f✝ : X✝ ⟶ Y
X : Scheme
U : Opens ↑↑X.toPresheafedSpace
hU : IsCompact ↑U
f : ↑(X.presheaf.obj (op U))
s : Set ↑(Scheme.affineOpens X)
hs : Set.Finite s
e : ↑U = ⋃ (i : ↑(Scheme.affineOpens X)) (_ : i ∈ s), ↑↑i
g : ↑s → ↑(Scheme.affineOpens X) :=
fun V => { val := ↑↑V ⊓ Scheme.basicOpen X f, property := (_ : ↑↑V ⊓ Scheme.basicOpen X f ∈ Scheme.affineOpens X) }
this : Finite ↑s
i : Opens ↑↑X.toPresheafedSpace
hi : i ∈ Scheme.affineOpens X
j : ↑(Scheme.affineOpens X)
hj : j ∈ s
hj' : g { val := j, property := hj } = { val := i, property := hi }
⊢ ↑↑(g { val := j, property := hj }) ⊆
↑↑j ∩ ↑(RingedSpace.basicOpen (LocallyRingedSpace.toRingedSpace X.toLocallyRingedSpace) f)
[PROOFSTEP]
exact Set.Subset.rfl
[GOAL]
X Y : Scheme
f : X ⟶ Y
⊢ AffineTargetMorphismProperty.IsLocal affineProperty
[PROOFSTEP]
constructor
[GOAL]
case RespectsIso
X Y : Scheme
f : X ⟶ Y
⊢ MorphismProperty.RespectsIso (AffineTargetMorphismProperty.toProperty affineProperty)
[PROOFSTEP]
apply AffineTargetMorphismProperty.respectsIso_mk
[GOAL]
case RespectsIso.h₁
X Y : Scheme
f : X ⟶ Y
⊢ ∀ {X Y Z : Scheme} (e : X ≅ Y) (f : Y ⟶ Z) [inst : IsAffine Z], affineProperty f → affineProperty (e.hom ≫ f)
[PROOFSTEP]
rintro X Y Z e _ _ H
[GOAL]
case RespectsIso.h₂
X Y : Scheme
f : X ⟶ Y
⊢ ∀ {X Y Z : Scheme} (e : Y ≅ Z) (f : X ⟶ Y) [h : IsAffine Y], affineProperty f → affineProperty (f ≫ e.hom)
[PROOFSTEP]
rintro X Y Z e _ _ H
[GOAL]
case RespectsIso.h₁
X✝ Y✝ : Scheme
f : X✝ ⟶ Y✝
X Y Z : Scheme
e : X ≅ Y
f✝ : Y ⟶ Z
inst✝ : IsAffine Z
H : affineProperty f✝
⊢ affineProperty (e.hom ≫ f✝)
case RespectsIso.h₂
X✝ Y✝ : Scheme
f : X✝ ⟶ Y✝
X Y Z : Scheme
e : Y ≅ Z
f✝ : X ⟶ Y
h✝ : IsAffine Y
H : affineProperty f✝
⊢ affineProperty (f✝ ≫ e.hom)
[PROOFSTEP]
exacts [@Homeomorph.compactSpace _ _ _ _ H (TopCat.homeoOfIso (asIso e.inv.1.base)), H]
[GOAL]
case toBasicOpen
X Y : Scheme
f : X ⟶ Y
⊢ ∀ {X Y : Scheme} [inst : IsAffine Y] (f : X ⟶ Y) (r : ↑(Y.presheaf.obj (op ⊤))),
affineProperty f → affineProperty (f ∣_ Scheme.basicOpen Y r)
[PROOFSTEP]
introv H
[GOAL]
case toBasicOpen
X✝ Y✝ : Scheme
f✝ : X✝ ⟶ Y✝
X Y : Scheme
inst✝ : IsAffine Y
f : X ⟶ Y
r : ↑(Y.presheaf.obj (op ⊤))
H : affineProperty f
⊢ affineProperty (f ∣_ Scheme.basicOpen Y r)
[PROOFSTEP]
dsimp [affineProperty] at H ⊢
[GOAL]
case toBasicOpen
X✝ Y✝ : Scheme
f✝ : X✝ ⟶ Y✝
X Y : Scheme
inst✝ : IsAffine Y
f : X ⟶ Y
r : ↑(Y.presheaf.obj (op ⊤))
H : CompactSpace ↑↑X.toPresheafedSpace
⊢ CompactSpace ↑((Opens.toTopCat ↑X.toPresheafedSpace).obj ((Opens.map f.val.base).obj (Scheme.basicOpen Y r)))
[PROOFSTEP]
change CompactSpace ((Opens.map f.val.base).obj (Y.basicOpen r))
[GOAL]
case toBasicOpen
X✝ Y✝ : Scheme
f✝ : X✝ ⟶ Y✝
X Y : Scheme
inst✝ : IsAffine Y
f : X ⟶ Y
r : ↑(Y.presheaf.obj (op ⊤))
H : CompactSpace ↑↑X.toPresheafedSpace
⊢ CompactSpace { x // x ∈ (Opens.map f.val.base).obj (Scheme.basicOpen Y r) }
[PROOFSTEP]
rw [Scheme.preimage_basicOpen f r]
[GOAL]
case toBasicOpen
X✝ Y✝ : Scheme
f✝ : X✝ ⟶ Y✝
X Y : Scheme
inst✝ : IsAffine Y
f : X ⟶ Y
r : ↑(Y.presheaf.obj (op ⊤))
H : CompactSpace ↑↑X.toPresheafedSpace
⊢ CompactSpace { x // x ∈ Scheme.basicOpen X (↑(NatTrans.app f.val.c (op ⊤)) r) }
[PROOFSTEP]
erw [← isCompact_iff_compactSpace]
[GOAL]
case toBasicOpen
X✝ Y✝ : Scheme
f✝ : X✝ ⟶ Y✝
X Y : Scheme
inst✝ : IsAffine Y
f : X ⟶ Y
r : ↑(Y.presheaf.obj (op ⊤))
H : CompactSpace ↑↑X.toPresheafedSpace
⊢ IsCompact ↑(Scheme.basicOpen X (↑(NatTrans.app f.val.c (op ⊤)) r))
[PROOFSTEP]
rw [← isCompact_univ_iff] at H
[GOAL]
case toBasicOpen
X✝ Y✝ : Scheme
f✝ : X✝ ⟶ Y✝
X Y : Scheme
inst✝ : IsAffine Y
f : X ⟶ Y
r : ↑(Y.presheaf.obj (op ⊤))
H : IsCompact Set.univ
⊢ IsCompact ↑(Scheme.basicOpen X (↑(NatTrans.app f.val.c (op ⊤)) r))
[PROOFSTEP]
apply isCompact_basicOpen
[GOAL]
case toBasicOpen.hU
X✝ Y✝ : Scheme
f✝ : X✝ ⟶ Y✝
X Y : Scheme
inst✝ : IsAffine Y
f : X ⟶ Y
r : ↑(Y.presheaf.obj (op ⊤))
H : IsCompact Set.univ
⊢ IsCompact ↑((Opens.map f.val.base).obj ⊤)
[PROOFSTEP]
exact H
[GOAL]
case ofBasicOpenCover
X Y : Scheme
f : X ⟶ Y
⊢ ∀ {X Y : Scheme} [inst : IsAffine Y] (f : X ⟶ Y) (s : Finset ↑(Y.presheaf.obj (op ⊤))),
Ideal.span ↑s = ⊤ → (∀ (r : { x // x ∈ s }), affineProperty (f ∣_ Scheme.basicOpen Y ↑r)) → affineProperty f
[PROOFSTEP]
rintro X Y H f S hS hS'
[GOAL]
case ofBasicOpenCover
X✝ Y✝ : Scheme
f✝ : X✝ ⟶ Y✝
X Y : Scheme
H : IsAffine Y
f : X ⟶ Y
S : Finset ↑(Y.presheaf.obj (op ⊤))
hS : Ideal.span ↑S = ⊤
hS' : ∀ (r : { x // x ∈ S }), affineProperty (f ∣_ Scheme.basicOpen Y ↑r)
⊢ affineProperty f
[PROOFSTEP]
rw [← IsAffineOpen.basicOpen_union_eq_self_iff] at hS
[GOAL]
case ofBasicOpenCover
X✝ Y✝ : Scheme
f✝ : X✝ ⟶ Y✝
X Y : Scheme
H : IsAffine Y
f : X ⟶ Y
S : Finset ↑(Y.presheaf.obj (op ⊤))
hS : ⨆ (f : ↑↑S), Scheme.basicOpen Y ↑f = ⊤
hS' : ∀ (r : { x // x ∈ S }), affineProperty (f ∣_ Scheme.basicOpen Y ↑r)
⊢ affineProperty f
case ofBasicOpenCover.hU
X✝ Y✝ : Scheme
f✝ : X✝ ⟶ Y✝
X Y : Scheme
H : IsAffine Y
f : X ⟶ Y
S : Finset ↑(Y.presheaf.obj (op ⊤))
hS : Ideal.span ↑S = ⊤
hS' : ∀ (r : { x // x ∈ S }), affineProperty (f ∣_ Scheme.basicOpen Y ↑r)
⊢ IsAffineOpen ⊤
[PROOFSTEP]
delta QuasiCompact.affineProperty
[GOAL]
case ofBasicOpenCover
X✝ Y✝ : Scheme
f✝ : X✝ ⟶ Y✝
X Y : Scheme
H : IsAffine Y
f : X ⟶ Y
S : Finset ↑(Y.presheaf.obj (op ⊤))
hS : ⨆ (f : ↑↑S), Scheme.basicOpen Y ↑f = ⊤
hS' : ∀ (r : { x // x ∈ S }), affineProperty (f ∣_ Scheme.basicOpen Y ↑r)
⊢ CompactSpace ↑↑X.toPresheafedSpace
case ofBasicOpenCover.hU
X✝ Y✝ : Scheme
f✝ : X✝ ⟶ Y✝
X Y : Scheme
H : IsAffine Y
f : X ⟶ Y
S : Finset ↑(Y.presheaf.obj (op ⊤))
hS : Ideal.span ↑S = ⊤
hS' : ∀ (r : { x // x ∈ S }), affineProperty (f ∣_ Scheme.basicOpen Y ↑r)
⊢ IsAffineOpen ⊤
[PROOFSTEP]
rw [← isCompact_univ_iff]
[GOAL]
case ofBasicOpenCover
X✝ Y✝ : Scheme
f✝ : X✝ ⟶ Y✝
X Y : Scheme
H : IsAffine Y
f : X ⟶ Y
S : Finset ↑(Y.presheaf.obj (op ⊤))
hS : ⨆ (f : ↑↑S), Scheme.basicOpen Y ↑f = ⊤
hS' : ∀ (r : { x // x ∈ S }), affineProperty (f ∣_ Scheme.basicOpen Y ↑r)
⊢ IsCompact Set.univ
case ofBasicOpenCover.hU
X✝ Y✝ : Scheme
f✝ : X✝ ⟶ Y✝
X Y : Scheme
H : IsAffine Y
f : X ⟶ Y
S : Finset ↑(Y.presheaf.obj (op ⊤))
hS : Ideal.span ↑S = ⊤
hS' : ∀ (r : { x // x ∈ S }), affineProperty (f ∣_ Scheme.basicOpen Y ↑r)
⊢ IsAffineOpen ⊤
[PROOFSTEP]
change IsCompact ((Opens.map f.val.base).obj ⊤).1
[GOAL]
case ofBasicOpenCover
X✝ Y✝ : Scheme
f✝ : X✝ ⟶ Y✝
X Y : Scheme
H : IsAffine Y
f : X ⟶ Y
S : Finset ↑(Y.presheaf.obj (op ⊤))
hS : ⨆ (f : ↑↑S), Scheme.basicOpen Y ↑f = ⊤
hS' : ∀ (r : { x // x ∈ S }), affineProperty (f ∣_ Scheme.basicOpen Y ↑r)
⊢ IsCompact ((Opens.map f.val.base).obj ⊤).carrier
case ofBasicOpenCover.hU
X✝ Y✝ : Scheme
f✝ : X✝ ⟶ Y✝
X Y : Scheme
H : IsAffine Y
f : X ⟶ Y
S : Finset ↑(Y.presheaf.obj (op ⊤))
hS : Ideal.span ↑S = ⊤
hS' : ∀ (r : { x // x ∈ S }), affineProperty (f ∣_ Scheme.basicOpen Y ↑r)
⊢ IsAffineOpen ⊤
[PROOFSTEP]
rw [← hS]
[GOAL]
case ofBasicOpenCover
X✝ Y✝ : Scheme
f✝ : X✝ ⟶ Y✝
X Y : Scheme
H : IsAffine Y
f : X ⟶ Y
S : Finset ↑(Y.presheaf.obj (op ⊤))
hS : ⨆ (f : ↑↑S), Scheme.basicOpen Y ↑f = ⊤
hS' : ∀ (r : { x // x ∈ S }), affineProperty (f ∣_ Scheme.basicOpen Y ↑r)
⊢ IsCompact ((Opens.map f.val.base).obj (⨆ (f : ↑↑S), Scheme.basicOpen Y ↑f)).carrier
case ofBasicOpenCover.hU
X✝ Y✝ : Scheme
f✝ : X✝ ⟶ Y✝
X Y : Scheme
H : IsAffine Y
f : X ⟶ Y
S : Finset ↑(Y.presheaf.obj (op ⊤))
hS : Ideal.span ↑S = ⊤
hS' : ∀ (r : { x // x ∈ S }), affineProperty (f ∣_ Scheme.basicOpen Y ↑r)
⊢ IsAffineOpen ⊤
[PROOFSTEP]
dsimp [Opens.map]
[GOAL]
case ofBasicOpenCover
X✝ Y✝ : Scheme
f✝ : X✝ ⟶ Y✝
X Y : Scheme
H : IsAffine Y
f : X ⟶ Y
S : Finset ↑(Y.presheaf.obj (op ⊤))
hS : ⨆ (f : ↑↑S), Scheme.basicOpen Y ↑f = ⊤
hS' : ∀ (r : { x // x ∈ S }), affineProperty (f ∣_ Scheme.basicOpen Y ↑r)
⊢ IsCompact (↑f.val.base ⁻¹' ↑(⨆ (f : { x // x ∈ S }), Scheme.basicOpen Y ↑f))
case ofBasicOpenCover.hU
X✝ Y✝ : Scheme
f✝ : X✝ ⟶ Y✝
X Y : Scheme
H : IsAffine Y
f : X ⟶ Y
S : Finset ↑(Y.presheaf.obj (op ⊤))
hS : Ideal.span ↑S = ⊤
hS' : ∀ (r : { x // x ∈ S }), affineProperty (f ∣_ Scheme.basicOpen Y ↑r)
⊢ IsAffineOpen ⊤
[PROOFSTEP]
simp only [Opens.iSup_mk, Opens.carrier_eq_coe, Opens.coe_mk, Set.preimage_iUnion]
[GOAL]
case ofBasicOpenCover
X✝ Y✝ : Scheme
f✝ : X✝ ⟶ Y✝
X Y : Scheme
H : IsAffine Y
f : X ⟶ Y
S : Finset ↑(Y.presheaf.obj (op ⊤))
hS : ⨆ (f : ↑↑S), Scheme.basicOpen Y ↑f = ⊤
hS' : ∀ (r : { x // x ∈ S }), affineProperty (f ∣_ Scheme.basicOpen Y ↑r)
⊢ IsCompact (⋃ (i : { x // x ∈ S }), ↑f.val.base ⁻¹' ↑(Scheme.basicOpen Y ↑i))
case ofBasicOpenCover.hU
X✝ Y✝ : Scheme
f✝ : X✝ ⟶ Y✝
X Y : Scheme
H : IsAffine Y
f : X ⟶ Y
S : Finset ↑(Y.presheaf.obj (op ⊤))
hS : Ideal.span ↑S = ⊤
hS' : ∀ (r : { x // x ∈ S }), affineProperty (f ∣_ Scheme.basicOpen Y ↑r)
⊢ IsAffineOpen ⊤
[PROOFSTEP]
exacts [isCompact_iUnion fun i => isCompact_iff_compactSpace.mpr (hS' i), topIsAffineOpen _]
[GOAL]
X Y : Scheme
f : X ⟶ Y
⊢ AffineTargetMorphismProperty.StableUnderBaseChange affineProperty
[PROOFSTEP]
intro X Y S _ _ f g h
[GOAL]
X✝ Y✝ : Scheme
f✝ : X✝ ⟶ Y✝
X Y S : Scheme
inst✝¹ : IsAffine S
inst✝ : IsAffine X
f : X ⟶ S
g : Y ⟶ S
h : affineProperty g
⊢ affineProperty pullback.fst
[PROOFSTEP]
rw [QuasiCompact.affineProperty] at h ⊢
[GOAL]
X✝ Y✝ : Scheme
f✝ : X✝ ⟶ Y✝
X Y S : Scheme
inst✝¹ : IsAffine S
inst✝ : IsAffine X
f : X ⟶ S
g : Y ⟶ S
h : CompactSpace ↑↑Y.toPresheafedSpace
⊢ CompactSpace ↑↑(pullback f g).toLocallyRingedSpace.toSheafedSpace.toPresheafedSpace
[PROOFSTEP]
skip
[GOAL]
X✝ Y✝ : Scheme
f✝ : X✝ ⟶ Y✝
X Y S : Scheme
inst✝¹ : IsAffine S
inst✝ : IsAffine X
f : X ⟶ S
g : Y ⟶ S
h : CompactSpace ↑↑Y.toPresheafedSpace
⊢ CompactSpace ↑↑(pullback f g).toLocallyRingedSpace.toSheafedSpace.toPresheafedSpace
[PROOFSTEP]
let 𝒰 := Scheme.Pullback.openCoverOfRight Y.affineCover.finiteSubcover f g
[GOAL]
X✝ Y✝ : Scheme
f✝ : X✝ ⟶ Y✝
X Y S : Scheme
inst✝¹ : IsAffine S
inst✝ : IsAffine X
f : X ⟶ S
g : Y ⟶ S
h : CompactSpace ↑↑Y.toPresheafedSpace
𝒰 : Scheme.OpenCover (pullback f g) :=
Scheme.Pullback.openCoverOfRight (Scheme.OpenCover.finiteSubcover (Scheme.affineCover Y)) f g
⊢ CompactSpace ↑↑(pullback f g).toLocallyRingedSpace.toSheafedSpace.toPresheafedSpace
[PROOFSTEP]
have : Finite 𝒰.J := by dsimp; infer_instance
[GOAL]
X✝ Y✝ : Scheme
f✝ : X✝ ⟶ Y✝
X Y S : Scheme
inst✝¹ : IsAffine S
inst✝ : IsAffine X
f : X ⟶ S
g : Y ⟶ S
h : CompactSpace ↑↑Y.toPresheafedSpace
𝒰 : Scheme.OpenCover (pullback f g) :=
Scheme.Pullback.openCoverOfRight (Scheme.OpenCover.finiteSubcover (Scheme.affineCover Y)) f g
⊢ Finite 𝒰.J
[PROOFSTEP]
dsimp
[GOAL]
X✝ Y✝ : Scheme
f✝ : X✝ ⟶ Y✝
X Y S : Scheme
inst✝¹ : IsAffine S
inst✝ : IsAffine X
f : X ⟶ S
g : Y ⟶ S
h : CompactSpace ↑↑Y.toPresheafedSpace
𝒰 : Scheme.OpenCover (pullback f g) :=
Scheme.Pullback.openCoverOfRight (Scheme.OpenCover.finiteSubcover (Scheme.affineCover Y)) f g
⊢ Finite (Scheme.OpenCover.finiteSubcover (Scheme.affineCover Y)).J
[PROOFSTEP]
infer_instance
[GOAL]
X✝ Y✝ : Scheme
f✝ : X✝ ⟶ Y✝
X Y S : Scheme
inst✝¹ : IsAffine S
inst✝ : IsAffine X
f : X ⟶ S
g : Y ⟶ S
h : CompactSpace ↑↑Y.toPresheafedSpace
𝒰 : Scheme.OpenCover (pullback f g) :=
Scheme.Pullback.openCoverOfRight (Scheme.OpenCover.finiteSubcover (Scheme.affineCover Y)) f g
this : Finite 𝒰.J
⊢ CompactSpace ↑↑(pullback f g).toLocallyRingedSpace.toSheafedSpace.toPresheafedSpace
[PROOFSTEP]
have : ∀ i, CompactSpace (𝒰.obj i).carrier := by intro i; dsimp; infer_instance
[GOAL]
X✝ Y✝ : Scheme
f✝ : X✝ ⟶ Y✝
X Y S : Scheme
inst✝¹ : IsAffine S
inst✝ : IsAffine X
f : X ⟶ S
g : Y ⟶ S
h : CompactSpace ↑↑Y.toPresheafedSpace
𝒰 : Scheme.OpenCover (pullback f g) :=
Scheme.Pullback.openCoverOfRight (Scheme.OpenCover.finiteSubcover (Scheme.affineCover Y)) f g
this : Finite 𝒰.J
⊢ ∀ (i : 𝒰.J), CompactSpace ↑↑(Scheme.OpenCover.obj 𝒰 i).toPresheafedSpace
[PROOFSTEP]
intro i
[GOAL]
X✝ Y✝ : Scheme
f✝ : X✝ ⟶ Y✝
X Y S : Scheme
inst✝¹ : IsAffine S
inst✝ : IsAffine X
f : X ⟶ S
g : Y ⟶ S
h : CompactSpace ↑↑Y.toPresheafedSpace
𝒰 : Scheme.OpenCover (pullback f g) :=
Scheme.Pullback.openCoverOfRight (Scheme.OpenCover.finiteSubcover (Scheme.affineCover Y)) f g
this : Finite 𝒰.J
i : 𝒰.J
⊢ CompactSpace ↑↑(Scheme.OpenCover.obj 𝒰 i).toPresheafedSpace
[PROOFSTEP]
dsimp
[GOAL]
X✝ Y✝ : Scheme
f✝ : X✝ ⟶ Y✝
X Y S : Scheme
inst✝¹ : IsAffine S
inst✝ : IsAffine X
f : X ⟶ S
g : Y ⟶ S
h : CompactSpace ↑↑Y.toPresheafedSpace
𝒰 : Scheme.OpenCover (pullback f g) :=
Scheme.Pullback.openCoverOfRight (Scheme.OpenCover.finiteSubcover (Scheme.affineCover Y)) f g
this : Finite 𝒰.J
i : 𝒰.J
⊢ CompactSpace
↑↑(pullback f
(Scheme.OpenCover.map (Scheme.affineCover Y) (Scheme.OpenCover.f (Scheme.affineCover Y) ↑i) ≫
g)).toLocallyRingedSpace.toSheafedSpace.toPresheafedSpace
[PROOFSTEP]
infer_instance
[GOAL]
X✝ Y✝ : Scheme
f✝ : X✝ ⟶ Y✝
X Y S : Scheme
inst✝¹ : IsAffine S
inst✝ : IsAffine X
f : X ⟶ S
g : Y ⟶ S
h : CompactSpace ↑↑Y.toPresheafedSpace
𝒰 : Scheme.OpenCover (pullback f g) :=
Scheme.Pullback.openCoverOfRight (Scheme.OpenCover.finiteSubcover (Scheme.affineCover Y)) f g
this✝ : Finite 𝒰.J
this : ∀ (i : 𝒰.J), CompactSpace ↑↑(Scheme.OpenCover.obj 𝒰 i).toPresheafedSpace
⊢ CompactSpace ↑↑(pullback f g).toLocallyRingedSpace.toSheafedSpace.toPresheafedSpace
[PROOFSTEP]
exact 𝒰.compactSpace
[GOAL]
X Y : Scheme
f : X ⟶ Y
Z : Scheme
P : Opens ↑↑X.toPresheafedSpace → Prop
S : Opens ↑↑X.toPresheafedSpace
hS : IsCompact S.carrier
h₁ : P ⊥
h₂ : ∀ (S : Opens ↑↑X.toPresheafedSpace), IsCompact S.carrier → ∀ (U : ↑(Scheme.affineOpens X)), P S → P (S ⊔ ↑U)
⊢ P S
[PROOFSTEP]
classical
obtain ⟨s, hs, hs'⟩ := (isCompact_open_iff_eq_finset_affine_union S.1).mp ⟨hS, S.2⟩
replace hs' : S = iSup fun i : s => (i : Opens X.carrier) := by ext1; simpa using hs'
subst hs'
apply @Set.Finite.induction_on _ _ _ hs
· convert h₁; rw [iSup_eq_bot]; rintro ⟨_, h⟩; exact h.elim
· intro x s _ hs h₄
have : IsCompact (⨆ i : s, (i : Opens X.carrier)).1 := by
refine' ((isCompact_open_iff_eq_finset_affine_union _).mpr _).1; exact ⟨s, hs, by simp⟩
convert h₂ _ this x h₄
rw [iSup_subtype, sup_comm]
conv_rhs => rw [iSup_subtype]
exact iSup_insert
[GOAL]
X Y : Scheme
f : X ⟶ Y
Z : Scheme
P : Opens ↑↑X.toPresheafedSpace → Prop
S : Opens ↑↑X.toPresheafedSpace
hS : IsCompact S.carrier
h₁ : P ⊥
h₂ : ∀ (S : Opens ↑↑X.toPresheafedSpace), IsCompact S.carrier → ∀ (U : ↑(Scheme.affineOpens X)), P S → P (S ⊔ ↑U)
⊢ P S
[PROOFSTEP]
obtain ⟨s, hs, hs'⟩ := (isCompact_open_iff_eq_finset_affine_union S.1).mp ⟨hS, S.2⟩
[GOAL]
case intro.intro
X Y : Scheme
f : X ⟶ Y
Z : Scheme
P : Opens ↑↑X.toPresheafedSpace → Prop
S : Opens ↑↑X.toPresheafedSpace
hS : IsCompact S.carrier
h₁ : P ⊥
h₂ : ∀ (S : Opens ↑↑X.toPresheafedSpace), IsCompact S.carrier → ∀ (U : ↑(Scheme.affineOpens X)), P S → P (S ⊔ ↑U)
s : Set ↑(Scheme.affineOpens X)
hs : Set.Finite s
hs' : S.carrier = ⋃ (i : ↑(Scheme.affineOpens X)) (_ : i ∈ s), ↑↑i
⊢ P S
[PROOFSTEP]
replace hs' : S = iSup fun i : s => (i : Opens X.carrier) := by ext1; simpa using hs'
[GOAL]
X Y : Scheme
f : X ⟶ Y
Z : Scheme
P : Opens ↑↑X.toPresheafedSpace → Prop
S : Opens ↑↑X.toPresheafedSpace
hS : IsCompact S.carrier
h₁ : P ⊥
h₂ : ∀ (S : Opens ↑↑X.toPresheafedSpace), IsCompact S.carrier → ∀ (U : ↑(Scheme.affineOpens X)), P S → P (S ⊔ ↑U)
s : Set ↑(Scheme.affineOpens X)
hs : Set.Finite s
hs' : S.carrier = ⋃ (i : ↑(Scheme.affineOpens X)) (_ : i ∈ s), ↑↑i
⊢ S = ⨆ (i : ↑s), ↑↑i
[PROOFSTEP]
ext1
[GOAL]
case h
X Y : Scheme
f : X ⟶ Y
Z : Scheme
P : Opens ↑↑X.toPresheafedSpace → Prop
S : Opens ↑↑X.toPresheafedSpace
hS : IsCompact S.carrier
h₁ : P ⊥
h₂ : ∀ (S : Opens ↑↑X.toPresheafedSpace), IsCompact S.carrier → ∀ (U : ↑(Scheme.affineOpens X)), P S → P (S ⊔ ↑U)
s : Set ↑(Scheme.affineOpens X)
hs : Set.Finite s
hs' : S.carrier = ⋃ (i : ↑(Scheme.affineOpens X)) (_ : i ∈ s), ↑↑i
⊢ ↑S = ↑(⨆ (i : ↑s), ↑↑i)
[PROOFSTEP]
simpa using hs'
[GOAL]
case intro.intro
X Y : Scheme
f : X ⟶ Y
Z : Scheme
P : Opens ↑↑X.toPresheafedSpace → Prop
S : Opens ↑↑X.toPresheafedSpace
hS : IsCompact S.carrier
h₁ : P ⊥
h₂ : ∀ (S : Opens ↑↑X.toPresheafedSpace), IsCompact S.carrier → ∀ (U : ↑(Scheme.affineOpens X)), P S → P (S ⊔ ↑U)
s : Set ↑(Scheme.affineOpens X)
hs : Set.Finite s
hs' : S = ⨆ (i : ↑s), ↑↑i
⊢ P S
[PROOFSTEP]
subst hs'
[GOAL]
case intro.intro
X Y : Scheme
f : X ⟶ Y
Z : Scheme
P : Opens ↑↑X.toPresheafedSpace → Prop
h₁ : P ⊥
h₂ : ∀ (S : Opens ↑↑X.toPresheafedSpace), IsCompact S.carrier → ∀ (U : ↑(Scheme.affineOpens X)), P S → P (S ⊔ ↑U)
s : Set ↑(Scheme.affineOpens X)
hs : Set.Finite s
hS : IsCompact (⨆ (i : ↑s), ↑↑i).carrier
⊢ P (⨆ (i : ↑s), ↑↑i)
[PROOFSTEP]
apply @Set.Finite.induction_on _ _ _ hs
[GOAL]
case intro.intro.H0
X Y : Scheme
f : X ⟶ Y
Z : Scheme
P : Opens ↑↑X.toPresheafedSpace → Prop
h₁ : P ⊥
h₂ : ∀ (S : Opens ↑↑X.toPresheafedSpace), IsCompact S.carrier → ∀ (U : ↑(Scheme.affineOpens X)), P S → P (S ⊔ ↑U)
s : Set ↑(Scheme.affineOpens X)
hs : Set.Finite s
hS : IsCompact (⨆ (i : ↑s), ↑↑i).carrier
⊢ P (⨆ (i : ↑∅), ↑↑i)
[PROOFSTEP]
convert h₁
[GOAL]
case h.e'_1
X Y : Scheme
f : X ⟶ Y
Z : Scheme
P : Opens ↑↑X.toPresheafedSpace → Prop
h₁ : P ⊥
h₂ : ∀ (S : Opens ↑↑X.toPresheafedSpace), IsCompact S.carrier → ∀ (U : ↑(Scheme.affineOpens X)), P S → P (S ⊔ ↑U)
s : Set ↑(Scheme.affineOpens X)
hs : Set.Finite s
hS : IsCompact (⨆ (i : ↑s), ↑↑i).carrier
⊢ ⨆ (i : ↑∅), ↑↑i = ⊥
[PROOFSTEP]
rw [iSup_eq_bot]
[GOAL]
case h.e'_1
X Y : Scheme
f : X ⟶ Y
Z : Scheme
P : Opens ↑↑X.toPresheafedSpace → Prop
h₁ : P ⊥
h₂ : ∀ (S : Opens ↑↑X.toPresheafedSpace), IsCompact S.carrier → ∀ (U : ↑(Scheme.affineOpens X)), P S → P (S ⊔ ↑U)
s : Set ↑(Scheme.affineOpens X)
hs : Set.Finite s
hS : IsCompact (⨆ (i : ↑s), ↑↑i).carrier
⊢ ∀ (i : ↑∅), ↑↑i = ⊥
[PROOFSTEP]
rintro ⟨_, h⟩
[GOAL]
case h.e'_1.mk
X Y : Scheme
f : X ⟶ Y
Z : Scheme
P : Opens ↑↑X.toPresheafedSpace → Prop
h₁ : P ⊥
h₂ : ∀ (S : Opens ↑↑X.toPresheafedSpace), IsCompact S.carrier → ∀ (U : ↑(Scheme.affineOpens X)), P S → P (S ⊔ ↑U)
s : Set ↑(Scheme.affineOpens X)
hs : Set.Finite s
hS : IsCompact (⨆ (i : ↑s), ↑↑i).carrier
val✝ : ↑(Scheme.affineOpens X)
h : val✝ ∈ ∅
⊢ ↑↑{ val := val✝, property := h } = ⊥
[PROOFSTEP]
exact h.elim
[GOAL]
case intro.intro.H1
X Y : Scheme
f : X ⟶ Y
Z : Scheme
P : Opens ↑↑X.toPresheafedSpace → Prop
h₁ : P ⊥
h₂ : ∀ (S : Opens ↑↑X.toPresheafedSpace), IsCompact S.carrier → ∀ (U : ↑(Scheme.affineOpens X)), P S → P (S ⊔ ↑U)
s : Set ↑(Scheme.affineOpens X)
hs : Set.Finite s
hS : IsCompact (⨆ (i : ↑s), ↑↑i).carrier
⊢ ∀ {a : ↑(Scheme.affineOpens X)} {s : Set ↑(Scheme.affineOpens X)},
¬a ∈ s → Set.Finite s → P (⨆ (i : ↑s), ↑↑i) → P (⨆ (i : ↑(insert a s)), ↑↑i)
[PROOFSTEP]
intro x s _ hs h₄
[GOAL]
case intro.intro.H1
X Y : Scheme
f : X ⟶ Y
Z : Scheme
P : Opens ↑↑X.toPresheafedSpace → Prop
h₁ : P ⊥
h₂ : ∀ (S : Opens ↑↑X.toPresheafedSpace), IsCompact S.carrier → ∀ (U : ↑(Scheme.affineOpens X)), P S → P (S ⊔ ↑U)
s✝ : Set ↑(Scheme.affineOpens X)
hs✝ : Set.Finite s✝
hS : IsCompact (⨆ (i : ↑s✝), ↑↑i).carrier
x : ↑(Scheme.affineOpens X)
s : Set ↑(Scheme.affineOpens X)
a✝ : ¬x ∈ s
hs : Set.Finite s
h₄ : P (⨆ (i : ↑s), ↑↑i)
⊢ P (⨆ (i : ↑(insert x s)), ↑↑i)
[PROOFSTEP]
have : IsCompact (⨆ i : s, (i : Opens X.carrier)).1 := by
refine' ((isCompact_open_iff_eq_finset_affine_union _).mpr _).1; exact ⟨s, hs, by simp⟩
[GOAL]
X Y : Scheme
f : X ⟶ Y
Z : Scheme
P : Opens ↑↑X.toPresheafedSpace → Prop
h₁ : P ⊥
h₂ : ∀ (S : Opens ↑↑X.toPresheafedSpace), IsCompact S.carrier → ∀ (U : ↑(Scheme.affineOpens X)), P S → P (S ⊔ ↑U)
s✝ : Set ↑(Scheme.affineOpens X)
hs✝ : Set.Finite s✝
hS : IsCompact (⨆ (i : ↑s✝), ↑↑i).carrier
x : ↑(Scheme.affineOpens X)
s : Set ↑(Scheme.affineOpens X)
a✝ : ¬x ∈ s
hs : Set.Finite s
h₄ : P (⨆ (i : ↑s), ↑↑i)
⊢ IsCompact (⨆ (i : ↑s), ↑↑i).carrier
[PROOFSTEP]
refine' ((isCompact_open_iff_eq_finset_affine_union _).mpr _).1
[GOAL]
X Y : Scheme
f : X ⟶ Y
Z : Scheme
P : Opens ↑↑X.toPresheafedSpace → Prop
h₁ : P ⊥
h₂ : ∀ (S : Opens ↑↑X.toPresheafedSpace), IsCompact S.carrier → ∀ (U : ↑(Scheme.affineOpens X)), P S → P (S ⊔ ↑U)
s✝ : Set ↑(Scheme.affineOpens X)
hs✝ : Set.Finite s✝
hS : IsCompact (⨆ (i : ↑s✝), ↑↑i).carrier
x : ↑(Scheme.affineOpens X)
s : Set ↑(Scheme.affineOpens X)
a✝ : ¬x ∈ s
hs : Set.Finite s
h₄ : P (⨆ (i : ↑s), ↑↑i)
⊢ ∃ s_1, Set.Finite s_1 ∧ (⨆ (i : ↑s), ↑↑i).carrier = ⋃ (i : ↑(Scheme.affineOpens X)) (_ : i ∈ s_1), ↑↑i
[PROOFSTEP]
exact ⟨s, hs, by simp⟩
[GOAL]
X Y : Scheme
f : X ⟶ Y
Z : Scheme
P : Opens ↑↑X.toPresheafedSpace → Prop
h₁ : P ⊥
h₂ : ∀ (S : Opens ↑↑X.toPresheafedSpace), IsCompact S.carrier → ∀ (U : ↑(Scheme.affineOpens X)), P S → P (S ⊔ ↑U)
s✝ : Set ↑(Scheme.affineOpens X)
hs✝ : Set.Finite s✝
hS : IsCompact (⨆ (i : ↑s✝), ↑↑i).carrier
x : ↑(Scheme.affineOpens X)
s : Set ↑(Scheme.affineOpens X)
a✝ : ¬x ∈ s
hs : Set.Finite s
h₄ : P (⨆ (i : ↑s), ↑↑i)
⊢ (⨆ (i : ↑s), ↑↑i).carrier = ⋃ (i : ↑(Scheme.affineOpens X)) (_ : i ∈ s), ↑↑i
[PROOFSTEP]
simp
[GOAL]
case intro.intro.H1
X Y : Scheme
f : X ⟶ Y
Z : Scheme
P : Opens ↑↑X.toPresheafedSpace → Prop
h₁ : P ⊥
h₂ : ∀ (S : Opens ↑↑X.toPresheafedSpace), IsCompact S.carrier → ∀ (U : ↑(Scheme.affineOpens X)), P S → P (S ⊔ ↑U)
s✝ : Set ↑(Scheme.affineOpens X)
hs✝ : Set.Finite s✝
hS : IsCompact (⨆ (i : ↑s✝), ↑↑i).carrier
x : ↑(Scheme.affineOpens X)
s : Set ↑(Scheme.affineOpens X)
a✝ : ¬x ∈ s
hs : Set.Finite s
h₄ : P (⨆ (i : ↑s), ↑↑i)
this : IsCompact (⨆ (i : ↑s), ↑↑i).carrier
⊢ P (⨆ (i : ↑(insert x s)), ↑↑i)
[PROOFSTEP]
convert h₂ _ this x h₄
[GOAL]
case h.e'_1
X Y : Scheme
f : X ⟶ Y
Z : Scheme
P : Opens ↑↑X.toPresheafedSpace → Prop
h₁ : P ⊥
h₂ : ∀ (S : Opens ↑↑X.toPresheafedSpace), IsCompact S.carrier → ∀ (U : ↑(Scheme.affineOpens X)), P S → P (S ⊔ ↑U)
s✝ : Set ↑(Scheme.affineOpens X)
hs✝ : Set.Finite s✝
hS : IsCompact (⨆ (i : ↑s✝), ↑↑i).carrier
x : ↑(Scheme.affineOpens X)
s : Set ↑(Scheme.affineOpens X)
a✝ : ¬x ∈ s
hs : Set.Finite s
h₄ : P (⨆ (i : ↑s), ↑↑i)
this : IsCompact (⨆ (i : ↑s), ↑↑i).carrier
⊢ ⨆ (i : ↑(insert x s)), ↑↑i = (⨆ (i : ↑s), ↑↑i) ⊔ ↑x
[PROOFSTEP]
rw [iSup_subtype, sup_comm]
[GOAL]
case h.e'_1
X Y : Scheme
f : X ⟶ Y
Z : Scheme
P : Opens ↑↑X.toPresheafedSpace → Prop
h₁ : P ⊥
h₂ : ∀ (S : Opens ↑↑X.toPresheafedSpace), IsCompact S.carrier → ∀ (U : ↑(Scheme.affineOpens X)), P S → P (S ⊔ ↑U)
s✝ : Set ↑(Scheme.affineOpens X)
hs✝ : Set.Finite s✝
hS : IsCompact (⨆ (i : ↑s✝), ↑↑i).carrier
x : ↑(Scheme.affineOpens X)
s : Set ↑(Scheme.affineOpens X)
a✝ : ¬x ∈ s
hs : Set.Finite s
h₄ : P (⨆ (i : ↑s), ↑↑i)
this : IsCompact (⨆ (i : ↑s), ↑↑i).carrier
⊢ ⨆ (i : ↑(Scheme.affineOpens X)) (h : i ∈ insert x s), ↑↑{ val := i, property := h } = ↑x ⊔ ⨆ (i : ↑s), ↑↑i
[PROOFSTEP]
conv_rhs => rw [iSup_subtype]
[GOAL]
X Y : Scheme
f : X ⟶ Y
Z : Scheme
P : Opens ↑↑X.toPresheafedSpace → Prop
h₁ : P ⊥
h₂ : ∀ (S : Opens ↑↑X.toPresheafedSpace), IsCompact S.carrier → ∀ (U : ↑(Scheme.affineOpens X)), P S → P (S ⊔ ↑U)
s✝ : Set ↑(Scheme.affineOpens X)
hs✝ : Set.Finite s✝
hS : IsCompact (⨆ (i : ↑s✝), ↑↑i).carrier
x : ↑(Scheme.affineOpens X)
s : Set ↑(Scheme.affineOpens X)
a✝ : ¬x ∈ s
hs : Set.Finite s
h₄ : P (⨆ (i : ↑s), ↑↑i)
this : IsCompact (⨆ (i : ↑s), ↑↑i).carrier
| ↑x ⊔ ⨆ (i : ↑s), ↑↑i
[PROOFSTEP]
rw [iSup_subtype]
[GOAL]
X Y : Scheme
f : X ⟶ Y
Z : Scheme
P : Opens ↑↑X.toPresheafedSpace → Prop
h₁ : P ⊥
h₂ : ∀ (S : Opens ↑↑X.toPresheafedSpace), IsCompact S.carrier → ∀ (U : ↑(Scheme.affineOpens X)), P S → P (S ⊔ ↑U)
s✝ : Set ↑(Scheme.affineOpens X)
hs✝ : Set.Finite s✝
hS : IsCompact (⨆ (i : ↑s✝), ↑↑i).carrier
x : ↑(Scheme.affineOpens X)
s : Set ↑(Scheme.affineOpens X)
a✝ : ¬x ∈ s
hs : Set.Finite s
h₄ : P (⨆ (i : ↑s), ↑↑i)
this : IsCompact (⨆ (i : ↑s), ↑↑i).carrier
| ↑x ⊔ ⨆ (i : ↑s), ↑↑i
[PROOFSTEP]
rw [iSup_subtype]
[GOAL]
X Y : Scheme
f : X ⟶ Y
Z : Scheme
P : Opens ↑↑X.toPresheafedSpace → Prop
h₁ : P ⊥
h₂ : ∀ (S : Opens ↑↑X.toPresheafedSpace), IsCompact S.carrier → ∀ (U : ↑(Scheme.affineOpens X)), P S → P (S ⊔ ↑U)
s✝ : Set ↑(Scheme.affineOpens X)
hs✝ : Set.Finite s✝
hS : IsCompact (⨆ (i : ↑s✝), ↑↑i).carrier
x : ↑(Scheme.affineOpens X)
s : Set ↑(Scheme.affineOpens X)
a✝ : ¬x ∈ s
hs : Set.Finite s
h₄ : P (⨆ (i : ↑s), ↑↑i)
this : IsCompact (⨆ (i : ↑s), ↑↑i).carrier
| ↑x ⊔ ⨆ (i : ↑s), ↑↑i
[PROOFSTEP]
rw [iSup_subtype]
[GOAL]
case h.e'_1
X Y : Scheme
f : X ⟶ Y
Z : Scheme
P : Opens ↑↑X.toPresheafedSpace → Prop
h₁ : P ⊥
h₂ : ∀ (S : Opens ↑↑X.toPresheafedSpace), IsCompact S.carrier → ∀ (U : ↑(Scheme.affineOpens X)), P S → P (S ⊔ ↑U)
s✝ : Set ↑(Scheme.affineOpens X)
hs✝ : Set.Finite s✝
hS : IsCompact (⨆ (i : ↑s✝), ↑↑i).carrier
x : ↑(Scheme.affineOpens X)
s : Set ↑(Scheme.affineOpens X)
a✝ : ¬x ∈ s
hs : Set.Finite s
h₄ : P (⨆ (i : ↑s), ↑↑i)
this : IsCompact (⨆ (i : ↑s), ↑↑i).carrier
⊢ ⨆ (i : ↑(Scheme.affineOpens X)) (h : i ∈ insert x s), ↑↑{ val := i, property := h } =
↑x ⊔ ⨆ (i : ↑(Scheme.affineOpens X)) (h : i ∈ s), ↑↑{ val := i, property := h }
[PROOFSTEP]
exact iSup_insert
[GOAL]
X✝ Y : Scheme
f✝ : X✝ ⟶ Y
Z : Scheme
X : Scheme
U : Opens ↑↑X.toPresheafedSpace
hU : IsAffineOpen U
x f : ↑(X.presheaf.obj (op U))
H : x |_ Scheme.basicOpen X f = 0
⊢ ∃ n, f ^ n * x = 0
[PROOFSTEP]
rw [← map_zero (X.presheaf.map (homOfLE <| X.basicOpen_le f : X.basicOpen f ⟶ U).op)] at H
[GOAL]
X✝ Y : Scheme
f✝ : X✝ ⟶ Y
Z : Scheme
X : Scheme
U : Opens ↑↑X.toPresheafedSpace
hU : IsAffineOpen U
x f : ↑(X.presheaf.obj (op U))
H : x |_ Scheme.basicOpen X f = ↑(X.presheaf.map (homOfLE (_ : Scheme.basicOpen X f ≤ U)).op) 0
⊢ ∃ n, f ^ n * x = 0
[PROOFSTEP]
obtain ⟨⟨_, n, rfl⟩, e⟩ := (isLocalization_basicOpen hU f).eq_iff_exists'.mp H
[GOAL]
case intro.mk.intro
X✝ Y : Scheme
f✝ : X✝ ⟶ Y
Z : Scheme
X : Scheme
U : Opens ↑↑X.toPresheafedSpace
hU : IsAffineOpen U
x f : ↑(X.presheaf.obj (op U))
H : x |_ Scheme.basicOpen X f = ↑(X.presheaf.map (homOfLE (_ : Scheme.basicOpen X f ≤ U)).op) 0
n : ℕ
e :
↑{ val := (fun x x_1 => x ^ x_1) f n,
property := (_ : ∃ y, (fun x x_1 => x ^ x_1) f y = (fun x x_1 => x ^ x_1) f n) } *
x =
↑{ val := (fun x x_1 => x ^ x_1) f n,
property := (_ : ∃ y, (fun x x_1 => x ^ x_1) f y = (fun x x_1 => x ^ x_1) f n) } *
0
⊢ ∃ n, f ^ n * x = 0
[PROOFSTEP]
exact ⟨n, by simpa [mul_comm x] using e⟩
[GOAL]
X✝ Y : Scheme
f✝ : X✝ ⟶ Y
Z : Scheme
X : Scheme
U : Opens ↑↑X.toPresheafedSpace
hU : IsAffineOpen U
x f : ↑(X.presheaf.obj (op U))
H : x |_ Scheme.basicOpen X f = ↑(X.presheaf.map (homOfLE (_ : Scheme.basicOpen X f ≤ U)).op) 0
n : ℕ
e :
↑{ val := (fun x x_1 => x ^ x_1) f n,
property := (_ : ∃ y, (fun x x_1 => x ^ x_1) f y = (fun x x_1 => x ^ x_1) f n) } *
x =
↑{ val := (fun x x_1 => x ^ x_1) f n,
property := (_ : ∃ y, (fun x x_1 => x ^ x_1) f y = (fun x x_1 => x ^ x_1) f n) } *
0
⊢ f ^ n * x = 0
[PROOFSTEP]
simpa [mul_comm x] using e
[GOAL]
X✝ Y : Scheme
f✝ : X✝ ⟶ Y
Z X : Scheme
U : Opens ↑↑X.toPresheafedSpace
hU : IsCompact U.carrier
x f : ↑(X.presheaf.obj (op U))
H : x |_ Scheme.basicOpen X f = 0
⊢ ∃ n, f ^ n * x = 0
[PROOFSTEP]
obtain ⟨s, hs, e⟩ := (isCompact_open_iff_eq_finset_affine_union U.1).mp ⟨hU, U.2⟩
[GOAL]
case intro.intro
X✝ Y : Scheme
f✝ : X✝ ⟶ Y
Z X : Scheme
U : Opens ↑↑X.toPresheafedSpace
hU : IsCompact U.carrier
x f : ↑(X.presheaf.obj (op U))
H : x |_ Scheme.basicOpen X f = 0
s : Set ↑(Scheme.affineOpens X)
hs : Set.Finite s
e : U.carrier = ⋃ (i : ↑(Scheme.affineOpens X)) (_ : i ∈ s), ↑↑i
⊢ ∃ n, f ^ n * x = 0
[PROOFSTEP]
replace e : U = iSup fun i : s => (i : Opens X.carrier)
[GOAL]
case e
X✝ Y : Scheme
f✝ : X✝ ⟶ Y
Z X : Scheme
U : Opens ↑↑X.toPresheafedSpace
hU : IsCompact U.carrier
x f : ↑(X.presheaf.obj (op U))
H : x |_ Scheme.basicOpen X f = 0
s : Set ↑(Scheme.affineOpens X)
hs : Set.Finite s
e : U.carrier = ⋃ (i : ↑(Scheme.affineOpens X)) (_ : i ∈ s), ↑↑i
⊢ U = ⨆ (i : ↑s), ↑↑i
[PROOFSTEP]
ext1
[GOAL]
case e.h
X✝ Y : Scheme
f✝ : X✝ ⟶ Y
Z X : Scheme
U : Opens ↑↑X.toPresheafedSpace
hU : IsCompact U.carrier
x f : ↑(X.presheaf.obj (op U))
H : x |_ Scheme.basicOpen X f = 0
s : Set ↑(Scheme.affineOpens X)
hs : Set.Finite s
e : U.carrier = ⋃ (i : ↑(Scheme.affineOpens X)) (_ : i ∈ s), ↑↑i
⊢ ↑U = ↑(⨆ (i : ↑s), ↑↑i)
[PROOFSTEP]
simpa using e
[GOAL]
case intro.intro
X✝ Y : Scheme
f✝ : X✝ ⟶ Y
Z X : Scheme
U : Opens ↑↑X.toPresheafedSpace
hU : IsCompact U.carrier
x f : ↑(X.presheaf.obj (op U))
H : x |_ Scheme.basicOpen X f = 0
s : Set ↑(Scheme.affineOpens X)
hs : Set.Finite s
e : U = ⨆ (i : ↑s), ↑↑i
⊢ ∃ n, f ^ n * x = 0
[PROOFSTEP]
have h₁ : ∀ i : s, i.1.1 ≤ U := by
intro i
change (i : Opens X.carrier) ≤ U
rw [e]
-- porting note: `exact le_iSup _ _` no longer works
exact le_iSup (fun (i : s) => (i : Opens (X.toPresheafedSpace))) _
[GOAL]
X✝ Y : Scheme
f✝ : X✝ ⟶ Y
Z X : Scheme
U : Opens ↑↑X.toPresheafedSpace
hU : IsCompact U.carrier
x f : ↑(X.presheaf.obj (op U))
H : x |_ Scheme.basicOpen X f = 0
s : Set ↑(Scheme.affineOpens X)
hs : Set.Finite s
e : U = ⨆ (i : ↑s), ↑↑i
⊢ ∀ (i : ↑s), ↑↑i ≤ U
[PROOFSTEP]
intro i
[GOAL]
X✝ Y : Scheme
f✝ : X✝ ⟶ Y
Z X : Scheme
U : Opens ↑↑X.toPresheafedSpace
hU : IsCompact U.carrier
x f : ↑(X.presheaf.obj (op U))
H : x |_ Scheme.basicOpen X f = 0
s : Set ↑(Scheme.affineOpens X)
hs : Set.Finite s
e : U = ⨆ (i : ↑s), ↑↑i
i : ↑s
⊢ ↑↑i ≤ U
[PROOFSTEP]
change (i : Opens X.carrier) ≤ U
[GOAL]
X✝ Y : Scheme
f✝ : X✝ ⟶ Y
Z X : Scheme
U : Opens ↑↑X.toPresheafedSpace
hU : IsCompact U.carrier
x f : ↑(X.presheaf.obj (op U))
H : x |_ Scheme.basicOpen X f = 0
s : Set ↑(Scheme.affineOpens X)
hs : Set.Finite s
e : U = ⨆ (i : ↑s), ↑↑i
i : ↑s
⊢ ↑↑i ≤ U
[PROOFSTEP]
rw [e]
-- porting note: `exact le_iSup _ _` no longer works
[GOAL]
X✝ Y : Scheme
f✝ : X✝ ⟶ Y
Z X : Scheme
U : Opens ↑↑X.toPresheafedSpace
hU : IsCompact U.carrier
x f : ↑(X.presheaf.obj (op U))
H : x |_ Scheme.basicOpen X f = 0
s : Set ↑(Scheme.affineOpens X)
hs : Set.Finite s
e : U = ⨆ (i : ↑s), ↑↑i
i : ↑s
⊢ ↑↑i ≤ ⨆ (i : ↑s), ↑↑i
[PROOFSTEP]
exact le_iSup (fun (i : s) => (i : Opens (X.toPresheafedSpace))) _
[GOAL]
case intro.intro
X✝ Y : Scheme
f✝ : X✝ ⟶ Y
Z X : Scheme
U : Opens ↑↑X.toPresheafedSpace
hU : IsCompact U.carrier
x f : ↑(X.presheaf.obj (op U))
H : x |_ Scheme.basicOpen X f = 0
s : Set ↑(Scheme.affineOpens X)
hs : Set.Finite s
e : U = ⨆ (i : ↑s), ↑↑i
h₁ : ∀ (i : ↑s), ↑↑i ≤ U
⊢ ∃ n, f ^ n * x = 0
[PROOFSTEP]
have H' := fun i : s =>
exists_pow_mul_eq_zero_of_res_basicOpen_eq_zero_of_isAffineOpen X i.1.2 (X.presheaf.map (homOfLE (h₁ i)).op x)
(X.presheaf.map (homOfLE (h₁ i)).op f) ?_
[GOAL]
case intro.intro.refine_2
X✝ Y : Scheme
f✝ : X✝ ⟶ Y
Z X : Scheme
U : Opens ↑↑X.toPresheafedSpace
hU : IsCompact U.carrier
x f : ↑(X.presheaf.obj (op U))
H : x |_ Scheme.basicOpen X f = 0
s : Set ↑(Scheme.affineOpens X)
hs : Set.Finite s
e : U = ⨆ (i : ↑s), ↑↑i
h₁ : ∀ (i : ↑s), ↑↑i ≤ U
H' :
∀ (i : ↑s),
∃ n, ↑(X.presheaf.map (homOfLE (_ : ↑↑i ≤ U)).op) f ^ n * ↑(X.presheaf.map (homOfLE (_ : ↑↑i ≤ U)).op) x = 0
⊢ ∃ n, f ^ n * x = 0
case intro.intro.refine_1
X✝ Y : Scheme
f✝ : X✝ ⟶ Y
Z X : Scheme
U : Opens ↑↑X.toPresheafedSpace
hU : IsCompact U.carrier
x f : ↑(X.presheaf.obj (op U))
H : x |_ Scheme.basicOpen X f = 0
s : Set ↑(Scheme.affineOpens X)
hs : Set.Finite s
e : U = ⨆ (i : ↑s), ↑↑i
h₁ : ∀ (i : ↑s), ↑↑i ≤ U
i : ↑s
⊢ ↑(X.presheaf.map (homOfLE (_ : ↑↑i ≤ U)).op) x |_
Scheme.basicOpen X (↑(X.presheaf.map (homOfLE (_ : ↑↑i ≤ U)).op) f) =
0
[PROOFSTEP]
swap
[GOAL]
case intro.intro.refine_1
X✝ Y : Scheme
f✝ : X✝ ⟶ Y
Z X : Scheme
U : Opens ↑↑X.toPresheafedSpace
hU : IsCompact U.carrier
x f : ↑(X.presheaf.obj (op U))
H : x |_ Scheme.basicOpen X f = 0
s : Set ↑(Scheme.affineOpens X)
hs : Set.Finite s
e : U = ⨆ (i : ↑s), ↑↑i
h₁ : ∀ (i : ↑s), ↑↑i ≤ U
i : ↑s
⊢ ↑(X.presheaf.map (homOfLE (_ : ↑↑i ≤ U)).op) x |_
Scheme.basicOpen X (↑(X.presheaf.map (homOfLE (_ : ↑↑i ≤ U)).op) f) =
0
[PROOFSTEP]
delta TopCat.Presheaf.restrictOpen TopCat.Presheaf.restrict at H ⊢
[GOAL]
case intro.intro.refine_1
X✝ Y : Scheme
f✝ : X✝ ⟶ Y
Z X : Scheme
U : Opens ↑↑X.toPresheafedSpace
hU : IsCompact U.carrier
x f : ↑(X.presheaf.obj (op U))
H : ↑(X.presheaf.map (homOfLE (_ : ∀ ⦃a : ↑↑X.toPresheafedSpace⦄, a ∈ ↑(Scheme.basicOpen X f) → a ∈ ↑U)).op) x = 0
s : Set ↑(Scheme.affineOpens X)
hs : Set.Finite s
e : U = ⨆ (i : ↑s), ↑↑i
h₁ : ∀ (i : ↑s), ↑↑i ≤ U
i : ↑s
⊢ ↑(X.presheaf.map
(homOfLE
(_ :
∀ ⦃a : ↑↑X.toPresheafedSpace⦄,
a ∈ ↑(Scheme.basicOpen X (↑(X.presheaf.map (homOfLE (_ : ↑↑i ≤ U)).op) f)) → a ∈ ↑↑↑i)).op)
(↑(X.presheaf.map (homOfLE (_ : ↑↑i ≤ U)).op) x) =
0
[PROOFSTEP]
convert congr_arg (X.presheaf.map (homOfLE _).op) H
[GOAL]
case h.e'_2
X✝ Y : Scheme
f✝ : X✝ ⟶ Y
Z X : Scheme
U : Opens ↑↑X.toPresheafedSpace
hU : IsCompact U.carrier
x f : ↑(X.presheaf.obj (op U))
H : ↑(X.presheaf.map (homOfLE (_ : ∀ ⦃a : ↑↑X.toPresheafedSpace⦄, a ∈ ↑(Scheme.basicOpen X f) → a ∈ ↑U)).op) x = 0
s : Set ↑(Scheme.affineOpens X)
hs : Set.Finite s
e : U = ⨆ (i : ↑s), ↑↑i
h₁ : ∀ (i : ↑s), ↑↑i ≤ U
i : ↑s
⊢ ↑(X.presheaf.map
(homOfLE
(_ :
∀ ⦃a : ↑↑X.toPresheafedSpace⦄,
a ∈ ↑(Scheme.basicOpen X (↑(X.presheaf.map (homOfLE (_ : ↑↑i ≤ U)).op) f)) → a ∈ ↑↑↑i)).op)
(↑(X.presheaf.map (homOfLE (_ : ↑↑i ≤ U)).op) x) =
↑(X.presheaf.map (homOfLE ?intro.intro.refine_1.convert_1).op)
(↑(X.presheaf.map (homOfLE (_ : ∀ ⦃a : ↑↑X.toPresheafedSpace⦄, a ∈ ↑(Scheme.basicOpen X f) → a ∈ ↑U)).op) x)
[PROOFSTEP]
simp only [← comp_apply, ← Functor.map_comp]
[GOAL]
case h.e'_2
X✝ Y : Scheme
f✝ : X✝ ⟶ Y
Z X : Scheme
U : Opens ↑↑X.toPresheafedSpace
hU : IsCompact U.carrier
x f : ↑(X.presheaf.obj (op U))
H : ↑(X.presheaf.map (homOfLE (_ : ∀ ⦃a : ↑↑X.toPresheafedSpace⦄, a ∈ ↑(Scheme.basicOpen X f) → a ∈ ↑U)).op) x = 0
s : Set ↑(Scheme.affineOpens X)
hs : Set.Finite s
e : U = ⨆ (i : ↑s), ↑↑i
h₁ : ∀ (i : ↑s), ↑↑i ≤ U
i : ↑s
⊢ ↑(X.presheaf.map
((homOfLE (_ : ↑↑i ≤ U)).op ≫
(homOfLE
(_ :
∀ ⦃a : ↑↑X.toPresheafedSpace⦄,
a ∈ ↑(Scheme.basicOpen X (↑(X.presheaf.map (homOfLE (_ : ↑↑i ≤ U)).op) f)) → a ∈ ↑↑↑i)).op))
x =
↑(X.presheaf.map
((homOfLE (_ : ∀ ⦃a : ↑↑X.toPresheafedSpace⦄, a ∈ ↑(Scheme.basicOpen X f) → a ∈ ↑U)).op ≫
(homOfLE ?intro.intro.refine_1.convert_1).op))
x
[PROOFSTEP]
rfl
[GOAL]
case h.e'_3
X✝ Y : Scheme
f✝ : X✝ ⟶ Y
Z X : Scheme
U : Opens ↑↑X.toPresheafedSpace
hU : IsCompact U.carrier
x f : ↑(X.presheaf.obj (op U))
H : ↑(X.presheaf.map (homOfLE (_ : ∀ ⦃a : ↑↑X.toPresheafedSpace⦄, a ∈ ↑(Scheme.basicOpen X f) → a ∈ ↑U)).op) x = 0
s : Set ↑(Scheme.affineOpens X)
hs : Set.Finite s
e : U = ⨆ (i : ↑s), ↑↑i
h₁ : ∀ (i : ↑s), ↑↑i ≤ U
i : ↑s
⊢ 0 = ↑(X.presheaf.map (homOfLE ?intro.intro.refine_1.convert_1).op) 0
[PROOFSTEP]
rw [map_zero]
[GOAL]
case intro.intro.refine_1.convert_1
X✝ Y : Scheme
f✝ : X✝ ⟶ Y
Z X : Scheme
U : Opens ↑↑X.toPresheafedSpace
hU : IsCompact U.carrier
x f : ↑(X.presheaf.obj (op U))
H : ↑(X.presheaf.map (homOfLE (_ : ∀ ⦃a : ↑↑X.toPresheafedSpace⦄, a ∈ ↑(Scheme.basicOpen X f) → a ∈ ↑U)).op) x = 0
s : Set ↑(Scheme.affineOpens X)
hs : Set.Finite s
e : U = ⨆ (i : ↑s), ↑↑i
h₁ : ∀ (i : ↑s), ↑↑i ≤ U
i : ↑s
⊢ Scheme.basicOpen X (↑(X.presheaf.map (homOfLE (_ : ↑↑i ≤ U)).op) f) ≤ Scheme.basicOpen X f
[PROOFSTEP]
simp only [Scheme.basicOpen_res, ge_iff_le, inf_le_right]
[GOAL]
case intro.intro.refine_2
X✝ Y : Scheme
f✝ : X✝ ⟶ Y
Z X : Scheme
U : Opens ↑↑X.toPresheafedSpace
hU : IsCompact U.carrier
x f : ↑(X.presheaf.obj (op U))
H : x |_ Scheme.basicOpen X f = 0
s : Set ↑(Scheme.affineOpens X)
hs : Set.Finite s
e : U = ⨆ (i : ↑s), ↑↑i
h₁ : ∀ (i : ↑s), ↑↑i ≤ U
H' :
∀ (i : ↑s),
∃ n, ↑(X.presheaf.map (homOfLE (_ : ↑↑i ≤ U)).op) f ^ n * ↑(X.presheaf.map (homOfLE (_ : ↑↑i ≤ U)).op) x = 0
⊢ ∃ n, f ^ n * x = 0
[PROOFSTEP]
choose n hn using H'
[GOAL]
case intro.intro.refine_2
X✝ Y : Scheme
f✝ : X✝ ⟶ Y
Z X : Scheme
U : Opens ↑↑X.toPresheafedSpace
hU : IsCompact U.carrier
x f : ↑(X.presheaf.obj (op U))
H : x |_ Scheme.basicOpen X f = 0
s : Set ↑(Scheme.affineOpens X)
hs : Set.Finite s
e : U = ⨆ (i : ↑s), ↑↑i
h₁ : ∀ (i : ↑s), ↑↑i ≤ U
n : ↑s → ℕ
hn :
∀ (i : ↑s), ↑(X.presheaf.map (homOfLE (_ : ↑↑i ≤ U)).op) f ^ n i * ↑(X.presheaf.map (homOfLE (_ : ↑↑i ≤ U)).op) x = 0
⊢ ∃ n, f ^ n * x = 0
[PROOFSTEP]
haveI := hs.to_subtype
[GOAL]
case intro.intro.refine_2
X✝ Y : Scheme
f✝ : X✝ ⟶ Y
Z X : Scheme
U : Opens ↑↑X.toPresheafedSpace
hU : IsCompact U.carrier
x f : ↑(X.presheaf.obj (op U))
H : x |_ Scheme.basicOpen X f = 0
s : Set ↑(Scheme.affineOpens X)
hs : Set.Finite s
e : U = ⨆ (i : ↑s), ↑↑i
h₁ : ∀ (i : ↑s), ↑↑i ≤ U
n : ↑s → ℕ
hn :
∀ (i : ↑s), ↑(X.presheaf.map (homOfLE (_ : ↑↑i ≤ U)).op) f ^ n i * ↑(X.presheaf.map (homOfLE (_ : ↑↑i ≤ U)).op) x = 0
this : Finite ↑s
⊢ ∃ n, f ^ n * x = 0
[PROOFSTEP]
cases nonempty_fintype s
[GOAL]
case intro.intro.refine_2.intro
X✝ Y : Scheme
f✝ : X✝ ⟶ Y
Z X : Scheme
U : Opens ↑↑X.toPresheafedSpace
hU : IsCompact U.carrier
x f : ↑(X.presheaf.obj (op U))
H : x |_ Scheme.basicOpen X f = 0
s : Set ↑(Scheme.affineOpens X)
hs : Set.Finite s
e : U = ⨆ (i : ↑s), ↑↑i
h₁ : ∀ (i : ↑s), ↑↑i ≤ U
n : ↑s → ℕ
hn :
∀ (i : ↑s), ↑(X.presheaf.map (homOfLE (_ : ↑↑i ≤ U)).op) f ^ n i * ↑(X.presheaf.map (homOfLE (_ : ↑↑i ≤ U)).op) x = 0
this : Finite ↑s
val✝ : Fintype ↑s
⊢ ∃ n, f ^ n * x = 0
[PROOFSTEP]
use Finset.univ.sup n
[GOAL]
case h
X✝ Y : Scheme
f✝ : X✝ ⟶ Y
Z X : Scheme
U : Opens ↑↑X.toPresheafedSpace
hU : IsCompact U.carrier
x f : ↑(X.presheaf.obj (op U))
H : x |_ Scheme.basicOpen X f = 0
s : Set ↑(Scheme.affineOpens X)
hs : Set.Finite s
e : U = ⨆ (i : ↑s), ↑↑i
h₁ : ∀ (i : ↑s), ↑↑i ≤ U
n : ↑s → ℕ
hn :
∀ (i : ↑s), ↑(X.presheaf.map (homOfLE (_ : ↑↑i ≤ U)).op) f ^ n i * ↑(X.presheaf.map (homOfLE (_ : ↑↑i ≤ U)).op) x = 0
this : Finite ↑s
val✝ : Fintype ↑s
⊢ f ^ Finset.sup Finset.univ n * x = 0
[PROOFSTEP]
suffices ∀ i : s, X.presheaf.map (homOfLE (h₁ i)).op (f ^ Finset.univ.sup n * x) = 0
by
subst e
apply TopCat.Sheaf.eq_of_locally_eq X.sheaf fun i : s => (i : Opens X.carrier)
intro i
rw [map_zero]
apply this
[GOAL]
X✝ Y : Scheme
f✝ : X✝ ⟶ Y
Z X : Scheme
U : Opens ↑↑X.toPresheafedSpace
hU : IsCompact U.carrier
x f : ↑(X.presheaf.obj (op U))
H : x |_ Scheme.basicOpen X f = 0
s : Set ↑(Scheme.affineOpens X)
hs : Set.Finite s
e : U = ⨆ (i : ↑s), ↑↑i
h₁ : ∀ (i : ↑s), ↑↑i ≤ U
n : ↑s → ℕ
hn :
∀ (i : ↑s), ↑(X.presheaf.map (homOfLE (_ : ↑↑i ≤ U)).op) f ^ n i * ↑(X.presheaf.map (homOfLE (_ : ↑↑i ≤ U)).op) x = 0
this✝ : Finite ↑s
val✝ : Fintype ↑s
this : ∀ (i : ↑s), ↑(X.presheaf.map (homOfLE (_ : ↑↑i ≤ U)).op) (f ^ Finset.sup Finset.univ n * x) = 0
⊢ f ^ Finset.sup Finset.univ n * x = 0
[PROOFSTEP]
subst e
[GOAL]
X✝ Y : Scheme
f✝ : X✝ ⟶ Y
Z X : Scheme
s : Set ↑(Scheme.affineOpens X)
hs : Set.Finite s
n : ↑s → ℕ
this✝ : Finite ↑s
val✝ : Fintype ↑s
hU : IsCompact (⨆ (i : ↑s), ↑↑i).carrier
x f : ↑(X.presheaf.obj (op (⨆ (i : ↑s), ↑↑i)))
H : x |_ Scheme.basicOpen X f = 0
h₁ : ∀ (i : ↑s), ↑↑i ≤ ⨆ (i : ↑s), ↑↑i
hn :
∀ (i : ↑s),
↑(X.presheaf.map (homOfLE (_ : ↑↑i ≤ ⨆ (i : ↑s), ↑↑i)).op) f ^ n i *
↑(X.presheaf.map (homOfLE (_ : ↑↑i ≤ ⨆ (i : ↑s), ↑↑i)).op) x =
0
this : ∀ (i : ↑s), ↑(X.presheaf.map (homOfLE (_ : ↑↑i ≤ ⨆ (i : ↑s), ↑↑i)).op) (f ^ Finset.sup Finset.univ n * x) = 0
⊢ f ^ Finset.sup Finset.univ n * x = 0
[PROOFSTEP]
apply TopCat.Sheaf.eq_of_locally_eq X.sheaf fun i : s => (i : Opens X.carrier)
[GOAL]
case h
X✝ Y : Scheme
f✝ : X✝ ⟶ Y
Z X : Scheme
s : Set ↑(Scheme.affineOpens X)
hs : Set.Finite s
n : ↑s → ℕ
this✝ : Finite ↑s
val✝ : Fintype ↑s
hU : IsCompact (⨆ (i : ↑s), ↑↑i).carrier
x f : ↑(X.presheaf.obj (op (⨆ (i : ↑s), ↑↑i)))
H : x |_ Scheme.basicOpen X f = 0
h₁ : ∀ (i : ↑s), ↑↑i ≤ ⨆ (i : ↑s), ↑↑i
hn :
∀ (i : ↑s),
↑(X.presheaf.map (homOfLE (_ : ↑↑i ≤ ⨆ (i : ↑s), ↑↑i)).op) f ^ n i *
↑(X.presheaf.map (homOfLE (_ : ↑↑i ≤ ⨆ (i : ↑s), ↑↑i)).op) x =
0
this : ∀ (i : ↑s), ↑(X.presheaf.map (homOfLE (_ : ↑↑i ≤ ⨆ (i : ↑s), ↑↑i)).op) (f ^ Finset.sup Finset.univ n * x) = 0
⊢ ∀ (i : ↑s),
↑((Scheme.sheaf X).val.map (Opens.leSupr (fun i => ↑↑i) i).op) (f ^ Finset.sup Finset.univ n * x) =
↑((Scheme.sheaf X).val.map (Opens.leSupr (fun i => ↑↑i) i).op) 0
[PROOFSTEP]
intro i
[GOAL]
case h
X✝ Y : Scheme
f✝ : X✝ ⟶ Y
Z X : Scheme
s : Set ↑(Scheme.affineOpens X)
hs : Set.Finite s
n : ↑s → ℕ
this✝ : Finite ↑s
val✝ : Fintype ↑s
hU : IsCompact (⨆ (i : ↑s), ↑↑i).carrier
x f : ↑(X.presheaf.obj (op (⨆ (i : ↑s), ↑↑i)))
H : x |_ Scheme.basicOpen X f = 0
h₁ : ∀ (i : ↑s), ↑↑i ≤ ⨆ (i : ↑s), ↑↑i
hn :
∀ (i : ↑s),
↑(X.presheaf.map (homOfLE (_ : ↑↑i ≤ ⨆ (i : ↑s), ↑↑i)).op) f ^ n i *
↑(X.presheaf.map (homOfLE (_ : ↑↑i ≤ ⨆ (i : ↑s), ↑↑i)).op) x =
0
this : ∀ (i : ↑s), ↑(X.presheaf.map (homOfLE (_ : ↑↑i ≤ ⨆ (i : ↑s), ↑↑i)).op) (f ^ Finset.sup Finset.univ n * x) = 0
i : ↑s
⊢ ↑((Scheme.sheaf X).val.map (Opens.leSupr (fun i => ↑↑i) i).op) (f ^ Finset.sup Finset.univ n * x) =
↑((Scheme.sheaf X).val.map (Opens.leSupr (fun i => ↑↑i) i).op) 0
[PROOFSTEP]
rw [map_zero]
[GOAL]
case h
X✝ Y : Scheme
f✝ : X✝ ⟶ Y
Z X : Scheme
s : Set ↑(Scheme.affineOpens X)
hs : Set.Finite s
n : ↑s → ℕ
this✝ : Finite ↑s
val✝ : Fintype ↑s
hU : IsCompact (⨆ (i : ↑s), ↑↑i).carrier
x f : ↑(X.presheaf.obj (op (⨆ (i : ↑s), ↑↑i)))
H : x |_ Scheme.basicOpen X f = 0
h₁ : ∀ (i : ↑s), ↑↑i ≤ ⨆ (i : ↑s), ↑↑i
hn :
∀ (i : ↑s),
↑(X.presheaf.map (homOfLE (_ : ↑↑i ≤ ⨆ (i : ↑s), ↑↑i)).op) f ^ n i *
↑(X.presheaf.map (homOfLE (_ : ↑↑i ≤ ⨆ (i : ↑s), ↑↑i)).op) x =
0
this : ∀ (i : ↑s), ↑(X.presheaf.map (homOfLE (_ : ↑↑i ≤ ⨆ (i : ↑s), ↑↑i)).op) (f ^ Finset.sup Finset.univ n * x) = 0
i : ↑s
⊢ ↑((Scheme.sheaf X).val.map (Opens.leSupr (fun i => ↑↑i) i).op) (f ^ Finset.sup Finset.univ n * x) = 0
[PROOFSTEP]
apply this
[GOAL]
case h
X✝ Y : Scheme
f✝ : X✝ ⟶ Y
Z X : Scheme
U : Opens ↑↑X.toPresheafedSpace
hU : IsCompact U.carrier
x f : ↑(X.presheaf.obj (op U))
H : x |_ Scheme.basicOpen X f = 0
s : Set ↑(Scheme.affineOpens X)
hs : Set.Finite s
e : U = ⨆ (i : ↑s), ↑↑i
h₁ : ∀ (i : ↑s), ↑↑i ≤ U
n : ↑s → ℕ
hn :
∀ (i : ↑s), ↑(X.presheaf.map (homOfLE (_ : ↑↑i ≤ U)).op) f ^ n i * ↑(X.presheaf.map (homOfLE (_ : ↑↑i ≤ U)).op) x = 0
this : Finite ↑s
val✝ : Fintype ↑s
⊢ ∀ (i : ↑s), ↑(X.presheaf.map (homOfLE (_ : ↑↑i ≤ U)).op) (f ^ Finset.sup Finset.univ n * x) = 0
[PROOFSTEP]
intro i
[GOAL]
case h
X✝ Y : Scheme
f✝ : X✝ ⟶ Y
Z X : Scheme
U : Opens ↑↑X.toPresheafedSpace
hU : IsCompact U.carrier
x f : ↑(X.presheaf.obj (op U))
H : x |_ Scheme.basicOpen X f = 0
s : Set ↑(Scheme.affineOpens X)
hs : Set.Finite s
e : U = ⨆ (i : ↑s), ↑↑i
h₁ : ∀ (i : ↑s), ↑↑i ≤ U
n : ↑s → ℕ
hn :
∀ (i : ↑s), ↑(X.presheaf.map (homOfLE (_ : ↑↑i ≤ U)).op) f ^ n i * ↑(X.presheaf.map (homOfLE (_ : ↑↑i ≤ U)).op) x = 0
this : Finite ↑s
val✝ : Fintype ↑s
i : ↑s
⊢ ↑(X.presheaf.map (homOfLE (_ : ↑↑i ≤ U)).op) (f ^ Finset.sup Finset.univ n * x) = 0
[PROOFSTEP]
replace hn := congr_arg (fun x => X.presheaf.map (homOfLE (h₁ i)).op (f ^ (Finset.univ.sup n - n i)) * x) (hn i)
[GOAL]
case h
X✝ Y : Scheme
f✝ : X✝ ⟶ Y
Z X : Scheme
U : Opens ↑↑X.toPresheafedSpace
hU : IsCompact U.carrier
x f : ↑(X.presheaf.obj (op U))
H : x |_ Scheme.basicOpen X f = 0
s : Set ↑(Scheme.affineOpens X)
hs : Set.Finite s
e : U = ⨆ (i : ↑s), ↑↑i
h₁ : ∀ (i : ↑s), ↑↑i ≤ U
n : ↑s → ℕ
this : Finite ↑s
val✝ : Fintype ↑s
i : ↑s
hn :
(fun x => ↑(X.presheaf.map (homOfLE (_ : ↑↑i ≤ U)).op) (f ^ (Finset.sup Finset.univ n - n i)) * x)
(↑(X.presheaf.map (homOfLE (_ : ↑↑i ≤ U)).op) f ^ n i * ↑(X.presheaf.map (homOfLE (_ : ↑↑i ≤ U)).op) x) =
(fun x => ↑(X.presheaf.map (homOfLE (_ : ↑↑i ≤ U)).op) (f ^ (Finset.sup Finset.univ n - n i)) * x) 0
⊢ ↑(X.presheaf.map (homOfLE (_ : ↑↑i ≤ U)).op) (f ^ Finset.sup Finset.univ n * x) = 0
[PROOFSTEP]
dsimp at hn
[GOAL]
case h
X✝ Y : Scheme
f✝ : X✝ ⟶ Y
Z X : Scheme
U : Opens ↑↑X.toPresheafedSpace
hU : IsCompact U.carrier
x f : ↑(X.presheaf.obj (op U))
H : x |_ Scheme.basicOpen X f = 0
s : Set ↑(Scheme.affineOpens X)
hs : Set.Finite s
e : U = ⨆ (i : ↑s), ↑↑i
h₁ : ∀ (i : ↑s), ↑↑i ≤ U
n : ↑s → ℕ
this : Finite ↑s
val✝ : Fintype ↑s
i : ↑s
hn :
↑(X.presheaf.map (homOfLE (_ : ↑↑i ≤ U)).op) (f ^ (Finset.sup Finset.univ n - n i)) *
(↑(X.presheaf.map (homOfLE (_ : ↑↑i ≤ U)).op) f ^ n i * ↑(X.presheaf.map (homOfLE (_ : ↑↑i ≤ U)).op) x) =
↑(X.presheaf.map (homOfLE (_ : ↑↑i ≤ U)).op) (f ^ (Finset.sup Finset.univ n - n i)) * 0
⊢ ↑(X.presheaf.map (homOfLE (_ : ↑↑i ≤ U)).op) (f ^ Finset.sup Finset.univ n * x) = 0
[PROOFSTEP]
simp only [← map_mul, ← map_pow] at hn
[GOAL]
case h
X✝ Y : Scheme
f✝ : X✝ ⟶ Y
Z X : Scheme
U : Opens ↑↑X.toPresheafedSpace
hU : IsCompact U.carrier
x f : ↑(X.presheaf.obj (op U))
H : x |_ Scheme.basicOpen X f = 0
s : Set ↑(Scheme.affineOpens X)
hs : Set.Finite s
e : U = ⨆ (i : ↑s), ↑↑i
h₁ : ∀ (i : ↑s), ↑↑i ≤ U
n : ↑s → ℕ
this : Finite ↑s
val✝ : Fintype ↑s
i : ↑s
hn :
↑(X.presheaf.map (homOfLE (_ : ↑↑i ≤ U)).op) (f ^ (Finset.sup Finset.univ n - n i) * (f ^ n i * x)) =
↑(X.presheaf.map (homOfLE (_ : ↑↑i ≤ U)).op) (f ^ (Finset.sup Finset.univ n - n i)) * 0
⊢ ↑(X.presheaf.map (homOfLE (_ : ↑↑i ≤ U)).op) (f ^ Finset.sup Finset.univ n * x) = 0
[PROOFSTEP]
rwa [mul_zero, ← mul_assoc, ← pow_add, tsub_add_cancel_of_le] at hn
[GOAL]
case h
X✝ Y : Scheme
f✝ : X✝ ⟶ Y
Z X : Scheme
U : Opens ↑↑X.toPresheafedSpace
hU : IsCompact U.carrier
x f : ↑(X.presheaf.obj (op U))
H : x |_ Scheme.basicOpen X f = 0
s : Set ↑(Scheme.affineOpens X)
hs : Set.Finite s
e : U = ⨆ (i : ↑s), ↑↑i
h₁ : ∀ (i : ↑s), ↑↑i ≤ U
n : ↑s → ℕ
this : Finite ↑s
val✝ : Fintype ↑s
i : ↑s
hn : ↑(X.presheaf.map (homOfLE (_ : ↑↑i ≤ U)).op) (f ^ (Finset.sup Finset.univ n - n i + n i) * x) = 0
⊢ n i ≤ Finset.sup Finset.univ n
[PROOFSTEP]
apply Finset.le_sup (Finset.mem_univ i)
|
## This replication file
## makes the following tables and figures:
# Figure 5
# Packages
library(MatchIt)
library(lmtest)
library(sandwich)
library(tidyverse)
##
# Figure 5
##
#savepdf <- function(file, width=7.28, height=3.54){ # 3.46, 1.77
# fname <- paste("",file,".pdf",sep="")
# pdf(fname, width=width, height=height,
# pointsize=7)
# par(mgp=c(2.2,0.45,0), tcl=-0.4, mar=c(3.3,3.6,1.1,1.1))
#}
#savepdf("output/Fukumoto_Fig5")
p_path <- "../Fukumoto2021/preprocess/preprocess.0406.R"
m_path <- "../Fukumoto2021/matching_results/alt_att_0406_1000_1000.RDS"
source("gen.0406.att_beta.R")
plot(YMD[part.adj], average.C2, col = 2, lty = 2,
xlim=c(YMD[part.adj[1]], YMD[part.adj[length(part.adj)]]),
xlab="Day", ylab="Outcome",
type="l",
main = paste("Outcome (April 6): ", reduction*100, "% mitigating effect"),
ylim=c(0,1))
lines(YMD[part.adj], average.C2, col = 2, lty = 1)
lines(YMD[part.adj], average.T2, lty = 1)
#lines(YMD[part.adj], average.all.T2[part.adj], lty = 2)
abline(v=YMD[part.adj[8]], col = "turquoise") # April 6
plot(YMD[part.adj], summary.Match.out2[, 1],
xlim=c(YMD[part.adj[1]], YMD[part.adj[length(part.adj)]]),
ylim=c(-0.5,0.5),
main = "ATT",
xlab="Day", ylab="ATT",
type="l")
polygon(c(YMD[part.adj], rev(YMD[part.adj])), c(UB2, rev(LB2)), col = shadecolor, border = NA)
lines(YMD[part.adj], summary.Match.out2[, 1], col = 1, lty = 1, lwd = 3)
abline(v=YMD[part.adj[8]], col = "turquoise") # April 6
abline(0,0,lty=2)
# Fukumoto_Fig5_a_source
Fukumoto_Fig5_a <- cbind(as.character(YMD[part.adj]), average.C2, average.T2, average.all.T2[part.adj])
colnames(Fukumoto_Fig5_a) <- c("date", "outcome average in matched control municipalities", "outcome average in matched treated municipalities",
"outcome average in all treated municipalities")
write.csv(Fukumoto_Fig5_a, "../output/Fukumoto_Fig5_a_beta_source.csv", row.names = FALSE)
# Fukumoto_Fig5_b_source
Fukumoto_Fig5_b <- cbind(as.character(YMD[part.adj]), summary.Match.out2[, 1], LB2, UB2)
colnames(Fukumoto_Fig5_b) <- c("date", "ATC", "lower bound of 95% confidence interval",
"upper bound of 95% confidence interval")
write.csv(Fukumoto_Fig5_b, "../output/Fukumoto_Fig5_b_beta_source.csv", row.names = FALSE)
|
Humans vs. Zombies Rule 1: Dont Be A Dick
|
[STATEMENT]
lemma Eats_Subset_E [intro!]:
"insert (x SUBS z) (insert (y IN z) H) \<turnstile> C \<Longrightarrow> insert (Eats x y SUBS z) H \<turnstile> C"
[PROOF STATE]
proof (prove)
goal (1 subgoal):
1. insert (x SUBS z) (insert (y IN z) H) \<turnstile> C \<Longrightarrow> insert (Eats x y SUBS z) H \<turnstile> C
[PROOF STEP]
by (metis Conj_E Eats_Subset_Iff Iff_MP_left') |
/-
Copyright (c) 2022 Joël Riou. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joël Riou
! This file was ported from Lean 3 source module algebraic_topology.dold_kan.faces
! leanprover-community/mathlib commit 70fd9563a21e7b963887c9360bd29b2393e6225a
! Please do not edit these lines, except to modify the commit id
! if you have ported upstream changes.
-/
import Mathbin.AlgebraicTopology.DoldKan.Homotopies
import Mathbin.Tactic.RingExp
/-!
# Study of face maps for the Dold-Kan correspondence
TODO (@joelriou) continue adding the various files referenced below
In this file, we obtain the technical lemmas that are used in the file
`projections.lean` in order to get basic properties of the endomorphisms
`P q : K[X] ⟶ K[X]` with respect to face maps (see `homotopies.lean` for the
role of these endomorphisms in the overall strategy of proof).
The main lemma in this file is `higher_faces_vanish.induction`. It is based
on two technical lemmas `higher_faces_vanish.comp_Hσ_eq` and
`higher_faces_vanish.comp_Hσ_eq_zero`.
-/
open Nat
open CategoryTheory
open CategoryTheory.Limits
open CategoryTheory.Category
open CategoryTheory.Preadditive
open CategoryTheory.SimplicialObject
open Simplicial DoldKan
namespace AlgebraicTopology
namespace DoldKan
variable {C : Type _} [Category C] [Preadditive C]
variable {X : SimplicialObject C}
/-- A morphism `φ : Y ⟶ X _[n+1]` satisfies `higher_faces_vanish q φ`
when the compositions `φ ≫ X.δ j` are `0` for `j ≥ max 1 (n+2-q)`. When `q ≤ n+1`,
it basically means that the composition `φ ≫ X.δ j` are `0` for the `q` highest
possible values of a nonzero `j`. Otherwise, when `q ≥ n+2`, all the compositions
`φ ≫ X.δ j` for nonzero `j` vanish. See also the lemma `comp_P_eq_self_iff` in
`projections.lean` which states that `higher_faces_vanish q φ` is equivalent to
the identity `φ ≫ (P q).f (n+1) = φ`. -/
def HigherFacesVanish {Y : C} {n : ℕ} (q : ℕ) (φ : Y ⟶ X _[n + 1]) : Prop :=
∀ j : Fin (n + 1), n + 1 ≤ (j : ℕ) + q → φ ≫ X.δ j.succ = 0
#align algebraic_topology.dold_kan.higher_faces_vanish AlgebraicTopology.DoldKan.HigherFacesVanish
namespace HigherFacesVanish
@[reassoc.1]
theorem comp_δ_eq_zero {Y : C} {n : ℕ} {q : ℕ} {φ : Y ⟶ X _[n + 1]} (v : HigherFacesVanish q φ)
(j : Fin (n + 2)) (hj₁ : j ≠ 0) (hj₂ : n + 2 ≤ (j : ℕ) + q) : φ ≫ X.δ j = 0 :=
by
obtain ⟨i, hi⟩ := Fin.eq_succ_of_ne_zero hj₁
subst hi
apply v i
rw [← @Nat.add_le_add_iff_right 1, add_assoc]
simpa only [Fin.val_succ, add_assoc, add_comm 1] using hj₂
#align algebraic_topology.dold_kan.higher_faces_vanish.comp_δ_eq_zero AlgebraicTopology.DoldKan.HigherFacesVanish.comp_δ_eq_zero
theorem of_succ {Y : C} {n q : ℕ} {φ : Y ⟶ X _[n + 1]} (v : HigherFacesVanish (q + 1) φ) :
HigherFacesVanish q φ := fun j hj => v j (by simpa only [← add_assoc] using le_add_right hj)
#align algebraic_topology.dold_kan.higher_faces_vanish.of_succ AlgebraicTopology.DoldKan.HigherFacesVanish.of_succ
theorem of_comp {Y Z : C} {q n : ℕ} {φ : Y ⟶ X _[n + 1]} (v : HigherFacesVanish q φ) (f : Z ⟶ Y) :
HigherFacesVanish q (f ≫ φ) := fun j hj => by rw [assoc, v j hj, comp_zero]
#align algebraic_topology.dold_kan.higher_faces_vanish.of_comp AlgebraicTopology.DoldKan.HigherFacesVanish.of_comp
theorem comp_hσ_eq {Y : C} {n a q : ℕ} {φ : Y ⟶ X _[n + 1]} (v : HigherFacesVanish q φ)
(hnaq : n = a + q) :
φ ≫ (hσ q).f (n + 1) =
-φ ≫
X.δ ⟨a + 1, Nat.succ_lt_succ (Nat.lt_succ_iff.mpr (Nat.le.intro hnaq.symm))⟩ ≫
X.σ ⟨a, Nat.lt_succ_iff.mpr (Nat.le.intro hnaq.symm)⟩ :=
by
have hnaq_shift : ∀ d : ℕ, n + d = a + d + q :=
by
intro d
rw [add_assoc, add_comm d, ← add_assoc, hnaq]
rw [Hσ, Homotopy.nullHomotopicMap'_f (c_mk (n + 2) (n + 1) rfl) (c_mk (n + 1) n rfl),
hσ'_eq hnaq (c_mk (n + 1) n rfl), hσ'_eq (hnaq_shift 1) (c_mk (n + 2) (n + 1) rfl)]
simp only [alternating_face_map_complex.obj_d_eq, eq_to_hom_refl, comp_id, comp_sum, sum_comp,
comp_add]
simp only [comp_zsmul, zsmul_comp, ← assoc, ← mul_zsmul]
-- cleaning up the first sum
rw [← Fin.sum_congr' _ (hnaq_shift 2).symm, Fin.sum_trunc]
swap
· rintro ⟨k, hk⟩
suffices φ ≫ X.δ (⟨a + 2 + k, by linarith⟩ : Fin (n + 2)) = 0 by
simp only [this, Fin.natAdd_mk, Fin.cast_mk, zero_comp, smul_zero]
convert v ⟨a + k + 1, by linarith⟩
(by
rw [Fin.val_mk]
linarith)
rw [Nat.succ_eq_add_one]
linarith
-- cleaning up the second sum
rw [← Fin.sum_congr' _ (hnaq_shift 3).symm, @Fin.sum_trunc _ _ (a + 3)]
swap
· rintro ⟨k, hk⟩
rw [assoc, X.δ_comp_σ_of_gt', v.comp_δ_eq_zero_assoc, zero_comp, zsmul_zero]
· intro h
rw [Fin.pred_eq_iff_eq_succ, Fin.ext_iff] at h
dsimp at h
linarith
· dsimp
simp only [Fin.coe_pred, Fin.val_mk, succ_add_sub_one]
linarith
· dsimp
linarith
-- leaving out three specific terms
conv_lhs =>
congr
skip
rw [Fin.sum_univ_castSucc, Fin.sum_univ_castSucc]
rw [Fin.sum_univ_castSucc]
simp only [Fin.last, Fin.castLe_mk, Fin.coe_cast, Fin.cast_mk, Fin.coe_castLe, Fin.val_mk,
Fin.castSucc_mk, Fin.coe_castSucc]
/- the purpose of the following `simplif` is to create three subgoals in order
to finish the proof -/
have simplif :
∀ a b c d e f : Y ⟶ X _[n + 1], b = f → d + e = 0 → c + a = 0 → a + b + (c + d + e) = f :=
by
intro a b c d e f h1 h2 h3
rw [add_assoc c d e, h2, add_zero, add_comm a b, add_assoc, add_comm a c, h3, add_zero, h1]
apply simplif
· -- b=f
rw [← pow_add, Odd.neg_one_pow, neg_smul, one_zsmul]
use a
linarith
· -- d+e = 0
rw [assoc, assoc, X.δ_comp_σ_self' (Fin.castSucc_mk _ _ _).symm,
X.δ_comp_σ_succ' (Fin.succ_mk _ _ _).symm]
simp only [comp_id, pow_add _ (a + 1) 1, pow_one, mul_neg, mul_one, neg_smul, add_right_neg]
· -- c+a = 0
rw [← Finset.sum_add_distrib]
apply Finset.sum_eq_zero
rintro ⟨i, hi⟩ h₀
have hia : (⟨i, by linarith⟩ : Fin (n + 2)) ≤ Fin.castSucc (⟨a, by linarith⟩ : Fin (n + 1)) :=
by simpa only [Fin.le_iff_val_le_val, Fin.val_mk, Fin.castSucc_mk, ← lt_succ_iff] using hi
simp only [Fin.val_mk, Fin.castLe_mk, Fin.castSucc_mk, Fin.succ_mk, assoc, Fin.cast_mk, ←
δ_comp_σ_of_le X hia, add_eq_zero_iff_eq_neg, ← neg_zsmul]
congr
ring
#align algebraic_topology.dold_kan.higher_faces_vanish.comp_Hσ_eq AlgebraicTopology.DoldKan.HigherFacesVanish.comp_hσ_eq
theorem comp_hσ_eq_zero {Y : C} {n q : ℕ} {φ : Y ⟶ X _[n + 1]} (v : HigherFacesVanish q φ)
(hqn : n < q) : φ ≫ (hσ q).f (n + 1) = 0 :=
by
simp only [Hσ, Homotopy.nullHomotopicMap'_f (c_mk (n + 2) (n + 1) rfl) (c_mk (n + 1) n rfl)]
rw [hσ'_eq_zero hqn (c_mk (n + 1) n rfl), comp_zero, zero_add]
by_cases hqn' : n + 1 < q
· rw [hσ'_eq_zero hqn' (c_mk (n + 2) (n + 1) rfl), zero_comp, comp_zero]
· simp only [hσ'_eq (show n + 1 = 0 + q by linarith) (c_mk (n + 2) (n + 1) rfl), pow_zero,
Fin.mk_zero, one_zsmul, eq_to_hom_refl, comp_id, comp_sum,
alternating_face_map_complex.obj_d_eq]
rw [← Fin.sum_congr' _ (show 2 + (n + 1) = n + 1 + 2 by linarith), Fin.sum_trunc]
· simp only [Fin.sum_univ_castSucc, Fin.sum_univ_zero, zero_add, Fin.last, Fin.castLe_mk,
Fin.cast_mk, Fin.castSucc_mk]
simp only [Fin.mk_zero, Fin.val_zero, pow_zero, one_zsmul, Fin.mk_one, Fin.val_one, pow_one,
neg_smul, comp_neg]
erw [δ_comp_σ_self, δ_comp_σ_succ, add_right_neg]
· intro j
rw [comp_zsmul, comp_zsmul, δ_comp_σ_of_gt', v.comp_δ_eq_zero_assoc, zero_comp, zsmul_zero]
· intro h
rw [Fin.pred_eq_iff_eq_succ, Fin.ext_iff] at h
dsimp at h
linarith
· dsimp
simp only [Fin.cast_natAdd, Fin.coe_pred, Fin.coe_addNat, add_succ_sub_one]
linarith
· rw [Fin.lt_iff_val_lt_val]
dsimp
linarith
#align algebraic_topology.dold_kan.higher_faces_vanish.comp_Hσ_eq_zero AlgebraicTopology.DoldKan.HigherFacesVanish.comp_hσ_eq_zero
theorem induction {Y : C} {n q : ℕ} {φ : Y ⟶ X _[n + 1]} (v : HigherFacesVanish q φ) :
HigherFacesVanish (q + 1) (φ ≫ (𝟙 _ + hσ q).f (n + 1)) :=
by
intro j hj₁
dsimp
simp only [comp_add, add_comp, comp_id]
-- when n < q, the result follows immediately from the assumption
by_cases hqn : n < q
· rw [v.comp_Hσ_eq_zero hqn, zero_comp, add_zero, v j (by linarith)]
-- we now assume that n≥q, and write n=a+q
cases' Nat.le.dest (not_lt.mp hqn) with a ha
rw [v.comp_Hσ_eq (show n = a + q by linarith), neg_comp, add_neg_eq_zero, assoc, assoc]
cases' n with m hm
-- the boundary case n=0
·
simpa only [Nat.eq_zero_of_add_eq_zero_left ha, Fin.eq_zero j, Fin.mk_zero, Fin.mk_one,
δ_comp_σ_succ, comp_id]
-- in the other case, we need to write n as m+1
-- then, we first consider the particular case j = a
by_cases hj₂ : a = (j : ℕ)
· simp only [hj₂, Fin.eta, δ_comp_σ_succ, comp_id]
congr
ext
simp only [Fin.val_succ, Fin.val_mk]
-- now, we assume j ≠ a (i.e. a < j)
have haj : a < j := (Ne.le_iff_lt hj₂).mp (by linarith)
have hj₃ := j.is_lt
have ham : a ≤ m := by
by_contra
rw [not_le, ← Nat.succ_le_iff] at h
linarith
rw [X.δ_comp_σ_of_gt', j.pred_succ]
swap
· rw [Fin.lt_iff_val_lt_val]
simpa only [Fin.val_mk, Fin.val_succ, add_lt_add_iff_right] using haj
obtain ham' | ham'' := ham.lt_or_eq
· -- case where `a<m`
rw [← X.δ_comp_δ''_assoc]
swap
· rw [Fin.le_iff_val_le_val]
dsimp
linarith
simp only [← assoc, v j (by linarith), zero_comp]
· -- in the last case, a=m, q=1 and j=a+1
rw [X.δ_comp_δ_self'_assoc]
swap
· ext
dsimp
have hq : q = 1 := by rw [← add_left_inj a, ha, ham'', add_comm]
linarith
simp only [← assoc, v j (by linarith), zero_comp]
#align algebraic_topology.dold_kan.higher_faces_vanish.induction AlgebraicTopology.DoldKan.HigherFacesVanish.induction
end HigherFacesVanish
end DoldKan
end AlgebraicTopology
|
Inductive nat : Set :=
| O : nat
| S : nat -> nat.
Fixpoint plus (m n : nat) : nat :=
match m with
| O => match n with
| O => O
| S n' => S n'
end
| S m' => match n with
| O => S m'
| S n' => S (S (plus m' n'))
end
end.
Theorem S_injective : forall m n : nat, S m = S n -> m = n.
injection 1.
trivial.
Qed.
Theorem O_left_id : forall m : nat, plus O m = m.
intros.
case m.
(* m = O *)
simpl.
trivial.
(* m = S m' *)
intros.
simpl.
trivial.
Qed.
Theorem O_right_id : forall m : nat, plus m O = m.
intros.
case m.
(* m = O *)
simpl.
trivial.
(* m = S m' *)
intros.
simpl.
trivial.
Qed.
Theorem O_id : forall m : nat, plus O m = m /\ plus m O = m.
intros.
refine (conj _ _).
apply O_left_id.
apply O_right_id.
Qed. |
"""
WSVarScoreTestBaseObs
A base per-observation object for the score test of within-subject variance
linear mixed model data instance without information on X1 or W1.
Contains base variables for testing
H0: β1 = 0 and τ1 = 0, H1: β1 ≠ 0 or τ1 ≠ 0,
for the full model of WiSER (with parameters β = [β1, β2],
τ = [τ1, τ2], and Lγ),
We make use of the fitted null model.
"""
struct WSVarScoreTestBaseObs{T <: BlasReal}
nullObs :: WSVarLmmObs{T} # is it redundant?
n :: Int
p :: Int
q :: Int
l :: Int
# working arrays
A_21_β2β1_pre :: Matrix{T} # becomes A_21_β2β1 when right-multiplied by X1
A_21_τ2τ1_pre :: Matrix{T} # becomes A_21_τ2τ1 when right-multiplied by W1
A_21_Lγτ1_pre :: Matrix{T} # becomes A_21_Lγτ1 when right-multiplied by W1
A_21_β2β1_rowsum :: Vector{T} # for time-invariant X1, length p
A_21_τ2τ1_rowsum :: Vector{T} # for time-invariant W1, length l
A_21_Lγτ1_rowsum :: Vector{T} # for time-invariant W1, length q◺
end
function WSVarScoreTestBaseObs(nullObs::WSVarLmmObs{T}) where T <: BlasReal
n, p, q, l = size(nullObs.Xt, 2), size(nullObs.Xt, 1),
size(nullObs.Zt, 1), size(nullObs.Wt, 1)
q◺ = ◺(q)
A_21_β2β1_pre = Matrix{T}(undef, p, n)
A_21_τ2τ1_pre = Matrix{T}(undef, l, n)
A_21_Lγτ1_pre = Matrix{T}(undef, q◺, n)
# construct A_21_β2β1_pre
# X2t * Vinv, Vinv = Dinv - UUt
X2t_U = nullObs.Xt * transpose(nullObs.Ut)
mul!(A_21_β2β1_pre, -X2t_U, nullObs.Ut)
@inbounds @simd for j in 1:n
for i in 1:p
A_21_β2β1_pre[i, j] += nullObs.Xt[i, j] *
nullObs.Dinv[j] # first term
end
end
# construct A_21_τ2τ1_pre
# W2t * D * Vinv .* Vinv * D, this is no longer symmetric.
# compute W2t * D * Vinv .* Vinv
mul!(A_21_τ2τ1_pre, nullObs.Wt_D_Ut_kr_Utt, nullObs.Ut_kr_Ut) # third term
@inbounds @simd for j in 1:n
for i in 1:l
# first term
A_21_τ2τ1_pre[i, j] += nullObs.Wt_D_Dinv[i, j] * nullObs.Dinv[j]
# second term
A_21_τ2τ1_pre[i, j] += -2 * nullObs.Wt_D_sqrtdiagDinv_UUt[i, j] *
nullObs.sqrtDinv_UUt[j]
end
end
# right-multiply by D.
@inbounds @simd for j in 1:n
for i in 1:l
A_21_τ2τ1_pre[i, j] = A_21_τ2τ1_pre[i, j] * nullObs.expwτ[j]
end
end
# construct A_21_Lγτ1_pre
# 2 * Cq' * (L'Z'(V^-1) ⊙ Z'(V^-1)) * Diagonal(expwτ)
# nullObs.storage_q◺n is always Cq' * (L'Z'(V^-1) ⊙ Z'(V^-1)).
A_21_Lγτ1_pre .= 2 * nullObs.storage_q◺n
@inbounds @simd for j in 1:n
for i in 1:q◺
A_21_Lγτ1_pre[i, j] = A_21_Lγτ1_pre[i, j] * nullObs.expwτ[j]
end
end
A_21_β2β1_rowsum = reshape(sum(A_21_β2β1_pre; dims=2), :)
A_21_τ2τ1_rowsum = reshape(sum(A_21_τ2τ1_pre; dims=2), :)
A_21_Lγτ1_rowsum = reshape(sum(A_21_Lγτ1_pre; dims=2), :)
WSVarScoreTestBaseObs{T}(nullObs, n, p, q, l, A_21_β2β1_pre, A_21_τ2τ1_pre,
A_21_Lγτ1_pre, A_21_β2β1_rowsum, A_21_τ2τ1_rowsum, A_21_Lγτ1_rowsum)
end
"""
WSVarScoreTestObs
A base per-observation object for the score test of
within-subject variance linear mixed model data instance.
H0: β1 = 0 and τ1 = 0, H1: β1 ≠ 0 or τ1 ≠ 0,
for the full model of WiSER (with parameters β = [β1, β2], τ = [τ1, τ2], and Lγ).
We make use of the fitted null model.
"""
struct WSVarScoreTestObs{T <: BlasReal}
# data
testbaseobs :: WSVarScoreTestBaseObs{T}
r_X1 :: Int # number of test variables in X1
r_W1 :: Int # number of test variables in W1
r :: Int # number of total test variables, = r_X1 + r_W1.
#X1t :: Matrix{T} # test variables for X
#W1t :: Matrix{T} # test variables for W
# working arrays
ψ_1 :: Vector{T}
ψ_β1 :: AbstractVector{T}
ψ_τ1 :: AbstractVector{T}
# ψ_β2 = testbaseobs.nullObs.∇β
# ψ_τ2 = testbaseobs.nullObs.∇τ
# ψ_Lγ = vech(testbaseobs.nullObs.∇Lγ),
# be consistent with order of variables in WiSER.sandwich!()
A_21_β2β1 :: Matrix{T} # p x r_X1, testbaseobs.A_21_β2β1_pre * X1
A_21_τ2τ1 :: Matrix{T} # l x r_W1, testbaseobs.A_21_τ2τ1_pre * W1
A_21_Lγτ1 :: Matrix{T} # q◺ x r_W1, testbaseobs.A_21_Lγτ1_pre * W1
end
function WSVarScoreTestObs(testbaseobs::WSVarScoreTestBaseObs{T},
r_X1, r_W1
#X1obs::AbstractMatrix{T}, W1obs::AbstractMatrix{T}
) where {T <: BlasReal}
n, p, q, l = testbaseobs.n, testbaseobs.p, testbaseobs.q, testbaseobs.l
q◺ = ◺(q)
#r_X1 = size(X1obs, 2)
#r_W1 = size(W1obs, 2)
r = r_X1 + r_W1
#X1t = Matrix{T}(undef, r_X1, n)
#W1t = Matrix{T}(undef, r_W1, n)
ψ_1 = Vector{T}(undef, r)
ψ_β1 = @view(ψ_1[1:r_X1])
ψ_τ1 = @view(ψ_1[r_X1+1:end])
A_21_β2β1 = Matrix{T}(undef, p, r_X1)
A_21_τ2τ1 = Matrix{T}(undef, l, r_W1)
A_21_Lγτ1 = Matrix{T}(undef, q◺, r_W1)
WSVarScoreTestObs{T}(testbaseobs, r_X1, r_W1, r, #X1t, W1t,
ψ_1, ψ_β1, ψ_τ1, A_21_β2β1, A_21_τ2τ1, A_21_Lγτ1)
end
"""
update!(testobs::WSVarScoreTestObs, X1obs, W1obs)
Updates WSVarScoreTestObs based on the test data.
"""
function update!(testobs::WSVarScoreTestObs,
X1obs::Union{Nothing, AbstractVecOrMat{T}},
W1obs::Union{Nothing, AbstractVecOrMat{T}}
) where {T <: BlasReal}
testbaseobs = testobs.testbaseobs
if testobs.r_X1 > 0
mul!(testobs.ψ_β1,
transpose(X1obs),
reshape(testbaseobs.nullObs.Dinv_r -
transpose(testbaseobs.nullObs.rt_UUt), :
)
)
mul!(testobs.A_21_β2β1, testbaseobs.A_21_β2β1_pre, X1obs)
end
if testobs.r_W1 > 0
mul!(testobs.ψ_τ1, -transpose(W1obs), testbaseobs.nullObs.diagDVRV)
mul!(testobs.A_21_τ2τ1, testbaseobs.A_21_τ2τ1_pre, W1obs)
mul!(testobs.A_21_Lγτ1, testbaseobs.A_21_Lγτ1_pre, W1obs)
end
end
"""
WSVarScoreTest
A base object for the score test of within-subject vLMM data instance.
H0: β1 = 0 and τ1 = 0, H1: β1 ≠ 0 or τ1 ≠ 0,
for the full model of WiSER (with parameters β = [β1, β2], τ = [τ1, τ2], and Lγ).
We make use of the fitted null model.
"""
struct WSVarScoreTest{T <: BlasReal}
nullmodel :: WSVarLmmModel{T}
testobsvec :: Vector{WSVarScoreTestObs{T}}
# dimensions
p :: Int # #mean parameters in linear regression
q :: Int # #random effects
l :: Int # #parameters for modeling WS variability
m :: Int # #individuals/clusters
nsum :: Int # #observations (summed across individuals)
r_X1 :: Int # #test variables in X1
r_W1 :: Int # #test variables in W1
r :: Int # #total test variables, = r_X1 + r_W1.
# working arrays
ψ_1 :: Vector{T} # length-r vector, sum_i testobs.ψ_1
ψ_1obs :: Matrix{T} # r x m
ψ_2obs :: Matrix{T} # (p + l + q◺) x m
B_11 :: Matrix{T} # r x r matrix.
B_21 :: Matrix{T} # (p + l + q◺) x r matrix.
A_21 :: Matrix{T} # (p + l + q◺) x r matrix.
# B_22 = nullmodel.B
# inv(A_22) = nullmodel.Ainv
AinvBAinv :: AbstractMatrix{T} # r x r.
tmp_sr :: AbstractMatrix{T} # p + l + q◺ x r.
tmp_srx1 :: AbstractMatrix{T} # p + l + q◺ x r_X1.
tmp_srw1 :: AbstractMatrix{T} # p + l + q◺ x r_W1.
tmp_rr :: AbstractMatrix{T} # r x r.
tmp_rx1rx1 :: AbstractMatrix{T} # r_X1 x r_X1.
tmp_rw1rw1 :: AbstractMatrix{T} # r_W1 x r_W1.
tmp_r :: AbstractVector{T}
tmp_rx1 :: AbstractVector{T}
tmp_rw1 :: AbstractVector{T}
end
function WSVarScoreTest(nullmodel::WSVarLmmModel{T},
r_X1::Int, r_W1::Int;
testbaseobsvec::Union{Vector{WSVarScoreTestBaseObs{T}}, Nothing} = nothing
) where {T <: BlasReal}
@assert r_X1 >= 0 && r_W1 >= 0
@assert r_X1 > 0 || r_W1 > 0
if testbaseobsvec === nothing
testbaseobsvec = [WSVarScoreTestBaseObs(obs) for obs in nullmodel.data]
end
testobsvec = [WSVarScoreTestObs(testbaseobs, r_X1, r_W1) for
testbaseobs in testbaseobsvec]
@assert nullmodel.isfitted[1] "Please fit the model first."
p, q, l, m, nsum = nullmodel.p, nullmodel.q,
nullmodel.l, nullmodel.m, nullmodel.nsum
q◺ = ◺(q)
r = r_X1 + r_W1
ψ_1 = Vector{T}(undef, r)
ψ_1obs = Matrix{T}(undef, r, m)
ψ_2obs = Matrix{T}(undef, p + l + q◺, m)
fill!(ψ_2obs, zero(T))
for (i, obs) in enumerate(nullmodel.data)
ψ_β2 = @view ψ_2obs[1:p, i]
ψ_τ2 = @view ψ_2obs[(p + 1):(p + l), i]
ψ_Lγ = @view ψ_2obs[(p + l + 1):end, i]
ψ_β2 .= obs.∇β
ψ_τ2 .= obs.∇τ
offset = 1
@inbounds for j in 1:q, i in j:q
ψ_Lγ[offset] = obs.∇Lγ[i, j]
offset += 1
end
end
B_11 = Matrix{T}(undef, r, r)
B_21 = Matrix{T}(undef, p + l + q◺, r)
A_21 = Matrix{T}(undef, p + l + q◺, r)
AinvBAinv = nullmodel.Ainv * nullmodel.B * nullmodel.Ainv
tmp_sr = Matrix{T}(undef, p + l + q◺, r)
tmp_srx1 = Matrix{T}(undef, p + l + q◺, r_X1)
tmp_srw1 = Matrix{T}(undef, p + l + q◺, r_W1)
tmp_rr = Matrix{T}(undef, r, r)
tmp_rx1rx1 = Matrix{T}(undef, r_X1, r_X1)
tmp_rw1rw1 = Matrix{T}(undef, r_W1, r_W1)
tmp_r = Vector{T}(undef, r)
tmp_rx1 = Vector{T}(undef, r_X1)
tmp_rw1 = Vector{T}(undef, r_W1)
WSVarScoreTest{T}(nullmodel, testobsvec, p, q, l, m, nsum, r_X1, r_W1, r,
ψ_1, ψ_1obs, ψ_2obs, B_11, B_21, A_21,
AinvBAinv, tmp_sr, tmp_srx1, tmp_srw1, tmp_rr, tmp_rx1rx1, tmp_rw1rw1,
tmp_r, tmp_rx1, tmp_rw1
)
end
"""
test!(st::WSVarScoreTest, X1Vec, W1Vec)
Performs the score test, and returns the three p-values on the
time-variant test data
"""
function test!(st::WSVarScoreTest,
X1vec::Union{Nothing, Vector{<:AbstractVecOrMat{T}}},
W1vec::Union{Nothing, Vector{<:AbstractVecOrMat{T}}}) where {T <: BlasReal}
if X1vec === nothing
@assert st.r_X1 == 0
X1vec = repeat([nothing], length(st.testobsvec))
elseif W1vec === nothing
@assert st.r_W1 == 0
W1vec = repeat([nothing], length(st.testobsvec))
end
nm = st.nullmodel
p, q, l, m, nsum = nm.p, nm.q, nm.l, nm.m, nm.nsum
r_X1, r_W1, r = st.r_X1, st.r_W1, st.r
# update testobsvec first
for (testobs, X1obs, W1obs) in zip(st.testobsvec, X1vec, W1vec)
if r_X1 > 0
@assert size(X1obs, 1) == testobs.testbaseobs.n
@assert size(X1obs, 2) == st.r_X1
end
if r_W1 > 0
@assert size(W1obs, 1) == testobs.testbaseobs.n
@assert size(W1obs, 2) == st.r_W1
end
update!(testobs, X1obs, W1obs)
end
# build ψ_1: sum_i testobs.ψ_1
fill!(st.ψ_1, zero(T))
for (i, testobs) in enumerate(st.testobsvec)
st.ψ_1obs[:, i] .= testobs.ψ_1
st.ψ_1 .+= testobs.ψ_1
end
# build B_11: using BLAS.syr!()
fill!(st.B_11, zero(T))
BLAS.syrk!('U', 'N', one(T), st.ψ_1obs, zero(T), st.B_11)
copytri!(st.B_11, 'U')
lmul!(one(T) / m, st.B_11)
# build B_21
mul!(st.B_21, st.ψ_2obs, transpose(st.ψ_1obs), one(T) / m, zero(T))
# build A_21: 1/m sum_i Ai_21.
fill!(st.A_21, zero(T))
A_21_β2β1 = @view st.A_21[1 : p , 1 : r_X1]
A_21_τ2τ1 = @view st.A_21[p + 1 : p + l, r_X1 + 1 : r]
A_21_Lγτ1 = @view st.A_21[p + l + 1 : end , r_X1 + 1 : r]
if r_X1 > 0
for testobs in st.testobsvec
A_21_β2β1 .+= testobs.A_21_β2β1
end
end
if r_W1 > 0
for testobs in st.testobsvec
A_21_τ2τ1 .+= testobs.A_21_τ2τ1
A_21_Lγτ1 .+= testobs.A_21_Lγτ1
end
end
lmul!(1 / m, st.A_21)
pvalues!(st)
end
|
[STATEMENT]
lemma prj_chine:
shows "\<lbrakk>\<lbrakk>TfTgh.p\<^sub>1 \<star> TTfgh_TfTgh.chine\<rbrakk>\<rbrakk> = \<lbrakk>\<lbrakk>Tfg.p\<^sub>1 \<star> TTfgh.p\<^sub>1\<rbrakk>\<rbrakk>"
and "\<lbrakk>\<lbrakk>Tgh.p\<^sub>1 \<star> TfTgh.p\<^sub>0 \<star> TTfgh_TfTgh.chine\<rbrakk>\<rbrakk> = \<lbrakk>\<lbrakk>Tfg.p\<^sub>0 \<star> TTfgh.p\<^sub>1\<rbrakk>\<rbrakk>"
and "\<lbrakk>\<lbrakk>Tgh.p\<^sub>0 \<star> TfTgh.p\<^sub>0 \<star> TTfgh_TfTgh.chine\<rbrakk>\<rbrakk> = \<lbrakk>\<lbrakk>TTfgh.p\<^sub>0\<rbrakk>\<rbrakk>"
[PROOF STATE]
proof (prove)
goal (1 subgoal):
1. \<lbrakk>\<lbrakk>f\<^sub>0gh\<^sub>1.p\<^sub>1 \<star> TTfgh_TfTgh.chine\<rbrakk>\<rbrakk> = \<lbrakk>\<lbrakk>f\<^sub>0g\<^sub>1.p\<^sub>1 \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1\<rbrakk>\<rbrakk> &&& \<lbrakk>\<lbrakk>g\<^sub>0h\<^sub>1.p\<^sub>1 \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine\<rbrakk>\<rbrakk> = \<lbrakk>\<lbrakk>f\<^sub>0g\<^sub>1.p\<^sub>0 \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1\<rbrakk>\<rbrakk> &&& \<lbrakk>\<lbrakk>g\<^sub>0h\<^sub>1.p\<^sub>0 \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine\<rbrakk>\<rbrakk> = \<lbrakk>\<lbrakk>fg\<^sub>0h\<^sub>1.p\<^sub>0\<rbrakk>\<rbrakk>
[PROOF STEP]
proof -
[PROOF STATE]
proof (state)
goal (3 subgoals):
1. \<lbrakk>\<lbrakk>f\<^sub>0gh\<^sub>1.p\<^sub>1 \<star> TTfgh_TfTgh.chine\<rbrakk>\<rbrakk> = \<lbrakk>\<lbrakk>f\<^sub>0g\<^sub>1.p\<^sub>1 \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1\<rbrakk>\<rbrakk>
2. \<lbrakk>\<lbrakk>g\<^sub>0h\<^sub>1.p\<^sub>1 \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine\<rbrakk>\<rbrakk> = \<lbrakk>\<lbrakk>f\<^sub>0g\<^sub>1.p\<^sub>0 \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1\<rbrakk>\<rbrakk>
3. \<lbrakk>\<lbrakk>g\<^sub>0h\<^sub>1.p\<^sub>0 \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine\<rbrakk>\<rbrakk> = \<lbrakk>\<lbrakk>fg\<^sub>0h\<^sub>1.p\<^sub>0\<rbrakk>\<rbrakk>
[PROOF STEP]
have 1: "ide TfTgh.p\<^sub>1"
[PROOF STATE]
proof (prove)
goal (1 subgoal):
1. ide f\<^sub>0gh\<^sub>1.p\<^sub>1
[PROOF STEP]
by (simp add: TfTgh.composable)
[PROOF STATE]
proof (state)
this:
ide f\<^sub>0gh\<^sub>1.p\<^sub>1
goal (3 subgoals):
1. \<lbrakk>\<lbrakk>f\<^sub>0gh\<^sub>1.p\<^sub>1 \<star> TTfgh_TfTgh.chine\<rbrakk>\<rbrakk> = \<lbrakk>\<lbrakk>f\<^sub>0g\<^sub>1.p\<^sub>1 \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1\<rbrakk>\<rbrakk>
2. \<lbrakk>\<lbrakk>g\<^sub>0h\<^sub>1.p\<^sub>1 \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine\<rbrakk>\<rbrakk> = \<lbrakk>\<lbrakk>f\<^sub>0g\<^sub>1.p\<^sub>0 \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1\<rbrakk>\<rbrakk>
3. \<lbrakk>\<lbrakk>g\<^sub>0h\<^sub>1.p\<^sub>0 \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine\<rbrakk>\<rbrakk> = \<lbrakk>\<lbrakk>fg\<^sub>0h\<^sub>1.p\<^sub>0\<rbrakk>\<rbrakk>
[PROOF STEP]
have 2: "ide TTfgh_TfTgh.chine"
[PROOF STATE]
proof (prove)
goal (1 subgoal):
1. ide TTfgh_TfTgh.chine
[PROOF STEP]
by simp
[PROOF STATE]
proof (state)
this:
ide TTfgh_TfTgh.chine
goal (3 subgoals):
1. \<lbrakk>\<lbrakk>f\<^sub>0gh\<^sub>1.p\<^sub>1 \<star> TTfgh_TfTgh.chine\<rbrakk>\<rbrakk> = \<lbrakk>\<lbrakk>f\<^sub>0g\<^sub>1.p\<^sub>1 \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1\<rbrakk>\<rbrakk>
2. \<lbrakk>\<lbrakk>g\<^sub>0h\<^sub>1.p\<^sub>1 \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine\<rbrakk>\<rbrakk> = \<lbrakk>\<lbrakk>f\<^sub>0g\<^sub>1.p\<^sub>0 \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1\<rbrakk>\<rbrakk>
3. \<lbrakk>\<lbrakk>g\<^sub>0h\<^sub>1.p\<^sub>0 \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine\<rbrakk>\<rbrakk> = \<lbrakk>\<lbrakk>fg\<^sub>0h\<^sub>1.p\<^sub>0\<rbrakk>\<rbrakk>
[PROOF STEP]
have 3: "src TfTgh.p\<^sub>1 = trg TTfgh_TfTgh.chine"
[PROOF STATE]
proof (prove)
goal (1 subgoal):
1. src f\<^sub>0gh\<^sub>1.p\<^sub>1 = trg TTfgh_TfTgh.chine
[PROOF STEP]
using TTfgh_TfTgh.chine_in_hom(1)
[PROOF STATE]
proof (prove)
using this:
\<guillemotleft>TTfgh_TfTgh.chine : src (tab\<^sub>0 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0) \<rightarrow> src ((tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0)\<guillemotright>
goal (1 subgoal):
1. src f\<^sub>0gh\<^sub>1.p\<^sub>1 = trg TTfgh_TfTgh.chine
[PROOF STEP]
by simp
[PROOF STATE]
proof (state)
this:
src f\<^sub>0gh\<^sub>1.p\<^sub>1 = trg TTfgh_TfTgh.chine
goal (3 subgoals):
1. \<lbrakk>\<lbrakk>f\<^sub>0gh\<^sub>1.p\<^sub>1 \<star> TTfgh_TfTgh.chine\<rbrakk>\<rbrakk> = \<lbrakk>\<lbrakk>f\<^sub>0g\<^sub>1.p\<^sub>1 \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1\<rbrakk>\<rbrakk>
2. \<lbrakk>\<lbrakk>g\<^sub>0h\<^sub>1.p\<^sub>1 \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine\<rbrakk>\<rbrakk> = \<lbrakk>\<lbrakk>f\<^sub>0g\<^sub>1.p\<^sub>0 \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1\<rbrakk>\<rbrakk>
3. \<lbrakk>\<lbrakk>g\<^sub>0h\<^sub>1.p\<^sub>0 \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine\<rbrakk>\<rbrakk> = \<lbrakk>\<lbrakk>fg\<^sub>0h\<^sub>1.p\<^sub>0\<rbrakk>\<rbrakk>
[PROOF STEP]
have 4: "src (tab\<^sub>1 f) = trg TfTgh.p\<^sub>1"
[PROOF STATE]
proof (prove)
goal (1 subgoal):
1. src (tab\<^sub>1 f) = trg f\<^sub>0gh\<^sub>1.p\<^sub>1
[PROOF STEP]
using TfTgh.leg1_simps(2)
[PROOF STATE]
proof (prove)
using this:
hseq (tab\<^sub>1 f) f\<^sub>0gh\<^sub>1.p\<^sub>1
goal (1 subgoal):
1. src (tab\<^sub>1 f) = trg f\<^sub>0gh\<^sub>1.p\<^sub>1
[PROOF STEP]
by blast
[PROOF STATE]
proof (state)
this:
src (tab\<^sub>1 f) = trg f\<^sub>0gh\<^sub>1.p\<^sub>1
goal (3 subgoals):
1. \<lbrakk>\<lbrakk>f\<^sub>0gh\<^sub>1.p\<^sub>1 \<star> TTfgh_TfTgh.chine\<rbrakk>\<rbrakk> = \<lbrakk>\<lbrakk>f\<^sub>0g\<^sub>1.p\<^sub>1 \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1\<rbrakk>\<rbrakk>
2. \<lbrakk>\<lbrakk>g\<^sub>0h\<^sub>1.p\<^sub>1 \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine\<rbrakk>\<rbrakk> = \<lbrakk>\<lbrakk>f\<^sub>0g\<^sub>1.p\<^sub>0 \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1\<rbrakk>\<rbrakk>
3. \<lbrakk>\<lbrakk>g\<^sub>0h\<^sub>1.p\<^sub>0 \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine\<rbrakk>\<rbrakk> = \<lbrakk>\<lbrakk>fg\<^sub>0h\<^sub>1.p\<^sub>0\<rbrakk>\<rbrakk>
[PROOF STEP]
define u\<^sub>f where "u\<^sub>f = g \<star> h \<star> tab\<^sub>0 h \<star> TTfgh.p\<^sub>0"
[PROOF STATE]
proof (state)
this:
u\<^sub>f = g \<star> h \<star> tab\<^sub>0 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0
goal (3 subgoals):
1. \<lbrakk>\<lbrakk>f\<^sub>0gh\<^sub>1.p\<^sub>1 \<star> TTfgh_TfTgh.chine\<rbrakk>\<rbrakk> = \<lbrakk>\<lbrakk>f\<^sub>0g\<^sub>1.p\<^sub>1 \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1\<rbrakk>\<rbrakk>
2. \<lbrakk>\<lbrakk>g\<^sub>0h\<^sub>1.p\<^sub>1 \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine\<rbrakk>\<rbrakk> = \<lbrakk>\<lbrakk>f\<^sub>0g\<^sub>1.p\<^sub>0 \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1\<rbrakk>\<rbrakk>
3. \<lbrakk>\<lbrakk>g\<^sub>0h\<^sub>1.p\<^sub>0 \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine\<rbrakk>\<rbrakk> = \<lbrakk>\<lbrakk>fg\<^sub>0h\<^sub>1.p\<^sub>0\<rbrakk>\<rbrakk>
[PROOF STEP]
define w\<^sub>f where "w\<^sub>f = Tfg.p\<^sub>1 \<star> TTfgh.p\<^sub>1"
[PROOF STATE]
proof (state)
this:
w\<^sub>f = f\<^sub>0g\<^sub>1.p\<^sub>1 \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1
goal (3 subgoals):
1. \<lbrakk>\<lbrakk>f\<^sub>0gh\<^sub>1.p\<^sub>1 \<star> TTfgh_TfTgh.chine\<rbrakk>\<rbrakk> = \<lbrakk>\<lbrakk>f\<^sub>0g\<^sub>1.p\<^sub>1 \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1\<rbrakk>\<rbrakk>
2. \<lbrakk>\<lbrakk>g\<^sub>0h\<^sub>1.p\<^sub>1 \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine\<rbrakk>\<rbrakk> = \<lbrakk>\<lbrakk>f\<^sub>0g\<^sub>1.p\<^sub>0 \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1\<rbrakk>\<rbrakk>
3. \<lbrakk>\<lbrakk>g\<^sub>0h\<^sub>1.p\<^sub>0 \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine\<rbrakk>\<rbrakk> = \<lbrakk>\<lbrakk>fg\<^sub>0h\<^sub>1.p\<^sub>0\<rbrakk>\<rbrakk>
[PROOF STEP]
define w\<^sub>f' where "w\<^sub>f' = TfTgh.p\<^sub>1 \<star> TTfgh_TfTgh.chine"
[PROOF STATE]
proof (state)
this:
w\<^sub>f' = f\<^sub>0gh\<^sub>1.p\<^sub>1 \<star> TTfgh_TfTgh.chine
goal (3 subgoals):
1. \<lbrakk>\<lbrakk>f\<^sub>0gh\<^sub>1.p\<^sub>1 \<star> TTfgh_TfTgh.chine\<rbrakk>\<rbrakk> = \<lbrakk>\<lbrakk>f\<^sub>0g\<^sub>1.p\<^sub>1 \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1\<rbrakk>\<rbrakk>
2. \<lbrakk>\<lbrakk>g\<^sub>0h\<^sub>1.p\<^sub>1 \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine\<rbrakk>\<rbrakk> = \<lbrakk>\<lbrakk>f\<^sub>0g\<^sub>1.p\<^sub>0 \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1\<rbrakk>\<rbrakk>
3. \<lbrakk>\<lbrakk>g\<^sub>0h\<^sub>1.p\<^sub>0 \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine\<rbrakk>\<rbrakk> = \<lbrakk>\<lbrakk>fg\<^sub>0h\<^sub>1.p\<^sub>0\<rbrakk>\<rbrakk>
[PROOF STEP]
define \<theta>\<^sub>f
where "\<theta>\<^sub>f = (g \<star> \<a>[h, tab\<^sub>0 h, TTfgh.p\<^sub>0]) \<cdot> (g \<star> h.tab \<star> TTfgh.p\<^sub>0) \<cdot> (g \<star> fg\<^sub>0h\<^sub>1.\<phi>) \<cdot>
\<a>[g, tab\<^sub>0 g \<star> Tfg.p\<^sub>0, TTfgh.p\<^sub>1] \<cdot> (\<a>[g, tab\<^sub>0 g, Tfg.p\<^sub>0] \<star> TTfgh.p\<^sub>1) \<cdot>
((g.tab \<star> Tfg.p\<^sub>0) \<star> TTfgh.p\<^sub>1) \<cdot> (f\<^sub>0g\<^sub>1.\<phi> \<star> TTfgh.p\<^sub>1) \<cdot>
\<a>\<^sup>-\<^sup>1[tab\<^sub>0 f, Tfg.p\<^sub>1, TTfgh.p\<^sub>1]"
[PROOF STATE]
proof (state)
this:
\<theta>\<^sub>f = (g \<star> \<a>[h, tab\<^sub>0 h, fg\<^sub>0h\<^sub>1.p\<^sub>0]) \<cdot> (g \<star> h.tab \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0) \<cdot> (g \<star> fg\<^sub>0h\<^sub>1.\<phi>) \<cdot> \<a>[g, tab\<^sub>0 g \<star> f\<^sub>0g\<^sub>1.p\<^sub>0, fg\<^sub>0h\<^sub>1.p\<^sub>1] \<cdot> (\<a>[g, tab\<^sub>0 g, f\<^sub>0g\<^sub>1.p\<^sub>0] \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> ((g.tab \<star> f\<^sub>0g\<^sub>1.p\<^sub>0) \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> (f\<^sub>0g\<^sub>1.\<phi> \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>0 f, f\<^sub>0g\<^sub>1.p\<^sub>1, fg\<^sub>0h\<^sub>1.p\<^sub>1]
goal (3 subgoals):
1. \<lbrakk>\<lbrakk>f\<^sub>0gh\<^sub>1.p\<^sub>1 \<star> TTfgh_TfTgh.chine\<rbrakk>\<rbrakk> = \<lbrakk>\<lbrakk>f\<^sub>0g\<^sub>1.p\<^sub>1 \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1\<rbrakk>\<rbrakk>
2. \<lbrakk>\<lbrakk>g\<^sub>0h\<^sub>1.p\<^sub>1 \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine\<rbrakk>\<rbrakk> = \<lbrakk>\<lbrakk>f\<^sub>0g\<^sub>1.p\<^sub>0 \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1\<rbrakk>\<rbrakk>
3. \<lbrakk>\<lbrakk>g\<^sub>0h\<^sub>1.p\<^sub>0 \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine\<rbrakk>\<rbrakk> = \<lbrakk>\<lbrakk>fg\<^sub>0h\<^sub>1.p\<^sub>0\<rbrakk>\<rbrakk>
[PROOF STEP]
define \<theta>\<^sub>f'
where "\<theta>\<^sub>f' = (g \<star> h \<star> TTfgh_TfTgh.the_\<theta>) \<cdot>
can (\<^bold>\<langle>g\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> ((\<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>Tgh.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TfTgh.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>)
(((\<^bold>\<langle>g\<^bold>\<rangle> \<^bold>\<star> (\<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>Tgh.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TfTgh.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) \<cdot>
(((g \<star> h.tab \<star> Tgh.p\<^sub>0) \<star> TfTgh.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot>
(((g \<star> g\<^sub>0h\<^sub>1.\<phi>) \<star> TfTgh.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot>
((\<a>[g, tab\<^sub>0 g, Tgh.p\<^sub>1] \<star> TfTgh.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot>
(((g.tab \<star> Tgh.p\<^sub>1) \<star> TfTgh.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot>
(f\<^sub>0gh\<^sub>1.\<phi> \<star> TTfgh_TfTgh.chine) \<cdot>
\<a>\<^sup>-\<^sup>1[tab\<^sub>0 f, TfTgh.p\<^sub>1, TTfgh_TfTgh.chine]"
[PROOF STATE]
proof (state)
this:
\<theta>\<^sub>f' = (g \<star> h \<star> TTfgh_TfTgh.the_\<theta>) \<cdot> can (\<^bold>\<langle>g\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> ((\<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) (((\<^bold>\<langle>g\<^bold>\<rangle> \<^bold>\<star> (\<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) \<cdot> (((g \<star> h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (((g \<star> g\<^sub>0h\<^sub>1.\<phi>) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((\<a>[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (((g.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>1) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (f\<^sub>0gh\<^sub>1.\<phi> \<star> TTfgh_TfTgh.chine) \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>0 f, f\<^sub>0gh\<^sub>1.p\<^sub>1, TTfgh_TfTgh.chine]
goal (3 subgoals):
1. \<lbrakk>\<lbrakk>f\<^sub>0gh\<^sub>1.p\<^sub>1 \<star> TTfgh_TfTgh.chine\<rbrakk>\<rbrakk> = \<lbrakk>\<lbrakk>f\<^sub>0g\<^sub>1.p\<^sub>1 \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1\<rbrakk>\<rbrakk>
2. \<lbrakk>\<lbrakk>g\<^sub>0h\<^sub>1.p\<^sub>1 \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine\<rbrakk>\<rbrakk> = \<lbrakk>\<lbrakk>f\<^sub>0g\<^sub>1.p\<^sub>0 \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1\<rbrakk>\<rbrakk>
3. \<lbrakk>\<lbrakk>g\<^sub>0h\<^sub>1.p\<^sub>0 \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine\<rbrakk>\<rbrakk> = \<lbrakk>\<lbrakk>fg\<^sub>0h\<^sub>1.p\<^sub>0\<rbrakk>\<rbrakk>
[PROOF STEP]
define \<beta>\<^sub>f
where "\<beta>\<^sub>f = \<a>[tab\<^sub>1 f, TfTgh.p\<^sub>1, TTfgh_TfTgh.chine] \<cdot>
TTfgh_TfTgh.the_\<nu> \<cdot>
\<a>\<^sup>-\<^sup>1[tab\<^sub>1 f, Tfg.p\<^sub>1, TTfgh.p\<^sub>1]"
[PROOF STATE]
proof (state)
this:
\<beta>\<^sub>f = \<a>[tab\<^sub>1 f, f\<^sub>0gh\<^sub>1.p\<^sub>1, TTfgh_TfTgh.chine] \<cdot> TTfgh_TfTgh.the_\<nu> \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>1 f, f\<^sub>0g\<^sub>1.p\<^sub>1, fg\<^sub>0h\<^sub>1.p\<^sub>1]
goal (3 subgoals):
1. \<lbrakk>\<lbrakk>f\<^sub>0gh\<^sub>1.p\<^sub>1 \<star> TTfgh_TfTgh.chine\<rbrakk>\<rbrakk> = \<lbrakk>\<lbrakk>f\<^sub>0g\<^sub>1.p\<^sub>1 \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1\<rbrakk>\<rbrakk>
2. \<lbrakk>\<lbrakk>g\<^sub>0h\<^sub>1.p\<^sub>1 \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine\<rbrakk>\<rbrakk> = \<lbrakk>\<lbrakk>f\<^sub>0g\<^sub>1.p\<^sub>0 \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1\<rbrakk>\<rbrakk>
3. \<lbrakk>\<lbrakk>g\<^sub>0h\<^sub>1.p\<^sub>0 \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine\<rbrakk>\<rbrakk> = \<lbrakk>\<lbrakk>fg\<^sub>0h\<^sub>1.p\<^sub>0\<rbrakk>\<rbrakk>
[PROOF STEP]
have w\<^sub>f: "ide w\<^sub>f"
[PROOF STATE]
proof (prove)
goal (1 subgoal):
1. ide w\<^sub>f
[PROOF STEP]
using w\<^sub>f_def fg\<^sub>0h\<^sub>1.p\<^sub>1_simps
[PROOF STATE]
proof (prove)
using this:
w\<^sub>f = f\<^sub>0g\<^sub>1.p\<^sub>1 \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1
trg fg\<^sub>0h\<^sub>1.p\<^sub>1 = src (tab\<^sub>0 g \<star> f\<^sub>0g\<^sub>1.p\<^sub>0)
goal (1 subgoal):
1. ide w\<^sub>f
[PROOF STEP]
by auto
[PROOF STATE]
proof (state)
this:
ide w\<^sub>f
goal (3 subgoals):
1. \<lbrakk>\<lbrakk>f\<^sub>0gh\<^sub>1.p\<^sub>1 \<star> TTfgh_TfTgh.chine\<rbrakk>\<rbrakk> = \<lbrakk>\<lbrakk>f\<^sub>0g\<^sub>1.p\<^sub>1 \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1\<rbrakk>\<rbrakk>
2. \<lbrakk>\<lbrakk>g\<^sub>0h\<^sub>1.p\<^sub>1 \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine\<rbrakk>\<rbrakk> = \<lbrakk>\<lbrakk>f\<^sub>0g\<^sub>1.p\<^sub>0 \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1\<rbrakk>\<rbrakk>
3. \<lbrakk>\<lbrakk>g\<^sub>0h\<^sub>1.p\<^sub>0 \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine\<rbrakk>\<rbrakk> = \<lbrakk>\<lbrakk>fg\<^sub>0h\<^sub>1.p\<^sub>0\<rbrakk>\<rbrakk>
[PROOF STEP]
have w\<^sub>f_is_map: "is_left_adjoint w\<^sub>f"
[PROOF STATE]
proof (prove)
goal (1 subgoal):
1. is_left_adjoint w\<^sub>f
[PROOF STEP]
using w\<^sub>f_def fg\<^sub>0h\<^sub>1.p\<^sub>1_simps
[PROOF STATE]
proof (prove)
using this:
w\<^sub>f = f\<^sub>0g\<^sub>1.p\<^sub>1 \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1
trg fg\<^sub>0h\<^sub>1.p\<^sub>1 = src (tab\<^sub>0 g \<star> f\<^sub>0g\<^sub>1.p\<^sub>0)
goal (1 subgoal):
1. is_left_adjoint w\<^sub>f
[PROOF STEP]
by (simp add: left_adjoints_compose)
[PROOF STATE]
proof (state)
this:
is_left_adjoint w\<^sub>f
goal (3 subgoals):
1. \<lbrakk>\<lbrakk>f\<^sub>0gh\<^sub>1.p\<^sub>1 \<star> TTfgh_TfTgh.chine\<rbrakk>\<rbrakk> = \<lbrakk>\<lbrakk>f\<^sub>0g\<^sub>1.p\<^sub>1 \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1\<rbrakk>\<rbrakk>
2. \<lbrakk>\<lbrakk>g\<^sub>0h\<^sub>1.p\<^sub>1 \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine\<rbrakk>\<rbrakk> = \<lbrakk>\<lbrakk>f\<^sub>0g\<^sub>1.p\<^sub>0 \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1\<rbrakk>\<rbrakk>
3. \<lbrakk>\<lbrakk>g\<^sub>0h\<^sub>1.p\<^sub>0 \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine\<rbrakk>\<rbrakk> = \<lbrakk>\<lbrakk>fg\<^sub>0h\<^sub>1.p\<^sub>0\<rbrakk>\<rbrakk>
[PROOF STEP]
have w\<^sub>f': "ide w\<^sub>f'"
[PROOF STATE]
proof (prove)
goal (1 subgoal):
1. ide w\<^sub>f'
[PROOF STEP]
unfolding w\<^sub>f'_def
[PROOF STATE]
proof (prove)
goal (1 subgoal):
1. ide (f\<^sub>0gh\<^sub>1.p\<^sub>1 \<star> TTfgh_TfTgh.chine)
[PROOF STEP]
by simp
[PROOF STATE]
proof (state)
this:
ide w\<^sub>f'
goal (3 subgoals):
1. \<lbrakk>\<lbrakk>f\<^sub>0gh\<^sub>1.p\<^sub>1 \<star> TTfgh_TfTgh.chine\<rbrakk>\<rbrakk> = \<lbrakk>\<lbrakk>f\<^sub>0g\<^sub>1.p\<^sub>1 \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1\<rbrakk>\<rbrakk>
2. \<lbrakk>\<lbrakk>g\<^sub>0h\<^sub>1.p\<^sub>1 \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine\<rbrakk>\<rbrakk> = \<lbrakk>\<lbrakk>f\<^sub>0g\<^sub>1.p\<^sub>0 \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1\<rbrakk>\<rbrakk>
3. \<lbrakk>\<lbrakk>g\<^sub>0h\<^sub>1.p\<^sub>0 \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine\<rbrakk>\<rbrakk> = \<lbrakk>\<lbrakk>fg\<^sub>0h\<^sub>1.p\<^sub>0\<rbrakk>\<rbrakk>
[PROOF STEP]
have w\<^sub>f'_is_map: "is_left_adjoint w\<^sub>f'"
[PROOF STATE]
proof (prove)
goal (1 subgoal):
1. is_left_adjoint w\<^sub>f'
[PROOF STEP]
unfolding w\<^sub>f'_def
[PROOF STATE]
proof (prove)
goal (1 subgoal):
1. is_left_adjoint (f\<^sub>0gh\<^sub>1.p\<^sub>1 \<star> TTfgh_TfTgh.chine)
[PROOF STEP]
using 3 TTfgh_TfTgh.is_map f\<^sub>0gh\<^sub>1.leg1_is_map
[PROOF STATE]
proof (prove)
using this:
src f\<^sub>0gh\<^sub>1.p\<^sub>1 = trg TTfgh_TfTgh.chine
is_left_adjoint TTfgh_TfTgh.chine
is_left_adjoint f\<^sub>0gh\<^sub>1.p\<^sub>1
goal (1 subgoal):
1. is_left_adjoint (f\<^sub>0gh\<^sub>1.p\<^sub>1 \<star> TTfgh_TfTgh.chine)
[PROOF STEP]
by (simp add: left_adjoints_compose)
[PROOF STATE]
proof (state)
this:
is_left_adjoint w\<^sub>f'
goal (3 subgoals):
1. \<lbrakk>\<lbrakk>f\<^sub>0gh\<^sub>1.p\<^sub>1 \<star> TTfgh_TfTgh.chine\<rbrakk>\<rbrakk> = \<lbrakk>\<lbrakk>f\<^sub>0g\<^sub>1.p\<^sub>1 \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1\<rbrakk>\<rbrakk>
2. \<lbrakk>\<lbrakk>g\<^sub>0h\<^sub>1.p\<^sub>1 \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine\<rbrakk>\<rbrakk> = \<lbrakk>\<lbrakk>f\<^sub>0g\<^sub>1.p\<^sub>0 \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1\<rbrakk>\<rbrakk>
3. \<lbrakk>\<lbrakk>g\<^sub>0h\<^sub>1.p\<^sub>0 \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine\<rbrakk>\<rbrakk> = \<lbrakk>\<lbrakk>fg\<^sub>0h\<^sub>1.p\<^sub>0\<rbrakk>\<rbrakk>
[PROOF STEP]
have \<theta>\<^sub>f: "\<guillemotleft>\<theta>\<^sub>f : tab\<^sub>0 f \<star> w\<^sub>f \<Rightarrow> u\<^sub>f\<guillemotright>"
[PROOF STATE]
proof (prove)
goal (1 subgoal):
1. \<guillemotleft>\<theta>\<^sub>f : tab\<^sub>0 f \<star> w\<^sub>f \<rightarrow> u\<^sub>f\<guillemotright>
[PROOF STEP]
proof (unfold \<theta>\<^sub>f_def w\<^sub>f_def u\<^sub>f_def, intro comp_in_homI)
[PROOF STATE]
proof (state)
goal (8 subgoals):
1. \<guillemotleft>\<a>\<^sup>-\<^sup>1[tab\<^sub>0 f, f\<^sub>0g\<^sub>1.p\<^sub>1, fg\<^sub>0h\<^sub>1.p\<^sub>1] : tab\<^sub>0 f \<star> f\<^sub>0g\<^sub>1.p\<^sub>1 \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1 \<rightarrow> ?b6\<guillemotright>
2. \<guillemotleft>f\<^sub>0g\<^sub>1.\<phi> \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1 : ?b6 \<rightarrow> ?b5\<guillemotright>
3. \<guillemotleft>(g.tab \<star> f\<^sub>0g\<^sub>1.p\<^sub>0) \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1 : ?b5 \<rightarrow> ?b4\<guillemotright>
4. \<guillemotleft>\<a>[g, tab\<^sub>0 g, f\<^sub>0g\<^sub>1.p\<^sub>0] \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1 : ?b4 \<rightarrow> ?b3\<guillemotright>
5. \<guillemotleft>\<a>[g, tab\<^sub>0 g \<star> f\<^sub>0g\<^sub>1.p\<^sub>0, fg\<^sub>0h\<^sub>1.p\<^sub>1] : ?b3 \<rightarrow> ?b2\<guillemotright>
6. \<guillemotleft>g \<star> fg\<^sub>0h\<^sub>1.\<phi> : ?b2 \<rightarrow> ?b1\<guillemotright>
7. \<guillemotleft>g \<star> h.tab \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0 : ?b1 \<rightarrow> ?b\<guillemotright>
8. \<guillemotleft>g \<star> \<a>[h, tab\<^sub>0 h, fg\<^sub>0h\<^sub>1.p\<^sub>0] : ?b \<rightarrow> g \<star> h \<star> tab\<^sub>0 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0\<guillemotright>
[PROOF STEP]
show "\<guillemotleft>\<a>\<^sup>-\<^sup>1[tab\<^sub>0 f, Tfg.p\<^sub>1, TTfgh.p\<^sub>1] :
tab\<^sub>0 f \<star> Tfg.p\<^sub>1 \<star> TTfgh.p\<^sub>1 \<Rightarrow> (tab\<^sub>0 f \<star> Tfg.p\<^sub>1) \<star> TTfgh.p\<^sub>1\<guillemotright>"
[PROOF STATE]
proof (prove)
goal (1 subgoal):
1. \<guillemotleft>\<a>\<^sup>-\<^sup>1[tab\<^sub>0 f, f\<^sub>0g\<^sub>1.p\<^sub>1, fg\<^sub>0h\<^sub>1.p\<^sub>1] : tab\<^sub>0 f \<star> f\<^sub>0g\<^sub>1.p\<^sub>1 \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1 \<rightarrow> (tab\<^sub>0 f \<star> f\<^sub>0g\<^sub>1.p\<^sub>1) \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1\<guillemotright>
[PROOF STEP]
using f\<^sub>0g\<^sub>1.leg1_simps fg\<^sub>0h\<^sub>1.p\<^sub>1_simps f\<^sub>0g\<^sub>1.cospan g\<^sub>0h\<^sub>1.cospan
[PROOF STATE]
proof (prove)
using this:
ide f\<^sub>0g\<^sub>1.p\<^sub>1
arr f\<^sub>0g\<^sub>1.p\<^sub>1
src f\<^sub>0g\<^sub>1.p\<^sub>1 = src f\<^sub>0g\<^sub>1.p\<^sub>0
trg f\<^sub>0g\<^sub>1.p\<^sub>1 = trg ((tab\<^sub>0 f)\<^sup>* \<star> tab\<^sub>1 g)
local.dom f\<^sub>0g\<^sub>1.p\<^sub>1 = f\<^sub>0g\<^sub>1.p\<^sub>1
cod f\<^sub>0g\<^sub>1.p\<^sub>1 = f\<^sub>0g\<^sub>1.p\<^sub>1
trg fg\<^sub>0h\<^sub>1.p\<^sub>1 = src (tab\<^sub>0 g \<star> f\<^sub>0g\<^sub>1.p\<^sub>0)
trg (tab\<^sub>1 g) = trg (tab\<^sub>0 f)
trg (tab\<^sub>1 h) = trg (tab\<^sub>0 g)
goal (1 subgoal):
1. \<guillemotleft>\<a>\<^sup>-\<^sup>1[tab\<^sub>0 f, f\<^sub>0g\<^sub>1.p\<^sub>1, fg\<^sub>0h\<^sub>1.p\<^sub>1] : tab\<^sub>0 f \<star> f\<^sub>0g\<^sub>1.p\<^sub>1 \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1 \<rightarrow> (tab\<^sub>0 f \<star> f\<^sub>0g\<^sub>1.p\<^sub>1) \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1\<guillemotright>
[PROOF STEP]
by auto
[PROOF STATE]
proof (state)
this:
\<guillemotleft>\<a>\<^sup>-\<^sup>1[tab\<^sub>0 f, f\<^sub>0g\<^sub>1.p\<^sub>1, fg\<^sub>0h\<^sub>1.p\<^sub>1] : tab\<^sub>0 f \<star> f\<^sub>0g\<^sub>1.p\<^sub>1 \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1 \<rightarrow> (tab\<^sub>0 f \<star> f\<^sub>0g\<^sub>1.p\<^sub>1) \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1\<guillemotright>
goal (7 subgoals):
1. \<guillemotleft>f\<^sub>0g\<^sub>1.\<phi> \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1 : (tab\<^sub>0 f \<star> f\<^sub>0g\<^sub>1.p\<^sub>1) \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1 \<rightarrow> ?b5\<guillemotright>
2. \<guillemotleft>(g.tab \<star> f\<^sub>0g\<^sub>1.p\<^sub>0) \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1 : ?b5 \<rightarrow> ?b4\<guillemotright>
3. \<guillemotleft>\<a>[g, tab\<^sub>0 g, f\<^sub>0g\<^sub>1.p\<^sub>0] \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1 : ?b4 \<rightarrow> ?b3\<guillemotright>
4. \<guillemotleft>\<a>[g, tab\<^sub>0 g \<star> f\<^sub>0g\<^sub>1.p\<^sub>0, fg\<^sub>0h\<^sub>1.p\<^sub>1] : ?b3 \<rightarrow> ?b2\<guillemotright>
5. \<guillemotleft>g \<star> fg\<^sub>0h\<^sub>1.\<phi> : ?b2 \<rightarrow> ?b1\<guillemotright>
6. \<guillemotleft>g \<star> h.tab \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0 : ?b1 \<rightarrow> ?b\<guillemotright>
7. \<guillemotleft>g \<star> \<a>[h, tab\<^sub>0 h, fg\<^sub>0h\<^sub>1.p\<^sub>0] : ?b \<rightarrow> g \<star> h \<star> tab\<^sub>0 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0\<guillemotright>
[PROOF STEP]
show "\<guillemotleft>f\<^sub>0g\<^sub>1.\<phi> \<star> TTfgh.p\<^sub>1 :
(tab\<^sub>0 f \<star> Tfg.p\<^sub>1) \<star> TTfgh.p\<^sub>1 \<Rightarrow> (tab\<^sub>1 g \<star> Tfg.p\<^sub>0) \<star> TTfgh.p\<^sub>1\<guillemotright>"
[PROOF STATE]
proof (prove)
goal (1 subgoal):
1. \<guillemotleft>f\<^sub>0g\<^sub>1.\<phi> \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1 : (tab\<^sub>0 f \<star> f\<^sub>0g\<^sub>1.p\<^sub>1) \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1 \<rightarrow> (tab\<^sub>1 g \<star> f\<^sub>0g\<^sub>1.p\<^sub>0) \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1\<guillemotright>
[PROOF STEP]
using f\<^sub>0g\<^sub>1.\<phi>_in_hom(2) Tfg.\<rho>\<sigma>.T0.antipar(1)
[PROOF STATE]
proof (prove)
using this:
\<guillemotleft>f\<^sub>0g\<^sub>1.\<phi> : tab\<^sub>0 f \<star> f\<^sub>0g\<^sub>1.p\<^sub>1 \<rightarrow> tab\<^sub>1 g \<star> f\<^sub>0g\<^sub>1.p\<^sub>0\<guillemotright>
trg (tab\<^sub>0 g \<star> f\<^sub>0g\<^sub>1.p\<^sub>0)\<^sup>* = src (tab\<^sub>0 g \<star> f\<^sub>0g\<^sub>1.p\<^sub>0)
goal (1 subgoal):
1. \<guillemotleft>f\<^sub>0g\<^sub>1.\<phi> \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1 : (tab\<^sub>0 f \<star> f\<^sub>0g\<^sub>1.p\<^sub>1) \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1 \<rightarrow> (tab\<^sub>1 g \<star> f\<^sub>0g\<^sub>1.p\<^sub>0) \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1\<guillemotright>
[PROOF STEP]
by (intro hcomp_in_vhom, auto)
[PROOF STATE]
proof (state)
this:
\<guillemotleft>f\<^sub>0g\<^sub>1.\<phi> \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1 : (tab\<^sub>0 f \<star> f\<^sub>0g\<^sub>1.p\<^sub>1) \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1 \<rightarrow> (tab\<^sub>1 g \<star> f\<^sub>0g\<^sub>1.p\<^sub>0) \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1\<guillemotright>
goal (6 subgoals):
1. \<guillemotleft>(g.tab \<star> f\<^sub>0g\<^sub>1.p\<^sub>0) \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1 : (tab\<^sub>1 g \<star> f\<^sub>0g\<^sub>1.p\<^sub>0) \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1 \<rightarrow> ?b4\<guillemotright>
2. \<guillemotleft>\<a>[g, tab\<^sub>0 g, f\<^sub>0g\<^sub>1.p\<^sub>0] \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1 : ?b4 \<rightarrow> ?b3\<guillemotright>
3. \<guillemotleft>\<a>[g, tab\<^sub>0 g \<star> f\<^sub>0g\<^sub>1.p\<^sub>0, fg\<^sub>0h\<^sub>1.p\<^sub>1] : ?b3 \<rightarrow> ?b2\<guillemotright>
4. \<guillemotleft>g \<star> fg\<^sub>0h\<^sub>1.\<phi> : ?b2 \<rightarrow> ?b1\<guillemotright>
5. \<guillemotleft>g \<star> h.tab \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0 : ?b1 \<rightarrow> ?b\<guillemotright>
6. \<guillemotleft>g \<star> \<a>[h, tab\<^sub>0 h, fg\<^sub>0h\<^sub>1.p\<^sub>0] : ?b \<rightarrow> g \<star> h \<star> tab\<^sub>0 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0\<guillemotright>
[PROOF STEP]
show "\<guillemotleft>(g.tab \<star> Tfg.p\<^sub>0) \<star> TTfgh.p\<^sub>1 :
(tab\<^sub>1 g \<star> Tfg.p\<^sub>0) \<star> TTfgh.p\<^sub>1 \<Rightarrow> ((g \<star> tab\<^sub>0 g) \<star> Tfg.p\<^sub>0) \<star> TTfgh.p\<^sub>1\<guillemotright>"
[PROOF STATE]
proof (prove)
goal (1 subgoal):
1. \<guillemotleft>(g.tab \<star> f\<^sub>0g\<^sub>1.p\<^sub>0) \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1 : (tab\<^sub>1 g \<star> f\<^sub>0g\<^sub>1.p\<^sub>0) \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1 \<rightarrow> ((g \<star> tab\<^sub>0 g) \<star> f\<^sub>0g\<^sub>1.p\<^sub>0) \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1\<guillemotright>
[PROOF STEP]
using Tfg.\<rho>\<sigma>.T0.antipar(1)
[PROOF STATE]
proof (prove)
using this:
trg (tab\<^sub>0 g \<star> f\<^sub>0g\<^sub>1.p\<^sub>0)\<^sup>* = src (tab\<^sub>0 g \<star> f\<^sub>0g\<^sub>1.p\<^sub>0)
goal (1 subgoal):
1. \<guillemotleft>(g.tab \<star> f\<^sub>0g\<^sub>1.p\<^sub>0) \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1 : (tab\<^sub>1 g \<star> f\<^sub>0g\<^sub>1.p\<^sub>0) \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1 \<rightarrow> ((g \<star> tab\<^sub>0 g) \<star> f\<^sub>0g\<^sub>1.p\<^sub>0) \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1\<guillemotright>
[PROOF STEP]
by (intro hcomp_in_vhom, auto)
[PROOF STATE]
proof (state)
this:
\<guillemotleft>(g.tab \<star> f\<^sub>0g\<^sub>1.p\<^sub>0) \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1 : (tab\<^sub>1 g \<star> f\<^sub>0g\<^sub>1.p\<^sub>0) \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1 \<rightarrow> ((g \<star> tab\<^sub>0 g) \<star> f\<^sub>0g\<^sub>1.p\<^sub>0) \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1\<guillemotright>
goal (5 subgoals):
1. \<guillemotleft>\<a>[g, tab\<^sub>0 g, f\<^sub>0g\<^sub>1.p\<^sub>0] \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1 : ((g \<star> tab\<^sub>0 g) \<star> f\<^sub>0g\<^sub>1.p\<^sub>0) \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1 \<rightarrow> ?b3\<guillemotright>
2. \<guillemotleft>\<a>[g, tab\<^sub>0 g \<star> f\<^sub>0g\<^sub>1.p\<^sub>0, fg\<^sub>0h\<^sub>1.p\<^sub>1] : ?b3 \<rightarrow> ?b2\<guillemotright>
3. \<guillemotleft>g \<star> fg\<^sub>0h\<^sub>1.\<phi> : ?b2 \<rightarrow> ?b1\<guillemotright>
4. \<guillemotleft>g \<star> h.tab \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0 : ?b1 \<rightarrow> ?b\<guillemotright>
5. \<guillemotleft>g \<star> \<a>[h, tab\<^sub>0 h, fg\<^sub>0h\<^sub>1.p\<^sub>0] : ?b \<rightarrow> g \<star> h \<star> tab\<^sub>0 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0\<guillemotright>
[PROOF STEP]
show "\<guillemotleft>\<a>[g, tab\<^sub>0 g, Tfg.p\<^sub>0] \<star> TTfgh.p\<^sub>1 :
((g \<star> tab\<^sub>0 g) \<star> Tfg.p\<^sub>0) \<star> TTfgh.p\<^sub>1 \<Rightarrow> (g \<star> tab\<^sub>0 g \<star> Tfg.p\<^sub>0) \<star> TTfgh.p\<^sub>1\<guillemotright>"
[PROOF STATE]
proof (prove)
goal (1 subgoal):
1. \<guillemotleft>\<a>[g, tab\<^sub>0 g, f\<^sub>0g\<^sub>1.p\<^sub>0] \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1 : ((g \<star> tab\<^sub>0 g) \<star> f\<^sub>0g\<^sub>1.p\<^sub>0) \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1 \<rightarrow> (g \<star> tab\<^sub>0 g \<star> f\<^sub>0g\<^sub>1.p\<^sub>0) \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1\<guillemotright>
[PROOF STEP]
using fg\<^sub>0h\<^sub>1.p\<^sub>1_simps
[PROOF STATE]
proof (prove)
using this:
trg fg\<^sub>0h\<^sub>1.p\<^sub>1 = src (tab\<^sub>0 g \<star> f\<^sub>0g\<^sub>1.p\<^sub>0)
goal (1 subgoal):
1. \<guillemotleft>\<a>[g, tab\<^sub>0 g, f\<^sub>0g\<^sub>1.p\<^sub>0] \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1 : ((g \<star> tab\<^sub>0 g) \<star> f\<^sub>0g\<^sub>1.p\<^sub>0) \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1 \<rightarrow> (g \<star> tab\<^sub>0 g \<star> f\<^sub>0g\<^sub>1.p\<^sub>0) \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1\<guillemotright>
[PROOF STEP]
by (intro hcomp_in_vhom, auto)
[PROOF STATE]
proof (state)
this:
\<guillemotleft>\<a>[g, tab\<^sub>0 g, f\<^sub>0g\<^sub>1.p\<^sub>0] \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1 : ((g \<star> tab\<^sub>0 g) \<star> f\<^sub>0g\<^sub>1.p\<^sub>0) \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1 \<rightarrow> (g \<star> tab\<^sub>0 g \<star> f\<^sub>0g\<^sub>1.p\<^sub>0) \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1\<guillemotright>
goal (4 subgoals):
1. \<guillemotleft>\<a>[g, tab\<^sub>0 g \<star> f\<^sub>0g\<^sub>1.p\<^sub>0, fg\<^sub>0h\<^sub>1.p\<^sub>1] : (g \<star> tab\<^sub>0 g \<star> f\<^sub>0g\<^sub>1.p\<^sub>0) \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1 \<rightarrow> ?b2\<guillemotright>
2. \<guillemotleft>g \<star> fg\<^sub>0h\<^sub>1.\<phi> : ?b2 \<rightarrow> ?b1\<guillemotright>
3. \<guillemotleft>g \<star> h.tab \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0 : ?b1 \<rightarrow> ?b\<guillemotright>
4. \<guillemotleft>g \<star> \<a>[h, tab\<^sub>0 h, fg\<^sub>0h\<^sub>1.p\<^sub>0] : ?b \<rightarrow> g \<star> h \<star> tab\<^sub>0 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0\<guillemotright>
[PROOF STEP]
show "\<guillemotleft>\<a>[g, tab\<^sub>0 g \<star> Tfg.p\<^sub>0, TTfgh.p\<^sub>1] :
(g \<star> tab\<^sub>0 g \<star> Tfg.p\<^sub>0) \<star> TTfgh.p\<^sub>1 \<Rightarrow> g \<star> (tab\<^sub>0 g \<star> Tfg.p\<^sub>0) \<star> TTfgh.p\<^sub>1\<guillemotright>"
[PROOF STATE]
proof (prove)
goal (1 subgoal):
1. \<guillemotleft>\<a>[g, tab\<^sub>0 g \<star> f\<^sub>0g\<^sub>1.p\<^sub>0, fg\<^sub>0h\<^sub>1.p\<^sub>1] : (g \<star> tab\<^sub>0 g \<star> f\<^sub>0g\<^sub>1.p\<^sub>0) \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1 \<rightarrow> g \<star> (tab\<^sub>0 g \<star> f\<^sub>0g\<^sub>1.p\<^sub>0) \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1\<guillemotright>
[PROOF STEP]
using fg\<^sub>0h\<^sub>1.p\<^sub>1_simps
[PROOF STATE]
proof (prove)
using this:
trg fg\<^sub>0h\<^sub>1.p\<^sub>1 = src (tab\<^sub>0 g \<star> f\<^sub>0g\<^sub>1.p\<^sub>0)
goal (1 subgoal):
1. \<guillemotleft>\<a>[g, tab\<^sub>0 g \<star> f\<^sub>0g\<^sub>1.p\<^sub>0, fg\<^sub>0h\<^sub>1.p\<^sub>1] : (g \<star> tab\<^sub>0 g \<star> f\<^sub>0g\<^sub>1.p\<^sub>0) \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1 \<rightarrow> g \<star> (tab\<^sub>0 g \<star> f\<^sub>0g\<^sub>1.p\<^sub>0) \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1\<guillemotright>
[PROOF STEP]
by auto
[PROOF STATE]
proof (state)
this:
\<guillemotleft>\<a>[g, tab\<^sub>0 g \<star> f\<^sub>0g\<^sub>1.p\<^sub>0, fg\<^sub>0h\<^sub>1.p\<^sub>1] : (g \<star> tab\<^sub>0 g \<star> f\<^sub>0g\<^sub>1.p\<^sub>0) \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1 \<rightarrow> g \<star> (tab\<^sub>0 g \<star> f\<^sub>0g\<^sub>1.p\<^sub>0) \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1\<guillemotright>
goal (3 subgoals):
1. \<guillemotleft>g \<star> fg\<^sub>0h\<^sub>1.\<phi> : g \<star> (tab\<^sub>0 g \<star> f\<^sub>0g\<^sub>1.p\<^sub>0) \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1 \<rightarrow> ?b1\<guillemotright>
2. \<guillemotleft>g \<star> h.tab \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0 : ?b1 \<rightarrow> ?b\<guillemotright>
3. \<guillemotleft>g \<star> \<a>[h, tab\<^sub>0 h, fg\<^sub>0h\<^sub>1.p\<^sub>0] : ?b \<rightarrow> g \<star> h \<star> tab\<^sub>0 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0\<guillemotright>
[PROOF STEP]
show "\<guillemotleft>g \<star> fg\<^sub>0h\<^sub>1.\<phi> : g \<star> (tab\<^sub>0 g \<star> Tfg.p\<^sub>0) \<star> TTfgh.p\<^sub>1 \<Rightarrow> g \<star> tab\<^sub>1 h \<star> TTfgh.p\<^sub>0\<guillemotright>"
[PROOF STATE]
proof (prove)
goal (1 subgoal):
1. \<guillemotleft>g \<star> fg\<^sub>0h\<^sub>1.\<phi> : g \<star> (tab\<^sub>0 g \<star> f\<^sub>0g\<^sub>1.p\<^sub>0) \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1 \<rightarrow> g \<star> tab\<^sub>1 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0\<guillemotright>
[PROOF STEP]
using fg\<^sub>0h\<^sub>1.\<phi>_in_hom fg\<^sub>0h\<^sub>1.p\<^sub>1_simps
[PROOF STATE]
proof (prove)
using this:
\<guillemotleft>fg\<^sub>0h\<^sub>1.\<phi> : src fg\<^sub>0h\<^sub>1.p\<^sub>0 \<rightarrow> trg (tab\<^sub>1 h)\<guillemotright>
\<guillemotleft>fg\<^sub>0h\<^sub>1.\<phi> : (tab\<^sub>0 g \<star> f\<^sub>0g\<^sub>1.p\<^sub>0) \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1 \<rightarrow> tab\<^sub>1 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0\<guillemotright>
trg fg\<^sub>0h\<^sub>1.p\<^sub>1 = src (tab\<^sub>0 g \<star> f\<^sub>0g\<^sub>1.p\<^sub>0)
goal (1 subgoal):
1. \<guillemotleft>g \<star> fg\<^sub>0h\<^sub>1.\<phi> : g \<star> (tab\<^sub>0 g \<star> f\<^sub>0g\<^sub>1.p\<^sub>0) \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1 \<rightarrow> g \<star> tab\<^sub>1 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0\<guillemotright>
[PROOF STEP]
by (intro hcomp_in_vhom, auto)
[PROOF STATE]
proof (state)
this:
\<guillemotleft>g \<star> fg\<^sub>0h\<^sub>1.\<phi> : g \<star> (tab\<^sub>0 g \<star> f\<^sub>0g\<^sub>1.p\<^sub>0) \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1 \<rightarrow> g \<star> tab\<^sub>1 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0\<guillemotright>
goal (2 subgoals):
1. \<guillemotleft>g \<star> h.tab \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0 : g \<star> tab\<^sub>1 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0 \<rightarrow> ?b\<guillemotright>
2. \<guillemotleft>g \<star> \<a>[h, tab\<^sub>0 h, fg\<^sub>0h\<^sub>1.p\<^sub>0] : ?b \<rightarrow> g \<star> h \<star> tab\<^sub>0 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0\<guillemotright>
[PROOF STEP]
show "\<guillemotleft>g \<star> h.tab \<star> TTfgh.p\<^sub>0 : g \<star> tab\<^sub>1 h \<star> TTfgh.p\<^sub>0 \<Rightarrow> g \<star> (h \<star> tab\<^sub>0 h) \<star> TTfgh.p\<^sub>0\<guillemotright>"
[PROOF STATE]
proof (prove)
goal (1 subgoal):
1. \<guillemotleft>g \<star> h.tab \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0 : g \<star> tab\<^sub>1 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0 \<rightarrow> g \<star> (h \<star> tab\<^sub>0 h) \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0\<guillemotright>
[PROOF STEP]
using gh fg\<^sub>0h\<^sub>1.\<phi>_in_hom fg\<^sub>0h\<^sub>1.p\<^sub>1_simps
[PROOF STATE]
proof (prove)
using this:
src g = trg h
\<guillemotleft>fg\<^sub>0h\<^sub>1.\<phi> : src fg\<^sub>0h\<^sub>1.p\<^sub>0 \<rightarrow> trg (tab\<^sub>1 h)\<guillemotright>
\<guillemotleft>fg\<^sub>0h\<^sub>1.\<phi> : (tab\<^sub>0 g \<star> f\<^sub>0g\<^sub>1.p\<^sub>0) \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1 \<rightarrow> tab\<^sub>1 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0\<guillemotright>
trg fg\<^sub>0h\<^sub>1.p\<^sub>1 = src (tab\<^sub>0 g \<star> f\<^sub>0g\<^sub>1.p\<^sub>0)
goal (1 subgoal):
1. \<guillemotleft>g \<star> h.tab \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0 : g \<star> tab\<^sub>1 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0 \<rightarrow> g \<star> (h \<star> tab\<^sub>0 h) \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0\<guillemotright>
[PROOF STEP]
by (intro hcomp_in_vhom, auto)
[PROOF STATE]
proof (state)
this:
\<guillemotleft>g \<star> h.tab \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0 : g \<star> tab\<^sub>1 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0 \<rightarrow> g \<star> (h \<star> tab\<^sub>0 h) \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0\<guillemotright>
goal (1 subgoal):
1. \<guillemotleft>g \<star> \<a>[h, tab\<^sub>0 h, fg\<^sub>0h\<^sub>1.p\<^sub>0] : g \<star> (h \<star> tab\<^sub>0 h) \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0 \<rightarrow> g \<star> h \<star> tab\<^sub>0 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0\<guillemotright>
[PROOF STEP]
show "\<guillemotleft>g \<star> \<a>[h, tab\<^sub>0 h, TTfgh.p\<^sub>0] :
g \<star> (h \<star> tab\<^sub>0 h) \<star> TTfgh.p\<^sub>0 \<Rightarrow> g \<star> h \<star> tab\<^sub>0 h \<star> TTfgh.p\<^sub>0\<guillemotright>"
[PROOF STATE]
proof (prove)
goal (1 subgoal):
1. \<guillemotleft>g \<star> \<a>[h, tab\<^sub>0 h, fg\<^sub>0h\<^sub>1.p\<^sub>0] : g \<star> (h \<star> tab\<^sub>0 h) \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0 \<rightarrow> g \<star> h \<star> tab\<^sub>0 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0\<guillemotright>
[PROOF STEP]
using gh fg\<^sub>0h\<^sub>1.\<phi>_in_hom fg\<^sub>0h\<^sub>1.p\<^sub>1_simps
[PROOF STATE]
proof (prove)
using this:
src g = trg h
\<guillemotleft>fg\<^sub>0h\<^sub>1.\<phi> : src fg\<^sub>0h\<^sub>1.p\<^sub>0 \<rightarrow> trg (tab\<^sub>1 h)\<guillemotright>
\<guillemotleft>fg\<^sub>0h\<^sub>1.\<phi> : (tab\<^sub>0 g \<star> f\<^sub>0g\<^sub>1.p\<^sub>0) \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1 \<rightarrow> tab\<^sub>1 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0\<guillemotright>
trg fg\<^sub>0h\<^sub>1.p\<^sub>1 = src (tab\<^sub>0 g \<star> f\<^sub>0g\<^sub>1.p\<^sub>0)
goal (1 subgoal):
1. \<guillemotleft>g \<star> \<a>[h, tab\<^sub>0 h, fg\<^sub>0h\<^sub>1.p\<^sub>0] : g \<star> (h \<star> tab\<^sub>0 h) \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0 \<rightarrow> g \<star> h \<star> tab\<^sub>0 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0\<guillemotright>
[PROOF STEP]
by (intro hcomp_in_vhom, auto)
[PROOF STATE]
proof (state)
this:
\<guillemotleft>g \<star> \<a>[h, tab\<^sub>0 h, fg\<^sub>0h\<^sub>1.p\<^sub>0] : g \<star> (h \<star> tab\<^sub>0 h) \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0 \<rightarrow> g \<star> h \<star> tab\<^sub>0 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0\<guillemotright>
goal:
No subgoals!
[PROOF STEP]
qed
[PROOF STATE]
proof (state)
this:
\<guillemotleft>\<theta>\<^sub>f : tab\<^sub>0 f \<star> w\<^sub>f \<rightarrow> u\<^sub>f\<guillemotright>
goal (3 subgoals):
1. \<lbrakk>\<lbrakk>f\<^sub>0gh\<^sub>1.p\<^sub>1 \<star> TTfgh_TfTgh.chine\<rbrakk>\<rbrakk> = \<lbrakk>\<lbrakk>f\<^sub>0g\<^sub>1.p\<^sub>1 \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1\<rbrakk>\<rbrakk>
2. \<lbrakk>\<lbrakk>g\<^sub>0h\<^sub>1.p\<^sub>1 \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine\<rbrakk>\<rbrakk> = \<lbrakk>\<lbrakk>f\<^sub>0g\<^sub>1.p\<^sub>0 \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1\<rbrakk>\<rbrakk>
3. \<lbrakk>\<lbrakk>g\<^sub>0h\<^sub>1.p\<^sub>0 \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine\<rbrakk>\<rbrakk> = \<lbrakk>\<lbrakk>fg\<^sub>0h\<^sub>1.p\<^sub>0\<rbrakk>\<rbrakk>
[PROOF STEP]
have \<theta>\<^sub>f': "\<guillemotleft>\<theta>\<^sub>f' : tab\<^sub>0 f \<star> w\<^sub>f' \<Rightarrow> u\<^sub>f\<guillemotright>"
[PROOF STATE]
proof (prove)
goal (1 subgoal):
1. \<guillemotleft>\<theta>\<^sub>f' : tab\<^sub>0 f \<star> w\<^sub>f' \<rightarrow> u\<^sub>f\<guillemotright>
[PROOF STEP]
proof (unfold \<theta>\<^sub>f'_def w\<^sub>f'_def u\<^sub>f_def, intro comp_in_homI)
[PROOF STATE]
proof (state)
goal (8 subgoals):
1. \<guillemotleft>\<a>\<^sup>-\<^sup>1[tab\<^sub>0 f, f\<^sub>0gh\<^sub>1.p\<^sub>1, TTfgh_TfTgh.chine] : tab\<^sub>0 f \<star> f\<^sub>0gh\<^sub>1.p\<^sub>1 \<star> TTfgh_TfTgh.chine \<rightarrow> ?b6\<guillemotright>
2. \<guillemotleft>f\<^sub>0gh\<^sub>1.\<phi> \<star> TTfgh_TfTgh.chine : ?b6 \<rightarrow> ?b5\<guillemotright>
3. \<guillemotleft>((g.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>1) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine : ?b5 \<rightarrow> ?b4\<guillemotright>
4. \<guillemotleft>(\<a>[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine : ?b4 \<rightarrow> ?b3\<guillemotright>
5. \<guillemotleft>((g \<star> g\<^sub>0h\<^sub>1.\<phi>) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine : ?b3 \<rightarrow> ?b2\<guillemotright>
6. \<guillemotleft>((g \<star> h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine : ?b2 \<rightarrow> ?b1\<guillemotright>
7. \<guillemotleft>can (\<^bold>\<langle>g\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> ((\<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) (((\<^bold>\<langle>g\<^bold>\<rangle> \<^bold>\<star> (\<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) : ?b1 \<rightarrow> ?b\<guillemotright>
8. \<guillemotleft>g \<star> h \<star> TTfgh_TfTgh.the_\<theta> : ?b \<rightarrow> g \<star> h \<star> tab\<^sub>0 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0\<guillemotright>
[PROOF STEP]
show "\<guillemotleft>\<a>\<^sup>-\<^sup>1[tab\<^sub>0 f, TfTgh.p\<^sub>1, TTfgh_TfTgh.chine] :
tab\<^sub>0 f \<star> TfTgh.p\<^sub>1 \<star> TTfgh_TfTgh.chine \<Rightarrow> (tab\<^sub>0 f \<star> TfTgh.p\<^sub>1) \<star> TTfgh_TfTgh.chine\<guillemotright>"
[PROOF STATE]
proof (prove)
goal (1 subgoal):
1. \<guillemotleft>\<a>\<^sup>-\<^sup>1[tab\<^sub>0 f, f\<^sub>0gh\<^sub>1.p\<^sub>1, TTfgh_TfTgh.chine] : tab\<^sub>0 f \<star> f\<^sub>0gh\<^sub>1.p\<^sub>1 \<star> TTfgh_TfTgh.chine \<rightarrow> (tab\<^sub>0 f \<star> f\<^sub>0gh\<^sub>1.p\<^sub>1) \<star> TTfgh_TfTgh.chine\<guillemotright>
[PROOF STEP]
using "1" "2" "3" "4" assoc'_in_hom(2) f.ide_u f.leg1_simps(3)
[PROOF STATE]
proof (prove)
using this:
ide f\<^sub>0gh\<^sub>1.p\<^sub>1
ide TTfgh_TfTgh.chine
src f\<^sub>0gh\<^sub>1.p\<^sub>1 = trg TTfgh_TfTgh.chine
src (tab\<^sub>1 f) = trg f\<^sub>0gh\<^sub>1.p\<^sub>1
\<lbrakk>ide ?f; ide ?g; ide ?h; src ?f = trg ?g; src ?g = trg ?h\<rbrakk> \<Longrightarrow> \<guillemotleft>\<a>\<^sup>-\<^sup>1[?f, ?g, ?h] : local.dom ?f \<star> local.dom ?g \<star> local.dom ?h \<rightarrow> (cod ?f \<star> cod ?g) \<star> cod ?h\<guillemotright>
ide (tab\<^sub>0 f)
src (tab\<^sub>1 f) = src (tab\<^sub>0 f)
goal (1 subgoal):
1. \<guillemotleft>\<a>\<^sup>-\<^sup>1[tab\<^sub>0 f, f\<^sub>0gh\<^sub>1.p\<^sub>1, TTfgh_TfTgh.chine] : tab\<^sub>0 f \<star> f\<^sub>0gh\<^sub>1.p\<^sub>1 \<star> TTfgh_TfTgh.chine \<rightarrow> (tab\<^sub>0 f \<star> f\<^sub>0gh\<^sub>1.p\<^sub>1) \<star> TTfgh_TfTgh.chine\<guillemotright>
[PROOF STEP]
by auto
[PROOF STATE]
proof (state)
this:
\<guillemotleft>\<a>\<^sup>-\<^sup>1[tab\<^sub>0 f, f\<^sub>0gh\<^sub>1.p\<^sub>1, TTfgh_TfTgh.chine] : tab\<^sub>0 f \<star> f\<^sub>0gh\<^sub>1.p\<^sub>1 \<star> TTfgh_TfTgh.chine \<rightarrow> (tab\<^sub>0 f \<star> f\<^sub>0gh\<^sub>1.p\<^sub>1) \<star> TTfgh_TfTgh.chine\<guillemotright>
goal (7 subgoals):
1. \<guillemotleft>f\<^sub>0gh\<^sub>1.\<phi> \<star> TTfgh_TfTgh.chine : (tab\<^sub>0 f \<star> f\<^sub>0gh\<^sub>1.p\<^sub>1) \<star> TTfgh_TfTgh.chine \<rightarrow> ?b5\<guillemotright>
2. \<guillemotleft>((g.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>1) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine : ?b5 \<rightarrow> ?b4\<guillemotright>
3. \<guillemotleft>(\<a>[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine : ?b4 \<rightarrow> ?b3\<guillemotright>
4. \<guillemotleft>((g \<star> g\<^sub>0h\<^sub>1.\<phi>) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine : ?b3 \<rightarrow> ?b2\<guillemotright>
5. \<guillemotleft>((g \<star> h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine : ?b2 \<rightarrow> ?b1\<guillemotright>
6. \<guillemotleft>can (\<^bold>\<langle>g\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> ((\<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) (((\<^bold>\<langle>g\<^bold>\<rangle> \<^bold>\<star> (\<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) : ?b1 \<rightarrow> ?b\<guillemotright>
7. \<guillemotleft>g \<star> h \<star> TTfgh_TfTgh.the_\<theta> : ?b \<rightarrow> g \<star> h \<star> tab\<^sub>0 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0\<guillemotright>
[PROOF STEP]
show "\<guillemotleft>f\<^sub>0gh\<^sub>1.\<phi> \<star> TTfgh_TfTgh.chine :
(tab\<^sub>0 f \<star> TfTgh.p\<^sub>1) \<star> TTfgh_TfTgh.chine \<Rightarrow>
((tab\<^sub>1 g \<star> Tgh.p\<^sub>1) \<star> TfTgh.p\<^sub>0) \<star> TTfgh_TfTgh.chine\<guillemotright>"
[PROOF STATE]
proof (prove)
goal (1 subgoal):
1. \<guillemotleft>f\<^sub>0gh\<^sub>1.\<phi> \<star> TTfgh_TfTgh.chine : (tab\<^sub>0 f \<star> f\<^sub>0gh\<^sub>1.p\<^sub>1) \<star> TTfgh_TfTgh.chine \<rightarrow> ((tab\<^sub>1 g \<star> g\<^sub>0h\<^sub>1.p\<^sub>1) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine\<guillemotright>
[PROOF STEP]
using f\<^sub>0gh\<^sub>1.\<phi>_in_hom(2)
[PROOF STATE]
proof (prove)
using this:
\<guillemotleft>f\<^sub>0gh\<^sub>1.\<phi> : tab\<^sub>0 f \<star> f\<^sub>0gh\<^sub>1.p\<^sub>1 \<rightarrow> (tab\<^sub>1 g \<star> g\<^sub>0h\<^sub>1.p\<^sub>1) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0\<guillemotright>
goal (1 subgoal):
1. \<guillemotleft>f\<^sub>0gh\<^sub>1.\<phi> \<star> TTfgh_TfTgh.chine : (tab\<^sub>0 f \<star> f\<^sub>0gh\<^sub>1.p\<^sub>1) \<star> TTfgh_TfTgh.chine \<rightarrow> ((tab\<^sub>1 g \<star> g\<^sub>0h\<^sub>1.p\<^sub>1) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine\<guillemotright>
[PROOF STEP]
by (intro hcomp_in_vhom, auto)
[PROOF STATE]
proof (state)
this:
\<guillemotleft>f\<^sub>0gh\<^sub>1.\<phi> \<star> TTfgh_TfTgh.chine : (tab\<^sub>0 f \<star> f\<^sub>0gh\<^sub>1.p\<^sub>1) \<star> TTfgh_TfTgh.chine \<rightarrow> ((tab\<^sub>1 g \<star> g\<^sub>0h\<^sub>1.p\<^sub>1) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine\<guillemotright>
goal (6 subgoals):
1. \<guillemotleft>((g.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>1) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine : ((tab\<^sub>1 g \<star> g\<^sub>0h\<^sub>1.p\<^sub>1) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine \<rightarrow> ?b4\<guillemotright>
2. \<guillemotleft>(\<a>[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine : ?b4 \<rightarrow> ?b3\<guillemotright>
3. \<guillemotleft>((g \<star> g\<^sub>0h\<^sub>1.\<phi>) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine : ?b3 \<rightarrow> ?b2\<guillemotright>
4. \<guillemotleft>((g \<star> h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine : ?b2 \<rightarrow> ?b1\<guillemotright>
5. \<guillemotleft>can (\<^bold>\<langle>g\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> ((\<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) (((\<^bold>\<langle>g\<^bold>\<rangle> \<^bold>\<star> (\<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) : ?b1 \<rightarrow> ?b\<guillemotright>
6. \<guillemotleft>g \<star> h \<star> TTfgh_TfTgh.the_\<theta> : ?b \<rightarrow> g \<star> h \<star> tab\<^sub>0 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0\<guillemotright>
[PROOF STEP]
show "\<guillemotleft>((g.tab \<star> Tgh.p\<^sub>1) \<star> TfTgh.p\<^sub>0) \<star> TTfgh_TfTgh.chine :
((tab\<^sub>1 g \<star> Tgh.p\<^sub>1) \<star> TfTgh.p\<^sub>0) \<star> TTfgh_TfTgh.chine
\<Rightarrow> (((g \<star> tab\<^sub>0 g) \<star> Tgh.p\<^sub>1) \<star> TfTgh.p\<^sub>0) \<star> TTfgh_TfTgh.chine\<guillemotright>"
[PROOF STATE]
proof (prove)
goal (1 subgoal):
1. \<guillemotleft>((g.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>1) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine : ((tab\<^sub>1 g \<star> g\<^sub>0h\<^sub>1.p\<^sub>1) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine \<rightarrow> (((g \<star> tab\<^sub>0 g) \<star> g\<^sub>0h\<^sub>1.p\<^sub>1) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine\<guillemotright>
[PROOF STEP]
using f\<^sub>0gh\<^sub>1.cospan g\<^sub>0h\<^sub>1.cospan
[PROOF STATE]
proof (prove)
using this:
trg (tab\<^sub>1 g \<star> g\<^sub>0h\<^sub>1.p\<^sub>1) = trg (tab\<^sub>0 f)
trg (tab\<^sub>1 h) = trg (tab\<^sub>0 g)
goal (1 subgoal):
1. \<guillemotleft>((g.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>1) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine : ((tab\<^sub>1 g \<star> g\<^sub>0h\<^sub>1.p\<^sub>1) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine \<rightarrow> (((g \<star> tab\<^sub>0 g) \<star> g\<^sub>0h\<^sub>1.p\<^sub>1) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine\<guillemotright>
[PROOF STEP]
by (intro hcomp_in_vhom, auto)
[PROOF STATE]
proof (state)
this:
\<guillemotleft>((g.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>1) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine : ((tab\<^sub>1 g \<star> g\<^sub>0h\<^sub>1.p\<^sub>1) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine \<rightarrow> (((g \<star> tab\<^sub>0 g) \<star> g\<^sub>0h\<^sub>1.p\<^sub>1) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine\<guillemotright>
goal (5 subgoals):
1. \<guillemotleft>(\<a>[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine : (((g \<star> tab\<^sub>0 g) \<star> g\<^sub>0h\<^sub>1.p\<^sub>1) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine \<rightarrow> ?b3\<guillemotright>
2. \<guillemotleft>((g \<star> g\<^sub>0h\<^sub>1.\<phi>) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine : ?b3 \<rightarrow> ?b2\<guillemotright>
3. \<guillemotleft>((g \<star> h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine : ?b2 \<rightarrow> ?b1\<guillemotright>
4. \<guillemotleft>can (\<^bold>\<langle>g\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> ((\<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) (((\<^bold>\<langle>g\<^bold>\<rangle> \<^bold>\<star> (\<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) : ?b1 \<rightarrow> ?b\<guillemotright>
5. \<guillemotleft>g \<star> h \<star> TTfgh_TfTgh.the_\<theta> : ?b \<rightarrow> g \<star> h \<star> tab\<^sub>0 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0\<guillemotright>
[PROOF STEP]
show "\<guillemotleft>(\<a>[g, tab\<^sub>0 g, Tgh.p\<^sub>1] \<star> TfTgh.p\<^sub>0) \<star> TTfgh_TfTgh.chine :
(((g \<star> tab\<^sub>0 g) \<star> Tgh.p\<^sub>1) \<star> TfTgh.p\<^sub>0) \<star> TTfgh_TfTgh.chine
\<Rightarrow> ((g \<star> tab\<^sub>0 g \<star> Tgh.p\<^sub>1) \<star> TfTgh.p\<^sub>0) \<star> TTfgh_TfTgh.chine\<guillemotright>"
[PROOF STATE]
proof (prove)
goal (1 subgoal):
1. \<guillemotleft>(\<a>[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine : (((g \<star> tab\<^sub>0 g) \<star> g\<^sub>0h\<^sub>1.p\<^sub>1) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine \<rightarrow> ((g \<star> tab\<^sub>0 g \<star> g\<^sub>0h\<^sub>1.p\<^sub>1) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine\<guillemotright>
[PROOF STEP]
using f\<^sub>0gh\<^sub>1.cospan g\<^sub>0h\<^sub>1.cospan
[PROOF STATE]
proof (prove)
using this:
trg (tab\<^sub>1 g \<star> g\<^sub>0h\<^sub>1.p\<^sub>1) = trg (tab\<^sub>0 f)
trg (tab\<^sub>1 h) = trg (tab\<^sub>0 g)
goal (1 subgoal):
1. \<guillemotleft>(\<a>[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine : (((g \<star> tab\<^sub>0 g) \<star> g\<^sub>0h\<^sub>1.p\<^sub>1) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine \<rightarrow> ((g \<star> tab\<^sub>0 g \<star> g\<^sub>0h\<^sub>1.p\<^sub>1) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine\<guillemotright>
[PROOF STEP]
by (intro hcomp_in_vhom, auto)
[PROOF STATE]
proof (state)
this:
\<guillemotleft>(\<a>[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine : (((g \<star> tab\<^sub>0 g) \<star> g\<^sub>0h\<^sub>1.p\<^sub>1) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine \<rightarrow> ((g \<star> tab\<^sub>0 g \<star> g\<^sub>0h\<^sub>1.p\<^sub>1) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine\<guillemotright>
goal (4 subgoals):
1. \<guillemotleft>((g \<star> g\<^sub>0h\<^sub>1.\<phi>) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine : ((g \<star> tab\<^sub>0 g \<star> g\<^sub>0h\<^sub>1.p\<^sub>1) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine \<rightarrow> ?b2\<guillemotright>
2. \<guillemotleft>((g \<star> h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine : ?b2 \<rightarrow> ?b1\<guillemotright>
3. \<guillemotleft>can (\<^bold>\<langle>g\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> ((\<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) (((\<^bold>\<langle>g\<^bold>\<rangle> \<^bold>\<star> (\<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) : ?b1 \<rightarrow> ?b\<guillemotright>
4. \<guillemotleft>g \<star> h \<star> TTfgh_TfTgh.the_\<theta> : ?b \<rightarrow> g \<star> h \<star> tab\<^sub>0 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0\<guillemotright>
[PROOF STEP]
show "\<guillemotleft>((g \<star> g\<^sub>0h\<^sub>1.\<phi>) \<star> TfTgh.p\<^sub>0) \<star> TTfgh_TfTgh.chine :
((g \<star> tab\<^sub>0 g \<star> Tgh.p\<^sub>1) \<star> TfTgh.p\<^sub>0) \<star> TTfgh_TfTgh.chine
\<Rightarrow> ((g \<star> tab\<^sub>1 h \<star> Tgh.p\<^sub>0) \<star> TfTgh.p\<^sub>0) \<star> TTfgh_TfTgh.chine\<guillemotright>"
[PROOF STATE]
proof (prove)
goal (1 subgoal):
1. \<guillemotleft>((g \<star> g\<^sub>0h\<^sub>1.\<phi>) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine : ((g \<star> tab\<^sub>0 g \<star> g\<^sub>0h\<^sub>1.p\<^sub>1) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine \<rightarrow> ((g \<star> tab\<^sub>1 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine\<guillemotright>
[PROOF STEP]
using f\<^sub>0gh\<^sub>1.cospan g\<^sub>0h\<^sub>1.cospan g\<^sub>0h\<^sub>1.\<phi>_in_hom(2)
[PROOF STATE]
proof (prove)
using this:
trg (tab\<^sub>1 g \<star> g\<^sub>0h\<^sub>1.p\<^sub>1) = trg (tab\<^sub>0 f)
trg (tab\<^sub>1 h) = trg (tab\<^sub>0 g)
\<guillemotleft>g\<^sub>0h\<^sub>1.\<phi> : tab\<^sub>0 g \<star> g\<^sub>0h\<^sub>1.p\<^sub>1 \<rightarrow> tab\<^sub>1 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0\<guillemotright>
goal (1 subgoal):
1. \<guillemotleft>((g \<star> g\<^sub>0h\<^sub>1.\<phi>) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine : ((g \<star> tab\<^sub>0 g \<star> g\<^sub>0h\<^sub>1.p\<^sub>1) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine \<rightarrow> ((g \<star> tab\<^sub>1 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine\<guillemotright>
[PROOF STEP]
by (intro hcomp_in_vhom, auto)
[PROOF STATE]
proof (state)
this:
\<guillemotleft>((g \<star> g\<^sub>0h\<^sub>1.\<phi>) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine : ((g \<star> tab\<^sub>0 g \<star> g\<^sub>0h\<^sub>1.p\<^sub>1) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine \<rightarrow> ((g \<star> tab\<^sub>1 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine\<guillemotright>
goal (3 subgoals):
1. \<guillemotleft>((g \<star> h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine : ((g \<star> tab\<^sub>1 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine \<rightarrow> ?b1\<guillemotright>
2. \<guillemotleft>can (\<^bold>\<langle>g\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> ((\<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) (((\<^bold>\<langle>g\<^bold>\<rangle> \<^bold>\<star> (\<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) : ?b1 \<rightarrow> ?b\<guillemotright>
3. \<guillemotleft>g \<star> h \<star> TTfgh_TfTgh.the_\<theta> : ?b \<rightarrow> g \<star> h \<star> tab\<^sub>0 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0\<guillemotright>
[PROOF STEP]
show "\<guillemotleft>((g \<star> h.tab \<star> Tgh.p\<^sub>0) \<star> TfTgh.p\<^sub>0) \<star> TTfgh_TfTgh.chine :
((g \<star> tab\<^sub>1 h \<star> Tgh.p\<^sub>0) \<star> TfTgh.p\<^sub>0) \<star> TTfgh_TfTgh.chine
\<Rightarrow> ((g \<star> (h \<star> tab\<^sub>0 h) \<star> Tgh.p\<^sub>0) \<star> TfTgh.p\<^sub>0) \<star> TTfgh_TfTgh.chine\<guillemotright>"
[PROOF STATE]
proof (prove)
goal (1 subgoal):
1. \<guillemotleft>((g \<star> h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine : ((g \<star> tab\<^sub>1 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine \<rightarrow> ((g \<star> (h \<star> tab\<^sub>0 h) \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine\<guillemotright>
[PROOF STEP]
using f\<^sub>0gh\<^sub>1.cospan g\<^sub>0h\<^sub>1.cospan
[PROOF STATE]
proof (prove)
using this:
trg (tab\<^sub>1 g \<star> g\<^sub>0h\<^sub>1.p\<^sub>1) = trg (tab\<^sub>0 f)
trg (tab\<^sub>1 h) = trg (tab\<^sub>0 g)
goal (1 subgoal):
1. \<guillemotleft>((g \<star> h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine : ((g \<star> tab\<^sub>1 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine \<rightarrow> ((g \<star> (h \<star> tab\<^sub>0 h) \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine\<guillemotright>
[PROOF STEP]
by (intro hcomp_in_vhom, auto)
[PROOF STATE]
proof (state)
this:
\<guillemotleft>((g \<star> h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine : ((g \<star> tab\<^sub>1 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine \<rightarrow> ((g \<star> (h \<star> tab\<^sub>0 h) \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine\<guillemotright>
goal (2 subgoals):
1. \<guillemotleft>can (\<^bold>\<langle>g\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> ((\<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) (((\<^bold>\<langle>g\<^bold>\<rangle> \<^bold>\<star> (\<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) : ((g \<star> (h \<star> tab\<^sub>0 h) \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine \<rightarrow> ?b\<guillemotright>
2. \<guillemotleft>g \<star> h \<star> TTfgh_TfTgh.the_\<theta> : ?b \<rightarrow> g \<star> h \<star> tab\<^sub>0 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0\<guillemotright>
[PROOF STEP]
show "\<guillemotleft>can (\<^bold>\<langle>g\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> ((\<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>Tgh.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TfTgh.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>)
(((\<^bold>\<langle>g\<^bold>\<rangle> \<^bold>\<star> (\<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>Tgh.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TfTgh.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) :
((g \<star> (h \<star> tab\<^sub>0 h) \<star> Tgh.p\<^sub>0) \<star> TfTgh.p\<^sub>0) \<star> TTfgh_TfTgh.chine
\<Rightarrow> g \<star> h \<star> ((tab\<^sub>0 h \<star> Tgh.p\<^sub>0) \<star> TfTgh.p\<^sub>0) \<star> TTfgh_TfTgh.chine\<guillemotright>"
[PROOF STATE]
proof (prove)
goal (1 subgoal):
1. \<guillemotleft>can (\<^bold>\<langle>g\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> ((\<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) (((\<^bold>\<langle>g\<^bold>\<rangle> \<^bold>\<star> (\<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) : ((g \<star> (h \<star> tab\<^sub>0 h) \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine \<rightarrow> g \<star> h \<star> ((tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine\<guillemotright>
[PROOF STEP]
using f\<^sub>0gh\<^sub>1.cospan g\<^sub>0h\<^sub>1.cospan
[PROOF STATE]
proof (prove)
using this:
trg (tab\<^sub>1 g \<star> g\<^sub>0h\<^sub>1.p\<^sub>1) = trg (tab\<^sub>0 f)
trg (tab\<^sub>1 h) = trg (tab\<^sub>0 g)
goal (1 subgoal):
1. \<guillemotleft>can (\<^bold>\<langle>g\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> ((\<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) (((\<^bold>\<langle>g\<^bold>\<rangle> \<^bold>\<star> (\<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) : ((g \<star> (h \<star> tab\<^sub>0 h) \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine \<rightarrow> g \<star> h \<star> ((tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine\<guillemotright>
[PROOF STEP]
by auto
[PROOF STATE]
proof (state)
this:
\<guillemotleft>can (\<^bold>\<langle>g\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> ((\<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) (((\<^bold>\<langle>g\<^bold>\<rangle> \<^bold>\<star> (\<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) : ((g \<star> (h \<star> tab\<^sub>0 h) \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine \<rightarrow> g \<star> h \<star> ((tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine\<guillemotright>
goal (1 subgoal):
1. \<guillemotleft>g \<star> h \<star> TTfgh_TfTgh.the_\<theta> : g \<star> h \<star> ((tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine \<rightarrow> g \<star> h \<star> tab\<^sub>0 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0\<guillemotright>
[PROOF STEP]
show "\<guillemotleft>g \<star> h \<star> TTfgh_TfTgh.the_\<theta> :
g \<star> h \<star> ((tab\<^sub>0 h \<star> Tgh.p\<^sub>0) \<star> TfTgh.p\<^sub>0) \<star> TTfgh_TfTgh.chine
\<Rightarrow> g \<star> h \<star> tab\<^sub>0 h \<star> TTfgh.p\<^sub>0\<guillemotright>"
[PROOF STATE]
proof (prove)
goal (1 subgoal):
1. \<guillemotleft>g \<star> h \<star> TTfgh_TfTgh.the_\<theta> : g \<star> h \<star> ((tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine \<rightarrow> g \<star> h \<star> tab\<^sub>0 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0\<guillemotright>
[PROOF STEP]
using f\<^sub>0gh\<^sub>1.cospan g\<^sub>0h\<^sub>1.cospan TTfgh_TfTgh.the_\<theta>_in_hom
[PROOF STATE]
proof (prove)
using this:
trg (tab\<^sub>1 g \<star> g\<^sub>0h\<^sub>1.p\<^sub>1) = trg (tab\<^sub>0 f)
trg (tab\<^sub>1 h) = trg (tab\<^sub>0 g)
\<guillemotleft>TTfgh_TfTgh.the_\<theta> : src (tab\<^sub>0 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0) \<rightarrow> trg (tab\<^sub>0 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0)\<guillemotright>
\<guillemotleft>TTfgh_TfTgh.the_\<theta> : ((tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine \<rightarrow> tab\<^sub>0 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0\<guillemotright>
goal (1 subgoal):
1. \<guillemotleft>g \<star> h \<star> TTfgh_TfTgh.the_\<theta> : g \<star> h \<star> ((tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine \<rightarrow> g \<star> h \<star> tab\<^sub>0 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0\<guillemotright>
[PROOF STEP]
by (intro hcomp_in_vhom, auto)
[PROOF STATE]
proof (state)
this:
\<guillemotleft>g \<star> h \<star> TTfgh_TfTgh.the_\<theta> : g \<star> h \<star> ((tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine \<rightarrow> g \<star> h \<star> tab\<^sub>0 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0\<guillemotright>
goal:
No subgoals!
[PROOF STEP]
qed
[PROOF STATE]
proof (state)
this:
\<guillemotleft>\<theta>\<^sub>f' : tab\<^sub>0 f \<star> w\<^sub>f' \<rightarrow> u\<^sub>f\<guillemotright>
goal (3 subgoals):
1. \<lbrakk>\<lbrakk>f\<^sub>0gh\<^sub>1.p\<^sub>1 \<star> TTfgh_TfTgh.chine\<rbrakk>\<rbrakk> = \<lbrakk>\<lbrakk>f\<^sub>0g\<^sub>1.p\<^sub>1 \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1\<rbrakk>\<rbrakk>
2. \<lbrakk>\<lbrakk>g\<^sub>0h\<^sub>1.p\<^sub>1 \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine\<rbrakk>\<rbrakk> = \<lbrakk>\<lbrakk>f\<^sub>0g\<^sub>1.p\<^sub>0 \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1\<rbrakk>\<rbrakk>
3. \<lbrakk>\<lbrakk>g\<^sub>0h\<^sub>1.p\<^sub>0 \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine\<rbrakk>\<rbrakk> = \<lbrakk>\<lbrakk>fg\<^sub>0h\<^sub>1.p\<^sub>0\<rbrakk>\<rbrakk>
[PROOF STEP]
have \<beta>\<^sub>f: "\<guillemotleft>\<beta>\<^sub>f : tab\<^sub>1 f \<star> w\<^sub>f \<Rightarrow> tab\<^sub>1 f \<star> w\<^sub>f'\<guillemotright>"
[PROOF STATE]
proof (prove)
goal (1 subgoal):
1. \<guillemotleft>\<beta>\<^sub>f : tab\<^sub>1 f \<star> w\<^sub>f \<rightarrow> tab\<^sub>1 f \<star> w\<^sub>f'\<guillemotright>
[PROOF STEP]
proof (unfold \<beta>\<^sub>f_def w\<^sub>f_def w\<^sub>f'_def, intro comp_in_homI)
[PROOF STATE]
proof (state)
goal (3 subgoals):
1. \<guillemotleft>\<a>\<^sup>-\<^sup>1[tab\<^sub>1 f, f\<^sub>0g\<^sub>1.p\<^sub>1, fg\<^sub>0h\<^sub>1.p\<^sub>1] : tab\<^sub>1 f \<star> f\<^sub>0g\<^sub>1.p\<^sub>1 \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1 \<rightarrow> ?b1\<guillemotright>
2. \<guillemotleft>TTfgh_TfTgh.the_\<nu> : ?b1 \<rightarrow> ?b\<guillemotright>
3. \<guillemotleft>\<a>[tab\<^sub>1 f, f\<^sub>0gh\<^sub>1.p\<^sub>1, TTfgh_TfTgh.chine] : ?b \<rightarrow> tab\<^sub>1 f \<star> f\<^sub>0gh\<^sub>1.p\<^sub>1 \<star> TTfgh_TfTgh.chine\<guillemotright>
[PROOF STEP]
show "\<guillemotleft>\<a>\<^sup>-\<^sup>1[tab\<^sub>1 f, Tfg.p\<^sub>1, TTfgh.p\<^sub>1] :
tab\<^sub>1 f \<star> Tfg.p\<^sub>1 \<star> TTfgh.p\<^sub>1 \<Rightarrow> (tab\<^sub>1 f \<star> Tfg.p\<^sub>1) \<star> TTfgh.p\<^sub>1\<guillemotright>"
[PROOF STATE]
proof (prove)
goal (1 subgoal):
1. \<guillemotleft>\<a>\<^sup>-\<^sup>1[tab\<^sub>1 f, f\<^sub>0g\<^sub>1.p\<^sub>1, fg\<^sub>0h\<^sub>1.p\<^sub>1] : tab\<^sub>1 f \<star> f\<^sub>0g\<^sub>1.p\<^sub>1 \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1 \<rightarrow> (tab\<^sub>1 f \<star> f\<^sub>0g\<^sub>1.p\<^sub>1) \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1\<guillemotright>
[PROOF STEP]
using TTfgh.leg1_in_hom(2) assoc'_in_hom
[PROOF STATE]
proof (prove)
using this:
\<guillemotleft>(tab\<^sub>1 f \<star> f\<^sub>0g\<^sub>1.p\<^sub>1) \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1 : (tab\<^sub>1 f \<star> f\<^sub>0g\<^sub>1.p\<^sub>1) \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1 \<rightarrow> (tab\<^sub>1 f \<star> f\<^sub>0g\<^sub>1.p\<^sub>1) \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1\<guillemotright>
\<lbrakk>ide ?f; ide ?g; ide ?h; src ?f = trg ?g; src ?g = trg ?h\<rbrakk> \<Longrightarrow> \<guillemotleft>\<a>\<^sup>-\<^sup>1[?f, ?g, ?h] : src ?h \<rightarrow> trg ?f\<guillemotright>
\<lbrakk>ide ?f; ide ?g; ide ?h; src ?f = trg ?g; src ?g = trg ?h\<rbrakk> \<Longrightarrow> \<guillemotleft>\<a>\<^sup>-\<^sup>1[?f, ?g, ?h] : local.dom ?f \<star> local.dom ?g \<star> local.dom ?h \<rightarrow> (cod ?f \<star> cod ?g) \<star> cod ?h\<guillemotright>
goal (1 subgoal):
1. \<guillemotleft>\<a>\<^sup>-\<^sup>1[tab\<^sub>1 f, f\<^sub>0g\<^sub>1.p\<^sub>1, fg\<^sub>0h\<^sub>1.p\<^sub>1] : tab\<^sub>1 f \<star> f\<^sub>0g\<^sub>1.p\<^sub>1 \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1 \<rightarrow> (tab\<^sub>1 f \<star> f\<^sub>0g\<^sub>1.p\<^sub>1) \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1\<guillemotright>
[PROOF STEP]
by auto
[PROOF STATE]
proof (state)
this:
\<guillemotleft>\<a>\<^sup>-\<^sup>1[tab\<^sub>1 f, f\<^sub>0g\<^sub>1.p\<^sub>1, fg\<^sub>0h\<^sub>1.p\<^sub>1] : tab\<^sub>1 f \<star> f\<^sub>0g\<^sub>1.p\<^sub>1 \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1 \<rightarrow> (tab\<^sub>1 f \<star> f\<^sub>0g\<^sub>1.p\<^sub>1) \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1\<guillemotright>
goal (2 subgoals):
1. \<guillemotleft>TTfgh_TfTgh.the_\<nu> : (tab\<^sub>1 f \<star> f\<^sub>0g\<^sub>1.p\<^sub>1) \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1 \<rightarrow> ?b\<guillemotright>
2. \<guillemotleft>\<a>[tab\<^sub>1 f, f\<^sub>0gh\<^sub>1.p\<^sub>1, TTfgh_TfTgh.chine] : ?b \<rightarrow> tab\<^sub>1 f \<star> f\<^sub>0gh\<^sub>1.p\<^sub>1 \<star> TTfgh_TfTgh.chine\<guillemotright>
[PROOF STEP]
show "\<guillemotleft>TTfgh_TfTgh.the_\<nu> :
(tab\<^sub>1 f \<star> Tfg.p\<^sub>1) \<star> TTfgh.p\<^sub>1 \<Rightarrow> (tab\<^sub>1 f \<star> TfTgh.p\<^sub>1) \<star> TTfgh_TfTgh.chine\<guillemotright>"
[PROOF STATE]
proof (prove)
goal (1 subgoal):
1. \<guillemotleft>TTfgh_TfTgh.the_\<nu> : (tab\<^sub>1 f \<star> f\<^sub>0g\<^sub>1.p\<^sub>1) \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1 \<rightarrow> (tab\<^sub>1 f \<star> f\<^sub>0gh\<^sub>1.p\<^sub>1) \<star> TTfgh_TfTgh.chine\<guillemotright>
[PROOF STEP]
using TTfgh_TfTgh.the_\<nu>_in_hom TTfgh_TfTgh.the_\<nu>_props
[PROOF STATE]
proof (prove)
using this:
\<guillemotleft>TTfgh_TfTgh.the_\<nu> : src ((tab\<^sub>1 f \<star> f\<^sub>0g\<^sub>1.p\<^sub>1) \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<rightarrow> trg ((tab\<^sub>1 f \<star> f\<^sub>0g\<^sub>1.p\<^sub>1) \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1)\<guillemotright>
\<guillemotleft>TTfgh_TfTgh.the_\<nu> : (tab\<^sub>1 f \<star> f\<^sub>0g\<^sub>1.p\<^sub>1) \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1 \<rightarrow> (tab\<^sub>1 f \<star> f\<^sub>0gh\<^sub>1.p\<^sub>1) \<star> TTfgh_TfTgh.chine\<guillemotright>
\<guillemotleft>TTfgh_TfTgh.the_\<nu> : (tab\<^sub>1 f \<star> f\<^sub>0g\<^sub>1.p\<^sub>1) \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1 \<rightarrow> (tab\<^sub>1 f \<star> f\<^sub>0gh\<^sub>1.p\<^sub>1) \<star> TTfgh_TfTgh.chine\<guillemotright>
local.iso TTfgh_TfTgh.the_\<nu>
goal (1 subgoal):
1. \<guillemotleft>TTfgh_TfTgh.the_\<nu> : (tab\<^sub>1 f \<star> f\<^sub>0g\<^sub>1.p\<^sub>1) \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1 \<rightarrow> (tab\<^sub>1 f \<star> f\<^sub>0gh\<^sub>1.p\<^sub>1) \<star> TTfgh_TfTgh.chine\<guillemotright>
[PROOF STEP]
by simp
[PROOF STATE]
proof (state)
this:
\<guillemotleft>TTfgh_TfTgh.the_\<nu> : (tab\<^sub>1 f \<star> f\<^sub>0g\<^sub>1.p\<^sub>1) \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1 \<rightarrow> (tab\<^sub>1 f \<star> f\<^sub>0gh\<^sub>1.p\<^sub>1) \<star> TTfgh_TfTgh.chine\<guillemotright>
goal (1 subgoal):
1. \<guillemotleft>\<a>[tab\<^sub>1 f, f\<^sub>0gh\<^sub>1.p\<^sub>1, TTfgh_TfTgh.chine] : (tab\<^sub>1 f \<star> f\<^sub>0gh\<^sub>1.p\<^sub>1) \<star> TTfgh_TfTgh.chine \<rightarrow> tab\<^sub>1 f \<star> f\<^sub>0gh\<^sub>1.p\<^sub>1 \<star> TTfgh_TfTgh.chine\<guillemotright>
[PROOF STEP]
show "\<guillemotleft>\<a>[tab\<^sub>1 f, TfTgh.p\<^sub>1, TTfgh_TfTgh.chine] :
(tab\<^sub>1 f \<star> TfTgh.p\<^sub>1) \<star> TTfgh_TfTgh.chine \<Rightarrow> tab\<^sub>1 f \<star> TfTgh.p\<^sub>1 \<star> TTfgh_TfTgh.chine\<guillemotright>"
[PROOF STATE]
proof (prove)
goal (1 subgoal):
1. \<guillemotleft>\<a>[tab\<^sub>1 f, f\<^sub>0gh\<^sub>1.p\<^sub>1, TTfgh_TfTgh.chine] : (tab\<^sub>1 f \<star> f\<^sub>0gh\<^sub>1.p\<^sub>1) \<star> TTfgh_TfTgh.chine \<rightarrow> tab\<^sub>1 f \<star> f\<^sub>0gh\<^sub>1.p\<^sub>1 \<star> TTfgh_TfTgh.chine\<guillemotright>
[PROOF STEP]
using 1 2 3 4
[PROOF STATE]
proof (prove)
using this:
ide f\<^sub>0gh\<^sub>1.p\<^sub>1
ide TTfgh_TfTgh.chine
src f\<^sub>0gh\<^sub>1.p\<^sub>1 = trg TTfgh_TfTgh.chine
src (tab\<^sub>1 f) = trg f\<^sub>0gh\<^sub>1.p\<^sub>1
goal (1 subgoal):
1. \<guillemotleft>\<a>[tab\<^sub>1 f, f\<^sub>0gh\<^sub>1.p\<^sub>1, TTfgh_TfTgh.chine] : (tab\<^sub>1 f \<star> f\<^sub>0gh\<^sub>1.p\<^sub>1) \<star> TTfgh_TfTgh.chine \<rightarrow> tab\<^sub>1 f \<star> f\<^sub>0gh\<^sub>1.p\<^sub>1 \<star> TTfgh_TfTgh.chine\<guillemotright>
[PROOF STEP]
by auto
[PROOF STATE]
proof (state)
this:
\<guillemotleft>\<a>[tab\<^sub>1 f, f\<^sub>0gh\<^sub>1.p\<^sub>1, TTfgh_TfTgh.chine] : (tab\<^sub>1 f \<star> f\<^sub>0gh\<^sub>1.p\<^sub>1) \<star> TTfgh_TfTgh.chine \<rightarrow> tab\<^sub>1 f \<star> f\<^sub>0gh\<^sub>1.p\<^sub>1 \<star> TTfgh_TfTgh.chine\<guillemotright>
goal:
No subgoals!
[PROOF STEP]
qed
[PROOF STATE]
proof (state)
this:
\<guillemotleft>\<beta>\<^sub>f : tab\<^sub>1 f \<star> w\<^sub>f \<rightarrow> tab\<^sub>1 f \<star> w\<^sub>f'\<guillemotright>
goal (3 subgoals):
1. \<lbrakk>\<lbrakk>f\<^sub>0gh\<^sub>1.p\<^sub>1 \<star> TTfgh_TfTgh.chine\<rbrakk>\<rbrakk> = \<lbrakk>\<lbrakk>f\<^sub>0g\<^sub>1.p\<^sub>1 \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1\<rbrakk>\<rbrakk>
2. \<lbrakk>\<lbrakk>g\<^sub>0h\<^sub>1.p\<^sub>1 \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine\<rbrakk>\<rbrakk> = \<lbrakk>\<lbrakk>f\<^sub>0g\<^sub>1.p\<^sub>0 \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1\<rbrakk>\<rbrakk>
3. \<lbrakk>\<lbrakk>g\<^sub>0h\<^sub>1.p\<^sub>0 \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine\<rbrakk>\<rbrakk> = \<lbrakk>\<lbrakk>fg\<^sub>0h\<^sub>1.p\<^sub>0\<rbrakk>\<rbrakk>
[PROOF STEP]
have iso_\<beta>\<^sub>f: "iso \<beta>\<^sub>f"
[PROOF STATE]
proof (prove)
goal (1 subgoal):
1. local.iso \<beta>\<^sub>f
[PROOF STEP]
unfolding \<beta>\<^sub>f_def
[PROOF STATE]
proof (prove)
goal (1 subgoal):
1. local.iso (\<a>[tab\<^sub>1 f, f\<^sub>0gh\<^sub>1.p\<^sub>1, TTfgh_TfTgh.chine] \<cdot> TTfgh_TfTgh.the_\<nu> \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>1 f, f\<^sub>0g\<^sub>1.p\<^sub>1, fg\<^sub>0h\<^sub>1.p\<^sub>1])
[PROOF STEP]
using 1 2 3 4 \<beta>\<^sub>f \<beta>\<^sub>f_def isos_compose
[PROOF STATE]
proof (prove)
using this:
ide f\<^sub>0gh\<^sub>1.p\<^sub>1
ide TTfgh_TfTgh.chine
src f\<^sub>0gh\<^sub>1.p\<^sub>1 = trg TTfgh_TfTgh.chine
src (tab\<^sub>1 f) = trg f\<^sub>0gh\<^sub>1.p\<^sub>1
\<guillemotleft>\<beta>\<^sub>f : tab\<^sub>1 f \<star> w\<^sub>f \<rightarrow> tab\<^sub>1 f \<star> w\<^sub>f'\<guillemotright>
\<beta>\<^sub>f = \<a>[tab\<^sub>1 f, f\<^sub>0gh\<^sub>1.p\<^sub>1, TTfgh_TfTgh.chine] \<cdot> TTfgh_TfTgh.the_\<nu> \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>1 f, f\<^sub>0g\<^sub>1.p\<^sub>1, fg\<^sub>0h\<^sub>1.p\<^sub>1]
\<lbrakk>local.iso ?f; local.iso ?f'; seq ?f' ?f\<rbrakk> \<Longrightarrow> local.iso (?f' \<cdot> ?f)
goal (1 subgoal):
1. local.iso (\<a>[tab\<^sub>1 f, f\<^sub>0gh\<^sub>1.p\<^sub>1, TTfgh_TfTgh.chine] \<cdot> TTfgh_TfTgh.the_\<nu> \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>1 f, f\<^sub>0g\<^sub>1.p\<^sub>1, fg\<^sub>0h\<^sub>1.p\<^sub>1])
[PROOF STEP]
apply (intro isos_compose)
[PROOF STATE]
proof (prove)
goal (5 subgoals):
1. \<lbrakk>ide f\<^sub>0gh\<^sub>1.p\<^sub>1; ide TTfgh_TfTgh.chine; src f\<^sub>0gh\<^sub>1.p\<^sub>1 = trg TTfgh_TfTgh.chine; src (tab\<^sub>1 f) = trg f\<^sub>0gh\<^sub>1.p\<^sub>1; \<guillemotleft>\<beta>\<^sub>f : tab\<^sub>1 f \<star> w\<^sub>f \<rightarrow> tab\<^sub>1 f \<star> w\<^sub>f'\<guillemotright>; \<beta>\<^sub>f = \<a>[tab\<^sub>1 f, f\<^sub>0gh\<^sub>1.p\<^sub>1, TTfgh_TfTgh.chine] \<cdot> TTfgh_TfTgh.the_\<nu> \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>1 f, f\<^sub>0g\<^sub>1.p\<^sub>1, fg\<^sub>0h\<^sub>1.p\<^sub>1]; \<And>f f'. \<lbrakk>local.iso f; local.iso f'; seq f' f\<rbrakk> \<Longrightarrow> local.iso (f' \<cdot> f)\<rbrakk> \<Longrightarrow> local.iso \<a>\<^sup>-\<^sup>1[tab\<^sub>1 f, f\<^sub>0g\<^sub>1.p\<^sub>1, fg\<^sub>0h\<^sub>1.p\<^sub>1]
2. \<lbrakk>ide f\<^sub>0gh\<^sub>1.p\<^sub>1; ide TTfgh_TfTgh.chine; src f\<^sub>0gh\<^sub>1.p\<^sub>1 = trg TTfgh_TfTgh.chine; src (tab\<^sub>1 f) = trg f\<^sub>0gh\<^sub>1.p\<^sub>1; \<guillemotleft>\<beta>\<^sub>f : tab\<^sub>1 f \<star> w\<^sub>f \<rightarrow> tab\<^sub>1 f \<star> w\<^sub>f'\<guillemotright>; \<beta>\<^sub>f = \<a>[tab\<^sub>1 f, f\<^sub>0gh\<^sub>1.p\<^sub>1, TTfgh_TfTgh.chine] \<cdot> TTfgh_TfTgh.the_\<nu> \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>1 f, f\<^sub>0g\<^sub>1.p\<^sub>1, fg\<^sub>0h\<^sub>1.p\<^sub>1]; \<And>f f'. \<lbrakk>local.iso f; local.iso f'; seq f' f\<rbrakk> \<Longrightarrow> local.iso (f' \<cdot> f)\<rbrakk> \<Longrightarrow> local.iso TTfgh_TfTgh.the_\<nu>
3. \<lbrakk>ide f\<^sub>0gh\<^sub>1.p\<^sub>1; ide TTfgh_TfTgh.chine; src f\<^sub>0gh\<^sub>1.p\<^sub>1 = trg TTfgh_TfTgh.chine; src (tab\<^sub>1 f) = trg f\<^sub>0gh\<^sub>1.p\<^sub>1; \<guillemotleft>\<beta>\<^sub>f : tab\<^sub>1 f \<star> w\<^sub>f \<rightarrow> tab\<^sub>1 f \<star> w\<^sub>f'\<guillemotright>; \<beta>\<^sub>f = \<a>[tab\<^sub>1 f, f\<^sub>0gh\<^sub>1.p\<^sub>1, TTfgh_TfTgh.chine] \<cdot> TTfgh_TfTgh.the_\<nu> \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>1 f, f\<^sub>0g\<^sub>1.p\<^sub>1, fg\<^sub>0h\<^sub>1.p\<^sub>1]; \<And>f f'. \<lbrakk>local.iso f; local.iso f'; seq f' f\<rbrakk> \<Longrightarrow> local.iso (f' \<cdot> f)\<rbrakk> \<Longrightarrow> seq TTfgh_TfTgh.the_\<nu> \<a>\<^sup>-\<^sup>1[tab\<^sub>1 f, f\<^sub>0g\<^sub>1.p\<^sub>1, fg\<^sub>0h\<^sub>1.p\<^sub>1]
4. \<lbrakk>ide f\<^sub>0gh\<^sub>1.p\<^sub>1; ide TTfgh_TfTgh.chine; src f\<^sub>0gh\<^sub>1.p\<^sub>1 = trg TTfgh_TfTgh.chine; src (tab\<^sub>1 f) = trg f\<^sub>0gh\<^sub>1.p\<^sub>1; \<guillemotleft>\<beta>\<^sub>f : tab\<^sub>1 f \<star> w\<^sub>f \<rightarrow> tab\<^sub>1 f \<star> w\<^sub>f'\<guillemotright>; \<beta>\<^sub>f = \<a>[tab\<^sub>1 f, f\<^sub>0gh\<^sub>1.p\<^sub>1, TTfgh_TfTgh.chine] \<cdot> TTfgh_TfTgh.the_\<nu> \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>1 f, f\<^sub>0g\<^sub>1.p\<^sub>1, fg\<^sub>0h\<^sub>1.p\<^sub>1]; \<And>f f'. \<lbrakk>local.iso f; local.iso f'; seq f' f\<rbrakk> \<Longrightarrow> local.iso (f' \<cdot> f)\<rbrakk> \<Longrightarrow> local.iso \<a>[tab\<^sub>1 f, f\<^sub>0gh\<^sub>1.p\<^sub>1, TTfgh_TfTgh.chine]
5. \<lbrakk>ide f\<^sub>0gh\<^sub>1.p\<^sub>1; ide TTfgh_TfTgh.chine; src f\<^sub>0gh\<^sub>1.p\<^sub>1 = trg TTfgh_TfTgh.chine; src (tab\<^sub>1 f) = trg f\<^sub>0gh\<^sub>1.p\<^sub>1; \<guillemotleft>\<beta>\<^sub>f : tab\<^sub>1 f \<star> w\<^sub>f \<rightarrow> tab\<^sub>1 f \<star> w\<^sub>f'\<guillemotright>; \<beta>\<^sub>f = \<a>[tab\<^sub>1 f, f\<^sub>0gh\<^sub>1.p\<^sub>1, TTfgh_TfTgh.chine] \<cdot> TTfgh_TfTgh.the_\<nu> \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>1 f, f\<^sub>0g\<^sub>1.p\<^sub>1, fg\<^sub>0h\<^sub>1.p\<^sub>1]; \<And>f f'. \<lbrakk>local.iso f; local.iso f'; seq f' f\<rbrakk> \<Longrightarrow> local.iso (f' \<cdot> f)\<rbrakk> \<Longrightarrow> seq \<a>[tab\<^sub>1 f, f\<^sub>0gh\<^sub>1.p\<^sub>1, TTfgh_TfTgh.chine] (TTfgh_TfTgh.the_\<nu> \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>1 f, f\<^sub>0g\<^sub>1.p\<^sub>1, fg\<^sub>0h\<^sub>1.p\<^sub>1])
[PROOF STEP]
apply (metis TTfgh.composable TTfgh.leg1_in_hom(2) Tfg.\<rho>\<sigma>.T0.antipar(2)
Tfg.\<rho>\<sigma>.T0.ide_right Tfg.\<rho>\<sigma>.leg1_in_hom(2) Tfg_Hfg.u_simps(3)
f.T0.antipar(2) f.T0.ide_right f.ide_leg1 f\<^sub>0g\<^sub>1.cospan g.ide_leg1
h.ide_leg1 h.leg1_simps(4) hcomp_in_vhomE ide_hcomp
iso_assoc' tab\<^sub>1_simps(1))
[PROOF STATE]
proof (prove)
goal (4 subgoals):
1. \<lbrakk>ide f\<^sub>0gh\<^sub>1.p\<^sub>1; ide TTfgh_TfTgh.chine; src f\<^sub>0gh\<^sub>1.p\<^sub>1 = trg TTfgh_TfTgh.chine; src (tab\<^sub>1 f) = trg f\<^sub>0gh\<^sub>1.p\<^sub>1; \<guillemotleft>\<beta>\<^sub>f : tab\<^sub>1 f \<star> w\<^sub>f \<rightarrow> tab\<^sub>1 f \<star> w\<^sub>f'\<guillemotright>; \<beta>\<^sub>f = \<a>[tab\<^sub>1 f, f\<^sub>0gh\<^sub>1.p\<^sub>1, TTfgh_TfTgh.chine] \<cdot> TTfgh_TfTgh.the_\<nu> \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>1 f, f\<^sub>0g\<^sub>1.p\<^sub>1, fg\<^sub>0h\<^sub>1.p\<^sub>1]; \<And>f f'. \<lbrakk>local.iso f; local.iso f'; seq f' f\<rbrakk> \<Longrightarrow> local.iso (f' \<cdot> f)\<rbrakk> \<Longrightarrow> local.iso TTfgh_TfTgh.the_\<nu>
2. \<lbrakk>ide f\<^sub>0gh\<^sub>1.p\<^sub>1; ide TTfgh_TfTgh.chine; src f\<^sub>0gh\<^sub>1.p\<^sub>1 = trg TTfgh_TfTgh.chine; src (tab\<^sub>1 f) = trg f\<^sub>0gh\<^sub>1.p\<^sub>1; \<guillemotleft>\<beta>\<^sub>f : tab\<^sub>1 f \<star> w\<^sub>f \<rightarrow> tab\<^sub>1 f \<star> w\<^sub>f'\<guillemotright>; \<beta>\<^sub>f = \<a>[tab\<^sub>1 f, f\<^sub>0gh\<^sub>1.p\<^sub>1, TTfgh_TfTgh.chine] \<cdot> TTfgh_TfTgh.the_\<nu> \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>1 f, f\<^sub>0g\<^sub>1.p\<^sub>1, fg\<^sub>0h\<^sub>1.p\<^sub>1]; \<And>f f'. \<lbrakk>local.iso f; local.iso f'; seq f' f\<rbrakk> \<Longrightarrow> local.iso (f' \<cdot> f)\<rbrakk> \<Longrightarrow> seq TTfgh_TfTgh.the_\<nu> \<a>\<^sup>-\<^sup>1[tab\<^sub>1 f, f\<^sub>0g\<^sub>1.p\<^sub>1, fg\<^sub>0h\<^sub>1.p\<^sub>1]
3. \<lbrakk>ide f\<^sub>0gh\<^sub>1.p\<^sub>1; ide TTfgh_TfTgh.chine; src f\<^sub>0gh\<^sub>1.p\<^sub>1 = trg TTfgh_TfTgh.chine; src (tab\<^sub>1 f) = trg f\<^sub>0gh\<^sub>1.p\<^sub>1; \<guillemotleft>\<beta>\<^sub>f : tab\<^sub>1 f \<star> w\<^sub>f \<rightarrow> tab\<^sub>1 f \<star> w\<^sub>f'\<guillemotright>; \<beta>\<^sub>f = \<a>[tab\<^sub>1 f, f\<^sub>0gh\<^sub>1.p\<^sub>1, TTfgh_TfTgh.chine] \<cdot> TTfgh_TfTgh.the_\<nu> \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>1 f, f\<^sub>0g\<^sub>1.p\<^sub>1, fg\<^sub>0h\<^sub>1.p\<^sub>1]; \<And>f f'. \<lbrakk>local.iso f; local.iso f'; seq f' f\<rbrakk> \<Longrightarrow> local.iso (f' \<cdot> f)\<rbrakk> \<Longrightarrow> local.iso \<a>[tab\<^sub>1 f, f\<^sub>0gh\<^sub>1.p\<^sub>1, TTfgh_TfTgh.chine]
4. \<lbrakk>ide f\<^sub>0gh\<^sub>1.p\<^sub>1; ide TTfgh_TfTgh.chine; src f\<^sub>0gh\<^sub>1.p\<^sub>1 = trg TTfgh_TfTgh.chine; src (tab\<^sub>1 f) = trg f\<^sub>0gh\<^sub>1.p\<^sub>1; \<guillemotleft>\<beta>\<^sub>f : tab\<^sub>1 f \<star> w\<^sub>f \<rightarrow> tab\<^sub>1 f \<star> w\<^sub>f'\<guillemotright>; \<beta>\<^sub>f = \<a>[tab\<^sub>1 f, f\<^sub>0gh\<^sub>1.p\<^sub>1, TTfgh_TfTgh.chine] \<cdot> TTfgh_TfTgh.the_\<nu> \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>1 f, f\<^sub>0g\<^sub>1.p\<^sub>1, fg\<^sub>0h\<^sub>1.p\<^sub>1]; \<And>f f'. \<lbrakk>local.iso f; local.iso f'; seq f' f\<rbrakk> \<Longrightarrow> local.iso (f' \<cdot> f)\<rbrakk> \<Longrightarrow> seq \<a>[tab\<^sub>1 f, f\<^sub>0gh\<^sub>1.p\<^sub>1, TTfgh_TfTgh.chine] (TTfgh_TfTgh.the_\<nu> \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>1 f, f\<^sub>0g\<^sub>1.p\<^sub>1, fg\<^sub>0h\<^sub>1.p\<^sub>1])
[PROOF STEP]
using TTfgh_TfTgh.the_\<nu>_props(2) f.ide_leg1 iso_assoc
[PROOF STATE]
proof (prove)
using this:
local.iso TTfgh_TfTgh.the_\<nu>
ide (tab\<^sub>1 f)
\<lbrakk>ide ?f; ide ?g; ide ?h; src ?f = trg ?g; src ?g = trg ?h\<rbrakk> \<Longrightarrow> local.iso \<a>[?f, ?g, ?h]
goal (4 subgoals):
1. \<lbrakk>ide f\<^sub>0gh\<^sub>1.p\<^sub>1; ide TTfgh_TfTgh.chine; src f\<^sub>0gh\<^sub>1.p\<^sub>1 = trg TTfgh_TfTgh.chine; src (tab\<^sub>1 f) = trg f\<^sub>0gh\<^sub>1.p\<^sub>1; \<guillemotleft>\<beta>\<^sub>f : tab\<^sub>1 f \<star> w\<^sub>f \<rightarrow> tab\<^sub>1 f \<star> w\<^sub>f'\<guillemotright>; \<beta>\<^sub>f = \<a>[tab\<^sub>1 f, f\<^sub>0gh\<^sub>1.p\<^sub>1, TTfgh_TfTgh.chine] \<cdot> TTfgh_TfTgh.the_\<nu> \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>1 f, f\<^sub>0g\<^sub>1.p\<^sub>1, fg\<^sub>0h\<^sub>1.p\<^sub>1]; \<And>f f'. \<lbrakk>local.iso f; local.iso f'; seq f' f\<rbrakk> \<Longrightarrow> local.iso (f' \<cdot> f)\<rbrakk> \<Longrightarrow> local.iso TTfgh_TfTgh.the_\<nu>
2. \<lbrakk>ide f\<^sub>0gh\<^sub>1.p\<^sub>1; ide TTfgh_TfTgh.chine; src f\<^sub>0gh\<^sub>1.p\<^sub>1 = trg TTfgh_TfTgh.chine; src (tab\<^sub>1 f) = trg f\<^sub>0gh\<^sub>1.p\<^sub>1; \<guillemotleft>\<beta>\<^sub>f : tab\<^sub>1 f \<star> w\<^sub>f \<rightarrow> tab\<^sub>1 f \<star> w\<^sub>f'\<guillemotright>; \<beta>\<^sub>f = \<a>[tab\<^sub>1 f, f\<^sub>0gh\<^sub>1.p\<^sub>1, TTfgh_TfTgh.chine] \<cdot> TTfgh_TfTgh.the_\<nu> \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>1 f, f\<^sub>0g\<^sub>1.p\<^sub>1, fg\<^sub>0h\<^sub>1.p\<^sub>1]; \<And>f f'. \<lbrakk>local.iso f; local.iso f'; seq f' f\<rbrakk> \<Longrightarrow> local.iso (f' \<cdot> f)\<rbrakk> \<Longrightarrow> seq TTfgh_TfTgh.the_\<nu> \<a>\<^sup>-\<^sup>1[tab\<^sub>1 f, f\<^sub>0g\<^sub>1.p\<^sub>1, fg\<^sub>0h\<^sub>1.p\<^sub>1]
3. \<lbrakk>ide f\<^sub>0gh\<^sub>1.p\<^sub>1; ide TTfgh_TfTgh.chine; src f\<^sub>0gh\<^sub>1.p\<^sub>1 = trg TTfgh_TfTgh.chine; src (tab\<^sub>1 f) = trg f\<^sub>0gh\<^sub>1.p\<^sub>1; \<guillemotleft>\<beta>\<^sub>f : tab\<^sub>1 f \<star> w\<^sub>f \<rightarrow> tab\<^sub>1 f \<star> w\<^sub>f'\<guillemotright>; \<beta>\<^sub>f = \<a>[tab\<^sub>1 f, f\<^sub>0gh\<^sub>1.p\<^sub>1, TTfgh_TfTgh.chine] \<cdot> TTfgh_TfTgh.the_\<nu> \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>1 f, f\<^sub>0g\<^sub>1.p\<^sub>1, fg\<^sub>0h\<^sub>1.p\<^sub>1]; \<And>f f'. \<lbrakk>local.iso f; local.iso f'; seq f' f\<rbrakk> \<Longrightarrow> local.iso (f' \<cdot> f)\<rbrakk> \<Longrightarrow> local.iso \<a>[tab\<^sub>1 f, f\<^sub>0gh\<^sub>1.p\<^sub>1, TTfgh_TfTgh.chine]
4. \<lbrakk>ide f\<^sub>0gh\<^sub>1.p\<^sub>1; ide TTfgh_TfTgh.chine; src f\<^sub>0gh\<^sub>1.p\<^sub>1 = trg TTfgh_TfTgh.chine; src (tab\<^sub>1 f) = trg f\<^sub>0gh\<^sub>1.p\<^sub>1; \<guillemotleft>\<beta>\<^sub>f : tab\<^sub>1 f \<star> w\<^sub>f \<rightarrow> tab\<^sub>1 f \<star> w\<^sub>f'\<guillemotright>; \<beta>\<^sub>f = \<a>[tab\<^sub>1 f, f\<^sub>0gh\<^sub>1.p\<^sub>1, TTfgh_TfTgh.chine] \<cdot> TTfgh_TfTgh.the_\<nu> \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>1 f, f\<^sub>0g\<^sub>1.p\<^sub>1, fg\<^sub>0h\<^sub>1.p\<^sub>1]; \<And>f f'. \<lbrakk>local.iso f; local.iso f'; seq f' f\<rbrakk> \<Longrightarrow> local.iso (f' \<cdot> f)\<rbrakk> \<Longrightarrow> seq \<a>[tab\<^sub>1 f, f\<^sub>0gh\<^sub>1.p\<^sub>1, TTfgh_TfTgh.chine] (TTfgh_TfTgh.the_\<nu> \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>1 f, f\<^sub>0g\<^sub>1.p\<^sub>1, fg\<^sub>0h\<^sub>1.p\<^sub>1])
[PROOF STEP]
by blast+
[PROOF STATE]
proof (state)
this:
local.iso \<beta>\<^sub>f
goal (3 subgoals):
1. \<lbrakk>\<lbrakk>f\<^sub>0gh\<^sub>1.p\<^sub>1 \<star> TTfgh_TfTgh.chine\<rbrakk>\<rbrakk> = \<lbrakk>\<lbrakk>f\<^sub>0g\<^sub>1.p\<^sub>1 \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1\<rbrakk>\<rbrakk>
2. \<lbrakk>\<lbrakk>g\<^sub>0h\<^sub>1.p\<^sub>1 \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine\<rbrakk>\<rbrakk> = \<lbrakk>\<lbrakk>f\<^sub>0g\<^sub>1.p\<^sub>0 \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1\<rbrakk>\<rbrakk>
3. \<lbrakk>\<lbrakk>g\<^sub>0h\<^sub>1.p\<^sub>0 \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine\<rbrakk>\<rbrakk> = \<lbrakk>\<lbrakk>fg\<^sub>0h\<^sub>1.p\<^sub>0\<rbrakk>\<rbrakk>
[PROOF STEP]
have u\<^sub>f: "ide u\<^sub>f"
[PROOF STATE]
proof (prove)
goal (1 subgoal):
1. ide u\<^sub>f
[PROOF STEP]
using \<theta>\<^sub>f ide_cod
[PROOF STATE]
proof (prove)
using this:
\<guillemotleft>\<theta>\<^sub>f : tab\<^sub>0 f \<star> w\<^sub>f \<rightarrow> u\<^sub>f\<guillemotright>
arr ?f \<Longrightarrow> ide (cod ?f)
goal (1 subgoal):
1. ide u\<^sub>f
[PROOF STEP]
by blast
[PROOF STATE]
proof (state)
this:
ide u\<^sub>f
goal (3 subgoals):
1. \<lbrakk>\<lbrakk>f\<^sub>0gh\<^sub>1.p\<^sub>1 \<star> TTfgh_TfTgh.chine\<rbrakk>\<rbrakk> = \<lbrakk>\<lbrakk>f\<^sub>0g\<^sub>1.p\<^sub>1 \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1\<rbrakk>\<rbrakk>
2. \<lbrakk>\<lbrakk>g\<^sub>0h\<^sub>1.p\<^sub>1 \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine\<rbrakk>\<rbrakk> = \<lbrakk>\<lbrakk>f\<^sub>0g\<^sub>1.p\<^sub>0 \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1\<rbrakk>\<rbrakk>
3. \<lbrakk>\<lbrakk>g\<^sub>0h\<^sub>1.p\<^sub>0 \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine\<rbrakk>\<rbrakk> = \<lbrakk>\<lbrakk>fg\<^sub>0h\<^sub>1.p\<^sub>0\<rbrakk>\<rbrakk>
[PROOF STEP]
have w\<^sub>f_in_hhom: "in_hhom w\<^sub>f (src u\<^sub>f) (src (tab\<^sub>0 f))"
[PROOF STATE]
proof (prove)
goal (1 subgoal):
1. \<guillemotleft>w\<^sub>f : src u\<^sub>f \<rightarrow> src (tab\<^sub>0 f)\<guillemotright>
[PROOF STEP]
using u\<^sub>f w\<^sub>f u\<^sub>f_def w\<^sub>f_def
[PROOF STATE]
proof (prove)
using this:
ide u\<^sub>f
ide w\<^sub>f
u\<^sub>f = g \<star> h \<star> tab\<^sub>0 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0
w\<^sub>f = f\<^sub>0g\<^sub>1.p\<^sub>1 \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1
goal (1 subgoal):
1. \<guillemotleft>w\<^sub>f : src u\<^sub>f \<rightarrow> src (tab\<^sub>0 f)\<guillemotright>
[PROOF STEP]
by simp
[PROOF STATE]
proof (state)
this:
\<guillemotleft>w\<^sub>f : src u\<^sub>f \<rightarrow> src (tab\<^sub>0 f)\<guillemotright>
goal (3 subgoals):
1. \<lbrakk>\<lbrakk>f\<^sub>0gh\<^sub>1.p\<^sub>1 \<star> TTfgh_TfTgh.chine\<rbrakk>\<rbrakk> = \<lbrakk>\<lbrakk>f\<^sub>0g\<^sub>1.p\<^sub>1 \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1\<rbrakk>\<rbrakk>
2. \<lbrakk>\<lbrakk>g\<^sub>0h\<^sub>1.p\<^sub>1 \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine\<rbrakk>\<rbrakk> = \<lbrakk>\<lbrakk>f\<^sub>0g\<^sub>1.p\<^sub>0 \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1\<rbrakk>\<rbrakk>
3. \<lbrakk>\<lbrakk>g\<^sub>0h\<^sub>1.p\<^sub>0 \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine\<rbrakk>\<rbrakk> = \<lbrakk>\<lbrakk>fg\<^sub>0h\<^sub>1.p\<^sub>0\<rbrakk>\<rbrakk>
[PROOF STEP]
have w\<^sub>f'_in_hhom: "in_hhom w\<^sub>f' (src u\<^sub>f) (src (tab\<^sub>0 f))"
[PROOF STATE]
proof (prove)
goal (1 subgoal):
1. \<guillemotleft>w\<^sub>f' : src u\<^sub>f \<rightarrow> src (tab\<^sub>0 f)\<guillemotright>
[PROOF STEP]
using u\<^sub>f w\<^sub>f' w\<^sub>f'_def u\<^sub>f_def
[PROOF STATE]
proof (prove)
using this:
ide u\<^sub>f
ide w\<^sub>f'
w\<^sub>f' = f\<^sub>0gh\<^sub>1.p\<^sub>1 \<star> TTfgh_TfTgh.chine
u\<^sub>f = g \<star> h \<star> tab\<^sub>0 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0
goal (1 subgoal):
1. \<guillemotleft>w\<^sub>f' : src u\<^sub>f \<rightarrow> src (tab\<^sub>0 f)\<guillemotright>
[PROOF STEP]
by simp
[PROOF STATE]
proof (state)
this:
\<guillemotleft>w\<^sub>f' : src u\<^sub>f \<rightarrow> src (tab\<^sub>0 f)\<guillemotright>
goal (3 subgoals):
1. \<lbrakk>\<lbrakk>f\<^sub>0gh\<^sub>1.p\<^sub>1 \<star> TTfgh_TfTgh.chine\<rbrakk>\<rbrakk> = \<lbrakk>\<lbrakk>f\<^sub>0g\<^sub>1.p\<^sub>1 \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1\<rbrakk>\<rbrakk>
2. \<lbrakk>\<lbrakk>g\<^sub>0h\<^sub>1.p\<^sub>1 \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine\<rbrakk>\<rbrakk> = \<lbrakk>\<lbrakk>f\<^sub>0g\<^sub>1.p\<^sub>0 \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1\<rbrakk>\<rbrakk>
3. \<lbrakk>\<lbrakk>g\<^sub>0h\<^sub>1.p\<^sub>0 \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine\<rbrakk>\<rbrakk> = \<lbrakk>\<lbrakk>fg\<^sub>0h\<^sub>1.p\<^sub>0\<rbrakk>\<rbrakk>
[PROOF STEP]
have 5: "\<exists>!\<gamma>. \<guillemotleft>\<gamma> : w\<^sub>f \<Rightarrow> w\<^sub>f'\<guillemotright> \<and> \<beta>\<^sub>f = tab\<^sub>1 f \<star> \<gamma> \<and> \<theta>\<^sub>f = \<theta>\<^sub>f' \<cdot> (tab\<^sub>0 f \<star> \<gamma>)"
[PROOF STATE]
proof (prove)
goal (1 subgoal):
1. \<exists>!\<gamma>. \<guillemotleft>\<gamma> : w\<^sub>f \<rightarrow> w\<^sub>f'\<guillemotright> \<and> \<beta>\<^sub>f = tab\<^sub>1 f \<star> \<gamma> \<and> \<theta>\<^sub>f = \<theta>\<^sub>f' \<cdot> (tab\<^sub>0 f \<star> \<gamma>)
[PROOF STEP]
proof -
[PROOF STATE]
proof (state)
goal (1 subgoal):
1. \<exists>!\<gamma>. \<guillemotleft>\<gamma> : w\<^sub>f \<rightarrow> w\<^sub>f'\<guillemotright> \<and> \<beta>\<^sub>f = tab\<^sub>1 f \<star> \<gamma> \<and> \<theta>\<^sub>f = \<theta>\<^sub>f' \<cdot> (tab\<^sub>0 f \<star> \<gamma>)
[PROOF STEP]
have eq\<^sub>f: "f.composite_cell w\<^sub>f \<theta>\<^sub>f = f.composite_cell w\<^sub>f' \<theta>\<^sub>f' \<cdot> \<beta>\<^sub>f"
[PROOF STATE]
proof (prove)
goal (1 subgoal):
1. (f \<star> \<theta>\<^sub>f) \<cdot> \<a>[f, tab\<^sub>0 f, w\<^sub>f] \<cdot> (f.tab \<star> w\<^sub>f) = ((f \<star> \<theta>\<^sub>f') \<cdot> \<a>[f, tab\<^sub>0 f, w\<^sub>f'] \<cdot> (f.tab \<star> w\<^sub>f')) \<cdot> \<beta>\<^sub>f
[PROOF STEP]
proof -
[PROOF STATE]
proof (state)
goal (1 subgoal):
1. (f \<star> \<theta>\<^sub>f) \<cdot> \<a>[f, tab\<^sub>0 f, w\<^sub>f] \<cdot> (f.tab \<star> w\<^sub>f) = ((f \<star> \<theta>\<^sub>f') \<cdot> \<a>[f, tab\<^sub>0 f, w\<^sub>f'] \<cdot> (f.tab \<star> w\<^sub>f')) \<cdot> \<beta>\<^sub>f
[PROOF STEP]
have "f.composite_cell w\<^sub>f \<theta>\<^sub>f =
((f \<star> g \<star> \<a>[h, tab\<^sub>0 h, fg\<^sub>0h\<^sub>1.p\<^sub>0]) \<cdot>
(f \<star> g \<star> h.tab \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0) \<cdot>
(f \<star> g \<star> fg\<^sub>0h\<^sub>1.\<phi>) \<cdot>
(f \<star> \<a>[g, tab\<^sub>0 g \<star> f\<^sub>0g\<^sub>1.p\<^sub>0, fg\<^sub>0h\<^sub>1.p\<^sub>1]) \<cdot>
(f \<star> \<a>[g, tab\<^sub>0 g, f\<^sub>0g\<^sub>1.p\<^sub>0] \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot>
(f \<star> (g.tab \<star> f\<^sub>0g\<^sub>1.p\<^sub>0) \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot>
(f \<star> f\<^sub>0g\<^sub>1.\<phi> \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot>
(f \<star> \<a>\<^sup>-\<^sup>1[tab\<^sub>0 f, f\<^sub>0g\<^sub>1.p\<^sub>1, fg\<^sub>0h\<^sub>1.p\<^sub>1])) \<cdot>
\<a>[f, tab\<^sub>0 f, f\<^sub>0g\<^sub>1.p\<^sub>1 \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1] \<cdot>
(f.tab \<star> f\<^sub>0g\<^sub>1.p\<^sub>1 \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1)"
[PROOF STATE]
proof (prove)
goal (1 subgoal):
1. (f \<star> \<theta>\<^sub>f) \<cdot> \<a>[f, tab\<^sub>0 f, w\<^sub>f] \<cdot> (f.tab \<star> w\<^sub>f) = ((f \<star> g \<star> \<a>[h, tab\<^sub>0 h, fg\<^sub>0h\<^sub>1.p\<^sub>0]) \<cdot> (f \<star> g \<star> h.tab \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0) \<cdot> (f \<star> g \<star> fg\<^sub>0h\<^sub>1.\<phi>) \<cdot> (f \<star> \<a>[g, tab\<^sub>0 g \<star> f\<^sub>0g\<^sub>1.p\<^sub>0, fg\<^sub>0h\<^sub>1.p\<^sub>1]) \<cdot> (f \<star> \<a>[g, tab\<^sub>0 g, f\<^sub>0g\<^sub>1.p\<^sub>0] \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> (f \<star> (g.tab \<star> f\<^sub>0g\<^sub>1.p\<^sub>0) \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> (f \<star> f\<^sub>0g\<^sub>1.\<phi> \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> (f \<star> \<a>\<^sup>-\<^sup>1[tab\<^sub>0 f, f\<^sub>0g\<^sub>1.p\<^sub>1, fg\<^sub>0h\<^sub>1.p\<^sub>1])) \<cdot> \<a>[f, tab\<^sub>0 f, f\<^sub>0g\<^sub>1.p\<^sub>1 \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1] \<cdot> (f.tab \<star> f\<^sub>0g\<^sub>1.p\<^sub>1 \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1)
[PROOF STEP]
unfolding w\<^sub>f_def \<theta>\<^sub>f_def
[PROOF STATE]
proof (prove)
goal (1 subgoal):
1. (f \<star> (g \<star> \<a>[h, tab\<^sub>0 h, fg\<^sub>0h\<^sub>1.p\<^sub>0]) \<cdot> (g \<star> h.tab \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0) \<cdot> (g \<star> fg\<^sub>0h\<^sub>1.\<phi>) \<cdot> \<a>[g, tab\<^sub>0 g \<star> f\<^sub>0g\<^sub>1.p\<^sub>0, fg\<^sub>0h\<^sub>1.p\<^sub>1] \<cdot> (\<a>[g, tab\<^sub>0 g, f\<^sub>0g\<^sub>1.p\<^sub>0] \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> ((g.tab \<star> f\<^sub>0g\<^sub>1.p\<^sub>0) \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> (f\<^sub>0g\<^sub>1.\<phi> \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>0 f, f\<^sub>0g\<^sub>1.p\<^sub>1, fg\<^sub>0h\<^sub>1.p\<^sub>1]) \<cdot> \<a>[f, tab\<^sub>0 f, f\<^sub>0g\<^sub>1.p\<^sub>1 \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1] \<cdot> (f.tab \<star> f\<^sub>0g\<^sub>1.p\<^sub>1 \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) = ((f \<star> g \<star> \<a>[h, tab\<^sub>0 h, fg\<^sub>0h\<^sub>1.p\<^sub>0]) \<cdot> (f \<star> g \<star> h.tab \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0) \<cdot> (f \<star> g \<star> fg\<^sub>0h\<^sub>1.\<phi>) \<cdot> (f \<star> \<a>[g, tab\<^sub>0 g \<star> f\<^sub>0g\<^sub>1.p\<^sub>0, fg\<^sub>0h\<^sub>1.p\<^sub>1]) \<cdot> (f \<star> \<a>[g, tab\<^sub>0 g, f\<^sub>0g\<^sub>1.p\<^sub>0] \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> (f \<star> (g.tab \<star> f\<^sub>0g\<^sub>1.p\<^sub>0) \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> (f \<star> f\<^sub>0g\<^sub>1.\<phi> \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> (f \<star> \<a>\<^sup>-\<^sup>1[tab\<^sub>0 f, f\<^sub>0g\<^sub>1.p\<^sub>1, fg\<^sub>0h\<^sub>1.p\<^sub>1])) \<cdot> \<a>[f, tab\<^sub>0 f, f\<^sub>0g\<^sub>1.p\<^sub>1 \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1] \<cdot> (f.tab \<star> f\<^sub>0g\<^sub>1.p\<^sub>1 \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1)
[PROOF STEP]
using fg\<^sub>0h\<^sub>1.p\<^sub>1_simps Tgh.composable whisker_left
[PROOF STATE]
proof (prove)
using this:
trg fg\<^sub>0h\<^sub>1.p\<^sub>1 = src (tab\<^sub>0 g \<star> f\<^sub>0g\<^sub>1.p\<^sub>0)
src g = trg h
\<lbrakk>ide ?f; seq ?\<nu> ?\<mu>\<rbrakk> \<Longrightarrow> ?f \<star> ?\<nu> \<cdot> ?\<mu> = (?f \<star> ?\<nu>) \<cdot> (?f \<star> ?\<mu>)
goal (1 subgoal):
1. (f \<star> (g \<star> \<a>[h, tab\<^sub>0 h, fg\<^sub>0h\<^sub>1.p\<^sub>0]) \<cdot> (g \<star> h.tab \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0) \<cdot> (g \<star> fg\<^sub>0h\<^sub>1.\<phi>) \<cdot> \<a>[g, tab\<^sub>0 g \<star> f\<^sub>0g\<^sub>1.p\<^sub>0, fg\<^sub>0h\<^sub>1.p\<^sub>1] \<cdot> (\<a>[g, tab\<^sub>0 g, f\<^sub>0g\<^sub>1.p\<^sub>0] \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> ((g.tab \<star> f\<^sub>0g\<^sub>1.p\<^sub>0) \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> (f\<^sub>0g\<^sub>1.\<phi> \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>0 f, f\<^sub>0g\<^sub>1.p\<^sub>1, fg\<^sub>0h\<^sub>1.p\<^sub>1]) \<cdot> \<a>[f, tab\<^sub>0 f, f\<^sub>0g\<^sub>1.p\<^sub>1 \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1] \<cdot> (f.tab \<star> f\<^sub>0g\<^sub>1.p\<^sub>1 \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) = ((f \<star> g \<star> \<a>[h, tab\<^sub>0 h, fg\<^sub>0h\<^sub>1.p\<^sub>0]) \<cdot> (f \<star> g \<star> h.tab \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0) \<cdot> (f \<star> g \<star> fg\<^sub>0h\<^sub>1.\<phi>) \<cdot> (f \<star> \<a>[g, tab\<^sub>0 g \<star> f\<^sub>0g\<^sub>1.p\<^sub>0, fg\<^sub>0h\<^sub>1.p\<^sub>1]) \<cdot> (f \<star> \<a>[g, tab\<^sub>0 g, f\<^sub>0g\<^sub>1.p\<^sub>0] \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> (f \<star> (g.tab \<star> f\<^sub>0g\<^sub>1.p\<^sub>0) \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> (f \<star> f\<^sub>0g\<^sub>1.\<phi> \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> (f \<star> \<a>\<^sup>-\<^sup>1[tab\<^sub>0 f, f\<^sub>0g\<^sub>1.p\<^sub>1, fg\<^sub>0h\<^sub>1.p\<^sub>1])) \<cdot> \<a>[f, tab\<^sub>0 f, f\<^sub>0g\<^sub>1.p\<^sub>1 \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1] \<cdot> (f.tab \<star> f\<^sub>0g\<^sub>1.p\<^sub>1 \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1)
[PROOF STEP]
by simp
[PROOF STATE]
proof (state)
this:
(f \<star> \<theta>\<^sub>f) \<cdot> \<a>[f, tab\<^sub>0 f, w\<^sub>f] \<cdot> (f.tab \<star> w\<^sub>f) = ((f \<star> g \<star> \<a>[h, tab\<^sub>0 h, fg\<^sub>0h\<^sub>1.p\<^sub>0]) \<cdot> (f \<star> g \<star> h.tab \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0) \<cdot> (f \<star> g \<star> fg\<^sub>0h\<^sub>1.\<phi>) \<cdot> (f \<star> \<a>[g, tab\<^sub>0 g \<star> f\<^sub>0g\<^sub>1.p\<^sub>0, fg\<^sub>0h\<^sub>1.p\<^sub>1]) \<cdot> (f \<star> \<a>[g, tab\<^sub>0 g, f\<^sub>0g\<^sub>1.p\<^sub>0] \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> (f \<star> (g.tab \<star> f\<^sub>0g\<^sub>1.p\<^sub>0) \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> (f \<star> f\<^sub>0g\<^sub>1.\<phi> \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> (f \<star> \<a>\<^sup>-\<^sup>1[tab\<^sub>0 f, f\<^sub>0g\<^sub>1.p\<^sub>1, fg\<^sub>0h\<^sub>1.p\<^sub>1])) \<cdot> \<a>[f, tab\<^sub>0 f, f\<^sub>0g\<^sub>1.p\<^sub>1 \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1] \<cdot> (f.tab \<star> f\<^sub>0g\<^sub>1.p\<^sub>1 \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1)
goal (1 subgoal):
1. (f \<star> \<theta>\<^sub>f) \<cdot> \<a>[f, tab\<^sub>0 f, w\<^sub>f] \<cdot> (f.tab \<star> w\<^sub>f) = ((f \<star> \<theta>\<^sub>f') \<cdot> \<a>[f, tab\<^sub>0 f, w\<^sub>f'] \<cdot> (f.tab \<star> w\<^sub>f')) \<cdot> \<beta>\<^sub>f
[PROOF STEP]
also
[PROOF STATE]
proof (state)
this:
(f \<star> \<theta>\<^sub>f) \<cdot> \<a>[f, tab\<^sub>0 f, w\<^sub>f] \<cdot> (f.tab \<star> w\<^sub>f) = ((f \<star> g \<star> \<a>[h, tab\<^sub>0 h, fg\<^sub>0h\<^sub>1.p\<^sub>0]) \<cdot> (f \<star> g \<star> h.tab \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0) \<cdot> (f \<star> g \<star> fg\<^sub>0h\<^sub>1.\<phi>) \<cdot> (f \<star> \<a>[g, tab\<^sub>0 g \<star> f\<^sub>0g\<^sub>1.p\<^sub>0, fg\<^sub>0h\<^sub>1.p\<^sub>1]) \<cdot> (f \<star> \<a>[g, tab\<^sub>0 g, f\<^sub>0g\<^sub>1.p\<^sub>0] \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> (f \<star> (g.tab \<star> f\<^sub>0g\<^sub>1.p\<^sub>0) \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> (f \<star> f\<^sub>0g\<^sub>1.\<phi> \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> (f \<star> \<a>\<^sup>-\<^sup>1[tab\<^sub>0 f, f\<^sub>0g\<^sub>1.p\<^sub>1, fg\<^sub>0h\<^sub>1.p\<^sub>1])) \<cdot> \<a>[f, tab\<^sub>0 f, f\<^sub>0g\<^sub>1.p\<^sub>1 \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1] \<cdot> (f.tab \<star> f\<^sub>0g\<^sub>1.p\<^sub>1 \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1)
goal (1 subgoal):
1. (f \<star> \<theta>\<^sub>f) \<cdot> \<a>[f, tab\<^sub>0 f, w\<^sub>f] \<cdot> (f.tab \<star> w\<^sub>f) = ((f \<star> \<theta>\<^sub>f') \<cdot> \<a>[f, tab\<^sub>0 f, w\<^sub>f'] \<cdot> (f.tab \<star> w\<^sub>f')) \<cdot> \<beta>\<^sub>f
[PROOF STEP]
have "... =
(f \<star> g \<star> \<a>[h, tab\<^sub>0 h, TTfgh.p\<^sub>0]) \<cdot>
(f \<star> g \<star> h.tab \<star> TTfgh.p\<^sub>0) \<cdot>
(f \<star> g \<star> fg\<^sub>0h\<^sub>1.\<phi>) \<cdot>
(f \<star> \<a>[g, tab\<^sub>0 g \<star> Tfg.p\<^sub>0, TTfgh.p\<^sub>1]) \<cdot>
(f \<star> \<a>[g, tab\<^sub>0 g, Tfg.p\<^sub>0] \<star> TTfgh.p\<^sub>1) \<cdot>
(f \<star> (g.tab \<star> Tfg.p\<^sub>0) \<star> TTfgh.p\<^sub>1) \<cdot>
(f \<star> f\<^sub>0g\<^sub>1.\<phi> \<star> TTfgh.p\<^sub>1) \<cdot>
(f \<star> \<a>\<^sup>-\<^sup>1[tab\<^sub>0 f, Tfg.p\<^sub>1, TTfgh.p\<^sub>1]) \<cdot>
\<a>[f, tab\<^sub>0 f, Tfg.p\<^sub>1 \<star> TTfgh.p\<^sub>1] \<cdot>
(f.tab \<star> Tfg.p\<^sub>1 \<star> TTfgh.p\<^sub>1)"
[PROOF STATE]
proof (prove)
goal (1 subgoal):
1. ((f \<star> g \<star> \<a>[h, tab\<^sub>0 h, fg\<^sub>0h\<^sub>1.p\<^sub>0]) \<cdot> (f \<star> g \<star> h.tab \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0) \<cdot> (f \<star> g \<star> fg\<^sub>0h\<^sub>1.\<phi>) \<cdot> (f \<star> \<a>[g, tab\<^sub>0 g \<star> f\<^sub>0g\<^sub>1.p\<^sub>0, fg\<^sub>0h\<^sub>1.p\<^sub>1]) \<cdot> (f \<star> \<a>[g, tab\<^sub>0 g, f\<^sub>0g\<^sub>1.p\<^sub>0] \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> (f \<star> (g.tab \<star> f\<^sub>0g\<^sub>1.p\<^sub>0) \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> (f \<star> f\<^sub>0g\<^sub>1.\<phi> \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> (f \<star> \<a>\<^sup>-\<^sup>1[tab\<^sub>0 f, f\<^sub>0g\<^sub>1.p\<^sub>1, fg\<^sub>0h\<^sub>1.p\<^sub>1])) \<cdot> \<a>[f, tab\<^sub>0 f, f\<^sub>0g\<^sub>1.p\<^sub>1 \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1] \<cdot> (f.tab \<star> f\<^sub>0g\<^sub>1.p\<^sub>1 \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) = (f \<star> g \<star> \<a>[h, tab\<^sub>0 h, fg\<^sub>0h\<^sub>1.p\<^sub>0]) \<cdot> (f \<star> g \<star> h.tab \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0) \<cdot> (f \<star> g \<star> fg\<^sub>0h\<^sub>1.\<phi>) \<cdot> (f \<star> \<a>[g, tab\<^sub>0 g \<star> f\<^sub>0g\<^sub>1.p\<^sub>0, fg\<^sub>0h\<^sub>1.p\<^sub>1]) \<cdot> (f \<star> \<a>[g, tab\<^sub>0 g, f\<^sub>0g\<^sub>1.p\<^sub>0] \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> (f \<star> (g.tab \<star> f\<^sub>0g\<^sub>1.p\<^sub>0) \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> (f \<star> f\<^sub>0g\<^sub>1.\<phi> \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> (f \<star> \<a>\<^sup>-\<^sup>1[tab\<^sub>0 f, f\<^sub>0g\<^sub>1.p\<^sub>1, fg\<^sub>0h\<^sub>1.p\<^sub>1]) \<cdot> \<a>[f, tab\<^sub>0 f, f\<^sub>0g\<^sub>1.p\<^sub>1 \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1] \<cdot> (f.tab \<star> f\<^sub>0g\<^sub>1.p\<^sub>1 \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1)
[PROOF STEP]
using comp_assoc
[PROOF STATE]
proof (prove)
using this:
(?h \<cdot> ?g) \<cdot> ?f = ?h \<cdot> ?g \<cdot> ?f
goal (1 subgoal):
1. ((f \<star> g \<star> \<a>[h, tab\<^sub>0 h, fg\<^sub>0h\<^sub>1.p\<^sub>0]) \<cdot> (f \<star> g \<star> h.tab \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0) \<cdot> (f \<star> g \<star> fg\<^sub>0h\<^sub>1.\<phi>) \<cdot> (f \<star> \<a>[g, tab\<^sub>0 g \<star> f\<^sub>0g\<^sub>1.p\<^sub>0, fg\<^sub>0h\<^sub>1.p\<^sub>1]) \<cdot> (f \<star> \<a>[g, tab\<^sub>0 g, f\<^sub>0g\<^sub>1.p\<^sub>0] \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> (f \<star> (g.tab \<star> f\<^sub>0g\<^sub>1.p\<^sub>0) \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> (f \<star> f\<^sub>0g\<^sub>1.\<phi> \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> (f \<star> \<a>\<^sup>-\<^sup>1[tab\<^sub>0 f, f\<^sub>0g\<^sub>1.p\<^sub>1, fg\<^sub>0h\<^sub>1.p\<^sub>1])) \<cdot> \<a>[f, tab\<^sub>0 f, f\<^sub>0g\<^sub>1.p\<^sub>1 \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1] \<cdot> (f.tab \<star> f\<^sub>0g\<^sub>1.p\<^sub>1 \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) = (f \<star> g \<star> \<a>[h, tab\<^sub>0 h, fg\<^sub>0h\<^sub>1.p\<^sub>0]) \<cdot> (f \<star> g \<star> h.tab \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0) \<cdot> (f \<star> g \<star> fg\<^sub>0h\<^sub>1.\<phi>) \<cdot> (f \<star> \<a>[g, tab\<^sub>0 g \<star> f\<^sub>0g\<^sub>1.p\<^sub>0, fg\<^sub>0h\<^sub>1.p\<^sub>1]) \<cdot> (f \<star> \<a>[g, tab\<^sub>0 g, f\<^sub>0g\<^sub>1.p\<^sub>0] \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> (f \<star> (g.tab \<star> f\<^sub>0g\<^sub>1.p\<^sub>0) \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> (f \<star> f\<^sub>0g\<^sub>1.\<phi> \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> (f \<star> \<a>\<^sup>-\<^sup>1[tab\<^sub>0 f, f\<^sub>0g\<^sub>1.p\<^sub>1, fg\<^sub>0h\<^sub>1.p\<^sub>1]) \<cdot> \<a>[f, tab\<^sub>0 f, f\<^sub>0g\<^sub>1.p\<^sub>1 \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1] \<cdot> (f.tab \<star> f\<^sub>0g\<^sub>1.p\<^sub>1 \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1)
[PROOF STEP]
by simp
[PROOF STATE]
proof (state)
this:
((f \<star> g \<star> \<a>[h, tab\<^sub>0 h, fg\<^sub>0h\<^sub>1.p\<^sub>0]) \<cdot> (f \<star> g \<star> h.tab \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0) \<cdot> (f \<star> g \<star> fg\<^sub>0h\<^sub>1.\<phi>) \<cdot> (f \<star> \<a>[g, tab\<^sub>0 g \<star> f\<^sub>0g\<^sub>1.p\<^sub>0, fg\<^sub>0h\<^sub>1.p\<^sub>1]) \<cdot> (f \<star> \<a>[g, tab\<^sub>0 g, f\<^sub>0g\<^sub>1.p\<^sub>0] \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> (f \<star> (g.tab \<star> f\<^sub>0g\<^sub>1.p\<^sub>0) \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> (f \<star> f\<^sub>0g\<^sub>1.\<phi> \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> (f \<star> \<a>\<^sup>-\<^sup>1[tab\<^sub>0 f, f\<^sub>0g\<^sub>1.p\<^sub>1, fg\<^sub>0h\<^sub>1.p\<^sub>1])) \<cdot> \<a>[f, tab\<^sub>0 f, f\<^sub>0g\<^sub>1.p\<^sub>1 \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1] \<cdot> (f.tab \<star> f\<^sub>0g\<^sub>1.p\<^sub>1 \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) = (f \<star> g \<star> \<a>[h, tab\<^sub>0 h, fg\<^sub>0h\<^sub>1.p\<^sub>0]) \<cdot> (f \<star> g \<star> h.tab \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0) \<cdot> (f \<star> g \<star> fg\<^sub>0h\<^sub>1.\<phi>) \<cdot> (f \<star> \<a>[g, tab\<^sub>0 g \<star> f\<^sub>0g\<^sub>1.p\<^sub>0, fg\<^sub>0h\<^sub>1.p\<^sub>1]) \<cdot> (f \<star> \<a>[g, tab\<^sub>0 g, f\<^sub>0g\<^sub>1.p\<^sub>0] \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> (f \<star> (g.tab \<star> f\<^sub>0g\<^sub>1.p\<^sub>0) \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> (f \<star> f\<^sub>0g\<^sub>1.\<phi> \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> (f \<star> \<a>\<^sup>-\<^sup>1[tab\<^sub>0 f, f\<^sub>0g\<^sub>1.p\<^sub>1, fg\<^sub>0h\<^sub>1.p\<^sub>1]) \<cdot> \<a>[f, tab\<^sub>0 f, f\<^sub>0g\<^sub>1.p\<^sub>1 \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1] \<cdot> (f.tab \<star> f\<^sub>0g\<^sub>1.p\<^sub>1 \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1)
goal (1 subgoal):
1. (f \<star> \<theta>\<^sub>f) \<cdot> \<a>[f, tab\<^sub>0 f, w\<^sub>f] \<cdot> (f.tab \<star> w\<^sub>f) = ((f \<star> \<theta>\<^sub>f') \<cdot> \<a>[f, tab\<^sub>0 f, w\<^sub>f'] \<cdot> (f.tab \<star> w\<^sub>f')) \<cdot> \<beta>\<^sub>f
[PROOF STEP]
also
[PROOF STATE]
proof (state)
this:
((f \<star> g \<star> \<a>[h, tab\<^sub>0 h, fg\<^sub>0h\<^sub>1.p\<^sub>0]) \<cdot> (f \<star> g \<star> h.tab \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0) \<cdot> (f \<star> g \<star> fg\<^sub>0h\<^sub>1.\<phi>) \<cdot> (f \<star> \<a>[g, tab\<^sub>0 g \<star> f\<^sub>0g\<^sub>1.p\<^sub>0, fg\<^sub>0h\<^sub>1.p\<^sub>1]) \<cdot> (f \<star> \<a>[g, tab\<^sub>0 g, f\<^sub>0g\<^sub>1.p\<^sub>0] \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> (f \<star> (g.tab \<star> f\<^sub>0g\<^sub>1.p\<^sub>0) \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> (f \<star> f\<^sub>0g\<^sub>1.\<phi> \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> (f \<star> \<a>\<^sup>-\<^sup>1[tab\<^sub>0 f, f\<^sub>0g\<^sub>1.p\<^sub>1, fg\<^sub>0h\<^sub>1.p\<^sub>1])) \<cdot> \<a>[f, tab\<^sub>0 f, f\<^sub>0g\<^sub>1.p\<^sub>1 \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1] \<cdot> (f.tab \<star> f\<^sub>0g\<^sub>1.p\<^sub>1 \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) = (f \<star> g \<star> \<a>[h, tab\<^sub>0 h, fg\<^sub>0h\<^sub>1.p\<^sub>0]) \<cdot> (f \<star> g \<star> h.tab \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0) \<cdot> (f \<star> g \<star> fg\<^sub>0h\<^sub>1.\<phi>) \<cdot> (f \<star> \<a>[g, tab\<^sub>0 g \<star> f\<^sub>0g\<^sub>1.p\<^sub>0, fg\<^sub>0h\<^sub>1.p\<^sub>1]) \<cdot> (f \<star> \<a>[g, tab\<^sub>0 g, f\<^sub>0g\<^sub>1.p\<^sub>0] \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> (f \<star> (g.tab \<star> f\<^sub>0g\<^sub>1.p\<^sub>0) \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> (f \<star> f\<^sub>0g\<^sub>1.\<phi> \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> (f \<star> \<a>\<^sup>-\<^sup>1[tab\<^sub>0 f, f\<^sub>0g\<^sub>1.p\<^sub>1, fg\<^sub>0h\<^sub>1.p\<^sub>1]) \<cdot> \<a>[f, tab\<^sub>0 f, f\<^sub>0g\<^sub>1.p\<^sub>1 \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1] \<cdot> (f.tab \<star> f\<^sub>0g\<^sub>1.p\<^sub>1 \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1)
goal (1 subgoal):
1. (f \<star> \<theta>\<^sub>f) \<cdot> \<a>[f, tab\<^sub>0 f, w\<^sub>f] \<cdot> (f.tab \<star> w\<^sub>f) = ((f \<star> \<theta>\<^sub>f') \<cdot> \<a>[f, tab\<^sub>0 f, w\<^sub>f'] \<cdot> (f.tab \<star> w\<^sub>f')) \<cdot> \<beta>\<^sub>f
[PROOF STEP]
have "... =
(\<a>[f, g, h \<star> tab\<^sub>0 h \<star> TTfgh.p\<^sub>0] \<cdot>
\<a>\<^sup>-\<^sup>1[f, g, h \<star> tab\<^sub>0 h \<star> TTfgh.p\<^sub>0] \<cdot>
(f \<star> g \<star> \<a>[h, tab\<^sub>0 h, TTfgh.p\<^sub>0])) \<cdot>
(f \<star> g \<star> h.tab \<star> TTfgh.p\<^sub>0) \<cdot>
(f \<star> g \<star> fg\<^sub>0h\<^sub>1.\<phi>) \<cdot>
(f \<star> \<a>[g, tab\<^sub>0 g \<star> Tfg.p\<^sub>0, TTfgh.p\<^sub>1]) \<cdot>
(f \<star> \<a>[g, tab\<^sub>0 g, Tfg.p\<^sub>0] \<star> TTfgh.p\<^sub>1) \<cdot>
(f \<star> (g.tab \<star> Tfg.p\<^sub>0) \<star> TTfgh.p\<^sub>1) \<cdot>
(f \<star> f\<^sub>0g\<^sub>1.\<phi> \<star> TTfgh.p\<^sub>1) \<cdot>
(f \<star> \<a>\<^sup>-\<^sup>1[tab\<^sub>0 f, Tfg.p\<^sub>1, TTfgh.p\<^sub>1]) \<cdot>
\<a>[f, tab\<^sub>0 f, Tfg.p\<^sub>1 \<star> TTfgh.p\<^sub>1] \<cdot>
(f.tab \<star> Tfg.p\<^sub>1 \<star> TTfgh.p\<^sub>1)"
[PROOF STATE]
proof (prove)
goal (1 subgoal):
1. (f \<star> g \<star> \<a>[h, tab\<^sub>0 h, fg\<^sub>0h\<^sub>1.p\<^sub>0]) \<cdot> (f \<star> g \<star> h.tab \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0) \<cdot> (f \<star> g \<star> fg\<^sub>0h\<^sub>1.\<phi>) \<cdot> (f \<star> \<a>[g, tab\<^sub>0 g \<star> f\<^sub>0g\<^sub>1.p\<^sub>0, fg\<^sub>0h\<^sub>1.p\<^sub>1]) \<cdot> (f \<star> \<a>[g, tab\<^sub>0 g, f\<^sub>0g\<^sub>1.p\<^sub>0] \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> (f \<star> (g.tab \<star> f\<^sub>0g\<^sub>1.p\<^sub>0) \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> (f \<star> f\<^sub>0g\<^sub>1.\<phi> \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> (f \<star> \<a>\<^sup>-\<^sup>1[tab\<^sub>0 f, f\<^sub>0g\<^sub>1.p\<^sub>1, fg\<^sub>0h\<^sub>1.p\<^sub>1]) \<cdot> \<a>[f, tab\<^sub>0 f, f\<^sub>0g\<^sub>1.p\<^sub>1 \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1] \<cdot> (f.tab \<star> f\<^sub>0g\<^sub>1.p\<^sub>1 \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) = (\<a>[f, g, h \<star> tab\<^sub>0 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0] \<cdot> \<a>\<^sup>-\<^sup>1[f, g, h \<star> tab\<^sub>0 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0] \<cdot> (f \<star> g \<star> \<a>[h, tab\<^sub>0 h, fg\<^sub>0h\<^sub>1.p\<^sub>0])) \<cdot> (f \<star> g \<star> h.tab \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0) \<cdot> (f \<star> g \<star> fg\<^sub>0h\<^sub>1.\<phi>) \<cdot> (f \<star> \<a>[g, tab\<^sub>0 g \<star> f\<^sub>0g\<^sub>1.p\<^sub>0, fg\<^sub>0h\<^sub>1.p\<^sub>1]) \<cdot> (f \<star> \<a>[g, tab\<^sub>0 g, f\<^sub>0g\<^sub>1.p\<^sub>0] \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> (f \<star> (g.tab \<star> f\<^sub>0g\<^sub>1.p\<^sub>0) \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> (f \<star> f\<^sub>0g\<^sub>1.\<phi> \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> (f \<star> \<a>\<^sup>-\<^sup>1[tab\<^sub>0 f, f\<^sub>0g\<^sub>1.p\<^sub>1, fg\<^sub>0h\<^sub>1.p\<^sub>1]) \<cdot> \<a>[f, tab\<^sub>0 f, f\<^sub>0g\<^sub>1.p\<^sub>1 \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1] \<cdot> (f.tab \<star> f\<^sub>0g\<^sub>1.p\<^sub>1 \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1)
[PROOF STEP]
proof -
[PROOF STATE]
proof (state)
goal (1 subgoal):
1. (f \<star> g \<star> \<a>[h, tab\<^sub>0 h, fg\<^sub>0h\<^sub>1.p\<^sub>0]) \<cdot> (f \<star> g \<star> h.tab \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0) \<cdot> (f \<star> g \<star> fg\<^sub>0h\<^sub>1.\<phi>) \<cdot> (f \<star> \<a>[g, tab\<^sub>0 g \<star> f\<^sub>0g\<^sub>1.p\<^sub>0, fg\<^sub>0h\<^sub>1.p\<^sub>1]) \<cdot> (f \<star> \<a>[g, tab\<^sub>0 g, f\<^sub>0g\<^sub>1.p\<^sub>0] \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> (f \<star> (g.tab \<star> f\<^sub>0g\<^sub>1.p\<^sub>0) \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> (f \<star> f\<^sub>0g\<^sub>1.\<phi> \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> (f \<star> \<a>\<^sup>-\<^sup>1[tab\<^sub>0 f, f\<^sub>0g\<^sub>1.p\<^sub>1, fg\<^sub>0h\<^sub>1.p\<^sub>1]) \<cdot> \<a>[f, tab\<^sub>0 f, f\<^sub>0g\<^sub>1.p\<^sub>1 \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1] \<cdot> (f.tab \<star> f\<^sub>0g\<^sub>1.p\<^sub>1 \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) = (\<a>[f, g, h \<star> tab\<^sub>0 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0] \<cdot> \<a>\<^sup>-\<^sup>1[f, g, h \<star> tab\<^sub>0 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0] \<cdot> (f \<star> g \<star> \<a>[h, tab\<^sub>0 h, fg\<^sub>0h\<^sub>1.p\<^sub>0])) \<cdot> (f \<star> g \<star> h.tab \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0) \<cdot> (f \<star> g \<star> fg\<^sub>0h\<^sub>1.\<phi>) \<cdot> (f \<star> \<a>[g, tab\<^sub>0 g \<star> f\<^sub>0g\<^sub>1.p\<^sub>0, fg\<^sub>0h\<^sub>1.p\<^sub>1]) \<cdot> (f \<star> \<a>[g, tab\<^sub>0 g, f\<^sub>0g\<^sub>1.p\<^sub>0] \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> (f \<star> (g.tab \<star> f\<^sub>0g\<^sub>1.p\<^sub>0) \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> (f \<star> f\<^sub>0g\<^sub>1.\<phi> \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> (f \<star> \<a>\<^sup>-\<^sup>1[tab\<^sub>0 f, f\<^sub>0g\<^sub>1.p\<^sub>1, fg\<^sub>0h\<^sub>1.p\<^sub>1]) \<cdot> \<a>[f, tab\<^sub>0 f, f\<^sub>0g\<^sub>1.p\<^sub>1 \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1] \<cdot> (f.tab \<star> f\<^sub>0g\<^sub>1.p\<^sub>1 \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1)
[PROOF STEP]
have "(\<a>[f, g, h \<star> tab\<^sub>0 h \<star> TTfgh.p\<^sub>0] \<cdot>
\<a>\<^sup>-\<^sup>1[f, g, h \<star> tab\<^sub>0 h \<star> TTfgh.p\<^sub>0]) \<cdot>
(f \<star> g \<star> \<a>[h, tab\<^sub>0 h, TTfgh.p\<^sub>0]) =
f \<star> g \<star> \<a>[h, tab\<^sub>0 h, TTfgh.p\<^sub>0]"
[PROOF STATE]
proof (prove)
goal (1 subgoal):
1. (\<a>[f, g, h \<star> tab\<^sub>0 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0] \<cdot> \<a>\<^sup>-\<^sup>1[f, g, h \<star> tab\<^sub>0 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0]) \<cdot> (f \<star> g \<star> \<a>[h, tab\<^sub>0 h, fg\<^sub>0h\<^sub>1.p\<^sub>0]) = f \<star> g \<star> \<a>[h, tab\<^sub>0 h, fg\<^sub>0h\<^sub>1.p\<^sub>0]
[PROOF STEP]
using fg gh fg\<^sub>0h\<^sub>1.p\<^sub>1_simps comp_cod_arr comp_assoc_assoc'
[PROOF STATE]
proof (prove)
using this:
src f = trg g
src g = trg h
trg fg\<^sub>0h\<^sub>1.p\<^sub>1 = src (tab\<^sub>0 g \<star> f\<^sub>0g\<^sub>1.p\<^sub>0)
\<lbrakk>arr ?f; cod ?f = ?b\<rbrakk> \<Longrightarrow> ?b \<cdot> ?f = ?f
\<lbrakk>ide ?f; ide ?g; ide ?h; src ?f = trg ?g; src ?g = trg ?h\<rbrakk> \<Longrightarrow> \<a>[?f, ?g, ?h] \<cdot> \<a>\<^sup>-\<^sup>1[?f, ?g, ?h] = ?f \<star> ?g \<star> ?h
\<lbrakk>ide ?f; ide ?g; ide ?h; src ?f = trg ?g; src ?g = trg ?h\<rbrakk> \<Longrightarrow> \<a>\<^sup>-\<^sup>1[?f, ?g, ?h] \<cdot> \<a>[?f, ?g, ?h] = (?f \<star> ?g) \<star> ?h
goal (1 subgoal):
1. (\<a>[f, g, h \<star> tab\<^sub>0 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0] \<cdot> \<a>\<^sup>-\<^sup>1[f, g, h \<star> tab\<^sub>0 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0]) \<cdot> (f \<star> g \<star> \<a>[h, tab\<^sub>0 h, fg\<^sub>0h\<^sub>1.p\<^sub>0]) = f \<star> g \<star> \<a>[h, tab\<^sub>0 h, fg\<^sub>0h\<^sub>1.p\<^sub>0]
[PROOF STEP]
by simp
[PROOF STATE]
proof (state)
this:
(\<a>[f, g, h \<star> tab\<^sub>0 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0] \<cdot> \<a>\<^sup>-\<^sup>1[f, g, h \<star> tab\<^sub>0 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0]) \<cdot> (f \<star> g \<star> \<a>[h, tab\<^sub>0 h, fg\<^sub>0h\<^sub>1.p\<^sub>0]) = f \<star> g \<star> \<a>[h, tab\<^sub>0 h, fg\<^sub>0h\<^sub>1.p\<^sub>0]
goal (1 subgoal):
1. (f \<star> g \<star> \<a>[h, tab\<^sub>0 h, fg\<^sub>0h\<^sub>1.p\<^sub>0]) \<cdot> (f \<star> g \<star> h.tab \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0) \<cdot> (f \<star> g \<star> fg\<^sub>0h\<^sub>1.\<phi>) \<cdot> (f \<star> \<a>[g, tab\<^sub>0 g \<star> f\<^sub>0g\<^sub>1.p\<^sub>0, fg\<^sub>0h\<^sub>1.p\<^sub>1]) \<cdot> (f \<star> \<a>[g, tab\<^sub>0 g, f\<^sub>0g\<^sub>1.p\<^sub>0] \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> (f \<star> (g.tab \<star> f\<^sub>0g\<^sub>1.p\<^sub>0) \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> (f \<star> f\<^sub>0g\<^sub>1.\<phi> \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> (f \<star> \<a>\<^sup>-\<^sup>1[tab\<^sub>0 f, f\<^sub>0g\<^sub>1.p\<^sub>1, fg\<^sub>0h\<^sub>1.p\<^sub>1]) \<cdot> \<a>[f, tab\<^sub>0 f, f\<^sub>0g\<^sub>1.p\<^sub>1 \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1] \<cdot> (f.tab \<star> f\<^sub>0g\<^sub>1.p\<^sub>1 \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) = (\<a>[f, g, h \<star> tab\<^sub>0 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0] \<cdot> \<a>\<^sup>-\<^sup>1[f, g, h \<star> tab\<^sub>0 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0] \<cdot> (f \<star> g \<star> \<a>[h, tab\<^sub>0 h, fg\<^sub>0h\<^sub>1.p\<^sub>0])) \<cdot> (f \<star> g \<star> h.tab \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0) \<cdot> (f \<star> g \<star> fg\<^sub>0h\<^sub>1.\<phi>) \<cdot> (f \<star> \<a>[g, tab\<^sub>0 g \<star> f\<^sub>0g\<^sub>1.p\<^sub>0, fg\<^sub>0h\<^sub>1.p\<^sub>1]) \<cdot> (f \<star> \<a>[g, tab\<^sub>0 g, f\<^sub>0g\<^sub>1.p\<^sub>0] \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> (f \<star> (g.tab \<star> f\<^sub>0g\<^sub>1.p\<^sub>0) \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> (f \<star> f\<^sub>0g\<^sub>1.\<phi> \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> (f \<star> \<a>\<^sup>-\<^sup>1[tab\<^sub>0 f, f\<^sub>0g\<^sub>1.p\<^sub>1, fg\<^sub>0h\<^sub>1.p\<^sub>1]) \<cdot> \<a>[f, tab\<^sub>0 f, f\<^sub>0g\<^sub>1.p\<^sub>1 \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1] \<cdot> (f.tab \<star> f\<^sub>0g\<^sub>1.p\<^sub>1 \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1)
[PROOF STEP]
thus ?thesis
[PROOF STATE]
proof (prove)
using this:
(\<a>[f, g, h \<star> tab\<^sub>0 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0] \<cdot> \<a>\<^sup>-\<^sup>1[f, g, h \<star> tab\<^sub>0 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0]) \<cdot> (f \<star> g \<star> \<a>[h, tab\<^sub>0 h, fg\<^sub>0h\<^sub>1.p\<^sub>0]) = f \<star> g \<star> \<a>[h, tab\<^sub>0 h, fg\<^sub>0h\<^sub>1.p\<^sub>0]
goal (1 subgoal):
1. (f \<star> g \<star> \<a>[h, tab\<^sub>0 h, fg\<^sub>0h\<^sub>1.p\<^sub>0]) \<cdot> (f \<star> g \<star> h.tab \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0) \<cdot> (f \<star> g \<star> fg\<^sub>0h\<^sub>1.\<phi>) \<cdot> (f \<star> \<a>[g, tab\<^sub>0 g \<star> f\<^sub>0g\<^sub>1.p\<^sub>0, fg\<^sub>0h\<^sub>1.p\<^sub>1]) \<cdot> (f \<star> \<a>[g, tab\<^sub>0 g, f\<^sub>0g\<^sub>1.p\<^sub>0] \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> (f \<star> (g.tab \<star> f\<^sub>0g\<^sub>1.p\<^sub>0) \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> (f \<star> f\<^sub>0g\<^sub>1.\<phi> \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> (f \<star> \<a>\<^sup>-\<^sup>1[tab\<^sub>0 f, f\<^sub>0g\<^sub>1.p\<^sub>1, fg\<^sub>0h\<^sub>1.p\<^sub>1]) \<cdot> \<a>[f, tab\<^sub>0 f, f\<^sub>0g\<^sub>1.p\<^sub>1 \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1] \<cdot> (f.tab \<star> f\<^sub>0g\<^sub>1.p\<^sub>1 \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) = (\<a>[f, g, h \<star> tab\<^sub>0 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0] \<cdot> \<a>\<^sup>-\<^sup>1[f, g, h \<star> tab\<^sub>0 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0] \<cdot> (f \<star> g \<star> \<a>[h, tab\<^sub>0 h, fg\<^sub>0h\<^sub>1.p\<^sub>0])) \<cdot> (f \<star> g \<star> h.tab \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0) \<cdot> (f \<star> g \<star> fg\<^sub>0h\<^sub>1.\<phi>) \<cdot> (f \<star> \<a>[g, tab\<^sub>0 g \<star> f\<^sub>0g\<^sub>1.p\<^sub>0, fg\<^sub>0h\<^sub>1.p\<^sub>1]) \<cdot> (f \<star> \<a>[g, tab\<^sub>0 g, f\<^sub>0g\<^sub>1.p\<^sub>0] \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> (f \<star> (g.tab \<star> f\<^sub>0g\<^sub>1.p\<^sub>0) \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> (f \<star> f\<^sub>0g\<^sub>1.\<phi> \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> (f \<star> \<a>\<^sup>-\<^sup>1[tab\<^sub>0 f, f\<^sub>0g\<^sub>1.p\<^sub>1, fg\<^sub>0h\<^sub>1.p\<^sub>1]) \<cdot> \<a>[f, tab\<^sub>0 f, f\<^sub>0g\<^sub>1.p\<^sub>1 \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1] \<cdot> (f.tab \<star> f\<^sub>0g\<^sub>1.p\<^sub>1 \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1)
[PROOF STEP]
using comp_assoc
[PROOF STATE]
proof (prove)
using this:
(\<a>[f, g, h \<star> tab\<^sub>0 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0] \<cdot> \<a>\<^sup>-\<^sup>1[f, g, h \<star> tab\<^sub>0 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0]) \<cdot> (f \<star> g \<star> \<a>[h, tab\<^sub>0 h, fg\<^sub>0h\<^sub>1.p\<^sub>0]) = f \<star> g \<star> \<a>[h, tab\<^sub>0 h, fg\<^sub>0h\<^sub>1.p\<^sub>0]
(?h \<cdot> ?g) \<cdot> ?f = ?h \<cdot> ?g \<cdot> ?f
goal (1 subgoal):
1. (f \<star> g \<star> \<a>[h, tab\<^sub>0 h, fg\<^sub>0h\<^sub>1.p\<^sub>0]) \<cdot> (f \<star> g \<star> h.tab \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0) \<cdot> (f \<star> g \<star> fg\<^sub>0h\<^sub>1.\<phi>) \<cdot> (f \<star> \<a>[g, tab\<^sub>0 g \<star> f\<^sub>0g\<^sub>1.p\<^sub>0, fg\<^sub>0h\<^sub>1.p\<^sub>1]) \<cdot> (f \<star> \<a>[g, tab\<^sub>0 g, f\<^sub>0g\<^sub>1.p\<^sub>0] \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> (f \<star> (g.tab \<star> f\<^sub>0g\<^sub>1.p\<^sub>0) \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> (f \<star> f\<^sub>0g\<^sub>1.\<phi> \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> (f \<star> \<a>\<^sup>-\<^sup>1[tab\<^sub>0 f, f\<^sub>0g\<^sub>1.p\<^sub>1, fg\<^sub>0h\<^sub>1.p\<^sub>1]) \<cdot> \<a>[f, tab\<^sub>0 f, f\<^sub>0g\<^sub>1.p\<^sub>1 \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1] \<cdot> (f.tab \<star> f\<^sub>0g\<^sub>1.p\<^sub>1 \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) = (\<a>[f, g, h \<star> tab\<^sub>0 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0] \<cdot> \<a>\<^sup>-\<^sup>1[f, g, h \<star> tab\<^sub>0 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0] \<cdot> (f \<star> g \<star> \<a>[h, tab\<^sub>0 h, fg\<^sub>0h\<^sub>1.p\<^sub>0])) \<cdot> (f \<star> g \<star> h.tab \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0) \<cdot> (f \<star> g \<star> fg\<^sub>0h\<^sub>1.\<phi>) \<cdot> (f \<star> \<a>[g, tab\<^sub>0 g \<star> f\<^sub>0g\<^sub>1.p\<^sub>0, fg\<^sub>0h\<^sub>1.p\<^sub>1]) \<cdot> (f \<star> \<a>[g, tab\<^sub>0 g, f\<^sub>0g\<^sub>1.p\<^sub>0] \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> (f \<star> (g.tab \<star> f\<^sub>0g\<^sub>1.p\<^sub>0) \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> (f \<star> f\<^sub>0g\<^sub>1.\<phi> \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> (f \<star> \<a>\<^sup>-\<^sup>1[tab\<^sub>0 f, f\<^sub>0g\<^sub>1.p\<^sub>1, fg\<^sub>0h\<^sub>1.p\<^sub>1]) \<cdot> \<a>[f, tab\<^sub>0 f, f\<^sub>0g\<^sub>1.p\<^sub>1 \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1] \<cdot> (f.tab \<star> f\<^sub>0g\<^sub>1.p\<^sub>1 \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1)
[PROOF STEP]
by simp
[PROOF STATE]
proof (state)
this:
(f \<star> g \<star> \<a>[h, tab\<^sub>0 h, fg\<^sub>0h\<^sub>1.p\<^sub>0]) \<cdot> (f \<star> g \<star> h.tab \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0) \<cdot> (f \<star> g \<star> fg\<^sub>0h\<^sub>1.\<phi>) \<cdot> (f \<star> \<a>[g, tab\<^sub>0 g \<star> f\<^sub>0g\<^sub>1.p\<^sub>0, fg\<^sub>0h\<^sub>1.p\<^sub>1]) \<cdot> (f \<star> \<a>[g, tab\<^sub>0 g, f\<^sub>0g\<^sub>1.p\<^sub>0] \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> (f \<star> (g.tab \<star> f\<^sub>0g\<^sub>1.p\<^sub>0) \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> (f \<star> f\<^sub>0g\<^sub>1.\<phi> \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> (f \<star> \<a>\<^sup>-\<^sup>1[tab\<^sub>0 f, f\<^sub>0g\<^sub>1.p\<^sub>1, fg\<^sub>0h\<^sub>1.p\<^sub>1]) \<cdot> \<a>[f, tab\<^sub>0 f, f\<^sub>0g\<^sub>1.p\<^sub>1 \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1] \<cdot> (f.tab \<star> f\<^sub>0g\<^sub>1.p\<^sub>1 \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) = (\<a>[f, g, h \<star> tab\<^sub>0 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0] \<cdot> \<a>\<^sup>-\<^sup>1[f, g, h \<star> tab\<^sub>0 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0] \<cdot> (f \<star> g \<star> \<a>[h, tab\<^sub>0 h, fg\<^sub>0h\<^sub>1.p\<^sub>0])) \<cdot> (f \<star> g \<star> h.tab \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0) \<cdot> (f \<star> g \<star> fg\<^sub>0h\<^sub>1.\<phi>) \<cdot> (f \<star> \<a>[g, tab\<^sub>0 g \<star> f\<^sub>0g\<^sub>1.p\<^sub>0, fg\<^sub>0h\<^sub>1.p\<^sub>1]) \<cdot> (f \<star> \<a>[g, tab\<^sub>0 g, f\<^sub>0g\<^sub>1.p\<^sub>0] \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> (f \<star> (g.tab \<star> f\<^sub>0g\<^sub>1.p\<^sub>0) \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> (f \<star> f\<^sub>0g\<^sub>1.\<phi> \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> (f \<star> \<a>\<^sup>-\<^sup>1[tab\<^sub>0 f, f\<^sub>0g\<^sub>1.p\<^sub>1, fg\<^sub>0h\<^sub>1.p\<^sub>1]) \<cdot> \<a>[f, tab\<^sub>0 f, f\<^sub>0g\<^sub>1.p\<^sub>1 \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1] \<cdot> (f.tab \<star> f\<^sub>0g\<^sub>1.p\<^sub>1 \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1)
goal:
No subgoals!
[PROOF STEP]
qed
[PROOF STATE]
proof (state)
this:
(f \<star> g \<star> \<a>[h, tab\<^sub>0 h, fg\<^sub>0h\<^sub>1.p\<^sub>0]) \<cdot> (f \<star> g \<star> h.tab \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0) \<cdot> (f \<star> g \<star> fg\<^sub>0h\<^sub>1.\<phi>) \<cdot> (f \<star> \<a>[g, tab\<^sub>0 g \<star> f\<^sub>0g\<^sub>1.p\<^sub>0, fg\<^sub>0h\<^sub>1.p\<^sub>1]) \<cdot> (f \<star> \<a>[g, tab\<^sub>0 g, f\<^sub>0g\<^sub>1.p\<^sub>0] \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> (f \<star> (g.tab \<star> f\<^sub>0g\<^sub>1.p\<^sub>0) \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> (f \<star> f\<^sub>0g\<^sub>1.\<phi> \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> (f \<star> \<a>\<^sup>-\<^sup>1[tab\<^sub>0 f, f\<^sub>0g\<^sub>1.p\<^sub>1, fg\<^sub>0h\<^sub>1.p\<^sub>1]) \<cdot> \<a>[f, tab\<^sub>0 f, f\<^sub>0g\<^sub>1.p\<^sub>1 \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1] \<cdot> (f.tab \<star> f\<^sub>0g\<^sub>1.p\<^sub>1 \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) = (\<a>[f, g, h \<star> tab\<^sub>0 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0] \<cdot> \<a>\<^sup>-\<^sup>1[f, g, h \<star> tab\<^sub>0 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0] \<cdot> (f \<star> g \<star> \<a>[h, tab\<^sub>0 h, fg\<^sub>0h\<^sub>1.p\<^sub>0])) \<cdot> (f \<star> g \<star> h.tab \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0) \<cdot> (f \<star> g \<star> fg\<^sub>0h\<^sub>1.\<phi>) \<cdot> (f \<star> \<a>[g, tab\<^sub>0 g \<star> f\<^sub>0g\<^sub>1.p\<^sub>0, fg\<^sub>0h\<^sub>1.p\<^sub>1]) \<cdot> (f \<star> \<a>[g, tab\<^sub>0 g, f\<^sub>0g\<^sub>1.p\<^sub>0] \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> (f \<star> (g.tab \<star> f\<^sub>0g\<^sub>1.p\<^sub>0) \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> (f \<star> f\<^sub>0g\<^sub>1.\<phi> \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> (f \<star> \<a>\<^sup>-\<^sup>1[tab\<^sub>0 f, f\<^sub>0g\<^sub>1.p\<^sub>1, fg\<^sub>0h\<^sub>1.p\<^sub>1]) \<cdot> \<a>[f, tab\<^sub>0 f, f\<^sub>0g\<^sub>1.p\<^sub>1 \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1] \<cdot> (f.tab \<star> f\<^sub>0g\<^sub>1.p\<^sub>1 \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1)
goal (1 subgoal):
1. (f \<star> \<theta>\<^sub>f) \<cdot> \<a>[f, tab\<^sub>0 f, w\<^sub>f] \<cdot> (f.tab \<star> w\<^sub>f) = ((f \<star> \<theta>\<^sub>f') \<cdot> \<a>[f, tab\<^sub>0 f, w\<^sub>f'] \<cdot> (f.tab \<star> w\<^sub>f')) \<cdot> \<beta>\<^sub>f
[PROOF STEP]
also
[PROOF STATE]
proof (state)
this:
(f \<star> g \<star> \<a>[h, tab\<^sub>0 h, fg\<^sub>0h\<^sub>1.p\<^sub>0]) \<cdot> (f \<star> g \<star> h.tab \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0) \<cdot> (f \<star> g \<star> fg\<^sub>0h\<^sub>1.\<phi>) \<cdot> (f \<star> \<a>[g, tab\<^sub>0 g \<star> f\<^sub>0g\<^sub>1.p\<^sub>0, fg\<^sub>0h\<^sub>1.p\<^sub>1]) \<cdot> (f \<star> \<a>[g, tab\<^sub>0 g, f\<^sub>0g\<^sub>1.p\<^sub>0] \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> (f \<star> (g.tab \<star> f\<^sub>0g\<^sub>1.p\<^sub>0) \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> (f \<star> f\<^sub>0g\<^sub>1.\<phi> \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> (f \<star> \<a>\<^sup>-\<^sup>1[tab\<^sub>0 f, f\<^sub>0g\<^sub>1.p\<^sub>1, fg\<^sub>0h\<^sub>1.p\<^sub>1]) \<cdot> \<a>[f, tab\<^sub>0 f, f\<^sub>0g\<^sub>1.p\<^sub>1 \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1] \<cdot> (f.tab \<star> f\<^sub>0g\<^sub>1.p\<^sub>1 \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) = (\<a>[f, g, h \<star> tab\<^sub>0 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0] \<cdot> \<a>\<^sup>-\<^sup>1[f, g, h \<star> tab\<^sub>0 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0] \<cdot> (f \<star> g \<star> \<a>[h, tab\<^sub>0 h, fg\<^sub>0h\<^sub>1.p\<^sub>0])) \<cdot> (f \<star> g \<star> h.tab \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0) \<cdot> (f \<star> g \<star> fg\<^sub>0h\<^sub>1.\<phi>) \<cdot> (f \<star> \<a>[g, tab\<^sub>0 g \<star> f\<^sub>0g\<^sub>1.p\<^sub>0, fg\<^sub>0h\<^sub>1.p\<^sub>1]) \<cdot> (f \<star> \<a>[g, tab\<^sub>0 g, f\<^sub>0g\<^sub>1.p\<^sub>0] \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> (f \<star> (g.tab \<star> f\<^sub>0g\<^sub>1.p\<^sub>0) \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> (f \<star> f\<^sub>0g\<^sub>1.\<phi> \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> (f \<star> \<a>\<^sup>-\<^sup>1[tab\<^sub>0 f, f\<^sub>0g\<^sub>1.p\<^sub>1, fg\<^sub>0h\<^sub>1.p\<^sub>1]) \<cdot> \<a>[f, tab\<^sub>0 f, f\<^sub>0g\<^sub>1.p\<^sub>1 \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1] \<cdot> (f.tab \<star> f\<^sub>0g\<^sub>1.p\<^sub>1 \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1)
goal (1 subgoal):
1. (f \<star> \<theta>\<^sub>f) \<cdot> \<a>[f, tab\<^sub>0 f, w\<^sub>f] \<cdot> (f.tab \<star> w\<^sub>f) = ((f \<star> \<theta>\<^sub>f') \<cdot> \<a>[f, tab\<^sub>0 f, w\<^sub>f'] \<cdot> (f.tab \<star> w\<^sub>f')) \<cdot> \<beta>\<^sub>f
[PROOF STEP]
have "... =
\<a>[f, g, h \<star> tab\<^sub>0 h \<star> TTfgh.p\<^sub>0] \<cdot>
(\<a>\<^sup>-\<^sup>1[f, g, h \<star> tab\<^sub>0 h \<star> TTfgh.p\<^sub>0] \<cdot>
(f \<star> g \<star> \<a>[h, tab\<^sub>0 h, TTfgh.p\<^sub>0])) \<cdot>
(f \<star> g \<star> h.tab \<star> TTfgh.p\<^sub>0) \<cdot>
(f \<star> g \<star> fg\<^sub>0h\<^sub>1.\<phi>) \<cdot>
(f \<star> \<a>[g, tab\<^sub>0 g \<star> Tfg.p\<^sub>0, TTfgh.p\<^sub>1]) \<cdot>
(f \<star> \<a>[g, tab\<^sub>0 g, Tfg.p\<^sub>0] \<star> TTfgh.p\<^sub>1) \<cdot>
(f \<star> (g.tab \<star> Tfg.p\<^sub>0) \<star> TTfgh.p\<^sub>1) \<cdot>
(f \<star> f\<^sub>0g\<^sub>1.\<phi> \<star> TTfgh.p\<^sub>1) \<cdot>
(f \<star> \<a>\<^sup>-\<^sup>1[tab\<^sub>0 f, Tfg.p\<^sub>1, TTfgh.p\<^sub>1]) \<cdot>
\<a>[f, tab\<^sub>0 f, Tfg.p\<^sub>1 \<star> TTfgh.p\<^sub>1] \<cdot>
(f.tab \<star> Tfg.p\<^sub>1 \<star> TTfgh.p\<^sub>1)"
[PROOF STATE]
proof (prove)
goal (1 subgoal):
1. (\<a>[f, g, h \<star> tab\<^sub>0 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0] \<cdot> \<a>\<^sup>-\<^sup>1[f, g, h \<star> tab\<^sub>0 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0] \<cdot> (f \<star> g \<star> \<a>[h, tab\<^sub>0 h, fg\<^sub>0h\<^sub>1.p\<^sub>0])) \<cdot> (f \<star> g \<star> h.tab \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0) \<cdot> (f \<star> g \<star> fg\<^sub>0h\<^sub>1.\<phi>) \<cdot> (f \<star> \<a>[g, tab\<^sub>0 g \<star> f\<^sub>0g\<^sub>1.p\<^sub>0, fg\<^sub>0h\<^sub>1.p\<^sub>1]) \<cdot> (f \<star> \<a>[g, tab\<^sub>0 g, f\<^sub>0g\<^sub>1.p\<^sub>0] \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> (f \<star> (g.tab \<star> f\<^sub>0g\<^sub>1.p\<^sub>0) \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> (f \<star> f\<^sub>0g\<^sub>1.\<phi> \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> (f \<star> \<a>\<^sup>-\<^sup>1[tab\<^sub>0 f, f\<^sub>0g\<^sub>1.p\<^sub>1, fg\<^sub>0h\<^sub>1.p\<^sub>1]) \<cdot> \<a>[f, tab\<^sub>0 f, f\<^sub>0g\<^sub>1.p\<^sub>1 \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1] \<cdot> (f.tab \<star> f\<^sub>0g\<^sub>1.p\<^sub>1 \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) = \<a>[f, g, h \<star> tab\<^sub>0 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0] \<cdot> (\<a>\<^sup>-\<^sup>1[f, g, h \<star> tab\<^sub>0 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0] \<cdot> (f \<star> g \<star> \<a>[h, tab\<^sub>0 h, fg\<^sub>0h\<^sub>1.p\<^sub>0])) \<cdot> (f \<star> g \<star> h.tab \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0) \<cdot> (f \<star> g \<star> fg\<^sub>0h\<^sub>1.\<phi>) \<cdot> (f \<star> \<a>[g, tab\<^sub>0 g \<star> f\<^sub>0g\<^sub>1.p\<^sub>0, fg\<^sub>0h\<^sub>1.p\<^sub>1]) \<cdot> (f \<star> \<a>[g, tab\<^sub>0 g, f\<^sub>0g\<^sub>1.p\<^sub>0] \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> (f \<star> (g.tab \<star> f\<^sub>0g\<^sub>1.p\<^sub>0) \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> (f \<star> f\<^sub>0g\<^sub>1.\<phi> \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> (f \<star> \<a>\<^sup>-\<^sup>1[tab\<^sub>0 f, f\<^sub>0g\<^sub>1.p\<^sub>1, fg\<^sub>0h\<^sub>1.p\<^sub>1]) \<cdot> \<a>[f, tab\<^sub>0 f, f\<^sub>0g\<^sub>1.p\<^sub>1 \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1] \<cdot> (f.tab \<star> f\<^sub>0g\<^sub>1.p\<^sub>1 \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1)
[PROOF STEP]
using comp_assoc
[PROOF STATE]
proof (prove)
using this:
(?h \<cdot> ?g) \<cdot> ?f = ?h \<cdot> ?g \<cdot> ?f
goal (1 subgoal):
1. (\<a>[f, g, h \<star> tab\<^sub>0 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0] \<cdot> \<a>\<^sup>-\<^sup>1[f, g, h \<star> tab\<^sub>0 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0] \<cdot> (f \<star> g \<star> \<a>[h, tab\<^sub>0 h, fg\<^sub>0h\<^sub>1.p\<^sub>0])) \<cdot> (f \<star> g \<star> h.tab \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0) \<cdot> (f \<star> g \<star> fg\<^sub>0h\<^sub>1.\<phi>) \<cdot> (f \<star> \<a>[g, tab\<^sub>0 g \<star> f\<^sub>0g\<^sub>1.p\<^sub>0, fg\<^sub>0h\<^sub>1.p\<^sub>1]) \<cdot> (f \<star> \<a>[g, tab\<^sub>0 g, f\<^sub>0g\<^sub>1.p\<^sub>0] \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> (f \<star> (g.tab \<star> f\<^sub>0g\<^sub>1.p\<^sub>0) \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> (f \<star> f\<^sub>0g\<^sub>1.\<phi> \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> (f \<star> \<a>\<^sup>-\<^sup>1[tab\<^sub>0 f, f\<^sub>0g\<^sub>1.p\<^sub>1, fg\<^sub>0h\<^sub>1.p\<^sub>1]) \<cdot> \<a>[f, tab\<^sub>0 f, f\<^sub>0g\<^sub>1.p\<^sub>1 \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1] \<cdot> (f.tab \<star> f\<^sub>0g\<^sub>1.p\<^sub>1 \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) = \<a>[f, g, h \<star> tab\<^sub>0 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0] \<cdot> (\<a>\<^sup>-\<^sup>1[f, g, h \<star> tab\<^sub>0 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0] \<cdot> (f \<star> g \<star> \<a>[h, tab\<^sub>0 h, fg\<^sub>0h\<^sub>1.p\<^sub>0])) \<cdot> (f \<star> g \<star> h.tab \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0) \<cdot> (f \<star> g \<star> fg\<^sub>0h\<^sub>1.\<phi>) \<cdot> (f \<star> \<a>[g, tab\<^sub>0 g \<star> f\<^sub>0g\<^sub>1.p\<^sub>0, fg\<^sub>0h\<^sub>1.p\<^sub>1]) \<cdot> (f \<star> \<a>[g, tab\<^sub>0 g, f\<^sub>0g\<^sub>1.p\<^sub>0] \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> (f \<star> (g.tab \<star> f\<^sub>0g\<^sub>1.p\<^sub>0) \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> (f \<star> f\<^sub>0g\<^sub>1.\<phi> \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> (f \<star> \<a>\<^sup>-\<^sup>1[tab\<^sub>0 f, f\<^sub>0g\<^sub>1.p\<^sub>1, fg\<^sub>0h\<^sub>1.p\<^sub>1]) \<cdot> \<a>[f, tab\<^sub>0 f, f\<^sub>0g\<^sub>1.p\<^sub>1 \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1] \<cdot> (f.tab \<star> f\<^sub>0g\<^sub>1.p\<^sub>1 \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1)
[PROOF STEP]
by presburger
[PROOF STATE]
proof (state)
this:
(\<a>[f, g, h \<star> tab\<^sub>0 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0] \<cdot> \<a>\<^sup>-\<^sup>1[f, g, h \<star> tab\<^sub>0 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0] \<cdot> (f \<star> g \<star> \<a>[h, tab\<^sub>0 h, fg\<^sub>0h\<^sub>1.p\<^sub>0])) \<cdot> (f \<star> g \<star> h.tab \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0) \<cdot> (f \<star> g \<star> fg\<^sub>0h\<^sub>1.\<phi>) \<cdot> (f \<star> \<a>[g, tab\<^sub>0 g \<star> f\<^sub>0g\<^sub>1.p\<^sub>0, fg\<^sub>0h\<^sub>1.p\<^sub>1]) \<cdot> (f \<star> \<a>[g, tab\<^sub>0 g, f\<^sub>0g\<^sub>1.p\<^sub>0] \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> (f \<star> (g.tab \<star> f\<^sub>0g\<^sub>1.p\<^sub>0) \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> (f \<star> f\<^sub>0g\<^sub>1.\<phi> \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> (f \<star> \<a>\<^sup>-\<^sup>1[tab\<^sub>0 f, f\<^sub>0g\<^sub>1.p\<^sub>1, fg\<^sub>0h\<^sub>1.p\<^sub>1]) \<cdot> \<a>[f, tab\<^sub>0 f, f\<^sub>0g\<^sub>1.p\<^sub>1 \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1] \<cdot> (f.tab \<star> f\<^sub>0g\<^sub>1.p\<^sub>1 \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) = \<a>[f, g, h \<star> tab\<^sub>0 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0] \<cdot> (\<a>\<^sup>-\<^sup>1[f, g, h \<star> tab\<^sub>0 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0] \<cdot> (f \<star> g \<star> \<a>[h, tab\<^sub>0 h, fg\<^sub>0h\<^sub>1.p\<^sub>0])) \<cdot> (f \<star> g \<star> h.tab \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0) \<cdot> (f \<star> g \<star> fg\<^sub>0h\<^sub>1.\<phi>) \<cdot> (f \<star> \<a>[g, tab\<^sub>0 g \<star> f\<^sub>0g\<^sub>1.p\<^sub>0, fg\<^sub>0h\<^sub>1.p\<^sub>1]) \<cdot> (f \<star> \<a>[g, tab\<^sub>0 g, f\<^sub>0g\<^sub>1.p\<^sub>0] \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> (f \<star> (g.tab \<star> f\<^sub>0g\<^sub>1.p\<^sub>0) \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> (f \<star> f\<^sub>0g\<^sub>1.\<phi> \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> (f \<star> \<a>\<^sup>-\<^sup>1[tab\<^sub>0 f, f\<^sub>0g\<^sub>1.p\<^sub>1, fg\<^sub>0h\<^sub>1.p\<^sub>1]) \<cdot> \<a>[f, tab\<^sub>0 f, f\<^sub>0g\<^sub>1.p\<^sub>1 \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1] \<cdot> (f.tab \<star> f\<^sub>0g\<^sub>1.p\<^sub>1 \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1)
goal (1 subgoal):
1. (f \<star> \<theta>\<^sub>f) \<cdot> \<a>[f, tab\<^sub>0 f, w\<^sub>f] \<cdot> (f.tab \<star> w\<^sub>f) = ((f \<star> \<theta>\<^sub>f') \<cdot> \<a>[f, tab\<^sub>0 f, w\<^sub>f'] \<cdot> (f.tab \<star> w\<^sub>f')) \<cdot> \<beta>\<^sub>f
[PROOF STEP]
also
[PROOF STATE]
proof (state)
this:
(\<a>[f, g, h \<star> tab\<^sub>0 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0] \<cdot> \<a>\<^sup>-\<^sup>1[f, g, h \<star> tab\<^sub>0 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0] \<cdot> (f \<star> g \<star> \<a>[h, tab\<^sub>0 h, fg\<^sub>0h\<^sub>1.p\<^sub>0])) \<cdot> (f \<star> g \<star> h.tab \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0) \<cdot> (f \<star> g \<star> fg\<^sub>0h\<^sub>1.\<phi>) \<cdot> (f \<star> \<a>[g, tab\<^sub>0 g \<star> f\<^sub>0g\<^sub>1.p\<^sub>0, fg\<^sub>0h\<^sub>1.p\<^sub>1]) \<cdot> (f \<star> \<a>[g, tab\<^sub>0 g, f\<^sub>0g\<^sub>1.p\<^sub>0] \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> (f \<star> (g.tab \<star> f\<^sub>0g\<^sub>1.p\<^sub>0) \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> (f \<star> f\<^sub>0g\<^sub>1.\<phi> \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> (f \<star> \<a>\<^sup>-\<^sup>1[tab\<^sub>0 f, f\<^sub>0g\<^sub>1.p\<^sub>1, fg\<^sub>0h\<^sub>1.p\<^sub>1]) \<cdot> \<a>[f, tab\<^sub>0 f, f\<^sub>0g\<^sub>1.p\<^sub>1 \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1] \<cdot> (f.tab \<star> f\<^sub>0g\<^sub>1.p\<^sub>1 \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) = \<a>[f, g, h \<star> tab\<^sub>0 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0] \<cdot> (\<a>\<^sup>-\<^sup>1[f, g, h \<star> tab\<^sub>0 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0] \<cdot> (f \<star> g \<star> \<a>[h, tab\<^sub>0 h, fg\<^sub>0h\<^sub>1.p\<^sub>0])) \<cdot> (f \<star> g \<star> h.tab \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0) \<cdot> (f \<star> g \<star> fg\<^sub>0h\<^sub>1.\<phi>) \<cdot> (f \<star> \<a>[g, tab\<^sub>0 g \<star> f\<^sub>0g\<^sub>1.p\<^sub>0, fg\<^sub>0h\<^sub>1.p\<^sub>1]) \<cdot> (f \<star> \<a>[g, tab\<^sub>0 g, f\<^sub>0g\<^sub>1.p\<^sub>0] \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> (f \<star> (g.tab \<star> f\<^sub>0g\<^sub>1.p\<^sub>0) \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> (f \<star> f\<^sub>0g\<^sub>1.\<phi> \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> (f \<star> \<a>\<^sup>-\<^sup>1[tab\<^sub>0 f, f\<^sub>0g\<^sub>1.p\<^sub>1, fg\<^sub>0h\<^sub>1.p\<^sub>1]) \<cdot> \<a>[f, tab\<^sub>0 f, f\<^sub>0g\<^sub>1.p\<^sub>1 \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1] \<cdot> (f.tab \<star> f\<^sub>0g\<^sub>1.p\<^sub>1 \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1)
goal (1 subgoal):
1. (f \<star> \<theta>\<^sub>f) \<cdot> \<a>[f, tab\<^sub>0 f, w\<^sub>f] \<cdot> (f.tab \<star> w\<^sub>f) = ((f \<star> \<theta>\<^sub>f') \<cdot> \<a>[f, tab\<^sub>0 f, w\<^sub>f'] \<cdot> (f.tab \<star> w\<^sub>f')) \<cdot> \<beta>\<^sub>f
[PROOF STEP]
have "... =
\<a>[f, g, h \<star> tab\<^sub>0 h \<star> TTfgh.p\<^sub>0] \<cdot>
((f \<star> g) \<star> \<a>[h, tab\<^sub>0 h, TTfgh.p\<^sub>0]) \<cdot>
(\<a>\<^sup>-\<^sup>1[f, g, (h \<star> tab\<^sub>0 h) \<star> TTfgh.p\<^sub>0] \<cdot>
(f \<star> g \<star> h.tab \<star> TTfgh.p\<^sub>0)) \<cdot>
(f \<star> g \<star> fg\<^sub>0h\<^sub>1.\<phi>) \<cdot>
(f \<star> \<a>[g, tab\<^sub>0 g \<star> Tfg.p\<^sub>0, TTfgh.p\<^sub>1]) \<cdot>
(f \<star> \<a>[g, tab\<^sub>0 g, Tfg.p\<^sub>0] \<star> TTfgh.p\<^sub>1) \<cdot>
(f \<star> (g.tab \<star> Tfg.p\<^sub>0) \<star> TTfgh.p\<^sub>1) \<cdot>
(f \<star> f\<^sub>0g\<^sub>1.\<phi> \<star> TTfgh.p\<^sub>1) \<cdot>
(f \<star> \<a>\<^sup>-\<^sup>1[tab\<^sub>0 f, Tfg.p\<^sub>1, TTfgh.p\<^sub>1]) \<cdot>
\<a>[f, tab\<^sub>0 f, Tfg.p\<^sub>1 \<star> TTfgh.p\<^sub>1] \<cdot>
(f.tab \<star> Tfg.p\<^sub>1 \<star> TTfgh.p\<^sub>1)"
[PROOF STATE]
proof (prove)
goal (1 subgoal):
1. \<a>[f, g, h \<star> tab\<^sub>0 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0] \<cdot> (\<a>\<^sup>-\<^sup>1[f, g, h \<star> tab\<^sub>0 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0] \<cdot> (f \<star> g \<star> \<a>[h, tab\<^sub>0 h, fg\<^sub>0h\<^sub>1.p\<^sub>0])) \<cdot> (f \<star> g \<star> h.tab \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0) \<cdot> (f \<star> g \<star> fg\<^sub>0h\<^sub>1.\<phi>) \<cdot> (f \<star> \<a>[g, tab\<^sub>0 g \<star> f\<^sub>0g\<^sub>1.p\<^sub>0, fg\<^sub>0h\<^sub>1.p\<^sub>1]) \<cdot> (f \<star> \<a>[g, tab\<^sub>0 g, f\<^sub>0g\<^sub>1.p\<^sub>0] \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> (f \<star> (g.tab \<star> f\<^sub>0g\<^sub>1.p\<^sub>0) \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> (f \<star> f\<^sub>0g\<^sub>1.\<phi> \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> (f \<star> \<a>\<^sup>-\<^sup>1[tab\<^sub>0 f, f\<^sub>0g\<^sub>1.p\<^sub>1, fg\<^sub>0h\<^sub>1.p\<^sub>1]) \<cdot> \<a>[f, tab\<^sub>0 f, f\<^sub>0g\<^sub>1.p\<^sub>1 \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1] \<cdot> (f.tab \<star> f\<^sub>0g\<^sub>1.p\<^sub>1 \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) = \<a>[f, g, h \<star> tab\<^sub>0 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0] \<cdot> ((f \<star> g) \<star> \<a>[h, tab\<^sub>0 h, fg\<^sub>0h\<^sub>1.p\<^sub>0]) \<cdot> (\<a>\<^sup>-\<^sup>1[f, g, (h \<star> tab\<^sub>0 h) \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0] \<cdot> (f \<star> g \<star> h.tab \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0)) \<cdot> (f \<star> g \<star> fg\<^sub>0h\<^sub>1.\<phi>) \<cdot> (f \<star> \<a>[g, tab\<^sub>0 g \<star> f\<^sub>0g\<^sub>1.p\<^sub>0, fg\<^sub>0h\<^sub>1.p\<^sub>1]) \<cdot> (f \<star> \<a>[g, tab\<^sub>0 g, f\<^sub>0g\<^sub>1.p\<^sub>0] \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> (f \<star> (g.tab \<star> f\<^sub>0g\<^sub>1.p\<^sub>0) \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> (f \<star> f\<^sub>0g\<^sub>1.\<phi> \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> (f \<star> \<a>\<^sup>-\<^sup>1[tab\<^sub>0 f, f\<^sub>0g\<^sub>1.p\<^sub>1, fg\<^sub>0h\<^sub>1.p\<^sub>1]) \<cdot> \<a>[f, tab\<^sub>0 f, f\<^sub>0g\<^sub>1.p\<^sub>1 \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1] \<cdot> (f.tab \<star> f\<^sub>0g\<^sub>1.p\<^sub>1 \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1)
[PROOF STEP]
using fg gh fg\<^sub>0h\<^sub>1.p\<^sub>0_simps fg\<^sub>0h\<^sub>1.p\<^sub>1_simps comp_assoc
assoc'_naturality [of f g "\<a>[h, tab\<^sub>0 h, TTfgh.p\<^sub>0]"]
[PROOF STATE]
proof (prove)
using this:
src f = trg g
src g = trg h
trg fg\<^sub>0h\<^sub>1.p\<^sub>0 = src (tab\<^sub>1 h)
trg fg\<^sub>0h\<^sub>1.p\<^sub>1 = src (tab\<^sub>0 g \<star> f\<^sub>0g\<^sub>1.p\<^sub>0)
(?h \<cdot> ?g) \<cdot> ?f = ?h \<cdot> ?g \<cdot> ?f
\<lbrakk>arr f; arr g; arr \<a>[h, tab\<^sub>0 h, fg\<^sub>0h\<^sub>1.p\<^sub>0]; src f = trg g; src g = trg \<a>[h, tab\<^sub>0 h, fg\<^sub>0h\<^sub>1.p\<^sub>0]\<rbrakk> \<Longrightarrow> \<a>\<^sup>-\<^sup>1[cod f, cod g, cod \<a>[h, tab\<^sub>0 h, fg\<^sub>0h\<^sub>1.p\<^sub>0]] \<cdot> (f \<star> g \<star> \<a>[h, tab\<^sub>0 h, fg\<^sub>0h\<^sub>1.p\<^sub>0]) = ((f \<star> g) \<star> \<a>[h, tab\<^sub>0 h, fg\<^sub>0h\<^sub>1.p\<^sub>0]) \<cdot> \<a>\<^sup>-\<^sup>1[local.dom f, local.dom g, local.dom \<a>[h, tab\<^sub>0 h, fg\<^sub>0h\<^sub>1.p\<^sub>0]]
goal (1 subgoal):
1. \<a>[f, g, h \<star> tab\<^sub>0 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0] \<cdot> (\<a>\<^sup>-\<^sup>1[f, g, h \<star> tab\<^sub>0 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0] \<cdot> (f \<star> g \<star> \<a>[h, tab\<^sub>0 h, fg\<^sub>0h\<^sub>1.p\<^sub>0])) \<cdot> (f \<star> g \<star> h.tab \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0) \<cdot> (f \<star> g \<star> fg\<^sub>0h\<^sub>1.\<phi>) \<cdot> (f \<star> \<a>[g, tab\<^sub>0 g \<star> f\<^sub>0g\<^sub>1.p\<^sub>0, fg\<^sub>0h\<^sub>1.p\<^sub>1]) \<cdot> (f \<star> \<a>[g, tab\<^sub>0 g, f\<^sub>0g\<^sub>1.p\<^sub>0] \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> (f \<star> (g.tab \<star> f\<^sub>0g\<^sub>1.p\<^sub>0) \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> (f \<star> f\<^sub>0g\<^sub>1.\<phi> \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> (f \<star> \<a>\<^sup>-\<^sup>1[tab\<^sub>0 f, f\<^sub>0g\<^sub>1.p\<^sub>1, fg\<^sub>0h\<^sub>1.p\<^sub>1]) \<cdot> \<a>[f, tab\<^sub>0 f, f\<^sub>0g\<^sub>1.p\<^sub>1 \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1] \<cdot> (f.tab \<star> f\<^sub>0g\<^sub>1.p\<^sub>1 \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) = \<a>[f, g, h \<star> tab\<^sub>0 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0] \<cdot> ((f \<star> g) \<star> \<a>[h, tab\<^sub>0 h, fg\<^sub>0h\<^sub>1.p\<^sub>0]) \<cdot> (\<a>\<^sup>-\<^sup>1[f, g, (h \<star> tab\<^sub>0 h) \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0] \<cdot> (f \<star> g \<star> h.tab \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0)) \<cdot> (f \<star> g \<star> fg\<^sub>0h\<^sub>1.\<phi>) \<cdot> (f \<star> \<a>[g, tab\<^sub>0 g \<star> f\<^sub>0g\<^sub>1.p\<^sub>0, fg\<^sub>0h\<^sub>1.p\<^sub>1]) \<cdot> (f \<star> \<a>[g, tab\<^sub>0 g, f\<^sub>0g\<^sub>1.p\<^sub>0] \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> (f \<star> (g.tab \<star> f\<^sub>0g\<^sub>1.p\<^sub>0) \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> (f \<star> f\<^sub>0g\<^sub>1.\<phi> \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> (f \<star> \<a>\<^sup>-\<^sup>1[tab\<^sub>0 f, f\<^sub>0g\<^sub>1.p\<^sub>1, fg\<^sub>0h\<^sub>1.p\<^sub>1]) \<cdot> \<a>[f, tab\<^sub>0 f, f\<^sub>0g\<^sub>1.p\<^sub>1 \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1] \<cdot> (f.tab \<star> f\<^sub>0g\<^sub>1.p\<^sub>1 \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1)
[PROOF STEP]
by simp
[PROOF STATE]
proof (state)
this:
\<a>[f, g, h \<star> tab\<^sub>0 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0] \<cdot> (\<a>\<^sup>-\<^sup>1[f, g, h \<star> tab\<^sub>0 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0] \<cdot> (f \<star> g \<star> \<a>[h, tab\<^sub>0 h, fg\<^sub>0h\<^sub>1.p\<^sub>0])) \<cdot> (f \<star> g \<star> h.tab \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0) \<cdot> (f \<star> g \<star> fg\<^sub>0h\<^sub>1.\<phi>) \<cdot> (f \<star> \<a>[g, tab\<^sub>0 g \<star> f\<^sub>0g\<^sub>1.p\<^sub>0, fg\<^sub>0h\<^sub>1.p\<^sub>1]) \<cdot> (f \<star> \<a>[g, tab\<^sub>0 g, f\<^sub>0g\<^sub>1.p\<^sub>0] \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> (f \<star> (g.tab \<star> f\<^sub>0g\<^sub>1.p\<^sub>0) \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> (f \<star> f\<^sub>0g\<^sub>1.\<phi> \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> (f \<star> \<a>\<^sup>-\<^sup>1[tab\<^sub>0 f, f\<^sub>0g\<^sub>1.p\<^sub>1, fg\<^sub>0h\<^sub>1.p\<^sub>1]) \<cdot> \<a>[f, tab\<^sub>0 f, f\<^sub>0g\<^sub>1.p\<^sub>1 \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1] \<cdot> (f.tab \<star> f\<^sub>0g\<^sub>1.p\<^sub>1 \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) = \<a>[f, g, h \<star> tab\<^sub>0 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0] \<cdot> ((f \<star> g) \<star> \<a>[h, tab\<^sub>0 h, fg\<^sub>0h\<^sub>1.p\<^sub>0]) \<cdot> (\<a>\<^sup>-\<^sup>1[f, g, (h \<star> tab\<^sub>0 h) \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0] \<cdot> (f \<star> g \<star> h.tab \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0)) \<cdot> (f \<star> g \<star> fg\<^sub>0h\<^sub>1.\<phi>) \<cdot> (f \<star> \<a>[g, tab\<^sub>0 g \<star> f\<^sub>0g\<^sub>1.p\<^sub>0, fg\<^sub>0h\<^sub>1.p\<^sub>1]) \<cdot> (f \<star> \<a>[g, tab\<^sub>0 g, f\<^sub>0g\<^sub>1.p\<^sub>0] \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> (f \<star> (g.tab \<star> f\<^sub>0g\<^sub>1.p\<^sub>0) \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> (f \<star> f\<^sub>0g\<^sub>1.\<phi> \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> (f \<star> \<a>\<^sup>-\<^sup>1[tab\<^sub>0 f, f\<^sub>0g\<^sub>1.p\<^sub>1, fg\<^sub>0h\<^sub>1.p\<^sub>1]) \<cdot> \<a>[f, tab\<^sub>0 f, f\<^sub>0g\<^sub>1.p\<^sub>1 \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1] \<cdot> (f.tab \<star> f\<^sub>0g\<^sub>1.p\<^sub>1 \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1)
goal (1 subgoal):
1. (f \<star> \<theta>\<^sub>f) \<cdot> \<a>[f, tab\<^sub>0 f, w\<^sub>f] \<cdot> (f.tab \<star> w\<^sub>f) = ((f \<star> \<theta>\<^sub>f') \<cdot> \<a>[f, tab\<^sub>0 f, w\<^sub>f'] \<cdot> (f.tab \<star> w\<^sub>f')) \<cdot> \<beta>\<^sub>f
[PROOF STEP]
also
[PROOF STATE]
proof (state)
this:
\<a>[f, g, h \<star> tab\<^sub>0 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0] \<cdot> (\<a>\<^sup>-\<^sup>1[f, g, h \<star> tab\<^sub>0 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0] \<cdot> (f \<star> g \<star> \<a>[h, tab\<^sub>0 h, fg\<^sub>0h\<^sub>1.p\<^sub>0])) \<cdot> (f \<star> g \<star> h.tab \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0) \<cdot> (f \<star> g \<star> fg\<^sub>0h\<^sub>1.\<phi>) \<cdot> (f \<star> \<a>[g, tab\<^sub>0 g \<star> f\<^sub>0g\<^sub>1.p\<^sub>0, fg\<^sub>0h\<^sub>1.p\<^sub>1]) \<cdot> (f \<star> \<a>[g, tab\<^sub>0 g, f\<^sub>0g\<^sub>1.p\<^sub>0] \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> (f \<star> (g.tab \<star> f\<^sub>0g\<^sub>1.p\<^sub>0) \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> (f \<star> f\<^sub>0g\<^sub>1.\<phi> \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> (f \<star> \<a>\<^sup>-\<^sup>1[tab\<^sub>0 f, f\<^sub>0g\<^sub>1.p\<^sub>1, fg\<^sub>0h\<^sub>1.p\<^sub>1]) \<cdot> \<a>[f, tab\<^sub>0 f, f\<^sub>0g\<^sub>1.p\<^sub>1 \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1] \<cdot> (f.tab \<star> f\<^sub>0g\<^sub>1.p\<^sub>1 \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) = \<a>[f, g, h \<star> tab\<^sub>0 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0] \<cdot> ((f \<star> g) \<star> \<a>[h, tab\<^sub>0 h, fg\<^sub>0h\<^sub>1.p\<^sub>0]) \<cdot> (\<a>\<^sup>-\<^sup>1[f, g, (h \<star> tab\<^sub>0 h) \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0] \<cdot> (f \<star> g \<star> h.tab \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0)) \<cdot> (f \<star> g \<star> fg\<^sub>0h\<^sub>1.\<phi>) \<cdot> (f \<star> \<a>[g, tab\<^sub>0 g \<star> f\<^sub>0g\<^sub>1.p\<^sub>0, fg\<^sub>0h\<^sub>1.p\<^sub>1]) \<cdot> (f \<star> \<a>[g, tab\<^sub>0 g, f\<^sub>0g\<^sub>1.p\<^sub>0] \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> (f \<star> (g.tab \<star> f\<^sub>0g\<^sub>1.p\<^sub>0) \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> (f \<star> f\<^sub>0g\<^sub>1.\<phi> \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> (f \<star> \<a>\<^sup>-\<^sup>1[tab\<^sub>0 f, f\<^sub>0g\<^sub>1.p\<^sub>1, fg\<^sub>0h\<^sub>1.p\<^sub>1]) \<cdot> \<a>[f, tab\<^sub>0 f, f\<^sub>0g\<^sub>1.p\<^sub>1 \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1] \<cdot> (f.tab \<star> f\<^sub>0g\<^sub>1.p\<^sub>1 \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1)
goal (1 subgoal):
1. (f \<star> \<theta>\<^sub>f) \<cdot> \<a>[f, tab\<^sub>0 f, w\<^sub>f] \<cdot> (f.tab \<star> w\<^sub>f) = ((f \<star> \<theta>\<^sub>f') \<cdot> \<a>[f, tab\<^sub>0 f, w\<^sub>f'] \<cdot> (f.tab \<star> w\<^sub>f')) \<cdot> \<beta>\<^sub>f
[PROOF STEP]
have "... =
\<a>[f, g, h \<star> tab\<^sub>0 h \<star> TTfgh.p\<^sub>0] \<cdot>
((f \<star> g) \<star> \<a>[h, tab\<^sub>0 h, TTfgh.p\<^sub>0]) \<cdot>
((f \<star> g) \<star> h.tab \<star> TTfgh.p\<^sub>0) \<cdot>
(\<a>\<^sup>-\<^sup>1[f, g, tab\<^sub>1 h \<star> TTfgh.p\<^sub>0] \<cdot>
(f \<star> g \<star> fg\<^sub>0h\<^sub>1.\<phi>)) \<cdot>
(f \<star> \<a>[g, tab\<^sub>0 g \<star> Tfg.p\<^sub>0, TTfgh.p\<^sub>1]) \<cdot>
(f \<star> \<a>[g, tab\<^sub>0 g, Tfg.p\<^sub>0] \<star> TTfgh.p\<^sub>1) \<cdot>
(f \<star> (g.tab \<star> Tfg.p\<^sub>0) \<star> TTfgh.p\<^sub>1) \<cdot>
(f \<star> f\<^sub>0g\<^sub>1.\<phi> \<star> TTfgh.p\<^sub>1) \<cdot>
(f \<star> \<a>\<^sup>-\<^sup>1[tab\<^sub>0 f, Tfg.p\<^sub>1, TTfgh.p\<^sub>1]) \<cdot>
\<a>[f, tab\<^sub>0 f, Tfg.p\<^sub>1 \<star> TTfgh.p\<^sub>1] \<cdot>
(f.tab \<star> Tfg.p\<^sub>1 \<star> TTfgh.p\<^sub>1)"
[PROOF STATE]
proof (prove)
goal (1 subgoal):
1. \<a>[f, g, h \<star> tab\<^sub>0 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0] \<cdot> ((f \<star> g) \<star> \<a>[h, tab\<^sub>0 h, fg\<^sub>0h\<^sub>1.p\<^sub>0]) \<cdot> (\<a>\<^sup>-\<^sup>1[f, g, (h \<star> tab\<^sub>0 h) \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0] \<cdot> (f \<star> g \<star> h.tab \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0)) \<cdot> (f \<star> g \<star> fg\<^sub>0h\<^sub>1.\<phi>) \<cdot> (f \<star> \<a>[g, tab\<^sub>0 g \<star> f\<^sub>0g\<^sub>1.p\<^sub>0, fg\<^sub>0h\<^sub>1.p\<^sub>1]) \<cdot> (f \<star> \<a>[g, tab\<^sub>0 g, f\<^sub>0g\<^sub>1.p\<^sub>0] \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> (f \<star> (g.tab \<star> f\<^sub>0g\<^sub>1.p\<^sub>0) \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> (f \<star> f\<^sub>0g\<^sub>1.\<phi> \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> (f \<star> \<a>\<^sup>-\<^sup>1[tab\<^sub>0 f, f\<^sub>0g\<^sub>1.p\<^sub>1, fg\<^sub>0h\<^sub>1.p\<^sub>1]) \<cdot> \<a>[f, tab\<^sub>0 f, f\<^sub>0g\<^sub>1.p\<^sub>1 \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1] \<cdot> (f.tab \<star> f\<^sub>0g\<^sub>1.p\<^sub>1 \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) = \<a>[f, g, h \<star> tab\<^sub>0 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0] \<cdot> ((f \<star> g) \<star> \<a>[h, tab\<^sub>0 h, fg\<^sub>0h\<^sub>1.p\<^sub>0]) \<cdot> ((f \<star> g) \<star> h.tab \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0) \<cdot> (\<a>\<^sup>-\<^sup>1[f, g, tab\<^sub>1 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0] \<cdot> (f \<star> g \<star> fg\<^sub>0h\<^sub>1.\<phi>)) \<cdot> (f \<star> \<a>[g, tab\<^sub>0 g \<star> f\<^sub>0g\<^sub>1.p\<^sub>0, fg\<^sub>0h\<^sub>1.p\<^sub>1]) \<cdot> (f \<star> \<a>[g, tab\<^sub>0 g, f\<^sub>0g\<^sub>1.p\<^sub>0] \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> (f \<star> (g.tab \<star> f\<^sub>0g\<^sub>1.p\<^sub>0) \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> (f \<star> f\<^sub>0g\<^sub>1.\<phi> \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> (f \<star> \<a>\<^sup>-\<^sup>1[tab\<^sub>0 f, f\<^sub>0g\<^sub>1.p\<^sub>1, fg\<^sub>0h\<^sub>1.p\<^sub>1]) \<cdot> \<a>[f, tab\<^sub>0 f, f\<^sub>0g\<^sub>1.p\<^sub>1 \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1] \<cdot> (f.tab \<star> f\<^sub>0g\<^sub>1.p\<^sub>1 \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1)
[PROOF STEP]
using fg gh fg\<^sub>0h\<^sub>1.p\<^sub>0_simps fg\<^sub>0h\<^sub>1.p\<^sub>1_simps comp_assoc
assoc'_naturality [of f g "h.tab \<star> TTfgh.p\<^sub>0"]
[PROOF STATE]
proof (prove)
using this:
src f = trg g
src g = trg h
trg fg\<^sub>0h\<^sub>1.p\<^sub>0 = src (tab\<^sub>1 h)
trg fg\<^sub>0h\<^sub>1.p\<^sub>1 = src (tab\<^sub>0 g \<star> f\<^sub>0g\<^sub>1.p\<^sub>0)
(?h \<cdot> ?g) \<cdot> ?f = ?h \<cdot> ?g \<cdot> ?f
\<lbrakk>arr f; arr g; hseq h.tab fg\<^sub>0h\<^sub>1.p\<^sub>0; src f = trg g; src g = trg (h.tab \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0)\<rbrakk> \<Longrightarrow> \<a>\<^sup>-\<^sup>1[cod f, cod g, cod (h.tab \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0)] \<cdot> (f \<star> g \<star> h.tab \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0) = ((f \<star> g) \<star> h.tab \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0) \<cdot> \<a>\<^sup>-\<^sup>1[local.dom f, local.dom g, local.dom (h.tab \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0)]
goal (1 subgoal):
1. \<a>[f, g, h \<star> tab\<^sub>0 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0] \<cdot> ((f \<star> g) \<star> \<a>[h, tab\<^sub>0 h, fg\<^sub>0h\<^sub>1.p\<^sub>0]) \<cdot> (\<a>\<^sup>-\<^sup>1[f, g, (h \<star> tab\<^sub>0 h) \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0] \<cdot> (f \<star> g \<star> h.tab \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0)) \<cdot> (f \<star> g \<star> fg\<^sub>0h\<^sub>1.\<phi>) \<cdot> (f \<star> \<a>[g, tab\<^sub>0 g \<star> f\<^sub>0g\<^sub>1.p\<^sub>0, fg\<^sub>0h\<^sub>1.p\<^sub>1]) \<cdot> (f \<star> \<a>[g, tab\<^sub>0 g, f\<^sub>0g\<^sub>1.p\<^sub>0] \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> (f \<star> (g.tab \<star> f\<^sub>0g\<^sub>1.p\<^sub>0) \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> (f \<star> f\<^sub>0g\<^sub>1.\<phi> \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> (f \<star> \<a>\<^sup>-\<^sup>1[tab\<^sub>0 f, f\<^sub>0g\<^sub>1.p\<^sub>1, fg\<^sub>0h\<^sub>1.p\<^sub>1]) \<cdot> \<a>[f, tab\<^sub>0 f, f\<^sub>0g\<^sub>1.p\<^sub>1 \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1] \<cdot> (f.tab \<star> f\<^sub>0g\<^sub>1.p\<^sub>1 \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) = \<a>[f, g, h \<star> tab\<^sub>0 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0] \<cdot> ((f \<star> g) \<star> \<a>[h, tab\<^sub>0 h, fg\<^sub>0h\<^sub>1.p\<^sub>0]) \<cdot> ((f \<star> g) \<star> h.tab \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0) \<cdot> (\<a>\<^sup>-\<^sup>1[f, g, tab\<^sub>1 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0] \<cdot> (f \<star> g \<star> fg\<^sub>0h\<^sub>1.\<phi>)) \<cdot> (f \<star> \<a>[g, tab\<^sub>0 g \<star> f\<^sub>0g\<^sub>1.p\<^sub>0, fg\<^sub>0h\<^sub>1.p\<^sub>1]) \<cdot> (f \<star> \<a>[g, tab\<^sub>0 g, f\<^sub>0g\<^sub>1.p\<^sub>0] \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> (f \<star> (g.tab \<star> f\<^sub>0g\<^sub>1.p\<^sub>0) \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> (f \<star> f\<^sub>0g\<^sub>1.\<phi> \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> (f \<star> \<a>\<^sup>-\<^sup>1[tab\<^sub>0 f, f\<^sub>0g\<^sub>1.p\<^sub>1, fg\<^sub>0h\<^sub>1.p\<^sub>1]) \<cdot> \<a>[f, tab\<^sub>0 f, f\<^sub>0g\<^sub>1.p\<^sub>1 \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1] \<cdot> (f.tab \<star> f\<^sub>0g\<^sub>1.p\<^sub>1 \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1)
[PROOF STEP]
by simp
[PROOF STATE]
proof (state)
this:
\<a>[f, g, h \<star> tab\<^sub>0 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0] \<cdot> ((f \<star> g) \<star> \<a>[h, tab\<^sub>0 h, fg\<^sub>0h\<^sub>1.p\<^sub>0]) \<cdot> (\<a>\<^sup>-\<^sup>1[f, g, (h \<star> tab\<^sub>0 h) \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0] \<cdot> (f \<star> g \<star> h.tab \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0)) \<cdot> (f \<star> g \<star> fg\<^sub>0h\<^sub>1.\<phi>) \<cdot> (f \<star> \<a>[g, tab\<^sub>0 g \<star> f\<^sub>0g\<^sub>1.p\<^sub>0, fg\<^sub>0h\<^sub>1.p\<^sub>1]) \<cdot> (f \<star> \<a>[g, tab\<^sub>0 g, f\<^sub>0g\<^sub>1.p\<^sub>0] \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> (f \<star> (g.tab \<star> f\<^sub>0g\<^sub>1.p\<^sub>0) \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> (f \<star> f\<^sub>0g\<^sub>1.\<phi> \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> (f \<star> \<a>\<^sup>-\<^sup>1[tab\<^sub>0 f, f\<^sub>0g\<^sub>1.p\<^sub>1, fg\<^sub>0h\<^sub>1.p\<^sub>1]) \<cdot> \<a>[f, tab\<^sub>0 f, f\<^sub>0g\<^sub>1.p\<^sub>1 \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1] \<cdot> (f.tab \<star> f\<^sub>0g\<^sub>1.p\<^sub>1 \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) = \<a>[f, g, h \<star> tab\<^sub>0 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0] \<cdot> ((f \<star> g) \<star> \<a>[h, tab\<^sub>0 h, fg\<^sub>0h\<^sub>1.p\<^sub>0]) \<cdot> ((f \<star> g) \<star> h.tab \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0) \<cdot> (\<a>\<^sup>-\<^sup>1[f, g, tab\<^sub>1 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0] \<cdot> (f \<star> g \<star> fg\<^sub>0h\<^sub>1.\<phi>)) \<cdot> (f \<star> \<a>[g, tab\<^sub>0 g \<star> f\<^sub>0g\<^sub>1.p\<^sub>0, fg\<^sub>0h\<^sub>1.p\<^sub>1]) \<cdot> (f \<star> \<a>[g, tab\<^sub>0 g, f\<^sub>0g\<^sub>1.p\<^sub>0] \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> (f \<star> (g.tab \<star> f\<^sub>0g\<^sub>1.p\<^sub>0) \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> (f \<star> f\<^sub>0g\<^sub>1.\<phi> \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> (f \<star> \<a>\<^sup>-\<^sup>1[tab\<^sub>0 f, f\<^sub>0g\<^sub>1.p\<^sub>1, fg\<^sub>0h\<^sub>1.p\<^sub>1]) \<cdot> \<a>[f, tab\<^sub>0 f, f\<^sub>0g\<^sub>1.p\<^sub>1 \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1] \<cdot> (f.tab \<star> f\<^sub>0g\<^sub>1.p\<^sub>1 \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1)
goal (1 subgoal):
1. (f \<star> \<theta>\<^sub>f) \<cdot> \<a>[f, tab\<^sub>0 f, w\<^sub>f] \<cdot> (f.tab \<star> w\<^sub>f) = ((f \<star> \<theta>\<^sub>f') \<cdot> \<a>[f, tab\<^sub>0 f, w\<^sub>f'] \<cdot> (f.tab \<star> w\<^sub>f')) \<cdot> \<beta>\<^sub>f
[PROOF STEP]
also
[PROOF STATE]
proof (state)
this:
\<a>[f, g, h \<star> tab\<^sub>0 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0] \<cdot> ((f \<star> g) \<star> \<a>[h, tab\<^sub>0 h, fg\<^sub>0h\<^sub>1.p\<^sub>0]) \<cdot> (\<a>\<^sup>-\<^sup>1[f, g, (h \<star> tab\<^sub>0 h) \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0] \<cdot> (f \<star> g \<star> h.tab \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0)) \<cdot> (f \<star> g \<star> fg\<^sub>0h\<^sub>1.\<phi>) \<cdot> (f \<star> \<a>[g, tab\<^sub>0 g \<star> f\<^sub>0g\<^sub>1.p\<^sub>0, fg\<^sub>0h\<^sub>1.p\<^sub>1]) \<cdot> (f \<star> \<a>[g, tab\<^sub>0 g, f\<^sub>0g\<^sub>1.p\<^sub>0] \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> (f \<star> (g.tab \<star> f\<^sub>0g\<^sub>1.p\<^sub>0) \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> (f \<star> f\<^sub>0g\<^sub>1.\<phi> \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> (f \<star> \<a>\<^sup>-\<^sup>1[tab\<^sub>0 f, f\<^sub>0g\<^sub>1.p\<^sub>1, fg\<^sub>0h\<^sub>1.p\<^sub>1]) \<cdot> \<a>[f, tab\<^sub>0 f, f\<^sub>0g\<^sub>1.p\<^sub>1 \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1] \<cdot> (f.tab \<star> f\<^sub>0g\<^sub>1.p\<^sub>1 \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) = \<a>[f, g, h \<star> tab\<^sub>0 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0] \<cdot> ((f \<star> g) \<star> \<a>[h, tab\<^sub>0 h, fg\<^sub>0h\<^sub>1.p\<^sub>0]) \<cdot> ((f \<star> g) \<star> h.tab \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0) \<cdot> (\<a>\<^sup>-\<^sup>1[f, g, tab\<^sub>1 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0] \<cdot> (f \<star> g \<star> fg\<^sub>0h\<^sub>1.\<phi>)) \<cdot> (f \<star> \<a>[g, tab\<^sub>0 g \<star> f\<^sub>0g\<^sub>1.p\<^sub>0, fg\<^sub>0h\<^sub>1.p\<^sub>1]) \<cdot> (f \<star> \<a>[g, tab\<^sub>0 g, f\<^sub>0g\<^sub>1.p\<^sub>0] \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> (f \<star> (g.tab \<star> f\<^sub>0g\<^sub>1.p\<^sub>0) \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> (f \<star> f\<^sub>0g\<^sub>1.\<phi> \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> (f \<star> \<a>\<^sup>-\<^sup>1[tab\<^sub>0 f, f\<^sub>0g\<^sub>1.p\<^sub>1, fg\<^sub>0h\<^sub>1.p\<^sub>1]) \<cdot> \<a>[f, tab\<^sub>0 f, f\<^sub>0g\<^sub>1.p\<^sub>1 \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1] \<cdot> (f.tab \<star> f\<^sub>0g\<^sub>1.p\<^sub>1 \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1)
goal (1 subgoal):
1. (f \<star> \<theta>\<^sub>f) \<cdot> \<a>[f, tab\<^sub>0 f, w\<^sub>f] \<cdot> (f.tab \<star> w\<^sub>f) = ((f \<star> \<theta>\<^sub>f') \<cdot> \<a>[f, tab\<^sub>0 f, w\<^sub>f'] \<cdot> (f.tab \<star> w\<^sub>f')) \<cdot> \<beta>\<^sub>f
[PROOF STEP]
have "... =
\<a>[f, g, h \<star> tab\<^sub>0 h \<star> TTfgh.p\<^sub>0] \<cdot>
((f \<star> g) \<star> \<a>[h, tab\<^sub>0 h, TTfgh.p\<^sub>0]) \<cdot>
((f \<star> g) \<star> h.tab \<star> TTfgh.p\<^sub>0) \<cdot>
((f \<star> g) \<star> fg\<^sub>0h\<^sub>1.\<phi>) \<cdot>
\<a>\<^sup>-\<^sup>1[f, g, (tab\<^sub>0 g \<star> Tfg.p\<^sub>0) \<star> TTfgh.p\<^sub>1] \<cdot>
(f \<star> \<a>[g, tab\<^sub>0 g \<star> Tfg.p\<^sub>0, TTfgh.p\<^sub>1]) \<cdot>
(f \<star> \<a>[g, tab\<^sub>0 g, Tfg.p\<^sub>0] \<star> TTfgh.p\<^sub>1) \<cdot>
(f \<star> (g.tab \<star> Tfg.p\<^sub>0) \<star> TTfgh.p\<^sub>1) \<cdot>
(f \<star> f\<^sub>0g\<^sub>1.\<phi> \<star> TTfgh.p\<^sub>1) \<cdot>
((f \<star> \<a>\<^sup>-\<^sup>1[tab\<^sub>0 f, Tfg.p\<^sub>1, TTfgh.p\<^sub>1]) \<cdot>
\<a>[f, tab\<^sub>0 f, Tfg.p\<^sub>1 \<star> TTfgh.p\<^sub>1]) \<cdot>
(f.tab \<star> Tfg.p\<^sub>1 \<star> TTfgh.p\<^sub>1)"
[PROOF STATE]
proof (prove)
goal (1 subgoal):
1. \<a>[f, g, h \<star> tab\<^sub>0 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0] \<cdot> ((f \<star> g) \<star> \<a>[h, tab\<^sub>0 h, fg\<^sub>0h\<^sub>1.p\<^sub>0]) \<cdot> ((f \<star> g) \<star> h.tab \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0) \<cdot> (\<a>\<^sup>-\<^sup>1[f, g, tab\<^sub>1 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0] \<cdot> (f \<star> g \<star> fg\<^sub>0h\<^sub>1.\<phi>)) \<cdot> (f \<star> \<a>[g, tab\<^sub>0 g \<star> f\<^sub>0g\<^sub>1.p\<^sub>0, fg\<^sub>0h\<^sub>1.p\<^sub>1]) \<cdot> (f \<star> \<a>[g, tab\<^sub>0 g, f\<^sub>0g\<^sub>1.p\<^sub>0] \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> (f \<star> (g.tab \<star> f\<^sub>0g\<^sub>1.p\<^sub>0) \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> (f \<star> f\<^sub>0g\<^sub>1.\<phi> \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> (f \<star> \<a>\<^sup>-\<^sup>1[tab\<^sub>0 f, f\<^sub>0g\<^sub>1.p\<^sub>1, fg\<^sub>0h\<^sub>1.p\<^sub>1]) \<cdot> \<a>[f, tab\<^sub>0 f, f\<^sub>0g\<^sub>1.p\<^sub>1 \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1] \<cdot> (f.tab \<star> f\<^sub>0g\<^sub>1.p\<^sub>1 \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) = \<a>[f, g, h \<star> tab\<^sub>0 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0] \<cdot> ((f \<star> g) \<star> \<a>[h, tab\<^sub>0 h, fg\<^sub>0h\<^sub>1.p\<^sub>0]) \<cdot> ((f \<star> g) \<star> h.tab \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0) \<cdot> ((f \<star> g) \<star> fg\<^sub>0h\<^sub>1.\<phi>) \<cdot> \<a>\<^sup>-\<^sup>1[f, g, (tab\<^sub>0 g \<star> f\<^sub>0g\<^sub>1.p\<^sub>0) \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1] \<cdot> (f \<star> \<a>[g, tab\<^sub>0 g \<star> f\<^sub>0g\<^sub>1.p\<^sub>0, fg\<^sub>0h\<^sub>1.p\<^sub>1]) \<cdot> (f \<star> \<a>[g, tab\<^sub>0 g, f\<^sub>0g\<^sub>1.p\<^sub>0] \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> (f \<star> (g.tab \<star> f\<^sub>0g\<^sub>1.p\<^sub>0) \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> (f \<star> f\<^sub>0g\<^sub>1.\<phi> \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> ((f \<star> \<a>\<^sup>-\<^sup>1[tab\<^sub>0 f, f\<^sub>0g\<^sub>1.p\<^sub>1, fg\<^sub>0h\<^sub>1.p\<^sub>1]) \<cdot> \<a>[f, tab\<^sub>0 f, f\<^sub>0g\<^sub>1.p\<^sub>1 \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1]) \<cdot> (f.tab \<star> f\<^sub>0g\<^sub>1.p\<^sub>1 \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1)
[PROOF STEP]
using fg gh fg\<^sub>0h\<^sub>1.p\<^sub>0_simps fg\<^sub>0h\<^sub>1.p\<^sub>1_simps comp_assoc
assoc'_naturality [of f g fg\<^sub>0h\<^sub>1.\<phi>]
[PROOF STATE]
proof (prove)
using this:
src f = trg g
src g = trg h
trg fg\<^sub>0h\<^sub>1.p\<^sub>0 = src (tab\<^sub>1 h)
trg fg\<^sub>0h\<^sub>1.p\<^sub>1 = src (tab\<^sub>0 g \<star> f\<^sub>0g\<^sub>1.p\<^sub>0)
(?h \<cdot> ?g) \<cdot> ?f = ?h \<cdot> ?g \<cdot> ?f
\<lbrakk>arr f; arr g; arr fg\<^sub>0h\<^sub>1.\<phi>; src f = trg g; src g = trg fg\<^sub>0h\<^sub>1.\<phi>\<rbrakk> \<Longrightarrow> \<a>\<^sup>-\<^sup>1[cod f, cod g, cod fg\<^sub>0h\<^sub>1.\<phi>] \<cdot> (f \<star> g \<star> fg\<^sub>0h\<^sub>1.\<phi>) = ((f \<star> g) \<star> fg\<^sub>0h\<^sub>1.\<phi>) \<cdot> \<a>\<^sup>-\<^sup>1[local.dom f, local.dom g, local.dom fg\<^sub>0h\<^sub>1.\<phi>]
goal (1 subgoal):
1. \<a>[f, g, h \<star> tab\<^sub>0 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0] \<cdot> ((f \<star> g) \<star> \<a>[h, tab\<^sub>0 h, fg\<^sub>0h\<^sub>1.p\<^sub>0]) \<cdot> ((f \<star> g) \<star> h.tab \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0) \<cdot> (\<a>\<^sup>-\<^sup>1[f, g, tab\<^sub>1 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0] \<cdot> (f \<star> g \<star> fg\<^sub>0h\<^sub>1.\<phi>)) \<cdot> (f \<star> \<a>[g, tab\<^sub>0 g \<star> f\<^sub>0g\<^sub>1.p\<^sub>0, fg\<^sub>0h\<^sub>1.p\<^sub>1]) \<cdot> (f \<star> \<a>[g, tab\<^sub>0 g, f\<^sub>0g\<^sub>1.p\<^sub>0] \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> (f \<star> (g.tab \<star> f\<^sub>0g\<^sub>1.p\<^sub>0) \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> (f \<star> f\<^sub>0g\<^sub>1.\<phi> \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> (f \<star> \<a>\<^sup>-\<^sup>1[tab\<^sub>0 f, f\<^sub>0g\<^sub>1.p\<^sub>1, fg\<^sub>0h\<^sub>1.p\<^sub>1]) \<cdot> \<a>[f, tab\<^sub>0 f, f\<^sub>0g\<^sub>1.p\<^sub>1 \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1] \<cdot> (f.tab \<star> f\<^sub>0g\<^sub>1.p\<^sub>1 \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) = \<a>[f, g, h \<star> tab\<^sub>0 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0] \<cdot> ((f \<star> g) \<star> \<a>[h, tab\<^sub>0 h, fg\<^sub>0h\<^sub>1.p\<^sub>0]) \<cdot> ((f \<star> g) \<star> h.tab \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0) \<cdot> ((f \<star> g) \<star> fg\<^sub>0h\<^sub>1.\<phi>) \<cdot> \<a>\<^sup>-\<^sup>1[f, g, (tab\<^sub>0 g \<star> f\<^sub>0g\<^sub>1.p\<^sub>0) \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1] \<cdot> (f \<star> \<a>[g, tab\<^sub>0 g \<star> f\<^sub>0g\<^sub>1.p\<^sub>0, fg\<^sub>0h\<^sub>1.p\<^sub>1]) \<cdot> (f \<star> \<a>[g, tab\<^sub>0 g, f\<^sub>0g\<^sub>1.p\<^sub>0] \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> (f \<star> (g.tab \<star> f\<^sub>0g\<^sub>1.p\<^sub>0) \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> (f \<star> f\<^sub>0g\<^sub>1.\<phi> \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> ((f \<star> \<a>\<^sup>-\<^sup>1[tab\<^sub>0 f, f\<^sub>0g\<^sub>1.p\<^sub>1, fg\<^sub>0h\<^sub>1.p\<^sub>1]) \<cdot> \<a>[f, tab\<^sub>0 f, f\<^sub>0g\<^sub>1.p\<^sub>1 \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1]) \<cdot> (f.tab \<star> f\<^sub>0g\<^sub>1.p\<^sub>1 \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1)
[PROOF STEP]
by simp
[PROOF STATE]
proof (state)
this:
\<a>[f, g, h \<star> tab\<^sub>0 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0] \<cdot> ((f \<star> g) \<star> \<a>[h, tab\<^sub>0 h, fg\<^sub>0h\<^sub>1.p\<^sub>0]) \<cdot> ((f \<star> g) \<star> h.tab \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0) \<cdot> (\<a>\<^sup>-\<^sup>1[f, g, tab\<^sub>1 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0] \<cdot> (f \<star> g \<star> fg\<^sub>0h\<^sub>1.\<phi>)) \<cdot> (f \<star> \<a>[g, tab\<^sub>0 g \<star> f\<^sub>0g\<^sub>1.p\<^sub>0, fg\<^sub>0h\<^sub>1.p\<^sub>1]) \<cdot> (f \<star> \<a>[g, tab\<^sub>0 g, f\<^sub>0g\<^sub>1.p\<^sub>0] \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> (f \<star> (g.tab \<star> f\<^sub>0g\<^sub>1.p\<^sub>0) \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> (f \<star> f\<^sub>0g\<^sub>1.\<phi> \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> (f \<star> \<a>\<^sup>-\<^sup>1[tab\<^sub>0 f, f\<^sub>0g\<^sub>1.p\<^sub>1, fg\<^sub>0h\<^sub>1.p\<^sub>1]) \<cdot> \<a>[f, tab\<^sub>0 f, f\<^sub>0g\<^sub>1.p\<^sub>1 \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1] \<cdot> (f.tab \<star> f\<^sub>0g\<^sub>1.p\<^sub>1 \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) = \<a>[f, g, h \<star> tab\<^sub>0 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0] \<cdot> ((f \<star> g) \<star> \<a>[h, tab\<^sub>0 h, fg\<^sub>0h\<^sub>1.p\<^sub>0]) \<cdot> ((f \<star> g) \<star> h.tab \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0) \<cdot> ((f \<star> g) \<star> fg\<^sub>0h\<^sub>1.\<phi>) \<cdot> \<a>\<^sup>-\<^sup>1[f, g, (tab\<^sub>0 g \<star> f\<^sub>0g\<^sub>1.p\<^sub>0) \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1] \<cdot> (f \<star> \<a>[g, tab\<^sub>0 g \<star> f\<^sub>0g\<^sub>1.p\<^sub>0, fg\<^sub>0h\<^sub>1.p\<^sub>1]) \<cdot> (f \<star> \<a>[g, tab\<^sub>0 g, f\<^sub>0g\<^sub>1.p\<^sub>0] \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> (f \<star> (g.tab \<star> f\<^sub>0g\<^sub>1.p\<^sub>0) \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> (f \<star> f\<^sub>0g\<^sub>1.\<phi> \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> ((f \<star> \<a>\<^sup>-\<^sup>1[tab\<^sub>0 f, f\<^sub>0g\<^sub>1.p\<^sub>1, fg\<^sub>0h\<^sub>1.p\<^sub>1]) \<cdot> \<a>[f, tab\<^sub>0 f, f\<^sub>0g\<^sub>1.p\<^sub>1 \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1]) \<cdot> (f.tab \<star> f\<^sub>0g\<^sub>1.p\<^sub>1 \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1)
goal (1 subgoal):
1. (f \<star> \<theta>\<^sub>f) \<cdot> \<a>[f, tab\<^sub>0 f, w\<^sub>f] \<cdot> (f.tab \<star> w\<^sub>f) = ((f \<star> \<theta>\<^sub>f') \<cdot> \<a>[f, tab\<^sub>0 f, w\<^sub>f'] \<cdot> (f.tab \<star> w\<^sub>f')) \<cdot> \<beta>\<^sub>f
[PROOF STEP]
also
[PROOF STATE]
proof (state)
this:
\<a>[f, g, h \<star> tab\<^sub>0 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0] \<cdot> ((f \<star> g) \<star> \<a>[h, tab\<^sub>0 h, fg\<^sub>0h\<^sub>1.p\<^sub>0]) \<cdot> ((f \<star> g) \<star> h.tab \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0) \<cdot> (\<a>\<^sup>-\<^sup>1[f, g, tab\<^sub>1 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0] \<cdot> (f \<star> g \<star> fg\<^sub>0h\<^sub>1.\<phi>)) \<cdot> (f \<star> \<a>[g, tab\<^sub>0 g \<star> f\<^sub>0g\<^sub>1.p\<^sub>0, fg\<^sub>0h\<^sub>1.p\<^sub>1]) \<cdot> (f \<star> \<a>[g, tab\<^sub>0 g, f\<^sub>0g\<^sub>1.p\<^sub>0] \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> (f \<star> (g.tab \<star> f\<^sub>0g\<^sub>1.p\<^sub>0) \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> (f \<star> f\<^sub>0g\<^sub>1.\<phi> \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> (f \<star> \<a>\<^sup>-\<^sup>1[tab\<^sub>0 f, f\<^sub>0g\<^sub>1.p\<^sub>1, fg\<^sub>0h\<^sub>1.p\<^sub>1]) \<cdot> \<a>[f, tab\<^sub>0 f, f\<^sub>0g\<^sub>1.p\<^sub>1 \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1] \<cdot> (f.tab \<star> f\<^sub>0g\<^sub>1.p\<^sub>1 \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) = \<a>[f, g, h \<star> tab\<^sub>0 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0] \<cdot> ((f \<star> g) \<star> \<a>[h, tab\<^sub>0 h, fg\<^sub>0h\<^sub>1.p\<^sub>0]) \<cdot> ((f \<star> g) \<star> h.tab \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0) \<cdot> ((f \<star> g) \<star> fg\<^sub>0h\<^sub>1.\<phi>) \<cdot> \<a>\<^sup>-\<^sup>1[f, g, (tab\<^sub>0 g \<star> f\<^sub>0g\<^sub>1.p\<^sub>0) \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1] \<cdot> (f \<star> \<a>[g, tab\<^sub>0 g \<star> f\<^sub>0g\<^sub>1.p\<^sub>0, fg\<^sub>0h\<^sub>1.p\<^sub>1]) \<cdot> (f \<star> \<a>[g, tab\<^sub>0 g, f\<^sub>0g\<^sub>1.p\<^sub>0] \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> (f \<star> (g.tab \<star> f\<^sub>0g\<^sub>1.p\<^sub>0) \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> (f \<star> f\<^sub>0g\<^sub>1.\<phi> \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> ((f \<star> \<a>\<^sup>-\<^sup>1[tab\<^sub>0 f, f\<^sub>0g\<^sub>1.p\<^sub>1, fg\<^sub>0h\<^sub>1.p\<^sub>1]) \<cdot> \<a>[f, tab\<^sub>0 f, f\<^sub>0g\<^sub>1.p\<^sub>1 \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1]) \<cdot> (f.tab \<star> f\<^sub>0g\<^sub>1.p\<^sub>1 \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1)
goal (1 subgoal):
1. (f \<star> \<theta>\<^sub>f) \<cdot> \<a>[f, tab\<^sub>0 f, w\<^sub>f] \<cdot> (f.tab \<star> w\<^sub>f) = ((f \<star> \<theta>\<^sub>f') \<cdot> \<a>[f, tab\<^sub>0 f, w\<^sub>f'] \<cdot> (f.tab \<star> w\<^sub>f')) \<cdot> \<beta>\<^sub>f
[PROOF STEP]
have "... =
\<a>[f, g, h \<star> tab\<^sub>0 h \<star> TTfgh.p\<^sub>0] \<cdot>
((f \<star> g) \<star> \<a>[h, tab\<^sub>0 h, TTfgh.p\<^sub>0]) \<cdot>
((f \<star> g) \<star> h.tab \<star> TTfgh.p\<^sub>0) \<cdot>
((f \<star> g) \<star> fg\<^sub>0h\<^sub>1.\<phi>) \<cdot>
\<a>\<^sup>-\<^sup>1[f, g, (tab\<^sub>0 g \<star> Tfg.p\<^sub>0) \<star> TTfgh.p\<^sub>1] \<cdot>
(f \<star> \<a>[g, tab\<^sub>0 g \<star> Tfg.p\<^sub>0, TTfgh.p\<^sub>1]) \<cdot>
(f \<star> \<a>[g, tab\<^sub>0 g, Tfg.p\<^sub>0] \<star> TTfgh.p\<^sub>1) \<cdot>
(f \<star> (g.tab \<star> Tfg.p\<^sub>0) \<star> TTfgh.p\<^sub>1) \<cdot>
((f \<star> f\<^sub>0g\<^sub>1.\<phi> \<star> TTfgh.p\<^sub>1) \<cdot>
\<a>[f, tab\<^sub>0 f \<star> Tfg.p\<^sub>1, TTfgh.p\<^sub>1]) \<cdot>
(\<a>[f, tab\<^sub>0 f, Tfg.p\<^sub>1] \<star> TTfgh.p\<^sub>1) \<cdot>
\<a>\<^sup>-\<^sup>1[f \<star> tab\<^sub>0 f, Tfg.p\<^sub>1, TTfgh.p\<^sub>1] \<cdot>
(f.tab \<star> Tfg.p\<^sub>1 \<star> TTfgh.p\<^sub>1)"
[PROOF STATE]
proof (prove)
goal (1 subgoal):
1. \<a>[f, g, h \<star> tab\<^sub>0 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0] \<cdot> ((f \<star> g) \<star> \<a>[h, tab\<^sub>0 h, fg\<^sub>0h\<^sub>1.p\<^sub>0]) \<cdot> ((f \<star> g) \<star> h.tab \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0) \<cdot> ((f \<star> g) \<star> fg\<^sub>0h\<^sub>1.\<phi>) \<cdot> \<a>\<^sup>-\<^sup>1[f, g, (tab\<^sub>0 g \<star> f\<^sub>0g\<^sub>1.p\<^sub>0) \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1] \<cdot> (f \<star> \<a>[g, tab\<^sub>0 g \<star> f\<^sub>0g\<^sub>1.p\<^sub>0, fg\<^sub>0h\<^sub>1.p\<^sub>1]) \<cdot> (f \<star> \<a>[g, tab\<^sub>0 g, f\<^sub>0g\<^sub>1.p\<^sub>0] \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> (f \<star> (g.tab \<star> f\<^sub>0g\<^sub>1.p\<^sub>0) \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> (f \<star> f\<^sub>0g\<^sub>1.\<phi> \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> ((f \<star> \<a>\<^sup>-\<^sup>1[tab\<^sub>0 f, f\<^sub>0g\<^sub>1.p\<^sub>1, fg\<^sub>0h\<^sub>1.p\<^sub>1]) \<cdot> \<a>[f, tab\<^sub>0 f, f\<^sub>0g\<^sub>1.p\<^sub>1 \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1]) \<cdot> (f.tab \<star> f\<^sub>0g\<^sub>1.p\<^sub>1 \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) = \<a>[f, g, h \<star> tab\<^sub>0 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0] \<cdot> ((f \<star> g) \<star> \<a>[h, tab\<^sub>0 h, fg\<^sub>0h\<^sub>1.p\<^sub>0]) \<cdot> ((f \<star> g) \<star> h.tab \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0) \<cdot> ((f \<star> g) \<star> fg\<^sub>0h\<^sub>1.\<phi>) \<cdot> \<a>\<^sup>-\<^sup>1[f, g, (tab\<^sub>0 g \<star> f\<^sub>0g\<^sub>1.p\<^sub>0) \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1] \<cdot> (f \<star> \<a>[g, tab\<^sub>0 g \<star> f\<^sub>0g\<^sub>1.p\<^sub>0, fg\<^sub>0h\<^sub>1.p\<^sub>1]) \<cdot> (f \<star> \<a>[g, tab\<^sub>0 g, f\<^sub>0g\<^sub>1.p\<^sub>0] \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> (f \<star> (g.tab \<star> f\<^sub>0g\<^sub>1.p\<^sub>0) \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> ((f \<star> f\<^sub>0g\<^sub>1.\<phi> \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> \<a>[f, tab\<^sub>0 f \<star> f\<^sub>0g\<^sub>1.p\<^sub>1, fg\<^sub>0h\<^sub>1.p\<^sub>1]) \<cdot> (\<a>[f, tab\<^sub>0 f, f\<^sub>0g\<^sub>1.p\<^sub>1] \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> \<a>\<^sup>-\<^sup>1[f \<star> tab\<^sub>0 f, f\<^sub>0g\<^sub>1.p\<^sub>1, fg\<^sub>0h\<^sub>1.p\<^sub>1] \<cdot> (f.tab \<star> f\<^sub>0g\<^sub>1.p\<^sub>1 \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1)
[PROOF STEP]
proof -
[PROOF STATE]
proof (state)
goal (1 subgoal):
1. \<a>[f, g, h \<star> tab\<^sub>0 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0] \<cdot> ((f \<star> g) \<star> \<a>[h, tab\<^sub>0 h, fg\<^sub>0h\<^sub>1.p\<^sub>0]) \<cdot> ((f \<star> g) \<star> h.tab \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0) \<cdot> ((f \<star> g) \<star> fg\<^sub>0h\<^sub>1.\<phi>) \<cdot> \<a>\<^sup>-\<^sup>1[f, g, (tab\<^sub>0 g \<star> f\<^sub>0g\<^sub>1.p\<^sub>0) \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1] \<cdot> (f \<star> \<a>[g, tab\<^sub>0 g \<star> f\<^sub>0g\<^sub>1.p\<^sub>0, fg\<^sub>0h\<^sub>1.p\<^sub>1]) \<cdot> (f \<star> \<a>[g, tab\<^sub>0 g, f\<^sub>0g\<^sub>1.p\<^sub>0] \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> (f \<star> (g.tab \<star> f\<^sub>0g\<^sub>1.p\<^sub>0) \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> (f \<star> f\<^sub>0g\<^sub>1.\<phi> \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> ((f \<star> \<a>\<^sup>-\<^sup>1[tab\<^sub>0 f, f\<^sub>0g\<^sub>1.p\<^sub>1, fg\<^sub>0h\<^sub>1.p\<^sub>1]) \<cdot> \<a>[f, tab\<^sub>0 f, f\<^sub>0g\<^sub>1.p\<^sub>1 \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1]) \<cdot> (f.tab \<star> f\<^sub>0g\<^sub>1.p\<^sub>1 \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) = \<a>[f, g, h \<star> tab\<^sub>0 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0] \<cdot> ((f \<star> g) \<star> \<a>[h, tab\<^sub>0 h, fg\<^sub>0h\<^sub>1.p\<^sub>0]) \<cdot> ((f \<star> g) \<star> h.tab \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0) \<cdot> ((f \<star> g) \<star> fg\<^sub>0h\<^sub>1.\<phi>) \<cdot> \<a>\<^sup>-\<^sup>1[f, g, (tab\<^sub>0 g \<star> f\<^sub>0g\<^sub>1.p\<^sub>0) \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1] \<cdot> (f \<star> \<a>[g, tab\<^sub>0 g \<star> f\<^sub>0g\<^sub>1.p\<^sub>0, fg\<^sub>0h\<^sub>1.p\<^sub>1]) \<cdot> (f \<star> \<a>[g, tab\<^sub>0 g, f\<^sub>0g\<^sub>1.p\<^sub>0] \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> (f \<star> (g.tab \<star> f\<^sub>0g\<^sub>1.p\<^sub>0) \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> ((f \<star> f\<^sub>0g\<^sub>1.\<phi> \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> \<a>[f, tab\<^sub>0 f \<star> f\<^sub>0g\<^sub>1.p\<^sub>1, fg\<^sub>0h\<^sub>1.p\<^sub>1]) \<cdot> (\<a>[f, tab\<^sub>0 f, f\<^sub>0g\<^sub>1.p\<^sub>1] \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> \<a>\<^sup>-\<^sup>1[f \<star> tab\<^sub>0 f, f\<^sub>0g\<^sub>1.p\<^sub>1, fg\<^sub>0h\<^sub>1.p\<^sub>1] \<cdot> (f.tab \<star> f\<^sub>0g\<^sub>1.p\<^sub>1 \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1)
[PROOF STEP]
have "(f \<star> \<a>\<^sup>-\<^sup>1[tab\<^sub>0 f, Tfg.p\<^sub>1, TTfgh.p\<^sub>1]) \<cdot> \<a>[f, tab\<^sub>0 f, Tfg.p\<^sub>1 \<star> TTfgh.p\<^sub>1] =
\<a>[f, tab\<^sub>0 f \<star> Tfg.p\<^sub>1, TTfgh.p\<^sub>1] \<cdot> (\<a>[f, tab\<^sub>0 f, Tfg.p\<^sub>1] \<star> TTfgh.p\<^sub>1) \<cdot>
\<a>\<^sup>-\<^sup>1[f \<star> tab\<^sub>0 f, Tfg.p\<^sub>1, TTfgh.p\<^sub>1]"
[PROOF STATE]
proof (prove)
goal (1 subgoal):
1. (f \<star> \<a>\<^sup>-\<^sup>1[tab\<^sub>0 f, f\<^sub>0g\<^sub>1.p\<^sub>1, fg\<^sub>0h\<^sub>1.p\<^sub>1]) \<cdot> \<a>[f, tab\<^sub>0 f, f\<^sub>0g\<^sub>1.p\<^sub>1 \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1] = \<a>[f, tab\<^sub>0 f \<star> f\<^sub>0g\<^sub>1.p\<^sub>1, fg\<^sub>0h\<^sub>1.p\<^sub>1] \<cdot> (\<a>[f, tab\<^sub>0 f, f\<^sub>0g\<^sub>1.p\<^sub>1] \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> \<a>\<^sup>-\<^sup>1[f \<star> tab\<^sub>0 f, f\<^sub>0g\<^sub>1.p\<^sub>1, fg\<^sub>0h\<^sub>1.p\<^sub>1]
[PROOF STEP]
proof -
[PROOF STATE]
proof (state)
goal (1 subgoal):
1. (f \<star> \<a>\<^sup>-\<^sup>1[tab\<^sub>0 f, f\<^sub>0g\<^sub>1.p\<^sub>1, fg\<^sub>0h\<^sub>1.p\<^sub>1]) \<cdot> \<a>[f, tab\<^sub>0 f, f\<^sub>0g\<^sub>1.p\<^sub>1 \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1] = \<a>[f, tab\<^sub>0 f \<star> f\<^sub>0g\<^sub>1.p\<^sub>1, fg\<^sub>0h\<^sub>1.p\<^sub>1] \<cdot> (\<a>[f, tab\<^sub>0 f, f\<^sub>0g\<^sub>1.p\<^sub>1] \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> \<a>\<^sup>-\<^sup>1[f \<star> tab\<^sub>0 f, f\<^sub>0g\<^sub>1.p\<^sub>1, fg\<^sub>0h\<^sub>1.p\<^sub>1]
[PROOF STEP]
have "(f \<star> \<a>\<^sup>-\<^sup>1[tab\<^sub>0 f, Tfg.p\<^sub>1, TTfgh.p\<^sub>1]) \<cdot> \<a>[f, tab\<^sub>0 f, Tfg.p\<^sub>1 \<star> TTfgh.p\<^sub>1] =
\<lbrace>(\<^bold>\<langle>f\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<a>\<^sup>-\<^sup>1\<^bold>[\<^bold>\<langle>tab\<^sub>0 f\<^bold>\<rangle>, \<^bold>\<langle>Tfg.p\<^sub>1\<^bold>\<rangle>, \<^bold>\<langle>TTfgh.p\<^sub>1\<^bold>\<rangle>\<^bold>]) \<^bold>\<cdot>
\<^bold>\<a>\<^bold>[\<^bold>\<langle>f\<^bold>\<rangle>, \<^bold>\<langle>tab\<^sub>0 f\<^bold>\<rangle>, \<^bold>\<langle>Tfg.p\<^sub>1\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>TTfgh.p\<^sub>1\<^bold>\<rangle>\<^bold>]\<rbrace>"
[PROOF STATE]
proof (prove)
goal (1 subgoal):
1. (f \<star> \<a>\<^sup>-\<^sup>1[tab\<^sub>0 f, f\<^sub>0g\<^sub>1.p\<^sub>1, fg\<^sub>0h\<^sub>1.p\<^sub>1]) \<cdot> \<a>[f, tab\<^sub>0 f, f\<^sub>0g\<^sub>1.p\<^sub>1 \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1] = \<lbrace>(\<^bold>\<langle>f\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<a>\<^sup>-\<^sup>1\<^bold>[\<^bold>\<langle>tab\<^sub>0 f\<^bold>\<rangle>, \<^bold>\<langle>f\<^sub>0g\<^sub>1.p\<^sub>1\<^bold>\<rangle>, \<^bold>\<langle>fg\<^sub>0h\<^sub>1.p\<^sub>1\<^bold>\<rangle>\<^bold>]) \<^bold>\<cdot> \<^bold>\<a>\<^bold>[\<^bold>\<langle>f\<^bold>\<rangle>, \<^bold>\<langle>tab\<^sub>0 f\<^bold>\<rangle>, \<^bold>\<langle>f\<^sub>0g\<^sub>1.p\<^sub>1\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>fg\<^sub>0h\<^sub>1.p\<^sub>1\<^bold>\<rangle>\<^bold>]\<rbrace>
[PROOF STEP]
using fg gh fg\<^sub>0h\<^sub>1.p\<^sub>0_simps fg\<^sub>0h\<^sub>1.p\<^sub>1_simps f\<^sub>0g\<^sub>1.p\<^sub>0_simps f\<^sub>0g\<^sub>1.p\<^sub>1_simps
\<a>'_def \<alpha>_def
[PROOF STATE]
proof (prove)
using this:
src f = trg g
src g = trg h
trg fg\<^sub>0h\<^sub>1.p\<^sub>0 = src (tab\<^sub>1 h)
trg fg\<^sub>0h\<^sub>1.p\<^sub>1 = src (tab\<^sub>0 g \<star> f\<^sub>0g\<^sub>1.p\<^sub>0)
trg f\<^sub>0g\<^sub>1.p\<^sub>0 = src (tab\<^sub>1 g)
trg f\<^sub>0g\<^sub>1.p\<^sub>1 = src (tab\<^sub>0 f)
\<a>\<^sup>-\<^sup>1[?\<mu>, ?\<nu>, ?\<tau>] \<equiv> \<alpha>' (?\<mu>, ?\<nu>, ?\<tau>)
\<alpha> ?\<mu>\<nu>\<tau> \<equiv> \<a>[fst ?\<mu>\<nu>\<tau>, fst (snd ?\<mu>\<nu>\<tau>), snd (snd ?\<mu>\<nu>\<tau>)]
goal (1 subgoal):
1. (f \<star> \<a>\<^sup>-\<^sup>1[tab\<^sub>0 f, f\<^sub>0g\<^sub>1.p\<^sub>1, fg\<^sub>0h\<^sub>1.p\<^sub>1]) \<cdot> \<a>[f, tab\<^sub>0 f, f\<^sub>0g\<^sub>1.p\<^sub>1 \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1] = \<lbrace>(\<^bold>\<langle>f\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<a>\<^sup>-\<^sup>1\<^bold>[\<^bold>\<langle>tab\<^sub>0 f\<^bold>\<rangle>, \<^bold>\<langle>f\<^sub>0g\<^sub>1.p\<^sub>1\<^bold>\<rangle>, \<^bold>\<langle>fg\<^sub>0h\<^sub>1.p\<^sub>1\<^bold>\<rangle>\<^bold>]) \<^bold>\<cdot> \<^bold>\<a>\<^bold>[\<^bold>\<langle>f\<^bold>\<rangle>, \<^bold>\<langle>tab\<^sub>0 f\<^bold>\<rangle>, \<^bold>\<langle>f\<^sub>0g\<^sub>1.p\<^sub>1\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>fg\<^sub>0h\<^sub>1.p\<^sub>1\<^bold>\<rangle>\<^bold>]\<rbrace>
[PROOF STEP]
by simp
[PROOF STATE]
proof (state)
this:
(f \<star> \<a>\<^sup>-\<^sup>1[tab\<^sub>0 f, f\<^sub>0g\<^sub>1.p\<^sub>1, fg\<^sub>0h\<^sub>1.p\<^sub>1]) \<cdot> \<a>[f, tab\<^sub>0 f, f\<^sub>0g\<^sub>1.p\<^sub>1 \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1] = \<lbrace>(\<^bold>\<langle>f\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<a>\<^sup>-\<^sup>1\<^bold>[\<^bold>\<langle>tab\<^sub>0 f\<^bold>\<rangle>, \<^bold>\<langle>f\<^sub>0g\<^sub>1.p\<^sub>1\<^bold>\<rangle>, \<^bold>\<langle>fg\<^sub>0h\<^sub>1.p\<^sub>1\<^bold>\<rangle>\<^bold>]) \<^bold>\<cdot> \<^bold>\<a>\<^bold>[\<^bold>\<langle>f\<^bold>\<rangle>, \<^bold>\<langle>tab\<^sub>0 f\<^bold>\<rangle>, \<^bold>\<langle>f\<^sub>0g\<^sub>1.p\<^sub>1\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>fg\<^sub>0h\<^sub>1.p\<^sub>1\<^bold>\<rangle>\<^bold>]\<rbrace>
goal (1 subgoal):
1. (f \<star> \<a>\<^sup>-\<^sup>1[tab\<^sub>0 f, f\<^sub>0g\<^sub>1.p\<^sub>1, fg\<^sub>0h\<^sub>1.p\<^sub>1]) \<cdot> \<a>[f, tab\<^sub>0 f, f\<^sub>0g\<^sub>1.p\<^sub>1 \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1] = \<a>[f, tab\<^sub>0 f \<star> f\<^sub>0g\<^sub>1.p\<^sub>1, fg\<^sub>0h\<^sub>1.p\<^sub>1] \<cdot> (\<a>[f, tab\<^sub>0 f, f\<^sub>0g\<^sub>1.p\<^sub>1] \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> \<a>\<^sup>-\<^sup>1[f \<star> tab\<^sub>0 f, f\<^sub>0g\<^sub>1.p\<^sub>1, fg\<^sub>0h\<^sub>1.p\<^sub>1]
[PROOF STEP]
also
[PROOF STATE]
proof (state)
this:
(f \<star> \<a>\<^sup>-\<^sup>1[tab\<^sub>0 f, f\<^sub>0g\<^sub>1.p\<^sub>1, fg\<^sub>0h\<^sub>1.p\<^sub>1]) \<cdot> \<a>[f, tab\<^sub>0 f, f\<^sub>0g\<^sub>1.p\<^sub>1 \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1] = \<lbrace>(\<^bold>\<langle>f\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<a>\<^sup>-\<^sup>1\<^bold>[\<^bold>\<langle>tab\<^sub>0 f\<^bold>\<rangle>, \<^bold>\<langle>f\<^sub>0g\<^sub>1.p\<^sub>1\<^bold>\<rangle>, \<^bold>\<langle>fg\<^sub>0h\<^sub>1.p\<^sub>1\<^bold>\<rangle>\<^bold>]) \<^bold>\<cdot> \<^bold>\<a>\<^bold>[\<^bold>\<langle>f\<^bold>\<rangle>, \<^bold>\<langle>tab\<^sub>0 f\<^bold>\<rangle>, \<^bold>\<langle>f\<^sub>0g\<^sub>1.p\<^sub>1\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>fg\<^sub>0h\<^sub>1.p\<^sub>1\<^bold>\<rangle>\<^bold>]\<rbrace>
goal (1 subgoal):
1. (f \<star> \<a>\<^sup>-\<^sup>1[tab\<^sub>0 f, f\<^sub>0g\<^sub>1.p\<^sub>1, fg\<^sub>0h\<^sub>1.p\<^sub>1]) \<cdot> \<a>[f, tab\<^sub>0 f, f\<^sub>0g\<^sub>1.p\<^sub>1 \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1] = \<a>[f, tab\<^sub>0 f \<star> f\<^sub>0g\<^sub>1.p\<^sub>1, fg\<^sub>0h\<^sub>1.p\<^sub>1] \<cdot> (\<a>[f, tab\<^sub>0 f, f\<^sub>0g\<^sub>1.p\<^sub>1] \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> \<a>\<^sup>-\<^sup>1[f \<star> tab\<^sub>0 f, f\<^sub>0g\<^sub>1.p\<^sub>1, fg\<^sub>0h\<^sub>1.p\<^sub>1]
[PROOF STEP]
have "... =
\<lbrace>\<^bold>\<a>\<^bold>[\<^bold>\<langle>f\<^bold>\<rangle>, \<^bold>\<langle>tab\<^sub>0 f\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>Tfg.p\<^sub>1\<^bold>\<rangle>, \<^bold>\<langle>TTfgh.p\<^sub>1\<^bold>\<rangle>\<^bold>] \<^bold>\<cdot>
(\<^bold>\<a>\<^bold>[\<^bold>\<langle>f\<^bold>\<rangle>, \<^bold>\<langle>tab\<^sub>0 f\<^bold>\<rangle>, \<^bold>\<langle>Tfg.p\<^sub>1\<^bold>\<rangle>\<^bold>] \<^bold>\<star> \<^bold>\<langle>TTfgh.p\<^sub>1\<^bold>\<rangle>) \<^bold>\<cdot>
\<^bold>\<a>\<^sup>-\<^sup>1\<^bold>[\<^bold>\<langle>f\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>tab\<^sub>0 f\<^bold>\<rangle>, \<^bold>\<langle>Tfg.p\<^sub>1\<^bold>\<rangle>, \<^bold>\<langle>TTfgh.p\<^sub>1\<^bold>\<rangle>\<^bold>]\<rbrace>"
[PROOF STATE]
proof (prove)
goal (1 subgoal):
1. \<lbrace>(\<^bold>\<langle>f\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<a>\<^sup>-\<^sup>1\<^bold>[\<^bold>\<langle>tab\<^sub>0 f\<^bold>\<rangle>, \<^bold>\<langle>f\<^sub>0g\<^sub>1.p\<^sub>1\<^bold>\<rangle>, \<^bold>\<langle>fg\<^sub>0h\<^sub>1.p\<^sub>1\<^bold>\<rangle>\<^bold>]) \<^bold>\<cdot> \<^bold>\<a>\<^bold>[\<^bold>\<langle>f\<^bold>\<rangle>, \<^bold>\<langle>tab\<^sub>0 f\<^bold>\<rangle>, \<^bold>\<langle>f\<^sub>0g\<^sub>1.p\<^sub>1\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>fg\<^sub>0h\<^sub>1.p\<^sub>1\<^bold>\<rangle>\<^bold>]\<rbrace> = \<lbrace>\<^bold>\<a>\<^bold>[\<^bold>\<langle>f\<^bold>\<rangle>, \<^bold>\<langle>tab\<^sub>0 f\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>f\<^sub>0g\<^sub>1.p\<^sub>1\<^bold>\<rangle>, \<^bold>\<langle>fg\<^sub>0h\<^sub>1.p\<^sub>1\<^bold>\<rangle>\<^bold>] \<^bold>\<cdot> (\<^bold>\<a>\<^bold>[\<^bold>\<langle>f\<^bold>\<rangle>, \<^bold>\<langle>tab\<^sub>0 f\<^bold>\<rangle>, \<^bold>\<langle>f\<^sub>0g\<^sub>1.p\<^sub>1\<^bold>\<rangle>\<^bold>] \<^bold>\<star> \<^bold>\<langle>fg\<^sub>0h\<^sub>1.p\<^sub>1\<^bold>\<rangle>) \<^bold>\<cdot> \<^bold>\<a>\<^sup>-\<^sup>1\<^bold>[\<^bold>\<langle>f\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>tab\<^sub>0 f\<^bold>\<rangle>, \<^bold>\<langle>f\<^sub>0g\<^sub>1.p\<^sub>1\<^bold>\<rangle>, \<^bold>\<langle>fg\<^sub>0h\<^sub>1.p\<^sub>1\<^bold>\<rangle>\<^bold>]\<rbrace>
[PROOF STEP]
using fg gh fg\<^sub>0h\<^sub>1.p\<^sub>0_simps fg\<^sub>0h\<^sub>1.p\<^sub>1_simps f\<^sub>0g\<^sub>1.p\<^sub>0_simps f\<^sub>0g\<^sub>1.p\<^sub>1_simps
[PROOF STATE]
proof (prove)
using this:
src f = trg g
src g = trg h
trg fg\<^sub>0h\<^sub>1.p\<^sub>0 = src (tab\<^sub>1 h)
trg fg\<^sub>0h\<^sub>1.p\<^sub>1 = src (tab\<^sub>0 g \<star> f\<^sub>0g\<^sub>1.p\<^sub>0)
trg f\<^sub>0g\<^sub>1.p\<^sub>0 = src (tab\<^sub>1 g)
trg f\<^sub>0g\<^sub>1.p\<^sub>1 = src (tab\<^sub>0 f)
goal (1 subgoal):
1. \<lbrace>(\<^bold>\<langle>f\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<a>\<^sup>-\<^sup>1\<^bold>[\<^bold>\<langle>tab\<^sub>0 f\<^bold>\<rangle>, \<^bold>\<langle>f\<^sub>0g\<^sub>1.p\<^sub>1\<^bold>\<rangle>, \<^bold>\<langle>fg\<^sub>0h\<^sub>1.p\<^sub>1\<^bold>\<rangle>\<^bold>]) \<^bold>\<cdot> \<^bold>\<a>\<^bold>[\<^bold>\<langle>f\<^bold>\<rangle>, \<^bold>\<langle>tab\<^sub>0 f\<^bold>\<rangle>, \<^bold>\<langle>f\<^sub>0g\<^sub>1.p\<^sub>1\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>fg\<^sub>0h\<^sub>1.p\<^sub>1\<^bold>\<rangle>\<^bold>]\<rbrace> = \<lbrace>\<^bold>\<a>\<^bold>[\<^bold>\<langle>f\<^bold>\<rangle>, \<^bold>\<langle>tab\<^sub>0 f\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>f\<^sub>0g\<^sub>1.p\<^sub>1\<^bold>\<rangle>, \<^bold>\<langle>fg\<^sub>0h\<^sub>1.p\<^sub>1\<^bold>\<rangle>\<^bold>] \<^bold>\<cdot> (\<^bold>\<a>\<^bold>[\<^bold>\<langle>f\<^bold>\<rangle>, \<^bold>\<langle>tab\<^sub>0 f\<^bold>\<rangle>, \<^bold>\<langle>f\<^sub>0g\<^sub>1.p\<^sub>1\<^bold>\<rangle>\<^bold>] \<^bold>\<star> \<^bold>\<langle>fg\<^sub>0h\<^sub>1.p\<^sub>1\<^bold>\<rangle>) \<^bold>\<cdot> \<^bold>\<a>\<^sup>-\<^sup>1\<^bold>[\<^bold>\<langle>f\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>tab\<^sub>0 f\<^bold>\<rangle>, \<^bold>\<langle>f\<^sub>0g\<^sub>1.p\<^sub>1\<^bold>\<rangle>, \<^bold>\<langle>fg\<^sub>0h\<^sub>1.p\<^sub>1\<^bold>\<rangle>\<^bold>]\<rbrace>
[PROOF STEP]
by (intro E.eval_eqI, simp_all)
[PROOF STATE]
proof (state)
this:
\<lbrace>(\<^bold>\<langle>f\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<a>\<^sup>-\<^sup>1\<^bold>[\<^bold>\<langle>tab\<^sub>0 f\<^bold>\<rangle>, \<^bold>\<langle>f\<^sub>0g\<^sub>1.p\<^sub>1\<^bold>\<rangle>, \<^bold>\<langle>fg\<^sub>0h\<^sub>1.p\<^sub>1\<^bold>\<rangle>\<^bold>]) \<^bold>\<cdot> \<^bold>\<a>\<^bold>[\<^bold>\<langle>f\<^bold>\<rangle>, \<^bold>\<langle>tab\<^sub>0 f\<^bold>\<rangle>, \<^bold>\<langle>f\<^sub>0g\<^sub>1.p\<^sub>1\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>fg\<^sub>0h\<^sub>1.p\<^sub>1\<^bold>\<rangle>\<^bold>]\<rbrace> = \<lbrace>\<^bold>\<a>\<^bold>[\<^bold>\<langle>f\<^bold>\<rangle>, \<^bold>\<langle>tab\<^sub>0 f\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>f\<^sub>0g\<^sub>1.p\<^sub>1\<^bold>\<rangle>, \<^bold>\<langle>fg\<^sub>0h\<^sub>1.p\<^sub>1\<^bold>\<rangle>\<^bold>] \<^bold>\<cdot> (\<^bold>\<a>\<^bold>[\<^bold>\<langle>f\<^bold>\<rangle>, \<^bold>\<langle>tab\<^sub>0 f\<^bold>\<rangle>, \<^bold>\<langle>f\<^sub>0g\<^sub>1.p\<^sub>1\<^bold>\<rangle>\<^bold>] \<^bold>\<star> \<^bold>\<langle>fg\<^sub>0h\<^sub>1.p\<^sub>1\<^bold>\<rangle>) \<^bold>\<cdot> \<^bold>\<a>\<^sup>-\<^sup>1\<^bold>[\<^bold>\<langle>f\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>tab\<^sub>0 f\<^bold>\<rangle>, \<^bold>\<langle>f\<^sub>0g\<^sub>1.p\<^sub>1\<^bold>\<rangle>, \<^bold>\<langle>fg\<^sub>0h\<^sub>1.p\<^sub>1\<^bold>\<rangle>\<^bold>]\<rbrace>
goal (1 subgoal):
1. (f \<star> \<a>\<^sup>-\<^sup>1[tab\<^sub>0 f, f\<^sub>0g\<^sub>1.p\<^sub>1, fg\<^sub>0h\<^sub>1.p\<^sub>1]) \<cdot> \<a>[f, tab\<^sub>0 f, f\<^sub>0g\<^sub>1.p\<^sub>1 \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1] = \<a>[f, tab\<^sub>0 f \<star> f\<^sub>0g\<^sub>1.p\<^sub>1, fg\<^sub>0h\<^sub>1.p\<^sub>1] \<cdot> (\<a>[f, tab\<^sub>0 f, f\<^sub>0g\<^sub>1.p\<^sub>1] \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> \<a>\<^sup>-\<^sup>1[f \<star> tab\<^sub>0 f, f\<^sub>0g\<^sub>1.p\<^sub>1, fg\<^sub>0h\<^sub>1.p\<^sub>1]
[PROOF STEP]
also
[PROOF STATE]
proof (state)
this:
\<lbrace>(\<^bold>\<langle>f\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<a>\<^sup>-\<^sup>1\<^bold>[\<^bold>\<langle>tab\<^sub>0 f\<^bold>\<rangle>, \<^bold>\<langle>f\<^sub>0g\<^sub>1.p\<^sub>1\<^bold>\<rangle>, \<^bold>\<langle>fg\<^sub>0h\<^sub>1.p\<^sub>1\<^bold>\<rangle>\<^bold>]) \<^bold>\<cdot> \<^bold>\<a>\<^bold>[\<^bold>\<langle>f\<^bold>\<rangle>, \<^bold>\<langle>tab\<^sub>0 f\<^bold>\<rangle>, \<^bold>\<langle>f\<^sub>0g\<^sub>1.p\<^sub>1\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>fg\<^sub>0h\<^sub>1.p\<^sub>1\<^bold>\<rangle>\<^bold>]\<rbrace> = \<lbrace>\<^bold>\<a>\<^bold>[\<^bold>\<langle>f\<^bold>\<rangle>, \<^bold>\<langle>tab\<^sub>0 f\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>f\<^sub>0g\<^sub>1.p\<^sub>1\<^bold>\<rangle>, \<^bold>\<langle>fg\<^sub>0h\<^sub>1.p\<^sub>1\<^bold>\<rangle>\<^bold>] \<^bold>\<cdot> (\<^bold>\<a>\<^bold>[\<^bold>\<langle>f\<^bold>\<rangle>, \<^bold>\<langle>tab\<^sub>0 f\<^bold>\<rangle>, \<^bold>\<langle>f\<^sub>0g\<^sub>1.p\<^sub>1\<^bold>\<rangle>\<^bold>] \<^bold>\<star> \<^bold>\<langle>fg\<^sub>0h\<^sub>1.p\<^sub>1\<^bold>\<rangle>) \<^bold>\<cdot> \<^bold>\<a>\<^sup>-\<^sup>1\<^bold>[\<^bold>\<langle>f\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>tab\<^sub>0 f\<^bold>\<rangle>, \<^bold>\<langle>f\<^sub>0g\<^sub>1.p\<^sub>1\<^bold>\<rangle>, \<^bold>\<langle>fg\<^sub>0h\<^sub>1.p\<^sub>1\<^bold>\<rangle>\<^bold>]\<rbrace>
goal (1 subgoal):
1. (f \<star> \<a>\<^sup>-\<^sup>1[tab\<^sub>0 f, f\<^sub>0g\<^sub>1.p\<^sub>1, fg\<^sub>0h\<^sub>1.p\<^sub>1]) \<cdot> \<a>[f, tab\<^sub>0 f, f\<^sub>0g\<^sub>1.p\<^sub>1 \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1] = \<a>[f, tab\<^sub>0 f \<star> f\<^sub>0g\<^sub>1.p\<^sub>1, fg\<^sub>0h\<^sub>1.p\<^sub>1] \<cdot> (\<a>[f, tab\<^sub>0 f, f\<^sub>0g\<^sub>1.p\<^sub>1] \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> \<a>\<^sup>-\<^sup>1[f \<star> tab\<^sub>0 f, f\<^sub>0g\<^sub>1.p\<^sub>1, fg\<^sub>0h\<^sub>1.p\<^sub>1]
[PROOF STEP]
have "... = \<a>[f, tab\<^sub>0 f \<star> Tfg.p\<^sub>1, TTfgh.p\<^sub>1] \<cdot> (\<a>[f, tab\<^sub>0 f, Tfg.p\<^sub>1] \<star> TTfgh.p\<^sub>1) \<cdot>
\<a>\<^sup>-\<^sup>1[f \<star> tab\<^sub>0 f, Tfg.p\<^sub>1, TTfgh.p\<^sub>1]"
[PROOF STATE]
proof (prove)
goal (1 subgoal):
1. \<lbrace>\<^bold>\<a>\<^bold>[\<^bold>\<langle>f\<^bold>\<rangle>, \<^bold>\<langle>tab\<^sub>0 f\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>f\<^sub>0g\<^sub>1.p\<^sub>1\<^bold>\<rangle>, \<^bold>\<langle>fg\<^sub>0h\<^sub>1.p\<^sub>1\<^bold>\<rangle>\<^bold>] \<^bold>\<cdot> (\<^bold>\<a>\<^bold>[\<^bold>\<langle>f\<^bold>\<rangle>, \<^bold>\<langle>tab\<^sub>0 f\<^bold>\<rangle>, \<^bold>\<langle>f\<^sub>0g\<^sub>1.p\<^sub>1\<^bold>\<rangle>\<^bold>] \<^bold>\<star> \<^bold>\<langle>fg\<^sub>0h\<^sub>1.p\<^sub>1\<^bold>\<rangle>) \<^bold>\<cdot> \<^bold>\<a>\<^sup>-\<^sup>1\<^bold>[\<^bold>\<langle>f\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>tab\<^sub>0 f\<^bold>\<rangle>, \<^bold>\<langle>f\<^sub>0g\<^sub>1.p\<^sub>1\<^bold>\<rangle>, \<^bold>\<langle>fg\<^sub>0h\<^sub>1.p\<^sub>1\<^bold>\<rangle>\<^bold>]\<rbrace> = \<a>[f, tab\<^sub>0 f \<star> f\<^sub>0g\<^sub>1.p\<^sub>1, fg\<^sub>0h\<^sub>1.p\<^sub>1] \<cdot> (\<a>[f, tab\<^sub>0 f, f\<^sub>0g\<^sub>1.p\<^sub>1] \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> \<a>\<^sup>-\<^sup>1[f \<star> tab\<^sub>0 f, f\<^sub>0g\<^sub>1.p\<^sub>1, fg\<^sub>0h\<^sub>1.p\<^sub>1]
[PROOF STEP]
using fg gh fg\<^sub>0h\<^sub>1.p\<^sub>0_simps fg\<^sub>0h\<^sub>1.p\<^sub>1_simps f\<^sub>0g\<^sub>1.p\<^sub>0_simps f\<^sub>0g\<^sub>1.p\<^sub>1_simps
\<a>'_def \<alpha>_def
[PROOF STATE]
proof (prove)
using this:
src f = trg g
src g = trg h
trg fg\<^sub>0h\<^sub>1.p\<^sub>0 = src (tab\<^sub>1 h)
trg fg\<^sub>0h\<^sub>1.p\<^sub>1 = src (tab\<^sub>0 g \<star> f\<^sub>0g\<^sub>1.p\<^sub>0)
trg f\<^sub>0g\<^sub>1.p\<^sub>0 = src (tab\<^sub>1 g)
trg f\<^sub>0g\<^sub>1.p\<^sub>1 = src (tab\<^sub>0 f)
\<a>\<^sup>-\<^sup>1[?\<mu>, ?\<nu>, ?\<tau>] \<equiv> \<alpha>' (?\<mu>, ?\<nu>, ?\<tau>)
\<alpha> ?\<mu>\<nu>\<tau> \<equiv> \<a>[fst ?\<mu>\<nu>\<tau>, fst (snd ?\<mu>\<nu>\<tau>), snd (snd ?\<mu>\<nu>\<tau>)]
goal (1 subgoal):
1. \<lbrace>\<^bold>\<a>\<^bold>[\<^bold>\<langle>f\<^bold>\<rangle>, \<^bold>\<langle>tab\<^sub>0 f\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>f\<^sub>0g\<^sub>1.p\<^sub>1\<^bold>\<rangle>, \<^bold>\<langle>fg\<^sub>0h\<^sub>1.p\<^sub>1\<^bold>\<rangle>\<^bold>] \<^bold>\<cdot> (\<^bold>\<a>\<^bold>[\<^bold>\<langle>f\<^bold>\<rangle>, \<^bold>\<langle>tab\<^sub>0 f\<^bold>\<rangle>, \<^bold>\<langle>f\<^sub>0g\<^sub>1.p\<^sub>1\<^bold>\<rangle>\<^bold>] \<^bold>\<star> \<^bold>\<langle>fg\<^sub>0h\<^sub>1.p\<^sub>1\<^bold>\<rangle>) \<^bold>\<cdot> \<^bold>\<a>\<^sup>-\<^sup>1\<^bold>[\<^bold>\<langle>f\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>tab\<^sub>0 f\<^bold>\<rangle>, \<^bold>\<langle>f\<^sub>0g\<^sub>1.p\<^sub>1\<^bold>\<rangle>, \<^bold>\<langle>fg\<^sub>0h\<^sub>1.p\<^sub>1\<^bold>\<rangle>\<^bold>]\<rbrace> = \<a>[f, tab\<^sub>0 f \<star> f\<^sub>0g\<^sub>1.p\<^sub>1, fg\<^sub>0h\<^sub>1.p\<^sub>1] \<cdot> (\<a>[f, tab\<^sub>0 f, f\<^sub>0g\<^sub>1.p\<^sub>1] \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> \<a>\<^sup>-\<^sup>1[f \<star> tab\<^sub>0 f, f\<^sub>0g\<^sub>1.p\<^sub>1, fg\<^sub>0h\<^sub>1.p\<^sub>1]
[PROOF STEP]
by simp
[PROOF STATE]
proof (state)
this:
\<lbrace>\<^bold>\<a>\<^bold>[\<^bold>\<langle>f\<^bold>\<rangle>, \<^bold>\<langle>tab\<^sub>0 f\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>f\<^sub>0g\<^sub>1.p\<^sub>1\<^bold>\<rangle>, \<^bold>\<langle>fg\<^sub>0h\<^sub>1.p\<^sub>1\<^bold>\<rangle>\<^bold>] \<^bold>\<cdot> (\<^bold>\<a>\<^bold>[\<^bold>\<langle>f\<^bold>\<rangle>, \<^bold>\<langle>tab\<^sub>0 f\<^bold>\<rangle>, \<^bold>\<langle>f\<^sub>0g\<^sub>1.p\<^sub>1\<^bold>\<rangle>\<^bold>] \<^bold>\<star> \<^bold>\<langle>fg\<^sub>0h\<^sub>1.p\<^sub>1\<^bold>\<rangle>) \<^bold>\<cdot> \<^bold>\<a>\<^sup>-\<^sup>1\<^bold>[\<^bold>\<langle>f\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>tab\<^sub>0 f\<^bold>\<rangle>, \<^bold>\<langle>f\<^sub>0g\<^sub>1.p\<^sub>1\<^bold>\<rangle>, \<^bold>\<langle>fg\<^sub>0h\<^sub>1.p\<^sub>1\<^bold>\<rangle>\<^bold>]\<rbrace> = \<a>[f, tab\<^sub>0 f \<star> f\<^sub>0g\<^sub>1.p\<^sub>1, fg\<^sub>0h\<^sub>1.p\<^sub>1] \<cdot> (\<a>[f, tab\<^sub>0 f, f\<^sub>0g\<^sub>1.p\<^sub>1] \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> \<a>\<^sup>-\<^sup>1[f \<star> tab\<^sub>0 f, f\<^sub>0g\<^sub>1.p\<^sub>1, fg\<^sub>0h\<^sub>1.p\<^sub>1]
goal (1 subgoal):
1. (f \<star> \<a>\<^sup>-\<^sup>1[tab\<^sub>0 f, f\<^sub>0g\<^sub>1.p\<^sub>1, fg\<^sub>0h\<^sub>1.p\<^sub>1]) \<cdot> \<a>[f, tab\<^sub>0 f, f\<^sub>0g\<^sub>1.p\<^sub>1 \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1] = \<a>[f, tab\<^sub>0 f \<star> f\<^sub>0g\<^sub>1.p\<^sub>1, fg\<^sub>0h\<^sub>1.p\<^sub>1] \<cdot> (\<a>[f, tab\<^sub>0 f, f\<^sub>0g\<^sub>1.p\<^sub>1] \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> \<a>\<^sup>-\<^sup>1[f \<star> tab\<^sub>0 f, f\<^sub>0g\<^sub>1.p\<^sub>1, fg\<^sub>0h\<^sub>1.p\<^sub>1]
[PROOF STEP]
finally
[PROOF STATE]
proof (chain)
picking this:
(f \<star> \<a>\<^sup>-\<^sup>1[tab\<^sub>0 f, f\<^sub>0g\<^sub>1.p\<^sub>1, fg\<^sub>0h\<^sub>1.p\<^sub>1]) \<cdot> \<a>[f, tab\<^sub>0 f, f\<^sub>0g\<^sub>1.p\<^sub>1 \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1] = \<a>[f, tab\<^sub>0 f \<star> f\<^sub>0g\<^sub>1.p\<^sub>1, fg\<^sub>0h\<^sub>1.p\<^sub>1] \<cdot> (\<a>[f, tab\<^sub>0 f, f\<^sub>0g\<^sub>1.p\<^sub>1] \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> \<a>\<^sup>-\<^sup>1[f \<star> tab\<^sub>0 f, f\<^sub>0g\<^sub>1.p\<^sub>1, fg\<^sub>0h\<^sub>1.p\<^sub>1]
[PROOF STEP]
show ?thesis
[PROOF STATE]
proof (prove)
using this:
(f \<star> \<a>\<^sup>-\<^sup>1[tab\<^sub>0 f, f\<^sub>0g\<^sub>1.p\<^sub>1, fg\<^sub>0h\<^sub>1.p\<^sub>1]) \<cdot> \<a>[f, tab\<^sub>0 f, f\<^sub>0g\<^sub>1.p\<^sub>1 \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1] = \<a>[f, tab\<^sub>0 f \<star> f\<^sub>0g\<^sub>1.p\<^sub>1, fg\<^sub>0h\<^sub>1.p\<^sub>1] \<cdot> (\<a>[f, tab\<^sub>0 f, f\<^sub>0g\<^sub>1.p\<^sub>1] \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> \<a>\<^sup>-\<^sup>1[f \<star> tab\<^sub>0 f, f\<^sub>0g\<^sub>1.p\<^sub>1, fg\<^sub>0h\<^sub>1.p\<^sub>1]
goal (1 subgoal):
1. (f \<star> \<a>\<^sup>-\<^sup>1[tab\<^sub>0 f, f\<^sub>0g\<^sub>1.p\<^sub>1, fg\<^sub>0h\<^sub>1.p\<^sub>1]) \<cdot> \<a>[f, tab\<^sub>0 f, f\<^sub>0g\<^sub>1.p\<^sub>1 \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1] = \<a>[f, tab\<^sub>0 f \<star> f\<^sub>0g\<^sub>1.p\<^sub>1, fg\<^sub>0h\<^sub>1.p\<^sub>1] \<cdot> (\<a>[f, tab\<^sub>0 f, f\<^sub>0g\<^sub>1.p\<^sub>1] \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> \<a>\<^sup>-\<^sup>1[f \<star> tab\<^sub>0 f, f\<^sub>0g\<^sub>1.p\<^sub>1, fg\<^sub>0h\<^sub>1.p\<^sub>1]
[PROOF STEP]
by blast
[PROOF STATE]
proof (state)
this:
(f \<star> \<a>\<^sup>-\<^sup>1[tab\<^sub>0 f, f\<^sub>0g\<^sub>1.p\<^sub>1, fg\<^sub>0h\<^sub>1.p\<^sub>1]) \<cdot> \<a>[f, tab\<^sub>0 f, f\<^sub>0g\<^sub>1.p\<^sub>1 \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1] = \<a>[f, tab\<^sub>0 f \<star> f\<^sub>0g\<^sub>1.p\<^sub>1, fg\<^sub>0h\<^sub>1.p\<^sub>1] \<cdot> (\<a>[f, tab\<^sub>0 f, f\<^sub>0g\<^sub>1.p\<^sub>1] \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> \<a>\<^sup>-\<^sup>1[f \<star> tab\<^sub>0 f, f\<^sub>0g\<^sub>1.p\<^sub>1, fg\<^sub>0h\<^sub>1.p\<^sub>1]
goal:
No subgoals!
[PROOF STEP]
qed
[PROOF STATE]
proof (state)
this:
(f \<star> \<a>\<^sup>-\<^sup>1[tab\<^sub>0 f, f\<^sub>0g\<^sub>1.p\<^sub>1, fg\<^sub>0h\<^sub>1.p\<^sub>1]) \<cdot> \<a>[f, tab\<^sub>0 f, f\<^sub>0g\<^sub>1.p\<^sub>1 \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1] = \<a>[f, tab\<^sub>0 f \<star> f\<^sub>0g\<^sub>1.p\<^sub>1, fg\<^sub>0h\<^sub>1.p\<^sub>1] \<cdot> (\<a>[f, tab\<^sub>0 f, f\<^sub>0g\<^sub>1.p\<^sub>1] \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> \<a>\<^sup>-\<^sup>1[f \<star> tab\<^sub>0 f, f\<^sub>0g\<^sub>1.p\<^sub>1, fg\<^sub>0h\<^sub>1.p\<^sub>1]
goal (1 subgoal):
1. \<a>[f, g, h \<star> tab\<^sub>0 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0] \<cdot> ((f \<star> g) \<star> \<a>[h, tab\<^sub>0 h, fg\<^sub>0h\<^sub>1.p\<^sub>0]) \<cdot> ((f \<star> g) \<star> h.tab \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0) \<cdot> ((f \<star> g) \<star> fg\<^sub>0h\<^sub>1.\<phi>) \<cdot> \<a>\<^sup>-\<^sup>1[f, g, (tab\<^sub>0 g \<star> f\<^sub>0g\<^sub>1.p\<^sub>0) \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1] \<cdot> (f \<star> \<a>[g, tab\<^sub>0 g \<star> f\<^sub>0g\<^sub>1.p\<^sub>0, fg\<^sub>0h\<^sub>1.p\<^sub>1]) \<cdot> (f \<star> \<a>[g, tab\<^sub>0 g, f\<^sub>0g\<^sub>1.p\<^sub>0] \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> (f \<star> (g.tab \<star> f\<^sub>0g\<^sub>1.p\<^sub>0) \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> (f \<star> f\<^sub>0g\<^sub>1.\<phi> \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> ((f \<star> \<a>\<^sup>-\<^sup>1[tab\<^sub>0 f, f\<^sub>0g\<^sub>1.p\<^sub>1, fg\<^sub>0h\<^sub>1.p\<^sub>1]) \<cdot> \<a>[f, tab\<^sub>0 f, f\<^sub>0g\<^sub>1.p\<^sub>1 \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1]) \<cdot> (f.tab \<star> f\<^sub>0g\<^sub>1.p\<^sub>1 \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) = \<a>[f, g, h \<star> tab\<^sub>0 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0] \<cdot> ((f \<star> g) \<star> \<a>[h, tab\<^sub>0 h, fg\<^sub>0h\<^sub>1.p\<^sub>0]) \<cdot> ((f \<star> g) \<star> h.tab \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0) \<cdot> ((f \<star> g) \<star> fg\<^sub>0h\<^sub>1.\<phi>) \<cdot> \<a>\<^sup>-\<^sup>1[f, g, (tab\<^sub>0 g \<star> f\<^sub>0g\<^sub>1.p\<^sub>0) \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1] \<cdot> (f \<star> \<a>[g, tab\<^sub>0 g \<star> f\<^sub>0g\<^sub>1.p\<^sub>0, fg\<^sub>0h\<^sub>1.p\<^sub>1]) \<cdot> (f \<star> \<a>[g, tab\<^sub>0 g, f\<^sub>0g\<^sub>1.p\<^sub>0] \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> (f \<star> (g.tab \<star> f\<^sub>0g\<^sub>1.p\<^sub>0) \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> ((f \<star> f\<^sub>0g\<^sub>1.\<phi> \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> \<a>[f, tab\<^sub>0 f \<star> f\<^sub>0g\<^sub>1.p\<^sub>1, fg\<^sub>0h\<^sub>1.p\<^sub>1]) \<cdot> (\<a>[f, tab\<^sub>0 f, f\<^sub>0g\<^sub>1.p\<^sub>1] \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> \<a>\<^sup>-\<^sup>1[f \<star> tab\<^sub>0 f, f\<^sub>0g\<^sub>1.p\<^sub>1, fg\<^sub>0h\<^sub>1.p\<^sub>1] \<cdot> (f.tab \<star> f\<^sub>0g\<^sub>1.p\<^sub>1 \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1)
[PROOF STEP]
thus ?thesis
[PROOF STATE]
proof (prove)
using this:
(f \<star> \<a>\<^sup>-\<^sup>1[tab\<^sub>0 f, f\<^sub>0g\<^sub>1.p\<^sub>1, fg\<^sub>0h\<^sub>1.p\<^sub>1]) \<cdot> \<a>[f, tab\<^sub>0 f, f\<^sub>0g\<^sub>1.p\<^sub>1 \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1] = \<a>[f, tab\<^sub>0 f \<star> f\<^sub>0g\<^sub>1.p\<^sub>1, fg\<^sub>0h\<^sub>1.p\<^sub>1] \<cdot> (\<a>[f, tab\<^sub>0 f, f\<^sub>0g\<^sub>1.p\<^sub>1] \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> \<a>\<^sup>-\<^sup>1[f \<star> tab\<^sub>0 f, f\<^sub>0g\<^sub>1.p\<^sub>1, fg\<^sub>0h\<^sub>1.p\<^sub>1]
goal (1 subgoal):
1. \<a>[f, g, h \<star> tab\<^sub>0 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0] \<cdot> ((f \<star> g) \<star> \<a>[h, tab\<^sub>0 h, fg\<^sub>0h\<^sub>1.p\<^sub>0]) \<cdot> ((f \<star> g) \<star> h.tab \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0) \<cdot> ((f \<star> g) \<star> fg\<^sub>0h\<^sub>1.\<phi>) \<cdot> \<a>\<^sup>-\<^sup>1[f, g, (tab\<^sub>0 g \<star> f\<^sub>0g\<^sub>1.p\<^sub>0) \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1] \<cdot> (f \<star> \<a>[g, tab\<^sub>0 g \<star> f\<^sub>0g\<^sub>1.p\<^sub>0, fg\<^sub>0h\<^sub>1.p\<^sub>1]) \<cdot> (f \<star> \<a>[g, tab\<^sub>0 g, f\<^sub>0g\<^sub>1.p\<^sub>0] \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> (f \<star> (g.tab \<star> f\<^sub>0g\<^sub>1.p\<^sub>0) \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> (f \<star> f\<^sub>0g\<^sub>1.\<phi> \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> ((f \<star> \<a>\<^sup>-\<^sup>1[tab\<^sub>0 f, f\<^sub>0g\<^sub>1.p\<^sub>1, fg\<^sub>0h\<^sub>1.p\<^sub>1]) \<cdot> \<a>[f, tab\<^sub>0 f, f\<^sub>0g\<^sub>1.p\<^sub>1 \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1]) \<cdot> (f.tab \<star> f\<^sub>0g\<^sub>1.p\<^sub>1 \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) = \<a>[f, g, h \<star> tab\<^sub>0 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0] \<cdot> ((f \<star> g) \<star> \<a>[h, tab\<^sub>0 h, fg\<^sub>0h\<^sub>1.p\<^sub>0]) \<cdot> ((f \<star> g) \<star> h.tab \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0) \<cdot> ((f \<star> g) \<star> fg\<^sub>0h\<^sub>1.\<phi>) \<cdot> \<a>\<^sup>-\<^sup>1[f, g, (tab\<^sub>0 g \<star> f\<^sub>0g\<^sub>1.p\<^sub>0) \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1] \<cdot> (f \<star> \<a>[g, tab\<^sub>0 g \<star> f\<^sub>0g\<^sub>1.p\<^sub>0, fg\<^sub>0h\<^sub>1.p\<^sub>1]) \<cdot> (f \<star> \<a>[g, tab\<^sub>0 g, f\<^sub>0g\<^sub>1.p\<^sub>0] \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> (f \<star> (g.tab \<star> f\<^sub>0g\<^sub>1.p\<^sub>0) \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> ((f \<star> f\<^sub>0g\<^sub>1.\<phi> \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> \<a>[f, tab\<^sub>0 f \<star> f\<^sub>0g\<^sub>1.p\<^sub>1, fg\<^sub>0h\<^sub>1.p\<^sub>1]) \<cdot> (\<a>[f, tab\<^sub>0 f, f\<^sub>0g\<^sub>1.p\<^sub>1] \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> \<a>\<^sup>-\<^sup>1[f \<star> tab\<^sub>0 f, f\<^sub>0g\<^sub>1.p\<^sub>1, fg\<^sub>0h\<^sub>1.p\<^sub>1] \<cdot> (f.tab \<star> f\<^sub>0g\<^sub>1.p\<^sub>1 \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1)
[PROOF STEP]
using comp_assoc
[PROOF STATE]
proof (prove)
using this:
(f \<star> \<a>\<^sup>-\<^sup>1[tab\<^sub>0 f, f\<^sub>0g\<^sub>1.p\<^sub>1, fg\<^sub>0h\<^sub>1.p\<^sub>1]) \<cdot> \<a>[f, tab\<^sub>0 f, f\<^sub>0g\<^sub>1.p\<^sub>1 \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1] = \<a>[f, tab\<^sub>0 f \<star> f\<^sub>0g\<^sub>1.p\<^sub>1, fg\<^sub>0h\<^sub>1.p\<^sub>1] \<cdot> (\<a>[f, tab\<^sub>0 f, f\<^sub>0g\<^sub>1.p\<^sub>1] \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> \<a>\<^sup>-\<^sup>1[f \<star> tab\<^sub>0 f, f\<^sub>0g\<^sub>1.p\<^sub>1, fg\<^sub>0h\<^sub>1.p\<^sub>1]
(?h \<cdot> ?g) \<cdot> ?f = ?h \<cdot> ?g \<cdot> ?f
goal (1 subgoal):
1. \<a>[f, g, h \<star> tab\<^sub>0 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0] \<cdot> ((f \<star> g) \<star> \<a>[h, tab\<^sub>0 h, fg\<^sub>0h\<^sub>1.p\<^sub>0]) \<cdot> ((f \<star> g) \<star> h.tab \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0) \<cdot> ((f \<star> g) \<star> fg\<^sub>0h\<^sub>1.\<phi>) \<cdot> \<a>\<^sup>-\<^sup>1[f, g, (tab\<^sub>0 g \<star> f\<^sub>0g\<^sub>1.p\<^sub>0) \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1] \<cdot> (f \<star> \<a>[g, tab\<^sub>0 g \<star> f\<^sub>0g\<^sub>1.p\<^sub>0, fg\<^sub>0h\<^sub>1.p\<^sub>1]) \<cdot> (f \<star> \<a>[g, tab\<^sub>0 g, f\<^sub>0g\<^sub>1.p\<^sub>0] \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> (f \<star> (g.tab \<star> f\<^sub>0g\<^sub>1.p\<^sub>0) \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> (f \<star> f\<^sub>0g\<^sub>1.\<phi> \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> ((f \<star> \<a>\<^sup>-\<^sup>1[tab\<^sub>0 f, f\<^sub>0g\<^sub>1.p\<^sub>1, fg\<^sub>0h\<^sub>1.p\<^sub>1]) \<cdot> \<a>[f, tab\<^sub>0 f, f\<^sub>0g\<^sub>1.p\<^sub>1 \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1]) \<cdot> (f.tab \<star> f\<^sub>0g\<^sub>1.p\<^sub>1 \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) = \<a>[f, g, h \<star> tab\<^sub>0 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0] \<cdot> ((f \<star> g) \<star> \<a>[h, tab\<^sub>0 h, fg\<^sub>0h\<^sub>1.p\<^sub>0]) \<cdot> ((f \<star> g) \<star> h.tab \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0) \<cdot> ((f \<star> g) \<star> fg\<^sub>0h\<^sub>1.\<phi>) \<cdot> \<a>\<^sup>-\<^sup>1[f, g, (tab\<^sub>0 g \<star> f\<^sub>0g\<^sub>1.p\<^sub>0) \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1] \<cdot> (f \<star> \<a>[g, tab\<^sub>0 g \<star> f\<^sub>0g\<^sub>1.p\<^sub>0, fg\<^sub>0h\<^sub>1.p\<^sub>1]) \<cdot> (f \<star> \<a>[g, tab\<^sub>0 g, f\<^sub>0g\<^sub>1.p\<^sub>0] \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> (f \<star> (g.tab \<star> f\<^sub>0g\<^sub>1.p\<^sub>0) \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> ((f \<star> f\<^sub>0g\<^sub>1.\<phi> \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> \<a>[f, tab\<^sub>0 f \<star> f\<^sub>0g\<^sub>1.p\<^sub>1, fg\<^sub>0h\<^sub>1.p\<^sub>1]) \<cdot> (\<a>[f, tab\<^sub>0 f, f\<^sub>0g\<^sub>1.p\<^sub>1] \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> \<a>\<^sup>-\<^sup>1[f \<star> tab\<^sub>0 f, f\<^sub>0g\<^sub>1.p\<^sub>1, fg\<^sub>0h\<^sub>1.p\<^sub>1] \<cdot> (f.tab \<star> f\<^sub>0g\<^sub>1.p\<^sub>1 \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1)
[PROOF STEP]
by presburger
[PROOF STATE]
proof (state)
this:
\<a>[f, g, h \<star> tab\<^sub>0 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0] \<cdot> ((f \<star> g) \<star> \<a>[h, tab\<^sub>0 h, fg\<^sub>0h\<^sub>1.p\<^sub>0]) \<cdot> ((f \<star> g) \<star> h.tab \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0) \<cdot> ((f \<star> g) \<star> fg\<^sub>0h\<^sub>1.\<phi>) \<cdot> \<a>\<^sup>-\<^sup>1[f, g, (tab\<^sub>0 g \<star> f\<^sub>0g\<^sub>1.p\<^sub>0) \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1] \<cdot> (f \<star> \<a>[g, tab\<^sub>0 g \<star> f\<^sub>0g\<^sub>1.p\<^sub>0, fg\<^sub>0h\<^sub>1.p\<^sub>1]) \<cdot> (f \<star> \<a>[g, tab\<^sub>0 g, f\<^sub>0g\<^sub>1.p\<^sub>0] \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> (f \<star> (g.tab \<star> f\<^sub>0g\<^sub>1.p\<^sub>0) \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> (f \<star> f\<^sub>0g\<^sub>1.\<phi> \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> ((f \<star> \<a>\<^sup>-\<^sup>1[tab\<^sub>0 f, f\<^sub>0g\<^sub>1.p\<^sub>1, fg\<^sub>0h\<^sub>1.p\<^sub>1]) \<cdot> \<a>[f, tab\<^sub>0 f, f\<^sub>0g\<^sub>1.p\<^sub>1 \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1]) \<cdot> (f.tab \<star> f\<^sub>0g\<^sub>1.p\<^sub>1 \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) = \<a>[f, g, h \<star> tab\<^sub>0 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0] \<cdot> ((f \<star> g) \<star> \<a>[h, tab\<^sub>0 h, fg\<^sub>0h\<^sub>1.p\<^sub>0]) \<cdot> ((f \<star> g) \<star> h.tab \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0) \<cdot> ((f \<star> g) \<star> fg\<^sub>0h\<^sub>1.\<phi>) \<cdot> \<a>\<^sup>-\<^sup>1[f, g, (tab\<^sub>0 g \<star> f\<^sub>0g\<^sub>1.p\<^sub>0) \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1] \<cdot> (f \<star> \<a>[g, tab\<^sub>0 g \<star> f\<^sub>0g\<^sub>1.p\<^sub>0, fg\<^sub>0h\<^sub>1.p\<^sub>1]) \<cdot> (f \<star> \<a>[g, tab\<^sub>0 g, f\<^sub>0g\<^sub>1.p\<^sub>0] \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> (f \<star> (g.tab \<star> f\<^sub>0g\<^sub>1.p\<^sub>0) \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> ((f \<star> f\<^sub>0g\<^sub>1.\<phi> \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> \<a>[f, tab\<^sub>0 f \<star> f\<^sub>0g\<^sub>1.p\<^sub>1, fg\<^sub>0h\<^sub>1.p\<^sub>1]) \<cdot> (\<a>[f, tab\<^sub>0 f, f\<^sub>0g\<^sub>1.p\<^sub>1] \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> \<a>\<^sup>-\<^sup>1[f \<star> tab\<^sub>0 f, f\<^sub>0g\<^sub>1.p\<^sub>1, fg\<^sub>0h\<^sub>1.p\<^sub>1] \<cdot> (f.tab \<star> f\<^sub>0g\<^sub>1.p\<^sub>1 \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1)
goal:
No subgoals!
[PROOF STEP]
qed
[PROOF STATE]
proof (state)
this:
\<a>[f, g, h \<star> tab\<^sub>0 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0] \<cdot> ((f \<star> g) \<star> \<a>[h, tab\<^sub>0 h, fg\<^sub>0h\<^sub>1.p\<^sub>0]) \<cdot> ((f \<star> g) \<star> h.tab \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0) \<cdot> ((f \<star> g) \<star> fg\<^sub>0h\<^sub>1.\<phi>) \<cdot> \<a>\<^sup>-\<^sup>1[f, g, (tab\<^sub>0 g \<star> f\<^sub>0g\<^sub>1.p\<^sub>0) \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1] \<cdot> (f \<star> \<a>[g, tab\<^sub>0 g \<star> f\<^sub>0g\<^sub>1.p\<^sub>0, fg\<^sub>0h\<^sub>1.p\<^sub>1]) \<cdot> (f \<star> \<a>[g, tab\<^sub>0 g, f\<^sub>0g\<^sub>1.p\<^sub>0] \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> (f \<star> (g.tab \<star> f\<^sub>0g\<^sub>1.p\<^sub>0) \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> (f \<star> f\<^sub>0g\<^sub>1.\<phi> \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> ((f \<star> \<a>\<^sup>-\<^sup>1[tab\<^sub>0 f, f\<^sub>0g\<^sub>1.p\<^sub>1, fg\<^sub>0h\<^sub>1.p\<^sub>1]) \<cdot> \<a>[f, tab\<^sub>0 f, f\<^sub>0g\<^sub>1.p\<^sub>1 \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1]) \<cdot> (f.tab \<star> f\<^sub>0g\<^sub>1.p\<^sub>1 \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) = \<a>[f, g, h \<star> tab\<^sub>0 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0] \<cdot> ((f \<star> g) \<star> \<a>[h, tab\<^sub>0 h, fg\<^sub>0h\<^sub>1.p\<^sub>0]) \<cdot> ((f \<star> g) \<star> h.tab \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0) \<cdot> ((f \<star> g) \<star> fg\<^sub>0h\<^sub>1.\<phi>) \<cdot> \<a>\<^sup>-\<^sup>1[f, g, (tab\<^sub>0 g \<star> f\<^sub>0g\<^sub>1.p\<^sub>0) \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1] \<cdot> (f \<star> \<a>[g, tab\<^sub>0 g \<star> f\<^sub>0g\<^sub>1.p\<^sub>0, fg\<^sub>0h\<^sub>1.p\<^sub>1]) \<cdot> (f \<star> \<a>[g, tab\<^sub>0 g, f\<^sub>0g\<^sub>1.p\<^sub>0] \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> (f \<star> (g.tab \<star> f\<^sub>0g\<^sub>1.p\<^sub>0) \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> ((f \<star> f\<^sub>0g\<^sub>1.\<phi> \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> \<a>[f, tab\<^sub>0 f \<star> f\<^sub>0g\<^sub>1.p\<^sub>1, fg\<^sub>0h\<^sub>1.p\<^sub>1]) \<cdot> (\<a>[f, tab\<^sub>0 f, f\<^sub>0g\<^sub>1.p\<^sub>1] \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> \<a>\<^sup>-\<^sup>1[f \<star> tab\<^sub>0 f, f\<^sub>0g\<^sub>1.p\<^sub>1, fg\<^sub>0h\<^sub>1.p\<^sub>1] \<cdot> (f.tab \<star> f\<^sub>0g\<^sub>1.p\<^sub>1 \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1)
goal (1 subgoal):
1. (f \<star> \<theta>\<^sub>f) \<cdot> \<a>[f, tab\<^sub>0 f, w\<^sub>f] \<cdot> (f.tab \<star> w\<^sub>f) = ((f \<star> \<theta>\<^sub>f') \<cdot> \<a>[f, tab\<^sub>0 f, w\<^sub>f'] \<cdot> (f.tab \<star> w\<^sub>f')) \<cdot> \<beta>\<^sub>f
[PROOF STEP]
also
[PROOF STATE]
proof (state)
this:
\<a>[f, g, h \<star> tab\<^sub>0 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0] \<cdot> ((f \<star> g) \<star> \<a>[h, tab\<^sub>0 h, fg\<^sub>0h\<^sub>1.p\<^sub>0]) \<cdot> ((f \<star> g) \<star> h.tab \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0) \<cdot> ((f \<star> g) \<star> fg\<^sub>0h\<^sub>1.\<phi>) \<cdot> \<a>\<^sup>-\<^sup>1[f, g, (tab\<^sub>0 g \<star> f\<^sub>0g\<^sub>1.p\<^sub>0) \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1] \<cdot> (f \<star> \<a>[g, tab\<^sub>0 g \<star> f\<^sub>0g\<^sub>1.p\<^sub>0, fg\<^sub>0h\<^sub>1.p\<^sub>1]) \<cdot> (f \<star> \<a>[g, tab\<^sub>0 g, f\<^sub>0g\<^sub>1.p\<^sub>0] \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> (f \<star> (g.tab \<star> f\<^sub>0g\<^sub>1.p\<^sub>0) \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> (f \<star> f\<^sub>0g\<^sub>1.\<phi> \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> ((f \<star> \<a>\<^sup>-\<^sup>1[tab\<^sub>0 f, f\<^sub>0g\<^sub>1.p\<^sub>1, fg\<^sub>0h\<^sub>1.p\<^sub>1]) \<cdot> \<a>[f, tab\<^sub>0 f, f\<^sub>0g\<^sub>1.p\<^sub>1 \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1]) \<cdot> (f.tab \<star> f\<^sub>0g\<^sub>1.p\<^sub>1 \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) = \<a>[f, g, h \<star> tab\<^sub>0 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0] \<cdot> ((f \<star> g) \<star> \<a>[h, tab\<^sub>0 h, fg\<^sub>0h\<^sub>1.p\<^sub>0]) \<cdot> ((f \<star> g) \<star> h.tab \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0) \<cdot> ((f \<star> g) \<star> fg\<^sub>0h\<^sub>1.\<phi>) \<cdot> \<a>\<^sup>-\<^sup>1[f, g, (tab\<^sub>0 g \<star> f\<^sub>0g\<^sub>1.p\<^sub>0) \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1] \<cdot> (f \<star> \<a>[g, tab\<^sub>0 g \<star> f\<^sub>0g\<^sub>1.p\<^sub>0, fg\<^sub>0h\<^sub>1.p\<^sub>1]) \<cdot> (f \<star> \<a>[g, tab\<^sub>0 g, f\<^sub>0g\<^sub>1.p\<^sub>0] \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> (f \<star> (g.tab \<star> f\<^sub>0g\<^sub>1.p\<^sub>0) \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> ((f \<star> f\<^sub>0g\<^sub>1.\<phi> \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> \<a>[f, tab\<^sub>0 f \<star> f\<^sub>0g\<^sub>1.p\<^sub>1, fg\<^sub>0h\<^sub>1.p\<^sub>1]) \<cdot> (\<a>[f, tab\<^sub>0 f, f\<^sub>0g\<^sub>1.p\<^sub>1] \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> \<a>\<^sup>-\<^sup>1[f \<star> tab\<^sub>0 f, f\<^sub>0g\<^sub>1.p\<^sub>1, fg\<^sub>0h\<^sub>1.p\<^sub>1] \<cdot> (f.tab \<star> f\<^sub>0g\<^sub>1.p\<^sub>1 \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1)
goal (1 subgoal):
1. (f \<star> \<theta>\<^sub>f) \<cdot> \<a>[f, tab\<^sub>0 f, w\<^sub>f] \<cdot> (f.tab \<star> w\<^sub>f) = ((f \<star> \<theta>\<^sub>f') \<cdot> \<a>[f, tab\<^sub>0 f, w\<^sub>f'] \<cdot> (f.tab \<star> w\<^sub>f')) \<cdot> \<beta>\<^sub>f
[PROOF STEP]
have "... =
\<a>[f, g, h \<star> tab\<^sub>0 h \<star> TTfgh.p\<^sub>0] \<cdot>
((f \<star> g) \<star> \<a>[h, tab\<^sub>0 h, TTfgh.p\<^sub>0]) \<cdot>
((f \<star> g) \<star> h.tab \<star> TTfgh.p\<^sub>0) \<cdot>
((f \<star> g) \<star> fg\<^sub>0h\<^sub>1.\<phi>) \<cdot>
\<a>\<^sup>-\<^sup>1[f, g, (tab\<^sub>0 g \<star> Tfg.p\<^sub>0) \<star> TTfgh.p\<^sub>1] \<cdot>
(f \<star> \<a>[g, tab\<^sub>0 g \<star> Tfg.p\<^sub>0, TTfgh.p\<^sub>1]) \<cdot>
(f \<star> \<a>[g, tab\<^sub>0 g, Tfg.p\<^sub>0] \<star> TTfgh.p\<^sub>1) \<cdot>
((f \<star> (g.tab \<star> Tfg.p\<^sub>0) \<star> TTfgh.p\<^sub>1) \<cdot>
\<a>[f, tab\<^sub>1 g \<star> Tfg.p\<^sub>0, TTfgh.p\<^sub>1]) \<cdot>
((f \<star> f\<^sub>0g\<^sub>1.\<phi>) \<star> TTfgh.p\<^sub>1) \<cdot>
(\<a>[f, tab\<^sub>0 f, Tfg.p\<^sub>1] \<star> TTfgh.p\<^sub>1) \<cdot>
\<a>\<^sup>-\<^sup>1[f \<star> tab\<^sub>0 f, Tfg.p\<^sub>1, TTfgh.p\<^sub>1] \<cdot>
(f.tab \<star> Tfg.p\<^sub>1 \<star> TTfgh.p\<^sub>1)"
[PROOF STATE]
proof (prove)
goal (1 subgoal):
1. \<a>[f, g, h \<star> tab\<^sub>0 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0] \<cdot> ((f \<star> g) \<star> \<a>[h, tab\<^sub>0 h, fg\<^sub>0h\<^sub>1.p\<^sub>0]) \<cdot> ((f \<star> g) \<star> h.tab \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0) \<cdot> ((f \<star> g) \<star> fg\<^sub>0h\<^sub>1.\<phi>) \<cdot> \<a>\<^sup>-\<^sup>1[f, g, (tab\<^sub>0 g \<star> f\<^sub>0g\<^sub>1.p\<^sub>0) \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1] \<cdot> (f \<star> \<a>[g, tab\<^sub>0 g \<star> f\<^sub>0g\<^sub>1.p\<^sub>0, fg\<^sub>0h\<^sub>1.p\<^sub>1]) \<cdot> (f \<star> \<a>[g, tab\<^sub>0 g, f\<^sub>0g\<^sub>1.p\<^sub>0] \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> (f \<star> (g.tab \<star> f\<^sub>0g\<^sub>1.p\<^sub>0) \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> ((f \<star> f\<^sub>0g\<^sub>1.\<phi> \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> \<a>[f, tab\<^sub>0 f \<star> f\<^sub>0g\<^sub>1.p\<^sub>1, fg\<^sub>0h\<^sub>1.p\<^sub>1]) \<cdot> (\<a>[f, tab\<^sub>0 f, f\<^sub>0g\<^sub>1.p\<^sub>1] \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> \<a>\<^sup>-\<^sup>1[f \<star> tab\<^sub>0 f, f\<^sub>0g\<^sub>1.p\<^sub>1, fg\<^sub>0h\<^sub>1.p\<^sub>1] \<cdot> (f.tab \<star> f\<^sub>0g\<^sub>1.p\<^sub>1 \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) = \<a>[f, g, h \<star> tab\<^sub>0 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0] \<cdot> ((f \<star> g) \<star> \<a>[h, tab\<^sub>0 h, fg\<^sub>0h\<^sub>1.p\<^sub>0]) \<cdot> ((f \<star> g) \<star> h.tab \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0) \<cdot> ((f \<star> g) \<star> fg\<^sub>0h\<^sub>1.\<phi>) \<cdot> \<a>\<^sup>-\<^sup>1[f, g, (tab\<^sub>0 g \<star> f\<^sub>0g\<^sub>1.p\<^sub>0) \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1] \<cdot> (f \<star> \<a>[g, tab\<^sub>0 g \<star> f\<^sub>0g\<^sub>1.p\<^sub>0, fg\<^sub>0h\<^sub>1.p\<^sub>1]) \<cdot> (f \<star> \<a>[g, tab\<^sub>0 g, f\<^sub>0g\<^sub>1.p\<^sub>0] \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> ((f \<star> (g.tab \<star> f\<^sub>0g\<^sub>1.p\<^sub>0) \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> \<a>[f, tab\<^sub>1 g \<star> f\<^sub>0g\<^sub>1.p\<^sub>0, fg\<^sub>0h\<^sub>1.p\<^sub>1]) \<cdot> ((f \<star> f\<^sub>0g\<^sub>1.\<phi>) \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> (\<a>[f, tab\<^sub>0 f, f\<^sub>0g\<^sub>1.p\<^sub>1] \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> \<a>\<^sup>-\<^sup>1[f \<star> tab\<^sub>0 f, f\<^sub>0g\<^sub>1.p\<^sub>1, fg\<^sub>0h\<^sub>1.p\<^sub>1] \<cdot> (f.tab \<star> f\<^sub>0g\<^sub>1.p\<^sub>1 \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1)
[PROOF STEP]
proof -
[PROOF STATE]
proof (state)
goal (1 subgoal):
1. \<a>[f, g, h \<star> tab\<^sub>0 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0] \<cdot> ((f \<star> g) \<star> \<a>[h, tab\<^sub>0 h, fg\<^sub>0h\<^sub>1.p\<^sub>0]) \<cdot> ((f \<star> g) \<star> h.tab \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0) \<cdot> ((f \<star> g) \<star> fg\<^sub>0h\<^sub>1.\<phi>) \<cdot> \<a>\<^sup>-\<^sup>1[f, g, (tab\<^sub>0 g \<star> f\<^sub>0g\<^sub>1.p\<^sub>0) \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1] \<cdot> (f \<star> \<a>[g, tab\<^sub>0 g \<star> f\<^sub>0g\<^sub>1.p\<^sub>0, fg\<^sub>0h\<^sub>1.p\<^sub>1]) \<cdot> (f \<star> \<a>[g, tab\<^sub>0 g, f\<^sub>0g\<^sub>1.p\<^sub>0] \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> (f \<star> (g.tab \<star> f\<^sub>0g\<^sub>1.p\<^sub>0) \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> ((f \<star> f\<^sub>0g\<^sub>1.\<phi> \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> \<a>[f, tab\<^sub>0 f \<star> f\<^sub>0g\<^sub>1.p\<^sub>1, fg\<^sub>0h\<^sub>1.p\<^sub>1]) \<cdot> (\<a>[f, tab\<^sub>0 f, f\<^sub>0g\<^sub>1.p\<^sub>1] \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> \<a>\<^sup>-\<^sup>1[f \<star> tab\<^sub>0 f, f\<^sub>0g\<^sub>1.p\<^sub>1, fg\<^sub>0h\<^sub>1.p\<^sub>1] \<cdot> (f.tab \<star> f\<^sub>0g\<^sub>1.p\<^sub>1 \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) = \<a>[f, g, h \<star> tab\<^sub>0 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0] \<cdot> ((f \<star> g) \<star> \<a>[h, tab\<^sub>0 h, fg\<^sub>0h\<^sub>1.p\<^sub>0]) \<cdot> ((f \<star> g) \<star> h.tab \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0) \<cdot> ((f \<star> g) \<star> fg\<^sub>0h\<^sub>1.\<phi>) \<cdot> \<a>\<^sup>-\<^sup>1[f, g, (tab\<^sub>0 g \<star> f\<^sub>0g\<^sub>1.p\<^sub>0) \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1] \<cdot> (f \<star> \<a>[g, tab\<^sub>0 g \<star> f\<^sub>0g\<^sub>1.p\<^sub>0, fg\<^sub>0h\<^sub>1.p\<^sub>1]) \<cdot> (f \<star> \<a>[g, tab\<^sub>0 g, f\<^sub>0g\<^sub>1.p\<^sub>0] \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> ((f \<star> (g.tab \<star> f\<^sub>0g\<^sub>1.p\<^sub>0) \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> \<a>[f, tab\<^sub>1 g \<star> f\<^sub>0g\<^sub>1.p\<^sub>0, fg\<^sub>0h\<^sub>1.p\<^sub>1]) \<cdot> ((f \<star> f\<^sub>0g\<^sub>1.\<phi>) \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> (\<a>[f, tab\<^sub>0 f, f\<^sub>0g\<^sub>1.p\<^sub>1] \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> \<a>\<^sup>-\<^sup>1[f \<star> tab\<^sub>0 f, f\<^sub>0g\<^sub>1.p\<^sub>1, fg\<^sub>0h\<^sub>1.p\<^sub>1] \<cdot> (f.tab \<star> f\<^sub>0g\<^sub>1.p\<^sub>1 \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1)
[PROOF STEP]
have "(f \<star> f\<^sub>0g\<^sub>1.\<phi> \<star> TTfgh.p\<^sub>1) \<cdot> \<a>[f, tab\<^sub>0 f \<star> Tfg.p\<^sub>1, TTfgh.p\<^sub>1] =
\<a>[f, tab\<^sub>1 g \<star> Tfg.p\<^sub>0, TTfgh.p\<^sub>1] \<cdot> ((f \<star> f\<^sub>0g\<^sub>1.\<phi>) \<star> TTfgh.p\<^sub>1)"
[PROOF STATE]
proof (prove)
goal (1 subgoal):
1. (f \<star> f\<^sub>0g\<^sub>1.\<phi> \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> \<a>[f, tab\<^sub>0 f \<star> f\<^sub>0g\<^sub>1.p\<^sub>1, fg\<^sub>0h\<^sub>1.p\<^sub>1] = \<a>[f, tab\<^sub>1 g \<star> f\<^sub>0g\<^sub>1.p\<^sub>0, fg\<^sub>0h\<^sub>1.p\<^sub>1] \<cdot> ((f \<star> f\<^sub>0g\<^sub>1.\<phi>) \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1)
[PROOF STEP]
using fg gh fg\<^sub>0h\<^sub>1.p\<^sub>0_simps fg\<^sub>0h\<^sub>1.p\<^sub>1_simps f\<^sub>0g\<^sub>1.\<phi>_in_hom
assoc_naturality [of f f\<^sub>0g\<^sub>1.\<phi> TTfgh.p\<^sub>1]
[PROOF STATE]
proof (prove)
using this:
src f = trg g
src g = trg h
trg fg\<^sub>0h\<^sub>1.p\<^sub>0 = src (tab\<^sub>1 h)
trg fg\<^sub>0h\<^sub>1.p\<^sub>1 = src (tab\<^sub>0 g \<star> f\<^sub>0g\<^sub>1.p\<^sub>0)
\<guillemotleft>f\<^sub>0g\<^sub>1.\<phi> : src f\<^sub>0g\<^sub>1.p\<^sub>0 \<rightarrow> trg (tab\<^sub>1 g)\<guillemotright>
\<guillemotleft>f\<^sub>0g\<^sub>1.\<phi> : tab\<^sub>0 f \<star> f\<^sub>0g\<^sub>1.p\<^sub>1 \<rightarrow> tab\<^sub>1 g \<star> f\<^sub>0g\<^sub>1.p\<^sub>0\<guillemotright>
\<lbrakk>arr f; arr f\<^sub>0g\<^sub>1.\<phi>; arr fg\<^sub>0h\<^sub>1.p\<^sub>1; src f = trg f\<^sub>0g\<^sub>1.\<phi>; src f\<^sub>0g\<^sub>1.\<phi> = trg fg\<^sub>0h\<^sub>1.p\<^sub>1\<rbrakk> \<Longrightarrow> \<a>[cod f, cod f\<^sub>0g\<^sub>1.\<phi>, cod fg\<^sub>0h\<^sub>1.p\<^sub>1] \<cdot> ((f \<star> f\<^sub>0g\<^sub>1.\<phi>) \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) = (f \<star> f\<^sub>0g\<^sub>1.\<phi> \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> \<a>[local.dom f, local.dom f\<^sub>0g\<^sub>1.\<phi>, local.dom fg\<^sub>0h\<^sub>1.p\<^sub>1]
goal (1 subgoal):
1. (f \<star> f\<^sub>0g\<^sub>1.\<phi> \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> \<a>[f, tab\<^sub>0 f \<star> f\<^sub>0g\<^sub>1.p\<^sub>1, fg\<^sub>0h\<^sub>1.p\<^sub>1] = \<a>[f, tab\<^sub>1 g \<star> f\<^sub>0g\<^sub>1.p\<^sub>0, fg\<^sub>0h\<^sub>1.p\<^sub>1] \<cdot> ((f \<star> f\<^sub>0g\<^sub>1.\<phi>) \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1)
[PROOF STEP]
by simp
[PROOF STATE]
proof (state)
this:
(f \<star> f\<^sub>0g\<^sub>1.\<phi> \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> \<a>[f, tab\<^sub>0 f \<star> f\<^sub>0g\<^sub>1.p\<^sub>1, fg\<^sub>0h\<^sub>1.p\<^sub>1] = \<a>[f, tab\<^sub>1 g \<star> f\<^sub>0g\<^sub>1.p\<^sub>0, fg\<^sub>0h\<^sub>1.p\<^sub>1] \<cdot> ((f \<star> f\<^sub>0g\<^sub>1.\<phi>) \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1)
goal (1 subgoal):
1. \<a>[f, g, h \<star> tab\<^sub>0 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0] \<cdot> ((f \<star> g) \<star> \<a>[h, tab\<^sub>0 h, fg\<^sub>0h\<^sub>1.p\<^sub>0]) \<cdot> ((f \<star> g) \<star> h.tab \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0) \<cdot> ((f \<star> g) \<star> fg\<^sub>0h\<^sub>1.\<phi>) \<cdot> \<a>\<^sup>-\<^sup>1[f, g, (tab\<^sub>0 g \<star> f\<^sub>0g\<^sub>1.p\<^sub>0) \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1] \<cdot> (f \<star> \<a>[g, tab\<^sub>0 g \<star> f\<^sub>0g\<^sub>1.p\<^sub>0, fg\<^sub>0h\<^sub>1.p\<^sub>1]) \<cdot> (f \<star> \<a>[g, tab\<^sub>0 g, f\<^sub>0g\<^sub>1.p\<^sub>0] \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> (f \<star> (g.tab \<star> f\<^sub>0g\<^sub>1.p\<^sub>0) \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> ((f \<star> f\<^sub>0g\<^sub>1.\<phi> \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> \<a>[f, tab\<^sub>0 f \<star> f\<^sub>0g\<^sub>1.p\<^sub>1, fg\<^sub>0h\<^sub>1.p\<^sub>1]) \<cdot> (\<a>[f, tab\<^sub>0 f, f\<^sub>0g\<^sub>1.p\<^sub>1] \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> \<a>\<^sup>-\<^sup>1[f \<star> tab\<^sub>0 f, f\<^sub>0g\<^sub>1.p\<^sub>1, fg\<^sub>0h\<^sub>1.p\<^sub>1] \<cdot> (f.tab \<star> f\<^sub>0g\<^sub>1.p\<^sub>1 \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) = \<a>[f, g, h \<star> tab\<^sub>0 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0] \<cdot> ((f \<star> g) \<star> \<a>[h, tab\<^sub>0 h, fg\<^sub>0h\<^sub>1.p\<^sub>0]) \<cdot> ((f \<star> g) \<star> h.tab \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0) \<cdot> ((f \<star> g) \<star> fg\<^sub>0h\<^sub>1.\<phi>) \<cdot> \<a>\<^sup>-\<^sup>1[f, g, (tab\<^sub>0 g \<star> f\<^sub>0g\<^sub>1.p\<^sub>0) \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1] \<cdot> (f \<star> \<a>[g, tab\<^sub>0 g \<star> f\<^sub>0g\<^sub>1.p\<^sub>0, fg\<^sub>0h\<^sub>1.p\<^sub>1]) \<cdot> (f \<star> \<a>[g, tab\<^sub>0 g, f\<^sub>0g\<^sub>1.p\<^sub>0] \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> ((f \<star> (g.tab \<star> f\<^sub>0g\<^sub>1.p\<^sub>0) \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> \<a>[f, tab\<^sub>1 g \<star> f\<^sub>0g\<^sub>1.p\<^sub>0, fg\<^sub>0h\<^sub>1.p\<^sub>1]) \<cdot> ((f \<star> f\<^sub>0g\<^sub>1.\<phi>) \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> (\<a>[f, tab\<^sub>0 f, f\<^sub>0g\<^sub>1.p\<^sub>1] \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> \<a>\<^sup>-\<^sup>1[f \<star> tab\<^sub>0 f, f\<^sub>0g\<^sub>1.p\<^sub>1, fg\<^sub>0h\<^sub>1.p\<^sub>1] \<cdot> (f.tab \<star> f\<^sub>0g\<^sub>1.p\<^sub>1 \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1)
[PROOF STEP]
thus ?thesis
[PROOF STATE]
proof (prove)
using this:
(f \<star> f\<^sub>0g\<^sub>1.\<phi> \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> \<a>[f, tab\<^sub>0 f \<star> f\<^sub>0g\<^sub>1.p\<^sub>1, fg\<^sub>0h\<^sub>1.p\<^sub>1] = \<a>[f, tab\<^sub>1 g \<star> f\<^sub>0g\<^sub>1.p\<^sub>0, fg\<^sub>0h\<^sub>1.p\<^sub>1] \<cdot> ((f \<star> f\<^sub>0g\<^sub>1.\<phi>) \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1)
goal (1 subgoal):
1. \<a>[f, g, h \<star> tab\<^sub>0 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0] \<cdot> ((f \<star> g) \<star> \<a>[h, tab\<^sub>0 h, fg\<^sub>0h\<^sub>1.p\<^sub>0]) \<cdot> ((f \<star> g) \<star> h.tab \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0) \<cdot> ((f \<star> g) \<star> fg\<^sub>0h\<^sub>1.\<phi>) \<cdot> \<a>\<^sup>-\<^sup>1[f, g, (tab\<^sub>0 g \<star> f\<^sub>0g\<^sub>1.p\<^sub>0) \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1] \<cdot> (f \<star> \<a>[g, tab\<^sub>0 g \<star> f\<^sub>0g\<^sub>1.p\<^sub>0, fg\<^sub>0h\<^sub>1.p\<^sub>1]) \<cdot> (f \<star> \<a>[g, tab\<^sub>0 g, f\<^sub>0g\<^sub>1.p\<^sub>0] \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> (f \<star> (g.tab \<star> f\<^sub>0g\<^sub>1.p\<^sub>0) \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> ((f \<star> f\<^sub>0g\<^sub>1.\<phi> \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> \<a>[f, tab\<^sub>0 f \<star> f\<^sub>0g\<^sub>1.p\<^sub>1, fg\<^sub>0h\<^sub>1.p\<^sub>1]) \<cdot> (\<a>[f, tab\<^sub>0 f, f\<^sub>0g\<^sub>1.p\<^sub>1] \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> \<a>\<^sup>-\<^sup>1[f \<star> tab\<^sub>0 f, f\<^sub>0g\<^sub>1.p\<^sub>1, fg\<^sub>0h\<^sub>1.p\<^sub>1] \<cdot> (f.tab \<star> f\<^sub>0g\<^sub>1.p\<^sub>1 \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) = \<a>[f, g, h \<star> tab\<^sub>0 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0] \<cdot> ((f \<star> g) \<star> \<a>[h, tab\<^sub>0 h, fg\<^sub>0h\<^sub>1.p\<^sub>0]) \<cdot> ((f \<star> g) \<star> h.tab \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0) \<cdot> ((f \<star> g) \<star> fg\<^sub>0h\<^sub>1.\<phi>) \<cdot> \<a>\<^sup>-\<^sup>1[f, g, (tab\<^sub>0 g \<star> f\<^sub>0g\<^sub>1.p\<^sub>0) \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1] \<cdot> (f \<star> \<a>[g, tab\<^sub>0 g \<star> f\<^sub>0g\<^sub>1.p\<^sub>0, fg\<^sub>0h\<^sub>1.p\<^sub>1]) \<cdot> (f \<star> \<a>[g, tab\<^sub>0 g, f\<^sub>0g\<^sub>1.p\<^sub>0] \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> ((f \<star> (g.tab \<star> f\<^sub>0g\<^sub>1.p\<^sub>0) \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> \<a>[f, tab\<^sub>1 g \<star> f\<^sub>0g\<^sub>1.p\<^sub>0, fg\<^sub>0h\<^sub>1.p\<^sub>1]) \<cdot> ((f \<star> f\<^sub>0g\<^sub>1.\<phi>) \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> (\<a>[f, tab\<^sub>0 f, f\<^sub>0g\<^sub>1.p\<^sub>1] \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> \<a>\<^sup>-\<^sup>1[f \<star> tab\<^sub>0 f, f\<^sub>0g\<^sub>1.p\<^sub>1, fg\<^sub>0h\<^sub>1.p\<^sub>1] \<cdot> (f.tab \<star> f\<^sub>0g\<^sub>1.p\<^sub>1 \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1)
[PROOF STEP]
using comp_assoc
[PROOF STATE]
proof (prove)
using this:
(f \<star> f\<^sub>0g\<^sub>1.\<phi> \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> \<a>[f, tab\<^sub>0 f \<star> f\<^sub>0g\<^sub>1.p\<^sub>1, fg\<^sub>0h\<^sub>1.p\<^sub>1] = \<a>[f, tab\<^sub>1 g \<star> f\<^sub>0g\<^sub>1.p\<^sub>0, fg\<^sub>0h\<^sub>1.p\<^sub>1] \<cdot> ((f \<star> f\<^sub>0g\<^sub>1.\<phi>) \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1)
(?h \<cdot> ?g) \<cdot> ?f = ?h \<cdot> ?g \<cdot> ?f
goal (1 subgoal):
1. \<a>[f, g, h \<star> tab\<^sub>0 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0] \<cdot> ((f \<star> g) \<star> \<a>[h, tab\<^sub>0 h, fg\<^sub>0h\<^sub>1.p\<^sub>0]) \<cdot> ((f \<star> g) \<star> h.tab \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0) \<cdot> ((f \<star> g) \<star> fg\<^sub>0h\<^sub>1.\<phi>) \<cdot> \<a>\<^sup>-\<^sup>1[f, g, (tab\<^sub>0 g \<star> f\<^sub>0g\<^sub>1.p\<^sub>0) \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1] \<cdot> (f \<star> \<a>[g, tab\<^sub>0 g \<star> f\<^sub>0g\<^sub>1.p\<^sub>0, fg\<^sub>0h\<^sub>1.p\<^sub>1]) \<cdot> (f \<star> \<a>[g, tab\<^sub>0 g, f\<^sub>0g\<^sub>1.p\<^sub>0] \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> (f \<star> (g.tab \<star> f\<^sub>0g\<^sub>1.p\<^sub>0) \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> ((f \<star> f\<^sub>0g\<^sub>1.\<phi> \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> \<a>[f, tab\<^sub>0 f \<star> f\<^sub>0g\<^sub>1.p\<^sub>1, fg\<^sub>0h\<^sub>1.p\<^sub>1]) \<cdot> (\<a>[f, tab\<^sub>0 f, f\<^sub>0g\<^sub>1.p\<^sub>1] \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> \<a>\<^sup>-\<^sup>1[f \<star> tab\<^sub>0 f, f\<^sub>0g\<^sub>1.p\<^sub>1, fg\<^sub>0h\<^sub>1.p\<^sub>1] \<cdot> (f.tab \<star> f\<^sub>0g\<^sub>1.p\<^sub>1 \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) = \<a>[f, g, h \<star> tab\<^sub>0 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0] \<cdot> ((f \<star> g) \<star> \<a>[h, tab\<^sub>0 h, fg\<^sub>0h\<^sub>1.p\<^sub>0]) \<cdot> ((f \<star> g) \<star> h.tab \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0) \<cdot> ((f \<star> g) \<star> fg\<^sub>0h\<^sub>1.\<phi>) \<cdot> \<a>\<^sup>-\<^sup>1[f, g, (tab\<^sub>0 g \<star> f\<^sub>0g\<^sub>1.p\<^sub>0) \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1] \<cdot> (f \<star> \<a>[g, tab\<^sub>0 g \<star> f\<^sub>0g\<^sub>1.p\<^sub>0, fg\<^sub>0h\<^sub>1.p\<^sub>1]) \<cdot> (f \<star> \<a>[g, tab\<^sub>0 g, f\<^sub>0g\<^sub>1.p\<^sub>0] \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> ((f \<star> (g.tab \<star> f\<^sub>0g\<^sub>1.p\<^sub>0) \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> \<a>[f, tab\<^sub>1 g \<star> f\<^sub>0g\<^sub>1.p\<^sub>0, fg\<^sub>0h\<^sub>1.p\<^sub>1]) \<cdot> ((f \<star> f\<^sub>0g\<^sub>1.\<phi>) \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> (\<a>[f, tab\<^sub>0 f, f\<^sub>0g\<^sub>1.p\<^sub>1] \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> \<a>\<^sup>-\<^sup>1[f \<star> tab\<^sub>0 f, f\<^sub>0g\<^sub>1.p\<^sub>1, fg\<^sub>0h\<^sub>1.p\<^sub>1] \<cdot> (f.tab \<star> f\<^sub>0g\<^sub>1.p\<^sub>1 \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1)
[PROOF STEP]
by presburger
[PROOF STATE]
proof (state)
this:
\<a>[f, g, h \<star> tab\<^sub>0 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0] \<cdot> ((f \<star> g) \<star> \<a>[h, tab\<^sub>0 h, fg\<^sub>0h\<^sub>1.p\<^sub>0]) \<cdot> ((f \<star> g) \<star> h.tab \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0) \<cdot> ((f \<star> g) \<star> fg\<^sub>0h\<^sub>1.\<phi>) \<cdot> \<a>\<^sup>-\<^sup>1[f, g, (tab\<^sub>0 g \<star> f\<^sub>0g\<^sub>1.p\<^sub>0) \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1] \<cdot> (f \<star> \<a>[g, tab\<^sub>0 g \<star> f\<^sub>0g\<^sub>1.p\<^sub>0, fg\<^sub>0h\<^sub>1.p\<^sub>1]) \<cdot> (f \<star> \<a>[g, tab\<^sub>0 g, f\<^sub>0g\<^sub>1.p\<^sub>0] \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> (f \<star> (g.tab \<star> f\<^sub>0g\<^sub>1.p\<^sub>0) \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> ((f \<star> f\<^sub>0g\<^sub>1.\<phi> \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> \<a>[f, tab\<^sub>0 f \<star> f\<^sub>0g\<^sub>1.p\<^sub>1, fg\<^sub>0h\<^sub>1.p\<^sub>1]) \<cdot> (\<a>[f, tab\<^sub>0 f, f\<^sub>0g\<^sub>1.p\<^sub>1] \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> \<a>\<^sup>-\<^sup>1[f \<star> tab\<^sub>0 f, f\<^sub>0g\<^sub>1.p\<^sub>1, fg\<^sub>0h\<^sub>1.p\<^sub>1] \<cdot> (f.tab \<star> f\<^sub>0g\<^sub>1.p\<^sub>1 \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) = \<a>[f, g, h \<star> tab\<^sub>0 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0] \<cdot> ((f \<star> g) \<star> \<a>[h, tab\<^sub>0 h, fg\<^sub>0h\<^sub>1.p\<^sub>0]) \<cdot> ((f \<star> g) \<star> h.tab \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0) \<cdot> ((f \<star> g) \<star> fg\<^sub>0h\<^sub>1.\<phi>) \<cdot> \<a>\<^sup>-\<^sup>1[f, g, (tab\<^sub>0 g \<star> f\<^sub>0g\<^sub>1.p\<^sub>0) \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1] \<cdot> (f \<star> \<a>[g, tab\<^sub>0 g \<star> f\<^sub>0g\<^sub>1.p\<^sub>0, fg\<^sub>0h\<^sub>1.p\<^sub>1]) \<cdot> (f \<star> \<a>[g, tab\<^sub>0 g, f\<^sub>0g\<^sub>1.p\<^sub>0] \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> ((f \<star> (g.tab \<star> f\<^sub>0g\<^sub>1.p\<^sub>0) \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> \<a>[f, tab\<^sub>1 g \<star> f\<^sub>0g\<^sub>1.p\<^sub>0, fg\<^sub>0h\<^sub>1.p\<^sub>1]) \<cdot> ((f \<star> f\<^sub>0g\<^sub>1.\<phi>) \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> (\<a>[f, tab\<^sub>0 f, f\<^sub>0g\<^sub>1.p\<^sub>1] \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> \<a>\<^sup>-\<^sup>1[f \<star> tab\<^sub>0 f, f\<^sub>0g\<^sub>1.p\<^sub>1, fg\<^sub>0h\<^sub>1.p\<^sub>1] \<cdot> (f.tab \<star> f\<^sub>0g\<^sub>1.p\<^sub>1 \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1)
goal:
No subgoals!
[PROOF STEP]
qed
[PROOF STATE]
proof (state)
this:
\<a>[f, g, h \<star> tab\<^sub>0 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0] \<cdot> ((f \<star> g) \<star> \<a>[h, tab\<^sub>0 h, fg\<^sub>0h\<^sub>1.p\<^sub>0]) \<cdot> ((f \<star> g) \<star> h.tab \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0) \<cdot> ((f \<star> g) \<star> fg\<^sub>0h\<^sub>1.\<phi>) \<cdot> \<a>\<^sup>-\<^sup>1[f, g, (tab\<^sub>0 g \<star> f\<^sub>0g\<^sub>1.p\<^sub>0) \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1] \<cdot> (f \<star> \<a>[g, tab\<^sub>0 g \<star> f\<^sub>0g\<^sub>1.p\<^sub>0, fg\<^sub>0h\<^sub>1.p\<^sub>1]) \<cdot> (f \<star> \<a>[g, tab\<^sub>0 g, f\<^sub>0g\<^sub>1.p\<^sub>0] \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> (f \<star> (g.tab \<star> f\<^sub>0g\<^sub>1.p\<^sub>0) \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> ((f \<star> f\<^sub>0g\<^sub>1.\<phi> \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> \<a>[f, tab\<^sub>0 f \<star> f\<^sub>0g\<^sub>1.p\<^sub>1, fg\<^sub>0h\<^sub>1.p\<^sub>1]) \<cdot> (\<a>[f, tab\<^sub>0 f, f\<^sub>0g\<^sub>1.p\<^sub>1] \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> \<a>\<^sup>-\<^sup>1[f \<star> tab\<^sub>0 f, f\<^sub>0g\<^sub>1.p\<^sub>1, fg\<^sub>0h\<^sub>1.p\<^sub>1] \<cdot> (f.tab \<star> f\<^sub>0g\<^sub>1.p\<^sub>1 \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) = \<a>[f, g, h \<star> tab\<^sub>0 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0] \<cdot> ((f \<star> g) \<star> \<a>[h, tab\<^sub>0 h, fg\<^sub>0h\<^sub>1.p\<^sub>0]) \<cdot> ((f \<star> g) \<star> h.tab \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0) \<cdot> ((f \<star> g) \<star> fg\<^sub>0h\<^sub>1.\<phi>) \<cdot> \<a>\<^sup>-\<^sup>1[f, g, (tab\<^sub>0 g \<star> f\<^sub>0g\<^sub>1.p\<^sub>0) \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1] \<cdot> (f \<star> \<a>[g, tab\<^sub>0 g \<star> f\<^sub>0g\<^sub>1.p\<^sub>0, fg\<^sub>0h\<^sub>1.p\<^sub>1]) \<cdot> (f \<star> \<a>[g, tab\<^sub>0 g, f\<^sub>0g\<^sub>1.p\<^sub>0] \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> ((f \<star> (g.tab \<star> f\<^sub>0g\<^sub>1.p\<^sub>0) \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> \<a>[f, tab\<^sub>1 g \<star> f\<^sub>0g\<^sub>1.p\<^sub>0, fg\<^sub>0h\<^sub>1.p\<^sub>1]) \<cdot> ((f \<star> f\<^sub>0g\<^sub>1.\<phi>) \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> (\<a>[f, tab\<^sub>0 f, f\<^sub>0g\<^sub>1.p\<^sub>1] \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> \<a>\<^sup>-\<^sup>1[f \<star> tab\<^sub>0 f, f\<^sub>0g\<^sub>1.p\<^sub>1, fg\<^sub>0h\<^sub>1.p\<^sub>1] \<cdot> (f.tab \<star> f\<^sub>0g\<^sub>1.p\<^sub>1 \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1)
goal (1 subgoal):
1. (f \<star> \<theta>\<^sub>f) \<cdot> \<a>[f, tab\<^sub>0 f, w\<^sub>f] \<cdot> (f.tab \<star> w\<^sub>f) = ((f \<star> \<theta>\<^sub>f') \<cdot> \<a>[f, tab\<^sub>0 f, w\<^sub>f'] \<cdot> (f.tab \<star> w\<^sub>f')) \<cdot> \<beta>\<^sub>f
[PROOF STEP]
also
[PROOF STATE]
proof (state)
this:
\<a>[f, g, h \<star> tab\<^sub>0 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0] \<cdot> ((f \<star> g) \<star> \<a>[h, tab\<^sub>0 h, fg\<^sub>0h\<^sub>1.p\<^sub>0]) \<cdot> ((f \<star> g) \<star> h.tab \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0) \<cdot> ((f \<star> g) \<star> fg\<^sub>0h\<^sub>1.\<phi>) \<cdot> \<a>\<^sup>-\<^sup>1[f, g, (tab\<^sub>0 g \<star> f\<^sub>0g\<^sub>1.p\<^sub>0) \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1] \<cdot> (f \<star> \<a>[g, tab\<^sub>0 g \<star> f\<^sub>0g\<^sub>1.p\<^sub>0, fg\<^sub>0h\<^sub>1.p\<^sub>1]) \<cdot> (f \<star> \<a>[g, tab\<^sub>0 g, f\<^sub>0g\<^sub>1.p\<^sub>0] \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> (f \<star> (g.tab \<star> f\<^sub>0g\<^sub>1.p\<^sub>0) \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> ((f \<star> f\<^sub>0g\<^sub>1.\<phi> \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> \<a>[f, tab\<^sub>0 f \<star> f\<^sub>0g\<^sub>1.p\<^sub>1, fg\<^sub>0h\<^sub>1.p\<^sub>1]) \<cdot> (\<a>[f, tab\<^sub>0 f, f\<^sub>0g\<^sub>1.p\<^sub>1] \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> \<a>\<^sup>-\<^sup>1[f \<star> tab\<^sub>0 f, f\<^sub>0g\<^sub>1.p\<^sub>1, fg\<^sub>0h\<^sub>1.p\<^sub>1] \<cdot> (f.tab \<star> f\<^sub>0g\<^sub>1.p\<^sub>1 \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) = \<a>[f, g, h \<star> tab\<^sub>0 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0] \<cdot> ((f \<star> g) \<star> \<a>[h, tab\<^sub>0 h, fg\<^sub>0h\<^sub>1.p\<^sub>0]) \<cdot> ((f \<star> g) \<star> h.tab \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0) \<cdot> ((f \<star> g) \<star> fg\<^sub>0h\<^sub>1.\<phi>) \<cdot> \<a>\<^sup>-\<^sup>1[f, g, (tab\<^sub>0 g \<star> f\<^sub>0g\<^sub>1.p\<^sub>0) \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1] \<cdot> (f \<star> \<a>[g, tab\<^sub>0 g \<star> f\<^sub>0g\<^sub>1.p\<^sub>0, fg\<^sub>0h\<^sub>1.p\<^sub>1]) \<cdot> (f \<star> \<a>[g, tab\<^sub>0 g, f\<^sub>0g\<^sub>1.p\<^sub>0] \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> ((f \<star> (g.tab \<star> f\<^sub>0g\<^sub>1.p\<^sub>0) \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> \<a>[f, tab\<^sub>1 g \<star> f\<^sub>0g\<^sub>1.p\<^sub>0, fg\<^sub>0h\<^sub>1.p\<^sub>1]) \<cdot> ((f \<star> f\<^sub>0g\<^sub>1.\<phi>) \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> (\<a>[f, tab\<^sub>0 f, f\<^sub>0g\<^sub>1.p\<^sub>1] \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> \<a>\<^sup>-\<^sup>1[f \<star> tab\<^sub>0 f, f\<^sub>0g\<^sub>1.p\<^sub>1, fg\<^sub>0h\<^sub>1.p\<^sub>1] \<cdot> (f.tab \<star> f\<^sub>0g\<^sub>1.p\<^sub>1 \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1)
goal (1 subgoal):
1. (f \<star> \<theta>\<^sub>f) \<cdot> \<a>[f, tab\<^sub>0 f, w\<^sub>f] \<cdot> (f.tab \<star> w\<^sub>f) = ((f \<star> \<theta>\<^sub>f') \<cdot> \<a>[f, tab\<^sub>0 f, w\<^sub>f'] \<cdot> (f.tab \<star> w\<^sub>f')) \<cdot> \<beta>\<^sub>f
[PROOF STEP]
have "... =
\<a>[f, g, h \<star> tab\<^sub>0 h \<star> TTfgh.p\<^sub>0] \<cdot>
((f \<star> g) \<star> \<a>[h, tab\<^sub>0 h, TTfgh.p\<^sub>0]) \<cdot>
((f \<star> g) \<star> h.tab \<star> TTfgh.p\<^sub>0) \<cdot>
((f \<star> g) \<star> fg\<^sub>0h\<^sub>1.\<phi>) \<cdot>
\<a>\<^sup>-\<^sup>1[f, g, (tab\<^sub>0 g \<star> Tfg.p\<^sub>0) \<star> TTfgh.p\<^sub>1] \<cdot>
(f \<star> \<a>[g, tab\<^sub>0 g \<star> Tfg.p\<^sub>0, TTfgh.p\<^sub>1]) \<cdot>
(f \<star> \<a>[g, tab\<^sub>0 g, Tfg.p\<^sub>0] \<star> TTfgh.p\<^sub>1) \<cdot>
\<a>[f, (g \<star> tab\<^sub>0 g) \<star> Tfg.p\<^sub>0, TTfgh.p\<^sub>1] \<cdot>
((f \<star> (g.tab \<star> Tfg.p\<^sub>0)) \<star> TTfgh.p\<^sub>1) \<cdot>
((f \<star> f\<^sub>0g\<^sub>1.\<phi>) \<star> TTfgh.p\<^sub>1) \<cdot>
(\<a>[f, tab\<^sub>0 f, Tfg.p\<^sub>1] \<star> TTfgh.p\<^sub>1) \<cdot>
\<a>\<^sup>-\<^sup>1[f \<star> tab\<^sub>0 f, Tfg.p\<^sub>1, TTfgh.p\<^sub>1] \<cdot>
(f.tab \<star> Tfg.p\<^sub>1 \<star> TTfgh.p\<^sub>1)"
[PROOF STATE]
proof (prove)
goal (1 subgoal):
1. \<a>[f, g, h \<star> tab\<^sub>0 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0] \<cdot> ((f \<star> g) \<star> \<a>[h, tab\<^sub>0 h, fg\<^sub>0h\<^sub>1.p\<^sub>0]) \<cdot> ((f \<star> g) \<star> h.tab \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0) \<cdot> ((f \<star> g) \<star> fg\<^sub>0h\<^sub>1.\<phi>) \<cdot> \<a>\<^sup>-\<^sup>1[f, g, (tab\<^sub>0 g \<star> f\<^sub>0g\<^sub>1.p\<^sub>0) \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1] \<cdot> (f \<star> \<a>[g, tab\<^sub>0 g \<star> f\<^sub>0g\<^sub>1.p\<^sub>0, fg\<^sub>0h\<^sub>1.p\<^sub>1]) \<cdot> (f \<star> \<a>[g, tab\<^sub>0 g, f\<^sub>0g\<^sub>1.p\<^sub>0] \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> ((f \<star> (g.tab \<star> f\<^sub>0g\<^sub>1.p\<^sub>0) \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> \<a>[f, tab\<^sub>1 g \<star> f\<^sub>0g\<^sub>1.p\<^sub>0, fg\<^sub>0h\<^sub>1.p\<^sub>1]) \<cdot> ((f \<star> f\<^sub>0g\<^sub>1.\<phi>) \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> (\<a>[f, tab\<^sub>0 f, f\<^sub>0g\<^sub>1.p\<^sub>1] \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> \<a>\<^sup>-\<^sup>1[f \<star> tab\<^sub>0 f, f\<^sub>0g\<^sub>1.p\<^sub>1, fg\<^sub>0h\<^sub>1.p\<^sub>1] \<cdot> (f.tab \<star> f\<^sub>0g\<^sub>1.p\<^sub>1 \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) = \<a>[f, g, h \<star> tab\<^sub>0 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0] \<cdot> ((f \<star> g) \<star> \<a>[h, tab\<^sub>0 h, fg\<^sub>0h\<^sub>1.p\<^sub>0]) \<cdot> ((f \<star> g) \<star> h.tab \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0) \<cdot> ((f \<star> g) \<star> fg\<^sub>0h\<^sub>1.\<phi>) \<cdot> \<a>\<^sup>-\<^sup>1[f, g, (tab\<^sub>0 g \<star> f\<^sub>0g\<^sub>1.p\<^sub>0) \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1] \<cdot> (f \<star> \<a>[g, tab\<^sub>0 g \<star> f\<^sub>0g\<^sub>1.p\<^sub>0, fg\<^sub>0h\<^sub>1.p\<^sub>1]) \<cdot> (f \<star> \<a>[g, tab\<^sub>0 g, f\<^sub>0g\<^sub>1.p\<^sub>0] \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> \<a>[f, (g \<star> tab\<^sub>0 g) \<star> f\<^sub>0g\<^sub>1.p\<^sub>0, fg\<^sub>0h\<^sub>1.p\<^sub>1] \<cdot> ((f \<star> g.tab \<star> f\<^sub>0g\<^sub>1.p\<^sub>0) \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> ((f \<star> f\<^sub>0g\<^sub>1.\<phi>) \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> (\<a>[f, tab\<^sub>0 f, f\<^sub>0g\<^sub>1.p\<^sub>1] \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> \<a>\<^sup>-\<^sup>1[f \<star> tab\<^sub>0 f, f\<^sub>0g\<^sub>1.p\<^sub>1, fg\<^sub>0h\<^sub>1.p\<^sub>1] \<cdot> (f.tab \<star> f\<^sub>0g\<^sub>1.p\<^sub>1 \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1)
[PROOF STEP]
proof -
[PROOF STATE]
proof (state)
goal (1 subgoal):
1. \<a>[f, g, h \<star> tab\<^sub>0 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0] \<cdot> ((f \<star> g) \<star> \<a>[h, tab\<^sub>0 h, fg\<^sub>0h\<^sub>1.p\<^sub>0]) \<cdot> ((f \<star> g) \<star> h.tab \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0) \<cdot> ((f \<star> g) \<star> fg\<^sub>0h\<^sub>1.\<phi>) \<cdot> \<a>\<^sup>-\<^sup>1[f, g, (tab\<^sub>0 g \<star> f\<^sub>0g\<^sub>1.p\<^sub>0) \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1] \<cdot> (f \<star> \<a>[g, tab\<^sub>0 g \<star> f\<^sub>0g\<^sub>1.p\<^sub>0, fg\<^sub>0h\<^sub>1.p\<^sub>1]) \<cdot> (f \<star> \<a>[g, tab\<^sub>0 g, f\<^sub>0g\<^sub>1.p\<^sub>0] \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> ((f \<star> (g.tab \<star> f\<^sub>0g\<^sub>1.p\<^sub>0) \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> \<a>[f, tab\<^sub>1 g \<star> f\<^sub>0g\<^sub>1.p\<^sub>0, fg\<^sub>0h\<^sub>1.p\<^sub>1]) \<cdot> ((f \<star> f\<^sub>0g\<^sub>1.\<phi>) \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> (\<a>[f, tab\<^sub>0 f, f\<^sub>0g\<^sub>1.p\<^sub>1] \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> \<a>\<^sup>-\<^sup>1[f \<star> tab\<^sub>0 f, f\<^sub>0g\<^sub>1.p\<^sub>1, fg\<^sub>0h\<^sub>1.p\<^sub>1] \<cdot> (f.tab \<star> f\<^sub>0g\<^sub>1.p\<^sub>1 \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) = \<a>[f, g, h \<star> tab\<^sub>0 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0] \<cdot> ((f \<star> g) \<star> \<a>[h, tab\<^sub>0 h, fg\<^sub>0h\<^sub>1.p\<^sub>0]) \<cdot> ((f \<star> g) \<star> h.tab \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0) \<cdot> ((f \<star> g) \<star> fg\<^sub>0h\<^sub>1.\<phi>) \<cdot> \<a>\<^sup>-\<^sup>1[f, g, (tab\<^sub>0 g \<star> f\<^sub>0g\<^sub>1.p\<^sub>0) \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1] \<cdot> (f \<star> \<a>[g, tab\<^sub>0 g \<star> f\<^sub>0g\<^sub>1.p\<^sub>0, fg\<^sub>0h\<^sub>1.p\<^sub>1]) \<cdot> (f \<star> \<a>[g, tab\<^sub>0 g, f\<^sub>0g\<^sub>1.p\<^sub>0] \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> \<a>[f, (g \<star> tab\<^sub>0 g) \<star> f\<^sub>0g\<^sub>1.p\<^sub>0, fg\<^sub>0h\<^sub>1.p\<^sub>1] \<cdot> ((f \<star> g.tab \<star> f\<^sub>0g\<^sub>1.p\<^sub>0) \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> ((f \<star> f\<^sub>0g\<^sub>1.\<phi>) \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> (\<a>[f, tab\<^sub>0 f, f\<^sub>0g\<^sub>1.p\<^sub>1] \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> \<a>\<^sup>-\<^sup>1[f \<star> tab\<^sub>0 f, f\<^sub>0g\<^sub>1.p\<^sub>1, fg\<^sub>0h\<^sub>1.p\<^sub>1] \<cdot> (f.tab \<star> f\<^sub>0g\<^sub>1.p\<^sub>1 \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1)
[PROOF STEP]
have "(f \<star> (g.tab \<star> Tfg.p\<^sub>0) \<star> TTfgh.p\<^sub>1) \<cdot> \<a>[f, tab\<^sub>1 g \<star> Tfg.p\<^sub>0, TTfgh.p\<^sub>1] =
\<a>[f, (g \<star> tab\<^sub>0 g) \<star> Tfg.p\<^sub>0, TTfgh.p\<^sub>1] \<cdot> ((f \<star> (g.tab \<star> Tfg.p\<^sub>0)) \<star> TTfgh.p\<^sub>1)"
[PROOF STATE]
proof (prove)
goal (1 subgoal):
1. (f \<star> (g.tab \<star> f\<^sub>0g\<^sub>1.p\<^sub>0) \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> \<a>[f, tab\<^sub>1 g \<star> f\<^sub>0g\<^sub>1.p\<^sub>0, fg\<^sub>0h\<^sub>1.p\<^sub>1] = \<a>[f, (g \<star> tab\<^sub>0 g) \<star> f\<^sub>0g\<^sub>1.p\<^sub>0, fg\<^sub>0h\<^sub>1.p\<^sub>1] \<cdot> ((f \<star> g.tab \<star> f\<^sub>0g\<^sub>1.p\<^sub>0) \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1)
[PROOF STEP]
using fg gh fg\<^sub>0h\<^sub>1.p\<^sub>0_simps fg\<^sub>0h\<^sub>1.p\<^sub>1_simps f\<^sub>0g\<^sub>1.\<phi>_in_hom
assoc_naturality [of f "g.tab \<star> Tfg.p\<^sub>0" TTfgh.p\<^sub>1]
[PROOF STATE]
proof (prove)
using this:
src f = trg g
src g = trg h
trg fg\<^sub>0h\<^sub>1.p\<^sub>0 = src (tab\<^sub>1 h)
trg fg\<^sub>0h\<^sub>1.p\<^sub>1 = src (tab\<^sub>0 g \<star> f\<^sub>0g\<^sub>1.p\<^sub>0)
\<guillemotleft>f\<^sub>0g\<^sub>1.\<phi> : src f\<^sub>0g\<^sub>1.p\<^sub>0 \<rightarrow> trg (tab\<^sub>1 g)\<guillemotright>
\<guillemotleft>f\<^sub>0g\<^sub>1.\<phi> : tab\<^sub>0 f \<star> f\<^sub>0g\<^sub>1.p\<^sub>1 \<rightarrow> tab\<^sub>1 g \<star> f\<^sub>0g\<^sub>1.p\<^sub>0\<guillemotright>
\<lbrakk>arr f; hseq g.tab f\<^sub>0g\<^sub>1.p\<^sub>0; arr fg\<^sub>0h\<^sub>1.p\<^sub>1; src f = trg (g.tab \<star> f\<^sub>0g\<^sub>1.p\<^sub>0); src (g.tab \<star> f\<^sub>0g\<^sub>1.p\<^sub>0) = trg fg\<^sub>0h\<^sub>1.p\<^sub>1\<rbrakk> \<Longrightarrow> \<a>[cod f, cod (g.tab \<star> f\<^sub>0g\<^sub>1.p\<^sub>0), cod fg\<^sub>0h\<^sub>1.p\<^sub>1] \<cdot> ((f \<star> g.tab \<star> f\<^sub>0g\<^sub>1.p\<^sub>0) \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) = (f \<star> (g.tab \<star> f\<^sub>0g\<^sub>1.p\<^sub>0) \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> \<a>[local.dom f, local.dom (g.tab \<star> f\<^sub>0g\<^sub>1.p\<^sub>0), local.dom fg\<^sub>0h\<^sub>1.p\<^sub>1]
goal (1 subgoal):
1. (f \<star> (g.tab \<star> f\<^sub>0g\<^sub>1.p\<^sub>0) \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> \<a>[f, tab\<^sub>1 g \<star> f\<^sub>0g\<^sub>1.p\<^sub>0, fg\<^sub>0h\<^sub>1.p\<^sub>1] = \<a>[f, (g \<star> tab\<^sub>0 g) \<star> f\<^sub>0g\<^sub>1.p\<^sub>0, fg\<^sub>0h\<^sub>1.p\<^sub>1] \<cdot> ((f \<star> g.tab \<star> f\<^sub>0g\<^sub>1.p\<^sub>0) \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1)
[PROOF STEP]
by simp
[PROOF STATE]
proof (state)
this:
(f \<star> (g.tab \<star> f\<^sub>0g\<^sub>1.p\<^sub>0) \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> \<a>[f, tab\<^sub>1 g \<star> f\<^sub>0g\<^sub>1.p\<^sub>0, fg\<^sub>0h\<^sub>1.p\<^sub>1] = \<a>[f, (g \<star> tab\<^sub>0 g) \<star> f\<^sub>0g\<^sub>1.p\<^sub>0, fg\<^sub>0h\<^sub>1.p\<^sub>1] \<cdot> ((f \<star> g.tab \<star> f\<^sub>0g\<^sub>1.p\<^sub>0) \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1)
goal (1 subgoal):
1. \<a>[f, g, h \<star> tab\<^sub>0 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0] \<cdot> ((f \<star> g) \<star> \<a>[h, tab\<^sub>0 h, fg\<^sub>0h\<^sub>1.p\<^sub>0]) \<cdot> ((f \<star> g) \<star> h.tab \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0) \<cdot> ((f \<star> g) \<star> fg\<^sub>0h\<^sub>1.\<phi>) \<cdot> \<a>\<^sup>-\<^sup>1[f, g, (tab\<^sub>0 g \<star> f\<^sub>0g\<^sub>1.p\<^sub>0) \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1] \<cdot> (f \<star> \<a>[g, tab\<^sub>0 g \<star> f\<^sub>0g\<^sub>1.p\<^sub>0, fg\<^sub>0h\<^sub>1.p\<^sub>1]) \<cdot> (f \<star> \<a>[g, tab\<^sub>0 g, f\<^sub>0g\<^sub>1.p\<^sub>0] \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> ((f \<star> (g.tab \<star> f\<^sub>0g\<^sub>1.p\<^sub>0) \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> \<a>[f, tab\<^sub>1 g \<star> f\<^sub>0g\<^sub>1.p\<^sub>0, fg\<^sub>0h\<^sub>1.p\<^sub>1]) \<cdot> ((f \<star> f\<^sub>0g\<^sub>1.\<phi>) \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> (\<a>[f, tab\<^sub>0 f, f\<^sub>0g\<^sub>1.p\<^sub>1] \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> \<a>\<^sup>-\<^sup>1[f \<star> tab\<^sub>0 f, f\<^sub>0g\<^sub>1.p\<^sub>1, fg\<^sub>0h\<^sub>1.p\<^sub>1] \<cdot> (f.tab \<star> f\<^sub>0g\<^sub>1.p\<^sub>1 \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) = \<a>[f, g, h \<star> tab\<^sub>0 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0] \<cdot> ((f \<star> g) \<star> \<a>[h, tab\<^sub>0 h, fg\<^sub>0h\<^sub>1.p\<^sub>0]) \<cdot> ((f \<star> g) \<star> h.tab \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0) \<cdot> ((f \<star> g) \<star> fg\<^sub>0h\<^sub>1.\<phi>) \<cdot> \<a>\<^sup>-\<^sup>1[f, g, (tab\<^sub>0 g \<star> f\<^sub>0g\<^sub>1.p\<^sub>0) \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1] \<cdot> (f \<star> \<a>[g, tab\<^sub>0 g \<star> f\<^sub>0g\<^sub>1.p\<^sub>0, fg\<^sub>0h\<^sub>1.p\<^sub>1]) \<cdot> (f \<star> \<a>[g, tab\<^sub>0 g, f\<^sub>0g\<^sub>1.p\<^sub>0] \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> \<a>[f, (g \<star> tab\<^sub>0 g) \<star> f\<^sub>0g\<^sub>1.p\<^sub>0, fg\<^sub>0h\<^sub>1.p\<^sub>1] \<cdot> ((f \<star> g.tab \<star> f\<^sub>0g\<^sub>1.p\<^sub>0) \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> ((f \<star> f\<^sub>0g\<^sub>1.\<phi>) \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> (\<a>[f, tab\<^sub>0 f, f\<^sub>0g\<^sub>1.p\<^sub>1] \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> \<a>\<^sup>-\<^sup>1[f \<star> tab\<^sub>0 f, f\<^sub>0g\<^sub>1.p\<^sub>1, fg\<^sub>0h\<^sub>1.p\<^sub>1] \<cdot> (f.tab \<star> f\<^sub>0g\<^sub>1.p\<^sub>1 \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1)
[PROOF STEP]
thus ?thesis
[PROOF STATE]
proof (prove)
using this:
(f \<star> (g.tab \<star> f\<^sub>0g\<^sub>1.p\<^sub>0) \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> \<a>[f, tab\<^sub>1 g \<star> f\<^sub>0g\<^sub>1.p\<^sub>0, fg\<^sub>0h\<^sub>1.p\<^sub>1] = \<a>[f, (g \<star> tab\<^sub>0 g) \<star> f\<^sub>0g\<^sub>1.p\<^sub>0, fg\<^sub>0h\<^sub>1.p\<^sub>1] \<cdot> ((f \<star> g.tab \<star> f\<^sub>0g\<^sub>1.p\<^sub>0) \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1)
goal (1 subgoal):
1. \<a>[f, g, h \<star> tab\<^sub>0 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0] \<cdot> ((f \<star> g) \<star> \<a>[h, tab\<^sub>0 h, fg\<^sub>0h\<^sub>1.p\<^sub>0]) \<cdot> ((f \<star> g) \<star> h.tab \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0) \<cdot> ((f \<star> g) \<star> fg\<^sub>0h\<^sub>1.\<phi>) \<cdot> \<a>\<^sup>-\<^sup>1[f, g, (tab\<^sub>0 g \<star> f\<^sub>0g\<^sub>1.p\<^sub>0) \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1] \<cdot> (f \<star> \<a>[g, tab\<^sub>0 g \<star> f\<^sub>0g\<^sub>1.p\<^sub>0, fg\<^sub>0h\<^sub>1.p\<^sub>1]) \<cdot> (f \<star> \<a>[g, tab\<^sub>0 g, f\<^sub>0g\<^sub>1.p\<^sub>0] \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> ((f \<star> (g.tab \<star> f\<^sub>0g\<^sub>1.p\<^sub>0) \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> \<a>[f, tab\<^sub>1 g \<star> f\<^sub>0g\<^sub>1.p\<^sub>0, fg\<^sub>0h\<^sub>1.p\<^sub>1]) \<cdot> ((f \<star> f\<^sub>0g\<^sub>1.\<phi>) \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> (\<a>[f, tab\<^sub>0 f, f\<^sub>0g\<^sub>1.p\<^sub>1] \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> \<a>\<^sup>-\<^sup>1[f \<star> tab\<^sub>0 f, f\<^sub>0g\<^sub>1.p\<^sub>1, fg\<^sub>0h\<^sub>1.p\<^sub>1] \<cdot> (f.tab \<star> f\<^sub>0g\<^sub>1.p\<^sub>1 \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) = \<a>[f, g, h \<star> tab\<^sub>0 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0] \<cdot> ((f \<star> g) \<star> \<a>[h, tab\<^sub>0 h, fg\<^sub>0h\<^sub>1.p\<^sub>0]) \<cdot> ((f \<star> g) \<star> h.tab \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0) \<cdot> ((f \<star> g) \<star> fg\<^sub>0h\<^sub>1.\<phi>) \<cdot> \<a>\<^sup>-\<^sup>1[f, g, (tab\<^sub>0 g \<star> f\<^sub>0g\<^sub>1.p\<^sub>0) \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1] \<cdot> (f \<star> \<a>[g, tab\<^sub>0 g \<star> f\<^sub>0g\<^sub>1.p\<^sub>0, fg\<^sub>0h\<^sub>1.p\<^sub>1]) \<cdot> (f \<star> \<a>[g, tab\<^sub>0 g, f\<^sub>0g\<^sub>1.p\<^sub>0] \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> \<a>[f, (g \<star> tab\<^sub>0 g) \<star> f\<^sub>0g\<^sub>1.p\<^sub>0, fg\<^sub>0h\<^sub>1.p\<^sub>1] \<cdot> ((f \<star> g.tab \<star> f\<^sub>0g\<^sub>1.p\<^sub>0) \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> ((f \<star> f\<^sub>0g\<^sub>1.\<phi>) \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> (\<a>[f, tab\<^sub>0 f, f\<^sub>0g\<^sub>1.p\<^sub>1] \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> \<a>\<^sup>-\<^sup>1[f \<star> tab\<^sub>0 f, f\<^sub>0g\<^sub>1.p\<^sub>1, fg\<^sub>0h\<^sub>1.p\<^sub>1] \<cdot> (f.tab \<star> f\<^sub>0g\<^sub>1.p\<^sub>1 \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1)
[PROOF STEP]
using comp_assoc
[PROOF STATE]
proof (prove)
using this:
(f \<star> (g.tab \<star> f\<^sub>0g\<^sub>1.p\<^sub>0) \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> \<a>[f, tab\<^sub>1 g \<star> f\<^sub>0g\<^sub>1.p\<^sub>0, fg\<^sub>0h\<^sub>1.p\<^sub>1] = \<a>[f, (g \<star> tab\<^sub>0 g) \<star> f\<^sub>0g\<^sub>1.p\<^sub>0, fg\<^sub>0h\<^sub>1.p\<^sub>1] \<cdot> ((f \<star> g.tab \<star> f\<^sub>0g\<^sub>1.p\<^sub>0) \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1)
(?h \<cdot> ?g) \<cdot> ?f = ?h \<cdot> ?g \<cdot> ?f
goal (1 subgoal):
1. \<a>[f, g, h \<star> tab\<^sub>0 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0] \<cdot> ((f \<star> g) \<star> \<a>[h, tab\<^sub>0 h, fg\<^sub>0h\<^sub>1.p\<^sub>0]) \<cdot> ((f \<star> g) \<star> h.tab \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0) \<cdot> ((f \<star> g) \<star> fg\<^sub>0h\<^sub>1.\<phi>) \<cdot> \<a>\<^sup>-\<^sup>1[f, g, (tab\<^sub>0 g \<star> f\<^sub>0g\<^sub>1.p\<^sub>0) \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1] \<cdot> (f \<star> \<a>[g, tab\<^sub>0 g \<star> f\<^sub>0g\<^sub>1.p\<^sub>0, fg\<^sub>0h\<^sub>1.p\<^sub>1]) \<cdot> (f \<star> \<a>[g, tab\<^sub>0 g, f\<^sub>0g\<^sub>1.p\<^sub>0] \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> ((f \<star> (g.tab \<star> f\<^sub>0g\<^sub>1.p\<^sub>0) \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> \<a>[f, tab\<^sub>1 g \<star> f\<^sub>0g\<^sub>1.p\<^sub>0, fg\<^sub>0h\<^sub>1.p\<^sub>1]) \<cdot> ((f \<star> f\<^sub>0g\<^sub>1.\<phi>) \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> (\<a>[f, tab\<^sub>0 f, f\<^sub>0g\<^sub>1.p\<^sub>1] \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> \<a>\<^sup>-\<^sup>1[f \<star> tab\<^sub>0 f, f\<^sub>0g\<^sub>1.p\<^sub>1, fg\<^sub>0h\<^sub>1.p\<^sub>1] \<cdot> (f.tab \<star> f\<^sub>0g\<^sub>1.p\<^sub>1 \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) = \<a>[f, g, h \<star> tab\<^sub>0 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0] \<cdot> ((f \<star> g) \<star> \<a>[h, tab\<^sub>0 h, fg\<^sub>0h\<^sub>1.p\<^sub>0]) \<cdot> ((f \<star> g) \<star> h.tab \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0) \<cdot> ((f \<star> g) \<star> fg\<^sub>0h\<^sub>1.\<phi>) \<cdot> \<a>\<^sup>-\<^sup>1[f, g, (tab\<^sub>0 g \<star> f\<^sub>0g\<^sub>1.p\<^sub>0) \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1] \<cdot> (f \<star> \<a>[g, tab\<^sub>0 g \<star> f\<^sub>0g\<^sub>1.p\<^sub>0, fg\<^sub>0h\<^sub>1.p\<^sub>1]) \<cdot> (f \<star> \<a>[g, tab\<^sub>0 g, f\<^sub>0g\<^sub>1.p\<^sub>0] \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> \<a>[f, (g \<star> tab\<^sub>0 g) \<star> f\<^sub>0g\<^sub>1.p\<^sub>0, fg\<^sub>0h\<^sub>1.p\<^sub>1] \<cdot> ((f \<star> g.tab \<star> f\<^sub>0g\<^sub>1.p\<^sub>0) \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> ((f \<star> f\<^sub>0g\<^sub>1.\<phi>) \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> (\<a>[f, tab\<^sub>0 f, f\<^sub>0g\<^sub>1.p\<^sub>1] \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> \<a>\<^sup>-\<^sup>1[f \<star> tab\<^sub>0 f, f\<^sub>0g\<^sub>1.p\<^sub>1, fg\<^sub>0h\<^sub>1.p\<^sub>1] \<cdot> (f.tab \<star> f\<^sub>0g\<^sub>1.p\<^sub>1 \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1)
[PROOF STEP]
by presburger
[PROOF STATE]
proof (state)
this:
\<a>[f, g, h \<star> tab\<^sub>0 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0] \<cdot> ((f \<star> g) \<star> \<a>[h, tab\<^sub>0 h, fg\<^sub>0h\<^sub>1.p\<^sub>0]) \<cdot> ((f \<star> g) \<star> h.tab \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0) \<cdot> ((f \<star> g) \<star> fg\<^sub>0h\<^sub>1.\<phi>) \<cdot> \<a>\<^sup>-\<^sup>1[f, g, (tab\<^sub>0 g \<star> f\<^sub>0g\<^sub>1.p\<^sub>0) \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1] \<cdot> (f \<star> \<a>[g, tab\<^sub>0 g \<star> f\<^sub>0g\<^sub>1.p\<^sub>0, fg\<^sub>0h\<^sub>1.p\<^sub>1]) \<cdot> (f \<star> \<a>[g, tab\<^sub>0 g, f\<^sub>0g\<^sub>1.p\<^sub>0] \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> ((f \<star> (g.tab \<star> f\<^sub>0g\<^sub>1.p\<^sub>0) \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> \<a>[f, tab\<^sub>1 g \<star> f\<^sub>0g\<^sub>1.p\<^sub>0, fg\<^sub>0h\<^sub>1.p\<^sub>1]) \<cdot> ((f \<star> f\<^sub>0g\<^sub>1.\<phi>) \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> (\<a>[f, tab\<^sub>0 f, f\<^sub>0g\<^sub>1.p\<^sub>1] \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> \<a>\<^sup>-\<^sup>1[f \<star> tab\<^sub>0 f, f\<^sub>0g\<^sub>1.p\<^sub>1, fg\<^sub>0h\<^sub>1.p\<^sub>1] \<cdot> (f.tab \<star> f\<^sub>0g\<^sub>1.p\<^sub>1 \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) = \<a>[f, g, h \<star> tab\<^sub>0 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0] \<cdot> ((f \<star> g) \<star> \<a>[h, tab\<^sub>0 h, fg\<^sub>0h\<^sub>1.p\<^sub>0]) \<cdot> ((f \<star> g) \<star> h.tab \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0) \<cdot> ((f \<star> g) \<star> fg\<^sub>0h\<^sub>1.\<phi>) \<cdot> \<a>\<^sup>-\<^sup>1[f, g, (tab\<^sub>0 g \<star> f\<^sub>0g\<^sub>1.p\<^sub>0) \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1] \<cdot> (f \<star> \<a>[g, tab\<^sub>0 g \<star> f\<^sub>0g\<^sub>1.p\<^sub>0, fg\<^sub>0h\<^sub>1.p\<^sub>1]) \<cdot> (f \<star> \<a>[g, tab\<^sub>0 g, f\<^sub>0g\<^sub>1.p\<^sub>0] \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> \<a>[f, (g \<star> tab\<^sub>0 g) \<star> f\<^sub>0g\<^sub>1.p\<^sub>0, fg\<^sub>0h\<^sub>1.p\<^sub>1] \<cdot> ((f \<star> g.tab \<star> f\<^sub>0g\<^sub>1.p\<^sub>0) \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> ((f \<star> f\<^sub>0g\<^sub>1.\<phi>) \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> (\<a>[f, tab\<^sub>0 f, f\<^sub>0g\<^sub>1.p\<^sub>1] \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> \<a>\<^sup>-\<^sup>1[f \<star> tab\<^sub>0 f, f\<^sub>0g\<^sub>1.p\<^sub>1, fg\<^sub>0h\<^sub>1.p\<^sub>1] \<cdot> (f.tab \<star> f\<^sub>0g\<^sub>1.p\<^sub>1 \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1)
goal:
No subgoals!
[PROOF STEP]
qed
[PROOF STATE]
proof (state)
this:
\<a>[f, g, h \<star> tab\<^sub>0 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0] \<cdot> ((f \<star> g) \<star> \<a>[h, tab\<^sub>0 h, fg\<^sub>0h\<^sub>1.p\<^sub>0]) \<cdot> ((f \<star> g) \<star> h.tab \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0) \<cdot> ((f \<star> g) \<star> fg\<^sub>0h\<^sub>1.\<phi>) \<cdot> \<a>\<^sup>-\<^sup>1[f, g, (tab\<^sub>0 g \<star> f\<^sub>0g\<^sub>1.p\<^sub>0) \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1] \<cdot> (f \<star> \<a>[g, tab\<^sub>0 g \<star> f\<^sub>0g\<^sub>1.p\<^sub>0, fg\<^sub>0h\<^sub>1.p\<^sub>1]) \<cdot> (f \<star> \<a>[g, tab\<^sub>0 g, f\<^sub>0g\<^sub>1.p\<^sub>0] \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> ((f \<star> (g.tab \<star> f\<^sub>0g\<^sub>1.p\<^sub>0) \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> \<a>[f, tab\<^sub>1 g \<star> f\<^sub>0g\<^sub>1.p\<^sub>0, fg\<^sub>0h\<^sub>1.p\<^sub>1]) \<cdot> ((f \<star> f\<^sub>0g\<^sub>1.\<phi>) \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> (\<a>[f, tab\<^sub>0 f, f\<^sub>0g\<^sub>1.p\<^sub>1] \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> \<a>\<^sup>-\<^sup>1[f \<star> tab\<^sub>0 f, f\<^sub>0g\<^sub>1.p\<^sub>1, fg\<^sub>0h\<^sub>1.p\<^sub>1] \<cdot> (f.tab \<star> f\<^sub>0g\<^sub>1.p\<^sub>1 \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) = \<a>[f, g, h \<star> tab\<^sub>0 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0] \<cdot> ((f \<star> g) \<star> \<a>[h, tab\<^sub>0 h, fg\<^sub>0h\<^sub>1.p\<^sub>0]) \<cdot> ((f \<star> g) \<star> h.tab \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0) \<cdot> ((f \<star> g) \<star> fg\<^sub>0h\<^sub>1.\<phi>) \<cdot> \<a>\<^sup>-\<^sup>1[f, g, (tab\<^sub>0 g \<star> f\<^sub>0g\<^sub>1.p\<^sub>0) \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1] \<cdot> (f \<star> \<a>[g, tab\<^sub>0 g \<star> f\<^sub>0g\<^sub>1.p\<^sub>0, fg\<^sub>0h\<^sub>1.p\<^sub>1]) \<cdot> (f \<star> \<a>[g, tab\<^sub>0 g, f\<^sub>0g\<^sub>1.p\<^sub>0] \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> \<a>[f, (g \<star> tab\<^sub>0 g) \<star> f\<^sub>0g\<^sub>1.p\<^sub>0, fg\<^sub>0h\<^sub>1.p\<^sub>1] \<cdot> ((f \<star> g.tab \<star> f\<^sub>0g\<^sub>1.p\<^sub>0) \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> ((f \<star> f\<^sub>0g\<^sub>1.\<phi>) \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> (\<a>[f, tab\<^sub>0 f, f\<^sub>0g\<^sub>1.p\<^sub>1] \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> \<a>\<^sup>-\<^sup>1[f \<star> tab\<^sub>0 f, f\<^sub>0g\<^sub>1.p\<^sub>1, fg\<^sub>0h\<^sub>1.p\<^sub>1] \<cdot> (f.tab \<star> f\<^sub>0g\<^sub>1.p\<^sub>1 \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1)
goal (1 subgoal):
1. (f \<star> \<theta>\<^sub>f) \<cdot> \<a>[f, tab\<^sub>0 f, w\<^sub>f] \<cdot> (f.tab \<star> w\<^sub>f) = ((f \<star> \<theta>\<^sub>f') \<cdot> \<a>[f, tab\<^sub>0 f, w\<^sub>f'] \<cdot> (f.tab \<star> w\<^sub>f')) \<cdot> \<beta>\<^sub>f
[PROOF STEP]
also
[PROOF STATE]
proof (state)
this:
\<a>[f, g, h \<star> tab\<^sub>0 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0] \<cdot> ((f \<star> g) \<star> \<a>[h, tab\<^sub>0 h, fg\<^sub>0h\<^sub>1.p\<^sub>0]) \<cdot> ((f \<star> g) \<star> h.tab \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0) \<cdot> ((f \<star> g) \<star> fg\<^sub>0h\<^sub>1.\<phi>) \<cdot> \<a>\<^sup>-\<^sup>1[f, g, (tab\<^sub>0 g \<star> f\<^sub>0g\<^sub>1.p\<^sub>0) \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1] \<cdot> (f \<star> \<a>[g, tab\<^sub>0 g \<star> f\<^sub>0g\<^sub>1.p\<^sub>0, fg\<^sub>0h\<^sub>1.p\<^sub>1]) \<cdot> (f \<star> \<a>[g, tab\<^sub>0 g, f\<^sub>0g\<^sub>1.p\<^sub>0] \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> ((f \<star> (g.tab \<star> f\<^sub>0g\<^sub>1.p\<^sub>0) \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> \<a>[f, tab\<^sub>1 g \<star> f\<^sub>0g\<^sub>1.p\<^sub>0, fg\<^sub>0h\<^sub>1.p\<^sub>1]) \<cdot> ((f \<star> f\<^sub>0g\<^sub>1.\<phi>) \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> (\<a>[f, tab\<^sub>0 f, f\<^sub>0g\<^sub>1.p\<^sub>1] \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> \<a>\<^sup>-\<^sup>1[f \<star> tab\<^sub>0 f, f\<^sub>0g\<^sub>1.p\<^sub>1, fg\<^sub>0h\<^sub>1.p\<^sub>1] \<cdot> (f.tab \<star> f\<^sub>0g\<^sub>1.p\<^sub>1 \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) = \<a>[f, g, h \<star> tab\<^sub>0 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0] \<cdot> ((f \<star> g) \<star> \<a>[h, tab\<^sub>0 h, fg\<^sub>0h\<^sub>1.p\<^sub>0]) \<cdot> ((f \<star> g) \<star> h.tab \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0) \<cdot> ((f \<star> g) \<star> fg\<^sub>0h\<^sub>1.\<phi>) \<cdot> \<a>\<^sup>-\<^sup>1[f, g, (tab\<^sub>0 g \<star> f\<^sub>0g\<^sub>1.p\<^sub>0) \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1] \<cdot> (f \<star> \<a>[g, tab\<^sub>0 g \<star> f\<^sub>0g\<^sub>1.p\<^sub>0, fg\<^sub>0h\<^sub>1.p\<^sub>1]) \<cdot> (f \<star> \<a>[g, tab\<^sub>0 g, f\<^sub>0g\<^sub>1.p\<^sub>0] \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> \<a>[f, (g \<star> tab\<^sub>0 g) \<star> f\<^sub>0g\<^sub>1.p\<^sub>0, fg\<^sub>0h\<^sub>1.p\<^sub>1] \<cdot> ((f \<star> g.tab \<star> f\<^sub>0g\<^sub>1.p\<^sub>0) \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> ((f \<star> f\<^sub>0g\<^sub>1.\<phi>) \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> (\<a>[f, tab\<^sub>0 f, f\<^sub>0g\<^sub>1.p\<^sub>1] \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> \<a>\<^sup>-\<^sup>1[f \<star> tab\<^sub>0 f, f\<^sub>0g\<^sub>1.p\<^sub>1, fg\<^sub>0h\<^sub>1.p\<^sub>1] \<cdot> (f.tab \<star> f\<^sub>0g\<^sub>1.p\<^sub>1 \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1)
goal (1 subgoal):
1. (f \<star> \<theta>\<^sub>f) \<cdot> \<a>[f, tab\<^sub>0 f, w\<^sub>f] \<cdot> (f.tab \<star> w\<^sub>f) = ((f \<star> \<theta>\<^sub>f') \<cdot> \<a>[f, tab\<^sub>0 f, w\<^sub>f'] \<cdot> (f.tab \<star> w\<^sub>f')) \<cdot> \<beta>\<^sub>f
[PROOF STEP]
have "... =
\<a>[f, g, h \<star> tab\<^sub>0 h \<star> TTfgh.p\<^sub>0] \<cdot>
((f \<star> g) \<star> \<a>[h, tab\<^sub>0 h, TTfgh.p\<^sub>0]) \<cdot>
((f \<star> g) \<star> h.tab \<star> TTfgh.p\<^sub>0) \<cdot>
((f \<star> g) \<star> fg\<^sub>0h\<^sub>1.\<phi>) \<cdot>
\<a>\<^sup>-\<^sup>1[f, g, (tab\<^sub>0 g \<star> Tfg.p\<^sub>0) \<star> TTfgh.p\<^sub>1] \<cdot>
(f \<star> \<a>[g, tab\<^sub>0 g \<star> Tfg.p\<^sub>0, TTfgh.p\<^sub>1]) \<cdot>
((f \<star> \<a>[g, tab\<^sub>0 g, Tfg.p\<^sub>0] \<star> TTfgh.p\<^sub>1) \<cdot>
\<a>[f, (g \<star> tab\<^sub>0 g) \<star> Tfg.p\<^sub>0, TTfgh.p\<^sub>1]) \<cdot>
((f \<star> (g.tab \<star> Tfg.p\<^sub>0)) \<star> TTfgh.p\<^sub>1) \<cdot>
((f \<star> f\<^sub>0g\<^sub>1.\<phi>) \<star> TTfgh.p\<^sub>1) \<cdot>
(\<a>[f, tab\<^sub>0 f, Tfg.p\<^sub>1] \<star> TTfgh.p\<^sub>1) \<cdot>
((f.tab \<star> Tfg.p\<^sub>1) \<star> TTfgh.p\<^sub>1) \<cdot>
\<a>\<^sup>-\<^sup>1[tab\<^sub>1 f, Tfg.p\<^sub>1, TTfgh.p\<^sub>1]"
[PROOF STATE]
proof (prove)
goal (1 subgoal):
1. \<a>[f, g, h \<star> tab\<^sub>0 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0] \<cdot> ((f \<star> g) \<star> \<a>[h, tab\<^sub>0 h, fg\<^sub>0h\<^sub>1.p\<^sub>0]) \<cdot> ((f \<star> g) \<star> h.tab \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0) \<cdot> ((f \<star> g) \<star> fg\<^sub>0h\<^sub>1.\<phi>) \<cdot> \<a>\<^sup>-\<^sup>1[f, g, (tab\<^sub>0 g \<star> f\<^sub>0g\<^sub>1.p\<^sub>0) \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1] \<cdot> (f \<star> \<a>[g, tab\<^sub>0 g \<star> f\<^sub>0g\<^sub>1.p\<^sub>0, fg\<^sub>0h\<^sub>1.p\<^sub>1]) \<cdot> (f \<star> \<a>[g, tab\<^sub>0 g, f\<^sub>0g\<^sub>1.p\<^sub>0] \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> \<a>[f, (g \<star> tab\<^sub>0 g) \<star> f\<^sub>0g\<^sub>1.p\<^sub>0, fg\<^sub>0h\<^sub>1.p\<^sub>1] \<cdot> ((f \<star> g.tab \<star> f\<^sub>0g\<^sub>1.p\<^sub>0) \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> ((f \<star> f\<^sub>0g\<^sub>1.\<phi>) \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> (\<a>[f, tab\<^sub>0 f, f\<^sub>0g\<^sub>1.p\<^sub>1] \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> \<a>\<^sup>-\<^sup>1[f \<star> tab\<^sub>0 f, f\<^sub>0g\<^sub>1.p\<^sub>1, fg\<^sub>0h\<^sub>1.p\<^sub>1] \<cdot> (f.tab \<star> f\<^sub>0g\<^sub>1.p\<^sub>1 \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) = \<a>[f, g, h \<star> tab\<^sub>0 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0] \<cdot> ((f \<star> g) \<star> \<a>[h, tab\<^sub>0 h, fg\<^sub>0h\<^sub>1.p\<^sub>0]) \<cdot> ((f \<star> g) \<star> h.tab \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0) \<cdot> ((f \<star> g) \<star> fg\<^sub>0h\<^sub>1.\<phi>) \<cdot> \<a>\<^sup>-\<^sup>1[f, g, (tab\<^sub>0 g \<star> f\<^sub>0g\<^sub>1.p\<^sub>0) \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1] \<cdot> (f \<star> \<a>[g, tab\<^sub>0 g \<star> f\<^sub>0g\<^sub>1.p\<^sub>0, fg\<^sub>0h\<^sub>1.p\<^sub>1]) \<cdot> ((f \<star> \<a>[g, tab\<^sub>0 g, f\<^sub>0g\<^sub>1.p\<^sub>0] \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> \<a>[f, (g \<star> tab\<^sub>0 g) \<star> f\<^sub>0g\<^sub>1.p\<^sub>0, fg\<^sub>0h\<^sub>1.p\<^sub>1]) \<cdot> ((f \<star> g.tab \<star> f\<^sub>0g\<^sub>1.p\<^sub>0) \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> ((f \<star> f\<^sub>0g\<^sub>1.\<phi>) \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> (\<a>[f, tab\<^sub>0 f, f\<^sub>0g\<^sub>1.p\<^sub>1] \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> ((f.tab \<star> f\<^sub>0g\<^sub>1.p\<^sub>1) \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>1 f, f\<^sub>0g\<^sub>1.p\<^sub>1, fg\<^sub>0h\<^sub>1.p\<^sub>1]
[PROOF STEP]
using fg\<^sub>0h\<^sub>1.p\<^sub>1_simps assoc'_naturality [of f.tab Tfg.p\<^sub>1 TTfgh.p\<^sub>1] comp_assoc
[PROOF STATE]
proof (prove)
using this:
trg fg\<^sub>0h\<^sub>1.p\<^sub>1 = src (tab\<^sub>0 g \<star> f\<^sub>0g\<^sub>1.p\<^sub>0)
\<lbrakk>arr f.tab; arr f\<^sub>0g\<^sub>1.p\<^sub>1; arr fg\<^sub>0h\<^sub>1.p\<^sub>1; src f.tab = trg f\<^sub>0g\<^sub>1.p\<^sub>1; src f\<^sub>0g\<^sub>1.p\<^sub>1 = trg fg\<^sub>0h\<^sub>1.p\<^sub>1\<rbrakk> \<Longrightarrow> \<a>\<^sup>-\<^sup>1[cod f.tab, cod f\<^sub>0g\<^sub>1.p\<^sub>1, cod fg\<^sub>0h\<^sub>1.p\<^sub>1] \<cdot> (f.tab \<star> f\<^sub>0g\<^sub>1.p\<^sub>1 \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) = ((f.tab \<star> f\<^sub>0g\<^sub>1.p\<^sub>1) \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> \<a>\<^sup>-\<^sup>1[local.dom f.tab, local.dom f\<^sub>0g\<^sub>1.p\<^sub>1, local.dom fg\<^sub>0h\<^sub>1.p\<^sub>1]
(?h \<cdot> ?g) \<cdot> ?f = ?h \<cdot> ?g \<cdot> ?f
goal (1 subgoal):
1. \<a>[f, g, h \<star> tab\<^sub>0 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0] \<cdot> ((f \<star> g) \<star> \<a>[h, tab\<^sub>0 h, fg\<^sub>0h\<^sub>1.p\<^sub>0]) \<cdot> ((f \<star> g) \<star> h.tab \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0) \<cdot> ((f \<star> g) \<star> fg\<^sub>0h\<^sub>1.\<phi>) \<cdot> \<a>\<^sup>-\<^sup>1[f, g, (tab\<^sub>0 g \<star> f\<^sub>0g\<^sub>1.p\<^sub>0) \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1] \<cdot> (f \<star> \<a>[g, tab\<^sub>0 g \<star> f\<^sub>0g\<^sub>1.p\<^sub>0, fg\<^sub>0h\<^sub>1.p\<^sub>1]) \<cdot> (f \<star> \<a>[g, tab\<^sub>0 g, f\<^sub>0g\<^sub>1.p\<^sub>0] \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> \<a>[f, (g \<star> tab\<^sub>0 g) \<star> f\<^sub>0g\<^sub>1.p\<^sub>0, fg\<^sub>0h\<^sub>1.p\<^sub>1] \<cdot> ((f \<star> g.tab \<star> f\<^sub>0g\<^sub>1.p\<^sub>0) \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> ((f \<star> f\<^sub>0g\<^sub>1.\<phi>) \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> (\<a>[f, tab\<^sub>0 f, f\<^sub>0g\<^sub>1.p\<^sub>1] \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> \<a>\<^sup>-\<^sup>1[f \<star> tab\<^sub>0 f, f\<^sub>0g\<^sub>1.p\<^sub>1, fg\<^sub>0h\<^sub>1.p\<^sub>1] \<cdot> (f.tab \<star> f\<^sub>0g\<^sub>1.p\<^sub>1 \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) = \<a>[f, g, h \<star> tab\<^sub>0 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0] \<cdot> ((f \<star> g) \<star> \<a>[h, tab\<^sub>0 h, fg\<^sub>0h\<^sub>1.p\<^sub>0]) \<cdot> ((f \<star> g) \<star> h.tab \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0) \<cdot> ((f \<star> g) \<star> fg\<^sub>0h\<^sub>1.\<phi>) \<cdot> \<a>\<^sup>-\<^sup>1[f, g, (tab\<^sub>0 g \<star> f\<^sub>0g\<^sub>1.p\<^sub>0) \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1] \<cdot> (f \<star> \<a>[g, tab\<^sub>0 g \<star> f\<^sub>0g\<^sub>1.p\<^sub>0, fg\<^sub>0h\<^sub>1.p\<^sub>1]) \<cdot> ((f \<star> \<a>[g, tab\<^sub>0 g, f\<^sub>0g\<^sub>1.p\<^sub>0] \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> \<a>[f, (g \<star> tab\<^sub>0 g) \<star> f\<^sub>0g\<^sub>1.p\<^sub>0, fg\<^sub>0h\<^sub>1.p\<^sub>1]) \<cdot> ((f \<star> g.tab \<star> f\<^sub>0g\<^sub>1.p\<^sub>0) \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> ((f \<star> f\<^sub>0g\<^sub>1.\<phi>) \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> (\<a>[f, tab\<^sub>0 f, f\<^sub>0g\<^sub>1.p\<^sub>1] \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> ((f.tab \<star> f\<^sub>0g\<^sub>1.p\<^sub>1) \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>1 f, f\<^sub>0g\<^sub>1.p\<^sub>1, fg\<^sub>0h\<^sub>1.p\<^sub>1]
[PROOF STEP]
by simp
[PROOF STATE]
proof (state)
this:
\<a>[f, g, h \<star> tab\<^sub>0 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0] \<cdot> ((f \<star> g) \<star> \<a>[h, tab\<^sub>0 h, fg\<^sub>0h\<^sub>1.p\<^sub>0]) \<cdot> ((f \<star> g) \<star> h.tab \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0) \<cdot> ((f \<star> g) \<star> fg\<^sub>0h\<^sub>1.\<phi>) \<cdot> \<a>\<^sup>-\<^sup>1[f, g, (tab\<^sub>0 g \<star> f\<^sub>0g\<^sub>1.p\<^sub>0) \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1] \<cdot> (f \<star> \<a>[g, tab\<^sub>0 g \<star> f\<^sub>0g\<^sub>1.p\<^sub>0, fg\<^sub>0h\<^sub>1.p\<^sub>1]) \<cdot> (f \<star> \<a>[g, tab\<^sub>0 g, f\<^sub>0g\<^sub>1.p\<^sub>0] \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> \<a>[f, (g \<star> tab\<^sub>0 g) \<star> f\<^sub>0g\<^sub>1.p\<^sub>0, fg\<^sub>0h\<^sub>1.p\<^sub>1] \<cdot> ((f \<star> g.tab \<star> f\<^sub>0g\<^sub>1.p\<^sub>0) \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> ((f \<star> f\<^sub>0g\<^sub>1.\<phi>) \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> (\<a>[f, tab\<^sub>0 f, f\<^sub>0g\<^sub>1.p\<^sub>1] \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> \<a>\<^sup>-\<^sup>1[f \<star> tab\<^sub>0 f, f\<^sub>0g\<^sub>1.p\<^sub>1, fg\<^sub>0h\<^sub>1.p\<^sub>1] \<cdot> (f.tab \<star> f\<^sub>0g\<^sub>1.p\<^sub>1 \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) = \<a>[f, g, h \<star> tab\<^sub>0 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0] \<cdot> ((f \<star> g) \<star> \<a>[h, tab\<^sub>0 h, fg\<^sub>0h\<^sub>1.p\<^sub>0]) \<cdot> ((f \<star> g) \<star> h.tab \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0) \<cdot> ((f \<star> g) \<star> fg\<^sub>0h\<^sub>1.\<phi>) \<cdot> \<a>\<^sup>-\<^sup>1[f, g, (tab\<^sub>0 g \<star> f\<^sub>0g\<^sub>1.p\<^sub>0) \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1] \<cdot> (f \<star> \<a>[g, tab\<^sub>0 g \<star> f\<^sub>0g\<^sub>1.p\<^sub>0, fg\<^sub>0h\<^sub>1.p\<^sub>1]) \<cdot> ((f \<star> \<a>[g, tab\<^sub>0 g, f\<^sub>0g\<^sub>1.p\<^sub>0] \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> \<a>[f, (g \<star> tab\<^sub>0 g) \<star> f\<^sub>0g\<^sub>1.p\<^sub>0, fg\<^sub>0h\<^sub>1.p\<^sub>1]) \<cdot> ((f \<star> g.tab \<star> f\<^sub>0g\<^sub>1.p\<^sub>0) \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> ((f \<star> f\<^sub>0g\<^sub>1.\<phi>) \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> (\<a>[f, tab\<^sub>0 f, f\<^sub>0g\<^sub>1.p\<^sub>1] \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> ((f.tab \<star> f\<^sub>0g\<^sub>1.p\<^sub>1) \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>1 f, f\<^sub>0g\<^sub>1.p\<^sub>1, fg\<^sub>0h\<^sub>1.p\<^sub>1]
goal (1 subgoal):
1. (f \<star> \<theta>\<^sub>f) \<cdot> \<a>[f, tab\<^sub>0 f, w\<^sub>f] \<cdot> (f.tab \<star> w\<^sub>f) = ((f \<star> \<theta>\<^sub>f') \<cdot> \<a>[f, tab\<^sub>0 f, w\<^sub>f'] \<cdot> (f.tab \<star> w\<^sub>f')) \<cdot> \<beta>\<^sub>f
[PROOF STEP]
also
[PROOF STATE]
proof (state)
this:
\<a>[f, g, h \<star> tab\<^sub>0 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0] \<cdot> ((f \<star> g) \<star> \<a>[h, tab\<^sub>0 h, fg\<^sub>0h\<^sub>1.p\<^sub>0]) \<cdot> ((f \<star> g) \<star> h.tab \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0) \<cdot> ((f \<star> g) \<star> fg\<^sub>0h\<^sub>1.\<phi>) \<cdot> \<a>\<^sup>-\<^sup>1[f, g, (tab\<^sub>0 g \<star> f\<^sub>0g\<^sub>1.p\<^sub>0) \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1] \<cdot> (f \<star> \<a>[g, tab\<^sub>0 g \<star> f\<^sub>0g\<^sub>1.p\<^sub>0, fg\<^sub>0h\<^sub>1.p\<^sub>1]) \<cdot> (f \<star> \<a>[g, tab\<^sub>0 g, f\<^sub>0g\<^sub>1.p\<^sub>0] \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> \<a>[f, (g \<star> tab\<^sub>0 g) \<star> f\<^sub>0g\<^sub>1.p\<^sub>0, fg\<^sub>0h\<^sub>1.p\<^sub>1] \<cdot> ((f \<star> g.tab \<star> f\<^sub>0g\<^sub>1.p\<^sub>0) \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> ((f \<star> f\<^sub>0g\<^sub>1.\<phi>) \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> (\<a>[f, tab\<^sub>0 f, f\<^sub>0g\<^sub>1.p\<^sub>1] \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> \<a>\<^sup>-\<^sup>1[f \<star> tab\<^sub>0 f, f\<^sub>0g\<^sub>1.p\<^sub>1, fg\<^sub>0h\<^sub>1.p\<^sub>1] \<cdot> (f.tab \<star> f\<^sub>0g\<^sub>1.p\<^sub>1 \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) = \<a>[f, g, h \<star> tab\<^sub>0 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0] \<cdot> ((f \<star> g) \<star> \<a>[h, tab\<^sub>0 h, fg\<^sub>0h\<^sub>1.p\<^sub>0]) \<cdot> ((f \<star> g) \<star> h.tab \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0) \<cdot> ((f \<star> g) \<star> fg\<^sub>0h\<^sub>1.\<phi>) \<cdot> \<a>\<^sup>-\<^sup>1[f, g, (tab\<^sub>0 g \<star> f\<^sub>0g\<^sub>1.p\<^sub>0) \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1] \<cdot> (f \<star> \<a>[g, tab\<^sub>0 g \<star> f\<^sub>0g\<^sub>1.p\<^sub>0, fg\<^sub>0h\<^sub>1.p\<^sub>1]) \<cdot> ((f \<star> \<a>[g, tab\<^sub>0 g, f\<^sub>0g\<^sub>1.p\<^sub>0] \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> \<a>[f, (g \<star> tab\<^sub>0 g) \<star> f\<^sub>0g\<^sub>1.p\<^sub>0, fg\<^sub>0h\<^sub>1.p\<^sub>1]) \<cdot> ((f \<star> g.tab \<star> f\<^sub>0g\<^sub>1.p\<^sub>0) \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> ((f \<star> f\<^sub>0g\<^sub>1.\<phi>) \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> (\<a>[f, tab\<^sub>0 f, f\<^sub>0g\<^sub>1.p\<^sub>1] \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> ((f.tab \<star> f\<^sub>0g\<^sub>1.p\<^sub>1) \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>1 f, f\<^sub>0g\<^sub>1.p\<^sub>1, fg\<^sub>0h\<^sub>1.p\<^sub>1]
goal (1 subgoal):
1. (f \<star> \<theta>\<^sub>f) \<cdot> \<a>[f, tab\<^sub>0 f, w\<^sub>f] \<cdot> (f.tab \<star> w\<^sub>f) = ((f \<star> \<theta>\<^sub>f') \<cdot> \<a>[f, tab\<^sub>0 f, w\<^sub>f'] \<cdot> (f.tab \<star> w\<^sub>f')) \<cdot> \<beta>\<^sub>f
[PROOF STEP]
have "... =
\<a>[f, g, h \<star> tab\<^sub>0 h \<star> TTfgh.p\<^sub>0] \<cdot>
((f \<star> g) \<star> \<a>[h, tab\<^sub>0 h, TTfgh.p\<^sub>0]) \<cdot>
((f \<star> g) \<star> h.tab \<star> TTfgh.p\<^sub>0) \<cdot>
((f \<star> g) \<star> fg\<^sub>0h\<^sub>1.\<phi>) \<cdot>
\<a>\<^sup>-\<^sup>1[f, g, (tab\<^sub>0 g \<star> Tfg.p\<^sub>0) \<star> TTfgh.p\<^sub>1] \<cdot>
(f \<star> \<a>[g, tab\<^sub>0 g \<star> Tfg.p\<^sub>0, TTfgh.p\<^sub>1]) \<cdot>
(f \<star> \<a>[g, tab\<^sub>0 g, Tfg.p\<^sub>0] \<star> TTfgh.p\<^sub>1) \<cdot>
\<a>[f, (g \<star> tab\<^sub>0 g) \<star> Tfg.p\<^sub>0, TTfgh.p\<^sub>1] \<cdot>
((((f \<star> \<a>\<^sup>-\<^sup>1[g, tab\<^sub>0 g, Tfg.p\<^sub>0]) \<star> TTfgh.p\<^sub>1) \<cdot>
((f \<star> \<a>[g, tab\<^sub>0 g, Tfg.p\<^sub>0]) \<star> TTfgh.p\<^sub>1)) \<cdot>
((f \<star> (g.tab \<star> Tfg.p\<^sub>0)) \<star> TTfgh.p\<^sub>1)) \<cdot>
((f \<star> f\<^sub>0g\<^sub>1.\<phi>) \<star> TTfgh.p\<^sub>1) \<cdot>
(\<a>[f, tab\<^sub>0 f, Tfg.p\<^sub>1] \<star> TTfgh.p\<^sub>1) \<cdot>
((f.tab \<star> Tfg.p\<^sub>1) \<star> TTfgh.p\<^sub>1) \<cdot>
\<a>\<^sup>-\<^sup>1[tab\<^sub>1 f, Tfg.p\<^sub>1, TTfgh.p\<^sub>1]"
[PROOF STATE]
proof (prove)
goal (1 subgoal):
1. \<a>[f, g, h \<star> tab\<^sub>0 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0] \<cdot> ((f \<star> g) \<star> \<a>[h, tab\<^sub>0 h, fg\<^sub>0h\<^sub>1.p\<^sub>0]) \<cdot> ((f \<star> g) \<star> h.tab \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0) \<cdot> ((f \<star> g) \<star> fg\<^sub>0h\<^sub>1.\<phi>) \<cdot> \<a>\<^sup>-\<^sup>1[f, g, (tab\<^sub>0 g \<star> f\<^sub>0g\<^sub>1.p\<^sub>0) \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1] \<cdot> (f \<star> \<a>[g, tab\<^sub>0 g \<star> f\<^sub>0g\<^sub>1.p\<^sub>0, fg\<^sub>0h\<^sub>1.p\<^sub>1]) \<cdot> ((f \<star> \<a>[g, tab\<^sub>0 g, f\<^sub>0g\<^sub>1.p\<^sub>0] \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> \<a>[f, (g \<star> tab\<^sub>0 g) \<star> f\<^sub>0g\<^sub>1.p\<^sub>0, fg\<^sub>0h\<^sub>1.p\<^sub>1]) \<cdot> ((f \<star> g.tab \<star> f\<^sub>0g\<^sub>1.p\<^sub>0) \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> ((f \<star> f\<^sub>0g\<^sub>1.\<phi>) \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> (\<a>[f, tab\<^sub>0 f, f\<^sub>0g\<^sub>1.p\<^sub>1] \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> ((f.tab \<star> f\<^sub>0g\<^sub>1.p\<^sub>1) \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>1 f, f\<^sub>0g\<^sub>1.p\<^sub>1, fg\<^sub>0h\<^sub>1.p\<^sub>1] = \<a>[f, g, h \<star> tab\<^sub>0 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0] \<cdot> ((f \<star> g) \<star> \<a>[h, tab\<^sub>0 h, fg\<^sub>0h\<^sub>1.p\<^sub>0]) \<cdot> ((f \<star> g) \<star> h.tab \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0) \<cdot> ((f \<star> g) \<star> fg\<^sub>0h\<^sub>1.\<phi>) \<cdot> \<a>\<^sup>-\<^sup>1[f, g, (tab\<^sub>0 g \<star> f\<^sub>0g\<^sub>1.p\<^sub>0) \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1] \<cdot> (f \<star> \<a>[g, tab\<^sub>0 g \<star> f\<^sub>0g\<^sub>1.p\<^sub>0, fg\<^sub>0h\<^sub>1.p\<^sub>1]) \<cdot> (f \<star> \<a>[g, tab\<^sub>0 g, f\<^sub>0g\<^sub>1.p\<^sub>0] \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> \<a>[f, (g \<star> tab\<^sub>0 g) \<star> f\<^sub>0g\<^sub>1.p\<^sub>0, fg\<^sub>0h\<^sub>1.p\<^sub>1] \<cdot> ((((f \<star> \<a>\<^sup>-\<^sup>1[g, tab\<^sub>0 g, f\<^sub>0g\<^sub>1.p\<^sub>0]) \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> ((f \<star> \<a>[g, tab\<^sub>0 g, f\<^sub>0g\<^sub>1.p\<^sub>0]) \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1)) \<cdot> ((f \<star> g.tab \<star> f\<^sub>0g\<^sub>1.p\<^sub>0) \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1)) \<cdot> ((f \<star> f\<^sub>0g\<^sub>1.\<phi>) \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> (\<a>[f, tab\<^sub>0 f, f\<^sub>0g\<^sub>1.p\<^sub>1] \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> ((f.tab \<star> f\<^sub>0g\<^sub>1.p\<^sub>1) \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>1 f, f\<^sub>0g\<^sub>1.p\<^sub>1, fg\<^sub>0h\<^sub>1.p\<^sub>1]
[PROOF STEP]
proof -
[PROOF STATE]
proof (state)
goal (1 subgoal):
1. \<a>[f, g, h \<star> tab\<^sub>0 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0] \<cdot> ((f \<star> g) \<star> \<a>[h, tab\<^sub>0 h, fg\<^sub>0h\<^sub>1.p\<^sub>0]) \<cdot> ((f \<star> g) \<star> h.tab \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0) \<cdot> ((f \<star> g) \<star> fg\<^sub>0h\<^sub>1.\<phi>) \<cdot> \<a>\<^sup>-\<^sup>1[f, g, (tab\<^sub>0 g \<star> f\<^sub>0g\<^sub>1.p\<^sub>0) \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1] \<cdot> (f \<star> \<a>[g, tab\<^sub>0 g \<star> f\<^sub>0g\<^sub>1.p\<^sub>0, fg\<^sub>0h\<^sub>1.p\<^sub>1]) \<cdot> ((f \<star> \<a>[g, tab\<^sub>0 g, f\<^sub>0g\<^sub>1.p\<^sub>0] \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> \<a>[f, (g \<star> tab\<^sub>0 g) \<star> f\<^sub>0g\<^sub>1.p\<^sub>0, fg\<^sub>0h\<^sub>1.p\<^sub>1]) \<cdot> ((f \<star> g.tab \<star> f\<^sub>0g\<^sub>1.p\<^sub>0) \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> ((f \<star> f\<^sub>0g\<^sub>1.\<phi>) \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> (\<a>[f, tab\<^sub>0 f, f\<^sub>0g\<^sub>1.p\<^sub>1] \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> ((f.tab \<star> f\<^sub>0g\<^sub>1.p\<^sub>1) \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>1 f, f\<^sub>0g\<^sub>1.p\<^sub>1, fg\<^sub>0h\<^sub>1.p\<^sub>1] = \<a>[f, g, h \<star> tab\<^sub>0 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0] \<cdot> ((f \<star> g) \<star> \<a>[h, tab\<^sub>0 h, fg\<^sub>0h\<^sub>1.p\<^sub>0]) \<cdot> ((f \<star> g) \<star> h.tab \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0) \<cdot> ((f \<star> g) \<star> fg\<^sub>0h\<^sub>1.\<phi>) \<cdot> \<a>\<^sup>-\<^sup>1[f, g, (tab\<^sub>0 g \<star> f\<^sub>0g\<^sub>1.p\<^sub>0) \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1] \<cdot> (f \<star> \<a>[g, tab\<^sub>0 g \<star> f\<^sub>0g\<^sub>1.p\<^sub>0, fg\<^sub>0h\<^sub>1.p\<^sub>1]) \<cdot> (f \<star> \<a>[g, tab\<^sub>0 g, f\<^sub>0g\<^sub>1.p\<^sub>0] \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> \<a>[f, (g \<star> tab\<^sub>0 g) \<star> f\<^sub>0g\<^sub>1.p\<^sub>0, fg\<^sub>0h\<^sub>1.p\<^sub>1] \<cdot> ((((f \<star> \<a>\<^sup>-\<^sup>1[g, tab\<^sub>0 g, f\<^sub>0g\<^sub>1.p\<^sub>0]) \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> ((f \<star> \<a>[g, tab\<^sub>0 g, f\<^sub>0g\<^sub>1.p\<^sub>0]) \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1)) \<cdot> ((f \<star> g.tab \<star> f\<^sub>0g\<^sub>1.p\<^sub>0) \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1)) \<cdot> ((f \<star> f\<^sub>0g\<^sub>1.\<phi>) \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> (\<a>[f, tab\<^sub>0 f, f\<^sub>0g\<^sub>1.p\<^sub>1] \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> ((f.tab \<star> f\<^sub>0g\<^sub>1.p\<^sub>1) \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>1 f, f\<^sub>0g\<^sub>1.p\<^sub>1, fg\<^sub>0h\<^sub>1.p\<^sub>1]
[PROOF STEP]
have "(((f \<star> \<a>\<^sup>-\<^sup>1[g, tab\<^sub>0 g, Tfg.p\<^sub>0]) \<star> TTfgh.p\<^sub>1) \<cdot>
((f \<star> \<a>[g, tab\<^sub>0 g, Tfg.p\<^sub>0]) \<star> TTfgh.p\<^sub>1)) \<cdot>
((f \<star> (g.tab \<star> Tfg.p\<^sub>0)) \<star> TTfgh.p\<^sub>1) =
(f \<star> (g.tab \<star> Tfg.p\<^sub>0)) \<star> TTfgh.p\<^sub>1"
[PROOF STATE]
proof (prove)
goal (1 subgoal):
1. (((f \<star> \<a>\<^sup>-\<^sup>1[g, tab\<^sub>0 g, f\<^sub>0g\<^sub>1.p\<^sub>0]) \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> ((f \<star> \<a>[g, tab\<^sub>0 g, f\<^sub>0g\<^sub>1.p\<^sub>0]) \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1)) \<cdot> ((f \<star> g.tab \<star> f\<^sub>0g\<^sub>1.p\<^sub>0) \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) = (f \<star> g.tab \<star> f\<^sub>0g\<^sub>1.p\<^sub>0) \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1
[PROOF STEP]
using fg gh fg\<^sub>0h\<^sub>1.p\<^sub>1_simps comp_cod_arr whisker_right comp_assoc_assoc'
whisker_left [of f "\<a>\<^sup>-\<^sup>1[g, tab\<^sub>0 g, Tfg.p\<^sub>0]" "\<a>[g, tab\<^sub>0 g, Tfg.p\<^sub>0]"]
[PROOF STATE]
proof (prove)
using this:
src f = trg g
src g = trg h
trg fg\<^sub>0h\<^sub>1.p\<^sub>1 = src (tab\<^sub>0 g \<star> f\<^sub>0g\<^sub>1.p\<^sub>0)
\<lbrakk>arr ?f; cod ?f = ?b\<rbrakk> \<Longrightarrow> ?b \<cdot> ?f = ?f
\<lbrakk>ide ?f; seq ?\<nu> ?\<mu>\<rbrakk> \<Longrightarrow> ?\<nu> \<cdot> ?\<mu> \<star> ?f = (?\<nu> \<star> ?f) \<cdot> (?\<mu> \<star> ?f)
\<lbrakk>ide ?f; ide ?g; ide ?h; src ?f = trg ?g; src ?g = trg ?h\<rbrakk> \<Longrightarrow> \<a>[?f, ?g, ?h] \<cdot> \<a>\<^sup>-\<^sup>1[?f, ?g, ?h] = ?f \<star> ?g \<star> ?h
\<lbrakk>ide ?f; ide ?g; ide ?h; src ?f = trg ?g; src ?g = trg ?h\<rbrakk> \<Longrightarrow> \<a>\<^sup>-\<^sup>1[?f, ?g, ?h] \<cdot> \<a>[?f, ?g, ?h] = (?f \<star> ?g) \<star> ?h
\<lbrakk>ide f; seq \<a>\<^sup>-\<^sup>1[g, tab\<^sub>0 g, f\<^sub>0g\<^sub>1.p\<^sub>0] \<a>[g, tab\<^sub>0 g, f\<^sub>0g\<^sub>1.p\<^sub>0]\<rbrakk> \<Longrightarrow> f \<star> \<a>\<^sup>-\<^sup>1[g, tab\<^sub>0 g, f\<^sub>0g\<^sub>1.p\<^sub>0] \<cdot> \<a>[g, tab\<^sub>0 g, f\<^sub>0g\<^sub>1.p\<^sub>0] = (f \<star> \<a>\<^sup>-\<^sup>1[g, tab\<^sub>0 g, f\<^sub>0g\<^sub>1.p\<^sub>0]) \<cdot> (f \<star> \<a>[g, tab\<^sub>0 g, f\<^sub>0g\<^sub>1.p\<^sub>0])
goal (1 subgoal):
1. (((f \<star> \<a>\<^sup>-\<^sup>1[g, tab\<^sub>0 g, f\<^sub>0g\<^sub>1.p\<^sub>0]) \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> ((f \<star> \<a>[g, tab\<^sub>0 g, f\<^sub>0g\<^sub>1.p\<^sub>0]) \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1)) \<cdot> ((f \<star> g.tab \<star> f\<^sub>0g\<^sub>1.p\<^sub>0) \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) = (f \<star> g.tab \<star> f\<^sub>0g\<^sub>1.p\<^sub>0) \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1
[PROOF STEP]
by simp
[PROOF STATE]
proof (state)
this:
(((f \<star> \<a>\<^sup>-\<^sup>1[g, tab\<^sub>0 g, f\<^sub>0g\<^sub>1.p\<^sub>0]) \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> ((f \<star> \<a>[g, tab\<^sub>0 g, f\<^sub>0g\<^sub>1.p\<^sub>0]) \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1)) \<cdot> ((f \<star> g.tab \<star> f\<^sub>0g\<^sub>1.p\<^sub>0) \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) = (f \<star> g.tab \<star> f\<^sub>0g\<^sub>1.p\<^sub>0) \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1
goal (1 subgoal):
1. \<a>[f, g, h \<star> tab\<^sub>0 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0] \<cdot> ((f \<star> g) \<star> \<a>[h, tab\<^sub>0 h, fg\<^sub>0h\<^sub>1.p\<^sub>0]) \<cdot> ((f \<star> g) \<star> h.tab \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0) \<cdot> ((f \<star> g) \<star> fg\<^sub>0h\<^sub>1.\<phi>) \<cdot> \<a>\<^sup>-\<^sup>1[f, g, (tab\<^sub>0 g \<star> f\<^sub>0g\<^sub>1.p\<^sub>0) \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1] \<cdot> (f \<star> \<a>[g, tab\<^sub>0 g \<star> f\<^sub>0g\<^sub>1.p\<^sub>0, fg\<^sub>0h\<^sub>1.p\<^sub>1]) \<cdot> ((f \<star> \<a>[g, tab\<^sub>0 g, f\<^sub>0g\<^sub>1.p\<^sub>0] \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> \<a>[f, (g \<star> tab\<^sub>0 g) \<star> f\<^sub>0g\<^sub>1.p\<^sub>0, fg\<^sub>0h\<^sub>1.p\<^sub>1]) \<cdot> ((f \<star> g.tab \<star> f\<^sub>0g\<^sub>1.p\<^sub>0) \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> ((f \<star> f\<^sub>0g\<^sub>1.\<phi>) \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> (\<a>[f, tab\<^sub>0 f, f\<^sub>0g\<^sub>1.p\<^sub>1] \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> ((f.tab \<star> f\<^sub>0g\<^sub>1.p\<^sub>1) \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>1 f, f\<^sub>0g\<^sub>1.p\<^sub>1, fg\<^sub>0h\<^sub>1.p\<^sub>1] = \<a>[f, g, h \<star> tab\<^sub>0 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0] \<cdot> ((f \<star> g) \<star> \<a>[h, tab\<^sub>0 h, fg\<^sub>0h\<^sub>1.p\<^sub>0]) \<cdot> ((f \<star> g) \<star> h.tab \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0) \<cdot> ((f \<star> g) \<star> fg\<^sub>0h\<^sub>1.\<phi>) \<cdot> \<a>\<^sup>-\<^sup>1[f, g, (tab\<^sub>0 g \<star> f\<^sub>0g\<^sub>1.p\<^sub>0) \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1] \<cdot> (f \<star> \<a>[g, tab\<^sub>0 g \<star> f\<^sub>0g\<^sub>1.p\<^sub>0, fg\<^sub>0h\<^sub>1.p\<^sub>1]) \<cdot> (f \<star> \<a>[g, tab\<^sub>0 g, f\<^sub>0g\<^sub>1.p\<^sub>0] \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> \<a>[f, (g \<star> tab\<^sub>0 g) \<star> f\<^sub>0g\<^sub>1.p\<^sub>0, fg\<^sub>0h\<^sub>1.p\<^sub>1] \<cdot> ((((f \<star> \<a>\<^sup>-\<^sup>1[g, tab\<^sub>0 g, f\<^sub>0g\<^sub>1.p\<^sub>0]) \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> ((f \<star> \<a>[g, tab\<^sub>0 g, f\<^sub>0g\<^sub>1.p\<^sub>0]) \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1)) \<cdot> ((f \<star> g.tab \<star> f\<^sub>0g\<^sub>1.p\<^sub>0) \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1)) \<cdot> ((f \<star> f\<^sub>0g\<^sub>1.\<phi>) \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> (\<a>[f, tab\<^sub>0 f, f\<^sub>0g\<^sub>1.p\<^sub>1] \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> ((f.tab \<star> f\<^sub>0g\<^sub>1.p\<^sub>1) \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>1 f, f\<^sub>0g\<^sub>1.p\<^sub>1, fg\<^sub>0h\<^sub>1.p\<^sub>1]
[PROOF STEP]
thus ?thesis
[PROOF STATE]
proof (prove)
using this:
(((f \<star> \<a>\<^sup>-\<^sup>1[g, tab\<^sub>0 g, f\<^sub>0g\<^sub>1.p\<^sub>0]) \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> ((f \<star> \<a>[g, tab\<^sub>0 g, f\<^sub>0g\<^sub>1.p\<^sub>0]) \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1)) \<cdot> ((f \<star> g.tab \<star> f\<^sub>0g\<^sub>1.p\<^sub>0) \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) = (f \<star> g.tab \<star> f\<^sub>0g\<^sub>1.p\<^sub>0) \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1
goal (1 subgoal):
1. \<a>[f, g, h \<star> tab\<^sub>0 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0] \<cdot> ((f \<star> g) \<star> \<a>[h, tab\<^sub>0 h, fg\<^sub>0h\<^sub>1.p\<^sub>0]) \<cdot> ((f \<star> g) \<star> h.tab \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0) \<cdot> ((f \<star> g) \<star> fg\<^sub>0h\<^sub>1.\<phi>) \<cdot> \<a>\<^sup>-\<^sup>1[f, g, (tab\<^sub>0 g \<star> f\<^sub>0g\<^sub>1.p\<^sub>0) \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1] \<cdot> (f \<star> \<a>[g, tab\<^sub>0 g \<star> f\<^sub>0g\<^sub>1.p\<^sub>0, fg\<^sub>0h\<^sub>1.p\<^sub>1]) \<cdot> ((f \<star> \<a>[g, tab\<^sub>0 g, f\<^sub>0g\<^sub>1.p\<^sub>0] \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> \<a>[f, (g \<star> tab\<^sub>0 g) \<star> f\<^sub>0g\<^sub>1.p\<^sub>0, fg\<^sub>0h\<^sub>1.p\<^sub>1]) \<cdot> ((f \<star> g.tab \<star> f\<^sub>0g\<^sub>1.p\<^sub>0) \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> ((f \<star> f\<^sub>0g\<^sub>1.\<phi>) \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> (\<a>[f, tab\<^sub>0 f, f\<^sub>0g\<^sub>1.p\<^sub>1] \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> ((f.tab \<star> f\<^sub>0g\<^sub>1.p\<^sub>1) \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>1 f, f\<^sub>0g\<^sub>1.p\<^sub>1, fg\<^sub>0h\<^sub>1.p\<^sub>1] = \<a>[f, g, h \<star> tab\<^sub>0 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0] \<cdot> ((f \<star> g) \<star> \<a>[h, tab\<^sub>0 h, fg\<^sub>0h\<^sub>1.p\<^sub>0]) \<cdot> ((f \<star> g) \<star> h.tab \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0) \<cdot> ((f \<star> g) \<star> fg\<^sub>0h\<^sub>1.\<phi>) \<cdot> \<a>\<^sup>-\<^sup>1[f, g, (tab\<^sub>0 g \<star> f\<^sub>0g\<^sub>1.p\<^sub>0) \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1] \<cdot> (f \<star> \<a>[g, tab\<^sub>0 g \<star> f\<^sub>0g\<^sub>1.p\<^sub>0, fg\<^sub>0h\<^sub>1.p\<^sub>1]) \<cdot> (f \<star> \<a>[g, tab\<^sub>0 g, f\<^sub>0g\<^sub>1.p\<^sub>0] \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> \<a>[f, (g \<star> tab\<^sub>0 g) \<star> f\<^sub>0g\<^sub>1.p\<^sub>0, fg\<^sub>0h\<^sub>1.p\<^sub>1] \<cdot> ((((f \<star> \<a>\<^sup>-\<^sup>1[g, tab\<^sub>0 g, f\<^sub>0g\<^sub>1.p\<^sub>0]) \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> ((f \<star> \<a>[g, tab\<^sub>0 g, f\<^sub>0g\<^sub>1.p\<^sub>0]) \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1)) \<cdot> ((f \<star> g.tab \<star> f\<^sub>0g\<^sub>1.p\<^sub>0) \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1)) \<cdot> ((f \<star> f\<^sub>0g\<^sub>1.\<phi>) \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> (\<a>[f, tab\<^sub>0 f, f\<^sub>0g\<^sub>1.p\<^sub>1] \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> ((f.tab \<star> f\<^sub>0g\<^sub>1.p\<^sub>1) \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>1 f, f\<^sub>0g\<^sub>1.p\<^sub>1, fg\<^sub>0h\<^sub>1.p\<^sub>1]
[PROOF STEP]
using comp_assoc
[PROOF STATE]
proof (prove)
using this:
(((f \<star> \<a>\<^sup>-\<^sup>1[g, tab\<^sub>0 g, f\<^sub>0g\<^sub>1.p\<^sub>0]) \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> ((f \<star> \<a>[g, tab\<^sub>0 g, f\<^sub>0g\<^sub>1.p\<^sub>0]) \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1)) \<cdot> ((f \<star> g.tab \<star> f\<^sub>0g\<^sub>1.p\<^sub>0) \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) = (f \<star> g.tab \<star> f\<^sub>0g\<^sub>1.p\<^sub>0) \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1
(?h \<cdot> ?g) \<cdot> ?f = ?h \<cdot> ?g \<cdot> ?f
goal (1 subgoal):
1. \<a>[f, g, h \<star> tab\<^sub>0 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0] \<cdot> ((f \<star> g) \<star> \<a>[h, tab\<^sub>0 h, fg\<^sub>0h\<^sub>1.p\<^sub>0]) \<cdot> ((f \<star> g) \<star> h.tab \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0) \<cdot> ((f \<star> g) \<star> fg\<^sub>0h\<^sub>1.\<phi>) \<cdot> \<a>\<^sup>-\<^sup>1[f, g, (tab\<^sub>0 g \<star> f\<^sub>0g\<^sub>1.p\<^sub>0) \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1] \<cdot> (f \<star> \<a>[g, tab\<^sub>0 g \<star> f\<^sub>0g\<^sub>1.p\<^sub>0, fg\<^sub>0h\<^sub>1.p\<^sub>1]) \<cdot> ((f \<star> \<a>[g, tab\<^sub>0 g, f\<^sub>0g\<^sub>1.p\<^sub>0] \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> \<a>[f, (g \<star> tab\<^sub>0 g) \<star> f\<^sub>0g\<^sub>1.p\<^sub>0, fg\<^sub>0h\<^sub>1.p\<^sub>1]) \<cdot> ((f \<star> g.tab \<star> f\<^sub>0g\<^sub>1.p\<^sub>0) \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> ((f \<star> f\<^sub>0g\<^sub>1.\<phi>) \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> (\<a>[f, tab\<^sub>0 f, f\<^sub>0g\<^sub>1.p\<^sub>1] \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> ((f.tab \<star> f\<^sub>0g\<^sub>1.p\<^sub>1) \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>1 f, f\<^sub>0g\<^sub>1.p\<^sub>1, fg\<^sub>0h\<^sub>1.p\<^sub>1] = \<a>[f, g, h \<star> tab\<^sub>0 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0] \<cdot> ((f \<star> g) \<star> \<a>[h, tab\<^sub>0 h, fg\<^sub>0h\<^sub>1.p\<^sub>0]) \<cdot> ((f \<star> g) \<star> h.tab \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0) \<cdot> ((f \<star> g) \<star> fg\<^sub>0h\<^sub>1.\<phi>) \<cdot> \<a>\<^sup>-\<^sup>1[f, g, (tab\<^sub>0 g \<star> f\<^sub>0g\<^sub>1.p\<^sub>0) \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1] \<cdot> (f \<star> \<a>[g, tab\<^sub>0 g \<star> f\<^sub>0g\<^sub>1.p\<^sub>0, fg\<^sub>0h\<^sub>1.p\<^sub>1]) \<cdot> (f \<star> \<a>[g, tab\<^sub>0 g, f\<^sub>0g\<^sub>1.p\<^sub>0] \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> \<a>[f, (g \<star> tab\<^sub>0 g) \<star> f\<^sub>0g\<^sub>1.p\<^sub>0, fg\<^sub>0h\<^sub>1.p\<^sub>1] \<cdot> ((((f \<star> \<a>\<^sup>-\<^sup>1[g, tab\<^sub>0 g, f\<^sub>0g\<^sub>1.p\<^sub>0]) \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> ((f \<star> \<a>[g, tab\<^sub>0 g, f\<^sub>0g\<^sub>1.p\<^sub>0]) \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1)) \<cdot> ((f \<star> g.tab \<star> f\<^sub>0g\<^sub>1.p\<^sub>0) \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1)) \<cdot> ((f \<star> f\<^sub>0g\<^sub>1.\<phi>) \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> (\<a>[f, tab\<^sub>0 f, f\<^sub>0g\<^sub>1.p\<^sub>1] \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> ((f.tab \<star> f\<^sub>0g\<^sub>1.p\<^sub>1) \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>1 f, f\<^sub>0g\<^sub>1.p\<^sub>1, fg\<^sub>0h\<^sub>1.p\<^sub>1]
[PROOF STEP]
by simp
[PROOF STATE]
proof (state)
this:
\<a>[f, g, h \<star> tab\<^sub>0 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0] \<cdot> ((f \<star> g) \<star> \<a>[h, tab\<^sub>0 h, fg\<^sub>0h\<^sub>1.p\<^sub>0]) \<cdot> ((f \<star> g) \<star> h.tab \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0) \<cdot> ((f \<star> g) \<star> fg\<^sub>0h\<^sub>1.\<phi>) \<cdot> \<a>\<^sup>-\<^sup>1[f, g, (tab\<^sub>0 g \<star> f\<^sub>0g\<^sub>1.p\<^sub>0) \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1] \<cdot> (f \<star> \<a>[g, tab\<^sub>0 g \<star> f\<^sub>0g\<^sub>1.p\<^sub>0, fg\<^sub>0h\<^sub>1.p\<^sub>1]) \<cdot> ((f \<star> \<a>[g, tab\<^sub>0 g, f\<^sub>0g\<^sub>1.p\<^sub>0] \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> \<a>[f, (g \<star> tab\<^sub>0 g) \<star> f\<^sub>0g\<^sub>1.p\<^sub>0, fg\<^sub>0h\<^sub>1.p\<^sub>1]) \<cdot> ((f \<star> g.tab \<star> f\<^sub>0g\<^sub>1.p\<^sub>0) \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> ((f \<star> f\<^sub>0g\<^sub>1.\<phi>) \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> (\<a>[f, tab\<^sub>0 f, f\<^sub>0g\<^sub>1.p\<^sub>1] \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> ((f.tab \<star> f\<^sub>0g\<^sub>1.p\<^sub>1) \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>1 f, f\<^sub>0g\<^sub>1.p\<^sub>1, fg\<^sub>0h\<^sub>1.p\<^sub>1] = \<a>[f, g, h \<star> tab\<^sub>0 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0] \<cdot> ((f \<star> g) \<star> \<a>[h, tab\<^sub>0 h, fg\<^sub>0h\<^sub>1.p\<^sub>0]) \<cdot> ((f \<star> g) \<star> h.tab \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0) \<cdot> ((f \<star> g) \<star> fg\<^sub>0h\<^sub>1.\<phi>) \<cdot> \<a>\<^sup>-\<^sup>1[f, g, (tab\<^sub>0 g \<star> f\<^sub>0g\<^sub>1.p\<^sub>0) \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1] \<cdot> (f \<star> \<a>[g, tab\<^sub>0 g \<star> f\<^sub>0g\<^sub>1.p\<^sub>0, fg\<^sub>0h\<^sub>1.p\<^sub>1]) \<cdot> (f \<star> \<a>[g, tab\<^sub>0 g, f\<^sub>0g\<^sub>1.p\<^sub>0] \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> \<a>[f, (g \<star> tab\<^sub>0 g) \<star> f\<^sub>0g\<^sub>1.p\<^sub>0, fg\<^sub>0h\<^sub>1.p\<^sub>1] \<cdot> ((((f \<star> \<a>\<^sup>-\<^sup>1[g, tab\<^sub>0 g, f\<^sub>0g\<^sub>1.p\<^sub>0]) \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> ((f \<star> \<a>[g, tab\<^sub>0 g, f\<^sub>0g\<^sub>1.p\<^sub>0]) \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1)) \<cdot> ((f \<star> g.tab \<star> f\<^sub>0g\<^sub>1.p\<^sub>0) \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1)) \<cdot> ((f \<star> f\<^sub>0g\<^sub>1.\<phi>) \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> (\<a>[f, tab\<^sub>0 f, f\<^sub>0g\<^sub>1.p\<^sub>1] \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> ((f.tab \<star> f\<^sub>0g\<^sub>1.p\<^sub>1) \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>1 f, f\<^sub>0g\<^sub>1.p\<^sub>1, fg\<^sub>0h\<^sub>1.p\<^sub>1]
goal:
No subgoals!
[PROOF STEP]
qed
[PROOF STATE]
proof (state)
this:
\<a>[f, g, h \<star> tab\<^sub>0 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0] \<cdot> ((f \<star> g) \<star> \<a>[h, tab\<^sub>0 h, fg\<^sub>0h\<^sub>1.p\<^sub>0]) \<cdot> ((f \<star> g) \<star> h.tab \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0) \<cdot> ((f \<star> g) \<star> fg\<^sub>0h\<^sub>1.\<phi>) \<cdot> \<a>\<^sup>-\<^sup>1[f, g, (tab\<^sub>0 g \<star> f\<^sub>0g\<^sub>1.p\<^sub>0) \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1] \<cdot> (f \<star> \<a>[g, tab\<^sub>0 g \<star> f\<^sub>0g\<^sub>1.p\<^sub>0, fg\<^sub>0h\<^sub>1.p\<^sub>1]) \<cdot> ((f \<star> \<a>[g, tab\<^sub>0 g, f\<^sub>0g\<^sub>1.p\<^sub>0] \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> \<a>[f, (g \<star> tab\<^sub>0 g) \<star> f\<^sub>0g\<^sub>1.p\<^sub>0, fg\<^sub>0h\<^sub>1.p\<^sub>1]) \<cdot> ((f \<star> g.tab \<star> f\<^sub>0g\<^sub>1.p\<^sub>0) \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> ((f \<star> f\<^sub>0g\<^sub>1.\<phi>) \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> (\<a>[f, tab\<^sub>0 f, f\<^sub>0g\<^sub>1.p\<^sub>1] \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> ((f.tab \<star> f\<^sub>0g\<^sub>1.p\<^sub>1) \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>1 f, f\<^sub>0g\<^sub>1.p\<^sub>1, fg\<^sub>0h\<^sub>1.p\<^sub>1] = \<a>[f, g, h \<star> tab\<^sub>0 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0] \<cdot> ((f \<star> g) \<star> \<a>[h, tab\<^sub>0 h, fg\<^sub>0h\<^sub>1.p\<^sub>0]) \<cdot> ((f \<star> g) \<star> h.tab \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0) \<cdot> ((f \<star> g) \<star> fg\<^sub>0h\<^sub>1.\<phi>) \<cdot> \<a>\<^sup>-\<^sup>1[f, g, (tab\<^sub>0 g \<star> f\<^sub>0g\<^sub>1.p\<^sub>0) \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1] \<cdot> (f \<star> \<a>[g, tab\<^sub>0 g \<star> f\<^sub>0g\<^sub>1.p\<^sub>0, fg\<^sub>0h\<^sub>1.p\<^sub>1]) \<cdot> (f \<star> \<a>[g, tab\<^sub>0 g, f\<^sub>0g\<^sub>1.p\<^sub>0] \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> \<a>[f, (g \<star> tab\<^sub>0 g) \<star> f\<^sub>0g\<^sub>1.p\<^sub>0, fg\<^sub>0h\<^sub>1.p\<^sub>1] \<cdot> ((((f \<star> \<a>\<^sup>-\<^sup>1[g, tab\<^sub>0 g, f\<^sub>0g\<^sub>1.p\<^sub>0]) \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> ((f \<star> \<a>[g, tab\<^sub>0 g, f\<^sub>0g\<^sub>1.p\<^sub>0]) \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1)) \<cdot> ((f \<star> g.tab \<star> f\<^sub>0g\<^sub>1.p\<^sub>0) \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1)) \<cdot> ((f \<star> f\<^sub>0g\<^sub>1.\<phi>) \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> (\<a>[f, tab\<^sub>0 f, f\<^sub>0g\<^sub>1.p\<^sub>1] \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> ((f.tab \<star> f\<^sub>0g\<^sub>1.p\<^sub>1) \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>1 f, f\<^sub>0g\<^sub>1.p\<^sub>1, fg\<^sub>0h\<^sub>1.p\<^sub>1]
goal (1 subgoal):
1. (f \<star> \<theta>\<^sub>f) \<cdot> \<a>[f, tab\<^sub>0 f, w\<^sub>f] \<cdot> (f.tab \<star> w\<^sub>f) = ((f \<star> \<theta>\<^sub>f') \<cdot> \<a>[f, tab\<^sub>0 f, w\<^sub>f'] \<cdot> (f.tab \<star> w\<^sub>f')) \<cdot> \<beta>\<^sub>f
[PROOF STEP]
also
[PROOF STATE]
proof (state)
this:
\<a>[f, g, h \<star> tab\<^sub>0 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0] \<cdot> ((f \<star> g) \<star> \<a>[h, tab\<^sub>0 h, fg\<^sub>0h\<^sub>1.p\<^sub>0]) \<cdot> ((f \<star> g) \<star> h.tab \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0) \<cdot> ((f \<star> g) \<star> fg\<^sub>0h\<^sub>1.\<phi>) \<cdot> \<a>\<^sup>-\<^sup>1[f, g, (tab\<^sub>0 g \<star> f\<^sub>0g\<^sub>1.p\<^sub>0) \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1] \<cdot> (f \<star> \<a>[g, tab\<^sub>0 g \<star> f\<^sub>0g\<^sub>1.p\<^sub>0, fg\<^sub>0h\<^sub>1.p\<^sub>1]) \<cdot> ((f \<star> \<a>[g, tab\<^sub>0 g, f\<^sub>0g\<^sub>1.p\<^sub>0] \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> \<a>[f, (g \<star> tab\<^sub>0 g) \<star> f\<^sub>0g\<^sub>1.p\<^sub>0, fg\<^sub>0h\<^sub>1.p\<^sub>1]) \<cdot> ((f \<star> g.tab \<star> f\<^sub>0g\<^sub>1.p\<^sub>0) \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> ((f \<star> f\<^sub>0g\<^sub>1.\<phi>) \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> (\<a>[f, tab\<^sub>0 f, f\<^sub>0g\<^sub>1.p\<^sub>1] \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> ((f.tab \<star> f\<^sub>0g\<^sub>1.p\<^sub>1) \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>1 f, f\<^sub>0g\<^sub>1.p\<^sub>1, fg\<^sub>0h\<^sub>1.p\<^sub>1] = \<a>[f, g, h \<star> tab\<^sub>0 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0] \<cdot> ((f \<star> g) \<star> \<a>[h, tab\<^sub>0 h, fg\<^sub>0h\<^sub>1.p\<^sub>0]) \<cdot> ((f \<star> g) \<star> h.tab \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0) \<cdot> ((f \<star> g) \<star> fg\<^sub>0h\<^sub>1.\<phi>) \<cdot> \<a>\<^sup>-\<^sup>1[f, g, (tab\<^sub>0 g \<star> f\<^sub>0g\<^sub>1.p\<^sub>0) \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1] \<cdot> (f \<star> \<a>[g, tab\<^sub>0 g \<star> f\<^sub>0g\<^sub>1.p\<^sub>0, fg\<^sub>0h\<^sub>1.p\<^sub>1]) \<cdot> (f \<star> \<a>[g, tab\<^sub>0 g, f\<^sub>0g\<^sub>1.p\<^sub>0] \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> \<a>[f, (g \<star> tab\<^sub>0 g) \<star> f\<^sub>0g\<^sub>1.p\<^sub>0, fg\<^sub>0h\<^sub>1.p\<^sub>1] \<cdot> ((((f \<star> \<a>\<^sup>-\<^sup>1[g, tab\<^sub>0 g, f\<^sub>0g\<^sub>1.p\<^sub>0]) \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> ((f \<star> \<a>[g, tab\<^sub>0 g, f\<^sub>0g\<^sub>1.p\<^sub>0]) \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1)) \<cdot> ((f \<star> g.tab \<star> f\<^sub>0g\<^sub>1.p\<^sub>0) \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1)) \<cdot> ((f \<star> f\<^sub>0g\<^sub>1.\<phi>) \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> (\<a>[f, tab\<^sub>0 f, f\<^sub>0g\<^sub>1.p\<^sub>1] \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> ((f.tab \<star> f\<^sub>0g\<^sub>1.p\<^sub>1) \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>1 f, f\<^sub>0g\<^sub>1.p\<^sub>1, fg\<^sub>0h\<^sub>1.p\<^sub>1]
goal (1 subgoal):
1. (f \<star> \<theta>\<^sub>f) \<cdot> \<a>[f, tab\<^sub>0 f, w\<^sub>f] \<cdot> (f.tab \<star> w\<^sub>f) = ((f \<star> \<theta>\<^sub>f') \<cdot> \<a>[f, tab\<^sub>0 f, w\<^sub>f'] \<cdot> (f.tab \<star> w\<^sub>f')) \<cdot> \<beta>\<^sub>f
[PROOF STEP]
have "... =
\<a>[f, g, h \<star> tab\<^sub>0 h \<star> TTfgh.p\<^sub>0] \<cdot>
((f \<star> g) \<star> \<a>[h, tab\<^sub>0 h, TTfgh.p\<^sub>0]) \<cdot>
((f \<star> g) \<star> h.tab \<star> TTfgh.p\<^sub>0) \<cdot>
((f \<star> g) \<star> fg\<^sub>0h\<^sub>1.\<phi>) \<cdot>
\<a>\<^sup>-\<^sup>1[f, g, (tab\<^sub>0 g \<star> Tfg.p\<^sub>0) \<star> TTfgh.p\<^sub>1] \<cdot>
(f \<star> \<a>[g, tab\<^sub>0 g \<star> Tfg.p\<^sub>0, TTfgh.p\<^sub>1]) \<cdot>
(f \<star> \<a>[g, tab\<^sub>0 g, Tfg.p\<^sub>0] \<star> TTfgh.p\<^sub>1) \<cdot>
\<a>[f, (g \<star> tab\<^sub>0 g) \<star> Tfg.p\<^sub>0, TTfgh.p\<^sub>1] \<cdot>
((f \<star> \<a>\<^sup>-\<^sup>1[g, tab\<^sub>0 g, Tfg.p\<^sub>0]) \<star> TTfgh.p\<^sub>1) \<cdot>
((f \<star> \<a>[g, tab\<^sub>0 g, Tfg.p\<^sub>0]) \<star> TTfgh.p\<^sub>1) \<cdot>
((f \<star> (g.tab \<star> Tfg.p\<^sub>0)) \<star> TTfgh.p\<^sub>1) \<cdot>
((f \<star> f\<^sub>0g\<^sub>1.\<phi>) \<star> TTfgh.p\<^sub>1) \<cdot>
(\<a>[f, tab\<^sub>0 f, Tfg.p\<^sub>1] \<star> TTfgh.p\<^sub>1) \<cdot>
((f.tab \<star> Tfg.p\<^sub>1) \<star> TTfgh.p\<^sub>1) \<cdot>
\<a>\<^sup>-\<^sup>1[tab\<^sub>1 f, Tfg.p\<^sub>1, TTfgh.p\<^sub>1]"
[PROOF STATE]
proof (prove)
goal (1 subgoal):
1. \<a>[f, g, h \<star> tab\<^sub>0 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0] \<cdot> ((f \<star> g) \<star> \<a>[h, tab\<^sub>0 h, fg\<^sub>0h\<^sub>1.p\<^sub>0]) \<cdot> ((f \<star> g) \<star> h.tab \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0) \<cdot> ((f \<star> g) \<star> fg\<^sub>0h\<^sub>1.\<phi>) \<cdot> \<a>\<^sup>-\<^sup>1[f, g, (tab\<^sub>0 g \<star> f\<^sub>0g\<^sub>1.p\<^sub>0) \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1] \<cdot> (f \<star> \<a>[g, tab\<^sub>0 g \<star> f\<^sub>0g\<^sub>1.p\<^sub>0, fg\<^sub>0h\<^sub>1.p\<^sub>1]) \<cdot> (f \<star> \<a>[g, tab\<^sub>0 g, f\<^sub>0g\<^sub>1.p\<^sub>0] \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> \<a>[f, (g \<star> tab\<^sub>0 g) \<star> f\<^sub>0g\<^sub>1.p\<^sub>0, fg\<^sub>0h\<^sub>1.p\<^sub>1] \<cdot> ((((f \<star> \<a>\<^sup>-\<^sup>1[g, tab\<^sub>0 g, f\<^sub>0g\<^sub>1.p\<^sub>0]) \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> ((f \<star> \<a>[g, tab\<^sub>0 g, f\<^sub>0g\<^sub>1.p\<^sub>0]) \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1)) \<cdot> ((f \<star> g.tab \<star> f\<^sub>0g\<^sub>1.p\<^sub>0) \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1)) \<cdot> ((f \<star> f\<^sub>0g\<^sub>1.\<phi>) \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> (\<a>[f, tab\<^sub>0 f, f\<^sub>0g\<^sub>1.p\<^sub>1] \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> ((f.tab \<star> f\<^sub>0g\<^sub>1.p\<^sub>1) \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>1 f, f\<^sub>0g\<^sub>1.p\<^sub>1, fg\<^sub>0h\<^sub>1.p\<^sub>1] = \<a>[f, g, h \<star> tab\<^sub>0 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0] \<cdot> ((f \<star> g) \<star> \<a>[h, tab\<^sub>0 h, fg\<^sub>0h\<^sub>1.p\<^sub>0]) \<cdot> ((f \<star> g) \<star> h.tab \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0) \<cdot> ((f \<star> g) \<star> fg\<^sub>0h\<^sub>1.\<phi>) \<cdot> \<a>\<^sup>-\<^sup>1[f, g, (tab\<^sub>0 g \<star> f\<^sub>0g\<^sub>1.p\<^sub>0) \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1] \<cdot> (f \<star> \<a>[g, tab\<^sub>0 g \<star> f\<^sub>0g\<^sub>1.p\<^sub>0, fg\<^sub>0h\<^sub>1.p\<^sub>1]) \<cdot> (f \<star> \<a>[g, tab\<^sub>0 g, f\<^sub>0g\<^sub>1.p\<^sub>0] \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> \<a>[f, (g \<star> tab\<^sub>0 g) \<star> f\<^sub>0g\<^sub>1.p\<^sub>0, fg\<^sub>0h\<^sub>1.p\<^sub>1] \<cdot> ((f \<star> \<a>\<^sup>-\<^sup>1[g, tab\<^sub>0 g, f\<^sub>0g\<^sub>1.p\<^sub>0]) \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> ((f \<star> \<a>[g, tab\<^sub>0 g, f\<^sub>0g\<^sub>1.p\<^sub>0]) \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> ((f \<star> g.tab \<star> f\<^sub>0g\<^sub>1.p\<^sub>0) \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> ((f \<star> f\<^sub>0g\<^sub>1.\<phi>) \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> (\<a>[f, tab\<^sub>0 f, f\<^sub>0g\<^sub>1.p\<^sub>1] \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> ((f.tab \<star> f\<^sub>0g\<^sub>1.p\<^sub>1) \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>1 f, f\<^sub>0g\<^sub>1.p\<^sub>1, fg\<^sub>0h\<^sub>1.p\<^sub>1]
[PROOF STEP]
using comp_assoc
[PROOF STATE]
proof (prove)
using this:
(?h \<cdot> ?g) \<cdot> ?f = ?h \<cdot> ?g \<cdot> ?f
goal (1 subgoal):
1. \<a>[f, g, h \<star> tab\<^sub>0 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0] \<cdot> ((f \<star> g) \<star> \<a>[h, tab\<^sub>0 h, fg\<^sub>0h\<^sub>1.p\<^sub>0]) \<cdot> ((f \<star> g) \<star> h.tab \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0) \<cdot> ((f \<star> g) \<star> fg\<^sub>0h\<^sub>1.\<phi>) \<cdot> \<a>\<^sup>-\<^sup>1[f, g, (tab\<^sub>0 g \<star> f\<^sub>0g\<^sub>1.p\<^sub>0) \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1] \<cdot> (f \<star> \<a>[g, tab\<^sub>0 g \<star> f\<^sub>0g\<^sub>1.p\<^sub>0, fg\<^sub>0h\<^sub>1.p\<^sub>1]) \<cdot> (f \<star> \<a>[g, tab\<^sub>0 g, f\<^sub>0g\<^sub>1.p\<^sub>0] \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> \<a>[f, (g \<star> tab\<^sub>0 g) \<star> f\<^sub>0g\<^sub>1.p\<^sub>0, fg\<^sub>0h\<^sub>1.p\<^sub>1] \<cdot> ((((f \<star> \<a>\<^sup>-\<^sup>1[g, tab\<^sub>0 g, f\<^sub>0g\<^sub>1.p\<^sub>0]) \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> ((f \<star> \<a>[g, tab\<^sub>0 g, f\<^sub>0g\<^sub>1.p\<^sub>0]) \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1)) \<cdot> ((f \<star> g.tab \<star> f\<^sub>0g\<^sub>1.p\<^sub>0) \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1)) \<cdot> ((f \<star> f\<^sub>0g\<^sub>1.\<phi>) \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> (\<a>[f, tab\<^sub>0 f, f\<^sub>0g\<^sub>1.p\<^sub>1] \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> ((f.tab \<star> f\<^sub>0g\<^sub>1.p\<^sub>1) \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>1 f, f\<^sub>0g\<^sub>1.p\<^sub>1, fg\<^sub>0h\<^sub>1.p\<^sub>1] = \<a>[f, g, h \<star> tab\<^sub>0 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0] \<cdot> ((f \<star> g) \<star> \<a>[h, tab\<^sub>0 h, fg\<^sub>0h\<^sub>1.p\<^sub>0]) \<cdot> ((f \<star> g) \<star> h.tab \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0) \<cdot> ((f \<star> g) \<star> fg\<^sub>0h\<^sub>1.\<phi>) \<cdot> \<a>\<^sup>-\<^sup>1[f, g, (tab\<^sub>0 g \<star> f\<^sub>0g\<^sub>1.p\<^sub>0) \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1] \<cdot> (f \<star> \<a>[g, tab\<^sub>0 g \<star> f\<^sub>0g\<^sub>1.p\<^sub>0, fg\<^sub>0h\<^sub>1.p\<^sub>1]) \<cdot> (f \<star> \<a>[g, tab\<^sub>0 g, f\<^sub>0g\<^sub>1.p\<^sub>0] \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> \<a>[f, (g \<star> tab\<^sub>0 g) \<star> f\<^sub>0g\<^sub>1.p\<^sub>0, fg\<^sub>0h\<^sub>1.p\<^sub>1] \<cdot> ((f \<star> \<a>\<^sup>-\<^sup>1[g, tab\<^sub>0 g, f\<^sub>0g\<^sub>1.p\<^sub>0]) \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> ((f \<star> \<a>[g, tab\<^sub>0 g, f\<^sub>0g\<^sub>1.p\<^sub>0]) \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> ((f \<star> g.tab \<star> f\<^sub>0g\<^sub>1.p\<^sub>0) \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> ((f \<star> f\<^sub>0g\<^sub>1.\<phi>) \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> (\<a>[f, tab\<^sub>0 f, f\<^sub>0g\<^sub>1.p\<^sub>1] \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> ((f.tab \<star> f\<^sub>0g\<^sub>1.p\<^sub>1) \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>1 f, f\<^sub>0g\<^sub>1.p\<^sub>1, fg\<^sub>0h\<^sub>1.p\<^sub>1]
[PROOF STEP]
by presburger
[PROOF STATE]
proof (state)
this:
\<a>[f, g, h \<star> tab\<^sub>0 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0] \<cdot> ((f \<star> g) \<star> \<a>[h, tab\<^sub>0 h, fg\<^sub>0h\<^sub>1.p\<^sub>0]) \<cdot> ((f \<star> g) \<star> h.tab \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0) \<cdot> ((f \<star> g) \<star> fg\<^sub>0h\<^sub>1.\<phi>) \<cdot> \<a>\<^sup>-\<^sup>1[f, g, (tab\<^sub>0 g \<star> f\<^sub>0g\<^sub>1.p\<^sub>0) \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1] \<cdot> (f \<star> \<a>[g, tab\<^sub>0 g \<star> f\<^sub>0g\<^sub>1.p\<^sub>0, fg\<^sub>0h\<^sub>1.p\<^sub>1]) \<cdot> (f \<star> \<a>[g, tab\<^sub>0 g, f\<^sub>0g\<^sub>1.p\<^sub>0] \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> \<a>[f, (g \<star> tab\<^sub>0 g) \<star> f\<^sub>0g\<^sub>1.p\<^sub>0, fg\<^sub>0h\<^sub>1.p\<^sub>1] \<cdot> ((((f \<star> \<a>\<^sup>-\<^sup>1[g, tab\<^sub>0 g, f\<^sub>0g\<^sub>1.p\<^sub>0]) \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> ((f \<star> \<a>[g, tab\<^sub>0 g, f\<^sub>0g\<^sub>1.p\<^sub>0]) \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1)) \<cdot> ((f \<star> g.tab \<star> f\<^sub>0g\<^sub>1.p\<^sub>0) \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1)) \<cdot> ((f \<star> f\<^sub>0g\<^sub>1.\<phi>) \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> (\<a>[f, tab\<^sub>0 f, f\<^sub>0g\<^sub>1.p\<^sub>1] \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> ((f.tab \<star> f\<^sub>0g\<^sub>1.p\<^sub>1) \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>1 f, f\<^sub>0g\<^sub>1.p\<^sub>1, fg\<^sub>0h\<^sub>1.p\<^sub>1] = \<a>[f, g, h \<star> tab\<^sub>0 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0] \<cdot> ((f \<star> g) \<star> \<a>[h, tab\<^sub>0 h, fg\<^sub>0h\<^sub>1.p\<^sub>0]) \<cdot> ((f \<star> g) \<star> h.tab \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0) \<cdot> ((f \<star> g) \<star> fg\<^sub>0h\<^sub>1.\<phi>) \<cdot> \<a>\<^sup>-\<^sup>1[f, g, (tab\<^sub>0 g \<star> f\<^sub>0g\<^sub>1.p\<^sub>0) \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1] \<cdot> (f \<star> \<a>[g, tab\<^sub>0 g \<star> f\<^sub>0g\<^sub>1.p\<^sub>0, fg\<^sub>0h\<^sub>1.p\<^sub>1]) \<cdot> (f \<star> \<a>[g, tab\<^sub>0 g, f\<^sub>0g\<^sub>1.p\<^sub>0] \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> \<a>[f, (g \<star> tab\<^sub>0 g) \<star> f\<^sub>0g\<^sub>1.p\<^sub>0, fg\<^sub>0h\<^sub>1.p\<^sub>1] \<cdot> ((f \<star> \<a>\<^sup>-\<^sup>1[g, tab\<^sub>0 g, f\<^sub>0g\<^sub>1.p\<^sub>0]) \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> ((f \<star> \<a>[g, tab\<^sub>0 g, f\<^sub>0g\<^sub>1.p\<^sub>0]) \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> ((f \<star> g.tab \<star> f\<^sub>0g\<^sub>1.p\<^sub>0) \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> ((f \<star> f\<^sub>0g\<^sub>1.\<phi>) \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> (\<a>[f, tab\<^sub>0 f, f\<^sub>0g\<^sub>1.p\<^sub>1] \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> ((f.tab \<star> f\<^sub>0g\<^sub>1.p\<^sub>1) \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>1 f, f\<^sub>0g\<^sub>1.p\<^sub>1, fg\<^sub>0h\<^sub>1.p\<^sub>1]
goal (1 subgoal):
1. (f \<star> \<theta>\<^sub>f) \<cdot> \<a>[f, tab\<^sub>0 f, w\<^sub>f] \<cdot> (f.tab \<star> w\<^sub>f) = ((f \<star> \<theta>\<^sub>f') \<cdot> \<a>[f, tab\<^sub>0 f, w\<^sub>f'] \<cdot> (f.tab \<star> w\<^sub>f')) \<cdot> \<beta>\<^sub>f
[PROOF STEP]
also
[PROOF STATE]
proof (state)
this:
\<a>[f, g, h \<star> tab\<^sub>0 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0] \<cdot> ((f \<star> g) \<star> \<a>[h, tab\<^sub>0 h, fg\<^sub>0h\<^sub>1.p\<^sub>0]) \<cdot> ((f \<star> g) \<star> h.tab \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0) \<cdot> ((f \<star> g) \<star> fg\<^sub>0h\<^sub>1.\<phi>) \<cdot> \<a>\<^sup>-\<^sup>1[f, g, (tab\<^sub>0 g \<star> f\<^sub>0g\<^sub>1.p\<^sub>0) \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1] \<cdot> (f \<star> \<a>[g, tab\<^sub>0 g \<star> f\<^sub>0g\<^sub>1.p\<^sub>0, fg\<^sub>0h\<^sub>1.p\<^sub>1]) \<cdot> (f \<star> \<a>[g, tab\<^sub>0 g, f\<^sub>0g\<^sub>1.p\<^sub>0] \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> \<a>[f, (g \<star> tab\<^sub>0 g) \<star> f\<^sub>0g\<^sub>1.p\<^sub>0, fg\<^sub>0h\<^sub>1.p\<^sub>1] \<cdot> ((((f \<star> \<a>\<^sup>-\<^sup>1[g, tab\<^sub>0 g, f\<^sub>0g\<^sub>1.p\<^sub>0]) \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> ((f \<star> \<a>[g, tab\<^sub>0 g, f\<^sub>0g\<^sub>1.p\<^sub>0]) \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1)) \<cdot> ((f \<star> g.tab \<star> f\<^sub>0g\<^sub>1.p\<^sub>0) \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1)) \<cdot> ((f \<star> f\<^sub>0g\<^sub>1.\<phi>) \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> (\<a>[f, tab\<^sub>0 f, f\<^sub>0g\<^sub>1.p\<^sub>1] \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> ((f.tab \<star> f\<^sub>0g\<^sub>1.p\<^sub>1) \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>1 f, f\<^sub>0g\<^sub>1.p\<^sub>1, fg\<^sub>0h\<^sub>1.p\<^sub>1] = \<a>[f, g, h \<star> tab\<^sub>0 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0] \<cdot> ((f \<star> g) \<star> \<a>[h, tab\<^sub>0 h, fg\<^sub>0h\<^sub>1.p\<^sub>0]) \<cdot> ((f \<star> g) \<star> h.tab \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0) \<cdot> ((f \<star> g) \<star> fg\<^sub>0h\<^sub>1.\<phi>) \<cdot> \<a>\<^sup>-\<^sup>1[f, g, (tab\<^sub>0 g \<star> f\<^sub>0g\<^sub>1.p\<^sub>0) \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1] \<cdot> (f \<star> \<a>[g, tab\<^sub>0 g \<star> f\<^sub>0g\<^sub>1.p\<^sub>0, fg\<^sub>0h\<^sub>1.p\<^sub>1]) \<cdot> (f \<star> \<a>[g, tab\<^sub>0 g, f\<^sub>0g\<^sub>1.p\<^sub>0] \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> \<a>[f, (g \<star> tab\<^sub>0 g) \<star> f\<^sub>0g\<^sub>1.p\<^sub>0, fg\<^sub>0h\<^sub>1.p\<^sub>1] \<cdot> ((f \<star> \<a>\<^sup>-\<^sup>1[g, tab\<^sub>0 g, f\<^sub>0g\<^sub>1.p\<^sub>0]) \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> ((f \<star> \<a>[g, tab\<^sub>0 g, f\<^sub>0g\<^sub>1.p\<^sub>0]) \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> ((f \<star> g.tab \<star> f\<^sub>0g\<^sub>1.p\<^sub>0) \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> ((f \<star> f\<^sub>0g\<^sub>1.\<phi>) \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> (\<a>[f, tab\<^sub>0 f, f\<^sub>0g\<^sub>1.p\<^sub>1] \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> ((f.tab \<star> f\<^sub>0g\<^sub>1.p\<^sub>1) \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>1 f, f\<^sub>0g\<^sub>1.p\<^sub>1, fg\<^sub>0h\<^sub>1.p\<^sub>1]
goal (1 subgoal):
1. (f \<star> \<theta>\<^sub>f) \<cdot> \<a>[f, tab\<^sub>0 f, w\<^sub>f] \<cdot> (f.tab \<star> w\<^sub>f) = ((f \<star> \<theta>\<^sub>f') \<cdot> \<a>[f, tab\<^sub>0 f, w\<^sub>f'] \<cdot> (f.tab \<star> w\<^sub>f')) \<cdot> \<beta>\<^sub>f
[PROOF STEP]
have "... =
\<a>[f, g, h \<star> tab\<^sub>0 h \<star> TTfgh.p\<^sub>0] \<cdot>
((f \<star> g) \<star> \<a>[h, tab\<^sub>0 h, TTfgh.p\<^sub>0]) \<cdot>
((f \<star> g) \<star> h.tab \<star> TTfgh.p\<^sub>0) \<cdot>
((f \<star> g) \<star> fg\<^sub>0h\<^sub>1.\<phi>) \<cdot>
\<a>\<^sup>-\<^sup>1[f, g, (tab\<^sub>0 g \<star> Tfg.p\<^sub>0) \<star> TTfgh.p\<^sub>1] \<cdot>
(f \<star> \<a>[g, tab\<^sub>0 g \<star> Tfg.p\<^sub>0, TTfgh.p\<^sub>1]) \<cdot>
(f \<star> \<a>[g, tab\<^sub>0 g, Tfg.p\<^sub>0] \<star> TTfgh.p\<^sub>1) \<cdot>
\<a>[f, (g \<star> tab\<^sub>0 g) \<star> Tfg.p\<^sub>0, TTfgh.p\<^sub>1] \<cdot>
((f \<star> \<a>\<^sup>-\<^sup>1[g, tab\<^sub>0 g, Tfg.p\<^sub>0]) \<star> TTfgh.p\<^sub>1) \<cdot>
(((\<a>[f, g, tab\<^sub>0 g \<star> Tfg.p\<^sub>0] \<star> TTfgh.p\<^sub>1) \<cdot>
(\<a>\<^sup>-\<^sup>1[f, g, tab\<^sub>0 g \<star> Tfg.p\<^sub>0] \<star> TTfgh.p\<^sub>1)) \<cdot>
((f \<star> \<a>[g, tab\<^sub>0 g, Tfg.p\<^sub>0]) \<star> TTfgh.p\<^sub>1)) \<cdot>
((f \<star> (g.tab \<star> Tfg.p\<^sub>0)) \<star> TTfgh.p\<^sub>1) \<cdot>
((f \<star> f\<^sub>0g\<^sub>1.\<phi>) \<star> TTfgh.p\<^sub>1) \<cdot>
(\<a>[f, tab\<^sub>0 f, Tfg.p\<^sub>1] \<star> TTfgh.p\<^sub>1) \<cdot>
((f.tab \<star> Tfg.p\<^sub>1) \<star> TTfgh.p\<^sub>1) \<cdot>
\<a>\<^sup>-\<^sup>1[tab\<^sub>1 f, Tfg.p\<^sub>1, TTfgh.p\<^sub>1]"
[PROOF STATE]
proof (prove)
goal (1 subgoal):
1. \<a>[f, g, h \<star> tab\<^sub>0 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0] \<cdot> ((f \<star> g) \<star> \<a>[h, tab\<^sub>0 h, fg\<^sub>0h\<^sub>1.p\<^sub>0]) \<cdot> ((f \<star> g) \<star> h.tab \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0) \<cdot> ((f \<star> g) \<star> fg\<^sub>0h\<^sub>1.\<phi>) \<cdot> \<a>\<^sup>-\<^sup>1[f, g, (tab\<^sub>0 g \<star> f\<^sub>0g\<^sub>1.p\<^sub>0) \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1] \<cdot> (f \<star> \<a>[g, tab\<^sub>0 g \<star> f\<^sub>0g\<^sub>1.p\<^sub>0, fg\<^sub>0h\<^sub>1.p\<^sub>1]) \<cdot> (f \<star> \<a>[g, tab\<^sub>0 g, f\<^sub>0g\<^sub>1.p\<^sub>0] \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> \<a>[f, (g \<star> tab\<^sub>0 g) \<star> f\<^sub>0g\<^sub>1.p\<^sub>0, fg\<^sub>0h\<^sub>1.p\<^sub>1] \<cdot> ((f \<star> \<a>\<^sup>-\<^sup>1[g, tab\<^sub>0 g, f\<^sub>0g\<^sub>1.p\<^sub>0]) \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> ((f \<star> \<a>[g, tab\<^sub>0 g, f\<^sub>0g\<^sub>1.p\<^sub>0]) \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> ((f \<star> g.tab \<star> f\<^sub>0g\<^sub>1.p\<^sub>0) \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> ((f \<star> f\<^sub>0g\<^sub>1.\<phi>) \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> (\<a>[f, tab\<^sub>0 f, f\<^sub>0g\<^sub>1.p\<^sub>1] \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> ((f.tab \<star> f\<^sub>0g\<^sub>1.p\<^sub>1) \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>1 f, f\<^sub>0g\<^sub>1.p\<^sub>1, fg\<^sub>0h\<^sub>1.p\<^sub>1] = \<a>[f, g, h \<star> tab\<^sub>0 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0] \<cdot> ((f \<star> g) \<star> \<a>[h, tab\<^sub>0 h, fg\<^sub>0h\<^sub>1.p\<^sub>0]) \<cdot> ((f \<star> g) \<star> h.tab \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0) \<cdot> ((f \<star> g) \<star> fg\<^sub>0h\<^sub>1.\<phi>) \<cdot> \<a>\<^sup>-\<^sup>1[f, g, (tab\<^sub>0 g \<star> f\<^sub>0g\<^sub>1.p\<^sub>0) \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1] \<cdot> (f \<star> \<a>[g, tab\<^sub>0 g \<star> f\<^sub>0g\<^sub>1.p\<^sub>0, fg\<^sub>0h\<^sub>1.p\<^sub>1]) \<cdot> (f \<star> \<a>[g, tab\<^sub>0 g, f\<^sub>0g\<^sub>1.p\<^sub>0] \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> \<a>[f, (g \<star> tab\<^sub>0 g) \<star> f\<^sub>0g\<^sub>1.p\<^sub>0, fg\<^sub>0h\<^sub>1.p\<^sub>1] \<cdot> ((f \<star> \<a>\<^sup>-\<^sup>1[g, tab\<^sub>0 g, f\<^sub>0g\<^sub>1.p\<^sub>0]) \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> (((\<a>[f, g, tab\<^sub>0 g \<star> f\<^sub>0g\<^sub>1.p\<^sub>0] \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> (\<a>\<^sup>-\<^sup>1[f, g, tab\<^sub>0 g \<star> f\<^sub>0g\<^sub>1.p\<^sub>0] \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1)) \<cdot> ((f \<star> \<a>[g, tab\<^sub>0 g, f\<^sub>0g\<^sub>1.p\<^sub>0]) \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1)) \<cdot> ((f \<star> g.tab \<star> f\<^sub>0g\<^sub>1.p\<^sub>0) \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> ((f \<star> f\<^sub>0g\<^sub>1.\<phi>) \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> (\<a>[f, tab\<^sub>0 f, f\<^sub>0g\<^sub>1.p\<^sub>1] \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> ((f.tab \<star> f\<^sub>0g\<^sub>1.p\<^sub>1) \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>1 f, f\<^sub>0g\<^sub>1.p\<^sub>1, fg\<^sub>0h\<^sub>1.p\<^sub>1]
[PROOF STEP]
proof -
[PROOF STATE]
proof (state)
goal (1 subgoal):
1. \<a>[f, g, h \<star> tab\<^sub>0 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0] \<cdot> ((f \<star> g) \<star> \<a>[h, tab\<^sub>0 h, fg\<^sub>0h\<^sub>1.p\<^sub>0]) \<cdot> ((f \<star> g) \<star> h.tab \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0) \<cdot> ((f \<star> g) \<star> fg\<^sub>0h\<^sub>1.\<phi>) \<cdot> \<a>\<^sup>-\<^sup>1[f, g, (tab\<^sub>0 g \<star> f\<^sub>0g\<^sub>1.p\<^sub>0) \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1] \<cdot> (f \<star> \<a>[g, tab\<^sub>0 g \<star> f\<^sub>0g\<^sub>1.p\<^sub>0, fg\<^sub>0h\<^sub>1.p\<^sub>1]) \<cdot> (f \<star> \<a>[g, tab\<^sub>0 g, f\<^sub>0g\<^sub>1.p\<^sub>0] \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> \<a>[f, (g \<star> tab\<^sub>0 g) \<star> f\<^sub>0g\<^sub>1.p\<^sub>0, fg\<^sub>0h\<^sub>1.p\<^sub>1] \<cdot> ((f \<star> \<a>\<^sup>-\<^sup>1[g, tab\<^sub>0 g, f\<^sub>0g\<^sub>1.p\<^sub>0]) \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> ((f \<star> \<a>[g, tab\<^sub>0 g, f\<^sub>0g\<^sub>1.p\<^sub>0]) \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> ((f \<star> g.tab \<star> f\<^sub>0g\<^sub>1.p\<^sub>0) \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> ((f \<star> f\<^sub>0g\<^sub>1.\<phi>) \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> (\<a>[f, tab\<^sub>0 f, f\<^sub>0g\<^sub>1.p\<^sub>1] \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> ((f.tab \<star> f\<^sub>0g\<^sub>1.p\<^sub>1) \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>1 f, f\<^sub>0g\<^sub>1.p\<^sub>1, fg\<^sub>0h\<^sub>1.p\<^sub>1] = \<a>[f, g, h \<star> tab\<^sub>0 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0] \<cdot> ((f \<star> g) \<star> \<a>[h, tab\<^sub>0 h, fg\<^sub>0h\<^sub>1.p\<^sub>0]) \<cdot> ((f \<star> g) \<star> h.tab \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0) \<cdot> ((f \<star> g) \<star> fg\<^sub>0h\<^sub>1.\<phi>) \<cdot> \<a>\<^sup>-\<^sup>1[f, g, (tab\<^sub>0 g \<star> f\<^sub>0g\<^sub>1.p\<^sub>0) \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1] \<cdot> (f \<star> \<a>[g, tab\<^sub>0 g \<star> f\<^sub>0g\<^sub>1.p\<^sub>0, fg\<^sub>0h\<^sub>1.p\<^sub>1]) \<cdot> (f \<star> \<a>[g, tab\<^sub>0 g, f\<^sub>0g\<^sub>1.p\<^sub>0] \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> \<a>[f, (g \<star> tab\<^sub>0 g) \<star> f\<^sub>0g\<^sub>1.p\<^sub>0, fg\<^sub>0h\<^sub>1.p\<^sub>1] \<cdot> ((f \<star> \<a>\<^sup>-\<^sup>1[g, tab\<^sub>0 g, f\<^sub>0g\<^sub>1.p\<^sub>0]) \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> (((\<a>[f, g, tab\<^sub>0 g \<star> f\<^sub>0g\<^sub>1.p\<^sub>0] \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> (\<a>\<^sup>-\<^sup>1[f, g, tab\<^sub>0 g \<star> f\<^sub>0g\<^sub>1.p\<^sub>0] \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1)) \<cdot> ((f \<star> \<a>[g, tab\<^sub>0 g, f\<^sub>0g\<^sub>1.p\<^sub>0]) \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1)) \<cdot> ((f \<star> g.tab \<star> f\<^sub>0g\<^sub>1.p\<^sub>0) \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> ((f \<star> f\<^sub>0g\<^sub>1.\<phi>) \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> (\<a>[f, tab\<^sub>0 f, f\<^sub>0g\<^sub>1.p\<^sub>1] \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> ((f.tab \<star> f\<^sub>0g\<^sub>1.p\<^sub>1) \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>1 f, f\<^sub>0g\<^sub>1.p\<^sub>1, fg\<^sub>0h\<^sub>1.p\<^sub>1]
[PROOF STEP]
have "((\<a>[f, g, tab\<^sub>0 g \<star> Tfg.p\<^sub>0] \<star> TTfgh.p\<^sub>1) \<cdot>
(\<a>\<^sup>-\<^sup>1[f, g, tab\<^sub>0 g \<star> Tfg.p\<^sub>0] \<star> TTfgh.p\<^sub>1)) \<cdot>
((f \<star> \<a>[g, tab\<^sub>0 g, Tfg.p\<^sub>0]) \<star> TTfgh.p\<^sub>1) =
(f \<star> \<a>[g, tab\<^sub>0 g, Tfg.p\<^sub>0]) \<star> TTfgh.p\<^sub>1"
[PROOF STATE]
proof (prove)
goal (1 subgoal):
1. ((\<a>[f, g, tab\<^sub>0 g \<star> f\<^sub>0g\<^sub>1.p\<^sub>0] \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> (\<a>\<^sup>-\<^sup>1[f, g, tab\<^sub>0 g \<star> f\<^sub>0g\<^sub>1.p\<^sub>0] \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1)) \<cdot> ((f \<star> \<a>[g, tab\<^sub>0 g, f\<^sub>0g\<^sub>1.p\<^sub>0]) \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) = (f \<star> \<a>[g, tab\<^sub>0 g, f\<^sub>0g\<^sub>1.p\<^sub>0]) \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1
[PROOF STEP]
using fg fg\<^sub>0h\<^sub>1.p\<^sub>1_simps comp_cod_arr comp_assoc_assoc'
whisker_right
[of TTfgh.p\<^sub>1 "\<a>[f, g, tab\<^sub>0 g \<star> Tfg.p\<^sub>0]" "\<a>\<^sup>-\<^sup>1[f, g, tab\<^sub>0 g \<star> Tfg.p\<^sub>0]"]
[PROOF STATE]
proof (prove)
using this:
src f = trg g
trg fg\<^sub>0h\<^sub>1.p\<^sub>1 = src (tab\<^sub>0 g \<star> f\<^sub>0g\<^sub>1.p\<^sub>0)
\<lbrakk>arr ?f; cod ?f = ?b\<rbrakk> \<Longrightarrow> ?b \<cdot> ?f = ?f
\<lbrakk>ide ?f; ide ?g; ide ?h; src ?f = trg ?g; src ?g = trg ?h\<rbrakk> \<Longrightarrow> \<a>[?f, ?g, ?h] \<cdot> \<a>\<^sup>-\<^sup>1[?f, ?g, ?h] = ?f \<star> ?g \<star> ?h
\<lbrakk>ide ?f; ide ?g; ide ?h; src ?f = trg ?g; src ?g = trg ?h\<rbrakk> \<Longrightarrow> \<a>\<^sup>-\<^sup>1[?f, ?g, ?h] \<cdot> \<a>[?f, ?g, ?h] = (?f \<star> ?g) \<star> ?h
\<lbrakk>ide fg\<^sub>0h\<^sub>1.p\<^sub>1; seq \<a>[f, g, tab\<^sub>0 g \<star> f\<^sub>0g\<^sub>1.p\<^sub>0] \<a>\<^sup>-\<^sup>1[f, g, tab\<^sub>0 g \<star> f\<^sub>0g\<^sub>1.p\<^sub>0]\<rbrakk> \<Longrightarrow> \<a>[f, g, tab\<^sub>0 g \<star> f\<^sub>0g\<^sub>1.p\<^sub>0] \<cdot> \<a>\<^sup>-\<^sup>1[f, g, tab\<^sub>0 g \<star> f\<^sub>0g\<^sub>1.p\<^sub>0] \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1 = (\<a>[f, g, tab\<^sub>0 g \<star> f\<^sub>0g\<^sub>1.p\<^sub>0] \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> (\<a>\<^sup>-\<^sup>1[f, g, tab\<^sub>0 g \<star> f\<^sub>0g\<^sub>1.p\<^sub>0] \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1)
goal (1 subgoal):
1. ((\<a>[f, g, tab\<^sub>0 g \<star> f\<^sub>0g\<^sub>1.p\<^sub>0] \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> (\<a>\<^sup>-\<^sup>1[f, g, tab\<^sub>0 g \<star> f\<^sub>0g\<^sub>1.p\<^sub>0] \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1)) \<cdot> ((f \<star> \<a>[g, tab\<^sub>0 g, f\<^sub>0g\<^sub>1.p\<^sub>0]) \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) = (f \<star> \<a>[g, tab\<^sub>0 g, f\<^sub>0g\<^sub>1.p\<^sub>0]) \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1
[PROOF STEP]
by simp
[PROOF STATE]
proof (state)
this:
((\<a>[f, g, tab\<^sub>0 g \<star> f\<^sub>0g\<^sub>1.p\<^sub>0] \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> (\<a>\<^sup>-\<^sup>1[f, g, tab\<^sub>0 g \<star> f\<^sub>0g\<^sub>1.p\<^sub>0] \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1)) \<cdot> ((f \<star> \<a>[g, tab\<^sub>0 g, f\<^sub>0g\<^sub>1.p\<^sub>0]) \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) = (f \<star> \<a>[g, tab\<^sub>0 g, f\<^sub>0g\<^sub>1.p\<^sub>0]) \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1
goal (1 subgoal):
1. \<a>[f, g, h \<star> tab\<^sub>0 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0] \<cdot> ((f \<star> g) \<star> \<a>[h, tab\<^sub>0 h, fg\<^sub>0h\<^sub>1.p\<^sub>0]) \<cdot> ((f \<star> g) \<star> h.tab \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0) \<cdot> ((f \<star> g) \<star> fg\<^sub>0h\<^sub>1.\<phi>) \<cdot> \<a>\<^sup>-\<^sup>1[f, g, (tab\<^sub>0 g \<star> f\<^sub>0g\<^sub>1.p\<^sub>0) \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1] \<cdot> (f \<star> \<a>[g, tab\<^sub>0 g \<star> f\<^sub>0g\<^sub>1.p\<^sub>0, fg\<^sub>0h\<^sub>1.p\<^sub>1]) \<cdot> (f \<star> \<a>[g, tab\<^sub>0 g, f\<^sub>0g\<^sub>1.p\<^sub>0] \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> \<a>[f, (g \<star> tab\<^sub>0 g) \<star> f\<^sub>0g\<^sub>1.p\<^sub>0, fg\<^sub>0h\<^sub>1.p\<^sub>1] \<cdot> ((f \<star> \<a>\<^sup>-\<^sup>1[g, tab\<^sub>0 g, f\<^sub>0g\<^sub>1.p\<^sub>0]) \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> ((f \<star> \<a>[g, tab\<^sub>0 g, f\<^sub>0g\<^sub>1.p\<^sub>0]) \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> ((f \<star> g.tab \<star> f\<^sub>0g\<^sub>1.p\<^sub>0) \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> ((f \<star> f\<^sub>0g\<^sub>1.\<phi>) \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> (\<a>[f, tab\<^sub>0 f, f\<^sub>0g\<^sub>1.p\<^sub>1] \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> ((f.tab \<star> f\<^sub>0g\<^sub>1.p\<^sub>1) \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>1 f, f\<^sub>0g\<^sub>1.p\<^sub>1, fg\<^sub>0h\<^sub>1.p\<^sub>1] = \<a>[f, g, h \<star> tab\<^sub>0 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0] \<cdot> ((f \<star> g) \<star> \<a>[h, tab\<^sub>0 h, fg\<^sub>0h\<^sub>1.p\<^sub>0]) \<cdot> ((f \<star> g) \<star> h.tab \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0) \<cdot> ((f \<star> g) \<star> fg\<^sub>0h\<^sub>1.\<phi>) \<cdot> \<a>\<^sup>-\<^sup>1[f, g, (tab\<^sub>0 g \<star> f\<^sub>0g\<^sub>1.p\<^sub>0) \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1] \<cdot> (f \<star> \<a>[g, tab\<^sub>0 g \<star> f\<^sub>0g\<^sub>1.p\<^sub>0, fg\<^sub>0h\<^sub>1.p\<^sub>1]) \<cdot> (f \<star> \<a>[g, tab\<^sub>0 g, f\<^sub>0g\<^sub>1.p\<^sub>0] \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> \<a>[f, (g \<star> tab\<^sub>0 g) \<star> f\<^sub>0g\<^sub>1.p\<^sub>0, fg\<^sub>0h\<^sub>1.p\<^sub>1] \<cdot> ((f \<star> \<a>\<^sup>-\<^sup>1[g, tab\<^sub>0 g, f\<^sub>0g\<^sub>1.p\<^sub>0]) \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> (((\<a>[f, g, tab\<^sub>0 g \<star> f\<^sub>0g\<^sub>1.p\<^sub>0] \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> (\<a>\<^sup>-\<^sup>1[f, g, tab\<^sub>0 g \<star> f\<^sub>0g\<^sub>1.p\<^sub>0] \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1)) \<cdot> ((f \<star> \<a>[g, tab\<^sub>0 g, f\<^sub>0g\<^sub>1.p\<^sub>0]) \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1)) \<cdot> ((f \<star> g.tab \<star> f\<^sub>0g\<^sub>1.p\<^sub>0) \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> ((f \<star> f\<^sub>0g\<^sub>1.\<phi>) \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> (\<a>[f, tab\<^sub>0 f, f\<^sub>0g\<^sub>1.p\<^sub>1] \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> ((f.tab \<star> f\<^sub>0g\<^sub>1.p\<^sub>1) \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>1 f, f\<^sub>0g\<^sub>1.p\<^sub>1, fg\<^sub>0h\<^sub>1.p\<^sub>1]
[PROOF STEP]
thus ?thesis
[PROOF STATE]
proof (prove)
using this:
((\<a>[f, g, tab\<^sub>0 g \<star> f\<^sub>0g\<^sub>1.p\<^sub>0] \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> (\<a>\<^sup>-\<^sup>1[f, g, tab\<^sub>0 g \<star> f\<^sub>0g\<^sub>1.p\<^sub>0] \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1)) \<cdot> ((f \<star> \<a>[g, tab\<^sub>0 g, f\<^sub>0g\<^sub>1.p\<^sub>0]) \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) = (f \<star> \<a>[g, tab\<^sub>0 g, f\<^sub>0g\<^sub>1.p\<^sub>0]) \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1
goal (1 subgoal):
1. \<a>[f, g, h \<star> tab\<^sub>0 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0] \<cdot> ((f \<star> g) \<star> \<a>[h, tab\<^sub>0 h, fg\<^sub>0h\<^sub>1.p\<^sub>0]) \<cdot> ((f \<star> g) \<star> h.tab \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0) \<cdot> ((f \<star> g) \<star> fg\<^sub>0h\<^sub>1.\<phi>) \<cdot> \<a>\<^sup>-\<^sup>1[f, g, (tab\<^sub>0 g \<star> f\<^sub>0g\<^sub>1.p\<^sub>0) \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1] \<cdot> (f \<star> \<a>[g, tab\<^sub>0 g \<star> f\<^sub>0g\<^sub>1.p\<^sub>0, fg\<^sub>0h\<^sub>1.p\<^sub>1]) \<cdot> (f \<star> \<a>[g, tab\<^sub>0 g, f\<^sub>0g\<^sub>1.p\<^sub>0] \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> \<a>[f, (g \<star> tab\<^sub>0 g) \<star> f\<^sub>0g\<^sub>1.p\<^sub>0, fg\<^sub>0h\<^sub>1.p\<^sub>1] \<cdot> ((f \<star> \<a>\<^sup>-\<^sup>1[g, tab\<^sub>0 g, f\<^sub>0g\<^sub>1.p\<^sub>0]) \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> ((f \<star> \<a>[g, tab\<^sub>0 g, f\<^sub>0g\<^sub>1.p\<^sub>0]) \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> ((f \<star> g.tab \<star> f\<^sub>0g\<^sub>1.p\<^sub>0) \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> ((f \<star> f\<^sub>0g\<^sub>1.\<phi>) \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> (\<a>[f, tab\<^sub>0 f, f\<^sub>0g\<^sub>1.p\<^sub>1] \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> ((f.tab \<star> f\<^sub>0g\<^sub>1.p\<^sub>1) \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>1 f, f\<^sub>0g\<^sub>1.p\<^sub>1, fg\<^sub>0h\<^sub>1.p\<^sub>1] = \<a>[f, g, h \<star> tab\<^sub>0 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0] \<cdot> ((f \<star> g) \<star> \<a>[h, tab\<^sub>0 h, fg\<^sub>0h\<^sub>1.p\<^sub>0]) \<cdot> ((f \<star> g) \<star> h.tab \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0) \<cdot> ((f \<star> g) \<star> fg\<^sub>0h\<^sub>1.\<phi>) \<cdot> \<a>\<^sup>-\<^sup>1[f, g, (tab\<^sub>0 g \<star> f\<^sub>0g\<^sub>1.p\<^sub>0) \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1] \<cdot> (f \<star> \<a>[g, tab\<^sub>0 g \<star> f\<^sub>0g\<^sub>1.p\<^sub>0, fg\<^sub>0h\<^sub>1.p\<^sub>1]) \<cdot> (f \<star> \<a>[g, tab\<^sub>0 g, f\<^sub>0g\<^sub>1.p\<^sub>0] \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> \<a>[f, (g \<star> tab\<^sub>0 g) \<star> f\<^sub>0g\<^sub>1.p\<^sub>0, fg\<^sub>0h\<^sub>1.p\<^sub>1] \<cdot> ((f \<star> \<a>\<^sup>-\<^sup>1[g, tab\<^sub>0 g, f\<^sub>0g\<^sub>1.p\<^sub>0]) \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> (((\<a>[f, g, tab\<^sub>0 g \<star> f\<^sub>0g\<^sub>1.p\<^sub>0] \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> (\<a>\<^sup>-\<^sup>1[f, g, tab\<^sub>0 g \<star> f\<^sub>0g\<^sub>1.p\<^sub>0] \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1)) \<cdot> ((f \<star> \<a>[g, tab\<^sub>0 g, f\<^sub>0g\<^sub>1.p\<^sub>0]) \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1)) \<cdot> ((f \<star> g.tab \<star> f\<^sub>0g\<^sub>1.p\<^sub>0) \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> ((f \<star> f\<^sub>0g\<^sub>1.\<phi>) \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> (\<a>[f, tab\<^sub>0 f, f\<^sub>0g\<^sub>1.p\<^sub>1] \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> ((f.tab \<star> f\<^sub>0g\<^sub>1.p\<^sub>1) \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>1 f, f\<^sub>0g\<^sub>1.p\<^sub>1, fg\<^sub>0h\<^sub>1.p\<^sub>1]
[PROOF STEP]
using comp_assoc
[PROOF STATE]
proof (prove)
using this:
((\<a>[f, g, tab\<^sub>0 g \<star> f\<^sub>0g\<^sub>1.p\<^sub>0] \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> (\<a>\<^sup>-\<^sup>1[f, g, tab\<^sub>0 g \<star> f\<^sub>0g\<^sub>1.p\<^sub>0] \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1)) \<cdot> ((f \<star> \<a>[g, tab\<^sub>0 g, f\<^sub>0g\<^sub>1.p\<^sub>0]) \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) = (f \<star> \<a>[g, tab\<^sub>0 g, f\<^sub>0g\<^sub>1.p\<^sub>0]) \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1
(?h \<cdot> ?g) \<cdot> ?f = ?h \<cdot> ?g \<cdot> ?f
goal (1 subgoal):
1. \<a>[f, g, h \<star> tab\<^sub>0 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0] \<cdot> ((f \<star> g) \<star> \<a>[h, tab\<^sub>0 h, fg\<^sub>0h\<^sub>1.p\<^sub>0]) \<cdot> ((f \<star> g) \<star> h.tab \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0) \<cdot> ((f \<star> g) \<star> fg\<^sub>0h\<^sub>1.\<phi>) \<cdot> \<a>\<^sup>-\<^sup>1[f, g, (tab\<^sub>0 g \<star> f\<^sub>0g\<^sub>1.p\<^sub>0) \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1] \<cdot> (f \<star> \<a>[g, tab\<^sub>0 g \<star> f\<^sub>0g\<^sub>1.p\<^sub>0, fg\<^sub>0h\<^sub>1.p\<^sub>1]) \<cdot> (f \<star> \<a>[g, tab\<^sub>0 g, f\<^sub>0g\<^sub>1.p\<^sub>0] \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> \<a>[f, (g \<star> tab\<^sub>0 g) \<star> f\<^sub>0g\<^sub>1.p\<^sub>0, fg\<^sub>0h\<^sub>1.p\<^sub>1] \<cdot> ((f \<star> \<a>\<^sup>-\<^sup>1[g, tab\<^sub>0 g, f\<^sub>0g\<^sub>1.p\<^sub>0]) \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> ((f \<star> \<a>[g, tab\<^sub>0 g, f\<^sub>0g\<^sub>1.p\<^sub>0]) \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> ((f \<star> g.tab \<star> f\<^sub>0g\<^sub>1.p\<^sub>0) \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> ((f \<star> f\<^sub>0g\<^sub>1.\<phi>) \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> (\<a>[f, tab\<^sub>0 f, f\<^sub>0g\<^sub>1.p\<^sub>1] \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> ((f.tab \<star> f\<^sub>0g\<^sub>1.p\<^sub>1) \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>1 f, f\<^sub>0g\<^sub>1.p\<^sub>1, fg\<^sub>0h\<^sub>1.p\<^sub>1] = \<a>[f, g, h \<star> tab\<^sub>0 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0] \<cdot> ((f \<star> g) \<star> \<a>[h, tab\<^sub>0 h, fg\<^sub>0h\<^sub>1.p\<^sub>0]) \<cdot> ((f \<star> g) \<star> h.tab \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0) \<cdot> ((f \<star> g) \<star> fg\<^sub>0h\<^sub>1.\<phi>) \<cdot> \<a>\<^sup>-\<^sup>1[f, g, (tab\<^sub>0 g \<star> f\<^sub>0g\<^sub>1.p\<^sub>0) \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1] \<cdot> (f \<star> \<a>[g, tab\<^sub>0 g \<star> f\<^sub>0g\<^sub>1.p\<^sub>0, fg\<^sub>0h\<^sub>1.p\<^sub>1]) \<cdot> (f \<star> \<a>[g, tab\<^sub>0 g, f\<^sub>0g\<^sub>1.p\<^sub>0] \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> \<a>[f, (g \<star> tab\<^sub>0 g) \<star> f\<^sub>0g\<^sub>1.p\<^sub>0, fg\<^sub>0h\<^sub>1.p\<^sub>1] \<cdot> ((f \<star> \<a>\<^sup>-\<^sup>1[g, tab\<^sub>0 g, f\<^sub>0g\<^sub>1.p\<^sub>0]) \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> (((\<a>[f, g, tab\<^sub>0 g \<star> f\<^sub>0g\<^sub>1.p\<^sub>0] \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> (\<a>\<^sup>-\<^sup>1[f, g, tab\<^sub>0 g \<star> f\<^sub>0g\<^sub>1.p\<^sub>0] \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1)) \<cdot> ((f \<star> \<a>[g, tab\<^sub>0 g, f\<^sub>0g\<^sub>1.p\<^sub>0]) \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1)) \<cdot> ((f \<star> g.tab \<star> f\<^sub>0g\<^sub>1.p\<^sub>0) \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> ((f \<star> f\<^sub>0g\<^sub>1.\<phi>) \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> (\<a>[f, tab\<^sub>0 f, f\<^sub>0g\<^sub>1.p\<^sub>1] \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> ((f.tab \<star> f\<^sub>0g\<^sub>1.p\<^sub>1) \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>1 f, f\<^sub>0g\<^sub>1.p\<^sub>1, fg\<^sub>0h\<^sub>1.p\<^sub>1]
[PROOF STEP]
by simp
[PROOF STATE]
proof (state)
this:
\<a>[f, g, h \<star> tab\<^sub>0 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0] \<cdot> ((f \<star> g) \<star> \<a>[h, tab\<^sub>0 h, fg\<^sub>0h\<^sub>1.p\<^sub>0]) \<cdot> ((f \<star> g) \<star> h.tab \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0) \<cdot> ((f \<star> g) \<star> fg\<^sub>0h\<^sub>1.\<phi>) \<cdot> \<a>\<^sup>-\<^sup>1[f, g, (tab\<^sub>0 g \<star> f\<^sub>0g\<^sub>1.p\<^sub>0) \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1] \<cdot> (f \<star> \<a>[g, tab\<^sub>0 g \<star> f\<^sub>0g\<^sub>1.p\<^sub>0, fg\<^sub>0h\<^sub>1.p\<^sub>1]) \<cdot> (f \<star> \<a>[g, tab\<^sub>0 g, f\<^sub>0g\<^sub>1.p\<^sub>0] \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> \<a>[f, (g \<star> tab\<^sub>0 g) \<star> f\<^sub>0g\<^sub>1.p\<^sub>0, fg\<^sub>0h\<^sub>1.p\<^sub>1] \<cdot> ((f \<star> \<a>\<^sup>-\<^sup>1[g, tab\<^sub>0 g, f\<^sub>0g\<^sub>1.p\<^sub>0]) \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> ((f \<star> \<a>[g, tab\<^sub>0 g, f\<^sub>0g\<^sub>1.p\<^sub>0]) \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> ((f \<star> g.tab \<star> f\<^sub>0g\<^sub>1.p\<^sub>0) \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> ((f \<star> f\<^sub>0g\<^sub>1.\<phi>) \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> (\<a>[f, tab\<^sub>0 f, f\<^sub>0g\<^sub>1.p\<^sub>1] \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> ((f.tab \<star> f\<^sub>0g\<^sub>1.p\<^sub>1) \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>1 f, f\<^sub>0g\<^sub>1.p\<^sub>1, fg\<^sub>0h\<^sub>1.p\<^sub>1] = \<a>[f, g, h \<star> tab\<^sub>0 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0] \<cdot> ((f \<star> g) \<star> \<a>[h, tab\<^sub>0 h, fg\<^sub>0h\<^sub>1.p\<^sub>0]) \<cdot> ((f \<star> g) \<star> h.tab \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0) \<cdot> ((f \<star> g) \<star> fg\<^sub>0h\<^sub>1.\<phi>) \<cdot> \<a>\<^sup>-\<^sup>1[f, g, (tab\<^sub>0 g \<star> f\<^sub>0g\<^sub>1.p\<^sub>0) \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1] \<cdot> (f \<star> \<a>[g, tab\<^sub>0 g \<star> f\<^sub>0g\<^sub>1.p\<^sub>0, fg\<^sub>0h\<^sub>1.p\<^sub>1]) \<cdot> (f \<star> \<a>[g, tab\<^sub>0 g, f\<^sub>0g\<^sub>1.p\<^sub>0] \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> \<a>[f, (g \<star> tab\<^sub>0 g) \<star> f\<^sub>0g\<^sub>1.p\<^sub>0, fg\<^sub>0h\<^sub>1.p\<^sub>1] \<cdot> ((f \<star> \<a>\<^sup>-\<^sup>1[g, tab\<^sub>0 g, f\<^sub>0g\<^sub>1.p\<^sub>0]) \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> (((\<a>[f, g, tab\<^sub>0 g \<star> f\<^sub>0g\<^sub>1.p\<^sub>0] \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> (\<a>\<^sup>-\<^sup>1[f, g, tab\<^sub>0 g \<star> f\<^sub>0g\<^sub>1.p\<^sub>0] \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1)) \<cdot> ((f \<star> \<a>[g, tab\<^sub>0 g, f\<^sub>0g\<^sub>1.p\<^sub>0]) \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1)) \<cdot> ((f \<star> g.tab \<star> f\<^sub>0g\<^sub>1.p\<^sub>0) \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> ((f \<star> f\<^sub>0g\<^sub>1.\<phi>) \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> (\<a>[f, tab\<^sub>0 f, f\<^sub>0g\<^sub>1.p\<^sub>1] \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> ((f.tab \<star> f\<^sub>0g\<^sub>1.p\<^sub>1) \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>1 f, f\<^sub>0g\<^sub>1.p\<^sub>1, fg\<^sub>0h\<^sub>1.p\<^sub>1]
goal:
No subgoals!
[PROOF STEP]
qed
[PROOF STATE]
proof (state)
this:
\<a>[f, g, h \<star> tab\<^sub>0 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0] \<cdot> ((f \<star> g) \<star> \<a>[h, tab\<^sub>0 h, fg\<^sub>0h\<^sub>1.p\<^sub>0]) \<cdot> ((f \<star> g) \<star> h.tab \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0) \<cdot> ((f \<star> g) \<star> fg\<^sub>0h\<^sub>1.\<phi>) \<cdot> \<a>\<^sup>-\<^sup>1[f, g, (tab\<^sub>0 g \<star> f\<^sub>0g\<^sub>1.p\<^sub>0) \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1] \<cdot> (f \<star> \<a>[g, tab\<^sub>0 g \<star> f\<^sub>0g\<^sub>1.p\<^sub>0, fg\<^sub>0h\<^sub>1.p\<^sub>1]) \<cdot> (f \<star> \<a>[g, tab\<^sub>0 g, f\<^sub>0g\<^sub>1.p\<^sub>0] \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> \<a>[f, (g \<star> tab\<^sub>0 g) \<star> f\<^sub>0g\<^sub>1.p\<^sub>0, fg\<^sub>0h\<^sub>1.p\<^sub>1] \<cdot> ((f \<star> \<a>\<^sup>-\<^sup>1[g, tab\<^sub>0 g, f\<^sub>0g\<^sub>1.p\<^sub>0]) \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> ((f \<star> \<a>[g, tab\<^sub>0 g, f\<^sub>0g\<^sub>1.p\<^sub>0]) \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> ((f \<star> g.tab \<star> f\<^sub>0g\<^sub>1.p\<^sub>0) \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> ((f \<star> f\<^sub>0g\<^sub>1.\<phi>) \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> (\<a>[f, tab\<^sub>0 f, f\<^sub>0g\<^sub>1.p\<^sub>1] \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> ((f.tab \<star> f\<^sub>0g\<^sub>1.p\<^sub>1) \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>1 f, f\<^sub>0g\<^sub>1.p\<^sub>1, fg\<^sub>0h\<^sub>1.p\<^sub>1] = \<a>[f, g, h \<star> tab\<^sub>0 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0] \<cdot> ((f \<star> g) \<star> \<a>[h, tab\<^sub>0 h, fg\<^sub>0h\<^sub>1.p\<^sub>0]) \<cdot> ((f \<star> g) \<star> h.tab \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0) \<cdot> ((f \<star> g) \<star> fg\<^sub>0h\<^sub>1.\<phi>) \<cdot> \<a>\<^sup>-\<^sup>1[f, g, (tab\<^sub>0 g \<star> f\<^sub>0g\<^sub>1.p\<^sub>0) \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1] \<cdot> (f \<star> \<a>[g, tab\<^sub>0 g \<star> f\<^sub>0g\<^sub>1.p\<^sub>0, fg\<^sub>0h\<^sub>1.p\<^sub>1]) \<cdot> (f \<star> \<a>[g, tab\<^sub>0 g, f\<^sub>0g\<^sub>1.p\<^sub>0] \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> \<a>[f, (g \<star> tab\<^sub>0 g) \<star> f\<^sub>0g\<^sub>1.p\<^sub>0, fg\<^sub>0h\<^sub>1.p\<^sub>1] \<cdot> ((f \<star> \<a>\<^sup>-\<^sup>1[g, tab\<^sub>0 g, f\<^sub>0g\<^sub>1.p\<^sub>0]) \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> (((\<a>[f, g, tab\<^sub>0 g \<star> f\<^sub>0g\<^sub>1.p\<^sub>0] \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> (\<a>\<^sup>-\<^sup>1[f, g, tab\<^sub>0 g \<star> f\<^sub>0g\<^sub>1.p\<^sub>0] \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1)) \<cdot> ((f \<star> \<a>[g, tab\<^sub>0 g, f\<^sub>0g\<^sub>1.p\<^sub>0]) \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1)) \<cdot> ((f \<star> g.tab \<star> f\<^sub>0g\<^sub>1.p\<^sub>0) \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> ((f \<star> f\<^sub>0g\<^sub>1.\<phi>) \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> (\<a>[f, tab\<^sub>0 f, f\<^sub>0g\<^sub>1.p\<^sub>1] \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> ((f.tab \<star> f\<^sub>0g\<^sub>1.p\<^sub>1) \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>1 f, f\<^sub>0g\<^sub>1.p\<^sub>1, fg\<^sub>0h\<^sub>1.p\<^sub>1]
goal (1 subgoal):
1. (f \<star> \<theta>\<^sub>f) \<cdot> \<a>[f, tab\<^sub>0 f, w\<^sub>f] \<cdot> (f.tab \<star> w\<^sub>f) = ((f \<star> \<theta>\<^sub>f') \<cdot> \<a>[f, tab\<^sub>0 f, w\<^sub>f'] \<cdot> (f.tab \<star> w\<^sub>f')) \<cdot> \<beta>\<^sub>f
[PROOF STEP]
also
[PROOF STATE]
proof (state)
this:
\<a>[f, g, h \<star> tab\<^sub>0 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0] \<cdot> ((f \<star> g) \<star> \<a>[h, tab\<^sub>0 h, fg\<^sub>0h\<^sub>1.p\<^sub>0]) \<cdot> ((f \<star> g) \<star> h.tab \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0) \<cdot> ((f \<star> g) \<star> fg\<^sub>0h\<^sub>1.\<phi>) \<cdot> \<a>\<^sup>-\<^sup>1[f, g, (tab\<^sub>0 g \<star> f\<^sub>0g\<^sub>1.p\<^sub>0) \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1] \<cdot> (f \<star> \<a>[g, tab\<^sub>0 g \<star> f\<^sub>0g\<^sub>1.p\<^sub>0, fg\<^sub>0h\<^sub>1.p\<^sub>1]) \<cdot> (f \<star> \<a>[g, tab\<^sub>0 g, f\<^sub>0g\<^sub>1.p\<^sub>0] \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> \<a>[f, (g \<star> tab\<^sub>0 g) \<star> f\<^sub>0g\<^sub>1.p\<^sub>0, fg\<^sub>0h\<^sub>1.p\<^sub>1] \<cdot> ((f \<star> \<a>\<^sup>-\<^sup>1[g, tab\<^sub>0 g, f\<^sub>0g\<^sub>1.p\<^sub>0]) \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> ((f \<star> \<a>[g, tab\<^sub>0 g, f\<^sub>0g\<^sub>1.p\<^sub>0]) \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> ((f \<star> g.tab \<star> f\<^sub>0g\<^sub>1.p\<^sub>0) \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> ((f \<star> f\<^sub>0g\<^sub>1.\<phi>) \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> (\<a>[f, tab\<^sub>0 f, f\<^sub>0g\<^sub>1.p\<^sub>1] \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> ((f.tab \<star> f\<^sub>0g\<^sub>1.p\<^sub>1) \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>1 f, f\<^sub>0g\<^sub>1.p\<^sub>1, fg\<^sub>0h\<^sub>1.p\<^sub>1] = \<a>[f, g, h \<star> tab\<^sub>0 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0] \<cdot> ((f \<star> g) \<star> \<a>[h, tab\<^sub>0 h, fg\<^sub>0h\<^sub>1.p\<^sub>0]) \<cdot> ((f \<star> g) \<star> h.tab \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0) \<cdot> ((f \<star> g) \<star> fg\<^sub>0h\<^sub>1.\<phi>) \<cdot> \<a>\<^sup>-\<^sup>1[f, g, (tab\<^sub>0 g \<star> f\<^sub>0g\<^sub>1.p\<^sub>0) \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1] \<cdot> (f \<star> \<a>[g, tab\<^sub>0 g \<star> f\<^sub>0g\<^sub>1.p\<^sub>0, fg\<^sub>0h\<^sub>1.p\<^sub>1]) \<cdot> (f \<star> \<a>[g, tab\<^sub>0 g, f\<^sub>0g\<^sub>1.p\<^sub>0] \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> \<a>[f, (g \<star> tab\<^sub>0 g) \<star> f\<^sub>0g\<^sub>1.p\<^sub>0, fg\<^sub>0h\<^sub>1.p\<^sub>1] \<cdot> ((f \<star> \<a>\<^sup>-\<^sup>1[g, tab\<^sub>0 g, f\<^sub>0g\<^sub>1.p\<^sub>0]) \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> (((\<a>[f, g, tab\<^sub>0 g \<star> f\<^sub>0g\<^sub>1.p\<^sub>0] \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> (\<a>\<^sup>-\<^sup>1[f, g, tab\<^sub>0 g \<star> f\<^sub>0g\<^sub>1.p\<^sub>0] \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1)) \<cdot> ((f \<star> \<a>[g, tab\<^sub>0 g, f\<^sub>0g\<^sub>1.p\<^sub>0]) \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1)) \<cdot> ((f \<star> g.tab \<star> f\<^sub>0g\<^sub>1.p\<^sub>0) \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> ((f \<star> f\<^sub>0g\<^sub>1.\<phi>) \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> (\<a>[f, tab\<^sub>0 f, f\<^sub>0g\<^sub>1.p\<^sub>1] \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> ((f.tab \<star> f\<^sub>0g\<^sub>1.p\<^sub>1) \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>1 f, f\<^sub>0g\<^sub>1.p\<^sub>1, fg\<^sub>0h\<^sub>1.p\<^sub>1]
goal (1 subgoal):
1. (f \<star> \<theta>\<^sub>f) \<cdot> \<a>[f, tab\<^sub>0 f, w\<^sub>f] \<cdot> (f.tab \<star> w\<^sub>f) = ((f \<star> \<theta>\<^sub>f') \<cdot> \<a>[f, tab\<^sub>0 f, w\<^sub>f'] \<cdot> (f.tab \<star> w\<^sub>f')) \<cdot> \<beta>\<^sub>f
[PROOF STEP]
have "... =
\<a>[f, g, h \<star> tab\<^sub>0 h \<star> TTfgh.p\<^sub>0] \<cdot>
((f \<star> g) \<star> \<a>[h, tab\<^sub>0 h, TTfgh.p\<^sub>0]) \<cdot>
((f \<star> g) \<star> h.tab \<star> TTfgh.p\<^sub>0) \<cdot>
((f \<star> g) \<star> fg\<^sub>0h\<^sub>1.\<phi>) \<cdot>
\<a>\<^sup>-\<^sup>1[f, g, (tab\<^sub>0 g \<star> Tfg.p\<^sub>0) \<star> TTfgh.p\<^sub>1] \<cdot>
(f \<star> \<a>[g, tab\<^sub>0 g \<star> Tfg.p\<^sub>0, TTfgh.p\<^sub>1]) \<cdot>
(f \<star> \<a>[g, tab\<^sub>0 g, Tfg.p\<^sub>0] \<star> TTfgh.p\<^sub>1) \<cdot>
\<a>[f, (g \<star> tab\<^sub>0 g) \<star> Tfg.p\<^sub>0, TTfgh.p\<^sub>1] \<cdot>
((f \<star> \<a>\<^sup>-\<^sup>1[g, tab\<^sub>0 g, Tfg.p\<^sub>0]) \<star> TTfgh.p\<^sub>1) \<cdot>
(\<a>[f, g, tab\<^sub>0 g \<star> Tfg.p\<^sub>0] \<star> TTfgh.p\<^sub>1) \<cdot>
((\<a>\<^sup>-\<^sup>1[f, g, tab\<^sub>0 g \<star> Tfg.p\<^sub>0] \<star> TTfgh.p\<^sub>1) \<cdot>
((f \<star> \<a>[g, tab\<^sub>0 g, Tfg.p\<^sub>0]) \<star> TTfgh.p\<^sub>1) \<cdot>
((f \<star> (g.tab \<star> Tfg.p\<^sub>0)) \<star> TTfgh.p\<^sub>1) \<cdot>
((f \<star> f\<^sub>0g\<^sub>1.\<phi>) \<star> TTfgh.p\<^sub>1) \<cdot>
(\<a>[f, tab\<^sub>0 f, Tfg.p\<^sub>1] \<star> TTfgh.p\<^sub>1) \<cdot>
((f.tab \<star> Tfg.p\<^sub>1) \<star> TTfgh.p\<^sub>1)) \<cdot>
\<a>\<^sup>-\<^sup>1[tab\<^sub>1 f, Tfg.p\<^sub>1, TTfgh.p\<^sub>1]"
[PROOF STATE]
proof (prove)
goal (1 subgoal):
1. \<a>[f, g, h \<star> tab\<^sub>0 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0] \<cdot> ((f \<star> g) \<star> \<a>[h, tab\<^sub>0 h, fg\<^sub>0h\<^sub>1.p\<^sub>0]) \<cdot> ((f \<star> g) \<star> h.tab \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0) \<cdot> ((f \<star> g) \<star> fg\<^sub>0h\<^sub>1.\<phi>) \<cdot> \<a>\<^sup>-\<^sup>1[f, g, (tab\<^sub>0 g \<star> f\<^sub>0g\<^sub>1.p\<^sub>0) \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1] \<cdot> (f \<star> \<a>[g, tab\<^sub>0 g \<star> f\<^sub>0g\<^sub>1.p\<^sub>0, fg\<^sub>0h\<^sub>1.p\<^sub>1]) \<cdot> (f \<star> \<a>[g, tab\<^sub>0 g, f\<^sub>0g\<^sub>1.p\<^sub>0] \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> \<a>[f, (g \<star> tab\<^sub>0 g) \<star> f\<^sub>0g\<^sub>1.p\<^sub>0, fg\<^sub>0h\<^sub>1.p\<^sub>1] \<cdot> ((f \<star> \<a>\<^sup>-\<^sup>1[g, tab\<^sub>0 g, f\<^sub>0g\<^sub>1.p\<^sub>0]) \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> (((\<a>[f, g, tab\<^sub>0 g \<star> f\<^sub>0g\<^sub>1.p\<^sub>0] \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> (\<a>\<^sup>-\<^sup>1[f, g, tab\<^sub>0 g \<star> f\<^sub>0g\<^sub>1.p\<^sub>0] \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1)) \<cdot> ((f \<star> \<a>[g, tab\<^sub>0 g, f\<^sub>0g\<^sub>1.p\<^sub>0]) \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1)) \<cdot> ((f \<star> g.tab \<star> f\<^sub>0g\<^sub>1.p\<^sub>0) \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> ((f \<star> f\<^sub>0g\<^sub>1.\<phi>) \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> (\<a>[f, tab\<^sub>0 f, f\<^sub>0g\<^sub>1.p\<^sub>1] \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> ((f.tab \<star> f\<^sub>0g\<^sub>1.p\<^sub>1) \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>1 f, f\<^sub>0g\<^sub>1.p\<^sub>1, fg\<^sub>0h\<^sub>1.p\<^sub>1] = \<a>[f, g, h \<star> tab\<^sub>0 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0] \<cdot> ((f \<star> g) \<star> \<a>[h, tab\<^sub>0 h, fg\<^sub>0h\<^sub>1.p\<^sub>0]) \<cdot> ((f \<star> g) \<star> h.tab \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0) \<cdot> ((f \<star> g) \<star> fg\<^sub>0h\<^sub>1.\<phi>) \<cdot> \<a>\<^sup>-\<^sup>1[f, g, (tab\<^sub>0 g \<star> f\<^sub>0g\<^sub>1.p\<^sub>0) \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1] \<cdot> (f \<star> \<a>[g, tab\<^sub>0 g \<star> f\<^sub>0g\<^sub>1.p\<^sub>0, fg\<^sub>0h\<^sub>1.p\<^sub>1]) \<cdot> (f \<star> \<a>[g, tab\<^sub>0 g, f\<^sub>0g\<^sub>1.p\<^sub>0] \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> \<a>[f, (g \<star> tab\<^sub>0 g) \<star> f\<^sub>0g\<^sub>1.p\<^sub>0, fg\<^sub>0h\<^sub>1.p\<^sub>1] \<cdot> ((f \<star> \<a>\<^sup>-\<^sup>1[g, tab\<^sub>0 g, f\<^sub>0g\<^sub>1.p\<^sub>0]) \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> (\<a>[f, g, tab\<^sub>0 g \<star> f\<^sub>0g\<^sub>1.p\<^sub>0] \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> ((\<a>\<^sup>-\<^sup>1[f, g, tab\<^sub>0 g \<star> f\<^sub>0g\<^sub>1.p\<^sub>0] \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> ((f \<star> \<a>[g, tab\<^sub>0 g, f\<^sub>0g\<^sub>1.p\<^sub>0]) \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> ((f \<star> g.tab \<star> f\<^sub>0g\<^sub>1.p\<^sub>0) \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> ((f \<star> f\<^sub>0g\<^sub>1.\<phi>) \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> (\<a>[f, tab\<^sub>0 f, f\<^sub>0g\<^sub>1.p\<^sub>1] \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> ((f.tab \<star> f\<^sub>0g\<^sub>1.p\<^sub>1) \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1)) \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>1 f, f\<^sub>0g\<^sub>1.p\<^sub>1, fg\<^sub>0h\<^sub>1.p\<^sub>1]
[PROOF STEP]
using comp_assoc
[PROOF STATE]
proof (prove)
using this:
(?h \<cdot> ?g) \<cdot> ?f = ?h \<cdot> ?g \<cdot> ?f
goal (1 subgoal):
1. \<a>[f, g, h \<star> tab\<^sub>0 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0] \<cdot> ((f \<star> g) \<star> \<a>[h, tab\<^sub>0 h, fg\<^sub>0h\<^sub>1.p\<^sub>0]) \<cdot> ((f \<star> g) \<star> h.tab \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0) \<cdot> ((f \<star> g) \<star> fg\<^sub>0h\<^sub>1.\<phi>) \<cdot> \<a>\<^sup>-\<^sup>1[f, g, (tab\<^sub>0 g \<star> f\<^sub>0g\<^sub>1.p\<^sub>0) \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1] \<cdot> (f \<star> \<a>[g, tab\<^sub>0 g \<star> f\<^sub>0g\<^sub>1.p\<^sub>0, fg\<^sub>0h\<^sub>1.p\<^sub>1]) \<cdot> (f \<star> \<a>[g, tab\<^sub>0 g, f\<^sub>0g\<^sub>1.p\<^sub>0] \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> \<a>[f, (g \<star> tab\<^sub>0 g) \<star> f\<^sub>0g\<^sub>1.p\<^sub>0, fg\<^sub>0h\<^sub>1.p\<^sub>1] \<cdot> ((f \<star> \<a>\<^sup>-\<^sup>1[g, tab\<^sub>0 g, f\<^sub>0g\<^sub>1.p\<^sub>0]) \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> (((\<a>[f, g, tab\<^sub>0 g \<star> f\<^sub>0g\<^sub>1.p\<^sub>0] \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> (\<a>\<^sup>-\<^sup>1[f, g, tab\<^sub>0 g \<star> f\<^sub>0g\<^sub>1.p\<^sub>0] \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1)) \<cdot> ((f \<star> \<a>[g, tab\<^sub>0 g, f\<^sub>0g\<^sub>1.p\<^sub>0]) \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1)) \<cdot> ((f \<star> g.tab \<star> f\<^sub>0g\<^sub>1.p\<^sub>0) \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> ((f \<star> f\<^sub>0g\<^sub>1.\<phi>) \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> (\<a>[f, tab\<^sub>0 f, f\<^sub>0g\<^sub>1.p\<^sub>1] \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> ((f.tab \<star> f\<^sub>0g\<^sub>1.p\<^sub>1) \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>1 f, f\<^sub>0g\<^sub>1.p\<^sub>1, fg\<^sub>0h\<^sub>1.p\<^sub>1] = \<a>[f, g, h \<star> tab\<^sub>0 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0] \<cdot> ((f \<star> g) \<star> \<a>[h, tab\<^sub>0 h, fg\<^sub>0h\<^sub>1.p\<^sub>0]) \<cdot> ((f \<star> g) \<star> h.tab \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0) \<cdot> ((f \<star> g) \<star> fg\<^sub>0h\<^sub>1.\<phi>) \<cdot> \<a>\<^sup>-\<^sup>1[f, g, (tab\<^sub>0 g \<star> f\<^sub>0g\<^sub>1.p\<^sub>0) \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1] \<cdot> (f \<star> \<a>[g, tab\<^sub>0 g \<star> f\<^sub>0g\<^sub>1.p\<^sub>0, fg\<^sub>0h\<^sub>1.p\<^sub>1]) \<cdot> (f \<star> \<a>[g, tab\<^sub>0 g, f\<^sub>0g\<^sub>1.p\<^sub>0] \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> \<a>[f, (g \<star> tab\<^sub>0 g) \<star> f\<^sub>0g\<^sub>1.p\<^sub>0, fg\<^sub>0h\<^sub>1.p\<^sub>1] \<cdot> ((f \<star> \<a>\<^sup>-\<^sup>1[g, tab\<^sub>0 g, f\<^sub>0g\<^sub>1.p\<^sub>0]) \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> (\<a>[f, g, tab\<^sub>0 g \<star> f\<^sub>0g\<^sub>1.p\<^sub>0] \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> ((\<a>\<^sup>-\<^sup>1[f, g, tab\<^sub>0 g \<star> f\<^sub>0g\<^sub>1.p\<^sub>0] \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> ((f \<star> \<a>[g, tab\<^sub>0 g, f\<^sub>0g\<^sub>1.p\<^sub>0]) \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> ((f \<star> g.tab \<star> f\<^sub>0g\<^sub>1.p\<^sub>0) \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> ((f \<star> f\<^sub>0g\<^sub>1.\<phi>) \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> (\<a>[f, tab\<^sub>0 f, f\<^sub>0g\<^sub>1.p\<^sub>1] \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> ((f.tab \<star> f\<^sub>0g\<^sub>1.p\<^sub>1) \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1)) \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>1 f, f\<^sub>0g\<^sub>1.p\<^sub>1, fg\<^sub>0h\<^sub>1.p\<^sub>1]
[PROOF STEP]
by presburger
[PROOF STATE]
proof (state)
this:
\<a>[f, g, h \<star> tab\<^sub>0 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0] \<cdot> ((f \<star> g) \<star> \<a>[h, tab\<^sub>0 h, fg\<^sub>0h\<^sub>1.p\<^sub>0]) \<cdot> ((f \<star> g) \<star> h.tab \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0) \<cdot> ((f \<star> g) \<star> fg\<^sub>0h\<^sub>1.\<phi>) \<cdot> \<a>\<^sup>-\<^sup>1[f, g, (tab\<^sub>0 g \<star> f\<^sub>0g\<^sub>1.p\<^sub>0) \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1] \<cdot> (f \<star> \<a>[g, tab\<^sub>0 g \<star> f\<^sub>0g\<^sub>1.p\<^sub>0, fg\<^sub>0h\<^sub>1.p\<^sub>1]) \<cdot> (f \<star> \<a>[g, tab\<^sub>0 g, f\<^sub>0g\<^sub>1.p\<^sub>0] \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> \<a>[f, (g \<star> tab\<^sub>0 g) \<star> f\<^sub>0g\<^sub>1.p\<^sub>0, fg\<^sub>0h\<^sub>1.p\<^sub>1] \<cdot> ((f \<star> \<a>\<^sup>-\<^sup>1[g, tab\<^sub>0 g, f\<^sub>0g\<^sub>1.p\<^sub>0]) \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> (((\<a>[f, g, tab\<^sub>0 g \<star> f\<^sub>0g\<^sub>1.p\<^sub>0] \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> (\<a>\<^sup>-\<^sup>1[f, g, tab\<^sub>0 g \<star> f\<^sub>0g\<^sub>1.p\<^sub>0] \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1)) \<cdot> ((f \<star> \<a>[g, tab\<^sub>0 g, f\<^sub>0g\<^sub>1.p\<^sub>0]) \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1)) \<cdot> ((f \<star> g.tab \<star> f\<^sub>0g\<^sub>1.p\<^sub>0) \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> ((f \<star> f\<^sub>0g\<^sub>1.\<phi>) \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> (\<a>[f, tab\<^sub>0 f, f\<^sub>0g\<^sub>1.p\<^sub>1] \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> ((f.tab \<star> f\<^sub>0g\<^sub>1.p\<^sub>1) \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>1 f, f\<^sub>0g\<^sub>1.p\<^sub>1, fg\<^sub>0h\<^sub>1.p\<^sub>1] = \<a>[f, g, h \<star> tab\<^sub>0 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0] \<cdot> ((f \<star> g) \<star> \<a>[h, tab\<^sub>0 h, fg\<^sub>0h\<^sub>1.p\<^sub>0]) \<cdot> ((f \<star> g) \<star> h.tab \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0) \<cdot> ((f \<star> g) \<star> fg\<^sub>0h\<^sub>1.\<phi>) \<cdot> \<a>\<^sup>-\<^sup>1[f, g, (tab\<^sub>0 g \<star> f\<^sub>0g\<^sub>1.p\<^sub>0) \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1] \<cdot> (f \<star> \<a>[g, tab\<^sub>0 g \<star> f\<^sub>0g\<^sub>1.p\<^sub>0, fg\<^sub>0h\<^sub>1.p\<^sub>1]) \<cdot> (f \<star> \<a>[g, tab\<^sub>0 g, f\<^sub>0g\<^sub>1.p\<^sub>0] \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> \<a>[f, (g \<star> tab\<^sub>0 g) \<star> f\<^sub>0g\<^sub>1.p\<^sub>0, fg\<^sub>0h\<^sub>1.p\<^sub>1] \<cdot> ((f \<star> \<a>\<^sup>-\<^sup>1[g, tab\<^sub>0 g, f\<^sub>0g\<^sub>1.p\<^sub>0]) \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> (\<a>[f, g, tab\<^sub>0 g \<star> f\<^sub>0g\<^sub>1.p\<^sub>0] \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> ((\<a>\<^sup>-\<^sup>1[f, g, tab\<^sub>0 g \<star> f\<^sub>0g\<^sub>1.p\<^sub>0] \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> ((f \<star> \<a>[g, tab\<^sub>0 g, f\<^sub>0g\<^sub>1.p\<^sub>0]) \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> ((f \<star> g.tab \<star> f\<^sub>0g\<^sub>1.p\<^sub>0) \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> ((f \<star> f\<^sub>0g\<^sub>1.\<phi>) \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> (\<a>[f, tab\<^sub>0 f, f\<^sub>0g\<^sub>1.p\<^sub>1] \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> ((f.tab \<star> f\<^sub>0g\<^sub>1.p\<^sub>1) \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1)) \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>1 f, f\<^sub>0g\<^sub>1.p\<^sub>1, fg\<^sub>0h\<^sub>1.p\<^sub>1]
goal (1 subgoal):
1. (f \<star> \<theta>\<^sub>f) \<cdot> \<a>[f, tab\<^sub>0 f, w\<^sub>f] \<cdot> (f.tab \<star> w\<^sub>f) = ((f \<star> \<theta>\<^sub>f') \<cdot> \<a>[f, tab\<^sub>0 f, w\<^sub>f'] \<cdot> (f.tab \<star> w\<^sub>f')) \<cdot> \<beta>\<^sub>f
[PROOF STEP]
also
[PROOF STATE]
proof (state)
this:
\<a>[f, g, h \<star> tab\<^sub>0 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0] \<cdot> ((f \<star> g) \<star> \<a>[h, tab\<^sub>0 h, fg\<^sub>0h\<^sub>1.p\<^sub>0]) \<cdot> ((f \<star> g) \<star> h.tab \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0) \<cdot> ((f \<star> g) \<star> fg\<^sub>0h\<^sub>1.\<phi>) \<cdot> \<a>\<^sup>-\<^sup>1[f, g, (tab\<^sub>0 g \<star> f\<^sub>0g\<^sub>1.p\<^sub>0) \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1] \<cdot> (f \<star> \<a>[g, tab\<^sub>0 g \<star> f\<^sub>0g\<^sub>1.p\<^sub>0, fg\<^sub>0h\<^sub>1.p\<^sub>1]) \<cdot> (f \<star> \<a>[g, tab\<^sub>0 g, f\<^sub>0g\<^sub>1.p\<^sub>0] \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> \<a>[f, (g \<star> tab\<^sub>0 g) \<star> f\<^sub>0g\<^sub>1.p\<^sub>0, fg\<^sub>0h\<^sub>1.p\<^sub>1] \<cdot> ((f \<star> \<a>\<^sup>-\<^sup>1[g, tab\<^sub>0 g, f\<^sub>0g\<^sub>1.p\<^sub>0]) \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> (((\<a>[f, g, tab\<^sub>0 g \<star> f\<^sub>0g\<^sub>1.p\<^sub>0] \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> (\<a>\<^sup>-\<^sup>1[f, g, tab\<^sub>0 g \<star> f\<^sub>0g\<^sub>1.p\<^sub>0] \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1)) \<cdot> ((f \<star> \<a>[g, tab\<^sub>0 g, f\<^sub>0g\<^sub>1.p\<^sub>0]) \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1)) \<cdot> ((f \<star> g.tab \<star> f\<^sub>0g\<^sub>1.p\<^sub>0) \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> ((f \<star> f\<^sub>0g\<^sub>1.\<phi>) \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> (\<a>[f, tab\<^sub>0 f, f\<^sub>0g\<^sub>1.p\<^sub>1] \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> ((f.tab \<star> f\<^sub>0g\<^sub>1.p\<^sub>1) \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>1 f, f\<^sub>0g\<^sub>1.p\<^sub>1, fg\<^sub>0h\<^sub>1.p\<^sub>1] = \<a>[f, g, h \<star> tab\<^sub>0 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0] \<cdot> ((f \<star> g) \<star> \<a>[h, tab\<^sub>0 h, fg\<^sub>0h\<^sub>1.p\<^sub>0]) \<cdot> ((f \<star> g) \<star> h.tab \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0) \<cdot> ((f \<star> g) \<star> fg\<^sub>0h\<^sub>1.\<phi>) \<cdot> \<a>\<^sup>-\<^sup>1[f, g, (tab\<^sub>0 g \<star> f\<^sub>0g\<^sub>1.p\<^sub>0) \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1] \<cdot> (f \<star> \<a>[g, tab\<^sub>0 g \<star> f\<^sub>0g\<^sub>1.p\<^sub>0, fg\<^sub>0h\<^sub>1.p\<^sub>1]) \<cdot> (f \<star> \<a>[g, tab\<^sub>0 g, f\<^sub>0g\<^sub>1.p\<^sub>0] \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> \<a>[f, (g \<star> tab\<^sub>0 g) \<star> f\<^sub>0g\<^sub>1.p\<^sub>0, fg\<^sub>0h\<^sub>1.p\<^sub>1] \<cdot> ((f \<star> \<a>\<^sup>-\<^sup>1[g, tab\<^sub>0 g, f\<^sub>0g\<^sub>1.p\<^sub>0]) \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> (\<a>[f, g, tab\<^sub>0 g \<star> f\<^sub>0g\<^sub>1.p\<^sub>0] \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> ((\<a>\<^sup>-\<^sup>1[f, g, tab\<^sub>0 g \<star> f\<^sub>0g\<^sub>1.p\<^sub>0] \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> ((f \<star> \<a>[g, tab\<^sub>0 g, f\<^sub>0g\<^sub>1.p\<^sub>0]) \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> ((f \<star> g.tab \<star> f\<^sub>0g\<^sub>1.p\<^sub>0) \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> ((f \<star> f\<^sub>0g\<^sub>1.\<phi>) \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> (\<a>[f, tab\<^sub>0 f, f\<^sub>0g\<^sub>1.p\<^sub>1] \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> ((f.tab \<star> f\<^sub>0g\<^sub>1.p\<^sub>1) \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1)) \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>1 f, f\<^sub>0g\<^sub>1.p\<^sub>1, fg\<^sub>0h\<^sub>1.p\<^sub>1]
goal (1 subgoal):
1. (f \<star> \<theta>\<^sub>f) \<cdot> \<a>[f, tab\<^sub>0 f, w\<^sub>f] \<cdot> (f.tab \<star> w\<^sub>f) = ((f \<star> \<theta>\<^sub>f') \<cdot> \<a>[f, tab\<^sub>0 f, w\<^sub>f'] \<cdot> (f.tab \<star> w\<^sub>f')) \<cdot> \<beta>\<^sub>f
[PROOF STEP]
have "... =
\<a>[f, g, h \<star> tab\<^sub>0 h \<star> TTfgh.p\<^sub>0] \<cdot>
((f \<star> g) \<star> \<a>[h, tab\<^sub>0 h, TTfgh.p\<^sub>0]) \<cdot>
((f \<star> g) \<star> h.tab \<star> TTfgh.p\<^sub>0) \<cdot>
((f \<star> g) \<star> fg\<^sub>0h\<^sub>1.\<phi>) \<cdot>
(\<a>\<^sup>-\<^sup>1[f, g, (tab\<^sub>0 g \<star> Tfg.p\<^sub>0) \<star> TTfgh.p\<^sub>1] \<cdot>
(f \<star> \<a>[g, tab\<^sub>0 g \<star> Tfg.p\<^sub>0, TTfgh.p\<^sub>1]) \<cdot>
(f \<star> \<a>[g, tab\<^sub>0 g, Tfg.p\<^sub>0] \<star> TTfgh.p\<^sub>1) \<cdot>
\<a>[f, (g \<star> tab\<^sub>0 g) \<star> Tfg.p\<^sub>0, TTfgh.p\<^sub>1] \<cdot>
((f \<star> \<a>\<^sup>-\<^sup>1[g, tab\<^sub>0 g, Tfg.p\<^sub>0]) \<star> TTfgh.p\<^sub>1) \<cdot>
(\<a>[f, g, tab\<^sub>0 g \<star> Tfg.p\<^sub>0] \<star> TTfgh.p\<^sub>1)) \<cdot>
(\<a>\<^sup>-\<^sup>1[f, g, tab\<^sub>0 g \<star> Tfg.p\<^sub>0] \<cdot>
(f \<star> \<a>[g, tab\<^sub>0 g, Tfg.p\<^sub>0]) \<cdot>
(f \<star> (g.tab \<star> Tfg.p\<^sub>0)) \<cdot>
(f \<star> f\<^sub>0g\<^sub>1.\<phi>) \<cdot>
\<a>[f, tab\<^sub>0 f, Tfg.p\<^sub>1] \<cdot>
(f.tab \<star> Tfg.p\<^sub>1)
\<star> TTfgh.p\<^sub>1) \<cdot>
\<a>\<^sup>-\<^sup>1[tab\<^sub>1 f, Tfg.p\<^sub>1, TTfgh.p\<^sub>1]"
[PROOF STATE]
proof (prove)
goal (1 subgoal):
1. \<a>[f, g, h \<star> tab\<^sub>0 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0] \<cdot> ((f \<star> g) \<star> \<a>[h, tab\<^sub>0 h, fg\<^sub>0h\<^sub>1.p\<^sub>0]) \<cdot> ((f \<star> g) \<star> h.tab \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0) \<cdot> ((f \<star> g) \<star> fg\<^sub>0h\<^sub>1.\<phi>) \<cdot> \<a>\<^sup>-\<^sup>1[f, g, (tab\<^sub>0 g \<star> f\<^sub>0g\<^sub>1.p\<^sub>0) \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1] \<cdot> (f \<star> \<a>[g, tab\<^sub>0 g \<star> f\<^sub>0g\<^sub>1.p\<^sub>0, fg\<^sub>0h\<^sub>1.p\<^sub>1]) \<cdot> (f \<star> \<a>[g, tab\<^sub>0 g, f\<^sub>0g\<^sub>1.p\<^sub>0] \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> \<a>[f, (g \<star> tab\<^sub>0 g) \<star> f\<^sub>0g\<^sub>1.p\<^sub>0, fg\<^sub>0h\<^sub>1.p\<^sub>1] \<cdot> ((f \<star> \<a>\<^sup>-\<^sup>1[g, tab\<^sub>0 g, f\<^sub>0g\<^sub>1.p\<^sub>0]) \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> (\<a>[f, g, tab\<^sub>0 g \<star> f\<^sub>0g\<^sub>1.p\<^sub>0] \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> ((\<a>\<^sup>-\<^sup>1[f, g, tab\<^sub>0 g \<star> f\<^sub>0g\<^sub>1.p\<^sub>0] \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> ((f \<star> \<a>[g, tab\<^sub>0 g, f\<^sub>0g\<^sub>1.p\<^sub>0]) \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> ((f \<star> g.tab \<star> f\<^sub>0g\<^sub>1.p\<^sub>0) \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> ((f \<star> f\<^sub>0g\<^sub>1.\<phi>) \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> (\<a>[f, tab\<^sub>0 f, f\<^sub>0g\<^sub>1.p\<^sub>1] \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> ((f.tab \<star> f\<^sub>0g\<^sub>1.p\<^sub>1) \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1)) \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>1 f, f\<^sub>0g\<^sub>1.p\<^sub>1, fg\<^sub>0h\<^sub>1.p\<^sub>1] = \<a>[f, g, h \<star> tab\<^sub>0 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0] \<cdot> ((f \<star> g) \<star> \<a>[h, tab\<^sub>0 h, fg\<^sub>0h\<^sub>1.p\<^sub>0]) \<cdot> ((f \<star> g) \<star> h.tab \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0) \<cdot> ((f \<star> g) \<star> fg\<^sub>0h\<^sub>1.\<phi>) \<cdot> (\<a>\<^sup>-\<^sup>1[f, g, (tab\<^sub>0 g \<star> f\<^sub>0g\<^sub>1.p\<^sub>0) \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1] \<cdot> (f \<star> \<a>[g, tab\<^sub>0 g \<star> f\<^sub>0g\<^sub>1.p\<^sub>0, fg\<^sub>0h\<^sub>1.p\<^sub>1]) \<cdot> (f \<star> \<a>[g, tab\<^sub>0 g, f\<^sub>0g\<^sub>1.p\<^sub>0] \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> \<a>[f, (g \<star> tab\<^sub>0 g) \<star> f\<^sub>0g\<^sub>1.p\<^sub>0, fg\<^sub>0h\<^sub>1.p\<^sub>1] \<cdot> ((f \<star> \<a>\<^sup>-\<^sup>1[g, tab\<^sub>0 g, f\<^sub>0g\<^sub>1.p\<^sub>0]) \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> (\<a>[f, g, tab\<^sub>0 g \<star> f\<^sub>0g\<^sub>1.p\<^sub>0] \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1)) \<cdot> (\<a>\<^sup>-\<^sup>1[f, g, tab\<^sub>0 g \<star> f\<^sub>0g\<^sub>1.p\<^sub>0] \<cdot> (f \<star> \<a>[g, tab\<^sub>0 g, f\<^sub>0g\<^sub>1.p\<^sub>0]) \<cdot> (f \<star> g.tab \<star> f\<^sub>0g\<^sub>1.p\<^sub>0) \<cdot> (f \<star> f\<^sub>0g\<^sub>1.\<phi>) \<cdot> \<a>[f, tab\<^sub>0 f, f\<^sub>0g\<^sub>1.p\<^sub>1] \<cdot> (f.tab \<star> f\<^sub>0g\<^sub>1.p\<^sub>1) \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>1 f, f\<^sub>0g\<^sub>1.p\<^sub>1, fg\<^sub>0h\<^sub>1.p\<^sub>1]
[PROOF STEP]
using fg gh fg\<^sub>0h\<^sub>1.p\<^sub>0_simps fg\<^sub>0h\<^sub>1.p\<^sub>1_simps f\<^sub>0g\<^sub>1.p\<^sub>0_simps f\<^sub>0g\<^sub>1.p\<^sub>1_simps
whisker_right comp_assoc
[PROOF STATE]
proof (prove)
using this:
src f = trg g
src g = trg h
trg fg\<^sub>0h\<^sub>1.p\<^sub>0 = src (tab\<^sub>1 h)
trg fg\<^sub>0h\<^sub>1.p\<^sub>1 = src (tab\<^sub>0 g \<star> f\<^sub>0g\<^sub>1.p\<^sub>0)
trg f\<^sub>0g\<^sub>1.p\<^sub>0 = src (tab\<^sub>1 g)
trg f\<^sub>0g\<^sub>1.p\<^sub>1 = src (tab\<^sub>0 f)
\<lbrakk>ide ?f; seq ?\<nu> ?\<mu>\<rbrakk> \<Longrightarrow> ?\<nu> \<cdot> ?\<mu> \<star> ?f = (?\<nu> \<star> ?f) \<cdot> (?\<mu> \<star> ?f)
(?h \<cdot> ?g) \<cdot> ?f = ?h \<cdot> ?g \<cdot> ?f
goal (1 subgoal):
1. \<a>[f, g, h \<star> tab\<^sub>0 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0] \<cdot> ((f \<star> g) \<star> \<a>[h, tab\<^sub>0 h, fg\<^sub>0h\<^sub>1.p\<^sub>0]) \<cdot> ((f \<star> g) \<star> h.tab \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0) \<cdot> ((f \<star> g) \<star> fg\<^sub>0h\<^sub>1.\<phi>) \<cdot> \<a>\<^sup>-\<^sup>1[f, g, (tab\<^sub>0 g \<star> f\<^sub>0g\<^sub>1.p\<^sub>0) \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1] \<cdot> (f \<star> \<a>[g, tab\<^sub>0 g \<star> f\<^sub>0g\<^sub>1.p\<^sub>0, fg\<^sub>0h\<^sub>1.p\<^sub>1]) \<cdot> (f \<star> \<a>[g, tab\<^sub>0 g, f\<^sub>0g\<^sub>1.p\<^sub>0] \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> \<a>[f, (g \<star> tab\<^sub>0 g) \<star> f\<^sub>0g\<^sub>1.p\<^sub>0, fg\<^sub>0h\<^sub>1.p\<^sub>1] \<cdot> ((f \<star> \<a>\<^sup>-\<^sup>1[g, tab\<^sub>0 g, f\<^sub>0g\<^sub>1.p\<^sub>0]) \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> (\<a>[f, g, tab\<^sub>0 g \<star> f\<^sub>0g\<^sub>1.p\<^sub>0] \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> ((\<a>\<^sup>-\<^sup>1[f, g, tab\<^sub>0 g \<star> f\<^sub>0g\<^sub>1.p\<^sub>0] \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> ((f \<star> \<a>[g, tab\<^sub>0 g, f\<^sub>0g\<^sub>1.p\<^sub>0]) \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> ((f \<star> g.tab \<star> f\<^sub>0g\<^sub>1.p\<^sub>0) \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> ((f \<star> f\<^sub>0g\<^sub>1.\<phi>) \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> (\<a>[f, tab\<^sub>0 f, f\<^sub>0g\<^sub>1.p\<^sub>1] \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> ((f.tab \<star> f\<^sub>0g\<^sub>1.p\<^sub>1) \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1)) \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>1 f, f\<^sub>0g\<^sub>1.p\<^sub>1, fg\<^sub>0h\<^sub>1.p\<^sub>1] = \<a>[f, g, h \<star> tab\<^sub>0 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0] \<cdot> ((f \<star> g) \<star> \<a>[h, tab\<^sub>0 h, fg\<^sub>0h\<^sub>1.p\<^sub>0]) \<cdot> ((f \<star> g) \<star> h.tab \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0) \<cdot> ((f \<star> g) \<star> fg\<^sub>0h\<^sub>1.\<phi>) \<cdot> (\<a>\<^sup>-\<^sup>1[f, g, (tab\<^sub>0 g \<star> f\<^sub>0g\<^sub>1.p\<^sub>0) \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1] \<cdot> (f \<star> \<a>[g, tab\<^sub>0 g \<star> f\<^sub>0g\<^sub>1.p\<^sub>0, fg\<^sub>0h\<^sub>1.p\<^sub>1]) \<cdot> (f \<star> \<a>[g, tab\<^sub>0 g, f\<^sub>0g\<^sub>1.p\<^sub>0] \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> \<a>[f, (g \<star> tab\<^sub>0 g) \<star> f\<^sub>0g\<^sub>1.p\<^sub>0, fg\<^sub>0h\<^sub>1.p\<^sub>1] \<cdot> ((f \<star> \<a>\<^sup>-\<^sup>1[g, tab\<^sub>0 g, f\<^sub>0g\<^sub>1.p\<^sub>0]) \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> (\<a>[f, g, tab\<^sub>0 g \<star> f\<^sub>0g\<^sub>1.p\<^sub>0] \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1)) \<cdot> (\<a>\<^sup>-\<^sup>1[f, g, tab\<^sub>0 g \<star> f\<^sub>0g\<^sub>1.p\<^sub>0] \<cdot> (f \<star> \<a>[g, tab\<^sub>0 g, f\<^sub>0g\<^sub>1.p\<^sub>0]) \<cdot> (f \<star> g.tab \<star> f\<^sub>0g\<^sub>1.p\<^sub>0) \<cdot> (f \<star> f\<^sub>0g\<^sub>1.\<phi>) \<cdot> \<a>[f, tab\<^sub>0 f, f\<^sub>0g\<^sub>1.p\<^sub>1] \<cdot> (f.tab \<star> f\<^sub>0g\<^sub>1.p\<^sub>1) \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>1 f, f\<^sub>0g\<^sub>1.p\<^sub>1, fg\<^sub>0h\<^sub>1.p\<^sub>1]
[PROOF STEP]
by simp
[PROOF STATE]
proof (state)
this:
\<a>[f, g, h \<star> tab\<^sub>0 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0] \<cdot> ((f \<star> g) \<star> \<a>[h, tab\<^sub>0 h, fg\<^sub>0h\<^sub>1.p\<^sub>0]) \<cdot> ((f \<star> g) \<star> h.tab \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0) \<cdot> ((f \<star> g) \<star> fg\<^sub>0h\<^sub>1.\<phi>) \<cdot> \<a>\<^sup>-\<^sup>1[f, g, (tab\<^sub>0 g \<star> f\<^sub>0g\<^sub>1.p\<^sub>0) \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1] \<cdot> (f \<star> \<a>[g, tab\<^sub>0 g \<star> f\<^sub>0g\<^sub>1.p\<^sub>0, fg\<^sub>0h\<^sub>1.p\<^sub>1]) \<cdot> (f \<star> \<a>[g, tab\<^sub>0 g, f\<^sub>0g\<^sub>1.p\<^sub>0] \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> \<a>[f, (g \<star> tab\<^sub>0 g) \<star> f\<^sub>0g\<^sub>1.p\<^sub>0, fg\<^sub>0h\<^sub>1.p\<^sub>1] \<cdot> ((f \<star> \<a>\<^sup>-\<^sup>1[g, tab\<^sub>0 g, f\<^sub>0g\<^sub>1.p\<^sub>0]) \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> (\<a>[f, g, tab\<^sub>0 g \<star> f\<^sub>0g\<^sub>1.p\<^sub>0] \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> ((\<a>\<^sup>-\<^sup>1[f, g, tab\<^sub>0 g \<star> f\<^sub>0g\<^sub>1.p\<^sub>0] \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> ((f \<star> \<a>[g, tab\<^sub>0 g, f\<^sub>0g\<^sub>1.p\<^sub>0]) \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> ((f \<star> g.tab \<star> f\<^sub>0g\<^sub>1.p\<^sub>0) \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> ((f \<star> f\<^sub>0g\<^sub>1.\<phi>) \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> (\<a>[f, tab\<^sub>0 f, f\<^sub>0g\<^sub>1.p\<^sub>1] \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> ((f.tab \<star> f\<^sub>0g\<^sub>1.p\<^sub>1) \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1)) \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>1 f, f\<^sub>0g\<^sub>1.p\<^sub>1, fg\<^sub>0h\<^sub>1.p\<^sub>1] = \<a>[f, g, h \<star> tab\<^sub>0 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0] \<cdot> ((f \<star> g) \<star> \<a>[h, tab\<^sub>0 h, fg\<^sub>0h\<^sub>1.p\<^sub>0]) \<cdot> ((f \<star> g) \<star> h.tab \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0) \<cdot> ((f \<star> g) \<star> fg\<^sub>0h\<^sub>1.\<phi>) \<cdot> (\<a>\<^sup>-\<^sup>1[f, g, (tab\<^sub>0 g \<star> f\<^sub>0g\<^sub>1.p\<^sub>0) \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1] \<cdot> (f \<star> \<a>[g, tab\<^sub>0 g \<star> f\<^sub>0g\<^sub>1.p\<^sub>0, fg\<^sub>0h\<^sub>1.p\<^sub>1]) \<cdot> (f \<star> \<a>[g, tab\<^sub>0 g, f\<^sub>0g\<^sub>1.p\<^sub>0] \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> \<a>[f, (g \<star> tab\<^sub>0 g) \<star> f\<^sub>0g\<^sub>1.p\<^sub>0, fg\<^sub>0h\<^sub>1.p\<^sub>1] \<cdot> ((f \<star> \<a>\<^sup>-\<^sup>1[g, tab\<^sub>0 g, f\<^sub>0g\<^sub>1.p\<^sub>0]) \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> (\<a>[f, g, tab\<^sub>0 g \<star> f\<^sub>0g\<^sub>1.p\<^sub>0] \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1)) \<cdot> (\<a>\<^sup>-\<^sup>1[f, g, tab\<^sub>0 g \<star> f\<^sub>0g\<^sub>1.p\<^sub>0] \<cdot> (f \<star> \<a>[g, tab\<^sub>0 g, f\<^sub>0g\<^sub>1.p\<^sub>0]) \<cdot> (f \<star> g.tab \<star> f\<^sub>0g\<^sub>1.p\<^sub>0) \<cdot> (f \<star> f\<^sub>0g\<^sub>1.\<phi>) \<cdot> \<a>[f, tab\<^sub>0 f, f\<^sub>0g\<^sub>1.p\<^sub>1] \<cdot> (f.tab \<star> f\<^sub>0g\<^sub>1.p\<^sub>1) \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>1 f, f\<^sub>0g\<^sub>1.p\<^sub>1, fg\<^sub>0h\<^sub>1.p\<^sub>1]
goal (1 subgoal):
1. (f \<star> \<theta>\<^sub>f) \<cdot> \<a>[f, tab\<^sub>0 f, w\<^sub>f] \<cdot> (f.tab \<star> w\<^sub>f) = ((f \<star> \<theta>\<^sub>f') \<cdot> \<a>[f, tab\<^sub>0 f, w\<^sub>f'] \<cdot> (f.tab \<star> w\<^sub>f')) \<cdot> \<beta>\<^sub>f
[PROOF STEP]
also
[PROOF STATE]
proof (state)
this:
\<a>[f, g, h \<star> tab\<^sub>0 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0] \<cdot> ((f \<star> g) \<star> \<a>[h, tab\<^sub>0 h, fg\<^sub>0h\<^sub>1.p\<^sub>0]) \<cdot> ((f \<star> g) \<star> h.tab \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0) \<cdot> ((f \<star> g) \<star> fg\<^sub>0h\<^sub>1.\<phi>) \<cdot> \<a>\<^sup>-\<^sup>1[f, g, (tab\<^sub>0 g \<star> f\<^sub>0g\<^sub>1.p\<^sub>0) \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1] \<cdot> (f \<star> \<a>[g, tab\<^sub>0 g \<star> f\<^sub>0g\<^sub>1.p\<^sub>0, fg\<^sub>0h\<^sub>1.p\<^sub>1]) \<cdot> (f \<star> \<a>[g, tab\<^sub>0 g, f\<^sub>0g\<^sub>1.p\<^sub>0] \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> \<a>[f, (g \<star> tab\<^sub>0 g) \<star> f\<^sub>0g\<^sub>1.p\<^sub>0, fg\<^sub>0h\<^sub>1.p\<^sub>1] \<cdot> ((f \<star> \<a>\<^sup>-\<^sup>1[g, tab\<^sub>0 g, f\<^sub>0g\<^sub>1.p\<^sub>0]) \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> (\<a>[f, g, tab\<^sub>0 g \<star> f\<^sub>0g\<^sub>1.p\<^sub>0] \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> ((\<a>\<^sup>-\<^sup>1[f, g, tab\<^sub>0 g \<star> f\<^sub>0g\<^sub>1.p\<^sub>0] \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> ((f \<star> \<a>[g, tab\<^sub>0 g, f\<^sub>0g\<^sub>1.p\<^sub>0]) \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> ((f \<star> g.tab \<star> f\<^sub>0g\<^sub>1.p\<^sub>0) \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> ((f \<star> f\<^sub>0g\<^sub>1.\<phi>) \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> (\<a>[f, tab\<^sub>0 f, f\<^sub>0g\<^sub>1.p\<^sub>1] \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> ((f.tab \<star> f\<^sub>0g\<^sub>1.p\<^sub>1) \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1)) \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>1 f, f\<^sub>0g\<^sub>1.p\<^sub>1, fg\<^sub>0h\<^sub>1.p\<^sub>1] = \<a>[f, g, h \<star> tab\<^sub>0 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0] \<cdot> ((f \<star> g) \<star> \<a>[h, tab\<^sub>0 h, fg\<^sub>0h\<^sub>1.p\<^sub>0]) \<cdot> ((f \<star> g) \<star> h.tab \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0) \<cdot> ((f \<star> g) \<star> fg\<^sub>0h\<^sub>1.\<phi>) \<cdot> (\<a>\<^sup>-\<^sup>1[f, g, (tab\<^sub>0 g \<star> f\<^sub>0g\<^sub>1.p\<^sub>0) \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1] \<cdot> (f \<star> \<a>[g, tab\<^sub>0 g \<star> f\<^sub>0g\<^sub>1.p\<^sub>0, fg\<^sub>0h\<^sub>1.p\<^sub>1]) \<cdot> (f \<star> \<a>[g, tab\<^sub>0 g, f\<^sub>0g\<^sub>1.p\<^sub>0] \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> \<a>[f, (g \<star> tab\<^sub>0 g) \<star> f\<^sub>0g\<^sub>1.p\<^sub>0, fg\<^sub>0h\<^sub>1.p\<^sub>1] \<cdot> ((f \<star> \<a>\<^sup>-\<^sup>1[g, tab\<^sub>0 g, f\<^sub>0g\<^sub>1.p\<^sub>0]) \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> (\<a>[f, g, tab\<^sub>0 g \<star> f\<^sub>0g\<^sub>1.p\<^sub>0] \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1)) \<cdot> (\<a>\<^sup>-\<^sup>1[f, g, tab\<^sub>0 g \<star> f\<^sub>0g\<^sub>1.p\<^sub>0] \<cdot> (f \<star> \<a>[g, tab\<^sub>0 g, f\<^sub>0g\<^sub>1.p\<^sub>0]) \<cdot> (f \<star> g.tab \<star> f\<^sub>0g\<^sub>1.p\<^sub>0) \<cdot> (f \<star> f\<^sub>0g\<^sub>1.\<phi>) \<cdot> \<a>[f, tab\<^sub>0 f, f\<^sub>0g\<^sub>1.p\<^sub>1] \<cdot> (f.tab \<star> f\<^sub>0g\<^sub>1.p\<^sub>1) \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>1 f, f\<^sub>0g\<^sub>1.p\<^sub>1, fg\<^sub>0h\<^sub>1.p\<^sub>1]
goal (1 subgoal):
1. (f \<star> \<theta>\<^sub>f) \<cdot> \<a>[f, tab\<^sub>0 f, w\<^sub>f] \<cdot> (f.tab \<star> w\<^sub>f) = ((f \<star> \<theta>\<^sub>f') \<cdot> \<a>[f, tab\<^sub>0 f, w\<^sub>f'] \<cdot> (f.tab \<star> w\<^sub>f')) \<cdot> \<beta>\<^sub>f
[PROOF STEP]
have "... =
\<a>[f, g, h \<star> tab\<^sub>0 h \<star> TTfgh.p\<^sub>0] \<cdot>
((f \<star> g) \<star> \<a>[h, tab\<^sub>0 h, TTfgh.p\<^sub>0]) \<cdot>
((f \<star> g) \<star> h.tab \<star> TTfgh.p\<^sub>0) \<cdot>
((f \<star> g) \<star> fg\<^sub>0h\<^sub>1.\<phi>) \<cdot>
\<a>[f \<star> g, tab\<^sub>0 g \<star> Tfg.p\<^sub>0, TTfgh.p\<^sub>1] \<cdot>
(\<a>\<^sup>-\<^sup>1[f, g, tab\<^sub>0 g \<star> Tfg.p\<^sub>0] \<cdot>
(f \<star> \<a>[g, tab\<^sub>0 g, Tfg.p\<^sub>0]) \<cdot>
(f \<star> (g.tab \<star> Tfg.p\<^sub>0)) \<cdot>
(f \<star> f\<^sub>0g\<^sub>1.\<phi>) \<cdot>
\<a>[f, tab\<^sub>0 f, Tfg.p\<^sub>1] \<cdot>
(f.tab \<star> Tfg.p\<^sub>1)
\<star> TTfgh.p\<^sub>1) \<cdot>
\<a>\<^sup>-\<^sup>1[tab\<^sub>1 f, Tfg.p\<^sub>1, TTfgh.p\<^sub>1]"
[PROOF STATE]
proof (prove)
goal (1 subgoal):
1. \<a>[f, g, h \<star> tab\<^sub>0 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0] \<cdot> ((f \<star> g) \<star> \<a>[h, tab\<^sub>0 h, fg\<^sub>0h\<^sub>1.p\<^sub>0]) \<cdot> ((f \<star> g) \<star> h.tab \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0) \<cdot> ((f \<star> g) \<star> fg\<^sub>0h\<^sub>1.\<phi>) \<cdot> (\<a>\<^sup>-\<^sup>1[f, g, (tab\<^sub>0 g \<star> f\<^sub>0g\<^sub>1.p\<^sub>0) \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1] \<cdot> (f \<star> \<a>[g, tab\<^sub>0 g \<star> f\<^sub>0g\<^sub>1.p\<^sub>0, fg\<^sub>0h\<^sub>1.p\<^sub>1]) \<cdot> (f \<star> \<a>[g, tab\<^sub>0 g, f\<^sub>0g\<^sub>1.p\<^sub>0] \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> \<a>[f, (g \<star> tab\<^sub>0 g) \<star> f\<^sub>0g\<^sub>1.p\<^sub>0, fg\<^sub>0h\<^sub>1.p\<^sub>1] \<cdot> ((f \<star> \<a>\<^sup>-\<^sup>1[g, tab\<^sub>0 g, f\<^sub>0g\<^sub>1.p\<^sub>0]) \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> (\<a>[f, g, tab\<^sub>0 g \<star> f\<^sub>0g\<^sub>1.p\<^sub>0] \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1)) \<cdot> (\<a>\<^sup>-\<^sup>1[f, g, tab\<^sub>0 g \<star> f\<^sub>0g\<^sub>1.p\<^sub>0] \<cdot> (f \<star> \<a>[g, tab\<^sub>0 g, f\<^sub>0g\<^sub>1.p\<^sub>0]) \<cdot> (f \<star> g.tab \<star> f\<^sub>0g\<^sub>1.p\<^sub>0) \<cdot> (f \<star> f\<^sub>0g\<^sub>1.\<phi>) \<cdot> \<a>[f, tab\<^sub>0 f, f\<^sub>0g\<^sub>1.p\<^sub>1] \<cdot> (f.tab \<star> f\<^sub>0g\<^sub>1.p\<^sub>1) \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>1 f, f\<^sub>0g\<^sub>1.p\<^sub>1, fg\<^sub>0h\<^sub>1.p\<^sub>1] = \<a>[f, g, h \<star> tab\<^sub>0 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0] \<cdot> ((f \<star> g) \<star> \<a>[h, tab\<^sub>0 h, fg\<^sub>0h\<^sub>1.p\<^sub>0]) \<cdot> ((f \<star> g) \<star> h.tab \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0) \<cdot> ((f \<star> g) \<star> fg\<^sub>0h\<^sub>1.\<phi>) \<cdot> \<a>[f \<star> g, tab\<^sub>0 g \<star> f\<^sub>0g\<^sub>1.p\<^sub>0, fg\<^sub>0h\<^sub>1.p\<^sub>1] \<cdot> (\<a>\<^sup>-\<^sup>1[f, g, tab\<^sub>0 g \<star> f\<^sub>0g\<^sub>1.p\<^sub>0] \<cdot> (f \<star> \<a>[g, tab\<^sub>0 g, f\<^sub>0g\<^sub>1.p\<^sub>0]) \<cdot> (f \<star> g.tab \<star> f\<^sub>0g\<^sub>1.p\<^sub>0) \<cdot> (f \<star> f\<^sub>0g\<^sub>1.\<phi>) \<cdot> \<a>[f, tab\<^sub>0 f, f\<^sub>0g\<^sub>1.p\<^sub>1] \<cdot> (f.tab \<star> f\<^sub>0g\<^sub>1.p\<^sub>1) \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>1 f, f\<^sub>0g\<^sub>1.p\<^sub>1, fg\<^sub>0h\<^sub>1.p\<^sub>1]
[PROOF STEP]
proof -
[PROOF STATE]
proof (state)
goal (1 subgoal):
1. \<a>[f, g, h \<star> tab\<^sub>0 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0] \<cdot> ((f \<star> g) \<star> \<a>[h, tab\<^sub>0 h, fg\<^sub>0h\<^sub>1.p\<^sub>0]) \<cdot> ((f \<star> g) \<star> h.tab \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0) \<cdot> ((f \<star> g) \<star> fg\<^sub>0h\<^sub>1.\<phi>) \<cdot> (\<a>\<^sup>-\<^sup>1[f, g, (tab\<^sub>0 g \<star> f\<^sub>0g\<^sub>1.p\<^sub>0) \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1] \<cdot> (f \<star> \<a>[g, tab\<^sub>0 g \<star> f\<^sub>0g\<^sub>1.p\<^sub>0, fg\<^sub>0h\<^sub>1.p\<^sub>1]) \<cdot> (f \<star> \<a>[g, tab\<^sub>0 g, f\<^sub>0g\<^sub>1.p\<^sub>0] \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> \<a>[f, (g \<star> tab\<^sub>0 g) \<star> f\<^sub>0g\<^sub>1.p\<^sub>0, fg\<^sub>0h\<^sub>1.p\<^sub>1] \<cdot> ((f \<star> \<a>\<^sup>-\<^sup>1[g, tab\<^sub>0 g, f\<^sub>0g\<^sub>1.p\<^sub>0]) \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> (\<a>[f, g, tab\<^sub>0 g \<star> f\<^sub>0g\<^sub>1.p\<^sub>0] \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1)) \<cdot> (\<a>\<^sup>-\<^sup>1[f, g, tab\<^sub>0 g \<star> f\<^sub>0g\<^sub>1.p\<^sub>0] \<cdot> (f \<star> \<a>[g, tab\<^sub>0 g, f\<^sub>0g\<^sub>1.p\<^sub>0]) \<cdot> (f \<star> g.tab \<star> f\<^sub>0g\<^sub>1.p\<^sub>0) \<cdot> (f \<star> f\<^sub>0g\<^sub>1.\<phi>) \<cdot> \<a>[f, tab\<^sub>0 f, f\<^sub>0g\<^sub>1.p\<^sub>1] \<cdot> (f.tab \<star> f\<^sub>0g\<^sub>1.p\<^sub>1) \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>1 f, f\<^sub>0g\<^sub>1.p\<^sub>1, fg\<^sub>0h\<^sub>1.p\<^sub>1] = \<a>[f, g, h \<star> tab\<^sub>0 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0] \<cdot> ((f \<star> g) \<star> \<a>[h, tab\<^sub>0 h, fg\<^sub>0h\<^sub>1.p\<^sub>0]) \<cdot> ((f \<star> g) \<star> h.tab \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0) \<cdot> ((f \<star> g) \<star> fg\<^sub>0h\<^sub>1.\<phi>) \<cdot> \<a>[f \<star> g, tab\<^sub>0 g \<star> f\<^sub>0g\<^sub>1.p\<^sub>0, fg\<^sub>0h\<^sub>1.p\<^sub>1] \<cdot> (\<a>\<^sup>-\<^sup>1[f, g, tab\<^sub>0 g \<star> f\<^sub>0g\<^sub>1.p\<^sub>0] \<cdot> (f \<star> \<a>[g, tab\<^sub>0 g, f\<^sub>0g\<^sub>1.p\<^sub>0]) \<cdot> (f \<star> g.tab \<star> f\<^sub>0g\<^sub>1.p\<^sub>0) \<cdot> (f \<star> f\<^sub>0g\<^sub>1.\<phi>) \<cdot> \<a>[f, tab\<^sub>0 f, f\<^sub>0g\<^sub>1.p\<^sub>1] \<cdot> (f.tab \<star> f\<^sub>0g\<^sub>1.p\<^sub>1) \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>1 f, f\<^sub>0g\<^sub>1.p\<^sub>1, fg\<^sub>0h\<^sub>1.p\<^sub>1]
[PROOF STEP]
have "\<a>\<^sup>-\<^sup>1[f, g, (tab\<^sub>0 g \<star> Tfg.p\<^sub>0) \<star> TTfgh.p\<^sub>1] \<cdot>
(f \<star> \<a>[g, tab\<^sub>0 g \<star> Tfg.p\<^sub>0, TTfgh.p\<^sub>1]) \<cdot>
(f \<star> \<a>[g, tab\<^sub>0 g, Tfg.p\<^sub>0] \<star> TTfgh.p\<^sub>1) \<cdot>
\<a>[f, (g \<star> tab\<^sub>0 g) \<star> Tfg.p\<^sub>0, TTfgh.p\<^sub>1] \<cdot>
((f \<star> \<a>\<^sup>-\<^sup>1[g, tab\<^sub>0 g, Tfg.p\<^sub>0]) \<star> TTfgh.p\<^sub>1) \<cdot>
(\<a>[f, g, tab\<^sub>0 g \<star> Tfg.p\<^sub>0] \<star> TTfgh.p\<^sub>1) =
\<lbrace>\<^bold>\<a>\<^sup>-\<^sup>1\<^bold>[\<^bold>\<langle>f\<^bold>\<rangle>, \<^bold>\<langle>g\<^bold>\<rangle>, (\<^bold>\<langle>tab\<^sub>0 g\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>Tfg.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TTfgh.p\<^sub>1\<^bold>\<rangle>\<^bold>] \<^bold>\<cdot>
(\<^bold>\<langle>f\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<a>\<^bold>[\<^bold>\<langle>g\<^bold>\<rangle>, \<^bold>\<langle>tab\<^sub>0 g\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>Tfg.p\<^sub>0\<^bold>\<rangle>, \<^bold>\<langle>TTfgh.p\<^sub>1\<^bold>\<rangle>\<^bold>]) \<^bold>\<cdot>
(\<^bold>\<langle>f\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<a>\<^bold>[\<^bold>\<langle>g\<^bold>\<rangle>, \<^bold>\<langle>tab\<^sub>0 g\<^bold>\<rangle>, \<^bold>\<langle>Tfg.p\<^sub>0\<^bold>\<rangle>\<^bold>] \<^bold>\<star> \<^bold>\<langle>TTfgh.p\<^sub>1\<^bold>\<rangle>) \<^bold>\<cdot>
\<^bold>\<a>\<^bold>[\<^bold>\<langle>f\<^bold>\<rangle>, (\<^bold>\<langle>g\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>tab\<^sub>0 g\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>Tfg.p\<^sub>0\<^bold>\<rangle>, \<^bold>\<langle>TTfgh.p\<^sub>1\<^bold>\<rangle>\<^bold>] \<^bold>\<cdot>
((\<^bold>\<langle>f\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<a>\<^sup>-\<^sup>1\<^bold>[\<^bold>\<langle>g\<^bold>\<rangle>, \<^bold>\<langle>tab\<^sub>0 g\<^bold>\<rangle>, \<^bold>\<langle>Tfg.p\<^sub>0\<^bold>\<rangle>\<^bold>]) \<^bold>\<star> \<^bold>\<langle>TTfgh.p\<^sub>1\<^bold>\<rangle>) \<^bold>\<cdot>
(\<^bold>\<a>\<^bold>[\<^bold>\<langle>f\<^bold>\<rangle>, \<^bold>\<langle>g\<^bold>\<rangle>, \<^bold>\<langle>tab\<^sub>0 g\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>Tfg.p\<^sub>0\<^bold>\<rangle>\<^bold>] \<^bold>\<star> \<^bold>\<langle>TTfgh.p\<^sub>1\<^bold>\<rangle>)\<rbrace>"
[PROOF STATE]
proof (prove)
goal (1 subgoal):
1. \<a>\<^sup>-\<^sup>1[f, g, (tab\<^sub>0 g \<star> f\<^sub>0g\<^sub>1.p\<^sub>0) \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1] \<cdot> (f \<star> \<a>[g, tab\<^sub>0 g \<star> f\<^sub>0g\<^sub>1.p\<^sub>0, fg\<^sub>0h\<^sub>1.p\<^sub>1]) \<cdot> (f \<star> \<a>[g, tab\<^sub>0 g, f\<^sub>0g\<^sub>1.p\<^sub>0] \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> \<a>[f, (g \<star> tab\<^sub>0 g) \<star> f\<^sub>0g\<^sub>1.p\<^sub>0, fg\<^sub>0h\<^sub>1.p\<^sub>1] \<cdot> ((f \<star> \<a>\<^sup>-\<^sup>1[g, tab\<^sub>0 g, f\<^sub>0g\<^sub>1.p\<^sub>0]) \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> (\<a>[f, g, tab\<^sub>0 g \<star> f\<^sub>0g\<^sub>1.p\<^sub>0] \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) = \<lbrace>\<^bold>\<a>\<^sup>-\<^sup>1\<^bold>[\<^bold>\<langle>f\<^bold>\<rangle>, \<^bold>\<langle>g\<^bold>\<rangle>, (\<^bold>\<langle>tab\<^sub>0 g\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>f\<^sub>0g\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>fg\<^sub>0h\<^sub>1.p\<^sub>1\<^bold>\<rangle>\<^bold>] \<^bold>\<cdot> (\<^bold>\<langle>f\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<a>\<^bold>[\<^bold>\<langle>g\<^bold>\<rangle>, \<^bold>\<langle>tab\<^sub>0 g\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>f\<^sub>0g\<^sub>1.p\<^sub>0\<^bold>\<rangle>, \<^bold>\<langle>fg\<^sub>0h\<^sub>1.p\<^sub>1\<^bold>\<rangle>\<^bold>]) \<^bold>\<cdot> (\<^bold>\<langle>f\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<a>\<^bold>[\<^bold>\<langle>g\<^bold>\<rangle>, \<^bold>\<langle>tab\<^sub>0 g\<^bold>\<rangle>, \<^bold>\<langle>f\<^sub>0g\<^sub>1.p\<^sub>0\<^bold>\<rangle>\<^bold>] \<^bold>\<star> \<^bold>\<langle>fg\<^sub>0h\<^sub>1.p\<^sub>1\<^bold>\<rangle>) \<^bold>\<cdot> \<^bold>\<a>\<^bold>[\<^bold>\<langle>f\<^bold>\<rangle>, (\<^bold>\<langle>g\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>tab\<^sub>0 g\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0g\<^sub>1.p\<^sub>0\<^bold>\<rangle>, \<^bold>\<langle>fg\<^sub>0h\<^sub>1.p\<^sub>1\<^bold>\<rangle>\<^bold>] \<^bold>\<cdot> ((\<^bold>\<langle>f\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<a>\<^sup>-\<^sup>1\<^bold>[\<^bold>\<langle>g\<^bold>\<rangle>, \<^bold>\<langle>tab\<^sub>0 g\<^bold>\<rangle>, \<^bold>\<langle>f\<^sub>0g\<^sub>1.p\<^sub>0\<^bold>\<rangle>\<^bold>]) \<^bold>\<star> \<^bold>\<langle>fg\<^sub>0h\<^sub>1.p\<^sub>1\<^bold>\<rangle>) \<^bold>\<cdot> (\<^bold>\<a>\<^bold>[\<^bold>\<langle>f\<^bold>\<rangle>, \<^bold>\<langle>g\<^bold>\<rangle>, \<^bold>\<langle>tab\<^sub>0 g\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>f\<^sub>0g\<^sub>1.p\<^sub>0\<^bold>\<rangle>\<^bold>] \<^bold>\<star> \<^bold>\<langle>fg\<^sub>0h\<^sub>1.p\<^sub>1\<^bold>\<rangle>)\<rbrace>
[PROOF STEP]
using fg gh fg\<^sub>0h\<^sub>1.p\<^sub>0_simps fg\<^sub>0h\<^sub>1.p\<^sub>1_simps f\<^sub>0g\<^sub>1.p\<^sub>0_simps f\<^sub>0g\<^sub>1.p\<^sub>1_simps
\<a>'_def \<alpha>_def
[PROOF STATE]
proof (prove)
using this:
src f = trg g
src g = trg h
trg fg\<^sub>0h\<^sub>1.p\<^sub>0 = src (tab\<^sub>1 h)
trg fg\<^sub>0h\<^sub>1.p\<^sub>1 = src (tab\<^sub>0 g \<star> f\<^sub>0g\<^sub>1.p\<^sub>0)
trg f\<^sub>0g\<^sub>1.p\<^sub>0 = src (tab\<^sub>1 g)
trg f\<^sub>0g\<^sub>1.p\<^sub>1 = src (tab\<^sub>0 f)
\<a>\<^sup>-\<^sup>1[?\<mu>, ?\<nu>, ?\<tau>] \<equiv> \<alpha>' (?\<mu>, ?\<nu>, ?\<tau>)
\<alpha> ?\<mu>\<nu>\<tau> \<equiv> \<a>[fst ?\<mu>\<nu>\<tau>, fst (snd ?\<mu>\<nu>\<tau>), snd (snd ?\<mu>\<nu>\<tau>)]
goal (1 subgoal):
1. \<a>\<^sup>-\<^sup>1[f, g, (tab\<^sub>0 g \<star> f\<^sub>0g\<^sub>1.p\<^sub>0) \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1] \<cdot> (f \<star> \<a>[g, tab\<^sub>0 g \<star> f\<^sub>0g\<^sub>1.p\<^sub>0, fg\<^sub>0h\<^sub>1.p\<^sub>1]) \<cdot> (f \<star> \<a>[g, tab\<^sub>0 g, f\<^sub>0g\<^sub>1.p\<^sub>0] \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> \<a>[f, (g \<star> tab\<^sub>0 g) \<star> f\<^sub>0g\<^sub>1.p\<^sub>0, fg\<^sub>0h\<^sub>1.p\<^sub>1] \<cdot> ((f \<star> \<a>\<^sup>-\<^sup>1[g, tab\<^sub>0 g, f\<^sub>0g\<^sub>1.p\<^sub>0]) \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> (\<a>[f, g, tab\<^sub>0 g \<star> f\<^sub>0g\<^sub>1.p\<^sub>0] \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) = \<lbrace>\<^bold>\<a>\<^sup>-\<^sup>1\<^bold>[\<^bold>\<langle>f\<^bold>\<rangle>, \<^bold>\<langle>g\<^bold>\<rangle>, (\<^bold>\<langle>tab\<^sub>0 g\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>f\<^sub>0g\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>fg\<^sub>0h\<^sub>1.p\<^sub>1\<^bold>\<rangle>\<^bold>] \<^bold>\<cdot> (\<^bold>\<langle>f\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<a>\<^bold>[\<^bold>\<langle>g\<^bold>\<rangle>, \<^bold>\<langle>tab\<^sub>0 g\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>f\<^sub>0g\<^sub>1.p\<^sub>0\<^bold>\<rangle>, \<^bold>\<langle>fg\<^sub>0h\<^sub>1.p\<^sub>1\<^bold>\<rangle>\<^bold>]) \<^bold>\<cdot> (\<^bold>\<langle>f\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<a>\<^bold>[\<^bold>\<langle>g\<^bold>\<rangle>, \<^bold>\<langle>tab\<^sub>0 g\<^bold>\<rangle>, \<^bold>\<langle>f\<^sub>0g\<^sub>1.p\<^sub>0\<^bold>\<rangle>\<^bold>] \<^bold>\<star> \<^bold>\<langle>fg\<^sub>0h\<^sub>1.p\<^sub>1\<^bold>\<rangle>) \<^bold>\<cdot> \<^bold>\<a>\<^bold>[\<^bold>\<langle>f\<^bold>\<rangle>, (\<^bold>\<langle>g\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>tab\<^sub>0 g\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0g\<^sub>1.p\<^sub>0\<^bold>\<rangle>, \<^bold>\<langle>fg\<^sub>0h\<^sub>1.p\<^sub>1\<^bold>\<rangle>\<^bold>] \<^bold>\<cdot> ((\<^bold>\<langle>f\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<a>\<^sup>-\<^sup>1\<^bold>[\<^bold>\<langle>g\<^bold>\<rangle>, \<^bold>\<langle>tab\<^sub>0 g\<^bold>\<rangle>, \<^bold>\<langle>f\<^sub>0g\<^sub>1.p\<^sub>0\<^bold>\<rangle>\<^bold>]) \<^bold>\<star> \<^bold>\<langle>fg\<^sub>0h\<^sub>1.p\<^sub>1\<^bold>\<rangle>) \<^bold>\<cdot> (\<^bold>\<a>\<^bold>[\<^bold>\<langle>f\<^bold>\<rangle>, \<^bold>\<langle>g\<^bold>\<rangle>, \<^bold>\<langle>tab\<^sub>0 g\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>f\<^sub>0g\<^sub>1.p\<^sub>0\<^bold>\<rangle>\<^bold>] \<^bold>\<star> \<^bold>\<langle>fg\<^sub>0h\<^sub>1.p\<^sub>1\<^bold>\<rangle>)\<rbrace>
[PROOF STEP]
by simp
[PROOF STATE]
proof (state)
this:
\<a>\<^sup>-\<^sup>1[f, g, (tab\<^sub>0 g \<star> f\<^sub>0g\<^sub>1.p\<^sub>0) \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1] \<cdot> (f \<star> \<a>[g, tab\<^sub>0 g \<star> f\<^sub>0g\<^sub>1.p\<^sub>0, fg\<^sub>0h\<^sub>1.p\<^sub>1]) \<cdot> (f \<star> \<a>[g, tab\<^sub>0 g, f\<^sub>0g\<^sub>1.p\<^sub>0] \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> \<a>[f, (g \<star> tab\<^sub>0 g) \<star> f\<^sub>0g\<^sub>1.p\<^sub>0, fg\<^sub>0h\<^sub>1.p\<^sub>1] \<cdot> ((f \<star> \<a>\<^sup>-\<^sup>1[g, tab\<^sub>0 g, f\<^sub>0g\<^sub>1.p\<^sub>0]) \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> (\<a>[f, g, tab\<^sub>0 g \<star> f\<^sub>0g\<^sub>1.p\<^sub>0] \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) = \<lbrace>\<^bold>\<a>\<^sup>-\<^sup>1\<^bold>[\<^bold>\<langle>f\<^bold>\<rangle>, \<^bold>\<langle>g\<^bold>\<rangle>, (\<^bold>\<langle>tab\<^sub>0 g\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>f\<^sub>0g\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>fg\<^sub>0h\<^sub>1.p\<^sub>1\<^bold>\<rangle>\<^bold>] \<^bold>\<cdot> (\<^bold>\<langle>f\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<a>\<^bold>[\<^bold>\<langle>g\<^bold>\<rangle>, \<^bold>\<langle>tab\<^sub>0 g\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>f\<^sub>0g\<^sub>1.p\<^sub>0\<^bold>\<rangle>, \<^bold>\<langle>fg\<^sub>0h\<^sub>1.p\<^sub>1\<^bold>\<rangle>\<^bold>]) \<^bold>\<cdot> (\<^bold>\<langle>f\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<a>\<^bold>[\<^bold>\<langle>g\<^bold>\<rangle>, \<^bold>\<langle>tab\<^sub>0 g\<^bold>\<rangle>, \<^bold>\<langle>f\<^sub>0g\<^sub>1.p\<^sub>0\<^bold>\<rangle>\<^bold>] \<^bold>\<star> \<^bold>\<langle>fg\<^sub>0h\<^sub>1.p\<^sub>1\<^bold>\<rangle>) \<^bold>\<cdot> \<^bold>\<a>\<^bold>[\<^bold>\<langle>f\<^bold>\<rangle>, (\<^bold>\<langle>g\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>tab\<^sub>0 g\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0g\<^sub>1.p\<^sub>0\<^bold>\<rangle>, \<^bold>\<langle>fg\<^sub>0h\<^sub>1.p\<^sub>1\<^bold>\<rangle>\<^bold>] \<^bold>\<cdot> ((\<^bold>\<langle>f\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<a>\<^sup>-\<^sup>1\<^bold>[\<^bold>\<langle>g\<^bold>\<rangle>, \<^bold>\<langle>tab\<^sub>0 g\<^bold>\<rangle>, \<^bold>\<langle>f\<^sub>0g\<^sub>1.p\<^sub>0\<^bold>\<rangle>\<^bold>]) \<^bold>\<star> \<^bold>\<langle>fg\<^sub>0h\<^sub>1.p\<^sub>1\<^bold>\<rangle>) \<^bold>\<cdot> (\<^bold>\<a>\<^bold>[\<^bold>\<langle>f\<^bold>\<rangle>, \<^bold>\<langle>g\<^bold>\<rangle>, \<^bold>\<langle>tab\<^sub>0 g\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>f\<^sub>0g\<^sub>1.p\<^sub>0\<^bold>\<rangle>\<^bold>] \<^bold>\<star> \<^bold>\<langle>fg\<^sub>0h\<^sub>1.p\<^sub>1\<^bold>\<rangle>)\<rbrace>
goal (1 subgoal):
1. \<a>[f, g, h \<star> tab\<^sub>0 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0] \<cdot> ((f \<star> g) \<star> \<a>[h, tab\<^sub>0 h, fg\<^sub>0h\<^sub>1.p\<^sub>0]) \<cdot> ((f \<star> g) \<star> h.tab \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0) \<cdot> ((f \<star> g) \<star> fg\<^sub>0h\<^sub>1.\<phi>) \<cdot> (\<a>\<^sup>-\<^sup>1[f, g, (tab\<^sub>0 g \<star> f\<^sub>0g\<^sub>1.p\<^sub>0) \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1] \<cdot> (f \<star> \<a>[g, tab\<^sub>0 g \<star> f\<^sub>0g\<^sub>1.p\<^sub>0, fg\<^sub>0h\<^sub>1.p\<^sub>1]) \<cdot> (f \<star> \<a>[g, tab\<^sub>0 g, f\<^sub>0g\<^sub>1.p\<^sub>0] \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> \<a>[f, (g \<star> tab\<^sub>0 g) \<star> f\<^sub>0g\<^sub>1.p\<^sub>0, fg\<^sub>0h\<^sub>1.p\<^sub>1] \<cdot> ((f \<star> \<a>\<^sup>-\<^sup>1[g, tab\<^sub>0 g, f\<^sub>0g\<^sub>1.p\<^sub>0]) \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> (\<a>[f, g, tab\<^sub>0 g \<star> f\<^sub>0g\<^sub>1.p\<^sub>0] \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1)) \<cdot> (\<a>\<^sup>-\<^sup>1[f, g, tab\<^sub>0 g \<star> f\<^sub>0g\<^sub>1.p\<^sub>0] \<cdot> (f \<star> \<a>[g, tab\<^sub>0 g, f\<^sub>0g\<^sub>1.p\<^sub>0]) \<cdot> (f \<star> g.tab \<star> f\<^sub>0g\<^sub>1.p\<^sub>0) \<cdot> (f \<star> f\<^sub>0g\<^sub>1.\<phi>) \<cdot> \<a>[f, tab\<^sub>0 f, f\<^sub>0g\<^sub>1.p\<^sub>1] \<cdot> (f.tab \<star> f\<^sub>0g\<^sub>1.p\<^sub>1) \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>1 f, f\<^sub>0g\<^sub>1.p\<^sub>1, fg\<^sub>0h\<^sub>1.p\<^sub>1] = \<a>[f, g, h \<star> tab\<^sub>0 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0] \<cdot> ((f \<star> g) \<star> \<a>[h, tab\<^sub>0 h, fg\<^sub>0h\<^sub>1.p\<^sub>0]) \<cdot> ((f \<star> g) \<star> h.tab \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0) \<cdot> ((f \<star> g) \<star> fg\<^sub>0h\<^sub>1.\<phi>) \<cdot> \<a>[f \<star> g, tab\<^sub>0 g \<star> f\<^sub>0g\<^sub>1.p\<^sub>0, fg\<^sub>0h\<^sub>1.p\<^sub>1] \<cdot> (\<a>\<^sup>-\<^sup>1[f, g, tab\<^sub>0 g \<star> f\<^sub>0g\<^sub>1.p\<^sub>0] \<cdot> (f \<star> \<a>[g, tab\<^sub>0 g, f\<^sub>0g\<^sub>1.p\<^sub>0]) \<cdot> (f \<star> g.tab \<star> f\<^sub>0g\<^sub>1.p\<^sub>0) \<cdot> (f \<star> f\<^sub>0g\<^sub>1.\<phi>) \<cdot> \<a>[f, tab\<^sub>0 f, f\<^sub>0g\<^sub>1.p\<^sub>1] \<cdot> (f.tab \<star> f\<^sub>0g\<^sub>1.p\<^sub>1) \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>1 f, f\<^sub>0g\<^sub>1.p\<^sub>1, fg\<^sub>0h\<^sub>1.p\<^sub>1]
[PROOF STEP]
also
[PROOF STATE]
proof (state)
this:
\<a>\<^sup>-\<^sup>1[f, g, (tab\<^sub>0 g \<star> f\<^sub>0g\<^sub>1.p\<^sub>0) \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1] \<cdot> (f \<star> \<a>[g, tab\<^sub>0 g \<star> f\<^sub>0g\<^sub>1.p\<^sub>0, fg\<^sub>0h\<^sub>1.p\<^sub>1]) \<cdot> (f \<star> \<a>[g, tab\<^sub>0 g, f\<^sub>0g\<^sub>1.p\<^sub>0] \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> \<a>[f, (g \<star> tab\<^sub>0 g) \<star> f\<^sub>0g\<^sub>1.p\<^sub>0, fg\<^sub>0h\<^sub>1.p\<^sub>1] \<cdot> ((f \<star> \<a>\<^sup>-\<^sup>1[g, tab\<^sub>0 g, f\<^sub>0g\<^sub>1.p\<^sub>0]) \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> (\<a>[f, g, tab\<^sub>0 g \<star> f\<^sub>0g\<^sub>1.p\<^sub>0] \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) = \<lbrace>\<^bold>\<a>\<^sup>-\<^sup>1\<^bold>[\<^bold>\<langle>f\<^bold>\<rangle>, \<^bold>\<langle>g\<^bold>\<rangle>, (\<^bold>\<langle>tab\<^sub>0 g\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>f\<^sub>0g\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>fg\<^sub>0h\<^sub>1.p\<^sub>1\<^bold>\<rangle>\<^bold>] \<^bold>\<cdot> (\<^bold>\<langle>f\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<a>\<^bold>[\<^bold>\<langle>g\<^bold>\<rangle>, \<^bold>\<langle>tab\<^sub>0 g\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>f\<^sub>0g\<^sub>1.p\<^sub>0\<^bold>\<rangle>, \<^bold>\<langle>fg\<^sub>0h\<^sub>1.p\<^sub>1\<^bold>\<rangle>\<^bold>]) \<^bold>\<cdot> (\<^bold>\<langle>f\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<a>\<^bold>[\<^bold>\<langle>g\<^bold>\<rangle>, \<^bold>\<langle>tab\<^sub>0 g\<^bold>\<rangle>, \<^bold>\<langle>f\<^sub>0g\<^sub>1.p\<^sub>0\<^bold>\<rangle>\<^bold>] \<^bold>\<star> \<^bold>\<langle>fg\<^sub>0h\<^sub>1.p\<^sub>1\<^bold>\<rangle>) \<^bold>\<cdot> \<^bold>\<a>\<^bold>[\<^bold>\<langle>f\<^bold>\<rangle>, (\<^bold>\<langle>g\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>tab\<^sub>0 g\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0g\<^sub>1.p\<^sub>0\<^bold>\<rangle>, \<^bold>\<langle>fg\<^sub>0h\<^sub>1.p\<^sub>1\<^bold>\<rangle>\<^bold>] \<^bold>\<cdot> ((\<^bold>\<langle>f\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<a>\<^sup>-\<^sup>1\<^bold>[\<^bold>\<langle>g\<^bold>\<rangle>, \<^bold>\<langle>tab\<^sub>0 g\<^bold>\<rangle>, \<^bold>\<langle>f\<^sub>0g\<^sub>1.p\<^sub>0\<^bold>\<rangle>\<^bold>]) \<^bold>\<star> \<^bold>\<langle>fg\<^sub>0h\<^sub>1.p\<^sub>1\<^bold>\<rangle>) \<^bold>\<cdot> (\<^bold>\<a>\<^bold>[\<^bold>\<langle>f\<^bold>\<rangle>, \<^bold>\<langle>g\<^bold>\<rangle>, \<^bold>\<langle>tab\<^sub>0 g\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>f\<^sub>0g\<^sub>1.p\<^sub>0\<^bold>\<rangle>\<^bold>] \<^bold>\<star> \<^bold>\<langle>fg\<^sub>0h\<^sub>1.p\<^sub>1\<^bold>\<rangle>)\<rbrace>
goal (1 subgoal):
1. \<a>[f, g, h \<star> tab\<^sub>0 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0] \<cdot> ((f \<star> g) \<star> \<a>[h, tab\<^sub>0 h, fg\<^sub>0h\<^sub>1.p\<^sub>0]) \<cdot> ((f \<star> g) \<star> h.tab \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0) \<cdot> ((f \<star> g) \<star> fg\<^sub>0h\<^sub>1.\<phi>) \<cdot> (\<a>\<^sup>-\<^sup>1[f, g, (tab\<^sub>0 g \<star> f\<^sub>0g\<^sub>1.p\<^sub>0) \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1] \<cdot> (f \<star> \<a>[g, tab\<^sub>0 g \<star> f\<^sub>0g\<^sub>1.p\<^sub>0, fg\<^sub>0h\<^sub>1.p\<^sub>1]) \<cdot> (f \<star> \<a>[g, tab\<^sub>0 g, f\<^sub>0g\<^sub>1.p\<^sub>0] \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> \<a>[f, (g \<star> tab\<^sub>0 g) \<star> f\<^sub>0g\<^sub>1.p\<^sub>0, fg\<^sub>0h\<^sub>1.p\<^sub>1] \<cdot> ((f \<star> \<a>\<^sup>-\<^sup>1[g, tab\<^sub>0 g, f\<^sub>0g\<^sub>1.p\<^sub>0]) \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> (\<a>[f, g, tab\<^sub>0 g \<star> f\<^sub>0g\<^sub>1.p\<^sub>0] \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1)) \<cdot> (\<a>\<^sup>-\<^sup>1[f, g, tab\<^sub>0 g \<star> f\<^sub>0g\<^sub>1.p\<^sub>0] \<cdot> (f \<star> \<a>[g, tab\<^sub>0 g, f\<^sub>0g\<^sub>1.p\<^sub>0]) \<cdot> (f \<star> g.tab \<star> f\<^sub>0g\<^sub>1.p\<^sub>0) \<cdot> (f \<star> f\<^sub>0g\<^sub>1.\<phi>) \<cdot> \<a>[f, tab\<^sub>0 f, f\<^sub>0g\<^sub>1.p\<^sub>1] \<cdot> (f.tab \<star> f\<^sub>0g\<^sub>1.p\<^sub>1) \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>1 f, f\<^sub>0g\<^sub>1.p\<^sub>1, fg\<^sub>0h\<^sub>1.p\<^sub>1] = \<a>[f, g, h \<star> tab\<^sub>0 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0] \<cdot> ((f \<star> g) \<star> \<a>[h, tab\<^sub>0 h, fg\<^sub>0h\<^sub>1.p\<^sub>0]) \<cdot> ((f \<star> g) \<star> h.tab \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0) \<cdot> ((f \<star> g) \<star> fg\<^sub>0h\<^sub>1.\<phi>) \<cdot> \<a>[f \<star> g, tab\<^sub>0 g \<star> f\<^sub>0g\<^sub>1.p\<^sub>0, fg\<^sub>0h\<^sub>1.p\<^sub>1] \<cdot> (\<a>\<^sup>-\<^sup>1[f, g, tab\<^sub>0 g \<star> f\<^sub>0g\<^sub>1.p\<^sub>0] \<cdot> (f \<star> \<a>[g, tab\<^sub>0 g, f\<^sub>0g\<^sub>1.p\<^sub>0]) \<cdot> (f \<star> g.tab \<star> f\<^sub>0g\<^sub>1.p\<^sub>0) \<cdot> (f \<star> f\<^sub>0g\<^sub>1.\<phi>) \<cdot> \<a>[f, tab\<^sub>0 f, f\<^sub>0g\<^sub>1.p\<^sub>1] \<cdot> (f.tab \<star> f\<^sub>0g\<^sub>1.p\<^sub>1) \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>1 f, f\<^sub>0g\<^sub>1.p\<^sub>1, fg\<^sub>0h\<^sub>1.p\<^sub>1]
[PROOF STEP]
have "... = \<lbrace>\<^bold>\<a>\<^bold>[\<^bold>\<langle>f\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>g\<^bold>\<rangle>, \<^bold>\<langle>tab\<^sub>0 g\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>Tfg.p\<^sub>0\<^bold>\<rangle>, \<^bold>\<langle>TTfgh.p\<^sub>1\<^bold>\<rangle>\<^bold>]\<rbrace>"
[PROOF STATE]
proof (prove)
goal (1 subgoal):
1. \<lbrace>\<^bold>\<a>\<^sup>-\<^sup>1\<^bold>[\<^bold>\<langle>f\<^bold>\<rangle>, \<^bold>\<langle>g\<^bold>\<rangle>, (\<^bold>\<langle>tab\<^sub>0 g\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>f\<^sub>0g\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>fg\<^sub>0h\<^sub>1.p\<^sub>1\<^bold>\<rangle>\<^bold>] \<^bold>\<cdot> (\<^bold>\<langle>f\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<a>\<^bold>[\<^bold>\<langle>g\<^bold>\<rangle>, \<^bold>\<langle>tab\<^sub>0 g\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>f\<^sub>0g\<^sub>1.p\<^sub>0\<^bold>\<rangle>, \<^bold>\<langle>fg\<^sub>0h\<^sub>1.p\<^sub>1\<^bold>\<rangle>\<^bold>]) \<^bold>\<cdot> (\<^bold>\<langle>f\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<a>\<^bold>[\<^bold>\<langle>g\<^bold>\<rangle>, \<^bold>\<langle>tab\<^sub>0 g\<^bold>\<rangle>, \<^bold>\<langle>f\<^sub>0g\<^sub>1.p\<^sub>0\<^bold>\<rangle>\<^bold>] \<^bold>\<star> \<^bold>\<langle>fg\<^sub>0h\<^sub>1.p\<^sub>1\<^bold>\<rangle>) \<^bold>\<cdot> \<^bold>\<a>\<^bold>[\<^bold>\<langle>f\<^bold>\<rangle>, (\<^bold>\<langle>g\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>tab\<^sub>0 g\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0g\<^sub>1.p\<^sub>0\<^bold>\<rangle>, \<^bold>\<langle>fg\<^sub>0h\<^sub>1.p\<^sub>1\<^bold>\<rangle>\<^bold>] \<^bold>\<cdot> ((\<^bold>\<langle>f\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<a>\<^sup>-\<^sup>1\<^bold>[\<^bold>\<langle>g\<^bold>\<rangle>, \<^bold>\<langle>tab\<^sub>0 g\<^bold>\<rangle>, \<^bold>\<langle>f\<^sub>0g\<^sub>1.p\<^sub>0\<^bold>\<rangle>\<^bold>]) \<^bold>\<star> \<^bold>\<langle>fg\<^sub>0h\<^sub>1.p\<^sub>1\<^bold>\<rangle>) \<^bold>\<cdot> (\<^bold>\<a>\<^bold>[\<^bold>\<langle>f\<^bold>\<rangle>, \<^bold>\<langle>g\<^bold>\<rangle>, \<^bold>\<langle>tab\<^sub>0 g\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>f\<^sub>0g\<^sub>1.p\<^sub>0\<^bold>\<rangle>\<^bold>] \<^bold>\<star> \<^bold>\<langle>fg\<^sub>0h\<^sub>1.p\<^sub>1\<^bold>\<rangle>)\<rbrace> = \<lbrace>\<^bold>\<a>\<^bold>[\<^bold>\<langle>f\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>g\<^bold>\<rangle>, \<^bold>\<langle>tab\<^sub>0 g\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>f\<^sub>0g\<^sub>1.p\<^sub>0\<^bold>\<rangle>, \<^bold>\<langle>fg\<^sub>0h\<^sub>1.p\<^sub>1\<^bold>\<rangle>\<^bold>]\<rbrace>
[PROOF STEP]
using fg gh fg\<^sub>0h\<^sub>1.p\<^sub>0_simps fg\<^sub>0h\<^sub>1.p\<^sub>1_simps f\<^sub>0g\<^sub>1.p\<^sub>0_simps f\<^sub>0g\<^sub>1.p\<^sub>1_simps
[PROOF STATE]
proof (prove)
using this:
src f = trg g
src g = trg h
trg fg\<^sub>0h\<^sub>1.p\<^sub>0 = src (tab\<^sub>1 h)
trg fg\<^sub>0h\<^sub>1.p\<^sub>1 = src (tab\<^sub>0 g \<star> f\<^sub>0g\<^sub>1.p\<^sub>0)
trg f\<^sub>0g\<^sub>1.p\<^sub>0 = src (tab\<^sub>1 g)
trg f\<^sub>0g\<^sub>1.p\<^sub>1 = src (tab\<^sub>0 f)
goal (1 subgoal):
1. \<lbrace>\<^bold>\<a>\<^sup>-\<^sup>1\<^bold>[\<^bold>\<langle>f\<^bold>\<rangle>, \<^bold>\<langle>g\<^bold>\<rangle>, (\<^bold>\<langle>tab\<^sub>0 g\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>f\<^sub>0g\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>fg\<^sub>0h\<^sub>1.p\<^sub>1\<^bold>\<rangle>\<^bold>] \<^bold>\<cdot> (\<^bold>\<langle>f\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<a>\<^bold>[\<^bold>\<langle>g\<^bold>\<rangle>, \<^bold>\<langle>tab\<^sub>0 g\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>f\<^sub>0g\<^sub>1.p\<^sub>0\<^bold>\<rangle>, \<^bold>\<langle>fg\<^sub>0h\<^sub>1.p\<^sub>1\<^bold>\<rangle>\<^bold>]) \<^bold>\<cdot> (\<^bold>\<langle>f\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<a>\<^bold>[\<^bold>\<langle>g\<^bold>\<rangle>, \<^bold>\<langle>tab\<^sub>0 g\<^bold>\<rangle>, \<^bold>\<langle>f\<^sub>0g\<^sub>1.p\<^sub>0\<^bold>\<rangle>\<^bold>] \<^bold>\<star> \<^bold>\<langle>fg\<^sub>0h\<^sub>1.p\<^sub>1\<^bold>\<rangle>) \<^bold>\<cdot> \<^bold>\<a>\<^bold>[\<^bold>\<langle>f\<^bold>\<rangle>, (\<^bold>\<langle>g\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>tab\<^sub>0 g\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0g\<^sub>1.p\<^sub>0\<^bold>\<rangle>, \<^bold>\<langle>fg\<^sub>0h\<^sub>1.p\<^sub>1\<^bold>\<rangle>\<^bold>] \<^bold>\<cdot> ((\<^bold>\<langle>f\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<a>\<^sup>-\<^sup>1\<^bold>[\<^bold>\<langle>g\<^bold>\<rangle>, \<^bold>\<langle>tab\<^sub>0 g\<^bold>\<rangle>, \<^bold>\<langle>f\<^sub>0g\<^sub>1.p\<^sub>0\<^bold>\<rangle>\<^bold>]) \<^bold>\<star> \<^bold>\<langle>fg\<^sub>0h\<^sub>1.p\<^sub>1\<^bold>\<rangle>) \<^bold>\<cdot> (\<^bold>\<a>\<^bold>[\<^bold>\<langle>f\<^bold>\<rangle>, \<^bold>\<langle>g\<^bold>\<rangle>, \<^bold>\<langle>tab\<^sub>0 g\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>f\<^sub>0g\<^sub>1.p\<^sub>0\<^bold>\<rangle>\<^bold>] \<^bold>\<star> \<^bold>\<langle>fg\<^sub>0h\<^sub>1.p\<^sub>1\<^bold>\<rangle>)\<rbrace> = \<lbrace>\<^bold>\<a>\<^bold>[\<^bold>\<langle>f\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>g\<^bold>\<rangle>, \<^bold>\<langle>tab\<^sub>0 g\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>f\<^sub>0g\<^sub>1.p\<^sub>0\<^bold>\<rangle>, \<^bold>\<langle>fg\<^sub>0h\<^sub>1.p\<^sub>1\<^bold>\<rangle>\<^bold>]\<rbrace>
[PROOF STEP]
by (intro E.eval_eqI, auto)
[PROOF STATE]
proof (state)
this:
\<lbrace>\<^bold>\<a>\<^sup>-\<^sup>1\<^bold>[\<^bold>\<langle>f\<^bold>\<rangle>, \<^bold>\<langle>g\<^bold>\<rangle>, (\<^bold>\<langle>tab\<^sub>0 g\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>f\<^sub>0g\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>fg\<^sub>0h\<^sub>1.p\<^sub>1\<^bold>\<rangle>\<^bold>] \<^bold>\<cdot> (\<^bold>\<langle>f\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<a>\<^bold>[\<^bold>\<langle>g\<^bold>\<rangle>, \<^bold>\<langle>tab\<^sub>0 g\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>f\<^sub>0g\<^sub>1.p\<^sub>0\<^bold>\<rangle>, \<^bold>\<langle>fg\<^sub>0h\<^sub>1.p\<^sub>1\<^bold>\<rangle>\<^bold>]) \<^bold>\<cdot> (\<^bold>\<langle>f\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<a>\<^bold>[\<^bold>\<langle>g\<^bold>\<rangle>, \<^bold>\<langle>tab\<^sub>0 g\<^bold>\<rangle>, \<^bold>\<langle>f\<^sub>0g\<^sub>1.p\<^sub>0\<^bold>\<rangle>\<^bold>] \<^bold>\<star> \<^bold>\<langle>fg\<^sub>0h\<^sub>1.p\<^sub>1\<^bold>\<rangle>) \<^bold>\<cdot> \<^bold>\<a>\<^bold>[\<^bold>\<langle>f\<^bold>\<rangle>, (\<^bold>\<langle>g\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>tab\<^sub>0 g\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0g\<^sub>1.p\<^sub>0\<^bold>\<rangle>, \<^bold>\<langle>fg\<^sub>0h\<^sub>1.p\<^sub>1\<^bold>\<rangle>\<^bold>] \<^bold>\<cdot> ((\<^bold>\<langle>f\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<a>\<^sup>-\<^sup>1\<^bold>[\<^bold>\<langle>g\<^bold>\<rangle>, \<^bold>\<langle>tab\<^sub>0 g\<^bold>\<rangle>, \<^bold>\<langle>f\<^sub>0g\<^sub>1.p\<^sub>0\<^bold>\<rangle>\<^bold>]) \<^bold>\<star> \<^bold>\<langle>fg\<^sub>0h\<^sub>1.p\<^sub>1\<^bold>\<rangle>) \<^bold>\<cdot> (\<^bold>\<a>\<^bold>[\<^bold>\<langle>f\<^bold>\<rangle>, \<^bold>\<langle>g\<^bold>\<rangle>, \<^bold>\<langle>tab\<^sub>0 g\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>f\<^sub>0g\<^sub>1.p\<^sub>0\<^bold>\<rangle>\<^bold>] \<^bold>\<star> \<^bold>\<langle>fg\<^sub>0h\<^sub>1.p\<^sub>1\<^bold>\<rangle>)\<rbrace> = \<lbrace>\<^bold>\<a>\<^bold>[\<^bold>\<langle>f\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>g\<^bold>\<rangle>, \<^bold>\<langle>tab\<^sub>0 g\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>f\<^sub>0g\<^sub>1.p\<^sub>0\<^bold>\<rangle>, \<^bold>\<langle>fg\<^sub>0h\<^sub>1.p\<^sub>1\<^bold>\<rangle>\<^bold>]\<rbrace>
goal (1 subgoal):
1. \<a>[f, g, h \<star> tab\<^sub>0 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0] \<cdot> ((f \<star> g) \<star> \<a>[h, tab\<^sub>0 h, fg\<^sub>0h\<^sub>1.p\<^sub>0]) \<cdot> ((f \<star> g) \<star> h.tab \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0) \<cdot> ((f \<star> g) \<star> fg\<^sub>0h\<^sub>1.\<phi>) \<cdot> (\<a>\<^sup>-\<^sup>1[f, g, (tab\<^sub>0 g \<star> f\<^sub>0g\<^sub>1.p\<^sub>0) \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1] \<cdot> (f \<star> \<a>[g, tab\<^sub>0 g \<star> f\<^sub>0g\<^sub>1.p\<^sub>0, fg\<^sub>0h\<^sub>1.p\<^sub>1]) \<cdot> (f \<star> \<a>[g, tab\<^sub>0 g, f\<^sub>0g\<^sub>1.p\<^sub>0] \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> \<a>[f, (g \<star> tab\<^sub>0 g) \<star> f\<^sub>0g\<^sub>1.p\<^sub>0, fg\<^sub>0h\<^sub>1.p\<^sub>1] \<cdot> ((f \<star> \<a>\<^sup>-\<^sup>1[g, tab\<^sub>0 g, f\<^sub>0g\<^sub>1.p\<^sub>0]) \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> (\<a>[f, g, tab\<^sub>0 g \<star> f\<^sub>0g\<^sub>1.p\<^sub>0] \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1)) \<cdot> (\<a>\<^sup>-\<^sup>1[f, g, tab\<^sub>0 g \<star> f\<^sub>0g\<^sub>1.p\<^sub>0] \<cdot> (f \<star> \<a>[g, tab\<^sub>0 g, f\<^sub>0g\<^sub>1.p\<^sub>0]) \<cdot> (f \<star> g.tab \<star> f\<^sub>0g\<^sub>1.p\<^sub>0) \<cdot> (f \<star> f\<^sub>0g\<^sub>1.\<phi>) \<cdot> \<a>[f, tab\<^sub>0 f, f\<^sub>0g\<^sub>1.p\<^sub>1] \<cdot> (f.tab \<star> f\<^sub>0g\<^sub>1.p\<^sub>1) \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>1 f, f\<^sub>0g\<^sub>1.p\<^sub>1, fg\<^sub>0h\<^sub>1.p\<^sub>1] = \<a>[f, g, h \<star> tab\<^sub>0 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0] \<cdot> ((f \<star> g) \<star> \<a>[h, tab\<^sub>0 h, fg\<^sub>0h\<^sub>1.p\<^sub>0]) \<cdot> ((f \<star> g) \<star> h.tab \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0) \<cdot> ((f \<star> g) \<star> fg\<^sub>0h\<^sub>1.\<phi>) \<cdot> \<a>[f \<star> g, tab\<^sub>0 g \<star> f\<^sub>0g\<^sub>1.p\<^sub>0, fg\<^sub>0h\<^sub>1.p\<^sub>1] \<cdot> (\<a>\<^sup>-\<^sup>1[f, g, tab\<^sub>0 g \<star> f\<^sub>0g\<^sub>1.p\<^sub>0] \<cdot> (f \<star> \<a>[g, tab\<^sub>0 g, f\<^sub>0g\<^sub>1.p\<^sub>0]) \<cdot> (f \<star> g.tab \<star> f\<^sub>0g\<^sub>1.p\<^sub>0) \<cdot> (f \<star> f\<^sub>0g\<^sub>1.\<phi>) \<cdot> \<a>[f, tab\<^sub>0 f, f\<^sub>0g\<^sub>1.p\<^sub>1] \<cdot> (f.tab \<star> f\<^sub>0g\<^sub>1.p\<^sub>1) \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>1 f, f\<^sub>0g\<^sub>1.p\<^sub>1, fg\<^sub>0h\<^sub>1.p\<^sub>1]
[PROOF STEP]
also
[PROOF STATE]
proof (state)
this:
\<lbrace>\<^bold>\<a>\<^sup>-\<^sup>1\<^bold>[\<^bold>\<langle>f\<^bold>\<rangle>, \<^bold>\<langle>g\<^bold>\<rangle>, (\<^bold>\<langle>tab\<^sub>0 g\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>f\<^sub>0g\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>fg\<^sub>0h\<^sub>1.p\<^sub>1\<^bold>\<rangle>\<^bold>] \<^bold>\<cdot> (\<^bold>\<langle>f\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<a>\<^bold>[\<^bold>\<langle>g\<^bold>\<rangle>, \<^bold>\<langle>tab\<^sub>0 g\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>f\<^sub>0g\<^sub>1.p\<^sub>0\<^bold>\<rangle>, \<^bold>\<langle>fg\<^sub>0h\<^sub>1.p\<^sub>1\<^bold>\<rangle>\<^bold>]) \<^bold>\<cdot> (\<^bold>\<langle>f\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<a>\<^bold>[\<^bold>\<langle>g\<^bold>\<rangle>, \<^bold>\<langle>tab\<^sub>0 g\<^bold>\<rangle>, \<^bold>\<langle>f\<^sub>0g\<^sub>1.p\<^sub>0\<^bold>\<rangle>\<^bold>] \<^bold>\<star> \<^bold>\<langle>fg\<^sub>0h\<^sub>1.p\<^sub>1\<^bold>\<rangle>) \<^bold>\<cdot> \<^bold>\<a>\<^bold>[\<^bold>\<langle>f\<^bold>\<rangle>, (\<^bold>\<langle>g\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>tab\<^sub>0 g\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0g\<^sub>1.p\<^sub>0\<^bold>\<rangle>, \<^bold>\<langle>fg\<^sub>0h\<^sub>1.p\<^sub>1\<^bold>\<rangle>\<^bold>] \<^bold>\<cdot> ((\<^bold>\<langle>f\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<a>\<^sup>-\<^sup>1\<^bold>[\<^bold>\<langle>g\<^bold>\<rangle>, \<^bold>\<langle>tab\<^sub>0 g\<^bold>\<rangle>, \<^bold>\<langle>f\<^sub>0g\<^sub>1.p\<^sub>0\<^bold>\<rangle>\<^bold>]) \<^bold>\<star> \<^bold>\<langle>fg\<^sub>0h\<^sub>1.p\<^sub>1\<^bold>\<rangle>) \<^bold>\<cdot> (\<^bold>\<a>\<^bold>[\<^bold>\<langle>f\<^bold>\<rangle>, \<^bold>\<langle>g\<^bold>\<rangle>, \<^bold>\<langle>tab\<^sub>0 g\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>f\<^sub>0g\<^sub>1.p\<^sub>0\<^bold>\<rangle>\<^bold>] \<^bold>\<star> \<^bold>\<langle>fg\<^sub>0h\<^sub>1.p\<^sub>1\<^bold>\<rangle>)\<rbrace> = \<lbrace>\<^bold>\<a>\<^bold>[\<^bold>\<langle>f\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>g\<^bold>\<rangle>, \<^bold>\<langle>tab\<^sub>0 g\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>f\<^sub>0g\<^sub>1.p\<^sub>0\<^bold>\<rangle>, \<^bold>\<langle>fg\<^sub>0h\<^sub>1.p\<^sub>1\<^bold>\<rangle>\<^bold>]\<rbrace>
goal (1 subgoal):
1. \<a>[f, g, h \<star> tab\<^sub>0 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0] \<cdot> ((f \<star> g) \<star> \<a>[h, tab\<^sub>0 h, fg\<^sub>0h\<^sub>1.p\<^sub>0]) \<cdot> ((f \<star> g) \<star> h.tab \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0) \<cdot> ((f \<star> g) \<star> fg\<^sub>0h\<^sub>1.\<phi>) \<cdot> (\<a>\<^sup>-\<^sup>1[f, g, (tab\<^sub>0 g \<star> f\<^sub>0g\<^sub>1.p\<^sub>0) \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1] \<cdot> (f \<star> \<a>[g, tab\<^sub>0 g \<star> f\<^sub>0g\<^sub>1.p\<^sub>0, fg\<^sub>0h\<^sub>1.p\<^sub>1]) \<cdot> (f \<star> \<a>[g, tab\<^sub>0 g, f\<^sub>0g\<^sub>1.p\<^sub>0] \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> \<a>[f, (g \<star> tab\<^sub>0 g) \<star> f\<^sub>0g\<^sub>1.p\<^sub>0, fg\<^sub>0h\<^sub>1.p\<^sub>1] \<cdot> ((f \<star> \<a>\<^sup>-\<^sup>1[g, tab\<^sub>0 g, f\<^sub>0g\<^sub>1.p\<^sub>0]) \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> (\<a>[f, g, tab\<^sub>0 g \<star> f\<^sub>0g\<^sub>1.p\<^sub>0] \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1)) \<cdot> (\<a>\<^sup>-\<^sup>1[f, g, tab\<^sub>0 g \<star> f\<^sub>0g\<^sub>1.p\<^sub>0] \<cdot> (f \<star> \<a>[g, tab\<^sub>0 g, f\<^sub>0g\<^sub>1.p\<^sub>0]) \<cdot> (f \<star> g.tab \<star> f\<^sub>0g\<^sub>1.p\<^sub>0) \<cdot> (f \<star> f\<^sub>0g\<^sub>1.\<phi>) \<cdot> \<a>[f, tab\<^sub>0 f, f\<^sub>0g\<^sub>1.p\<^sub>1] \<cdot> (f.tab \<star> f\<^sub>0g\<^sub>1.p\<^sub>1) \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>1 f, f\<^sub>0g\<^sub>1.p\<^sub>1, fg\<^sub>0h\<^sub>1.p\<^sub>1] = \<a>[f, g, h \<star> tab\<^sub>0 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0] \<cdot> ((f \<star> g) \<star> \<a>[h, tab\<^sub>0 h, fg\<^sub>0h\<^sub>1.p\<^sub>0]) \<cdot> ((f \<star> g) \<star> h.tab \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0) \<cdot> ((f \<star> g) \<star> fg\<^sub>0h\<^sub>1.\<phi>) \<cdot> \<a>[f \<star> g, tab\<^sub>0 g \<star> f\<^sub>0g\<^sub>1.p\<^sub>0, fg\<^sub>0h\<^sub>1.p\<^sub>1] \<cdot> (\<a>\<^sup>-\<^sup>1[f, g, tab\<^sub>0 g \<star> f\<^sub>0g\<^sub>1.p\<^sub>0] \<cdot> (f \<star> \<a>[g, tab\<^sub>0 g, f\<^sub>0g\<^sub>1.p\<^sub>0]) \<cdot> (f \<star> g.tab \<star> f\<^sub>0g\<^sub>1.p\<^sub>0) \<cdot> (f \<star> f\<^sub>0g\<^sub>1.\<phi>) \<cdot> \<a>[f, tab\<^sub>0 f, f\<^sub>0g\<^sub>1.p\<^sub>1] \<cdot> (f.tab \<star> f\<^sub>0g\<^sub>1.p\<^sub>1) \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>1 f, f\<^sub>0g\<^sub>1.p\<^sub>1, fg\<^sub>0h\<^sub>1.p\<^sub>1]
[PROOF STEP]
have "... = \<a>[f \<star> g, tab\<^sub>0 g \<star> Tfg.p\<^sub>0, TTfgh.p\<^sub>1]"
[PROOF STATE]
proof (prove)
goal (1 subgoal):
1. \<lbrace>\<^bold>\<a>\<^bold>[\<^bold>\<langle>f\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>g\<^bold>\<rangle>, \<^bold>\<langle>tab\<^sub>0 g\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>f\<^sub>0g\<^sub>1.p\<^sub>0\<^bold>\<rangle>, \<^bold>\<langle>fg\<^sub>0h\<^sub>1.p\<^sub>1\<^bold>\<rangle>\<^bold>]\<rbrace> = \<a>[f \<star> g, tab\<^sub>0 g \<star> f\<^sub>0g\<^sub>1.p\<^sub>0, fg\<^sub>0h\<^sub>1.p\<^sub>1]
[PROOF STEP]
using fg gh fg\<^sub>0h\<^sub>1.p\<^sub>0_simps fg\<^sub>0h\<^sub>1.p\<^sub>1_simps f\<^sub>0g\<^sub>1.p\<^sub>0_simps f\<^sub>0g\<^sub>1.p\<^sub>1_simps
\<a>'_def \<alpha>_def
[PROOF STATE]
proof (prove)
using this:
src f = trg g
src g = trg h
trg fg\<^sub>0h\<^sub>1.p\<^sub>0 = src (tab\<^sub>1 h)
trg fg\<^sub>0h\<^sub>1.p\<^sub>1 = src (tab\<^sub>0 g \<star> f\<^sub>0g\<^sub>1.p\<^sub>0)
trg f\<^sub>0g\<^sub>1.p\<^sub>0 = src (tab\<^sub>1 g)
trg f\<^sub>0g\<^sub>1.p\<^sub>1 = src (tab\<^sub>0 f)
\<a>\<^sup>-\<^sup>1[?\<mu>, ?\<nu>, ?\<tau>] \<equiv> \<alpha>' (?\<mu>, ?\<nu>, ?\<tau>)
\<alpha> ?\<mu>\<nu>\<tau> \<equiv> \<a>[fst ?\<mu>\<nu>\<tau>, fst (snd ?\<mu>\<nu>\<tau>), snd (snd ?\<mu>\<nu>\<tau>)]
goal (1 subgoal):
1. \<lbrace>\<^bold>\<a>\<^bold>[\<^bold>\<langle>f\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>g\<^bold>\<rangle>, \<^bold>\<langle>tab\<^sub>0 g\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>f\<^sub>0g\<^sub>1.p\<^sub>0\<^bold>\<rangle>, \<^bold>\<langle>fg\<^sub>0h\<^sub>1.p\<^sub>1\<^bold>\<rangle>\<^bold>]\<rbrace> = \<a>[f \<star> g, tab\<^sub>0 g \<star> f\<^sub>0g\<^sub>1.p\<^sub>0, fg\<^sub>0h\<^sub>1.p\<^sub>1]
[PROOF STEP]
by simp
[PROOF STATE]
proof (state)
this:
\<lbrace>\<^bold>\<a>\<^bold>[\<^bold>\<langle>f\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>g\<^bold>\<rangle>, \<^bold>\<langle>tab\<^sub>0 g\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>f\<^sub>0g\<^sub>1.p\<^sub>0\<^bold>\<rangle>, \<^bold>\<langle>fg\<^sub>0h\<^sub>1.p\<^sub>1\<^bold>\<rangle>\<^bold>]\<rbrace> = \<a>[f \<star> g, tab\<^sub>0 g \<star> f\<^sub>0g\<^sub>1.p\<^sub>0, fg\<^sub>0h\<^sub>1.p\<^sub>1]
goal (1 subgoal):
1. \<a>[f, g, h \<star> tab\<^sub>0 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0] \<cdot> ((f \<star> g) \<star> \<a>[h, tab\<^sub>0 h, fg\<^sub>0h\<^sub>1.p\<^sub>0]) \<cdot> ((f \<star> g) \<star> h.tab \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0) \<cdot> ((f \<star> g) \<star> fg\<^sub>0h\<^sub>1.\<phi>) \<cdot> (\<a>\<^sup>-\<^sup>1[f, g, (tab\<^sub>0 g \<star> f\<^sub>0g\<^sub>1.p\<^sub>0) \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1] \<cdot> (f \<star> \<a>[g, tab\<^sub>0 g \<star> f\<^sub>0g\<^sub>1.p\<^sub>0, fg\<^sub>0h\<^sub>1.p\<^sub>1]) \<cdot> (f \<star> \<a>[g, tab\<^sub>0 g, f\<^sub>0g\<^sub>1.p\<^sub>0] \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> \<a>[f, (g \<star> tab\<^sub>0 g) \<star> f\<^sub>0g\<^sub>1.p\<^sub>0, fg\<^sub>0h\<^sub>1.p\<^sub>1] \<cdot> ((f \<star> \<a>\<^sup>-\<^sup>1[g, tab\<^sub>0 g, f\<^sub>0g\<^sub>1.p\<^sub>0]) \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> (\<a>[f, g, tab\<^sub>0 g \<star> f\<^sub>0g\<^sub>1.p\<^sub>0] \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1)) \<cdot> (\<a>\<^sup>-\<^sup>1[f, g, tab\<^sub>0 g \<star> f\<^sub>0g\<^sub>1.p\<^sub>0] \<cdot> (f \<star> \<a>[g, tab\<^sub>0 g, f\<^sub>0g\<^sub>1.p\<^sub>0]) \<cdot> (f \<star> g.tab \<star> f\<^sub>0g\<^sub>1.p\<^sub>0) \<cdot> (f \<star> f\<^sub>0g\<^sub>1.\<phi>) \<cdot> \<a>[f, tab\<^sub>0 f, f\<^sub>0g\<^sub>1.p\<^sub>1] \<cdot> (f.tab \<star> f\<^sub>0g\<^sub>1.p\<^sub>1) \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>1 f, f\<^sub>0g\<^sub>1.p\<^sub>1, fg\<^sub>0h\<^sub>1.p\<^sub>1] = \<a>[f, g, h \<star> tab\<^sub>0 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0] \<cdot> ((f \<star> g) \<star> \<a>[h, tab\<^sub>0 h, fg\<^sub>0h\<^sub>1.p\<^sub>0]) \<cdot> ((f \<star> g) \<star> h.tab \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0) \<cdot> ((f \<star> g) \<star> fg\<^sub>0h\<^sub>1.\<phi>) \<cdot> \<a>[f \<star> g, tab\<^sub>0 g \<star> f\<^sub>0g\<^sub>1.p\<^sub>0, fg\<^sub>0h\<^sub>1.p\<^sub>1] \<cdot> (\<a>\<^sup>-\<^sup>1[f, g, tab\<^sub>0 g \<star> f\<^sub>0g\<^sub>1.p\<^sub>0] \<cdot> (f \<star> \<a>[g, tab\<^sub>0 g, f\<^sub>0g\<^sub>1.p\<^sub>0]) \<cdot> (f \<star> g.tab \<star> f\<^sub>0g\<^sub>1.p\<^sub>0) \<cdot> (f \<star> f\<^sub>0g\<^sub>1.\<phi>) \<cdot> \<a>[f, tab\<^sub>0 f, f\<^sub>0g\<^sub>1.p\<^sub>1] \<cdot> (f.tab \<star> f\<^sub>0g\<^sub>1.p\<^sub>1) \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>1 f, f\<^sub>0g\<^sub>1.p\<^sub>1, fg\<^sub>0h\<^sub>1.p\<^sub>1]
[PROOF STEP]
finally
[PROOF STATE]
proof (chain)
picking this:
\<a>\<^sup>-\<^sup>1[f, g, (tab\<^sub>0 g \<star> f\<^sub>0g\<^sub>1.p\<^sub>0) \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1] \<cdot> (f \<star> \<a>[g, tab\<^sub>0 g \<star> f\<^sub>0g\<^sub>1.p\<^sub>0, fg\<^sub>0h\<^sub>1.p\<^sub>1]) \<cdot> (f \<star> \<a>[g, tab\<^sub>0 g, f\<^sub>0g\<^sub>1.p\<^sub>0] \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> \<a>[f, (g \<star> tab\<^sub>0 g) \<star> f\<^sub>0g\<^sub>1.p\<^sub>0, fg\<^sub>0h\<^sub>1.p\<^sub>1] \<cdot> ((f \<star> \<a>\<^sup>-\<^sup>1[g, tab\<^sub>0 g, f\<^sub>0g\<^sub>1.p\<^sub>0]) \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> (\<a>[f, g, tab\<^sub>0 g \<star> f\<^sub>0g\<^sub>1.p\<^sub>0] \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) = \<a>[f \<star> g, tab\<^sub>0 g \<star> f\<^sub>0g\<^sub>1.p\<^sub>0, fg\<^sub>0h\<^sub>1.p\<^sub>1]
[PROOF STEP]
show ?thesis
[PROOF STATE]
proof (prove)
using this:
\<a>\<^sup>-\<^sup>1[f, g, (tab\<^sub>0 g \<star> f\<^sub>0g\<^sub>1.p\<^sub>0) \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1] \<cdot> (f \<star> \<a>[g, tab\<^sub>0 g \<star> f\<^sub>0g\<^sub>1.p\<^sub>0, fg\<^sub>0h\<^sub>1.p\<^sub>1]) \<cdot> (f \<star> \<a>[g, tab\<^sub>0 g, f\<^sub>0g\<^sub>1.p\<^sub>0] \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> \<a>[f, (g \<star> tab\<^sub>0 g) \<star> f\<^sub>0g\<^sub>1.p\<^sub>0, fg\<^sub>0h\<^sub>1.p\<^sub>1] \<cdot> ((f \<star> \<a>\<^sup>-\<^sup>1[g, tab\<^sub>0 g, f\<^sub>0g\<^sub>1.p\<^sub>0]) \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> (\<a>[f, g, tab\<^sub>0 g \<star> f\<^sub>0g\<^sub>1.p\<^sub>0] \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) = \<a>[f \<star> g, tab\<^sub>0 g \<star> f\<^sub>0g\<^sub>1.p\<^sub>0, fg\<^sub>0h\<^sub>1.p\<^sub>1]
goal (1 subgoal):
1. \<a>[f, g, h \<star> tab\<^sub>0 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0] \<cdot> ((f \<star> g) \<star> \<a>[h, tab\<^sub>0 h, fg\<^sub>0h\<^sub>1.p\<^sub>0]) \<cdot> ((f \<star> g) \<star> h.tab \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0) \<cdot> ((f \<star> g) \<star> fg\<^sub>0h\<^sub>1.\<phi>) \<cdot> (\<a>\<^sup>-\<^sup>1[f, g, (tab\<^sub>0 g \<star> f\<^sub>0g\<^sub>1.p\<^sub>0) \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1] \<cdot> (f \<star> \<a>[g, tab\<^sub>0 g \<star> f\<^sub>0g\<^sub>1.p\<^sub>0, fg\<^sub>0h\<^sub>1.p\<^sub>1]) \<cdot> (f \<star> \<a>[g, tab\<^sub>0 g, f\<^sub>0g\<^sub>1.p\<^sub>0] \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> \<a>[f, (g \<star> tab\<^sub>0 g) \<star> f\<^sub>0g\<^sub>1.p\<^sub>0, fg\<^sub>0h\<^sub>1.p\<^sub>1] \<cdot> ((f \<star> \<a>\<^sup>-\<^sup>1[g, tab\<^sub>0 g, f\<^sub>0g\<^sub>1.p\<^sub>0]) \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> (\<a>[f, g, tab\<^sub>0 g \<star> f\<^sub>0g\<^sub>1.p\<^sub>0] \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1)) \<cdot> (\<a>\<^sup>-\<^sup>1[f, g, tab\<^sub>0 g \<star> f\<^sub>0g\<^sub>1.p\<^sub>0] \<cdot> (f \<star> \<a>[g, tab\<^sub>0 g, f\<^sub>0g\<^sub>1.p\<^sub>0]) \<cdot> (f \<star> g.tab \<star> f\<^sub>0g\<^sub>1.p\<^sub>0) \<cdot> (f \<star> f\<^sub>0g\<^sub>1.\<phi>) \<cdot> \<a>[f, tab\<^sub>0 f, f\<^sub>0g\<^sub>1.p\<^sub>1] \<cdot> (f.tab \<star> f\<^sub>0g\<^sub>1.p\<^sub>1) \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>1 f, f\<^sub>0g\<^sub>1.p\<^sub>1, fg\<^sub>0h\<^sub>1.p\<^sub>1] = \<a>[f, g, h \<star> tab\<^sub>0 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0] \<cdot> ((f \<star> g) \<star> \<a>[h, tab\<^sub>0 h, fg\<^sub>0h\<^sub>1.p\<^sub>0]) \<cdot> ((f \<star> g) \<star> h.tab \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0) \<cdot> ((f \<star> g) \<star> fg\<^sub>0h\<^sub>1.\<phi>) \<cdot> \<a>[f \<star> g, tab\<^sub>0 g \<star> f\<^sub>0g\<^sub>1.p\<^sub>0, fg\<^sub>0h\<^sub>1.p\<^sub>1] \<cdot> (\<a>\<^sup>-\<^sup>1[f, g, tab\<^sub>0 g \<star> f\<^sub>0g\<^sub>1.p\<^sub>0] \<cdot> (f \<star> \<a>[g, tab\<^sub>0 g, f\<^sub>0g\<^sub>1.p\<^sub>0]) \<cdot> (f \<star> g.tab \<star> f\<^sub>0g\<^sub>1.p\<^sub>0) \<cdot> (f \<star> f\<^sub>0g\<^sub>1.\<phi>) \<cdot> \<a>[f, tab\<^sub>0 f, f\<^sub>0g\<^sub>1.p\<^sub>1] \<cdot> (f.tab \<star> f\<^sub>0g\<^sub>1.p\<^sub>1) \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>1 f, f\<^sub>0g\<^sub>1.p\<^sub>1, fg\<^sub>0h\<^sub>1.p\<^sub>1]
[PROOF STEP]
using comp_assoc
[PROOF STATE]
proof (prove)
using this:
\<a>\<^sup>-\<^sup>1[f, g, (tab\<^sub>0 g \<star> f\<^sub>0g\<^sub>1.p\<^sub>0) \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1] \<cdot> (f \<star> \<a>[g, tab\<^sub>0 g \<star> f\<^sub>0g\<^sub>1.p\<^sub>0, fg\<^sub>0h\<^sub>1.p\<^sub>1]) \<cdot> (f \<star> \<a>[g, tab\<^sub>0 g, f\<^sub>0g\<^sub>1.p\<^sub>0] \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> \<a>[f, (g \<star> tab\<^sub>0 g) \<star> f\<^sub>0g\<^sub>1.p\<^sub>0, fg\<^sub>0h\<^sub>1.p\<^sub>1] \<cdot> ((f \<star> \<a>\<^sup>-\<^sup>1[g, tab\<^sub>0 g, f\<^sub>0g\<^sub>1.p\<^sub>0]) \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> (\<a>[f, g, tab\<^sub>0 g \<star> f\<^sub>0g\<^sub>1.p\<^sub>0] \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) = \<a>[f \<star> g, tab\<^sub>0 g \<star> f\<^sub>0g\<^sub>1.p\<^sub>0, fg\<^sub>0h\<^sub>1.p\<^sub>1]
(?h \<cdot> ?g) \<cdot> ?f = ?h \<cdot> ?g \<cdot> ?f
goal (1 subgoal):
1. \<a>[f, g, h \<star> tab\<^sub>0 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0] \<cdot> ((f \<star> g) \<star> \<a>[h, tab\<^sub>0 h, fg\<^sub>0h\<^sub>1.p\<^sub>0]) \<cdot> ((f \<star> g) \<star> h.tab \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0) \<cdot> ((f \<star> g) \<star> fg\<^sub>0h\<^sub>1.\<phi>) \<cdot> (\<a>\<^sup>-\<^sup>1[f, g, (tab\<^sub>0 g \<star> f\<^sub>0g\<^sub>1.p\<^sub>0) \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1] \<cdot> (f \<star> \<a>[g, tab\<^sub>0 g \<star> f\<^sub>0g\<^sub>1.p\<^sub>0, fg\<^sub>0h\<^sub>1.p\<^sub>1]) \<cdot> (f \<star> \<a>[g, tab\<^sub>0 g, f\<^sub>0g\<^sub>1.p\<^sub>0] \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> \<a>[f, (g \<star> tab\<^sub>0 g) \<star> f\<^sub>0g\<^sub>1.p\<^sub>0, fg\<^sub>0h\<^sub>1.p\<^sub>1] \<cdot> ((f \<star> \<a>\<^sup>-\<^sup>1[g, tab\<^sub>0 g, f\<^sub>0g\<^sub>1.p\<^sub>0]) \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> (\<a>[f, g, tab\<^sub>0 g \<star> f\<^sub>0g\<^sub>1.p\<^sub>0] \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1)) \<cdot> (\<a>\<^sup>-\<^sup>1[f, g, tab\<^sub>0 g \<star> f\<^sub>0g\<^sub>1.p\<^sub>0] \<cdot> (f \<star> \<a>[g, tab\<^sub>0 g, f\<^sub>0g\<^sub>1.p\<^sub>0]) \<cdot> (f \<star> g.tab \<star> f\<^sub>0g\<^sub>1.p\<^sub>0) \<cdot> (f \<star> f\<^sub>0g\<^sub>1.\<phi>) \<cdot> \<a>[f, tab\<^sub>0 f, f\<^sub>0g\<^sub>1.p\<^sub>1] \<cdot> (f.tab \<star> f\<^sub>0g\<^sub>1.p\<^sub>1) \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>1 f, f\<^sub>0g\<^sub>1.p\<^sub>1, fg\<^sub>0h\<^sub>1.p\<^sub>1] = \<a>[f, g, h \<star> tab\<^sub>0 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0] \<cdot> ((f \<star> g) \<star> \<a>[h, tab\<^sub>0 h, fg\<^sub>0h\<^sub>1.p\<^sub>0]) \<cdot> ((f \<star> g) \<star> h.tab \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0) \<cdot> ((f \<star> g) \<star> fg\<^sub>0h\<^sub>1.\<phi>) \<cdot> \<a>[f \<star> g, tab\<^sub>0 g \<star> f\<^sub>0g\<^sub>1.p\<^sub>0, fg\<^sub>0h\<^sub>1.p\<^sub>1] \<cdot> (\<a>\<^sup>-\<^sup>1[f, g, tab\<^sub>0 g \<star> f\<^sub>0g\<^sub>1.p\<^sub>0] \<cdot> (f \<star> \<a>[g, tab\<^sub>0 g, f\<^sub>0g\<^sub>1.p\<^sub>0]) \<cdot> (f \<star> g.tab \<star> f\<^sub>0g\<^sub>1.p\<^sub>0) \<cdot> (f \<star> f\<^sub>0g\<^sub>1.\<phi>) \<cdot> \<a>[f, tab\<^sub>0 f, f\<^sub>0g\<^sub>1.p\<^sub>1] \<cdot> (f.tab \<star> f\<^sub>0g\<^sub>1.p\<^sub>1) \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>1 f, f\<^sub>0g\<^sub>1.p\<^sub>1, fg\<^sub>0h\<^sub>1.p\<^sub>1]
[PROOF STEP]
by presburger
[PROOF STATE]
proof (state)
this:
\<a>[f, g, h \<star> tab\<^sub>0 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0] \<cdot> ((f \<star> g) \<star> \<a>[h, tab\<^sub>0 h, fg\<^sub>0h\<^sub>1.p\<^sub>0]) \<cdot> ((f \<star> g) \<star> h.tab \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0) \<cdot> ((f \<star> g) \<star> fg\<^sub>0h\<^sub>1.\<phi>) \<cdot> (\<a>\<^sup>-\<^sup>1[f, g, (tab\<^sub>0 g \<star> f\<^sub>0g\<^sub>1.p\<^sub>0) \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1] \<cdot> (f \<star> \<a>[g, tab\<^sub>0 g \<star> f\<^sub>0g\<^sub>1.p\<^sub>0, fg\<^sub>0h\<^sub>1.p\<^sub>1]) \<cdot> (f \<star> \<a>[g, tab\<^sub>0 g, f\<^sub>0g\<^sub>1.p\<^sub>0] \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> \<a>[f, (g \<star> tab\<^sub>0 g) \<star> f\<^sub>0g\<^sub>1.p\<^sub>0, fg\<^sub>0h\<^sub>1.p\<^sub>1] \<cdot> ((f \<star> \<a>\<^sup>-\<^sup>1[g, tab\<^sub>0 g, f\<^sub>0g\<^sub>1.p\<^sub>0]) \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> (\<a>[f, g, tab\<^sub>0 g \<star> f\<^sub>0g\<^sub>1.p\<^sub>0] \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1)) \<cdot> (\<a>\<^sup>-\<^sup>1[f, g, tab\<^sub>0 g \<star> f\<^sub>0g\<^sub>1.p\<^sub>0] \<cdot> (f \<star> \<a>[g, tab\<^sub>0 g, f\<^sub>0g\<^sub>1.p\<^sub>0]) \<cdot> (f \<star> g.tab \<star> f\<^sub>0g\<^sub>1.p\<^sub>0) \<cdot> (f \<star> f\<^sub>0g\<^sub>1.\<phi>) \<cdot> \<a>[f, tab\<^sub>0 f, f\<^sub>0g\<^sub>1.p\<^sub>1] \<cdot> (f.tab \<star> f\<^sub>0g\<^sub>1.p\<^sub>1) \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>1 f, f\<^sub>0g\<^sub>1.p\<^sub>1, fg\<^sub>0h\<^sub>1.p\<^sub>1] = \<a>[f, g, h \<star> tab\<^sub>0 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0] \<cdot> ((f \<star> g) \<star> \<a>[h, tab\<^sub>0 h, fg\<^sub>0h\<^sub>1.p\<^sub>0]) \<cdot> ((f \<star> g) \<star> h.tab \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0) \<cdot> ((f \<star> g) \<star> fg\<^sub>0h\<^sub>1.\<phi>) \<cdot> \<a>[f \<star> g, tab\<^sub>0 g \<star> f\<^sub>0g\<^sub>1.p\<^sub>0, fg\<^sub>0h\<^sub>1.p\<^sub>1] \<cdot> (\<a>\<^sup>-\<^sup>1[f, g, tab\<^sub>0 g \<star> f\<^sub>0g\<^sub>1.p\<^sub>0] \<cdot> (f \<star> \<a>[g, tab\<^sub>0 g, f\<^sub>0g\<^sub>1.p\<^sub>0]) \<cdot> (f \<star> g.tab \<star> f\<^sub>0g\<^sub>1.p\<^sub>0) \<cdot> (f \<star> f\<^sub>0g\<^sub>1.\<phi>) \<cdot> \<a>[f, tab\<^sub>0 f, f\<^sub>0g\<^sub>1.p\<^sub>1] \<cdot> (f.tab \<star> f\<^sub>0g\<^sub>1.p\<^sub>1) \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>1 f, f\<^sub>0g\<^sub>1.p\<^sub>1, fg\<^sub>0h\<^sub>1.p\<^sub>1]
goal:
No subgoals!
[PROOF STEP]
qed
[PROOF STATE]
proof (state)
this:
\<a>[f, g, h \<star> tab\<^sub>0 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0] \<cdot> ((f \<star> g) \<star> \<a>[h, tab\<^sub>0 h, fg\<^sub>0h\<^sub>1.p\<^sub>0]) \<cdot> ((f \<star> g) \<star> h.tab \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0) \<cdot> ((f \<star> g) \<star> fg\<^sub>0h\<^sub>1.\<phi>) \<cdot> (\<a>\<^sup>-\<^sup>1[f, g, (tab\<^sub>0 g \<star> f\<^sub>0g\<^sub>1.p\<^sub>0) \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1] \<cdot> (f \<star> \<a>[g, tab\<^sub>0 g \<star> f\<^sub>0g\<^sub>1.p\<^sub>0, fg\<^sub>0h\<^sub>1.p\<^sub>1]) \<cdot> (f \<star> \<a>[g, tab\<^sub>0 g, f\<^sub>0g\<^sub>1.p\<^sub>0] \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> \<a>[f, (g \<star> tab\<^sub>0 g) \<star> f\<^sub>0g\<^sub>1.p\<^sub>0, fg\<^sub>0h\<^sub>1.p\<^sub>1] \<cdot> ((f \<star> \<a>\<^sup>-\<^sup>1[g, tab\<^sub>0 g, f\<^sub>0g\<^sub>1.p\<^sub>0]) \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> (\<a>[f, g, tab\<^sub>0 g \<star> f\<^sub>0g\<^sub>1.p\<^sub>0] \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1)) \<cdot> (\<a>\<^sup>-\<^sup>1[f, g, tab\<^sub>0 g \<star> f\<^sub>0g\<^sub>1.p\<^sub>0] \<cdot> (f \<star> \<a>[g, tab\<^sub>0 g, f\<^sub>0g\<^sub>1.p\<^sub>0]) \<cdot> (f \<star> g.tab \<star> f\<^sub>0g\<^sub>1.p\<^sub>0) \<cdot> (f \<star> f\<^sub>0g\<^sub>1.\<phi>) \<cdot> \<a>[f, tab\<^sub>0 f, f\<^sub>0g\<^sub>1.p\<^sub>1] \<cdot> (f.tab \<star> f\<^sub>0g\<^sub>1.p\<^sub>1) \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>1 f, f\<^sub>0g\<^sub>1.p\<^sub>1, fg\<^sub>0h\<^sub>1.p\<^sub>1] = \<a>[f, g, h \<star> tab\<^sub>0 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0] \<cdot> ((f \<star> g) \<star> \<a>[h, tab\<^sub>0 h, fg\<^sub>0h\<^sub>1.p\<^sub>0]) \<cdot> ((f \<star> g) \<star> h.tab \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0) \<cdot> ((f \<star> g) \<star> fg\<^sub>0h\<^sub>1.\<phi>) \<cdot> \<a>[f \<star> g, tab\<^sub>0 g \<star> f\<^sub>0g\<^sub>1.p\<^sub>0, fg\<^sub>0h\<^sub>1.p\<^sub>1] \<cdot> (\<a>\<^sup>-\<^sup>1[f, g, tab\<^sub>0 g \<star> f\<^sub>0g\<^sub>1.p\<^sub>0] \<cdot> (f \<star> \<a>[g, tab\<^sub>0 g, f\<^sub>0g\<^sub>1.p\<^sub>0]) \<cdot> (f \<star> g.tab \<star> f\<^sub>0g\<^sub>1.p\<^sub>0) \<cdot> (f \<star> f\<^sub>0g\<^sub>1.\<phi>) \<cdot> \<a>[f, tab\<^sub>0 f, f\<^sub>0g\<^sub>1.p\<^sub>1] \<cdot> (f.tab \<star> f\<^sub>0g\<^sub>1.p\<^sub>1) \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>1 f, f\<^sub>0g\<^sub>1.p\<^sub>1, fg\<^sub>0h\<^sub>1.p\<^sub>1]
goal (1 subgoal):
1. (f \<star> \<theta>\<^sub>f) \<cdot> \<a>[f, tab\<^sub>0 f, w\<^sub>f] \<cdot> (f.tab \<star> w\<^sub>f) = ((f \<star> \<theta>\<^sub>f') \<cdot> \<a>[f, tab\<^sub>0 f, w\<^sub>f'] \<cdot> (f.tab \<star> w\<^sub>f')) \<cdot> \<beta>\<^sub>f
[PROOF STEP]
also
[PROOF STATE]
proof (state)
this:
\<a>[f, g, h \<star> tab\<^sub>0 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0] \<cdot> ((f \<star> g) \<star> \<a>[h, tab\<^sub>0 h, fg\<^sub>0h\<^sub>1.p\<^sub>0]) \<cdot> ((f \<star> g) \<star> h.tab \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0) \<cdot> ((f \<star> g) \<star> fg\<^sub>0h\<^sub>1.\<phi>) \<cdot> (\<a>\<^sup>-\<^sup>1[f, g, (tab\<^sub>0 g \<star> f\<^sub>0g\<^sub>1.p\<^sub>0) \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1] \<cdot> (f \<star> \<a>[g, tab\<^sub>0 g \<star> f\<^sub>0g\<^sub>1.p\<^sub>0, fg\<^sub>0h\<^sub>1.p\<^sub>1]) \<cdot> (f \<star> \<a>[g, tab\<^sub>0 g, f\<^sub>0g\<^sub>1.p\<^sub>0] \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> \<a>[f, (g \<star> tab\<^sub>0 g) \<star> f\<^sub>0g\<^sub>1.p\<^sub>0, fg\<^sub>0h\<^sub>1.p\<^sub>1] \<cdot> ((f \<star> \<a>\<^sup>-\<^sup>1[g, tab\<^sub>0 g, f\<^sub>0g\<^sub>1.p\<^sub>0]) \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> (\<a>[f, g, tab\<^sub>0 g \<star> f\<^sub>0g\<^sub>1.p\<^sub>0] \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1)) \<cdot> (\<a>\<^sup>-\<^sup>1[f, g, tab\<^sub>0 g \<star> f\<^sub>0g\<^sub>1.p\<^sub>0] \<cdot> (f \<star> \<a>[g, tab\<^sub>0 g, f\<^sub>0g\<^sub>1.p\<^sub>0]) \<cdot> (f \<star> g.tab \<star> f\<^sub>0g\<^sub>1.p\<^sub>0) \<cdot> (f \<star> f\<^sub>0g\<^sub>1.\<phi>) \<cdot> \<a>[f, tab\<^sub>0 f, f\<^sub>0g\<^sub>1.p\<^sub>1] \<cdot> (f.tab \<star> f\<^sub>0g\<^sub>1.p\<^sub>1) \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>1 f, f\<^sub>0g\<^sub>1.p\<^sub>1, fg\<^sub>0h\<^sub>1.p\<^sub>1] = \<a>[f, g, h \<star> tab\<^sub>0 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0] \<cdot> ((f \<star> g) \<star> \<a>[h, tab\<^sub>0 h, fg\<^sub>0h\<^sub>1.p\<^sub>0]) \<cdot> ((f \<star> g) \<star> h.tab \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0) \<cdot> ((f \<star> g) \<star> fg\<^sub>0h\<^sub>1.\<phi>) \<cdot> \<a>[f \<star> g, tab\<^sub>0 g \<star> f\<^sub>0g\<^sub>1.p\<^sub>0, fg\<^sub>0h\<^sub>1.p\<^sub>1] \<cdot> (\<a>\<^sup>-\<^sup>1[f, g, tab\<^sub>0 g \<star> f\<^sub>0g\<^sub>1.p\<^sub>0] \<cdot> (f \<star> \<a>[g, tab\<^sub>0 g, f\<^sub>0g\<^sub>1.p\<^sub>0]) \<cdot> (f \<star> g.tab \<star> f\<^sub>0g\<^sub>1.p\<^sub>0) \<cdot> (f \<star> f\<^sub>0g\<^sub>1.\<phi>) \<cdot> \<a>[f, tab\<^sub>0 f, f\<^sub>0g\<^sub>1.p\<^sub>1] \<cdot> (f.tab \<star> f\<^sub>0g\<^sub>1.p\<^sub>1) \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>1 f, f\<^sub>0g\<^sub>1.p\<^sub>1, fg\<^sub>0h\<^sub>1.p\<^sub>1]
goal (1 subgoal):
1. (f \<star> \<theta>\<^sub>f) \<cdot> \<a>[f, tab\<^sub>0 f, w\<^sub>f] \<cdot> (f.tab \<star> w\<^sub>f) = ((f \<star> \<theta>\<^sub>f') \<cdot> \<a>[f, tab\<^sub>0 f, w\<^sub>f'] \<cdot> (f.tab \<star> w\<^sub>f')) \<cdot> \<beta>\<^sub>f
[PROOF STEP]
have "... =
\<a>[f, g, h \<star> tab\<^sub>0 h \<star> TTfgh.p\<^sub>0] \<cdot>
((\<a>[f \<star> g, h, tab\<^sub>0 h \<star> TTfgh.p\<^sub>0] \<cdot>
\<a>\<^sup>-\<^sup>1[f \<star> g, h, tab\<^sub>0 h \<star> TTfgh.p\<^sub>0]) \<cdot>
((f \<star> g) \<star> \<a>[h, tab\<^sub>0 h, TTfgh.p\<^sub>0])) \<cdot>
((f \<star> g) \<star> h.tab \<star> TTfgh.p\<^sub>0) \<cdot>
((f \<star> g) \<star> fg\<^sub>0h\<^sub>1.\<phi>) \<cdot>
\<a>[f \<star> g, tab\<^sub>0 g \<star> Tfg.p\<^sub>0, TTfgh.p\<^sub>1] \<cdot>
(\<a>\<^sup>-\<^sup>1[f, g, tab\<^sub>0 g \<star> Tfg.p\<^sub>0] \<cdot>
(f \<star> \<a>[g, tab\<^sub>0 g, Tfg.p\<^sub>0]) \<cdot>
(f \<star> (g.tab \<star> Tfg.p\<^sub>0)) \<cdot>
(f \<star> f\<^sub>0g\<^sub>1.\<phi>) \<cdot>
\<a>[f, tab\<^sub>0 f, Tfg.p\<^sub>1] \<cdot>
(f.tab \<star> Tfg.p\<^sub>1)
\<star> TTfgh.p\<^sub>1) \<cdot>
\<a>\<^sup>-\<^sup>1[tab\<^sub>1 f, Tfg.p\<^sub>1, TTfgh.p\<^sub>1]"
[PROOF STATE]
proof (prove)
goal (1 subgoal):
1. \<a>[f, g, h \<star> tab\<^sub>0 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0] \<cdot> ((f \<star> g) \<star> \<a>[h, tab\<^sub>0 h, fg\<^sub>0h\<^sub>1.p\<^sub>0]) \<cdot> ((f \<star> g) \<star> h.tab \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0) \<cdot> ((f \<star> g) \<star> fg\<^sub>0h\<^sub>1.\<phi>) \<cdot> \<a>[f \<star> g, tab\<^sub>0 g \<star> f\<^sub>0g\<^sub>1.p\<^sub>0, fg\<^sub>0h\<^sub>1.p\<^sub>1] \<cdot> (\<a>\<^sup>-\<^sup>1[f, g, tab\<^sub>0 g \<star> f\<^sub>0g\<^sub>1.p\<^sub>0] \<cdot> (f \<star> \<a>[g, tab\<^sub>0 g, f\<^sub>0g\<^sub>1.p\<^sub>0]) \<cdot> (f \<star> g.tab \<star> f\<^sub>0g\<^sub>1.p\<^sub>0) \<cdot> (f \<star> f\<^sub>0g\<^sub>1.\<phi>) \<cdot> \<a>[f, tab\<^sub>0 f, f\<^sub>0g\<^sub>1.p\<^sub>1] \<cdot> (f.tab \<star> f\<^sub>0g\<^sub>1.p\<^sub>1) \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>1 f, f\<^sub>0g\<^sub>1.p\<^sub>1, fg\<^sub>0h\<^sub>1.p\<^sub>1] = \<a>[f, g, h \<star> tab\<^sub>0 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0] \<cdot> ((\<a>[f \<star> g, h, tab\<^sub>0 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0] \<cdot> \<a>\<^sup>-\<^sup>1[f \<star> g, h, tab\<^sub>0 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0]) \<cdot> ((f \<star> g) \<star> \<a>[h, tab\<^sub>0 h, fg\<^sub>0h\<^sub>1.p\<^sub>0])) \<cdot> ((f \<star> g) \<star> h.tab \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0) \<cdot> ((f \<star> g) \<star> fg\<^sub>0h\<^sub>1.\<phi>) \<cdot> \<a>[f \<star> g, tab\<^sub>0 g \<star> f\<^sub>0g\<^sub>1.p\<^sub>0, fg\<^sub>0h\<^sub>1.p\<^sub>1] \<cdot> (\<a>\<^sup>-\<^sup>1[f, g, tab\<^sub>0 g \<star> f\<^sub>0g\<^sub>1.p\<^sub>0] \<cdot> (f \<star> \<a>[g, tab\<^sub>0 g, f\<^sub>0g\<^sub>1.p\<^sub>0]) \<cdot> (f \<star> g.tab \<star> f\<^sub>0g\<^sub>1.p\<^sub>0) \<cdot> (f \<star> f\<^sub>0g\<^sub>1.\<phi>) \<cdot> \<a>[f, tab\<^sub>0 f, f\<^sub>0g\<^sub>1.p\<^sub>1] \<cdot> (f.tab \<star> f\<^sub>0g\<^sub>1.p\<^sub>1) \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>1 f, f\<^sub>0g\<^sub>1.p\<^sub>1, fg\<^sub>0h\<^sub>1.p\<^sub>1]
[PROOF STEP]
proof -
[PROOF STATE]
proof (state)
goal (1 subgoal):
1. \<a>[f, g, h \<star> tab\<^sub>0 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0] \<cdot> ((f \<star> g) \<star> \<a>[h, tab\<^sub>0 h, fg\<^sub>0h\<^sub>1.p\<^sub>0]) \<cdot> ((f \<star> g) \<star> h.tab \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0) \<cdot> ((f \<star> g) \<star> fg\<^sub>0h\<^sub>1.\<phi>) \<cdot> \<a>[f \<star> g, tab\<^sub>0 g \<star> f\<^sub>0g\<^sub>1.p\<^sub>0, fg\<^sub>0h\<^sub>1.p\<^sub>1] \<cdot> (\<a>\<^sup>-\<^sup>1[f, g, tab\<^sub>0 g \<star> f\<^sub>0g\<^sub>1.p\<^sub>0] \<cdot> (f \<star> \<a>[g, tab\<^sub>0 g, f\<^sub>0g\<^sub>1.p\<^sub>0]) \<cdot> (f \<star> g.tab \<star> f\<^sub>0g\<^sub>1.p\<^sub>0) \<cdot> (f \<star> f\<^sub>0g\<^sub>1.\<phi>) \<cdot> \<a>[f, tab\<^sub>0 f, f\<^sub>0g\<^sub>1.p\<^sub>1] \<cdot> (f.tab \<star> f\<^sub>0g\<^sub>1.p\<^sub>1) \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>1 f, f\<^sub>0g\<^sub>1.p\<^sub>1, fg\<^sub>0h\<^sub>1.p\<^sub>1] = \<a>[f, g, h \<star> tab\<^sub>0 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0] \<cdot> ((\<a>[f \<star> g, h, tab\<^sub>0 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0] \<cdot> \<a>\<^sup>-\<^sup>1[f \<star> g, h, tab\<^sub>0 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0]) \<cdot> ((f \<star> g) \<star> \<a>[h, tab\<^sub>0 h, fg\<^sub>0h\<^sub>1.p\<^sub>0])) \<cdot> ((f \<star> g) \<star> h.tab \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0) \<cdot> ((f \<star> g) \<star> fg\<^sub>0h\<^sub>1.\<phi>) \<cdot> \<a>[f \<star> g, tab\<^sub>0 g \<star> f\<^sub>0g\<^sub>1.p\<^sub>0, fg\<^sub>0h\<^sub>1.p\<^sub>1] \<cdot> (\<a>\<^sup>-\<^sup>1[f, g, tab\<^sub>0 g \<star> f\<^sub>0g\<^sub>1.p\<^sub>0] \<cdot> (f \<star> \<a>[g, tab\<^sub>0 g, f\<^sub>0g\<^sub>1.p\<^sub>0]) \<cdot> (f \<star> g.tab \<star> f\<^sub>0g\<^sub>1.p\<^sub>0) \<cdot> (f \<star> f\<^sub>0g\<^sub>1.\<phi>) \<cdot> \<a>[f, tab\<^sub>0 f, f\<^sub>0g\<^sub>1.p\<^sub>1] \<cdot> (f.tab \<star> f\<^sub>0g\<^sub>1.p\<^sub>1) \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>1 f, f\<^sub>0g\<^sub>1.p\<^sub>1, fg\<^sub>0h\<^sub>1.p\<^sub>1]
[PROOF STEP]
have "(\<a>[f \<star> g, h, tab\<^sub>0 h \<star> TTfgh.p\<^sub>0] \<cdot>
\<a>\<^sup>-\<^sup>1[f \<star> g, h, tab\<^sub>0 h \<star> TTfgh.p\<^sub>0]) \<cdot>
((f \<star> g) \<star> \<a>[h, tab\<^sub>0 h, TTfgh.p\<^sub>0]) =
((f \<star> g) \<star> \<a>[h, tab\<^sub>0 h, TTfgh.p\<^sub>0])"
[PROOF STATE]
proof (prove)
goal (1 subgoal):
1. (\<a>[f \<star> g, h, tab\<^sub>0 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0] \<cdot> \<a>\<^sup>-\<^sup>1[f \<star> g, h, tab\<^sub>0 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0]) \<cdot> ((f \<star> g) \<star> \<a>[h, tab\<^sub>0 h, fg\<^sub>0h\<^sub>1.p\<^sub>0]) = (f \<star> g) \<star> \<a>[h, tab\<^sub>0 h, fg\<^sub>0h\<^sub>1.p\<^sub>0]
[PROOF STEP]
using fg gh fg\<^sub>0h\<^sub>1.p\<^sub>0_simps fg\<^sub>0h\<^sub>1.p\<^sub>1_simps f\<^sub>0g\<^sub>1.p\<^sub>0_simps f\<^sub>0g\<^sub>1.p\<^sub>1_simps
comp_cod_arr comp_assoc_assoc'
[PROOF STATE]
proof (prove)
using this:
src f = trg g
src g = trg h
trg fg\<^sub>0h\<^sub>1.p\<^sub>0 = src (tab\<^sub>1 h)
trg fg\<^sub>0h\<^sub>1.p\<^sub>1 = src (tab\<^sub>0 g \<star> f\<^sub>0g\<^sub>1.p\<^sub>0)
trg f\<^sub>0g\<^sub>1.p\<^sub>0 = src (tab\<^sub>1 g)
trg f\<^sub>0g\<^sub>1.p\<^sub>1 = src (tab\<^sub>0 f)
\<lbrakk>arr ?f; cod ?f = ?b\<rbrakk> \<Longrightarrow> ?b \<cdot> ?f = ?f
\<lbrakk>ide ?f; ide ?g; ide ?h; src ?f = trg ?g; src ?g = trg ?h\<rbrakk> \<Longrightarrow> \<a>[?f, ?g, ?h] \<cdot> \<a>\<^sup>-\<^sup>1[?f, ?g, ?h] = ?f \<star> ?g \<star> ?h
\<lbrakk>ide ?f; ide ?g; ide ?h; src ?f = trg ?g; src ?g = trg ?h\<rbrakk> \<Longrightarrow> \<a>\<^sup>-\<^sup>1[?f, ?g, ?h] \<cdot> \<a>[?f, ?g, ?h] = (?f \<star> ?g) \<star> ?h
goal (1 subgoal):
1. (\<a>[f \<star> g, h, tab\<^sub>0 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0] \<cdot> \<a>\<^sup>-\<^sup>1[f \<star> g, h, tab\<^sub>0 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0]) \<cdot> ((f \<star> g) \<star> \<a>[h, tab\<^sub>0 h, fg\<^sub>0h\<^sub>1.p\<^sub>0]) = (f \<star> g) \<star> \<a>[h, tab\<^sub>0 h, fg\<^sub>0h\<^sub>1.p\<^sub>0]
[PROOF STEP]
by simp
[PROOF STATE]
proof (state)
this:
(\<a>[f \<star> g, h, tab\<^sub>0 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0] \<cdot> \<a>\<^sup>-\<^sup>1[f \<star> g, h, tab\<^sub>0 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0]) \<cdot> ((f \<star> g) \<star> \<a>[h, tab\<^sub>0 h, fg\<^sub>0h\<^sub>1.p\<^sub>0]) = (f \<star> g) \<star> \<a>[h, tab\<^sub>0 h, fg\<^sub>0h\<^sub>1.p\<^sub>0]
goal (1 subgoal):
1. \<a>[f, g, h \<star> tab\<^sub>0 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0] \<cdot> ((f \<star> g) \<star> \<a>[h, tab\<^sub>0 h, fg\<^sub>0h\<^sub>1.p\<^sub>0]) \<cdot> ((f \<star> g) \<star> h.tab \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0) \<cdot> ((f \<star> g) \<star> fg\<^sub>0h\<^sub>1.\<phi>) \<cdot> \<a>[f \<star> g, tab\<^sub>0 g \<star> f\<^sub>0g\<^sub>1.p\<^sub>0, fg\<^sub>0h\<^sub>1.p\<^sub>1] \<cdot> (\<a>\<^sup>-\<^sup>1[f, g, tab\<^sub>0 g \<star> f\<^sub>0g\<^sub>1.p\<^sub>0] \<cdot> (f \<star> \<a>[g, tab\<^sub>0 g, f\<^sub>0g\<^sub>1.p\<^sub>0]) \<cdot> (f \<star> g.tab \<star> f\<^sub>0g\<^sub>1.p\<^sub>0) \<cdot> (f \<star> f\<^sub>0g\<^sub>1.\<phi>) \<cdot> \<a>[f, tab\<^sub>0 f, f\<^sub>0g\<^sub>1.p\<^sub>1] \<cdot> (f.tab \<star> f\<^sub>0g\<^sub>1.p\<^sub>1) \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>1 f, f\<^sub>0g\<^sub>1.p\<^sub>1, fg\<^sub>0h\<^sub>1.p\<^sub>1] = \<a>[f, g, h \<star> tab\<^sub>0 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0] \<cdot> ((\<a>[f \<star> g, h, tab\<^sub>0 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0] \<cdot> \<a>\<^sup>-\<^sup>1[f \<star> g, h, tab\<^sub>0 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0]) \<cdot> ((f \<star> g) \<star> \<a>[h, tab\<^sub>0 h, fg\<^sub>0h\<^sub>1.p\<^sub>0])) \<cdot> ((f \<star> g) \<star> h.tab \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0) \<cdot> ((f \<star> g) \<star> fg\<^sub>0h\<^sub>1.\<phi>) \<cdot> \<a>[f \<star> g, tab\<^sub>0 g \<star> f\<^sub>0g\<^sub>1.p\<^sub>0, fg\<^sub>0h\<^sub>1.p\<^sub>1] \<cdot> (\<a>\<^sup>-\<^sup>1[f, g, tab\<^sub>0 g \<star> f\<^sub>0g\<^sub>1.p\<^sub>0] \<cdot> (f \<star> \<a>[g, tab\<^sub>0 g, f\<^sub>0g\<^sub>1.p\<^sub>0]) \<cdot> (f \<star> g.tab \<star> f\<^sub>0g\<^sub>1.p\<^sub>0) \<cdot> (f \<star> f\<^sub>0g\<^sub>1.\<phi>) \<cdot> \<a>[f, tab\<^sub>0 f, f\<^sub>0g\<^sub>1.p\<^sub>1] \<cdot> (f.tab \<star> f\<^sub>0g\<^sub>1.p\<^sub>1) \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>1 f, f\<^sub>0g\<^sub>1.p\<^sub>1, fg\<^sub>0h\<^sub>1.p\<^sub>1]
[PROOF STEP]
thus ?thesis
[PROOF STATE]
proof (prove)
using this:
(\<a>[f \<star> g, h, tab\<^sub>0 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0] \<cdot> \<a>\<^sup>-\<^sup>1[f \<star> g, h, tab\<^sub>0 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0]) \<cdot> ((f \<star> g) \<star> \<a>[h, tab\<^sub>0 h, fg\<^sub>0h\<^sub>1.p\<^sub>0]) = (f \<star> g) \<star> \<a>[h, tab\<^sub>0 h, fg\<^sub>0h\<^sub>1.p\<^sub>0]
goal (1 subgoal):
1. \<a>[f, g, h \<star> tab\<^sub>0 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0] \<cdot> ((f \<star> g) \<star> \<a>[h, tab\<^sub>0 h, fg\<^sub>0h\<^sub>1.p\<^sub>0]) \<cdot> ((f \<star> g) \<star> h.tab \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0) \<cdot> ((f \<star> g) \<star> fg\<^sub>0h\<^sub>1.\<phi>) \<cdot> \<a>[f \<star> g, tab\<^sub>0 g \<star> f\<^sub>0g\<^sub>1.p\<^sub>0, fg\<^sub>0h\<^sub>1.p\<^sub>1] \<cdot> (\<a>\<^sup>-\<^sup>1[f, g, tab\<^sub>0 g \<star> f\<^sub>0g\<^sub>1.p\<^sub>0] \<cdot> (f \<star> \<a>[g, tab\<^sub>0 g, f\<^sub>0g\<^sub>1.p\<^sub>0]) \<cdot> (f \<star> g.tab \<star> f\<^sub>0g\<^sub>1.p\<^sub>0) \<cdot> (f \<star> f\<^sub>0g\<^sub>1.\<phi>) \<cdot> \<a>[f, tab\<^sub>0 f, f\<^sub>0g\<^sub>1.p\<^sub>1] \<cdot> (f.tab \<star> f\<^sub>0g\<^sub>1.p\<^sub>1) \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>1 f, f\<^sub>0g\<^sub>1.p\<^sub>1, fg\<^sub>0h\<^sub>1.p\<^sub>1] = \<a>[f, g, h \<star> tab\<^sub>0 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0] \<cdot> ((\<a>[f \<star> g, h, tab\<^sub>0 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0] \<cdot> \<a>\<^sup>-\<^sup>1[f \<star> g, h, tab\<^sub>0 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0]) \<cdot> ((f \<star> g) \<star> \<a>[h, tab\<^sub>0 h, fg\<^sub>0h\<^sub>1.p\<^sub>0])) \<cdot> ((f \<star> g) \<star> h.tab \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0) \<cdot> ((f \<star> g) \<star> fg\<^sub>0h\<^sub>1.\<phi>) \<cdot> \<a>[f \<star> g, tab\<^sub>0 g \<star> f\<^sub>0g\<^sub>1.p\<^sub>0, fg\<^sub>0h\<^sub>1.p\<^sub>1] \<cdot> (\<a>\<^sup>-\<^sup>1[f, g, tab\<^sub>0 g \<star> f\<^sub>0g\<^sub>1.p\<^sub>0] \<cdot> (f \<star> \<a>[g, tab\<^sub>0 g, f\<^sub>0g\<^sub>1.p\<^sub>0]) \<cdot> (f \<star> g.tab \<star> f\<^sub>0g\<^sub>1.p\<^sub>0) \<cdot> (f \<star> f\<^sub>0g\<^sub>1.\<phi>) \<cdot> \<a>[f, tab\<^sub>0 f, f\<^sub>0g\<^sub>1.p\<^sub>1] \<cdot> (f.tab \<star> f\<^sub>0g\<^sub>1.p\<^sub>1) \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>1 f, f\<^sub>0g\<^sub>1.p\<^sub>1, fg\<^sub>0h\<^sub>1.p\<^sub>1]
[PROOF STEP]
by simp
[PROOF STATE]
proof (state)
this:
\<a>[f, g, h \<star> tab\<^sub>0 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0] \<cdot> ((f \<star> g) \<star> \<a>[h, tab\<^sub>0 h, fg\<^sub>0h\<^sub>1.p\<^sub>0]) \<cdot> ((f \<star> g) \<star> h.tab \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0) \<cdot> ((f \<star> g) \<star> fg\<^sub>0h\<^sub>1.\<phi>) \<cdot> \<a>[f \<star> g, tab\<^sub>0 g \<star> f\<^sub>0g\<^sub>1.p\<^sub>0, fg\<^sub>0h\<^sub>1.p\<^sub>1] \<cdot> (\<a>\<^sup>-\<^sup>1[f, g, tab\<^sub>0 g \<star> f\<^sub>0g\<^sub>1.p\<^sub>0] \<cdot> (f \<star> \<a>[g, tab\<^sub>0 g, f\<^sub>0g\<^sub>1.p\<^sub>0]) \<cdot> (f \<star> g.tab \<star> f\<^sub>0g\<^sub>1.p\<^sub>0) \<cdot> (f \<star> f\<^sub>0g\<^sub>1.\<phi>) \<cdot> \<a>[f, tab\<^sub>0 f, f\<^sub>0g\<^sub>1.p\<^sub>1] \<cdot> (f.tab \<star> f\<^sub>0g\<^sub>1.p\<^sub>1) \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>1 f, f\<^sub>0g\<^sub>1.p\<^sub>1, fg\<^sub>0h\<^sub>1.p\<^sub>1] = \<a>[f, g, h \<star> tab\<^sub>0 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0] \<cdot> ((\<a>[f \<star> g, h, tab\<^sub>0 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0] \<cdot> \<a>\<^sup>-\<^sup>1[f \<star> g, h, tab\<^sub>0 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0]) \<cdot> ((f \<star> g) \<star> \<a>[h, tab\<^sub>0 h, fg\<^sub>0h\<^sub>1.p\<^sub>0])) \<cdot> ((f \<star> g) \<star> h.tab \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0) \<cdot> ((f \<star> g) \<star> fg\<^sub>0h\<^sub>1.\<phi>) \<cdot> \<a>[f \<star> g, tab\<^sub>0 g \<star> f\<^sub>0g\<^sub>1.p\<^sub>0, fg\<^sub>0h\<^sub>1.p\<^sub>1] \<cdot> (\<a>\<^sup>-\<^sup>1[f, g, tab\<^sub>0 g \<star> f\<^sub>0g\<^sub>1.p\<^sub>0] \<cdot> (f \<star> \<a>[g, tab\<^sub>0 g, f\<^sub>0g\<^sub>1.p\<^sub>0]) \<cdot> (f \<star> g.tab \<star> f\<^sub>0g\<^sub>1.p\<^sub>0) \<cdot> (f \<star> f\<^sub>0g\<^sub>1.\<phi>) \<cdot> \<a>[f, tab\<^sub>0 f, f\<^sub>0g\<^sub>1.p\<^sub>1] \<cdot> (f.tab \<star> f\<^sub>0g\<^sub>1.p\<^sub>1) \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>1 f, f\<^sub>0g\<^sub>1.p\<^sub>1, fg\<^sub>0h\<^sub>1.p\<^sub>1]
goal:
No subgoals!
[PROOF STEP]
qed
[PROOF STATE]
proof (state)
this:
\<a>[f, g, h \<star> tab\<^sub>0 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0] \<cdot> ((f \<star> g) \<star> \<a>[h, tab\<^sub>0 h, fg\<^sub>0h\<^sub>1.p\<^sub>0]) \<cdot> ((f \<star> g) \<star> h.tab \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0) \<cdot> ((f \<star> g) \<star> fg\<^sub>0h\<^sub>1.\<phi>) \<cdot> \<a>[f \<star> g, tab\<^sub>0 g \<star> f\<^sub>0g\<^sub>1.p\<^sub>0, fg\<^sub>0h\<^sub>1.p\<^sub>1] \<cdot> (\<a>\<^sup>-\<^sup>1[f, g, tab\<^sub>0 g \<star> f\<^sub>0g\<^sub>1.p\<^sub>0] \<cdot> (f \<star> \<a>[g, tab\<^sub>0 g, f\<^sub>0g\<^sub>1.p\<^sub>0]) \<cdot> (f \<star> g.tab \<star> f\<^sub>0g\<^sub>1.p\<^sub>0) \<cdot> (f \<star> f\<^sub>0g\<^sub>1.\<phi>) \<cdot> \<a>[f, tab\<^sub>0 f, f\<^sub>0g\<^sub>1.p\<^sub>1] \<cdot> (f.tab \<star> f\<^sub>0g\<^sub>1.p\<^sub>1) \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>1 f, f\<^sub>0g\<^sub>1.p\<^sub>1, fg\<^sub>0h\<^sub>1.p\<^sub>1] = \<a>[f, g, h \<star> tab\<^sub>0 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0] \<cdot> ((\<a>[f \<star> g, h, tab\<^sub>0 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0] \<cdot> \<a>\<^sup>-\<^sup>1[f \<star> g, h, tab\<^sub>0 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0]) \<cdot> ((f \<star> g) \<star> \<a>[h, tab\<^sub>0 h, fg\<^sub>0h\<^sub>1.p\<^sub>0])) \<cdot> ((f \<star> g) \<star> h.tab \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0) \<cdot> ((f \<star> g) \<star> fg\<^sub>0h\<^sub>1.\<phi>) \<cdot> \<a>[f \<star> g, tab\<^sub>0 g \<star> f\<^sub>0g\<^sub>1.p\<^sub>0, fg\<^sub>0h\<^sub>1.p\<^sub>1] \<cdot> (\<a>\<^sup>-\<^sup>1[f, g, tab\<^sub>0 g \<star> f\<^sub>0g\<^sub>1.p\<^sub>0] \<cdot> (f \<star> \<a>[g, tab\<^sub>0 g, f\<^sub>0g\<^sub>1.p\<^sub>0]) \<cdot> (f \<star> g.tab \<star> f\<^sub>0g\<^sub>1.p\<^sub>0) \<cdot> (f \<star> f\<^sub>0g\<^sub>1.\<phi>) \<cdot> \<a>[f, tab\<^sub>0 f, f\<^sub>0g\<^sub>1.p\<^sub>1] \<cdot> (f.tab \<star> f\<^sub>0g\<^sub>1.p\<^sub>1) \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>1 f, f\<^sub>0g\<^sub>1.p\<^sub>1, fg\<^sub>0h\<^sub>1.p\<^sub>1]
goal (1 subgoal):
1. (f \<star> \<theta>\<^sub>f) \<cdot> \<a>[f, tab\<^sub>0 f, w\<^sub>f] \<cdot> (f.tab \<star> w\<^sub>f) = ((f \<star> \<theta>\<^sub>f') \<cdot> \<a>[f, tab\<^sub>0 f, w\<^sub>f'] \<cdot> (f.tab \<star> w\<^sub>f')) \<cdot> \<beta>\<^sub>f
[PROOF STEP]
also
[PROOF STATE]
proof (state)
this:
\<a>[f, g, h \<star> tab\<^sub>0 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0] \<cdot> ((f \<star> g) \<star> \<a>[h, tab\<^sub>0 h, fg\<^sub>0h\<^sub>1.p\<^sub>0]) \<cdot> ((f \<star> g) \<star> h.tab \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0) \<cdot> ((f \<star> g) \<star> fg\<^sub>0h\<^sub>1.\<phi>) \<cdot> \<a>[f \<star> g, tab\<^sub>0 g \<star> f\<^sub>0g\<^sub>1.p\<^sub>0, fg\<^sub>0h\<^sub>1.p\<^sub>1] \<cdot> (\<a>\<^sup>-\<^sup>1[f, g, tab\<^sub>0 g \<star> f\<^sub>0g\<^sub>1.p\<^sub>0] \<cdot> (f \<star> \<a>[g, tab\<^sub>0 g, f\<^sub>0g\<^sub>1.p\<^sub>0]) \<cdot> (f \<star> g.tab \<star> f\<^sub>0g\<^sub>1.p\<^sub>0) \<cdot> (f \<star> f\<^sub>0g\<^sub>1.\<phi>) \<cdot> \<a>[f, tab\<^sub>0 f, f\<^sub>0g\<^sub>1.p\<^sub>1] \<cdot> (f.tab \<star> f\<^sub>0g\<^sub>1.p\<^sub>1) \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>1 f, f\<^sub>0g\<^sub>1.p\<^sub>1, fg\<^sub>0h\<^sub>1.p\<^sub>1] = \<a>[f, g, h \<star> tab\<^sub>0 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0] \<cdot> ((\<a>[f \<star> g, h, tab\<^sub>0 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0] \<cdot> \<a>\<^sup>-\<^sup>1[f \<star> g, h, tab\<^sub>0 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0]) \<cdot> ((f \<star> g) \<star> \<a>[h, tab\<^sub>0 h, fg\<^sub>0h\<^sub>1.p\<^sub>0])) \<cdot> ((f \<star> g) \<star> h.tab \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0) \<cdot> ((f \<star> g) \<star> fg\<^sub>0h\<^sub>1.\<phi>) \<cdot> \<a>[f \<star> g, tab\<^sub>0 g \<star> f\<^sub>0g\<^sub>1.p\<^sub>0, fg\<^sub>0h\<^sub>1.p\<^sub>1] \<cdot> (\<a>\<^sup>-\<^sup>1[f, g, tab\<^sub>0 g \<star> f\<^sub>0g\<^sub>1.p\<^sub>0] \<cdot> (f \<star> \<a>[g, tab\<^sub>0 g, f\<^sub>0g\<^sub>1.p\<^sub>0]) \<cdot> (f \<star> g.tab \<star> f\<^sub>0g\<^sub>1.p\<^sub>0) \<cdot> (f \<star> f\<^sub>0g\<^sub>1.\<phi>) \<cdot> \<a>[f, tab\<^sub>0 f, f\<^sub>0g\<^sub>1.p\<^sub>1] \<cdot> (f.tab \<star> f\<^sub>0g\<^sub>1.p\<^sub>1) \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>1 f, f\<^sub>0g\<^sub>1.p\<^sub>1, fg\<^sub>0h\<^sub>1.p\<^sub>1]
goal (1 subgoal):
1. (f \<star> \<theta>\<^sub>f) \<cdot> \<a>[f, tab\<^sub>0 f, w\<^sub>f] \<cdot> (f.tab \<star> w\<^sub>f) = ((f \<star> \<theta>\<^sub>f') \<cdot> \<a>[f, tab\<^sub>0 f, w\<^sub>f'] \<cdot> (f.tab \<star> w\<^sub>f')) \<cdot> \<beta>\<^sub>f
[PROOF STEP]
have "... =
\<a>[f, g, h \<star> tab\<^sub>0 h \<star> TTfgh.p\<^sub>0] \<cdot>
\<a>[f \<star> g, h, tab\<^sub>0 h \<star> TTfgh.p\<^sub>0] \<cdot>
(\<a>\<^sup>-\<^sup>1[f \<star> g, h, tab\<^sub>0 h \<star> TTfgh.p\<^sub>0] \<cdot>
((f \<star> g) \<star> \<a>[h, tab\<^sub>0 h, TTfgh.p\<^sub>0]) \<cdot>
((f \<star> g) \<star> h.tab \<star> TTfgh.p\<^sub>0) \<cdot>
((f \<star> g) \<star> fg\<^sub>0h\<^sub>1.\<phi>) \<cdot>
\<a>[f \<star> g, tab\<^sub>0 g \<star> Tfg.p\<^sub>0, TTfgh.p\<^sub>1] \<cdot>
(\<a>\<^sup>-\<^sup>1[f, g, tab\<^sub>0 g \<star> Tfg.p\<^sub>0] \<cdot>
(f \<star> \<a>[g, tab\<^sub>0 g, Tfg.p\<^sub>0]) \<cdot>
(f \<star> (g.tab \<star> Tfg.p\<^sub>0)) \<cdot>
(f \<star> f\<^sub>0g\<^sub>1.\<phi>) \<cdot>
\<a>[f, tab\<^sub>0 f, Tfg.p\<^sub>1] \<cdot>
(f.tab \<star> Tfg.p\<^sub>1)
\<star> TTfgh.p\<^sub>1)) \<cdot>
\<a>\<^sup>-\<^sup>1[tab\<^sub>1 f, Tfg.p\<^sub>1, TTfgh.p\<^sub>1]"
[PROOF STATE]
proof (prove)
goal (1 subgoal):
1. \<a>[f, g, h \<star> tab\<^sub>0 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0] \<cdot> ((\<a>[f \<star> g, h, tab\<^sub>0 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0] \<cdot> \<a>\<^sup>-\<^sup>1[f \<star> g, h, tab\<^sub>0 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0]) \<cdot> ((f \<star> g) \<star> \<a>[h, tab\<^sub>0 h, fg\<^sub>0h\<^sub>1.p\<^sub>0])) \<cdot> ((f \<star> g) \<star> h.tab \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0) \<cdot> ((f \<star> g) \<star> fg\<^sub>0h\<^sub>1.\<phi>) \<cdot> \<a>[f \<star> g, tab\<^sub>0 g \<star> f\<^sub>0g\<^sub>1.p\<^sub>0, fg\<^sub>0h\<^sub>1.p\<^sub>1] \<cdot> (\<a>\<^sup>-\<^sup>1[f, g, tab\<^sub>0 g \<star> f\<^sub>0g\<^sub>1.p\<^sub>0] \<cdot> (f \<star> \<a>[g, tab\<^sub>0 g, f\<^sub>0g\<^sub>1.p\<^sub>0]) \<cdot> (f \<star> g.tab \<star> f\<^sub>0g\<^sub>1.p\<^sub>0) \<cdot> (f \<star> f\<^sub>0g\<^sub>1.\<phi>) \<cdot> \<a>[f, tab\<^sub>0 f, f\<^sub>0g\<^sub>1.p\<^sub>1] \<cdot> (f.tab \<star> f\<^sub>0g\<^sub>1.p\<^sub>1) \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>1 f, f\<^sub>0g\<^sub>1.p\<^sub>1, fg\<^sub>0h\<^sub>1.p\<^sub>1] = \<a>[f, g, h \<star> tab\<^sub>0 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0] \<cdot> \<a>[f \<star> g, h, tab\<^sub>0 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0] \<cdot> (\<a>\<^sup>-\<^sup>1[f \<star> g, h, tab\<^sub>0 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0] \<cdot> ((f \<star> g) \<star> \<a>[h, tab\<^sub>0 h, fg\<^sub>0h\<^sub>1.p\<^sub>0]) \<cdot> ((f \<star> g) \<star> h.tab \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0) \<cdot> ((f \<star> g) \<star> fg\<^sub>0h\<^sub>1.\<phi>) \<cdot> \<a>[f \<star> g, tab\<^sub>0 g \<star> f\<^sub>0g\<^sub>1.p\<^sub>0, fg\<^sub>0h\<^sub>1.p\<^sub>1] \<cdot> (\<a>\<^sup>-\<^sup>1[f, g, tab\<^sub>0 g \<star> f\<^sub>0g\<^sub>1.p\<^sub>0] \<cdot> (f \<star> \<a>[g, tab\<^sub>0 g, f\<^sub>0g\<^sub>1.p\<^sub>0]) \<cdot> (f \<star> g.tab \<star> f\<^sub>0g\<^sub>1.p\<^sub>0) \<cdot> (f \<star> f\<^sub>0g\<^sub>1.\<phi>) \<cdot> \<a>[f, tab\<^sub>0 f, f\<^sub>0g\<^sub>1.p\<^sub>1] \<cdot> (f.tab \<star> f\<^sub>0g\<^sub>1.p\<^sub>1) \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1)) \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>1 f, f\<^sub>0g\<^sub>1.p\<^sub>1, fg\<^sub>0h\<^sub>1.p\<^sub>1]
[PROOF STEP]
using comp_assoc
[PROOF STATE]
proof (prove)
using this:
(?h \<cdot> ?g) \<cdot> ?f = ?h \<cdot> ?g \<cdot> ?f
goal (1 subgoal):
1. \<a>[f, g, h \<star> tab\<^sub>0 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0] \<cdot> ((\<a>[f \<star> g, h, tab\<^sub>0 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0] \<cdot> \<a>\<^sup>-\<^sup>1[f \<star> g, h, tab\<^sub>0 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0]) \<cdot> ((f \<star> g) \<star> \<a>[h, tab\<^sub>0 h, fg\<^sub>0h\<^sub>1.p\<^sub>0])) \<cdot> ((f \<star> g) \<star> h.tab \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0) \<cdot> ((f \<star> g) \<star> fg\<^sub>0h\<^sub>1.\<phi>) \<cdot> \<a>[f \<star> g, tab\<^sub>0 g \<star> f\<^sub>0g\<^sub>1.p\<^sub>0, fg\<^sub>0h\<^sub>1.p\<^sub>1] \<cdot> (\<a>\<^sup>-\<^sup>1[f, g, tab\<^sub>0 g \<star> f\<^sub>0g\<^sub>1.p\<^sub>0] \<cdot> (f \<star> \<a>[g, tab\<^sub>0 g, f\<^sub>0g\<^sub>1.p\<^sub>0]) \<cdot> (f \<star> g.tab \<star> f\<^sub>0g\<^sub>1.p\<^sub>0) \<cdot> (f \<star> f\<^sub>0g\<^sub>1.\<phi>) \<cdot> \<a>[f, tab\<^sub>0 f, f\<^sub>0g\<^sub>1.p\<^sub>1] \<cdot> (f.tab \<star> f\<^sub>0g\<^sub>1.p\<^sub>1) \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>1 f, f\<^sub>0g\<^sub>1.p\<^sub>1, fg\<^sub>0h\<^sub>1.p\<^sub>1] = \<a>[f, g, h \<star> tab\<^sub>0 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0] \<cdot> \<a>[f \<star> g, h, tab\<^sub>0 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0] \<cdot> (\<a>\<^sup>-\<^sup>1[f \<star> g, h, tab\<^sub>0 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0] \<cdot> ((f \<star> g) \<star> \<a>[h, tab\<^sub>0 h, fg\<^sub>0h\<^sub>1.p\<^sub>0]) \<cdot> ((f \<star> g) \<star> h.tab \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0) \<cdot> ((f \<star> g) \<star> fg\<^sub>0h\<^sub>1.\<phi>) \<cdot> \<a>[f \<star> g, tab\<^sub>0 g \<star> f\<^sub>0g\<^sub>1.p\<^sub>0, fg\<^sub>0h\<^sub>1.p\<^sub>1] \<cdot> (\<a>\<^sup>-\<^sup>1[f, g, tab\<^sub>0 g \<star> f\<^sub>0g\<^sub>1.p\<^sub>0] \<cdot> (f \<star> \<a>[g, tab\<^sub>0 g, f\<^sub>0g\<^sub>1.p\<^sub>0]) \<cdot> (f \<star> g.tab \<star> f\<^sub>0g\<^sub>1.p\<^sub>0) \<cdot> (f \<star> f\<^sub>0g\<^sub>1.\<phi>) \<cdot> \<a>[f, tab\<^sub>0 f, f\<^sub>0g\<^sub>1.p\<^sub>1] \<cdot> (f.tab \<star> f\<^sub>0g\<^sub>1.p\<^sub>1) \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1)) \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>1 f, f\<^sub>0g\<^sub>1.p\<^sub>1, fg\<^sub>0h\<^sub>1.p\<^sub>1]
[PROOF STEP]
by presburger
[PROOF STATE]
proof (state)
this:
\<a>[f, g, h \<star> tab\<^sub>0 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0] \<cdot> ((\<a>[f \<star> g, h, tab\<^sub>0 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0] \<cdot> \<a>\<^sup>-\<^sup>1[f \<star> g, h, tab\<^sub>0 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0]) \<cdot> ((f \<star> g) \<star> \<a>[h, tab\<^sub>0 h, fg\<^sub>0h\<^sub>1.p\<^sub>0])) \<cdot> ((f \<star> g) \<star> h.tab \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0) \<cdot> ((f \<star> g) \<star> fg\<^sub>0h\<^sub>1.\<phi>) \<cdot> \<a>[f \<star> g, tab\<^sub>0 g \<star> f\<^sub>0g\<^sub>1.p\<^sub>0, fg\<^sub>0h\<^sub>1.p\<^sub>1] \<cdot> (\<a>\<^sup>-\<^sup>1[f, g, tab\<^sub>0 g \<star> f\<^sub>0g\<^sub>1.p\<^sub>0] \<cdot> (f \<star> \<a>[g, tab\<^sub>0 g, f\<^sub>0g\<^sub>1.p\<^sub>0]) \<cdot> (f \<star> g.tab \<star> f\<^sub>0g\<^sub>1.p\<^sub>0) \<cdot> (f \<star> f\<^sub>0g\<^sub>1.\<phi>) \<cdot> \<a>[f, tab\<^sub>0 f, f\<^sub>0g\<^sub>1.p\<^sub>1] \<cdot> (f.tab \<star> f\<^sub>0g\<^sub>1.p\<^sub>1) \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>1 f, f\<^sub>0g\<^sub>1.p\<^sub>1, fg\<^sub>0h\<^sub>1.p\<^sub>1] = \<a>[f, g, h \<star> tab\<^sub>0 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0] \<cdot> \<a>[f \<star> g, h, tab\<^sub>0 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0] \<cdot> (\<a>\<^sup>-\<^sup>1[f \<star> g, h, tab\<^sub>0 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0] \<cdot> ((f \<star> g) \<star> \<a>[h, tab\<^sub>0 h, fg\<^sub>0h\<^sub>1.p\<^sub>0]) \<cdot> ((f \<star> g) \<star> h.tab \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0) \<cdot> ((f \<star> g) \<star> fg\<^sub>0h\<^sub>1.\<phi>) \<cdot> \<a>[f \<star> g, tab\<^sub>0 g \<star> f\<^sub>0g\<^sub>1.p\<^sub>0, fg\<^sub>0h\<^sub>1.p\<^sub>1] \<cdot> (\<a>\<^sup>-\<^sup>1[f, g, tab\<^sub>0 g \<star> f\<^sub>0g\<^sub>1.p\<^sub>0] \<cdot> (f \<star> \<a>[g, tab\<^sub>0 g, f\<^sub>0g\<^sub>1.p\<^sub>0]) \<cdot> (f \<star> g.tab \<star> f\<^sub>0g\<^sub>1.p\<^sub>0) \<cdot> (f \<star> f\<^sub>0g\<^sub>1.\<phi>) \<cdot> \<a>[f, tab\<^sub>0 f, f\<^sub>0g\<^sub>1.p\<^sub>1] \<cdot> (f.tab \<star> f\<^sub>0g\<^sub>1.p\<^sub>1) \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1)) \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>1 f, f\<^sub>0g\<^sub>1.p\<^sub>1, fg\<^sub>0h\<^sub>1.p\<^sub>1]
goal (1 subgoal):
1. (f \<star> \<theta>\<^sub>f) \<cdot> \<a>[f, tab\<^sub>0 f, w\<^sub>f] \<cdot> (f.tab \<star> w\<^sub>f) = ((f \<star> \<theta>\<^sub>f') \<cdot> \<a>[f, tab\<^sub>0 f, w\<^sub>f'] \<cdot> (f.tab \<star> w\<^sub>f')) \<cdot> \<beta>\<^sub>f
[PROOF STEP]
also
[PROOF STATE]
proof (state)
this:
\<a>[f, g, h \<star> tab\<^sub>0 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0] \<cdot> ((\<a>[f \<star> g, h, tab\<^sub>0 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0] \<cdot> \<a>\<^sup>-\<^sup>1[f \<star> g, h, tab\<^sub>0 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0]) \<cdot> ((f \<star> g) \<star> \<a>[h, tab\<^sub>0 h, fg\<^sub>0h\<^sub>1.p\<^sub>0])) \<cdot> ((f \<star> g) \<star> h.tab \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0) \<cdot> ((f \<star> g) \<star> fg\<^sub>0h\<^sub>1.\<phi>) \<cdot> \<a>[f \<star> g, tab\<^sub>0 g \<star> f\<^sub>0g\<^sub>1.p\<^sub>0, fg\<^sub>0h\<^sub>1.p\<^sub>1] \<cdot> (\<a>\<^sup>-\<^sup>1[f, g, tab\<^sub>0 g \<star> f\<^sub>0g\<^sub>1.p\<^sub>0] \<cdot> (f \<star> \<a>[g, tab\<^sub>0 g, f\<^sub>0g\<^sub>1.p\<^sub>0]) \<cdot> (f \<star> g.tab \<star> f\<^sub>0g\<^sub>1.p\<^sub>0) \<cdot> (f \<star> f\<^sub>0g\<^sub>1.\<phi>) \<cdot> \<a>[f, tab\<^sub>0 f, f\<^sub>0g\<^sub>1.p\<^sub>1] \<cdot> (f.tab \<star> f\<^sub>0g\<^sub>1.p\<^sub>1) \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>1 f, f\<^sub>0g\<^sub>1.p\<^sub>1, fg\<^sub>0h\<^sub>1.p\<^sub>1] = \<a>[f, g, h \<star> tab\<^sub>0 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0] \<cdot> \<a>[f \<star> g, h, tab\<^sub>0 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0] \<cdot> (\<a>\<^sup>-\<^sup>1[f \<star> g, h, tab\<^sub>0 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0] \<cdot> ((f \<star> g) \<star> \<a>[h, tab\<^sub>0 h, fg\<^sub>0h\<^sub>1.p\<^sub>0]) \<cdot> ((f \<star> g) \<star> h.tab \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0) \<cdot> ((f \<star> g) \<star> fg\<^sub>0h\<^sub>1.\<phi>) \<cdot> \<a>[f \<star> g, tab\<^sub>0 g \<star> f\<^sub>0g\<^sub>1.p\<^sub>0, fg\<^sub>0h\<^sub>1.p\<^sub>1] \<cdot> (\<a>\<^sup>-\<^sup>1[f, g, tab\<^sub>0 g \<star> f\<^sub>0g\<^sub>1.p\<^sub>0] \<cdot> (f \<star> \<a>[g, tab\<^sub>0 g, f\<^sub>0g\<^sub>1.p\<^sub>0]) \<cdot> (f \<star> g.tab \<star> f\<^sub>0g\<^sub>1.p\<^sub>0) \<cdot> (f \<star> f\<^sub>0g\<^sub>1.\<phi>) \<cdot> \<a>[f, tab\<^sub>0 f, f\<^sub>0g\<^sub>1.p\<^sub>1] \<cdot> (f.tab \<star> f\<^sub>0g\<^sub>1.p\<^sub>1) \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1)) \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>1 f, f\<^sub>0g\<^sub>1.p\<^sub>1, fg\<^sub>0h\<^sub>1.p\<^sub>1]
goal (1 subgoal):
1. (f \<star> \<theta>\<^sub>f) \<cdot> \<a>[f, tab\<^sub>0 f, w\<^sub>f] \<cdot> (f.tab \<star> w\<^sub>f) = ((f \<star> \<theta>\<^sub>f') \<cdot> \<a>[f, tab\<^sub>0 f, w\<^sub>f'] \<cdot> (f.tab \<star> w\<^sub>f')) \<cdot> \<beta>\<^sub>f
[PROOF STEP]
have "... =
\<a>[f, g, h \<star> tab\<^sub>0 h \<star> TTfgh.p\<^sub>0] \<cdot>
\<a>[f \<star> g, h, tab\<^sub>0 h \<star> TTfgh.p\<^sub>0] \<cdot>
TTfgh.tab \<cdot>
\<a>\<^sup>-\<^sup>1[tab\<^sub>1 f, Tfg.p\<^sub>1, TTfgh.p\<^sub>1]"
[PROOF STATE]
proof (prove)
goal (1 subgoal):
1. \<a>[f, g, h \<star> tab\<^sub>0 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0] \<cdot> \<a>[f \<star> g, h, tab\<^sub>0 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0] \<cdot> (\<a>\<^sup>-\<^sup>1[f \<star> g, h, tab\<^sub>0 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0] \<cdot> ((f \<star> g) \<star> \<a>[h, tab\<^sub>0 h, fg\<^sub>0h\<^sub>1.p\<^sub>0]) \<cdot> ((f \<star> g) \<star> h.tab \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0) \<cdot> ((f \<star> g) \<star> fg\<^sub>0h\<^sub>1.\<phi>) \<cdot> \<a>[f \<star> g, tab\<^sub>0 g \<star> f\<^sub>0g\<^sub>1.p\<^sub>0, fg\<^sub>0h\<^sub>1.p\<^sub>1] \<cdot> (\<a>\<^sup>-\<^sup>1[f, g, tab\<^sub>0 g \<star> f\<^sub>0g\<^sub>1.p\<^sub>0] \<cdot> (f \<star> \<a>[g, tab\<^sub>0 g, f\<^sub>0g\<^sub>1.p\<^sub>0]) \<cdot> (f \<star> g.tab \<star> f\<^sub>0g\<^sub>1.p\<^sub>0) \<cdot> (f \<star> f\<^sub>0g\<^sub>1.\<phi>) \<cdot> \<a>[f, tab\<^sub>0 f, f\<^sub>0g\<^sub>1.p\<^sub>1] \<cdot> (f.tab \<star> f\<^sub>0g\<^sub>1.p\<^sub>1) \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1)) \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>1 f, f\<^sub>0g\<^sub>1.p\<^sub>1, fg\<^sub>0h\<^sub>1.p\<^sub>1] = \<a>[f, g, h \<star> tab\<^sub>0 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0] \<cdot> \<a>[f \<star> g, h, tab\<^sub>0 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0] \<cdot> TTfgh.tab \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>1 f, f\<^sub>0g\<^sub>1.p\<^sub>1, fg\<^sub>0h\<^sub>1.p\<^sub>1]
[PROOF STEP]
using TTfgh.tab_def Tfg.\<rho>\<sigma>.tab_def
[PROOF STATE]
proof (prove)
using this:
TTfgh.tab \<equiv> \<a>\<^sup>-\<^sup>1[f \<star> g, h, tab\<^sub>0 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0] \<cdot> ((f \<star> g) \<star> \<a>[h, tab\<^sub>0 h, fg\<^sub>0h\<^sub>1.p\<^sub>0]) \<cdot> ((f \<star> g) \<star> h.tab \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0) \<cdot> ((f \<star> g) \<star> fg\<^sub>0h\<^sub>1.\<phi>) \<cdot> \<a>[f \<star> g, tab\<^sub>0 g \<star> f\<^sub>0g\<^sub>1.p\<^sub>0, fg\<^sub>0h\<^sub>1.p\<^sub>1] \<cdot> (Tfg.\<rho>\<sigma>.tab \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1)
Tfg.\<rho>\<sigma>.tab \<equiv> \<a>\<^sup>-\<^sup>1[f, g, tab\<^sub>0 g \<star> f\<^sub>0g\<^sub>1.p\<^sub>0] \<cdot> (f \<star> \<a>[g, tab\<^sub>0 g, f\<^sub>0g\<^sub>1.p\<^sub>0]) \<cdot> (f \<star> g.tab \<star> f\<^sub>0g\<^sub>1.p\<^sub>0) \<cdot> (f \<star> f\<^sub>0g\<^sub>1.\<phi>) \<cdot> \<a>[f, tab\<^sub>0 f, f\<^sub>0g\<^sub>1.p\<^sub>1] \<cdot> (f.tab \<star> f\<^sub>0g\<^sub>1.p\<^sub>1)
goal (1 subgoal):
1. \<a>[f, g, h \<star> tab\<^sub>0 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0] \<cdot> \<a>[f \<star> g, h, tab\<^sub>0 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0] \<cdot> (\<a>\<^sup>-\<^sup>1[f \<star> g, h, tab\<^sub>0 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0] \<cdot> ((f \<star> g) \<star> \<a>[h, tab\<^sub>0 h, fg\<^sub>0h\<^sub>1.p\<^sub>0]) \<cdot> ((f \<star> g) \<star> h.tab \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0) \<cdot> ((f \<star> g) \<star> fg\<^sub>0h\<^sub>1.\<phi>) \<cdot> \<a>[f \<star> g, tab\<^sub>0 g \<star> f\<^sub>0g\<^sub>1.p\<^sub>0, fg\<^sub>0h\<^sub>1.p\<^sub>1] \<cdot> (\<a>\<^sup>-\<^sup>1[f, g, tab\<^sub>0 g \<star> f\<^sub>0g\<^sub>1.p\<^sub>0] \<cdot> (f \<star> \<a>[g, tab\<^sub>0 g, f\<^sub>0g\<^sub>1.p\<^sub>0]) \<cdot> (f \<star> g.tab \<star> f\<^sub>0g\<^sub>1.p\<^sub>0) \<cdot> (f \<star> f\<^sub>0g\<^sub>1.\<phi>) \<cdot> \<a>[f, tab\<^sub>0 f, f\<^sub>0g\<^sub>1.p\<^sub>1] \<cdot> (f.tab \<star> f\<^sub>0g\<^sub>1.p\<^sub>1) \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1)) \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>1 f, f\<^sub>0g\<^sub>1.p\<^sub>1, fg\<^sub>0h\<^sub>1.p\<^sub>1] = \<a>[f, g, h \<star> tab\<^sub>0 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0] \<cdot> \<a>[f \<star> g, h, tab\<^sub>0 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0] \<cdot> TTfgh.tab \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>1 f, f\<^sub>0g\<^sub>1.p\<^sub>1, fg\<^sub>0h\<^sub>1.p\<^sub>1]
[PROOF STEP]
by simp
[PROOF STATE]
proof (state)
this:
\<a>[f, g, h \<star> tab\<^sub>0 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0] \<cdot> \<a>[f \<star> g, h, tab\<^sub>0 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0] \<cdot> (\<a>\<^sup>-\<^sup>1[f \<star> g, h, tab\<^sub>0 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0] \<cdot> ((f \<star> g) \<star> \<a>[h, tab\<^sub>0 h, fg\<^sub>0h\<^sub>1.p\<^sub>0]) \<cdot> ((f \<star> g) \<star> h.tab \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0) \<cdot> ((f \<star> g) \<star> fg\<^sub>0h\<^sub>1.\<phi>) \<cdot> \<a>[f \<star> g, tab\<^sub>0 g \<star> f\<^sub>0g\<^sub>1.p\<^sub>0, fg\<^sub>0h\<^sub>1.p\<^sub>1] \<cdot> (\<a>\<^sup>-\<^sup>1[f, g, tab\<^sub>0 g \<star> f\<^sub>0g\<^sub>1.p\<^sub>0] \<cdot> (f \<star> \<a>[g, tab\<^sub>0 g, f\<^sub>0g\<^sub>1.p\<^sub>0]) \<cdot> (f \<star> g.tab \<star> f\<^sub>0g\<^sub>1.p\<^sub>0) \<cdot> (f \<star> f\<^sub>0g\<^sub>1.\<phi>) \<cdot> \<a>[f, tab\<^sub>0 f, f\<^sub>0g\<^sub>1.p\<^sub>1] \<cdot> (f.tab \<star> f\<^sub>0g\<^sub>1.p\<^sub>1) \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1)) \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>1 f, f\<^sub>0g\<^sub>1.p\<^sub>1, fg\<^sub>0h\<^sub>1.p\<^sub>1] = \<a>[f, g, h \<star> tab\<^sub>0 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0] \<cdot> \<a>[f \<star> g, h, tab\<^sub>0 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0] \<cdot> TTfgh.tab \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>1 f, f\<^sub>0g\<^sub>1.p\<^sub>1, fg\<^sub>0h\<^sub>1.p\<^sub>1]
goal (1 subgoal):
1. (f \<star> \<theta>\<^sub>f) \<cdot> \<a>[f, tab\<^sub>0 f, w\<^sub>f] \<cdot> (f.tab \<star> w\<^sub>f) = ((f \<star> \<theta>\<^sub>f') \<cdot> \<a>[f, tab\<^sub>0 f, w\<^sub>f'] \<cdot> (f.tab \<star> w\<^sub>f')) \<cdot> \<beta>\<^sub>f
[PROOF STEP]
also
[PROOF STATE]
proof (state)
this:
\<a>[f, g, h \<star> tab\<^sub>0 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0] \<cdot> \<a>[f \<star> g, h, tab\<^sub>0 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0] \<cdot> (\<a>\<^sup>-\<^sup>1[f \<star> g, h, tab\<^sub>0 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0] \<cdot> ((f \<star> g) \<star> \<a>[h, tab\<^sub>0 h, fg\<^sub>0h\<^sub>1.p\<^sub>0]) \<cdot> ((f \<star> g) \<star> h.tab \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0) \<cdot> ((f \<star> g) \<star> fg\<^sub>0h\<^sub>1.\<phi>) \<cdot> \<a>[f \<star> g, tab\<^sub>0 g \<star> f\<^sub>0g\<^sub>1.p\<^sub>0, fg\<^sub>0h\<^sub>1.p\<^sub>1] \<cdot> (\<a>\<^sup>-\<^sup>1[f, g, tab\<^sub>0 g \<star> f\<^sub>0g\<^sub>1.p\<^sub>0] \<cdot> (f \<star> \<a>[g, tab\<^sub>0 g, f\<^sub>0g\<^sub>1.p\<^sub>0]) \<cdot> (f \<star> g.tab \<star> f\<^sub>0g\<^sub>1.p\<^sub>0) \<cdot> (f \<star> f\<^sub>0g\<^sub>1.\<phi>) \<cdot> \<a>[f, tab\<^sub>0 f, f\<^sub>0g\<^sub>1.p\<^sub>1] \<cdot> (f.tab \<star> f\<^sub>0g\<^sub>1.p\<^sub>1) \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1)) \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>1 f, f\<^sub>0g\<^sub>1.p\<^sub>1, fg\<^sub>0h\<^sub>1.p\<^sub>1] = \<a>[f, g, h \<star> tab\<^sub>0 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0] \<cdot> \<a>[f \<star> g, h, tab\<^sub>0 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0] \<cdot> TTfgh.tab \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>1 f, f\<^sub>0g\<^sub>1.p\<^sub>1, fg\<^sub>0h\<^sub>1.p\<^sub>1]
goal (1 subgoal):
1. (f \<star> \<theta>\<^sub>f) \<cdot> \<a>[f, tab\<^sub>0 f, w\<^sub>f] \<cdot> (f.tab \<star> w\<^sub>f) = ((f \<star> \<theta>\<^sub>f') \<cdot> \<a>[f, tab\<^sub>0 f, w\<^sub>f'] \<cdot> (f.tab \<star> w\<^sub>f')) \<cdot> \<beta>\<^sub>f
[PROOF STEP]
have "... =
\<a>[f, g, h \<star> tab\<^sub>0 h \<star> TTfgh.p\<^sub>0] \<cdot>
\<a>[f \<star> g, h, tab\<^sub>0 h \<star> TTfgh.p\<^sub>0] \<cdot>
((\<a>\<^sup>-\<^sup>1[f, g, h] \<star> tab\<^sub>0 h \<star> TTfgh.p\<^sub>0) \<cdot>
((f \<star> g \<star> h) \<star> TTfgh_TfTgh.the_\<theta>) \<cdot>
\<a>[f \<star> g \<star> h, (tab\<^sub>0 h \<star> Tgh.p\<^sub>0) \<star> TfTgh.p\<^sub>0, TTfgh_TfTgh.chine] \<cdot>
(\<a>\<^sup>-\<^sup>1[f, g \<star> h, (tab\<^sub>0 h \<star> Tgh.p\<^sub>0) \<star> TfTgh.p\<^sub>0] \<cdot>
(f \<star> \<a>[g \<star> h, tab\<^sub>0 h \<star> Tgh.p\<^sub>0, TfTgh.p\<^sub>0]) \<cdot>
(f \<star> \<a>\<^sup>-\<^sup>1[g, h, tab\<^sub>0 h \<star> Tgh.p\<^sub>0] \<cdot>
(g \<star> \<a>[h, tab\<^sub>0 h, Tgh.p\<^sub>0]) \<cdot>
(g \<star> h.tab \<star> Tgh.p\<^sub>0) \<cdot>
(g \<star> g\<^sub>0h\<^sub>1.\<phi>) \<cdot>
\<a>[g, tab\<^sub>0 g, Tgh.p\<^sub>1] \<cdot>
(g.tab \<star> Tgh.p\<^sub>1)
\<star> TfTgh.p\<^sub>0) \<cdot>
(f \<star> f\<^sub>0gh\<^sub>1.\<phi>) \<cdot>
\<a>[f, tab\<^sub>0 f, TfTgh.p\<^sub>1] \<cdot>
(f.tab \<star> TfTgh.p\<^sub>1)
\<star> TTfgh_TfTgh.chine) \<cdot>
TTfgh_TfTgh.the_\<nu>) \<cdot>
\<a>\<^sup>-\<^sup>1[tab\<^sub>1 f, Tfg.p\<^sub>1, TTfgh.p\<^sub>1]"
[PROOF STATE]
proof (prove)
goal (1 subgoal):
1. \<a>[f, g, h \<star> tab\<^sub>0 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0] \<cdot> \<a>[f \<star> g, h, tab\<^sub>0 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0] \<cdot> TTfgh.tab \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>1 f, f\<^sub>0g\<^sub>1.p\<^sub>1, fg\<^sub>0h\<^sub>1.p\<^sub>1] = \<a>[f, g, h \<star> tab\<^sub>0 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0] \<cdot> \<a>[f \<star> g, h, tab\<^sub>0 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0] \<cdot> ((\<a>\<^sup>-\<^sup>1[f, g, h] \<star> tab\<^sub>0 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0) \<cdot> ((f \<star> g \<star> h) \<star> TTfgh_TfTgh.the_\<theta>) \<cdot> \<a>[f \<star> g \<star> h, (tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine] \<cdot> (\<a>\<^sup>-\<^sup>1[f, g \<star> h, (tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0] \<cdot> (f \<star> \<a>[g \<star> h, tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0, f\<^sub>0gh\<^sub>1.p\<^sub>0]) \<cdot> (f \<star> \<a>\<^sup>-\<^sup>1[g, h, tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0] \<cdot> (g \<star> \<a>[h, tab\<^sub>0 h, g\<^sub>0h\<^sub>1.p\<^sub>0]) \<cdot> (g \<star> h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<cdot> (g \<star> g\<^sub>0h\<^sub>1.\<phi>) \<cdot> \<a>[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1] \<cdot> (g.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>1) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<cdot> (f \<star> f\<^sub>0gh\<^sub>1.\<phi>) \<cdot> \<a>[f, tab\<^sub>0 f, f\<^sub>0gh\<^sub>1.p\<^sub>1] \<cdot> (f.tab \<star> f\<^sub>0gh\<^sub>1.p\<^sub>1) \<star> TTfgh_TfTgh.chine) \<cdot> TTfgh_TfTgh.the_\<nu>) \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>1 f, f\<^sub>0g\<^sub>1.p\<^sub>1, fg\<^sub>0h\<^sub>1.p\<^sub>1]
[PROOF STEP]
using src_tab_eq TfTgh.tab_def Tgh.\<rho>\<sigma>.tab_def comp_assoc
[PROOF STATE]
proof (prove)
using this:
(\<a>\<^sup>-\<^sup>1[f, g, h] \<star> tab\<^sub>0 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0) \<cdot> (((f \<star> g \<star> h) \<star> TTfgh_TfTgh.the_\<theta>) \<cdot> \<a>[f \<star> g \<star> h, (tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine] \<cdot> (TfTgh.tab \<star> TTfgh_TfTgh.chine)) \<cdot> TTfgh_TfTgh.the_\<nu> = TTfgh.tab
TfTgh.tab \<equiv> \<a>\<^sup>-\<^sup>1[f, g \<star> h, (tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0] \<cdot> (f \<star> \<a>[g \<star> h, tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0, f\<^sub>0gh\<^sub>1.p\<^sub>0]) \<cdot> (f \<star> Tgh.\<rho>\<sigma>.tab \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<cdot> (f \<star> f\<^sub>0gh\<^sub>1.\<phi>) \<cdot> \<a>[f, tab\<^sub>0 f, f\<^sub>0gh\<^sub>1.p\<^sub>1] \<cdot> (f.tab \<star> f\<^sub>0gh\<^sub>1.p\<^sub>1)
Tgh.\<rho>\<sigma>.tab \<equiv> \<a>\<^sup>-\<^sup>1[g, h, tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0] \<cdot> (g \<star> \<a>[h, tab\<^sub>0 h, g\<^sub>0h\<^sub>1.p\<^sub>0]) \<cdot> (g \<star> h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<cdot> (g \<star> g\<^sub>0h\<^sub>1.\<phi>) \<cdot> \<a>[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1] \<cdot> (g.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>1)
(?h \<cdot> ?g) \<cdot> ?f = ?h \<cdot> ?g \<cdot> ?f
goal (1 subgoal):
1. \<a>[f, g, h \<star> tab\<^sub>0 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0] \<cdot> \<a>[f \<star> g, h, tab\<^sub>0 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0] \<cdot> TTfgh.tab \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>1 f, f\<^sub>0g\<^sub>1.p\<^sub>1, fg\<^sub>0h\<^sub>1.p\<^sub>1] = \<a>[f, g, h \<star> tab\<^sub>0 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0] \<cdot> \<a>[f \<star> g, h, tab\<^sub>0 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0] \<cdot> ((\<a>\<^sup>-\<^sup>1[f, g, h] \<star> tab\<^sub>0 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0) \<cdot> ((f \<star> g \<star> h) \<star> TTfgh_TfTgh.the_\<theta>) \<cdot> \<a>[f \<star> g \<star> h, (tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine] \<cdot> (\<a>\<^sup>-\<^sup>1[f, g \<star> h, (tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0] \<cdot> (f \<star> \<a>[g \<star> h, tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0, f\<^sub>0gh\<^sub>1.p\<^sub>0]) \<cdot> (f \<star> \<a>\<^sup>-\<^sup>1[g, h, tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0] \<cdot> (g \<star> \<a>[h, tab\<^sub>0 h, g\<^sub>0h\<^sub>1.p\<^sub>0]) \<cdot> (g \<star> h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<cdot> (g \<star> g\<^sub>0h\<^sub>1.\<phi>) \<cdot> \<a>[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1] \<cdot> (g.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>1) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<cdot> (f \<star> f\<^sub>0gh\<^sub>1.\<phi>) \<cdot> \<a>[f, tab\<^sub>0 f, f\<^sub>0gh\<^sub>1.p\<^sub>1] \<cdot> (f.tab \<star> f\<^sub>0gh\<^sub>1.p\<^sub>1) \<star> TTfgh_TfTgh.chine) \<cdot> TTfgh_TfTgh.the_\<nu>) \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>1 f, f\<^sub>0g\<^sub>1.p\<^sub>1, fg\<^sub>0h\<^sub>1.p\<^sub>1]
[PROOF STEP]
by simp
[PROOF STATE]
proof (state)
this:
\<a>[f, g, h \<star> tab\<^sub>0 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0] \<cdot> \<a>[f \<star> g, h, tab\<^sub>0 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0] \<cdot> TTfgh.tab \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>1 f, f\<^sub>0g\<^sub>1.p\<^sub>1, fg\<^sub>0h\<^sub>1.p\<^sub>1] = \<a>[f, g, h \<star> tab\<^sub>0 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0] \<cdot> \<a>[f \<star> g, h, tab\<^sub>0 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0] \<cdot> ((\<a>\<^sup>-\<^sup>1[f, g, h] \<star> tab\<^sub>0 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0) \<cdot> ((f \<star> g \<star> h) \<star> TTfgh_TfTgh.the_\<theta>) \<cdot> \<a>[f \<star> g \<star> h, (tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine] \<cdot> (\<a>\<^sup>-\<^sup>1[f, g \<star> h, (tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0] \<cdot> (f \<star> \<a>[g \<star> h, tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0, f\<^sub>0gh\<^sub>1.p\<^sub>0]) \<cdot> (f \<star> \<a>\<^sup>-\<^sup>1[g, h, tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0] \<cdot> (g \<star> \<a>[h, tab\<^sub>0 h, g\<^sub>0h\<^sub>1.p\<^sub>0]) \<cdot> (g \<star> h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<cdot> (g \<star> g\<^sub>0h\<^sub>1.\<phi>) \<cdot> \<a>[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1] \<cdot> (g.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>1) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<cdot> (f \<star> f\<^sub>0gh\<^sub>1.\<phi>) \<cdot> \<a>[f, tab\<^sub>0 f, f\<^sub>0gh\<^sub>1.p\<^sub>1] \<cdot> (f.tab \<star> f\<^sub>0gh\<^sub>1.p\<^sub>1) \<star> TTfgh_TfTgh.chine) \<cdot> TTfgh_TfTgh.the_\<nu>) \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>1 f, f\<^sub>0g\<^sub>1.p\<^sub>1, fg\<^sub>0h\<^sub>1.p\<^sub>1]
goal (1 subgoal):
1. (f \<star> \<theta>\<^sub>f) \<cdot> \<a>[f, tab\<^sub>0 f, w\<^sub>f] \<cdot> (f.tab \<star> w\<^sub>f) = ((f \<star> \<theta>\<^sub>f') \<cdot> \<a>[f, tab\<^sub>0 f, w\<^sub>f'] \<cdot> (f.tab \<star> w\<^sub>f')) \<cdot> \<beta>\<^sub>f
[PROOF STEP]
also
[PROOF STATE]
proof (state)
this:
\<a>[f, g, h \<star> tab\<^sub>0 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0] \<cdot> \<a>[f \<star> g, h, tab\<^sub>0 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0] \<cdot> TTfgh.tab \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>1 f, f\<^sub>0g\<^sub>1.p\<^sub>1, fg\<^sub>0h\<^sub>1.p\<^sub>1] = \<a>[f, g, h \<star> tab\<^sub>0 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0] \<cdot> \<a>[f \<star> g, h, tab\<^sub>0 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0] \<cdot> ((\<a>\<^sup>-\<^sup>1[f, g, h] \<star> tab\<^sub>0 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0) \<cdot> ((f \<star> g \<star> h) \<star> TTfgh_TfTgh.the_\<theta>) \<cdot> \<a>[f \<star> g \<star> h, (tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine] \<cdot> (\<a>\<^sup>-\<^sup>1[f, g \<star> h, (tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0] \<cdot> (f \<star> \<a>[g \<star> h, tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0, f\<^sub>0gh\<^sub>1.p\<^sub>0]) \<cdot> (f \<star> \<a>\<^sup>-\<^sup>1[g, h, tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0] \<cdot> (g \<star> \<a>[h, tab\<^sub>0 h, g\<^sub>0h\<^sub>1.p\<^sub>0]) \<cdot> (g \<star> h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<cdot> (g \<star> g\<^sub>0h\<^sub>1.\<phi>) \<cdot> \<a>[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1] \<cdot> (g.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>1) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<cdot> (f \<star> f\<^sub>0gh\<^sub>1.\<phi>) \<cdot> \<a>[f, tab\<^sub>0 f, f\<^sub>0gh\<^sub>1.p\<^sub>1] \<cdot> (f.tab \<star> f\<^sub>0gh\<^sub>1.p\<^sub>1) \<star> TTfgh_TfTgh.chine) \<cdot> TTfgh_TfTgh.the_\<nu>) \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>1 f, f\<^sub>0g\<^sub>1.p\<^sub>1, fg\<^sub>0h\<^sub>1.p\<^sub>1]
goal (1 subgoal):
1. (f \<star> \<theta>\<^sub>f) \<cdot> \<a>[f, tab\<^sub>0 f, w\<^sub>f] \<cdot> (f.tab \<star> w\<^sub>f) = ((f \<star> \<theta>\<^sub>f') \<cdot> \<a>[f, tab\<^sub>0 f, w\<^sub>f'] \<cdot> (f.tab \<star> w\<^sub>f')) \<cdot> \<beta>\<^sub>f
[PROOF STEP]
have "... =
\<a>[f, g, h \<star> tab\<^sub>0 h \<star> TTfgh.p\<^sub>0] \<cdot>
\<a>[f \<star> g, h, tab\<^sub>0 h \<star> TTfgh.p\<^sub>0] \<cdot>
((\<a>\<^sup>-\<^sup>1[f, g, h] \<star> tab\<^sub>0 h \<star> TTfgh.p\<^sub>0) \<cdot>
((f \<star> g \<star> h) \<star> TTfgh_TfTgh.the_\<theta>) \<cdot>
\<a>[f \<star> g \<star> h, (tab\<^sub>0 h \<star> Tgh.p\<^sub>0) \<star> TfTgh.p\<^sub>0, TTfgh_TfTgh.chine] \<cdot>
(\<a>\<^sup>-\<^sup>1[f, g \<star> h, (tab\<^sub>0 h \<star> Tgh.p\<^sub>0) \<star> TfTgh.p\<^sub>0] \<cdot>
(f \<star> \<a>[g \<star> h, tab\<^sub>0 h \<star> Tgh.p\<^sub>0, TfTgh.p\<^sub>0]) \<cdot>
((f \<star> \<a>\<^sup>-\<^sup>1[g, h, tab\<^sub>0 h \<star> Tgh.p\<^sub>0] \<star> TfTgh.p\<^sub>0) \<cdot>
(f \<star> (g \<star> \<a>[h, tab\<^sub>0 h, Tgh.p\<^sub>0]) \<star> TfTgh.p\<^sub>0) \<cdot>
(f \<star> (g \<star> h.tab \<star> Tgh.p\<^sub>0) \<star> TfTgh.p\<^sub>0) \<cdot>
(f \<star> (g \<star> g\<^sub>0h\<^sub>1.\<phi>) \<star> TfTgh.p\<^sub>0) \<cdot>
(f \<star> \<a>[g, tab\<^sub>0 g, Tgh.p\<^sub>1] \<star> TfTgh.p\<^sub>0) \<cdot>
(f \<star> (g.tab \<star> Tgh.p\<^sub>1) \<star> TfTgh.p\<^sub>0) \<cdot>
(f \<star> f\<^sub>0gh\<^sub>1.\<phi>)) \<cdot>
\<a>[f, tab\<^sub>0 f, TfTgh.p\<^sub>1] \<cdot>
(f.tab \<star> TfTgh.p\<^sub>1)
\<star> TTfgh_TfTgh.chine) \<cdot>
TTfgh_TfTgh.the_\<nu>) \<cdot>
\<a>\<^sup>-\<^sup>1[tab\<^sub>1 f, Tfg.p\<^sub>1, TTfgh.p\<^sub>1]"
[PROOF STATE]
proof (prove)
goal (1 subgoal):
1. \<a>[f, g, h \<star> tab\<^sub>0 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0] \<cdot> \<a>[f \<star> g, h, tab\<^sub>0 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0] \<cdot> ((\<a>\<^sup>-\<^sup>1[f, g, h] \<star> tab\<^sub>0 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0) \<cdot> ((f \<star> g \<star> h) \<star> TTfgh_TfTgh.the_\<theta>) \<cdot> \<a>[f \<star> g \<star> h, (tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine] \<cdot> (\<a>\<^sup>-\<^sup>1[f, g \<star> h, (tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0] \<cdot> (f \<star> \<a>[g \<star> h, tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0, f\<^sub>0gh\<^sub>1.p\<^sub>0]) \<cdot> (f \<star> \<a>\<^sup>-\<^sup>1[g, h, tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0] \<cdot> (g \<star> \<a>[h, tab\<^sub>0 h, g\<^sub>0h\<^sub>1.p\<^sub>0]) \<cdot> (g \<star> h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<cdot> (g \<star> g\<^sub>0h\<^sub>1.\<phi>) \<cdot> \<a>[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1] \<cdot> (g.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>1) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<cdot> (f \<star> f\<^sub>0gh\<^sub>1.\<phi>) \<cdot> \<a>[f, tab\<^sub>0 f, f\<^sub>0gh\<^sub>1.p\<^sub>1] \<cdot> (f.tab \<star> f\<^sub>0gh\<^sub>1.p\<^sub>1) \<star> TTfgh_TfTgh.chine) \<cdot> TTfgh_TfTgh.the_\<nu>) \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>1 f, f\<^sub>0g\<^sub>1.p\<^sub>1, fg\<^sub>0h\<^sub>1.p\<^sub>1] = \<a>[f, g, h \<star> tab\<^sub>0 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0] \<cdot> \<a>[f \<star> g, h, tab\<^sub>0 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0] \<cdot> ((\<a>\<^sup>-\<^sup>1[f, g, h] \<star> tab\<^sub>0 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0) \<cdot> ((f \<star> g \<star> h) \<star> TTfgh_TfTgh.the_\<theta>) \<cdot> \<a>[f \<star> g \<star> h, (tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine] \<cdot> (\<a>\<^sup>-\<^sup>1[f, g \<star> h, (tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0] \<cdot> (f \<star> \<a>[g \<star> h, tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0, f\<^sub>0gh\<^sub>1.p\<^sub>0]) \<cdot> ((f \<star> \<a>\<^sup>-\<^sup>1[g, h, tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<cdot> (f \<star> (g \<star> \<a>[h, tab\<^sub>0 h, g\<^sub>0h\<^sub>1.p\<^sub>0]) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<cdot> (f \<star> (g \<star> h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<cdot> (f \<star> (g \<star> g\<^sub>0h\<^sub>1.\<phi>) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<cdot> (f \<star> \<a>[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<cdot> (f \<star> (g.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>1) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<cdot> (f \<star> f\<^sub>0gh\<^sub>1.\<phi>)) \<cdot> \<a>[f, tab\<^sub>0 f, f\<^sub>0gh\<^sub>1.p\<^sub>1] \<cdot> (f.tab \<star> f\<^sub>0gh\<^sub>1.p\<^sub>1) \<star> TTfgh_TfTgh.chine) \<cdot> TTfgh_TfTgh.the_\<nu>) \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>1 f, f\<^sub>0g\<^sub>1.p\<^sub>1, fg\<^sub>0h\<^sub>1.p\<^sub>1]
[PROOF STEP]
proof -
[PROOF STATE]
proof (state)
goal (1 subgoal):
1. \<a>[f, g, h \<star> tab\<^sub>0 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0] \<cdot> \<a>[f \<star> g, h, tab\<^sub>0 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0] \<cdot> ((\<a>\<^sup>-\<^sup>1[f, g, h] \<star> tab\<^sub>0 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0) \<cdot> ((f \<star> g \<star> h) \<star> TTfgh_TfTgh.the_\<theta>) \<cdot> \<a>[f \<star> g \<star> h, (tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine] \<cdot> (\<a>\<^sup>-\<^sup>1[f, g \<star> h, (tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0] \<cdot> (f \<star> \<a>[g \<star> h, tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0, f\<^sub>0gh\<^sub>1.p\<^sub>0]) \<cdot> (f \<star> \<a>\<^sup>-\<^sup>1[g, h, tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0] \<cdot> (g \<star> \<a>[h, tab\<^sub>0 h, g\<^sub>0h\<^sub>1.p\<^sub>0]) \<cdot> (g \<star> h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<cdot> (g \<star> g\<^sub>0h\<^sub>1.\<phi>) \<cdot> \<a>[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1] \<cdot> (g.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>1) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<cdot> (f \<star> f\<^sub>0gh\<^sub>1.\<phi>) \<cdot> \<a>[f, tab\<^sub>0 f, f\<^sub>0gh\<^sub>1.p\<^sub>1] \<cdot> (f.tab \<star> f\<^sub>0gh\<^sub>1.p\<^sub>1) \<star> TTfgh_TfTgh.chine) \<cdot> TTfgh_TfTgh.the_\<nu>) \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>1 f, f\<^sub>0g\<^sub>1.p\<^sub>1, fg\<^sub>0h\<^sub>1.p\<^sub>1] = \<a>[f, g, h \<star> tab\<^sub>0 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0] \<cdot> \<a>[f \<star> g, h, tab\<^sub>0 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0] \<cdot> ((\<a>\<^sup>-\<^sup>1[f, g, h] \<star> tab\<^sub>0 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0) \<cdot> ((f \<star> g \<star> h) \<star> TTfgh_TfTgh.the_\<theta>) \<cdot> \<a>[f \<star> g \<star> h, (tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine] \<cdot> (\<a>\<^sup>-\<^sup>1[f, g \<star> h, (tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0] \<cdot> (f \<star> \<a>[g \<star> h, tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0, f\<^sub>0gh\<^sub>1.p\<^sub>0]) \<cdot> ((f \<star> \<a>\<^sup>-\<^sup>1[g, h, tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<cdot> (f \<star> (g \<star> \<a>[h, tab\<^sub>0 h, g\<^sub>0h\<^sub>1.p\<^sub>0]) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<cdot> (f \<star> (g \<star> h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<cdot> (f \<star> (g \<star> g\<^sub>0h\<^sub>1.\<phi>) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<cdot> (f \<star> \<a>[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<cdot> (f \<star> (g.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>1) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<cdot> (f \<star> f\<^sub>0gh\<^sub>1.\<phi>)) \<cdot> \<a>[f, tab\<^sub>0 f, f\<^sub>0gh\<^sub>1.p\<^sub>1] \<cdot> (f.tab \<star> f\<^sub>0gh\<^sub>1.p\<^sub>1) \<star> TTfgh_TfTgh.chine) \<cdot> TTfgh_TfTgh.the_\<nu>) \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>1 f, f\<^sub>0g\<^sub>1.p\<^sub>1, fg\<^sub>0h\<^sub>1.p\<^sub>1]
[PROOF STEP]
have "f \<star> \<a>\<^sup>-\<^sup>1[g, h, tab\<^sub>0 h \<star> Tgh.p\<^sub>0] \<cdot>
(g \<star> \<a>[h, tab\<^sub>0 h, Tgh.p\<^sub>0]) \<cdot>
(g \<star> h.tab \<star> Tgh.p\<^sub>0) \<cdot>
(g \<star> g\<^sub>0h\<^sub>1.\<phi>) \<cdot>
\<a>[g, tab\<^sub>0 g, Tgh.p\<^sub>1] \<cdot>
(g.tab \<star> Tgh.p\<^sub>1)
\<star> TfTgh.p\<^sub>0 =
(f \<star> \<a>\<^sup>-\<^sup>1[g, h, tab\<^sub>0 h \<star> Tgh.p\<^sub>0] \<star> TfTgh.p\<^sub>0) \<cdot>
(f \<star> (g \<star> \<a>[h, tab\<^sub>0 h, Tgh.p\<^sub>0]) \<star> TfTgh.p\<^sub>0) \<cdot>
(f \<star> (g \<star> h.tab \<star> Tgh.p\<^sub>0) \<star> TfTgh.p\<^sub>0) \<cdot>
(f \<star> (g \<star> g\<^sub>0h\<^sub>1.\<phi>) \<star> TfTgh.p\<^sub>0) \<cdot>
(f \<star> \<a>[g, tab\<^sub>0 g, Tgh.p\<^sub>1] \<star> TfTgh.p\<^sub>0) \<cdot>
(f \<star> (g.tab \<star> Tgh.p\<^sub>1) \<star> TfTgh.p\<^sub>0)"
[PROOF STATE]
proof (prove)
goal (1 subgoal):
1. f \<star> \<a>\<^sup>-\<^sup>1[g, h, tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0] \<cdot> (g \<star> \<a>[h, tab\<^sub>0 h, g\<^sub>0h\<^sub>1.p\<^sub>0]) \<cdot> (g \<star> h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<cdot> (g \<star> g\<^sub>0h\<^sub>1.\<phi>) \<cdot> \<a>[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1] \<cdot> (g.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>1) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 = (f \<star> \<a>\<^sup>-\<^sup>1[g, h, tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<cdot> (f \<star> (g \<star> \<a>[h, tab\<^sub>0 h, g\<^sub>0h\<^sub>1.p\<^sub>0]) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<cdot> (f \<star> (g \<star> h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<cdot> (f \<star> (g \<star> g\<^sub>0h\<^sub>1.\<phi>) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<cdot> (f \<star> \<a>[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<cdot> (f \<star> (g.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>1) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0)
[PROOF STEP]
using fg gh whisker_right whisker_left
[PROOF STATE]
proof (prove)
using this:
src f = trg g
src g = trg h
\<lbrakk>ide ?f; seq ?\<nu> ?\<mu>\<rbrakk> \<Longrightarrow> ?\<nu> \<cdot> ?\<mu> \<star> ?f = (?\<nu> \<star> ?f) \<cdot> (?\<mu> \<star> ?f)
\<lbrakk>ide ?f; seq ?\<nu> ?\<mu>\<rbrakk> \<Longrightarrow> ?f \<star> ?\<nu> \<cdot> ?\<mu> = (?f \<star> ?\<nu>) \<cdot> (?f \<star> ?\<mu>)
goal (1 subgoal):
1. f \<star> \<a>\<^sup>-\<^sup>1[g, h, tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0] \<cdot> (g \<star> \<a>[h, tab\<^sub>0 h, g\<^sub>0h\<^sub>1.p\<^sub>0]) \<cdot> (g \<star> h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<cdot> (g \<star> g\<^sub>0h\<^sub>1.\<phi>) \<cdot> \<a>[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1] \<cdot> (g.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>1) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 = (f \<star> \<a>\<^sup>-\<^sup>1[g, h, tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<cdot> (f \<star> (g \<star> \<a>[h, tab\<^sub>0 h, g\<^sub>0h\<^sub>1.p\<^sub>0]) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<cdot> (f \<star> (g \<star> h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<cdot> (f \<star> (g \<star> g\<^sub>0h\<^sub>1.\<phi>) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<cdot> (f \<star> \<a>[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<cdot> (f \<star> (g.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>1) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0)
[PROOF STEP]
by simp
[PROOF STATE]
proof (state)
this:
f \<star> \<a>\<^sup>-\<^sup>1[g, h, tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0] \<cdot> (g \<star> \<a>[h, tab\<^sub>0 h, g\<^sub>0h\<^sub>1.p\<^sub>0]) \<cdot> (g \<star> h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<cdot> (g \<star> g\<^sub>0h\<^sub>1.\<phi>) \<cdot> \<a>[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1] \<cdot> (g.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>1) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 = (f \<star> \<a>\<^sup>-\<^sup>1[g, h, tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<cdot> (f \<star> (g \<star> \<a>[h, tab\<^sub>0 h, g\<^sub>0h\<^sub>1.p\<^sub>0]) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<cdot> (f \<star> (g \<star> h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<cdot> (f \<star> (g \<star> g\<^sub>0h\<^sub>1.\<phi>) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<cdot> (f \<star> \<a>[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<cdot> (f \<star> (g.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>1) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0)
goal (1 subgoal):
1. \<a>[f, g, h \<star> tab\<^sub>0 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0] \<cdot> \<a>[f \<star> g, h, tab\<^sub>0 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0] \<cdot> ((\<a>\<^sup>-\<^sup>1[f, g, h] \<star> tab\<^sub>0 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0) \<cdot> ((f \<star> g \<star> h) \<star> TTfgh_TfTgh.the_\<theta>) \<cdot> \<a>[f \<star> g \<star> h, (tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine] \<cdot> (\<a>\<^sup>-\<^sup>1[f, g \<star> h, (tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0] \<cdot> (f \<star> \<a>[g \<star> h, tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0, f\<^sub>0gh\<^sub>1.p\<^sub>0]) \<cdot> (f \<star> \<a>\<^sup>-\<^sup>1[g, h, tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0] \<cdot> (g \<star> \<a>[h, tab\<^sub>0 h, g\<^sub>0h\<^sub>1.p\<^sub>0]) \<cdot> (g \<star> h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<cdot> (g \<star> g\<^sub>0h\<^sub>1.\<phi>) \<cdot> \<a>[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1] \<cdot> (g.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>1) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<cdot> (f \<star> f\<^sub>0gh\<^sub>1.\<phi>) \<cdot> \<a>[f, tab\<^sub>0 f, f\<^sub>0gh\<^sub>1.p\<^sub>1] \<cdot> (f.tab \<star> f\<^sub>0gh\<^sub>1.p\<^sub>1) \<star> TTfgh_TfTgh.chine) \<cdot> TTfgh_TfTgh.the_\<nu>) \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>1 f, f\<^sub>0g\<^sub>1.p\<^sub>1, fg\<^sub>0h\<^sub>1.p\<^sub>1] = \<a>[f, g, h \<star> tab\<^sub>0 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0] \<cdot> \<a>[f \<star> g, h, tab\<^sub>0 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0] \<cdot> ((\<a>\<^sup>-\<^sup>1[f, g, h] \<star> tab\<^sub>0 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0) \<cdot> ((f \<star> g \<star> h) \<star> TTfgh_TfTgh.the_\<theta>) \<cdot> \<a>[f \<star> g \<star> h, (tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine] \<cdot> (\<a>\<^sup>-\<^sup>1[f, g \<star> h, (tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0] \<cdot> (f \<star> \<a>[g \<star> h, tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0, f\<^sub>0gh\<^sub>1.p\<^sub>0]) \<cdot> ((f \<star> \<a>\<^sup>-\<^sup>1[g, h, tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<cdot> (f \<star> (g \<star> \<a>[h, tab\<^sub>0 h, g\<^sub>0h\<^sub>1.p\<^sub>0]) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<cdot> (f \<star> (g \<star> h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<cdot> (f \<star> (g \<star> g\<^sub>0h\<^sub>1.\<phi>) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<cdot> (f \<star> \<a>[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<cdot> (f \<star> (g.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>1) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<cdot> (f \<star> f\<^sub>0gh\<^sub>1.\<phi>)) \<cdot> \<a>[f, tab\<^sub>0 f, f\<^sub>0gh\<^sub>1.p\<^sub>1] \<cdot> (f.tab \<star> f\<^sub>0gh\<^sub>1.p\<^sub>1) \<star> TTfgh_TfTgh.chine) \<cdot> TTfgh_TfTgh.the_\<nu>) \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>1 f, f\<^sub>0g\<^sub>1.p\<^sub>1, fg\<^sub>0h\<^sub>1.p\<^sub>1]
[PROOF STEP]
thus ?thesis
[PROOF STATE]
proof (prove)
using this:
f \<star> \<a>\<^sup>-\<^sup>1[g, h, tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0] \<cdot> (g \<star> \<a>[h, tab\<^sub>0 h, g\<^sub>0h\<^sub>1.p\<^sub>0]) \<cdot> (g \<star> h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<cdot> (g \<star> g\<^sub>0h\<^sub>1.\<phi>) \<cdot> \<a>[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1] \<cdot> (g.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>1) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 = (f \<star> \<a>\<^sup>-\<^sup>1[g, h, tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<cdot> (f \<star> (g \<star> \<a>[h, tab\<^sub>0 h, g\<^sub>0h\<^sub>1.p\<^sub>0]) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<cdot> (f \<star> (g \<star> h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<cdot> (f \<star> (g \<star> g\<^sub>0h\<^sub>1.\<phi>) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<cdot> (f \<star> \<a>[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<cdot> (f \<star> (g.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>1) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0)
goal (1 subgoal):
1. \<a>[f, g, h \<star> tab\<^sub>0 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0] \<cdot> \<a>[f \<star> g, h, tab\<^sub>0 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0] \<cdot> ((\<a>\<^sup>-\<^sup>1[f, g, h] \<star> tab\<^sub>0 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0) \<cdot> ((f \<star> g \<star> h) \<star> TTfgh_TfTgh.the_\<theta>) \<cdot> \<a>[f \<star> g \<star> h, (tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine] \<cdot> (\<a>\<^sup>-\<^sup>1[f, g \<star> h, (tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0] \<cdot> (f \<star> \<a>[g \<star> h, tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0, f\<^sub>0gh\<^sub>1.p\<^sub>0]) \<cdot> (f \<star> \<a>\<^sup>-\<^sup>1[g, h, tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0] \<cdot> (g \<star> \<a>[h, tab\<^sub>0 h, g\<^sub>0h\<^sub>1.p\<^sub>0]) \<cdot> (g \<star> h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<cdot> (g \<star> g\<^sub>0h\<^sub>1.\<phi>) \<cdot> \<a>[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1] \<cdot> (g.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>1) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<cdot> (f \<star> f\<^sub>0gh\<^sub>1.\<phi>) \<cdot> \<a>[f, tab\<^sub>0 f, f\<^sub>0gh\<^sub>1.p\<^sub>1] \<cdot> (f.tab \<star> f\<^sub>0gh\<^sub>1.p\<^sub>1) \<star> TTfgh_TfTgh.chine) \<cdot> TTfgh_TfTgh.the_\<nu>) \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>1 f, f\<^sub>0g\<^sub>1.p\<^sub>1, fg\<^sub>0h\<^sub>1.p\<^sub>1] = \<a>[f, g, h \<star> tab\<^sub>0 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0] \<cdot> \<a>[f \<star> g, h, tab\<^sub>0 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0] \<cdot> ((\<a>\<^sup>-\<^sup>1[f, g, h] \<star> tab\<^sub>0 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0) \<cdot> ((f \<star> g \<star> h) \<star> TTfgh_TfTgh.the_\<theta>) \<cdot> \<a>[f \<star> g \<star> h, (tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine] \<cdot> (\<a>\<^sup>-\<^sup>1[f, g \<star> h, (tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0] \<cdot> (f \<star> \<a>[g \<star> h, tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0, f\<^sub>0gh\<^sub>1.p\<^sub>0]) \<cdot> ((f \<star> \<a>\<^sup>-\<^sup>1[g, h, tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<cdot> (f \<star> (g \<star> \<a>[h, tab\<^sub>0 h, g\<^sub>0h\<^sub>1.p\<^sub>0]) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<cdot> (f \<star> (g \<star> h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<cdot> (f \<star> (g \<star> g\<^sub>0h\<^sub>1.\<phi>) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<cdot> (f \<star> \<a>[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<cdot> (f \<star> (g.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>1) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<cdot> (f \<star> f\<^sub>0gh\<^sub>1.\<phi>)) \<cdot> \<a>[f, tab\<^sub>0 f, f\<^sub>0gh\<^sub>1.p\<^sub>1] \<cdot> (f.tab \<star> f\<^sub>0gh\<^sub>1.p\<^sub>1) \<star> TTfgh_TfTgh.chine) \<cdot> TTfgh_TfTgh.the_\<nu>) \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>1 f, f\<^sub>0g\<^sub>1.p\<^sub>1, fg\<^sub>0h\<^sub>1.p\<^sub>1]
[PROOF STEP]
using comp_assoc
[PROOF STATE]
proof (prove)
using this:
f \<star> \<a>\<^sup>-\<^sup>1[g, h, tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0] \<cdot> (g \<star> \<a>[h, tab\<^sub>0 h, g\<^sub>0h\<^sub>1.p\<^sub>0]) \<cdot> (g \<star> h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<cdot> (g \<star> g\<^sub>0h\<^sub>1.\<phi>) \<cdot> \<a>[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1] \<cdot> (g.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>1) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 = (f \<star> \<a>\<^sup>-\<^sup>1[g, h, tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<cdot> (f \<star> (g \<star> \<a>[h, tab\<^sub>0 h, g\<^sub>0h\<^sub>1.p\<^sub>0]) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<cdot> (f \<star> (g \<star> h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<cdot> (f \<star> (g \<star> g\<^sub>0h\<^sub>1.\<phi>) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<cdot> (f \<star> \<a>[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<cdot> (f \<star> (g.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>1) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0)
(?h \<cdot> ?g) \<cdot> ?f = ?h \<cdot> ?g \<cdot> ?f
goal (1 subgoal):
1. \<a>[f, g, h \<star> tab\<^sub>0 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0] \<cdot> \<a>[f \<star> g, h, tab\<^sub>0 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0] \<cdot> ((\<a>\<^sup>-\<^sup>1[f, g, h] \<star> tab\<^sub>0 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0) \<cdot> ((f \<star> g \<star> h) \<star> TTfgh_TfTgh.the_\<theta>) \<cdot> \<a>[f \<star> g \<star> h, (tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine] \<cdot> (\<a>\<^sup>-\<^sup>1[f, g \<star> h, (tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0] \<cdot> (f \<star> \<a>[g \<star> h, tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0, f\<^sub>0gh\<^sub>1.p\<^sub>0]) \<cdot> (f \<star> \<a>\<^sup>-\<^sup>1[g, h, tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0] \<cdot> (g \<star> \<a>[h, tab\<^sub>0 h, g\<^sub>0h\<^sub>1.p\<^sub>0]) \<cdot> (g \<star> h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<cdot> (g \<star> g\<^sub>0h\<^sub>1.\<phi>) \<cdot> \<a>[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1] \<cdot> (g.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>1) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<cdot> (f \<star> f\<^sub>0gh\<^sub>1.\<phi>) \<cdot> \<a>[f, tab\<^sub>0 f, f\<^sub>0gh\<^sub>1.p\<^sub>1] \<cdot> (f.tab \<star> f\<^sub>0gh\<^sub>1.p\<^sub>1) \<star> TTfgh_TfTgh.chine) \<cdot> TTfgh_TfTgh.the_\<nu>) \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>1 f, f\<^sub>0g\<^sub>1.p\<^sub>1, fg\<^sub>0h\<^sub>1.p\<^sub>1] = \<a>[f, g, h \<star> tab\<^sub>0 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0] \<cdot> \<a>[f \<star> g, h, tab\<^sub>0 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0] \<cdot> ((\<a>\<^sup>-\<^sup>1[f, g, h] \<star> tab\<^sub>0 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0) \<cdot> ((f \<star> g \<star> h) \<star> TTfgh_TfTgh.the_\<theta>) \<cdot> \<a>[f \<star> g \<star> h, (tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine] \<cdot> (\<a>\<^sup>-\<^sup>1[f, g \<star> h, (tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0] \<cdot> (f \<star> \<a>[g \<star> h, tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0, f\<^sub>0gh\<^sub>1.p\<^sub>0]) \<cdot> ((f \<star> \<a>\<^sup>-\<^sup>1[g, h, tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<cdot> (f \<star> (g \<star> \<a>[h, tab\<^sub>0 h, g\<^sub>0h\<^sub>1.p\<^sub>0]) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<cdot> (f \<star> (g \<star> h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<cdot> (f \<star> (g \<star> g\<^sub>0h\<^sub>1.\<phi>) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<cdot> (f \<star> \<a>[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<cdot> (f \<star> (g.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>1) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<cdot> (f \<star> f\<^sub>0gh\<^sub>1.\<phi>)) \<cdot> \<a>[f, tab\<^sub>0 f, f\<^sub>0gh\<^sub>1.p\<^sub>1] \<cdot> (f.tab \<star> f\<^sub>0gh\<^sub>1.p\<^sub>1) \<star> TTfgh_TfTgh.chine) \<cdot> TTfgh_TfTgh.the_\<nu>) \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>1 f, f\<^sub>0g\<^sub>1.p\<^sub>1, fg\<^sub>0h\<^sub>1.p\<^sub>1]
[PROOF STEP]
by presburger
[PROOF STATE]
proof (state)
this:
\<a>[f, g, h \<star> tab\<^sub>0 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0] \<cdot> \<a>[f \<star> g, h, tab\<^sub>0 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0] \<cdot> ((\<a>\<^sup>-\<^sup>1[f, g, h] \<star> tab\<^sub>0 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0) \<cdot> ((f \<star> g \<star> h) \<star> TTfgh_TfTgh.the_\<theta>) \<cdot> \<a>[f \<star> g \<star> h, (tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine] \<cdot> (\<a>\<^sup>-\<^sup>1[f, g \<star> h, (tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0] \<cdot> (f \<star> \<a>[g \<star> h, tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0, f\<^sub>0gh\<^sub>1.p\<^sub>0]) \<cdot> (f \<star> \<a>\<^sup>-\<^sup>1[g, h, tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0] \<cdot> (g \<star> \<a>[h, tab\<^sub>0 h, g\<^sub>0h\<^sub>1.p\<^sub>0]) \<cdot> (g \<star> h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<cdot> (g \<star> g\<^sub>0h\<^sub>1.\<phi>) \<cdot> \<a>[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1] \<cdot> (g.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>1) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<cdot> (f \<star> f\<^sub>0gh\<^sub>1.\<phi>) \<cdot> \<a>[f, tab\<^sub>0 f, f\<^sub>0gh\<^sub>1.p\<^sub>1] \<cdot> (f.tab \<star> f\<^sub>0gh\<^sub>1.p\<^sub>1) \<star> TTfgh_TfTgh.chine) \<cdot> TTfgh_TfTgh.the_\<nu>) \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>1 f, f\<^sub>0g\<^sub>1.p\<^sub>1, fg\<^sub>0h\<^sub>1.p\<^sub>1] = \<a>[f, g, h \<star> tab\<^sub>0 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0] \<cdot> \<a>[f \<star> g, h, tab\<^sub>0 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0] \<cdot> ((\<a>\<^sup>-\<^sup>1[f, g, h] \<star> tab\<^sub>0 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0) \<cdot> ((f \<star> g \<star> h) \<star> TTfgh_TfTgh.the_\<theta>) \<cdot> \<a>[f \<star> g \<star> h, (tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine] \<cdot> (\<a>\<^sup>-\<^sup>1[f, g \<star> h, (tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0] \<cdot> (f \<star> \<a>[g \<star> h, tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0, f\<^sub>0gh\<^sub>1.p\<^sub>0]) \<cdot> ((f \<star> \<a>\<^sup>-\<^sup>1[g, h, tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<cdot> (f \<star> (g \<star> \<a>[h, tab\<^sub>0 h, g\<^sub>0h\<^sub>1.p\<^sub>0]) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<cdot> (f \<star> (g \<star> h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<cdot> (f \<star> (g \<star> g\<^sub>0h\<^sub>1.\<phi>) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<cdot> (f \<star> \<a>[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<cdot> (f \<star> (g.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>1) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<cdot> (f \<star> f\<^sub>0gh\<^sub>1.\<phi>)) \<cdot> \<a>[f, tab\<^sub>0 f, f\<^sub>0gh\<^sub>1.p\<^sub>1] \<cdot> (f.tab \<star> f\<^sub>0gh\<^sub>1.p\<^sub>1) \<star> TTfgh_TfTgh.chine) \<cdot> TTfgh_TfTgh.the_\<nu>) \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>1 f, f\<^sub>0g\<^sub>1.p\<^sub>1, fg\<^sub>0h\<^sub>1.p\<^sub>1]
goal:
No subgoals!
[PROOF STEP]
qed
[PROOF STATE]
proof (state)
this:
\<a>[f, g, h \<star> tab\<^sub>0 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0] \<cdot> \<a>[f \<star> g, h, tab\<^sub>0 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0] \<cdot> ((\<a>\<^sup>-\<^sup>1[f, g, h] \<star> tab\<^sub>0 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0) \<cdot> ((f \<star> g \<star> h) \<star> TTfgh_TfTgh.the_\<theta>) \<cdot> \<a>[f \<star> g \<star> h, (tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine] \<cdot> (\<a>\<^sup>-\<^sup>1[f, g \<star> h, (tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0] \<cdot> (f \<star> \<a>[g \<star> h, tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0, f\<^sub>0gh\<^sub>1.p\<^sub>0]) \<cdot> (f \<star> \<a>\<^sup>-\<^sup>1[g, h, tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0] \<cdot> (g \<star> \<a>[h, tab\<^sub>0 h, g\<^sub>0h\<^sub>1.p\<^sub>0]) \<cdot> (g \<star> h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<cdot> (g \<star> g\<^sub>0h\<^sub>1.\<phi>) \<cdot> \<a>[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1] \<cdot> (g.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>1) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<cdot> (f \<star> f\<^sub>0gh\<^sub>1.\<phi>) \<cdot> \<a>[f, tab\<^sub>0 f, f\<^sub>0gh\<^sub>1.p\<^sub>1] \<cdot> (f.tab \<star> f\<^sub>0gh\<^sub>1.p\<^sub>1) \<star> TTfgh_TfTgh.chine) \<cdot> TTfgh_TfTgh.the_\<nu>) \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>1 f, f\<^sub>0g\<^sub>1.p\<^sub>1, fg\<^sub>0h\<^sub>1.p\<^sub>1] = \<a>[f, g, h \<star> tab\<^sub>0 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0] \<cdot> \<a>[f \<star> g, h, tab\<^sub>0 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0] \<cdot> ((\<a>\<^sup>-\<^sup>1[f, g, h] \<star> tab\<^sub>0 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0) \<cdot> ((f \<star> g \<star> h) \<star> TTfgh_TfTgh.the_\<theta>) \<cdot> \<a>[f \<star> g \<star> h, (tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine] \<cdot> (\<a>\<^sup>-\<^sup>1[f, g \<star> h, (tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0] \<cdot> (f \<star> \<a>[g \<star> h, tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0, f\<^sub>0gh\<^sub>1.p\<^sub>0]) \<cdot> ((f \<star> \<a>\<^sup>-\<^sup>1[g, h, tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<cdot> (f \<star> (g \<star> \<a>[h, tab\<^sub>0 h, g\<^sub>0h\<^sub>1.p\<^sub>0]) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<cdot> (f \<star> (g \<star> h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<cdot> (f \<star> (g \<star> g\<^sub>0h\<^sub>1.\<phi>) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<cdot> (f \<star> \<a>[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<cdot> (f \<star> (g.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>1) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<cdot> (f \<star> f\<^sub>0gh\<^sub>1.\<phi>)) \<cdot> \<a>[f, tab\<^sub>0 f, f\<^sub>0gh\<^sub>1.p\<^sub>1] \<cdot> (f.tab \<star> f\<^sub>0gh\<^sub>1.p\<^sub>1) \<star> TTfgh_TfTgh.chine) \<cdot> TTfgh_TfTgh.the_\<nu>) \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>1 f, f\<^sub>0g\<^sub>1.p\<^sub>1, fg\<^sub>0h\<^sub>1.p\<^sub>1]
goal (1 subgoal):
1. (f \<star> \<theta>\<^sub>f) \<cdot> \<a>[f, tab\<^sub>0 f, w\<^sub>f] \<cdot> (f.tab \<star> w\<^sub>f) = ((f \<star> \<theta>\<^sub>f') \<cdot> \<a>[f, tab\<^sub>0 f, w\<^sub>f'] \<cdot> (f.tab \<star> w\<^sub>f')) \<cdot> \<beta>\<^sub>f
[PROOF STEP]
also
[PROOF STATE]
proof (state)
this:
\<a>[f, g, h \<star> tab\<^sub>0 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0] \<cdot> \<a>[f \<star> g, h, tab\<^sub>0 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0] \<cdot> ((\<a>\<^sup>-\<^sup>1[f, g, h] \<star> tab\<^sub>0 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0) \<cdot> ((f \<star> g \<star> h) \<star> TTfgh_TfTgh.the_\<theta>) \<cdot> \<a>[f \<star> g \<star> h, (tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine] \<cdot> (\<a>\<^sup>-\<^sup>1[f, g \<star> h, (tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0] \<cdot> (f \<star> \<a>[g \<star> h, tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0, f\<^sub>0gh\<^sub>1.p\<^sub>0]) \<cdot> (f \<star> \<a>\<^sup>-\<^sup>1[g, h, tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0] \<cdot> (g \<star> \<a>[h, tab\<^sub>0 h, g\<^sub>0h\<^sub>1.p\<^sub>0]) \<cdot> (g \<star> h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<cdot> (g \<star> g\<^sub>0h\<^sub>1.\<phi>) \<cdot> \<a>[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1] \<cdot> (g.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>1) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<cdot> (f \<star> f\<^sub>0gh\<^sub>1.\<phi>) \<cdot> \<a>[f, tab\<^sub>0 f, f\<^sub>0gh\<^sub>1.p\<^sub>1] \<cdot> (f.tab \<star> f\<^sub>0gh\<^sub>1.p\<^sub>1) \<star> TTfgh_TfTgh.chine) \<cdot> TTfgh_TfTgh.the_\<nu>) \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>1 f, f\<^sub>0g\<^sub>1.p\<^sub>1, fg\<^sub>0h\<^sub>1.p\<^sub>1] = \<a>[f, g, h \<star> tab\<^sub>0 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0] \<cdot> \<a>[f \<star> g, h, tab\<^sub>0 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0] \<cdot> ((\<a>\<^sup>-\<^sup>1[f, g, h] \<star> tab\<^sub>0 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0) \<cdot> ((f \<star> g \<star> h) \<star> TTfgh_TfTgh.the_\<theta>) \<cdot> \<a>[f \<star> g \<star> h, (tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine] \<cdot> (\<a>\<^sup>-\<^sup>1[f, g \<star> h, (tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0] \<cdot> (f \<star> \<a>[g \<star> h, tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0, f\<^sub>0gh\<^sub>1.p\<^sub>0]) \<cdot> ((f \<star> \<a>\<^sup>-\<^sup>1[g, h, tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<cdot> (f \<star> (g \<star> \<a>[h, tab\<^sub>0 h, g\<^sub>0h\<^sub>1.p\<^sub>0]) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<cdot> (f \<star> (g \<star> h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<cdot> (f \<star> (g \<star> g\<^sub>0h\<^sub>1.\<phi>) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<cdot> (f \<star> \<a>[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<cdot> (f \<star> (g.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>1) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<cdot> (f \<star> f\<^sub>0gh\<^sub>1.\<phi>)) \<cdot> \<a>[f, tab\<^sub>0 f, f\<^sub>0gh\<^sub>1.p\<^sub>1] \<cdot> (f.tab \<star> f\<^sub>0gh\<^sub>1.p\<^sub>1) \<star> TTfgh_TfTgh.chine) \<cdot> TTfgh_TfTgh.the_\<nu>) \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>1 f, f\<^sub>0g\<^sub>1.p\<^sub>1, fg\<^sub>0h\<^sub>1.p\<^sub>1]
goal (1 subgoal):
1. (f \<star> \<theta>\<^sub>f) \<cdot> \<a>[f, tab\<^sub>0 f, w\<^sub>f] \<cdot> (f.tab \<star> w\<^sub>f) = ((f \<star> \<theta>\<^sub>f') \<cdot> \<a>[f, tab\<^sub>0 f, w\<^sub>f'] \<cdot> (f.tab \<star> w\<^sub>f')) \<cdot> \<beta>\<^sub>f
[PROOF STEP]
have "... =
\<a>[f, g, h \<star> tab\<^sub>0 h \<star> TTfgh.p\<^sub>0] \<cdot>
\<a>[f \<star> g, h, tab\<^sub>0 h \<star> TTfgh.p\<^sub>0] \<cdot>
(\<a>\<^sup>-\<^sup>1[f, g, h] \<star> tab\<^sub>0 h \<star> TTfgh.p\<^sub>0) \<cdot>
((f \<star> g \<star> h) \<star> TTfgh_TfTgh.the_\<theta>) \<cdot>
\<a>[f \<star> g \<star> h, (tab\<^sub>0 h \<star> Tgh.p\<^sub>0) \<star> TfTgh.p\<^sub>0, TTfgh_TfTgh.chine] \<cdot>
(\<a>\<^sup>-\<^sup>1[f, g \<star> h, (tab\<^sub>0 h \<star> Tgh.p\<^sub>0) \<star> TfTgh.p\<^sub>0] \<star> TTfgh_TfTgh.chine) \<cdot>
((f \<star> \<a>[g \<star> h, tab\<^sub>0 h \<star> Tgh.p\<^sub>0, TfTgh.p\<^sub>0]) \<star> TTfgh_TfTgh.chine) \<cdot>
((f \<star> \<a>\<^sup>-\<^sup>1[g, h, tab\<^sub>0 h \<star> Tgh.p\<^sub>0] \<star> TfTgh.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot>
((f \<star> (g \<star> \<a>[h, tab\<^sub>0 h, Tgh.p\<^sub>0]) \<star> TfTgh.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot>
((f \<star> (g \<star> h.tab \<star> Tgh.p\<^sub>0) \<star> TfTgh.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot>
((f \<star> (g \<star> g\<^sub>0h\<^sub>1.\<phi>) \<star> TfTgh.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot>
((f \<star> \<a>[g, tab\<^sub>0 g, Tgh.p\<^sub>1] \<star> TfTgh.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot>
((f \<star> (g.tab \<star> Tgh.p\<^sub>1) \<star> TfTgh.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot>
((f \<star> f\<^sub>0gh\<^sub>1.\<phi>) \<star> TTfgh_TfTgh.chine) \<cdot>
(\<a>[f, tab\<^sub>0 f, TfTgh.p\<^sub>1] \<star> TTfgh_TfTgh.chine) \<cdot>
((f.tab \<star> TfTgh.p\<^sub>1) \<star> TTfgh_TfTgh.chine) \<cdot>
TTfgh_TfTgh.the_\<nu> \<cdot>
\<a>\<^sup>-\<^sup>1[tab\<^sub>1 f, Tfg.p\<^sub>1, TTfgh.p\<^sub>1]"
[PROOF STATE]
proof (prove)
goal (1 subgoal):
1. \<a>[f, g, h \<star> tab\<^sub>0 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0] \<cdot> \<a>[f \<star> g, h, tab\<^sub>0 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0] \<cdot> ((\<a>\<^sup>-\<^sup>1[f, g, h] \<star> tab\<^sub>0 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0) \<cdot> ((f \<star> g \<star> h) \<star> TTfgh_TfTgh.the_\<theta>) \<cdot> \<a>[f \<star> g \<star> h, (tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine] \<cdot> (\<a>\<^sup>-\<^sup>1[f, g \<star> h, (tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0] \<cdot> (f \<star> \<a>[g \<star> h, tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0, f\<^sub>0gh\<^sub>1.p\<^sub>0]) \<cdot> ((f \<star> \<a>\<^sup>-\<^sup>1[g, h, tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<cdot> (f \<star> (g \<star> \<a>[h, tab\<^sub>0 h, g\<^sub>0h\<^sub>1.p\<^sub>0]) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<cdot> (f \<star> (g \<star> h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<cdot> (f \<star> (g \<star> g\<^sub>0h\<^sub>1.\<phi>) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<cdot> (f \<star> \<a>[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<cdot> (f \<star> (g.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>1) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<cdot> (f \<star> f\<^sub>0gh\<^sub>1.\<phi>)) \<cdot> \<a>[f, tab\<^sub>0 f, f\<^sub>0gh\<^sub>1.p\<^sub>1] \<cdot> (f.tab \<star> f\<^sub>0gh\<^sub>1.p\<^sub>1) \<star> TTfgh_TfTgh.chine) \<cdot> TTfgh_TfTgh.the_\<nu>) \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>1 f, f\<^sub>0g\<^sub>1.p\<^sub>1, fg\<^sub>0h\<^sub>1.p\<^sub>1] = \<a>[f, g, h \<star> tab\<^sub>0 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0] \<cdot> \<a>[f \<star> g, h, tab\<^sub>0 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0] \<cdot> (\<a>\<^sup>-\<^sup>1[f, g, h] \<star> tab\<^sub>0 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0) \<cdot> ((f \<star> g \<star> h) \<star> TTfgh_TfTgh.the_\<theta>) \<cdot> \<a>[f \<star> g \<star> h, (tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine] \<cdot> (\<a>\<^sup>-\<^sup>1[f, g \<star> h, (tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0] \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> \<a>[g \<star> h, tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0, f\<^sub>0gh\<^sub>1.p\<^sub>0]) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> \<a>\<^sup>-\<^sup>1[g, h, tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> (g \<star> \<a>[h, tab\<^sub>0 h, g\<^sub>0h\<^sub>1.p\<^sub>0]) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> (g \<star> h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> (g \<star> g\<^sub>0h\<^sub>1.\<phi>) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> \<a>[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> (g.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>1) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> f\<^sub>0gh\<^sub>1.\<phi>) \<star> TTfgh_TfTgh.chine) \<cdot> (\<a>[f, tab\<^sub>0 f, f\<^sub>0gh\<^sub>1.p\<^sub>1] \<star> TTfgh_TfTgh.chine) \<cdot> ((f.tab \<star> f\<^sub>0gh\<^sub>1.p\<^sub>1) \<star> TTfgh_TfTgh.chine) \<cdot> TTfgh_TfTgh.the_\<nu> \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>1 f, f\<^sub>0g\<^sub>1.p\<^sub>1, fg\<^sub>0h\<^sub>1.p\<^sub>1]
[PROOF STEP]
proof -
[PROOF STATE]
proof (state)
goal (1 subgoal):
1. \<a>[f, g, h \<star> tab\<^sub>0 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0] \<cdot> \<a>[f \<star> g, h, tab\<^sub>0 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0] \<cdot> ((\<a>\<^sup>-\<^sup>1[f, g, h] \<star> tab\<^sub>0 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0) \<cdot> ((f \<star> g \<star> h) \<star> TTfgh_TfTgh.the_\<theta>) \<cdot> \<a>[f \<star> g \<star> h, (tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine] \<cdot> (\<a>\<^sup>-\<^sup>1[f, g \<star> h, (tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0] \<cdot> (f \<star> \<a>[g \<star> h, tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0, f\<^sub>0gh\<^sub>1.p\<^sub>0]) \<cdot> ((f \<star> \<a>\<^sup>-\<^sup>1[g, h, tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<cdot> (f \<star> (g \<star> \<a>[h, tab\<^sub>0 h, g\<^sub>0h\<^sub>1.p\<^sub>0]) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<cdot> (f \<star> (g \<star> h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<cdot> (f \<star> (g \<star> g\<^sub>0h\<^sub>1.\<phi>) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<cdot> (f \<star> \<a>[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<cdot> (f \<star> (g.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>1) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<cdot> (f \<star> f\<^sub>0gh\<^sub>1.\<phi>)) \<cdot> \<a>[f, tab\<^sub>0 f, f\<^sub>0gh\<^sub>1.p\<^sub>1] \<cdot> (f.tab \<star> f\<^sub>0gh\<^sub>1.p\<^sub>1) \<star> TTfgh_TfTgh.chine) \<cdot> TTfgh_TfTgh.the_\<nu>) \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>1 f, f\<^sub>0g\<^sub>1.p\<^sub>1, fg\<^sub>0h\<^sub>1.p\<^sub>1] = \<a>[f, g, h \<star> tab\<^sub>0 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0] \<cdot> \<a>[f \<star> g, h, tab\<^sub>0 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0] \<cdot> (\<a>\<^sup>-\<^sup>1[f, g, h] \<star> tab\<^sub>0 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0) \<cdot> ((f \<star> g \<star> h) \<star> TTfgh_TfTgh.the_\<theta>) \<cdot> \<a>[f \<star> g \<star> h, (tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine] \<cdot> (\<a>\<^sup>-\<^sup>1[f, g \<star> h, (tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0] \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> \<a>[g \<star> h, tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0, f\<^sub>0gh\<^sub>1.p\<^sub>0]) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> \<a>\<^sup>-\<^sup>1[g, h, tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> (g \<star> \<a>[h, tab\<^sub>0 h, g\<^sub>0h\<^sub>1.p\<^sub>0]) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> (g \<star> h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> (g \<star> g\<^sub>0h\<^sub>1.\<phi>) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> \<a>[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> (g.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>1) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> f\<^sub>0gh\<^sub>1.\<phi>) \<star> TTfgh_TfTgh.chine) \<cdot> (\<a>[f, tab\<^sub>0 f, f\<^sub>0gh\<^sub>1.p\<^sub>1] \<star> TTfgh_TfTgh.chine) \<cdot> ((f.tab \<star> f\<^sub>0gh\<^sub>1.p\<^sub>1) \<star> TTfgh_TfTgh.chine) \<cdot> TTfgh_TfTgh.the_\<nu> \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>1 f, f\<^sub>0g\<^sub>1.p\<^sub>1, fg\<^sub>0h\<^sub>1.p\<^sub>1]
[PROOF STEP]
have "\<a>\<^sup>-\<^sup>1[f, g \<star> h, (tab\<^sub>0 h \<star> Tgh.p\<^sub>0) \<star> TfTgh.p\<^sub>0] \<cdot>
(f \<star> \<a>[g \<star> h, tab\<^sub>0 h \<star> Tgh.p\<^sub>0, TfTgh.p\<^sub>0]) \<cdot>
(f \<star> \<a>\<^sup>-\<^sup>1[g, h, tab\<^sub>0 h \<star> Tgh.p\<^sub>0] \<star> TfTgh.p\<^sub>0) \<cdot>
(f \<star> (g \<star> \<a>[h, tab\<^sub>0 h, Tgh.p\<^sub>0]) \<star> TfTgh.p\<^sub>0) \<cdot>
(f \<star> (g \<star> h.tab \<star> Tgh.p\<^sub>0) \<star> TfTgh.p\<^sub>0) \<cdot>
(f \<star> (g \<star> g\<^sub>0h\<^sub>1.\<phi>) \<star> TfTgh.p\<^sub>0) \<cdot>
(f \<star> \<a>[g, tab\<^sub>0 g, Tgh.p\<^sub>1] \<star> TfTgh.p\<^sub>0) \<cdot>
(f \<star> (g.tab \<star> Tgh.p\<^sub>1) \<star> TfTgh.p\<^sub>0) \<cdot>
(f \<star> f\<^sub>0gh\<^sub>1.\<phi>) \<cdot>
\<a>[f, tab\<^sub>0 f, TfTgh.p\<^sub>1] \<cdot>
(f.tab \<star> TfTgh.p\<^sub>1)
\<star> TTfgh_TfTgh.chine =
(\<a>\<^sup>-\<^sup>1[f, g \<star> h, (tab\<^sub>0 h \<star> Tgh.p\<^sub>0) \<star> TfTgh.p\<^sub>0] \<star> TTfgh_TfTgh.chine) \<cdot>
((f \<star> \<a>[g \<star> h, tab\<^sub>0 h \<star> Tgh.p\<^sub>0, TfTgh.p\<^sub>0]) \<star> TTfgh_TfTgh.chine) \<cdot>
((f \<star> \<a>\<^sup>-\<^sup>1[g, h, tab\<^sub>0 h \<star> Tgh.p\<^sub>0] \<star> TfTgh.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot>
((f \<star> (g \<star> \<a>[h, tab\<^sub>0 h, Tgh.p\<^sub>0]) \<star> TfTgh.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot>
((f \<star> (g \<star> h.tab \<star> Tgh.p\<^sub>0) \<star> TfTgh.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot>
((f \<star> (g \<star> g\<^sub>0h\<^sub>1.\<phi>) \<star> TfTgh.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot>
((f \<star> \<a>[g, tab\<^sub>0 g, Tgh.p\<^sub>1] \<star> TfTgh.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot>
((f \<star> (g.tab \<star> Tgh.p\<^sub>1) \<star> TfTgh.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot>
((f \<star> f\<^sub>0gh\<^sub>1.\<phi>) \<star> TTfgh_TfTgh.chine) \<cdot>
(\<a>[f, tab\<^sub>0 f, TfTgh.p\<^sub>1] \<star> TTfgh_TfTgh.chine) \<cdot>
((f.tab \<star> TfTgh.p\<^sub>1) \<star> TTfgh_TfTgh.chine)"
[PROOF STATE]
proof (prove)
goal (1 subgoal):
1. \<a>\<^sup>-\<^sup>1[f, g \<star> h, (tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0] \<cdot> (f \<star> \<a>[g \<star> h, tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0, f\<^sub>0gh\<^sub>1.p\<^sub>0]) \<cdot> (f \<star> \<a>\<^sup>-\<^sup>1[g, h, tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<cdot> (f \<star> (g \<star> \<a>[h, tab\<^sub>0 h, g\<^sub>0h\<^sub>1.p\<^sub>0]) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<cdot> (f \<star> (g \<star> h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<cdot> (f \<star> (g \<star> g\<^sub>0h\<^sub>1.\<phi>) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<cdot> (f \<star> \<a>[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<cdot> (f \<star> (g.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>1) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<cdot> (f \<star> f\<^sub>0gh\<^sub>1.\<phi>) \<cdot> \<a>[f, tab\<^sub>0 f, f\<^sub>0gh\<^sub>1.p\<^sub>1] \<cdot> (f.tab \<star> f\<^sub>0gh\<^sub>1.p\<^sub>1) \<star> TTfgh_TfTgh.chine = (\<a>\<^sup>-\<^sup>1[f, g \<star> h, (tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0] \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> \<a>[g \<star> h, tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0, f\<^sub>0gh\<^sub>1.p\<^sub>0]) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> \<a>\<^sup>-\<^sup>1[g, h, tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> (g \<star> \<a>[h, tab\<^sub>0 h, g\<^sub>0h\<^sub>1.p\<^sub>0]) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> (g \<star> h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> (g \<star> g\<^sub>0h\<^sub>1.\<phi>) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> \<a>[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> (g.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>1) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> f\<^sub>0gh\<^sub>1.\<phi>) \<star> TTfgh_TfTgh.chine) \<cdot> (\<a>[f, tab\<^sub>0 f, f\<^sub>0gh\<^sub>1.p\<^sub>1] \<star> TTfgh_TfTgh.chine) \<cdot> ((f.tab \<star> f\<^sub>0gh\<^sub>1.p\<^sub>1) \<star> TTfgh_TfTgh.chine)
[PROOF STEP]
proof -
[PROOF STATE]
proof (state)
goal (1 subgoal):
1. \<a>\<^sup>-\<^sup>1[f, g \<star> h, (tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0] \<cdot> (f \<star> \<a>[g \<star> h, tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0, f\<^sub>0gh\<^sub>1.p\<^sub>0]) \<cdot> (f \<star> \<a>\<^sup>-\<^sup>1[g, h, tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<cdot> (f \<star> (g \<star> \<a>[h, tab\<^sub>0 h, g\<^sub>0h\<^sub>1.p\<^sub>0]) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<cdot> (f \<star> (g \<star> h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<cdot> (f \<star> (g \<star> g\<^sub>0h\<^sub>1.\<phi>) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<cdot> (f \<star> \<a>[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<cdot> (f \<star> (g.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>1) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<cdot> (f \<star> f\<^sub>0gh\<^sub>1.\<phi>) \<cdot> \<a>[f, tab\<^sub>0 f, f\<^sub>0gh\<^sub>1.p\<^sub>1] \<cdot> (f.tab \<star> f\<^sub>0gh\<^sub>1.p\<^sub>1) \<star> TTfgh_TfTgh.chine = (\<a>\<^sup>-\<^sup>1[f, g \<star> h, (tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0] \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> \<a>[g \<star> h, tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0, f\<^sub>0gh\<^sub>1.p\<^sub>0]) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> \<a>\<^sup>-\<^sup>1[g, h, tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> (g \<star> \<a>[h, tab\<^sub>0 h, g\<^sub>0h\<^sub>1.p\<^sub>0]) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> (g \<star> h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> (g \<star> g\<^sub>0h\<^sub>1.\<phi>) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> \<a>[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> (g.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>1) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> f\<^sub>0gh\<^sub>1.\<phi>) \<star> TTfgh_TfTgh.chine) \<cdot> (\<a>[f, tab\<^sub>0 f, f\<^sub>0gh\<^sub>1.p\<^sub>1] \<star> TTfgh_TfTgh.chine) \<cdot> ((f.tab \<star> f\<^sub>0gh\<^sub>1.p\<^sub>1) \<star> TTfgh_TfTgh.chine)
[PROOF STEP]
have "arr (\<a>\<^sup>-\<^sup>1[f, g \<star> h, (tab\<^sub>0 h \<star> Tgh.p\<^sub>0) \<star> TfTgh.p\<^sub>0] \<cdot>
(f \<star> \<a>[g \<star> h, tab\<^sub>0 h \<star> Tgh.p\<^sub>0, TfTgh.p\<^sub>0]) \<cdot>
(f \<star> \<a>\<^sup>-\<^sup>1[g, h, tab\<^sub>0 h \<star> Tgh.p\<^sub>0] \<star> TfTgh.p\<^sub>0) \<cdot>
(f \<star> (g \<star> \<a>[h, tab\<^sub>0 h, Tgh.p\<^sub>0]) \<star> TfTgh.p\<^sub>0) \<cdot>
(f \<star> (g \<star> h.tab \<star> Tgh.p\<^sub>0) \<star> TfTgh.p\<^sub>0) \<cdot>
(f \<star> (g \<star> g\<^sub>0h\<^sub>1.\<phi>) \<star> TfTgh.p\<^sub>0) \<cdot>
(f \<star> \<a>[g, tab\<^sub>0 g, Tgh.p\<^sub>1] \<star> TfTgh.p\<^sub>0) \<cdot>
(f \<star> (g.tab \<star> Tgh.p\<^sub>1) \<star> TfTgh.p\<^sub>0) \<cdot>
(f \<star> f\<^sub>0gh\<^sub>1.\<phi>) \<cdot>
\<a>[f, tab\<^sub>0 f, TfTgh.p\<^sub>1] \<cdot>
(f.tab \<star> TfTgh.p\<^sub>1))"
[PROOF STATE]
proof (prove)
goal (1 subgoal):
1. seq \<a>\<^sup>-\<^sup>1[f, g \<star> h, (tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0] ((f \<star> \<a>[g \<star> h, tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0, f\<^sub>0gh\<^sub>1.p\<^sub>0]) \<cdot> (f \<star> \<a>\<^sup>-\<^sup>1[g, h, tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<cdot> (f \<star> (g \<star> \<a>[h, tab\<^sub>0 h, g\<^sub>0h\<^sub>1.p\<^sub>0]) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<cdot> (f \<star> (g \<star> h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<cdot> (f \<star> (g \<star> g\<^sub>0h\<^sub>1.\<phi>) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<cdot> (f \<star> \<a>[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<cdot> (f \<star> (g.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>1) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<cdot> (f \<star> f\<^sub>0gh\<^sub>1.\<phi>) \<cdot> \<a>[f, tab\<^sub>0 f, f\<^sub>0gh\<^sub>1.p\<^sub>1] \<cdot> (f.tab \<star> f\<^sub>0gh\<^sub>1.p\<^sub>1))
[PROOF STEP]
using fg gh
[PROOF STATE]
proof (prove)
using this:
src f = trg g
src g = trg h
goal (1 subgoal):
1. seq \<a>\<^sup>-\<^sup>1[f, g \<star> h, (tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0] ((f \<star> \<a>[g \<star> h, tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0, f\<^sub>0gh\<^sub>1.p\<^sub>0]) \<cdot> (f \<star> \<a>\<^sup>-\<^sup>1[g, h, tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<cdot> (f \<star> (g \<star> \<a>[h, tab\<^sub>0 h, g\<^sub>0h\<^sub>1.p\<^sub>0]) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<cdot> (f \<star> (g \<star> h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<cdot> (f \<star> (g \<star> g\<^sub>0h\<^sub>1.\<phi>) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<cdot> (f \<star> \<a>[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<cdot> (f \<star> (g.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>1) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<cdot> (f \<star> f\<^sub>0gh\<^sub>1.\<phi>) \<cdot> \<a>[f, tab\<^sub>0 f, f\<^sub>0gh\<^sub>1.p\<^sub>1] \<cdot> (f.tab \<star> f\<^sub>0gh\<^sub>1.p\<^sub>1))
[PROOF STEP]
by (intro seqI' comp_in_homI) auto
[PROOF STATE]
proof (state)
this:
seq \<a>\<^sup>-\<^sup>1[f, g \<star> h, (tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0] ((f \<star> \<a>[g \<star> h, tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0, f\<^sub>0gh\<^sub>1.p\<^sub>0]) \<cdot> (f \<star> \<a>\<^sup>-\<^sup>1[g, h, tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<cdot> (f \<star> (g \<star> \<a>[h, tab\<^sub>0 h, g\<^sub>0h\<^sub>1.p\<^sub>0]) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<cdot> (f \<star> (g \<star> h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<cdot> (f \<star> (g \<star> g\<^sub>0h\<^sub>1.\<phi>) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<cdot> (f \<star> \<a>[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<cdot> (f \<star> (g.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>1) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<cdot> (f \<star> f\<^sub>0gh\<^sub>1.\<phi>) \<cdot> \<a>[f, tab\<^sub>0 f, f\<^sub>0gh\<^sub>1.p\<^sub>1] \<cdot> (f.tab \<star> f\<^sub>0gh\<^sub>1.p\<^sub>1))
goal (1 subgoal):
1. \<a>\<^sup>-\<^sup>1[f, g \<star> h, (tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0] \<cdot> (f \<star> \<a>[g \<star> h, tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0, f\<^sub>0gh\<^sub>1.p\<^sub>0]) \<cdot> (f \<star> \<a>\<^sup>-\<^sup>1[g, h, tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<cdot> (f \<star> (g \<star> \<a>[h, tab\<^sub>0 h, g\<^sub>0h\<^sub>1.p\<^sub>0]) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<cdot> (f \<star> (g \<star> h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<cdot> (f \<star> (g \<star> g\<^sub>0h\<^sub>1.\<phi>) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<cdot> (f \<star> \<a>[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<cdot> (f \<star> (g.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>1) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<cdot> (f \<star> f\<^sub>0gh\<^sub>1.\<phi>) \<cdot> \<a>[f, tab\<^sub>0 f, f\<^sub>0gh\<^sub>1.p\<^sub>1] \<cdot> (f.tab \<star> f\<^sub>0gh\<^sub>1.p\<^sub>1) \<star> TTfgh_TfTgh.chine = (\<a>\<^sup>-\<^sup>1[f, g \<star> h, (tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0] \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> \<a>[g \<star> h, tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0, f\<^sub>0gh\<^sub>1.p\<^sub>0]) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> \<a>\<^sup>-\<^sup>1[g, h, tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> (g \<star> \<a>[h, tab\<^sub>0 h, g\<^sub>0h\<^sub>1.p\<^sub>0]) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> (g \<star> h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> (g \<star> g\<^sub>0h\<^sub>1.\<phi>) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> \<a>[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> (g.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>1) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> f\<^sub>0gh\<^sub>1.\<phi>) \<star> TTfgh_TfTgh.chine) \<cdot> (\<a>[f, tab\<^sub>0 f, f\<^sub>0gh\<^sub>1.p\<^sub>1] \<star> TTfgh_TfTgh.chine) \<cdot> ((f.tab \<star> f\<^sub>0gh\<^sub>1.p\<^sub>1) \<star> TTfgh_TfTgh.chine)
[PROOF STEP]
moreover
[PROOF STATE]
proof (state)
this:
seq \<a>\<^sup>-\<^sup>1[f, g \<star> h, (tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0] ((f \<star> \<a>[g \<star> h, tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0, f\<^sub>0gh\<^sub>1.p\<^sub>0]) \<cdot> (f \<star> \<a>\<^sup>-\<^sup>1[g, h, tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<cdot> (f \<star> (g \<star> \<a>[h, tab\<^sub>0 h, g\<^sub>0h\<^sub>1.p\<^sub>0]) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<cdot> (f \<star> (g \<star> h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<cdot> (f \<star> (g \<star> g\<^sub>0h\<^sub>1.\<phi>) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<cdot> (f \<star> \<a>[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<cdot> (f \<star> (g.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>1) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<cdot> (f \<star> f\<^sub>0gh\<^sub>1.\<phi>) \<cdot> \<a>[f, tab\<^sub>0 f, f\<^sub>0gh\<^sub>1.p\<^sub>1] \<cdot> (f.tab \<star> f\<^sub>0gh\<^sub>1.p\<^sub>1))
goal (1 subgoal):
1. \<a>\<^sup>-\<^sup>1[f, g \<star> h, (tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0] \<cdot> (f \<star> \<a>[g \<star> h, tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0, f\<^sub>0gh\<^sub>1.p\<^sub>0]) \<cdot> (f \<star> \<a>\<^sup>-\<^sup>1[g, h, tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<cdot> (f \<star> (g \<star> \<a>[h, tab\<^sub>0 h, g\<^sub>0h\<^sub>1.p\<^sub>0]) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<cdot> (f \<star> (g \<star> h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<cdot> (f \<star> (g \<star> g\<^sub>0h\<^sub>1.\<phi>) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<cdot> (f \<star> \<a>[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<cdot> (f \<star> (g.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>1) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<cdot> (f \<star> f\<^sub>0gh\<^sub>1.\<phi>) \<cdot> \<a>[f, tab\<^sub>0 f, f\<^sub>0gh\<^sub>1.p\<^sub>1] \<cdot> (f.tab \<star> f\<^sub>0gh\<^sub>1.p\<^sub>1) \<star> TTfgh_TfTgh.chine = (\<a>\<^sup>-\<^sup>1[f, g \<star> h, (tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0] \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> \<a>[g \<star> h, tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0, f\<^sub>0gh\<^sub>1.p\<^sub>0]) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> \<a>\<^sup>-\<^sup>1[g, h, tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> (g \<star> \<a>[h, tab\<^sub>0 h, g\<^sub>0h\<^sub>1.p\<^sub>0]) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> (g \<star> h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> (g \<star> g\<^sub>0h\<^sub>1.\<phi>) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> \<a>[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> (g.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>1) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> f\<^sub>0gh\<^sub>1.\<phi>) \<star> TTfgh_TfTgh.chine) \<cdot> (\<a>[f, tab\<^sub>0 f, f\<^sub>0gh\<^sub>1.p\<^sub>1] \<star> TTfgh_TfTgh.chine) \<cdot> ((f.tab \<star> f\<^sub>0gh\<^sub>1.p\<^sub>1) \<star> TTfgh_TfTgh.chine)
[PROOF STEP]
have "arr ((f \<star> \<a>[g \<star> h, tab\<^sub>0 h \<star> Tgh.p\<^sub>0, TfTgh.p\<^sub>0]) \<cdot>
(f \<star> \<a>\<^sup>-\<^sup>1[g, h, tab\<^sub>0 h \<star> Tgh.p\<^sub>0] \<star> TfTgh.p\<^sub>0) \<cdot>
(f \<star> (g \<star> \<a>[h, tab\<^sub>0 h, Tgh.p\<^sub>0]) \<star> TfTgh.p\<^sub>0) \<cdot>
(f \<star> (g \<star> h.tab \<star> Tgh.p\<^sub>0) \<star> TfTgh.p\<^sub>0) \<cdot>
(f \<star> (g \<star> g\<^sub>0h\<^sub>1.\<phi>) \<star> TfTgh.p\<^sub>0) \<cdot>
(f \<star> \<a>[g, tab\<^sub>0 g, Tgh.p\<^sub>1] \<star> TfTgh.p\<^sub>0) \<cdot>
(f \<star> (g.tab \<star> Tgh.p\<^sub>1) \<star> TfTgh.p\<^sub>0) \<cdot>
(f \<star> f\<^sub>0gh\<^sub>1.\<phi>) \<cdot>
\<a>[f, tab\<^sub>0 f, TfTgh.p\<^sub>1] \<cdot>
(f.tab \<star> TfTgh.p\<^sub>1))"
[PROOF STATE]
proof (prove)
goal (1 subgoal):
1. seq (f \<star> \<a>[g \<star> h, tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0, f\<^sub>0gh\<^sub>1.p\<^sub>0]) ((f \<star> \<a>\<^sup>-\<^sup>1[g, h, tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<cdot> (f \<star> (g \<star> \<a>[h, tab\<^sub>0 h, g\<^sub>0h\<^sub>1.p\<^sub>0]) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<cdot> (f \<star> (g \<star> h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<cdot> (f \<star> (g \<star> g\<^sub>0h\<^sub>1.\<phi>) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<cdot> (f \<star> \<a>[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<cdot> (f \<star> (g.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>1) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<cdot> (f \<star> f\<^sub>0gh\<^sub>1.\<phi>) \<cdot> \<a>[f, tab\<^sub>0 f, f\<^sub>0gh\<^sub>1.p\<^sub>1] \<cdot> (f.tab \<star> f\<^sub>0gh\<^sub>1.p\<^sub>1))
[PROOF STEP]
using calculation
[PROOF STATE]
proof (prove)
using this:
seq \<a>\<^sup>-\<^sup>1[f, g \<star> h, (tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0] ((f \<star> \<a>[g \<star> h, tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0, f\<^sub>0gh\<^sub>1.p\<^sub>0]) \<cdot> (f \<star> \<a>\<^sup>-\<^sup>1[g, h, tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<cdot> (f \<star> (g \<star> \<a>[h, tab\<^sub>0 h, g\<^sub>0h\<^sub>1.p\<^sub>0]) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<cdot> (f \<star> (g \<star> h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<cdot> (f \<star> (g \<star> g\<^sub>0h\<^sub>1.\<phi>) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<cdot> (f \<star> \<a>[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<cdot> (f \<star> (g.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>1) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<cdot> (f \<star> f\<^sub>0gh\<^sub>1.\<phi>) \<cdot> \<a>[f, tab\<^sub>0 f, f\<^sub>0gh\<^sub>1.p\<^sub>1] \<cdot> (f.tab \<star> f\<^sub>0gh\<^sub>1.p\<^sub>1))
goal (1 subgoal):
1. seq (f \<star> \<a>[g \<star> h, tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0, f\<^sub>0gh\<^sub>1.p\<^sub>0]) ((f \<star> \<a>\<^sup>-\<^sup>1[g, h, tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<cdot> (f \<star> (g \<star> \<a>[h, tab\<^sub>0 h, g\<^sub>0h\<^sub>1.p\<^sub>0]) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<cdot> (f \<star> (g \<star> h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<cdot> (f \<star> (g \<star> g\<^sub>0h\<^sub>1.\<phi>) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<cdot> (f \<star> \<a>[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<cdot> (f \<star> (g.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>1) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<cdot> (f \<star> f\<^sub>0gh\<^sub>1.\<phi>) \<cdot> \<a>[f, tab\<^sub>0 f, f\<^sub>0gh\<^sub>1.p\<^sub>1] \<cdot> (f.tab \<star> f\<^sub>0gh\<^sub>1.p\<^sub>1))
[PROOF STEP]
by blast
[PROOF STATE]
proof (state)
this:
seq (f \<star> \<a>[g \<star> h, tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0, f\<^sub>0gh\<^sub>1.p\<^sub>0]) ((f \<star> \<a>\<^sup>-\<^sup>1[g, h, tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<cdot> (f \<star> (g \<star> \<a>[h, tab\<^sub>0 h, g\<^sub>0h\<^sub>1.p\<^sub>0]) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<cdot> (f \<star> (g \<star> h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<cdot> (f \<star> (g \<star> g\<^sub>0h\<^sub>1.\<phi>) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<cdot> (f \<star> \<a>[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<cdot> (f \<star> (g.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>1) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<cdot> (f \<star> f\<^sub>0gh\<^sub>1.\<phi>) \<cdot> \<a>[f, tab\<^sub>0 f, f\<^sub>0gh\<^sub>1.p\<^sub>1] \<cdot> (f.tab \<star> f\<^sub>0gh\<^sub>1.p\<^sub>1))
goal (1 subgoal):
1. \<a>\<^sup>-\<^sup>1[f, g \<star> h, (tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0] \<cdot> (f \<star> \<a>[g \<star> h, tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0, f\<^sub>0gh\<^sub>1.p\<^sub>0]) \<cdot> (f \<star> \<a>\<^sup>-\<^sup>1[g, h, tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<cdot> (f \<star> (g \<star> \<a>[h, tab\<^sub>0 h, g\<^sub>0h\<^sub>1.p\<^sub>0]) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<cdot> (f \<star> (g \<star> h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<cdot> (f \<star> (g \<star> g\<^sub>0h\<^sub>1.\<phi>) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<cdot> (f \<star> \<a>[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<cdot> (f \<star> (g.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>1) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<cdot> (f \<star> f\<^sub>0gh\<^sub>1.\<phi>) \<cdot> \<a>[f, tab\<^sub>0 f, f\<^sub>0gh\<^sub>1.p\<^sub>1] \<cdot> (f.tab \<star> f\<^sub>0gh\<^sub>1.p\<^sub>1) \<star> TTfgh_TfTgh.chine = (\<a>\<^sup>-\<^sup>1[f, g \<star> h, (tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0] \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> \<a>[g \<star> h, tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0, f\<^sub>0gh\<^sub>1.p\<^sub>0]) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> \<a>\<^sup>-\<^sup>1[g, h, tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> (g \<star> \<a>[h, tab\<^sub>0 h, g\<^sub>0h\<^sub>1.p\<^sub>0]) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> (g \<star> h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> (g \<star> g\<^sub>0h\<^sub>1.\<phi>) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> \<a>[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> (g.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>1) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> f\<^sub>0gh\<^sub>1.\<phi>) \<star> TTfgh_TfTgh.chine) \<cdot> (\<a>[f, tab\<^sub>0 f, f\<^sub>0gh\<^sub>1.p\<^sub>1] \<star> TTfgh_TfTgh.chine) \<cdot> ((f.tab \<star> f\<^sub>0gh\<^sub>1.p\<^sub>1) \<star> TTfgh_TfTgh.chine)
[PROOF STEP]
moreover
[PROOF STATE]
proof (state)
this:
seq (f \<star> \<a>[g \<star> h, tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0, f\<^sub>0gh\<^sub>1.p\<^sub>0]) ((f \<star> \<a>\<^sup>-\<^sup>1[g, h, tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<cdot> (f \<star> (g \<star> \<a>[h, tab\<^sub>0 h, g\<^sub>0h\<^sub>1.p\<^sub>0]) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<cdot> (f \<star> (g \<star> h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<cdot> (f \<star> (g \<star> g\<^sub>0h\<^sub>1.\<phi>) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<cdot> (f \<star> \<a>[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<cdot> (f \<star> (g.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>1) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<cdot> (f \<star> f\<^sub>0gh\<^sub>1.\<phi>) \<cdot> \<a>[f, tab\<^sub>0 f, f\<^sub>0gh\<^sub>1.p\<^sub>1] \<cdot> (f.tab \<star> f\<^sub>0gh\<^sub>1.p\<^sub>1))
goal (1 subgoal):
1. \<a>\<^sup>-\<^sup>1[f, g \<star> h, (tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0] \<cdot> (f \<star> \<a>[g \<star> h, tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0, f\<^sub>0gh\<^sub>1.p\<^sub>0]) \<cdot> (f \<star> \<a>\<^sup>-\<^sup>1[g, h, tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<cdot> (f \<star> (g \<star> \<a>[h, tab\<^sub>0 h, g\<^sub>0h\<^sub>1.p\<^sub>0]) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<cdot> (f \<star> (g \<star> h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<cdot> (f \<star> (g \<star> g\<^sub>0h\<^sub>1.\<phi>) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<cdot> (f \<star> \<a>[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<cdot> (f \<star> (g.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>1) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<cdot> (f \<star> f\<^sub>0gh\<^sub>1.\<phi>) \<cdot> \<a>[f, tab\<^sub>0 f, f\<^sub>0gh\<^sub>1.p\<^sub>1] \<cdot> (f.tab \<star> f\<^sub>0gh\<^sub>1.p\<^sub>1) \<star> TTfgh_TfTgh.chine = (\<a>\<^sup>-\<^sup>1[f, g \<star> h, (tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0] \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> \<a>[g \<star> h, tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0, f\<^sub>0gh\<^sub>1.p\<^sub>0]) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> \<a>\<^sup>-\<^sup>1[g, h, tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> (g \<star> \<a>[h, tab\<^sub>0 h, g\<^sub>0h\<^sub>1.p\<^sub>0]) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> (g \<star> h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> (g \<star> g\<^sub>0h\<^sub>1.\<phi>) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> \<a>[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> (g.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>1) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> f\<^sub>0gh\<^sub>1.\<phi>) \<star> TTfgh_TfTgh.chine) \<cdot> (\<a>[f, tab\<^sub>0 f, f\<^sub>0gh\<^sub>1.p\<^sub>1] \<star> TTfgh_TfTgh.chine) \<cdot> ((f.tab \<star> f\<^sub>0gh\<^sub>1.p\<^sub>1) \<star> TTfgh_TfTgh.chine)
[PROOF STEP]
have "arr ((f \<star> \<a>\<^sup>-\<^sup>1[g, h, tab\<^sub>0 h \<star> Tgh.p\<^sub>0] \<star> TfTgh.p\<^sub>0) \<cdot>
(f \<star> (g \<star> \<a>[h, tab\<^sub>0 h, Tgh.p\<^sub>0]) \<star> TfTgh.p\<^sub>0) \<cdot>
(f \<star> (g \<star> h.tab \<star> Tgh.p\<^sub>0) \<star> TfTgh.p\<^sub>0) \<cdot>
(f \<star> (g \<star> g\<^sub>0h\<^sub>1.\<phi>) \<star> TfTgh.p\<^sub>0) \<cdot>
(f \<star> \<a>[g, tab\<^sub>0 g, Tgh.p\<^sub>1] \<star> TfTgh.p\<^sub>0) \<cdot>
(f \<star> (g.tab \<star> Tgh.p\<^sub>1) \<star> TfTgh.p\<^sub>0) \<cdot>
(f \<star> f\<^sub>0gh\<^sub>1.\<phi>) \<cdot>
\<a>[f, tab\<^sub>0 f, TfTgh.p\<^sub>1] \<cdot>
(f.tab \<star> TfTgh.p\<^sub>1))"
[PROOF STATE]
proof (prove)
goal (1 subgoal):
1. seq (f \<star> \<a>\<^sup>-\<^sup>1[g, h, tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) ((f \<star> (g \<star> \<a>[h, tab\<^sub>0 h, g\<^sub>0h\<^sub>1.p\<^sub>0]) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<cdot> (f \<star> (g \<star> h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<cdot> (f \<star> (g \<star> g\<^sub>0h\<^sub>1.\<phi>) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<cdot> (f \<star> \<a>[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<cdot> (f \<star> (g.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>1) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<cdot> (f \<star> f\<^sub>0gh\<^sub>1.\<phi>) \<cdot> \<a>[f, tab\<^sub>0 f, f\<^sub>0gh\<^sub>1.p\<^sub>1] \<cdot> (f.tab \<star> f\<^sub>0gh\<^sub>1.p\<^sub>1))
[PROOF STEP]
using calculation
[PROOF STATE]
proof (prove)
using this:
seq \<a>\<^sup>-\<^sup>1[f, g \<star> h, (tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0] ((f \<star> \<a>[g \<star> h, tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0, f\<^sub>0gh\<^sub>1.p\<^sub>0]) \<cdot> (f \<star> \<a>\<^sup>-\<^sup>1[g, h, tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<cdot> (f \<star> (g \<star> \<a>[h, tab\<^sub>0 h, g\<^sub>0h\<^sub>1.p\<^sub>0]) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<cdot> (f \<star> (g \<star> h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<cdot> (f \<star> (g \<star> g\<^sub>0h\<^sub>1.\<phi>) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<cdot> (f \<star> \<a>[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<cdot> (f \<star> (g.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>1) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<cdot> (f \<star> f\<^sub>0gh\<^sub>1.\<phi>) \<cdot> \<a>[f, tab\<^sub>0 f, f\<^sub>0gh\<^sub>1.p\<^sub>1] \<cdot> (f.tab \<star> f\<^sub>0gh\<^sub>1.p\<^sub>1))
seq (f \<star> \<a>[g \<star> h, tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0, f\<^sub>0gh\<^sub>1.p\<^sub>0]) ((f \<star> \<a>\<^sup>-\<^sup>1[g, h, tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<cdot> (f \<star> (g \<star> \<a>[h, tab\<^sub>0 h, g\<^sub>0h\<^sub>1.p\<^sub>0]) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<cdot> (f \<star> (g \<star> h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<cdot> (f \<star> (g \<star> g\<^sub>0h\<^sub>1.\<phi>) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<cdot> (f \<star> \<a>[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<cdot> (f \<star> (g.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>1) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<cdot> (f \<star> f\<^sub>0gh\<^sub>1.\<phi>) \<cdot> \<a>[f, tab\<^sub>0 f, f\<^sub>0gh\<^sub>1.p\<^sub>1] \<cdot> (f.tab \<star> f\<^sub>0gh\<^sub>1.p\<^sub>1))
goal (1 subgoal):
1. seq (f \<star> \<a>\<^sup>-\<^sup>1[g, h, tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) ((f \<star> (g \<star> \<a>[h, tab\<^sub>0 h, g\<^sub>0h\<^sub>1.p\<^sub>0]) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<cdot> (f \<star> (g \<star> h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<cdot> (f \<star> (g \<star> g\<^sub>0h\<^sub>1.\<phi>) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<cdot> (f \<star> \<a>[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<cdot> (f \<star> (g.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>1) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<cdot> (f \<star> f\<^sub>0gh\<^sub>1.\<phi>) \<cdot> \<a>[f, tab\<^sub>0 f, f\<^sub>0gh\<^sub>1.p\<^sub>1] \<cdot> (f.tab \<star> f\<^sub>0gh\<^sub>1.p\<^sub>1))
[PROOF STEP]
by blast
[PROOF STATE]
proof (state)
this:
seq (f \<star> \<a>\<^sup>-\<^sup>1[g, h, tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) ((f \<star> (g \<star> \<a>[h, tab\<^sub>0 h, g\<^sub>0h\<^sub>1.p\<^sub>0]) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<cdot> (f \<star> (g \<star> h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<cdot> (f \<star> (g \<star> g\<^sub>0h\<^sub>1.\<phi>) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<cdot> (f \<star> \<a>[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<cdot> (f \<star> (g.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>1) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<cdot> (f \<star> f\<^sub>0gh\<^sub>1.\<phi>) \<cdot> \<a>[f, tab\<^sub>0 f, f\<^sub>0gh\<^sub>1.p\<^sub>1] \<cdot> (f.tab \<star> f\<^sub>0gh\<^sub>1.p\<^sub>1))
goal (1 subgoal):
1. \<a>\<^sup>-\<^sup>1[f, g \<star> h, (tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0] \<cdot> (f \<star> \<a>[g \<star> h, tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0, f\<^sub>0gh\<^sub>1.p\<^sub>0]) \<cdot> (f \<star> \<a>\<^sup>-\<^sup>1[g, h, tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<cdot> (f \<star> (g \<star> \<a>[h, tab\<^sub>0 h, g\<^sub>0h\<^sub>1.p\<^sub>0]) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<cdot> (f \<star> (g \<star> h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<cdot> (f \<star> (g \<star> g\<^sub>0h\<^sub>1.\<phi>) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<cdot> (f \<star> \<a>[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<cdot> (f \<star> (g.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>1) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<cdot> (f \<star> f\<^sub>0gh\<^sub>1.\<phi>) \<cdot> \<a>[f, tab\<^sub>0 f, f\<^sub>0gh\<^sub>1.p\<^sub>1] \<cdot> (f.tab \<star> f\<^sub>0gh\<^sub>1.p\<^sub>1) \<star> TTfgh_TfTgh.chine = (\<a>\<^sup>-\<^sup>1[f, g \<star> h, (tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0] \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> \<a>[g \<star> h, tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0, f\<^sub>0gh\<^sub>1.p\<^sub>0]) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> \<a>\<^sup>-\<^sup>1[g, h, tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> (g \<star> \<a>[h, tab\<^sub>0 h, g\<^sub>0h\<^sub>1.p\<^sub>0]) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> (g \<star> h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> (g \<star> g\<^sub>0h\<^sub>1.\<phi>) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> \<a>[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> (g.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>1) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> f\<^sub>0gh\<^sub>1.\<phi>) \<star> TTfgh_TfTgh.chine) \<cdot> (\<a>[f, tab\<^sub>0 f, f\<^sub>0gh\<^sub>1.p\<^sub>1] \<star> TTfgh_TfTgh.chine) \<cdot> ((f.tab \<star> f\<^sub>0gh\<^sub>1.p\<^sub>1) \<star> TTfgh_TfTgh.chine)
[PROOF STEP]
moreover
[PROOF STATE]
proof (state)
this:
seq (f \<star> \<a>\<^sup>-\<^sup>1[g, h, tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) ((f \<star> (g \<star> \<a>[h, tab\<^sub>0 h, g\<^sub>0h\<^sub>1.p\<^sub>0]) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<cdot> (f \<star> (g \<star> h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<cdot> (f \<star> (g \<star> g\<^sub>0h\<^sub>1.\<phi>) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<cdot> (f \<star> \<a>[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<cdot> (f \<star> (g.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>1) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<cdot> (f \<star> f\<^sub>0gh\<^sub>1.\<phi>) \<cdot> \<a>[f, tab\<^sub>0 f, f\<^sub>0gh\<^sub>1.p\<^sub>1] \<cdot> (f.tab \<star> f\<^sub>0gh\<^sub>1.p\<^sub>1))
goal (1 subgoal):
1. \<a>\<^sup>-\<^sup>1[f, g \<star> h, (tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0] \<cdot> (f \<star> \<a>[g \<star> h, tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0, f\<^sub>0gh\<^sub>1.p\<^sub>0]) \<cdot> (f \<star> \<a>\<^sup>-\<^sup>1[g, h, tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<cdot> (f \<star> (g \<star> \<a>[h, tab\<^sub>0 h, g\<^sub>0h\<^sub>1.p\<^sub>0]) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<cdot> (f \<star> (g \<star> h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<cdot> (f \<star> (g \<star> g\<^sub>0h\<^sub>1.\<phi>) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<cdot> (f \<star> \<a>[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<cdot> (f \<star> (g.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>1) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<cdot> (f \<star> f\<^sub>0gh\<^sub>1.\<phi>) \<cdot> \<a>[f, tab\<^sub>0 f, f\<^sub>0gh\<^sub>1.p\<^sub>1] \<cdot> (f.tab \<star> f\<^sub>0gh\<^sub>1.p\<^sub>1) \<star> TTfgh_TfTgh.chine = (\<a>\<^sup>-\<^sup>1[f, g \<star> h, (tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0] \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> \<a>[g \<star> h, tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0, f\<^sub>0gh\<^sub>1.p\<^sub>0]) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> \<a>\<^sup>-\<^sup>1[g, h, tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> (g \<star> \<a>[h, tab\<^sub>0 h, g\<^sub>0h\<^sub>1.p\<^sub>0]) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> (g \<star> h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> (g \<star> g\<^sub>0h\<^sub>1.\<phi>) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> \<a>[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> (g.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>1) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> f\<^sub>0gh\<^sub>1.\<phi>) \<star> TTfgh_TfTgh.chine) \<cdot> (\<a>[f, tab\<^sub>0 f, f\<^sub>0gh\<^sub>1.p\<^sub>1] \<star> TTfgh_TfTgh.chine) \<cdot> ((f.tab \<star> f\<^sub>0gh\<^sub>1.p\<^sub>1) \<star> TTfgh_TfTgh.chine)
[PROOF STEP]
have "arr ((f \<star> (g \<star> \<a>[h, tab\<^sub>0 h, Tgh.p\<^sub>0]) \<star> TfTgh.p\<^sub>0) \<cdot>
(f \<star> (g \<star> h.tab \<star> Tgh.p\<^sub>0) \<star> TfTgh.p\<^sub>0) \<cdot>
(f \<star> (g \<star> g\<^sub>0h\<^sub>1.\<phi>) \<star> TfTgh.p\<^sub>0) \<cdot>
(f \<star> \<a>[g, tab\<^sub>0 g, Tgh.p\<^sub>1] \<star> TfTgh.p\<^sub>0) \<cdot>
(f \<star> (g.tab \<star> Tgh.p\<^sub>1) \<star> TfTgh.p\<^sub>0) \<cdot>
(f \<star> f\<^sub>0gh\<^sub>1.\<phi>) \<cdot>
\<a>[f, tab\<^sub>0 f, TfTgh.p\<^sub>1] \<cdot>
(f.tab \<star> TfTgh.p\<^sub>1))"
[PROOF STATE]
proof (prove)
goal (1 subgoal):
1. seq (f \<star> (g \<star> \<a>[h, tab\<^sub>0 h, g\<^sub>0h\<^sub>1.p\<^sub>0]) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) ((f \<star> (g \<star> h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<cdot> (f \<star> (g \<star> g\<^sub>0h\<^sub>1.\<phi>) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<cdot> (f \<star> \<a>[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<cdot> (f \<star> (g.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>1) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<cdot> (f \<star> f\<^sub>0gh\<^sub>1.\<phi>) \<cdot> \<a>[f, tab\<^sub>0 f, f\<^sub>0gh\<^sub>1.p\<^sub>1] \<cdot> (f.tab \<star> f\<^sub>0gh\<^sub>1.p\<^sub>1))
[PROOF STEP]
using calculation
[PROOF STATE]
proof (prove)
using this:
seq \<a>\<^sup>-\<^sup>1[f, g \<star> h, (tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0] ((f \<star> \<a>[g \<star> h, tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0, f\<^sub>0gh\<^sub>1.p\<^sub>0]) \<cdot> (f \<star> \<a>\<^sup>-\<^sup>1[g, h, tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<cdot> (f \<star> (g \<star> \<a>[h, tab\<^sub>0 h, g\<^sub>0h\<^sub>1.p\<^sub>0]) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<cdot> (f \<star> (g \<star> h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<cdot> (f \<star> (g \<star> g\<^sub>0h\<^sub>1.\<phi>) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<cdot> (f \<star> \<a>[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<cdot> (f \<star> (g.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>1) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<cdot> (f \<star> f\<^sub>0gh\<^sub>1.\<phi>) \<cdot> \<a>[f, tab\<^sub>0 f, f\<^sub>0gh\<^sub>1.p\<^sub>1] \<cdot> (f.tab \<star> f\<^sub>0gh\<^sub>1.p\<^sub>1))
seq (f \<star> \<a>[g \<star> h, tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0, f\<^sub>0gh\<^sub>1.p\<^sub>0]) ((f \<star> \<a>\<^sup>-\<^sup>1[g, h, tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<cdot> (f \<star> (g \<star> \<a>[h, tab\<^sub>0 h, g\<^sub>0h\<^sub>1.p\<^sub>0]) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<cdot> (f \<star> (g \<star> h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<cdot> (f \<star> (g \<star> g\<^sub>0h\<^sub>1.\<phi>) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<cdot> (f \<star> \<a>[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<cdot> (f \<star> (g.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>1) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<cdot> (f \<star> f\<^sub>0gh\<^sub>1.\<phi>) \<cdot> \<a>[f, tab\<^sub>0 f, f\<^sub>0gh\<^sub>1.p\<^sub>1] \<cdot> (f.tab \<star> f\<^sub>0gh\<^sub>1.p\<^sub>1))
seq (f \<star> \<a>\<^sup>-\<^sup>1[g, h, tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) ((f \<star> (g \<star> \<a>[h, tab\<^sub>0 h, g\<^sub>0h\<^sub>1.p\<^sub>0]) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<cdot> (f \<star> (g \<star> h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<cdot> (f \<star> (g \<star> g\<^sub>0h\<^sub>1.\<phi>) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<cdot> (f \<star> \<a>[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<cdot> (f \<star> (g.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>1) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<cdot> (f \<star> f\<^sub>0gh\<^sub>1.\<phi>) \<cdot> \<a>[f, tab\<^sub>0 f, f\<^sub>0gh\<^sub>1.p\<^sub>1] \<cdot> (f.tab \<star> f\<^sub>0gh\<^sub>1.p\<^sub>1))
goal (1 subgoal):
1. seq (f \<star> (g \<star> \<a>[h, tab\<^sub>0 h, g\<^sub>0h\<^sub>1.p\<^sub>0]) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) ((f \<star> (g \<star> h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<cdot> (f \<star> (g \<star> g\<^sub>0h\<^sub>1.\<phi>) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<cdot> (f \<star> \<a>[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<cdot> (f \<star> (g.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>1) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<cdot> (f \<star> f\<^sub>0gh\<^sub>1.\<phi>) \<cdot> \<a>[f, tab\<^sub>0 f, f\<^sub>0gh\<^sub>1.p\<^sub>1] \<cdot> (f.tab \<star> f\<^sub>0gh\<^sub>1.p\<^sub>1))
[PROOF STEP]
by blast
[PROOF STATE]
proof (state)
this:
seq (f \<star> (g \<star> \<a>[h, tab\<^sub>0 h, g\<^sub>0h\<^sub>1.p\<^sub>0]) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) ((f \<star> (g \<star> h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<cdot> (f \<star> (g \<star> g\<^sub>0h\<^sub>1.\<phi>) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<cdot> (f \<star> \<a>[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<cdot> (f \<star> (g.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>1) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<cdot> (f \<star> f\<^sub>0gh\<^sub>1.\<phi>) \<cdot> \<a>[f, tab\<^sub>0 f, f\<^sub>0gh\<^sub>1.p\<^sub>1] \<cdot> (f.tab \<star> f\<^sub>0gh\<^sub>1.p\<^sub>1))
goal (1 subgoal):
1. \<a>\<^sup>-\<^sup>1[f, g \<star> h, (tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0] \<cdot> (f \<star> \<a>[g \<star> h, tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0, f\<^sub>0gh\<^sub>1.p\<^sub>0]) \<cdot> (f \<star> \<a>\<^sup>-\<^sup>1[g, h, tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<cdot> (f \<star> (g \<star> \<a>[h, tab\<^sub>0 h, g\<^sub>0h\<^sub>1.p\<^sub>0]) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<cdot> (f \<star> (g \<star> h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<cdot> (f \<star> (g \<star> g\<^sub>0h\<^sub>1.\<phi>) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<cdot> (f \<star> \<a>[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<cdot> (f \<star> (g.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>1) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<cdot> (f \<star> f\<^sub>0gh\<^sub>1.\<phi>) \<cdot> \<a>[f, tab\<^sub>0 f, f\<^sub>0gh\<^sub>1.p\<^sub>1] \<cdot> (f.tab \<star> f\<^sub>0gh\<^sub>1.p\<^sub>1) \<star> TTfgh_TfTgh.chine = (\<a>\<^sup>-\<^sup>1[f, g \<star> h, (tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0] \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> \<a>[g \<star> h, tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0, f\<^sub>0gh\<^sub>1.p\<^sub>0]) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> \<a>\<^sup>-\<^sup>1[g, h, tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> (g \<star> \<a>[h, tab\<^sub>0 h, g\<^sub>0h\<^sub>1.p\<^sub>0]) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> (g \<star> h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> (g \<star> g\<^sub>0h\<^sub>1.\<phi>) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> \<a>[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> (g.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>1) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> f\<^sub>0gh\<^sub>1.\<phi>) \<star> TTfgh_TfTgh.chine) \<cdot> (\<a>[f, tab\<^sub>0 f, f\<^sub>0gh\<^sub>1.p\<^sub>1] \<star> TTfgh_TfTgh.chine) \<cdot> ((f.tab \<star> f\<^sub>0gh\<^sub>1.p\<^sub>1) \<star> TTfgh_TfTgh.chine)
[PROOF STEP]
moreover
[PROOF STATE]
proof (state)
this:
seq (f \<star> (g \<star> \<a>[h, tab\<^sub>0 h, g\<^sub>0h\<^sub>1.p\<^sub>0]) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) ((f \<star> (g \<star> h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<cdot> (f \<star> (g \<star> g\<^sub>0h\<^sub>1.\<phi>) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<cdot> (f \<star> \<a>[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<cdot> (f \<star> (g.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>1) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<cdot> (f \<star> f\<^sub>0gh\<^sub>1.\<phi>) \<cdot> \<a>[f, tab\<^sub>0 f, f\<^sub>0gh\<^sub>1.p\<^sub>1] \<cdot> (f.tab \<star> f\<^sub>0gh\<^sub>1.p\<^sub>1))
goal (1 subgoal):
1. \<a>\<^sup>-\<^sup>1[f, g \<star> h, (tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0] \<cdot> (f \<star> \<a>[g \<star> h, tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0, f\<^sub>0gh\<^sub>1.p\<^sub>0]) \<cdot> (f \<star> \<a>\<^sup>-\<^sup>1[g, h, tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<cdot> (f \<star> (g \<star> \<a>[h, tab\<^sub>0 h, g\<^sub>0h\<^sub>1.p\<^sub>0]) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<cdot> (f \<star> (g \<star> h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<cdot> (f \<star> (g \<star> g\<^sub>0h\<^sub>1.\<phi>) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<cdot> (f \<star> \<a>[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<cdot> (f \<star> (g.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>1) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<cdot> (f \<star> f\<^sub>0gh\<^sub>1.\<phi>) \<cdot> \<a>[f, tab\<^sub>0 f, f\<^sub>0gh\<^sub>1.p\<^sub>1] \<cdot> (f.tab \<star> f\<^sub>0gh\<^sub>1.p\<^sub>1) \<star> TTfgh_TfTgh.chine = (\<a>\<^sup>-\<^sup>1[f, g \<star> h, (tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0] \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> \<a>[g \<star> h, tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0, f\<^sub>0gh\<^sub>1.p\<^sub>0]) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> \<a>\<^sup>-\<^sup>1[g, h, tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> (g \<star> \<a>[h, tab\<^sub>0 h, g\<^sub>0h\<^sub>1.p\<^sub>0]) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> (g \<star> h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> (g \<star> g\<^sub>0h\<^sub>1.\<phi>) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> \<a>[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> (g.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>1) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> f\<^sub>0gh\<^sub>1.\<phi>) \<star> TTfgh_TfTgh.chine) \<cdot> (\<a>[f, tab\<^sub>0 f, f\<^sub>0gh\<^sub>1.p\<^sub>1] \<star> TTfgh_TfTgh.chine) \<cdot> ((f.tab \<star> f\<^sub>0gh\<^sub>1.p\<^sub>1) \<star> TTfgh_TfTgh.chine)
[PROOF STEP]
have "arr ((f \<star> (g \<star> h.tab \<star> Tgh.p\<^sub>0) \<star> TfTgh.p\<^sub>0) \<cdot>
(f \<star> (g \<star> g\<^sub>0h\<^sub>1.\<phi>) \<star> TfTgh.p\<^sub>0) \<cdot>
(f \<star> \<a>[g, tab\<^sub>0 g, Tgh.p\<^sub>1] \<star> TfTgh.p\<^sub>0) \<cdot>
(f \<star> (g.tab \<star> Tgh.p\<^sub>1) \<star> TfTgh.p\<^sub>0) \<cdot>
(f \<star> f\<^sub>0gh\<^sub>1.\<phi>) \<cdot>
\<a>[f, tab\<^sub>0 f, TfTgh.p\<^sub>1] \<cdot>
(f.tab \<star> TfTgh.p\<^sub>1))"
[PROOF STATE]
proof (prove)
goal (1 subgoal):
1. seq (f \<star> (g \<star> h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) ((f \<star> (g \<star> g\<^sub>0h\<^sub>1.\<phi>) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<cdot> (f \<star> \<a>[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<cdot> (f \<star> (g.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>1) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<cdot> (f \<star> f\<^sub>0gh\<^sub>1.\<phi>) \<cdot> \<a>[f, tab\<^sub>0 f, f\<^sub>0gh\<^sub>1.p\<^sub>1] \<cdot> (f.tab \<star> f\<^sub>0gh\<^sub>1.p\<^sub>1))
[PROOF STEP]
using calculation
[PROOF STATE]
proof (prove)
using this:
seq \<a>\<^sup>-\<^sup>1[f, g \<star> h, (tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0] ((f \<star> \<a>[g \<star> h, tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0, f\<^sub>0gh\<^sub>1.p\<^sub>0]) \<cdot> (f \<star> \<a>\<^sup>-\<^sup>1[g, h, tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<cdot> (f \<star> (g \<star> \<a>[h, tab\<^sub>0 h, g\<^sub>0h\<^sub>1.p\<^sub>0]) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<cdot> (f \<star> (g \<star> h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<cdot> (f \<star> (g \<star> g\<^sub>0h\<^sub>1.\<phi>) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<cdot> (f \<star> \<a>[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<cdot> (f \<star> (g.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>1) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<cdot> (f \<star> f\<^sub>0gh\<^sub>1.\<phi>) \<cdot> \<a>[f, tab\<^sub>0 f, f\<^sub>0gh\<^sub>1.p\<^sub>1] \<cdot> (f.tab \<star> f\<^sub>0gh\<^sub>1.p\<^sub>1))
seq (f \<star> \<a>[g \<star> h, tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0, f\<^sub>0gh\<^sub>1.p\<^sub>0]) ((f \<star> \<a>\<^sup>-\<^sup>1[g, h, tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<cdot> (f \<star> (g \<star> \<a>[h, tab\<^sub>0 h, g\<^sub>0h\<^sub>1.p\<^sub>0]) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<cdot> (f \<star> (g \<star> h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<cdot> (f \<star> (g \<star> g\<^sub>0h\<^sub>1.\<phi>) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<cdot> (f \<star> \<a>[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<cdot> (f \<star> (g.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>1) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<cdot> (f \<star> f\<^sub>0gh\<^sub>1.\<phi>) \<cdot> \<a>[f, tab\<^sub>0 f, f\<^sub>0gh\<^sub>1.p\<^sub>1] \<cdot> (f.tab \<star> f\<^sub>0gh\<^sub>1.p\<^sub>1))
seq (f \<star> \<a>\<^sup>-\<^sup>1[g, h, tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) ((f \<star> (g \<star> \<a>[h, tab\<^sub>0 h, g\<^sub>0h\<^sub>1.p\<^sub>0]) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<cdot> (f \<star> (g \<star> h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<cdot> (f \<star> (g \<star> g\<^sub>0h\<^sub>1.\<phi>) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<cdot> (f \<star> \<a>[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<cdot> (f \<star> (g.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>1) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<cdot> (f \<star> f\<^sub>0gh\<^sub>1.\<phi>) \<cdot> \<a>[f, tab\<^sub>0 f, f\<^sub>0gh\<^sub>1.p\<^sub>1] \<cdot> (f.tab \<star> f\<^sub>0gh\<^sub>1.p\<^sub>1))
seq (f \<star> (g \<star> \<a>[h, tab\<^sub>0 h, g\<^sub>0h\<^sub>1.p\<^sub>0]) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) ((f \<star> (g \<star> h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<cdot> (f \<star> (g \<star> g\<^sub>0h\<^sub>1.\<phi>) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<cdot> (f \<star> \<a>[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<cdot> (f \<star> (g.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>1) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<cdot> (f \<star> f\<^sub>0gh\<^sub>1.\<phi>) \<cdot> \<a>[f, tab\<^sub>0 f, f\<^sub>0gh\<^sub>1.p\<^sub>1] \<cdot> (f.tab \<star> f\<^sub>0gh\<^sub>1.p\<^sub>1))
goal (1 subgoal):
1. seq (f \<star> (g \<star> h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) ((f \<star> (g \<star> g\<^sub>0h\<^sub>1.\<phi>) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<cdot> (f \<star> \<a>[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<cdot> (f \<star> (g.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>1) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<cdot> (f \<star> f\<^sub>0gh\<^sub>1.\<phi>) \<cdot> \<a>[f, tab\<^sub>0 f, f\<^sub>0gh\<^sub>1.p\<^sub>1] \<cdot> (f.tab \<star> f\<^sub>0gh\<^sub>1.p\<^sub>1))
[PROOF STEP]
by blast
[PROOF STATE]
proof (state)
this:
seq (f \<star> (g \<star> h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) ((f \<star> (g \<star> g\<^sub>0h\<^sub>1.\<phi>) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<cdot> (f \<star> \<a>[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<cdot> (f \<star> (g.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>1) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<cdot> (f \<star> f\<^sub>0gh\<^sub>1.\<phi>) \<cdot> \<a>[f, tab\<^sub>0 f, f\<^sub>0gh\<^sub>1.p\<^sub>1] \<cdot> (f.tab \<star> f\<^sub>0gh\<^sub>1.p\<^sub>1))
goal (1 subgoal):
1. \<a>\<^sup>-\<^sup>1[f, g \<star> h, (tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0] \<cdot> (f \<star> \<a>[g \<star> h, tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0, f\<^sub>0gh\<^sub>1.p\<^sub>0]) \<cdot> (f \<star> \<a>\<^sup>-\<^sup>1[g, h, tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<cdot> (f \<star> (g \<star> \<a>[h, tab\<^sub>0 h, g\<^sub>0h\<^sub>1.p\<^sub>0]) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<cdot> (f \<star> (g \<star> h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<cdot> (f \<star> (g \<star> g\<^sub>0h\<^sub>1.\<phi>) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<cdot> (f \<star> \<a>[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<cdot> (f \<star> (g.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>1) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<cdot> (f \<star> f\<^sub>0gh\<^sub>1.\<phi>) \<cdot> \<a>[f, tab\<^sub>0 f, f\<^sub>0gh\<^sub>1.p\<^sub>1] \<cdot> (f.tab \<star> f\<^sub>0gh\<^sub>1.p\<^sub>1) \<star> TTfgh_TfTgh.chine = (\<a>\<^sup>-\<^sup>1[f, g \<star> h, (tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0] \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> \<a>[g \<star> h, tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0, f\<^sub>0gh\<^sub>1.p\<^sub>0]) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> \<a>\<^sup>-\<^sup>1[g, h, tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> (g \<star> \<a>[h, tab\<^sub>0 h, g\<^sub>0h\<^sub>1.p\<^sub>0]) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> (g \<star> h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> (g \<star> g\<^sub>0h\<^sub>1.\<phi>) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> \<a>[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> (g.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>1) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> f\<^sub>0gh\<^sub>1.\<phi>) \<star> TTfgh_TfTgh.chine) \<cdot> (\<a>[f, tab\<^sub>0 f, f\<^sub>0gh\<^sub>1.p\<^sub>1] \<star> TTfgh_TfTgh.chine) \<cdot> ((f.tab \<star> f\<^sub>0gh\<^sub>1.p\<^sub>1) \<star> TTfgh_TfTgh.chine)
[PROOF STEP]
moreover
[PROOF STATE]
proof (state)
this:
seq (f \<star> (g \<star> h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) ((f \<star> (g \<star> g\<^sub>0h\<^sub>1.\<phi>) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<cdot> (f \<star> \<a>[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<cdot> (f \<star> (g.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>1) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<cdot> (f \<star> f\<^sub>0gh\<^sub>1.\<phi>) \<cdot> \<a>[f, tab\<^sub>0 f, f\<^sub>0gh\<^sub>1.p\<^sub>1] \<cdot> (f.tab \<star> f\<^sub>0gh\<^sub>1.p\<^sub>1))
goal (1 subgoal):
1. \<a>\<^sup>-\<^sup>1[f, g \<star> h, (tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0] \<cdot> (f \<star> \<a>[g \<star> h, tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0, f\<^sub>0gh\<^sub>1.p\<^sub>0]) \<cdot> (f \<star> \<a>\<^sup>-\<^sup>1[g, h, tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<cdot> (f \<star> (g \<star> \<a>[h, tab\<^sub>0 h, g\<^sub>0h\<^sub>1.p\<^sub>0]) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<cdot> (f \<star> (g \<star> h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<cdot> (f \<star> (g \<star> g\<^sub>0h\<^sub>1.\<phi>) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<cdot> (f \<star> \<a>[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<cdot> (f \<star> (g.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>1) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<cdot> (f \<star> f\<^sub>0gh\<^sub>1.\<phi>) \<cdot> \<a>[f, tab\<^sub>0 f, f\<^sub>0gh\<^sub>1.p\<^sub>1] \<cdot> (f.tab \<star> f\<^sub>0gh\<^sub>1.p\<^sub>1) \<star> TTfgh_TfTgh.chine = (\<a>\<^sup>-\<^sup>1[f, g \<star> h, (tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0] \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> \<a>[g \<star> h, tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0, f\<^sub>0gh\<^sub>1.p\<^sub>0]) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> \<a>\<^sup>-\<^sup>1[g, h, tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> (g \<star> \<a>[h, tab\<^sub>0 h, g\<^sub>0h\<^sub>1.p\<^sub>0]) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> (g \<star> h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> (g \<star> g\<^sub>0h\<^sub>1.\<phi>) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> \<a>[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> (g.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>1) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> f\<^sub>0gh\<^sub>1.\<phi>) \<star> TTfgh_TfTgh.chine) \<cdot> (\<a>[f, tab\<^sub>0 f, f\<^sub>0gh\<^sub>1.p\<^sub>1] \<star> TTfgh_TfTgh.chine) \<cdot> ((f.tab \<star> f\<^sub>0gh\<^sub>1.p\<^sub>1) \<star> TTfgh_TfTgh.chine)
[PROOF STEP]
have "arr ((f \<star> (g \<star> g\<^sub>0h\<^sub>1.\<phi>) \<star> TfTgh.p\<^sub>0) \<cdot>
(f \<star> \<a>[g, tab\<^sub>0 g, Tgh.p\<^sub>1] \<star> TfTgh.p\<^sub>0) \<cdot>
(f \<star> (g.tab \<star> Tgh.p\<^sub>1) \<star> TfTgh.p\<^sub>0) \<cdot>
(f \<star> f\<^sub>0gh\<^sub>1.\<phi>) \<cdot>
\<a>[f, tab\<^sub>0 f, TfTgh.p\<^sub>1] \<cdot>
(f.tab \<star> TfTgh.p\<^sub>1))"
[PROOF STATE]
proof (prove)
goal (1 subgoal):
1. seq (f \<star> (g \<star> g\<^sub>0h\<^sub>1.\<phi>) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) ((f \<star> \<a>[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<cdot> (f \<star> (g.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>1) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<cdot> (f \<star> f\<^sub>0gh\<^sub>1.\<phi>) \<cdot> \<a>[f, tab\<^sub>0 f, f\<^sub>0gh\<^sub>1.p\<^sub>1] \<cdot> (f.tab \<star> f\<^sub>0gh\<^sub>1.p\<^sub>1))
[PROOF STEP]
using calculation
[PROOF STATE]
proof (prove)
using this:
seq \<a>\<^sup>-\<^sup>1[f, g \<star> h, (tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0] ((f \<star> \<a>[g \<star> h, tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0, f\<^sub>0gh\<^sub>1.p\<^sub>0]) \<cdot> (f \<star> \<a>\<^sup>-\<^sup>1[g, h, tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<cdot> (f \<star> (g \<star> \<a>[h, tab\<^sub>0 h, g\<^sub>0h\<^sub>1.p\<^sub>0]) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<cdot> (f \<star> (g \<star> h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<cdot> (f \<star> (g \<star> g\<^sub>0h\<^sub>1.\<phi>) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<cdot> (f \<star> \<a>[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<cdot> (f \<star> (g.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>1) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<cdot> (f \<star> f\<^sub>0gh\<^sub>1.\<phi>) \<cdot> \<a>[f, tab\<^sub>0 f, f\<^sub>0gh\<^sub>1.p\<^sub>1] \<cdot> (f.tab \<star> f\<^sub>0gh\<^sub>1.p\<^sub>1))
seq (f \<star> \<a>[g \<star> h, tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0, f\<^sub>0gh\<^sub>1.p\<^sub>0]) ((f \<star> \<a>\<^sup>-\<^sup>1[g, h, tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<cdot> (f \<star> (g \<star> \<a>[h, tab\<^sub>0 h, g\<^sub>0h\<^sub>1.p\<^sub>0]) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<cdot> (f \<star> (g \<star> h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<cdot> (f \<star> (g \<star> g\<^sub>0h\<^sub>1.\<phi>) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<cdot> (f \<star> \<a>[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<cdot> (f \<star> (g.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>1) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<cdot> (f \<star> f\<^sub>0gh\<^sub>1.\<phi>) \<cdot> \<a>[f, tab\<^sub>0 f, f\<^sub>0gh\<^sub>1.p\<^sub>1] \<cdot> (f.tab \<star> f\<^sub>0gh\<^sub>1.p\<^sub>1))
seq (f \<star> \<a>\<^sup>-\<^sup>1[g, h, tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) ((f \<star> (g \<star> \<a>[h, tab\<^sub>0 h, g\<^sub>0h\<^sub>1.p\<^sub>0]) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<cdot> (f \<star> (g \<star> h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<cdot> (f \<star> (g \<star> g\<^sub>0h\<^sub>1.\<phi>) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<cdot> (f \<star> \<a>[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<cdot> (f \<star> (g.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>1) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<cdot> (f \<star> f\<^sub>0gh\<^sub>1.\<phi>) \<cdot> \<a>[f, tab\<^sub>0 f, f\<^sub>0gh\<^sub>1.p\<^sub>1] \<cdot> (f.tab \<star> f\<^sub>0gh\<^sub>1.p\<^sub>1))
seq (f \<star> (g \<star> \<a>[h, tab\<^sub>0 h, g\<^sub>0h\<^sub>1.p\<^sub>0]) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) ((f \<star> (g \<star> h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<cdot> (f \<star> (g \<star> g\<^sub>0h\<^sub>1.\<phi>) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<cdot> (f \<star> \<a>[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<cdot> (f \<star> (g.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>1) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<cdot> (f \<star> f\<^sub>0gh\<^sub>1.\<phi>) \<cdot> \<a>[f, tab\<^sub>0 f, f\<^sub>0gh\<^sub>1.p\<^sub>1] \<cdot> (f.tab \<star> f\<^sub>0gh\<^sub>1.p\<^sub>1))
seq (f \<star> (g \<star> h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) ((f \<star> (g \<star> g\<^sub>0h\<^sub>1.\<phi>) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<cdot> (f \<star> \<a>[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<cdot> (f \<star> (g.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>1) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<cdot> (f \<star> f\<^sub>0gh\<^sub>1.\<phi>) \<cdot> \<a>[f, tab\<^sub>0 f, f\<^sub>0gh\<^sub>1.p\<^sub>1] \<cdot> (f.tab \<star> f\<^sub>0gh\<^sub>1.p\<^sub>1))
goal (1 subgoal):
1. seq (f \<star> (g \<star> g\<^sub>0h\<^sub>1.\<phi>) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) ((f \<star> \<a>[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<cdot> (f \<star> (g.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>1) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<cdot> (f \<star> f\<^sub>0gh\<^sub>1.\<phi>) \<cdot> \<a>[f, tab\<^sub>0 f, f\<^sub>0gh\<^sub>1.p\<^sub>1] \<cdot> (f.tab \<star> f\<^sub>0gh\<^sub>1.p\<^sub>1))
[PROOF STEP]
by blast
[PROOF STATE]
proof (state)
this:
seq (f \<star> (g \<star> g\<^sub>0h\<^sub>1.\<phi>) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) ((f \<star> \<a>[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<cdot> (f \<star> (g.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>1) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<cdot> (f \<star> f\<^sub>0gh\<^sub>1.\<phi>) \<cdot> \<a>[f, tab\<^sub>0 f, f\<^sub>0gh\<^sub>1.p\<^sub>1] \<cdot> (f.tab \<star> f\<^sub>0gh\<^sub>1.p\<^sub>1))
goal (1 subgoal):
1. \<a>\<^sup>-\<^sup>1[f, g \<star> h, (tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0] \<cdot> (f \<star> \<a>[g \<star> h, tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0, f\<^sub>0gh\<^sub>1.p\<^sub>0]) \<cdot> (f \<star> \<a>\<^sup>-\<^sup>1[g, h, tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<cdot> (f \<star> (g \<star> \<a>[h, tab\<^sub>0 h, g\<^sub>0h\<^sub>1.p\<^sub>0]) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<cdot> (f \<star> (g \<star> h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<cdot> (f \<star> (g \<star> g\<^sub>0h\<^sub>1.\<phi>) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<cdot> (f \<star> \<a>[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<cdot> (f \<star> (g.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>1) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<cdot> (f \<star> f\<^sub>0gh\<^sub>1.\<phi>) \<cdot> \<a>[f, tab\<^sub>0 f, f\<^sub>0gh\<^sub>1.p\<^sub>1] \<cdot> (f.tab \<star> f\<^sub>0gh\<^sub>1.p\<^sub>1) \<star> TTfgh_TfTgh.chine = (\<a>\<^sup>-\<^sup>1[f, g \<star> h, (tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0] \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> \<a>[g \<star> h, tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0, f\<^sub>0gh\<^sub>1.p\<^sub>0]) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> \<a>\<^sup>-\<^sup>1[g, h, tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> (g \<star> \<a>[h, tab\<^sub>0 h, g\<^sub>0h\<^sub>1.p\<^sub>0]) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> (g \<star> h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> (g \<star> g\<^sub>0h\<^sub>1.\<phi>) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> \<a>[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> (g.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>1) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> f\<^sub>0gh\<^sub>1.\<phi>) \<star> TTfgh_TfTgh.chine) \<cdot> (\<a>[f, tab\<^sub>0 f, f\<^sub>0gh\<^sub>1.p\<^sub>1] \<star> TTfgh_TfTgh.chine) \<cdot> ((f.tab \<star> f\<^sub>0gh\<^sub>1.p\<^sub>1) \<star> TTfgh_TfTgh.chine)
[PROOF STEP]
moreover
[PROOF STATE]
proof (state)
this:
seq (f \<star> (g \<star> g\<^sub>0h\<^sub>1.\<phi>) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) ((f \<star> \<a>[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<cdot> (f \<star> (g.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>1) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<cdot> (f \<star> f\<^sub>0gh\<^sub>1.\<phi>) \<cdot> \<a>[f, tab\<^sub>0 f, f\<^sub>0gh\<^sub>1.p\<^sub>1] \<cdot> (f.tab \<star> f\<^sub>0gh\<^sub>1.p\<^sub>1))
goal (1 subgoal):
1. \<a>\<^sup>-\<^sup>1[f, g \<star> h, (tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0] \<cdot> (f \<star> \<a>[g \<star> h, tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0, f\<^sub>0gh\<^sub>1.p\<^sub>0]) \<cdot> (f \<star> \<a>\<^sup>-\<^sup>1[g, h, tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<cdot> (f \<star> (g \<star> \<a>[h, tab\<^sub>0 h, g\<^sub>0h\<^sub>1.p\<^sub>0]) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<cdot> (f \<star> (g \<star> h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<cdot> (f \<star> (g \<star> g\<^sub>0h\<^sub>1.\<phi>) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<cdot> (f \<star> \<a>[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<cdot> (f \<star> (g.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>1) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<cdot> (f \<star> f\<^sub>0gh\<^sub>1.\<phi>) \<cdot> \<a>[f, tab\<^sub>0 f, f\<^sub>0gh\<^sub>1.p\<^sub>1] \<cdot> (f.tab \<star> f\<^sub>0gh\<^sub>1.p\<^sub>1) \<star> TTfgh_TfTgh.chine = (\<a>\<^sup>-\<^sup>1[f, g \<star> h, (tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0] \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> \<a>[g \<star> h, tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0, f\<^sub>0gh\<^sub>1.p\<^sub>0]) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> \<a>\<^sup>-\<^sup>1[g, h, tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> (g \<star> \<a>[h, tab\<^sub>0 h, g\<^sub>0h\<^sub>1.p\<^sub>0]) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> (g \<star> h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> (g \<star> g\<^sub>0h\<^sub>1.\<phi>) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> \<a>[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> (g.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>1) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> f\<^sub>0gh\<^sub>1.\<phi>) \<star> TTfgh_TfTgh.chine) \<cdot> (\<a>[f, tab\<^sub>0 f, f\<^sub>0gh\<^sub>1.p\<^sub>1] \<star> TTfgh_TfTgh.chine) \<cdot> ((f.tab \<star> f\<^sub>0gh\<^sub>1.p\<^sub>1) \<star> TTfgh_TfTgh.chine)
[PROOF STEP]
have "arr ((f \<star> \<a>[g, tab\<^sub>0 g, Tgh.p\<^sub>1] \<star> TfTgh.p\<^sub>0) \<cdot>
(f \<star> (g.tab \<star> Tgh.p\<^sub>1) \<star> TfTgh.p\<^sub>0) \<cdot>
(f \<star> f\<^sub>0gh\<^sub>1.\<phi>) \<cdot>
\<a>[f, tab\<^sub>0 f, TfTgh.p\<^sub>1] \<cdot>
(f.tab \<star> TfTgh.p\<^sub>1))"
[PROOF STATE]
proof (prove)
goal (1 subgoal):
1. seq (f \<star> \<a>[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) ((f \<star> (g.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>1) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<cdot> (f \<star> f\<^sub>0gh\<^sub>1.\<phi>) \<cdot> \<a>[f, tab\<^sub>0 f, f\<^sub>0gh\<^sub>1.p\<^sub>1] \<cdot> (f.tab \<star> f\<^sub>0gh\<^sub>1.p\<^sub>1))
[PROOF STEP]
using calculation
[PROOF STATE]
proof (prove)
using this:
seq \<a>\<^sup>-\<^sup>1[f, g \<star> h, (tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0] ((f \<star> \<a>[g \<star> h, tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0, f\<^sub>0gh\<^sub>1.p\<^sub>0]) \<cdot> (f \<star> \<a>\<^sup>-\<^sup>1[g, h, tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<cdot> (f \<star> (g \<star> \<a>[h, tab\<^sub>0 h, g\<^sub>0h\<^sub>1.p\<^sub>0]) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<cdot> (f \<star> (g \<star> h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<cdot> (f \<star> (g \<star> g\<^sub>0h\<^sub>1.\<phi>) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<cdot> (f \<star> \<a>[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<cdot> (f \<star> (g.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>1) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<cdot> (f \<star> f\<^sub>0gh\<^sub>1.\<phi>) \<cdot> \<a>[f, tab\<^sub>0 f, f\<^sub>0gh\<^sub>1.p\<^sub>1] \<cdot> (f.tab \<star> f\<^sub>0gh\<^sub>1.p\<^sub>1))
seq (f \<star> \<a>[g \<star> h, tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0, f\<^sub>0gh\<^sub>1.p\<^sub>0]) ((f \<star> \<a>\<^sup>-\<^sup>1[g, h, tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<cdot> (f \<star> (g \<star> \<a>[h, tab\<^sub>0 h, g\<^sub>0h\<^sub>1.p\<^sub>0]) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<cdot> (f \<star> (g \<star> h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<cdot> (f \<star> (g \<star> g\<^sub>0h\<^sub>1.\<phi>) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<cdot> (f \<star> \<a>[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<cdot> (f \<star> (g.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>1) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<cdot> (f \<star> f\<^sub>0gh\<^sub>1.\<phi>) \<cdot> \<a>[f, tab\<^sub>0 f, f\<^sub>0gh\<^sub>1.p\<^sub>1] \<cdot> (f.tab \<star> f\<^sub>0gh\<^sub>1.p\<^sub>1))
seq (f \<star> \<a>\<^sup>-\<^sup>1[g, h, tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) ((f \<star> (g \<star> \<a>[h, tab\<^sub>0 h, g\<^sub>0h\<^sub>1.p\<^sub>0]) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<cdot> (f \<star> (g \<star> h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<cdot> (f \<star> (g \<star> g\<^sub>0h\<^sub>1.\<phi>) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<cdot> (f \<star> \<a>[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<cdot> (f \<star> (g.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>1) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<cdot> (f \<star> f\<^sub>0gh\<^sub>1.\<phi>) \<cdot> \<a>[f, tab\<^sub>0 f, f\<^sub>0gh\<^sub>1.p\<^sub>1] \<cdot> (f.tab \<star> f\<^sub>0gh\<^sub>1.p\<^sub>1))
seq (f \<star> (g \<star> \<a>[h, tab\<^sub>0 h, g\<^sub>0h\<^sub>1.p\<^sub>0]) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) ((f \<star> (g \<star> h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<cdot> (f \<star> (g \<star> g\<^sub>0h\<^sub>1.\<phi>) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<cdot> (f \<star> \<a>[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<cdot> (f \<star> (g.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>1) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<cdot> (f \<star> f\<^sub>0gh\<^sub>1.\<phi>) \<cdot> \<a>[f, tab\<^sub>0 f, f\<^sub>0gh\<^sub>1.p\<^sub>1] \<cdot> (f.tab \<star> f\<^sub>0gh\<^sub>1.p\<^sub>1))
seq (f \<star> (g \<star> h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) ((f \<star> (g \<star> g\<^sub>0h\<^sub>1.\<phi>) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<cdot> (f \<star> \<a>[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<cdot> (f \<star> (g.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>1) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<cdot> (f \<star> f\<^sub>0gh\<^sub>1.\<phi>) \<cdot> \<a>[f, tab\<^sub>0 f, f\<^sub>0gh\<^sub>1.p\<^sub>1] \<cdot> (f.tab \<star> f\<^sub>0gh\<^sub>1.p\<^sub>1))
seq (f \<star> (g \<star> g\<^sub>0h\<^sub>1.\<phi>) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) ((f \<star> \<a>[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<cdot> (f \<star> (g.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>1) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<cdot> (f \<star> f\<^sub>0gh\<^sub>1.\<phi>) \<cdot> \<a>[f, tab\<^sub>0 f, f\<^sub>0gh\<^sub>1.p\<^sub>1] \<cdot> (f.tab \<star> f\<^sub>0gh\<^sub>1.p\<^sub>1))
goal (1 subgoal):
1. seq (f \<star> \<a>[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) ((f \<star> (g.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>1) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<cdot> (f \<star> f\<^sub>0gh\<^sub>1.\<phi>) \<cdot> \<a>[f, tab\<^sub>0 f, f\<^sub>0gh\<^sub>1.p\<^sub>1] \<cdot> (f.tab \<star> f\<^sub>0gh\<^sub>1.p\<^sub>1))
[PROOF STEP]
by blast
[PROOF STATE]
proof (state)
this:
seq (f \<star> \<a>[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) ((f \<star> (g.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>1) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<cdot> (f \<star> f\<^sub>0gh\<^sub>1.\<phi>) \<cdot> \<a>[f, tab\<^sub>0 f, f\<^sub>0gh\<^sub>1.p\<^sub>1] \<cdot> (f.tab \<star> f\<^sub>0gh\<^sub>1.p\<^sub>1))
goal (1 subgoal):
1. \<a>\<^sup>-\<^sup>1[f, g \<star> h, (tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0] \<cdot> (f \<star> \<a>[g \<star> h, tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0, f\<^sub>0gh\<^sub>1.p\<^sub>0]) \<cdot> (f \<star> \<a>\<^sup>-\<^sup>1[g, h, tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<cdot> (f \<star> (g \<star> \<a>[h, tab\<^sub>0 h, g\<^sub>0h\<^sub>1.p\<^sub>0]) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<cdot> (f \<star> (g \<star> h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<cdot> (f \<star> (g \<star> g\<^sub>0h\<^sub>1.\<phi>) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<cdot> (f \<star> \<a>[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<cdot> (f \<star> (g.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>1) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<cdot> (f \<star> f\<^sub>0gh\<^sub>1.\<phi>) \<cdot> \<a>[f, tab\<^sub>0 f, f\<^sub>0gh\<^sub>1.p\<^sub>1] \<cdot> (f.tab \<star> f\<^sub>0gh\<^sub>1.p\<^sub>1) \<star> TTfgh_TfTgh.chine = (\<a>\<^sup>-\<^sup>1[f, g \<star> h, (tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0] \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> \<a>[g \<star> h, tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0, f\<^sub>0gh\<^sub>1.p\<^sub>0]) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> \<a>\<^sup>-\<^sup>1[g, h, tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> (g \<star> \<a>[h, tab\<^sub>0 h, g\<^sub>0h\<^sub>1.p\<^sub>0]) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> (g \<star> h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> (g \<star> g\<^sub>0h\<^sub>1.\<phi>) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> \<a>[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> (g.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>1) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> f\<^sub>0gh\<^sub>1.\<phi>) \<star> TTfgh_TfTgh.chine) \<cdot> (\<a>[f, tab\<^sub>0 f, f\<^sub>0gh\<^sub>1.p\<^sub>1] \<star> TTfgh_TfTgh.chine) \<cdot> ((f.tab \<star> f\<^sub>0gh\<^sub>1.p\<^sub>1) \<star> TTfgh_TfTgh.chine)
[PROOF STEP]
moreover
[PROOF STATE]
proof (state)
this:
seq (f \<star> \<a>[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) ((f \<star> (g.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>1) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<cdot> (f \<star> f\<^sub>0gh\<^sub>1.\<phi>) \<cdot> \<a>[f, tab\<^sub>0 f, f\<^sub>0gh\<^sub>1.p\<^sub>1] \<cdot> (f.tab \<star> f\<^sub>0gh\<^sub>1.p\<^sub>1))
goal (1 subgoal):
1. \<a>\<^sup>-\<^sup>1[f, g \<star> h, (tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0] \<cdot> (f \<star> \<a>[g \<star> h, tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0, f\<^sub>0gh\<^sub>1.p\<^sub>0]) \<cdot> (f \<star> \<a>\<^sup>-\<^sup>1[g, h, tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<cdot> (f \<star> (g \<star> \<a>[h, tab\<^sub>0 h, g\<^sub>0h\<^sub>1.p\<^sub>0]) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<cdot> (f \<star> (g \<star> h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<cdot> (f \<star> (g \<star> g\<^sub>0h\<^sub>1.\<phi>) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<cdot> (f \<star> \<a>[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<cdot> (f \<star> (g.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>1) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<cdot> (f \<star> f\<^sub>0gh\<^sub>1.\<phi>) \<cdot> \<a>[f, tab\<^sub>0 f, f\<^sub>0gh\<^sub>1.p\<^sub>1] \<cdot> (f.tab \<star> f\<^sub>0gh\<^sub>1.p\<^sub>1) \<star> TTfgh_TfTgh.chine = (\<a>\<^sup>-\<^sup>1[f, g \<star> h, (tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0] \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> \<a>[g \<star> h, tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0, f\<^sub>0gh\<^sub>1.p\<^sub>0]) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> \<a>\<^sup>-\<^sup>1[g, h, tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> (g \<star> \<a>[h, tab\<^sub>0 h, g\<^sub>0h\<^sub>1.p\<^sub>0]) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> (g \<star> h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> (g \<star> g\<^sub>0h\<^sub>1.\<phi>) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> \<a>[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> (g.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>1) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> f\<^sub>0gh\<^sub>1.\<phi>) \<star> TTfgh_TfTgh.chine) \<cdot> (\<a>[f, tab\<^sub>0 f, f\<^sub>0gh\<^sub>1.p\<^sub>1] \<star> TTfgh_TfTgh.chine) \<cdot> ((f.tab \<star> f\<^sub>0gh\<^sub>1.p\<^sub>1) \<star> TTfgh_TfTgh.chine)
[PROOF STEP]
have "arr ((f \<star> (g.tab \<star> Tgh.p\<^sub>1) \<star> TfTgh.p\<^sub>0) \<cdot>
(f \<star> f\<^sub>0gh\<^sub>1.\<phi>) \<cdot>
\<a>[f, tab\<^sub>0 f, TfTgh.p\<^sub>1] \<cdot>
(f.tab \<star> TfTgh.p\<^sub>1))"
[PROOF STATE]
proof (prove)
goal (1 subgoal):
1. seq (f \<star> (g.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>1) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) ((f \<star> f\<^sub>0gh\<^sub>1.\<phi>) \<cdot> \<a>[f, tab\<^sub>0 f, f\<^sub>0gh\<^sub>1.p\<^sub>1] \<cdot> (f.tab \<star> f\<^sub>0gh\<^sub>1.p\<^sub>1))
[PROOF STEP]
using calculation
[PROOF STATE]
proof (prove)
using this:
seq \<a>\<^sup>-\<^sup>1[f, g \<star> h, (tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0] ((f \<star> \<a>[g \<star> h, tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0, f\<^sub>0gh\<^sub>1.p\<^sub>0]) \<cdot> (f \<star> \<a>\<^sup>-\<^sup>1[g, h, tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<cdot> (f \<star> (g \<star> \<a>[h, tab\<^sub>0 h, g\<^sub>0h\<^sub>1.p\<^sub>0]) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<cdot> (f \<star> (g \<star> h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<cdot> (f \<star> (g \<star> g\<^sub>0h\<^sub>1.\<phi>) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<cdot> (f \<star> \<a>[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<cdot> (f \<star> (g.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>1) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<cdot> (f \<star> f\<^sub>0gh\<^sub>1.\<phi>) \<cdot> \<a>[f, tab\<^sub>0 f, f\<^sub>0gh\<^sub>1.p\<^sub>1] \<cdot> (f.tab \<star> f\<^sub>0gh\<^sub>1.p\<^sub>1))
seq (f \<star> \<a>[g \<star> h, tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0, f\<^sub>0gh\<^sub>1.p\<^sub>0]) ((f \<star> \<a>\<^sup>-\<^sup>1[g, h, tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<cdot> (f \<star> (g \<star> \<a>[h, tab\<^sub>0 h, g\<^sub>0h\<^sub>1.p\<^sub>0]) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<cdot> (f \<star> (g \<star> h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<cdot> (f \<star> (g \<star> g\<^sub>0h\<^sub>1.\<phi>) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<cdot> (f \<star> \<a>[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<cdot> (f \<star> (g.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>1) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<cdot> (f \<star> f\<^sub>0gh\<^sub>1.\<phi>) \<cdot> \<a>[f, tab\<^sub>0 f, f\<^sub>0gh\<^sub>1.p\<^sub>1] \<cdot> (f.tab \<star> f\<^sub>0gh\<^sub>1.p\<^sub>1))
seq (f \<star> \<a>\<^sup>-\<^sup>1[g, h, tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) ((f \<star> (g \<star> \<a>[h, tab\<^sub>0 h, g\<^sub>0h\<^sub>1.p\<^sub>0]) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<cdot> (f \<star> (g \<star> h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<cdot> (f \<star> (g \<star> g\<^sub>0h\<^sub>1.\<phi>) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<cdot> (f \<star> \<a>[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<cdot> (f \<star> (g.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>1) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<cdot> (f \<star> f\<^sub>0gh\<^sub>1.\<phi>) \<cdot> \<a>[f, tab\<^sub>0 f, f\<^sub>0gh\<^sub>1.p\<^sub>1] \<cdot> (f.tab \<star> f\<^sub>0gh\<^sub>1.p\<^sub>1))
seq (f \<star> (g \<star> \<a>[h, tab\<^sub>0 h, g\<^sub>0h\<^sub>1.p\<^sub>0]) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) ((f \<star> (g \<star> h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<cdot> (f \<star> (g \<star> g\<^sub>0h\<^sub>1.\<phi>) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<cdot> (f \<star> \<a>[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<cdot> (f \<star> (g.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>1) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<cdot> (f \<star> f\<^sub>0gh\<^sub>1.\<phi>) \<cdot> \<a>[f, tab\<^sub>0 f, f\<^sub>0gh\<^sub>1.p\<^sub>1] \<cdot> (f.tab \<star> f\<^sub>0gh\<^sub>1.p\<^sub>1))
seq (f \<star> (g \<star> h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) ((f \<star> (g \<star> g\<^sub>0h\<^sub>1.\<phi>) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<cdot> (f \<star> \<a>[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<cdot> (f \<star> (g.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>1) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<cdot> (f \<star> f\<^sub>0gh\<^sub>1.\<phi>) \<cdot> \<a>[f, tab\<^sub>0 f, f\<^sub>0gh\<^sub>1.p\<^sub>1] \<cdot> (f.tab \<star> f\<^sub>0gh\<^sub>1.p\<^sub>1))
seq (f \<star> (g \<star> g\<^sub>0h\<^sub>1.\<phi>) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) ((f \<star> \<a>[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<cdot> (f \<star> (g.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>1) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<cdot> (f \<star> f\<^sub>0gh\<^sub>1.\<phi>) \<cdot> \<a>[f, tab\<^sub>0 f, f\<^sub>0gh\<^sub>1.p\<^sub>1] \<cdot> (f.tab \<star> f\<^sub>0gh\<^sub>1.p\<^sub>1))
seq (f \<star> \<a>[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) ((f \<star> (g.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>1) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<cdot> (f \<star> f\<^sub>0gh\<^sub>1.\<phi>) \<cdot> \<a>[f, tab\<^sub>0 f, f\<^sub>0gh\<^sub>1.p\<^sub>1] \<cdot> (f.tab \<star> f\<^sub>0gh\<^sub>1.p\<^sub>1))
goal (1 subgoal):
1. seq (f \<star> (g.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>1) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) ((f \<star> f\<^sub>0gh\<^sub>1.\<phi>) \<cdot> \<a>[f, tab\<^sub>0 f, f\<^sub>0gh\<^sub>1.p\<^sub>1] \<cdot> (f.tab \<star> f\<^sub>0gh\<^sub>1.p\<^sub>1))
[PROOF STEP]
by blast
[PROOF STATE]
proof (state)
this:
seq (f \<star> (g.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>1) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) ((f \<star> f\<^sub>0gh\<^sub>1.\<phi>) \<cdot> \<a>[f, tab\<^sub>0 f, f\<^sub>0gh\<^sub>1.p\<^sub>1] \<cdot> (f.tab \<star> f\<^sub>0gh\<^sub>1.p\<^sub>1))
goal (1 subgoal):
1. \<a>\<^sup>-\<^sup>1[f, g \<star> h, (tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0] \<cdot> (f \<star> \<a>[g \<star> h, tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0, f\<^sub>0gh\<^sub>1.p\<^sub>0]) \<cdot> (f \<star> \<a>\<^sup>-\<^sup>1[g, h, tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<cdot> (f \<star> (g \<star> \<a>[h, tab\<^sub>0 h, g\<^sub>0h\<^sub>1.p\<^sub>0]) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<cdot> (f \<star> (g \<star> h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<cdot> (f \<star> (g \<star> g\<^sub>0h\<^sub>1.\<phi>) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<cdot> (f \<star> \<a>[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<cdot> (f \<star> (g.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>1) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<cdot> (f \<star> f\<^sub>0gh\<^sub>1.\<phi>) \<cdot> \<a>[f, tab\<^sub>0 f, f\<^sub>0gh\<^sub>1.p\<^sub>1] \<cdot> (f.tab \<star> f\<^sub>0gh\<^sub>1.p\<^sub>1) \<star> TTfgh_TfTgh.chine = (\<a>\<^sup>-\<^sup>1[f, g \<star> h, (tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0] \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> \<a>[g \<star> h, tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0, f\<^sub>0gh\<^sub>1.p\<^sub>0]) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> \<a>\<^sup>-\<^sup>1[g, h, tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> (g \<star> \<a>[h, tab\<^sub>0 h, g\<^sub>0h\<^sub>1.p\<^sub>0]) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> (g \<star> h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> (g \<star> g\<^sub>0h\<^sub>1.\<phi>) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> \<a>[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> (g.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>1) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> f\<^sub>0gh\<^sub>1.\<phi>) \<star> TTfgh_TfTgh.chine) \<cdot> (\<a>[f, tab\<^sub>0 f, f\<^sub>0gh\<^sub>1.p\<^sub>1] \<star> TTfgh_TfTgh.chine) \<cdot> ((f.tab \<star> f\<^sub>0gh\<^sub>1.p\<^sub>1) \<star> TTfgh_TfTgh.chine)
[PROOF STEP]
moreover
[PROOF STATE]
proof (state)
this:
seq (f \<star> (g.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>1) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) ((f \<star> f\<^sub>0gh\<^sub>1.\<phi>) \<cdot> \<a>[f, tab\<^sub>0 f, f\<^sub>0gh\<^sub>1.p\<^sub>1] \<cdot> (f.tab \<star> f\<^sub>0gh\<^sub>1.p\<^sub>1))
goal (1 subgoal):
1. \<a>\<^sup>-\<^sup>1[f, g \<star> h, (tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0] \<cdot> (f \<star> \<a>[g \<star> h, tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0, f\<^sub>0gh\<^sub>1.p\<^sub>0]) \<cdot> (f \<star> \<a>\<^sup>-\<^sup>1[g, h, tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<cdot> (f \<star> (g \<star> \<a>[h, tab\<^sub>0 h, g\<^sub>0h\<^sub>1.p\<^sub>0]) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<cdot> (f \<star> (g \<star> h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<cdot> (f \<star> (g \<star> g\<^sub>0h\<^sub>1.\<phi>) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<cdot> (f \<star> \<a>[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<cdot> (f \<star> (g.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>1) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<cdot> (f \<star> f\<^sub>0gh\<^sub>1.\<phi>) \<cdot> \<a>[f, tab\<^sub>0 f, f\<^sub>0gh\<^sub>1.p\<^sub>1] \<cdot> (f.tab \<star> f\<^sub>0gh\<^sub>1.p\<^sub>1) \<star> TTfgh_TfTgh.chine = (\<a>\<^sup>-\<^sup>1[f, g \<star> h, (tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0] \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> \<a>[g \<star> h, tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0, f\<^sub>0gh\<^sub>1.p\<^sub>0]) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> \<a>\<^sup>-\<^sup>1[g, h, tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> (g \<star> \<a>[h, tab\<^sub>0 h, g\<^sub>0h\<^sub>1.p\<^sub>0]) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> (g \<star> h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> (g \<star> g\<^sub>0h\<^sub>1.\<phi>) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> \<a>[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> (g.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>1) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> f\<^sub>0gh\<^sub>1.\<phi>) \<star> TTfgh_TfTgh.chine) \<cdot> (\<a>[f, tab\<^sub>0 f, f\<^sub>0gh\<^sub>1.p\<^sub>1] \<star> TTfgh_TfTgh.chine) \<cdot> ((f.tab \<star> f\<^sub>0gh\<^sub>1.p\<^sub>1) \<star> TTfgh_TfTgh.chine)
[PROOF STEP]
have "arr ((f \<star> f\<^sub>0gh\<^sub>1.\<phi>) \<cdot>
\<a>[f, tab\<^sub>0 f, TfTgh.p\<^sub>1] \<cdot>
(f.tab \<star> TfTgh.p\<^sub>1))"
[PROOF STATE]
proof (prove)
goal (1 subgoal):
1. seq (f \<star> f\<^sub>0gh\<^sub>1.\<phi>) (\<a>[f, tab\<^sub>0 f, f\<^sub>0gh\<^sub>1.p\<^sub>1] \<cdot> (f.tab \<star> f\<^sub>0gh\<^sub>1.p\<^sub>1))
[PROOF STEP]
using calculation
[PROOF STATE]
proof (prove)
using this:
seq \<a>\<^sup>-\<^sup>1[f, g \<star> h, (tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0] ((f \<star> \<a>[g \<star> h, tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0, f\<^sub>0gh\<^sub>1.p\<^sub>0]) \<cdot> (f \<star> \<a>\<^sup>-\<^sup>1[g, h, tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<cdot> (f \<star> (g \<star> \<a>[h, tab\<^sub>0 h, g\<^sub>0h\<^sub>1.p\<^sub>0]) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<cdot> (f \<star> (g \<star> h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<cdot> (f \<star> (g \<star> g\<^sub>0h\<^sub>1.\<phi>) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<cdot> (f \<star> \<a>[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<cdot> (f \<star> (g.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>1) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<cdot> (f \<star> f\<^sub>0gh\<^sub>1.\<phi>) \<cdot> \<a>[f, tab\<^sub>0 f, f\<^sub>0gh\<^sub>1.p\<^sub>1] \<cdot> (f.tab \<star> f\<^sub>0gh\<^sub>1.p\<^sub>1))
seq (f \<star> \<a>[g \<star> h, tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0, f\<^sub>0gh\<^sub>1.p\<^sub>0]) ((f \<star> \<a>\<^sup>-\<^sup>1[g, h, tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<cdot> (f \<star> (g \<star> \<a>[h, tab\<^sub>0 h, g\<^sub>0h\<^sub>1.p\<^sub>0]) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<cdot> (f \<star> (g \<star> h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<cdot> (f \<star> (g \<star> g\<^sub>0h\<^sub>1.\<phi>) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<cdot> (f \<star> \<a>[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<cdot> (f \<star> (g.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>1) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<cdot> (f \<star> f\<^sub>0gh\<^sub>1.\<phi>) \<cdot> \<a>[f, tab\<^sub>0 f, f\<^sub>0gh\<^sub>1.p\<^sub>1] \<cdot> (f.tab \<star> f\<^sub>0gh\<^sub>1.p\<^sub>1))
seq (f \<star> \<a>\<^sup>-\<^sup>1[g, h, tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) ((f \<star> (g \<star> \<a>[h, tab\<^sub>0 h, g\<^sub>0h\<^sub>1.p\<^sub>0]) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<cdot> (f \<star> (g \<star> h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<cdot> (f \<star> (g \<star> g\<^sub>0h\<^sub>1.\<phi>) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<cdot> (f \<star> \<a>[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<cdot> (f \<star> (g.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>1) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<cdot> (f \<star> f\<^sub>0gh\<^sub>1.\<phi>) \<cdot> \<a>[f, tab\<^sub>0 f, f\<^sub>0gh\<^sub>1.p\<^sub>1] \<cdot> (f.tab \<star> f\<^sub>0gh\<^sub>1.p\<^sub>1))
seq (f \<star> (g \<star> \<a>[h, tab\<^sub>0 h, g\<^sub>0h\<^sub>1.p\<^sub>0]) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) ((f \<star> (g \<star> h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<cdot> (f \<star> (g \<star> g\<^sub>0h\<^sub>1.\<phi>) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<cdot> (f \<star> \<a>[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<cdot> (f \<star> (g.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>1) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<cdot> (f \<star> f\<^sub>0gh\<^sub>1.\<phi>) \<cdot> \<a>[f, tab\<^sub>0 f, f\<^sub>0gh\<^sub>1.p\<^sub>1] \<cdot> (f.tab \<star> f\<^sub>0gh\<^sub>1.p\<^sub>1))
seq (f \<star> (g \<star> h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) ((f \<star> (g \<star> g\<^sub>0h\<^sub>1.\<phi>) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<cdot> (f \<star> \<a>[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<cdot> (f \<star> (g.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>1) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<cdot> (f \<star> f\<^sub>0gh\<^sub>1.\<phi>) \<cdot> \<a>[f, tab\<^sub>0 f, f\<^sub>0gh\<^sub>1.p\<^sub>1] \<cdot> (f.tab \<star> f\<^sub>0gh\<^sub>1.p\<^sub>1))
seq (f \<star> (g \<star> g\<^sub>0h\<^sub>1.\<phi>) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) ((f \<star> \<a>[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<cdot> (f \<star> (g.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>1) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<cdot> (f \<star> f\<^sub>0gh\<^sub>1.\<phi>) \<cdot> \<a>[f, tab\<^sub>0 f, f\<^sub>0gh\<^sub>1.p\<^sub>1] \<cdot> (f.tab \<star> f\<^sub>0gh\<^sub>1.p\<^sub>1))
seq (f \<star> \<a>[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) ((f \<star> (g.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>1) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<cdot> (f \<star> f\<^sub>0gh\<^sub>1.\<phi>) \<cdot> \<a>[f, tab\<^sub>0 f, f\<^sub>0gh\<^sub>1.p\<^sub>1] \<cdot> (f.tab \<star> f\<^sub>0gh\<^sub>1.p\<^sub>1))
seq (f \<star> (g.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>1) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) ((f \<star> f\<^sub>0gh\<^sub>1.\<phi>) \<cdot> \<a>[f, tab\<^sub>0 f, f\<^sub>0gh\<^sub>1.p\<^sub>1] \<cdot> (f.tab \<star> f\<^sub>0gh\<^sub>1.p\<^sub>1))
goal (1 subgoal):
1. seq (f \<star> f\<^sub>0gh\<^sub>1.\<phi>) (\<a>[f, tab\<^sub>0 f, f\<^sub>0gh\<^sub>1.p\<^sub>1] \<cdot> (f.tab \<star> f\<^sub>0gh\<^sub>1.p\<^sub>1))
[PROOF STEP]
by blast
[PROOF STATE]
proof (state)
this:
seq (f \<star> f\<^sub>0gh\<^sub>1.\<phi>) (\<a>[f, tab\<^sub>0 f, f\<^sub>0gh\<^sub>1.p\<^sub>1] \<cdot> (f.tab \<star> f\<^sub>0gh\<^sub>1.p\<^sub>1))
goal (1 subgoal):
1. \<a>\<^sup>-\<^sup>1[f, g \<star> h, (tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0] \<cdot> (f \<star> \<a>[g \<star> h, tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0, f\<^sub>0gh\<^sub>1.p\<^sub>0]) \<cdot> (f \<star> \<a>\<^sup>-\<^sup>1[g, h, tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<cdot> (f \<star> (g \<star> \<a>[h, tab\<^sub>0 h, g\<^sub>0h\<^sub>1.p\<^sub>0]) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<cdot> (f \<star> (g \<star> h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<cdot> (f \<star> (g \<star> g\<^sub>0h\<^sub>1.\<phi>) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<cdot> (f \<star> \<a>[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<cdot> (f \<star> (g.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>1) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<cdot> (f \<star> f\<^sub>0gh\<^sub>1.\<phi>) \<cdot> \<a>[f, tab\<^sub>0 f, f\<^sub>0gh\<^sub>1.p\<^sub>1] \<cdot> (f.tab \<star> f\<^sub>0gh\<^sub>1.p\<^sub>1) \<star> TTfgh_TfTgh.chine = (\<a>\<^sup>-\<^sup>1[f, g \<star> h, (tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0] \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> \<a>[g \<star> h, tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0, f\<^sub>0gh\<^sub>1.p\<^sub>0]) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> \<a>\<^sup>-\<^sup>1[g, h, tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> (g \<star> \<a>[h, tab\<^sub>0 h, g\<^sub>0h\<^sub>1.p\<^sub>0]) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> (g \<star> h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> (g \<star> g\<^sub>0h\<^sub>1.\<phi>) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> \<a>[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> (g.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>1) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> f\<^sub>0gh\<^sub>1.\<phi>) \<star> TTfgh_TfTgh.chine) \<cdot> (\<a>[f, tab\<^sub>0 f, f\<^sub>0gh\<^sub>1.p\<^sub>1] \<star> TTfgh_TfTgh.chine) \<cdot> ((f.tab \<star> f\<^sub>0gh\<^sub>1.p\<^sub>1) \<star> TTfgh_TfTgh.chine)
[PROOF STEP]
moreover
[PROOF STATE]
proof (state)
this:
seq (f \<star> f\<^sub>0gh\<^sub>1.\<phi>) (\<a>[f, tab\<^sub>0 f, f\<^sub>0gh\<^sub>1.p\<^sub>1] \<cdot> (f.tab \<star> f\<^sub>0gh\<^sub>1.p\<^sub>1))
goal (1 subgoal):
1. \<a>\<^sup>-\<^sup>1[f, g \<star> h, (tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0] \<cdot> (f \<star> \<a>[g \<star> h, tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0, f\<^sub>0gh\<^sub>1.p\<^sub>0]) \<cdot> (f \<star> \<a>\<^sup>-\<^sup>1[g, h, tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<cdot> (f \<star> (g \<star> \<a>[h, tab\<^sub>0 h, g\<^sub>0h\<^sub>1.p\<^sub>0]) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<cdot> (f \<star> (g \<star> h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<cdot> (f \<star> (g \<star> g\<^sub>0h\<^sub>1.\<phi>) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<cdot> (f \<star> \<a>[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<cdot> (f \<star> (g.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>1) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<cdot> (f \<star> f\<^sub>0gh\<^sub>1.\<phi>) \<cdot> \<a>[f, tab\<^sub>0 f, f\<^sub>0gh\<^sub>1.p\<^sub>1] \<cdot> (f.tab \<star> f\<^sub>0gh\<^sub>1.p\<^sub>1) \<star> TTfgh_TfTgh.chine = (\<a>\<^sup>-\<^sup>1[f, g \<star> h, (tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0] \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> \<a>[g \<star> h, tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0, f\<^sub>0gh\<^sub>1.p\<^sub>0]) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> \<a>\<^sup>-\<^sup>1[g, h, tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> (g \<star> \<a>[h, tab\<^sub>0 h, g\<^sub>0h\<^sub>1.p\<^sub>0]) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> (g \<star> h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> (g \<star> g\<^sub>0h\<^sub>1.\<phi>) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> \<a>[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> (g.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>1) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> f\<^sub>0gh\<^sub>1.\<phi>) \<star> TTfgh_TfTgh.chine) \<cdot> (\<a>[f, tab\<^sub>0 f, f\<^sub>0gh\<^sub>1.p\<^sub>1] \<star> TTfgh_TfTgh.chine) \<cdot> ((f.tab \<star> f\<^sub>0gh\<^sub>1.p\<^sub>1) \<star> TTfgh_TfTgh.chine)
[PROOF STEP]
have "arr (\<a>[f, tab\<^sub>0 f, TfTgh.p\<^sub>1] \<cdot>
(f.tab \<star> TfTgh.p\<^sub>1))"
[PROOF STATE]
proof (prove)
goal (1 subgoal):
1. seq \<a>[f, tab\<^sub>0 f, f\<^sub>0gh\<^sub>1.p\<^sub>1] (f.tab \<star> f\<^sub>0gh\<^sub>1.p\<^sub>1)
[PROOF STEP]
using calculation
[PROOF STATE]
proof (prove)
using this:
seq \<a>\<^sup>-\<^sup>1[f, g \<star> h, (tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0] ((f \<star> \<a>[g \<star> h, tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0, f\<^sub>0gh\<^sub>1.p\<^sub>0]) \<cdot> (f \<star> \<a>\<^sup>-\<^sup>1[g, h, tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<cdot> (f \<star> (g \<star> \<a>[h, tab\<^sub>0 h, g\<^sub>0h\<^sub>1.p\<^sub>0]) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<cdot> (f \<star> (g \<star> h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<cdot> (f \<star> (g \<star> g\<^sub>0h\<^sub>1.\<phi>) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<cdot> (f \<star> \<a>[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<cdot> (f \<star> (g.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>1) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<cdot> (f \<star> f\<^sub>0gh\<^sub>1.\<phi>) \<cdot> \<a>[f, tab\<^sub>0 f, f\<^sub>0gh\<^sub>1.p\<^sub>1] \<cdot> (f.tab \<star> f\<^sub>0gh\<^sub>1.p\<^sub>1))
seq (f \<star> \<a>[g \<star> h, tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0, f\<^sub>0gh\<^sub>1.p\<^sub>0]) ((f \<star> \<a>\<^sup>-\<^sup>1[g, h, tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<cdot> (f \<star> (g \<star> \<a>[h, tab\<^sub>0 h, g\<^sub>0h\<^sub>1.p\<^sub>0]) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<cdot> (f \<star> (g \<star> h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<cdot> (f \<star> (g \<star> g\<^sub>0h\<^sub>1.\<phi>) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<cdot> (f \<star> \<a>[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<cdot> (f \<star> (g.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>1) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<cdot> (f \<star> f\<^sub>0gh\<^sub>1.\<phi>) \<cdot> \<a>[f, tab\<^sub>0 f, f\<^sub>0gh\<^sub>1.p\<^sub>1] \<cdot> (f.tab \<star> f\<^sub>0gh\<^sub>1.p\<^sub>1))
seq (f \<star> \<a>\<^sup>-\<^sup>1[g, h, tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) ((f \<star> (g \<star> \<a>[h, tab\<^sub>0 h, g\<^sub>0h\<^sub>1.p\<^sub>0]) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<cdot> (f \<star> (g \<star> h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<cdot> (f \<star> (g \<star> g\<^sub>0h\<^sub>1.\<phi>) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<cdot> (f \<star> \<a>[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<cdot> (f \<star> (g.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>1) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<cdot> (f \<star> f\<^sub>0gh\<^sub>1.\<phi>) \<cdot> \<a>[f, tab\<^sub>0 f, f\<^sub>0gh\<^sub>1.p\<^sub>1] \<cdot> (f.tab \<star> f\<^sub>0gh\<^sub>1.p\<^sub>1))
seq (f \<star> (g \<star> \<a>[h, tab\<^sub>0 h, g\<^sub>0h\<^sub>1.p\<^sub>0]) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) ((f \<star> (g \<star> h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<cdot> (f \<star> (g \<star> g\<^sub>0h\<^sub>1.\<phi>) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<cdot> (f \<star> \<a>[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<cdot> (f \<star> (g.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>1) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<cdot> (f \<star> f\<^sub>0gh\<^sub>1.\<phi>) \<cdot> \<a>[f, tab\<^sub>0 f, f\<^sub>0gh\<^sub>1.p\<^sub>1] \<cdot> (f.tab \<star> f\<^sub>0gh\<^sub>1.p\<^sub>1))
seq (f \<star> (g \<star> h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) ((f \<star> (g \<star> g\<^sub>0h\<^sub>1.\<phi>) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<cdot> (f \<star> \<a>[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<cdot> (f \<star> (g.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>1) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<cdot> (f \<star> f\<^sub>0gh\<^sub>1.\<phi>) \<cdot> \<a>[f, tab\<^sub>0 f, f\<^sub>0gh\<^sub>1.p\<^sub>1] \<cdot> (f.tab \<star> f\<^sub>0gh\<^sub>1.p\<^sub>1))
seq (f \<star> (g \<star> g\<^sub>0h\<^sub>1.\<phi>) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) ((f \<star> \<a>[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<cdot> (f \<star> (g.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>1) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<cdot> (f \<star> f\<^sub>0gh\<^sub>1.\<phi>) \<cdot> \<a>[f, tab\<^sub>0 f, f\<^sub>0gh\<^sub>1.p\<^sub>1] \<cdot> (f.tab \<star> f\<^sub>0gh\<^sub>1.p\<^sub>1))
seq (f \<star> \<a>[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) ((f \<star> (g.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>1) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<cdot> (f \<star> f\<^sub>0gh\<^sub>1.\<phi>) \<cdot> \<a>[f, tab\<^sub>0 f, f\<^sub>0gh\<^sub>1.p\<^sub>1] \<cdot> (f.tab \<star> f\<^sub>0gh\<^sub>1.p\<^sub>1))
seq (f \<star> (g.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>1) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) ((f \<star> f\<^sub>0gh\<^sub>1.\<phi>) \<cdot> \<a>[f, tab\<^sub>0 f, f\<^sub>0gh\<^sub>1.p\<^sub>1] \<cdot> (f.tab \<star> f\<^sub>0gh\<^sub>1.p\<^sub>1))
seq (f \<star> f\<^sub>0gh\<^sub>1.\<phi>) (\<a>[f, tab\<^sub>0 f, f\<^sub>0gh\<^sub>1.p\<^sub>1] \<cdot> (f.tab \<star> f\<^sub>0gh\<^sub>1.p\<^sub>1))
goal (1 subgoal):
1. seq \<a>[f, tab\<^sub>0 f, f\<^sub>0gh\<^sub>1.p\<^sub>1] (f.tab \<star> f\<^sub>0gh\<^sub>1.p\<^sub>1)
[PROOF STEP]
by blast
[PROOF STATE]
proof (state)
this:
seq \<a>[f, tab\<^sub>0 f, f\<^sub>0gh\<^sub>1.p\<^sub>1] (f.tab \<star> f\<^sub>0gh\<^sub>1.p\<^sub>1)
goal (1 subgoal):
1. \<a>\<^sup>-\<^sup>1[f, g \<star> h, (tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0] \<cdot> (f \<star> \<a>[g \<star> h, tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0, f\<^sub>0gh\<^sub>1.p\<^sub>0]) \<cdot> (f \<star> \<a>\<^sup>-\<^sup>1[g, h, tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<cdot> (f \<star> (g \<star> \<a>[h, tab\<^sub>0 h, g\<^sub>0h\<^sub>1.p\<^sub>0]) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<cdot> (f \<star> (g \<star> h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<cdot> (f \<star> (g \<star> g\<^sub>0h\<^sub>1.\<phi>) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<cdot> (f \<star> \<a>[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<cdot> (f \<star> (g.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>1) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<cdot> (f \<star> f\<^sub>0gh\<^sub>1.\<phi>) \<cdot> \<a>[f, tab\<^sub>0 f, f\<^sub>0gh\<^sub>1.p\<^sub>1] \<cdot> (f.tab \<star> f\<^sub>0gh\<^sub>1.p\<^sub>1) \<star> TTfgh_TfTgh.chine = (\<a>\<^sup>-\<^sup>1[f, g \<star> h, (tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0] \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> \<a>[g \<star> h, tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0, f\<^sub>0gh\<^sub>1.p\<^sub>0]) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> \<a>\<^sup>-\<^sup>1[g, h, tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> (g \<star> \<a>[h, tab\<^sub>0 h, g\<^sub>0h\<^sub>1.p\<^sub>0]) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> (g \<star> h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> (g \<star> g\<^sub>0h\<^sub>1.\<phi>) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> \<a>[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> (g.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>1) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> f\<^sub>0gh\<^sub>1.\<phi>) \<star> TTfgh_TfTgh.chine) \<cdot> (\<a>[f, tab\<^sub>0 f, f\<^sub>0gh\<^sub>1.p\<^sub>1] \<star> TTfgh_TfTgh.chine) \<cdot> ((f.tab \<star> f\<^sub>0gh\<^sub>1.p\<^sub>1) \<star> TTfgh_TfTgh.chine)
[PROOF STEP]
ultimately
[PROOF STATE]
proof (chain)
picking this:
seq \<a>\<^sup>-\<^sup>1[f, g \<star> h, (tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0] ((f \<star> \<a>[g \<star> h, tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0, f\<^sub>0gh\<^sub>1.p\<^sub>0]) \<cdot> (f \<star> \<a>\<^sup>-\<^sup>1[g, h, tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<cdot> (f \<star> (g \<star> \<a>[h, tab\<^sub>0 h, g\<^sub>0h\<^sub>1.p\<^sub>0]) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<cdot> (f \<star> (g \<star> h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<cdot> (f \<star> (g \<star> g\<^sub>0h\<^sub>1.\<phi>) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<cdot> (f \<star> \<a>[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<cdot> (f \<star> (g.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>1) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<cdot> (f \<star> f\<^sub>0gh\<^sub>1.\<phi>) \<cdot> \<a>[f, tab\<^sub>0 f, f\<^sub>0gh\<^sub>1.p\<^sub>1] \<cdot> (f.tab \<star> f\<^sub>0gh\<^sub>1.p\<^sub>1))
seq (f \<star> \<a>[g \<star> h, tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0, f\<^sub>0gh\<^sub>1.p\<^sub>0]) ((f \<star> \<a>\<^sup>-\<^sup>1[g, h, tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<cdot> (f \<star> (g \<star> \<a>[h, tab\<^sub>0 h, g\<^sub>0h\<^sub>1.p\<^sub>0]) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<cdot> (f \<star> (g \<star> h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<cdot> (f \<star> (g \<star> g\<^sub>0h\<^sub>1.\<phi>) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<cdot> (f \<star> \<a>[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<cdot> (f \<star> (g.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>1) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<cdot> (f \<star> f\<^sub>0gh\<^sub>1.\<phi>) \<cdot> \<a>[f, tab\<^sub>0 f, f\<^sub>0gh\<^sub>1.p\<^sub>1] \<cdot> (f.tab \<star> f\<^sub>0gh\<^sub>1.p\<^sub>1))
seq (f \<star> \<a>\<^sup>-\<^sup>1[g, h, tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) ((f \<star> (g \<star> \<a>[h, tab\<^sub>0 h, g\<^sub>0h\<^sub>1.p\<^sub>0]) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<cdot> (f \<star> (g \<star> h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<cdot> (f \<star> (g \<star> g\<^sub>0h\<^sub>1.\<phi>) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<cdot> (f \<star> \<a>[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<cdot> (f \<star> (g.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>1) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<cdot> (f \<star> f\<^sub>0gh\<^sub>1.\<phi>) \<cdot> \<a>[f, tab\<^sub>0 f, f\<^sub>0gh\<^sub>1.p\<^sub>1] \<cdot> (f.tab \<star> f\<^sub>0gh\<^sub>1.p\<^sub>1))
seq (f \<star> (g \<star> \<a>[h, tab\<^sub>0 h, g\<^sub>0h\<^sub>1.p\<^sub>0]) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) ((f \<star> (g \<star> h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<cdot> (f \<star> (g \<star> g\<^sub>0h\<^sub>1.\<phi>) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<cdot> (f \<star> \<a>[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<cdot> (f \<star> (g.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>1) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<cdot> (f \<star> f\<^sub>0gh\<^sub>1.\<phi>) \<cdot> \<a>[f, tab\<^sub>0 f, f\<^sub>0gh\<^sub>1.p\<^sub>1] \<cdot> (f.tab \<star> f\<^sub>0gh\<^sub>1.p\<^sub>1))
seq (f \<star> (g \<star> h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) ((f \<star> (g \<star> g\<^sub>0h\<^sub>1.\<phi>) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<cdot> (f \<star> \<a>[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<cdot> (f \<star> (g.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>1) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<cdot> (f \<star> f\<^sub>0gh\<^sub>1.\<phi>) \<cdot> \<a>[f, tab\<^sub>0 f, f\<^sub>0gh\<^sub>1.p\<^sub>1] \<cdot> (f.tab \<star> f\<^sub>0gh\<^sub>1.p\<^sub>1))
seq (f \<star> (g \<star> g\<^sub>0h\<^sub>1.\<phi>) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) ((f \<star> \<a>[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<cdot> (f \<star> (g.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>1) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<cdot> (f \<star> f\<^sub>0gh\<^sub>1.\<phi>) \<cdot> \<a>[f, tab\<^sub>0 f, f\<^sub>0gh\<^sub>1.p\<^sub>1] \<cdot> (f.tab \<star> f\<^sub>0gh\<^sub>1.p\<^sub>1))
seq (f \<star> \<a>[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) ((f \<star> (g.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>1) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<cdot> (f \<star> f\<^sub>0gh\<^sub>1.\<phi>) \<cdot> \<a>[f, tab\<^sub>0 f, f\<^sub>0gh\<^sub>1.p\<^sub>1] \<cdot> (f.tab \<star> f\<^sub>0gh\<^sub>1.p\<^sub>1))
seq (f \<star> (g.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>1) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) ((f \<star> f\<^sub>0gh\<^sub>1.\<phi>) \<cdot> \<a>[f, tab\<^sub>0 f, f\<^sub>0gh\<^sub>1.p\<^sub>1] \<cdot> (f.tab \<star> f\<^sub>0gh\<^sub>1.p\<^sub>1))
seq (f \<star> f\<^sub>0gh\<^sub>1.\<phi>) (\<a>[f, tab\<^sub>0 f, f\<^sub>0gh\<^sub>1.p\<^sub>1] \<cdot> (f.tab \<star> f\<^sub>0gh\<^sub>1.p\<^sub>1))
seq \<a>[f, tab\<^sub>0 f, f\<^sub>0gh\<^sub>1.p\<^sub>1] (f.tab \<star> f\<^sub>0gh\<^sub>1.p\<^sub>1)
[PROOF STEP]
show ?thesis
[PROOF STATE]
proof (prove)
using this:
seq \<a>\<^sup>-\<^sup>1[f, g \<star> h, (tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0] ((f \<star> \<a>[g \<star> h, tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0, f\<^sub>0gh\<^sub>1.p\<^sub>0]) \<cdot> (f \<star> \<a>\<^sup>-\<^sup>1[g, h, tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<cdot> (f \<star> (g \<star> \<a>[h, tab\<^sub>0 h, g\<^sub>0h\<^sub>1.p\<^sub>0]) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<cdot> (f \<star> (g \<star> h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<cdot> (f \<star> (g \<star> g\<^sub>0h\<^sub>1.\<phi>) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<cdot> (f \<star> \<a>[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<cdot> (f \<star> (g.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>1) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<cdot> (f \<star> f\<^sub>0gh\<^sub>1.\<phi>) \<cdot> \<a>[f, tab\<^sub>0 f, f\<^sub>0gh\<^sub>1.p\<^sub>1] \<cdot> (f.tab \<star> f\<^sub>0gh\<^sub>1.p\<^sub>1))
seq (f \<star> \<a>[g \<star> h, tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0, f\<^sub>0gh\<^sub>1.p\<^sub>0]) ((f \<star> \<a>\<^sup>-\<^sup>1[g, h, tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<cdot> (f \<star> (g \<star> \<a>[h, tab\<^sub>0 h, g\<^sub>0h\<^sub>1.p\<^sub>0]) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<cdot> (f \<star> (g \<star> h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<cdot> (f \<star> (g \<star> g\<^sub>0h\<^sub>1.\<phi>) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<cdot> (f \<star> \<a>[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<cdot> (f \<star> (g.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>1) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<cdot> (f \<star> f\<^sub>0gh\<^sub>1.\<phi>) \<cdot> \<a>[f, tab\<^sub>0 f, f\<^sub>0gh\<^sub>1.p\<^sub>1] \<cdot> (f.tab \<star> f\<^sub>0gh\<^sub>1.p\<^sub>1))
seq (f \<star> \<a>\<^sup>-\<^sup>1[g, h, tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) ((f \<star> (g \<star> \<a>[h, tab\<^sub>0 h, g\<^sub>0h\<^sub>1.p\<^sub>0]) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<cdot> (f \<star> (g \<star> h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<cdot> (f \<star> (g \<star> g\<^sub>0h\<^sub>1.\<phi>) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<cdot> (f \<star> \<a>[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<cdot> (f \<star> (g.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>1) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<cdot> (f \<star> f\<^sub>0gh\<^sub>1.\<phi>) \<cdot> \<a>[f, tab\<^sub>0 f, f\<^sub>0gh\<^sub>1.p\<^sub>1] \<cdot> (f.tab \<star> f\<^sub>0gh\<^sub>1.p\<^sub>1))
seq (f \<star> (g \<star> \<a>[h, tab\<^sub>0 h, g\<^sub>0h\<^sub>1.p\<^sub>0]) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) ((f \<star> (g \<star> h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<cdot> (f \<star> (g \<star> g\<^sub>0h\<^sub>1.\<phi>) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<cdot> (f \<star> \<a>[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<cdot> (f \<star> (g.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>1) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<cdot> (f \<star> f\<^sub>0gh\<^sub>1.\<phi>) \<cdot> \<a>[f, tab\<^sub>0 f, f\<^sub>0gh\<^sub>1.p\<^sub>1] \<cdot> (f.tab \<star> f\<^sub>0gh\<^sub>1.p\<^sub>1))
seq (f \<star> (g \<star> h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) ((f \<star> (g \<star> g\<^sub>0h\<^sub>1.\<phi>) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<cdot> (f \<star> \<a>[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<cdot> (f \<star> (g.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>1) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<cdot> (f \<star> f\<^sub>0gh\<^sub>1.\<phi>) \<cdot> \<a>[f, tab\<^sub>0 f, f\<^sub>0gh\<^sub>1.p\<^sub>1] \<cdot> (f.tab \<star> f\<^sub>0gh\<^sub>1.p\<^sub>1))
seq (f \<star> (g \<star> g\<^sub>0h\<^sub>1.\<phi>) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) ((f \<star> \<a>[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<cdot> (f \<star> (g.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>1) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<cdot> (f \<star> f\<^sub>0gh\<^sub>1.\<phi>) \<cdot> \<a>[f, tab\<^sub>0 f, f\<^sub>0gh\<^sub>1.p\<^sub>1] \<cdot> (f.tab \<star> f\<^sub>0gh\<^sub>1.p\<^sub>1))
seq (f \<star> \<a>[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) ((f \<star> (g.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>1) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<cdot> (f \<star> f\<^sub>0gh\<^sub>1.\<phi>) \<cdot> \<a>[f, tab\<^sub>0 f, f\<^sub>0gh\<^sub>1.p\<^sub>1] \<cdot> (f.tab \<star> f\<^sub>0gh\<^sub>1.p\<^sub>1))
seq (f \<star> (g.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>1) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) ((f \<star> f\<^sub>0gh\<^sub>1.\<phi>) \<cdot> \<a>[f, tab\<^sub>0 f, f\<^sub>0gh\<^sub>1.p\<^sub>1] \<cdot> (f.tab \<star> f\<^sub>0gh\<^sub>1.p\<^sub>1))
seq (f \<star> f\<^sub>0gh\<^sub>1.\<phi>) (\<a>[f, tab\<^sub>0 f, f\<^sub>0gh\<^sub>1.p\<^sub>1] \<cdot> (f.tab \<star> f\<^sub>0gh\<^sub>1.p\<^sub>1))
seq \<a>[f, tab\<^sub>0 f, f\<^sub>0gh\<^sub>1.p\<^sub>1] (f.tab \<star> f\<^sub>0gh\<^sub>1.p\<^sub>1)
goal (1 subgoal):
1. \<a>\<^sup>-\<^sup>1[f, g \<star> h, (tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0] \<cdot> (f \<star> \<a>[g \<star> h, tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0, f\<^sub>0gh\<^sub>1.p\<^sub>0]) \<cdot> (f \<star> \<a>\<^sup>-\<^sup>1[g, h, tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<cdot> (f \<star> (g \<star> \<a>[h, tab\<^sub>0 h, g\<^sub>0h\<^sub>1.p\<^sub>0]) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<cdot> (f \<star> (g \<star> h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<cdot> (f \<star> (g \<star> g\<^sub>0h\<^sub>1.\<phi>) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<cdot> (f \<star> \<a>[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<cdot> (f \<star> (g.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>1) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<cdot> (f \<star> f\<^sub>0gh\<^sub>1.\<phi>) \<cdot> \<a>[f, tab\<^sub>0 f, f\<^sub>0gh\<^sub>1.p\<^sub>1] \<cdot> (f.tab \<star> f\<^sub>0gh\<^sub>1.p\<^sub>1) \<star> TTfgh_TfTgh.chine = (\<a>\<^sup>-\<^sup>1[f, g \<star> h, (tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0] \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> \<a>[g \<star> h, tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0, f\<^sub>0gh\<^sub>1.p\<^sub>0]) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> \<a>\<^sup>-\<^sup>1[g, h, tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> (g \<star> \<a>[h, tab\<^sub>0 h, g\<^sub>0h\<^sub>1.p\<^sub>0]) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> (g \<star> h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> (g \<star> g\<^sub>0h\<^sub>1.\<phi>) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> \<a>[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> (g.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>1) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> f\<^sub>0gh\<^sub>1.\<phi>) \<star> TTfgh_TfTgh.chine) \<cdot> (\<a>[f, tab\<^sub>0 f, f\<^sub>0gh\<^sub>1.p\<^sub>1] \<star> TTfgh_TfTgh.chine) \<cdot> ((f.tab \<star> f\<^sub>0gh\<^sub>1.p\<^sub>1) \<star> TTfgh_TfTgh.chine)
[PROOF STEP]
using whisker_right [of TTfgh_TfTgh.chine] TTfgh_TfTgh.is_ide
[PROOF STATE]
proof (prove)
using this:
seq \<a>\<^sup>-\<^sup>1[f, g \<star> h, (tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0] ((f \<star> \<a>[g \<star> h, tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0, f\<^sub>0gh\<^sub>1.p\<^sub>0]) \<cdot> (f \<star> \<a>\<^sup>-\<^sup>1[g, h, tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<cdot> (f \<star> (g \<star> \<a>[h, tab\<^sub>0 h, g\<^sub>0h\<^sub>1.p\<^sub>0]) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<cdot> (f \<star> (g \<star> h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<cdot> (f \<star> (g \<star> g\<^sub>0h\<^sub>1.\<phi>) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<cdot> (f \<star> \<a>[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<cdot> (f \<star> (g.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>1) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<cdot> (f \<star> f\<^sub>0gh\<^sub>1.\<phi>) \<cdot> \<a>[f, tab\<^sub>0 f, f\<^sub>0gh\<^sub>1.p\<^sub>1] \<cdot> (f.tab \<star> f\<^sub>0gh\<^sub>1.p\<^sub>1))
seq (f \<star> \<a>[g \<star> h, tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0, f\<^sub>0gh\<^sub>1.p\<^sub>0]) ((f \<star> \<a>\<^sup>-\<^sup>1[g, h, tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<cdot> (f \<star> (g \<star> \<a>[h, tab\<^sub>0 h, g\<^sub>0h\<^sub>1.p\<^sub>0]) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<cdot> (f \<star> (g \<star> h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<cdot> (f \<star> (g \<star> g\<^sub>0h\<^sub>1.\<phi>) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<cdot> (f \<star> \<a>[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<cdot> (f \<star> (g.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>1) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<cdot> (f \<star> f\<^sub>0gh\<^sub>1.\<phi>) \<cdot> \<a>[f, tab\<^sub>0 f, f\<^sub>0gh\<^sub>1.p\<^sub>1] \<cdot> (f.tab \<star> f\<^sub>0gh\<^sub>1.p\<^sub>1))
seq (f \<star> \<a>\<^sup>-\<^sup>1[g, h, tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) ((f \<star> (g \<star> \<a>[h, tab\<^sub>0 h, g\<^sub>0h\<^sub>1.p\<^sub>0]) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<cdot> (f \<star> (g \<star> h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<cdot> (f \<star> (g \<star> g\<^sub>0h\<^sub>1.\<phi>) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<cdot> (f \<star> \<a>[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<cdot> (f \<star> (g.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>1) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<cdot> (f \<star> f\<^sub>0gh\<^sub>1.\<phi>) \<cdot> \<a>[f, tab\<^sub>0 f, f\<^sub>0gh\<^sub>1.p\<^sub>1] \<cdot> (f.tab \<star> f\<^sub>0gh\<^sub>1.p\<^sub>1))
seq (f \<star> (g \<star> \<a>[h, tab\<^sub>0 h, g\<^sub>0h\<^sub>1.p\<^sub>0]) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) ((f \<star> (g \<star> h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<cdot> (f \<star> (g \<star> g\<^sub>0h\<^sub>1.\<phi>) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<cdot> (f \<star> \<a>[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<cdot> (f \<star> (g.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>1) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<cdot> (f \<star> f\<^sub>0gh\<^sub>1.\<phi>) \<cdot> \<a>[f, tab\<^sub>0 f, f\<^sub>0gh\<^sub>1.p\<^sub>1] \<cdot> (f.tab \<star> f\<^sub>0gh\<^sub>1.p\<^sub>1))
seq (f \<star> (g \<star> h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) ((f \<star> (g \<star> g\<^sub>0h\<^sub>1.\<phi>) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<cdot> (f \<star> \<a>[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<cdot> (f \<star> (g.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>1) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<cdot> (f \<star> f\<^sub>0gh\<^sub>1.\<phi>) \<cdot> \<a>[f, tab\<^sub>0 f, f\<^sub>0gh\<^sub>1.p\<^sub>1] \<cdot> (f.tab \<star> f\<^sub>0gh\<^sub>1.p\<^sub>1))
seq (f \<star> (g \<star> g\<^sub>0h\<^sub>1.\<phi>) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) ((f \<star> \<a>[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<cdot> (f \<star> (g.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>1) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<cdot> (f \<star> f\<^sub>0gh\<^sub>1.\<phi>) \<cdot> \<a>[f, tab\<^sub>0 f, f\<^sub>0gh\<^sub>1.p\<^sub>1] \<cdot> (f.tab \<star> f\<^sub>0gh\<^sub>1.p\<^sub>1))
seq (f \<star> \<a>[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) ((f \<star> (g.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>1) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<cdot> (f \<star> f\<^sub>0gh\<^sub>1.\<phi>) \<cdot> \<a>[f, tab\<^sub>0 f, f\<^sub>0gh\<^sub>1.p\<^sub>1] \<cdot> (f.tab \<star> f\<^sub>0gh\<^sub>1.p\<^sub>1))
seq (f \<star> (g.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>1) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) ((f \<star> f\<^sub>0gh\<^sub>1.\<phi>) \<cdot> \<a>[f, tab\<^sub>0 f, f\<^sub>0gh\<^sub>1.p\<^sub>1] \<cdot> (f.tab \<star> f\<^sub>0gh\<^sub>1.p\<^sub>1))
seq (f \<star> f\<^sub>0gh\<^sub>1.\<phi>) (\<a>[f, tab\<^sub>0 f, f\<^sub>0gh\<^sub>1.p\<^sub>1] \<cdot> (f.tab \<star> f\<^sub>0gh\<^sub>1.p\<^sub>1))
seq \<a>[f, tab\<^sub>0 f, f\<^sub>0gh\<^sub>1.p\<^sub>1] (f.tab \<star> f\<^sub>0gh\<^sub>1.p\<^sub>1)
\<lbrakk>ide TTfgh_TfTgh.chine; seq ?\<nu> ?\<mu>\<rbrakk> \<Longrightarrow> ?\<nu> \<cdot> ?\<mu> \<star> TTfgh_TfTgh.chine = (?\<nu> \<star> TTfgh_TfTgh.chine) \<cdot> (?\<mu> \<star> TTfgh_TfTgh.chine)
ide TTfgh_TfTgh.chine
goal (1 subgoal):
1. \<a>\<^sup>-\<^sup>1[f, g \<star> h, (tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0] \<cdot> (f \<star> \<a>[g \<star> h, tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0, f\<^sub>0gh\<^sub>1.p\<^sub>0]) \<cdot> (f \<star> \<a>\<^sup>-\<^sup>1[g, h, tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<cdot> (f \<star> (g \<star> \<a>[h, tab\<^sub>0 h, g\<^sub>0h\<^sub>1.p\<^sub>0]) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<cdot> (f \<star> (g \<star> h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<cdot> (f \<star> (g \<star> g\<^sub>0h\<^sub>1.\<phi>) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<cdot> (f \<star> \<a>[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<cdot> (f \<star> (g.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>1) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<cdot> (f \<star> f\<^sub>0gh\<^sub>1.\<phi>) \<cdot> \<a>[f, tab\<^sub>0 f, f\<^sub>0gh\<^sub>1.p\<^sub>1] \<cdot> (f.tab \<star> f\<^sub>0gh\<^sub>1.p\<^sub>1) \<star> TTfgh_TfTgh.chine = (\<a>\<^sup>-\<^sup>1[f, g \<star> h, (tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0] \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> \<a>[g \<star> h, tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0, f\<^sub>0gh\<^sub>1.p\<^sub>0]) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> \<a>\<^sup>-\<^sup>1[g, h, tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> (g \<star> \<a>[h, tab\<^sub>0 h, g\<^sub>0h\<^sub>1.p\<^sub>0]) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> (g \<star> h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> (g \<star> g\<^sub>0h\<^sub>1.\<phi>) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> \<a>[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> (g.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>1) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> f\<^sub>0gh\<^sub>1.\<phi>) \<star> TTfgh_TfTgh.chine) \<cdot> (\<a>[f, tab\<^sub>0 f, f\<^sub>0gh\<^sub>1.p\<^sub>1] \<star> TTfgh_TfTgh.chine) \<cdot> ((f.tab \<star> f\<^sub>0gh\<^sub>1.p\<^sub>1) \<star> TTfgh_TfTgh.chine)
[PROOF STEP]
by presburger
[PROOF STATE]
proof (state)
this:
\<a>\<^sup>-\<^sup>1[f, g \<star> h, (tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0] \<cdot> (f \<star> \<a>[g \<star> h, tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0, f\<^sub>0gh\<^sub>1.p\<^sub>0]) \<cdot> (f \<star> \<a>\<^sup>-\<^sup>1[g, h, tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<cdot> (f \<star> (g \<star> \<a>[h, tab\<^sub>0 h, g\<^sub>0h\<^sub>1.p\<^sub>0]) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<cdot> (f \<star> (g \<star> h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<cdot> (f \<star> (g \<star> g\<^sub>0h\<^sub>1.\<phi>) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<cdot> (f \<star> \<a>[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<cdot> (f \<star> (g.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>1) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<cdot> (f \<star> f\<^sub>0gh\<^sub>1.\<phi>) \<cdot> \<a>[f, tab\<^sub>0 f, f\<^sub>0gh\<^sub>1.p\<^sub>1] \<cdot> (f.tab \<star> f\<^sub>0gh\<^sub>1.p\<^sub>1) \<star> TTfgh_TfTgh.chine = (\<a>\<^sup>-\<^sup>1[f, g \<star> h, (tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0] \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> \<a>[g \<star> h, tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0, f\<^sub>0gh\<^sub>1.p\<^sub>0]) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> \<a>\<^sup>-\<^sup>1[g, h, tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> (g \<star> \<a>[h, tab\<^sub>0 h, g\<^sub>0h\<^sub>1.p\<^sub>0]) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> (g \<star> h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> (g \<star> g\<^sub>0h\<^sub>1.\<phi>) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> \<a>[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> (g.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>1) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> f\<^sub>0gh\<^sub>1.\<phi>) \<star> TTfgh_TfTgh.chine) \<cdot> (\<a>[f, tab\<^sub>0 f, f\<^sub>0gh\<^sub>1.p\<^sub>1] \<star> TTfgh_TfTgh.chine) \<cdot> ((f.tab \<star> f\<^sub>0gh\<^sub>1.p\<^sub>1) \<star> TTfgh_TfTgh.chine)
goal:
No subgoals!
[PROOF STEP]
qed
[PROOF STATE]
proof (state)
this:
\<a>\<^sup>-\<^sup>1[f, g \<star> h, (tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0] \<cdot> (f \<star> \<a>[g \<star> h, tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0, f\<^sub>0gh\<^sub>1.p\<^sub>0]) \<cdot> (f \<star> \<a>\<^sup>-\<^sup>1[g, h, tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<cdot> (f \<star> (g \<star> \<a>[h, tab\<^sub>0 h, g\<^sub>0h\<^sub>1.p\<^sub>0]) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<cdot> (f \<star> (g \<star> h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<cdot> (f \<star> (g \<star> g\<^sub>0h\<^sub>1.\<phi>) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<cdot> (f \<star> \<a>[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<cdot> (f \<star> (g.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>1) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<cdot> (f \<star> f\<^sub>0gh\<^sub>1.\<phi>) \<cdot> \<a>[f, tab\<^sub>0 f, f\<^sub>0gh\<^sub>1.p\<^sub>1] \<cdot> (f.tab \<star> f\<^sub>0gh\<^sub>1.p\<^sub>1) \<star> TTfgh_TfTgh.chine = (\<a>\<^sup>-\<^sup>1[f, g \<star> h, (tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0] \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> \<a>[g \<star> h, tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0, f\<^sub>0gh\<^sub>1.p\<^sub>0]) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> \<a>\<^sup>-\<^sup>1[g, h, tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> (g \<star> \<a>[h, tab\<^sub>0 h, g\<^sub>0h\<^sub>1.p\<^sub>0]) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> (g \<star> h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> (g \<star> g\<^sub>0h\<^sub>1.\<phi>) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> \<a>[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> (g.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>1) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> f\<^sub>0gh\<^sub>1.\<phi>) \<star> TTfgh_TfTgh.chine) \<cdot> (\<a>[f, tab\<^sub>0 f, f\<^sub>0gh\<^sub>1.p\<^sub>1] \<star> TTfgh_TfTgh.chine) \<cdot> ((f.tab \<star> f\<^sub>0gh\<^sub>1.p\<^sub>1) \<star> TTfgh_TfTgh.chine)
goal (1 subgoal):
1. \<a>[f, g, h \<star> tab\<^sub>0 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0] \<cdot> \<a>[f \<star> g, h, tab\<^sub>0 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0] \<cdot> ((\<a>\<^sup>-\<^sup>1[f, g, h] \<star> tab\<^sub>0 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0) \<cdot> ((f \<star> g \<star> h) \<star> TTfgh_TfTgh.the_\<theta>) \<cdot> \<a>[f \<star> g \<star> h, (tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine] \<cdot> (\<a>\<^sup>-\<^sup>1[f, g \<star> h, (tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0] \<cdot> (f \<star> \<a>[g \<star> h, tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0, f\<^sub>0gh\<^sub>1.p\<^sub>0]) \<cdot> ((f \<star> \<a>\<^sup>-\<^sup>1[g, h, tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<cdot> (f \<star> (g \<star> \<a>[h, tab\<^sub>0 h, g\<^sub>0h\<^sub>1.p\<^sub>0]) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<cdot> (f \<star> (g \<star> h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<cdot> (f \<star> (g \<star> g\<^sub>0h\<^sub>1.\<phi>) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<cdot> (f \<star> \<a>[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<cdot> (f \<star> (g.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>1) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<cdot> (f \<star> f\<^sub>0gh\<^sub>1.\<phi>)) \<cdot> \<a>[f, tab\<^sub>0 f, f\<^sub>0gh\<^sub>1.p\<^sub>1] \<cdot> (f.tab \<star> f\<^sub>0gh\<^sub>1.p\<^sub>1) \<star> TTfgh_TfTgh.chine) \<cdot> TTfgh_TfTgh.the_\<nu>) \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>1 f, f\<^sub>0g\<^sub>1.p\<^sub>1, fg\<^sub>0h\<^sub>1.p\<^sub>1] = \<a>[f, g, h \<star> tab\<^sub>0 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0] \<cdot> \<a>[f \<star> g, h, tab\<^sub>0 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0] \<cdot> (\<a>\<^sup>-\<^sup>1[f, g, h] \<star> tab\<^sub>0 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0) \<cdot> ((f \<star> g \<star> h) \<star> TTfgh_TfTgh.the_\<theta>) \<cdot> \<a>[f \<star> g \<star> h, (tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine] \<cdot> (\<a>\<^sup>-\<^sup>1[f, g \<star> h, (tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0] \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> \<a>[g \<star> h, tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0, f\<^sub>0gh\<^sub>1.p\<^sub>0]) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> \<a>\<^sup>-\<^sup>1[g, h, tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> (g \<star> \<a>[h, tab\<^sub>0 h, g\<^sub>0h\<^sub>1.p\<^sub>0]) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> (g \<star> h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> (g \<star> g\<^sub>0h\<^sub>1.\<phi>) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> \<a>[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> (g.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>1) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> f\<^sub>0gh\<^sub>1.\<phi>) \<star> TTfgh_TfTgh.chine) \<cdot> (\<a>[f, tab\<^sub>0 f, f\<^sub>0gh\<^sub>1.p\<^sub>1] \<star> TTfgh_TfTgh.chine) \<cdot> ((f.tab \<star> f\<^sub>0gh\<^sub>1.p\<^sub>1) \<star> TTfgh_TfTgh.chine) \<cdot> TTfgh_TfTgh.the_\<nu> \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>1 f, f\<^sub>0g\<^sub>1.p\<^sub>1, fg\<^sub>0h\<^sub>1.p\<^sub>1]
[PROOF STEP]
thus ?thesis
[PROOF STATE]
proof (prove)
using this:
\<a>\<^sup>-\<^sup>1[f, g \<star> h, (tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0] \<cdot> (f \<star> \<a>[g \<star> h, tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0, f\<^sub>0gh\<^sub>1.p\<^sub>0]) \<cdot> (f \<star> \<a>\<^sup>-\<^sup>1[g, h, tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<cdot> (f \<star> (g \<star> \<a>[h, tab\<^sub>0 h, g\<^sub>0h\<^sub>1.p\<^sub>0]) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<cdot> (f \<star> (g \<star> h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<cdot> (f \<star> (g \<star> g\<^sub>0h\<^sub>1.\<phi>) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<cdot> (f \<star> \<a>[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<cdot> (f \<star> (g.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>1) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<cdot> (f \<star> f\<^sub>0gh\<^sub>1.\<phi>) \<cdot> \<a>[f, tab\<^sub>0 f, f\<^sub>0gh\<^sub>1.p\<^sub>1] \<cdot> (f.tab \<star> f\<^sub>0gh\<^sub>1.p\<^sub>1) \<star> TTfgh_TfTgh.chine = (\<a>\<^sup>-\<^sup>1[f, g \<star> h, (tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0] \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> \<a>[g \<star> h, tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0, f\<^sub>0gh\<^sub>1.p\<^sub>0]) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> \<a>\<^sup>-\<^sup>1[g, h, tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> (g \<star> \<a>[h, tab\<^sub>0 h, g\<^sub>0h\<^sub>1.p\<^sub>0]) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> (g \<star> h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> (g \<star> g\<^sub>0h\<^sub>1.\<phi>) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> \<a>[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> (g.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>1) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> f\<^sub>0gh\<^sub>1.\<phi>) \<star> TTfgh_TfTgh.chine) \<cdot> (\<a>[f, tab\<^sub>0 f, f\<^sub>0gh\<^sub>1.p\<^sub>1] \<star> TTfgh_TfTgh.chine) \<cdot> ((f.tab \<star> f\<^sub>0gh\<^sub>1.p\<^sub>1) \<star> TTfgh_TfTgh.chine)
goal (1 subgoal):
1. \<a>[f, g, h \<star> tab\<^sub>0 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0] \<cdot> \<a>[f \<star> g, h, tab\<^sub>0 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0] \<cdot> ((\<a>\<^sup>-\<^sup>1[f, g, h] \<star> tab\<^sub>0 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0) \<cdot> ((f \<star> g \<star> h) \<star> TTfgh_TfTgh.the_\<theta>) \<cdot> \<a>[f \<star> g \<star> h, (tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine] \<cdot> (\<a>\<^sup>-\<^sup>1[f, g \<star> h, (tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0] \<cdot> (f \<star> \<a>[g \<star> h, tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0, f\<^sub>0gh\<^sub>1.p\<^sub>0]) \<cdot> ((f \<star> \<a>\<^sup>-\<^sup>1[g, h, tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<cdot> (f \<star> (g \<star> \<a>[h, tab\<^sub>0 h, g\<^sub>0h\<^sub>1.p\<^sub>0]) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<cdot> (f \<star> (g \<star> h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<cdot> (f \<star> (g \<star> g\<^sub>0h\<^sub>1.\<phi>) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<cdot> (f \<star> \<a>[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<cdot> (f \<star> (g.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>1) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<cdot> (f \<star> f\<^sub>0gh\<^sub>1.\<phi>)) \<cdot> \<a>[f, tab\<^sub>0 f, f\<^sub>0gh\<^sub>1.p\<^sub>1] \<cdot> (f.tab \<star> f\<^sub>0gh\<^sub>1.p\<^sub>1) \<star> TTfgh_TfTgh.chine) \<cdot> TTfgh_TfTgh.the_\<nu>) \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>1 f, f\<^sub>0g\<^sub>1.p\<^sub>1, fg\<^sub>0h\<^sub>1.p\<^sub>1] = \<a>[f, g, h \<star> tab\<^sub>0 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0] \<cdot> \<a>[f \<star> g, h, tab\<^sub>0 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0] \<cdot> (\<a>\<^sup>-\<^sup>1[f, g, h] \<star> tab\<^sub>0 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0) \<cdot> ((f \<star> g \<star> h) \<star> TTfgh_TfTgh.the_\<theta>) \<cdot> \<a>[f \<star> g \<star> h, (tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine] \<cdot> (\<a>\<^sup>-\<^sup>1[f, g \<star> h, (tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0] \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> \<a>[g \<star> h, tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0, f\<^sub>0gh\<^sub>1.p\<^sub>0]) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> \<a>\<^sup>-\<^sup>1[g, h, tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> (g \<star> \<a>[h, tab\<^sub>0 h, g\<^sub>0h\<^sub>1.p\<^sub>0]) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> (g \<star> h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> (g \<star> g\<^sub>0h\<^sub>1.\<phi>) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> \<a>[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> (g.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>1) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> f\<^sub>0gh\<^sub>1.\<phi>) \<star> TTfgh_TfTgh.chine) \<cdot> (\<a>[f, tab\<^sub>0 f, f\<^sub>0gh\<^sub>1.p\<^sub>1] \<star> TTfgh_TfTgh.chine) \<cdot> ((f.tab \<star> f\<^sub>0gh\<^sub>1.p\<^sub>1) \<star> TTfgh_TfTgh.chine) \<cdot> TTfgh_TfTgh.the_\<nu> \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>1 f, f\<^sub>0g\<^sub>1.p\<^sub>1, fg\<^sub>0h\<^sub>1.p\<^sub>1]
[PROOF STEP]
using comp_assoc
[PROOF STATE]
proof (prove)
using this:
\<a>\<^sup>-\<^sup>1[f, g \<star> h, (tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0] \<cdot> (f \<star> \<a>[g \<star> h, tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0, f\<^sub>0gh\<^sub>1.p\<^sub>0]) \<cdot> (f \<star> \<a>\<^sup>-\<^sup>1[g, h, tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<cdot> (f \<star> (g \<star> \<a>[h, tab\<^sub>0 h, g\<^sub>0h\<^sub>1.p\<^sub>0]) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<cdot> (f \<star> (g \<star> h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<cdot> (f \<star> (g \<star> g\<^sub>0h\<^sub>1.\<phi>) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<cdot> (f \<star> \<a>[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<cdot> (f \<star> (g.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>1) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<cdot> (f \<star> f\<^sub>0gh\<^sub>1.\<phi>) \<cdot> \<a>[f, tab\<^sub>0 f, f\<^sub>0gh\<^sub>1.p\<^sub>1] \<cdot> (f.tab \<star> f\<^sub>0gh\<^sub>1.p\<^sub>1) \<star> TTfgh_TfTgh.chine = (\<a>\<^sup>-\<^sup>1[f, g \<star> h, (tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0] \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> \<a>[g \<star> h, tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0, f\<^sub>0gh\<^sub>1.p\<^sub>0]) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> \<a>\<^sup>-\<^sup>1[g, h, tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> (g \<star> \<a>[h, tab\<^sub>0 h, g\<^sub>0h\<^sub>1.p\<^sub>0]) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> (g \<star> h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> (g \<star> g\<^sub>0h\<^sub>1.\<phi>) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> \<a>[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> (g.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>1) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> f\<^sub>0gh\<^sub>1.\<phi>) \<star> TTfgh_TfTgh.chine) \<cdot> (\<a>[f, tab\<^sub>0 f, f\<^sub>0gh\<^sub>1.p\<^sub>1] \<star> TTfgh_TfTgh.chine) \<cdot> ((f.tab \<star> f\<^sub>0gh\<^sub>1.p\<^sub>1) \<star> TTfgh_TfTgh.chine)
(?h \<cdot> ?g) \<cdot> ?f = ?h \<cdot> ?g \<cdot> ?f
goal (1 subgoal):
1. \<a>[f, g, h \<star> tab\<^sub>0 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0] \<cdot> \<a>[f \<star> g, h, tab\<^sub>0 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0] \<cdot> ((\<a>\<^sup>-\<^sup>1[f, g, h] \<star> tab\<^sub>0 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0) \<cdot> ((f \<star> g \<star> h) \<star> TTfgh_TfTgh.the_\<theta>) \<cdot> \<a>[f \<star> g \<star> h, (tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine] \<cdot> (\<a>\<^sup>-\<^sup>1[f, g \<star> h, (tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0] \<cdot> (f \<star> \<a>[g \<star> h, tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0, f\<^sub>0gh\<^sub>1.p\<^sub>0]) \<cdot> ((f \<star> \<a>\<^sup>-\<^sup>1[g, h, tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<cdot> (f \<star> (g \<star> \<a>[h, tab\<^sub>0 h, g\<^sub>0h\<^sub>1.p\<^sub>0]) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<cdot> (f \<star> (g \<star> h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<cdot> (f \<star> (g \<star> g\<^sub>0h\<^sub>1.\<phi>) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<cdot> (f \<star> \<a>[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<cdot> (f \<star> (g.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>1) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<cdot> (f \<star> f\<^sub>0gh\<^sub>1.\<phi>)) \<cdot> \<a>[f, tab\<^sub>0 f, f\<^sub>0gh\<^sub>1.p\<^sub>1] \<cdot> (f.tab \<star> f\<^sub>0gh\<^sub>1.p\<^sub>1) \<star> TTfgh_TfTgh.chine) \<cdot> TTfgh_TfTgh.the_\<nu>) \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>1 f, f\<^sub>0g\<^sub>1.p\<^sub>1, fg\<^sub>0h\<^sub>1.p\<^sub>1] = \<a>[f, g, h \<star> tab\<^sub>0 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0] \<cdot> \<a>[f \<star> g, h, tab\<^sub>0 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0] \<cdot> (\<a>\<^sup>-\<^sup>1[f, g, h] \<star> tab\<^sub>0 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0) \<cdot> ((f \<star> g \<star> h) \<star> TTfgh_TfTgh.the_\<theta>) \<cdot> \<a>[f \<star> g \<star> h, (tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine] \<cdot> (\<a>\<^sup>-\<^sup>1[f, g \<star> h, (tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0] \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> \<a>[g \<star> h, tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0, f\<^sub>0gh\<^sub>1.p\<^sub>0]) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> \<a>\<^sup>-\<^sup>1[g, h, tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> (g \<star> \<a>[h, tab\<^sub>0 h, g\<^sub>0h\<^sub>1.p\<^sub>0]) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> (g \<star> h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> (g \<star> g\<^sub>0h\<^sub>1.\<phi>) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> \<a>[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> (g.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>1) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> f\<^sub>0gh\<^sub>1.\<phi>) \<star> TTfgh_TfTgh.chine) \<cdot> (\<a>[f, tab\<^sub>0 f, f\<^sub>0gh\<^sub>1.p\<^sub>1] \<star> TTfgh_TfTgh.chine) \<cdot> ((f.tab \<star> f\<^sub>0gh\<^sub>1.p\<^sub>1) \<star> TTfgh_TfTgh.chine) \<cdot> TTfgh_TfTgh.the_\<nu> \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>1 f, f\<^sub>0g\<^sub>1.p\<^sub>1, fg\<^sub>0h\<^sub>1.p\<^sub>1]
[PROOF STEP]
by presburger
[PROOF STATE]
proof (state)
this:
\<a>[f, g, h \<star> tab\<^sub>0 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0] \<cdot> \<a>[f \<star> g, h, tab\<^sub>0 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0] \<cdot> ((\<a>\<^sup>-\<^sup>1[f, g, h] \<star> tab\<^sub>0 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0) \<cdot> ((f \<star> g \<star> h) \<star> TTfgh_TfTgh.the_\<theta>) \<cdot> \<a>[f \<star> g \<star> h, (tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine] \<cdot> (\<a>\<^sup>-\<^sup>1[f, g \<star> h, (tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0] \<cdot> (f \<star> \<a>[g \<star> h, tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0, f\<^sub>0gh\<^sub>1.p\<^sub>0]) \<cdot> ((f \<star> \<a>\<^sup>-\<^sup>1[g, h, tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<cdot> (f \<star> (g \<star> \<a>[h, tab\<^sub>0 h, g\<^sub>0h\<^sub>1.p\<^sub>0]) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<cdot> (f \<star> (g \<star> h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<cdot> (f \<star> (g \<star> g\<^sub>0h\<^sub>1.\<phi>) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<cdot> (f \<star> \<a>[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<cdot> (f \<star> (g.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>1) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<cdot> (f \<star> f\<^sub>0gh\<^sub>1.\<phi>)) \<cdot> \<a>[f, tab\<^sub>0 f, f\<^sub>0gh\<^sub>1.p\<^sub>1] \<cdot> (f.tab \<star> f\<^sub>0gh\<^sub>1.p\<^sub>1) \<star> TTfgh_TfTgh.chine) \<cdot> TTfgh_TfTgh.the_\<nu>) \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>1 f, f\<^sub>0g\<^sub>1.p\<^sub>1, fg\<^sub>0h\<^sub>1.p\<^sub>1] = \<a>[f, g, h \<star> tab\<^sub>0 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0] \<cdot> \<a>[f \<star> g, h, tab\<^sub>0 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0] \<cdot> (\<a>\<^sup>-\<^sup>1[f, g, h] \<star> tab\<^sub>0 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0) \<cdot> ((f \<star> g \<star> h) \<star> TTfgh_TfTgh.the_\<theta>) \<cdot> \<a>[f \<star> g \<star> h, (tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine] \<cdot> (\<a>\<^sup>-\<^sup>1[f, g \<star> h, (tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0] \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> \<a>[g \<star> h, tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0, f\<^sub>0gh\<^sub>1.p\<^sub>0]) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> \<a>\<^sup>-\<^sup>1[g, h, tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> (g \<star> \<a>[h, tab\<^sub>0 h, g\<^sub>0h\<^sub>1.p\<^sub>0]) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> (g \<star> h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> (g \<star> g\<^sub>0h\<^sub>1.\<phi>) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> \<a>[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> (g.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>1) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> f\<^sub>0gh\<^sub>1.\<phi>) \<star> TTfgh_TfTgh.chine) \<cdot> (\<a>[f, tab\<^sub>0 f, f\<^sub>0gh\<^sub>1.p\<^sub>1] \<star> TTfgh_TfTgh.chine) \<cdot> ((f.tab \<star> f\<^sub>0gh\<^sub>1.p\<^sub>1) \<star> TTfgh_TfTgh.chine) \<cdot> TTfgh_TfTgh.the_\<nu> \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>1 f, f\<^sub>0g\<^sub>1.p\<^sub>1, fg\<^sub>0h\<^sub>1.p\<^sub>1]
goal:
No subgoals!
[PROOF STEP]
qed
[PROOF STATE]
proof (state)
this:
\<a>[f, g, h \<star> tab\<^sub>0 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0] \<cdot> \<a>[f \<star> g, h, tab\<^sub>0 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0] \<cdot> ((\<a>\<^sup>-\<^sup>1[f, g, h] \<star> tab\<^sub>0 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0) \<cdot> ((f \<star> g \<star> h) \<star> TTfgh_TfTgh.the_\<theta>) \<cdot> \<a>[f \<star> g \<star> h, (tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine] \<cdot> (\<a>\<^sup>-\<^sup>1[f, g \<star> h, (tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0] \<cdot> (f \<star> \<a>[g \<star> h, tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0, f\<^sub>0gh\<^sub>1.p\<^sub>0]) \<cdot> ((f \<star> \<a>\<^sup>-\<^sup>1[g, h, tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<cdot> (f \<star> (g \<star> \<a>[h, tab\<^sub>0 h, g\<^sub>0h\<^sub>1.p\<^sub>0]) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<cdot> (f \<star> (g \<star> h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<cdot> (f \<star> (g \<star> g\<^sub>0h\<^sub>1.\<phi>) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<cdot> (f \<star> \<a>[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<cdot> (f \<star> (g.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>1) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<cdot> (f \<star> f\<^sub>0gh\<^sub>1.\<phi>)) \<cdot> \<a>[f, tab\<^sub>0 f, f\<^sub>0gh\<^sub>1.p\<^sub>1] \<cdot> (f.tab \<star> f\<^sub>0gh\<^sub>1.p\<^sub>1) \<star> TTfgh_TfTgh.chine) \<cdot> TTfgh_TfTgh.the_\<nu>) \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>1 f, f\<^sub>0g\<^sub>1.p\<^sub>1, fg\<^sub>0h\<^sub>1.p\<^sub>1] = \<a>[f, g, h \<star> tab\<^sub>0 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0] \<cdot> \<a>[f \<star> g, h, tab\<^sub>0 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0] \<cdot> (\<a>\<^sup>-\<^sup>1[f, g, h] \<star> tab\<^sub>0 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0) \<cdot> ((f \<star> g \<star> h) \<star> TTfgh_TfTgh.the_\<theta>) \<cdot> \<a>[f \<star> g \<star> h, (tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine] \<cdot> (\<a>\<^sup>-\<^sup>1[f, g \<star> h, (tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0] \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> \<a>[g \<star> h, tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0, f\<^sub>0gh\<^sub>1.p\<^sub>0]) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> \<a>\<^sup>-\<^sup>1[g, h, tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> (g \<star> \<a>[h, tab\<^sub>0 h, g\<^sub>0h\<^sub>1.p\<^sub>0]) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> (g \<star> h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> (g \<star> g\<^sub>0h\<^sub>1.\<phi>) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> \<a>[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> (g.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>1) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> f\<^sub>0gh\<^sub>1.\<phi>) \<star> TTfgh_TfTgh.chine) \<cdot> (\<a>[f, tab\<^sub>0 f, f\<^sub>0gh\<^sub>1.p\<^sub>1] \<star> TTfgh_TfTgh.chine) \<cdot> ((f.tab \<star> f\<^sub>0gh\<^sub>1.p\<^sub>1) \<star> TTfgh_TfTgh.chine) \<cdot> TTfgh_TfTgh.the_\<nu> \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>1 f, f\<^sub>0g\<^sub>1.p\<^sub>1, fg\<^sub>0h\<^sub>1.p\<^sub>1]
goal (1 subgoal):
1. (f \<star> \<theta>\<^sub>f) \<cdot> \<a>[f, tab\<^sub>0 f, w\<^sub>f] \<cdot> (f.tab \<star> w\<^sub>f) = ((f \<star> \<theta>\<^sub>f') \<cdot> \<a>[f, tab\<^sub>0 f, w\<^sub>f'] \<cdot> (f.tab \<star> w\<^sub>f')) \<cdot> \<beta>\<^sub>f
[PROOF STEP]
also
[PROOF STATE]
proof (state)
this:
\<a>[f, g, h \<star> tab\<^sub>0 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0] \<cdot> \<a>[f \<star> g, h, tab\<^sub>0 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0] \<cdot> ((\<a>\<^sup>-\<^sup>1[f, g, h] \<star> tab\<^sub>0 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0) \<cdot> ((f \<star> g \<star> h) \<star> TTfgh_TfTgh.the_\<theta>) \<cdot> \<a>[f \<star> g \<star> h, (tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine] \<cdot> (\<a>\<^sup>-\<^sup>1[f, g \<star> h, (tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0] \<cdot> (f \<star> \<a>[g \<star> h, tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0, f\<^sub>0gh\<^sub>1.p\<^sub>0]) \<cdot> ((f \<star> \<a>\<^sup>-\<^sup>1[g, h, tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<cdot> (f \<star> (g \<star> \<a>[h, tab\<^sub>0 h, g\<^sub>0h\<^sub>1.p\<^sub>0]) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<cdot> (f \<star> (g \<star> h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<cdot> (f \<star> (g \<star> g\<^sub>0h\<^sub>1.\<phi>) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<cdot> (f \<star> \<a>[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<cdot> (f \<star> (g.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>1) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<cdot> (f \<star> f\<^sub>0gh\<^sub>1.\<phi>)) \<cdot> \<a>[f, tab\<^sub>0 f, f\<^sub>0gh\<^sub>1.p\<^sub>1] \<cdot> (f.tab \<star> f\<^sub>0gh\<^sub>1.p\<^sub>1) \<star> TTfgh_TfTgh.chine) \<cdot> TTfgh_TfTgh.the_\<nu>) \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>1 f, f\<^sub>0g\<^sub>1.p\<^sub>1, fg\<^sub>0h\<^sub>1.p\<^sub>1] = \<a>[f, g, h \<star> tab\<^sub>0 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0] \<cdot> \<a>[f \<star> g, h, tab\<^sub>0 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0] \<cdot> (\<a>\<^sup>-\<^sup>1[f, g, h] \<star> tab\<^sub>0 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0) \<cdot> ((f \<star> g \<star> h) \<star> TTfgh_TfTgh.the_\<theta>) \<cdot> \<a>[f \<star> g \<star> h, (tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine] \<cdot> (\<a>\<^sup>-\<^sup>1[f, g \<star> h, (tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0] \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> \<a>[g \<star> h, tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0, f\<^sub>0gh\<^sub>1.p\<^sub>0]) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> \<a>\<^sup>-\<^sup>1[g, h, tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> (g \<star> \<a>[h, tab\<^sub>0 h, g\<^sub>0h\<^sub>1.p\<^sub>0]) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> (g \<star> h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> (g \<star> g\<^sub>0h\<^sub>1.\<phi>) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> \<a>[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> (g.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>1) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> f\<^sub>0gh\<^sub>1.\<phi>) \<star> TTfgh_TfTgh.chine) \<cdot> (\<a>[f, tab\<^sub>0 f, f\<^sub>0gh\<^sub>1.p\<^sub>1] \<star> TTfgh_TfTgh.chine) \<cdot> ((f.tab \<star> f\<^sub>0gh\<^sub>1.p\<^sub>1) \<star> TTfgh_TfTgh.chine) \<cdot> TTfgh_TfTgh.the_\<nu> \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>1 f, f\<^sub>0g\<^sub>1.p\<^sub>1, fg\<^sub>0h\<^sub>1.p\<^sub>1]
goal (1 subgoal):
1. (f \<star> \<theta>\<^sub>f) \<cdot> \<a>[f, tab\<^sub>0 f, w\<^sub>f] \<cdot> (f.tab \<star> w\<^sub>f) = ((f \<star> \<theta>\<^sub>f') \<cdot> \<a>[f, tab\<^sub>0 f, w\<^sub>f'] \<cdot> (f.tab \<star> w\<^sub>f')) \<cdot> \<beta>\<^sub>f
[PROOF STEP]
have "... =
\<a>[f, g, h \<star> tab\<^sub>0 h \<star> TTfgh.p\<^sub>0] \<cdot>
\<a>[f \<star> g, h, tab\<^sub>0 h \<star> TTfgh.p\<^sub>0] \<cdot>
(\<a>\<^sup>-\<^sup>1[f, g, h] \<star> tab\<^sub>0 h \<star> TTfgh.p\<^sub>0) \<cdot>
((f \<star> g \<star> h) \<star> TTfgh_TfTgh.the_\<theta>) \<cdot>
\<a>[f \<star> g \<star> h, (tab\<^sub>0 h \<star> Tgh.p\<^sub>0) \<star> TfTgh.p\<^sub>0, TTfgh_TfTgh.chine] \<cdot>
(\<a>\<^sup>-\<^sup>1[f, g \<star> h, (tab\<^sub>0 h \<star> Tgh.p\<^sub>0) \<star> TfTgh.p\<^sub>0] \<star> TTfgh_TfTgh.chine) \<cdot>
((f \<star> \<a>[g \<star> h, tab\<^sub>0 h \<star> Tgh.p\<^sub>0, TfTgh.p\<^sub>0]) \<star> TTfgh_TfTgh.chine) \<cdot>
((f \<star> \<a>\<^sup>-\<^sup>1[g, h, tab\<^sub>0 h \<star> Tgh.p\<^sub>0] \<star> TfTgh.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot>
((f \<star> (g \<star> \<a>[h, tab\<^sub>0 h, Tgh.p\<^sub>0]) \<star> TfTgh.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot>
((f \<star> (g \<star> h.tab \<star> Tgh.p\<^sub>0) \<star> TfTgh.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot>
((f \<star> (g \<star> g\<^sub>0h\<^sub>1.\<phi>) \<star> TfTgh.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot>
((f \<star> \<a>[g, tab\<^sub>0 g, Tgh.p\<^sub>1] \<star> TfTgh.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot>
((f \<star> (g.tab \<star> Tgh.p\<^sub>1) \<star> TfTgh.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot>
((f \<star> f\<^sub>0gh\<^sub>1.\<phi>) \<star> TTfgh_TfTgh.chine) \<cdot>
(\<a>[f, tab\<^sub>0 f, TfTgh.p\<^sub>1] \<star> TTfgh_TfTgh.chine) \<cdot>
(((f.tab \<star> TfTgh.p\<^sub>1) \<star> TTfgh_TfTgh.chine) \<cdot>
\<a>\<^sup>-\<^sup>1[tab\<^sub>1 f, TfTgh.p\<^sub>1, TTfgh_TfTgh.chine] \<cdot>
\<a>[tab\<^sub>1 f, TfTgh.p\<^sub>1, TTfgh_TfTgh.chine]) \<cdot>
TTfgh_TfTgh.the_\<nu> \<cdot>
\<a>\<^sup>-\<^sup>1[tab\<^sub>1 f, Tfg.p\<^sub>1, TTfgh.p\<^sub>1]"
[PROOF STATE]
proof (prove)
goal (1 subgoal):
1. \<a>[f, g, h \<star> tab\<^sub>0 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0] \<cdot> \<a>[f \<star> g, h, tab\<^sub>0 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0] \<cdot> (\<a>\<^sup>-\<^sup>1[f, g, h] \<star> tab\<^sub>0 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0) \<cdot> ((f \<star> g \<star> h) \<star> TTfgh_TfTgh.the_\<theta>) \<cdot> \<a>[f \<star> g \<star> h, (tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine] \<cdot> (\<a>\<^sup>-\<^sup>1[f, g \<star> h, (tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0] \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> \<a>[g \<star> h, tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0, f\<^sub>0gh\<^sub>1.p\<^sub>0]) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> \<a>\<^sup>-\<^sup>1[g, h, tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> (g \<star> \<a>[h, tab\<^sub>0 h, g\<^sub>0h\<^sub>1.p\<^sub>0]) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> (g \<star> h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> (g \<star> g\<^sub>0h\<^sub>1.\<phi>) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> \<a>[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> (g.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>1) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> f\<^sub>0gh\<^sub>1.\<phi>) \<star> TTfgh_TfTgh.chine) \<cdot> (\<a>[f, tab\<^sub>0 f, f\<^sub>0gh\<^sub>1.p\<^sub>1] \<star> TTfgh_TfTgh.chine) \<cdot> ((f.tab \<star> f\<^sub>0gh\<^sub>1.p\<^sub>1) \<star> TTfgh_TfTgh.chine) \<cdot> TTfgh_TfTgh.the_\<nu> \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>1 f, f\<^sub>0g\<^sub>1.p\<^sub>1, fg\<^sub>0h\<^sub>1.p\<^sub>1] = \<a>[f, g, h \<star> tab\<^sub>0 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0] \<cdot> \<a>[f \<star> g, h, tab\<^sub>0 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0] \<cdot> (\<a>\<^sup>-\<^sup>1[f, g, h] \<star> tab\<^sub>0 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0) \<cdot> ((f \<star> g \<star> h) \<star> TTfgh_TfTgh.the_\<theta>) \<cdot> \<a>[f \<star> g \<star> h, (tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine] \<cdot> (\<a>\<^sup>-\<^sup>1[f, g \<star> h, (tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0] \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> \<a>[g \<star> h, tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0, f\<^sub>0gh\<^sub>1.p\<^sub>0]) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> \<a>\<^sup>-\<^sup>1[g, h, tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> (g \<star> \<a>[h, tab\<^sub>0 h, g\<^sub>0h\<^sub>1.p\<^sub>0]) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> (g \<star> h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> (g \<star> g\<^sub>0h\<^sub>1.\<phi>) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> \<a>[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> (g.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>1) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> f\<^sub>0gh\<^sub>1.\<phi>) \<star> TTfgh_TfTgh.chine) \<cdot> (\<a>[f, tab\<^sub>0 f, f\<^sub>0gh\<^sub>1.p\<^sub>1] \<star> TTfgh_TfTgh.chine) \<cdot> (((f.tab \<star> f\<^sub>0gh\<^sub>1.p\<^sub>1) \<star> TTfgh_TfTgh.chine) \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>1 f, f\<^sub>0gh\<^sub>1.p\<^sub>1, TTfgh_TfTgh.chine] \<cdot> \<a>[tab\<^sub>1 f, f\<^sub>0gh\<^sub>1.p\<^sub>1, TTfgh_TfTgh.chine]) \<cdot> TTfgh_TfTgh.the_\<nu> \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>1 f, f\<^sub>0g\<^sub>1.p\<^sub>1, fg\<^sub>0h\<^sub>1.p\<^sub>1]
[PROOF STEP]
proof -
[PROOF STATE]
proof (state)
goal (1 subgoal):
1. \<a>[f, g, h \<star> tab\<^sub>0 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0] \<cdot> \<a>[f \<star> g, h, tab\<^sub>0 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0] \<cdot> (\<a>\<^sup>-\<^sup>1[f, g, h] \<star> tab\<^sub>0 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0) \<cdot> ((f \<star> g \<star> h) \<star> TTfgh_TfTgh.the_\<theta>) \<cdot> \<a>[f \<star> g \<star> h, (tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine] \<cdot> (\<a>\<^sup>-\<^sup>1[f, g \<star> h, (tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0] \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> \<a>[g \<star> h, tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0, f\<^sub>0gh\<^sub>1.p\<^sub>0]) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> \<a>\<^sup>-\<^sup>1[g, h, tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> (g \<star> \<a>[h, tab\<^sub>0 h, g\<^sub>0h\<^sub>1.p\<^sub>0]) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> (g \<star> h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> (g \<star> g\<^sub>0h\<^sub>1.\<phi>) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> \<a>[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> (g.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>1) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> f\<^sub>0gh\<^sub>1.\<phi>) \<star> TTfgh_TfTgh.chine) \<cdot> (\<a>[f, tab\<^sub>0 f, f\<^sub>0gh\<^sub>1.p\<^sub>1] \<star> TTfgh_TfTgh.chine) \<cdot> ((f.tab \<star> f\<^sub>0gh\<^sub>1.p\<^sub>1) \<star> TTfgh_TfTgh.chine) \<cdot> TTfgh_TfTgh.the_\<nu> \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>1 f, f\<^sub>0g\<^sub>1.p\<^sub>1, fg\<^sub>0h\<^sub>1.p\<^sub>1] = \<a>[f, g, h \<star> tab\<^sub>0 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0] \<cdot> \<a>[f \<star> g, h, tab\<^sub>0 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0] \<cdot> (\<a>\<^sup>-\<^sup>1[f, g, h] \<star> tab\<^sub>0 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0) \<cdot> ((f \<star> g \<star> h) \<star> TTfgh_TfTgh.the_\<theta>) \<cdot> \<a>[f \<star> g \<star> h, (tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine] \<cdot> (\<a>\<^sup>-\<^sup>1[f, g \<star> h, (tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0] \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> \<a>[g \<star> h, tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0, f\<^sub>0gh\<^sub>1.p\<^sub>0]) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> \<a>\<^sup>-\<^sup>1[g, h, tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> (g \<star> \<a>[h, tab\<^sub>0 h, g\<^sub>0h\<^sub>1.p\<^sub>0]) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> (g \<star> h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> (g \<star> g\<^sub>0h\<^sub>1.\<phi>) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> \<a>[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> (g.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>1) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> f\<^sub>0gh\<^sub>1.\<phi>) \<star> TTfgh_TfTgh.chine) \<cdot> (\<a>[f, tab\<^sub>0 f, f\<^sub>0gh\<^sub>1.p\<^sub>1] \<star> TTfgh_TfTgh.chine) \<cdot> (((f.tab \<star> f\<^sub>0gh\<^sub>1.p\<^sub>1) \<star> TTfgh_TfTgh.chine) \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>1 f, f\<^sub>0gh\<^sub>1.p\<^sub>1, TTfgh_TfTgh.chine] \<cdot> \<a>[tab\<^sub>1 f, f\<^sub>0gh\<^sub>1.p\<^sub>1, TTfgh_TfTgh.chine]) \<cdot> TTfgh_TfTgh.the_\<nu> \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>1 f, f\<^sub>0g\<^sub>1.p\<^sub>1, fg\<^sub>0h\<^sub>1.p\<^sub>1]
[PROOF STEP]
have "((f.tab \<star> TfTgh.p\<^sub>1) \<star> TTfgh_TfTgh.chine) \<cdot>
\<a>\<^sup>-\<^sup>1[tab\<^sub>1 f, TfTgh.p\<^sub>1, TTfgh_TfTgh.chine] \<cdot>
\<a>[tab\<^sub>1 f, TfTgh.p\<^sub>1, TTfgh_TfTgh.chine] =
(f.tab \<star> TfTgh.p\<^sub>1) \<star> TTfgh_TfTgh.chine"
[PROOF STATE]
proof (prove)
goal (1 subgoal):
1. ((f.tab \<star> f\<^sub>0gh\<^sub>1.p\<^sub>1) \<star> TTfgh_TfTgh.chine) \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>1 f, f\<^sub>0gh\<^sub>1.p\<^sub>1, TTfgh_TfTgh.chine] \<cdot> \<a>[tab\<^sub>1 f, f\<^sub>0gh\<^sub>1.p\<^sub>1, TTfgh_TfTgh.chine] = (f.tab \<star> f\<^sub>0gh\<^sub>1.p\<^sub>1) \<star> TTfgh_TfTgh.chine
[PROOF STEP]
using fg gh fg\<^sub>0h\<^sub>1.p\<^sub>0_simps fg\<^sub>0h\<^sub>1.p\<^sub>1_simps f\<^sub>0g\<^sub>1.p\<^sub>0_simps f\<^sub>0g\<^sub>1.p\<^sub>1_simps
comp_arr_dom comp_assoc_assoc'
[PROOF STATE]
proof (prove)
using this:
src f = trg g
src g = trg h
trg fg\<^sub>0h\<^sub>1.p\<^sub>0 = src (tab\<^sub>1 h)
trg fg\<^sub>0h\<^sub>1.p\<^sub>1 = src (tab\<^sub>0 g \<star> f\<^sub>0g\<^sub>1.p\<^sub>0)
trg f\<^sub>0g\<^sub>1.p\<^sub>0 = src (tab\<^sub>1 g)
trg f\<^sub>0g\<^sub>1.p\<^sub>1 = src (tab\<^sub>0 f)
\<lbrakk>arr ?f; local.dom ?f = ?a\<rbrakk> \<Longrightarrow> ?f \<cdot> ?a = ?f
\<lbrakk>ide ?f; ide ?g; ide ?h; src ?f = trg ?g; src ?g = trg ?h\<rbrakk> \<Longrightarrow> \<a>[?f, ?g, ?h] \<cdot> \<a>\<^sup>-\<^sup>1[?f, ?g, ?h] = ?f \<star> ?g \<star> ?h
\<lbrakk>ide ?f; ide ?g; ide ?h; src ?f = trg ?g; src ?g = trg ?h\<rbrakk> \<Longrightarrow> \<a>\<^sup>-\<^sup>1[?f, ?g, ?h] \<cdot> \<a>[?f, ?g, ?h] = (?f \<star> ?g) \<star> ?h
goal (1 subgoal):
1. ((f.tab \<star> f\<^sub>0gh\<^sub>1.p\<^sub>1) \<star> TTfgh_TfTgh.chine) \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>1 f, f\<^sub>0gh\<^sub>1.p\<^sub>1, TTfgh_TfTgh.chine] \<cdot> \<a>[tab\<^sub>1 f, f\<^sub>0gh\<^sub>1.p\<^sub>1, TTfgh_TfTgh.chine] = (f.tab \<star> f\<^sub>0gh\<^sub>1.p\<^sub>1) \<star> TTfgh_TfTgh.chine
[PROOF STEP]
by simp
[PROOF STATE]
proof (state)
this:
((f.tab \<star> f\<^sub>0gh\<^sub>1.p\<^sub>1) \<star> TTfgh_TfTgh.chine) \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>1 f, f\<^sub>0gh\<^sub>1.p\<^sub>1, TTfgh_TfTgh.chine] \<cdot> \<a>[tab\<^sub>1 f, f\<^sub>0gh\<^sub>1.p\<^sub>1, TTfgh_TfTgh.chine] = (f.tab \<star> f\<^sub>0gh\<^sub>1.p\<^sub>1) \<star> TTfgh_TfTgh.chine
goal (1 subgoal):
1. \<a>[f, g, h \<star> tab\<^sub>0 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0] \<cdot> \<a>[f \<star> g, h, tab\<^sub>0 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0] \<cdot> (\<a>\<^sup>-\<^sup>1[f, g, h] \<star> tab\<^sub>0 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0) \<cdot> ((f \<star> g \<star> h) \<star> TTfgh_TfTgh.the_\<theta>) \<cdot> \<a>[f \<star> g \<star> h, (tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine] \<cdot> (\<a>\<^sup>-\<^sup>1[f, g \<star> h, (tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0] \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> \<a>[g \<star> h, tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0, f\<^sub>0gh\<^sub>1.p\<^sub>0]) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> \<a>\<^sup>-\<^sup>1[g, h, tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> (g \<star> \<a>[h, tab\<^sub>0 h, g\<^sub>0h\<^sub>1.p\<^sub>0]) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> (g \<star> h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> (g \<star> g\<^sub>0h\<^sub>1.\<phi>) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> \<a>[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> (g.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>1) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> f\<^sub>0gh\<^sub>1.\<phi>) \<star> TTfgh_TfTgh.chine) \<cdot> (\<a>[f, tab\<^sub>0 f, f\<^sub>0gh\<^sub>1.p\<^sub>1] \<star> TTfgh_TfTgh.chine) \<cdot> ((f.tab \<star> f\<^sub>0gh\<^sub>1.p\<^sub>1) \<star> TTfgh_TfTgh.chine) \<cdot> TTfgh_TfTgh.the_\<nu> \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>1 f, f\<^sub>0g\<^sub>1.p\<^sub>1, fg\<^sub>0h\<^sub>1.p\<^sub>1] = \<a>[f, g, h \<star> tab\<^sub>0 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0] \<cdot> \<a>[f \<star> g, h, tab\<^sub>0 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0] \<cdot> (\<a>\<^sup>-\<^sup>1[f, g, h] \<star> tab\<^sub>0 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0) \<cdot> ((f \<star> g \<star> h) \<star> TTfgh_TfTgh.the_\<theta>) \<cdot> \<a>[f \<star> g \<star> h, (tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine] \<cdot> (\<a>\<^sup>-\<^sup>1[f, g \<star> h, (tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0] \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> \<a>[g \<star> h, tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0, f\<^sub>0gh\<^sub>1.p\<^sub>0]) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> \<a>\<^sup>-\<^sup>1[g, h, tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> (g \<star> \<a>[h, tab\<^sub>0 h, g\<^sub>0h\<^sub>1.p\<^sub>0]) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> (g \<star> h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> (g \<star> g\<^sub>0h\<^sub>1.\<phi>) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> \<a>[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> (g.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>1) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> f\<^sub>0gh\<^sub>1.\<phi>) \<star> TTfgh_TfTgh.chine) \<cdot> (\<a>[f, tab\<^sub>0 f, f\<^sub>0gh\<^sub>1.p\<^sub>1] \<star> TTfgh_TfTgh.chine) \<cdot> (((f.tab \<star> f\<^sub>0gh\<^sub>1.p\<^sub>1) \<star> TTfgh_TfTgh.chine) \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>1 f, f\<^sub>0gh\<^sub>1.p\<^sub>1, TTfgh_TfTgh.chine] \<cdot> \<a>[tab\<^sub>1 f, f\<^sub>0gh\<^sub>1.p\<^sub>1, TTfgh_TfTgh.chine]) \<cdot> TTfgh_TfTgh.the_\<nu> \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>1 f, f\<^sub>0g\<^sub>1.p\<^sub>1, fg\<^sub>0h\<^sub>1.p\<^sub>1]
[PROOF STEP]
thus ?thesis
[PROOF STATE]
proof (prove)
using this:
((f.tab \<star> f\<^sub>0gh\<^sub>1.p\<^sub>1) \<star> TTfgh_TfTgh.chine) \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>1 f, f\<^sub>0gh\<^sub>1.p\<^sub>1, TTfgh_TfTgh.chine] \<cdot> \<a>[tab\<^sub>1 f, f\<^sub>0gh\<^sub>1.p\<^sub>1, TTfgh_TfTgh.chine] = (f.tab \<star> f\<^sub>0gh\<^sub>1.p\<^sub>1) \<star> TTfgh_TfTgh.chine
goal (1 subgoal):
1. \<a>[f, g, h \<star> tab\<^sub>0 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0] \<cdot> \<a>[f \<star> g, h, tab\<^sub>0 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0] \<cdot> (\<a>\<^sup>-\<^sup>1[f, g, h] \<star> tab\<^sub>0 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0) \<cdot> ((f \<star> g \<star> h) \<star> TTfgh_TfTgh.the_\<theta>) \<cdot> \<a>[f \<star> g \<star> h, (tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine] \<cdot> (\<a>\<^sup>-\<^sup>1[f, g \<star> h, (tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0] \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> \<a>[g \<star> h, tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0, f\<^sub>0gh\<^sub>1.p\<^sub>0]) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> \<a>\<^sup>-\<^sup>1[g, h, tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> (g \<star> \<a>[h, tab\<^sub>0 h, g\<^sub>0h\<^sub>1.p\<^sub>0]) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> (g \<star> h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> (g \<star> g\<^sub>0h\<^sub>1.\<phi>) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> \<a>[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> (g.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>1) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> f\<^sub>0gh\<^sub>1.\<phi>) \<star> TTfgh_TfTgh.chine) \<cdot> (\<a>[f, tab\<^sub>0 f, f\<^sub>0gh\<^sub>1.p\<^sub>1] \<star> TTfgh_TfTgh.chine) \<cdot> ((f.tab \<star> f\<^sub>0gh\<^sub>1.p\<^sub>1) \<star> TTfgh_TfTgh.chine) \<cdot> TTfgh_TfTgh.the_\<nu> \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>1 f, f\<^sub>0g\<^sub>1.p\<^sub>1, fg\<^sub>0h\<^sub>1.p\<^sub>1] = \<a>[f, g, h \<star> tab\<^sub>0 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0] \<cdot> \<a>[f \<star> g, h, tab\<^sub>0 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0] \<cdot> (\<a>\<^sup>-\<^sup>1[f, g, h] \<star> tab\<^sub>0 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0) \<cdot> ((f \<star> g \<star> h) \<star> TTfgh_TfTgh.the_\<theta>) \<cdot> \<a>[f \<star> g \<star> h, (tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine] \<cdot> (\<a>\<^sup>-\<^sup>1[f, g \<star> h, (tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0] \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> \<a>[g \<star> h, tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0, f\<^sub>0gh\<^sub>1.p\<^sub>0]) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> \<a>\<^sup>-\<^sup>1[g, h, tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> (g \<star> \<a>[h, tab\<^sub>0 h, g\<^sub>0h\<^sub>1.p\<^sub>0]) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> (g \<star> h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> (g \<star> g\<^sub>0h\<^sub>1.\<phi>) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> \<a>[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> (g.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>1) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> f\<^sub>0gh\<^sub>1.\<phi>) \<star> TTfgh_TfTgh.chine) \<cdot> (\<a>[f, tab\<^sub>0 f, f\<^sub>0gh\<^sub>1.p\<^sub>1] \<star> TTfgh_TfTgh.chine) \<cdot> (((f.tab \<star> f\<^sub>0gh\<^sub>1.p\<^sub>1) \<star> TTfgh_TfTgh.chine) \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>1 f, f\<^sub>0gh\<^sub>1.p\<^sub>1, TTfgh_TfTgh.chine] \<cdot> \<a>[tab\<^sub>1 f, f\<^sub>0gh\<^sub>1.p\<^sub>1, TTfgh_TfTgh.chine]) \<cdot> TTfgh_TfTgh.the_\<nu> \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>1 f, f\<^sub>0g\<^sub>1.p\<^sub>1, fg\<^sub>0h\<^sub>1.p\<^sub>1]
[PROOF STEP]
by simp
[PROOF STATE]
proof (state)
this:
\<a>[f, g, h \<star> tab\<^sub>0 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0] \<cdot> \<a>[f \<star> g, h, tab\<^sub>0 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0] \<cdot> (\<a>\<^sup>-\<^sup>1[f, g, h] \<star> tab\<^sub>0 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0) \<cdot> ((f \<star> g \<star> h) \<star> TTfgh_TfTgh.the_\<theta>) \<cdot> \<a>[f \<star> g \<star> h, (tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine] \<cdot> (\<a>\<^sup>-\<^sup>1[f, g \<star> h, (tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0] \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> \<a>[g \<star> h, tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0, f\<^sub>0gh\<^sub>1.p\<^sub>0]) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> \<a>\<^sup>-\<^sup>1[g, h, tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> (g \<star> \<a>[h, tab\<^sub>0 h, g\<^sub>0h\<^sub>1.p\<^sub>0]) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> (g \<star> h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> (g \<star> g\<^sub>0h\<^sub>1.\<phi>) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> \<a>[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> (g.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>1) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> f\<^sub>0gh\<^sub>1.\<phi>) \<star> TTfgh_TfTgh.chine) \<cdot> (\<a>[f, tab\<^sub>0 f, f\<^sub>0gh\<^sub>1.p\<^sub>1] \<star> TTfgh_TfTgh.chine) \<cdot> ((f.tab \<star> f\<^sub>0gh\<^sub>1.p\<^sub>1) \<star> TTfgh_TfTgh.chine) \<cdot> TTfgh_TfTgh.the_\<nu> \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>1 f, f\<^sub>0g\<^sub>1.p\<^sub>1, fg\<^sub>0h\<^sub>1.p\<^sub>1] = \<a>[f, g, h \<star> tab\<^sub>0 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0] \<cdot> \<a>[f \<star> g, h, tab\<^sub>0 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0] \<cdot> (\<a>\<^sup>-\<^sup>1[f, g, h] \<star> tab\<^sub>0 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0) \<cdot> ((f \<star> g \<star> h) \<star> TTfgh_TfTgh.the_\<theta>) \<cdot> \<a>[f \<star> g \<star> h, (tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine] \<cdot> (\<a>\<^sup>-\<^sup>1[f, g \<star> h, (tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0] \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> \<a>[g \<star> h, tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0, f\<^sub>0gh\<^sub>1.p\<^sub>0]) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> \<a>\<^sup>-\<^sup>1[g, h, tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> (g \<star> \<a>[h, tab\<^sub>0 h, g\<^sub>0h\<^sub>1.p\<^sub>0]) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> (g \<star> h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> (g \<star> g\<^sub>0h\<^sub>1.\<phi>) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> \<a>[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> (g.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>1) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> f\<^sub>0gh\<^sub>1.\<phi>) \<star> TTfgh_TfTgh.chine) \<cdot> (\<a>[f, tab\<^sub>0 f, f\<^sub>0gh\<^sub>1.p\<^sub>1] \<star> TTfgh_TfTgh.chine) \<cdot> (((f.tab \<star> f\<^sub>0gh\<^sub>1.p\<^sub>1) \<star> TTfgh_TfTgh.chine) \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>1 f, f\<^sub>0gh\<^sub>1.p\<^sub>1, TTfgh_TfTgh.chine] \<cdot> \<a>[tab\<^sub>1 f, f\<^sub>0gh\<^sub>1.p\<^sub>1, TTfgh_TfTgh.chine]) \<cdot> TTfgh_TfTgh.the_\<nu> \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>1 f, f\<^sub>0g\<^sub>1.p\<^sub>1, fg\<^sub>0h\<^sub>1.p\<^sub>1]
goal:
No subgoals!
[PROOF STEP]
qed
[PROOF STATE]
proof (state)
this:
\<a>[f, g, h \<star> tab\<^sub>0 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0] \<cdot> \<a>[f \<star> g, h, tab\<^sub>0 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0] \<cdot> (\<a>\<^sup>-\<^sup>1[f, g, h] \<star> tab\<^sub>0 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0) \<cdot> ((f \<star> g \<star> h) \<star> TTfgh_TfTgh.the_\<theta>) \<cdot> \<a>[f \<star> g \<star> h, (tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine] \<cdot> (\<a>\<^sup>-\<^sup>1[f, g \<star> h, (tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0] \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> \<a>[g \<star> h, tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0, f\<^sub>0gh\<^sub>1.p\<^sub>0]) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> \<a>\<^sup>-\<^sup>1[g, h, tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> (g \<star> \<a>[h, tab\<^sub>0 h, g\<^sub>0h\<^sub>1.p\<^sub>0]) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> (g \<star> h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> (g \<star> g\<^sub>0h\<^sub>1.\<phi>) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> \<a>[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> (g.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>1) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> f\<^sub>0gh\<^sub>1.\<phi>) \<star> TTfgh_TfTgh.chine) \<cdot> (\<a>[f, tab\<^sub>0 f, f\<^sub>0gh\<^sub>1.p\<^sub>1] \<star> TTfgh_TfTgh.chine) \<cdot> ((f.tab \<star> f\<^sub>0gh\<^sub>1.p\<^sub>1) \<star> TTfgh_TfTgh.chine) \<cdot> TTfgh_TfTgh.the_\<nu> \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>1 f, f\<^sub>0g\<^sub>1.p\<^sub>1, fg\<^sub>0h\<^sub>1.p\<^sub>1] = \<a>[f, g, h \<star> tab\<^sub>0 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0] \<cdot> \<a>[f \<star> g, h, tab\<^sub>0 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0] \<cdot> (\<a>\<^sup>-\<^sup>1[f, g, h] \<star> tab\<^sub>0 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0) \<cdot> ((f \<star> g \<star> h) \<star> TTfgh_TfTgh.the_\<theta>) \<cdot> \<a>[f \<star> g \<star> h, (tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine] \<cdot> (\<a>\<^sup>-\<^sup>1[f, g \<star> h, (tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0] \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> \<a>[g \<star> h, tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0, f\<^sub>0gh\<^sub>1.p\<^sub>0]) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> \<a>\<^sup>-\<^sup>1[g, h, tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> (g \<star> \<a>[h, tab\<^sub>0 h, g\<^sub>0h\<^sub>1.p\<^sub>0]) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> (g \<star> h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> (g \<star> g\<^sub>0h\<^sub>1.\<phi>) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> \<a>[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> (g.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>1) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> f\<^sub>0gh\<^sub>1.\<phi>) \<star> TTfgh_TfTgh.chine) \<cdot> (\<a>[f, tab\<^sub>0 f, f\<^sub>0gh\<^sub>1.p\<^sub>1] \<star> TTfgh_TfTgh.chine) \<cdot> (((f.tab \<star> f\<^sub>0gh\<^sub>1.p\<^sub>1) \<star> TTfgh_TfTgh.chine) \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>1 f, f\<^sub>0gh\<^sub>1.p\<^sub>1, TTfgh_TfTgh.chine] \<cdot> \<a>[tab\<^sub>1 f, f\<^sub>0gh\<^sub>1.p\<^sub>1, TTfgh_TfTgh.chine]) \<cdot> TTfgh_TfTgh.the_\<nu> \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>1 f, f\<^sub>0g\<^sub>1.p\<^sub>1, fg\<^sub>0h\<^sub>1.p\<^sub>1]
goal (1 subgoal):
1. (f \<star> \<theta>\<^sub>f) \<cdot> \<a>[f, tab\<^sub>0 f, w\<^sub>f] \<cdot> (f.tab \<star> w\<^sub>f) = ((f \<star> \<theta>\<^sub>f') \<cdot> \<a>[f, tab\<^sub>0 f, w\<^sub>f'] \<cdot> (f.tab \<star> w\<^sub>f')) \<cdot> \<beta>\<^sub>f
[PROOF STEP]
also
[PROOF STATE]
proof (state)
this:
\<a>[f, g, h \<star> tab\<^sub>0 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0] \<cdot> \<a>[f \<star> g, h, tab\<^sub>0 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0] \<cdot> (\<a>\<^sup>-\<^sup>1[f, g, h] \<star> tab\<^sub>0 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0) \<cdot> ((f \<star> g \<star> h) \<star> TTfgh_TfTgh.the_\<theta>) \<cdot> \<a>[f \<star> g \<star> h, (tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine] \<cdot> (\<a>\<^sup>-\<^sup>1[f, g \<star> h, (tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0] \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> \<a>[g \<star> h, tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0, f\<^sub>0gh\<^sub>1.p\<^sub>0]) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> \<a>\<^sup>-\<^sup>1[g, h, tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> (g \<star> \<a>[h, tab\<^sub>0 h, g\<^sub>0h\<^sub>1.p\<^sub>0]) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> (g \<star> h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> (g \<star> g\<^sub>0h\<^sub>1.\<phi>) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> \<a>[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> (g.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>1) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> f\<^sub>0gh\<^sub>1.\<phi>) \<star> TTfgh_TfTgh.chine) \<cdot> (\<a>[f, tab\<^sub>0 f, f\<^sub>0gh\<^sub>1.p\<^sub>1] \<star> TTfgh_TfTgh.chine) \<cdot> ((f.tab \<star> f\<^sub>0gh\<^sub>1.p\<^sub>1) \<star> TTfgh_TfTgh.chine) \<cdot> TTfgh_TfTgh.the_\<nu> \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>1 f, f\<^sub>0g\<^sub>1.p\<^sub>1, fg\<^sub>0h\<^sub>1.p\<^sub>1] = \<a>[f, g, h \<star> tab\<^sub>0 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0] \<cdot> \<a>[f \<star> g, h, tab\<^sub>0 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0] \<cdot> (\<a>\<^sup>-\<^sup>1[f, g, h] \<star> tab\<^sub>0 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0) \<cdot> ((f \<star> g \<star> h) \<star> TTfgh_TfTgh.the_\<theta>) \<cdot> \<a>[f \<star> g \<star> h, (tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine] \<cdot> (\<a>\<^sup>-\<^sup>1[f, g \<star> h, (tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0] \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> \<a>[g \<star> h, tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0, f\<^sub>0gh\<^sub>1.p\<^sub>0]) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> \<a>\<^sup>-\<^sup>1[g, h, tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> (g \<star> \<a>[h, tab\<^sub>0 h, g\<^sub>0h\<^sub>1.p\<^sub>0]) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> (g \<star> h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> (g \<star> g\<^sub>0h\<^sub>1.\<phi>) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> \<a>[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> (g.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>1) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> f\<^sub>0gh\<^sub>1.\<phi>) \<star> TTfgh_TfTgh.chine) \<cdot> (\<a>[f, tab\<^sub>0 f, f\<^sub>0gh\<^sub>1.p\<^sub>1] \<star> TTfgh_TfTgh.chine) \<cdot> (((f.tab \<star> f\<^sub>0gh\<^sub>1.p\<^sub>1) \<star> TTfgh_TfTgh.chine) \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>1 f, f\<^sub>0gh\<^sub>1.p\<^sub>1, TTfgh_TfTgh.chine] \<cdot> \<a>[tab\<^sub>1 f, f\<^sub>0gh\<^sub>1.p\<^sub>1, TTfgh_TfTgh.chine]) \<cdot> TTfgh_TfTgh.the_\<nu> \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>1 f, f\<^sub>0g\<^sub>1.p\<^sub>1, fg\<^sub>0h\<^sub>1.p\<^sub>1]
goal (1 subgoal):
1. (f \<star> \<theta>\<^sub>f) \<cdot> \<a>[f, tab\<^sub>0 f, w\<^sub>f] \<cdot> (f.tab \<star> w\<^sub>f) = ((f \<star> \<theta>\<^sub>f') \<cdot> \<a>[f, tab\<^sub>0 f, w\<^sub>f'] \<cdot> (f.tab \<star> w\<^sub>f')) \<cdot> \<beta>\<^sub>f
[PROOF STEP]
have "... =
\<a>[f, g, h \<star> tab\<^sub>0 h \<star> TTfgh.p\<^sub>0] \<cdot>
\<a>[f \<star> g, h, tab\<^sub>0 h \<star> TTfgh.p\<^sub>0] \<cdot>
(\<a>\<^sup>-\<^sup>1[f, g, h] \<star> tab\<^sub>0 h \<star> TTfgh.p\<^sub>0) \<cdot>
((f \<star> g \<star> h) \<star> TTfgh_TfTgh.the_\<theta>) \<cdot>
\<a>[f \<star> g \<star> h, (tab\<^sub>0 h \<star> Tgh.p\<^sub>0) \<star> TfTgh.p\<^sub>0, TTfgh_TfTgh.chine] \<cdot>
(\<a>\<^sup>-\<^sup>1[f, g \<star> h, (tab\<^sub>0 h \<star> Tgh.p\<^sub>0) \<star> TfTgh.p\<^sub>0] \<star> TTfgh_TfTgh.chine) \<cdot>
((f \<star> \<a>[g \<star> h, tab\<^sub>0 h \<star> Tgh.p\<^sub>0, TfTgh.p\<^sub>0]) \<star> TTfgh_TfTgh.chine) \<cdot>
((f \<star> \<a>\<^sup>-\<^sup>1[g, h, tab\<^sub>0 h \<star> Tgh.p\<^sub>0] \<star> TfTgh.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot>
((f \<star> (g \<star> \<a>[h, tab\<^sub>0 h, Tgh.p\<^sub>0]) \<star> TfTgh.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot>
((f \<star> (g \<star> h.tab \<star> Tgh.p\<^sub>0) \<star> TfTgh.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot>
((f \<star> (g \<star> g\<^sub>0h\<^sub>1.\<phi>) \<star> TfTgh.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot>
((f \<star> \<a>[g, tab\<^sub>0 g, Tgh.p\<^sub>1] \<star> TfTgh.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot>
((f \<star> (g.tab \<star> Tgh.p\<^sub>1) \<star> TfTgh.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot>
((f \<star> f\<^sub>0gh\<^sub>1.\<phi>) \<star> TTfgh_TfTgh.chine) \<cdot>
(\<a>[f, tab\<^sub>0 f, TfTgh.p\<^sub>1] \<star> TTfgh_TfTgh.chine) \<cdot>
(((f.tab \<star> TfTgh.p\<^sub>1) \<star> TTfgh_TfTgh.chine) \<cdot>
\<a>\<^sup>-\<^sup>1[tab\<^sub>1 f, TfTgh.p\<^sub>1, TTfgh_TfTgh.chine]) \<cdot>
\<a>[tab\<^sub>1 f, TfTgh.p\<^sub>1, TTfgh_TfTgh.chine] \<cdot>
TTfgh_TfTgh.the_\<nu> \<cdot>
\<a>\<^sup>-\<^sup>1[tab\<^sub>1 f, Tfg.p\<^sub>1, TTfgh.p\<^sub>1]"
[PROOF STATE]
proof (prove)
goal (1 subgoal):
1. \<a>[f, g, h \<star> tab\<^sub>0 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0] \<cdot> \<a>[f \<star> g, h, tab\<^sub>0 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0] \<cdot> (\<a>\<^sup>-\<^sup>1[f, g, h] \<star> tab\<^sub>0 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0) \<cdot> ((f \<star> g \<star> h) \<star> TTfgh_TfTgh.the_\<theta>) \<cdot> \<a>[f \<star> g \<star> h, (tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine] \<cdot> (\<a>\<^sup>-\<^sup>1[f, g \<star> h, (tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0] \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> \<a>[g \<star> h, tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0, f\<^sub>0gh\<^sub>1.p\<^sub>0]) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> \<a>\<^sup>-\<^sup>1[g, h, tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> (g \<star> \<a>[h, tab\<^sub>0 h, g\<^sub>0h\<^sub>1.p\<^sub>0]) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> (g \<star> h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> (g \<star> g\<^sub>0h\<^sub>1.\<phi>) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> \<a>[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> (g.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>1) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> f\<^sub>0gh\<^sub>1.\<phi>) \<star> TTfgh_TfTgh.chine) \<cdot> (\<a>[f, tab\<^sub>0 f, f\<^sub>0gh\<^sub>1.p\<^sub>1] \<star> TTfgh_TfTgh.chine) \<cdot> (((f.tab \<star> f\<^sub>0gh\<^sub>1.p\<^sub>1) \<star> TTfgh_TfTgh.chine) \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>1 f, f\<^sub>0gh\<^sub>1.p\<^sub>1, TTfgh_TfTgh.chine] \<cdot> \<a>[tab\<^sub>1 f, f\<^sub>0gh\<^sub>1.p\<^sub>1, TTfgh_TfTgh.chine]) \<cdot> TTfgh_TfTgh.the_\<nu> \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>1 f, f\<^sub>0g\<^sub>1.p\<^sub>1, fg\<^sub>0h\<^sub>1.p\<^sub>1] = \<a>[f, g, h \<star> tab\<^sub>0 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0] \<cdot> \<a>[f \<star> g, h, tab\<^sub>0 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0] \<cdot> (\<a>\<^sup>-\<^sup>1[f, g, h] \<star> tab\<^sub>0 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0) \<cdot> ((f \<star> g \<star> h) \<star> TTfgh_TfTgh.the_\<theta>) \<cdot> \<a>[f \<star> g \<star> h, (tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine] \<cdot> (\<a>\<^sup>-\<^sup>1[f, g \<star> h, (tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0] \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> \<a>[g \<star> h, tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0, f\<^sub>0gh\<^sub>1.p\<^sub>0]) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> \<a>\<^sup>-\<^sup>1[g, h, tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> (g \<star> \<a>[h, tab\<^sub>0 h, g\<^sub>0h\<^sub>1.p\<^sub>0]) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> (g \<star> h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> (g \<star> g\<^sub>0h\<^sub>1.\<phi>) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> \<a>[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> (g.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>1) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> f\<^sub>0gh\<^sub>1.\<phi>) \<star> TTfgh_TfTgh.chine) \<cdot> (\<a>[f, tab\<^sub>0 f, f\<^sub>0gh\<^sub>1.p\<^sub>1] \<star> TTfgh_TfTgh.chine) \<cdot> (((f.tab \<star> f\<^sub>0gh\<^sub>1.p\<^sub>1) \<star> TTfgh_TfTgh.chine) \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>1 f, f\<^sub>0gh\<^sub>1.p\<^sub>1, TTfgh_TfTgh.chine]) \<cdot> \<a>[tab\<^sub>1 f, f\<^sub>0gh\<^sub>1.p\<^sub>1, TTfgh_TfTgh.chine] \<cdot> TTfgh_TfTgh.the_\<nu> \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>1 f, f\<^sub>0g\<^sub>1.p\<^sub>1, fg\<^sub>0h\<^sub>1.p\<^sub>1]
[PROOF STEP]
using comp_assoc
[PROOF STATE]
proof (prove)
using this:
(?h \<cdot> ?g) \<cdot> ?f = ?h \<cdot> ?g \<cdot> ?f
goal (1 subgoal):
1. \<a>[f, g, h \<star> tab\<^sub>0 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0] \<cdot> \<a>[f \<star> g, h, tab\<^sub>0 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0] \<cdot> (\<a>\<^sup>-\<^sup>1[f, g, h] \<star> tab\<^sub>0 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0) \<cdot> ((f \<star> g \<star> h) \<star> TTfgh_TfTgh.the_\<theta>) \<cdot> \<a>[f \<star> g \<star> h, (tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine] \<cdot> (\<a>\<^sup>-\<^sup>1[f, g \<star> h, (tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0] \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> \<a>[g \<star> h, tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0, f\<^sub>0gh\<^sub>1.p\<^sub>0]) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> \<a>\<^sup>-\<^sup>1[g, h, tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> (g \<star> \<a>[h, tab\<^sub>0 h, g\<^sub>0h\<^sub>1.p\<^sub>0]) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> (g \<star> h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> (g \<star> g\<^sub>0h\<^sub>1.\<phi>) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> \<a>[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> (g.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>1) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> f\<^sub>0gh\<^sub>1.\<phi>) \<star> TTfgh_TfTgh.chine) \<cdot> (\<a>[f, tab\<^sub>0 f, f\<^sub>0gh\<^sub>1.p\<^sub>1] \<star> TTfgh_TfTgh.chine) \<cdot> (((f.tab \<star> f\<^sub>0gh\<^sub>1.p\<^sub>1) \<star> TTfgh_TfTgh.chine) \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>1 f, f\<^sub>0gh\<^sub>1.p\<^sub>1, TTfgh_TfTgh.chine] \<cdot> \<a>[tab\<^sub>1 f, f\<^sub>0gh\<^sub>1.p\<^sub>1, TTfgh_TfTgh.chine]) \<cdot> TTfgh_TfTgh.the_\<nu> \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>1 f, f\<^sub>0g\<^sub>1.p\<^sub>1, fg\<^sub>0h\<^sub>1.p\<^sub>1] = \<a>[f, g, h \<star> tab\<^sub>0 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0] \<cdot> \<a>[f \<star> g, h, tab\<^sub>0 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0] \<cdot> (\<a>\<^sup>-\<^sup>1[f, g, h] \<star> tab\<^sub>0 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0) \<cdot> ((f \<star> g \<star> h) \<star> TTfgh_TfTgh.the_\<theta>) \<cdot> \<a>[f \<star> g \<star> h, (tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine] \<cdot> (\<a>\<^sup>-\<^sup>1[f, g \<star> h, (tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0] \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> \<a>[g \<star> h, tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0, f\<^sub>0gh\<^sub>1.p\<^sub>0]) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> \<a>\<^sup>-\<^sup>1[g, h, tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> (g \<star> \<a>[h, tab\<^sub>0 h, g\<^sub>0h\<^sub>1.p\<^sub>0]) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> (g \<star> h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> (g \<star> g\<^sub>0h\<^sub>1.\<phi>) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> \<a>[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> (g.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>1) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> f\<^sub>0gh\<^sub>1.\<phi>) \<star> TTfgh_TfTgh.chine) \<cdot> (\<a>[f, tab\<^sub>0 f, f\<^sub>0gh\<^sub>1.p\<^sub>1] \<star> TTfgh_TfTgh.chine) \<cdot> (((f.tab \<star> f\<^sub>0gh\<^sub>1.p\<^sub>1) \<star> TTfgh_TfTgh.chine) \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>1 f, f\<^sub>0gh\<^sub>1.p\<^sub>1, TTfgh_TfTgh.chine]) \<cdot> \<a>[tab\<^sub>1 f, f\<^sub>0gh\<^sub>1.p\<^sub>1, TTfgh_TfTgh.chine] \<cdot> TTfgh_TfTgh.the_\<nu> \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>1 f, f\<^sub>0g\<^sub>1.p\<^sub>1, fg\<^sub>0h\<^sub>1.p\<^sub>1]
[PROOF STEP]
by presburger
[PROOF STATE]
proof (state)
this:
\<a>[f, g, h \<star> tab\<^sub>0 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0] \<cdot> \<a>[f \<star> g, h, tab\<^sub>0 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0] \<cdot> (\<a>\<^sup>-\<^sup>1[f, g, h] \<star> tab\<^sub>0 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0) \<cdot> ((f \<star> g \<star> h) \<star> TTfgh_TfTgh.the_\<theta>) \<cdot> \<a>[f \<star> g \<star> h, (tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine] \<cdot> (\<a>\<^sup>-\<^sup>1[f, g \<star> h, (tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0] \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> \<a>[g \<star> h, tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0, f\<^sub>0gh\<^sub>1.p\<^sub>0]) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> \<a>\<^sup>-\<^sup>1[g, h, tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> (g \<star> \<a>[h, tab\<^sub>0 h, g\<^sub>0h\<^sub>1.p\<^sub>0]) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> (g \<star> h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> (g \<star> g\<^sub>0h\<^sub>1.\<phi>) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> \<a>[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> (g.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>1) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> f\<^sub>0gh\<^sub>1.\<phi>) \<star> TTfgh_TfTgh.chine) \<cdot> (\<a>[f, tab\<^sub>0 f, f\<^sub>0gh\<^sub>1.p\<^sub>1] \<star> TTfgh_TfTgh.chine) \<cdot> (((f.tab \<star> f\<^sub>0gh\<^sub>1.p\<^sub>1) \<star> TTfgh_TfTgh.chine) \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>1 f, f\<^sub>0gh\<^sub>1.p\<^sub>1, TTfgh_TfTgh.chine] \<cdot> \<a>[tab\<^sub>1 f, f\<^sub>0gh\<^sub>1.p\<^sub>1, TTfgh_TfTgh.chine]) \<cdot> TTfgh_TfTgh.the_\<nu> \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>1 f, f\<^sub>0g\<^sub>1.p\<^sub>1, fg\<^sub>0h\<^sub>1.p\<^sub>1] = \<a>[f, g, h \<star> tab\<^sub>0 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0] \<cdot> \<a>[f \<star> g, h, tab\<^sub>0 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0] \<cdot> (\<a>\<^sup>-\<^sup>1[f, g, h] \<star> tab\<^sub>0 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0) \<cdot> ((f \<star> g \<star> h) \<star> TTfgh_TfTgh.the_\<theta>) \<cdot> \<a>[f \<star> g \<star> h, (tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine] \<cdot> (\<a>\<^sup>-\<^sup>1[f, g \<star> h, (tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0] \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> \<a>[g \<star> h, tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0, f\<^sub>0gh\<^sub>1.p\<^sub>0]) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> \<a>\<^sup>-\<^sup>1[g, h, tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> (g \<star> \<a>[h, tab\<^sub>0 h, g\<^sub>0h\<^sub>1.p\<^sub>0]) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> (g \<star> h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> (g \<star> g\<^sub>0h\<^sub>1.\<phi>) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> \<a>[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> (g.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>1) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> f\<^sub>0gh\<^sub>1.\<phi>) \<star> TTfgh_TfTgh.chine) \<cdot> (\<a>[f, tab\<^sub>0 f, f\<^sub>0gh\<^sub>1.p\<^sub>1] \<star> TTfgh_TfTgh.chine) \<cdot> (((f.tab \<star> f\<^sub>0gh\<^sub>1.p\<^sub>1) \<star> TTfgh_TfTgh.chine) \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>1 f, f\<^sub>0gh\<^sub>1.p\<^sub>1, TTfgh_TfTgh.chine]) \<cdot> \<a>[tab\<^sub>1 f, f\<^sub>0gh\<^sub>1.p\<^sub>1, TTfgh_TfTgh.chine] \<cdot> TTfgh_TfTgh.the_\<nu> \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>1 f, f\<^sub>0g\<^sub>1.p\<^sub>1, fg\<^sub>0h\<^sub>1.p\<^sub>1]
goal (1 subgoal):
1. (f \<star> \<theta>\<^sub>f) \<cdot> \<a>[f, tab\<^sub>0 f, w\<^sub>f] \<cdot> (f.tab \<star> w\<^sub>f) = ((f \<star> \<theta>\<^sub>f') \<cdot> \<a>[f, tab\<^sub>0 f, w\<^sub>f'] \<cdot> (f.tab \<star> w\<^sub>f')) \<cdot> \<beta>\<^sub>f
[PROOF STEP]
also
[PROOF STATE]
proof (state)
this:
\<a>[f, g, h \<star> tab\<^sub>0 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0] \<cdot> \<a>[f \<star> g, h, tab\<^sub>0 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0] \<cdot> (\<a>\<^sup>-\<^sup>1[f, g, h] \<star> tab\<^sub>0 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0) \<cdot> ((f \<star> g \<star> h) \<star> TTfgh_TfTgh.the_\<theta>) \<cdot> \<a>[f \<star> g \<star> h, (tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine] \<cdot> (\<a>\<^sup>-\<^sup>1[f, g \<star> h, (tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0] \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> \<a>[g \<star> h, tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0, f\<^sub>0gh\<^sub>1.p\<^sub>0]) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> \<a>\<^sup>-\<^sup>1[g, h, tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> (g \<star> \<a>[h, tab\<^sub>0 h, g\<^sub>0h\<^sub>1.p\<^sub>0]) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> (g \<star> h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> (g \<star> g\<^sub>0h\<^sub>1.\<phi>) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> \<a>[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> (g.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>1) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> f\<^sub>0gh\<^sub>1.\<phi>) \<star> TTfgh_TfTgh.chine) \<cdot> (\<a>[f, tab\<^sub>0 f, f\<^sub>0gh\<^sub>1.p\<^sub>1] \<star> TTfgh_TfTgh.chine) \<cdot> (((f.tab \<star> f\<^sub>0gh\<^sub>1.p\<^sub>1) \<star> TTfgh_TfTgh.chine) \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>1 f, f\<^sub>0gh\<^sub>1.p\<^sub>1, TTfgh_TfTgh.chine] \<cdot> \<a>[tab\<^sub>1 f, f\<^sub>0gh\<^sub>1.p\<^sub>1, TTfgh_TfTgh.chine]) \<cdot> TTfgh_TfTgh.the_\<nu> \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>1 f, f\<^sub>0g\<^sub>1.p\<^sub>1, fg\<^sub>0h\<^sub>1.p\<^sub>1] = \<a>[f, g, h \<star> tab\<^sub>0 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0] \<cdot> \<a>[f \<star> g, h, tab\<^sub>0 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0] \<cdot> (\<a>\<^sup>-\<^sup>1[f, g, h] \<star> tab\<^sub>0 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0) \<cdot> ((f \<star> g \<star> h) \<star> TTfgh_TfTgh.the_\<theta>) \<cdot> \<a>[f \<star> g \<star> h, (tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine] \<cdot> (\<a>\<^sup>-\<^sup>1[f, g \<star> h, (tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0] \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> \<a>[g \<star> h, tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0, f\<^sub>0gh\<^sub>1.p\<^sub>0]) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> \<a>\<^sup>-\<^sup>1[g, h, tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> (g \<star> \<a>[h, tab\<^sub>0 h, g\<^sub>0h\<^sub>1.p\<^sub>0]) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> (g \<star> h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> (g \<star> g\<^sub>0h\<^sub>1.\<phi>) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> \<a>[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> (g.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>1) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> f\<^sub>0gh\<^sub>1.\<phi>) \<star> TTfgh_TfTgh.chine) \<cdot> (\<a>[f, tab\<^sub>0 f, f\<^sub>0gh\<^sub>1.p\<^sub>1] \<star> TTfgh_TfTgh.chine) \<cdot> (((f.tab \<star> f\<^sub>0gh\<^sub>1.p\<^sub>1) \<star> TTfgh_TfTgh.chine) \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>1 f, f\<^sub>0gh\<^sub>1.p\<^sub>1, TTfgh_TfTgh.chine]) \<cdot> \<a>[tab\<^sub>1 f, f\<^sub>0gh\<^sub>1.p\<^sub>1, TTfgh_TfTgh.chine] \<cdot> TTfgh_TfTgh.the_\<nu> \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>1 f, f\<^sub>0g\<^sub>1.p\<^sub>1, fg\<^sub>0h\<^sub>1.p\<^sub>1]
goal (1 subgoal):
1. (f \<star> \<theta>\<^sub>f) \<cdot> \<a>[f, tab\<^sub>0 f, w\<^sub>f] \<cdot> (f.tab \<star> w\<^sub>f) = ((f \<star> \<theta>\<^sub>f') \<cdot> \<a>[f, tab\<^sub>0 f, w\<^sub>f'] \<cdot> (f.tab \<star> w\<^sub>f')) \<cdot> \<beta>\<^sub>f
[PROOF STEP]
have "... =
\<a>[f, g, h \<star> tab\<^sub>0 h \<star> TTfgh.p\<^sub>0] \<cdot>
\<a>[f \<star> g, h, tab\<^sub>0 h \<star> TTfgh.p\<^sub>0] \<cdot>
(\<a>\<^sup>-\<^sup>1[f, g, h] \<star> tab\<^sub>0 h \<star> TTfgh.p\<^sub>0) \<cdot>
((f \<star> g \<star> h) \<star> TTfgh_TfTgh.the_\<theta>) \<cdot>
\<a>[f \<star> g \<star> h, (tab\<^sub>0 h \<star> Tgh.p\<^sub>0) \<star> TfTgh.p\<^sub>0, TTfgh_TfTgh.chine] \<cdot>
(\<a>\<^sup>-\<^sup>1[f, g \<star> h, (tab\<^sub>0 h \<star> Tgh.p\<^sub>0) \<star> TfTgh.p\<^sub>0] \<star> TTfgh_TfTgh.chine) \<cdot>
((f \<star> \<a>[g \<star> h, tab\<^sub>0 h \<star> Tgh.p\<^sub>0, TfTgh.p\<^sub>0]) \<star> TTfgh_TfTgh.chine) \<cdot>
((f \<star> \<a>\<^sup>-\<^sup>1[g, h, tab\<^sub>0 h \<star> Tgh.p\<^sub>0] \<star> TfTgh.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot>
((f \<star> (g \<star> \<a>[h, tab\<^sub>0 h, Tgh.p\<^sub>0]) \<star> TfTgh.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot>
((f \<star> (g \<star> h.tab \<star> Tgh.p\<^sub>0) \<star> TfTgh.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot>
((f \<star> (g \<star> g\<^sub>0h\<^sub>1.\<phi>) \<star> TfTgh.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot>
((f \<star> \<a>[g, tab\<^sub>0 g, Tgh.p\<^sub>1] \<star> TfTgh.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot>
((f \<star> (g.tab \<star> Tgh.p\<^sub>1) \<star> TfTgh.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot>
((f \<star> f\<^sub>0gh\<^sub>1.\<phi>) \<star> TTfgh_TfTgh.chine) \<cdot>
(\<a>[f, tab\<^sub>0 f, TfTgh.p\<^sub>1] \<star> TTfgh_TfTgh.chine) \<cdot>
\<a>\<^sup>-\<^sup>1[f \<star> tab\<^sub>0 f, TfTgh.p\<^sub>1, TTfgh_TfTgh.chine] \<cdot>
(f.tab \<star> TfTgh.p\<^sub>1 \<star> TTfgh_TfTgh.chine) \<cdot>
\<a>[tab\<^sub>1 f, TfTgh.p\<^sub>1, TTfgh_TfTgh.chine] \<cdot>
TTfgh_TfTgh.the_\<nu> \<cdot>
\<a>\<^sup>-\<^sup>1[tab\<^sub>1 f, Tfg.p\<^sub>1, TTfgh.p\<^sub>1]"
[PROOF STATE]
proof (prove)
goal (1 subgoal):
1. \<a>[f, g, h \<star> tab\<^sub>0 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0] \<cdot> \<a>[f \<star> g, h, tab\<^sub>0 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0] \<cdot> (\<a>\<^sup>-\<^sup>1[f, g, h] \<star> tab\<^sub>0 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0) \<cdot> ((f \<star> g \<star> h) \<star> TTfgh_TfTgh.the_\<theta>) \<cdot> \<a>[f \<star> g \<star> h, (tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine] \<cdot> (\<a>\<^sup>-\<^sup>1[f, g \<star> h, (tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0] \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> \<a>[g \<star> h, tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0, f\<^sub>0gh\<^sub>1.p\<^sub>0]) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> \<a>\<^sup>-\<^sup>1[g, h, tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> (g \<star> \<a>[h, tab\<^sub>0 h, g\<^sub>0h\<^sub>1.p\<^sub>0]) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> (g \<star> h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> (g \<star> g\<^sub>0h\<^sub>1.\<phi>) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> \<a>[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> (g.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>1) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> f\<^sub>0gh\<^sub>1.\<phi>) \<star> TTfgh_TfTgh.chine) \<cdot> (\<a>[f, tab\<^sub>0 f, f\<^sub>0gh\<^sub>1.p\<^sub>1] \<star> TTfgh_TfTgh.chine) \<cdot> (((f.tab \<star> f\<^sub>0gh\<^sub>1.p\<^sub>1) \<star> TTfgh_TfTgh.chine) \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>1 f, f\<^sub>0gh\<^sub>1.p\<^sub>1, TTfgh_TfTgh.chine]) \<cdot> \<a>[tab\<^sub>1 f, f\<^sub>0gh\<^sub>1.p\<^sub>1, TTfgh_TfTgh.chine] \<cdot> TTfgh_TfTgh.the_\<nu> \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>1 f, f\<^sub>0g\<^sub>1.p\<^sub>1, fg\<^sub>0h\<^sub>1.p\<^sub>1] = \<a>[f, g, h \<star> tab\<^sub>0 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0] \<cdot> \<a>[f \<star> g, h, tab\<^sub>0 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0] \<cdot> (\<a>\<^sup>-\<^sup>1[f, g, h] \<star> tab\<^sub>0 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0) \<cdot> ((f \<star> g \<star> h) \<star> TTfgh_TfTgh.the_\<theta>) \<cdot> \<a>[f \<star> g \<star> h, (tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine] \<cdot> (\<a>\<^sup>-\<^sup>1[f, g \<star> h, (tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0] \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> \<a>[g \<star> h, tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0, f\<^sub>0gh\<^sub>1.p\<^sub>0]) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> \<a>\<^sup>-\<^sup>1[g, h, tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> (g \<star> \<a>[h, tab\<^sub>0 h, g\<^sub>0h\<^sub>1.p\<^sub>0]) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> (g \<star> h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> (g \<star> g\<^sub>0h\<^sub>1.\<phi>) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> \<a>[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> (g.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>1) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> f\<^sub>0gh\<^sub>1.\<phi>) \<star> TTfgh_TfTgh.chine) \<cdot> (\<a>[f, tab\<^sub>0 f, f\<^sub>0gh\<^sub>1.p\<^sub>1] \<star> TTfgh_TfTgh.chine) \<cdot> \<a>\<^sup>-\<^sup>1[f \<star> tab\<^sub>0 f, f\<^sub>0gh\<^sub>1.p\<^sub>1, TTfgh_TfTgh.chine] \<cdot> (f.tab \<star> f\<^sub>0gh\<^sub>1.p\<^sub>1 \<star> TTfgh_TfTgh.chine) \<cdot> \<a>[tab\<^sub>1 f, f\<^sub>0gh\<^sub>1.p\<^sub>1, TTfgh_TfTgh.chine] \<cdot> TTfgh_TfTgh.the_\<nu> \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>1 f, f\<^sub>0g\<^sub>1.p\<^sub>1, fg\<^sub>0h\<^sub>1.p\<^sub>1]
[PROOF STEP]
proof -
[PROOF STATE]
proof (state)
goal (1 subgoal):
1. \<a>[f, g, h \<star> tab\<^sub>0 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0] \<cdot> \<a>[f \<star> g, h, tab\<^sub>0 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0] \<cdot> (\<a>\<^sup>-\<^sup>1[f, g, h] \<star> tab\<^sub>0 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0) \<cdot> ((f \<star> g \<star> h) \<star> TTfgh_TfTgh.the_\<theta>) \<cdot> \<a>[f \<star> g \<star> h, (tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine] \<cdot> (\<a>\<^sup>-\<^sup>1[f, g \<star> h, (tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0] \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> \<a>[g \<star> h, tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0, f\<^sub>0gh\<^sub>1.p\<^sub>0]) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> \<a>\<^sup>-\<^sup>1[g, h, tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> (g \<star> \<a>[h, tab\<^sub>0 h, g\<^sub>0h\<^sub>1.p\<^sub>0]) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> (g \<star> h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> (g \<star> g\<^sub>0h\<^sub>1.\<phi>) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> \<a>[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> (g.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>1) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> f\<^sub>0gh\<^sub>1.\<phi>) \<star> TTfgh_TfTgh.chine) \<cdot> (\<a>[f, tab\<^sub>0 f, f\<^sub>0gh\<^sub>1.p\<^sub>1] \<star> TTfgh_TfTgh.chine) \<cdot> (((f.tab \<star> f\<^sub>0gh\<^sub>1.p\<^sub>1) \<star> TTfgh_TfTgh.chine) \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>1 f, f\<^sub>0gh\<^sub>1.p\<^sub>1, TTfgh_TfTgh.chine]) \<cdot> \<a>[tab\<^sub>1 f, f\<^sub>0gh\<^sub>1.p\<^sub>1, TTfgh_TfTgh.chine] \<cdot> TTfgh_TfTgh.the_\<nu> \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>1 f, f\<^sub>0g\<^sub>1.p\<^sub>1, fg\<^sub>0h\<^sub>1.p\<^sub>1] = \<a>[f, g, h \<star> tab\<^sub>0 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0] \<cdot> \<a>[f \<star> g, h, tab\<^sub>0 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0] \<cdot> (\<a>\<^sup>-\<^sup>1[f, g, h] \<star> tab\<^sub>0 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0) \<cdot> ((f \<star> g \<star> h) \<star> TTfgh_TfTgh.the_\<theta>) \<cdot> \<a>[f \<star> g \<star> h, (tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine] \<cdot> (\<a>\<^sup>-\<^sup>1[f, g \<star> h, (tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0] \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> \<a>[g \<star> h, tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0, f\<^sub>0gh\<^sub>1.p\<^sub>0]) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> \<a>\<^sup>-\<^sup>1[g, h, tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> (g \<star> \<a>[h, tab\<^sub>0 h, g\<^sub>0h\<^sub>1.p\<^sub>0]) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> (g \<star> h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> (g \<star> g\<^sub>0h\<^sub>1.\<phi>) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> \<a>[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> (g.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>1) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> f\<^sub>0gh\<^sub>1.\<phi>) \<star> TTfgh_TfTgh.chine) \<cdot> (\<a>[f, tab\<^sub>0 f, f\<^sub>0gh\<^sub>1.p\<^sub>1] \<star> TTfgh_TfTgh.chine) \<cdot> \<a>\<^sup>-\<^sup>1[f \<star> tab\<^sub>0 f, f\<^sub>0gh\<^sub>1.p\<^sub>1, TTfgh_TfTgh.chine] \<cdot> (f.tab \<star> f\<^sub>0gh\<^sub>1.p\<^sub>1 \<star> TTfgh_TfTgh.chine) \<cdot> \<a>[tab\<^sub>1 f, f\<^sub>0gh\<^sub>1.p\<^sub>1, TTfgh_TfTgh.chine] \<cdot> TTfgh_TfTgh.the_\<nu> \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>1 f, f\<^sub>0g\<^sub>1.p\<^sub>1, fg\<^sub>0h\<^sub>1.p\<^sub>1]
[PROOF STEP]
have "((f.tab \<star> TfTgh.p\<^sub>1) \<star> TTfgh_TfTgh.chine) \<cdot>
\<a>\<^sup>-\<^sup>1[tab\<^sub>1 f, TfTgh.p\<^sub>1, TTfgh_TfTgh.chine] =
\<a>\<^sup>-\<^sup>1[f \<star> tab\<^sub>0 f, TfTgh.p\<^sub>1, TTfgh_TfTgh.chine] \<cdot>
(f.tab \<star> TfTgh.p\<^sub>1 \<star> TTfgh_TfTgh.chine)"
[PROOF STATE]
proof (prove)
goal (1 subgoal):
1. ((f.tab \<star> f\<^sub>0gh\<^sub>1.p\<^sub>1) \<star> TTfgh_TfTgh.chine) \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>1 f, f\<^sub>0gh\<^sub>1.p\<^sub>1, TTfgh_TfTgh.chine] = \<a>\<^sup>-\<^sup>1[f \<star> tab\<^sub>0 f, f\<^sub>0gh\<^sub>1.p\<^sub>1, TTfgh_TfTgh.chine] \<cdot> (f.tab \<star> f\<^sub>0gh\<^sub>1.p\<^sub>1 \<star> TTfgh_TfTgh.chine)
[PROOF STEP]
using fg gh fg\<^sub>0h\<^sub>1.p\<^sub>0_simps fg\<^sub>0h\<^sub>1.p\<^sub>1_simps f\<^sub>0g\<^sub>1.p\<^sub>0_simps f\<^sub>0g\<^sub>1.p\<^sub>1_simps
assoc'_naturality [of f.tab TfTgh.p\<^sub>1 TTfgh_TfTgh.chine]
[PROOF STATE]
proof (prove)
using this:
src f = trg g
src g = trg h
trg fg\<^sub>0h\<^sub>1.p\<^sub>0 = src (tab\<^sub>1 h)
trg fg\<^sub>0h\<^sub>1.p\<^sub>1 = src (tab\<^sub>0 g \<star> f\<^sub>0g\<^sub>1.p\<^sub>0)
trg f\<^sub>0g\<^sub>1.p\<^sub>0 = src (tab\<^sub>1 g)
trg f\<^sub>0g\<^sub>1.p\<^sub>1 = src (tab\<^sub>0 f)
\<lbrakk>arr f.tab; arr f\<^sub>0gh\<^sub>1.p\<^sub>1; arr TTfgh_TfTgh.chine; src f.tab = trg f\<^sub>0gh\<^sub>1.p\<^sub>1; src f\<^sub>0gh\<^sub>1.p\<^sub>1 = trg TTfgh_TfTgh.chine\<rbrakk> \<Longrightarrow> \<a>\<^sup>-\<^sup>1[cod f.tab, cod f\<^sub>0gh\<^sub>1.p\<^sub>1, cod TTfgh_TfTgh.chine] \<cdot> (f.tab \<star> f\<^sub>0gh\<^sub>1.p\<^sub>1 \<star> TTfgh_TfTgh.chine) = ((f.tab \<star> f\<^sub>0gh\<^sub>1.p\<^sub>1) \<star> TTfgh_TfTgh.chine) \<cdot> \<a>\<^sup>-\<^sup>1[local.dom f.tab, local.dom f\<^sub>0gh\<^sub>1.p\<^sub>1, local.dom TTfgh_TfTgh.chine]
goal (1 subgoal):
1. ((f.tab \<star> f\<^sub>0gh\<^sub>1.p\<^sub>1) \<star> TTfgh_TfTgh.chine) \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>1 f, f\<^sub>0gh\<^sub>1.p\<^sub>1, TTfgh_TfTgh.chine] = \<a>\<^sup>-\<^sup>1[f \<star> tab\<^sub>0 f, f\<^sub>0gh\<^sub>1.p\<^sub>1, TTfgh_TfTgh.chine] \<cdot> (f.tab \<star> f\<^sub>0gh\<^sub>1.p\<^sub>1 \<star> TTfgh_TfTgh.chine)
[PROOF STEP]
by simp
[PROOF STATE]
proof (state)
this:
((f.tab \<star> f\<^sub>0gh\<^sub>1.p\<^sub>1) \<star> TTfgh_TfTgh.chine) \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>1 f, f\<^sub>0gh\<^sub>1.p\<^sub>1, TTfgh_TfTgh.chine] = \<a>\<^sup>-\<^sup>1[f \<star> tab\<^sub>0 f, f\<^sub>0gh\<^sub>1.p\<^sub>1, TTfgh_TfTgh.chine] \<cdot> (f.tab \<star> f\<^sub>0gh\<^sub>1.p\<^sub>1 \<star> TTfgh_TfTgh.chine)
goal (1 subgoal):
1. \<a>[f, g, h \<star> tab\<^sub>0 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0] \<cdot> \<a>[f \<star> g, h, tab\<^sub>0 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0] \<cdot> (\<a>\<^sup>-\<^sup>1[f, g, h] \<star> tab\<^sub>0 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0) \<cdot> ((f \<star> g \<star> h) \<star> TTfgh_TfTgh.the_\<theta>) \<cdot> \<a>[f \<star> g \<star> h, (tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine] \<cdot> (\<a>\<^sup>-\<^sup>1[f, g \<star> h, (tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0] \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> \<a>[g \<star> h, tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0, f\<^sub>0gh\<^sub>1.p\<^sub>0]) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> \<a>\<^sup>-\<^sup>1[g, h, tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> (g \<star> \<a>[h, tab\<^sub>0 h, g\<^sub>0h\<^sub>1.p\<^sub>0]) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> (g \<star> h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> (g \<star> g\<^sub>0h\<^sub>1.\<phi>) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> \<a>[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> (g.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>1) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> f\<^sub>0gh\<^sub>1.\<phi>) \<star> TTfgh_TfTgh.chine) \<cdot> (\<a>[f, tab\<^sub>0 f, f\<^sub>0gh\<^sub>1.p\<^sub>1] \<star> TTfgh_TfTgh.chine) \<cdot> (((f.tab \<star> f\<^sub>0gh\<^sub>1.p\<^sub>1) \<star> TTfgh_TfTgh.chine) \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>1 f, f\<^sub>0gh\<^sub>1.p\<^sub>1, TTfgh_TfTgh.chine]) \<cdot> \<a>[tab\<^sub>1 f, f\<^sub>0gh\<^sub>1.p\<^sub>1, TTfgh_TfTgh.chine] \<cdot> TTfgh_TfTgh.the_\<nu> \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>1 f, f\<^sub>0g\<^sub>1.p\<^sub>1, fg\<^sub>0h\<^sub>1.p\<^sub>1] = \<a>[f, g, h \<star> tab\<^sub>0 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0] \<cdot> \<a>[f \<star> g, h, tab\<^sub>0 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0] \<cdot> (\<a>\<^sup>-\<^sup>1[f, g, h] \<star> tab\<^sub>0 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0) \<cdot> ((f \<star> g \<star> h) \<star> TTfgh_TfTgh.the_\<theta>) \<cdot> \<a>[f \<star> g \<star> h, (tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine] \<cdot> (\<a>\<^sup>-\<^sup>1[f, g \<star> h, (tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0] \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> \<a>[g \<star> h, tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0, f\<^sub>0gh\<^sub>1.p\<^sub>0]) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> \<a>\<^sup>-\<^sup>1[g, h, tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> (g \<star> \<a>[h, tab\<^sub>0 h, g\<^sub>0h\<^sub>1.p\<^sub>0]) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> (g \<star> h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> (g \<star> g\<^sub>0h\<^sub>1.\<phi>) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> \<a>[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> (g.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>1) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> f\<^sub>0gh\<^sub>1.\<phi>) \<star> TTfgh_TfTgh.chine) \<cdot> (\<a>[f, tab\<^sub>0 f, f\<^sub>0gh\<^sub>1.p\<^sub>1] \<star> TTfgh_TfTgh.chine) \<cdot> \<a>\<^sup>-\<^sup>1[f \<star> tab\<^sub>0 f, f\<^sub>0gh\<^sub>1.p\<^sub>1, TTfgh_TfTgh.chine] \<cdot> (f.tab \<star> f\<^sub>0gh\<^sub>1.p\<^sub>1 \<star> TTfgh_TfTgh.chine) \<cdot> \<a>[tab\<^sub>1 f, f\<^sub>0gh\<^sub>1.p\<^sub>1, TTfgh_TfTgh.chine] \<cdot> TTfgh_TfTgh.the_\<nu> \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>1 f, f\<^sub>0g\<^sub>1.p\<^sub>1, fg\<^sub>0h\<^sub>1.p\<^sub>1]
[PROOF STEP]
thus ?thesis
[PROOF STATE]
proof (prove)
using this:
((f.tab \<star> f\<^sub>0gh\<^sub>1.p\<^sub>1) \<star> TTfgh_TfTgh.chine) \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>1 f, f\<^sub>0gh\<^sub>1.p\<^sub>1, TTfgh_TfTgh.chine] = \<a>\<^sup>-\<^sup>1[f \<star> tab\<^sub>0 f, f\<^sub>0gh\<^sub>1.p\<^sub>1, TTfgh_TfTgh.chine] \<cdot> (f.tab \<star> f\<^sub>0gh\<^sub>1.p\<^sub>1 \<star> TTfgh_TfTgh.chine)
goal (1 subgoal):
1. \<a>[f, g, h \<star> tab\<^sub>0 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0] \<cdot> \<a>[f \<star> g, h, tab\<^sub>0 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0] \<cdot> (\<a>\<^sup>-\<^sup>1[f, g, h] \<star> tab\<^sub>0 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0) \<cdot> ((f \<star> g \<star> h) \<star> TTfgh_TfTgh.the_\<theta>) \<cdot> \<a>[f \<star> g \<star> h, (tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine] \<cdot> (\<a>\<^sup>-\<^sup>1[f, g \<star> h, (tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0] \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> \<a>[g \<star> h, tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0, f\<^sub>0gh\<^sub>1.p\<^sub>0]) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> \<a>\<^sup>-\<^sup>1[g, h, tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> (g \<star> \<a>[h, tab\<^sub>0 h, g\<^sub>0h\<^sub>1.p\<^sub>0]) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> (g \<star> h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> (g \<star> g\<^sub>0h\<^sub>1.\<phi>) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> \<a>[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> (g.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>1) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> f\<^sub>0gh\<^sub>1.\<phi>) \<star> TTfgh_TfTgh.chine) \<cdot> (\<a>[f, tab\<^sub>0 f, f\<^sub>0gh\<^sub>1.p\<^sub>1] \<star> TTfgh_TfTgh.chine) \<cdot> (((f.tab \<star> f\<^sub>0gh\<^sub>1.p\<^sub>1) \<star> TTfgh_TfTgh.chine) \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>1 f, f\<^sub>0gh\<^sub>1.p\<^sub>1, TTfgh_TfTgh.chine]) \<cdot> \<a>[tab\<^sub>1 f, f\<^sub>0gh\<^sub>1.p\<^sub>1, TTfgh_TfTgh.chine] \<cdot> TTfgh_TfTgh.the_\<nu> \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>1 f, f\<^sub>0g\<^sub>1.p\<^sub>1, fg\<^sub>0h\<^sub>1.p\<^sub>1] = \<a>[f, g, h \<star> tab\<^sub>0 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0] \<cdot> \<a>[f \<star> g, h, tab\<^sub>0 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0] \<cdot> (\<a>\<^sup>-\<^sup>1[f, g, h] \<star> tab\<^sub>0 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0) \<cdot> ((f \<star> g \<star> h) \<star> TTfgh_TfTgh.the_\<theta>) \<cdot> \<a>[f \<star> g \<star> h, (tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine] \<cdot> (\<a>\<^sup>-\<^sup>1[f, g \<star> h, (tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0] \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> \<a>[g \<star> h, tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0, f\<^sub>0gh\<^sub>1.p\<^sub>0]) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> \<a>\<^sup>-\<^sup>1[g, h, tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> (g \<star> \<a>[h, tab\<^sub>0 h, g\<^sub>0h\<^sub>1.p\<^sub>0]) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> (g \<star> h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> (g \<star> g\<^sub>0h\<^sub>1.\<phi>) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> \<a>[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> (g.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>1) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> f\<^sub>0gh\<^sub>1.\<phi>) \<star> TTfgh_TfTgh.chine) \<cdot> (\<a>[f, tab\<^sub>0 f, f\<^sub>0gh\<^sub>1.p\<^sub>1] \<star> TTfgh_TfTgh.chine) \<cdot> \<a>\<^sup>-\<^sup>1[f \<star> tab\<^sub>0 f, f\<^sub>0gh\<^sub>1.p\<^sub>1, TTfgh_TfTgh.chine] \<cdot> (f.tab \<star> f\<^sub>0gh\<^sub>1.p\<^sub>1 \<star> TTfgh_TfTgh.chine) \<cdot> \<a>[tab\<^sub>1 f, f\<^sub>0gh\<^sub>1.p\<^sub>1, TTfgh_TfTgh.chine] \<cdot> TTfgh_TfTgh.the_\<nu> \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>1 f, f\<^sub>0g\<^sub>1.p\<^sub>1, fg\<^sub>0h\<^sub>1.p\<^sub>1]
[PROOF STEP]
using comp_assoc
[PROOF STATE]
proof (prove)
using this:
((f.tab \<star> f\<^sub>0gh\<^sub>1.p\<^sub>1) \<star> TTfgh_TfTgh.chine) \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>1 f, f\<^sub>0gh\<^sub>1.p\<^sub>1, TTfgh_TfTgh.chine] = \<a>\<^sup>-\<^sup>1[f \<star> tab\<^sub>0 f, f\<^sub>0gh\<^sub>1.p\<^sub>1, TTfgh_TfTgh.chine] \<cdot> (f.tab \<star> f\<^sub>0gh\<^sub>1.p\<^sub>1 \<star> TTfgh_TfTgh.chine)
(?h \<cdot> ?g) \<cdot> ?f = ?h \<cdot> ?g \<cdot> ?f
goal (1 subgoal):
1. \<a>[f, g, h \<star> tab\<^sub>0 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0] \<cdot> \<a>[f \<star> g, h, tab\<^sub>0 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0] \<cdot> (\<a>\<^sup>-\<^sup>1[f, g, h] \<star> tab\<^sub>0 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0) \<cdot> ((f \<star> g \<star> h) \<star> TTfgh_TfTgh.the_\<theta>) \<cdot> \<a>[f \<star> g \<star> h, (tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine] \<cdot> (\<a>\<^sup>-\<^sup>1[f, g \<star> h, (tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0] \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> \<a>[g \<star> h, tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0, f\<^sub>0gh\<^sub>1.p\<^sub>0]) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> \<a>\<^sup>-\<^sup>1[g, h, tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> (g \<star> \<a>[h, tab\<^sub>0 h, g\<^sub>0h\<^sub>1.p\<^sub>0]) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> (g \<star> h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> (g \<star> g\<^sub>0h\<^sub>1.\<phi>) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> \<a>[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> (g.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>1) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> f\<^sub>0gh\<^sub>1.\<phi>) \<star> TTfgh_TfTgh.chine) \<cdot> (\<a>[f, tab\<^sub>0 f, f\<^sub>0gh\<^sub>1.p\<^sub>1] \<star> TTfgh_TfTgh.chine) \<cdot> (((f.tab \<star> f\<^sub>0gh\<^sub>1.p\<^sub>1) \<star> TTfgh_TfTgh.chine) \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>1 f, f\<^sub>0gh\<^sub>1.p\<^sub>1, TTfgh_TfTgh.chine]) \<cdot> \<a>[tab\<^sub>1 f, f\<^sub>0gh\<^sub>1.p\<^sub>1, TTfgh_TfTgh.chine] \<cdot> TTfgh_TfTgh.the_\<nu> \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>1 f, f\<^sub>0g\<^sub>1.p\<^sub>1, fg\<^sub>0h\<^sub>1.p\<^sub>1] = \<a>[f, g, h \<star> tab\<^sub>0 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0] \<cdot> \<a>[f \<star> g, h, tab\<^sub>0 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0] \<cdot> (\<a>\<^sup>-\<^sup>1[f, g, h] \<star> tab\<^sub>0 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0) \<cdot> ((f \<star> g \<star> h) \<star> TTfgh_TfTgh.the_\<theta>) \<cdot> \<a>[f \<star> g \<star> h, (tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine] \<cdot> (\<a>\<^sup>-\<^sup>1[f, g \<star> h, (tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0] \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> \<a>[g \<star> h, tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0, f\<^sub>0gh\<^sub>1.p\<^sub>0]) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> \<a>\<^sup>-\<^sup>1[g, h, tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> (g \<star> \<a>[h, tab\<^sub>0 h, g\<^sub>0h\<^sub>1.p\<^sub>0]) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> (g \<star> h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> (g \<star> g\<^sub>0h\<^sub>1.\<phi>) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> \<a>[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> (g.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>1) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> f\<^sub>0gh\<^sub>1.\<phi>) \<star> TTfgh_TfTgh.chine) \<cdot> (\<a>[f, tab\<^sub>0 f, f\<^sub>0gh\<^sub>1.p\<^sub>1] \<star> TTfgh_TfTgh.chine) \<cdot> \<a>\<^sup>-\<^sup>1[f \<star> tab\<^sub>0 f, f\<^sub>0gh\<^sub>1.p\<^sub>1, TTfgh_TfTgh.chine] \<cdot> (f.tab \<star> f\<^sub>0gh\<^sub>1.p\<^sub>1 \<star> TTfgh_TfTgh.chine) \<cdot> \<a>[tab\<^sub>1 f, f\<^sub>0gh\<^sub>1.p\<^sub>1, TTfgh_TfTgh.chine] \<cdot> TTfgh_TfTgh.the_\<nu> \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>1 f, f\<^sub>0g\<^sub>1.p\<^sub>1, fg\<^sub>0h\<^sub>1.p\<^sub>1]
[PROOF STEP]
by presburger
[PROOF STATE]
proof (state)
this:
\<a>[f, g, h \<star> tab\<^sub>0 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0] \<cdot> \<a>[f \<star> g, h, tab\<^sub>0 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0] \<cdot> (\<a>\<^sup>-\<^sup>1[f, g, h] \<star> tab\<^sub>0 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0) \<cdot> ((f \<star> g \<star> h) \<star> TTfgh_TfTgh.the_\<theta>) \<cdot> \<a>[f \<star> g \<star> h, (tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine] \<cdot> (\<a>\<^sup>-\<^sup>1[f, g \<star> h, (tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0] \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> \<a>[g \<star> h, tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0, f\<^sub>0gh\<^sub>1.p\<^sub>0]) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> \<a>\<^sup>-\<^sup>1[g, h, tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> (g \<star> \<a>[h, tab\<^sub>0 h, g\<^sub>0h\<^sub>1.p\<^sub>0]) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> (g \<star> h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> (g \<star> g\<^sub>0h\<^sub>1.\<phi>) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> \<a>[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> (g.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>1) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> f\<^sub>0gh\<^sub>1.\<phi>) \<star> TTfgh_TfTgh.chine) \<cdot> (\<a>[f, tab\<^sub>0 f, f\<^sub>0gh\<^sub>1.p\<^sub>1] \<star> TTfgh_TfTgh.chine) \<cdot> (((f.tab \<star> f\<^sub>0gh\<^sub>1.p\<^sub>1) \<star> TTfgh_TfTgh.chine) \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>1 f, f\<^sub>0gh\<^sub>1.p\<^sub>1, TTfgh_TfTgh.chine]) \<cdot> \<a>[tab\<^sub>1 f, f\<^sub>0gh\<^sub>1.p\<^sub>1, TTfgh_TfTgh.chine] \<cdot> TTfgh_TfTgh.the_\<nu> \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>1 f, f\<^sub>0g\<^sub>1.p\<^sub>1, fg\<^sub>0h\<^sub>1.p\<^sub>1] = \<a>[f, g, h \<star> tab\<^sub>0 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0] \<cdot> \<a>[f \<star> g, h, tab\<^sub>0 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0] \<cdot> (\<a>\<^sup>-\<^sup>1[f, g, h] \<star> tab\<^sub>0 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0) \<cdot> ((f \<star> g \<star> h) \<star> TTfgh_TfTgh.the_\<theta>) \<cdot> \<a>[f \<star> g \<star> h, (tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine] \<cdot> (\<a>\<^sup>-\<^sup>1[f, g \<star> h, (tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0] \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> \<a>[g \<star> h, tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0, f\<^sub>0gh\<^sub>1.p\<^sub>0]) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> \<a>\<^sup>-\<^sup>1[g, h, tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> (g \<star> \<a>[h, tab\<^sub>0 h, g\<^sub>0h\<^sub>1.p\<^sub>0]) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> (g \<star> h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> (g \<star> g\<^sub>0h\<^sub>1.\<phi>) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> \<a>[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> (g.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>1) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> f\<^sub>0gh\<^sub>1.\<phi>) \<star> TTfgh_TfTgh.chine) \<cdot> (\<a>[f, tab\<^sub>0 f, f\<^sub>0gh\<^sub>1.p\<^sub>1] \<star> TTfgh_TfTgh.chine) \<cdot> \<a>\<^sup>-\<^sup>1[f \<star> tab\<^sub>0 f, f\<^sub>0gh\<^sub>1.p\<^sub>1, TTfgh_TfTgh.chine] \<cdot> (f.tab \<star> f\<^sub>0gh\<^sub>1.p\<^sub>1 \<star> TTfgh_TfTgh.chine) \<cdot> \<a>[tab\<^sub>1 f, f\<^sub>0gh\<^sub>1.p\<^sub>1, TTfgh_TfTgh.chine] \<cdot> TTfgh_TfTgh.the_\<nu> \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>1 f, f\<^sub>0g\<^sub>1.p\<^sub>1, fg\<^sub>0h\<^sub>1.p\<^sub>1]
goal:
No subgoals!
[PROOF STEP]
qed
[PROOF STATE]
proof (state)
this:
\<a>[f, g, h \<star> tab\<^sub>0 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0] \<cdot> \<a>[f \<star> g, h, tab\<^sub>0 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0] \<cdot> (\<a>\<^sup>-\<^sup>1[f, g, h] \<star> tab\<^sub>0 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0) \<cdot> ((f \<star> g \<star> h) \<star> TTfgh_TfTgh.the_\<theta>) \<cdot> \<a>[f \<star> g \<star> h, (tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine] \<cdot> (\<a>\<^sup>-\<^sup>1[f, g \<star> h, (tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0] \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> \<a>[g \<star> h, tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0, f\<^sub>0gh\<^sub>1.p\<^sub>0]) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> \<a>\<^sup>-\<^sup>1[g, h, tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> (g \<star> \<a>[h, tab\<^sub>0 h, g\<^sub>0h\<^sub>1.p\<^sub>0]) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> (g \<star> h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> (g \<star> g\<^sub>0h\<^sub>1.\<phi>) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> \<a>[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> (g.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>1) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> f\<^sub>0gh\<^sub>1.\<phi>) \<star> TTfgh_TfTgh.chine) \<cdot> (\<a>[f, tab\<^sub>0 f, f\<^sub>0gh\<^sub>1.p\<^sub>1] \<star> TTfgh_TfTgh.chine) \<cdot> (((f.tab \<star> f\<^sub>0gh\<^sub>1.p\<^sub>1) \<star> TTfgh_TfTgh.chine) \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>1 f, f\<^sub>0gh\<^sub>1.p\<^sub>1, TTfgh_TfTgh.chine]) \<cdot> \<a>[tab\<^sub>1 f, f\<^sub>0gh\<^sub>1.p\<^sub>1, TTfgh_TfTgh.chine] \<cdot> TTfgh_TfTgh.the_\<nu> \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>1 f, f\<^sub>0g\<^sub>1.p\<^sub>1, fg\<^sub>0h\<^sub>1.p\<^sub>1] = \<a>[f, g, h \<star> tab\<^sub>0 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0] \<cdot> \<a>[f \<star> g, h, tab\<^sub>0 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0] \<cdot> (\<a>\<^sup>-\<^sup>1[f, g, h] \<star> tab\<^sub>0 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0) \<cdot> ((f \<star> g \<star> h) \<star> TTfgh_TfTgh.the_\<theta>) \<cdot> \<a>[f \<star> g \<star> h, (tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine] \<cdot> (\<a>\<^sup>-\<^sup>1[f, g \<star> h, (tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0] \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> \<a>[g \<star> h, tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0, f\<^sub>0gh\<^sub>1.p\<^sub>0]) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> \<a>\<^sup>-\<^sup>1[g, h, tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> (g \<star> \<a>[h, tab\<^sub>0 h, g\<^sub>0h\<^sub>1.p\<^sub>0]) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> (g \<star> h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> (g \<star> g\<^sub>0h\<^sub>1.\<phi>) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> \<a>[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> (g.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>1) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> f\<^sub>0gh\<^sub>1.\<phi>) \<star> TTfgh_TfTgh.chine) \<cdot> (\<a>[f, tab\<^sub>0 f, f\<^sub>0gh\<^sub>1.p\<^sub>1] \<star> TTfgh_TfTgh.chine) \<cdot> \<a>\<^sup>-\<^sup>1[f \<star> tab\<^sub>0 f, f\<^sub>0gh\<^sub>1.p\<^sub>1, TTfgh_TfTgh.chine] \<cdot> (f.tab \<star> f\<^sub>0gh\<^sub>1.p\<^sub>1 \<star> TTfgh_TfTgh.chine) \<cdot> \<a>[tab\<^sub>1 f, f\<^sub>0gh\<^sub>1.p\<^sub>1, TTfgh_TfTgh.chine] \<cdot> TTfgh_TfTgh.the_\<nu> \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>1 f, f\<^sub>0g\<^sub>1.p\<^sub>1, fg\<^sub>0h\<^sub>1.p\<^sub>1]
goal (1 subgoal):
1. (f \<star> \<theta>\<^sub>f) \<cdot> \<a>[f, tab\<^sub>0 f, w\<^sub>f] \<cdot> (f.tab \<star> w\<^sub>f) = ((f \<star> \<theta>\<^sub>f') \<cdot> \<a>[f, tab\<^sub>0 f, w\<^sub>f'] \<cdot> (f.tab \<star> w\<^sub>f')) \<cdot> \<beta>\<^sub>f
[PROOF STEP]
also
[PROOF STATE]
proof (state)
this:
\<a>[f, g, h \<star> tab\<^sub>0 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0] \<cdot> \<a>[f \<star> g, h, tab\<^sub>0 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0] \<cdot> (\<a>\<^sup>-\<^sup>1[f, g, h] \<star> tab\<^sub>0 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0) \<cdot> ((f \<star> g \<star> h) \<star> TTfgh_TfTgh.the_\<theta>) \<cdot> \<a>[f \<star> g \<star> h, (tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine] \<cdot> (\<a>\<^sup>-\<^sup>1[f, g \<star> h, (tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0] \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> \<a>[g \<star> h, tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0, f\<^sub>0gh\<^sub>1.p\<^sub>0]) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> \<a>\<^sup>-\<^sup>1[g, h, tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> (g \<star> \<a>[h, tab\<^sub>0 h, g\<^sub>0h\<^sub>1.p\<^sub>0]) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> (g \<star> h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> (g \<star> g\<^sub>0h\<^sub>1.\<phi>) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> \<a>[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> (g.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>1) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> f\<^sub>0gh\<^sub>1.\<phi>) \<star> TTfgh_TfTgh.chine) \<cdot> (\<a>[f, tab\<^sub>0 f, f\<^sub>0gh\<^sub>1.p\<^sub>1] \<star> TTfgh_TfTgh.chine) \<cdot> (((f.tab \<star> f\<^sub>0gh\<^sub>1.p\<^sub>1) \<star> TTfgh_TfTgh.chine) \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>1 f, f\<^sub>0gh\<^sub>1.p\<^sub>1, TTfgh_TfTgh.chine]) \<cdot> \<a>[tab\<^sub>1 f, f\<^sub>0gh\<^sub>1.p\<^sub>1, TTfgh_TfTgh.chine] \<cdot> TTfgh_TfTgh.the_\<nu> \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>1 f, f\<^sub>0g\<^sub>1.p\<^sub>1, fg\<^sub>0h\<^sub>1.p\<^sub>1] = \<a>[f, g, h \<star> tab\<^sub>0 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0] \<cdot> \<a>[f \<star> g, h, tab\<^sub>0 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0] \<cdot> (\<a>\<^sup>-\<^sup>1[f, g, h] \<star> tab\<^sub>0 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0) \<cdot> ((f \<star> g \<star> h) \<star> TTfgh_TfTgh.the_\<theta>) \<cdot> \<a>[f \<star> g \<star> h, (tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine] \<cdot> (\<a>\<^sup>-\<^sup>1[f, g \<star> h, (tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0] \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> \<a>[g \<star> h, tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0, f\<^sub>0gh\<^sub>1.p\<^sub>0]) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> \<a>\<^sup>-\<^sup>1[g, h, tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> (g \<star> \<a>[h, tab\<^sub>0 h, g\<^sub>0h\<^sub>1.p\<^sub>0]) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> (g \<star> h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> (g \<star> g\<^sub>0h\<^sub>1.\<phi>) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> \<a>[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> (g.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>1) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> f\<^sub>0gh\<^sub>1.\<phi>) \<star> TTfgh_TfTgh.chine) \<cdot> (\<a>[f, tab\<^sub>0 f, f\<^sub>0gh\<^sub>1.p\<^sub>1] \<star> TTfgh_TfTgh.chine) \<cdot> \<a>\<^sup>-\<^sup>1[f \<star> tab\<^sub>0 f, f\<^sub>0gh\<^sub>1.p\<^sub>1, TTfgh_TfTgh.chine] \<cdot> (f.tab \<star> f\<^sub>0gh\<^sub>1.p\<^sub>1 \<star> TTfgh_TfTgh.chine) \<cdot> \<a>[tab\<^sub>1 f, f\<^sub>0gh\<^sub>1.p\<^sub>1, TTfgh_TfTgh.chine] \<cdot> TTfgh_TfTgh.the_\<nu> \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>1 f, f\<^sub>0g\<^sub>1.p\<^sub>1, fg\<^sub>0h\<^sub>1.p\<^sub>1]
goal (1 subgoal):
1. (f \<star> \<theta>\<^sub>f) \<cdot> \<a>[f, tab\<^sub>0 f, w\<^sub>f] \<cdot> (f.tab \<star> w\<^sub>f) = ((f \<star> \<theta>\<^sub>f') \<cdot> \<a>[f, tab\<^sub>0 f, w\<^sub>f'] \<cdot> (f.tab \<star> w\<^sub>f')) \<cdot> \<beta>\<^sub>f
[PROOF STEP]
have "... =
\<a>[f, g, h \<star> tab\<^sub>0 h \<star> TTfgh.p\<^sub>0] \<cdot>
\<a>[f \<star> g, h, tab\<^sub>0 h \<star> TTfgh.p\<^sub>0] \<cdot>
(\<a>\<^sup>-\<^sup>1[f, g, h] \<star> tab\<^sub>0 h \<star> TTfgh.p\<^sub>0) \<cdot>
((f \<star> g \<star> h) \<star> TTfgh_TfTgh.the_\<theta>) \<cdot>
\<a>[f \<star> g \<star> h, (tab\<^sub>0 h \<star> Tgh.p\<^sub>0) \<star> TfTgh.p\<^sub>0, TTfgh_TfTgh.chine] \<cdot>
(\<a>\<^sup>-\<^sup>1[f, g \<star> h, (tab\<^sub>0 h \<star> Tgh.p\<^sub>0) \<star> TfTgh.p\<^sub>0] \<star> TTfgh_TfTgh.chine) \<cdot>
((f \<star> \<a>[g \<star> h, tab\<^sub>0 h \<star> Tgh.p\<^sub>0, TfTgh.p\<^sub>0]) \<star> TTfgh_TfTgh.chine) \<cdot>
((f \<star> \<a>\<^sup>-\<^sup>1[g, h, tab\<^sub>0 h \<star> Tgh.p\<^sub>0] \<star> TfTgh.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot>
((f \<star> (g \<star> \<a>[h, tab\<^sub>0 h, Tgh.p\<^sub>0]) \<star> TfTgh.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot>
((f \<star> (g \<star> h.tab \<star> Tgh.p\<^sub>0) \<star> TfTgh.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot>
((f \<star> (g \<star> g\<^sub>0h\<^sub>1.\<phi>) \<star> TfTgh.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot>
((f \<star> \<a>[g, tab\<^sub>0 g, Tgh.p\<^sub>1] \<star> TfTgh.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot>
((f \<star> (g.tab \<star> Tgh.p\<^sub>1) \<star> TfTgh.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot>
((f \<star> f\<^sub>0gh\<^sub>1.\<phi>) \<star> TTfgh_TfTgh.chine) \<cdot>
(\<a>[f, tab\<^sub>0 f, TfTgh.p\<^sub>1] \<star> TTfgh_TfTgh.chine) \<cdot>
\<a>\<^sup>-\<^sup>1[f \<star> tab\<^sub>0 f, TfTgh.p\<^sub>1, TTfgh_TfTgh.chine] \<cdot>
((\<a>\<^sup>-\<^sup>1[f, tab\<^sub>0 f, TfTgh.p\<^sub>1 \<star> TTfgh_TfTgh.chine] \<cdot>
\<a>[f, tab\<^sub>0 f, TfTgh.p\<^sub>1 \<star> TTfgh_TfTgh.chine]) \<cdot>
(f.tab \<star> TfTgh.p\<^sub>1 \<star> TTfgh_TfTgh.chine)) \<cdot>
\<a>[tab\<^sub>1 f, TfTgh.p\<^sub>1, TTfgh_TfTgh.chine] \<cdot>
TTfgh_TfTgh.the_\<nu> \<cdot>
\<a>\<^sup>-\<^sup>1[tab\<^sub>1 f, Tfg.p\<^sub>1, TTfgh.p\<^sub>1]"
[PROOF STATE]
proof (prove)
goal (1 subgoal):
1. \<a>[f, g, h \<star> tab\<^sub>0 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0] \<cdot> \<a>[f \<star> g, h, tab\<^sub>0 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0] \<cdot> (\<a>\<^sup>-\<^sup>1[f, g, h] \<star> tab\<^sub>0 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0) \<cdot> ((f \<star> g \<star> h) \<star> TTfgh_TfTgh.the_\<theta>) \<cdot> \<a>[f \<star> g \<star> h, (tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine] \<cdot> (\<a>\<^sup>-\<^sup>1[f, g \<star> h, (tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0] \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> \<a>[g \<star> h, tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0, f\<^sub>0gh\<^sub>1.p\<^sub>0]) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> \<a>\<^sup>-\<^sup>1[g, h, tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> (g \<star> \<a>[h, tab\<^sub>0 h, g\<^sub>0h\<^sub>1.p\<^sub>0]) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> (g \<star> h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> (g \<star> g\<^sub>0h\<^sub>1.\<phi>) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> \<a>[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> (g.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>1) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> f\<^sub>0gh\<^sub>1.\<phi>) \<star> TTfgh_TfTgh.chine) \<cdot> (\<a>[f, tab\<^sub>0 f, f\<^sub>0gh\<^sub>1.p\<^sub>1] \<star> TTfgh_TfTgh.chine) \<cdot> \<a>\<^sup>-\<^sup>1[f \<star> tab\<^sub>0 f, f\<^sub>0gh\<^sub>1.p\<^sub>1, TTfgh_TfTgh.chine] \<cdot> (f.tab \<star> f\<^sub>0gh\<^sub>1.p\<^sub>1 \<star> TTfgh_TfTgh.chine) \<cdot> \<a>[tab\<^sub>1 f, f\<^sub>0gh\<^sub>1.p\<^sub>1, TTfgh_TfTgh.chine] \<cdot> TTfgh_TfTgh.the_\<nu> \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>1 f, f\<^sub>0g\<^sub>1.p\<^sub>1, fg\<^sub>0h\<^sub>1.p\<^sub>1] = \<a>[f, g, h \<star> tab\<^sub>0 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0] \<cdot> \<a>[f \<star> g, h, tab\<^sub>0 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0] \<cdot> (\<a>\<^sup>-\<^sup>1[f, g, h] \<star> tab\<^sub>0 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0) \<cdot> ((f \<star> g \<star> h) \<star> TTfgh_TfTgh.the_\<theta>) \<cdot> \<a>[f \<star> g \<star> h, (tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine] \<cdot> (\<a>\<^sup>-\<^sup>1[f, g \<star> h, (tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0] \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> \<a>[g \<star> h, tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0, f\<^sub>0gh\<^sub>1.p\<^sub>0]) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> \<a>\<^sup>-\<^sup>1[g, h, tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> (g \<star> \<a>[h, tab\<^sub>0 h, g\<^sub>0h\<^sub>1.p\<^sub>0]) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> (g \<star> h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> (g \<star> g\<^sub>0h\<^sub>1.\<phi>) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> \<a>[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> (g.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>1) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> f\<^sub>0gh\<^sub>1.\<phi>) \<star> TTfgh_TfTgh.chine) \<cdot> (\<a>[f, tab\<^sub>0 f, f\<^sub>0gh\<^sub>1.p\<^sub>1] \<star> TTfgh_TfTgh.chine) \<cdot> \<a>\<^sup>-\<^sup>1[f \<star> tab\<^sub>0 f, f\<^sub>0gh\<^sub>1.p\<^sub>1, TTfgh_TfTgh.chine] \<cdot> ((\<a>\<^sup>-\<^sup>1[f, tab\<^sub>0 f, f\<^sub>0gh\<^sub>1.p\<^sub>1 \<star> TTfgh_TfTgh.chine] \<cdot> \<a>[f, tab\<^sub>0 f, f\<^sub>0gh\<^sub>1.p\<^sub>1 \<star> TTfgh_TfTgh.chine]) \<cdot> (f.tab \<star> f\<^sub>0gh\<^sub>1.p\<^sub>1 \<star> TTfgh_TfTgh.chine)) \<cdot> \<a>[tab\<^sub>1 f, f\<^sub>0gh\<^sub>1.p\<^sub>1, TTfgh_TfTgh.chine] \<cdot> TTfgh_TfTgh.the_\<nu> \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>1 f, f\<^sub>0g\<^sub>1.p\<^sub>1, fg\<^sub>0h\<^sub>1.p\<^sub>1]
[PROOF STEP]
proof -
[PROOF STATE]
proof (state)
goal (1 subgoal):
1. \<a>[f, g, h \<star> tab\<^sub>0 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0] \<cdot> \<a>[f \<star> g, h, tab\<^sub>0 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0] \<cdot> (\<a>\<^sup>-\<^sup>1[f, g, h] \<star> tab\<^sub>0 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0) \<cdot> ((f \<star> g \<star> h) \<star> TTfgh_TfTgh.the_\<theta>) \<cdot> \<a>[f \<star> g \<star> h, (tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine] \<cdot> (\<a>\<^sup>-\<^sup>1[f, g \<star> h, (tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0] \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> \<a>[g \<star> h, tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0, f\<^sub>0gh\<^sub>1.p\<^sub>0]) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> \<a>\<^sup>-\<^sup>1[g, h, tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> (g \<star> \<a>[h, tab\<^sub>0 h, g\<^sub>0h\<^sub>1.p\<^sub>0]) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> (g \<star> h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> (g \<star> g\<^sub>0h\<^sub>1.\<phi>) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> \<a>[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> (g.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>1) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> f\<^sub>0gh\<^sub>1.\<phi>) \<star> TTfgh_TfTgh.chine) \<cdot> (\<a>[f, tab\<^sub>0 f, f\<^sub>0gh\<^sub>1.p\<^sub>1] \<star> TTfgh_TfTgh.chine) \<cdot> \<a>\<^sup>-\<^sup>1[f \<star> tab\<^sub>0 f, f\<^sub>0gh\<^sub>1.p\<^sub>1, TTfgh_TfTgh.chine] \<cdot> (f.tab \<star> f\<^sub>0gh\<^sub>1.p\<^sub>1 \<star> TTfgh_TfTgh.chine) \<cdot> \<a>[tab\<^sub>1 f, f\<^sub>0gh\<^sub>1.p\<^sub>1, TTfgh_TfTgh.chine] \<cdot> TTfgh_TfTgh.the_\<nu> \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>1 f, f\<^sub>0g\<^sub>1.p\<^sub>1, fg\<^sub>0h\<^sub>1.p\<^sub>1] = \<a>[f, g, h \<star> tab\<^sub>0 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0] \<cdot> \<a>[f \<star> g, h, tab\<^sub>0 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0] \<cdot> (\<a>\<^sup>-\<^sup>1[f, g, h] \<star> tab\<^sub>0 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0) \<cdot> ((f \<star> g \<star> h) \<star> TTfgh_TfTgh.the_\<theta>) \<cdot> \<a>[f \<star> g \<star> h, (tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine] \<cdot> (\<a>\<^sup>-\<^sup>1[f, g \<star> h, (tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0] \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> \<a>[g \<star> h, tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0, f\<^sub>0gh\<^sub>1.p\<^sub>0]) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> \<a>\<^sup>-\<^sup>1[g, h, tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> (g \<star> \<a>[h, tab\<^sub>0 h, g\<^sub>0h\<^sub>1.p\<^sub>0]) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> (g \<star> h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> (g \<star> g\<^sub>0h\<^sub>1.\<phi>) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> \<a>[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> (g.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>1) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> f\<^sub>0gh\<^sub>1.\<phi>) \<star> TTfgh_TfTgh.chine) \<cdot> (\<a>[f, tab\<^sub>0 f, f\<^sub>0gh\<^sub>1.p\<^sub>1] \<star> TTfgh_TfTgh.chine) \<cdot> \<a>\<^sup>-\<^sup>1[f \<star> tab\<^sub>0 f, f\<^sub>0gh\<^sub>1.p\<^sub>1, TTfgh_TfTgh.chine] \<cdot> ((\<a>\<^sup>-\<^sup>1[f, tab\<^sub>0 f, f\<^sub>0gh\<^sub>1.p\<^sub>1 \<star> TTfgh_TfTgh.chine] \<cdot> \<a>[f, tab\<^sub>0 f, f\<^sub>0gh\<^sub>1.p\<^sub>1 \<star> TTfgh_TfTgh.chine]) \<cdot> (f.tab \<star> f\<^sub>0gh\<^sub>1.p\<^sub>1 \<star> TTfgh_TfTgh.chine)) \<cdot> \<a>[tab\<^sub>1 f, f\<^sub>0gh\<^sub>1.p\<^sub>1, TTfgh_TfTgh.chine] \<cdot> TTfgh_TfTgh.the_\<nu> \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>1 f, f\<^sub>0g\<^sub>1.p\<^sub>1, fg\<^sub>0h\<^sub>1.p\<^sub>1]
[PROOF STEP]
have "(\<a>\<^sup>-\<^sup>1[f, tab\<^sub>0 f, TfTgh.p\<^sub>1 \<star> TTfgh_TfTgh.chine] \<cdot>
\<a>[f, tab\<^sub>0 f, TfTgh.p\<^sub>1 \<star> TTfgh_TfTgh.chine]) \<cdot>
(f.tab \<star> TfTgh.p\<^sub>1 \<star> TTfgh_TfTgh.chine) =
f.tab \<star> TfTgh.p\<^sub>1 \<star> TTfgh_TfTgh.chine"
[PROOF STATE]
proof (prove)
goal (1 subgoal):
1. (\<a>\<^sup>-\<^sup>1[f, tab\<^sub>0 f, f\<^sub>0gh\<^sub>1.p\<^sub>1 \<star> TTfgh_TfTgh.chine] \<cdot> \<a>[f, tab\<^sub>0 f, f\<^sub>0gh\<^sub>1.p\<^sub>1 \<star> TTfgh_TfTgh.chine]) \<cdot> (f.tab \<star> f\<^sub>0gh\<^sub>1.p\<^sub>1 \<star> TTfgh_TfTgh.chine) = f.tab \<star> f\<^sub>0gh\<^sub>1.p\<^sub>1 \<star> TTfgh_TfTgh.chine
[PROOF STEP]
using fg gh fg\<^sub>0h\<^sub>1.p\<^sub>0_simps fg\<^sub>0h\<^sub>1.p\<^sub>1_simps f\<^sub>0g\<^sub>1.p\<^sub>0_simps f\<^sub>0g\<^sub>1.p\<^sub>1_simps
comp_cod_arr comp_assoc_assoc'
[PROOF STATE]
proof (prove)
using this:
src f = trg g
src g = trg h
trg fg\<^sub>0h\<^sub>1.p\<^sub>0 = src (tab\<^sub>1 h)
trg fg\<^sub>0h\<^sub>1.p\<^sub>1 = src (tab\<^sub>0 g \<star> f\<^sub>0g\<^sub>1.p\<^sub>0)
trg f\<^sub>0g\<^sub>1.p\<^sub>0 = src (tab\<^sub>1 g)
trg f\<^sub>0g\<^sub>1.p\<^sub>1 = src (tab\<^sub>0 f)
\<lbrakk>arr ?f; cod ?f = ?b\<rbrakk> \<Longrightarrow> ?b \<cdot> ?f = ?f
\<lbrakk>ide ?f; ide ?g; ide ?h; src ?f = trg ?g; src ?g = trg ?h\<rbrakk> \<Longrightarrow> \<a>[?f, ?g, ?h] \<cdot> \<a>\<^sup>-\<^sup>1[?f, ?g, ?h] = ?f \<star> ?g \<star> ?h
\<lbrakk>ide ?f; ide ?g; ide ?h; src ?f = trg ?g; src ?g = trg ?h\<rbrakk> \<Longrightarrow> \<a>\<^sup>-\<^sup>1[?f, ?g, ?h] \<cdot> \<a>[?f, ?g, ?h] = (?f \<star> ?g) \<star> ?h
goal (1 subgoal):
1. (\<a>\<^sup>-\<^sup>1[f, tab\<^sub>0 f, f\<^sub>0gh\<^sub>1.p\<^sub>1 \<star> TTfgh_TfTgh.chine] \<cdot> \<a>[f, tab\<^sub>0 f, f\<^sub>0gh\<^sub>1.p\<^sub>1 \<star> TTfgh_TfTgh.chine]) \<cdot> (f.tab \<star> f\<^sub>0gh\<^sub>1.p\<^sub>1 \<star> TTfgh_TfTgh.chine) = f.tab \<star> f\<^sub>0gh\<^sub>1.p\<^sub>1 \<star> TTfgh_TfTgh.chine
[PROOF STEP]
by simp
[PROOF STATE]
proof (state)
this:
(\<a>\<^sup>-\<^sup>1[f, tab\<^sub>0 f, f\<^sub>0gh\<^sub>1.p\<^sub>1 \<star> TTfgh_TfTgh.chine] \<cdot> \<a>[f, tab\<^sub>0 f, f\<^sub>0gh\<^sub>1.p\<^sub>1 \<star> TTfgh_TfTgh.chine]) \<cdot> (f.tab \<star> f\<^sub>0gh\<^sub>1.p\<^sub>1 \<star> TTfgh_TfTgh.chine) = f.tab \<star> f\<^sub>0gh\<^sub>1.p\<^sub>1 \<star> TTfgh_TfTgh.chine
goal (1 subgoal):
1. \<a>[f, g, h \<star> tab\<^sub>0 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0] \<cdot> \<a>[f \<star> g, h, tab\<^sub>0 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0] \<cdot> (\<a>\<^sup>-\<^sup>1[f, g, h] \<star> tab\<^sub>0 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0) \<cdot> ((f \<star> g \<star> h) \<star> TTfgh_TfTgh.the_\<theta>) \<cdot> \<a>[f \<star> g \<star> h, (tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine] \<cdot> (\<a>\<^sup>-\<^sup>1[f, g \<star> h, (tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0] \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> \<a>[g \<star> h, tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0, f\<^sub>0gh\<^sub>1.p\<^sub>0]) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> \<a>\<^sup>-\<^sup>1[g, h, tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> (g \<star> \<a>[h, tab\<^sub>0 h, g\<^sub>0h\<^sub>1.p\<^sub>0]) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> (g \<star> h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> (g \<star> g\<^sub>0h\<^sub>1.\<phi>) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> \<a>[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> (g.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>1) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> f\<^sub>0gh\<^sub>1.\<phi>) \<star> TTfgh_TfTgh.chine) \<cdot> (\<a>[f, tab\<^sub>0 f, f\<^sub>0gh\<^sub>1.p\<^sub>1] \<star> TTfgh_TfTgh.chine) \<cdot> \<a>\<^sup>-\<^sup>1[f \<star> tab\<^sub>0 f, f\<^sub>0gh\<^sub>1.p\<^sub>1, TTfgh_TfTgh.chine] \<cdot> (f.tab \<star> f\<^sub>0gh\<^sub>1.p\<^sub>1 \<star> TTfgh_TfTgh.chine) \<cdot> \<a>[tab\<^sub>1 f, f\<^sub>0gh\<^sub>1.p\<^sub>1, TTfgh_TfTgh.chine] \<cdot> TTfgh_TfTgh.the_\<nu> \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>1 f, f\<^sub>0g\<^sub>1.p\<^sub>1, fg\<^sub>0h\<^sub>1.p\<^sub>1] = \<a>[f, g, h \<star> tab\<^sub>0 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0] \<cdot> \<a>[f \<star> g, h, tab\<^sub>0 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0] \<cdot> (\<a>\<^sup>-\<^sup>1[f, g, h] \<star> tab\<^sub>0 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0) \<cdot> ((f \<star> g \<star> h) \<star> TTfgh_TfTgh.the_\<theta>) \<cdot> \<a>[f \<star> g \<star> h, (tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine] \<cdot> (\<a>\<^sup>-\<^sup>1[f, g \<star> h, (tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0] \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> \<a>[g \<star> h, tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0, f\<^sub>0gh\<^sub>1.p\<^sub>0]) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> \<a>\<^sup>-\<^sup>1[g, h, tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> (g \<star> \<a>[h, tab\<^sub>0 h, g\<^sub>0h\<^sub>1.p\<^sub>0]) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> (g \<star> h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> (g \<star> g\<^sub>0h\<^sub>1.\<phi>) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> \<a>[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> (g.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>1) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> f\<^sub>0gh\<^sub>1.\<phi>) \<star> TTfgh_TfTgh.chine) \<cdot> (\<a>[f, tab\<^sub>0 f, f\<^sub>0gh\<^sub>1.p\<^sub>1] \<star> TTfgh_TfTgh.chine) \<cdot> \<a>\<^sup>-\<^sup>1[f \<star> tab\<^sub>0 f, f\<^sub>0gh\<^sub>1.p\<^sub>1, TTfgh_TfTgh.chine] \<cdot> ((\<a>\<^sup>-\<^sup>1[f, tab\<^sub>0 f, f\<^sub>0gh\<^sub>1.p\<^sub>1 \<star> TTfgh_TfTgh.chine] \<cdot> \<a>[f, tab\<^sub>0 f, f\<^sub>0gh\<^sub>1.p\<^sub>1 \<star> TTfgh_TfTgh.chine]) \<cdot> (f.tab \<star> f\<^sub>0gh\<^sub>1.p\<^sub>1 \<star> TTfgh_TfTgh.chine)) \<cdot> \<a>[tab\<^sub>1 f, f\<^sub>0gh\<^sub>1.p\<^sub>1, TTfgh_TfTgh.chine] \<cdot> TTfgh_TfTgh.the_\<nu> \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>1 f, f\<^sub>0g\<^sub>1.p\<^sub>1, fg\<^sub>0h\<^sub>1.p\<^sub>1]
[PROOF STEP]
thus ?thesis
[PROOF STATE]
proof (prove)
using this:
(\<a>\<^sup>-\<^sup>1[f, tab\<^sub>0 f, f\<^sub>0gh\<^sub>1.p\<^sub>1 \<star> TTfgh_TfTgh.chine] \<cdot> \<a>[f, tab\<^sub>0 f, f\<^sub>0gh\<^sub>1.p\<^sub>1 \<star> TTfgh_TfTgh.chine]) \<cdot> (f.tab \<star> f\<^sub>0gh\<^sub>1.p\<^sub>1 \<star> TTfgh_TfTgh.chine) = f.tab \<star> f\<^sub>0gh\<^sub>1.p\<^sub>1 \<star> TTfgh_TfTgh.chine
goal (1 subgoal):
1. \<a>[f, g, h \<star> tab\<^sub>0 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0] \<cdot> \<a>[f \<star> g, h, tab\<^sub>0 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0] \<cdot> (\<a>\<^sup>-\<^sup>1[f, g, h] \<star> tab\<^sub>0 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0) \<cdot> ((f \<star> g \<star> h) \<star> TTfgh_TfTgh.the_\<theta>) \<cdot> \<a>[f \<star> g \<star> h, (tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine] \<cdot> (\<a>\<^sup>-\<^sup>1[f, g \<star> h, (tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0] \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> \<a>[g \<star> h, tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0, f\<^sub>0gh\<^sub>1.p\<^sub>0]) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> \<a>\<^sup>-\<^sup>1[g, h, tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> (g \<star> \<a>[h, tab\<^sub>0 h, g\<^sub>0h\<^sub>1.p\<^sub>0]) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> (g \<star> h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> (g \<star> g\<^sub>0h\<^sub>1.\<phi>) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> \<a>[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> (g.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>1) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> f\<^sub>0gh\<^sub>1.\<phi>) \<star> TTfgh_TfTgh.chine) \<cdot> (\<a>[f, tab\<^sub>0 f, f\<^sub>0gh\<^sub>1.p\<^sub>1] \<star> TTfgh_TfTgh.chine) \<cdot> \<a>\<^sup>-\<^sup>1[f \<star> tab\<^sub>0 f, f\<^sub>0gh\<^sub>1.p\<^sub>1, TTfgh_TfTgh.chine] \<cdot> (f.tab \<star> f\<^sub>0gh\<^sub>1.p\<^sub>1 \<star> TTfgh_TfTgh.chine) \<cdot> \<a>[tab\<^sub>1 f, f\<^sub>0gh\<^sub>1.p\<^sub>1, TTfgh_TfTgh.chine] \<cdot> TTfgh_TfTgh.the_\<nu> \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>1 f, f\<^sub>0g\<^sub>1.p\<^sub>1, fg\<^sub>0h\<^sub>1.p\<^sub>1] = \<a>[f, g, h \<star> tab\<^sub>0 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0] \<cdot> \<a>[f \<star> g, h, tab\<^sub>0 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0] \<cdot> (\<a>\<^sup>-\<^sup>1[f, g, h] \<star> tab\<^sub>0 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0) \<cdot> ((f \<star> g \<star> h) \<star> TTfgh_TfTgh.the_\<theta>) \<cdot> \<a>[f \<star> g \<star> h, (tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine] \<cdot> (\<a>\<^sup>-\<^sup>1[f, g \<star> h, (tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0] \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> \<a>[g \<star> h, tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0, f\<^sub>0gh\<^sub>1.p\<^sub>0]) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> \<a>\<^sup>-\<^sup>1[g, h, tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> (g \<star> \<a>[h, tab\<^sub>0 h, g\<^sub>0h\<^sub>1.p\<^sub>0]) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> (g \<star> h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> (g \<star> g\<^sub>0h\<^sub>1.\<phi>) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> \<a>[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> (g.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>1) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> f\<^sub>0gh\<^sub>1.\<phi>) \<star> TTfgh_TfTgh.chine) \<cdot> (\<a>[f, tab\<^sub>0 f, f\<^sub>0gh\<^sub>1.p\<^sub>1] \<star> TTfgh_TfTgh.chine) \<cdot> \<a>\<^sup>-\<^sup>1[f \<star> tab\<^sub>0 f, f\<^sub>0gh\<^sub>1.p\<^sub>1, TTfgh_TfTgh.chine] \<cdot> ((\<a>\<^sup>-\<^sup>1[f, tab\<^sub>0 f, f\<^sub>0gh\<^sub>1.p\<^sub>1 \<star> TTfgh_TfTgh.chine] \<cdot> \<a>[f, tab\<^sub>0 f, f\<^sub>0gh\<^sub>1.p\<^sub>1 \<star> TTfgh_TfTgh.chine]) \<cdot> (f.tab \<star> f\<^sub>0gh\<^sub>1.p\<^sub>1 \<star> TTfgh_TfTgh.chine)) \<cdot> \<a>[tab\<^sub>1 f, f\<^sub>0gh\<^sub>1.p\<^sub>1, TTfgh_TfTgh.chine] \<cdot> TTfgh_TfTgh.the_\<nu> \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>1 f, f\<^sub>0g\<^sub>1.p\<^sub>1, fg\<^sub>0h\<^sub>1.p\<^sub>1]
[PROOF STEP]
using comp_assoc
[PROOF STATE]
proof (prove)
using this:
(\<a>\<^sup>-\<^sup>1[f, tab\<^sub>0 f, f\<^sub>0gh\<^sub>1.p\<^sub>1 \<star> TTfgh_TfTgh.chine] \<cdot> \<a>[f, tab\<^sub>0 f, f\<^sub>0gh\<^sub>1.p\<^sub>1 \<star> TTfgh_TfTgh.chine]) \<cdot> (f.tab \<star> f\<^sub>0gh\<^sub>1.p\<^sub>1 \<star> TTfgh_TfTgh.chine) = f.tab \<star> f\<^sub>0gh\<^sub>1.p\<^sub>1 \<star> TTfgh_TfTgh.chine
(?h \<cdot> ?g) \<cdot> ?f = ?h \<cdot> ?g \<cdot> ?f
goal (1 subgoal):
1. \<a>[f, g, h \<star> tab\<^sub>0 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0] \<cdot> \<a>[f \<star> g, h, tab\<^sub>0 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0] \<cdot> (\<a>\<^sup>-\<^sup>1[f, g, h] \<star> tab\<^sub>0 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0) \<cdot> ((f \<star> g \<star> h) \<star> TTfgh_TfTgh.the_\<theta>) \<cdot> \<a>[f \<star> g \<star> h, (tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine] \<cdot> (\<a>\<^sup>-\<^sup>1[f, g \<star> h, (tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0] \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> \<a>[g \<star> h, tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0, f\<^sub>0gh\<^sub>1.p\<^sub>0]) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> \<a>\<^sup>-\<^sup>1[g, h, tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> (g \<star> \<a>[h, tab\<^sub>0 h, g\<^sub>0h\<^sub>1.p\<^sub>0]) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> (g \<star> h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> (g \<star> g\<^sub>0h\<^sub>1.\<phi>) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> \<a>[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> (g.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>1) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> f\<^sub>0gh\<^sub>1.\<phi>) \<star> TTfgh_TfTgh.chine) \<cdot> (\<a>[f, tab\<^sub>0 f, f\<^sub>0gh\<^sub>1.p\<^sub>1] \<star> TTfgh_TfTgh.chine) \<cdot> \<a>\<^sup>-\<^sup>1[f \<star> tab\<^sub>0 f, f\<^sub>0gh\<^sub>1.p\<^sub>1, TTfgh_TfTgh.chine] \<cdot> (f.tab \<star> f\<^sub>0gh\<^sub>1.p\<^sub>1 \<star> TTfgh_TfTgh.chine) \<cdot> \<a>[tab\<^sub>1 f, f\<^sub>0gh\<^sub>1.p\<^sub>1, TTfgh_TfTgh.chine] \<cdot> TTfgh_TfTgh.the_\<nu> \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>1 f, f\<^sub>0g\<^sub>1.p\<^sub>1, fg\<^sub>0h\<^sub>1.p\<^sub>1] = \<a>[f, g, h \<star> tab\<^sub>0 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0] \<cdot> \<a>[f \<star> g, h, tab\<^sub>0 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0] \<cdot> (\<a>\<^sup>-\<^sup>1[f, g, h] \<star> tab\<^sub>0 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0) \<cdot> ((f \<star> g \<star> h) \<star> TTfgh_TfTgh.the_\<theta>) \<cdot> \<a>[f \<star> g \<star> h, (tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine] \<cdot> (\<a>\<^sup>-\<^sup>1[f, g \<star> h, (tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0] \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> \<a>[g \<star> h, tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0, f\<^sub>0gh\<^sub>1.p\<^sub>0]) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> \<a>\<^sup>-\<^sup>1[g, h, tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> (g \<star> \<a>[h, tab\<^sub>0 h, g\<^sub>0h\<^sub>1.p\<^sub>0]) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> (g \<star> h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> (g \<star> g\<^sub>0h\<^sub>1.\<phi>) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> \<a>[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> (g.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>1) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> f\<^sub>0gh\<^sub>1.\<phi>) \<star> TTfgh_TfTgh.chine) \<cdot> (\<a>[f, tab\<^sub>0 f, f\<^sub>0gh\<^sub>1.p\<^sub>1] \<star> TTfgh_TfTgh.chine) \<cdot> \<a>\<^sup>-\<^sup>1[f \<star> tab\<^sub>0 f, f\<^sub>0gh\<^sub>1.p\<^sub>1, TTfgh_TfTgh.chine] \<cdot> ((\<a>\<^sup>-\<^sup>1[f, tab\<^sub>0 f, f\<^sub>0gh\<^sub>1.p\<^sub>1 \<star> TTfgh_TfTgh.chine] \<cdot> \<a>[f, tab\<^sub>0 f, f\<^sub>0gh\<^sub>1.p\<^sub>1 \<star> TTfgh_TfTgh.chine]) \<cdot> (f.tab \<star> f\<^sub>0gh\<^sub>1.p\<^sub>1 \<star> TTfgh_TfTgh.chine)) \<cdot> \<a>[tab\<^sub>1 f, f\<^sub>0gh\<^sub>1.p\<^sub>1, TTfgh_TfTgh.chine] \<cdot> TTfgh_TfTgh.the_\<nu> \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>1 f, f\<^sub>0g\<^sub>1.p\<^sub>1, fg\<^sub>0h\<^sub>1.p\<^sub>1]
[PROOF STEP]
by simp
[PROOF STATE]
proof (state)
this:
\<a>[f, g, h \<star> tab\<^sub>0 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0] \<cdot> \<a>[f \<star> g, h, tab\<^sub>0 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0] \<cdot> (\<a>\<^sup>-\<^sup>1[f, g, h] \<star> tab\<^sub>0 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0) \<cdot> ((f \<star> g \<star> h) \<star> TTfgh_TfTgh.the_\<theta>) \<cdot> \<a>[f \<star> g \<star> h, (tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine] \<cdot> (\<a>\<^sup>-\<^sup>1[f, g \<star> h, (tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0] \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> \<a>[g \<star> h, tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0, f\<^sub>0gh\<^sub>1.p\<^sub>0]) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> \<a>\<^sup>-\<^sup>1[g, h, tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> (g \<star> \<a>[h, tab\<^sub>0 h, g\<^sub>0h\<^sub>1.p\<^sub>0]) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> (g \<star> h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> (g \<star> g\<^sub>0h\<^sub>1.\<phi>) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> \<a>[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> (g.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>1) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> f\<^sub>0gh\<^sub>1.\<phi>) \<star> TTfgh_TfTgh.chine) \<cdot> (\<a>[f, tab\<^sub>0 f, f\<^sub>0gh\<^sub>1.p\<^sub>1] \<star> TTfgh_TfTgh.chine) \<cdot> \<a>\<^sup>-\<^sup>1[f \<star> tab\<^sub>0 f, f\<^sub>0gh\<^sub>1.p\<^sub>1, TTfgh_TfTgh.chine] \<cdot> (f.tab \<star> f\<^sub>0gh\<^sub>1.p\<^sub>1 \<star> TTfgh_TfTgh.chine) \<cdot> \<a>[tab\<^sub>1 f, f\<^sub>0gh\<^sub>1.p\<^sub>1, TTfgh_TfTgh.chine] \<cdot> TTfgh_TfTgh.the_\<nu> \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>1 f, f\<^sub>0g\<^sub>1.p\<^sub>1, fg\<^sub>0h\<^sub>1.p\<^sub>1] = \<a>[f, g, h \<star> tab\<^sub>0 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0] \<cdot> \<a>[f \<star> g, h, tab\<^sub>0 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0] \<cdot> (\<a>\<^sup>-\<^sup>1[f, g, h] \<star> tab\<^sub>0 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0) \<cdot> ((f \<star> g \<star> h) \<star> TTfgh_TfTgh.the_\<theta>) \<cdot> \<a>[f \<star> g \<star> h, (tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine] \<cdot> (\<a>\<^sup>-\<^sup>1[f, g \<star> h, (tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0] \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> \<a>[g \<star> h, tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0, f\<^sub>0gh\<^sub>1.p\<^sub>0]) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> \<a>\<^sup>-\<^sup>1[g, h, tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> (g \<star> \<a>[h, tab\<^sub>0 h, g\<^sub>0h\<^sub>1.p\<^sub>0]) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> (g \<star> h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> (g \<star> g\<^sub>0h\<^sub>1.\<phi>) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> \<a>[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> (g.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>1) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> f\<^sub>0gh\<^sub>1.\<phi>) \<star> TTfgh_TfTgh.chine) \<cdot> (\<a>[f, tab\<^sub>0 f, f\<^sub>0gh\<^sub>1.p\<^sub>1] \<star> TTfgh_TfTgh.chine) \<cdot> \<a>\<^sup>-\<^sup>1[f \<star> tab\<^sub>0 f, f\<^sub>0gh\<^sub>1.p\<^sub>1, TTfgh_TfTgh.chine] \<cdot> ((\<a>\<^sup>-\<^sup>1[f, tab\<^sub>0 f, f\<^sub>0gh\<^sub>1.p\<^sub>1 \<star> TTfgh_TfTgh.chine] \<cdot> \<a>[f, tab\<^sub>0 f, f\<^sub>0gh\<^sub>1.p\<^sub>1 \<star> TTfgh_TfTgh.chine]) \<cdot> (f.tab \<star> f\<^sub>0gh\<^sub>1.p\<^sub>1 \<star> TTfgh_TfTgh.chine)) \<cdot> \<a>[tab\<^sub>1 f, f\<^sub>0gh\<^sub>1.p\<^sub>1, TTfgh_TfTgh.chine] \<cdot> TTfgh_TfTgh.the_\<nu> \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>1 f, f\<^sub>0g\<^sub>1.p\<^sub>1, fg\<^sub>0h\<^sub>1.p\<^sub>1]
goal:
No subgoals!
[PROOF STEP]
qed
[PROOF STATE]
proof (state)
this:
\<a>[f, g, h \<star> tab\<^sub>0 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0] \<cdot> \<a>[f \<star> g, h, tab\<^sub>0 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0] \<cdot> (\<a>\<^sup>-\<^sup>1[f, g, h] \<star> tab\<^sub>0 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0) \<cdot> ((f \<star> g \<star> h) \<star> TTfgh_TfTgh.the_\<theta>) \<cdot> \<a>[f \<star> g \<star> h, (tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine] \<cdot> (\<a>\<^sup>-\<^sup>1[f, g \<star> h, (tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0] \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> \<a>[g \<star> h, tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0, f\<^sub>0gh\<^sub>1.p\<^sub>0]) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> \<a>\<^sup>-\<^sup>1[g, h, tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> (g \<star> \<a>[h, tab\<^sub>0 h, g\<^sub>0h\<^sub>1.p\<^sub>0]) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> (g \<star> h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> (g \<star> g\<^sub>0h\<^sub>1.\<phi>) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> \<a>[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> (g.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>1) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> f\<^sub>0gh\<^sub>1.\<phi>) \<star> TTfgh_TfTgh.chine) \<cdot> (\<a>[f, tab\<^sub>0 f, f\<^sub>0gh\<^sub>1.p\<^sub>1] \<star> TTfgh_TfTgh.chine) \<cdot> \<a>\<^sup>-\<^sup>1[f \<star> tab\<^sub>0 f, f\<^sub>0gh\<^sub>1.p\<^sub>1, TTfgh_TfTgh.chine] \<cdot> (f.tab \<star> f\<^sub>0gh\<^sub>1.p\<^sub>1 \<star> TTfgh_TfTgh.chine) \<cdot> \<a>[tab\<^sub>1 f, f\<^sub>0gh\<^sub>1.p\<^sub>1, TTfgh_TfTgh.chine] \<cdot> TTfgh_TfTgh.the_\<nu> \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>1 f, f\<^sub>0g\<^sub>1.p\<^sub>1, fg\<^sub>0h\<^sub>1.p\<^sub>1] = \<a>[f, g, h \<star> tab\<^sub>0 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0] \<cdot> \<a>[f \<star> g, h, tab\<^sub>0 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0] \<cdot> (\<a>\<^sup>-\<^sup>1[f, g, h] \<star> tab\<^sub>0 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0) \<cdot> ((f \<star> g \<star> h) \<star> TTfgh_TfTgh.the_\<theta>) \<cdot> \<a>[f \<star> g \<star> h, (tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine] \<cdot> (\<a>\<^sup>-\<^sup>1[f, g \<star> h, (tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0] \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> \<a>[g \<star> h, tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0, f\<^sub>0gh\<^sub>1.p\<^sub>0]) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> \<a>\<^sup>-\<^sup>1[g, h, tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> (g \<star> \<a>[h, tab\<^sub>0 h, g\<^sub>0h\<^sub>1.p\<^sub>0]) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> (g \<star> h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> (g \<star> g\<^sub>0h\<^sub>1.\<phi>) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> \<a>[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> (g.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>1) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> f\<^sub>0gh\<^sub>1.\<phi>) \<star> TTfgh_TfTgh.chine) \<cdot> (\<a>[f, tab\<^sub>0 f, f\<^sub>0gh\<^sub>1.p\<^sub>1] \<star> TTfgh_TfTgh.chine) \<cdot> \<a>\<^sup>-\<^sup>1[f \<star> tab\<^sub>0 f, f\<^sub>0gh\<^sub>1.p\<^sub>1, TTfgh_TfTgh.chine] \<cdot> ((\<a>\<^sup>-\<^sup>1[f, tab\<^sub>0 f, f\<^sub>0gh\<^sub>1.p\<^sub>1 \<star> TTfgh_TfTgh.chine] \<cdot> \<a>[f, tab\<^sub>0 f, f\<^sub>0gh\<^sub>1.p\<^sub>1 \<star> TTfgh_TfTgh.chine]) \<cdot> (f.tab \<star> f\<^sub>0gh\<^sub>1.p\<^sub>1 \<star> TTfgh_TfTgh.chine)) \<cdot> \<a>[tab\<^sub>1 f, f\<^sub>0gh\<^sub>1.p\<^sub>1, TTfgh_TfTgh.chine] \<cdot> TTfgh_TfTgh.the_\<nu> \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>1 f, f\<^sub>0g\<^sub>1.p\<^sub>1, fg\<^sub>0h\<^sub>1.p\<^sub>1]
goal (1 subgoal):
1. (f \<star> \<theta>\<^sub>f) \<cdot> \<a>[f, tab\<^sub>0 f, w\<^sub>f] \<cdot> (f.tab \<star> w\<^sub>f) = ((f \<star> \<theta>\<^sub>f') \<cdot> \<a>[f, tab\<^sub>0 f, w\<^sub>f'] \<cdot> (f.tab \<star> w\<^sub>f')) \<cdot> \<beta>\<^sub>f
[PROOF STEP]
also
[PROOF STATE]
proof (state)
this:
\<a>[f, g, h \<star> tab\<^sub>0 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0] \<cdot> \<a>[f \<star> g, h, tab\<^sub>0 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0] \<cdot> (\<a>\<^sup>-\<^sup>1[f, g, h] \<star> tab\<^sub>0 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0) \<cdot> ((f \<star> g \<star> h) \<star> TTfgh_TfTgh.the_\<theta>) \<cdot> \<a>[f \<star> g \<star> h, (tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine] \<cdot> (\<a>\<^sup>-\<^sup>1[f, g \<star> h, (tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0] \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> \<a>[g \<star> h, tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0, f\<^sub>0gh\<^sub>1.p\<^sub>0]) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> \<a>\<^sup>-\<^sup>1[g, h, tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> (g \<star> \<a>[h, tab\<^sub>0 h, g\<^sub>0h\<^sub>1.p\<^sub>0]) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> (g \<star> h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> (g \<star> g\<^sub>0h\<^sub>1.\<phi>) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> \<a>[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> (g.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>1) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> f\<^sub>0gh\<^sub>1.\<phi>) \<star> TTfgh_TfTgh.chine) \<cdot> (\<a>[f, tab\<^sub>0 f, f\<^sub>0gh\<^sub>1.p\<^sub>1] \<star> TTfgh_TfTgh.chine) \<cdot> \<a>\<^sup>-\<^sup>1[f \<star> tab\<^sub>0 f, f\<^sub>0gh\<^sub>1.p\<^sub>1, TTfgh_TfTgh.chine] \<cdot> (f.tab \<star> f\<^sub>0gh\<^sub>1.p\<^sub>1 \<star> TTfgh_TfTgh.chine) \<cdot> \<a>[tab\<^sub>1 f, f\<^sub>0gh\<^sub>1.p\<^sub>1, TTfgh_TfTgh.chine] \<cdot> TTfgh_TfTgh.the_\<nu> \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>1 f, f\<^sub>0g\<^sub>1.p\<^sub>1, fg\<^sub>0h\<^sub>1.p\<^sub>1] = \<a>[f, g, h \<star> tab\<^sub>0 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0] \<cdot> \<a>[f \<star> g, h, tab\<^sub>0 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0] \<cdot> (\<a>\<^sup>-\<^sup>1[f, g, h] \<star> tab\<^sub>0 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0) \<cdot> ((f \<star> g \<star> h) \<star> TTfgh_TfTgh.the_\<theta>) \<cdot> \<a>[f \<star> g \<star> h, (tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine] \<cdot> (\<a>\<^sup>-\<^sup>1[f, g \<star> h, (tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0] \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> \<a>[g \<star> h, tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0, f\<^sub>0gh\<^sub>1.p\<^sub>0]) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> \<a>\<^sup>-\<^sup>1[g, h, tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> (g \<star> \<a>[h, tab\<^sub>0 h, g\<^sub>0h\<^sub>1.p\<^sub>0]) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> (g \<star> h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> (g \<star> g\<^sub>0h\<^sub>1.\<phi>) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> \<a>[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> (g.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>1) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> f\<^sub>0gh\<^sub>1.\<phi>) \<star> TTfgh_TfTgh.chine) \<cdot> (\<a>[f, tab\<^sub>0 f, f\<^sub>0gh\<^sub>1.p\<^sub>1] \<star> TTfgh_TfTgh.chine) \<cdot> \<a>\<^sup>-\<^sup>1[f \<star> tab\<^sub>0 f, f\<^sub>0gh\<^sub>1.p\<^sub>1, TTfgh_TfTgh.chine] \<cdot> ((\<a>\<^sup>-\<^sup>1[f, tab\<^sub>0 f, f\<^sub>0gh\<^sub>1.p\<^sub>1 \<star> TTfgh_TfTgh.chine] \<cdot> \<a>[f, tab\<^sub>0 f, f\<^sub>0gh\<^sub>1.p\<^sub>1 \<star> TTfgh_TfTgh.chine]) \<cdot> (f.tab \<star> f\<^sub>0gh\<^sub>1.p\<^sub>1 \<star> TTfgh_TfTgh.chine)) \<cdot> \<a>[tab\<^sub>1 f, f\<^sub>0gh\<^sub>1.p\<^sub>1, TTfgh_TfTgh.chine] \<cdot> TTfgh_TfTgh.the_\<nu> \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>1 f, f\<^sub>0g\<^sub>1.p\<^sub>1, fg\<^sub>0h\<^sub>1.p\<^sub>1]
goal (1 subgoal):
1. (f \<star> \<theta>\<^sub>f) \<cdot> \<a>[f, tab\<^sub>0 f, w\<^sub>f] \<cdot> (f.tab \<star> w\<^sub>f) = ((f \<star> \<theta>\<^sub>f') \<cdot> \<a>[f, tab\<^sub>0 f, w\<^sub>f'] \<cdot> (f.tab \<star> w\<^sub>f')) \<cdot> \<beta>\<^sub>f
[PROOF STEP]
have "... =
\<a>[f, g, h \<star> tab\<^sub>0 h \<star> TTfgh.p\<^sub>0] \<cdot>
\<a>[f \<star> g, h, tab\<^sub>0 h \<star> TTfgh.p\<^sub>0] \<cdot>
(\<a>\<^sup>-\<^sup>1[f, g, h] \<star> tab\<^sub>0 h \<star> TTfgh.p\<^sub>0) \<cdot>
((f \<star> g \<star> h) \<star> TTfgh_TfTgh.the_\<theta>) \<cdot>
\<a>[f \<star> g \<star> h, (tab\<^sub>0 h \<star> Tgh.p\<^sub>0) \<star> TfTgh.p\<^sub>0, TTfgh_TfTgh.chine] \<cdot>
(\<a>\<^sup>-\<^sup>1[f, g \<star> h, (tab\<^sub>0 h \<star> Tgh.p\<^sub>0) \<star> TfTgh.p\<^sub>0] \<star> TTfgh_TfTgh.chine) \<cdot>
((f \<star> \<a>[g \<star> h, tab\<^sub>0 h \<star> Tgh.p\<^sub>0, TfTgh.p\<^sub>0]) \<star> TTfgh_TfTgh.chine) \<cdot>
((f \<star> \<a>\<^sup>-\<^sup>1[g, h, tab\<^sub>0 h \<star> Tgh.p\<^sub>0] \<star> TfTgh.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot>
((f \<star> (g \<star> \<a>[h, tab\<^sub>0 h, Tgh.p\<^sub>0]) \<star> TfTgh.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot>
((f \<star> (g \<star> h.tab \<star> Tgh.p\<^sub>0) \<star> TfTgh.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot>
((f \<star> (g \<star> g\<^sub>0h\<^sub>1.\<phi>) \<star> TfTgh.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot>
((f \<star> \<a>[g, tab\<^sub>0 g, Tgh.p\<^sub>1] \<star> TfTgh.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot>
((f \<star> (g.tab \<star> Tgh.p\<^sub>1) \<star> TfTgh.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot>
((f \<star> f\<^sub>0gh\<^sub>1.\<phi>) \<star> TTfgh_TfTgh.chine) \<cdot>
((\<a>[f, tab\<^sub>0 f, TfTgh.p\<^sub>1] \<star> TTfgh_TfTgh.chine) \<cdot>
\<a>\<^sup>-\<^sup>1[f \<star> tab\<^sub>0 f, TfTgh.p\<^sub>1, TTfgh_TfTgh.chine] \<cdot>
\<a>\<^sup>-\<^sup>1[f, tab\<^sub>0 f, TfTgh.p\<^sub>1 \<star> TTfgh_TfTgh.chine]) \<cdot>
\<a>[f, tab\<^sub>0 f, TfTgh.p\<^sub>1 \<star> TTfgh_TfTgh.chine] \<cdot>
(f.tab \<star> TfTgh.p\<^sub>1 \<star> TTfgh_TfTgh.chine) \<cdot>
\<a>[tab\<^sub>1 f, TfTgh.p\<^sub>1, TTfgh_TfTgh.chine] \<cdot>
TTfgh_TfTgh.the_\<nu> \<cdot>
\<a>\<^sup>-\<^sup>1[tab\<^sub>1 f, Tfg.p\<^sub>1, TTfgh.p\<^sub>1]"
[PROOF STATE]
proof (prove)
goal (1 subgoal):
1. \<a>[f, g, h \<star> tab\<^sub>0 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0] \<cdot> \<a>[f \<star> g, h, tab\<^sub>0 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0] \<cdot> (\<a>\<^sup>-\<^sup>1[f, g, h] \<star> tab\<^sub>0 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0) \<cdot> ((f \<star> g \<star> h) \<star> TTfgh_TfTgh.the_\<theta>) \<cdot> \<a>[f \<star> g \<star> h, (tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine] \<cdot> (\<a>\<^sup>-\<^sup>1[f, g \<star> h, (tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0] \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> \<a>[g \<star> h, tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0, f\<^sub>0gh\<^sub>1.p\<^sub>0]) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> \<a>\<^sup>-\<^sup>1[g, h, tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> (g \<star> \<a>[h, tab\<^sub>0 h, g\<^sub>0h\<^sub>1.p\<^sub>0]) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> (g \<star> h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> (g \<star> g\<^sub>0h\<^sub>1.\<phi>) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> \<a>[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> (g.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>1) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> f\<^sub>0gh\<^sub>1.\<phi>) \<star> TTfgh_TfTgh.chine) \<cdot> (\<a>[f, tab\<^sub>0 f, f\<^sub>0gh\<^sub>1.p\<^sub>1] \<star> TTfgh_TfTgh.chine) \<cdot> \<a>\<^sup>-\<^sup>1[f \<star> tab\<^sub>0 f, f\<^sub>0gh\<^sub>1.p\<^sub>1, TTfgh_TfTgh.chine] \<cdot> ((\<a>\<^sup>-\<^sup>1[f, tab\<^sub>0 f, f\<^sub>0gh\<^sub>1.p\<^sub>1 \<star> TTfgh_TfTgh.chine] \<cdot> \<a>[f, tab\<^sub>0 f, f\<^sub>0gh\<^sub>1.p\<^sub>1 \<star> TTfgh_TfTgh.chine]) \<cdot> (f.tab \<star> f\<^sub>0gh\<^sub>1.p\<^sub>1 \<star> TTfgh_TfTgh.chine)) \<cdot> \<a>[tab\<^sub>1 f, f\<^sub>0gh\<^sub>1.p\<^sub>1, TTfgh_TfTgh.chine] \<cdot> TTfgh_TfTgh.the_\<nu> \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>1 f, f\<^sub>0g\<^sub>1.p\<^sub>1, fg\<^sub>0h\<^sub>1.p\<^sub>1] = \<a>[f, g, h \<star> tab\<^sub>0 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0] \<cdot> \<a>[f \<star> g, h, tab\<^sub>0 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0] \<cdot> (\<a>\<^sup>-\<^sup>1[f, g, h] \<star> tab\<^sub>0 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0) \<cdot> ((f \<star> g \<star> h) \<star> TTfgh_TfTgh.the_\<theta>) \<cdot> \<a>[f \<star> g \<star> h, (tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine] \<cdot> (\<a>\<^sup>-\<^sup>1[f, g \<star> h, (tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0] \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> \<a>[g \<star> h, tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0, f\<^sub>0gh\<^sub>1.p\<^sub>0]) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> \<a>\<^sup>-\<^sup>1[g, h, tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> (g \<star> \<a>[h, tab\<^sub>0 h, g\<^sub>0h\<^sub>1.p\<^sub>0]) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> (g \<star> h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> (g \<star> g\<^sub>0h\<^sub>1.\<phi>) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> \<a>[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> (g.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>1) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> f\<^sub>0gh\<^sub>1.\<phi>) \<star> TTfgh_TfTgh.chine) \<cdot> ((\<a>[f, tab\<^sub>0 f, f\<^sub>0gh\<^sub>1.p\<^sub>1] \<star> TTfgh_TfTgh.chine) \<cdot> \<a>\<^sup>-\<^sup>1[f \<star> tab\<^sub>0 f, f\<^sub>0gh\<^sub>1.p\<^sub>1, TTfgh_TfTgh.chine] \<cdot> \<a>\<^sup>-\<^sup>1[f, tab\<^sub>0 f, f\<^sub>0gh\<^sub>1.p\<^sub>1 \<star> TTfgh_TfTgh.chine]) \<cdot> \<a>[f, tab\<^sub>0 f, f\<^sub>0gh\<^sub>1.p\<^sub>1 \<star> TTfgh_TfTgh.chine] \<cdot> (f.tab \<star> f\<^sub>0gh\<^sub>1.p\<^sub>1 \<star> TTfgh_TfTgh.chine) \<cdot> \<a>[tab\<^sub>1 f, f\<^sub>0gh\<^sub>1.p\<^sub>1, TTfgh_TfTgh.chine] \<cdot> TTfgh_TfTgh.the_\<nu> \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>1 f, f\<^sub>0g\<^sub>1.p\<^sub>1, fg\<^sub>0h\<^sub>1.p\<^sub>1]
[PROOF STEP]
using comp_assoc
[PROOF STATE]
proof (prove)
using this:
(?h \<cdot> ?g) \<cdot> ?f = ?h \<cdot> ?g \<cdot> ?f
goal (1 subgoal):
1. \<a>[f, g, h \<star> tab\<^sub>0 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0] \<cdot> \<a>[f \<star> g, h, tab\<^sub>0 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0] \<cdot> (\<a>\<^sup>-\<^sup>1[f, g, h] \<star> tab\<^sub>0 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0) \<cdot> ((f \<star> g \<star> h) \<star> TTfgh_TfTgh.the_\<theta>) \<cdot> \<a>[f \<star> g \<star> h, (tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine] \<cdot> (\<a>\<^sup>-\<^sup>1[f, g \<star> h, (tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0] \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> \<a>[g \<star> h, tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0, f\<^sub>0gh\<^sub>1.p\<^sub>0]) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> \<a>\<^sup>-\<^sup>1[g, h, tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> (g \<star> \<a>[h, tab\<^sub>0 h, g\<^sub>0h\<^sub>1.p\<^sub>0]) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> (g \<star> h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> (g \<star> g\<^sub>0h\<^sub>1.\<phi>) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> \<a>[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> (g.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>1) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> f\<^sub>0gh\<^sub>1.\<phi>) \<star> TTfgh_TfTgh.chine) \<cdot> (\<a>[f, tab\<^sub>0 f, f\<^sub>0gh\<^sub>1.p\<^sub>1] \<star> TTfgh_TfTgh.chine) \<cdot> \<a>\<^sup>-\<^sup>1[f \<star> tab\<^sub>0 f, f\<^sub>0gh\<^sub>1.p\<^sub>1, TTfgh_TfTgh.chine] \<cdot> ((\<a>\<^sup>-\<^sup>1[f, tab\<^sub>0 f, f\<^sub>0gh\<^sub>1.p\<^sub>1 \<star> TTfgh_TfTgh.chine] \<cdot> \<a>[f, tab\<^sub>0 f, f\<^sub>0gh\<^sub>1.p\<^sub>1 \<star> TTfgh_TfTgh.chine]) \<cdot> (f.tab \<star> f\<^sub>0gh\<^sub>1.p\<^sub>1 \<star> TTfgh_TfTgh.chine)) \<cdot> \<a>[tab\<^sub>1 f, f\<^sub>0gh\<^sub>1.p\<^sub>1, TTfgh_TfTgh.chine] \<cdot> TTfgh_TfTgh.the_\<nu> \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>1 f, f\<^sub>0g\<^sub>1.p\<^sub>1, fg\<^sub>0h\<^sub>1.p\<^sub>1] = \<a>[f, g, h \<star> tab\<^sub>0 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0] \<cdot> \<a>[f \<star> g, h, tab\<^sub>0 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0] \<cdot> (\<a>\<^sup>-\<^sup>1[f, g, h] \<star> tab\<^sub>0 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0) \<cdot> ((f \<star> g \<star> h) \<star> TTfgh_TfTgh.the_\<theta>) \<cdot> \<a>[f \<star> g \<star> h, (tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine] \<cdot> (\<a>\<^sup>-\<^sup>1[f, g \<star> h, (tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0] \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> \<a>[g \<star> h, tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0, f\<^sub>0gh\<^sub>1.p\<^sub>0]) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> \<a>\<^sup>-\<^sup>1[g, h, tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> (g \<star> \<a>[h, tab\<^sub>0 h, g\<^sub>0h\<^sub>1.p\<^sub>0]) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> (g \<star> h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> (g \<star> g\<^sub>0h\<^sub>1.\<phi>) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> \<a>[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> (g.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>1) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> f\<^sub>0gh\<^sub>1.\<phi>) \<star> TTfgh_TfTgh.chine) \<cdot> ((\<a>[f, tab\<^sub>0 f, f\<^sub>0gh\<^sub>1.p\<^sub>1] \<star> TTfgh_TfTgh.chine) \<cdot> \<a>\<^sup>-\<^sup>1[f \<star> tab\<^sub>0 f, f\<^sub>0gh\<^sub>1.p\<^sub>1, TTfgh_TfTgh.chine] \<cdot> \<a>\<^sup>-\<^sup>1[f, tab\<^sub>0 f, f\<^sub>0gh\<^sub>1.p\<^sub>1 \<star> TTfgh_TfTgh.chine]) \<cdot> \<a>[f, tab\<^sub>0 f, f\<^sub>0gh\<^sub>1.p\<^sub>1 \<star> TTfgh_TfTgh.chine] \<cdot> (f.tab \<star> f\<^sub>0gh\<^sub>1.p\<^sub>1 \<star> TTfgh_TfTgh.chine) \<cdot> \<a>[tab\<^sub>1 f, f\<^sub>0gh\<^sub>1.p\<^sub>1, TTfgh_TfTgh.chine] \<cdot> TTfgh_TfTgh.the_\<nu> \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>1 f, f\<^sub>0g\<^sub>1.p\<^sub>1, fg\<^sub>0h\<^sub>1.p\<^sub>1]
[PROOF STEP]
by presburger
[PROOF STATE]
proof (state)
this:
\<a>[f, g, h \<star> tab\<^sub>0 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0] \<cdot> \<a>[f \<star> g, h, tab\<^sub>0 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0] \<cdot> (\<a>\<^sup>-\<^sup>1[f, g, h] \<star> tab\<^sub>0 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0) \<cdot> ((f \<star> g \<star> h) \<star> TTfgh_TfTgh.the_\<theta>) \<cdot> \<a>[f \<star> g \<star> h, (tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine] \<cdot> (\<a>\<^sup>-\<^sup>1[f, g \<star> h, (tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0] \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> \<a>[g \<star> h, tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0, f\<^sub>0gh\<^sub>1.p\<^sub>0]) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> \<a>\<^sup>-\<^sup>1[g, h, tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> (g \<star> \<a>[h, tab\<^sub>0 h, g\<^sub>0h\<^sub>1.p\<^sub>0]) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> (g \<star> h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> (g \<star> g\<^sub>0h\<^sub>1.\<phi>) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> \<a>[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> (g.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>1) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> f\<^sub>0gh\<^sub>1.\<phi>) \<star> TTfgh_TfTgh.chine) \<cdot> (\<a>[f, tab\<^sub>0 f, f\<^sub>0gh\<^sub>1.p\<^sub>1] \<star> TTfgh_TfTgh.chine) \<cdot> \<a>\<^sup>-\<^sup>1[f \<star> tab\<^sub>0 f, f\<^sub>0gh\<^sub>1.p\<^sub>1, TTfgh_TfTgh.chine] \<cdot> ((\<a>\<^sup>-\<^sup>1[f, tab\<^sub>0 f, f\<^sub>0gh\<^sub>1.p\<^sub>1 \<star> TTfgh_TfTgh.chine] \<cdot> \<a>[f, tab\<^sub>0 f, f\<^sub>0gh\<^sub>1.p\<^sub>1 \<star> TTfgh_TfTgh.chine]) \<cdot> (f.tab \<star> f\<^sub>0gh\<^sub>1.p\<^sub>1 \<star> TTfgh_TfTgh.chine)) \<cdot> \<a>[tab\<^sub>1 f, f\<^sub>0gh\<^sub>1.p\<^sub>1, TTfgh_TfTgh.chine] \<cdot> TTfgh_TfTgh.the_\<nu> \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>1 f, f\<^sub>0g\<^sub>1.p\<^sub>1, fg\<^sub>0h\<^sub>1.p\<^sub>1] = \<a>[f, g, h \<star> tab\<^sub>0 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0] \<cdot> \<a>[f \<star> g, h, tab\<^sub>0 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0] \<cdot> (\<a>\<^sup>-\<^sup>1[f, g, h] \<star> tab\<^sub>0 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0) \<cdot> ((f \<star> g \<star> h) \<star> TTfgh_TfTgh.the_\<theta>) \<cdot> \<a>[f \<star> g \<star> h, (tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine] \<cdot> (\<a>\<^sup>-\<^sup>1[f, g \<star> h, (tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0] \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> \<a>[g \<star> h, tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0, f\<^sub>0gh\<^sub>1.p\<^sub>0]) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> \<a>\<^sup>-\<^sup>1[g, h, tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> (g \<star> \<a>[h, tab\<^sub>0 h, g\<^sub>0h\<^sub>1.p\<^sub>0]) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> (g \<star> h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> (g \<star> g\<^sub>0h\<^sub>1.\<phi>) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> \<a>[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> (g.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>1) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> f\<^sub>0gh\<^sub>1.\<phi>) \<star> TTfgh_TfTgh.chine) \<cdot> ((\<a>[f, tab\<^sub>0 f, f\<^sub>0gh\<^sub>1.p\<^sub>1] \<star> TTfgh_TfTgh.chine) \<cdot> \<a>\<^sup>-\<^sup>1[f \<star> tab\<^sub>0 f, f\<^sub>0gh\<^sub>1.p\<^sub>1, TTfgh_TfTgh.chine] \<cdot> \<a>\<^sup>-\<^sup>1[f, tab\<^sub>0 f, f\<^sub>0gh\<^sub>1.p\<^sub>1 \<star> TTfgh_TfTgh.chine]) \<cdot> \<a>[f, tab\<^sub>0 f, f\<^sub>0gh\<^sub>1.p\<^sub>1 \<star> TTfgh_TfTgh.chine] \<cdot> (f.tab \<star> f\<^sub>0gh\<^sub>1.p\<^sub>1 \<star> TTfgh_TfTgh.chine) \<cdot> \<a>[tab\<^sub>1 f, f\<^sub>0gh\<^sub>1.p\<^sub>1, TTfgh_TfTgh.chine] \<cdot> TTfgh_TfTgh.the_\<nu> \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>1 f, f\<^sub>0g\<^sub>1.p\<^sub>1, fg\<^sub>0h\<^sub>1.p\<^sub>1]
goal (1 subgoal):
1. (f \<star> \<theta>\<^sub>f) \<cdot> \<a>[f, tab\<^sub>0 f, w\<^sub>f] \<cdot> (f.tab \<star> w\<^sub>f) = ((f \<star> \<theta>\<^sub>f') \<cdot> \<a>[f, tab\<^sub>0 f, w\<^sub>f'] \<cdot> (f.tab \<star> w\<^sub>f')) \<cdot> \<beta>\<^sub>f
[PROOF STEP]
also
[PROOF STATE]
proof (state)
this:
\<a>[f, g, h \<star> tab\<^sub>0 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0] \<cdot> \<a>[f \<star> g, h, tab\<^sub>0 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0] \<cdot> (\<a>\<^sup>-\<^sup>1[f, g, h] \<star> tab\<^sub>0 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0) \<cdot> ((f \<star> g \<star> h) \<star> TTfgh_TfTgh.the_\<theta>) \<cdot> \<a>[f \<star> g \<star> h, (tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine] \<cdot> (\<a>\<^sup>-\<^sup>1[f, g \<star> h, (tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0] \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> \<a>[g \<star> h, tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0, f\<^sub>0gh\<^sub>1.p\<^sub>0]) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> \<a>\<^sup>-\<^sup>1[g, h, tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> (g \<star> \<a>[h, tab\<^sub>0 h, g\<^sub>0h\<^sub>1.p\<^sub>0]) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> (g \<star> h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> (g \<star> g\<^sub>0h\<^sub>1.\<phi>) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> \<a>[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> (g.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>1) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> f\<^sub>0gh\<^sub>1.\<phi>) \<star> TTfgh_TfTgh.chine) \<cdot> (\<a>[f, tab\<^sub>0 f, f\<^sub>0gh\<^sub>1.p\<^sub>1] \<star> TTfgh_TfTgh.chine) \<cdot> \<a>\<^sup>-\<^sup>1[f \<star> tab\<^sub>0 f, f\<^sub>0gh\<^sub>1.p\<^sub>1, TTfgh_TfTgh.chine] \<cdot> ((\<a>\<^sup>-\<^sup>1[f, tab\<^sub>0 f, f\<^sub>0gh\<^sub>1.p\<^sub>1 \<star> TTfgh_TfTgh.chine] \<cdot> \<a>[f, tab\<^sub>0 f, f\<^sub>0gh\<^sub>1.p\<^sub>1 \<star> TTfgh_TfTgh.chine]) \<cdot> (f.tab \<star> f\<^sub>0gh\<^sub>1.p\<^sub>1 \<star> TTfgh_TfTgh.chine)) \<cdot> \<a>[tab\<^sub>1 f, f\<^sub>0gh\<^sub>1.p\<^sub>1, TTfgh_TfTgh.chine] \<cdot> TTfgh_TfTgh.the_\<nu> \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>1 f, f\<^sub>0g\<^sub>1.p\<^sub>1, fg\<^sub>0h\<^sub>1.p\<^sub>1] = \<a>[f, g, h \<star> tab\<^sub>0 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0] \<cdot> \<a>[f \<star> g, h, tab\<^sub>0 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0] \<cdot> (\<a>\<^sup>-\<^sup>1[f, g, h] \<star> tab\<^sub>0 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0) \<cdot> ((f \<star> g \<star> h) \<star> TTfgh_TfTgh.the_\<theta>) \<cdot> \<a>[f \<star> g \<star> h, (tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine] \<cdot> (\<a>\<^sup>-\<^sup>1[f, g \<star> h, (tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0] \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> \<a>[g \<star> h, tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0, f\<^sub>0gh\<^sub>1.p\<^sub>0]) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> \<a>\<^sup>-\<^sup>1[g, h, tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> (g \<star> \<a>[h, tab\<^sub>0 h, g\<^sub>0h\<^sub>1.p\<^sub>0]) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> (g \<star> h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> (g \<star> g\<^sub>0h\<^sub>1.\<phi>) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> \<a>[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> (g.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>1) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> f\<^sub>0gh\<^sub>1.\<phi>) \<star> TTfgh_TfTgh.chine) \<cdot> ((\<a>[f, tab\<^sub>0 f, f\<^sub>0gh\<^sub>1.p\<^sub>1] \<star> TTfgh_TfTgh.chine) \<cdot> \<a>\<^sup>-\<^sup>1[f \<star> tab\<^sub>0 f, f\<^sub>0gh\<^sub>1.p\<^sub>1, TTfgh_TfTgh.chine] \<cdot> \<a>\<^sup>-\<^sup>1[f, tab\<^sub>0 f, f\<^sub>0gh\<^sub>1.p\<^sub>1 \<star> TTfgh_TfTgh.chine]) \<cdot> \<a>[f, tab\<^sub>0 f, f\<^sub>0gh\<^sub>1.p\<^sub>1 \<star> TTfgh_TfTgh.chine] \<cdot> (f.tab \<star> f\<^sub>0gh\<^sub>1.p\<^sub>1 \<star> TTfgh_TfTgh.chine) \<cdot> \<a>[tab\<^sub>1 f, f\<^sub>0gh\<^sub>1.p\<^sub>1, TTfgh_TfTgh.chine] \<cdot> TTfgh_TfTgh.the_\<nu> \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>1 f, f\<^sub>0g\<^sub>1.p\<^sub>1, fg\<^sub>0h\<^sub>1.p\<^sub>1]
goal (1 subgoal):
1. (f \<star> \<theta>\<^sub>f) \<cdot> \<a>[f, tab\<^sub>0 f, w\<^sub>f] \<cdot> (f.tab \<star> w\<^sub>f) = ((f \<star> \<theta>\<^sub>f') \<cdot> \<a>[f, tab\<^sub>0 f, w\<^sub>f'] \<cdot> (f.tab \<star> w\<^sub>f')) \<cdot> \<beta>\<^sub>f
[PROOF STEP]
have "... =
\<a>[f, g, h \<star> tab\<^sub>0 h \<star> TTfgh.p\<^sub>0] \<cdot>
\<a>[f \<star> g, h, tab\<^sub>0 h \<star> TTfgh.p\<^sub>0] \<cdot>
(\<a>\<^sup>-\<^sup>1[f, g, h] \<star> tab\<^sub>0 h \<star> TTfgh.p\<^sub>0) \<cdot>
((f \<star> g \<star> h) \<star> TTfgh_TfTgh.the_\<theta>) \<cdot>
\<a>[f \<star> g \<star> h, (tab\<^sub>0 h \<star> Tgh.p\<^sub>0) \<star> TfTgh.p\<^sub>0, TTfgh_TfTgh.chine] \<cdot>
(\<a>\<^sup>-\<^sup>1[f, g \<star> h, (tab\<^sub>0 h \<star> Tgh.p\<^sub>0) \<star> TfTgh.p\<^sub>0] \<star> TTfgh_TfTgh.chine) \<cdot>
(((f \<star> \<a>[g \<star> h, tab\<^sub>0 h \<star> Tgh.p\<^sub>0, TfTgh.p\<^sub>0]) \<star> TTfgh_TfTgh.chine) \<cdot>
((f \<star> \<a>\<^sup>-\<^sup>1[g, h, tab\<^sub>0 h \<star> Tgh.p\<^sub>0] \<star> TfTgh.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot>
((f \<star> (g \<star> \<a>[h, tab\<^sub>0 h, Tgh.p\<^sub>0]) \<star> TfTgh.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot>
((f \<star> (g \<star> h.tab \<star> Tgh.p\<^sub>0) \<star> TfTgh.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot>
((f \<star> (g \<star> g\<^sub>0h\<^sub>1.\<phi>) \<star> TfTgh.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot>
((f \<star> \<a>[g, tab\<^sub>0 g, Tgh.p\<^sub>1] \<star> TfTgh.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot>
((f \<star> (g.tab \<star> Tgh.p\<^sub>1) \<star> TfTgh.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot>
((f \<star> f\<^sub>0gh\<^sub>1.\<phi>) \<star> TTfgh_TfTgh.chine) \<cdot>
\<a>\<^sup>-\<^sup>1[f, tab\<^sub>0 f \<star> TfTgh.p\<^sub>1, TTfgh_TfTgh.chine]) \<cdot>
(f \<star> \<a>\<^sup>-\<^sup>1[tab\<^sub>0 f, TfTgh.p\<^sub>1, TTfgh_TfTgh.chine]) \<cdot>
\<a>[f, tab\<^sub>0 f, TfTgh.p\<^sub>1 \<star> TTfgh_TfTgh.chine] \<cdot>
(f.tab \<star> TfTgh.p\<^sub>1 \<star> TTfgh_TfTgh.chine) \<cdot>
\<a>[tab\<^sub>1 f, TfTgh.p\<^sub>1, TTfgh_TfTgh.chine] \<cdot>
TTfgh_TfTgh.the_\<nu> \<cdot>
\<a>\<^sup>-\<^sup>1[tab\<^sub>1 f, Tfg.p\<^sub>1, TTfgh.p\<^sub>1]"
[PROOF STATE]
proof (prove)
goal (1 subgoal):
1. \<a>[f, g, h \<star> tab\<^sub>0 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0] \<cdot> \<a>[f \<star> g, h, tab\<^sub>0 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0] \<cdot> (\<a>\<^sup>-\<^sup>1[f, g, h] \<star> tab\<^sub>0 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0) \<cdot> ((f \<star> g \<star> h) \<star> TTfgh_TfTgh.the_\<theta>) \<cdot> \<a>[f \<star> g \<star> h, (tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine] \<cdot> (\<a>\<^sup>-\<^sup>1[f, g \<star> h, (tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0] \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> \<a>[g \<star> h, tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0, f\<^sub>0gh\<^sub>1.p\<^sub>0]) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> \<a>\<^sup>-\<^sup>1[g, h, tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> (g \<star> \<a>[h, tab\<^sub>0 h, g\<^sub>0h\<^sub>1.p\<^sub>0]) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> (g \<star> h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> (g \<star> g\<^sub>0h\<^sub>1.\<phi>) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> \<a>[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> (g.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>1) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> f\<^sub>0gh\<^sub>1.\<phi>) \<star> TTfgh_TfTgh.chine) \<cdot> ((\<a>[f, tab\<^sub>0 f, f\<^sub>0gh\<^sub>1.p\<^sub>1] \<star> TTfgh_TfTgh.chine) \<cdot> \<a>\<^sup>-\<^sup>1[f \<star> tab\<^sub>0 f, f\<^sub>0gh\<^sub>1.p\<^sub>1, TTfgh_TfTgh.chine] \<cdot> \<a>\<^sup>-\<^sup>1[f, tab\<^sub>0 f, f\<^sub>0gh\<^sub>1.p\<^sub>1 \<star> TTfgh_TfTgh.chine]) \<cdot> \<a>[f, tab\<^sub>0 f, f\<^sub>0gh\<^sub>1.p\<^sub>1 \<star> TTfgh_TfTgh.chine] \<cdot> (f.tab \<star> f\<^sub>0gh\<^sub>1.p\<^sub>1 \<star> TTfgh_TfTgh.chine) \<cdot> \<a>[tab\<^sub>1 f, f\<^sub>0gh\<^sub>1.p\<^sub>1, TTfgh_TfTgh.chine] \<cdot> TTfgh_TfTgh.the_\<nu> \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>1 f, f\<^sub>0g\<^sub>1.p\<^sub>1, fg\<^sub>0h\<^sub>1.p\<^sub>1] = \<a>[f, g, h \<star> tab\<^sub>0 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0] \<cdot> \<a>[f \<star> g, h, tab\<^sub>0 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0] \<cdot> (\<a>\<^sup>-\<^sup>1[f, g, h] \<star> tab\<^sub>0 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0) \<cdot> ((f \<star> g \<star> h) \<star> TTfgh_TfTgh.the_\<theta>) \<cdot> \<a>[f \<star> g \<star> h, (tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine] \<cdot> (\<a>\<^sup>-\<^sup>1[f, g \<star> h, (tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0] \<star> TTfgh_TfTgh.chine) \<cdot> (((f \<star> \<a>[g \<star> h, tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0, f\<^sub>0gh\<^sub>1.p\<^sub>0]) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> \<a>\<^sup>-\<^sup>1[g, h, tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> (g \<star> \<a>[h, tab\<^sub>0 h, g\<^sub>0h\<^sub>1.p\<^sub>0]) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> (g \<star> h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> (g \<star> g\<^sub>0h\<^sub>1.\<phi>) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> \<a>[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> (g.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>1) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> f\<^sub>0gh\<^sub>1.\<phi>) \<star> TTfgh_TfTgh.chine) \<cdot> \<a>\<^sup>-\<^sup>1[f, tab\<^sub>0 f \<star> f\<^sub>0gh\<^sub>1.p\<^sub>1, TTfgh_TfTgh.chine]) \<cdot> (f \<star> \<a>\<^sup>-\<^sup>1[tab\<^sub>0 f, f\<^sub>0gh\<^sub>1.p\<^sub>1, TTfgh_TfTgh.chine]) \<cdot> \<a>[f, tab\<^sub>0 f, f\<^sub>0gh\<^sub>1.p\<^sub>1 \<star> TTfgh_TfTgh.chine] \<cdot> (f.tab \<star> f\<^sub>0gh\<^sub>1.p\<^sub>1 \<star> TTfgh_TfTgh.chine) \<cdot> \<a>[tab\<^sub>1 f, f\<^sub>0gh\<^sub>1.p\<^sub>1, TTfgh_TfTgh.chine] \<cdot> TTfgh_TfTgh.the_\<nu> \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>1 f, f\<^sub>0g\<^sub>1.p\<^sub>1, fg\<^sub>0h\<^sub>1.p\<^sub>1]
[PROOF STEP]
proof -
[PROOF STATE]
proof (state)
goal (1 subgoal):
1. \<a>[f, g, h \<star> tab\<^sub>0 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0] \<cdot> \<a>[f \<star> g, h, tab\<^sub>0 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0] \<cdot> (\<a>\<^sup>-\<^sup>1[f, g, h] \<star> tab\<^sub>0 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0) \<cdot> ((f \<star> g \<star> h) \<star> TTfgh_TfTgh.the_\<theta>) \<cdot> \<a>[f \<star> g \<star> h, (tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine] \<cdot> (\<a>\<^sup>-\<^sup>1[f, g \<star> h, (tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0] \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> \<a>[g \<star> h, tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0, f\<^sub>0gh\<^sub>1.p\<^sub>0]) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> \<a>\<^sup>-\<^sup>1[g, h, tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> (g \<star> \<a>[h, tab\<^sub>0 h, g\<^sub>0h\<^sub>1.p\<^sub>0]) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> (g \<star> h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> (g \<star> g\<^sub>0h\<^sub>1.\<phi>) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> \<a>[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> (g.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>1) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> f\<^sub>0gh\<^sub>1.\<phi>) \<star> TTfgh_TfTgh.chine) \<cdot> ((\<a>[f, tab\<^sub>0 f, f\<^sub>0gh\<^sub>1.p\<^sub>1] \<star> TTfgh_TfTgh.chine) \<cdot> \<a>\<^sup>-\<^sup>1[f \<star> tab\<^sub>0 f, f\<^sub>0gh\<^sub>1.p\<^sub>1, TTfgh_TfTgh.chine] \<cdot> \<a>\<^sup>-\<^sup>1[f, tab\<^sub>0 f, f\<^sub>0gh\<^sub>1.p\<^sub>1 \<star> TTfgh_TfTgh.chine]) \<cdot> \<a>[f, tab\<^sub>0 f, f\<^sub>0gh\<^sub>1.p\<^sub>1 \<star> TTfgh_TfTgh.chine] \<cdot> (f.tab \<star> f\<^sub>0gh\<^sub>1.p\<^sub>1 \<star> TTfgh_TfTgh.chine) \<cdot> \<a>[tab\<^sub>1 f, f\<^sub>0gh\<^sub>1.p\<^sub>1, TTfgh_TfTgh.chine] \<cdot> TTfgh_TfTgh.the_\<nu> \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>1 f, f\<^sub>0g\<^sub>1.p\<^sub>1, fg\<^sub>0h\<^sub>1.p\<^sub>1] = \<a>[f, g, h \<star> tab\<^sub>0 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0] \<cdot> \<a>[f \<star> g, h, tab\<^sub>0 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0] \<cdot> (\<a>\<^sup>-\<^sup>1[f, g, h] \<star> tab\<^sub>0 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0) \<cdot> ((f \<star> g \<star> h) \<star> TTfgh_TfTgh.the_\<theta>) \<cdot> \<a>[f \<star> g \<star> h, (tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine] \<cdot> (\<a>\<^sup>-\<^sup>1[f, g \<star> h, (tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0] \<star> TTfgh_TfTgh.chine) \<cdot> (((f \<star> \<a>[g \<star> h, tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0, f\<^sub>0gh\<^sub>1.p\<^sub>0]) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> \<a>\<^sup>-\<^sup>1[g, h, tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> (g \<star> \<a>[h, tab\<^sub>0 h, g\<^sub>0h\<^sub>1.p\<^sub>0]) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> (g \<star> h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> (g \<star> g\<^sub>0h\<^sub>1.\<phi>) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> \<a>[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> (g.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>1) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> f\<^sub>0gh\<^sub>1.\<phi>) \<star> TTfgh_TfTgh.chine) \<cdot> \<a>\<^sup>-\<^sup>1[f, tab\<^sub>0 f \<star> f\<^sub>0gh\<^sub>1.p\<^sub>1, TTfgh_TfTgh.chine]) \<cdot> (f \<star> \<a>\<^sup>-\<^sup>1[tab\<^sub>0 f, f\<^sub>0gh\<^sub>1.p\<^sub>1, TTfgh_TfTgh.chine]) \<cdot> \<a>[f, tab\<^sub>0 f, f\<^sub>0gh\<^sub>1.p\<^sub>1 \<star> TTfgh_TfTgh.chine] \<cdot> (f.tab \<star> f\<^sub>0gh\<^sub>1.p\<^sub>1 \<star> TTfgh_TfTgh.chine) \<cdot> \<a>[tab\<^sub>1 f, f\<^sub>0gh\<^sub>1.p\<^sub>1, TTfgh_TfTgh.chine] \<cdot> TTfgh_TfTgh.the_\<nu> \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>1 f, f\<^sub>0g\<^sub>1.p\<^sub>1, fg\<^sub>0h\<^sub>1.p\<^sub>1]
[PROOF STEP]
have "(\<a>[f, tab\<^sub>0 f, TfTgh.p\<^sub>1] \<star> TTfgh_TfTgh.chine) \<cdot>
\<a>\<^sup>-\<^sup>1[f \<star> tab\<^sub>0 f, TfTgh.p\<^sub>1, TTfgh_TfTgh.chine] \<cdot>
\<a>\<^sup>-\<^sup>1[f, tab\<^sub>0 f, TfTgh.p\<^sub>1 \<star> TTfgh_TfTgh.chine] =
\<a>\<^sup>-\<^sup>1[f, tab\<^sub>0 f \<star> TfTgh.p\<^sub>1, TTfgh_TfTgh.chine] \<cdot>
(f \<star> \<a>\<^sup>-\<^sup>1[tab\<^sub>0 f, TfTgh.p\<^sub>1, TTfgh_TfTgh.chine])"
[PROOF STATE]
proof (prove)
goal (1 subgoal):
1. (\<a>[f, tab\<^sub>0 f, f\<^sub>0gh\<^sub>1.p\<^sub>1] \<star> TTfgh_TfTgh.chine) \<cdot> \<a>\<^sup>-\<^sup>1[f \<star> tab\<^sub>0 f, f\<^sub>0gh\<^sub>1.p\<^sub>1, TTfgh_TfTgh.chine] \<cdot> \<a>\<^sup>-\<^sup>1[f, tab\<^sub>0 f, f\<^sub>0gh\<^sub>1.p\<^sub>1 \<star> TTfgh_TfTgh.chine] = \<a>\<^sup>-\<^sup>1[f, tab\<^sub>0 f \<star> f\<^sub>0gh\<^sub>1.p\<^sub>1, TTfgh_TfTgh.chine] \<cdot> (f \<star> \<a>\<^sup>-\<^sup>1[tab\<^sub>0 f, f\<^sub>0gh\<^sub>1.p\<^sub>1, TTfgh_TfTgh.chine])
[PROOF STEP]
proof -
[PROOF STATE]
proof (state)
goal (1 subgoal):
1. (\<a>[f, tab\<^sub>0 f, f\<^sub>0gh\<^sub>1.p\<^sub>1] \<star> TTfgh_TfTgh.chine) \<cdot> \<a>\<^sup>-\<^sup>1[f \<star> tab\<^sub>0 f, f\<^sub>0gh\<^sub>1.p\<^sub>1, TTfgh_TfTgh.chine] \<cdot> \<a>\<^sup>-\<^sup>1[f, tab\<^sub>0 f, f\<^sub>0gh\<^sub>1.p\<^sub>1 \<star> TTfgh_TfTgh.chine] = \<a>\<^sup>-\<^sup>1[f, tab\<^sub>0 f \<star> f\<^sub>0gh\<^sub>1.p\<^sub>1, TTfgh_TfTgh.chine] \<cdot> (f \<star> \<a>\<^sup>-\<^sup>1[tab\<^sub>0 f, f\<^sub>0gh\<^sub>1.p\<^sub>1, TTfgh_TfTgh.chine])
[PROOF STEP]
have "(\<a>[f, tab\<^sub>0 f, TfTgh.p\<^sub>1] \<star> TTfgh_TfTgh.chine) \<cdot>
\<a>\<^sup>-\<^sup>1[f \<star> tab\<^sub>0 f, TfTgh.p\<^sub>1, TTfgh_TfTgh.chine] \<cdot>
\<a>\<^sup>-\<^sup>1[f, tab\<^sub>0 f, TfTgh.p\<^sub>1 \<star> TTfgh_TfTgh.chine] =
\<lbrace>(\<^bold>\<a>\<^bold>[\<^bold>\<langle>f\<^bold>\<rangle>, \<^bold>\<langle>tab\<^sub>0 f\<^bold>\<rangle>, \<^bold>\<langle>TfTgh.p\<^sub>1\<^bold>\<rangle>\<^bold>] \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) \<^bold>\<cdot>
\<^bold>\<a>\<^sup>-\<^sup>1\<^bold>[\<^bold>\<langle>f\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>tab\<^sub>0 f\<^bold>\<rangle>, \<^bold>\<langle>TfTgh.p\<^sub>1\<^bold>\<rangle>, \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>\<^bold>] \<^bold>\<cdot>
\<^bold>\<a>\<^sup>-\<^sup>1\<^bold>[\<^bold>\<langle>f\<^bold>\<rangle>, \<^bold>\<langle>tab\<^sub>0 f\<^bold>\<rangle>, \<^bold>\<langle>TfTgh.p\<^sub>1\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>\<^bold>]\<rbrace>"
[PROOF STATE]
proof (prove)
goal (1 subgoal):
1. (\<a>[f, tab\<^sub>0 f, f\<^sub>0gh\<^sub>1.p\<^sub>1] \<star> TTfgh_TfTgh.chine) \<cdot> \<a>\<^sup>-\<^sup>1[f \<star> tab\<^sub>0 f, f\<^sub>0gh\<^sub>1.p\<^sub>1, TTfgh_TfTgh.chine] \<cdot> \<a>\<^sup>-\<^sup>1[f, tab\<^sub>0 f, f\<^sub>0gh\<^sub>1.p\<^sub>1 \<star> TTfgh_TfTgh.chine] = \<lbrace>(\<^bold>\<a>\<^bold>[\<^bold>\<langle>f\<^bold>\<rangle>, \<^bold>\<langle>tab\<^sub>0 f\<^bold>\<rangle>, \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>1\<^bold>\<rangle>\<^bold>] \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) \<^bold>\<cdot> \<^bold>\<a>\<^sup>-\<^sup>1\<^bold>[\<^bold>\<langle>f\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>tab\<^sub>0 f\<^bold>\<rangle>, \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>1\<^bold>\<rangle>, \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>\<^bold>] \<^bold>\<cdot> \<^bold>\<a>\<^sup>-\<^sup>1\<^bold>[\<^bold>\<langle>f\<^bold>\<rangle>, \<^bold>\<langle>tab\<^sub>0 f\<^bold>\<rangle>, \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>1\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>\<^bold>]\<rbrace>
[PROOF STEP]
using \<a>'_def \<alpha>_def
[PROOF STATE]
proof (prove)
using this:
\<a>\<^sup>-\<^sup>1[?\<mu>, ?\<nu>, ?\<tau>] \<equiv> \<alpha>' (?\<mu>, ?\<nu>, ?\<tau>)
\<alpha> ?\<mu>\<nu>\<tau> \<equiv> \<a>[fst ?\<mu>\<nu>\<tau>, fst (snd ?\<mu>\<nu>\<tau>), snd (snd ?\<mu>\<nu>\<tau>)]
goal (1 subgoal):
1. (\<a>[f, tab\<^sub>0 f, f\<^sub>0gh\<^sub>1.p\<^sub>1] \<star> TTfgh_TfTgh.chine) \<cdot> \<a>\<^sup>-\<^sup>1[f \<star> tab\<^sub>0 f, f\<^sub>0gh\<^sub>1.p\<^sub>1, TTfgh_TfTgh.chine] \<cdot> \<a>\<^sup>-\<^sup>1[f, tab\<^sub>0 f, f\<^sub>0gh\<^sub>1.p\<^sub>1 \<star> TTfgh_TfTgh.chine] = \<lbrace>(\<^bold>\<a>\<^bold>[\<^bold>\<langle>f\<^bold>\<rangle>, \<^bold>\<langle>tab\<^sub>0 f\<^bold>\<rangle>, \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>1\<^bold>\<rangle>\<^bold>] \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) \<^bold>\<cdot> \<^bold>\<a>\<^sup>-\<^sup>1\<^bold>[\<^bold>\<langle>f\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>tab\<^sub>0 f\<^bold>\<rangle>, \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>1\<^bold>\<rangle>, \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>\<^bold>] \<^bold>\<cdot> \<^bold>\<a>\<^sup>-\<^sup>1\<^bold>[\<^bold>\<langle>f\<^bold>\<rangle>, \<^bold>\<langle>tab\<^sub>0 f\<^bold>\<rangle>, \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>1\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>\<^bold>]\<rbrace>
[PROOF STEP]
by simp
[PROOF STATE]
proof (state)
this:
(\<a>[f, tab\<^sub>0 f, f\<^sub>0gh\<^sub>1.p\<^sub>1] \<star> TTfgh_TfTgh.chine) \<cdot> \<a>\<^sup>-\<^sup>1[f \<star> tab\<^sub>0 f, f\<^sub>0gh\<^sub>1.p\<^sub>1, TTfgh_TfTgh.chine] \<cdot> \<a>\<^sup>-\<^sup>1[f, tab\<^sub>0 f, f\<^sub>0gh\<^sub>1.p\<^sub>1 \<star> TTfgh_TfTgh.chine] = \<lbrace>(\<^bold>\<a>\<^bold>[\<^bold>\<langle>f\<^bold>\<rangle>, \<^bold>\<langle>tab\<^sub>0 f\<^bold>\<rangle>, \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>1\<^bold>\<rangle>\<^bold>] \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) \<^bold>\<cdot> \<^bold>\<a>\<^sup>-\<^sup>1\<^bold>[\<^bold>\<langle>f\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>tab\<^sub>0 f\<^bold>\<rangle>, \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>1\<^bold>\<rangle>, \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>\<^bold>] \<^bold>\<cdot> \<^bold>\<a>\<^sup>-\<^sup>1\<^bold>[\<^bold>\<langle>f\<^bold>\<rangle>, \<^bold>\<langle>tab\<^sub>0 f\<^bold>\<rangle>, \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>1\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>\<^bold>]\<rbrace>
goal (1 subgoal):
1. (\<a>[f, tab\<^sub>0 f, f\<^sub>0gh\<^sub>1.p\<^sub>1] \<star> TTfgh_TfTgh.chine) \<cdot> \<a>\<^sup>-\<^sup>1[f \<star> tab\<^sub>0 f, f\<^sub>0gh\<^sub>1.p\<^sub>1, TTfgh_TfTgh.chine] \<cdot> \<a>\<^sup>-\<^sup>1[f, tab\<^sub>0 f, f\<^sub>0gh\<^sub>1.p\<^sub>1 \<star> TTfgh_TfTgh.chine] = \<a>\<^sup>-\<^sup>1[f, tab\<^sub>0 f \<star> f\<^sub>0gh\<^sub>1.p\<^sub>1, TTfgh_TfTgh.chine] \<cdot> (f \<star> \<a>\<^sup>-\<^sup>1[tab\<^sub>0 f, f\<^sub>0gh\<^sub>1.p\<^sub>1, TTfgh_TfTgh.chine])
[PROOF STEP]
also
[PROOF STATE]
proof (state)
this:
(\<a>[f, tab\<^sub>0 f, f\<^sub>0gh\<^sub>1.p\<^sub>1] \<star> TTfgh_TfTgh.chine) \<cdot> \<a>\<^sup>-\<^sup>1[f \<star> tab\<^sub>0 f, f\<^sub>0gh\<^sub>1.p\<^sub>1, TTfgh_TfTgh.chine] \<cdot> \<a>\<^sup>-\<^sup>1[f, tab\<^sub>0 f, f\<^sub>0gh\<^sub>1.p\<^sub>1 \<star> TTfgh_TfTgh.chine] = \<lbrace>(\<^bold>\<a>\<^bold>[\<^bold>\<langle>f\<^bold>\<rangle>, \<^bold>\<langle>tab\<^sub>0 f\<^bold>\<rangle>, \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>1\<^bold>\<rangle>\<^bold>] \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) \<^bold>\<cdot> \<^bold>\<a>\<^sup>-\<^sup>1\<^bold>[\<^bold>\<langle>f\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>tab\<^sub>0 f\<^bold>\<rangle>, \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>1\<^bold>\<rangle>, \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>\<^bold>] \<^bold>\<cdot> \<^bold>\<a>\<^sup>-\<^sup>1\<^bold>[\<^bold>\<langle>f\<^bold>\<rangle>, \<^bold>\<langle>tab\<^sub>0 f\<^bold>\<rangle>, \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>1\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>\<^bold>]\<rbrace>
goal (1 subgoal):
1. (\<a>[f, tab\<^sub>0 f, f\<^sub>0gh\<^sub>1.p\<^sub>1] \<star> TTfgh_TfTgh.chine) \<cdot> \<a>\<^sup>-\<^sup>1[f \<star> tab\<^sub>0 f, f\<^sub>0gh\<^sub>1.p\<^sub>1, TTfgh_TfTgh.chine] \<cdot> \<a>\<^sup>-\<^sup>1[f, tab\<^sub>0 f, f\<^sub>0gh\<^sub>1.p\<^sub>1 \<star> TTfgh_TfTgh.chine] = \<a>\<^sup>-\<^sup>1[f, tab\<^sub>0 f \<star> f\<^sub>0gh\<^sub>1.p\<^sub>1, TTfgh_TfTgh.chine] \<cdot> (f \<star> \<a>\<^sup>-\<^sup>1[tab\<^sub>0 f, f\<^sub>0gh\<^sub>1.p\<^sub>1, TTfgh_TfTgh.chine])
[PROOF STEP]
have "... = \<lbrace>\<^bold>\<a>\<^sup>-\<^sup>1\<^bold>[\<^bold>\<langle>f\<^bold>\<rangle>, \<^bold>\<langle>tab\<^sub>0 f\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>TfTgh.p\<^sub>1\<^bold>\<rangle>, \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>\<^bold>] \<^bold>\<cdot>
(\<^bold>\<langle>f\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<a>\<^sup>-\<^sup>1\<^bold>[\<^bold>\<langle>tab\<^sub>0 f\<^bold>\<rangle>, \<^bold>\<langle>TfTgh.p\<^sub>1\<^bold>\<rangle>, \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>\<^bold>])\<rbrace>"
[PROOF STATE]
proof (prove)
goal (1 subgoal):
1. \<lbrace>(\<^bold>\<a>\<^bold>[\<^bold>\<langle>f\<^bold>\<rangle>, \<^bold>\<langle>tab\<^sub>0 f\<^bold>\<rangle>, \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>1\<^bold>\<rangle>\<^bold>] \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) \<^bold>\<cdot> \<^bold>\<a>\<^sup>-\<^sup>1\<^bold>[\<^bold>\<langle>f\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>tab\<^sub>0 f\<^bold>\<rangle>, \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>1\<^bold>\<rangle>, \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>\<^bold>] \<^bold>\<cdot> \<^bold>\<a>\<^sup>-\<^sup>1\<^bold>[\<^bold>\<langle>f\<^bold>\<rangle>, \<^bold>\<langle>tab\<^sub>0 f\<^bold>\<rangle>, \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>1\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>\<^bold>]\<rbrace> = \<lbrace>\<^bold>\<a>\<^sup>-\<^sup>1\<^bold>[\<^bold>\<langle>f\<^bold>\<rangle>, \<^bold>\<langle>tab\<^sub>0 f\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>1\<^bold>\<rangle>, \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>\<^bold>] \<^bold>\<cdot> (\<^bold>\<langle>f\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<a>\<^sup>-\<^sup>1\<^bold>[\<^bold>\<langle>tab\<^sub>0 f\<^bold>\<rangle>, \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>1\<^bold>\<rangle>, \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>\<^bold>])\<rbrace>
[PROOF STEP]
using fg gh fg\<^sub>0h\<^sub>1.p\<^sub>0_simps fg\<^sub>0h\<^sub>1.p\<^sub>1_simps f\<^sub>0g\<^sub>1.p\<^sub>0_simps f\<^sub>0g\<^sub>1.p\<^sub>1_simps
[PROOF STATE]
proof (prove)
using this:
src f = trg g
src g = trg h
trg fg\<^sub>0h\<^sub>1.p\<^sub>0 = src (tab\<^sub>1 h)
trg fg\<^sub>0h\<^sub>1.p\<^sub>1 = src (tab\<^sub>0 g \<star> f\<^sub>0g\<^sub>1.p\<^sub>0)
trg f\<^sub>0g\<^sub>1.p\<^sub>0 = src (tab\<^sub>1 g)
trg f\<^sub>0g\<^sub>1.p\<^sub>1 = src (tab\<^sub>0 f)
goal (1 subgoal):
1. \<lbrace>(\<^bold>\<a>\<^bold>[\<^bold>\<langle>f\<^bold>\<rangle>, \<^bold>\<langle>tab\<^sub>0 f\<^bold>\<rangle>, \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>1\<^bold>\<rangle>\<^bold>] \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) \<^bold>\<cdot> \<^bold>\<a>\<^sup>-\<^sup>1\<^bold>[\<^bold>\<langle>f\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>tab\<^sub>0 f\<^bold>\<rangle>, \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>1\<^bold>\<rangle>, \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>\<^bold>] \<^bold>\<cdot> \<^bold>\<a>\<^sup>-\<^sup>1\<^bold>[\<^bold>\<langle>f\<^bold>\<rangle>, \<^bold>\<langle>tab\<^sub>0 f\<^bold>\<rangle>, \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>1\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>\<^bold>]\<rbrace> = \<lbrace>\<^bold>\<a>\<^sup>-\<^sup>1\<^bold>[\<^bold>\<langle>f\<^bold>\<rangle>, \<^bold>\<langle>tab\<^sub>0 f\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>1\<^bold>\<rangle>, \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>\<^bold>] \<^bold>\<cdot> (\<^bold>\<langle>f\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<a>\<^sup>-\<^sup>1\<^bold>[\<^bold>\<langle>tab\<^sub>0 f\<^bold>\<rangle>, \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>1\<^bold>\<rangle>, \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>\<^bold>])\<rbrace>
[PROOF STEP]
by (intro E.eval_eqI, auto)
[PROOF STATE]
proof (state)
this:
\<lbrace>(\<^bold>\<a>\<^bold>[\<^bold>\<langle>f\<^bold>\<rangle>, \<^bold>\<langle>tab\<^sub>0 f\<^bold>\<rangle>, \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>1\<^bold>\<rangle>\<^bold>] \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) \<^bold>\<cdot> \<^bold>\<a>\<^sup>-\<^sup>1\<^bold>[\<^bold>\<langle>f\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>tab\<^sub>0 f\<^bold>\<rangle>, \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>1\<^bold>\<rangle>, \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>\<^bold>] \<^bold>\<cdot> \<^bold>\<a>\<^sup>-\<^sup>1\<^bold>[\<^bold>\<langle>f\<^bold>\<rangle>, \<^bold>\<langle>tab\<^sub>0 f\<^bold>\<rangle>, \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>1\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>\<^bold>]\<rbrace> = \<lbrace>\<^bold>\<a>\<^sup>-\<^sup>1\<^bold>[\<^bold>\<langle>f\<^bold>\<rangle>, \<^bold>\<langle>tab\<^sub>0 f\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>1\<^bold>\<rangle>, \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>\<^bold>] \<^bold>\<cdot> (\<^bold>\<langle>f\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<a>\<^sup>-\<^sup>1\<^bold>[\<^bold>\<langle>tab\<^sub>0 f\<^bold>\<rangle>, \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>1\<^bold>\<rangle>, \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>\<^bold>])\<rbrace>
goal (1 subgoal):
1. (\<a>[f, tab\<^sub>0 f, f\<^sub>0gh\<^sub>1.p\<^sub>1] \<star> TTfgh_TfTgh.chine) \<cdot> \<a>\<^sup>-\<^sup>1[f \<star> tab\<^sub>0 f, f\<^sub>0gh\<^sub>1.p\<^sub>1, TTfgh_TfTgh.chine] \<cdot> \<a>\<^sup>-\<^sup>1[f, tab\<^sub>0 f, f\<^sub>0gh\<^sub>1.p\<^sub>1 \<star> TTfgh_TfTgh.chine] = \<a>\<^sup>-\<^sup>1[f, tab\<^sub>0 f \<star> f\<^sub>0gh\<^sub>1.p\<^sub>1, TTfgh_TfTgh.chine] \<cdot> (f \<star> \<a>\<^sup>-\<^sup>1[tab\<^sub>0 f, f\<^sub>0gh\<^sub>1.p\<^sub>1, TTfgh_TfTgh.chine])
[PROOF STEP]
also
[PROOF STATE]
proof (state)
this:
\<lbrace>(\<^bold>\<a>\<^bold>[\<^bold>\<langle>f\<^bold>\<rangle>, \<^bold>\<langle>tab\<^sub>0 f\<^bold>\<rangle>, \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>1\<^bold>\<rangle>\<^bold>] \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) \<^bold>\<cdot> \<^bold>\<a>\<^sup>-\<^sup>1\<^bold>[\<^bold>\<langle>f\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>tab\<^sub>0 f\<^bold>\<rangle>, \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>1\<^bold>\<rangle>, \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>\<^bold>] \<^bold>\<cdot> \<^bold>\<a>\<^sup>-\<^sup>1\<^bold>[\<^bold>\<langle>f\<^bold>\<rangle>, \<^bold>\<langle>tab\<^sub>0 f\<^bold>\<rangle>, \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>1\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>\<^bold>]\<rbrace> = \<lbrace>\<^bold>\<a>\<^sup>-\<^sup>1\<^bold>[\<^bold>\<langle>f\<^bold>\<rangle>, \<^bold>\<langle>tab\<^sub>0 f\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>1\<^bold>\<rangle>, \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>\<^bold>] \<^bold>\<cdot> (\<^bold>\<langle>f\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<a>\<^sup>-\<^sup>1\<^bold>[\<^bold>\<langle>tab\<^sub>0 f\<^bold>\<rangle>, \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>1\<^bold>\<rangle>, \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>\<^bold>])\<rbrace>
goal (1 subgoal):
1. (\<a>[f, tab\<^sub>0 f, f\<^sub>0gh\<^sub>1.p\<^sub>1] \<star> TTfgh_TfTgh.chine) \<cdot> \<a>\<^sup>-\<^sup>1[f \<star> tab\<^sub>0 f, f\<^sub>0gh\<^sub>1.p\<^sub>1, TTfgh_TfTgh.chine] \<cdot> \<a>\<^sup>-\<^sup>1[f, tab\<^sub>0 f, f\<^sub>0gh\<^sub>1.p\<^sub>1 \<star> TTfgh_TfTgh.chine] = \<a>\<^sup>-\<^sup>1[f, tab\<^sub>0 f \<star> f\<^sub>0gh\<^sub>1.p\<^sub>1, TTfgh_TfTgh.chine] \<cdot> (f \<star> \<a>\<^sup>-\<^sup>1[tab\<^sub>0 f, f\<^sub>0gh\<^sub>1.p\<^sub>1, TTfgh_TfTgh.chine])
[PROOF STEP]
have "... = \<a>\<^sup>-\<^sup>1[f, tab\<^sub>0 f \<star> TfTgh.p\<^sub>1, TTfgh_TfTgh.chine] \<cdot>
(f \<star> \<a>\<^sup>-\<^sup>1[tab\<^sub>0 f, TfTgh.p\<^sub>1, TTfgh_TfTgh.chine])"
[PROOF STATE]
proof (prove)
goal (1 subgoal):
1. \<lbrace>\<^bold>\<a>\<^sup>-\<^sup>1\<^bold>[\<^bold>\<langle>f\<^bold>\<rangle>, \<^bold>\<langle>tab\<^sub>0 f\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>1\<^bold>\<rangle>, \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>\<^bold>] \<^bold>\<cdot> (\<^bold>\<langle>f\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<a>\<^sup>-\<^sup>1\<^bold>[\<^bold>\<langle>tab\<^sub>0 f\<^bold>\<rangle>, \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>1\<^bold>\<rangle>, \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>\<^bold>])\<rbrace> = \<a>\<^sup>-\<^sup>1[f, tab\<^sub>0 f \<star> f\<^sub>0gh\<^sub>1.p\<^sub>1, TTfgh_TfTgh.chine] \<cdot> (f \<star> \<a>\<^sup>-\<^sup>1[tab\<^sub>0 f, f\<^sub>0gh\<^sub>1.p\<^sub>1, TTfgh_TfTgh.chine])
[PROOF STEP]
using \<a>'_def \<alpha>_def
[PROOF STATE]
proof (prove)
using this:
\<a>\<^sup>-\<^sup>1[?\<mu>, ?\<nu>, ?\<tau>] \<equiv> \<alpha>' (?\<mu>, ?\<nu>, ?\<tau>)
\<alpha> ?\<mu>\<nu>\<tau> \<equiv> \<a>[fst ?\<mu>\<nu>\<tau>, fst (snd ?\<mu>\<nu>\<tau>), snd (snd ?\<mu>\<nu>\<tau>)]
goal (1 subgoal):
1. \<lbrace>\<^bold>\<a>\<^sup>-\<^sup>1\<^bold>[\<^bold>\<langle>f\<^bold>\<rangle>, \<^bold>\<langle>tab\<^sub>0 f\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>1\<^bold>\<rangle>, \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>\<^bold>] \<^bold>\<cdot> (\<^bold>\<langle>f\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<a>\<^sup>-\<^sup>1\<^bold>[\<^bold>\<langle>tab\<^sub>0 f\<^bold>\<rangle>, \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>1\<^bold>\<rangle>, \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>\<^bold>])\<rbrace> = \<a>\<^sup>-\<^sup>1[f, tab\<^sub>0 f \<star> f\<^sub>0gh\<^sub>1.p\<^sub>1, TTfgh_TfTgh.chine] \<cdot> (f \<star> \<a>\<^sup>-\<^sup>1[tab\<^sub>0 f, f\<^sub>0gh\<^sub>1.p\<^sub>1, TTfgh_TfTgh.chine])
[PROOF STEP]
by simp
[PROOF STATE]
proof (state)
this:
\<lbrace>\<^bold>\<a>\<^sup>-\<^sup>1\<^bold>[\<^bold>\<langle>f\<^bold>\<rangle>, \<^bold>\<langle>tab\<^sub>0 f\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>1\<^bold>\<rangle>, \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>\<^bold>] \<^bold>\<cdot> (\<^bold>\<langle>f\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<a>\<^sup>-\<^sup>1\<^bold>[\<^bold>\<langle>tab\<^sub>0 f\<^bold>\<rangle>, \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>1\<^bold>\<rangle>, \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>\<^bold>])\<rbrace> = \<a>\<^sup>-\<^sup>1[f, tab\<^sub>0 f \<star> f\<^sub>0gh\<^sub>1.p\<^sub>1, TTfgh_TfTgh.chine] \<cdot> (f \<star> \<a>\<^sup>-\<^sup>1[tab\<^sub>0 f, f\<^sub>0gh\<^sub>1.p\<^sub>1, TTfgh_TfTgh.chine])
goal (1 subgoal):
1. (\<a>[f, tab\<^sub>0 f, f\<^sub>0gh\<^sub>1.p\<^sub>1] \<star> TTfgh_TfTgh.chine) \<cdot> \<a>\<^sup>-\<^sup>1[f \<star> tab\<^sub>0 f, f\<^sub>0gh\<^sub>1.p\<^sub>1, TTfgh_TfTgh.chine] \<cdot> \<a>\<^sup>-\<^sup>1[f, tab\<^sub>0 f, f\<^sub>0gh\<^sub>1.p\<^sub>1 \<star> TTfgh_TfTgh.chine] = \<a>\<^sup>-\<^sup>1[f, tab\<^sub>0 f \<star> f\<^sub>0gh\<^sub>1.p\<^sub>1, TTfgh_TfTgh.chine] \<cdot> (f \<star> \<a>\<^sup>-\<^sup>1[tab\<^sub>0 f, f\<^sub>0gh\<^sub>1.p\<^sub>1, TTfgh_TfTgh.chine])
[PROOF STEP]
finally
[PROOF STATE]
proof (chain)
picking this:
(\<a>[f, tab\<^sub>0 f, f\<^sub>0gh\<^sub>1.p\<^sub>1] \<star> TTfgh_TfTgh.chine) \<cdot> \<a>\<^sup>-\<^sup>1[f \<star> tab\<^sub>0 f, f\<^sub>0gh\<^sub>1.p\<^sub>1, TTfgh_TfTgh.chine] \<cdot> \<a>\<^sup>-\<^sup>1[f, tab\<^sub>0 f, f\<^sub>0gh\<^sub>1.p\<^sub>1 \<star> TTfgh_TfTgh.chine] = \<a>\<^sup>-\<^sup>1[f, tab\<^sub>0 f \<star> f\<^sub>0gh\<^sub>1.p\<^sub>1, TTfgh_TfTgh.chine] \<cdot> (f \<star> \<a>\<^sup>-\<^sup>1[tab\<^sub>0 f, f\<^sub>0gh\<^sub>1.p\<^sub>1, TTfgh_TfTgh.chine])
[PROOF STEP]
show ?thesis
[PROOF STATE]
proof (prove)
using this:
(\<a>[f, tab\<^sub>0 f, f\<^sub>0gh\<^sub>1.p\<^sub>1] \<star> TTfgh_TfTgh.chine) \<cdot> \<a>\<^sup>-\<^sup>1[f \<star> tab\<^sub>0 f, f\<^sub>0gh\<^sub>1.p\<^sub>1, TTfgh_TfTgh.chine] \<cdot> \<a>\<^sup>-\<^sup>1[f, tab\<^sub>0 f, f\<^sub>0gh\<^sub>1.p\<^sub>1 \<star> TTfgh_TfTgh.chine] = \<a>\<^sup>-\<^sup>1[f, tab\<^sub>0 f \<star> f\<^sub>0gh\<^sub>1.p\<^sub>1, TTfgh_TfTgh.chine] \<cdot> (f \<star> \<a>\<^sup>-\<^sup>1[tab\<^sub>0 f, f\<^sub>0gh\<^sub>1.p\<^sub>1, TTfgh_TfTgh.chine])
goal (1 subgoal):
1. (\<a>[f, tab\<^sub>0 f, f\<^sub>0gh\<^sub>1.p\<^sub>1] \<star> TTfgh_TfTgh.chine) \<cdot> \<a>\<^sup>-\<^sup>1[f \<star> tab\<^sub>0 f, f\<^sub>0gh\<^sub>1.p\<^sub>1, TTfgh_TfTgh.chine] \<cdot> \<a>\<^sup>-\<^sup>1[f, tab\<^sub>0 f, f\<^sub>0gh\<^sub>1.p\<^sub>1 \<star> TTfgh_TfTgh.chine] = \<a>\<^sup>-\<^sup>1[f, tab\<^sub>0 f \<star> f\<^sub>0gh\<^sub>1.p\<^sub>1, TTfgh_TfTgh.chine] \<cdot> (f \<star> \<a>\<^sup>-\<^sup>1[tab\<^sub>0 f, f\<^sub>0gh\<^sub>1.p\<^sub>1, TTfgh_TfTgh.chine])
[PROOF STEP]
by simp
[PROOF STATE]
proof (state)
this:
(\<a>[f, tab\<^sub>0 f, f\<^sub>0gh\<^sub>1.p\<^sub>1] \<star> TTfgh_TfTgh.chine) \<cdot> \<a>\<^sup>-\<^sup>1[f \<star> tab\<^sub>0 f, f\<^sub>0gh\<^sub>1.p\<^sub>1, TTfgh_TfTgh.chine] \<cdot> \<a>\<^sup>-\<^sup>1[f, tab\<^sub>0 f, f\<^sub>0gh\<^sub>1.p\<^sub>1 \<star> TTfgh_TfTgh.chine] = \<a>\<^sup>-\<^sup>1[f, tab\<^sub>0 f \<star> f\<^sub>0gh\<^sub>1.p\<^sub>1, TTfgh_TfTgh.chine] \<cdot> (f \<star> \<a>\<^sup>-\<^sup>1[tab\<^sub>0 f, f\<^sub>0gh\<^sub>1.p\<^sub>1, TTfgh_TfTgh.chine])
goal:
No subgoals!
[PROOF STEP]
qed
[PROOF STATE]
proof (state)
this:
(\<a>[f, tab\<^sub>0 f, f\<^sub>0gh\<^sub>1.p\<^sub>1] \<star> TTfgh_TfTgh.chine) \<cdot> \<a>\<^sup>-\<^sup>1[f \<star> tab\<^sub>0 f, f\<^sub>0gh\<^sub>1.p\<^sub>1, TTfgh_TfTgh.chine] \<cdot> \<a>\<^sup>-\<^sup>1[f, tab\<^sub>0 f, f\<^sub>0gh\<^sub>1.p\<^sub>1 \<star> TTfgh_TfTgh.chine] = \<a>\<^sup>-\<^sup>1[f, tab\<^sub>0 f \<star> f\<^sub>0gh\<^sub>1.p\<^sub>1, TTfgh_TfTgh.chine] \<cdot> (f \<star> \<a>\<^sup>-\<^sup>1[tab\<^sub>0 f, f\<^sub>0gh\<^sub>1.p\<^sub>1, TTfgh_TfTgh.chine])
goal (1 subgoal):
1. \<a>[f, g, h \<star> tab\<^sub>0 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0] \<cdot> \<a>[f \<star> g, h, tab\<^sub>0 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0] \<cdot> (\<a>\<^sup>-\<^sup>1[f, g, h] \<star> tab\<^sub>0 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0) \<cdot> ((f \<star> g \<star> h) \<star> TTfgh_TfTgh.the_\<theta>) \<cdot> \<a>[f \<star> g \<star> h, (tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine] \<cdot> (\<a>\<^sup>-\<^sup>1[f, g \<star> h, (tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0] \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> \<a>[g \<star> h, tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0, f\<^sub>0gh\<^sub>1.p\<^sub>0]) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> \<a>\<^sup>-\<^sup>1[g, h, tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> (g \<star> \<a>[h, tab\<^sub>0 h, g\<^sub>0h\<^sub>1.p\<^sub>0]) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> (g \<star> h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> (g \<star> g\<^sub>0h\<^sub>1.\<phi>) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> \<a>[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> (g.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>1) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> f\<^sub>0gh\<^sub>1.\<phi>) \<star> TTfgh_TfTgh.chine) \<cdot> ((\<a>[f, tab\<^sub>0 f, f\<^sub>0gh\<^sub>1.p\<^sub>1] \<star> TTfgh_TfTgh.chine) \<cdot> \<a>\<^sup>-\<^sup>1[f \<star> tab\<^sub>0 f, f\<^sub>0gh\<^sub>1.p\<^sub>1, TTfgh_TfTgh.chine] \<cdot> \<a>\<^sup>-\<^sup>1[f, tab\<^sub>0 f, f\<^sub>0gh\<^sub>1.p\<^sub>1 \<star> TTfgh_TfTgh.chine]) \<cdot> \<a>[f, tab\<^sub>0 f, f\<^sub>0gh\<^sub>1.p\<^sub>1 \<star> TTfgh_TfTgh.chine] \<cdot> (f.tab \<star> f\<^sub>0gh\<^sub>1.p\<^sub>1 \<star> TTfgh_TfTgh.chine) \<cdot> \<a>[tab\<^sub>1 f, f\<^sub>0gh\<^sub>1.p\<^sub>1, TTfgh_TfTgh.chine] \<cdot> TTfgh_TfTgh.the_\<nu> \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>1 f, f\<^sub>0g\<^sub>1.p\<^sub>1, fg\<^sub>0h\<^sub>1.p\<^sub>1] = \<a>[f, g, h \<star> tab\<^sub>0 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0] \<cdot> \<a>[f \<star> g, h, tab\<^sub>0 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0] \<cdot> (\<a>\<^sup>-\<^sup>1[f, g, h] \<star> tab\<^sub>0 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0) \<cdot> ((f \<star> g \<star> h) \<star> TTfgh_TfTgh.the_\<theta>) \<cdot> \<a>[f \<star> g \<star> h, (tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine] \<cdot> (\<a>\<^sup>-\<^sup>1[f, g \<star> h, (tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0] \<star> TTfgh_TfTgh.chine) \<cdot> (((f \<star> \<a>[g \<star> h, tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0, f\<^sub>0gh\<^sub>1.p\<^sub>0]) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> \<a>\<^sup>-\<^sup>1[g, h, tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> (g \<star> \<a>[h, tab\<^sub>0 h, g\<^sub>0h\<^sub>1.p\<^sub>0]) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> (g \<star> h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> (g \<star> g\<^sub>0h\<^sub>1.\<phi>) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> \<a>[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> (g.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>1) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> f\<^sub>0gh\<^sub>1.\<phi>) \<star> TTfgh_TfTgh.chine) \<cdot> \<a>\<^sup>-\<^sup>1[f, tab\<^sub>0 f \<star> f\<^sub>0gh\<^sub>1.p\<^sub>1, TTfgh_TfTgh.chine]) \<cdot> (f \<star> \<a>\<^sup>-\<^sup>1[tab\<^sub>0 f, f\<^sub>0gh\<^sub>1.p\<^sub>1, TTfgh_TfTgh.chine]) \<cdot> \<a>[f, tab\<^sub>0 f, f\<^sub>0gh\<^sub>1.p\<^sub>1 \<star> TTfgh_TfTgh.chine] \<cdot> (f.tab \<star> f\<^sub>0gh\<^sub>1.p\<^sub>1 \<star> TTfgh_TfTgh.chine) \<cdot> \<a>[tab\<^sub>1 f, f\<^sub>0gh\<^sub>1.p\<^sub>1, TTfgh_TfTgh.chine] \<cdot> TTfgh_TfTgh.the_\<nu> \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>1 f, f\<^sub>0g\<^sub>1.p\<^sub>1, fg\<^sub>0h\<^sub>1.p\<^sub>1]
[PROOF STEP]
thus ?thesis
[PROOF STATE]
proof (prove)
using this:
(\<a>[f, tab\<^sub>0 f, f\<^sub>0gh\<^sub>1.p\<^sub>1] \<star> TTfgh_TfTgh.chine) \<cdot> \<a>\<^sup>-\<^sup>1[f \<star> tab\<^sub>0 f, f\<^sub>0gh\<^sub>1.p\<^sub>1, TTfgh_TfTgh.chine] \<cdot> \<a>\<^sup>-\<^sup>1[f, tab\<^sub>0 f, f\<^sub>0gh\<^sub>1.p\<^sub>1 \<star> TTfgh_TfTgh.chine] = \<a>\<^sup>-\<^sup>1[f, tab\<^sub>0 f \<star> f\<^sub>0gh\<^sub>1.p\<^sub>1, TTfgh_TfTgh.chine] \<cdot> (f \<star> \<a>\<^sup>-\<^sup>1[tab\<^sub>0 f, f\<^sub>0gh\<^sub>1.p\<^sub>1, TTfgh_TfTgh.chine])
goal (1 subgoal):
1. \<a>[f, g, h \<star> tab\<^sub>0 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0] \<cdot> \<a>[f \<star> g, h, tab\<^sub>0 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0] \<cdot> (\<a>\<^sup>-\<^sup>1[f, g, h] \<star> tab\<^sub>0 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0) \<cdot> ((f \<star> g \<star> h) \<star> TTfgh_TfTgh.the_\<theta>) \<cdot> \<a>[f \<star> g \<star> h, (tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine] \<cdot> (\<a>\<^sup>-\<^sup>1[f, g \<star> h, (tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0] \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> \<a>[g \<star> h, tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0, f\<^sub>0gh\<^sub>1.p\<^sub>0]) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> \<a>\<^sup>-\<^sup>1[g, h, tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> (g \<star> \<a>[h, tab\<^sub>0 h, g\<^sub>0h\<^sub>1.p\<^sub>0]) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> (g \<star> h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> (g \<star> g\<^sub>0h\<^sub>1.\<phi>) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> \<a>[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> (g.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>1) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> f\<^sub>0gh\<^sub>1.\<phi>) \<star> TTfgh_TfTgh.chine) \<cdot> ((\<a>[f, tab\<^sub>0 f, f\<^sub>0gh\<^sub>1.p\<^sub>1] \<star> TTfgh_TfTgh.chine) \<cdot> \<a>\<^sup>-\<^sup>1[f \<star> tab\<^sub>0 f, f\<^sub>0gh\<^sub>1.p\<^sub>1, TTfgh_TfTgh.chine] \<cdot> \<a>\<^sup>-\<^sup>1[f, tab\<^sub>0 f, f\<^sub>0gh\<^sub>1.p\<^sub>1 \<star> TTfgh_TfTgh.chine]) \<cdot> \<a>[f, tab\<^sub>0 f, f\<^sub>0gh\<^sub>1.p\<^sub>1 \<star> TTfgh_TfTgh.chine] \<cdot> (f.tab \<star> f\<^sub>0gh\<^sub>1.p\<^sub>1 \<star> TTfgh_TfTgh.chine) \<cdot> \<a>[tab\<^sub>1 f, f\<^sub>0gh\<^sub>1.p\<^sub>1, TTfgh_TfTgh.chine] \<cdot> TTfgh_TfTgh.the_\<nu> \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>1 f, f\<^sub>0g\<^sub>1.p\<^sub>1, fg\<^sub>0h\<^sub>1.p\<^sub>1] = \<a>[f, g, h \<star> tab\<^sub>0 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0] \<cdot> \<a>[f \<star> g, h, tab\<^sub>0 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0] \<cdot> (\<a>\<^sup>-\<^sup>1[f, g, h] \<star> tab\<^sub>0 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0) \<cdot> ((f \<star> g \<star> h) \<star> TTfgh_TfTgh.the_\<theta>) \<cdot> \<a>[f \<star> g \<star> h, (tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine] \<cdot> (\<a>\<^sup>-\<^sup>1[f, g \<star> h, (tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0] \<star> TTfgh_TfTgh.chine) \<cdot> (((f \<star> \<a>[g \<star> h, tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0, f\<^sub>0gh\<^sub>1.p\<^sub>0]) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> \<a>\<^sup>-\<^sup>1[g, h, tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> (g \<star> \<a>[h, tab\<^sub>0 h, g\<^sub>0h\<^sub>1.p\<^sub>0]) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> (g \<star> h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> (g \<star> g\<^sub>0h\<^sub>1.\<phi>) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> \<a>[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> (g.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>1) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> f\<^sub>0gh\<^sub>1.\<phi>) \<star> TTfgh_TfTgh.chine) \<cdot> \<a>\<^sup>-\<^sup>1[f, tab\<^sub>0 f \<star> f\<^sub>0gh\<^sub>1.p\<^sub>1, TTfgh_TfTgh.chine]) \<cdot> (f \<star> \<a>\<^sup>-\<^sup>1[tab\<^sub>0 f, f\<^sub>0gh\<^sub>1.p\<^sub>1, TTfgh_TfTgh.chine]) \<cdot> \<a>[f, tab\<^sub>0 f, f\<^sub>0gh\<^sub>1.p\<^sub>1 \<star> TTfgh_TfTgh.chine] \<cdot> (f.tab \<star> f\<^sub>0gh\<^sub>1.p\<^sub>1 \<star> TTfgh_TfTgh.chine) \<cdot> \<a>[tab\<^sub>1 f, f\<^sub>0gh\<^sub>1.p\<^sub>1, TTfgh_TfTgh.chine] \<cdot> TTfgh_TfTgh.the_\<nu> \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>1 f, f\<^sub>0g\<^sub>1.p\<^sub>1, fg\<^sub>0h\<^sub>1.p\<^sub>1]
[PROOF STEP]
using comp_assoc
[PROOF STATE]
proof (prove)
using this:
(\<a>[f, tab\<^sub>0 f, f\<^sub>0gh\<^sub>1.p\<^sub>1] \<star> TTfgh_TfTgh.chine) \<cdot> \<a>\<^sup>-\<^sup>1[f \<star> tab\<^sub>0 f, f\<^sub>0gh\<^sub>1.p\<^sub>1, TTfgh_TfTgh.chine] \<cdot> \<a>\<^sup>-\<^sup>1[f, tab\<^sub>0 f, f\<^sub>0gh\<^sub>1.p\<^sub>1 \<star> TTfgh_TfTgh.chine] = \<a>\<^sup>-\<^sup>1[f, tab\<^sub>0 f \<star> f\<^sub>0gh\<^sub>1.p\<^sub>1, TTfgh_TfTgh.chine] \<cdot> (f \<star> \<a>\<^sup>-\<^sup>1[tab\<^sub>0 f, f\<^sub>0gh\<^sub>1.p\<^sub>1, TTfgh_TfTgh.chine])
(?h \<cdot> ?g) \<cdot> ?f = ?h \<cdot> ?g \<cdot> ?f
goal (1 subgoal):
1. \<a>[f, g, h \<star> tab\<^sub>0 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0] \<cdot> \<a>[f \<star> g, h, tab\<^sub>0 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0] \<cdot> (\<a>\<^sup>-\<^sup>1[f, g, h] \<star> tab\<^sub>0 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0) \<cdot> ((f \<star> g \<star> h) \<star> TTfgh_TfTgh.the_\<theta>) \<cdot> \<a>[f \<star> g \<star> h, (tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine] \<cdot> (\<a>\<^sup>-\<^sup>1[f, g \<star> h, (tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0] \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> \<a>[g \<star> h, tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0, f\<^sub>0gh\<^sub>1.p\<^sub>0]) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> \<a>\<^sup>-\<^sup>1[g, h, tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> (g \<star> \<a>[h, tab\<^sub>0 h, g\<^sub>0h\<^sub>1.p\<^sub>0]) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> (g \<star> h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> (g \<star> g\<^sub>0h\<^sub>1.\<phi>) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> \<a>[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> (g.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>1) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> f\<^sub>0gh\<^sub>1.\<phi>) \<star> TTfgh_TfTgh.chine) \<cdot> ((\<a>[f, tab\<^sub>0 f, f\<^sub>0gh\<^sub>1.p\<^sub>1] \<star> TTfgh_TfTgh.chine) \<cdot> \<a>\<^sup>-\<^sup>1[f \<star> tab\<^sub>0 f, f\<^sub>0gh\<^sub>1.p\<^sub>1, TTfgh_TfTgh.chine] \<cdot> \<a>\<^sup>-\<^sup>1[f, tab\<^sub>0 f, f\<^sub>0gh\<^sub>1.p\<^sub>1 \<star> TTfgh_TfTgh.chine]) \<cdot> \<a>[f, tab\<^sub>0 f, f\<^sub>0gh\<^sub>1.p\<^sub>1 \<star> TTfgh_TfTgh.chine] \<cdot> (f.tab \<star> f\<^sub>0gh\<^sub>1.p\<^sub>1 \<star> TTfgh_TfTgh.chine) \<cdot> \<a>[tab\<^sub>1 f, f\<^sub>0gh\<^sub>1.p\<^sub>1, TTfgh_TfTgh.chine] \<cdot> TTfgh_TfTgh.the_\<nu> \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>1 f, f\<^sub>0g\<^sub>1.p\<^sub>1, fg\<^sub>0h\<^sub>1.p\<^sub>1] = \<a>[f, g, h \<star> tab\<^sub>0 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0] \<cdot> \<a>[f \<star> g, h, tab\<^sub>0 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0] \<cdot> (\<a>\<^sup>-\<^sup>1[f, g, h] \<star> tab\<^sub>0 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0) \<cdot> ((f \<star> g \<star> h) \<star> TTfgh_TfTgh.the_\<theta>) \<cdot> \<a>[f \<star> g \<star> h, (tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine] \<cdot> (\<a>\<^sup>-\<^sup>1[f, g \<star> h, (tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0] \<star> TTfgh_TfTgh.chine) \<cdot> (((f \<star> \<a>[g \<star> h, tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0, f\<^sub>0gh\<^sub>1.p\<^sub>0]) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> \<a>\<^sup>-\<^sup>1[g, h, tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> (g \<star> \<a>[h, tab\<^sub>0 h, g\<^sub>0h\<^sub>1.p\<^sub>0]) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> (g \<star> h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> (g \<star> g\<^sub>0h\<^sub>1.\<phi>) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> \<a>[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> (g.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>1) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> f\<^sub>0gh\<^sub>1.\<phi>) \<star> TTfgh_TfTgh.chine) \<cdot> \<a>\<^sup>-\<^sup>1[f, tab\<^sub>0 f \<star> f\<^sub>0gh\<^sub>1.p\<^sub>1, TTfgh_TfTgh.chine]) \<cdot> (f \<star> \<a>\<^sup>-\<^sup>1[tab\<^sub>0 f, f\<^sub>0gh\<^sub>1.p\<^sub>1, TTfgh_TfTgh.chine]) \<cdot> \<a>[f, tab\<^sub>0 f, f\<^sub>0gh\<^sub>1.p\<^sub>1 \<star> TTfgh_TfTgh.chine] \<cdot> (f.tab \<star> f\<^sub>0gh\<^sub>1.p\<^sub>1 \<star> TTfgh_TfTgh.chine) \<cdot> \<a>[tab\<^sub>1 f, f\<^sub>0gh\<^sub>1.p\<^sub>1, TTfgh_TfTgh.chine] \<cdot> TTfgh_TfTgh.the_\<nu> \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>1 f, f\<^sub>0g\<^sub>1.p\<^sub>1, fg\<^sub>0h\<^sub>1.p\<^sub>1]
[PROOF STEP]
by presburger
[PROOF STATE]
proof (state)
this:
\<a>[f, g, h \<star> tab\<^sub>0 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0] \<cdot> \<a>[f \<star> g, h, tab\<^sub>0 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0] \<cdot> (\<a>\<^sup>-\<^sup>1[f, g, h] \<star> tab\<^sub>0 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0) \<cdot> ((f \<star> g \<star> h) \<star> TTfgh_TfTgh.the_\<theta>) \<cdot> \<a>[f \<star> g \<star> h, (tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine] \<cdot> (\<a>\<^sup>-\<^sup>1[f, g \<star> h, (tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0] \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> \<a>[g \<star> h, tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0, f\<^sub>0gh\<^sub>1.p\<^sub>0]) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> \<a>\<^sup>-\<^sup>1[g, h, tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> (g \<star> \<a>[h, tab\<^sub>0 h, g\<^sub>0h\<^sub>1.p\<^sub>0]) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> (g \<star> h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> (g \<star> g\<^sub>0h\<^sub>1.\<phi>) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> \<a>[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> (g.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>1) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> f\<^sub>0gh\<^sub>1.\<phi>) \<star> TTfgh_TfTgh.chine) \<cdot> ((\<a>[f, tab\<^sub>0 f, f\<^sub>0gh\<^sub>1.p\<^sub>1] \<star> TTfgh_TfTgh.chine) \<cdot> \<a>\<^sup>-\<^sup>1[f \<star> tab\<^sub>0 f, f\<^sub>0gh\<^sub>1.p\<^sub>1, TTfgh_TfTgh.chine] \<cdot> \<a>\<^sup>-\<^sup>1[f, tab\<^sub>0 f, f\<^sub>0gh\<^sub>1.p\<^sub>1 \<star> TTfgh_TfTgh.chine]) \<cdot> \<a>[f, tab\<^sub>0 f, f\<^sub>0gh\<^sub>1.p\<^sub>1 \<star> TTfgh_TfTgh.chine] \<cdot> (f.tab \<star> f\<^sub>0gh\<^sub>1.p\<^sub>1 \<star> TTfgh_TfTgh.chine) \<cdot> \<a>[tab\<^sub>1 f, f\<^sub>0gh\<^sub>1.p\<^sub>1, TTfgh_TfTgh.chine] \<cdot> TTfgh_TfTgh.the_\<nu> \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>1 f, f\<^sub>0g\<^sub>1.p\<^sub>1, fg\<^sub>0h\<^sub>1.p\<^sub>1] = \<a>[f, g, h \<star> tab\<^sub>0 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0] \<cdot> \<a>[f \<star> g, h, tab\<^sub>0 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0] \<cdot> (\<a>\<^sup>-\<^sup>1[f, g, h] \<star> tab\<^sub>0 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0) \<cdot> ((f \<star> g \<star> h) \<star> TTfgh_TfTgh.the_\<theta>) \<cdot> \<a>[f \<star> g \<star> h, (tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine] \<cdot> (\<a>\<^sup>-\<^sup>1[f, g \<star> h, (tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0] \<star> TTfgh_TfTgh.chine) \<cdot> (((f \<star> \<a>[g \<star> h, tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0, f\<^sub>0gh\<^sub>1.p\<^sub>0]) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> \<a>\<^sup>-\<^sup>1[g, h, tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> (g \<star> \<a>[h, tab\<^sub>0 h, g\<^sub>0h\<^sub>1.p\<^sub>0]) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> (g \<star> h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> (g \<star> g\<^sub>0h\<^sub>1.\<phi>) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> \<a>[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> (g.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>1) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> f\<^sub>0gh\<^sub>1.\<phi>) \<star> TTfgh_TfTgh.chine) \<cdot> \<a>\<^sup>-\<^sup>1[f, tab\<^sub>0 f \<star> f\<^sub>0gh\<^sub>1.p\<^sub>1, TTfgh_TfTgh.chine]) \<cdot> (f \<star> \<a>\<^sup>-\<^sup>1[tab\<^sub>0 f, f\<^sub>0gh\<^sub>1.p\<^sub>1, TTfgh_TfTgh.chine]) \<cdot> \<a>[f, tab\<^sub>0 f, f\<^sub>0gh\<^sub>1.p\<^sub>1 \<star> TTfgh_TfTgh.chine] \<cdot> (f.tab \<star> f\<^sub>0gh\<^sub>1.p\<^sub>1 \<star> TTfgh_TfTgh.chine) \<cdot> \<a>[tab\<^sub>1 f, f\<^sub>0gh\<^sub>1.p\<^sub>1, TTfgh_TfTgh.chine] \<cdot> TTfgh_TfTgh.the_\<nu> \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>1 f, f\<^sub>0g\<^sub>1.p\<^sub>1, fg\<^sub>0h\<^sub>1.p\<^sub>1]
goal:
No subgoals!
[PROOF STEP]
qed
[PROOF STATE]
proof (state)
this:
\<a>[f, g, h \<star> tab\<^sub>0 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0] \<cdot> \<a>[f \<star> g, h, tab\<^sub>0 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0] \<cdot> (\<a>\<^sup>-\<^sup>1[f, g, h] \<star> tab\<^sub>0 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0) \<cdot> ((f \<star> g \<star> h) \<star> TTfgh_TfTgh.the_\<theta>) \<cdot> \<a>[f \<star> g \<star> h, (tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine] \<cdot> (\<a>\<^sup>-\<^sup>1[f, g \<star> h, (tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0] \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> \<a>[g \<star> h, tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0, f\<^sub>0gh\<^sub>1.p\<^sub>0]) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> \<a>\<^sup>-\<^sup>1[g, h, tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> (g \<star> \<a>[h, tab\<^sub>0 h, g\<^sub>0h\<^sub>1.p\<^sub>0]) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> (g \<star> h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> (g \<star> g\<^sub>0h\<^sub>1.\<phi>) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> \<a>[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> (g.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>1) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> f\<^sub>0gh\<^sub>1.\<phi>) \<star> TTfgh_TfTgh.chine) \<cdot> ((\<a>[f, tab\<^sub>0 f, f\<^sub>0gh\<^sub>1.p\<^sub>1] \<star> TTfgh_TfTgh.chine) \<cdot> \<a>\<^sup>-\<^sup>1[f \<star> tab\<^sub>0 f, f\<^sub>0gh\<^sub>1.p\<^sub>1, TTfgh_TfTgh.chine] \<cdot> \<a>\<^sup>-\<^sup>1[f, tab\<^sub>0 f, f\<^sub>0gh\<^sub>1.p\<^sub>1 \<star> TTfgh_TfTgh.chine]) \<cdot> \<a>[f, tab\<^sub>0 f, f\<^sub>0gh\<^sub>1.p\<^sub>1 \<star> TTfgh_TfTgh.chine] \<cdot> (f.tab \<star> f\<^sub>0gh\<^sub>1.p\<^sub>1 \<star> TTfgh_TfTgh.chine) \<cdot> \<a>[tab\<^sub>1 f, f\<^sub>0gh\<^sub>1.p\<^sub>1, TTfgh_TfTgh.chine] \<cdot> TTfgh_TfTgh.the_\<nu> \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>1 f, f\<^sub>0g\<^sub>1.p\<^sub>1, fg\<^sub>0h\<^sub>1.p\<^sub>1] = \<a>[f, g, h \<star> tab\<^sub>0 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0] \<cdot> \<a>[f \<star> g, h, tab\<^sub>0 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0] \<cdot> (\<a>\<^sup>-\<^sup>1[f, g, h] \<star> tab\<^sub>0 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0) \<cdot> ((f \<star> g \<star> h) \<star> TTfgh_TfTgh.the_\<theta>) \<cdot> \<a>[f \<star> g \<star> h, (tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine] \<cdot> (\<a>\<^sup>-\<^sup>1[f, g \<star> h, (tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0] \<star> TTfgh_TfTgh.chine) \<cdot> (((f \<star> \<a>[g \<star> h, tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0, f\<^sub>0gh\<^sub>1.p\<^sub>0]) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> \<a>\<^sup>-\<^sup>1[g, h, tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> (g \<star> \<a>[h, tab\<^sub>0 h, g\<^sub>0h\<^sub>1.p\<^sub>0]) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> (g \<star> h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> (g \<star> g\<^sub>0h\<^sub>1.\<phi>) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> \<a>[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> (g.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>1) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> f\<^sub>0gh\<^sub>1.\<phi>) \<star> TTfgh_TfTgh.chine) \<cdot> \<a>\<^sup>-\<^sup>1[f, tab\<^sub>0 f \<star> f\<^sub>0gh\<^sub>1.p\<^sub>1, TTfgh_TfTgh.chine]) \<cdot> (f \<star> \<a>\<^sup>-\<^sup>1[tab\<^sub>0 f, f\<^sub>0gh\<^sub>1.p\<^sub>1, TTfgh_TfTgh.chine]) \<cdot> \<a>[f, tab\<^sub>0 f, f\<^sub>0gh\<^sub>1.p\<^sub>1 \<star> TTfgh_TfTgh.chine] \<cdot> (f.tab \<star> f\<^sub>0gh\<^sub>1.p\<^sub>1 \<star> TTfgh_TfTgh.chine) \<cdot> \<a>[tab\<^sub>1 f, f\<^sub>0gh\<^sub>1.p\<^sub>1, TTfgh_TfTgh.chine] \<cdot> TTfgh_TfTgh.the_\<nu> \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>1 f, f\<^sub>0g\<^sub>1.p\<^sub>1, fg\<^sub>0h\<^sub>1.p\<^sub>1]
goal (1 subgoal):
1. (f \<star> \<theta>\<^sub>f) \<cdot> \<a>[f, tab\<^sub>0 f, w\<^sub>f] \<cdot> (f.tab \<star> w\<^sub>f) = ((f \<star> \<theta>\<^sub>f') \<cdot> \<a>[f, tab\<^sub>0 f, w\<^sub>f'] \<cdot> (f.tab \<star> w\<^sub>f')) \<cdot> \<beta>\<^sub>f
[PROOF STEP]
also
[PROOF STATE]
proof (state)
this:
\<a>[f, g, h \<star> tab\<^sub>0 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0] \<cdot> \<a>[f \<star> g, h, tab\<^sub>0 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0] \<cdot> (\<a>\<^sup>-\<^sup>1[f, g, h] \<star> tab\<^sub>0 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0) \<cdot> ((f \<star> g \<star> h) \<star> TTfgh_TfTgh.the_\<theta>) \<cdot> \<a>[f \<star> g \<star> h, (tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine] \<cdot> (\<a>\<^sup>-\<^sup>1[f, g \<star> h, (tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0] \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> \<a>[g \<star> h, tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0, f\<^sub>0gh\<^sub>1.p\<^sub>0]) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> \<a>\<^sup>-\<^sup>1[g, h, tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> (g \<star> \<a>[h, tab\<^sub>0 h, g\<^sub>0h\<^sub>1.p\<^sub>0]) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> (g \<star> h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> (g \<star> g\<^sub>0h\<^sub>1.\<phi>) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> \<a>[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> (g.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>1) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> f\<^sub>0gh\<^sub>1.\<phi>) \<star> TTfgh_TfTgh.chine) \<cdot> ((\<a>[f, tab\<^sub>0 f, f\<^sub>0gh\<^sub>1.p\<^sub>1] \<star> TTfgh_TfTgh.chine) \<cdot> \<a>\<^sup>-\<^sup>1[f \<star> tab\<^sub>0 f, f\<^sub>0gh\<^sub>1.p\<^sub>1, TTfgh_TfTgh.chine] \<cdot> \<a>\<^sup>-\<^sup>1[f, tab\<^sub>0 f, f\<^sub>0gh\<^sub>1.p\<^sub>1 \<star> TTfgh_TfTgh.chine]) \<cdot> \<a>[f, tab\<^sub>0 f, f\<^sub>0gh\<^sub>1.p\<^sub>1 \<star> TTfgh_TfTgh.chine] \<cdot> (f.tab \<star> f\<^sub>0gh\<^sub>1.p\<^sub>1 \<star> TTfgh_TfTgh.chine) \<cdot> \<a>[tab\<^sub>1 f, f\<^sub>0gh\<^sub>1.p\<^sub>1, TTfgh_TfTgh.chine] \<cdot> TTfgh_TfTgh.the_\<nu> \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>1 f, f\<^sub>0g\<^sub>1.p\<^sub>1, fg\<^sub>0h\<^sub>1.p\<^sub>1] = \<a>[f, g, h \<star> tab\<^sub>0 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0] \<cdot> \<a>[f \<star> g, h, tab\<^sub>0 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0] \<cdot> (\<a>\<^sup>-\<^sup>1[f, g, h] \<star> tab\<^sub>0 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0) \<cdot> ((f \<star> g \<star> h) \<star> TTfgh_TfTgh.the_\<theta>) \<cdot> \<a>[f \<star> g \<star> h, (tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine] \<cdot> (\<a>\<^sup>-\<^sup>1[f, g \<star> h, (tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0] \<star> TTfgh_TfTgh.chine) \<cdot> (((f \<star> \<a>[g \<star> h, tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0, f\<^sub>0gh\<^sub>1.p\<^sub>0]) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> \<a>\<^sup>-\<^sup>1[g, h, tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> (g \<star> \<a>[h, tab\<^sub>0 h, g\<^sub>0h\<^sub>1.p\<^sub>0]) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> (g \<star> h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> (g \<star> g\<^sub>0h\<^sub>1.\<phi>) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> \<a>[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> (g.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>1) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> f\<^sub>0gh\<^sub>1.\<phi>) \<star> TTfgh_TfTgh.chine) \<cdot> \<a>\<^sup>-\<^sup>1[f, tab\<^sub>0 f \<star> f\<^sub>0gh\<^sub>1.p\<^sub>1, TTfgh_TfTgh.chine]) \<cdot> (f \<star> \<a>\<^sup>-\<^sup>1[tab\<^sub>0 f, f\<^sub>0gh\<^sub>1.p\<^sub>1, TTfgh_TfTgh.chine]) \<cdot> \<a>[f, tab\<^sub>0 f, f\<^sub>0gh\<^sub>1.p\<^sub>1 \<star> TTfgh_TfTgh.chine] \<cdot> (f.tab \<star> f\<^sub>0gh\<^sub>1.p\<^sub>1 \<star> TTfgh_TfTgh.chine) \<cdot> \<a>[tab\<^sub>1 f, f\<^sub>0gh\<^sub>1.p\<^sub>1, TTfgh_TfTgh.chine] \<cdot> TTfgh_TfTgh.the_\<nu> \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>1 f, f\<^sub>0g\<^sub>1.p\<^sub>1, fg\<^sub>0h\<^sub>1.p\<^sub>1]
goal (1 subgoal):
1. (f \<star> \<theta>\<^sub>f) \<cdot> \<a>[f, tab\<^sub>0 f, w\<^sub>f] \<cdot> (f.tab \<star> w\<^sub>f) = ((f \<star> \<theta>\<^sub>f') \<cdot> \<a>[f, tab\<^sub>0 f, w\<^sub>f'] \<cdot> (f.tab \<star> w\<^sub>f')) \<cdot> \<beta>\<^sub>f
[PROOF STEP]
have "... =
\<a>[f, g, h \<star> tab\<^sub>0 h \<star> TTfgh.p\<^sub>0] \<cdot>
\<a>[f \<star> g, h, tab\<^sub>0 h \<star> TTfgh.p\<^sub>0] \<cdot>
(\<a>\<^sup>-\<^sup>1[f, g, h] \<star> tab\<^sub>0 h \<star> TTfgh.p\<^sub>0) \<cdot>
((f \<star> g \<star> h) \<star> TTfgh_TfTgh.the_\<theta>) \<cdot>
\<a>[f \<star> g \<star> h, (tab\<^sub>0 h \<star> Tgh.p\<^sub>0) \<star> TfTgh.p\<^sub>0, TTfgh_TfTgh.chine] \<cdot>
(\<a>\<^sup>-\<^sup>1[f, g \<star> h, (tab\<^sub>0 h \<star> Tgh.p\<^sub>0) \<star> TfTgh.p\<^sub>0] \<star> TTfgh_TfTgh.chine) \<cdot>
\<a>\<^sup>-\<^sup>1[f, (g \<star> h) \<star> (tab\<^sub>0 h \<star> Tgh.p\<^sub>0) \<star> TfTgh.p\<^sub>0, TTfgh_TfTgh.chine] \<cdot>
(f \<star> \<a>[g \<star> h, tab\<^sub>0 h \<star> Tgh.p\<^sub>0, TfTgh.p\<^sub>0] \<star> TTfgh_TfTgh.chine) \<cdot>
(f \<star> (\<a>\<^sup>-\<^sup>1[g, h, tab\<^sub>0 h \<star> Tgh.p\<^sub>0] \<star> TfTgh.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot>
(f \<star> ((g \<star> \<a>[h, tab\<^sub>0 h, Tgh.p\<^sub>0]) \<star> TfTgh.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot>
((f \<star> ((g \<star> h.tab \<star> Tgh.p\<^sub>0) \<star> TfTgh.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot>
(f \<star> ((g \<star> g\<^sub>0h\<^sub>1.\<phi>) \<star> TfTgh.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot>
(f \<star> (\<a>[g, tab\<^sub>0 g, Tgh.p\<^sub>1] \<star> TfTgh.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot>
(f \<star> ((g.tab \<star> Tgh.p\<^sub>1) \<star> TfTgh.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot>
(f \<star> f\<^sub>0gh\<^sub>1.\<phi> \<star> TTfgh_TfTgh.chine) \<cdot>
(f \<star> \<a>\<^sup>-\<^sup>1[tab\<^sub>0 f, TfTgh.p\<^sub>1, TTfgh_TfTgh.chine])) \<cdot>
\<a>[f, tab\<^sub>0 f, TfTgh.p\<^sub>1 \<star> TTfgh_TfTgh.chine] \<cdot>
(f.tab \<star> TfTgh.p\<^sub>1 \<star> TTfgh_TfTgh.chine) \<cdot>
\<a>[tab\<^sub>1 f, TfTgh.p\<^sub>1, TTfgh_TfTgh.chine] \<cdot>
TTfgh_TfTgh.the_\<nu> \<cdot>
\<a>\<^sup>-\<^sup>1[tab\<^sub>1 f, Tfg.p\<^sub>1, TTfgh.p\<^sub>1]"
[PROOF STATE]
proof (prove)
goal (1 subgoal):
1. \<a>[f, g, h \<star> tab\<^sub>0 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0] \<cdot> \<a>[f \<star> g, h, tab\<^sub>0 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0] \<cdot> (\<a>\<^sup>-\<^sup>1[f, g, h] \<star> tab\<^sub>0 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0) \<cdot> ((f \<star> g \<star> h) \<star> TTfgh_TfTgh.the_\<theta>) \<cdot> \<a>[f \<star> g \<star> h, (tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine] \<cdot> (\<a>\<^sup>-\<^sup>1[f, g \<star> h, (tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0] \<star> TTfgh_TfTgh.chine) \<cdot> (((f \<star> \<a>[g \<star> h, tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0, f\<^sub>0gh\<^sub>1.p\<^sub>0]) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> \<a>\<^sup>-\<^sup>1[g, h, tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> (g \<star> \<a>[h, tab\<^sub>0 h, g\<^sub>0h\<^sub>1.p\<^sub>0]) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> (g \<star> h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> (g \<star> g\<^sub>0h\<^sub>1.\<phi>) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> \<a>[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> (g.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>1) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> f\<^sub>0gh\<^sub>1.\<phi>) \<star> TTfgh_TfTgh.chine) \<cdot> \<a>\<^sup>-\<^sup>1[f, tab\<^sub>0 f \<star> f\<^sub>0gh\<^sub>1.p\<^sub>1, TTfgh_TfTgh.chine]) \<cdot> (f \<star> \<a>\<^sup>-\<^sup>1[tab\<^sub>0 f, f\<^sub>0gh\<^sub>1.p\<^sub>1, TTfgh_TfTgh.chine]) \<cdot> \<a>[f, tab\<^sub>0 f, f\<^sub>0gh\<^sub>1.p\<^sub>1 \<star> TTfgh_TfTgh.chine] \<cdot> (f.tab \<star> f\<^sub>0gh\<^sub>1.p\<^sub>1 \<star> TTfgh_TfTgh.chine) \<cdot> \<a>[tab\<^sub>1 f, f\<^sub>0gh\<^sub>1.p\<^sub>1, TTfgh_TfTgh.chine] \<cdot> TTfgh_TfTgh.the_\<nu> \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>1 f, f\<^sub>0g\<^sub>1.p\<^sub>1, fg\<^sub>0h\<^sub>1.p\<^sub>1] = \<a>[f, g, h \<star> tab\<^sub>0 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0] \<cdot> \<a>[f \<star> g, h, tab\<^sub>0 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0] \<cdot> (\<a>\<^sup>-\<^sup>1[f, g, h] \<star> tab\<^sub>0 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0) \<cdot> ((f \<star> g \<star> h) \<star> TTfgh_TfTgh.the_\<theta>) \<cdot> \<a>[f \<star> g \<star> h, (tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine] \<cdot> (\<a>\<^sup>-\<^sup>1[f, g \<star> h, (tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0] \<star> TTfgh_TfTgh.chine) \<cdot> \<a>\<^sup>-\<^sup>1[f, (g \<star> h) \<star> (tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine] \<cdot> (f \<star> \<a>[g \<star> h, tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0, f\<^sub>0gh\<^sub>1.p\<^sub>0] \<star> TTfgh_TfTgh.chine) \<cdot> (f \<star> (\<a>\<^sup>-\<^sup>1[g, h, tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (f \<star> ((g \<star> \<a>[h, tab\<^sub>0 h, g\<^sub>0h\<^sub>1.p\<^sub>0]) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> ((g \<star> h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (f \<star> ((g \<star> g\<^sub>0h\<^sub>1.\<phi>) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (f \<star> (\<a>[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (f \<star> ((g.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>1) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (f \<star> f\<^sub>0gh\<^sub>1.\<phi> \<star> TTfgh_TfTgh.chine) \<cdot> (f \<star> \<a>\<^sup>-\<^sup>1[tab\<^sub>0 f, f\<^sub>0gh\<^sub>1.p\<^sub>1, TTfgh_TfTgh.chine])) \<cdot> \<a>[f, tab\<^sub>0 f, f\<^sub>0gh\<^sub>1.p\<^sub>1 \<star> TTfgh_TfTgh.chine] \<cdot> (f.tab \<star> f\<^sub>0gh\<^sub>1.p\<^sub>1 \<star> TTfgh_TfTgh.chine) \<cdot> \<a>[tab\<^sub>1 f, f\<^sub>0gh\<^sub>1.p\<^sub>1, TTfgh_TfTgh.chine] \<cdot> TTfgh_TfTgh.the_\<nu> \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>1 f, f\<^sub>0g\<^sub>1.p\<^sub>1, fg\<^sub>0h\<^sub>1.p\<^sub>1]
[PROOF STEP]
proof -
[PROOF STATE]
proof (state)
goal (1 subgoal):
1. \<a>[f, g, h \<star> tab\<^sub>0 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0] \<cdot> \<a>[f \<star> g, h, tab\<^sub>0 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0] \<cdot> (\<a>\<^sup>-\<^sup>1[f, g, h] \<star> tab\<^sub>0 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0) \<cdot> ((f \<star> g \<star> h) \<star> TTfgh_TfTgh.the_\<theta>) \<cdot> \<a>[f \<star> g \<star> h, (tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine] \<cdot> (\<a>\<^sup>-\<^sup>1[f, g \<star> h, (tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0] \<star> TTfgh_TfTgh.chine) \<cdot> (((f \<star> \<a>[g \<star> h, tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0, f\<^sub>0gh\<^sub>1.p\<^sub>0]) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> \<a>\<^sup>-\<^sup>1[g, h, tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> (g \<star> \<a>[h, tab\<^sub>0 h, g\<^sub>0h\<^sub>1.p\<^sub>0]) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> (g \<star> h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> (g \<star> g\<^sub>0h\<^sub>1.\<phi>) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> \<a>[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> (g.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>1) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> f\<^sub>0gh\<^sub>1.\<phi>) \<star> TTfgh_TfTgh.chine) \<cdot> \<a>\<^sup>-\<^sup>1[f, tab\<^sub>0 f \<star> f\<^sub>0gh\<^sub>1.p\<^sub>1, TTfgh_TfTgh.chine]) \<cdot> (f \<star> \<a>\<^sup>-\<^sup>1[tab\<^sub>0 f, f\<^sub>0gh\<^sub>1.p\<^sub>1, TTfgh_TfTgh.chine]) \<cdot> \<a>[f, tab\<^sub>0 f, f\<^sub>0gh\<^sub>1.p\<^sub>1 \<star> TTfgh_TfTgh.chine] \<cdot> (f.tab \<star> f\<^sub>0gh\<^sub>1.p\<^sub>1 \<star> TTfgh_TfTgh.chine) \<cdot> \<a>[tab\<^sub>1 f, f\<^sub>0gh\<^sub>1.p\<^sub>1, TTfgh_TfTgh.chine] \<cdot> TTfgh_TfTgh.the_\<nu> \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>1 f, f\<^sub>0g\<^sub>1.p\<^sub>1, fg\<^sub>0h\<^sub>1.p\<^sub>1] = \<a>[f, g, h \<star> tab\<^sub>0 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0] \<cdot> \<a>[f \<star> g, h, tab\<^sub>0 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0] \<cdot> (\<a>\<^sup>-\<^sup>1[f, g, h] \<star> tab\<^sub>0 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0) \<cdot> ((f \<star> g \<star> h) \<star> TTfgh_TfTgh.the_\<theta>) \<cdot> \<a>[f \<star> g \<star> h, (tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine] \<cdot> (\<a>\<^sup>-\<^sup>1[f, g \<star> h, (tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0] \<star> TTfgh_TfTgh.chine) \<cdot> \<a>\<^sup>-\<^sup>1[f, (g \<star> h) \<star> (tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine] \<cdot> (f \<star> \<a>[g \<star> h, tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0, f\<^sub>0gh\<^sub>1.p\<^sub>0] \<star> TTfgh_TfTgh.chine) \<cdot> (f \<star> (\<a>\<^sup>-\<^sup>1[g, h, tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (f \<star> ((g \<star> \<a>[h, tab\<^sub>0 h, g\<^sub>0h\<^sub>1.p\<^sub>0]) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> ((g \<star> h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (f \<star> ((g \<star> g\<^sub>0h\<^sub>1.\<phi>) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (f \<star> (\<a>[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (f \<star> ((g.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>1) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (f \<star> f\<^sub>0gh\<^sub>1.\<phi> \<star> TTfgh_TfTgh.chine) \<cdot> (f \<star> \<a>\<^sup>-\<^sup>1[tab\<^sub>0 f, f\<^sub>0gh\<^sub>1.p\<^sub>1, TTfgh_TfTgh.chine])) \<cdot> \<a>[f, tab\<^sub>0 f, f\<^sub>0gh\<^sub>1.p\<^sub>1 \<star> TTfgh_TfTgh.chine] \<cdot> (f.tab \<star> f\<^sub>0gh\<^sub>1.p\<^sub>1 \<star> TTfgh_TfTgh.chine) \<cdot> \<a>[tab\<^sub>1 f, f\<^sub>0gh\<^sub>1.p\<^sub>1, TTfgh_TfTgh.chine] \<cdot> TTfgh_TfTgh.the_\<nu> \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>1 f, f\<^sub>0g\<^sub>1.p\<^sub>1, fg\<^sub>0h\<^sub>1.p\<^sub>1]
[PROOF STEP]
have "((f \<star> \<a>[g \<star> h, tab\<^sub>0 h \<star> Tgh.p\<^sub>0, TfTgh.p\<^sub>0]) \<star> TTfgh_TfTgh.chine) \<cdot>
((f \<star> \<a>\<^sup>-\<^sup>1[g, h, tab\<^sub>0 h \<star> Tgh.p\<^sub>0] \<star> TfTgh.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot>
((f \<star> (g \<star> \<a>[h, tab\<^sub>0 h, Tgh.p\<^sub>0]) \<star> TfTgh.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot>
((f \<star> (g \<star> h.tab \<star> Tgh.p\<^sub>0) \<star> TfTgh.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot>
((f \<star> (g \<star> g\<^sub>0h\<^sub>1.\<phi>) \<star> TfTgh.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot>
((f \<star> \<a>[g, tab\<^sub>0 g, Tgh.p\<^sub>1] \<star> TfTgh.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot>
((f \<star> (g.tab \<star> Tgh.p\<^sub>1) \<star> TfTgh.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot>
((f \<star> f\<^sub>0gh\<^sub>1.\<phi>) \<star> TTfgh_TfTgh.chine) \<cdot>
\<a>\<^sup>-\<^sup>1[f, tab\<^sub>0 f \<star> TfTgh.p\<^sub>1, TTfgh_TfTgh.chine] =
\<a>\<^sup>-\<^sup>1[f, (g \<star> h) \<star> (tab\<^sub>0 h \<star> Tgh.p\<^sub>0) \<star> TfTgh.p\<^sub>0, TTfgh_TfTgh.chine] \<cdot>
(f \<star> \<a>[g \<star> h, tab\<^sub>0 h \<star> Tgh.p\<^sub>0, TfTgh.p\<^sub>0] \<star> TTfgh_TfTgh.chine) \<cdot>
(f \<star> (\<a>\<^sup>-\<^sup>1[g, h, tab\<^sub>0 h \<star> Tgh.p\<^sub>0] \<star> TfTgh.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot>
(f \<star> ((g \<star> \<a>[h, tab\<^sub>0 h, Tgh.p\<^sub>0]) \<star> TfTgh.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot>
(f \<star> ((g \<star> h.tab \<star> Tgh.p\<^sub>0) \<star> TfTgh.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot>
(f \<star> ((g \<star> g\<^sub>0h\<^sub>1.\<phi>) \<star> TfTgh.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot>
(f \<star> (\<a>[g, tab\<^sub>0 g, Tgh.p\<^sub>1] \<star> TfTgh.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot>
(f \<star> ((g.tab \<star> Tgh.p\<^sub>1) \<star> TfTgh.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot>
(f \<star> f\<^sub>0gh\<^sub>1.\<phi> \<star> TTfgh_TfTgh.chine)"
[PROOF STATE]
proof (prove)
goal (1 subgoal):
1. ((f \<star> \<a>[g \<star> h, tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0, f\<^sub>0gh\<^sub>1.p\<^sub>0]) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> \<a>\<^sup>-\<^sup>1[g, h, tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> (g \<star> \<a>[h, tab\<^sub>0 h, g\<^sub>0h\<^sub>1.p\<^sub>0]) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> (g \<star> h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> (g \<star> g\<^sub>0h\<^sub>1.\<phi>) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> \<a>[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> (g.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>1) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> f\<^sub>0gh\<^sub>1.\<phi>) \<star> TTfgh_TfTgh.chine) \<cdot> \<a>\<^sup>-\<^sup>1[f, tab\<^sub>0 f \<star> f\<^sub>0gh\<^sub>1.p\<^sub>1, TTfgh_TfTgh.chine] = \<a>\<^sup>-\<^sup>1[f, (g \<star> h) \<star> (tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine] \<cdot> (f \<star> \<a>[g \<star> h, tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0, f\<^sub>0gh\<^sub>1.p\<^sub>0] \<star> TTfgh_TfTgh.chine) \<cdot> (f \<star> (\<a>\<^sup>-\<^sup>1[g, h, tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (f \<star> ((g \<star> \<a>[h, tab\<^sub>0 h, g\<^sub>0h\<^sub>1.p\<^sub>0]) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (f \<star> ((g \<star> h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (f \<star> ((g \<star> g\<^sub>0h\<^sub>1.\<phi>) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (f \<star> (\<a>[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (f \<star> ((g.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>1) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (f \<star> f\<^sub>0gh\<^sub>1.\<phi> \<star> TTfgh_TfTgh.chine)
[PROOF STEP]
proof -
[PROOF STATE]
proof (state)
goal (1 subgoal):
1. ((f \<star> \<a>[g \<star> h, tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0, f\<^sub>0gh\<^sub>1.p\<^sub>0]) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> \<a>\<^sup>-\<^sup>1[g, h, tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> (g \<star> \<a>[h, tab\<^sub>0 h, g\<^sub>0h\<^sub>1.p\<^sub>0]) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> (g \<star> h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> (g \<star> g\<^sub>0h\<^sub>1.\<phi>) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> \<a>[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> (g.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>1) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> f\<^sub>0gh\<^sub>1.\<phi>) \<star> TTfgh_TfTgh.chine) \<cdot> \<a>\<^sup>-\<^sup>1[f, tab\<^sub>0 f \<star> f\<^sub>0gh\<^sub>1.p\<^sub>1, TTfgh_TfTgh.chine] = \<a>\<^sup>-\<^sup>1[f, (g \<star> h) \<star> (tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine] \<cdot> (f \<star> \<a>[g \<star> h, tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0, f\<^sub>0gh\<^sub>1.p\<^sub>0] \<star> TTfgh_TfTgh.chine) \<cdot> (f \<star> (\<a>\<^sup>-\<^sup>1[g, h, tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (f \<star> ((g \<star> \<a>[h, tab\<^sub>0 h, g\<^sub>0h\<^sub>1.p\<^sub>0]) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (f \<star> ((g \<star> h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (f \<star> ((g \<star> g\<^sub>0h\<^sub>1.\<phi>) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (f \<star> (\<a>[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (f \<star> ((g.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>1) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (f \<star> f\<^sub>0gh\<^sub>1.\<phi> \<star> TTfgh_TfTgh.chine)
[PROOF STEP]
have "((f \<star> \<a>[g \<star> h, tab\<^sub>0 h \<star> Tgh.p\<^sub>0, TfTgh.p\<^sub>0]) \<star> TTfgh_TfTgh.chine) \<cdot>
((f \<star> \<a>\<^sup>-\<^sup>1[g, h, tab\<^sub>0 h \<star> Tgh.p\<^sub>0] \<star> TfTgh.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot>
((f \<star> (g \<star> \<a>[h, tab\<^sub>0 h, Tgh.p\<^sub>0]) \<star> TfTgh.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot>
((f \<star> (g \<star> h.tab \<star> Tgh.p\<^sub>0) \<star> TfTgh.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot>
((f \<star> (g \<star> g\<^sub>0h\<^sub>1.\<phi>) \<star> TfTgh.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot>
((f \<star> \<a>[g, tab\<^sub>0 g, Tgh.p\<^sub>1] \<star> TfTgh.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot>
((f \<star> (g.tab \<star> Tgh.p\<^sub>1) \<star> TfTgh.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot>
((f \<star> f\<^sub>0gh\<^sub>1.\<phi>) \<star> TTfgh_TfTgh.chine) \<cdot>
\<a>\<^sup>-\<^sup>1[f, tab\<^sub>0 f \<star> TfTgh.p\<^sub>1, TTfgh_TfTgh.chine] =
((f \<star> \<a>[g \<star> h, tab\<^sub>0 h \<star> Tgh.p\<^sub>0, TfTgh.p\<^sub>0]) \<star> TTfgh_TfTgh.chine) \<cdot>
((f \<star> \<a>\<^sup>-\<^sup>1[g, h, tab\<^sub>0 h \<star> Tgh.p\<^sub>0] \<star> TfTgh.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot>
((f \<star> (g \<star> \<a>[h, tab\<^sub>0 h, Tgh.p\<^sub>0]) \<star> TfTgh.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot>
((f \<star> (g \<star> h.tab \<star> Tgh.p\<^sub>0) \<star> TfTgh.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot>
((f \<star> (g \<star> g\<^sub>0h\<^sub>1.\<phi>) \<star> TfTgh.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot>
((f \<star> \<a>[g, tab\<^sub>0 g, Tgh.p\<^sub>1] \<star> TfTgh.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot>
(((f \<star> (g.tab \<star> Tgh.p\<^sub>1) \<star> TfTgh.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot>
\<a>\<^sup>-\<^sup>1[f, (tab\<^sub>1 g \<star> Tgh.p\<^sub>1) \<star> TfTgh.p\<^sub>0, TTfgh_TfTgh.chine]) \<cdot>
(f \<star> f\<^sub>0gh\<^sub>1.\<phi> \<star> TTfgh_TfTgh.chine)"
[PROOF STATE]
proof (prove)
goal (1 subgoal):
1. ((f \<star> \<a>[g \<star> h, tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0, f\<^sub>0gh\<^sub>1.p\<^sub>0]) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> \<a>\<^sup>-\<^sup>1[g, h, tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> (g \<star> \<a>[h, tab\<^sub>0 h, g\<^sub>0h\<^sub>1.p\<^sub>0]) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> (g \<star> h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> (g \<star> g\<^sub>0h\<^sub>1.\<phi>) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> \<a>[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> (g.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>1) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> f\<^sub>0gh\<^sub>1.\<phi>) \<star> TTfgh_TfTgh.chine) \<cdot> \<a>\<^sup>-\<^sup>1[f, tab\<^sub>0 f \<star> f\<^sub>0gh\<^sub>1.p\<^sub>1, TTfgh_TfTgh.chine] = ((f \<star> \<a>[g \<star> h, tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0, f\<^sub>0gh\<^sub>1.p\<^sub>0]) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> \<a>\<^sup>-\<^sup>1[g, h, tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> (g \<star> \<a>[h, tab\<^sub>0 h, g\<^sub>0h\<^sub>1.p\<^sub>0]) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> (g \<star> h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> (g \<star> g\<^sub>0h\<^sub>1.\<phi>) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> \<a>[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (((f \<star> (g.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>1) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> \<a>\<^sup>-\<^sup>1[f, (tab\<^sub>1 g \<star> g\<^sub>0h\<^sub>1.p\<^sub>1) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine]) \<cdot> (f \<star> f\<^sub>0gh\<^sub>1.\<phi> \<star> TTfgh_TfTgh.chine)
[PROOF STEP]
using fg gh fg\<^sub>0h\<^sub>1.p\<^sub>0_simps fg\<^sub>0h\<^sub>1.p\<^sub>1_simps f\<^sub>0g\<^sub>1.p\<^sub>0_simps f\<^sub>0g\<^sub>1.p\<^sub>1_simps
assoc'_naturality [of f f\<^sub>0gh\<^sub>1.\<phi> TTfgh_TfTgh.chine] comp_assoc
[PROOF STATE]
proof (prove)
using this:
src f = trg g
src g = trg h
trg fg\<^sub>0h\<^sub>1.p\<^sub>0 = src (tab\<^sub>1 h)
trg fg\<^sub>0h\<^sub>1.p\<^sub>1 = src (tab\<^sub>0 g \<star> f\<^sub>0g\<^sub>1.p\<^sub>0)
trg f\<^sub>0g\<^sub>1.p\<^sub>0 = src (tab\<^sub>1 g)
trg f\<^sub>0g\<^sub>1.p\<^sub>1 = src (tab\<^sub>0 f)
\<lbrakk>arr f; arr f\<^sub>0gh\<^sub>1.\<phi>; arr TTfgh_TfTgh.chine; src f = trg f\<^sub>0gh\<^sub>1.\<phi>; src f\<^sub>0gh\<^sub>1.\<phi> = trg TTfgh_TfTgh.chine\<rbrakk> \<Longrightarrow> \<a>\<^sup>-\<^sup>1[cod f, cod f\<^sub>0gh\<^sub>1.\<phi>, cod TTfgh_TfTgh.chine] \<cdot> (f \<star> f\<^sub>0gh\<^sub>1.\<phi> \<star> TTfgh_TfTgh.chine) = ((f \<star> f\<^sub>0gh\<^sub>1.\<phi>) \<star> TTfgh_TfTgh.chine) \<cdot> \<a>\<^sup>-\<^sup>1[local.dom f, local.dom f\<^sub>0gh\<^sub>1.\<phi>, local.dom TTfgh_TfTgh.chine]
(?h \<cdot> ?g) \<cdot> ?f = ?h \<cdot> ?g \<cdot> ?f
goal (1 subgoal):
1. ((f \<star> \<a>[g \<star> h, tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0, f\<^sub>0gh\<^sub>1.p\<^sub>0]) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> \<a>\<^sup>-\<^sup>1[g, h, tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> (g \<star> \<a>[h, tab\<^sub>0 h, g\<^sub>0h\<^sub>1.p\<^sub>0]) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> (g \<star> h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> (g \<star> g\<^sub>0h\<^sub>1.\<phi>) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> \<a>[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> (g.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>1) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> f\<^sub>0gh\<^sub>1.\<phi>) \<star> TTfgh_TfTgh.chine) \<cdot> \<a>\<^sup>-\<^sup>1[f, tab\<^sub>0 f \<star> f\<^sub>0gh\<^sub>1.p\<^sub>1, TTfgh_TfTgh.chine] = ((f \<star> \<a>[g \<star> h, tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0, f\<^sub>0gh\<^sub>1.p\<^sub>0]) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> \<a>\<^sup>-\<^sup>1[g, h, tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> (g \<star> \<a>[h, tab\<^sub>0 h, g\<^sub>0h\<^sub>1.p\<^sub>0]) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> (g \<star> h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> (g \<star> g\<^sub>0h\<^sub>1.\<phi>) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> \<a>[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (((f \<star> (g.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>1) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> \<a>\<^sup>-\<^sup>1[f, (tab\<^sub>1 g \<star> g\<^sub>0h\<^sub>1.p\<^sub>1) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine]) \<cdot> (f \<star> f\<^sub>0gh\<^sub>1.\<phi> \<star> TTfgh_TfTgh.chine)
[PROOF STEP]
by simp
[PROOF STATE]
proof (state)
this:
((f \<star> \<a>[g \<star> h, tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0, f\<^sub>0gh\<^sub>1.p\<^sub>0]) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> \<a>\<^sup>-\<^sup>1[g, h, tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> (g \<star> \<a>[h, tab\<^sub>0 h, g\<^sub>0h\<^sub>1.p\<^sub>0]) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> (g \<star> h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> (g \<star> g\<^sub>0h\<^sub>1.\<phi>) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> \<a>[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> (g.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>1) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> f\<^sub>0gh\<^sub>1.\<phi>) \<star> TTfgh_TfTgh.chine) \<cdot> \<a>\<^sup>-\<^sup>1[f, tab\<^sub>0 f \<star> f\<^sub>0gh\<^sub>1.p\<^sub>1, TTfgh_TfTgh.chine] = ((f \<star> \<a>[g \<star> h, tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0, f\<^sub>0gh\<^sub>1.p\<^sub>0]) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> \<a>\<^sup>-\<^sup>1[g, h, tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> (g \<star> \<a>[h, tab\<^sub>0 h, g\<^sub>0h\<^sub>1.p\<^sub>0]) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> (g \<star> h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> (g \<star> g\<^sub>0h\<^sub>1.\<phi>) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> \<a>[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (((f \<star> (g.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>1) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> \<a>\<^sup>-\<^sup>1[f, (tab\<^sub>1 g \<star> g\<^sub>0h\<^sub>1.p\<^sub>1) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine]) \<cdot> (f \<star> f\<^sub>0gh\<^sub>1.\<phi> \<star> TTfgh_TfTgh.chine)
goal (1 subgoal):
1. ((f \<star> \<a>[g \<star> h, tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0, f\<^sub>0gh\<^sub>1.p\<^sub>0]) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> \<a>\<^sup>-\<^sup>1[g, h, tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> (g \<star> \<a>[h, tab\<^sub>0 h, g\<^sub>0h\<^sub>1.p\<^sub>0]) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> (g \<star> h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> (g \<star> g\<^sub>0h\<^sub>1.\<phi>) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> \<a>[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> (g.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>1) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> f\<^sub>0gh\<^sub>1.\<phi>) \<star> TTfgh_TfTgh.chine) \<cdot> \<a>\<^sup>-\<^sup>1[f, tab\<^sub>0 f \<star> f\<^sub>0gh\<^sub>1.p\<^sub>1, TTfgh_TfTgh.chine] = \<a>\<^sup>-\<^sup>1[f, (g \<star> h) \<star> (tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine] \<cdot> (f \<star> \<a>[g \<star> h, tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0, f\<^sub>0gh\<^sub>1.p\<^sub>0] \<star> TTfgh_TfTgh.chine) \<cdot> (f \<star> (\<a>\<^sup>-\<^sup>1[g, h, tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (f \<star> ((g \<star> \<a>[h, tab\<^sub>0 h, g\<^sub>0h\<^sub>1.p\<^sub>0]) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (f \<star> ((g \<star> h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (f \<star> ((g \<star> g\<^sub>0h\<^sub>1.\<phi>) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (f \<star> (\<a>[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (f \<star> ((g.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>1) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (f \<star> f\<^sub>0gh\<^sub>1.\<phi> \<star> TTfgh_TfTgh.chine)
[PROOF STEP]
also
[PROOF STATE]
proof (state)
this:
((f \<star> \<a>[g \<star> h, tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0, f\<^sub>0gh\<^sub>1.p\<^sub>0]) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> \<a>\<^sup>-\<^sup>1[g, h, tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> (g \<star> \<a>[h, tab\<^sub>0 h, g\<^sub>0h\<^sub>1.p\<^sub>0]) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> (g \<star> h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> (g \<star> g\<^sub>0h\<^sub>1.\<phi>) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> \<a>[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> (g.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>1) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> f\<^sub>0gh\<^sub>1.\<phi>) \<star> TTfgh_TfTgh.chine) \<cdot> \<a>\<^sup>-\<^sup>1[f, tab\<^sub>0 f \<star> f\<^sub>0gh\<^sub>1.p\<^sub>1, TTfgh_TfTgh.chine] = ((f \<star> \<a>[g \<star> h, tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0, f\<^sub>0gh\<^sub>1.p\<^sub>0]) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> \<a>\<^sup>-\<^sup>1[g, h, tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> (g \<star> \<a>[h, tab\<^sub>0 h, g\<^sub>0h\<^sub>1.p\<^sub>0]) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> (g \<star> h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> (g \<star> g\<^sub>0h\<^sub>1.\<phi>) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> \<a>[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (((f \<star> (g.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>1) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> \<a>\<^sup>-\<^sup>1[f, (tab\<^sub>1 g \<star> g\<^sub>0h\<^sub>1.p\<^sub>1) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine]) \<cdot> (f \<star> f\<^sub>0gh\<^sub>1.\<phi> \<star> TTfgh_TfTgh.chine)
goal (1 subgoal):
1. ((f \<star> \<a>[g \<star> h, tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0, f\<^sub>0gh\<^sub>1.p\<^sub>0]) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> \<a>\<^sup>-\<^sup>1[g, h, tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> (g \<star> \<a>[h, tab\<^sub>0 h, g\<^sub>0h\<^sub>1.p\<^sub>0]) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> (g \<star> h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> (g \<star> g\<^sub>0h\<^sub>1.\<phi>) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> \<a>[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> (g.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>1) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> f\<^sub>0gh\<^sub>1.\<phi>) \<star> TTfgh_TfTgh.chine) \<cdot> \<a>\<^sup>-\<^sup>1[f, tab\<^sub>0 f \<star> f\<^sub>0gh\<^sub>1.p\<^sub>1, TTfgh_TfTgh.chine] = \<a>\<^sup>-\<^sup>1[f, (g \<star> h) \<star> (tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine] \<cdot> (f \<star> \<a>[g \<star> h, tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0, f\<^sub>0gh\<^sub>1.p\<^sub>0] \<star> TTfgh_TfTgh.chine) \<cdot> (f \<star> (\<a>\<^sup>-\<^sup>1[g, h, tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (f \<star> ((g \<star> \<a>[h, tab\<^sub>0 h, g\<^sub>0h\<^sub>1.p\<^sub>0]) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (f \<star> ((g \<star> h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (f \<star> ((g \<star> g\<^sub>0h\<^sub>1.\<phi>) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (f \<star> (\<a>[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (f \<star> ((g.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>1) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (f \<star> f\<^sub>0gh\<^sub>1.\<phi> \<star> TTfgh_TfTgh.chine)
[PROOF STEP]
have "... =
((f \<star> \<a>[g \<star> h, tab\<^sub>0 h \<star> Tgh.p\<^sub>0, TfTgh.p\<^sub>0]) \<star> TTfgh_TfTgh.chine) \<cdot>
((f \<star> \<a>\<^sup>-\<^sup>1[g, h, tab\<^sub>0 h \<star> Tgh.p\<^sub>0] \<star> TfTgh.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot>
((f \<star> (g \<star> \<a>[h, tab\<^sub>0 h, Tgh.p\<^sub>0]) \<star> TfTgh.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot>
((f \<star> (g \<star> h.tab \<star> Tgh.p\<^sub>0) \<star> TfTgh.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot>
((f \<star> (g \<star> g\<^sub>0h\<^sub>1.\<phi>) \<star> TfTgh.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot>
(((f \<star> \<a>[g, tab\<^sub>0 g, Tgh.p\<^sub>1] \<star> TfTgh.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot>
\<a>\<^sup>-\<^sup>1[f, ((g \<star> tab\<^sub>0 g) \<star> Tgh.p\<^sub>1) \<star> TfTgh.p\<^sub>0, TTfgh_TfTgh.chine]) \<cdot>
(f \<star> ((g.tab \<star> Tgh.p\<^sub>1) \<star> TfTgh.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot>
(f \<star> f\<^sub>0gh\<^sub>1.\<phi> \<star> TTfgh_TfTgh.chine)"
[PROOF STATE]
proof (prove)
goal (1 subgoal):
1. ((f \<star> \<a>[g \<star> h, tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0, f\<^sub>0gh\<^sub>1.p\<^sub>0]) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> \<a>\<^sup>-\<^sup>1[g, h, tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> (g \<star> \<a>[h, tab\<^sub>0 h, g\<^sub>0h\<^sub>1.p\<^sub>0]) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> (g \<star> h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> (g \<star> g\<^sub>0h\<^sub>1.\<phi>) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> \<a>[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (((f \<star> (g.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>1) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> \<a>\<^sup>-\<^sup>1[f, (tab\<^sub>1 g \<star> g\<^sub>0h\<^sub>1.p\<^sub>1) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine]) \<cdot> (f \<star> f\<^sub>0gh\<^sub>1.\<phi> \<star> TTfgh_TfTgh.chine) = ((f \<star> \<a>[g \<star> h, tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0, f\<^sub>0gh\<^sub>1.p\<^sub>0]) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> \<a>\<^sup>-\<^sup>1[g, h, tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> (g \<star> \<a>[h, tab\<^sub>0 h, g\<^sub>0h\<^sub>1.p\<^sub>0]) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> (g \<star> h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> (g \<star> g\<^sub>0h\<^sub>1.\<phi>) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (((f \<star> \<a>[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> \<a>\<^sup>-\<^sup>1[f, ((g \<star> tab\<^sub>0 g) \<star> g\<^sub>0h\<^sub>1.p\<^sub>1) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine]) \<cdot> (f \<star> ((g.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>1) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (f \<star> f\<^sub>0gh\<^sub>1.\<phi> \<star> TTfgh_TfTgh.chine)
[PROOF STEP]
proof -
[PROOF STATE]
proof (state)
goal (1 subgoal):
1. ((f \<star> \<a>[g \<star> h, tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0, f\<^sub>0gh\<^sub>1.p\<^sub>0]) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> \<a>\<^sup>-\<^sup>1[g, h, tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> (g \<star> \<a>[h, tab\<^sub>0 h, g\<^sub>0h\<^sub>1.p\<^sub>0]) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> (g \<star> h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> (g \<star> g\<^sub>0h\<^sub>1.\<phi>) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> \<a>[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (((f \<star> (g.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>1) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> \<a>\<^sup>-\<^sup>1[f, (tab\<^sub>1 g \<star> g\<^sub>0h\<^sub>1.p\<^sub>1) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine]) \<cdot> (f \<star> f\<^sub>0gh\<^sub>1.\<phi> \<star> TTfgh_TfTgh.chine) = ((f \<star> \<a>[g \<star> h, tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0, f\<^sub>0gh\<^sub>1.p\<^sub>0]) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> \<a>\<^sup>-\<^sup>1[g, h, tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> (g \<star> \<a>[h, tab\<^sub>0 h, g\<^sub>0h\<^sub>1.p\<^sub>0]) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> (g \<star> h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> (g \<star> g\<^sub>0h\<^sub>1.\<phi>) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (((f \<star> \<a>[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> \<a>\<^sup>-\<^sup>1[f, ((g \<star> tab\<^sub>0 g) \<star> g\<^sub>0h\<^sub>1.p\<^sub>1) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine]) \<cdot> (f \<star> ((g.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>1) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (f \<star> f\<^sub>0gh\<^sub>1.\<phi> \<star> TTfgh_TfTgh.chine)
[PROOF STEP]
have "((f \<star> (g.tab \<star> Tgh.p\<^sub>1) \<star> TfTgh.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot>
\<a>\<^sup>-\<^sup>1[f, (tab\<^sub>1 g \<star> Tgh.p\<^sub>1) \<star> TfTgh.p\<^sub>0, TTfgh_TfTgh.chine] =
\<a>\<^sup>-\<^sup>1[f, ((g \<star> tab\<^sub>0 g) \<star> Tgh.p\<^sub>1) \<star> TfTgh.p\<^sub>0, TTfgh_TfTgh.chine] \<cdot>
(f \<star> ((g.tab \<star> Tgh.p\<^sub>1) \<star> TfTgh.p\<^sub>0) \<star> TTfgh_TfTgh.chine)"
[PROOF STATE]
proof (prove)
goal (1 subgoal):
1. ((f \<star> (g.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>1) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> \<a>\<^sup>-\<^sup>1[f, (tab\<^sub>1 g \<star> g\<^sub>0h\<^sub>1.p\<^sub>1) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine] = \<a>\<^sup>-\<^sup>1[f, ((g \<star> tab\<^sub>0 g) \<star> g\<^sub>0h\<^sub>1.p\<^sub>1) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine] \<cdot> (f \<star> ((g.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>1) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine)
[PROOF STEP]
using fg gh fg\<^sub>0h\<^sub>1.p\<^sub>0_simps fg\<^sub>0h\<^sub>1.p\<^sub>1_simps f\<^sub>0g\<^sub>1.p\<^sub>0_simps f\<^sub>0g\<^sub>1.p\<^sub>1_simps
assoc'_naturality [of f "(g.tab \<star> Tgh.p\<^sub>1) \<star> TfTgh.p\<^sub>0" TTfgh_TfTgh.chine]
[PROOF STATE]
proof (prove)
using this:
src f = trg g
src g = trg h
trg fg\<^sub>0h\<^sub>1.p\<^sub>0 = src (tab\<^sub>1 h)
trg fg\<^sub>0h\<^sub>1.p\<^sub>1 = src (tab\<^sub>0 g \<star> f\<^sub>0g\<^sub>1.p\<^sub>0)
trg f\<^sub>0g\<^sub>1.p\<^sub>0 = src (tab\<^sub>1 g)
trg f\<^sub>0g\<^sub>1.p\<^sub>1 = src (tab\<^sub>0 f)
\<lbrakk>arr f; hseq (g.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>1) f\<^sub>0gh\<^sub>1.p\<^sub>0; arr TTfgh_TfTgh.chine; src f = trg ((g.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>1) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0); src ((g.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>1) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) = trg TTfgh_TfTgh.chine\<rbrakk> \<Longrightarrow> \<a>\<^sup>-\<^sup>1[cod f, cod ((g.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>1) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0), cod TTfgh_TfTgh.chine] \<cdot> (f \<star> ((g.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>1) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) = ((f \<star> (g.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>1) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> \<a>\<^sup>-\<^sup>1[local.dom f, local.dom ((g.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>1) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0), local.dom TTfgh_TfTgh.chine]
goal (1 subgoal):
1. ((f \<star> (g.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>1) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> \<a>\<^sup>-\<^sup>1[f, (tab\<^sub>1 g \<star> g\<^sub>0h\<^sub>1.p\<^sub>1) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine] = \<a>\<^sup>-\<^sup>1[f, ((g \<star> tab\<^sub>0 g) \<star> g\<^sub>0h\<^sub>1.p\<^sub>1) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine] \<cdot> (f \<star> ((g.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>1) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine)
[PROOF STEP]
by simp
[PROOF STATE]
proof (state)
this:
((f \<star> (g.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>1) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> \<a>\<^sup>-\<^sup>1[f, (tab\<^sub>1 g \<star> g\<^sub>0h\<^sub>1.p\<^sub>1) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine] = \<a>\<^sup>-\<^sup>1[f, ((g \<star> tab\<^sub>0 g) \<star> g\<^sub>0h\<^sub>1.p\<^sub>1) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine] \<cdot> (f \<star> ((g.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>1) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine)
goal (1 subgoal):
1. ((f \<star> \<a>[g \<star> h, tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0, f\<^sub>0gh\<^sub>1.p\<^sub>0]) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> \<a>\<^sup>-\<^sup>1[g, h, tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> (g \<star> \<a>[h, tab\<^sub>0 h, g\<^sub>0h\<^sub>1.p\<^sub>0]) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> (g \<star> h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> (g \<star> g\<^sub>0h\<^sub>1.\<phi>) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> \<a>[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (((f \<star> (g.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>1) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> \<a>\<^sup>-\<^sup>1[f, (tab\<^sub>1 g \<star> g\<^sub>0h\<^sub>1.p\<^sub>1) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine]) \<cdot> (f \<star> f\<^sub>0gh\<^sub>1.\<phi> \<star> TTfgh_TfTgh.chine) = ((f \<star> \<a>[g \<star> h, tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0, f\<^sub>0gh\<^sub>1.p\<^sub>0]) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> \<a>\<^sup>-\<^sup>1[g, h, tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> (g \<star> \<a>[h, tab\<^sub>0 h, g\<^sub>0h\<^sub>1.p\<^sub>0]) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> (g \<star> h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> (g \<star> g\<^sub>0h\<^sub>1.\<phi>) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (((f \<star> \<a>[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> \<a>\<^sup>-\<^sup>1[f, ((g \<star> tab\<^sub>0 g) \<star> g\<^sub>0h\<^sub>1.p\<^sub>1) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine]) \<cdot> (f \<star> ((g.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>1) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (f \<star> f\<^sub>0gh\<^sub>1.\<phi> \<star> TTfgh_TfTgh.chine)
[PROOF STEP]
thus ?thesis
[PROOF STATE]
proof (prove)
using this:
((f \<star> (g.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>1) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> \<a>\<^sup>-\<^sup>1[f, (tab\<^sub>1 g \<star> g\<^sub>0h\<^sub>1.p\<^sub>1) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine] = \<a>\<^sup>-\<^sup>1[f, ((g \<star> tab\<^sub>0 g) \<star> g\<^sub>0h\<^sub>1.p\<^sub>1) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine] \<cdot> (f \<star> ((g.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>1) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine)
goal (1 subgoal):
1. ((f \<star> \<a>[g \<star> h, tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0, f\<^sub>0gh\<^sub>1.p\<^sub>0]) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> \<a>\<^sup>-\<^sup>1[g, h, tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> (g \<star> \<a>[h, tab\<^sub>0 h, g\<^sub>0h\<^sub>1.p\<^sub>0]) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> (g \<star> h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> (g \<star> g\<^sub>0h\<^sub>1.\<phi>) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> \<a>[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (((f \<star> (g.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>1) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> \<a>\<^sup>-\<^sup>1[f, (tab\<^sub>1 g \<star> g\<^sub>0h\<^sub>1.p\<^sub>1) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine]) \<cdot> (f \<star> f\<^sub>0gh\<^sub>1.\<phi> \<star> TTfgh_TfTgh.chine) = ((f \<star> \<a>[g \<star> h, tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0, f\<^sub>0gh\<^sub>1.p\<^sub>0]) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> \<a>\<^sup>-\<^sup>1[g, h, tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> (g \<star> \<a>[h, tab\<^sub>0 h, g\<^sub>0h\<^sub>1.p\<^sub>0]) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> (g \<star> h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> (g \<star> g\<^sub>0h\<^sub>1.\<phi>) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (((f \<star> \<a>[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> \<a>\<^sup>-\<^sup>1[f, ((g \<star> tab\<^sub>0 g) \<star> g\<^sub>0h\<^sub>1.p\<^sub>1) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine]) \<cdot> (f \<star> ((g.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>1) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (f \<star> f\<^sub>0gh\<^sub>1.\<phi> \<star> TTfgh_TfTgh.chine)
[PROOF STEP]
using comp_assoc
[PROOF STATE]
proof (prove)
using this:
((f \<star> (g.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>1) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> \<a>\<^sup>-\<^sup>1[f, (tab\<^sub>1 g \<star> g\<^sub>0h\<^sub>1.p\<^sub>1) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine] = \<a>\<^sup>-\<^sup>1[f, ((g \<star> tab\<^sub>0 g) \<star> g\<^sub>0h\<^sub>1.p\<^sub>1) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine] \<cdot> (f \<star> ((g.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>1) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine)
(?h \<cdot> ?g) \<cdot> ?f = ?h \<cdot> ?g \<cdot> ?f
goal (1 subgoal):
1. ((f \<star> \<a>[g \<star> h, tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0, f\<^sub>0gh\<^sub>1.p\<^sub>0]) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> \<a>\<^sup>-\<^sup>1[g, h, tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> (g \<star> \<a>[h, tab\<^sub>0 h, g\<^sub>0h\<^sub>1.p\<^sub>0]) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> (g \<star> h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> (g \<star> g\<^sub>0h\<^sub>1.\<phi>) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> \<a>[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (((f \<star> (g.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>1) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> \<a>\<^sup>-\<^sup>1[f, (tab\<^sub>1 g \<star> g\<^sub>0h\<^sub>1.p\<^sub>1) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine]) \<cdot> (f \<star> f\<^sub>0gh\<^sub>1.\<phi> \<star> TTfgh_TfTgh.chine) = ((f \<star> \<a>[g \<star> h, tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0, f\<^sub>0gh\<^sub>1.p\<^sub>0]) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> \<a>\<^sup>-\<^sup>1[g, h, tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> (g \<star> \<a>[h, tab\<^sub>0 h, g\<^sub>0h\<^sub>1.p\<^sub>0]) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> (g \<star> h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> (g \<star> g\<^sub>0h\<^sub>1.\<phi>) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (((f \<star> \<a>[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> \<a>\<^sup>-\<^sup>1[f, ((g \<star> tab\<^sub>0 g) \<star> g\<^sub>0h\<^sub>1.p\<^sub>1) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine]) \<cdot> (f \<star> ((g.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>1) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (f \<star> f\<^sub>0gh\<^sub>1.\<phi> \<star> TTfgh_TfTgh.chine)
[PROOF STEP]
by presburger
[PROOF STATE]
proof (state)
this:
((f \<star> \<a>[g \<star> h, tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0, f\<^sub>0gh\<^sub>1.p\<^sub>0]) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> \<a>\<^sup>-\<^sup>1[g, h, tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> (g \<star> \<a>[h, tab\<^sub>0 h, g\<^sub>0h\<^sub>1.p\<^sub>0]) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> (g \<star> h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> (g \<star> g\<^sub>0h\<^sub>1.\<phi>) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> \<a>[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (((f \<star> (g.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>1) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> \<a>\<^sup>-\<^sup>1[f, (tab\<^sub>1 g \<star> g\<^sub>0h\<^sub>1.p\<^sub>1) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine]) \<cdot> (f \<star> f\<^sub>0gh\<^sub>1.\<phi> \<star> TTfgh_TfTgh.chine) = ((f \<star> \<a>[g \<star> h, tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0, f\<^sub>0gh\<^sub>1.p\<^sub>0]) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> \<a>\<^sup>-\<^sup>1[g, h, tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> (g \<star> \<a>[h, tab\<^sub>0 h, g\<^sub>0h\<^sub>1.p\<^sub>0]) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> (g \<star> h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> (g \<star> g\<^sub>0h\<^sub>1.\<phi>) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (((f \<star> \<a>[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> \<a>\<^sup>-\<^sup>1[f, ((g \<star> tab\<^sub>0 g) \<star> g\<^sub>0h\<^sub>1.p\<^sub>1) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine]) \<cdot> (f \<star> ((g.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>1) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (f \<star> f\<^sub>0gh\<^sub>1.\<phi> \<star> TTfgh_TfTgh.chine)
goal:
No subgoals!
[PROOF STEP]
qed
[PROOF STATE]
proof (state)
this:
((f \<star> \<a>[g \<star> h, tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0, f\<^sub>0gh\<^sub>1.p\<^sub>0]) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> \<a>\<^sup>-\<^sup>1[g, h, tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> (g \<star> \<a>[h, tab\<^sub>0 h, g\<^sub>0h\<^sub>1.p\<^sub>0]) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> (g \<star> h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> (g \<star> g\<^sub>0h\<^sub>1.\<phi>) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> \<a>[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (((f \<star> (g.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>1) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> \<a>\<^sup>-\<^sup>1[f, (tab\<^sub>1 g \<star> g\<^sub>0h\<^sub>1.p\<^sub>1) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine]) \<cdot> (f \<star> f\<^sub>0gh\<^sub>1.\<phi> \<star> TTfgh_TfTgh.chine) = ((f \<star> \<a>[g \<star> h, tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0, f\<^sub>0gh\<^sub>1.p\<^sub>0]) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> \<a>\<^sup>-\<^sup>1[g, h, tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> (g \<star> \<a>[h, tab\<^sub>0 h, g\<^sub>0h\<^sub>1.p\<^sub>0]) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> (g \<star> h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> (g \<star> g\<^sub>0h\<^sub>1.\<phi>) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (((f \<star> \<a>[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> \<a>\<^sup>-\<^sup>1[f, ((g \<star> tab\<^sub>0 g) \<star> g\<^sub>0h\<^sub>1.p\<^sub>1) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine]) \<cdot> (f \<star> ((g.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>1) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (f \<star> f\<^sub>0gh\<^sub>1.\<phi> \<star> TTfgh_TfTgh.chine)
goal (1 subgoal):
1. ((f \<star> \<a>[g \<star> h, tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0, f\<^sub>0gh\<^sub>1.p\<^sub>0]) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> \<a>\<^sup>-\<^sup>1[g, h, tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> (g \<star> \<a>[h, tab\<^sub>0 h, g\<^sub>0h\<^sub>1.p\<^sub>0]) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> (g \<star> h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> (g \<star> g\<^sub>0h\<^sub>1.\<phi>) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> \<a>[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> (g.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>1) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> f\<^sub>0gh\<^sub>1.\<phi>) \<star> TTfgh_TfTgh.chine) \<cdot> \<a>\<^sup>-\<^sup>1[f, tab\<^sub>0 f \<star> f\<^sub>0gh\<^sub>1.p\<^sub>1, TTfgh_TfTgh.chine] = \<a>\<^sup>-\<^sup>1[f, (g \<star> h) \<star> (tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine] \<cdot> (f \<star> \<a>[g \<star> h, tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0, f\<^sub>0gh\<^sub>1.p\<^sub>0] \<star> TTfgh_TfTgh.chine) \<cdot> (f \<star> (\<a>\<^sup>-\<^sup>1[g, h, tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (f \<star> ((g \<star> \<a>[h, tab\<^sub>0 h, g\<^sub>0h\<^sub>1.p\<^sub>0]) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (f \<star> ((g \<star> h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (f \<star> ((g \<star> g\<^sub>0h\<^sub>1.\<phi>) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (f \<star> (\<a>[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (f \<star> ((g.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>1) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (f \<star> f\<^sub>0gh\<^sub>1.\<phi> \<star> TTfgh_TfTgh.chine)
[PROOF STEP]
also
[PROOF STATE]
proof (state)
this:
((f \<star> \<a>[g \<star> h, tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0, f\<^sub>0gh\<^sub>1.p\<^sub>0]) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> \<a>\<^sup>-\<^sup>1[g, h, tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> (g \<star> \<a>[h, tab\<^sub>0 h, g\<^sub>0h\<^sub>1.p\<^sub>0]) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> (g \<star> h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> (g \<star> g\<^sub>0h\<^sub>1.\<phi>) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> \<a>[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (((f \<star> (g.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>1) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> \<a>\<^sup>-\<^sup>1[f, (tab\<^sub>1 g \<star> g\<^sub>0h\<^sub>1.p\<^sub>1) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine]) \<cdot> (f \<star> f\<^sub>0gh\<^sub>1.\<phi> \<star> TTfgh_TfTgh.chine) = ((f \<star> \<a>[g \<star> h, tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0, f\<^sub>0gh\<^sub>1.p\<^sub>0]) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> \<a>\<^sup>-\<^sup>1[g, h, tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> (g \<star> \<a>[h, tab\<^sub>0 h, g\<^sub>0h\<^sub>1.p\<^sub>0]) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> (g \<star> h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> (g \<star> g\<^sub>0h\<^sub>1.\<phi>) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (((f \<star> \<a>[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> \<a>\<^sup>-\<^sup>1[f, ((g \<star> tab\<^sub>0 g) \<star> g\<^sub>0h\<^sub>1.p\<^sub>1) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine]) \<cdot> (f \<star> ((g.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>1) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (f \<star> f\<^sub>0gh\<^sub>1.\<phi> \<star> TTfgh_TfTgh.chine)
goal (1 subgoal):
1. ((f \<star> \<a>[g \<star> h, tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0, f\<^sub>0gh\<^sub>1.p\<^sub>0]) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> \<a>\<^sup>-\<^sup>1[g, h, tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> (g \<star> \<a>[h, tab\<^sub>0 h, g\<^sub>0h\<^sub>1.p\<^sub>0]) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> (g \<star> h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> (g \<star> g\<^sub>0h\<^sub>1.\<phi>) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> \<a>[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> (g.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>1) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> f\<^sub>0gh\<^sub>1.\<phi>) \<star> TTfgh_TfTgh.chine) \<cdot> \<a>\<^sup>-\<^sup>1[f, tab\<^sub>0 f \<star> f\<^sub>0gh\<^sub>1.p\<^sub>1, TTfgh_TfTgh.chine] = \<a>\<^sup>-\<^sup>1[f, (g \<star> h) \<star> (tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine] \<cdot> (f \<star> \<a>[g \<star> h, tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0, f\<^sub>0gh\<^sub>1.p\<^sub>0] \<star> TTfgh_TfTgh.chine) \<cdot> (f \<star> (\<a>\<^sup>-\<^sup>1[g, h, tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (f \<star> ((g \<star> \<a>[h, tab\<^sub>0 h, g\<^sub>0h\<^sub>1.p\<^sub>0]) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (f \<star> ((g \<star> h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (f \<star> ((g \<star> g\<^sub>0h\<^sub>1.\<phi>) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (f \<star> (\<a>[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (f \<star> ((g.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>1) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (f \<star> f\<^sub>0gh\<^sub>1.\<phi> \<star> TTfgh_TfTgh.chine)
[PROOF STEP]
have "... =
((f \<star> \<a>[g \<star> h, tab\<^sub>0 h \<star> Tgh.p\<^sub>0, TfTgh.p\<^sub>0]) \<star> TTfgh_TfTgh.chine) \<cdot>
((f \<star> \<a>\<^sup>-\<^sup>1[g, h, tab\<^sub>0 h \<star> Tgh.p\<^sub>0] \<star> TfTgh.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot>
((f \<star> (g \<star> \<a>[h, tab\<^sub>0 h, Tgh.p\<^sub>0]) \<star> TfTgh.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot>
((f \<star> (g \<star> h.tab \<star> Tgh.p\<^sub>0) \<star> TfTgh.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot>
(((f \<star> (g \<star> g\<^sub>0h\<^sub>1.\<phi>) \<star> TfTgh.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot>
\<a>\<^sup>-\<^sup>1[f, (g \<star> tab\<^sub>0 g \<star> Tgh.p\<^sub>1) \<star> TfTgh.p\<^sub>0, TTfgh_TfTgh.chine]) \<cdot>
(f \<star> (\<a>[g, tab\<^sub>0 g, Tgh.p\<^sub>1] \<star> TfTgh.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot>
(f \<star> ((g.tab \<star> Tgh.p\<^sub>1) \<star> TfTgh.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot>
(f \<star> f\<^sub>0gh\<^sub>1.\<phi> \<star> TTfgh_TfTgh.chine)"
[PROOF STATE]
proof (prove)
goal (1 subgoal):
1. ((f \<star> \<a>[g \<star> h, tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0, f\<^sub>0gh\<^sub>1.p\<^sub>0]) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> \<a>\<^sup>-\<^sup>1[g, h, tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> (g \<star> \<a>[h, tab\<^sub>0 h, g\<^sub>0h\<^sub>1.p\<^sub>0]) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> (g \<star> h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> (g \<star> g\<^sub>0h\<^sub>1.\<phi>) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (((f \<star> \<a>[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> \<a>\<^sup>-\<^sup>1[f, ((g \<star> tab\<^sub>0 g) \<star> g\<^sub>0h\<^sub>1.p\<^sub>1) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine]) \<cdot> (f \<star> ((g.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>1) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (f \<star> f\<^sub>0gh\<^sub>1.\<phi> \<star> TTfgh_TfTgh.chine) = ((f \<star> \<a>[g \<star> h, tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0, f\<^sub>0gh\<^sub>1.p\<^sub>0]) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> \<a>\<^sup>-\<^sup>1[g, h, tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> (g \<star> \<a>[h, tab\<^sub>0 h, g\<^sub>0h\<^sub>1.p\<^sub>0]) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> (g \<star> h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (((f \<star> (g \<star> g\<^sub>0h\<^sub>1.\<phi>) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> \<a>\<^sup>-\<^sup>1[f, (g \<star> tab\<^sub>0 g \<star> g\<^sub>0h\<^sub>1.p\<^sub>1) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine]) \<cdot> (f \<star> (\<a>[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (f \<star> ((g.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>1) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (f \<star> f\<^sub>0gh\<^sub>1.\<phi> \<star> TTfgh_TfTgh.chine)
[PROOF STEP]
proof -
[PROOF STATE]
proof (state)
goal (1 subgoal):
1. ((f \<star> \<a>[g \<star> h, tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0, f\<^sub>0gh\<^sub>1.p\<^sub>0]) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> \<a>\<^sup>-\<^sup>1[g, h, tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> (g \<star> \<a>[h, tab\<^sub>0 h, g\<^sub>0h\<^sub>1.p\<^sub>0]) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> (g \<star> h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> (g \<star> g\<^sub>0h\<^sub>1.\<phi>) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (((f \<star> \<a>[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> \<a>\<^sup>-\<^sup>1[f, ((g \<star> tab\<^sub>0 g) \<star> g\<^sub>0h\<^sub>1.p\<^sub>1) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine]) \<cdot> (f \<star> ((g.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>1) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (f \<star> f\<^sub>0gh\<^sub>1.\<phi> \<star> TTfgh_TfTgh.chine) = ((f \<star> \<a>[g \<star> h, tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0, f\<^sub>0gh\<^sub>1.p\<^sub>0]) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> \<a>\<^sup>-\<^sup>1[g, h, tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> (g \<star> \<a>[h, tab\<^sub>0 h, g\<^sub>0h\<^sub>1.p\<^sub>0]) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> (g \<star> h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (((f \<star> (g \<star> g\<^sub>0h\<^sub>1.\<phi>) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> \<a>\<^sup>-\<^sup>1[f, (g \<star> tab\<^sub>0 g \<star> g\<^sub>0h\<^sub>1.p\<^sub>1) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine]) \<cdot> (f \<star> (\<a>[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (f \<star> ((g.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>1) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (f \<star> f\<^sub>0gh\<^sub>1.\<phi> \<star> TTfgh_TfTgh.chine)
[PROOF STEP]
have "((f \<star> \<a>[g, tab\<^sub>0 g, Tgh.p\<^sub>1] \<star> TfTgh.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot>
\<a>\<^sup>-\<^sup>1[f, ((g \<star> tab\<^sub>0 g) \<star> Tgh.p\<^sub>1) \<star> TfTgh.p\<^sub>0, TTfgh_TfTgh.chine] =
\<a>\<^sup>-\<^sup>1[f, (g \<star> tab\<^sub>0 g \<star> Tgh.p\<^sub>1) \<star> TfTgh.p\<^sub>0, TTfgh_TfTgh.chine] \<cdot>
(f \<star> (\<a>[g, tab\<^sub>0 g, Tgh.p\<^sub>1] \<star> TfTgh.p\<^sub>0) \<star> TTfgh_TfTgh.chine)"
[PROOF STATE]
proof (prove)
goal (1 subgoal):
1. ((f \<star> \<a>[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> \<a>\<^sup>-\<^sup>1[f, ((g \<star> tab\<^sub>0 g) \<star> g\<^sub>0h\<^sub>1.p\<^sub>1) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine] = \<a>\<^sup>-\<^sup>1[f, (g \<star> tab\<^sub>0 g \<star> g\<^sub>0h\<^sub>1.p\<^sub>1) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine] \<cdot> (f \<star> (\<a>[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine)
[PROOF STEP]
using fg gh fg\<^sub>0h\<^sub>1.p\<^sub>0_simps fg\<^sub>0h\<^sub>1.p\<^sub>1_simps f\<^sub>0g\<^sub>1.p\<^sub>0_simps f\<^sub>0g\<^sub>1.p\<^sub>1_simps
assoc'_naturality
[of f "\<a>[g, tab\<^sub>0 g, Tgh.p\<^sub>1] \<star> TfTgh.p\<^sub>0" TTfgh_TfTgh.chine]
[PROOF STATE]
proof (prove)
using this:
src f = trg g
src g = trg h
trg fg\<^sub>0h\<^sub>1.p\<^sub>0 = src (tab\<^sub>1 h)
trg fg\<^sub>0h\<^sub>1.p\<^sub>1 = src (tab\<^sub>0 g \<star> f\<^sub>0g\<^sub>1.p\<^sub>0)
trg f\<^sub>0g\<^sub>1.p\<^sub>0 = src (tab\<^sub>1 g)
trg f\<^sub>0g\<^sub>1.p\<^sub>1 = src (tab\<^sub>0 f)
\<lbrakk>arr f; hseq \<a>[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1] f\<^sub>0gh\<^sub>1.p\<^sub>0; arr TTfgh_TfTgh.chine; src f = trg (\<a>[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0); src (\<a>[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) = trg TTfgh_TfTgh.chine\<rbrakk> \<Longrightarrow> \<a>\<^sup>-\<^sup>1[cod f, cod (\<a>[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0), cod TTfgh_TfTgh.chine] \<cdot> (f \<star> (\<a>[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) = ((f \<star> \<a>[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> \<a>\<^sup>-\<^sup>1[local.dom f, local.dom (\<a>[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0), local.dom TTfgh_TfTgh.chine]
goal (1 subgoal):
1. ((f \<star> \<a>[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> \<a>\<^sup>-\<^sup>1[f, ((g \<star> tab\<^sub>0 g) \<star> g\<^sub>0h\<^sub>1.p\<^sub>1) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine] = \<a>\<^sup>-\<^sup>1[f, (g \<star> tab\<^sub>0 g \<star> g\<^sub>0h\<^sub>1.p\<^sub>1) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine] \<cdot> (f \<star> (\<a>[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine)
[PROOF STEP]
by simp
[PROOF STATE]
proof (state)
this:
((f \<star> \<a>[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> \<a>\<^sup>-\<^sup>1[f, ((g \<star> tab\<^sub>0 g) \<star> g\<^sub>0h\<^sub>1.p\<^sub>1) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine] = \<a>\<^sup>-\<^sup>1[f, (g \<star> tab\<^sub>0 g \<star> g\<^sub>0h\<^sub>1.p\<^sub>1) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine] \<cdot> (f \<star> (\<a>[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine)
goal (1 subgoal):
1. ((f \<star> \<a>[g \<star> h, tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0, f\<^sub>0gh\<^sub>1.p\<^sub>0]) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> \<a>\<^sup>-\<^sup>1[g, h, tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> (g \<star> \<a>[h, tab\<^sub>0 h, g\<^sub>0h\<^sub>1.p\<^sub>0]) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> (g \<star> h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> (g \<star> g\<^sub>0h\<^sub>1.\<phi>) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (((f \<star> \<a>[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> \<a>\<^sup>-\<^sup>1[f, ((g \<star> tab\<^sub>0 g) \<star> g\<^sub>0h\<^sub>1.p\<^sub>1) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine]) \<cdot> (f \<star> ((g.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>1) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (f \<star> f\<^sub>0gh\<^sub>1.\<phi> \<star> TTfgh_TfTgh.chine) = ((f \<star> \<a>[g \<star> h, tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0, f\<^sub>0gh\<^sub>1.p\<^sub>0]) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> \<a>\<^sup>-\<^sup>1[g, h, tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> (g \<star> \<a>[h, tab\<^sub>0 h, g\<^sub>0h\<^sub>1.p\<^sub>0]) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> (g \<star> h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (((f \<star> (g \<star> g\<^sub>0h\<^sub>1.\<phi>) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> \<a>\<^sup>-\<^sup>1[f, (g \<star> tab\<^sub>0 g \<star> g\<^sub>0h\<^sub>1.p\<^sub>1) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine]) \<cdot> (f \<star> (\<a>[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (f \<star> ((g.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>1) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (f \<star> f\<^sub>0gh\<^sub>1.\<phi> \<star> TTfgh_TfTgh.chine)
[PROOF STEP]
thus ?thesis
[PROOF STATE]
proof (prove)
using this:
((f \<star> \<a>[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> \<a>\<^sup>-\<^sup>1[f, ((g \<star> tab\<^sub>0 g) \<star> g\<^sub>0h\<^sub>1.p\<^sub>1) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine] = \<a>\<^sup>-\<^sup>1[f, (g \<star> tab\<^sub>0 g \<star> g\<^sub>0h\<^sub>1.p\<^sub>1) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine] \<cdot> (f \<star> (\<a>[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine)
goal (1 subgoal):
1. ((f \<star> \<a>[g \<star> h, tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0, f\<^sub>0gh\<^sub>1.p\<^sub>0]) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> \<a>\<^sup>-\<^sup>1[g, h, tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> (g \<star> \<a>[h, tab\<^sub>0 h, g\<^sub>0h\<^sub>1.p\<^sub>0]) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> (g \<star> h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> (g \<star> g\<^sub>0h\<^sub>1.\<phi>) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (((f \<star> \<a>[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> \<a>\<^sup>-\<^sup>1[f, ((g \<star> tab\<^sub>0 g) \<star> g\<^sub>0h\<^sub>1.p\<^sub>1) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine]) \<cdot> (f \<star> ((g.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>1) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (f \<star> f\<^sub>0gh\<^sub>1.\<phi> \<star> TTfgh_TfTgh.chine) = ((f \<star> \<a>[g \<star> h, tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0, f\<^sub>0gh\<^sub>1.p\<^sub>0]) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> \<a>\<^sup>-\<^sup>1[g, h, tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> (g \<star> \<a>[h, tab\<^sub>0 h, g\<^sub>0h\<^sub>1.p\<^sub>0]) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> (g \<star> h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (((f \<star> (g \<star> g\<^sub>0h\<^sub>1.\<phi>) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> \<a>\<^sup>-\<^sup>1[f, (g \<star> tab\<^sub>0 g \<star> g\<^sub>0h\<^sub>1.p\<^sub>1) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine]) \<cdot> (f \<star> (\<a>[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (f \<star> ((g.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>1) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (f \<star> f\<^sub>0gh\<^sub>1.\<phi> \<star> TTfgh_TfTgh.chine)
[PROOF STEP]
using comp_assoc
[PROOF STATE]
proof (prove)
using this:
((f \<star> \<a>[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> \<a>\<^sup>-\<^sup>1[f, ((g \<star> tab\<^sub>0 g) \<star> g\<^sub>0h\<^sub>1.p\<^sub>1) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine] = \<a>\<^sup>-\<^sup>1[f, (g \<star> tab\<^sub>0 g \<star> g\<^sub>0h\<^sub>1.p\<^sub>1) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine] \<cdot> (f \<star> (\<a>[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine)
(?h \<cdot> ?g) \<cdot> ?f = ?h \<cdot> ?g \<cdot> ?f
goal (1 subgoal):
1. ((f \<star> \<a>[g \<star> h, tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0, f\<^sub>0gh\<^sub>1.p\<^sub>0]) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> \<a>\<^sup>-\<^sup>1[g, h, tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> (g \<star> \<a>[h, tab\<^sub>0 h, g\<^sub>0h\<^sub>1.p\<^sub>0]) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> (g \<star> h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> (g \<star> g\<^sub>0h\<^sub>1.\<phi>) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (((f \<star> \<a>[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> \<a>\<^sup>-\<^sup>1[f, ((g \<star> tab\<^sub>0 g) \<star> g\<^sub>0h\<^sub>1.p\<^sub>1) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine]) \<cdot> (f \<star> ((g.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>1) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (f \<star> f\<^sub>0gh\<^sub>1.\<phi> \<star> TTfgh_TfTgh.chine) = ((f \<star> \<a>[g \<star> h, tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0, f\<^sub>0gh\<^sub>1.p\<^sub>0]) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> \<a>\<^sup>-\<^sup>1[g, h, tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> (g \<star> \<a>[h, tab\<^sub>0 h, g\<^sub>0h\<^sub>1.p\<^sub>0]) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> (g \<star> h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (((f \<star> (g \<star> g\<^sub>0h\<^sub>1.\<phi>) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> \<a>\<^sup>-\<^sup>1[f, (g \<star> tab\<^sub>0 g \<star> g\<^sub>0h\<^sub>1.p\<^sub>1) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine]) \<cdot> (f \<star> (\<a>[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (f \<star> ((g.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>1) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (f \<star> f\<^sub>0gh\<^sub>1.\<phi> \<star> TTfgh_TfTgh.chine)
[PROOF STEP]
by presburger
[PROOF STATE]
proof (state)
this:
((f \<star> \<a>[g \<star> h, tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0, f\<^sub>0gh\<^sub>1.p\<^sub>0]) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> \<a>\<^sup>-\<^sup>1[g, h, tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> (g \<star> \<a>[h, tab\<^sub>0 h, g\<^sub>0h\<^sub>1.p\<^sub>0]) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> (g \<star> h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> (g \<star> g\<^sub>0h\<^sub>1.\<phi>) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (((f \<star> \<a>[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> \<a>\<^sup>-\<^sup>1[f, ((g \<star> tab\<^sub>0 g) \<star> g\<^sub>0h\<^sub>1.p\<^sub>1) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine]) \<cdot> (f \<star> ((g.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>1) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (f \<star> f\<^sub>0gh\<^sub>1.\<phi> \<star> TTfgh_TfTgh.chine) = ((f \<star> \<a>[g \<star> h, tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0, f\<^sub>0gh\<^sub>1.p\<^sub>0]) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> \<a>\<^sup>-\<^sup>1[g, h, tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> (g \<star> \<a>[h, tab\<^sub>0 h, g\<^sub>0h\<^sub>1.p\<^sub>0]) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> (g \<star> h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (((f \<star> (g \<star> g\<^sub>0h\<^sub>1.\<phi>) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> \<a>\<^sup>-\<^sup>1[f, (g \<star> tab\<^sub>0 g \<star> g\<^sub>0h\<^sub>1.p\<^sub>1) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine]) \<cdot> (f \<star> (\<a>[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (f \<star> ((g.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>1) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (f \<star> f\<^sub>0gh\<^sub>1.\<phi> \<star> TTfgh_TfTgh.chine)
goal:
No subgoals!
[PROOF STEP]
qed
[PROOF STATE]
proof (state)
this:
((f \<star> \<a>[g \<star> h, tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0, f\<^sub>0gh\<^sub>1.p\<^sub>0]) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> \<a>\<^sup>-\<^sup>1[g, h, tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> (g \<star> \<a>[h, tab\<^sub>0 h, g\<^sub>0h\<^sub>1.p\<^sub>0]) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> (g \<star> h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> (g \<star> g\<^sub>0h\<^sub>1.\<phi>) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (((f \<star> \<a>[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> \<a>\<^sup>-\<^sup>1[f, ((g \<star> tab\<^sub>0 g) \<star> g\<^sub>0h\<^sub>1.p\<^sub>1) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine]) \<cdot> (f \<star> ((g.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>1) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (f \<star> f\<^sub>0gh\<^sub>1.\<phi> \<star> TTfgh_TfTgh.chine) = ((f \<star> \<a>[g \<star> h, tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0, f\<^sub>0gh\<^sub>1.p\<^sub>0]) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> \<a>\<^sup>-\<^sup>1[g, h, tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> (g \<star> \<a>[h, tab\<^sub>0 h, g\<^sub>0h\<^sub>1.p\<^sub>0]) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> (g \<star> h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (((f \<star> (g \<star> g\<^sub>0h\<^sub>1.\<phi>) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> \<a>\<^sup>-\<^sup>1[f, (g \<star> tab\<^sub>0 g \<star> g\<^sub>0h\<^sub>1.p\<^sub>1) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine]) \<cdot> (f \<star> (\<a>[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (f \<star> ((g.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>1) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (f \<star> f\<^sub>0gh\<^sub>1.\<phi> \<star> TTfgh_TfTgh.chine)
goal (1 subgoal):
1. ((f \<star> \<a>[g \<star> h, tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0, f\<^sub>0gh\<^sub>1.p\<^sub>0]) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> \<a>\<^sup>-\<^sup>1[g, h, tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> (g \<star> \<a>[h, tab\<^sub>0 h, g\<^sub>0h\<^sub>1.p\<^sub>0]) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> (g \<star> h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> (g \<star> g\<^sub>0h\<^sub>1.\<phi>) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> \<a>[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> (g.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>1) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> f\<^sub>0gh\<^sub>1.\<phi>) \<star> TTfgh_TfTgh.chine) \<cdot> \<a>\<^sup>-\<^sup>1[f, tab\<^sub>0 f \<star> f\<^sub>0gh\<^sub>1.p\<^sub>1, TTfgh_TfTgh.chine] = \<a>\<^sup>-\<^sup>1[f, (g \<star> h) \<star> (tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine] \<cdot> (f \<star> \<a>[g \<star> h, tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0, f\<^sub>0gh\<^sub>1.p\<^sub>0] \<star> TTfgh_TfTgh.chine) \<cdot> (f \<star> (\<a>\<^sup>-\<^sup>1[g, h, tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (f \<star> ((g \<star> \<a>[h, tab\<^sub>0 h, g\<^sub>0h\<^sub>1.p\<^sub>0]) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (f \<star> ((g \<star> h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (f \<star> ((g \<star> g\<^sub>0h\<^sub>1.\<phi>) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (f \<star> (\<a>[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (f \<star> ((g.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>1) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (f \<star> f\<^sub>0gh\<^sub>1.\<phi> \<star> TTfgh_TfTgh.chine)
[PROOF STEP]
also
[PROOF STATE]
proof (state)
this:
((f \<star> \<a>[g \<star> h, tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0, f\<^sub>0gh\<^sub>1.p\<^sub>0]) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> \<a>\<^sup>-\<^sup>1[g, h, tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> (g \<star> \<a>[h, tab\<^sub>0 h, g\<^sub>0h\<^sub>1.p\<^sub>0]) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> (g \<star> h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> (g \<star> g\<^sub>0h\<^sub>1.\<phi>) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (((f \<star> \<a>[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> \<a>\<^sup>-\<^sup>1[f, ((g \<star> tab\<^sub>0 g) \<star> g\<^sub>0h\<^sub>1.p\<^sub>1) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine]) \<cdot> (f \<star> ((g.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>1) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (f \<star> f\<^sub>0gh\<^sub>1.\<phi> \<star> TTfgh_TfTgh.chine) = ((f \<star> \<a>[g \<star> h, tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0, f\<^sub>0gh\<^sub>1.p\<^sub>0]) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> \<a>\<^sup>-\<^sup>1[g, h, tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> (g \<star> \<a>[h, tab\<^sub>0 h, g\<^sub>0h\<^sub>1.p\<^sub>0]) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> (g \<star> h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (((f \<star> (g \<star> g\<^sub>0h\<^sub>1.\<phi>) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> \<a>\<^sup>-\<^sup>1[f, (g \<star> tab\<^sub>0 g \<star> g\<^sub>0h\<^sub>1.p\<^sub>1) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine]) \<cdot> (f \<star> (\<a>[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (f \<star> ((g.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>1) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (f \<star> f\<^sub>0gh\<^sub>1.\<phi> \<star> TTfgh_TfTgh.chine)
goal (1 subgoal):
1. ((f \<star> \<a>[g \<star> h, tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0, f\<^sub>0gh\<^sub>1.p\<^sub>0]) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> \<a>\<^sup>-\<^sup>1[g, h, tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> (g \<star> \<a>[h, tab\<^sub>0 h, g\<^sub>0h\<^sub>1.p\<^sub>0]) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> (g \<star> h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> (g \<star> g\<^sub>0h\<^sub>1.\<phi>) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> \<a>[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> (g.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>1) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> f\<^sub>0gh\<^sub>1.\<phi>) \<star> TTfgh_TfTgh.chine) \<cdot> \<a>\<^sup>-\<^sup>1[f, tab\<^sub>0 f \<star> f\<^sub>0gh\<^sub>1.p\<^sub>1, TTfgh_TfTgh.chine] = \<a>\<^sup>-\<^sup>1[f, (g \<star> h) \<star> (tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine] \<cdot> (f \<star> \<a>[g \<star> h, tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0, f\<^sub>0gh\<^sub>1.p\<^sub>0] \<star> TTfgh_TfTgh.chine) \<cdot> (f \<star> (\<a>\<^sup>-\<^sup>1[g, h, tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (f \<star> ((g \<star> \<a>[h, tab\<^sub>0 h, g\<^sub>0h\<^sub>1.p\<^sub>0]) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (f \<star> ((g \<star> h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (f \<star> ((g \<star> g\<^sub>0h\<^sub>1.\<phi>) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (f \<star> (\<a>[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (f \<star> ((g.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>1) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (f \<star> f\<^sub>0gh\<^sub>1.\<phi> \<star> TTfgh_TfTgh.chine)
[PROOF STEP]
have "... =
((f \<star> \<a>[g \<star> h, tab\<^sub>0 h \<star> Tgh.p\<^sub>0, TfTgh.p\<^sub>0]) \<star> TTfgh_TfTgh.chine) \<cdot>
((f \<star> \<a>\<^sup>-\<^sup>1[g, h, tab\<^sub>0 h \<star> Tgh.p\<^sub>0] \<star> TfTgh.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot>
((f \<star> (g \<star> \<a>[h, tab\<^sub>0 h, Tgh.p\<^sub>0]) \<star> TfTgh.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot>
(((f \<star> (g \<star> h.tab \<star> Tgh.p\<^sub>0) \<star> TfTgh.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot>
\<a>\<^sup>-\<^sup>1[f, (g \<star> tab\<^sub>1 h \<star> Tgh.p\<^sub>0) \<star> TfTgh.p\<^sub>0, TTfgh_TfTgh.chine]) \<cdot>
(f \<star> ((g \<star> g\<^sub>0h\<^sub>1.\<phi>) \<star> TfTgh.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot>
(f \<star> (\<a>[g, tab\<^sub>0 g, Tgh.p\<^sub>1] \<star> TfTgh.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot>
(f \<star> ((g.tab \<star> Tgh.p\<^sub>1) \<star> TfTgh.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot>
(f \<star> f\<^sub>0gh\<^sub>1.\<phi> \<star> TTfgh_TfTgh.chine)"
[PROOF STATE]
proof (prove)
goal (1 subgoal):
1. ((f \<star> \<a>[g \<star> h, tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0, f\<^sub>0gh\<^sub>1.p\<^sub>0]) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> \<a>\<^sup>-\<^sup>1[g, h, tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> (g \<star> \<a>[h, tab\<^sub>0 h, g\<^sub>0h\<^sub>1.p\<^sub>0]) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> (g \<star> h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (((f \<star> (g \<star> g\<^sub>0h\<^sub>1.\<phi>) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> \<a>\<^sup>-\<^sup>1[f, (g \<star> tab\<^sub>0 g \<star> g\<^sub>0h\<^sub>1.p\<^sub>1) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine]) \<cdot> (f \<star> (\<a>[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (f \<star> ((g.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>1) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (f \<star> f\<^sub>0gh\<^sub>1.\<phi> \<star> TTfgh_TfTgh.chine) = ((f \<star> \<a>[g \<star> h, tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0, f\<^sub>0gh\<^sub>1.p\<^sub>0]) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> \<a>\<^sup>-\<^sup>1[g, h, tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> (g \<star> \<a>[h, tab\<^sub>0 h, g\<^sub>0h\<^sub>1.p\<^sub>0]) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (((f \<star> (g \<star> h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> \<a>\<^sup>-\<^sup>1[f, (g \<star> tab\<^sub>1 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine]) \<cdot> (f \<star> ((g \<star> g\<^sub>0h\<^sub>1.\<phi>) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (f \<star> (\<a>[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (f \<star> ((g.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>1) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (f \<star> f\<^sub>0gh\<^sub>1.\<phi> \<star> TTfgh_TfTgh.chine)
[PROOF STEP]
proof -
[PROOF STATE]
proof (state)
goal (1 subgoal):
1. ((f \<star> \<a>[g \<star> h, tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0, f\<^sub>0gh\<^sub>1.p\<^sub>0]) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> \<a>\<^sup>-\<^sup>1[g, h, tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> (g \<star> \<a>[h, tab\<^sub>0 h, g\<^sub>0h\<^sub>1.p\<^sub>0]) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> (g \<star> h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (((f \<star> (g \<star> g\<^sub>0h\<^sub>1.\<phi>) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> \<a>\<^sup>-\<^sup>1[f, (g \<star> tab\<^sub>0 g \<star> g\<^sub>0h\<^sub>1.p\<^sub>1) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine]) \<cdot> (f \<star> (\<a>[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (f \<star> ((g.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>1) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (f \<star> f\<^sub>0gh\<^sub>1.\<phi> \<star> TTfgh_TfTgh.chine) = ((f \<star> \<a>[g \<star> h, tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0, f\<^sub>0gh\<^sub>1.p\<^sub>0]) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> \<a>\<^sup>-\<^sup>1[g, h, tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> (g \<star> \<a>[h, tab\<^sub>0 h, g\<^sub>0h\<^sub>1.p\<^sub>0]) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (((f \<star> (g \<star> h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> \<a>\<^sup>-\<^sup>1[f, (g \<star> tab\<^sub>1 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine]) \<cdot> (f \<star> ((g \<star> g\<^sub>0h\<^sub>1.\<phi>) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (f \<star> (\<a>[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (f \<star> ((g.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>1) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (f \<star> f\<^sub>0gh\<^sub>1.\<phi> \<star> TTfgh_TfTgh.chine)
[PROOF STEP]
have "((f \<star> (g \<star> g\<^sub>0h\<^sub>1.\<phi>) \<star> TfTgh.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot>
\<a>\<^sup>-\<^sup>1[f, (g \<star> tab\<^sub>0 g \<star> Tgh.p\<^sub>1) \<star> TfTgh.p\<^sub>0, TTfgh_TfTgh.chine] =
\<a>\<^sup>-\<^sup>1[f, (g \<star> tab\<^sub>1 h \<star> Tgh.p\<^sub>0) \<star> TfTgh.p\<^sub>0, TTfgh_TfTgh.chine] \<cdot>
(f \<star> ((g \<star> g\<^sub>0h\<^sub>1.\<phi>) \<star> TfTgh.p\<^sub>0) \<star> TTfgh_TfTgh.chine)"
[PROOF STATE]
proof (prove)
goal (1 subgoal):
1. ((f \<star> (g \<star> g\<^sub>0h\<^sub>1.\<phi>) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> \<a>\<^sup>-\<^sup>1[f, (g \<star> tab\<^sub>0 g \<star> g\<^sub>0h\<^sub>1.p\<^sub>1) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine] = \<a>\<^sup>-\<^sup>1[f, (g \<star> tab\<^sub>1 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine] \<cdot> (f \<star> ((g \<star> g\<^sub>0h\<^sub>1.\<phi>) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine)
[PROOF STEP]
using fg gh fg\<^sub>0h\<^sub>1.p\<^sub>0_simps fg\<^sub>0h\<^sub>1.p\<^sub>1_simps f\<^sub>0g\<^sub>1.p\<^sub>0_simps f\<^sub>0g\<^sub>1.p\<^sub>1_simps
assoc'_naturality [of f "(g \<star> g\<^sub>0h\<^sub>1.\<phi>) \<star> TfTgh.p\<^sub>0" TTfgh_TfTgh.chine]
[PROOF STATE]
proof (prove)
using this:
src f = trg g
src g = trg h
trg fg\<^sub>0h\<^sub>1.p\<^sub>0 = src (tab\<^sub>1 h)
trg fg\<^sub>0h\<^sub>1.p\<^sub>1 = src (tab\<^sub>0 g \<star> f\<^sub>0g\<^sub>1.p\<^sub>0)
trg f\<^sub>0g\<^sub>1.p\<^sub>0 = src (tab\<^sub>1 g)
trg f\<^sub>0g\<^sub>1.p\<^sub>1 = src (tab\<^sub>0 f)
\<lbrakk>arr f; hseq (g \<star> g\<^sub>0h\<^sub>1.\<phi>) f\<^sub>0gh\<^sub>1.p\<^sub>0; arr TTfgh_TfTgh.chine; src f = trg ((g \<star> g\<^sub>0h\<^sub>1.\<phi>) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0); src ((g \<star> g\<^sub>0h\<^sub>1.\<phi>) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) = trg TTfgh_TfTgh.chine\<rbrakk> \<Longrightarrow> \<a>\<^sup>-\<^sup>1[cod f, cod ((g \<star> g\<^sub>0h\<^sub>1.\<phi>) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0), cod TTfgh_TfTgh.chine] \<cdot> (f \<star> ((g \<star> g\<^sub>0h\<^sub>1.\<phi>) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) = ((f \<star> (g \<star> g\<^sub>0h\<^sub>1.\<phi>) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> \<a>\<^sup>-\<^sup>1[local.dom f, local.dom ((g \<star> g\<^sub>0h\<^sub>1.\<phi>) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0), local.dom TTfgh_TfTgh.chine]
goal (1 subgoal):
1. ((f \<star> (g \<star> g\<^sub>0h\<^sub>1.\<phi>) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> \<a>\<^sup>-\<^sup>1[f, (g \<star> tab\<^sub>0 g \<star> g\<^sub>0h\<^sub>1.p\<^sub>1) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine] = \<a>\<^sup>-\<^sup>1[f, (g \<star> tab\<^sub>1 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine] \<cdot> (f \<star> ((g \<star> g\<^sub>0h\<^sub>1.\<phi>) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine)
[PROOF STEP]
by simp
[PROOF STATE]
proof (state)
this:
((f \<star> (g \<star> g\<^sub>0h\<^sub>1.\<phi>) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> \<a>\<^sup>-\<^sup>1[f, (g \<star> tab\<^sub>0 g \<star> g\<^sub>0h\<^sub>1.p\<^sub>1) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine] = \<a>\<^sup>-\<^sup>1[f, (g \<star> tab\<^sub>1 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine] \<cdot> (f \<star> ((g \<star> g\<^sub>0h\<^sub>1.\<phi>) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine)
goal (1 subgoal):
1. ((f \<star> \<a>[g \<star> h, tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0, f\<^sub>0gh\<^sub>1.p\<^sub>0]) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> \<a>\<^sup>-\<^sup>1[g, h, tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> (g \<star> \<a>[h, tab\<^sub>0 h, g\<^sub>0h\<^sub>1.p\<^sub>0]) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> (g \<star> h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (((f \<star> (g \<star> g\<^sub>0h\<^sub>1.\<phi>) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> \<a>\<^sup>-\<^sup>1[f, (g \<star> tab\<^sub>0 g \<star> g\<^sub>0h\<^sub>1.p\<^sub>1) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine]) \<cdot> (f \<star> (\<a>[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (f \<star> ((g.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>1) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (f \<star> f\<^sub>0gh\<^sub>1.\<phi> \<star> TTfgh_TfTgh.chine) = ((f \<star> \<a>[g \<star> h, tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0, f\<^sub>0gh\<^sub>1.p\<^sub>0]) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> \<a>\<^sup>-\<^sup>1[g, h, tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> (g \<star> \<a>[h, tab\<^sub>0 h, g\<^sub>0h\<^sub>1.p\<^sub>0]) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (((f \<star> (g \<star> h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> \<a>\<^sup>-\<^sup>1[f, (g \<star> tab\<^sub>1 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine]) \<cdot> (f \<star> ((g \<star> g\<^sub>0h\<^sub>1.\<phi>) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (f \<star> (\<a>[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (f \<star> ((g.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>1) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (f \<star> f\<^sub>0gh\<^sub>1.\<phi> \<star> TTfgh_TfTgh.chine)
[PROOF STEP]
thus ?thesis
[PROOF STATE]
proof (prove)
using this:
((f \<star> (g \<star> g\<^sub>0h\<^sub>1.\<phi>) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> \<a>\<^sup>-\<^sup>1[f, (g \<star> tab\<^sub>0 g \<star> g\<^sub>0h\<^sub>1.p\<^sub>1) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine] = \<a>\<^sup>-\<^sup>1[f, (g \<star> tab\<^sub>1 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine] \<cdot> (f \<star> ((g \<star> g\<^sub>0h\<^sub>1.\<phi>) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine)
goal (1 subgoal):
1. ((f \<star> \<a>[g \<star> h, tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0, f\<^sub>0gh\<^sub>1.p\<^sub>0]) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> \<a>\<^sup>-\<^sup>1[g, h, tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> (g \<star> \<a>[h, tab\<^sub>0 h, g\<^sub>0h\<^sub>1.p\<^sub>0]) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> (g \<star> h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (((f \<star> (g \<star> g\<^sub>0h\<^sub>1.\<phi>) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> \<a>\<^sup>-\<^sup>1[f, (g \<star> tab\<^sub>0 g \<star> g\<^sub>0h\<^sub>1.p\<^sub>1) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine]) \<cdot> (f \<star> (\<a>[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (f \<star> ((g.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>1) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (f \<star> f\<^sub>0gh\<^sub>1.\<phi> \<star> TTfgh_TfTgh.chine) = ((f \<star> \<a>[g \<star> h, tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0, f\<^sub>0gh\<^sub>1.p\<^sub>0]) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> \<a>\<^sup>-\<^sup>1[g, h, tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> (g \<star> \<a>[h, tab\<^sub>0 h, g\<^sub>0h\<^sub>1.p\<^sub>0]) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (((f \<star> (g \<star> h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> \<a>\<^sup>-\<^sup>1[f, (g \<star> tab\<^sub>1 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine]) \<cdot> (f \<star> ((g \<star> g\<^sub>0h\<^sub>1.\<phi>) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (f \<star> (\<a>[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (f \<star> ((g.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>1) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (f \<star> f\<^sub>0gh\<^sub>1.\<phi> \<star> TTfgh_TfTgh.chine)
[PROOF STEP]
using comp_assoc
[PROOF STATE]
proof (prove)
using this:
((f \<star> (g \<star> g\<^sub>0h\<^sub>1.\<phi>) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> \<a>\<^sup>-\<^sup>1[f, (g \<star> tab\<^sub>0 g \<star> g\<^sub>0h\<^sub>1.p\<^sub>1) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine] = \<a>\<^sup>-\<^sup>1[f, (g \<star> tab\<^sub>1 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine] \<cdot> (f \<star> ((g \<star> g\<^sub>0h\<^sub>1.\<phi>) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine)
(?h \<cdot> ?g) \<cdot> ?f = ?h \<cdot> ?g \<cdot> ?f
goal (1 subgoal):
1. ((f \<star> \<a>[g \<star> h, tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0, f\<^sub>0gh\<^sub>1.p\<^sub>0]) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> \<a>\<^sup>-\<^sup>1[g, h, tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> (g \<star> \<a>[h, tab\<^sub>0 h, g\<^sub>0h\<^sub>1.p\<^sub>0]) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> (g \<star> h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (((f \<star> (g \<star> g\<^sub>0h\<^sub>1.\<phi>) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> \<a>\<^sup>-\<^sup>1[f, (g \<star> tab\<^sub>0 g \<star> g\<^sub>0h\<^sub>1.p\<^sub>1) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine]) \<cdot> (f \<star> (\<a>[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (f \<star> ((g.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>1) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (f \<star> f\<^sub>0gh\<^sub>1.\<phi> \<star> TTfgh_TfTgh.chine) = ((f \<star> \<a>[g \<star> h, tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0, f\<^sub>0gh\<^sub>1.p\<^sub>0]) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> \<a>\<^sup>-\<^sup>1[g, h, tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> (g \<star> \<a>[h, tab\<^sub>0 h, g\<^sub>0h\<^sub>1.p\<^sub>0]) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (((f \<star> (g \<star> h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> \<a>\<^sup>-\<^sup>1[f, (g \<star> tab\<^sub>1 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine]) \<cdot> (f \<star> ((g \<star> g\<^sub>0h\<^sub>1.\<phi>) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (f \<star> (\<a>[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (f \<star> ((g.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>1) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (f \<star> f\<^sub>0gh\<^sub>1.\<phi> \<star> TTfgh_TfTgh.chine)
[PROOF STEP]
by presburger
[PROOF STATE]
proof (state)
this:
((f \<star> \<a>[g \<star> h, tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0, f\<^sub>0gh\<^sub>1.p\<^sub>0]) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> \<a>\<^sup>-\<^sup>1[g, h, tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> (g \<star> \<a>[h, tab\<^sub>0 h, g\<^sub>0h\<^sub>1.p\<^sub>0]) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> (g \<star> h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (((f \<star> (g \<star> g\<^sub>0h\<^sub>1.\<phi>) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> \<a>\<^sup>-\<^sup>1[f, (g \<star> tab\<^sub>0 g \<star> g\<^sub>0h\<^sub>1.p\<^sub>1) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine]) \<cdot> (f \<star> (\<a>[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (f \<star> ((g.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>1) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (f \<star> f\<^sub>0gh\<^sub>1.\<phi> \<star> TTfgh_TfTgh.chine) = ((f \<star> \<a>[g \<star> h, tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0, f\<^sub>0gh\<^sub>1.p\<^sub>0]) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> \<a>\<^sup>-\<^sup>1[g, h, tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> (g \<star> \<a>[h, tab\<^sub>0 h, g\<^sub>0h\<^sub>1.p\<^sub>0]) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (((f \<star> (g \<star> h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> \<a>\<^sup>-\<^sup>1[f, (g \<star> tab\<^sub>1 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine]) \<cdot> (f \<star> ((g \<star> g\<^sub>0h\<^sub>1.\<phi>) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (f \<star> (\<a>[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (f \<star> ((g.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>1) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (f \<star> f\<^sub>0gh\<^sub>1.\<phi> \<star> TTfgh_TfTgh.chine)
goal:
No subgoals!
[PROOF STEP]
qed
[PROOF STATE]
proof (state)
this:
((f \<star> \<a>[g \<star> h, tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0, f\<^sub>0gh\<^sub>1.p\<^sub>0]) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> \<a>\<^sup>-\<^sup>1[g, h, tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> (g \<star> \<a>[h, tab\<^sub>0 h, g\<^sub>0h\<^sub>1.p\<^sub>0]) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> (g \<star> h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (((f \<star> (g \<star> g\<^sub>0h\<^sub>1.\<phi>) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> \<a>\<^sup>-\<^sup>1[f, (g \<star> tab\<^sub>0 g \<star> g\<^sub>0h\<^sub>1.p\<^sub>1) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine]) \<cdot> (f \<star> (\<a>[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (f \<star> ((g.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>1) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (f \<star> f\<^sub>0gh\<^sub>1.\<phi> \<star> TTfgh_TfTgh.chine) = ((f \<star> \<a>[g \<star> h, tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0, f\<^sub>0gh\<^sub>1.p\<^sub>0]) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> \<a>\<^sup>-\<^sup>1[g, h, tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> (g \<star> \<a>[h, tab\<^sub>0 h, g\<^sub>0h\<^sub>1.p\<^sub>0]) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (((f \<star> (g \<star> h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> \<a>\<^sup>-\<^sup>1[f, (g \<star> tab\<^sub>1 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine]) \<cdot> (f \<star> ((g \<star> g\<^sub>0h\<^sub>1.\<phi>) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (f \<star> (\<a>[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (f \<star> ((g.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>1) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (f \<star> f\<^sub>0gh\<^sub>1.\<phi> \<star> TTfgh_TfTgh.chine)
goal (1 subgoal):
1. ((f \<star> \<a>[g \<star> h, tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0, f\<^sub>0gh\<^sub>1.p\<^sub>0]) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> \<a>\<^sup>-\<^sup>1[g, h, tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> (g \<star> \<a>[h, tab\<^sub>0 h, g\<^sub>0h\<^sub>1.p\<^sub>0]) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> (g \<star> h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> (g \<star> g\<^sub>0h\<^sub>1.\<phi>) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> \<a>[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> (g.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>1) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> f\<^sub>0gh\<^sub>1.\<phi>) \<star> TTfgh_TfTgh.chine) \<cdot> \<a>\<^sup>-\<^sup>1[f, tab\<^sub>0 f \<star> f\<^sub>0gh\<^sub>1.p\<^sub>1, TTfgh_TfTgh.chine] = \<a>\<^sup>-\<^sup>1[f, (g \<star> h) \<star> (tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine] \<cdot> (f \<star> \<a>[g \<star> h, tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0, f\<^sub>0gh\<^sub>1.p\<^sub>0] \<star> TTfgh_TfTgh.chine) \<cdot> (f \<star> (\<a>\<^sup>-\<^sup>1[g, h, tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (f \<star> ((g \<star> \<a>[h, tab\<^sub>0 h, g\<^sub>0h\<^sub>1.p\<^sub>0]) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (f \<star> ((g \<star> h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (f \<star> ((g \<star> g\<^sub>0h\<^sub>1.\<phi>) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (f \<star> (\<a>[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (f \<star> ((g.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>1) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (f \<star> f\<^sub>0gh\<^sub>1.\<phi> \<star> TTfgh_TfTgh.chine)
[PROOF STEP]
also
[PROOF STATE]
proof (state)
this:
((f \<star> \<a>[g \<star> h, tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0, f\<^sub>0gh\<^sub>1.p\<^sub>0]) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> \<a>\<^sup>-\<^sup>1[g, h, tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> (g \<star> \<a>[h, tab\<^sub>0 h, g\<^sub>0h\<^sub>1.p\<^sub>0]) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> (g \<star> h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (((f \<star> (g \<star> g\<^sub>0h\<^sub>1.\<phi>) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> \<a>\<^sup>-\<^sup>1[f, (g \<star> tab\<^sub>0 g \<star> g\<^sub>0h\<^sub>1.p\<^sub>1) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine]) \<cdot> (f \<star> (\<a>[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (f \<star> ((g.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>1) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (f \<star> f\<^sub>0gh\<^sub>1.\<phi> \<star> TTfgh_TfTgh.chine) = ((f \<star> \<a>[g \<star> h, tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0, f\<^sub>0gh\<^sub>1.p\<^sub>0]) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> \<a>\<^sup>-\<^sup>1[g, h, tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> (g \<star> \<a>[h, tab\<^sub>0 h, g\<^sub>0h\<^sub>1.p\<^sub>0]) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (((f \<star> (g \<star> h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> \<a>\<^sup>-\<^sup>1[f, (g \<star> tab\<^sub>1 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine]) \<cdot> (f \<star> ((g \<star> g\<^sub>0h\<^sub>1.\<phi>) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (f \<star> (\<a>[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (f \<star> ((g.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>1) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (f \<star> f\<^sub>0gh\<^sub>1.\<phi> \<star> TTfgh_TfTgh.chine)
goal (1 subgoal):
1. ((f \<star> \<a>[g \<star> h, tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0, f\<^sub>0gh\<^sub>1.p\<^sub>0]) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> \<a>\<^sup>-\<^sup>1[g, h, tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> (g \<star> \<a>[h, tab\<^sub>0 h, g\<^sub>0h\<^sub>1.p\<^sub>0]) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> (g \<star> h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> (g \<star> g\<^sub>0h\<^sub>1.\<phi>) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> \<a>[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> (g.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>1) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> f\<^sub>0gh\<^sub>1.\<phi>) \<star> TTfgh_TfTgh.chine) \<cdot> \<a>\<^sup>-\<^sup>1[f, tab\<^sub>0 f \<star> f\<^sub>0gh\<^sub>1.p\<^sub>1, TTfgh_TfTgh.chine] = \<a>\<^sup>-\<^sup>1[f, (g \<star> h) \<star> (tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine] \<cdot> (f \<star> \<a>[g \<star> h, tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0, f\<^sub>0gh\<^sub>1.p\<^sub>0] \<star> TTfgh_TfTgh.chine) \<cdot> (f \<star> (\<a>\<^sup>-\<^sup>1[g, h, tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (f \<star> ((g \<star> \<a>[h, tab\<^sub>0 h, g\<^sub>0h\<^sub>1.p\<^sub>0]) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (f \<star> ((g \<star> h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (f \<star> ((g \<star> g\<^sub>0h\<^sub>1.\<phi>) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (f \<star> (\<a>[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (f \<star> ((g.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>1) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (f \<star> f\<^sub>0gh\<^sub>1.\<phi> \<star> TTfgh_TfTgh.chine)
[PROOF STEP]
have "... =
(((f \<star> \<a>[g \<star> h, tab\<^sub>0 h \<star> Tgh.p\<^sub>0, TfTgh.p\<^sub>0]) \<star> TTfgh_TfTgh.chine) \<cdot>
((f \<star> \<a>\<^sup>-\<^sup>1[g, h, tab\<^sub>0 h \<star> Tgh.p\<^sub>0] \<star> TfTgh.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot>
((f \<star> (g \<star> \<a>[h, tab\<^sub>0 h, Tgh.p\<^sub>0]) \<star> TfTgh.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot>
\<a>\<^sup>-\<^sup>1[f, (g \<star> (h \<star> tab\<^sub>0 h) \<star> Tgh.p\<^sub>0) \<star> TfTgh.p\<^sub>0, TTfgh_TfTgh.chine]) \<cdot>
(f \<star> ((g \<star> h.tab \<star> Tgh.p\<^sub>0) \<star> TfTgh.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot>
(f \<star> ((g \<star> g\<^sub>0h\<^sub>1.\<phi>) \<star> TfTgh.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot>
(f \<star> (\<a>[g, tab\<^sub>0 g, Tgh.p\<^sub>1] \<star> TfTgh.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot>
(f \<star> ((g.tab \<star> Tgh.p\<^sub>1) \<star> TfTgh.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot>
(f \<star> f\<^sub>0gh\<^sub>1.\<phi> \<star> TTfgh_TfTgh.chine)"
[PROOF STATE]
proof (prove)
goal (1 subgoal):
1. ((f \<star> \<a>[g \<star> h, tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0, f\<^sub>0gh\<^sub>1.p\<^sub>0]) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> \<a>\<^sup>-\<^sup>1[g, h, tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> (g \<star> \<a>[h, tab\<^sub>0 h, g\<^sub>0h\<^sub>1.p\<^sub>0]) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (((f \<star> (g \<star> h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> \<a>\<^sup>-\<^sup>1[f, (g \<star> tab\<^sub>1 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine]) \<cdot> (f \<star> ((g \<star> g\<^sub>0h\<^sub>1.\<phi>) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (f \<star> (\<a>[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (f \<star> ((g.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>1) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (f \<star> f\<^sub>0gh\<^sub>1.\<phi> \<star> TTfgh_TfTgh.chine) = (((f \<star> \<a>[g \<star> h, tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0, f\<^sub>0gh\<^sub>1.p\<^sub>0]) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> \<a>\<^sup>-\<^sup>1[g, h, tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> (g \<star> \<a>[h, tab\<^sub>0 h, g\<^sub>0h\<^sub>1.p\<^sub>0]) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> \<a>\<^sup>-\<^sup>1[f, (g \<star> (h \<star> tab\<^sub>0 h) \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine]) \<cdot> (f \<star> ((g \<star> h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (f \<star> ((g \<star> g\<^sub>0h\<^sub>1.\<phi>) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (f \<star> (\<a>[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (f \<star> ((g.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>1) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (f \<star> f\<^sub>0gh\<^sub>1.\<phi> \<star> TTfgh_TfTgh.chine)
[PROOF STEP]
proof -
[PROOF STATE]
proof (state)
goal (1 subgoal):
1. ((f \<star> \<a>[g \<star> h, tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0, f\<^sub>0gh\<^sub>1.p\<^sub>0]) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> \<a>\<^sup>-\<^sup>1[g, h, tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> (g \<star> \<a>[h, tab\<^sub>0 h, g\<^sub>0h\<^sub>1.p\<^sub>0]) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (((f \<star> (g \<star> h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> \<a>\<^sup>-\<^sup>1[f, (g \<star> tab\<^sub>1 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine]) \<cdot> (f \<star> ((g \<star> g\<^sub>0h\<^sub>1.\<phi>) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (f \<star> (\<a>[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (f \<star> ((g.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>1) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (f \<star> f\<^sub>0gh\<^sub>1.\<phi> \<star> TTfgh_TfTgh.chine) = (((f \<star> \<a>[g \<star> h, tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0, f\<^sub>0gh\<^sub>1.p\<^sub>0]) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> \<a>\<^sup>-\<^sup>1[g, h, tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> (g \<star> \<a>[h, tab\<^sub>0 h, g\<^sub>0h\<^sub>1.p\<^sub>0]) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> \<a>\<^sup>-\<^sup>1[f, (g \<star> (h \<star> tab\<^sub>0 h) \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine]) \<cdot> (f \<star> ((g \<star> h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (f \<star> ((g \<star> g\<^sub>0h\<^sub>1.\<phi>) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (f \<star> (\<a>[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (f \<star> ((g.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>1) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (f \<star> f\<^sub>0gh\<^sub>1.\<phi> \<star> TTfgh_TfTgh.chine)
[PROOF STEP]
have "((f \<star> (g \<star> h.tab \<star> Tgh.p\<^sub>0) \<star> TfTgh.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot>
\<a>\<^sup>-\<^sup>1[f, (g \<star> tab\<^sub>1 h \<star> Tgh.p\<^sub>0) \<star> TfTgh.p\<^sub>0, TTfgh_TfTgh.chine] =
\<a>\<^sup>-\<^sup>1[f, (g \<star> (h \<star> tab\<^sub>0 h) \<star> Tgh.p\<^sub>0) \<star> TfTgh.p\<^sub>0, TTfgh_TfTgh.chine] \<cdot>
(f \<star> ((g \<star> h.tab \<star> Tgh.p\<^sub>0) \<star> TfTgh.p\<^sub>0) \<star> TTfgh_TfTgh.chine)"
[PROOF STATE]
proof (prove)
goal (1 subgoal):
1. ((f \<star> (g \<star> h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> \<a>\<^sup>-\<^sup>1[f, (g \<star> tab\<^sub>1 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine] = \<a>\<^sup>-\<^sup>1[f, (g \<star> (h \<star> tab\<^sub>0 h) \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine] \<cdot> (f \<star> ((g \<star> h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine)
[PROOF STEP]
using fg gh fg\<^sub>0h\<^sub>1.p\<^sub>0_simps fg\<^sub>0h\<^sub>1.p\<^sub>1_simps f\<^sub>0g\<^sub>1.p\<^sub>0_simps f\<^sub>0g\<^sub>1.p\<^sub>1_simps
assoc'_naturality
[of f "(g \<star> h.tab \<star> Tgh.p\<^sub>0) \<star> TfTgh.p\<^sub>0" TTfgh_TfTgh.chine]
[PROOF STATE]
proof (prove)
using this:
src f = trg g
src g = trg h
trg fg\<^sub>0h\<^sub>1.p\<^sub>0 = src (tab\<^sub>1 h)
trg fg\<^sub>0h\<^sub>1.p\<^sub>1 = src (tab\<^sub>0 g \<star> f\<^sub>0g\<^sub>1.p\<^sub>0)
trg f\<^sub>0g\<^sub>1.p\<^sub>0 = src (tab\<^sub>1 g)
trg f\<^sub>0g\<^sub>1.p\<^sub>1 = src (tab\<^sub>0 f)
\<lbrakk>arr f; hseq (g \<star> h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) f\<^sub>0gh\<^sub>1.p\<^sub>0; arr TTfgh_TfTgh.chine; src f = trg ((g \<star> h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0); src ((g \<star> h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) = trg TTfgh_TfTgh.chine\<rbrakk> \<Longrightarrow> \<a>\<^sup>-\<^sup>1[cod f, cod ((g \<star> h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0), cod TTfgh_TfTgh.chine] \<cdot> (f \<star> ((g \<star> h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) = ((f \<star> (g \<star> h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> \<a>\<^sup>-\<^sup>1[local.dom f, local.dom ((g \<star> h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0), local.dom TTfgh_TfTgh.chine]
goal (1 subgoal):
1. ((f \<star> (g \<star> h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> \<a>\<^sup>-\<^sup>1[f, (g \<star> tab\<^sub>1 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine] = \<a>\<^sup>-\<^sup>1[f, (g \<star> (h \<star> tab\<^sub>0 h) \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine] \<cdot> (f \<star> ((g \<star> h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine)
[PROOF STEP]
by simp
[PROOF STATE]
proof (state)
this:
((f \<star> (g \<star> h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> \<a>\<^sup>-\<^sup>1[f, (g \<star> tab\<^sub>1 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine] = \<a>\<^sup>-\<^sup>1[f, (g \<star> (h \<star> tab\<^sub>0 h) \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine] \<cdot> (f \<star> ((g \<star> h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine)
goal (1 subgoal):
1. ((f \<star> \<a>[g \<star> h, tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0, f\<^sub>0gh\<^sub>1.p\<^sub>0]) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> \<a>\<^sup>-\<^sup>1[g, h, tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> (g \<star> \<a>[h, tab\<^sub>0 h, g\<^sub>0h\<^sub>1.p\<^sub>0]) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (((f \<star> (g \<star> h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> \<a>\<^sup>-\<^sup>1[f, (g \<star> tab\<^sub>1 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine]) \<cdot> (f \<star> ((g \<star> g\<^sub>0h\<^sub>1.\<phi>) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (f \<star> (\<a>[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (f \<star> ((g.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>1) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (f \<star> f\<^sub>0gh\<^sub>1.\<phi> \<star> TTfgh_TfTgh.chine) = (((f \<star> \<a>[g \<star> h, tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0, f\<^sub>0gh\<^sub>1.p\<^sub>0]) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> \<a>\<^sup>-\<^sup>1[g, h, tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> (g \<star> \<a>[h, tab\<^sub>0 h, g\<^sub>0h\<^sub>1.p\<^sub>0]) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> \<a>\<^sup>-\<^sup>1[f, (g \<star> (h \<star> tab\<^sub>0 h) \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine]) \<cdot> (f \<star> ((g \<star> h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (f \<star> ((g \<star> g\<^sub>0h\<^sub>1.\<phi>) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (f \<star> (\<a>[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (f \<star> ((g.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>1) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (f \<star> f\<^sub>0gh\<^sub>1.\<phi> \<star> TTfgh_TfTgh.chine)
[PROOF STEP]
thus ?thesis
[PROOF STATE]
proof (prove)
using this:
((f \<star> (g \<star> h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> \<a>\<^sup>-\<^sup>1[f, (g \<star> tab\<^sub>1 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine] = \<a>\<^sup>-\<^sup>1[f, (g \<star> (h \<star> tab\<^sub>0 h) \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine] \<cdot> (f \<star> ((g \<star> h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine)
goal (1 subgoal):
1. ((f \<star> \<a>[g \<star> h, tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0, f\<^sub>0gh\<^sub>1.p\<^sub>0]) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> \<a>\<^sup>-\<^sup>1[g, h, tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> (g \<star> \<a>[h, tab\<^sub>0 h, g\<^sub>0h\<^sub>1.p\<^sub>0]) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (((f \<star> (g \<star> h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> \<a>\<^sup>-\<^sup>1[f, (g \<star> tab\<^sub>1 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine]) \<cdot> (f \<star> ((g \<star> g\<^sub>0h\<^sub>1.\<phi>) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (f \<star> (\<a>[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (f \<star> ((g.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>1) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (f \<star> f\<^sub>0gh\<^sub>1.\<phi> \<star> TTfgh_TfTgh.chine) = (((f \<star> \<a>[g \<star> h, tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0, f\<^sub>0gh\<^sub>1.p\<^sub>0]) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> \<a>\<^sup>-\<^sup>1[g, h, tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> (g \<star> \<a>[h, tab\<^sub>0 h, g\<^sub>0h\<^sub>1.p\<^sub>0]) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> \<a>\<^sup>-\<^sup>1[f, (g \<star> (h \<star> tab\<^sub>0 h) \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine]) \<cdot> (f \<star> ((g \<star> h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (f \<star> ((g \<star> g\<^sub>0h\<^sub>1.\<phi>) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (f \<star> (\<a>[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (f \<star> ((g.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>1) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (f \<star> f\<^sub>0gh\<^sub>1.\<phi> \<star> TTfgh_TfTgh.chine)
[PROOF STEP]
using comp_assoc
[PROOF STATE]
proof (prove)
using this:
((f \<star> (g \<star> h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> \<a>\<^sup>-\<^sup>1[f, (g \<star> tab\<^sub>1 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine] = \<a>\<^sup>-\<^sup>1[f, (g \<star> (h \<star> tab\<^sub>0 h) \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine] \<cdot> (f \<star> ((g \<star> h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine)
(?h \<cdot> ?g) \<cdot> ?f = ?h \<cdot> ?g \<cdot> ?f
goal (1 subgoal):
1. ((f \<star> \<a>[g \<star> h, tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0, f\<^sub>0gh\<^sub>1.p\<^sub>0]) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> \<a>\<^sup>-\<^sup>1[g, h, tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> (g \<star> \<a>[h, tab\<^sub>0 h, g\<^sub>0h\<^sub>1.p\<^sub>0]) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (((f \<star> (g \<star> h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> \<a>\<^sup>-\<^sup>1[f, (g \<star> tab\<^sub>1 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine]) \<cdot> (f \<star> ((g \<star> g\<^sub>0h\<^sub>1.\<phi>) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (f \<star> (\<a>[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (f \<star> ((g.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>1) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (f \<star> f\<^sub>0gh\<^sub>1.\<phi> \<star> TTfgh_TfTgh.chine) = (((f \<star> \<a>[g \<star> h, tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0, f\<^sub>0gh\<^sub>1.p\<^sub>0]) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> \<a>\<^sup>-\<^sup>1[g, h, tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> (g \<star> \<a>[h, tab\<^sub>0 h, g\<^sub>0h\<^sub>1.p\<^sub>0]) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> \<a>\<^sup>-\<^sup>1[f, (g \<star> (h \<star> tab\<^sub>0 h) \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine]) \<cdot> (f \<star> ((g \<star> h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (f \<star> ((g \<star> g\<^sub>0h\<^sub>1.\<phi>) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (f \<star> (\<a>[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (f \<star> ((g.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>1) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (f \<star> f\<^sub>0gh\<^sub>1.\<phi> \<star> TTfgh_TfTgh.chine)
[PROOF STEP]
by presburger
[PROOF STATE]
proof (state)
this:
((f \<star> \<a>[g \<star> h, tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0, f\<^sub>0gh\<^sub>1.p\<^sub>0]) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> \<a>\<^sup>-\<^sup>1[g, h, tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> (g \<star> \<a>[h, tab\<^sub>0 h, g\<^sub>0h\<^sub>1.p\<^sub>0]) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (((f \<star> (g \<star> h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> \<a>\<^sup>-\<^sup>1[f, (g \<star> tab\<^sub>1 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine]) \<cdot> (f \<star> ((g \<star> g\<^sub>0h\<^sub>1.\<phi>) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (f \<star> (\<a>[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (f \<star> ((g.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>1) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (f \<star> f\<^sub>0gh\<^sub>1.\<phi> \<star> TTfgh_TfTgh.chine) = (((f \<star> \<a>[g \<star> h, tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0, f\<^sub>0gh\<^sub>1.p\<^sub>0]) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> \<a>\<^sup>-\<^sup>1[g, h, tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> (g \<star> \<a>[h, tab\<^sub>0 h, g\<^sub>0h\<^sub>1.p\<^sub>0]) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> \<a>\<^sup>-\<^sup>1[f, (g \<star> (h \<star> tab\<^sub>0 h) \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine]) \<cdot> (f \<star> ((g \<star> h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (f \<star> ((g \<star> g\<^sub>0h\<^sub>1.\<phi>) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (f \<star> (\<a>[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (f \<star> ((g.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>1) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (f \<star> f\<^sub>0gh\<^sub>1.\<phi> \<star> TTfgh_TfTgh.chine)
goal:
No subgoals!
[PROOF STEP]
qed
[PROOF STATE]
proof (state)
this:
((f \<star> \<a>[g \<star> h, tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0, f\<^sub>0gh\<^sub>1.p\<^sub>0]) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> \<a>\<^sup>-\<^sup>1[g, h, tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> (g \<star> \<a>[h, tab\<^sub>0 h, g\<^sub>0h\<^sub>1.p\<^sub>0]) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (((f \<star> (g \<star> h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> \<a>\<^sup>-\<^sup>1[f, (g \<star> tab\<^sub>1 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine]) \<cdot> (f \<star> ((g \<star> g\<^sub>0h\<^sub>1.\<phi>) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (f \<star> (\<a>[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (f \<star> ((g.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>1) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (f \<star> f\<^sub>0gh\<^sub>1.\<phi> \<star> TTfgh_TfTgh.chine) = (((f \<star> \<a>[g \<star> h, tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0, f\<^sub>0gh\<^sub>1.p\<^sub>0]) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> \<a>\<^sup>-\<^sup>1[g, h, tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> (g \<star> \<a>[h, tab\<^sub>0 h, g\<^sub>0h\<^sub>1.p\<^sub>0]) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> \<a>\<^sup>-\<^sup>1[f, (g \<star> (h \<star> tab\<^sub>0 h) \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine]) \<cdot> (f \<star> ((g \<star> h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (f \<star> ((g \<star> g\<^sub>0h\<^sub>1.\<phi>) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (f \<star> (\<a>[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (f \<star> ((g.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>1) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (f \<star> f\<^sub>0gh\<^sub>1.\<phi> \<star> TTfgh_TfTgh.chine)
goal (1 subgoal):
1. ((f \<star> \<a>[g \<star> h, tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0, f\<^sub>0gh\<^sub>1.p\<^sub>0]) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> \<a>\<^sup>-\<^sup>1[g, h, tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> (g \<star> \<a>[h, tab\<^sub>0 h, g\<^sub>0h\<^sub>1.p\<^sub>0]) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> (g \<star> h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> (g \<star> g\<^sub>0h\<^sub>1.\<phi>) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> \<a>[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> (g.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>1) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> f\<^sub>0gh\<^sub>1.\<phi>) \<star> TTfgh_TfTgh.chine) \<cdot> \<a>\<^sup>-\<^sup>1[f, tab\<^sub>0 f \<star> f\<^sub>0gh\<^sub>1.p\<^sub>1, TTfgh_TfTgh.chine] = \<a>\<^sup>-\<^sup>1[f, (g \<star> h) \<star> (tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine] \<cdot> (f \<star> \<a>[g \<star> h, tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0, f\<^sub>0gh\<^sub>1.p\<^sub>0] \<star> TTfgh_TfTgh.chine) \<cdot> (f \<star> (\<a>\<^sup>-\<^sup>1[g, h, tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (f \<star> ((g \<star> \<a>[h, tab\<^sub>0 h, g\<^sub>0h\<^sub>1.p\<^sub>0]) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (f \<star> ((g \<star> h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (f \<star> ((g \<star> g\<^sub>0h\<^sub>1.\<phi>) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (f \<star> (\<a>[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (f \<star> ((g.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>1) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (f \<star> f\<^sub>0gh\<^sub>1.\<phi> \<star> TTfgh_TfTgh.chine)
[PROOF STEP]
also
[PROOF STATE]
proof (state)
this:
((f \<star> \<a>[g \<star> h, tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0, f\<^sub>0gh\<^sub>1.p\<^sub>0]) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> \<a>\<^sup>-\<^sup>1[g, h, tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> (g \<star> \<a>[h, tab\<^sub>0 h, g\<^sub>0h\<^sub>1.p\<^sub>0]) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (((f \<star> (g \<star> h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> \<a>\<^sup>-\<^sup>1[f, (g \<star> tab\<^sub>1 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine]) \<cdot> (f \<star> ((g \<star> g\<^sub>0h\<^sub>1.\<phi>) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (f \<star> (\<a>[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (f \<star> ((g.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>1) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (f \<star> f\<^sub>0gh\<^sub>1.\<phi> \<star> TTfgh_TfTgh.chine) = (((f \<star> \<a>[g \<star> h, tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0, f\<^sub>0gh\<^sub>1.p\<^sub>0]) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> \<a>\<^sup>-\<^sup>1[g, h, tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> (g \<star> \<a>[h, tab\<^sub>0 h, g\<^sub>0h\<^sub>1.p\<^sub>0]) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> \<a>\<^sup>-\<^sup>1[f, (g \<star> (h \<star> tab\<^sub>0 h) \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine]) \<cdot> (f \<star> ((g \<star> h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (f \<star> ((g \<star> g\<^sub>0h\<^sub>1.\<phi>) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (f \<star> (\<a>[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (f \<star> ((g.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>1) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (f \<star> f\<^sub>0gh\<^sub>1.\<phi> \<star> TTfgh_TfTgh.chine)
goal (1 subgoal):
1. ((f \<star> \<a>[g \<star> h, tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0, f\<^sub>0gh\<^sub>1.p\<^sub>0]) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> \<a>\<^sup>-\<^sup>1[g, h, tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> (g \<star> \<a>[h, tab\<^sub>0 h, g\<^sub>0h\<^sub>1.p\<^sub>0]) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> (g \<star> h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> (g \<star> g\<^sub>0h\<^sub>1.\<phi>) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> \<a>[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> (g.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>1) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> f\<^sub>0gh\<^sub>1.\<phi>) \<star> TTfgh_TfTgh.chine) \<cdot> \<a>\<^sup>-\<^sup>1[f, tab\<^sub>0 f \<star> f\<^sub>0gh\<^sub>1.p\<^sub>1, TTfgh_TfTgh.chine] = \<a>\<^sup>-\<^sup>1[f, (g \<star> h) \<star> (tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine] \<cdot> (f \<star> \<a>[g \<star> h, tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0, f\<^sub>0gh\<^sub>1.p\<^sub>0] \<star> TTfgh_TfTgh.chine) \<cdot> (f \<star> (\<a>\<^sup>-\<^sup>1[g, h, tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (f \<star> ((g \<star> \<a>[h, tab\<^sub>0 h, g\<^sub>0h\<^sub>1.p\<^sub>0]) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (f \<star> ((g \<star> h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (f \<star> ((g \<star> g\<^sub>0h\<^sub>1.\<phi>) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (f \<star> (\<a>[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (f \<star> ((g.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>1) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (f \<star> f\<^sub>0gh\<^sub>1.\<phi> \<star> TTfgh_TfTgh.chine)
[PROOF STEP]
have "... =
\<a>\<^sup>-\<^sup>1[f, (g \<star> h) \<star> (tab\<^sub>0 h \<star> Tgh.p\<^sub>0) \<star> TfTgh.p\<^sub>0, TTfgh_TfTgh.chine] \<cdot>
(f \<star> \<a>[g \<star> h, tab\<^sub>0 h \<star> Tgh.p\<^sub>0, TfTgh.p\<^sub>0] \<star> TTfgh_TfTgh.chine) \<cdot>
(f \<star> (\<a>\<^sup>-\<^sup>1[g, h, tab\<^sub>0 h \<star> Tgh.p\<^sub>0] \<star> TfTgh.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot>
(f \<star> ((g \<star> \<a>[h, tab\<^sub>0 h, Tgh.p\<^sub>0]) \<star> TfTgh.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot>
(f \<star> ((g \<star> h.tab \<star> Tgh.p\<^sub>0) \<star> TfTgh.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot>
(f \<star> ((g \<star> g\<^sub>0h\<^sub>1.\<phi>) \<star> TfTgh.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot>
(f \<star> (\<a>[g, tab\<^sub>0 g, Tgh.p\<^sub>1] \<star> TfTgh.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot>
(f \<star> ((g.tab \<star> Tgh.p\<^sub>1) \<star> TfTgh.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot>
(f \<star> f\<^sub>0gh\<^sub>1.\<phi> \<star> TTfgh_TfTgh.chine)"
[PROOF STATE]
proof (prove)
goal (1 subgoal):
1. (((f \<star> \<a>[g \<star> h, tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0, f\<^sub>0gh\<^sub>1.p\<^sub>0]) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> \<a>\<^sup>-\<^sup>1[g, h, tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> (g \<star> \<a>[h, tab\<^sub>0 h, g\<^sub>0h\<^sub>1.p\<^sub>0]) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> \<a>\<^sup>-\<^sup>1[f, (g \<star> (h \<star> tab\<^sub>0 h) \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine]) \<cdot> (f \<star> ((g \<star> h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (f \<star> ((g \<star> g\<^sub>0h\<^sub>1.\<phi>) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (f \<star> (\<a>[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (f \<star> ((g.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>1) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (f \<star> f\<^sub>0gh\<^sub>1.\<phi> \<star> TTfgh_TfTgh.chine) = \<a>\<^sup>-\<^sup>1[f, (g \<star> h) \<star> (tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine] \<cdot> (f \<star> \<a>[g \<star> h, tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0, f\<^sub>0gh\<^sub>1.p\<^sub>0] \<star> TTfgh_TfTgh.chine) \<cdot> (f \<star> (\<a>\<^sup>-\<^sup>1[g, h, tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (f \<star> ((g \<star> \<a>[h, tab\<^sub>0 h, g\<^sub>0h\<^sub>1.p\<^sub>0]) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (f \<star> ((g \<star> h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (f \<star> ((g \<star> g\<^sub>0h\<^sub>1.\<phi>) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (f \<star> (\<a>[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (f \<star> ((g.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>1) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (f \<star> f\<^sub>0gh\<^sub>1.\<phi> \<star> TTfgh_TfTgh.chine)
[PROOF STEP]
proof -
[PROOF STATE]
proof (state)
goal (1 subgoal):
1. (((f \<star> \<a>[g \<star> h, tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0, f\<^sub>0gh\<^sub>1.p\<^sub>0]) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> \<a>\<^sup>-\<^sup>1[g, h, tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> (g \<star> \<a>[h, tab\<^sub>0 h, g\<^sub>0h\<^sub>1.p\<^sub>0]) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> \<a>\<^sup>-\<^sup>1[f, (g \<star> (h \<star> tab\<^sub>0 h) \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine]) \<cdot> (f \<star> ((g \<star> h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (f \<star> ((g \<star> g\<^sub>0h\<^sub>1.\<phi>) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (f \<star> (\<a>[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (f \<star> ((g.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>1) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (f \<star> f\<^sub>0gh\<^sub>1.\<phi> \<star> TTfgh_TfTgh.chine) = \<a>\<^sup>-\<^sup>1[f, (g \<star> h) \<star> (tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine] \<cdot> (f \<star> \<a>[g \<star> h, tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0, f\<^sub>0gh\<^sub>1.p\<^sub>0] \<star> TTfgh_TfTgh.chine) \<cdot> (f \<star> (\<a>\<^sup>-\<^sup>1[g, h, tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (f \<star> ((g \<star> \<a>[h, tab\<^sub>0 h, g\<^sub>0h\<^sub>1.p\<^sub>0]) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (f \<star> ((g \<star> h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (f \<star> ((g \<star> g\<^sub>0h\<^sub>1.\<phi>) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (f \<star> (\<a>[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (f \<star> ((g.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>1) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (f \<star> f\<^sub>0gh\<^sub>1.\<phi> \<star> TTfgh_TfTgh.chine)
[PROOF STEP]
have "((f \<star> \<a>[g \<star> h, tab\<^sub>0 h \<star> Tgh.p\<^sub>0, TfTgh.p\<^sub>0]) \<star> TTfgh_TfTgh.chine) \<cdot>
((f \<star> \<a>\<^sup>-\<^sup>1[g, h, tab\<^sub>0 h \<star> Tgh.p\<^sub>0] \<star> TfTgh.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot>
((f \<star> (g \<star> \<a>[h, tab\<^sub>0 h, Tgh.p\<^sub>0]) \<star> TfTgh.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot>
\<a>\<^sup>-\<^sup>1[f, (g \<star> (h \<star> tab\<^sub>0 h) \<star> Tgh.p\<^sub>0) \<star> TfTgh.p\<^sub>0, TTfgh_TfTgh.chine] =
\<lbrace>((\<^bold>\<langle>f\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<a>\<^bold>[\<^bold>\<langle>g\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>h\<^bold>\<rangle>, \<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>Tgh.p\<^sub>0\<^bold>\<rangle>, \<^bold>\<langle>TfTgh.p\<^sub>0\<^bold>\<rangle>\<^bold>])
\<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) \<^bold>\<cdot>
((\<^bold>\<langle>f\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<a>\<^sup>-\<^sup>1\<^bold>[\<^bold>\<langle>g\<^bold>\<rangle>, \<^bold>\<langle>h\<^bold>\<rangle>, \<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>Tgh.p\<^sub>0\<^bold>\<rangle>\<^bold>] \<^bold>\<star> \<^bold>\<langle>TfTgh.p\<^sub>0\<^bold>\<rangle>)
\<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) \<^bold>\<cdot>
((\<^bold>\<langle>f\<^bold>\<rangle> \<^bold>\<star> (\<^bold>\<langle>g\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<a>\<^bold>[\<^bold>\<langle>h\<^bold>\<rangle>, \<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle>, \<^bold>\<langle>Tgh.p\<^sub>0\<^bold>\<rangle>\<^bold>]) \<^bold>\<star> \<^bold>\<langle>TfTgh.p\<^sub>0\<^bold>\<rangle>)
\<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) \<^bold>\<cdot>
\<^bold>\<a>\<^sup>-\<^sup>1\<^bold>[\<^bold>\<langle>f\<^bold>\<rangle>, (\<^bold>\<langle>g\<^bold>\<rangle> \<^bold>\<star> (\<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>Tgh.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TfTgh.p\<^sub>0\<^bold>\<rangle>,
\<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>\<^bold>]\<rbrace>"
[PROOF STATE]
proof (prove)
goal (1 subgoal):
1. ((f \<star> \<a>[g \<star> h, tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0, f\<^sub>0gh\<^sub>1.p\<^sub>0]) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> \<a>\<^sup>-\<^sup>1[g, h, tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> (g \<star> \<a>[h, tab\<^sub>0 h, g\<^sub>0h\<^sub>1.p\<^sub>0]) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> \<a>\<^sup>-\<^sup>1[f, (g \<star> (h \<star> tab\<^sub>0 h) \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine] = \<lbrace>((\<^bold>\<langle>f\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<a>\<^bold>[\<^bold>\<langle>g\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>h\<^bold>\<rangle>, \<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>, \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>\<^bold>]) \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) \<^bold>\<cdot> ((\<^bold>\<langle>f\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<a>\<^sup>-\<^sup>1\<^bold>[\<^bold>\<langle>g\<^bold>\<rangle>, \<^bold>\<langle>h\<^bold>\<rangle>, \<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>\<^bold>] \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) \<^bold>\<cdot> ((\<^bold>\<langle>f\<^bold>\<rangle> \<^bold>\<star> (\<^bold>\<langle>g\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<a>\<^bold>[\<^bold>\<langle>h\<^bold>\<rangle>, \<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle>, \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>\<^bold>]) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) \<^bold>\<cdot> \<^bold>\<a>\<^sup>-\<^sup>1\<^bold>[\<^bold>\<langle>f\<^bold>\<rangle>, (\<^bold>\<langle>g\<^bold>\<rangle> \<^bold>\<star> (\<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>, \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>\<^bold>]\<rbrace>
[PROOF STEP]
using fg gh fg\<^sub>0h\<^sub>1.p\<^sub>0_simps fg\<^sub>0h\<^sub>1.p\<^sub>1_simps f\<^sub>0g\<^sub>1.p\<^sub>0_simps f\<^sub>0g\<^sub>1.p\<^sub>1_simps
\<a>'_def \<alpha>_def
[PROOF STATE]
proof (prove)
using this:
src f = trg g
src g = trg h
trg fg\<^sub>0h\<^sub>1.p\<^sub>0 = src (tab\<^sub>1 h)
trg fg\<^sub>0h\<^sub>1.p\<^sub>1 = src (tab\<^sub>0 g \<star> f\<^sub>0g\<^sub>1.p\<^sub>0)
trg f\<^sub>0g\<^sub>1.p\<^sub>0 = src (tab\<^sub>1 g)
trg f\<^sub>0g\<^sub>1.p\<^sub>1 = src (tab\<^sub>0 f)
\<a>\<^sup>-\<^sup>1[?\<mu>, ?\<nu>, ?\<tau>] \<equiv> \<alpha>' (?\<mu>, ?\<nu>, ?\<tau>)
\<alpha> ?\<mu>\<nu>\<tau> \<equiv> \<a>[fst ?\<mu>\<nu>\<tau>, fst (snd ?\<mu>\<nu>\<tau>), snd (snd ?\<mu>\<nu>\<tau>)]
goal (1 subgoal):
1. ((f \<star> \<a>[g \<star> h, tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0, f\<^sub>0gh\<^sub>1.p\<^sub>0]) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> \<a>\<^sup>-\<^sup>1[g, h, tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> (g \<star> \<a>[h, tab\<^sub>0 h, g\<^sub>0h\<^sub>1.p\<^sub>0]) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> \<a>\<^sup>-\<^sup>1[f, (g \<star> (h \<star> tab\<^sub>0 h) \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine] = \<lbrace>((\<^bold>\<langle>f\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<a>\<^bold>[\<^bold>\<langle>g\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>h\<^bold>\<rangle>, \<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>, \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>\<^bold>]) \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) \<^bold>\<cdot> ((\<^bold>\<langle>f\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<a>\<^sup>-\<^sup>1\<^bold>[\<^bold>\<langle>g\<^bold>\<rangle>, \<^bold>\<langle>h\<^bold>\<rangle>, \<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>\<^bold>] \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) \<^bold>\<cdot> ((\<^bold>\<langle>f\<^bold>\<rangle> \<^bold>\<star> (\<^bold>\<langle>g\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<a>\<^bold>[\<^bold>\<langle>h\<^bold>\<rangle>, \<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle>, \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>\<^bold>]) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) \<^bold>\<cdot> \<^bold>\<a>\<^sup>-\<^sup>1\<^bold>[\<^bold>\<langle>f\<^bold>\<rangle>, (\<^bold>\<langle>g\<^bold>\<rangle> \<^bold>\<star> (\<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>, \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>\<^bold>]\<rbrace>
[PROOF STEP]
by simp
[PROOF STATE]
proof (state)
this:
((f \<star> \<a>[g \<star> h, tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0, f\<^sub>0gh\<^sub>1.p\<^sub>0]) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> \<a>\<^sup>-\<^sup>1[g, h, tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> (g \<star> \<a>[h, tab\<^sub>0 h, g\<^sub>0h\<^sub>1.p\<^sub>0]) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> \<a>\<^sup>-\<^sup>1[f, (g \<star> (h \<star> tab\<^sub>0 h) \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine] = \<lbrace>((\<^bold>\<langle>f\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<a>\<^bold>[\<^bold>\<langle>g\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>h\<^bold>\<rangle>, \<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>, \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>\<^bold>]) \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) \<^bold>\<cdot> ((\<^bold>\<langle>f\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<a>\<^sup>-\<^sup>1\<^bold>[\<^bold>\<langle>g\<^bold>\<rangle>, \<^bold>\<langle>h\<^bold>\<rangle>, \<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>\<^bold>] \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) \<^bold>\<cdot> ((\<^bold>\<langle>f\<^bold>\<rangle> \<^bold>\<star> (\<^bold>\<langle>g\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<a>\<^bold>[\<^bold>\<langle>h\<^bold>\<rangle>, \<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle>, \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>\<^bold>]) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) \<^bold>\<cdot> \<^bold>\<a>\<^sup>-\<^sup>1\<^bold>[\<^bold>\<langle>f\<^bold>\<rangle>, (\<^bold>\<langle>g\<^bold>\<rangle> \<^bold>\<star> (\<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>, \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>\<^bold>]\<rbrace>
goal (1 subgoal):
1. (((f \<star> \<a>[g \<star> h, tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0, f\<^sub>0gh\<^sub>1.p\<^sub>0]) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> \<a>\<^sup>-\<^sup>1[g, h, tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> (g \<star> \<a>[h, tab\<^sub>0 h, g\<^sub>0h\<^sub>1.p\<^sub>0]) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> \<a>\<^sup>-\<^sup>1[f, (g \<star> (h \<star> tab\<^sub>0 h) \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine]) \<cdot> (f \<star> ((g \<star> h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (f \<star> ((g \<star> g\<^sub>0h\<^sub>1.\<phi>) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (f \<star> (\<a>[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (f \<star> ((g.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>1) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (f \<star> f\<^sub>0gh\<^sub>1.\<phi> \<star> TTfgh_TfTgh.chine) = \<a>\<^sup>-\<^sup>1[f, (g \<star> h) \<star> (tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine] \<cdot> (f \<star> \<a>[g \<star> h, tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0, f\<^sub>0gh\<^sub>1.p\<^sub>0] \<star> TTfgh_TfTgh.chine) \<cdot> (f \<star> (\<a>\<^sup>-\<^sup>1[g, h, tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (f \<star> ((g \<star> \<a>[h, tab\<^sub>0 h, g\<^sub>0h\<^sub>1.p\<^sub>0]) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (f \<star> ((g \<star> h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (f \<star> ((g \<star> g\<^sub>0h\<^sub>1.\<phi>) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (f \<star> (\<a>[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (f \<star> ((g.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>1) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (f \<star> f\<^sub>0gh\<^sub>1.\<phi> \<star> TTfgh_TfTgh.chine)
[PROOF STEP]
also
[PROOF STATE]
proof (state)
this:
((f \<star> \<a>[g \<star> h, tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0, f\<^sub>0gh\<^sub>1.p\<^sub>0]) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> \<a>\<^sup>-\<^sup>1[g, h, tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> (g \<star> \<a>[h, tab\<^sub>0 h, g\<^sub>0h\<^sub>1.p\<^sub>0]) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> \<a>\<^sup>-\<^sup>1[f, (g \<star> (h \<star> tab\<^sub>0 h) \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine] = \<lbrace>((\<^bold>\<langle>f\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<a>\<^bold>[\<^bold>\<langle>g\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>h\<^bold>\<rangle>, \<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>, \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>\<^bold>]) \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) \<^bold>\<cdot> ((\<^bold>\<langle>f\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<a>\<^sup>-\<^sup>1\<^bold>[\<^bold>\<langle>g\<^bold>\<rangle>, \<^bold>\<langle>h\<^bold>\<rangle>, \<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>\<^bold>] \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) \<^bold>\<cdot> ((\<^bold>\<langle>f\<^bold>\<rangle> \<^bold>\<star> (\<^bold>\<langle>g\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<a>\<^bold>[\<^bold>\<langle>h\<^bold>\<rangle>, \<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle>, \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>\<^bold>]) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) \<^bold>\<cdot> \<^bold>\<a>\<^sup>-\<^sup>1\<^bold>[\<^bold>\<langle>f\<^bold>\<rangle>, (\<^bold>\<langle>g\<^bold>\<rangle> \<^bold>\<star> (\<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>, \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>\<^bold>]\<rbrace>
goal (1 subgoal):
1. (((f \<star> \<a>[g \<star> h, tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0, f\<^sub>0gh\<^sub>1.p\<^sub>0]) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> \<a>\<^sup>-\<^sup>1[g, h, tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> (g \<star> \<a>[h, tab\<^sub>0 h, g\<^sub>0h\<^sub>1.p\<^sub>0]) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> \<a>\<^sup>-\<^sup>1[f, (g \<star> (h \<star> tab\<^sub>0 h) \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine]) \<cdot> (f \<star> ((g \<star> h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (f \<star> ((g \<star> g\<^sub>0h\<^sub>1.\<phi>) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (f \<star> (\<a>[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (f \<star> ((g.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>1) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (f \<star> f\<^sub>0gh\<^sub>1.\<phi> \<star> TTfgh_TfTgh.chine) = \<a>\<^sup>-\<^sup>1[f, (g \<star> h) \<star> (tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine] \<cdot> (f \<star> \<a>[g \<star> h, tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0, f\<^sub>0gh\<^sub>1.p\<^sub>0] \<star> TTfgh_TfTgh.chine) \<cdot> (f \<star> (\<a>\<^sup>-\<^sup>1[g, h, tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (f \<star> ((g \<star> \<a>[h, tab\<^sub>0 h, g\<^sub>0h\<^sub>1.p\<^sub>0]) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (f \<star> ((g \<star> h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (f \<star> ((g \<star> g\<^sub>0h\<^sub>1.\<phi>) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (f \<star> (\<a>[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (f \<star> ((g.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>1) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (f \<star> f\<^sub>0gh\<^sub>1.\<phi> \<star> TTfgh_TfTgh.chine)
[PROOF STEP]
have "... =
\<lbrace>\<^bold>\<a>\<^sup>-\<^sup>1\<^bold>[\<^bold>\<langle>f\<^bold>\<rangle>, (\<^bold>\<langle>g\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>h\<^bold>\<rangle>) \<^bold>\<star> (\<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>Tgh.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TfTgh.p\<^sub>0\<^bold>\<rangle>,
\<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>\<^bold>] \<^bold>\<cdot>
(\<^bold>\<langle>f\<^bold>\<rangle> \<^bold>\<star> (\<^bold>\<a>\<^bold>[\<^bold>\<langle>g\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>h\<^bold>\<rangle>, \<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>Tgh.p\<^sub>0\<^bold>\<rangle>, \<^bold>\<langle>TfTgh.p\<^sub>0\<^bold>\<rangle>\<^bold>])
\<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) \<^bold>\<cdot>
(\<^bold>\<langle>f\<^bold>\<rangle> \<^bold>\<star> (\<^bold>\<a>\<^sup>-\<^sup>1\<^bold>[\<^bold>\<langle>g\<^bold>\<rangle>, \<^bold>\<langle>h\<^bold>\<rangle>, \<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>Tgh.p\<^sub>0\<^bold>\<rangle>\<^bold>] \<^bold>\<star> \<^bold>\<langle>TfTgh.p\<^sub>0\<^bold>\<rangle>)
\<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) \<^bold>\<cdot>
(\<^bold>\<langle>f\<^bold>\<rangle> \<^bold>\<star> ((\<^bold>\<langle>g\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<a>\<^bold>[\<^bold>\<langle>h\<^bold>\<rangle>, \<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle>, \<^bold>\<langle>Tgh.p\<^sub>0\<^bold>\<rangle>\<^bold>]) \<^bold>\<star> \<^bold>\<langle>TfTgh.p\<^sub>0\<^bold>\<rangle>)
\<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>)\<rbrace>"
[PROOF STATE]
proof (prove)
goal (1 subgoal):
1. \<lbrace>((\<^bold>\<langle>f\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<a>\<^bold>[\<^bold>\<langle>g\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>h\<^bold>\<rangle>, \<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>, \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>\<^bold>]) \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) \<^bold>\<cdot> ((\<^bold>\<langle>f\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<a>\<^sup>-\<^sup>1\<^bold>[\<^bold>\<langle>g\<^bold>\<rangle>, \<^bold>\<langle>h\<^bold>\<rangle>, \<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>\<^bold>] \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) \<^bold>\<cdot> ((\<^bold>\<langle>f\<^bold>\<rangle> \<^bold>\<star> (\<^bold>\<langle>g\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<a>\<^bold>[\<^bold>\<langle>h\<^bold>\<rangle>, \<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle>, \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>\<^bold>]) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) \<^bold>\<cdot> \<^bold>\<a>\<^sup>-\<^sup>1\<^bold>[\<^bold>\<langle>f\<^bold>\<rangle>, (\<^bold>\<langle>g\<^bold>\<rangle> \<^bold>\<star> (\<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>, \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>\<^bold>]\<rbrace> = \<lbrace>\<^bold>\<a>\<^sup>-\<^sup>1\<^bold>[\<^bold>\<langle>f\<^bold>\<rangle>, (\<^bold>\<langle>g\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>h\<^bold>\<rangle>) \<^bold>\<star> (\<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>, \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>\<^bold>] \<^bold>\<cdot> (\<^bold>\<langle>f\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<a>\<^bold>[\<^bold>\<langle>g\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>h\<^bold>\<rangle>, \<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>, \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>\<^bold>] \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) \<^bold>\<cdot> (\<^bold>\<langle>f\<^bold>\<rangle> \<^bold>\<star> (\<^bold>\<a>\<^sup>-\<^sup>1\<^bold>[\<^bold>\<langle>g\<^bold>\<rangle>, \<^bold>\<langle>h\<^bold>\<rangle>, \<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>\<^bold>] \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) \<^bold>\<cdot> (\<^bold>\<langle>f\<^bold>\<rangle> \<^bold>\<star> ((\<^bold>\<langle>g\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<a>\<^bold>[\<^bold>\<langle>h\<^bold>\<rangle>, \<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle>, \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>\<^bold>]) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>)\<rbrace>
[PROOF STEP]
using fg gh fg\<^sub>0h\<^sub>1.p\<^sub>0_simps fg\<^sub>0h\<^sub>1.p\<^sub>1_simps f\<^sub>0g\<^sub>1.p\<^sub>0_simps f\<^sub>0g\<^sub>1.p\<^sub>1_simps
[PROOF STATE]
proof (prove)
using this:
src f = trg g
src g = trg h
trg fg\<^sub>0h\<^sub>1.p\<^sub>0 = src (tab\<^sub>1 h)
trg fg\<^sub>0h\<^sub>1.p\<^sub>1 = src (tab\<^sub>0 g \<star> f\<^sub>0g\<^sub>1.p\<^sub>0)
trg f\<^sub>0g\<^sub>1.p\<^sub>0 = src (tab\<^sub>1 g)
trg f\<^sub>0g\<^sub>1.p\<^sub>1 = src (tab\<^sub>0 f)
goal (1 subgoal):
1. \<lbrace>((\<^bold>\<langle>f\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<a>\<^bold>[\<^bold>\<langle>g\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>h\<^bold>\<rangle>, \<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>, \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>\<^bold>]) \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) \<^bold>\<cdot> ((\<^bold>\<langle>f\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<a>\<^sup>-\<^sup>1\<^bold>[\<^bold>\<langle>g\<^bold>\<rangle>, \<^bold>\<langle>h\<^bold>\<rangle>, \<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>\<^bold>] \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) \<^bold>\<cdot> ((\<^bold>\<langle>f\<^bold>\<rangle> \<^bold>\<star> (\<^bold>\<langle>g\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<a>\<^bold>[\<^bold>\<langle>h\<^bold>\<rangle>, \<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle>, \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>\<^bold>]) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) \<^bold>\<cdot> \<^bold>\<a>\<^sup>-\<^sup>1\<^bold>[\<^bold>\<langle>f\<^bold>\<rangle>, (\<^bold>\<langle>g\<^bold>\<rangle> \<^bold>\<star> (\<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>, \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>\<^bold>]\<rbrace> = \<lbrace>\<^bold>\<a>\<^sup>-\<^sup>1\<^bold>[\<^bold>\<langle>f\<^bold>\<rangle>, (\<^bold>\<langle>g\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>h\<^bold>\<rangle>) \<^bold>\<star> (\<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>, \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>\<^bold>] \<^bold>\<cdot> (\<^bold>\<langle>f\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<a>\<^bold>[\<^bold>\<langle>g\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>h\<^bold>\<rangle>, \<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>, \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>\<^bold>] \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) \<^bold>\<cdot> (\<^bold>\<langle>f\<^bold>\<rangle> \<^bold>\<star> (\<^bold>\<a>\<^sup>-\<^sup>1\<^bold>[\<^bold>\<langle>g\<^bold>\<rangle>, \<^bold>\<langle>h\<^bold>\<rangle>, \<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>\<^bold>] \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) \<^bold>\<cdot> (\<^bold>\<langle>f\<^bold>\<rangle> \<^bold>\<star> ((\<^bold>\<langle>g\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<a>\<^bold>[\<^bold>\<langle>h\<^bold>\<rangle>, \<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle>, \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>\<^bold>]) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>)\<rbrace>
[PROOF STEP]
apply (intro E.eval_eqI)
[PROOF STATE]
proof (prove)
goal (2 subgoals):
1. \<lbrakk>src f = trg g; src g = trg h; trg fg\<^sub>0h\<^sub>1.p\<^sub>0 = src (tab\<^sub>1 h); trg fg\<^sub>0h\<^sub>1.p\<^sub>1 = src (tab\<^sub>0 g \<star> f\<^sub>0g\<^sub>1.p\<^sub>0); trg f\<^sub>0g\<^sub>1.p\<^sub>0 = src (tab\<^sub>1 g); trg f\<^sub>0g\<^sub>1.p\<^sub>1 = src (tab\<^sub>0 f)\<rbrakk> \<Longrightarrow> E.VPar (((\<^bold>\<langle>f\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<a>\<^bold>[\<^bold>\<langle>g\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>h\<^bold>\<rangle>, \<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>, \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>\<^bold>]) \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) \<^bold>\<cdot> ((\<^bold>\<langle>f\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<a>\<^sup>-\<^sup>1\<^bold>[\<^bold>\<langle>g\<^bold>\<rangle>, \<^bold>\<langle>h\<^bold>\<rangle>, \<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>\<^bold>] \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) \<^bold>\<cdot> ((\<^bold>\<langle>f\<^bold>\<rangle> \<^bold>\<star> (\<^bold>\<langle>g\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<a>\<^bold>[\<^bold>\<langle>h\<^bold>\<rangle>, \<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle>, \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>\<^bold>]) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) \<^bold>\<cdot> \<^bold>\<a>\<^sup>-\<^sup>1\<^bold>[\<^bold>\<langle>f\<^bold>\<rangle>, (\<^bold>\<langle>g\<^bold>\<rangle> \<^bold>\<star> (\<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>, \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>\<^bold>]) (\<^bold>\<a>\<^sup>-\<^sup>1\<^bold>[\<^bold>\<langle>f\<^bold>\<rangle>, (\<^bold>\<langle>g\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>h\<^bold>\<rangle>) \<^bold>\<star> (\<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>, \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>\<^bold>] \<^bold>\<cdot> (\<^bold>\<langle>f\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<a>\<^bold>[\<^bold>\<langle>g\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>h\<^bold>\<rangle>, \<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>, \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>\<^bold>] \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) \<^bold>\<cdot> (\<^bold>\<langle>f\<^bold>\<rangle> \<^bold>\<star> (\<^bold>\<a>\<^sup>-\<^sup>1\<^bold>[\<^bold>\<langle>g\<^bold>\<rangle>, \<^bold>\<langle>h\<^bold>\<rangle>, \<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>\<^bold>] \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) \<^bold>\<cdot> (\<^bold>\<langle>f\<^bold>\<rangle> \<^bold>\<star> ((\<^bold>\<langle>g\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<a>\<^bold>[\<^bold>\<langle>h\<^bold>\<rangle>, \<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle>, \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>\<^bold>]) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>))
2. \<lbrakk>src f = trg g; src g = trg h; trg fg\<^sub>0h\<^sub>1.p\<^sub>0 = src (tab\<^sub>1 h); trg fg\<^sub>0h\<^sub>1.p\<^sub>1 = src (tab\<^sub>0 g \<star> f\<^sub>0g\<^sub>1.p\<^sub>0); trg f\<^sub>0g\<^sub>1.p\<^sub>0 = src (tab\<^sub>1 g); trg f\<^sub>0g\<^sub>1.p\<^sub>1 = src (tab\<^sub>0 f)\<rbrakk> \<Longrightarrow> \<^bold>\<lfloor>((\<^bold>\<langle>f\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<a>\<^bold>[\<^bold>\<langle>g\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>h\<^bold>\<rangle>, \<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>, \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>\<^bold>]) \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) \<^bold>\<cdot> ((\<^bold>\<langle>f\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<a>\<^sup>-\<^sup>1\<^bold>[\<^bold>\<langle>g\<^bold>\<rangle>, \<^bold>\<langle>h\<^bold>\<rangle>, \<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>\<^bold>] \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) \<^bold>\<cdot> ((\<^bold>\<langle>f\<^bold>\<rangle> \<^bold>\<star> (\<^bold>\<langle>g\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<a>\<^bold>[\<^bold>\<langle>h\<^bold>\<rangle>, \<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle>, \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>\<^bold>]) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) \<^bold>\<cdot> \<^bold>\<a>\<^sup>-\<^sup>1\<^bold>[\<^bold>\<langle>f\<^bold>\<rangle>, (\<^bold>\<langle>g\<^bold>\<rangle> \<^bold>\<star> (\<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>, \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>\<^bold>]\<^bold>\<rfloor> = \<^bold>\<lfloor>\<^bold>\<a>\<^sup>-\<^sup>1\<^bold>[\<^bold>\<langle>f\<^bold>\<rangle>, (\<^bold>\<langle>g\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>h\<^bold>\<rangle>) \<^bold>\<star> (\<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>, \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>\<^bold>] \<^bold>\<cdot> (\<^bold>\<langle>f\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<a>\<^bold>[\<^bold>\<langle>g\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>h\<^bold>\<rangle>, \<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>, \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>\<^bold>] \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) \<^bold>\<cdot> (\<^bold>\<langle>f\<^bold>\<rangle> \<^bold>\<star> (\<^bold>\<a>\<^sup>-\<^sup>1\<^bold>[\<^bold>\<langle>g\<^bold>\<rangle>, \<^bold>\<langle>h\<^bold>\<rangle>, \<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>\<^bold>] \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) \<^bold>\<cdot> (\<^bold>\<langle>f\<^bold>\<rangle> \<^bold>\<star> ((\<^bold>\<langle>g\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<a>\<^bold>[\<^bold>\<langle>h\<^bold>\<rangle>, \<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle>, \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>\<^bold>]) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>)\<^bold>\<rfloor>
[PROOF STEP]
by simp_all
[PROOF STATE]
proof (state)
this:
\<lbrace>((\<^bold>\<langle>f\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<a>\<^bold>[\<^bold>\<langle>g\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>h\<^bold>\<rangle>, \<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>, \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>\<^bold>]) \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) \<^bold>\<cdot> ((\<^bold>\<langle>f\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<a>\<^sup>-\<^sup>1\<^bold>[\<^bold>\<langle>g\<^bold>\<rangle>, \<^bold>\<langle>h\<^bold>\<rangle>, \<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>\<^bold>] \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) \<^bold>\<cdot> ((\<^bold>\<langle>f\<^bold>\<rangle> \<^bold>\<star> (\<^bold>\<langle>g\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<a>\<^bold>[\<^bold>\<langle>h\<^bold>\<rangle>, \<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle>, \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>\<^bold>]) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) \<^bold>\<cdot> \<^bold>\<a>\<^sup>-\<^sup>1\<^bold>[\<^bold>\<langle>f\<^bold>\<rangle>, (\<^bold>\<langle>g\<^bold>\<rangle> \<^bold>\<star> (\<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>, \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>\<^bold>]\<rbrace> = \<lbrace>\<^bold>\<a>\<^sup>-\<^sup>1\<^bold>[\<^bold>\<langle>f\<^bold>\<rangle>, (\<^bold>\<langle>g\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>h\<^bold>\<rangle>) \<^bold>\<star> (\<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>, \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>\<^bold>] \<^bold>\<cdot> (\<^bold>\<langle>f\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<a>\<^bold>[\<^bold>\<langle>g\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>h\<^bold>\<rangle>, \<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>, \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>\<^bold>] \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) \<^bold>\<cdot> (\<^bold>\<langle>f\<^bold>\<rangle> \<^bold>\<star> (\<^bold>\<a>\<^sup>-\<^sup>1\<^bold>[\<^bold>\<langle>g\<^bold>\<rangle>, \<^bold>\<langle>h\<^bold>\<rangle>, \<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>\<^bold>] \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) \<^bold>\<cdot> (\<^bold>\<langle>f\<^bold>\<rangle> \<^bold>\<star> ((\<^bold>\<langle>g\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<a>\<^bold>[\<^bold>\<langle>h\<^bold>\<rangle>, \<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle>, \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>\<^bold>]) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>)\<rbrace>
goal (1 subgoal):
1. (((f \<star> \<a>[g \<star> h, tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0, f\<^sub>0gh\<^sub>1.p\<^sub>0]) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> \<a>\<^sup>-\<^sup>1[g, h, tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> (g \<star> \<a>[h, tab\<^sub>0 h, g\<^sub>0h\<^sub>1.p\<^sub>0]) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> \<a>\<^sup>-\<^sup>1[f, (g \<star> (h \<star> tab\<^sub>0 h) \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine]) \<cdot> (f \<star> ((g \<star> h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (f \<star> ((g \<star> g\<^sub>0h\<^sub>1.\<phi>) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (f \<star> (\<a>[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (f \<star> ((g.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>1) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (f \<star> f\<^sub>0gh\<^sub>1.\<phi> \<star> TTfgh_TfTgh.chine) = \<a>\<^sup>-\<^sup>1[f, (g \<star> h) \<star> (tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine] \<cdot> (f \<star> \<a>[g \<star> h, tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0, f\<^sub>0gh\<^sub>1.p\<^sub>0] \<star> TTfgh_TfTgh.chine) \<cdot> (f \<star> (\<a>\<^sup>-\<^sup>1[g, h, tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (f \<star> ((g \<star> \<a>[h, tab\<^sub>0 h, g\<^sub>0h\<^sub>1.p\<^sub>0]) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (f \<star> ((g \<star> h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (f \<star> ((g \<star> g\<^sub>0h\<^sub>1.\<phi>) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (f \<star> (\<a>[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (f \<star> ((g.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>1) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (f \<star> f\<^sub>0gh\<^sub>1.\<phi> \<star> TTfgh_TfTgh.chine)
[PROOF STEP]
also
[PROOF STATE]
proof (state)
this:
\<lbrace>((\<^bold>\<langle>f\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<a>\<^bold>[\<^bold>\<langle>g\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>h\<^bold>\<rangle>, \<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>, \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>\<^bold>]) \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) \<^bold>\<cdot> ((\<^bold>\<langle>f\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<a>\<^sup>-\<^sup>1\<^bold>[\<^bold>\<langle>g\<^bold>\<rangle>, \<^bold>\<langle>h\<^bold>\<rangle>, \<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>\<^bold>] \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) \<^bold>\<cdot> ((\<^bold>\<langle>f\<^bold>\<rangle> \<^bold>\<star> (\<^bold>\<langle>g\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<a>\<^bold>[\<^bold>\<langle>h\<^bold>\<rangle>, \<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle>, \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>\<^bold>]) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) \<^bold>\<cdot> \<^bold>\<a>\<^sup>-\<^sup>1\<^bold>[\<^bold>\<langle>f\<^bold>\<rangle>, (\<^bold>\<langle>g\<^bold>\<rangle> \<^bold>\<star> (\<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>, \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>\<^bold>]\<rbrace> = \<lbrace>\<^bold>\<a>\<^sup>-\<^sup>1\<^bold>[\<^bold>\<langle>f\<^bold>\<rangle>, (\<^bold>\<langle>g\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>h\<^bold>\<rangle>) \<^bold>\<star> (\<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>, \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>\<^bold>] \<^bold>\<cdot> (\<^bold>\<langle>f\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<a>\<^bold>[\<^bold>\<langle>g\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>h\<^bold>\<rangle>, \<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>, \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>\<^bold>] \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) \<^bold>\<cdot> (\<^bold>\<langle>f\<^bold>\<rangle> \<^bold>\<star> (\<^bold>\<a>\<^sup>-\<^sup>1\<^bold>[\<^bold>\<langle>g\<^bold>\<rangle>, \<^bold>\<langle>h\<^bold>\<rangle>, \<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>\<^bold>] \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) \<^bold>\<cdot> (\<^bold>\<langle>f\<^bold>\<rangle> \<^bold>\<star> ((\<^bold>\<langle>g\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<a>\<^bold>[\<^bold>\<langle>h\<^bold>\<rangle>, \<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle>, \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>\<^bold>]) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>)\<rbrace>
goal (1 subgoal):
1. (((f \<star> \<a>[g \<star> h, tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0, f\<^sub>0gh\<^sub>1.p\<^sub>0]) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> \<a>\<^sup>-\<^sup>1[g, h, tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> (g \<star> \<a>[h, tab\<^sub>0 h, g\<^sub>0h\<^sub>1.p\<^sub>0]) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> \<a>\<^sup>-\<^sup>1[f, (g \<star> (h \<star> tab\<^sub>0 h) \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine]) \<cdot> (f \<star> ((g \<star> h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (f \<star> ((g \<star> g\<^sub>0h\<^sub>1.\<phi>) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (f \<star> (\<a>[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (f \<star> ((g.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>1) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (f \<star> f\<^sub>0gh\<^sub>1.\<phi> \<star> TTfgh_TfTgh.chine) = \<a>\<^sup>-\<^sup>1[f, (g \<star> h) \<star> (tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine] \<cdot> (f \<star> \<a>[g \<star> h, tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0, f\<^sub>0gh\<^sub>1.p\<^sub>0] \<star> TTfgh_TfTgh.chine) \<cdot> (f \<star> (\<a>\<^sup>-\<^sup>1[g, h, tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (f \<star> ((g \<star> \<a>[h, tab\<^sub>0 h, g\<^sub>0h\<^sub>1.p\<^sub>0]) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (f \<star> ((g \<star> h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (f \<star> ((g \<star> g\<^sub>0h\<^sub>1.\<phi>) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (f \<star> (\<a>[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (f \<star> ((g.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>1) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (f \<star> f\<^sub>0gh\<^sub>1.\<phi> \<star> TTfgh_TfTgh.chine)
[PROOF STEP]
have "... =
\<a>\<^sup>-\<^sup>1[f, (g \<star> h) \<star> (tab\<^sub>0 h \<star> Tgh.p\<^sub>0) \<star> TfTgh.p\<^sub>0, TTfgh_TfTgh.chine] \<cdot>
(f \<star> \<a>[g \<star> h, tab\<^sub>0 h \<star> Tgh.p\<^sub>0, TfTgh.p\<^sub>0] \<star> TTfgh_TfTgh.chine) \<cdot>
(f \<star> (\<a>\<^sup>-\<^sup>1[g, h, tab\<^sub>0 h \<star> Tgh.p\<^sub>0] \<star> TfTgh.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot>
(f \<star> ((g \<star> \<a>[h, tab\<^sub>0 h, Tgh.p\<^sub>0]) \<star> TfTgh.p\<^sub>0) \<star> TTfgh_TfTgh.chine)"
[PROOF STATE]
proof (prove)
goal (1 subgoal):
1. \<lbrace>\<^bold>\<a>\<^sup>-\<^sup>1\<^bold>[\<^bold>\<langle>f\<^bold>\<rangle>, (\<^bold>\<langle>g\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>h\<^bold>\<rangle>) \<^bold>\<star> (\<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>, \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>\<^bold>] \<^bold>\<cdot> (\<^bold>\<langle>f\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<a>\<^bold>[\<^bold>\<langle>g\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>h\<^bold>\<rangle>, \<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>, \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>\<^bold>] \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) \<^bold>\<cdot> (\<^bold>\<langle>f\<^bold>\<rangle> \<^bold>\<star> (\<^bold>\<a>\<^sup>-\<^sup>1\<^bold>[\<^bold>\<langle>g\<^bold>\<rangle>, \<^bold>\<langle>h\<^bold>\<rangle>, \<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>\<^bold>] \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) \<^bold>\<cdot> (\<^bold>\<langle>f\<^bold>\<rangle> \<^bold>\<star> ((\<^bold>\<langle>g\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<a>\<^bold>[\<^bold>\<langle>h\<^bold>\<rangle>, \<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle>, \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>\<^bold>]) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>)\<rbrace> = \<a>\<^sup>-\<^sup>1[f, (g \<star> h) \<star> (tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine] \<cdot> (f \<star> \<a>[g \<star> h, tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0, f\<^sub>0gh\<^sub>1.p\<^sub>0] \<star> TTfgh_TfTgh.chine) \<cdot> (f \<star> (\<a>\<^sup>-\<^sup>1[g, h, tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (f \<star> ((g \<star> \<a>[h, tab\<^sub>0 h, g\<^sub>0h\<^sub>1.p\<^sub>0]) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine)
[PROOF STEP]
using fg gh fg\<^sub>0h\<^sub>1.p\<^sub>0_simps fg\<^sub>0h\<^sub>1.p\<^sub>1_simps f\<^sub>0g\<^sub>1.p\<^sub>0_simps f\<^sub>0g\<^sub>1.p\<^sub>1_simps
\<a>'_def \<alpha>_def
[PROOF STATE]
proof (prove)
using this:
src f = trg g
src g = trg h
trg fg\<^sub>0h\<^sub>1.p\<^sub>0 = src (tab\<^sub>1 h)
trg fg\<^sub>0h\<^sub>1.p\<^sub>1 = src (tab\<^sub>0 g \<star> f\<^sub>0g\<^sub>1.p\<^sub>0)
trg f\<^sub>0g\<^sub>1.p\<^sub>0 = src (tab\<^sub>1 g)
trg f\<^sub>0g\<^sub>1.p\<^sub>1 = src (tab\<^sub>0 f)
\<a>\<^sup>-\<^sup>1[?\<mu>, ?\<nu>, ?\<tau>] \<equiv> \<alpha>' (?\<mu>, ?\<nu>, ?\<tau>)
\<alpha> ?\<mu>\<nu>\<tau> \<equiv> \<a>[fst ?\<mu>\<nu>\<tau>, fst (snd ?\<mu>\<nu>\<tau>), snd (snd ?\<mu>\<nu>\<tau>)]
goal (1 subgoal):
1. \<lbrace>\<^bold>\<a>\<^sup>-\<^sup>1\<^bold>[\<^bold>\<langle>f\<^bold>\<rangle>, (\<^bold>\<langle>g\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>h\<^bold>\<rangle>) \<^bold>\<star> (\<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>, \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>\<^bold>] \<^bold>\<cdot> (\<^bold>\<langle>f\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<a>\<^bold>[\<^bold>\<langle>g\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>h\<^bold>\<rangle>, \<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>, \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>\<^bold>] \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) \<^bold>\<cdot> (\<^bold>\<langle>f\<^bold>\<rangle> \<^bold>\<star> (\<^bold>\<a>\<^sup>-\<^sup>1\<^bold>[\<^bold>\<langle>g\<^bold>\<rangle>, \<^bold>\<langle>h\<^bold>\<rangle>, \<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>\<^bold>] \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) \<^bold>\<cdot> (\<^bold>\<langle>f\<^bold>\<rangle> \<^bold>\<star> ((\<^bold>\<langle>g\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<a>\<^bold>[\<^bold>\<langle>h\<^bold>\<rangle>, \<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle>, \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>\<^bold>]) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>)\<rbrace> = \<a>\<^sup>-\<^sup>1[f, (g \<star> h) \<star> (tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine] \<cdot> (f \<star> \<a>[g \<star> h, tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0, f\<^sub>0gh\<^sub>1.p\<^sub>0] \<star> TTfgh_TfTgh.chine) \<cdot> (f \<star> (\<a>\<^sup>-\<^sup>1[g, h, tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (f \<star> ((g \<star> \<a>[h, tab\<^sub>0 h, g\<^sub>0h\<^sub>1.p\<^sub>0]) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine)
[PROOF STEP]
by simp
[PROOF STATE]
proof (state)
this:
\<lbrace>\<^bold>\<a>\<^sup>-\<^sup>1\<^bold>[\<^bold>\<langle>f\<^bold>\<rangle>, (\<^bold>\<langle>g\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>h\<^bold>\<rangle>) \<^bold>\<star> (\<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>, \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>\<^bold>] \<^bold>\<cdot> (\<^bold>\<langle>f\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<a>\<^bold>[\<^bold>\<langle>g\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>h\<^bold>\<rangle>, \<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>, \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>\<^bold>] \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) \<^bold>\<cdot> (\<^bold>\<langle>f\<^bold>\<rangle> \<^bold>\<star> (\<^bold>\<a>\<^sup>-\<^sup>1\<^bold>[\<^bold>\<langle>g\<^bold>\<rangle>, \<^bold>\<langle>h\<^bold>\<rangle>, \<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>\<^bold>] \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) \<^bold>\<cdot> (\<^bold>\<langle>f\<^bold>\<rangle> \<^bold>\<star> ((\<^bold>\<langle>g\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<a>\<^bold>[\<^bold>\<langle>h\<^bold>\<rangle>, \<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle>, \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>\<^bold>]) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>)\<rbrace> = \<a>\<^sup>-\<^sup>1[f, (g \<star> h) \<star> (tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine] \<cdot> (f \<star> \<a>[g \<star> h, tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0, f\<^sub>0gh\<^sub>1.p\<^sub>0] \<star> TTfgh_TfTgh.chine) \<cdot> (f \<star> (\<a>\<^sup>-\<^sup>1[g, h, tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (f \<star> ((g \<star> \<a>[h, tab\<^sub>0 h, g\<^sub>0h\<^sub>1.p\<^sub>0]) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine)
goal (1 subgoal):
1. (((f \<star> \<a>[g \<star> h, tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0, f\<^sub>0gh\<^sub>1.p\<^sub>0]) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> \<a>\<^sup>-\<^sup>1[g, h, tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> (g \<star> \<a>[h, tab\<^sub>0 h, g\<^sub>0h\<^sub>1.p\<^sub>0]) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> \<a>\<^sup>-\<^sup>1[f, (g \<star> (h \<star> tab\<^sub>0 h) \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine]) \<cdot> (f \<star> ((g \<star> h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (f \<star> ((g \<star> g\<^sub>0h\<^sub>1.\<phi>) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (f \<star> (\<a>[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (f \<star> ((g.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>1) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (f \<star> f\<^sub>0gh\<^sub>1.\<phi> \<star> TTfgh_TfTgh.chine) = \<a>\<^sup>-\<^sup>1[f, (g \<star> h) \<star> (tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine] \<cdot> (f \<star> \<a>[g \<star> h, tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0, f\<^sub>0gh\<^sub>1.p\<^sub>0] \<star> TTfgh_TfTgh.chine) \<cdot> (f \<star> (\<a>\<^sup>-\<^sup>1[g, h, tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (f \<star> ((g \<star> \<a>[h, tab\<^sub>0 h, g\<^sub>0h\<^sub>1.p\<^sub>0]) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (f \<star> ((g \<star> h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (f \<star> ((g \<star> g\<^sub>0h\<^sub>1.\<phi>) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (f \<star> (\<a>[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (f \<star> ((g.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>1) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (f \<star> f\<^sub>0gh\<^sub>1.\<phi> \<star> TTfgh_TfTgh.chine)
[PROOF STEP]
finally
[PROOF STATE]
proof (chain)
picking this:
((f \<star> \<a>[g \<star> h, tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0, f\<^sub>0gh\<^sub>1.p\<^sub>0]) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> \<a>\<^sup>-\<^sup>1[g, h, tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> (g \<star> \<a>[h, tab\<^sub>0 h, g\<^sub>0h\<^sub>1.p\<^sub>0]) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> \<a>\<^sup>-\<^sup>1[f, (g \<star> (h \<star> tab\<^sub>0 h) \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine] = \<a>\<^sup>-\<^sup>1[f, (g \<star> h) \<star> (tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine] \<cdot> (f \<star> \<a>[g \<star> h, tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0, f\<^sub>0gh\<^sub>1.p\<^sub>0] \<star> TTfgh_TfTgh.chine) \<cdot> (f \<star> (\<a>\<^sup>-\<^sup>1[g, h, tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (f \<star> ((g \<star> \<a>[h, tab\<^sub>0 h, g\<^sub>0h\<^sub>1.p\<^sub>0]) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine)
[PROOF STEP]
show ?thesis
[PROOF STATE]
proof (prove)
using this:
((f \<star> \<a>[g \<star> h, tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0, f\<^sub>0gh\<^sub>1.p\<^sub>0]) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> \<a>\<^sup>-\<^sup>1[g, h, tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> (g \<star> \<a>[h, tab\<^sub>0 h, g\<^sub>0h\<^sub>1.p\<^sub>0]) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> \<a>\<^sup>-\<^sup>1[f, (g \<star> (h \<star> tab\<^sub>0 h) \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine] = \<a>\<^sup>-\<^sup>1[f, (g \<star> h) \<star> (tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine] \<cdot> (f \<star> \<a>[g \<star> h, tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0, f\<^sub>0gh\<^sub>1.p\<^sub>0] \<star> TTfgh_TfTgh.chine) \<cdot> (f \<star> (\<a>\<^sup>-\<^sup>1[g, h, tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (f \<star> ((g \<star> \<a>[h, tab\<^sub>0 h, g\<^sub>0h\<^sub>1.p\<^sub>0]) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine)
goal (1 subgoal):
1. (((f \<star> \<a>[g \<star> h, tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0, f\<^sub>0gh\<^sub>1.p\<^sub>0]) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> \<a>\<^sup>-\<^sup>1[g, h, tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> (g \<star> \<a>[h, tab\<^sub>0 h, g\<^sub>0h\<^sub>1.p\<^sub>0]) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> \<a>\<^sup>-\<^sup>1[f, (g \<star> (h \<star> tab\<^sub>0 h) \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine]) \<cdot> (f \<star> ((g \<star> h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (f \<star> ((g \<star> g\<^sub>0h\<^sub>1.\<phi>) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (f \<star> (\<a>[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (f \<star> ((g.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>1) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (f \<star> f\<^sub>0gh\<^sub>1.\<phi> \<star> TTfgh_TfTgh.chine) = \<a>\<^sup>-\<^sup>1[f, (g \<star> h) \<star> (tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine] \<cdot> (f \<star> \<a>[g \<star> h, tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0, f\<^sub>0gh\<^sub>1.p\<^sub>0] \<star> TTfgh_TfTgh.chine) \<cdot> (f \<star> (\<a>\<^sup>-\<^sup>1[g, h, tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (f \<star> ((g \<star> \<a>[h, tab\<^sub>0 h, g\<^sub>0h\<^sub>1.p\<^sub>0]) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (f \<star> ((g \<star> h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (f \<star> ((g \<star> g\<^sub>0h\<^sub>1.\<phi>) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (f \<star> (\<a>[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (f \<star> ((g.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>1) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (f \<star> f\<^sub>0gh\<^sub>1.\<phi> \<star> TTfgh_TfTgh.chine)
[PROOF STEP]
using comp_assoc
[PROOF STATE]
proof (prove)
using this:
((f \<star> \<a>[g \<star> h, tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0, f\<^sub>0gh\<^sub>1.p\<^sub>0]) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> \<a>\<^sup>-\<^sup>1[g, h, tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> (g \<star> \<a>[h, tab\<^sub>0 h, g\<^sub>0h\<^sub>1.p\<^sub>0]) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> \<a>\<^sup>-\<^sup>1[f, (g \<star> (h \<star> tab\<^sub>0 h) \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine] = \<a>\<^sup>-\<^sup>1[f, (g \<star> h) \<star> (tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine] \<cdot> (f \<star> \<a>[g \<star> h, tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0, f\<^sub>0gh\<^sub>1.p\<^sub>0] \<star> TTfgh_TfTgh.chine) \<cdot> (f \<star> (\<a>\<^sup>-\<^sup>1[g, h, tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (f \<star> ((g \<star> \<a>[h, tab\<^sub>0 h, g\<^sub>0h\<^sub>1.p\<^sub>0]) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine)
(?h \<cdot> ?g) \<cdot> ?f = ?h \<cdot> ?g \<cdot> ?f
goal (1 subgoal):
1. (((f \<star> \<a>[g \<star> h, tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0, f\<^sub>0gh\<^sub>1.p\<^sub>0]) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> \<a>\<^sup>-\<^sup>1[g, h, tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> (g \<star> \<a>[h, tab\<^sub>0 h, g\<^sub>0h\<^sub>1.p\<^sub>0]) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> \<a>\<^sup>-\<^sup>1[f, (g \<star> (h \<star> tab\<^sub>0 h) \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine]) \<cdot> (f \<star> ((g \<star> h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (f \<star> ((g \<star> g\<^sub>0h\<^sub>1.\<phi>) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (f \<star> (\<a>[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (f \<star> ((g.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>1) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (f \<star> f\<^sub>0gh\<^sub>1.\<phi> \<star> TTfgh_TfTgh.chine) = \<a>\<^sup>-\<^sup>1[f, (g \<star> h) \<star> (tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine] \<cdot> (f \<star> \<a>[g \<star> h, tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0, f\<^sub>0gh\<^sub>1.p\<^sub>0] \<star> TTfgh_TfTgh.chine) \<cdot> (f \<star> (\<a>\<^sup>-\<^sup>1[g, h, tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (f \<star> ((g \<star> \<a>[h, tab\<^sub>0 h, g\<^sub>0h\<^sub>1.p\<^sub>0]) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (f \<star> ((g \<star> h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (f \<star> ((g \<star> g\<^sub>0h\<^sub>1.\<phi>) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (f \<star> (\<a>[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (f \<star> ((g.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>1) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (f \<star> f\<^sub>0gh\<^sub>1.\<phi> \<star> TTfgh_TfTgh.chine)
[PROOF STEP]
by presburger
[PROOF STATE]
proof (state)
this:
(((f \<star> \<a>[g \<star> h, tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0, f\<^sub>0gh\<^sub>1.p\<^sub>0]) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> \<a>\<^sup>-\<^sup>1[g, h, tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> (g \<star> \<a>[h, tab\<^sub>0 h, g\<^sub>0h\<^sub>1.p\<^sub>0]) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> \<a>\<^sup>-\<^sup>1[f, (g \<star> (h \<star> tab\<^sub>0 h) \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine]) \<cdot> (f \<star> ((g \<star> h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (f \<star> ((g \<star> g\<^sub>0h\<^sub>1.\<phi>) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (f \<star> (\<a>[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (f \<star> ((g.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>1) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (f \<star> f\<^sub>0gh\<^sub>1.\<phi> \<star> TTfgh_TfTgh.chine) = \<a>\<^sup>-\<^sup>1[f, (g \<star> h) \<star> (tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine] \<cdot> (f \<star> \<a>[g \<star> h, tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0, f\<^sub>0gh\<^sub>1.p\<^sub>0] \<star> TTfgh_TfTgh.chine) \<cdot> (f \<star> (\<a>\<^sup>-\<^sup>1[g, h, tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (f \<star> ((g \<star> \<a>[h, tab\<^sub>0 h, g\<^sub>0h\<^sub>1.p\<^sub>0]) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (f \<star> ((g \<star> h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (f \<star> ((g \<star> g\<^sub>0h\<^sub>1.\<phi>) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (f \<star> (\<a>[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (f \<star> ((g.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>1) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (f \<star> f\<^sub>0gh\<^sub>1.\<phi> \<star> TTfgh_TfTgh.chine)
goal:
No subgoals!
[PROOF STEP]
qed
[PROOF STATE]
proof (state)
this:
(((f \<star> \<a>[g \<star> h, tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0, f\<^sub>0gh\<^sub>1.p\<^sub>0]) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> \<a>\<^sup>-\<^sup>1[g, h, tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> (g \<star> \<a>[h, tab\<^sub>0 h, g\<^sub>0h\<^sub>1.p\<^sub>0]) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> \<a>\<^sup>-\<^sup>1[f, (g \<star> (h \<star> tab\<^sub>0 h) \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine]) \<cdot> (f \<star> ((g \<star> h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (f \<star> ((g \<star> g\<^sub>0h\<^sub>1.\<phi>) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (f \<star> (\<a>[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (f \<star> ((g.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>1) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (f \<star> f\<^sub>0gh\<^sub>1.\<phi> \<star> TTfgh_TfTgh.chine) = \<a>\<^sup>-\<^sup>1[f, (g \<star> h) \<star> (tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine] \<cdot> (f \<star> \<a>[g \<star> h, tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0, f\<^sub>0gh\<^sub>1.p\<^sub>0] \<star> TTfgh_TfTgh.chine) \<cdot> (f \<star> (\<a>\<^sup>-\<^sup>1[g, h, tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (f \<star> ((g \<star> \<a>[h, tab\<^sub>0 h, g\<^sub>0h\<^sub>1.p\<^sub>0]) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (f \<star> ((g \<star> h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (f \<star> ((g \<star> g\<^sub>0h\<^sub>1.\<phi>) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (f \<star> (\<a>[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (f \<star> ((g.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>1) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (f \<star> f\<^sub>0gh\<^sub>1.\<phi> \<star> TTfgh_TfTgh.chine)
goal (1 subgoal):
1. ((f \<star> \<a>[g \<star> h, tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0, f\<^sub>0gh\<^sub>1.p\<^sub>0]) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> \<a>\<^sup>-\<^sup>1[g, h, tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> (g \<star> \<a>[h, tab\<^sub>0 h, g\<^sub>0h\<^sub>1.p\<^sub>0]) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> (g \<star> h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> (g \<star> g\<^sub>0h\<^sub>1.\<phi>) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> \<a>[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> (g.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>1) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> f\<^sub>0gh\<^sub>1.\<phi>) \<star> TTfgh_TfTgh.chine) \<cdot> \<a>\<^sup>-\<^sup>1[f, tab\<^sub>0 f \<star> f\<^sub>0gh\<^sub>1.p\<^sub>1, TTfgh_TfTgh.chine] = \<a>\<^sup>-\<^sup>1[f, (g \<star> h) \<star> (tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine] \<cdot> (f \<star> \<a>[g \<star> h, tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0, f\<^sub>0gh\<^sub>1.p\<^sub>0] \<star> TTfgh_TfTgh.chine) \<cdot> (f \<star> (\<a>\<^sup>-\<^sup>1[g, h, tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (f \<star> ((g \<star> \<a>[h, tab\<^sub>0 h, g\<^sub>0h\<^sub>1.p\<^sub>0]) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (f \<star> ((g \<star> h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (f \<star> ((g \<star> g\<^sub>0h\<^sub>1.\<phi>) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (f \<star> (\<a>[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (f \<star> ((g.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>1) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (f \<star> f\<^sub>0gh\<^sub>1.\<phi> \<star> TTfgh_TfTgh.chine)
[PROOF STEP]
finally
[PROOF STATE]
proof (chain)
picking this:
((f \<star> \<a>[g \<star> h, tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0, f\<^sub>0gh\<^sub>1.p\<^sub>0]) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> \<a>\<^sup>-\<^sup>1[g, h, tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> (g \<star> \<a>[h, tab\<^sub>0 h, g\<^sub>0h\<^sub>1.p\<^sub>0]) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> (g \<star> h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> (g \<star> g\<^sub>0h\<^sub>1.\<phi>) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> \<a>[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> (g.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>1) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> f\<^sub>0gh\<^sub>1.\<phi>) \<star> TTfgh_TfTgh.chine) \<cdot> \<a>\<^sup>-\<^sup>1[f, tab\<^sub>0 f \<star> f\<^sub>0gh\<^sub>1.p\<^sub>1, TTfgh_TfTgh.chine] = \<a>\<^sup>-\<^sup>1[f, (g \<star> h) \<star> (tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine] \<cdot> (f \<star> \<a>[g \<star> h, tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0, f\<^sub>0gh\<^sub>1.p\<^sub>0] \<star> TTfgh_TfTgh.chine) \<cdot> (f \<star> (\<a>\<^sup>-\<^sup>1[g, h, tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (f \<star> ((g \<star> \<a>[h, tab\<^sub>0 h, g\<^sub>0h\<^sub>1.p\<^sub>0]) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (f \<star> ((g \<star> h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (f \<star> ((g \<star> g\<^sub>0h\<^sub>1.\<phi>) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (f \<star> (\<a>[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (f \<star> ((g.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>1) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (f \<star> f\<^sub>0gh\<^sub>1.\<phi> \<star> TTfgh_TfTgh.chine)
[PROOF STEP]
show ?thesis
[PROOF STATE]
proof (prove)
using this:
((f \<star> \<a>[g \<star> h, tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0, f\<^sub>0gh\<^sub>1.p\<^sub>0]) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> \<a>\<^sup>-\<^sup>1[g, h, tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> (g \<star> \<a>[h, tab\<^sub>0 h, g\<^sub>0h\<^sub>1.p\<^sub>0]) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> (g \<star> h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> (g \<star> g\<^sub>0h\<^sub>1.\<phi>) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> \<a>[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> (g.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>1) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> f\<^sub>0gh\<^sub>1.\<phi>) \<star> TTfgh_TfTgh.chine) \<cdot> \<a>\<^sup>-\<^sup>1[f, tab\<^sub>0 f \<star> f\<^sub>0gh\<^sub>1.p\<^sub>1, TTfgh_TfTgh.chine] = \<a>\<^sup>-\<^sup>1[f, (g \<star> h) \<star> (tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine] \<cdot> (f \<star> \<a>[g \<star> h, tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0, f\<^sub>0gh\<^sub>1.p\<^sub>0] \<star> TTfgh_TfTgh.chine) \<cdot> (f \<star> (\<a>\<^sup>-\<^sup>1[g, h, tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (f \<star> ((g \<star> \<a>[h, tab\<^sub>0 h, g\<^sub>0h\<^sub>1.p\<^sub>0]) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (f \<star> ((g \<star> h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (f \<star> ((g \<star> g\<^sub>0h\<^sub>1.\<phi>) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (f \<star> (\<a>[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (f \<star> ((g.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>1) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (f \<star> f\<^sub>0gh\<^sub>1.\<phi> \<star> TTfgh_TfTgh.chine)
goal (1 subgoal):
1. ((f \<star> \<a>[g \<star> h, tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0, f\<^sub>0gh\<^sub>1.p\<^sub>0]) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> \<a>\<^sup>-\<^sup>1[g, h, tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> (g \<star> \<a>[h, tab\<^sub>0 h, g\<^sub>0h\<^sub>1.p\<^sub>0]) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> (g \<star> h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> (g \<star> g\<^sub>0h\<^sub>1.\<phi>) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> \<a>[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> (g.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>1) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> f\<^sub>0gh\<^sub>1.\<phi>) \<star> TTfgh_TfTgh.chine) \<cdot> \<a>\<^sup>-\<^sup>1[f, tab\<^sub>0 f \<star> f\<^sub>0gh\<^sub>1.p\<^sub>1, TTfgh_TfTgh.chine] = \<a>\<^sup>-\<^sup>1[f, (g \<star> h) \<star> (tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine] \<cdot> (f \<star> \<a>[g \<star> h, tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0, f\<^sub>0gh\<^sub>1.p\<^sub>0] \<star> TTfgh_TfTgh.chine) \<cdot> (f \<star> (\<a>\<^sup>-\<^sup>1[g, h, tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (f \<star> ((g \<star> \<a>[h, tab\<^sub>0 h, g\<^sub>0h\<^sub>1.p\<^sub>0]) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (f \<star> ((g \<star> h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (f \<star> ((g \<star> g\<^sub>0h\<^sub>1.\<phi>) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (f \<star> (\<a>[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (f \<star> ((g.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>1) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (f \<star> f\<^sub>0gh\<^sub>1.\<phi> \<star> TTfgh_TfTgh.chine)
[PROOF STEP]
using comp_assoc
[PROOF STATE]
proof (prove)
using this:
((f \<star> \<a>[g \<star> h, tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0, f\<^sub>0gh\<^sub>1.p\<^sub>0]) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> \<a>\<^sup>-\<^sup>1[g, h, tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> (g \<star> \<a>[h, tab\<^sub>0 h, g\<^sub>0h\<^sub>1.p\<^sub>0]) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> (g \<star> h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> (g \<star> g\<^sub>0h\<^sub>1.\<phi>) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> \<a>[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> (g.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>1) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> f\<^sub>0gh\<^sub>1.\<phi>) \<star> TTfgh_TfTgh.chine) \<cdot> \<a>\<^sup>-\<^sup>1[f, tab\<^sub>0 f \<star> f\<^sub>0gh\<^sub>1.p\<^sub>1, TTfgh_TfTgh.chine] = \<a>\<^sup>-\<^sup>1[f, (g \<star> h) \<star> (tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine] \<cdot> (f \<star> \<a>[g \<star> h, tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0, f\<^sub>0gh\<^sub>1.p\<^sub>0] \<star> TTfgh_TfTgh.chine) \<cdot> (f \<star> (\<a>\<^sup>-\<^sup>1[g, h, tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (f \<star> ((g \<star> \<a>[h, tab\<^sub>0 h, g\<^sub>0h\<^sub>1.p\<^sub>0]) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (f \<star> ((g \<star> h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (f \<star> ((g \<star> g\<^sub>0h\<^sub>1.\<phi>) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (f \<star> (\<a>[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (f \<star> ((g.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>1) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (f \<star> f\<^sub>0gh\<^sub>1.\<phi> \<star> TTfgh_TfTgh.chine)
(?h \<cdot> ?g) \<cdot> ?f = ?h \<cdot> ?g \<cdot> ?f
goal (1 subgoal):
1. ((f \<star> \<a>[g \<star> h, tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0, f\<^sub>0gh\<^sub>1.p\<^sub>0]) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> \<a>\<^sup>-\<^sup>1[g, h, tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> (g \<star> \<a>[h, tab\<^sub>0 h, g\<^sub>0h\<^sub>1.p\<^sub>0]) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> (g \<star> h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> (g \<star> g\<^sub>0h\<^sub>1.\<phi>) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> \<a>[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> (g.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>1) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> f\<^sub>0gh\<^sub>1.\<phi>) \<star> TTfgh_TfTgh.chine) \<cdot> \<a>\<^sup>-\<^sup>1[f, tab\<^sub>0 f \<star> f\<^sub>0gh\<^sub>1.p\<^sub>1, TTfgh_TfTgh.chine] = \<a>\<^sup>-\<^sup>1[f, (g \<star> h) \<star> (tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine] \<cdot> (f \<star> \<a>[g \<star> h, tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0, f\<^sub>0gh\<^sub>1.p\<^sub>0] \<star> TTfgh_TfTgh.chine) \<cdot> (f \<star> (\<a>\<^sup>-\<^sup>1[g, h, tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (f \<star> ((g \<star> \<a>[h, tab\<^sub>0 h, g\<^sub>0h\<^sub>1.p\<^sub>0]) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (f \<star> ((g \<star> h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (f \<star> ((g \<star> g\<^sub>0h\<^sub>1.\<phi>) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (f \<star> (\<a>[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (f \<star> ((g.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>1) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (f \<star> f\<^sub>0gh\<^sub>1.\<phi> \<star> TTfgh_TfTgh.chine)
[PROOF STEP]
by presburger
[PROOF STATE]
proof (state)
this:
((f \<star> \<a>[g \<star> h, tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0, f\<^sub>0gh\<^sub>1.p\<^sub>0]) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> \<a>\<^sup>-\<^sup>1[g, h, tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> (g \<star> \<a>[h, tab\<^sub>0 h, g\<^sub>0h\<^sub>1.p\<^sub>0]) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> (g \<star> h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> (g \<star> g\<^sub>0h\<^sub>1.\<phi>) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> \<a>[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> (g.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>1) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> f\<^sub>0gh\<^sub>1.\<phi>) \<star> TTfgh_TfTgh.chine) \<cdot> \<a>\<^sup>-\<^sup>1[f, tab\<^sub>0 f \<star> f\<^sub>0gh\<^sub>1.p\<^sub>1, TTfgh_TfTgh.chine] = \<a>\<^sup>-\<^sup>1[f, (g \<star> h) \<star> (tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine] \<cdot> (f \<star> \<a>[g \<star> h, tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0, f\<^sub>0gh\<^sub>1.p\<^sub>0] \<star> TTfgh_TfTgh.chine) \<cdot> (f \<star> (\<a>\<^sup>-\<^sup>1[g, h, tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (f \<star> ((g \<star> \<a>[h, tab\<^sub>0 h, g\<^sub>0h\<^sub>1.p\<^sub>0]) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (f \<star> ((g \<star> h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (f \<star> ((g \<star> g\<^sub>0h\<^sub>1.\<phi>) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (f \<star> (\<a>[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (f \<star> ((g.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>1) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (f \<star> f\<^sub>0gh\<^sub>1.\<phi> \<star> TTfgh_TfTgh.chine)
goal:
No subgoals!
[PROOF STEP]
qed
[PROOF STATE]
proof (state)
this:
((f \<star> \<a>[g \<star> h, tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0, f\<^sub>0gh\<^sub>1.p\<^sub>0]) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> \<a>\<^sup>-\<^sup>1[g, h, tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> (g \<star> \<a>[h, tab\<^sub>0 h, g\<^sub>0h\<^sub>1.p\<^sub>0]) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> (g \<star> h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> (g \<star> g\<^sub>0h\<^sub>1.\<phi>) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> \<a>[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> (g.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>1) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> f\<^sub>0gh\<^sub>1.\<phi>) \<star> TTfgh_TfTgh.chine) \<cdot> \<a>\<^sup>-\<^sup>1[f, tab\<^sub>0 f \<star> f\<^sub>0gh\<^sub>1.p\<^sub>1, TTfgh_TfTgh.chine] = \<a>\<^sup>-\<^sup>1[f, (g \<star> h) \<star> (tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine] \<cdot> (f \<star> \<a>[g \<star> h, tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0, f\<^sub>0gh\<^sub>1.p\<^sub>0] \<star> TTfgh_TfTgh.chine) \<cdot> (f \<star> (\<a>\<^sup>-\<^sup>1[g, h, tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (f \<star> ((g \<star> \<a>[h, tab\<^sub>0 h, g\<^sub>0h\<^sub>1.p\<^sub>0]) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (f \<star> ((g \<star> h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (f \<star> ((g \<star> g\<^sub>0h\<^sub>1.\<phi>) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (f \<star> (\<a>[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (f \<star> ((g.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>1) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (f \<star> f\<^sub>0gh\<^sub>1.\<phi> \<star> TTfgh_TfTgh.chine)
goal (1 subgoal):
1. \<a>[f, g, h \<star> tab\<^sub>0 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0] \<cdot> \<a>[f \<star> g, h, tab\<^sub>0 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0] \<cdot> (\<a>\<^sup>-\<^sup>1[f, g, h] \<star> tab\<^sub>0 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0) \<cdot> ((f \<star> g \<star> h) \<star> TTfgh_TfTgh.the_\<theta>) \<cdot> \<a>[f \<star> g \<star> h, (tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine] \<cdot> (\<a>\<^sup>-\<^sup>1[f, g \<star> h, (tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0] \<star> TTfgh_TfTgh.chine) \<cdot> (((f \<star> \<a>[g \<star> h, tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0, f\<^sub>0gh\<^sub>1.p\<^sub>0]) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> \<a>\<^sup>-\<^sup>1[g, h, tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> (g \<star> \<a>[h, tab\<^sub>0 h, g\<^sub>0h\<^sub>1.p\<^sub>0]) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> (g \<star> h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> (g \<star> g\<^sub>0h\<^sub>1.\<phi>) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> \<a>[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> (g.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>1) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> f\<^sub>0gh\<^sub>1.\<phi>) \<star> TTfgh_TfTgh.chine) \<cdot> \<a>\<^sup>-\<^sup>1[f, tab\<^sub>0 f \<star> f\<^sub>0gh\<^sub>1.p\<^sub>1, TTfgh_TfTgh.chine]) \<cdot> (f \<star> \<a>\<^sup>-\<^sup>1[tab\<^sub>0 f, f\<^sub>0gh\<^sub>1.p\<^sub>1, TTfgh_TfTgh.chine]) \<cdot> \<a>[f, tab\<^sub>0 f, f\<^sub>0gh\<^sub>1.p\<^sub>1 \<star> TTfgh_TfTgh.chine] \<cdot> (f.tab \<star> f\<^sub>0gh\<^sub>1.p\<^sub>1 \<star> TTfgh_TfTgh.chine) \<cdot> \<a>[tab\<^sub>1 f, f\<^sub>0gh\<^sub>1.p\<^sub>1, TTfgh_TfTgh.chine] \<cdot> TTfgh_TfTgh.the_\<nu> \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>1 f, f\<^sub>0g\<^sub>1.p\<^sub>1, fg\<^sub>0h\<^sub>1.p\<^sub>1] = \<a>[f, g, h \<star> tab\<^sub>0 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0] \<cdot> \<a>[f \<star> g, h, tab\<^sub>0 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0] \<cdot> (\<a>\<^sup>-\<^sup>1[f, g, h] \<star> tab\<^sub>0 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0) \<cdot> ((f \<star> g \<star> h) \<star> TTfgh_TfTgh.the_\<theta>) \<cdot> \<a>[f \<star> g \<star> h, (tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine] \<cdot> (\<a>\<^sup>-\<^sup>1[f, g \<star> h, (tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0] \<star> TTfgh_TfTgh.chine) \<cdot> \<a>\<^sup>-\<^sup>1[f, (g \<star> h) \<star> (tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine] \<cdot> (f \<star> \<a>[g \<star> h, tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0, f\<^sub>0gh\<^sub>1.p\<^sub>0] \<star> TTfgh_TfTgh.chine) \<cdot> (f \<star> (\<a>\<^sup>-\<^sup>1[g, h, tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (f \<star> ((g \<star> \<a>[h, tab\<^sub>0 h, g\<^sub>0h\<^sub>1.p\<^sub>0]) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> ((g \<star> h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (f \<star> ((g \<star> g\<^sub>0h\<^sub>1.\<phi>) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (f \<star> (\<a>[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (f \<star> ((g.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>1) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (f \<star> f\<^sub>0gh\<^sub>1.\<phi> \<star> TTfgh_TfTgh.chine) \<cdot> (f \<star> \<a>\<^sup>-\<^sup>1[tab\<^sub>0 f, f\<^sub>0gh\<^sub>1.p\<^sub>1, TTfgh_TfTgh.chine])) \<cdot> \<a>[f, tab\<^sub>0 f, f\<^sub>0gh\<^sub>1.p\<^sub>1 \<star> TTfgh_TfTgh.chine] \<cdot> (f.tab \<star> f\<^sub>0gh\<^sub>1.p\<^sub>1 \<star> TTfgh_TfTgh.chine) \<cdot> \<a>[tab\<^sub>1 f, f\<^sub>0gh\<^sub>1.p\<^sub>1, TTfgh_TfTgh.chine] \<cdot> TTfgh_TfTgh.the_\<nu> \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>1 f, f\<^sub>0g\<^sub>1.p\<^sub>1, fg\<^sub>0h\<^sub>1.p\<^sub>1]
[PROOF STEP]
thus ?thesis
[PROOF STATE]
proof (prove)
using this:
((f \<star> \<a>[g \<star> h, tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0, f\<^sub>0gh\<^sub>1.p\<^sub>0]) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> \<a>\<^sup>-\<^sup>1[g, h, tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> (g \<star> \<a>[h, tab\<^sub>0 h, g\<^sub>0h\<^sub>1.p\<^sub>0]) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> (g \<star> h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> (g \<star> g\<^sub>0h\<^sub>1.\<phi>) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> \<a>[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> (g.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>1) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> f\<^sub>0gh\<^sub>1.\<phi>) \<star> TTfgh_TfTgh.chine) \<cdot> \<a>\<^sup>-\<^sup>1[f, tab\<^sub>0 f \<star> f\<^sub>0gh\<^sub>1.p\<^sub>1, TTfgh_TfTgh.chine] = \<a>\<^sup>-\<^sup>1[f, (g \<star> h) \<star> (tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine] \<cdot> (f \<star> \<a>[g \<star> h, tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0, f\<^sub>0gh\<^sub>1.p\<^sub>0] \<star> TTfgh_TfTgh.chine) \<cdot> (f \<star> (\<a>\<^sup>-\<^sup>1[g, h, tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (f \<star> ((g \<star> \<a>[h, tab\<^sub>0 h, g\<^sub>0h\<^sub>1.p\<^sub>0]) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (f \<star> ((g \<star> h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (f \<star> ((g \<star> g\<^sub>0h\<^sub>1.\<phi>) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (f \<star> (\<a>[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (f \<star> ((g.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>1) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (f \<star> f\<^sub>0gh\<^sub>1.\<phi> \<star> TTfgh_TfTgh.chine)
goal (1 subgoal):
1. \<a>[f, g, h \<star> tab\<^sub>0 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0] \<cdot> \<a>[f \<star> g, h, tab\<^sub>0 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0] \<cdot> (\<a>\<^sup>-\<^sup>1[f, g, h] \<star> tab\<^sub>0 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0) \<cdot> ((f \<star> g \<star> h) \<star> TTfgh_TfTgh.the_\<theta>) \<cdot> \<a>[f \<star> g \<star> h, (tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine] \<cdot> (\<a>\<^sup>-\<^sup>1[f, g \<star> h, (tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0] \<star> TTfgh_TfTgh.chine) \<cdot> (((f \<star> \<a>[g \<star> h, tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0, f\<^sub>0gh\<^sub>1.p\<^sub>0]) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> \<a>\<^sup>-\<^sup>1[g, h, tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> (g \<star> \<a>[h, tab\<^sub>0 h, g\<^sub>0h\<^sub>1.p\<^sub>0]) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> (g \<star> h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> (g \<star> g\<^sub>0h\<^sub>1.\<phi>) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> \<a>[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> (g.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>1) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> f\<^sub>0gh\<^sub>1.\<phi>) \<star> TTfgh_TfTgh.chine) \<cdot> \<a>\<^sup>-\<^sup>1[f, tab\<^sub>0 f \<star> f\<^sub>0gh\<^sub>1.p\<^sub>1, TTfgh_TfTgh.chine]) \<cdot> (f \<star> \<a>\<^sup>-\<^sup>1[tab\<^sub>0 f, f\<^sub>0gh\<^sub>1.p\<^sub>1, TTfgh_TfTgh.chine]) \<cdot> \<a>[f, tab\<^sub>0 f, f\<^sub>0gh\<^sub>1.p\<^sub>1 \<star> TTfgh_TfTgh.chine] \<cdot> (f.tab \<star> f\<^sub>0gh\<^sub>1.p\<^sub>1 \<star> TTfgh_TfTgh.chine) \<cdot> \<a>[tab\<^sub>1 f, f\<^sub>0gh\<^sub>1.p\<^sub>1, TTfgh_TfTgh.chine] \<cdot> TTfgh_TfTgh.the_\<nu> \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>1 f, f\<^sub>0g\<^sub>1.p\<^sub>1, fg\<^sub>0h\<^sub>1.p\<^sub>1] = \<a>[f, g, h \<star> tab\<^sub>0 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0] \<cdot> \<a>[f \<star> g, h, tab\<^sub>0 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0] \<cdot> (\<a>\<^sup>-\<^sup>1[f, g, h] \<star> tab\<^sub>0 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0) \<cdot> ((f \<star> g \<star> h) \<star> TTfgh_TfTgh.the_\<theta>) \<cdot> \<a>[f \<star> g \<star> h, (tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine] \<cdot> (\<a>\<^sup>-\<^sup>1[f, g \<star> h, (tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0] \<star> TTfgh_TfTgh.chine) \<cdot> \<a>\<^sup>-\<^sup>1[f, (g \<star> h) \<star> (tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine] \<cdot> (f \<star> \<a>[g \<star> h, tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0, f\<^sub>0gh\<^sub>1.p\<^sub>0] \<star> TTfgh_TfTgh.chine) \<cdot> (f \<star> (\<a>\<^sup>-\<^sup>1[g, h, tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (f \<star> ((g \<star> \<a>[h, tab\<^sub>0 h, g\<^sub>0h\<^sub>1.p\<^sub>0]) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> ((g \<star> h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (f \<star> ((g \<star> g\<^sub>0h\<^sub>1.\<phi>) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (f \<star> (\<a>[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (f \<star> ((g.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>1) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (f \<star> f\<^sub>0gh\<^sub>1.\<phi> \<star> TTfgh_TfTgh.chine) \<cdot> (f \<star> \<a>\<^sup>-\<^sup>1[tab\<^sub>0 f, f\<^sub>0gh\<^sub>1.p\<^sub>1, TTfgh_TfTgh.chine])) \<cdot> \<a>[f, tab\<^sub>0 f, f\<^sub>0gh\<^sub>1.p\<^sub>1 \<star> TTfgh_TfTgh.chine] \<cdot> (f.tab \<star> f\<^sub>0gh\<^sub>1.p\<^sub>1 \<star> TTfgh_TfTgh.chine) \<cdot> \<a>[tab\<^sub>1 f, f\<^sub>0gh\<^sub>1.p\<^sub>1, TTfgh_TfTgh.chine] \<cdot> TTfgh_TfTgh.the_\<nu> \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>1 f, f\<^sub>0g\<^sub>1.p\<^sub>1, fg\<^sub>0h\<^sub>1.p\<^sub>1]
[PROOF STEP]
using comp_assoc
[PROOF STATE]
proof (prove)
using this:
((f \<star> \<a>[g \<star> h, tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0, f\<^sub>0gh\<^sub>1.p\<^sub>0]) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> \<a>\<^sup>-\<^sup>1[g, h, tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> (g \<star> \<a>[h, tab\<^sub>0 h, g\<^sub>0h\<^sub>1.p\<^sub>0]) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> (g \<star> h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> (g \<star> g\<^sub>0h\<^sub>1.\<phi>) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> \<a>[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> (g.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>1) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> f\<^sub>0gh\<^sub>1.\<phi>) \<star> TTfgh_TfTgh.chine) \<cdot> \<a>\<^sup>-\<^sup>1[f, tab\<^sub>0 f \<star> f\<^sub>0gh\<^sub>1.p\<^sub>1, TTfgh_TfTgh.chine] = \<a>\<^sup>-\<^sup>1[f, (g \<star> h) \<star> (tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine] \<cdot> (f \<star> \<a>[g \<star> h, tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0, f\<^sub>0gh\<^sub>1.p\<^sub>0] \<star> TTfgh_TfTgh.chine) \<cdot> (f \<star> (\<a>\<^sup>-\<^sup>1[g, h, tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (f \<star> ((g \<star> \<a>[h, tab\<^sub>0 h, g\<^sub>0h\<^sub>1.p\<^sub>0]) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (f \<star> ((g \<star> h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (f \<star> ((g \<star> g\<^sub>0h\<^sub>1.\<phi>) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (f \<star> (\<a>[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (f \<star> ((g.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>1) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (f \<star> f\<^sub>0gh\<^sub>1.\<phi> \<star> TTfgh_TfTgh.chine)
(?h \<cdot> ?g) \<cdot> ?f = ?h \<cdot> ?g \<cdot> ?f
goal (1 subgoal):
1. \<a>[f, g, h \<star> tab\<^sub>0 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0] \<cdot> \<a>[f \<star> g, h, tab\<^sub>0 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0] \<cdot> (\<a>\<^sup>-\<^sup>1[f, g, h] \<star> tab\<^sub>0 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0) \<cdot> ((f \<star> g \<star> h) \<star> TTfgh_TfTgh.the_\<theta>) \<cdot> \<a>[f \<star> g \<star> h, (tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine] \<cdot> (\<a>\<^sup>-\<^sup>1[f, g \<star> h, (tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0] \<star> TTfgh_TfTgh.chine) \<cdot> (((f \<star> \<a>[g \<star> h, tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0, f\<^sub>0gh\<^sub>1.p\<^sub>0]) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> \<a>\<^sup>-\<^sup>1[g, h, tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> (g \<star> \<a>[h, tab\<^sub>0 h, g\<^sub>0h\<^sub>1.p\<^sub>0]) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> (g \<star> h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> (g \<star> g\<^sub>0h\<^sub>1.\<phi>) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> \<a>[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> (g.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>1) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> f\<^sub>0gh\<^sub>1.\<phi>) \<star> TTfgh_TfTgh.chine) \<cdot> \<a>\<^sup>-\<^sup>1[f, tab\<^sub>0 f \<star> f\<^sub>0gh\<^sub>1.p\<^sub>1, TTfgh_TfTgh.chine]) \<cdot> (f \<star> \<a>\<^sup>-\<^sup>1[tab\<^sub>0 f, f\<^sub>0gh\<^sub>1.p\<^sub>1, TTfgh_TfTgh.chine]) \<cdot> \<a>[f, tab\<^sub>0 f, f\<^sub>0gh\<^sub>1.p\<^sub>1 \<star> TTfgh_TfTgh.chine] \<cdot> (f.tab \<star> f\<^sub>0gh\<^sub>1.p\<^sub>1 \<star> TTfgh_TfTgh.chine) \<cdot> \<a>[tab\<^sub>1 f, f\<^sub>0gh\<^sub>1.p\<^sub>1, TTfgh_TfTgh.chine] \<cdot> TTfgh_TfTgh.the_\<nu> \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>1 f, f\<^sub>0g\<^sub>1.p\<^sub>1, fg\<^sub>0h\<^sub>1.p\<^sub>1] = \<a>[f, g, h \<star> tab\<^sub>0 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0] \<cdot> \<a>[f \<star> g, h, tab\<^sub>0 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0] \<cdot> (\<a>\<^sup>-\<^sup>1[f, g, h] \<star> tab\<^sub>0 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0) \<cdot> ((f \<star> g \<star> h) \<star> TTfgh_TfTgh.the_\<theta>) \<cdot> \<a>[f \<star> g \<star> h, (tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine] \<cdot> (\<a>\<^sup>-\<^sup>1[f, g \<star> h, (tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0] \<star> TTfgh_TfTgh.chine) \<cdot> \<a>\<^sup>-\<^sup>1[f, (g \<star> h) \<star> (tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine] \<cdot> (f \<star> \<a>[g \<star> h, tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0, f\<^sub>0gh\<^sub>1.p\<^sub>0] \<star> TTfgh_TfTgh.chine) \<cdot> (f \<star> (\<a>\<^sup>-\<^sup>1[g, h, tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (f \<star> ((g \<star> \<a>[h, tab\<^sub>0 h, g\<^sub>0h\<^sub>1.p\<^sub>0]) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> ((g \<star> h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (f \<star> ((g \<star> g\<^sub>0h\<^sub>1.\<phi>) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (f \<star> (\<a>[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (f \<star> ((g.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>1) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (f \<star> f\<^sub>0gh\<^sub>1.\<phi> \<star> TTfgh_TfTgh.chine) \<cdot> (f \<star> \<a>\<^sup>-\<^sup>1[tab\<^sub>0 f, f\<^sub>0gh\<^sub>1.p\<^sub>1, TTfgh_TfTgh.chine])) \<cdot> \<a>[f, tab\<^sub>0 f, f\<^sub>0gh\<^sub>1.p\<^sub>1 \<star> TTfgh_TfTgh.chine] \<cdot> (f.tab \<star> f\<^sub>0gh\<^sub>1.p\<^sub>1 \<star> TTfgh_TfTgh.chine) \<cdot> \<a>[tab\<^sub>1 f, f\<^sub>0gh\<^sub>1.p\<^sub>1, TTfgh_TfTgh.chine] \<cdot> TTfgh_TfTgh.the_\<nu> \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>1 f, f\<^sub>0g\<^sub>1.p\<^sub>1, fg\<^sub>0h\<^sub>1.p\<^sub>1]
[PROOF STEP]
by presburger
[PROOF STATE]
proof (state)
this:
\<a>[f, g, h \<star> tab\<^sub>0 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0] \<cdot> \<a>[f \<star> g, h, tab\<^sub>0 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0] \<cdot> (\<a>\<^sup>-\<^sup>1[f, g, h] \<star> tab\<^sub>0 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0) \<cdot> ((f \<star> g \<star> h) \<star> TTfgh_TfTgh.the_\<theta>) \<cdot> \<a>[f \<star> g \<star> h, (tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine] \<cdot> (\<a>\<^sup>-\<^sup>1[f, g \<star> h, (tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0] \<star> TTfgh_TfTgh.chine) \<cdot> (((f \<star> \<a>[g \<star> h, tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0, f\<^sub>0gh\<^sub>1.p\<^sub>0]) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> \<a>\<^sup>-\<^sup>1[g, h, tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> (g \<star> \<a>[h, tab\<^sub>0 h, g\<^sub>0h\<^sub>1.p\<^sub>0]) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> (g \<star> h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> (g \<star> g\<^sub>0h\<^sub>1.\<phi>) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> \<a>[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> (g.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>1) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> f\<^sub>0gh\<^sub>1.\<phi>) \<star> TTfgh_TfTgh.chine) \<cdot> \<a>\<^sup>-\<^sup>1[f, tab\<^sub>0 f \<star> f\<^sub>0gh\<^sub>1.p\<^sub>1, TTfgh_TfTgh.chine]) \<cdot> (f \<star> \<a>\<^sup>-\<^sup>1[tab\<^sub>0 f, f\<^sub>0gh\<^sub>1.p\<^sub>1, TTfgh_TfTgh.chine]) \<cdot> \<a>[f, tab\<^sub>0 f, f\<^sub>0gh\<^sub>1.p\<^sub>1 \<star> TTfgh_TfTgh.chine] \<cdot> (f.tab \<star> f\<^sub>0gh\<^sub>1.p\<^sub>1 \<star> TTfgh_TfTgh.chine) \<cdot> \<a>[tab\<^sub>1 f, f\<^sub>0gh\<^sub>1.p\<^sub>1, TTfgh_TfTgh.chine] \<cdot> TTfgh_TfTgh.the_\<nu> \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>1 f, f\<^sub>0g\<^sub>1.p\<^sub>1, fg\<^sub>0h\<^sub>1.p\<^sub>1] = \<a>[f, g, h \<star> tab\<^sub>0 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0] \<cdot> \<a>[f \<star> g, h, tab\<^sub>0 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0] \<cdot> (\<a>\<^sup>-\<^sup>1[f, g, h] \<star> tab\<^sub>0 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0) \<cdot> ((f \<star> g \<star> h) \<star> TTfgh_TfTgh.the_\<theta>) \<cdot> \<a>[f \<star> g \<star> h, (tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine] \<cdot> (\<a>\<^sup>-\<^sup>1[f, g \<star> h, (tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0] \<star> TTfgh_TfTgh.chine) \<cdot> \<a>\<^sup>-\<^sup>1[f, (g \<star> h) \<star> (tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine] \<cdot> (f \<star> \<a>[g \<star> h, tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0, f\<^sub>0gh\<^sub>1.p\<^sub>0] \<star> TTfgh_TfTgh.chine) \<cdot> (f \<star> (\<a>\<^sup>-\<^sup>1[g, h, tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (f \<star> ((g \<star> \<a>[h, tab\<^sub>0 h, g\<^sub>0h\<^sub>1.p\<^sub>0]) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> ((g \<star> h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (f \<star> ((g \<star> g\<^sub>0h\<^sub>1.\<phi>) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (f \<star> (\<a>[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (f \<star> ((g.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>1) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (f \<star> f\<^sub>0gh\<^sub>1.\<phi> \<star> TTfgh_TfTgh.chine) \<cdot> (f \<star> \<a>\<^sup>-\<^sup>1[tab\<^sub>0 f, f\<^sub>0gh\<^sub>1.p\<^sub>1, TTfgh_TfTgh.chine])) \<cdot> \<a>[f, tab\<^sub>0 f, f\<^sub>0gh\<^sub>1.p\<^sub>1 \<star> TTfgh_TfTgh.chine] \<cdot> (f.tab \<star> f\<^sub>0gh\<^sub>1.p\<^sub>1 \<star> TTfgh_TfTgh.chine) \<cdot> \<a>[tab\<^sub>1 f, f\<^sub>0gh\<^sub>1.p\<^sub>1, TTfgh_TfTgh.chine] \<cdot> TTfgh_TfTgh.the_\<nu> \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>1 f, f\<^sub>0g\<^sub>1.p\<^sub>1, fg\<^sub>0h\<^sub>1.p\<^sub>1]
goal:
No subgoals!
[PROOF STEP]
qed
[PROOF STATE]
proof (state)
this:
\<a>[f, g, h \<star> tab\<^sub>0 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0] \<cdot> \<a>[f \<star> g, h, tab\<^sub>0 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0] \<cdot> (\<a>\<^sup>-\<^sup>1[f, g, h] \<star> tab\<^sub>0 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0) \<cdot> ((f \<star> g \<star> h) \<star> TTfgh_TfTgh.the_\<theta>) \<cdot> \<a>[f \<star> g \<star> h, (tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine] \<cdot> (\<a>\<^sup>-\<^sup>1[f, g \<star> h, (tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0] \<star> TTfgh_TfTgh.chine) \<cdot> (((f \<star> \<a>[g \<star> h, tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0, f\<^sub>0gh\<^sub>1.p\<^sub>0]) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> \<a>\<^sup>-\<^sup>1[g, h, tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> (g \<star> \<a>[h, tab\<^sub>0 h, g\<^sub>0h\<^sub>1.p\<^sub>0]) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> (g \<star> h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> (g \<star> g\<^sub>0h\<^sub>1.\<phi>) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> \<a>[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> (g.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>1) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> f\<^sub>0gh\<^sub>1.\<phi>) \<star> TTfgh_TfTgh.chine) \<cdot> \<a>\<^sup>-\<^sup>1[f, tab\<^sub>0 f \<star> f\<^sub>0gh\<^sub>1.p\<^sub>1, TTfgh_TfTgh.chine]) \<cdot> (f \<star> \<a>\<^sup>-\<^sup>1[tab\<^sub>0 f, f\<^sub>0gh\<^sub>1.p\<^sub>1, TTfgh_TfTgh.chine]) \<cdot> \<a>[f, tab\<^sub>0 f, f\<^sub>0gh\<^sub>1.p\<^sub>1 \<star> TTfgh_TfTgh.chine] \<cdot> (f.tab \<star> f\<^sub>0gh\<^sub>1.p\<^sub>1 \<star> TTfgh_TfTgh.chine) \<cdot> \<a>[tab\<^sub>1 f, f\<^sub>0gh\<^sub>1.p\<^sub>1, TTfgh_TfTgh.chine] \<cdot> TTfgh_TfTgh.the_\<nu> \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>1 f, f\<^sub>0g\<^sub>1.p\<^sub>1, fg\<^sub>0h\<^sub>1.p\<^sub>1] = \<a>[f, g, h \<star> tab\<^sub>0 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0] \<cdot> \<a>[f \<star> g, h, tab\<^sub>0 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0] \<cdot> (\<a>\<^sup>-\<^sup>1[f, g, h] \<star> tab\<^sub>0 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0) \<cdot> ((f \<star> g \<star> h) \<star> TTfgh_TfTgh.the_\<theta>) \<cdot> \<a>[f \<star> g \<star> h, (tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine] \<cdot> (\<a>\<^sup>-\<^sup>1[f, g \<star> h, (tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0] \<star> TTfgh_TfTgh.chine) \<cdot> \<a>\<^sup>-\<^sup>1[f, (g \<star> h) \<star> (tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine] \<cdot> (f \<star> \<a>[g \<star> h, tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0, f\<^sub>0gh\<^sub>1.p\<^sub>0] \<star> TTfgh_TfTgh.chine) \<cdot> (f \<star> (\<a>\<^sup>-\<^sup>1[g, h, tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (f \<star> ((g \<star> \<a>[h, tab\<^sub>0 h, g\<^sub>0h\<^sub>1.p\<^sub>0]) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> ((g \<star> h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (f \<star> ((g \<star> g\<^sub>0h\<^sub>1.\<phi>) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (f \<star> (\<a>[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (f \<star> ((g.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>1) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (f \<star> f\<^sub>0gh\<^sub>1.\<phi> \<star> TTfgh_TfTgh.chine) \<cdot> (f \<star> \<a>\<^sup>-\<^sup>1[tab\<^sub>0 f, f\<^sub>0gh\<^sub>1.p\<^sub>1, TTfgh_TfTgh.chine])) \<cdot> \<a>[f, tab\<^sub>0 f, f\<^sub>0gh\<^sub>1.p\<^sub>1 \<star> TTfgh_TfTgh.chine] \<cdot> (f.tab \<star> f\<^sub>0gh\<^sub>1.p\<^sub>1 \<star> TTfgh_TfTgh.chine) \<cdot> \<a>[tab\<^sub>1 f, f\<^sub>0gh\<^sub>1.p\<^sub>1, TTfgh_TfTgh.chine] \<cdot> TTfgh_TfTgh.the_\<nu> \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>1 f, f\<^sub>0g\<^sub>1.p\<^sub>1, fg\<^sub>0h\<^sub>1.p\<^sub>1]
goal (1 subgoal):
1. (f \<star> \<theta>\<^sub>f) \<cdot> \<a>[f, tab\<^sub>0 f, w\<^sub>f] \<cdot> (f.tab \<star> w\<^sub>f) = ((f \<star> \<theta>\<^sub>f') \<cdot> \<a>[f, tab\<^sub>0 f, w\<^sub>f'] \<cdot> (f.tab \<star> w\<^sub>f')) \<cdot> \<beta>\<^sub>f
[PROOF STEP]
also
[PROOF STATE]
proof (state)
this:
\<a>[f, g, h \<star> tab\<^sub>0 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0] \<cdot> \<a>[f \<star> g, h, tab\<^sub>0 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0] \<cdot> (\<a>\<^sup>-\<^sup>1[f, g, h] \<star> tab\<^sub>0 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0) \<cdot> ((f \<star> g \<star> h) \<star> TTfgh_TfTgh.the_\<theta>) \<cdot> \<a>[f \<star> g \<star> h, (tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine] \<cdot> (\<a>\<^sup>-\<^sup>1[f, g \<star> h, (tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0] \<star> TTfgh_TfTgh.chine) \<cdot> (((f \<star> \<a>[g \<star> h, tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0, f\<^sub>0gh\<^sub>1.p\<^sub>0]) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> \<a>\<^sup>-\<^sup>1[g, h, tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> (g \<star> \<a>[h, tab\<^sub>0 h, g\<^sub>0h\<^sub>1.p\<^sub>0]) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> (g \<star> h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> (g \<star> g\<^sub>0h\<^sub>1.\<phi>) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> \<a>[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> (g.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>1) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> f\<^sub>0gh\<^sub>1.\<phi>) \<star> TTfgh_TfTgh.chine) \<cdot> \<a>\<^sup>-\<^sup>1[f, tab\<^sub>0 f \<star> f\<^sub>0gh\<^sub>1.p\<^sub>1, TTfgh_TfTgh.chine]) \<cdot> (f \<star> \<a>\<^sup>-\<^sup>1[tab\<^sub>0 f, f\<^sub>0gh\<^sub>1.p\<^sub>1, TTfgh_TfTgh.chine]) \<cdot> \<a>[f, tab\<^sub>0 f, f\<^sub>0gh\<^sub>1.p\<^sub>1 \<star> TTfgh_TfTgh.chine] \<cdot> (f.tab \<star> f\<^sub>0gh\<^sub>1.p\<^sub>1 \<star> TTfgh_TfTgh.chine) \<cdot> \<a>[tab\<^sub>1 f, f\<^sub>0gh\<^sub>1.p\<^sub>1, TTfgh_TfTgh.chine] \<cdot> TTfgh_TfTgh.the_\<nu> \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>1 f, f\<^sub>0g\<^sub>1.p\<^sub>1, fg\<^sub>0h\<^sub>1.p\<^sub>1] = \<a>[f, g, h \<star> tab\<^sub>0 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0] \<cdot> \<a>[f \<star> g, h, tab\<^sub>0 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0] \<cdot> (\<a>\<^sup>-\<^sup>1[f, g, h] \<star> tab\<^sub>0 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0) \<cdot> ((f \<star> g \<star> h) \<star> TTfgh_TfTgh.the_\<theta>) \<cdot> \<a>[f \<star> g \<star> h, (tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine] \<cdot> (\<a>\<^sup>-\<^sup>1[f, g \<star> h, (tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0] \<star> TTfgh_TfTgh.chine) \<cdot> \<a>\<^sup>-\<^sup>1[f, (g \<star> h) \<star> (tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine] \<cdot> (f \<star> \<a>[g \<star> h, tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0, f\<^sub>0gh\<^sub>1.p\<^sub>0] \<star> TTfgh_TfTgh.chine) \<cdot> (f \<star> (\<a>\<^sup>-\<^sup>1[g, h, tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (f \<star> ((g \<star> \<a>[h, tab\<^sub>0 h, g\<^sub>0h\<^sub>1.p\<^sub>0]) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> ((g \<star> h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (f \<star> ((g \<star> g\<^sub>0h\<^sub>1.\<phi>) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (f \<star> (\<a>[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (f \<star> ((g.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>1) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (f \<star> f\<^sub>0gh\<^sub>1.\<phi> \<star> TTfgh_TfTgh.chine) \<cdot> (f \<star> \<a>\<^sup>-\<^sup>1[tab\<^sub>0 f, f\<^sub>0gh\<^sub>1.p\<^sub>1, TTfgh_TfTgh.chine])) \<cdot> \<a>[f, tab\<^sub>0 f, f\<^sub>0gh\<^sub>1.p\<^sub>1 \<star> TTfgh_TfTgh.chine] \<cdot> (f.tab \<star> f\<^sub>0gh\<^sub>1.p\<^sub>1 \<star> TTfgh_TfTgh.chine) \<cdot> \<a>[tab\<^sub>1 f, f\<^sub>0gh\<^sub>1.p\<^sub>1, TTfgh_TfTgh.chine] \<cdot> TTfgh_TfTgh.the_\<nu> \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>1 f, f\<^sub>0g\<^sub>1.p\<^sub>1, fg\<^sub>0h\<^sub>1.p\<^sub>1]
goal (1 subgoal):
1. (f \<star> \<theta>\<^sub>f) \<cdot> \<a>[f, tab\<^sub>0 f, w\<^sub>f] \<cdot> (f.tab \<star> w\<^sub>f) = ((f \<star> \<theta>\<^sub>f') \<cdot> \<a>[f, tab\<^sub>0 f, w\<^sub>f'] \<cdot> (f.tab \<star> w\<^sub>f')) \<cdot> \<beta>\<^sub>f
[PROOF STEP]
have "... =
(\<a>[f, g, h \<star> tab\<^sub>0 h \<star> TTfgh.p\<^sub>0] \<cdot>
\<a>[f \<star> g, h, tab\<^sub>0 h \<star> TTfgh.p\<^sub>0] \<cdot>
(\<a>\<^sup>-\<^sup>1[f, g, h] \<star> tab\<^sub>0 h \<star> TTfgh.p\<^sub>0) \<cdot>
((f \<star> g \<star> h) \<star> TTfgh_TfTgh.the_\<theta>)) \<cdot>
\<a>[f \<star> g \<star> h, (tab\<^sub>0 h \<star> Tgh.p\<^sub>0) \<star> TfTgh.p\<^sub>0, TTfgh_TfTgh.chine] \<cdot>
(\<a>\<^sup>-\<^sup>1[f, g \<star> h, (tab\<^sub>0 h \<star> Tgh.p\<^sub>0) \<star> TfTgh.p\<^sub>0] \<star> TTfgh_TfTgh.chine) \<cdot>
\<a>\<^sup>-\<^sup>1[f, (g \<star> h) \<star> (tab\<^sub>0 h \<star> Tgh.p\<^sub>0) \<star> TfTgh.p\<^sub>0, TTfgh_TfTgh.chine] \<cdot>
(f \<star> \<a>[g \<star> h, tab\<^sub>0 h \<star> Tgh.p\<^sub>0, TfTgh.p\<^sub>0] \<star> TTfgh_TfTgh.chine) \<cdot>
(f \<star> (\<a>\<^sup>-\<^sup>1[g, h, tab\<^sub>0 h \<star> Tgh.p\<^sub>0] \<star> TfTgh.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot>
(f \<star> ((g \<star> \<a>[h, tab\<^sub>0 h, Tgh.p\<^sub>0]) \<star> TfTgh.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot>
(f \<star> (((g \<star> h.tab \<star> Tgh.p\<^sub>0) \<star> TfTgh.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot>
(((g \<star> g\<^sub>0h\<^sub>1.\<phi>) \<star> TfTgh.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot>
((\<a>[g, tab\<^sub>0 g, Tgh.p\<^sub>1] \<star> TfTgh.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot>
(((g.tab \<star> Tgh.p\<^sub>1) \<star> TfTgh.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot>
(f\<^sub>0gh\<^sub>1.\<phi> \<star> TTfgh_TfTgh.chine) \<cdot>
\<a>\<^sup>-\<^sup>1[tab\<^sub>0 f, TfTgh.p\<^sub>1, TTfgh_TfTgh.chine]) \<cdot>
\<a>[f, tab\<^sub>0 f, TfTgh.p\<^sub>1 \<star> TTfgh_TfTgh.chine] \<cdot>
(f.tab \<star> TfTgh.p\<^sub>1 \<star> TTfgh_TfTgh.chine) \<cdot>
\<a>[tab\<^sub>1 f, TfTgh.p\<^sub>1, TTfgh_TfTgh.chine] \<cdot>
TTfgh_TfTgh.the_\<nu> \<cdot>
\<a>\<^sup>-\<^sup>1[tab\<^sub>1 f, Tfg.p\<^sub>1, TTfgh.p\<^sub>1]"
[PROOF STATE]
proof (prove)
goal (1 subgoal):
1. \<a>[f, g, h \<star> tab\<^sub>0 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0] \<cdot> \<a>[f \<star> g, h, tab\<^sub>0 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0] \<cdot> (\<a>\<^sup>-\<^sup>1[f, g, h] \<star> tab\<^sub>0 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0) \<cdot> ((f \<star> g \<star> h) \<star> TTfgh_TfTgh.the_\<theta>) \<cdot> \<a>[f \<star> g \<star> h, (tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine] \<cdot> (\<a>\<^sup>-\<^sup>1[f, g \<star> h, (tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0] \<star> TTfgh_TfTgh.chine) \<cdot> \<a>\<^sup>-\<^sup>1[f, (g \<star> h) \<star> (tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine] \<cdot> (f \<star> \<a>[g \<star> h, tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0, f\<^sub>0gh\<^sub>1.p\<^sub>0] \<star> TTfgh_TfTgh.chine) \<cdot> (f \<star> (\<a>\<^sup>-\<^sup>1[g, h, tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (f \<star> ((g \<star> \<a>[h, tab\<^sub>0 h, g\<^sub>0h\<^sub>1.p\<^sub>0]) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> ((g \<star> h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (f \<star> ((g \<star> g\<^sub>0h\<^sub>1.\<phi>) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (f \<star> (\<a>[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (f \<star> ((g.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>1) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (f \<star> f\<^sub>0gh\<^sub>1.\<phi> \<star> TTfgh_TfTgh.chine) \<cdot> (f \<star> \<a>\<^sup>-\<^sup>1[tab\<^sub>0 f, f\<^sub>0gh\<^sub>1.p\<^sub>1, TTfgh_TfTgh.chine])) \<cdot> \<a>[f, tab\<^sub>0 f, f\<^sub>0gh\<^sub>1.p\<^sub>1 \<star> TTfgh_TfTgh.chine] \<cdot> (f.tab \<star> f\<^sub>0gh\<^sub>1.p\<^sub>1 \<star> TTfgh_TfTgh.chine) \<cdot> \<a>[tab\<^sub>1 f, f\<^sub>0gh\<^sub>1.p\<^sub>1, TTfgh_TfTgh.chine] \<cdot> TTfgh_TfTgh.the_\<nu> \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>1 f, f\<^sub>0g\<^sub>1.p\<^sub>1, fg\<^sub>0h\<^sub>1.p\<^sub>1] = (\<a>[f, g, h \<star> tab\<^sub>0 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0] \<cdot> \<a>[f \<star> g, h, tab\<^sub>0 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0] \<cdot> (\<a>\<^sup>-\<^sup>1[f, g, h] \<star> tab\<^sub>0 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0) \<cdot> ((f \<star> g \<star> h) \<star> TTfgh_TfTgh.the_\<theta>)) \<cdot> \<a>[f \<star> g \<star> h, (tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine] \<cdot> (\<a>\<^sup>-\<^sup>1[f, g \<star> h, (tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0] \<star> TTfgh_TfTgh.chine) \<cdot> \<a>\<^sup>-\<^sup>1[f, (g \<star> h) \<star> (tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine] \<cdot> (f \<star> \<a>[g \<star> h, tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0, f\<^sub>0gh\<^sub>1.p\<^sub>0] \<star> TTfgh_TfTgh.chine) \<cdot> (f \<star> (\<a>\<^sup>-\<^sup>1[g, h, tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (f \<star> ((g \<star> \<a>[h, tab\<^sub>0 h, g\<^sub>0h\<^sub>1.p\<^sub>0]) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (f \<star> (((g \<star> h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (((g \<star> g\<^sub>0h\<^sub>1.\<phi>) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((\<a>[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (((g.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>1) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (f\<^sub>0gh\<^sub>1.\<phi> \<star> TTfgh_TfTgh.chine) \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>0 f, f\<^sub>0gh\<^sub>1.p\<^sub>1, TTfgh_TfTgh.chine]) \<cdot> \<a>[f, tab\<^sub>0 f, f\<^sub>0gh\<^sub>1.p\<^sub>1 \<star> TTfgh_TfTgh.chine] \<cdot> (f.tab \<star> f\<^sub>0gh\<^sub>1.p\<^sub>1 \<star> TTfgh_TfTgh.chine) \<cdot> \<a>[tab\<^sub>1 f, f\<^sub>0gh\<^sub>1.p\<^sub>1, TTfgh_TfTgh.chine] \<cdot> TTfgh_TfTgh.the_\<nu> \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>1 f, f\<^sub>0g\<^sub>1.p\<^sub>1, fg\<^sub>0h\<^sub>1.p\<^sub>1]
[PROOF STEP]
proof -
[PROOF STATE]
proof (state)
goal (1 subgoal):
1. \<a>[f, g, h \<star> tab\<^sub>0 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0] \<cdot> \<a>[f \<star> g, h, tab\<^sub>0 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0] \<cdot> (\<a>\<^sup>-\<^sup>1[f, g, h] \<star> tab\<^sub>0 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0) \<cdot> ((f \<star> g \<star> h) \<star> TTfgh_TfTgh.the_\<theta>) \<cdot> \<a>[f \<star> g \<star> h, (tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine] \<cdot> (\<a>\<^sup>-\<^sup>1[f, g \<star> h, (tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0] \<star> TTfgh_TfTgh.chine) \<cdot> \<a>\<^sup>-\<^sup>1[f, (g \<star> h) \<star> (tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine] \<cdot> (f \<star> \<a>[g \<star> h, tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0, f\<^sub>0gh\<^sub>1.p\<^sub>0] \<star> TTfgh_TfTgh.chine) \<cdot> (f \<star> (\<a>\<^sup>-\<^sup>1[g, h, tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (f \<star> ((g \<star> \<a>[h, tab\<^sub>0 h, g\<^sub>0h\<^sub>1.p\<^sub>0]) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> ((g \<star> h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (f \<star> ((g \<star> g\<^sub>0h\<^sub>1.\<phi>) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (f \<star> (\<a>[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (f \<star> ((g.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>1) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (f \<star> f\<^sub>0gh\<^sub>1.\<phi> \<star> TTfgh_TfTgh.chine) \<cdot> (f \<star> \<a>\<^sup>-\<^sup>1[tab\<^sub>0 f, f\<^sub>0gh\<^sub>1.p\<^sub>1, TTfgh_TfTgh.chine])) \<cdot> \<a>[f, tab\<^sub>0 f, f\<^sub>0gh\<^sub>1.p\<^sub>1 \<star> TTfgh_TfTgh.chine] \<cdot> (f.tab \<star> f\<^sub>0gh\<^sub>1.p\<^sub>1 \<star> TTfgh_TfTgh.chine) \<cdot> \<a>[tab\<^sub>1 f, f\<^sub>0gh\<^sub>1.p\<^sub>1, TTfgh_TfTgh.chine] \<cdot> TTfgh_TfTgh.the_\<nu> \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>1 f, f\<^sub>0g\<^sub>1.p\<^sub>1, fg\<^sub>0h\<^sub>1.p\<^sub>1] = (\<a>[f, g, h \<star> tab\<^sub>0 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0] \<cdot> \<a>[f \<star> g, h, tab\<^sub>0 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0] \<cdot> (\<a>\<^sup>-\<^sup>1[f, g, h] \<star> tab\<^sub>0 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0) \<cdot> ((f \<star> g \<star> h) \<star> TTfgh_TfTgh.the_\<theta>)) \<cdot> \<a>[f \<star> g \<star> h, (tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine] \<cdot> (\<a>\<^sup>-\<^sup>1[f, g \<star> h, (tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0] \<star> TTfgh_TfTgh.chine) \<cdot> \<a>\<^sup>-\<^sup>1[f, (g \<star> h) \<star> (tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine] \<cdot> (f \<star> \<a>[g \<star> h, tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0, f\<^sub>0gh\<^sub>1.p\<^sub>0] \<star> TTfgh_TfTgh.chine) \<cdot> (f \<star> (\<a>\<^sup>-\<^sup>1[g, h, tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (f \<star> ((g \<star> \<a>[h, tab\<^sub>0 h, g\<^sub>0h\<^sub>1.p\<^sub>0]) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (f \<star> (((g \<star> h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (((g \<star> g\<^sub>0h\<^sub>1.\<phi>) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((\<a>[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (((g.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>1) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (f\<^sub>0gh\<^sub>1.\<phi> \<star> TTfgh_TfTgh.chine) \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>0 f, f\<^sub>0gh\<^sub>1.p\<^sub>1, TTfgh_TfTgh.chine]) \<cdot> \<a>[f, tab\<^sub>0 f, f\<^sub>0gh\<^sub>1.p\<^sub>1 \<star> TTfgh_TfTgh.chine] \<cdot> (f.tab \<star> f\<^sub>0gh\<^sub>1.p\<^sub>1 \<star> TTfgh_TfTgh.chine) \<cdot> \<a>[tab\<^sub>1 f, f\<^sub>0gh\<^sub>1.p\<^sub>1, TTfgh_TfTgh.chine] \<cdot> TTfgh_TfTgh.the_\<nu> \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>1 f, f\<^sub>0g\<^sub>1.p\<^sub>1, fg\<^sub>0h\<^sub>1.p\<^sub>1]
[PROOF STEP]
have "(f \<star> ((g \<star> h.tab \<star> Tgh.p\<^sub>0) \<star> TfTgh.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot>
(f \<star> ((g \<star> g\<^sub>0h\<^sub>1.\<phi>) \<star> TfTgh.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot>
(f \<star> (\<a>[g, tab\<^sub>0 g, Tgh.p\<^sub>1] \<star> TfTgh.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot>
(f \<star> ((g.tab \<star> Tgh.p\<^sub>1) \<star> TfTgh.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot>
(f \<star> f\<^sub>0gh\<^sub>1.\<phi> \<star> TTfgh_TfTgh.chine) \<cdot>
(f \<star> \<a>\<^sup>-\<^sup>1[tab\<^sub>0 f, TfTgh.p\<^sub>1, TTfgh_TfTgh.chine]) =
(f \<star> (((g \<star> h.tab \<star> Tgh.p\<^sub>0) \<star> TfTgh.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot>
(((g \<star> g\<^sub>0h\<^sub>1.\<phi>) \<star> TfTgh.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot>
((\<a>[g, tab\<^sub>0 g, Tgh.p\<^sub>1] \<star> TfTgh.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot>
(((g.tab \<star> Tgh.p\<^sub>1) \<star> TfTgh.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot>
(f\<^sub>0gh\<^sub>1.\<phi> \<star> TTfgh_TfTgh.chine) \<cdot>
\<a>\<^sup>-\<^sup>1[tab\<^sub>0 f, TfTgh.p\<^sub>1, TTfgh_TfTgh.chine])"
[PROOF STATE]
proof (prove)
goal (1 subgoal):
1. (f \<star> ((g \<star> h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (f \<star> ((g \<star> g\<^sub>0h\<^sub>1.\<phi>) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (f \<star> (\<a>[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (f \<star> ((g.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>1) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (f \<star> f\<^sub>0gh\<^sub>1.\<phi> \<star> TTfgh_TfTgh.chine) \<cdot> (f \<star> \<a>\<^sup>-\<^sup>1[tab\<^sub>0 f, f\<^sub>0gh\<^sub>1.p\<^sub>1, TTfgh_TfTgh.chine]) = f \<star> (((g \<star> h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (((g \<star> g\<^sub>0h\<^sub>1.\<phi>) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((\<a>[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (((g.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>1) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (f\<^sub>0gh\<^sub>1.\<phi> \<star> TTfgh_TfTgh.chine) \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>0 f, f\<^sub>0gh\<^sub>1.p\<^sub>1, TTfgh_TfTgh.chine]
[PROOF STEP]
using fg gh whisker_left
[PROOF STATE]
proof (prove)
using this:
src f = trg g
src g = trg h
\<lbrakk>ide ?f; seq ?\<nu> ?\<mu>\<rbrakk> \<Longrightarrow> ?f \<star> ?\<nu> \<cdot> ?\<mu> = (?f \<star> ?\<nu>) \<cdot> (?f \<star> ?\<mu>)
goal (1 subgoal):
1. (f \<star> ((g \<star> h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (f \<star> ((g \<star> g\<^sub>0h\<^sub>1.\<phi>) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (f \<star> (\<a>[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (f \<star> ((g.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>1) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (f \<star> f\<^sub>0gh\<^sub>1.\<phi> \<star> TTfgh_TfTgh.chine) \<cdot> (f \<star> \<a>\<^sup>-\<^sup>1[tab\<^sub>0 f, f\<^sub>0gh\<^sub>1.p\<^sub>1, TTfgh_TfTgh.chine]) = f \<star> (((g \<star> h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (((g \<star> g\<^sub>0h\<^sub>1.\<phi>) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((\<a>[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (((g.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>1) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (f\<^sub>0gh\<^sub>1.\<phi> \<star> TTfgh_TfTgh.chine) \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>0 f, f\<^sub>0gh\<^sub>1.p\<^sub>1, TTfgh_TfTgh.chine]
[PROOF STEP]
by simp
[PROOF STATE]
proof (state)
this:
(f \<star> ((g \<star> h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (f \<star> ((g \<star> g\<^sub>0h\<^sub>1.\<phi>) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (f \<star> (\<a>[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (f \<star> ((g.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>1) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (f \<star> f\<^sub>0gh\<^sub>1.\<phi> \<star> TTfgh_TfTgh.chine) \<cdot> (f \<star> \<a>\<^sup>-\<^sup>1[tab\<^sub>0 f, f\<^sub>0gh\<^sub>1.p\<^sub>1, TTfgh_TfTgh.chine]) = f \<star> (((g \<star> h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (((g \<star> g\<^sub>0h\<^sub>1.\<phi>) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((\<a>[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (((g.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>1) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (f\<^sub>0gh\<^sub>1.\<phi> \<star> TTfgh_TfTgh.chine) \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>0 f, f\<^sub>0gh\<^sub>1.p\<^sub>1, TTfgh_TfTgh.chine]
goal (1 subgoal):
1. \<a>[f, g, h \<star> tab\<^sub>0 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0] \<cdot> \<a>[f \<star> g, h, tab\<^sub>0 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0] \<cdot> (\<a>\<^sup>-\<^sup>1[f, g, h] \<star> tab\<^sub>0 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0) \<cdot> ((f \<star> g \<star> h) \<star> TTfgh_TfTgh.the_\<theta>) \<cdot> \<a>[f \<star> g \<star> h, (tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine] \<cdot> (\<a>\<^sup>-\<^sup>1[f, g \<star> h, (tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0] \<star> TTfgh_TfTgh.chine) \<cdot> \<a>\<^sup>-\<^sup>1[f, (g \<star> h) \<star> (tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine] \<cdot> (f \<star> \<a>[g \<star> h, tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0, f\<^sub>0gh\<^sub>1.p\<^sub>0] \<star> TTfgh_TfTgh.chine) \<cdot> (f \<star> (\<a>\<^sup>-\<^sup>1[g, h, tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (f \<star> ((g \<star> \<a>[h, tab\<^sub>0 h, g\<^sub>0h\<^sub>1.p\<^sub>0]) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> ((g \<star> h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (f \<star> ((g \<star> g\<^sub>0h\<^sub>1.\<phi>) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (f \<star> (\<a>[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (f \<star> ((g.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>1) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (f \<star> f\<^sub>0gh\<^sub>1.\<phi> \<star> TTfgh_TfTgh.chine) \<cdot> (f \<star> \<a>\<^sup>-\<^sup>1[tab\<^sub>0 f, f\<^sub>0gh\<^sub>1.p\<^sub>1, TTfgh_TfTgh.chine])) \<cdot> \<a>[f, tab\<^sub>0 f, f\<^sub>0gh\<^sub>1.p\<^sub>1 \<star> TTfgh_TfTgh.chine] \<cdot> (f.tab \<star> f\<^sub>0gh\<^sub>1.p\<^sub>1 \<star> TTfgh_TfTgh.chine) \<cdot> \<a>[tab\<^sub>1 f, f\<^sub>0gh\<^sub>1.p\<^sub>1, TTfgh_TfTgh.chine] \<cdot> TTfgh_TfTgh.the_\<nu> \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>1 f, f\<^sub>0g\<^sub>1.p\<^sub>1, fg\<^sub>0h\<^sub>1.p\<^sub>1] = (\<a>[f, g, h \<star> tab\<^sub>0 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0] \<cdot> \<a>[f \<star> g, h, tab\<^sub>0 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0] \<cdot> (\<a>\<^sup>-\<^sup>1[f, g, h] \<star> tab\<^sub>0 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0) \<cdot> ((f \<star> g \<star> h) \<star> TTfgh_TfTgh.the_\<theta>)) \<cdot> \<a>[f \<star> g \<star> h, (tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine] \<cdot> (\<a>\<^sup>-\<^sup>1[f, g \<star> h, (tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0] \<star> TTfgh_TfTgh.chine) \<cdot> \<a>\<^sup>-\<^sup>1[f, (g \<star> h) \<star> (tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine] \<cdot> (f \<star> \<a>[g \<star> h, tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0, f\<^sub>0gh\<^sub>1.p\<^sub>0] \<star> TTfgh_TfTgh.chine) \<cdot> (f \<star> (\<a>\<^sup>-\<^sup>1[g, h, tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (f \<star> ((g \<star> \<a>[h, tab\<^sub>0 h, g\<^sub>0h\<^sub>1.p\<^sub>0]) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (f \<star> (((g \<star> h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (((g \<star> g\<^sub>0h\<^sub>1.\<phi>) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((\<a>[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (((g.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>1) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (f\<^sub>0gh\<^sub>1.\<phi> \<star> TTfgh_TfTgh.chine) \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>0 f, f\<^sub>0gh\<^sub>1.p\<^sub>1, TTfgh_TfTgh.chine]) \<cdot> \<a>[f, tab\<^sub>0 f, f\<^sub>0gh\<^sub>1.p\<^sub>1 \<star> TTfgh_TfTgh.chine] \<cdot> (f.tab \<star> f\<^sub>0gh\<^sub>1.p\<^sub>1 \<star> TTfgh_TfTgh.chine) \<cdot> \<a>[tab\<^sub>1 f, f\<^sub>0gh\<^sub>1.p\<^sub>1, TTfgh_TfTgh.chine] \<cdot> TTfgh_TfTgh.the_\<nu> \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>1 f, f\<^sub>0g\<^sub>1.p\<^sub>1, fg\<^sub>0h\<^sub>1.p\<^sub>1]
[PROOF STEP]
thus ?thesis
[PROOF STATE]
proof (prove)
using this:
(f \<star> ((g \<star> h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (f \<star> ((g \<star> g\<^sub>0h\<^sub>1.\<phi>) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (f \<star> (\<a>[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (f \<star> ((g.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>1) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (f \<star> f\<^sub>0gh\<^sub>1.\<phi> \<star> TTfgh_TfTgh.chine) \<cdot> (f \<star> \<a>\<^sup>-\<^sup>1[tab\<^sub>0 f, f\<^sub>0gh\<^sub>1.p\<^sub>1, TTfgh_TfTgh.chine]) = f \<star> (((g \<star> h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (((g \<star> g\<^sub>0h\<^sub>1.\<phi>) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((\<a>[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (((g.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>1) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (f\<^sub>0gh\<^sub>1.\<phi> \<star> TTfgh_TfTgh.chine) \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>0 f, f\<^sub>0gh\<^sub>1.p\<^sub>1, TTfgh_TfTgh.chine]
goal (1 subgoal):
1. \<a>[f, g, h \<star> tab\<^sub>0 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0] \<cdot> \<a>[f \<star> g, h, tab\<^sub>0 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0] \<cdot> (\<a>\<^sup>-\<^sup>1[f, g, h] \<star> tab\<^sub>0 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0) \<cdot> ((f \<star> g \<star> h) \<star> TTfgh_TfTgh.the_\<theta>) \<cdot> \<a>[f \<star> g \<star> h, (tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine] \<cdot> (\<a>\<^sup>-\<^sup>1[f, g \<star> h, (tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0] \<star> TTfgh_TfTgh.chine) \<cdot> \<a>\<^sup>-\<^sup>1[f, (g \<star> h) \<star> (tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine] \<cdot> (f \<star> \<a>[g \<star> h, tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0, f\<^sub>0gh\<^sub>1.p\<^sub>0] \<star> TTfgh_TfTgh.chine) \<cdot> (f \<star> (\<a>\<^sup>-\<^sup>1[g, h, tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (f \<star> ((g \<star> \<a>[h, tab\<^sub>0 h, g\<^sub>0h\<^sub>1.p\<^sub>0]) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> ((g \<star> h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (f \<star> ((g \<star> g\<^sub>0h\<^sub>1.\<phi>) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (f \<star> (\<a>[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (f \<star> ((g.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>1) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (f \<star> f\<^sub>0gh\<^sub>1.\<phi> \<star> TTfgh_TfTgh.chine) \<cdot> (f \<star> \<a>\<^sup>-\<^sup>1[tab\<^sub>0 f, f\<^sub>0gh\<^sub>1.p\<^sub>1, TTfgh_TfTgh.chine])) \<cdot> \<a>[f, tab\<^sub>0 f, f\<^sub>0gh\<^sub>1.p\<^sub>1 \<star> TTfgh_TfTgh.chine] \<cdot> (f.tab \<star> f\<^sub>0gh\<^sub>1.p\<^sub>1 \<star> TTfgh_TfTgh.chine) \<cdot> \<a>[tab\<^sub>1 f, f\<^sub>0gh\<^sub>1.p\<^sub>1, TTfgh_TfTgh.chine] \<cdot> TTfgh_TfTgh.the_\<nu> \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>1 f, f\<^sub>0g\<^sub>1.p\<^sub>1, fg\<^sub>0h\<^sub>1.p\<^sub>1] = (\<a>[f, g, h \<star> tab\<^sub>0 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0] \<cdot> \<a>[f \<star> g, h, tab\<^sub>0 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0] \<cdot> (\<a>\<^sup>-\<^sup>1[f, g, h] \<star> tab\<^sub>0 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0) \<cdot> ((f \<star> g \<star> h) \<star> TTfgh_TfTgh.the_\<theta>)) \<cdot> \<a>[f \<star> g \<star> h, (tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine] \<cdot> (\<a>\<^sup>-\<^sup>1[f, g \<star> h, (tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0] \<star> TTfgh_TfTgh.chine) \<cdot> \<a>\<^sup>-\<^sup>1[f, (g \<star> h) \<star> (tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine] \<cdot> (f \<star> \<a>[g \<star> h, tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0, f\<^sub>0gh\<^sub>1.p\<^sub>0] \<star> TTfgh_TfTgh.chine) \<cdot> (f \<star> (\<a>\<^sup>-\<^sup>1[g, h, tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (f \<star> ((g \<star> \<a>[h, tab\<^sub>0 h, g\<^sub>0h\<^sub>1.p\<^sub>0]) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (f \<star> (((g \<star> h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (((g \<star> g\<^sub>0h\<^sub>1.\<phi>) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((\<a>[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (((g.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>1) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (f\<^sub>0gh\<^sub>1.\<phi> \<star> TTfgh_TfTgh.chine) \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>0 f, f\<^sub>0gh\<^sub>1.p\<^sub>1, TTfgh_TfTgh.chine]) \<cdot> \<a>[f, tab\<^sub>0 f, f\<^sub>0gh\<^sub>1.p\<^sub>1 \<star> TTfgh_TfTgh.chine] \<cdot> (f.tab \<star> f\<^sub>0gh\<^sub>1.p\<^sub>1 \<star> TTfgh_TfTgh.chine) \<cdot> \<a>[tab\<^sub>1 f, f\<^sub>0gh\<^sub>1.p\<^sub>1, TTfgh_TfTgh.chine] \<cdot> TTfgh_TfTgh.the_\<nu> \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>1 f, f\<^sub>0g\<^sub>1.p\<^sub>1, fg\<^sub>0h\<^sub>1.p\<^sub>1]
[PROOF STEP]
using comp_assoc
[PROOF STATE]
proof (prove)
using this:
(f \<star> ((g \<star> h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (f \<star> ((g \<star> g\<^sub>0h\<^sub>1.\<phi>) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (f \<star> (\<a>[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (f \<star> ((g.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>1) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (f \<star> f\<^sub>0gh\<^sub>1.\<phi> \<star> TTfgh_TfTgh.chine) \<cdot> (f \<star> \<a>\<^sup>-\<^sup>1[tab\<^sub>0 f, f\<^sub>0gh\<^sub>1.p\<^sub>1, TTfgh_TfTgh.chine]) = f \<star> (((g \<star> h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (((g \<star> g\<^sub>0h\<^sub>1.\<phi>) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((\<a>[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (((g.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>1) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (f\<^sub>0gh\<^sub>1.\<phi> \<star> TTfgh_TfTgh.chine) \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>0 f, f\<^sub>0gh\<^sub>1.p\<^sub>1, TTfgh_TfTgh.chine]
(?h \<cdot> ?g) \<cdot> ?f = ?h \<cdot> ?g \<cdot> ?f
goal (1 subgoal):
1. \<a>[f, g, h \<star> tab\<^sub>0 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0] \<cdot> \<a>[f \<star> g, h, tab\<^sub>0 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0] \<cdot> (\<a>\<^sup>-\<^sup>1[f, g, h] \<star> tab\<^sub>0 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0) \<cdot> ((f \<star> g \<star> h) \<star> TTfgh_TfTgh.the_\<theta>) \<cdot> \<a>[f \<star> g \<star> h, (tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine] \<cdot> (\<a>\<^sup>-\<^sup>1[f, g \<star> h, (tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0] \<star> TTfgh_TfTgh.chine) \<cdot> \<a>\<^sup>-\<^sup>1[f, (g \<star> h) \<star> (tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine] \<cdot> (f \<star> \<a>[g \<star> h, tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0, f\<^sub>0gh\<^sub>1.p\<^sub>0] \<star> TTfgh_TfTgh.chine) \<cdot> (f \<star> (\<a>\<^sup>-\<^sup>1[g, h, tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (f \<star> ((g \<star> \<a>[h, tab\<^sub>0 h, g\<^sub>0h\<^sub>1.p\<^sub>0]) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> ((g \<star> h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (f \<star> ((g \<star> g\<^sub>0h\<^sub>1.\<phi>) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (f \<star> (\<a>[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (f \<star> ((g.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>1) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (f \<star> f\<^sub>0gh\<^sub>1.\<phi> \<star> TTfgh_TfTgh.chine) \<cdot> (f \<star> \<a>\<^sup>-\<^sup>1[tab\<^sub>0 f, f\<^sub>0gh\<^sub>1.p\<^sub>1, TTfgh_TfTgh.chine])) \<cdot> \<a>[f, tab\<^sub>0 f, f\<^sub>0gh\<^sub>1.p\<^sub>1 \<star> TTfgh_TfTgh.chine] \<cdot> (f.tab \<star> f\<^sub>0gh\<^sub>1.p\<^sub>1 \<star> TTfgh_TfTgh.chine) \<cdot> \<a>[tab\<^sub>1 f, f\<^sub>0gh\<^sub>1.p\<^sub>1, TTfgh_TfTgh.chine] \<cdot> TTfgh_TfTgh.the_\<nu> \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>1 f, f\<^sub>0g\<^sub>1.p\<^sub>1, fg\<^sub>0h\<^sub>1.p\<^sub>1] = (\<a>[f, g, h \<star> tab\<^sub>0 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0] \<cdot> \<a>[f \<star> g, h, tab\<^sub>0 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0] \<cdot> (\<a>\<^sup>-\<^sup>1[f, g, h] \<star> tab\<^sub>0 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0) \<cdot> ((f \<star> g \<star> h) \<star> TTfgh_TfTgh.the_\<theta>)) \<cdot> \<a>[f \<star> g \<star> h, (tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine] \<cdot> (\<a>\<^sup>-\<^sup>1[f, g \<star> h, (tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0] \<star> TTfgh_TfTgh.chine) \<cdot> \<a>\<^sup>-\<^sup>1[f, (g \<star> h) \<star> (tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine] \<cdot> (f \<star> \<a>[g \<star> h, tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0, f\<^sub>0gh\<^sub>1.p\<^sub>0] \<star> TTfgh_TfTgh.chine) \<cdot> (f \<star> (\<a>\<^sup>-\<^sup>1[g, h, tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (f \<star> ((g \<star> \<a>[h, tab\<^sub>0 h, g\<^sub>0h\<^sub>1.p\<^sub>0]) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (f \<star> (((g \<star> h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (((g \<star> g\<^sub>0h\<^sub>1.\<phi>) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((\<a>[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (((g.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>1) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (f\<^sub>0gh\<^sub>1.\<phi> \<star> TTfgh_TfTgh.chine) \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>0 f, f\<^sub>0gh\<^sub>1.p\<^sub>1, TTfgh_TfTgh.chine]) \<cdot> \<a>[f, tab\<^sub>0 f, f\<^sub>0gh\<^sub>1.p\<^sub>1 \<star> TTfgh_TfTgh.chine] \<cdot> (f.tab \<star> f\<^sub>0gh\<^sub>1.p\<^sub>1 \<star> TTfgh_TfTgh.chine) \<cdot> \<a>[tab\<^sub>1 f, f\<^sub>0gh\<^sub>1.p\<^sub>1, TTfgh_TfTgh.chine] \<cdot> TTfgh_TfTgh.the_\<nu> \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>1 f, f\<^sub>0g\<^sub>1.p\<^sub>1, fg\<^sub>0h\<^sub>1.p\<^sub>1]
[PROOF STEP]
by presburger
[PROOF STATE]
proof (state)
this:
\<a>[f, g, h \<star> tab\<^sub>0 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0] \<cdot> \<a>[f \<star> g, h, tab\<^sub>0 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0] \<cdot> (\<a>\<^sup>-\<^sup>1[f, g, h] \<star> tab\<^sub>0 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0) \<cdot> ((f \<star> g \<star> h) \<star> TTfgh_TfTgh.the_\<theta>) \<cdot> \<a>[f \<star> g \<star> h, (tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine] \<cdot> (\<a>\<^sup>-\<^sup>1[f, g \<star> h, (tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0] \<star> TTfgh_TfTgh.chine) \<cdot> \<a>\<^sup>-\<^sup>1[f, (g \<star> h) \<star> (tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine] \<cdot> (f \<star> \<a>[g \<star> h, tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0, f\<^sub>0gh\<^sub>1.p\<^sub>0] \<star> TTfgh_TfTgh.chine) \<cdot> (f \<star> (\<a>\<^sup>-\<^sup>1[g, h, tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (f \<star> ((g \<star> \<a>[h, tab\<^sub>0 h, g\<^sub>0h\<^sub>1.p\<^sub>0]) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> ((g \<star> h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (f \<star> ((g \<star> g\<^sub>0h\<^sub>1.\<phi>) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (f \<star> (\<a>[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (f \<star> ((g.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>1) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (f \<star> f\<^sub>0gh\<^sub>1.\<phi> \<star> TTfgh_TfTgh.chine) \<cdot> (f \<star> \<a>\<^sup>-\<^sup>1[tab\<^sub>0 f, f\<^sub>0gh\<^sub>1.p\<^sub>1, TTfgh_TfTgh.chine])) \<cdot> \<a>[f, tab\<^sub>0 f, f\<^sub>0gh\<^sub>1.p\<^sub>1 \<star> TTfgh_TfTgh.chine] \<cdot> (f.tab \<star> f\<^sub>0gh\<^sub>1.p\<^sub>1 \<star> TTfgh_TfTgh.chine) \<cdot> \<a>[tab\<^sub>1 f, f\<^sub>0gh\<^sub>1.p\<^sub>1, TTfgh_TfTgh.chine] \<cdot> TTfgh_TfTgh.the_\<nu> \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>1 f, f\<^sub>0g\<^sub>1.p\<^sub>1, fg\<^sub>0h\<^sub>1.p\<^sub>1] = (\<a>[f, g, h \<star> tab\<^sub>0 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0] \<cdot> \<a>[f \<star> g, h, tab\<^sub>0 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0] \<cdot> (\<a>\<^sup>-\<^sup>1[f, g, h] \<star> tab\<^sub>0 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0) \<cdot> ((f \<star> g \<star> h) \<star> TTfgh_TfTgh.the_\<theta>)) \<cdot> \<a>[f \<star> g \<star> h, (tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine] \<cdot> (\<a>\<^sup>-\<^sup>1[f, g \<star> h, (tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0] \<star> TTfgh_TfTgh.chine) \<cdot> \<a>\<^sup>-\<^sup>1[f, (g \<star> h) \<star> (tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine] \<cdot> (f \<star> \<a>[g \<star> h, tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0, f\<^sub>0gh\<^sub>1.p\<^sub>0] \<star> TTfgh_TfTgh.chine) \<cdot> (f \<star> (\<a>\<^sup>-\<^sup>1[g, h, tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (f \<star> ((g \<star> \<a>[h, tab\<^sub>0 h, g\<^sub>0h\<^sub>1.p\<^sub>0]) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (f \<star> (((g \<star> h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (((g \<star> g\<^sub>0h\<^sub>1.\<phi>) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((\<a>[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (((g.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>1) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (f\<^sub>0gh\<^sub>1.\<phi> \<star> TTfgh_TfTgh.chine) \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>0 f, f\<^sub>0gh\<^sub>1.p\<^sub>1, TTfgh_TfTgh.chine]) \<cdot> \<a>[f, tab\<^sub>0 f, f\<^sub>0gh\<^sub>1.p\<^sub>1 \<star> TTfgh_TfTgh.chine] \<cdot> (f.tab \<star> f\<^sub>0gh\<^sub>1.p\<^sub>1 \<star> TTfgh_TfTgh.chine) \<cdot> \<a>[tab\<^sub>1 f, f\<^sub>0gh\<^sub>1.p\<^sub>1, TTfgh_TfTgh.chine] \<cdot> TTfgh_TfTgh.the_\<nu> \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>1 f, f\<^sub>0g\<^sub>1.p\<^sub>1, fg\<^sub>0h\<^sub>1.p\<^sub>1]
goal:
No subgoals!
[PROOF STEP]
qed
[PROOF STATE]
proof (state)
this:
\<a>[f, g, h \<star> tab\<^sub>0 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0] \<cdot> \<a>[f \<star> g, h, tab\<^sub>0 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0] \<cdot> (\<a>\<^sup>-\<^sup>1[f, g, h] \<star> tab\<^sub>0 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0) \<cdot> ((f \<star> g \<star> h) \<star> TTfgh_TfTgh.the_\<theta>) \<cdot> \<a>[f \<star> g \<star> h, (tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine] \<cdot> (\<a>\<^sup>-\<^sup>1[f, g \<star> h, (tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0] \<star> TTfgh_TfTgh.chine) \<cdot> \<a>\<^sup>-\<^sup>1[f, (g \<star> h) \<star> (tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine] \<cdot> (f \<star> \<a>[g \<star> h, tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0, f\<^sub>0gh\<^sub>1.p\<^sub>0] \<star> TTfgh_TfTgh.chine) \<cdot> (f \<star> (\<a>\<^sup>-\<^sup>1[g, h, tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (f \<star> ((g \<star> \<a>[h, tab\<^sub>0 h, g\<^sub>0h\<^sub>1.p\<^sub>0]) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> ((g \<star> h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (f \<star> ((g \<star> g\<^sub>0h\<^sub>1.\<phi>) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (f \<star> (\<a>[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (f \<star> ((g.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>1) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (f \<star> f\<^sub>0gh\<^sub>1.\<phi> \<star> TTfgh_TfTgh.chine) \<cdot> (f \<star> \<a>\<^sup>-\<^sup>1[tab\<^sub>0 f, f\<^sub>0gh\<^sub>1.p\<^sub>1, TTfgh_TfTgh.chine])) \<cdot> \<a>[f, tab\<^sub>0 f, f\<^sub>0gh\<^sub>1.p\<^sub>1 \<star> TTfgh_TfTgh.chine] \<cdot> (f.tab \<star> f\<^sub>0gh\<^sub>1.p\<^sub>1 \<star> TTfgh_TfTgh.chine) \<cdot> \<a>[tab\<^sub>1 f, f\<^sub>0gh\<^sub>1.p\<^sub>1, TTfgh_TfTgh.chine] \<cdot> TTfgh_TfTgh.the_\<nu> \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>1 f, f\<^sub>0g\<^sub>1.p\<^sub>1, fg\<^sub>0h\<^sub>1.p\<^sub>1] = (\<a>[f, g, h \<star> tab\<^sub>0 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0] \<cdot> \<a>[f \<star> g, h, tab\<^sub>0 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0] \<cdot> (\<a>\<^sup>-\<^sup>1[f, g, h] \<star> tab\<^sub>0 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0) \<cdot> ((f \<star> g \<star> h) \<star> TTfgh_TfTgh.the_\<theta>)) \<cdot> \<a>[f \<star> g \<star> h, (tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine] \<cdot> (\<a>\<^sup>-\<^sup>1[f, g \<star> h, (tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0] \<star> TTfgh_TfTgh.chine) \<cdot> \<a>\<^sup>-\<^sup>1[f, (g \<star> h) \<star> (tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine] \<cdot> (f \<star> \<a>[g \<star> h, tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0, f\<^sub>0gh\<^sub>1.p\<^sub>0] \<star> TTfgh_TfTgh.chine) \<cdot> (f \<star> (\<a>\<^sup>-\<^sup>1[g, h, tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (f \<star> ((g \<star> \<a>[h, tab\<^sub>0 h, g\<^sub>0h\<^sub>1.p\<^sub>0]) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (f \<star> (((g \<star> h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (((g \<star> g\<^sub>0h\<^sub>1.\<phi>) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((\<a>[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (((g.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>1) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (f\<^sub>0gh\<^sub>1.\<phi> \<star> TTfgh_TfTgh.chine) \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>0 f, f\<^sub>0gh\<^sub>1.p\<^sub>1, TTfgh_TfTgh.chine]) \<cdot> \<a>[f, tab\<^sub>0 f, f\<^sub>0gh\<^sub>1.p\<^sub>1 \<star> TTfgh_TfTgh.chine] \<cdot> (f.tab \<star> f\<^sub>0gh\<^sub>1.p\<^sub>1 \<star> TTfgh_TfTgh.chine) \<cdot> \<a>[tab\<^sub>1 f, f\<^sub>0gh\<^sub>1.p\<^sub>1, TTfgh_TfTgh.chine] \<cdot> TTfgh_TfTgh.the_\<nu> \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>1 f, f\<^sub>0g\<^sub>1.p\<^sub>1, fg\<^sub>0h\<^sub>1.p\<^sub>1]
goal (1 subgoal):
1. (f \<star> \<theta>\<^sub>f) \<cdot> \<a>[f, tab\<^sub>0 f, w\<^sub>f] \<cdot> (f.tab \<star> w\<^sub>f) = ((f \<star> \<theta>\<^sub>f') \<cdot> \<a>[f, tab\<^sub>0 f, w\<^sub>f'] \<cdot> (f.tab \<star> w\<^sub>f')) \<cdot> \<beta>\<^sub>f
[PROOF STEP]
also
[PROOF STATE]
proof (state)
this:
\<a>[f, g, h \<star> tab\<^sub>0 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0] \<cdot> \<a>[f \<star> g, h, tab\<^sub>0 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0] \<cdot> (\<a>\<^sup>-\<^sup>1[f, g, h] \<star> tab\<^sub>0 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0) \<cdot> ((f \<star> g \<star> h) \<star> TTfgh_TfTgh.the_\<theta>) \<cdot> \<a>[f \<star> g \<star> h, (tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine] \<cdot> (\<a>\<^sup>-\<^sup>1[f, g \<star> h, (tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0] \<star> TTfgh_TfTgh.chine) \<cdot> \<a>\<^sup>-\<^sup>1[f, (g \<star> h) \<star> (tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine] \<cdot> (f \<star> \<a>[g \<star> h, tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0, f\<^sub>0gh\<^sub>1.p\<^sub>0] \<star> TTfgh_TfTgh.chine) \<cdot> (f \<star> (\<a>\<^sup>-\<^sup>1[g, h, tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (f \<star> ((g \<star> \<a>[h, tab\<^sub>0 h, g\<^sub>0h\<^sub>1.p\<^sub>0]) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((f \<star> ((g \<star> h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (f \<star> ((g \<star> g\<^sub>0h\<^sub>1.\<phi>) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (f \<star> (\<a>[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (f \<star> ((g.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>1) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (f \<star> f\<^sub>0gh\<^sub>1.\<phi> \<star> TTfgh_TfTgh.chine) \<cdot> (f \<star> \<a>\<^sup>-\<^sup>1[tab\<^sub>0 f, f\<^sub>0gh\<^sub>1.p\<^sub>1, TTfgh_TfTgh.chine])) \<cdot> \<a>[f, tab\<^sub>0 f, f\<^sub>0gh\<^sub>1.p\<^sub>1 \<star> TTfgh_TfTgh.chine] \<cdot> (f.tab \<star> f\<^sub>0gh\<^sub>1.p\<^sub>1 \<star> TTfgh_TfTgh.chine) \<cdot> \<a>[tab\<^sub>1 f, f\<^sub>0gh\<^sub>1.p\<^sub>1, TTfgh_TfTgh.chine] \<cdot> TTfgh_TfTgh.the_\<nu> \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>1 f, f\<^sub>0g\<^sub>1.p\<^sub>1, fg\<^sub>0h\<^sub>1.p\<^sub>1] = (\<a>[f, g, h \<star> tab\<^sub>0 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0] \<cdot> \<a>[f \<star> g, h, tab\<^sub>0 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0] \<cdot> (\<a>\<^sup>-\<^sup>1[f, g, h] \<star> tab\<^sub>0 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0) \<cdot> ((f \<star> g \<star> h) \<star> TTfgh_TfTgh.the_\<theta>)) \<cdot> \<a>[f \<star> g \<star> h, (tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine] \<cdot> (\<a>\<^sup>-\<^sup>1[f, g \<star> h, (tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0] \<star> TTfgh_TfTgh.chine) \<cdot> \<a>\<^sup>-\<^sup>1[f, (g \<star> h) \<star> (tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine] \<cdot> (f \<star> \<a>[g \<star> h, tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0, f\<^sub>0gh\<^sub>1.p\<^sub>0] \<star> TTfgh_TfTgh.chine) \<cdot> (f \<star> (\<a>\<^sup>-\<^sup>1[g, h, tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (f \<star> ((g \<star> \<a>[h, tab\<^sub>0 h, g\<^sub>0h\<^sub>1.p\<^sub>0]) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (f \<star> (((g \<star> h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (((g \<star> g\<^sub>0h\<^sub>1.\<phi>) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((\<a>[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (((g.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>1) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (f\<^sub>0gh\<^sub>1.\<phi> \<star> TTfgh_TfTgh.chine) \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>0 f, f\<^sub>0gh\<^sub>1.p\<^sub>1, TTfgh_TfTgh.chine]) \<cdot> \<a>[f, tab\<^sub>0 f, f\<^sub>0gh\<^sub>1.p\<^sub>1 \<star> TTfgh_TfTgh.chine] \<cdot> (f.tab \<star> f\<^sub>0gh\<^sub>1.p\<^sub>1 \<star> TTfgh_TfTgh.chine) \<cdot> \<a>[tab\<^sub>1 f, f\<^sub>0gh\<^sub>1.p\<^sub>1, TTfgh_TfTgh.chine] \<cdot> TTfgh_TfTgh.the_\<nu> \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>1 f, f\<^sub>0g\<^sub>1.p\<^sub>1, fg\<^sub>0h\<^sub>1.p\<^sub>1]
goal (1 subgoal):
1. (f \<star> \<theta>\<^sub>f) \<cdot> \<a>[f, tab\<^sub>0 f, w\<^sub>f] \<cdot> (f.tab \<star> w\<^sub>f) = ((f \<star> \<theta>\<^sub>f') \<cdot> \<a>[f, tab\<^sub>0 f, w\<^sub>f'] \<cdot> (f.tab \<star> w\<^sub>f')) \<cdot> \<beta>\<^sub>f
[PROOF STEP]
have "... =
(f \<star> g \<star> h \<star> TTfgh_TfTgh.the_\<theta>) \<cdot>
(\<a>[f, g, h \<star> ((tab\<^sub>0 h \<star> Tgh.p\<^sub>0) \<star> TfTgh.p\<^sub>0) \<star> TTfgh_TfTgh.chine] \<cdot>
\<a>[f \<star> g, h, ((tab\<^sub>0 h \<star> Tgh.p\<^sub>0) \<star> TfTgh.p\<^sub>0) \<star> TTfgh_TfTgh.chine] \<cdot>
(\<a>\<^sup>-\<^sup>1[f, g, h] \<star> ((tab\<^sub>0 h \<star> Tgh.p\<^sub>0) \<star> TfTgh.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot>
\<a>[f \<star> g \<star> h, (tab\<^sub>0 h \<star> Tgh.p\<^sub>0) \<star> TfTgh.p\<^sub>0, TTfgh_TfTgh.chine] \<cdot>
(\<a>\<^sup>-\<^sup>1[f, g \<star> h, (tab\<^sub>0 h \<star> Tgh.p\<^sub>0) \<star> TfTgh.p\<^sub>0] \<star> TTfgh_TfTgh.chine) \<cdot>
\<a>\<^sup>-\<^sup>1[f, (g \<star> h) \<star> (tab\<^sub>0 h \<star> Tgh.p\<^sub>0) \<star> TfTgh.p\<^sub>0, TTfgh_TfTgh.chine] \<cdot>
(f \<star> \<a>[g \<star> h, tab\<^sub>0 h \<star> Tgh.p\<^sub>0, TfTgh.p\<^sub>0] \<star> TTfgh_TfTgh.chine) \<cdot>
(f \<star> (\<a>\<^sup>-\<^sup>1[g, h, tab\<^sub>0 h \<star> Tgh.p\<^sub>0] \<star> TfTgh.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot>
(f \<star> ((g \<star> \<a>[h, tab\<^sub>0 h, Tgh.p\<^sub>0]) \<star> TfTgh.p\<^sub>0) \<star> TTfgh_TfTgh.chine)) \<cdot>
(f \<star> (((g \<star> h.tab \<star> Tgh.p\<^sub>0) \<star> TfTgh.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot>
(((g \<star> g\<^sub>0h\<^sub>1.\<phi>) \<star> TfTgh.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot>
((\<a>[g, tab\<^sub>0 g, Tgh.p\<^sub>1] \<star> TfTgh.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot>
(((g.tab \<star> Tgh.p\<^sub>1) \<star> TfTgh.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot>
(f\<^sub>0gh\<^sub>1.\<phi> \<star> TTfgh_TfTgh.chine) \<cdot>
\<a>\<^sup>-\<^sup>1[tab\<^sub>0 f, TfTgh.p\<^sub>1, TTfgh_TfTgh.chine]) \<cdot>
\<a>[f, tab\<^sub>0 f, TfTgh.p\<^sub>1 \<star> TTfgh_TfTgh.chine] \<cdot>
(f.tab \<star> TfTgh.p\<^sub>1 \<star> TTfgh_TfTgh.chine) \<cdot>
\<a>[tab\<^sub>1 f, TfTgh.p\<^sub>1, TTfgh_TfTgh.chine] \<cdot>
TTfgh_TfTgh.the_\<nu> \<cdot>
\<a>\<^sup>-\<^sup>1[tab\<^sub>1 f, Tfg.p\<^sub>1, TTfgh.p\<^sub>1]"
[PROOF STATE]
proof (prove)
goal (1 subgoal):
1. (\<a>[f, g, h \<star> tab\<^sub>0 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0] \<cdot> \<a>[f \<star> g, h, tab\<^sub>0 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0] \<cdot> (\<a>\<^sup>-\<^sup>1[f, g, h] \<star> tab\<^sub>0 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0) \<cdot> ((f \<star> g \<star> h) \<star> TTfgh_TfTgh.the_\<theta>)) \<cdot> \<a>[f \<star> g \<star> h, (tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine] \<cdot> (\<a>\<^sup>-\<^sup>1[f, g \<star> h, (tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0] \<star> TTfgh_TfTgh.chine) \<cdot> \<a>\<^sup>-\<^sup>1[f, (g \<star> h) \<star> (tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine] \<cdot> (f \<star> \<a>[g \<star> h, tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0, f\<^sub>0gh\<^sub>1.p\<^sub>0] \<star> TTfgh_TfTgh.chine) \<cdot> (f \<star> (\<a>\<^sup>-\<^sup>1[g, h, tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (f \<star> ((g \<star> \<a>[h, tab\<^sub>0 h, g\<^sub>0h\<^sub>1.p\<^sub>0]) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (f \<star> (((g \<star> h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (((g \<star> g\<^sub>0h\<^sub>1.\<phi>) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((\<a>[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (((g.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>1) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (f\<^sub>0gh\<^sub>1.\<phi> \<star> TTfgh_TfTgh.chine) \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>0 f, f\<^sub>0gh\<^sub>1.p\<^sub>1, TTfgh_TfTgh.chine]) \<cdot> \<a>[f, tab\<^sub>0 f, f\<^sub>0gh\<^sub>1.p\<^sub>1 \<star> TTfgh_TfTgh.chine] \<cdot> (f.tab \<star> f\<^sub>0gh\<^sub>1.p\<^sub>1 \<star> TTfgh_TfTgh.chine) \<cdot> \<a>[tab\<^sub>1 f, f\<^sub>0gh\<^sub>1.p\<^sub>1, TTfgh_TfTgh.chine] \<cdot> TTfgh_TfTgh.the_\<nu> \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>1 f, f\<^sub>0g\<^sub>1.p\<^sub>1, fg\<^sub>0h\<^sub>1.p\<^sub>1] = (f \<star> g \<star> h \<star> TTfgh_TfTgh.the_\<theta>) \<cdot> (\<a>[f, g, h \<star> ((tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine] \<cdot> \<a>[f \<star> g, h, ((tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine] \<cdot> (\<a>\<^sup>-\<^sup>1[f, g, h] \<star> ((tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> \<a>[f \<star> g \<star> h, (tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine] \<cdot> (\<a>\<^sup>-\<^sup>1[f, g \<star> h, (tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0] \<star> TTfgh_TfTgh.chine) \<cdot> \<a>\<^sup>-\<^sup>1[f, (g \<star> h) \<star> (tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine] \<cdot> (f \<star> \<a>[g \<star> h, tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0, f\<^sub>0gh\<^sub>1.p\<^sub>0] \<star> TTfgh_TfTgh.chine) \<cdot> (f \<star> (\<a>\<^sup>-\<^sup>1[g, h, tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (f \<star> ((g \<star> \<a>[h, tab\<^sub>0 h, g\<^sub>0h\<^sub>1.p\<^sub>0]) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine)) \<cdot> (f \<star> (((g \<star> h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (((g \<star> g\<^sub>0h\<^sub>1.\<phi>) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((\<a>[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (((g.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>1) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (f\<^sub>0gh\<^sub>1.\<phi> \<star> TTfgh_TfTgh.chine) \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>0 f, f\<^sub>0gh\<^sub>1.p\<^sub>1, TTfgh_TfTgh.chine]) \<cdot> \<a>[f, tab\<^sub>0 f, f\<^sub>0gh\<^sub>1.p\<^sub>1 \<star> TTfgh_TfTgh.chine] \<cdot> (f.tab \<star> f\<^sub>0gh\<^sub>1.p\<^sub>1 \<star> TTfgh_TfTgh.chine) \<cdot> \<a>[tab\<^sub>1 f, f\<^sub>0gh\<^sub>1.p\<^sub>1, TTfgh_TfTgh.chine] \<cdot> TTfgh_TfTgh.the_\<nu> \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>1 f, f\<^sub>0g\<^sub>1.p\<^sub>1, fg\<^sub>0h\<^sub>1.p\<^sub>1]
[PROOF STEP]
proof -
[PROOF STATE]
proof (state)
goal (1 subgoal):
1. (\<a>[f, g, h \<star> tab\<^sub>0 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0] \<cdot> \<a>[f \<star> g, h, tab\<^sub>0 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0] \<cdot> (\<a>\<^sup>-\<^sup>1[f, g, h] \<star> tab\<^sub>0 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0) \<cdot> ((f \<star> g \<star> h) \<star> TTfgh_TfTgh.the_\<theta>)) \<cdot> \<a>[f \<star> g \<star> h, (tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine] \<cdot> (\<a>\<^sup>-\<^sup>1[f, g \<star> h, (tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0] \<star> TTfgh_TfTgh.chine) \<cdot> \<a>\<^sup>-\<^sup>1[f, (g \<star> h) \<star> (tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine] \<cdot> (f \<star> \<a>[g \<star> h, tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0, f\<^sub>0gh\<^sub>1.p\<^sub>0] \<star> TTfgh_TfTgh.chine) \<cdot> (f \<star> (\<a>\<^sup>-\<^sup>1[g, h, tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (f \<star> ((g \<star> \<a>[h, tab\<^sub>0 h, g\<^sub>0h\<^sub>1.p\<^sub>0]) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (f \<star> (((g \<star> h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (((g \<star> g\<^sub>0h\<^sub>1.\<phi>) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((\<a>[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (((g.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>1) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (f\<^sub>0gh\<^sub>1.\<phi> \<star> TTfgh_TfTgh.chine) \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>0 f, f\<^sub>0gh\<^sub>1.p\<^sub>1, TTfgh_TfTgh.chine]) \<cdot> \<a>[f, tab\<^sub>0 f, f\<^sub>0gh\<^sub>1.p\<^sub>1 \<star> TTfgh_TfTgh.chine] \<cdot> (f.tab \<star> f\<^sub>0gh\<^sub>1.p\<^sub>1 \<star> TTfgh_TfTgh.chine) \<cdot> \<a>[tab\<^sub>1 f, f\<^sub>0gh\<^sub>1.p\<^sub>1, TTfgh_TfTgh.chine] \<cdot> TTfgh_TfTgh.the_\<nu> \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>1 f, f\<^sub>0g\<^sub>1.p\<^sub>1, fg\<^sub>0h\<^sub>1.p\<^sub>1] = (f \<star> g \<star> h \<star> TTfgh_TfTgh.the_\<theta>) \<cdot> (\<a>[f, g, h \<star> ((tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine] \<cdot> \<a>[f \<star> g, h, ((tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine] \<cdot> (\<a>\<^sup>-\<^sup>1[f, g, h] \<star> ((tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> \<a>[f \<star> g \<star> h, (tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine] \<cdot> (\<a>\<^sup>-\<^sup>1[f, g \<star> h, (tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0] \<star> TTfgh_TfTgh.chine) \<cdot> \<a>\<^sup>-\<^sup>1[f, (g \<star> h) \<star> (tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine] \<cdot> (f \<star> \<a>[g \<star> h, tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0, f\<^sub>0gh\<^sub>1.p\<^sub>0] \<star> TTfgh_TfTgh.chine) \<cdot> (f \<star> (\<a>\<^sup>-\<^sup>1[g, h, tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (f \<star> ((g \<star> \<a>[h, tab\<^sub>0 h, g\<^sub>0h\<^sub>1.p\<^sub>0]) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine)) \<cdot> (f \<star> (((g \<star> h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (((g \<star> g\<^sub>0h\<^sub>1.\<phi>) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((\<a>[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (((g.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>1) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (f\<^sub>0gh\<^sub>1.\<phi> \<star> TTfgh_TfTgh.chine) \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>0 f, f\<^sub>0gh\<^sub>1.p\<^sub>1, TTfgh_TfTgh.chine]) \<cdot> \<a>[f, tab\<^sub>0 f, f\<^sub>0gh\<^sub>1.p\<^sub>1 \<star> TTfgh_TfTgh.chine] \<cdot> (f.tab \<star> f\<^sub>0gh\<^sub>1.p\<^sub>1 \<star> TTfgh_TfTgh.chine) \<cdot> \<a>[tab\<^sub>1 f, f\<^sub>0gh\<^sub>1.p\<^sub>1, TTfgh_TfTgh.chine] \<cdot> TTfgh_TfTgh.the_\<nu> \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>1 f, f\<^sub>0g\<^sub>1.p\<^sub>1, fg\<^sub>0h\<^sub>1.p\<^sub>1]
[PROOF STEP]
have "\<a>[f, g, h \<star> tab\<^sub>0 h \<star> TTfgh.p\<^sub>0] \<cdot>
\<a>[f \<star> g, h, tab\<^sub>0 h \<star> TTfgh.p\<^sub>0] \<cdot>
(\<a>\<^sup>-\<^sup>1[f, g, h] \<star> tab\<^sub>0 h \<star> TTfgh.p\<^sub>0) \<cdot>
((f \<star> g \<star> h) \<star> TTfgh_TfTgh.the_\<theta>) =
\<a>[f, g, h \<star> tab\<^sub>0 h \<star> TTfgh.p\<^sub>0] \<cdot>
(\<a>[f \<star> g, h, tab\<^sub>0 h \<star> TTfgh.p\<^sub>0] \<cdot>
(((f \<star> g) \<star> h) \<star> TTfgh_TfTgh.the_\<theta>)) \<cdot>
(\<a>\<^sup>-\<^sup>1[f, g, h] \<star> ((tab\<^sub>0 h \<star> Tgh.p\<^sub>0) \<star> TfTgh.p\<^sub>0) \<star> TTfgh_TfTgh.chine)"
[PROOF STATE]
proof (prove)
goal (1 subgoal):
1. \<a>[f, g, h \<star> tab\<^sub>0 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0] \<cdot> \<a>[f \<star> g, h, tab\<^sub>0 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0] \<cdot> (\<a>\<^sup>-\<^sup>1[f, g, h] \<star> tab\<^sub>0 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0) \<cdot> ((f \<star> g \<star> h) \<star> TTfgh_TfTgh.the_\<theta>) = \<a>[f, g, h \<star> tab\<^sub>0 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0] \<cdot> (\<a>[f \<star> g, h, tab\<^sub>0 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0] \<cdot> (((f \<star> g) \<star> h) \<star> TTfgh_TfTgh.the_\<theta>)) \<cdot> (\<a>\<^sup>-\<^sup>1[f, g, h] \<star> ((tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine)
[PROOF STEP]
proof -
[PROOF STATE]
proof (state)
goal (1 subgoal):
1. \<a>[f, g, h \<star> tab\<^sub>0 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0] \<cdot> \<a>[f \<star> g, h, tab\<^sub>0 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0] \<cdot> (\<a>\<^sup>-\<^sup>1[f, g, h] \<star> tab\<^sub>0 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0) \<cdot> ((f \<star> g \<star> h) \<star> TTfgh_TfTgh.the_\<theta>) = \<a>[f, g, h \<star> tab\<^sub>0 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0] \<cdot> (\<a>[f \<star> g, h, tab\<^sub>0 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0] \<cdot> (((f \<star> g) \<star> h) \<star> TTfgh_TfTgh.the_\<theta>)) \<cdot> (\<a>\<^sup>-\<^sup>1[f, g, h] \<star> ((tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine)
[PROOF STEP]
have "(\<a>\<^sup>-\<^sup>1[f, g, h] \<star> tab\<^sub>0 h \<star> TTfgh.p\<^sub>0) \<cdot> ((f \<star> g \<star> h) \<star> TTfgh_TfTgh.the_\<theta>) =
\<a>\<^sup>-\<^sup>1[f, g, h] \<star> TTfgh_TfTgh.the_\<theta>"
[PROOF STATE]
proof (prove)
goal (1 subgoal):
1. (\<a>\<^sup>-\<^sup>1[f, g, h] \<star> tab\<^sub>0 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0) \<cdot> ((f \<star> g \<star> h) \<star> TTfgh_TfTgh.the_\<theta>) = \<a>\<^sup>-\<^sup>1[f, g, h] \<star> TTfgh_TfTgh.the_\<theta>
[PROOF STEP]
using fg gh comp_arr_dom comp_cod_arr
interchange [of "\<a>\<^sup>-\<^sup>1[f, g, h]" "f \<star> g \<star> h"
"tab\<^sub>0 h \<star> TTfgh.p\<^sub>0" TTfgh_TfTgh.the_\<theta>]
[PROOF STATE]
proof (prove)
using this:
src f = trg g
src g = trg h
\<lbrakk>arr ?f; local.dom ?f = ?a\<rbrakk> \<Longrightarrow> ?f \<cdot> ?a = ?f
\<lbrakk>arr ?f; cod ?f = ?b\<rbrakk> \<Longrightarrow> ?b \<cdot> ?f = ?f
\<lbrakk>seq \<a>\<^sup>-\<^sup>1[f, g, h] (f \<star> g \<star> h); seq (tab\<^sub>0 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0) TTfgh_TfTgh.the_\<theta>\<rbrakk> \<Longrightarrow> \<a>\<^sup>-\<^sup>1[f, g, h] \<cdot> (f \<star> g \<star> h) \<star> (tab\<^sub>0 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0) \<cdot> TTfgh_TfTgh.the_\<theta> = (\<a>\<^sup>-\<^sup>1[f, g, h] \<star> tab\<^sub>0 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0) \<cdot> ((f \<star> g \<star> h) \<star> TTfgh_TfTgh.the_\<theta>)
goal (1 subgoal):
1. (\<a>\<^sup>-\<^sup>1[f, g, h] \<star> tab\<^sub>0 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0) \<cdot> ((f \<star> g \<star> h) \<star> TTfgh_TfTgh.the_\<theta>) = \<a>\<^sup>-\<^sup>1[f, g, h] \<star> TTfgh_TfTgh.the_\<theta>
[PROOF STEP]
by simp
[PROOF STATE]
proof (state)
this:
(\<a>\<^sup>-\<^sup>1[f, g, h] \<star> tab\<^sub>0 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0) \<cdot> ((f \<star> g \<star> h) \<star> TTfgh_TfTgh.the_\<theta>) = \<a>\<^sup>-\<^sup>1[f, g, h] \<star> TTfgh_TfTgh.the_\<theta>
goal (1 subgoal):
1. \<a>[f, g, h \<star> tab\<^sub>0 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0] \<cdot> \<a>[f \<star> g, h, tab\<^sub>0 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0] \<cdot> (\<a>\<^sup>-\<^sup>1[f, g, h] \<star> tab\<^sub>0 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0) \<cdot> ((f \<star> g \<star> h) \<star> TTfgh_TfTgh.the_\<theta>) = \<a>[f, g, h \<star> tab\<^sub>0 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0] \<cdot> (\<a>[f \<star> g, h, tab\<^sub>0 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0] \<cdot> (((f \<star> g) \<star> h) \<star> TTfgh_TfTgh.the_\<theta>)) \<cdot> (\<a>\<^sup>-\<^sup>1[f, g, h] \<star> ((tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine)
[PROOF STEP]
also
[PROOF STATE]
proof (state)
this:
(\<a>\<^sup>-\<^sup>1[f, g, h] \<star> tab\<^sub>0 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0) \<cdot> ((f \<star> g \<star> h) \<star> TTfgh_TfTgh.the_\<theta>) = \<a>\<^sup>-\<^sup>1[f, g, h] \<star> TTfgh_TfTgh.the_\<theta>
goal (1 subgoal):
1. \<a>[f, g, h \<star> tab\<^sub>0 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0] \<cdot> \<a>[f \<star> g, h, tab\<^sub>0 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0] \<cdot> (\<a>\<^sup>-\<^sup>1[f, g, h] \<star> tab\<^sub>0 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0) \<cdot> ((f \<star> g \<star> h) \<star> TTfgh_TfTgh.the_\<theta>) = \<a>[f, g, h \<star> tab\<^sub>0 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0] \<cdot> (\<a>[f \<star> g, h, tab\<^sub>0 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0] \<cdot> (((f \<star> g) \<star> h) \<star> TTfgh_TfTgh.the_\<theta>)) \<cdot> (\<a>\<^sup>-\<^sup>1[f, g, h] \<star> ((tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine)
[PROOF STEP]
have "... = (((f \<star> g) \<star> h) \<star> TTfgh_TfTgh.the_\<theta>) \<cdot>
(\<a>\<^sup>-\<^sup>1[f, g, h] \<star> ((tab\<^sub>0 h \<star> Tgh.p\<^sub>0) \<star> TfTgh.p\<^sub>0) \<star> TTfgh_TfTgh.chine)"
[PROOF STATE]
proof (prove)
goal (1 subgoal):
1. \<a>\<^sup>-\<^sup>1[f, g, h] \<star> TTfgh_TfTgh.the_\<theta> = (((f \<star> g) \<star> h) \<star> TTfgh_TfTgh.the_\<theta>) \<cdot> (\<a>\<^sup>-\<^sup>1[f, g, h] \<star> ((tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine)
[PROOF STEP]
using fg gh comp_arr_dom comp_cod_arr
interchange [of "(f \<star> g) \<star> h" "\<a>\<^sup>-\<^sup>1[f, g, h]" TTfgh_TfTgh.the_\<theta>
"((tab\<^sub>0 h \<star> Tgh.p\<^sub>0) \<star> TfTgh.p\<^sub>0) \<star> TTfgh_TfTgh.chine"]
[PROOF STATE]
proof (prove)
using this:
src f = trg g
src g = trg h
\<lbrakk>arr ?f; local.dom ?f = ?a\<rbrakk> \<Longrightarrow> ?f \<cdot> ?a = ?f
\<lbrakk>arr ?f; cod ?f = ?b\<rbrakk> \<Longrightarrow> ?b \<cdot> ?f = ?f
\<lbrakk>seq ((f \<star> g) \<star> h) \<a>\<^sup>-\<^sup>1[f, g, h]; seq TTfgh_TfTgh.the_\<theta> (((tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine)\<rbrakk> \<Longrightarrow> ((f \<star> g) \<star> h) \<cdot> \<a>\<^sup>-\<^sup>1[f, g, h] \<star> TTfgh_TfTgh.the_\<theta> \<cdot> (((tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) = (((f \<star> g) \<star> h) \<star> TTfgh_TfTgh.the_\<theta>) \<cdot> (\<a>\<^sup>-\<^sup>1[f, g, h] \<star> ((tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine)
goal (1 subgoal):
1. \<a>\<^sup>-\<^sup>1[f, g, h] \<star> TTfgh_TfTgh.the_\<theta> = (((f \<star> g) \<star> h) \<star> TTfgh_TfTgh.the_\<theta>) \<cdot> (\<a>\<^sup>-\<^sup>1[f, g, h] \<star> ((tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine)
[PROOF STEP]
by simp
[PROOF STATE]
proof (state)
this:
\<a>\<^sup>-\<^sup>1[f, g, h] \<star> TTfgh_TfTgh.the_\<theta> = (((f \<star> g) \<star> h) \<star> TTfgh_TfTgh.the_\<theta>) \<cdot> (\<a>\<^sup>-\<^sup>1[f, g, h] \<star> ((tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine)
goal (1 subgoal):
1. \<a>[f, g, h \<star> tab\<^sub>0 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0] \<cdot> \<a>[f \<star> g, h, tab\<^sub>0 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0] \<cdot> (\<a>\<^sup>-\<^sup>1[f, g, h] \<star> tab\<^sub>0 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0) \<cdot> ((f \<star> g \<star> h) \<star> TTfgh_TfTgh.the_\<theta>) = \<a>[f, g, h \<star> tab\<^sub>0 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0] \<cdot> (\<a>[f \<star> g, h, tab\<^sub>0 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0] \<cdot> (((f \<star> g) \<star> h) \<star> TTfgh_TfTgh.the_\<theta>)) \<cdot> (\<a>\<^sup>-\<^sup>1[f, g, h] \<star> ((tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine)
[PROOF STEP]
finally
[PROOF STATE]
proof (chain)
picking this:
(\<a>\<^sup>-\<^sup>1[f, g, h] \<star> tab\<^sub>0 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0) \<cdot> ((f \<star> g \<star> h) \<star> TTfgh_TfTgh.the_\<theta>) = (((f \<star> g) \<star> h) \<star> TTfgh_TfTgh.the_\<theta>) \<cdot> (\<a>\<^sup>-\<^sup>1[f, g, h] \<star> ((tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine)
[PROOF STEP]
show ?thesis
[PROOF STATE]
proof (prove)
using this:
(\<a>\<^sup>-\<^sup>1[f, g, h] \<star> tab\<^sub>0 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0) \<cdot> ((f \<star> g \<star> h) \<star> TTfgh_TfTgh.the_\<theta>) = (((f \<star> g) \<star> h) \<star> TTfgh_TfTgh.the_\<theta>) \<cdot> (\<a>\<^sup>-\<^sup>1[f, g, h] \<star> ((tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine)
goal (1 subgoal):
1. \<a>[f, g, h \<star> tab\<^sub>0 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0] \<cdot> \<a>[f \<star> g, h, tab\<^sub>0 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0] \<cdot> (\<a>\<^sup>-\<^sup>1[f, g, h] \<star> tab\<^sub>0 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0) \<cdot> ((f \<star> g \<star> h) \<star> TTfgh_TfTgh.the_\<theta>) = \<a>[f, g, h \<star> tab\<^sub>0 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0] \<cdot> (\<a>[f \<star> g, h, tab\<^sub>0 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0] \<cdot> (((f \<star> g) \<star> h) \<star> TTfgh_TfTgh.the_\<theta>)) \<cdot> (\<a>\<^sup>-\<^sup>1[f, g, h] \<star> ((tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine)
[PROOF STEP]
using comp_assoc
[PROOF STATE]
proof (prove)
using this:
(\<a>\<^sup>-\<^sup>1[f, g, h] \<star> tab\<^sub>0 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0) \<cdot> ((f \<star> g \<star> h) \<star> TTfgh_TfTgh.the_\<theta>) = (((f \<star> g) \<star> h) \<star> TTfgh_TfTgh.the_\<theta>) \<cdot> (\<a>\<^sup>-\<^sup>1[f, g, h] \<star> ((tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine)
(?h \<cdot> ?g) \<cdot> ?f = ?h \<cdot> ?g \<cdot> ?f
goal (1 subgoal):
1. \<a>[f, g, h \<star> tab\<^sub>0 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0] \<cdot> \<a>[f \<star> g, h, tab\<^sub>0 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0] \<cdot> (\<a>\<^sup>-\<^sup>1[f, g, h] \<star> tab\<^sub>0 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0) \<cdot> ((f \<star> g \<star> h) \<star> TTfgh_TfTgh.the_\<theta>) = \<a>[f, g, h \<star> tab\<^sub>0 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0] \<cdot> (\<a>[f \<star> g, h, tab\<^sub>0 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0] \<cdot> (((f \<star> g) \<star> h) \<star> TTfgh_TfTgh.the_\<theta>)) \<cdot> (\<a>\<^sup>-\<^sup>1[f, g, h] \<star> ((tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine)
[PROOF STEP]
by presburger
[PROOF STATE]
proof (state)
this:
\<a>[f, g, h \<star> tab\<^sub>0 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0] \<cdot> \<a>[f \<star> g, h, tab\<^sub>0 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0] \<cdot> (\<a>\<^sup>-\<^sup>1[f, g, h] \<star> tab\<^sub>0 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0) \<cdot> ((f \<star> g \<star> h) \<star> TTfgh_TfTgh.the_\<theta>) = \<a>[f, g, h \<star> tab\<^sub>0 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0] \<cdot> (\<a>[f \<star> g, h, tab\<^sub>0 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0] \<cdot> (((f \<star> g) \<star> h) \<star> TTfgh_TfTgh.the_\<theta>)) \<cdot> (\<a>\<^sup>-\<^sup>1[f, g, h] \<star> ((tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine)
goal:
No subgoals!
[PROOF STEP]
qed
[PROOF STATE]
proof (state)
this:
\<a>[f, g, h \<star> tab\<^sub>0 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0] \<cdot> \<a>[f \<star> g, h, tab\<^sub>0 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0] \<cdot> (\<a>\<^sup>-\<^sup>1[f, g, h] \<star> tab\<^sub>0 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0) \<cdot> ((f \<star> g \<star> h) \<star> TTfgh_TfTgh.the_\<theta>) = \<a>[f, g, h \<star> tab\<^sub>0 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0] \<cdot> (\<a>[f \<star> g, h, tab\<^sub>0 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0] \<cdot> (((f \<star> g) \<star> h) \<star> TTfgh_TfTgh.the_\<theta>)) \<cdot> (\<a>\<^sup>-\<^sup>1[f, g, h] \<star> ((tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine)
goal (1 subgoal):
1. (\<a>[f, g, h \<star> tab\<^sub>0 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0] \<cdot> \<a>[f \<star> g, h, tab\<^sub>0 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0] \<cdot> (\<a>\<^sup>-\<^sup>1[f, g, h] \<star> tab\<^sub>0 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0) \<cdot> ((f \<star> g \<star> h) \<star> TTfgh_TfTgh.the_\<theta>)) \<cdot> \<a>[f \<star> g \<star> h, (tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine] \<cdot> (\<a>\<^sup>-\<^sup>1[f, g \<star> h, (tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0] \<star> TTfgh_TfTgh.chine) \<cdot> \<a>\<^sup>-\<^sup>1[f, (g \<star> h) \<star> (tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine] \<cdot> (f \<star> \<a>[g \<star> h, tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0, f\<^sub>0gh\<^sub>1.p\<^sub>0] \<star> TTfgh_TfTgh.chine) \<cdot> (f \<star> (\<a>\<^sup>-\<^sup>1[g, h, tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (f \<star> ((g \<star> \<a>[h, tab\<^sub>0 h, g\<^sub>0h\<^sub>1.p\<^sub>0]) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (f \<star> (((g \<star> h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (((g \<star> g\<^sub>0h\<^sub>1.\<phi>) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((\<a>[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (((g.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>1) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (f\<^sub>0gh\<^sub>1.\<phi> \<star> TTfgh_TfTgh.chine) \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>0 f, f\<^sub>0gh\<^sub>1.p\<^sub>1, TTfgh_TfTgh.chine]) \<cdot> \<a>[f, tab\<^sub>0 f, f\<^sub>0gh\<^sub>1.p\<^sub>1 \<star> TTfgh_TfTgh.chine] \<cdot> (f.tab \<star> f\<^sub>0gh\<^sub>1.p\<^sub>1 \<star> TTfgh_TfTgh.chine) \<cdot> \<a>[tab\<^sub>1 f, f\<^sub>0gh\<^sub>1.p\<^sub>1, TTfgh_TfTgh.chine] \<cdot> TTfgh_TfTgh.the_\<nu> \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>1 f, f\<^sub>0g\<^sub>1.p\<^sub>1, fg\<^sub>0h\<^sub>1.p\<^sub>1] = (f \<star> g \<star> h \<star> TTfgh_TfTgh.the_\<theta>) \<cdot> (\<a>[f, g, h \<star> ((tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine] \<cdot> \<a>[f \<star> g, h, ((tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine] \<cdot> (\<a>\<^sup>-\<^sup>1[f, g, h] \<star> ((tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> \<a>[f \<star> g \<star> h, (tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine] \<cdot> (\<a>\<^sup>-\<^sup>1[f, g \<star> h, (tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0] \<star> TTfgh_TfTgh.chine) \<cdot> \<a>\<^sup>-\<^sup>1[f, (g \<star> h) \<star> (tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine] \<cdot> (f \<star> \<a>[g \<star> h, tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0, f\<^sub>0gh\<^sub>1.p\<^sub>0] \<star> TTfgh_TfTgh.chine) \<cdot> (f \<star> (\<a>\<^sup>-\<^sup>1[g, h, tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (f \<star> ((g \<star> \<a>[h, tab\<^sub>0 h, g\<^sub>0h\<^sub>1.p\<^sub>0]) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine)) \<cdot> (f \<star> (((g \<star> h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (((g \<star> g\<^sub>0h\<^sub>1.\<phi>) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((\<a>[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (((g.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>1) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (f\<^sub>0gh\<^sub>1.\<phi> \<star> TTfgh_TfTgh.chine) \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>0 f, f\<^sub>0gh\<^sub>1.p\<^sub>1, TTfgh_TfTgh.chine]) \<cdot> \<a>[f, tab\<^sub>0 f, f\<^sub>0gh\<^sub>1.p\<^sub>1 \<star> TTfgh_TfTgh.chine] \<cdot> (f.tab \<star> f\<^sub>0gh\<^sub>1.p\<^sub>1 \<star> TTfgh_TfTgh.chine) \<cdot> \<a>[tab\<^sub>1 f, f\<^sub>0gh\<^sub>1.p\<^sub>1, TTfgh_TfTgh.chine] \<cdot> TTfgh_TfTgh.the_\<nu> \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>1 f, f\<^sub>0g\<^sub>1.p\<^sub>1, fg\<^sub>0h\<^sub>1.p\<^sub>1]
[PROOF STEP]
also
[PROOF STATE]
proof (state)
this:
\<a>[f, g, h \<star> tab\<^sub>0 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0] \<cdot> \<a>[f \<star> g, h, tab\<^sub>0 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0] \<cdot> (\<a>\<^sup>-\<^sup>1[f, g, h] \<star> tab\<^sub>0 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0) \<cdot> ((f \<star> g \<star> h) \<star> TTfgh_TfTgh.the_\<theta>) = \<a>[f, g, h \<star> tab\<^sub>0 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0] \<cdot> (\<a>[f \<star> g, h, tab\<^sub>0 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0] \<cdot> (((f \<star> g) \<star> h) \<star> TTfgh_TfTgh.the_\<theta>)) \<cdot> (\<a>\<^sup>-\<^sup>1[f, g, h] \<star> ((tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine)
goal (1 subgoal):
1. (\<a>[f, g, h \<star> tab\<^sub>0 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0] \<cdot> \<a>[f \<star> g, h, tab\<^sub>0 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0] \<cdot> (\<a>\<^sup>-\<^sup>1[f, g, h] \<star> tab\<^sub>0 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0) \<cdot> ((f \<star> g \<star> h) \<star> TTfgh_TfTgh.the_\<theta>)) \<cdot> \<a>[f \<star> g \<star> h, (tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine] \<cdot> (\<a>\<^sup>-\<^sup>1[f, g \<star> h, (tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0] \<star> TTfgh_TfTgh.chine) \<cdot> \<a>\<^sup>-\<^sup>1[f, (g \<star> h) \<star> (tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine] \<cdot> (f \<star> \<a>[g \<star> h, tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0, f\<^sub>0gh\<^sub>1.p\<^sub>0] \<star> TTfgh_TfTgh.chine) \<cdot> (f \<star> (\<a>\<^sup>-\<^sup>1[g, h, tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (f \<star> ((g \<star> \<a>[h, tab\<^sub>0 h, g\<^sub>0h\<^sub>1.p\<^sub>0]) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (f \<star> (((g \<star> h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (((g \<star> g\<^sub>0h\<^sub>1.\<phi>) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((\<a>[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (((g.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>1) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (f\<^sub>0gh\<^sub>1.\<phi> \<star> TTfgh_TfTgh.chine) \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>0 f, f\<^sub>0gh\<^sub>1.p\<^sub>1, TTfgh_TfTgh.chine]) \<cdot> \<a>[f, tab\<^sub>0 f, f\<^sub>0gh\<^sub>1.p\<^sub>1 \<star> TTfgh_TfTgh.chine] \<cdot> (f.tab \<star> f\<^sub>0gh\<^sub>1.p\<^sub>1 \<star> TTfgh_TfTgh.chine) \<cdot> \<a>[tab\<^sub>1 f, f\<^sub>0gh\<^sub>1.p\<^sub>1, TTfgh_TfTgh.chine] \<cdot> TTfgh_TfTgh.the_\<nu> \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>1 f, f\<^sub>0g\<^sub>1.p\<^sub>1, fg\<^sub>0h\<^sub>1.p\<^sub>1] = (f \<star> g \<star> h \<star> TTfgh_TfTgh.the_\<theta>) \<cdot> (\<a>[f, g, h \<star> ((tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine] \<cdot> \<a>[f \<star> g, h, ((tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine] \<cdot> (\<a>\<^sup>-\<^sup>1[f, g, h] \<star> ((tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> \<a>[f \<star> g \<star> h, (tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine] \<cdot> (\<a>\<^sup>-\<^sup>1[f, g \<star> h, (tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0] \<star> TTfgh_TfTgh.chine) \<cdot> \<a>\<^sup>-\<^sup>1[f, (g \<star> h) \<star> (tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine] \<cdot> (f \<star> \<a>[g \<star> h, tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0, f\<^sub>0gh\<^sub>1.p\<^sub>0] \<star> TTfgh_TfTgh.chine) \<cdot> (f \<star> (\<a>\<^sup>-\<^sup>1[g, h, tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (f \<star> ((g \<star> \<a>[h, tab\<^sub>0 h, g\<^sub>0h\<^sub>1.p\<^sub>0]) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine)) \<cdot> (f \<star> (((g \<star> h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (((g \<star> g\<^sub>0h\<^sub>1.\<phi>) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((\<a>[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (((g.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>1) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (f\<^sub>0gh\<^sub>1.\<phi> \<star> TTfgh_TfTgh.chine) \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>0 f, f\<^sub>0gh\<^sub>1.p\<^sub>1, TTfgh_TfTgh.chine]) \<cdot> \<a>[f, tab\<^sub>0 f, f\<^sub>0gh\<^sub>1.p\<^sub>1 \<star> TTfgh_TfTgh.chine] \<cdot> (f.tab \<star> f\<^sub>0gh\<^sub>1.p\<^sub>1 \<star> TTfgh_TfTgh.chine) \<cdot> \<a>[tab\<^sub>1 f, f\<^sub>0gh\<^sub>1.p\<^sub>1, TTfgh_TfTgh.chine] \<cdot> TTfgh_TfTgh.the_\<nu> \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>1 f, f\<^sub>0g\<^sub>1.p\<^sub>1, fg\<^sub>0h\<^sub>1.p\<^sub>1]
[PROOF STEP]
have "... =
(\<a>[f, g, h \<star> tab\<^sub>0 h \<star> TTfgh.p\<^sub>0] \<cdot>
((f \<star> g) \<star> h \<star> TTfgh_TfTgh.the_\<theta>)) \<cdot>
\<a>[f \<star> g, h, ((tab\<^sub>0 h \<star> Tgh.p\<^sub>0) \<star> TfTgh.p\<^sub>0) \<star> TTfgh_TfTgh.chine] \<cdot>
(\<a>\<^sup>-\<^sup>1[f, g, h] \<star> ((tab\<^sub>0 h \<star> Tgh.p\<^sub>0) \<star> TfTgh.p\<^sub>0) \<star> TTfgh_TfTgh.chine)"
[PROOF STATE]
proof (prove)
goal (1 subgoal):
1. \<a>[f, g, h \<star> tab\<^sub>0 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0] \<cdot> (\<a>[f \<star> g, h, tab\<^sub>0 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0] \<cdot> (((f \<star> g) \<star> h) \<star> TTfgh_TfTgh.the_\<theta>)) \<cdot> (\<a>\<^sup>-\<^sup>1[f, g, h] \<star> ((tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) = (\<a>[f, g, h \<star> tab\<^sub>0 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0] \<cdot> ((f \<star> g) \<star> h \<star> TTfgh_TfTgh.the_\<theta>)) \<cdot> \<a>[f \<star> g, h, ((tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine] \<cdot> (\<a>\<^sup>-\<^sup>1[f, g, h] \<star> ((tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine)
[PROOF STEP]
using fg gh assoc_naturality [of "f \<star> g" h TTfgh_TfTgh.the_\<theta>] comp_assoc
[PROOF STATE]
proof (prove)
using this:
src f = trg g
src g = trg h
\<lbrakk>hseq f g; arr h; arr TTfgh_TfTgh.the_\<theta>; src (f \<star> g) = trg h; src h = trg TTfgh_TfTgh.the_\<theta>\<rbrakk> \<Longrightarrow> \<a>[cod (f \<star> g), cod h, cod TTfgh_TfTgh.the_\<theta>] \<cdot> (((f \<star> g) \<star> h) \<star> TTfgh_TfTgh.the_\<theta>) = ((f \<star> g) \<star> h \<star> TTfgh_TfTgh.the_\<theta>) \<cdot> \<a>[local.dom (f \<star> g), local.dom h, local.dom TTfgh_TfTgh.the_\<theta>]
(?h \<cdot> ?g) \<cdot> ?f = ?h \<cdot> ?g \<cdot> ?f
goal (1 subgoal):
1. \<a>[f, g, h \<star> tab\<^sub>0 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0] \<cdot> (\<a>[f \<star> g, h, tab\<^sub>0 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0] \<cdot> (((f \<star> g) \<star> h) \<star> TTfgh_TfTgh.the_\<theta>)) \<cdot> (\<a>\<^sup>-\<^sup>1[f, g, h] \<star> ((tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) = (\<a>[f, g, h \<star> tab\<^sub>0 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0] \<cdot> ((f \<star> g) \<star> h \<star> TTfgh_TfTgh.the_\<theta>)) \<cdot> \<a>[f \<star> g, h, ((tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine] \<cdot> (\<a>\<^sup>-\<^sup>1[f, g, h] \<star> ((tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine)
[PROOF STEP]
by simp
[PROOF STATE]
proof (state)
this:
\<a>[f, g, h \<star> tab\<^sub>0 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0] \<cdot> (\<a>[f \<star> g, h, tab\<^sub>0 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0] \<cdot> (((f \<star> g) \<star> h) \<star> TTfgh_TfTgh.the_\<theta>)) \<cdot> (\<a>\<^sup>-\<^sup>1[f, g, h] \<star> ((tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) = (\<a>[f, g, h \<star> tab\<^sub>0 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0] \<cdot> ((f \<star> g) \<star> h \<star> TTfgh_TfTgh.the_\<theta>)) \<cdot> \<a>[f \<star> g, h, ((tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine] \<cdot> (\<a>\<^sup>-\<^sup>1[f, g, h] \<star> ((tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine)
goal (1 subgoal):
1. (\<a>[f, g, h \<star> tab\<^sub>0 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0] \<cdot> \<a>[f \<star> g, h, tab\<^sub>0 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0] \<cdot> (\<a>\<^sup>-\<^sup>1[f, g, h] \<star> tab\<^sub>0 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0) \<cdot> ((f \<star> g \<star> h) \<star> TTfgh_TfTgh.the_\<theta>)) \<cdot> \<a>[f \<star> g \<star> h, (tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine] \<cdot> (\<a>\<^sup>-\<^sup>1[f, g \<star> h, (tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0] \<star> TTfgh_TfTgh.chine) \<cdot> \<a>\<^sup>-\<^sup>1[f, (g \<star> h) \<star> (tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine] \<cdot> (f \<star> \<a>[g \<star> h, tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0, f\<^sub>0gh\<^sub>1.p\<^sub>0] \<star> TTfgh_TfTgh.chine) \<cdot> (f \<star> (\<a>\<^sup>-\<^sup>1[g, h, tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (f \<star> ((g \<star> \<a>[h, tab\<^sub>0 h, g\<^sub>0h\<^sub>1.p\<^sub>0]) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (f \<star> (((g \<star> h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (((g \<star> g\<^sub>0h\<^sub>1.\<phi>) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((\<a>[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (((g.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>1) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (f\<^sub>0gh\<^sub>1.\<phi> \<star> TTfgh_TfTgh.chine) \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>0 f, f\<^sub>0gh\<^sub>1.p\<^sub>1, TTfgh_TfTgh.chine]) \<cdot> \<a>[f, tab\<^sub>0 f, f\<^sub>0gh\<^sub>1.p\<^sub>1 \<star> TTfgh_TfTgh.chine] \<cdot> (f.tab \<star> f\<^sub>0gh\<^sub>1.p\<^sub>1 \<star> TTfgh_TfTgh.chine) \<cdot> \<a>[tab\<^sub>1 f, f\<^sub>0gh\<^sub>1.p\<^sub>1, TTfgh_TfTgh.chine] \<cdot> TTfgh_TfTgh.the_\<nu> \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>1 f, f\<^sub>0g\<^sub>1.p\<^sub>1, fg\<^sub>0h\<^sub>1.p\<^sub>1] = (f \<star> g \<star> h \<star> TTfgh_TfTgh.the_\<theta>) \<cdot> (\<a>[f, g, h \<star> ((tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine] \<cdot> \<a>[f \<star> g, h, ((tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine] \<cdot> (\<a>\<^sup>-\<^sup>1[f, g, h] \<star> ((tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> \<a>[f \<star> g \<star> h, (tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine] \<cdot> (\<a>\<^sup>-\<^sup>1[f, g \<star> h, (tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0] \<star> TTfgh_TfTgh.chine) \<cdot> \<a>\<^sup>-\<^sup>1[f, (g \<star> h) \<star> (tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine] \<cdot> (f \<star> \<a>[g \<star> h, tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0, f\<^sub>0gh\<^sub>1.p\<^sub>0] \<star> TTfgh_TfTgh.chine) \<cdot> (f \<star> (\<a>\<^sup>-\<^sup>1[g, h, tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (f \<star> ((g \<star> \<a>[h, tab\<^sub>0 h, g\<^sub>0h\<^sub>1.p\<^sub>0]) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine)) \<cdot> (f \<star> (((g \<star> h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (((g \<star> g\<^sub>0h\<^sub>1.\<phi>) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((\<a>[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (((g.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>1) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (f\<^sub>0gh\<^sub>1.\<phi> \<star> TTfgh_TfTgh.chine) \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>0 f, f\<^sub>0gh\<^sub>1.p\<^sub>1, TTfgh_TfTgh.chine]) \<cdot> \<a>[f, tab\<^sub>0 f, f\<^sub>0gh\<^sub>1.p\<^sub>1 \<star> TTfgh_TfTgh.chine] \<cdot> (f.tab \<star> f\<^sub>0gh\<^sub>1.p\<^sub>1 \<star> TTfgh_TfTgh.chine) \<cdot> \<a>[tab\<^sub>1 f, f\<^sub>0gh\<^sub>1.p\<^sub>1, TTfgh_TfTgh.chine] \<cdot> TTfgh_TfTgh.the_\<nu> \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>1 f, f\<^sub>0g\<^sub>1.p\<^sub>1, fg\<^sub>0h\<^sub>1.p\<^sub>1]
[PROOF STEP]
also
[PROOF STATE]
proof (state)
this:
\<a>[f, g, h \<star> tab\<^sub>0 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0] \<cdot> (\<a>[f \<star> g, h, tab\<^sub>0 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0] \<cdot> (((f \<star> g) \<star> h) \<star> TTfgh_TfTgh.the_\<theta>)) \<cdot> (\<a>\<^sup>-\<^sup>1[f, g, h] \<star> ((tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) = (\<a>[f, g, h \<star> tab\<^sub>0 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0] \<cdot> ((f \<star> g) \<star> h \<star> TTfgh_TfTgh.the_\<theta>)) \<cdot> \<a>[f \<star> g, h, ((tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine] \<cdot> (\<a>\<^sup>-\<^sup>1[f, g, h] \<star> ((tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine)
goal (1 subgoal):
1. (\<a>[f, g, h \<star> tab\<^sub>0 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0] \<cdot> \<a>[f \<star> g, h, tab\<^sub>0 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0] \<cdot> (\<a>\<^sup>-\<^sup>1[f, g, h] \<star> tab\<^sub>0 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0) \<cdot> ((f \<star> g \<star> h) \<star> TTfgh_TfTgh.the_\<theta>)) \<cdot> \<a>[f \<star> g \<star> h, (tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine] \<cdot> (\<a>\<^sup>-\<^sup>1[f, g \<star> h, (tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0] \<star> TTfgh_TfTgh.chine) \<cdot> \<a>\<^sup>-\<^sup>1[f, (g \<star> h) \<star> (tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine] \<cdot> (f \<star> \<a>[g \<star> h, tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0, f\<^sub>0gh\<^sub>1.p\<^sub>0] \<star> TTfgh_TfTgh.chine) \<cdot> (f \<star> (\<a>\<^sup>-\<^sup>1[g, h, tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (f \<star> ((g \<star> \<a>[h, tab\<^sub>0 h, g\<^sub>0h\<^sub>1.p\<^sub>0]) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (f \<star> (((g \<star> h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (((g \<star> g\<^sub>0h\<^sub>1.\<phi>) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((\<a>[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (((g.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>1) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (f\<^sub>0gh\<^sub>1.\<phi> \<star> TTfgh_TfTgh.chine) \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>0 f, f\<^sub>0gh\<^sub>1.p\<^sub>1, TTfgh_TfTgh.chine]) \<cdot> \<a>[f, tab\<^sub>0 f, f\<^sub>0gh\<^sub>1.p\<^sub>1 \<star> TTfgh_TfTgh.chine] \<cdot> (f.tab \<star> f\<^sub>0gh\<^sub>1.p\<^sub>1 \<star> TTfgh_TfTgh.chine) \<cdot> \<a>[tab\<^sub>1 f, f\<^sub>0gh\<^sub>1.p\<^sub>1, TTfgh_TfTgh.chine] \<cdot> TTfgh_TfTgh.the_\<nu> \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>1 f, f\<^sub>0g\<^sub>1.p\<^sub>1, fg\<^sub>0h\<^sub>1.p\<^sub>1] = (f \<star> g \<star> h \<star> TTfgh_TfTgh.the_\<theta>) \<cdot> (\<a>[f, g, h \<star> ((tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine] \<cdot> \<a>[f \<star> g, h, ((tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine] \<cdot> (\<a>\<^sup>-\<^sup>1[f, g, h] \<star> ((tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> \<a>[f \<star> g \<star> h, (tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine] \<cdot> (\<a>\<^sup>-\<^sup>1[f, g \<star> h, (tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0] \<star> TTfgh_TfTgh.chine) \<cdot> \<a>\<^sup>-\<^sup>1[f, (g \<star> h) \<star> (tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine] \<cdot> (f \<star> \<a>[g \<star> h, tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0, f\<^sub>0gh\<^sub>1.p\<^sub>0] \<star> TTfgh_TfTgh.chine) \<cdot> (f \<star> (\<a>\<^sup>-\<^sup>1[g, h, tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (f \<star> ((g \<star> \<a>[h, tab\<^sub>0 h, g\<^sub>0h\<^sub>1.p\<^sub>0]) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine)) \<cdot> (f \<star> (((g \<star> h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (((g \<star> g\<^sub>0h\<^sub>1.\<phi>) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((\<a>[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (((g.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>1) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (f\<^sub>0gh\<^sub>1.\<phi> \<star> TTfgh_TfTgh.chine) \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>0 f, f\<^sub>0gh\<^sub>1.p\<^sub>1, TTfgh_TfTgh.chine]) \<cdot> \<a>[f, tab\<^sub>0 f, f\<^sub>0gh\<^sub>1.p\<^sub>1 \<star> TTfgh_TfTgh.chine] \<cdot> (f.tab \<star> f\<^sub>0gh\<^sub>1.p\<^sub>1 \<star> TTfgh_TfTgh.chine) \<cdot> \<a>[tab\<^sub>1 f, f\<^sub>0gh\<^sub>1.p\<^sub>1, TTfgh_TfTgh.chine] \<cdot> TTfgh_TfTgh.the_\<nu> \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>1 f, f\<^sub>0g\<^sub>1.p\<^sub>1, fg\<^sub>0h\<^sub>1.p\<^sub>1]
[PROOF STEP]
have "... =
(f \<star> g \<star> h \<star> TTfgh_TfTgh.the_\<theta>) \<cdot>
\<a>[f, g, h \<star> ((tab\<^sub>0 h \<star> Tgh.p\<^sub>0) \<star> TfTgh.p\<^sub>0) \<star> TTfgh_TfTgh.chine] \<cdot>
\<a>[f \<star> g, h, ((tab\<^sub>0 h \<star> Tgh.p\<^sub>0) \<star> TfTgh.p\<^sub>0) \<star> TTfgh_TfTgh.chine] \<cdot>
(\<a>\<^sup>-\<^sup>1[f, g, h] \<star> ((tab\<^sub>0 h \<star> Tgh.p\<^sub>0) \<star> TfTgh.p\<^sub>0) \<star> TTfgh_TfTgh.chine)"
[PROOF STATE]
proof (prove)
goal (1 subgoal):
1. (\<a>[f, g, h \<star> tab\<^sub>0 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0] \<cdot> ((f \<star> g) \<star> h \<star> TTfgh_TfTgh.the_\<theta>)) \<cdot> \<a>[f \<star> g, h, ((tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine] \<cdot> (\<a>\<^sup>-\<^sup>1[f, g, h] \<star> ((tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) = (f \<star> g \<star> h \<star> TTfgh_TfTgh.the_\<theta>) \<cdot> \<a>[f, g, h \<star> ((tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine] \<cdot> \<a>[f \<star> g, h, ((tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine] \<cdot> (\<a>\<^sup>-\<^sup>1[f, g, h] \<star> ((tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine)
[PROOF STEP]
using fg gh assoc_naturality [of f g "h \<star> TTfgh_TfTgh.the_\<theta>"] comp_assoc
[PROOF STATE]
proof (prove)
using this:
src f = trg g
src g = trg h
\<lbrakk>arr f; arr g; hseq h TTfgh_TfTgh.the_\<theta>; src f = trg g; src g = trg (h \<star> TTfgh_TfTgh.the_\<theta>)\<rbrakk> \<Longrightarrow> \<a>[cod f, cod g, cod (h \<star> TTfgh_TfTgh.the_\<theta>)] \<cdot> ((f \<star> g) \<star> h \<star> TTfgh_TfTgh.the_\<theta>) = (f \<star> g \<star> h \<star> TTfgh_TfTgh.the_\<theta>) \<cdot> \<a>[local.dom f, local.dom g, local.dom (h \<star> TTfgh_TfTgh.the_\<theta>)]
(?h \<cdot> ?g) \<cdot> ?f = ?h \<cdot> ?g \<cdot> ?f
goal (1 subgoal):
1. (\<a>[f, g, h \<star> tab\<^sub>0 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0] \<cdot> ((f \<star> g) \<star> h \<star> TTfgh_TfTgh.the_\<theta>)) \<cdot> \<a>[f \<star> g, h, ((tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine] \<cdot> (\<a>\<^sup>-\<^sup>1[f, g, h] \<star> ((tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) = (f \<star> g \<star> h \<star> TTfgh_TfTgh.the_\<theta>) \<cdot> \<a>[f, g, h \<star> ((tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine] \<cdot> \<a>[f \<star> g, h, ((tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine] \<cdot> (\<a>\<^sup>-\<^sup>1[f, g, h] \<star> ((tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine)
[PROOF STEP]
by simp
[PROOF STATE]
proof (state)
this:
(\<a>[f, g, h \<star> tab\<^sub>0 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0] \<cdot> ((f \<star> g) \<star> h \<star> TTfgh_TfTgh.the_\<theta>)) \<cdot> \<a>[f \<star> g, h, ((tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine] \<cdot> (\<a>\<^sup>-\<^sup>1[f, g, h] \<star> ((tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) = (f \<star> g \<star> h \<star> TTfgh_TfTgh.the_\<theta>) \<cdot> \<a>[f, g, h \<star> ((tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine] \<cdot> \<a>[f \<star> g, h, ((tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine] \<cdot> (\<a>\<^sup>-\<^sup>1[f, g, h] \<star> ((tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine)
goal (1 subgoal):
1. (\<a>[f, g, h \<star> tab\<^sub>0 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0] \<cdot> \<a>[f \<star> g, h, tab\<^sub>0 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0] \<cdot> (\<a>\<^sup>-\<^sup>1[f, g, h] \<star> tab\<^sub>0 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0) \<cdot> ((f \<star> g \<star> h) \<star> TTfgh_TfTgh.the_\<theta>)) \<cdot> \<a>[f \<star> g \<star> h, (tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine] \<cdot> (\<a>\<^sup>-\<^sup>1[f, g \<star> h, (tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0] \<star> TTfgh_TfTgh.chine) \<cdot> \<a>\<^sup>-\<^sup>1[f, (g \<star> h) \<star> (tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine] \<cdot> (f \<star> \<a>[g \<star> h, tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0, f\<^sub>0gh\<^sub>1.p\<^sub>0] \<star> TTfgh_TfTgh.chine) \<cdot> (f \<star> (\<a>\<^sup>-\<^sup>1[g, h, tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (f \<star> ((g \<star> \<a>[h, tab\<^sub>0 h, g\<^sub>0h\<^sub>1.p\<^sub>0]) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (f \<star> (((g \<star> h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (((g \<star> g\<^sub>0h\<^sub>1.\<phi>) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((\<a>[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (((g.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>1) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (f\<^sub>0gh\<^sub>1.\<phi> \<star> TTfgh_TfTgh.chine) \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>0 f, f\<^sub>0gh\<^sub>1.p\<^sub>1, TTfgh_TfTgh.chine]) \<cdot> \<a>[f, tab\<^sub>0 f, f\<^sub>0gh\<^sub>1.p\<^sub>1 \<star> TTfgh_TfTgh.chine] \<cdot> (f.tab \<star> f\<^sub>0gh\<^sub>1.p\<^sub>1 \<star> TTfgh_TfTgh.chine) \<cdot> \<a>[tab\<^sub>1 f, f\<^sub>0gh\<^sub>1.p\<^sub>1, TTfgh_TfTgh.chine] \<cdot> TTfgh_TfTgh.the_\<nu> \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>1 f, f\<^sub>0g\<^sub>1.p\<^sub>1, fg\<^sub>0h\<^sub>1.p\<^sub>1] = (f \<star> g \<star> h \<star> TTfgh_TfTgh.the_\<theta>) \<cdot> (\<a>[f, g, h \<star> ((tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine] \<cdot> \<a>[f \<star> g, h, ((tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine] \<cdot> (\<a>\<^sup>-\<^sup>1[f, g, h] \<star> ((tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> \<a>[f \<star> g \<star> h, (tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine] \<cdot> (\<a>\<^sup>-\<^sup>1[f, g \<star> h, (tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0] \<star> TTfgh_TfTgh.chine) \<cdot> \<a>\<^sup>-\<^sup>1[f, (g \<star> h) \<star> (tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine] \<cdot> (f \<star> \<a>[g \<star> h, tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0, f\<^sub>0gh\<^sub>1.p\<^sub>0] \<star> TTfgh_TfTgh.chine) \<cdot> (f \<star> (\<a>\<^sup>-\<^sup>1[g, h, tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (f \<star> ((g \<star> \<a>[h, tab\<^sub>0 h, g\<^sub>0h\<^sub>1.p\<^sub>0]) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine)) \<cdot> (f \<star> (((g \<star> h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (((g \<star> g\<^sub>0h\<^sub>1.\<phi>) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((\<a>[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (((g.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>1) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (f\<^sub>0gh\<^sub>1.\<phi> \<star> TTfgh_TfTgh.chine) \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>0 f, f\<^sub>0gh\<^sub>1.p\<^sub>1, TTfgh_TfTgh.chine]) \<cdot> \<a>[f, tab\<^sub>0 f, f\<^sub>0gh\<^sub>1.p\<^sub>1 \<star> TTfgh_TfTgh.chine] \<cdot> (f.tab \<star> f\<^sub>0gh\<^sub>1.p\<^sub>1 \<star> TTfgh_TfTgh.chine) \<cdot> \<a>[tab\<^sub>1 f, f\<^sub>0gh\<^sub>1.p\<^sub>1, TTfgh_TfTgh.chine] \<cdot> TTfgh_TfTgh.the_\<nu> \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>1 f, f\<^sub>0g\<^sub>1.p\<^sub>1, fg\<^sub>0h\<^sub>1.p\<^sub>1]
[PROOF STEP]
finally
[PROOF STATE]
proof (chain)
picking this:
\<a>[f, g, h \<star> tab\<^sub>0 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0] \<cdot> \<a>[f \<star> g, h, tab\<^sub>0 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0] \<cdot> (\<a>\<^sup>-\<^sup>1[f, g, h] \<star> tab\<^sub>0 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0) \<cdot> ((f \<star> g \<star> h) \<star> TTfgh_TfTgh.the_\<theta>) = (f \<star> g \<star> h \<star> TTfgh_TfTgh.the_\<theta>) \<cdot> \<a>[f, g, h \<star> ((tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine] \<cdot> \<a>[f \<star> g, h, ((tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine] \<cdot> (\<a>\<^sup>-\<^sup>1[f, g, h] \<star> ((tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine)
[PROOF STEP]
show ?thesis
[PROOF STATE]
proof (prove)
using this:
\<a>[f, g, h \<star> tab\<^sub>0 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0] \<cdot> \<a>[f \<star> g, h, tab\<^sub>0 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0] \<cdot> (\<a>\<^sup>-\<^sup>1[f, g, h] \<star> tab\<^sub>0 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0) \<cdot> ((f \<star> g \<star> h) \<star> TTfgh_TfTgh.the_\<theta>) = (f \<star> g \<star> h \<star> TTfgh_TfTgh.the_\<theta>) \<cdot> \<a>[f, g, h \<star> ((tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine] \<cdot> \<a>[f \<star> g, h, ((tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine] \<cdot> (\<a>\<^sup>-\<^sup>1[f, g, h] \<star> ((tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine)
goal (1 subgoal):
1. (\<a>[f, g, h \<star> tab\<^sub>0 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0] \<cdot> \<a>[f \<star> g, h, tab\<^sub>0 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0] \<cdot> (\<a>\<^sup>-\<^sup>1[f, g, h] \<star> tab\<^sub>0 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0) \<cdot> ((f \<star> g \<star> h) \<star> TTfgh_TfTgh.the_\<theta>)) \<cdot> \<a>[f \<star> g \<star> h, (tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine] \<cdot> (\<a>\<^sup>-\<^sup>1[f, g \<star> h, (tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0] \<star> TTfgh_TfTgh.chine) \<cdot> \<a>\<^sup>-\<^sup>1[f, (g \<star> h) \<star> (tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine] \<cdot> (f \<star> \<a>[g \<star> h, tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0, f\<^sub>0gh\<^sub>1.p\<^sub>0] \<star> TTfgh_TfTgh.chine) \<cdot> (f \<star> (\<a>\<^sup>-\<^sup>1[g, h, tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (f \<star> ((g \<star> \<a>[h, tab\<^sub>0 h, g\<^sub>0h\<^sub>1.p\<^sub>0]) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (f \<star> (((g \<star> h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (((g \<star> g\<^sub>0h\<^sub>1.\<phi>) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((\<a>[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (((g.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>1) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (f\<^sub>0gh\<^sub>1.\<phi> \<star> TTfgh_TfTgh.chine) \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>0 f, f\<^sub>0gh\<^sub>1.p\<^sub>1, TTfgh_TfTgh.chine]) \<cdot> \<a>[f, tab\<^sub>0 f, f\<^sub>0gh\<^sub>1.p\<^sub>1 \<star> TTfgh_TfTgh.chine] \<cdot> (f.tab \<star> f\<^sub>0gh\<^sub>1.p\<^sub>1 \<star> TTfgh_TfTgh.chine) \<cdot> \<a>[tab\<^sub>1 f, f\<^sub>0gh\<^sub>1.p\<^sub>1, TTfgh_TfTgh.chine] \<cdot> TTfgh_TfTgh.the_\<nu> \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>1 f, f\<^sub>0g\<^sub>1.p\<^sub>1, fg\<^sub>0h\<^sub>1.p\<^sub>1] = (f \<star> g \<star> h \<star> TTfgh_TfTgh.the_\<theta>) \<cdot> (\<a>[f, g, h \<star> ((tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine] \<cdot> \<a>[f \<star> g, h, ((tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine] \<cdot> (\<a>\<^sup>-\<^sup>1[f, g, h] \<star> ((tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> \<a>[f \<star> g \<star> h, (tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine] \<cdot> (\<a>\<^sup>-\<^sup>1[f, g \<star> h, (tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0] \<star> TTfgh_TfTgh.chine) \<cdot> \<a>\<^sup>-\<^sup>1[f, (g \<star> h) \<star> (tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine] \<cdot> (f \<star> \<a>[g \<star> h, tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0, f\<^sub>0gh\<^sub>1.p\<^sub>0] \<star> TTfgh_TfTgh.chine) \<cdot> (f \<star> (\<a>\<^sup>-\<^sup>1[g, h, tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (f \<star> ((g \<star> \<a>[h, tab\<^sub>0 h, g\<^sub>0h\<^sub>1.p\<^sub>0]) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine)) \<cdot> (f \<star> (((g \<star> h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (((g \<star> g\<^sub>0h\<^sub>1.\<phi>) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((\<a>[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (((g.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>1) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (f\<^sub>0gh\<^sub>1.\<phi> \<star> TTfgh_TfTgh.chine) \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>0 f, f\<^sub>0gh\<^sub>1.p\<^sub>1, TTfgh_TfTgh.chine]) \<cdot> \<a>[f, tab\<^sub>0 f, f\<^sub>0gh\<^sub>1.p\<^sub>1 \<star> TTfgh_TfTgh.chine] \<cdot> (f.tab \<star> f\<^sub>0gh\<^sub>1.p\<^sub>1 \<star> TTfgh_TfTgh.chine) \<cdot> \<a>[tab\<^sub>1 f, f\<^sub>0gh\<^sub>1.p\<^sub>1, TTfgh_TfTgh.chine] \<cdot> TTfgh_TfTgh.the_\<nu> \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>1 f, f\<^sub>0g\<^sub>1.p\<^sub>1, fg\<^sub>0h\<^sub>1.p\<^sub>1]
[PROOF STEP]
using comp_assoc
[PROOF STATE]
proof (prove)
using this:
\<a>[f, g, h \<star> tab\<^sub>0 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0] \<cdot> \<a>[f \<star> g, h, tab\<^sub>0 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0] \<cdot> (\<a>\<^sup>-\<^sup>1[f, g, h] \<star> tab\<^sub>0 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0) \<cdot> ((f \<star> g \<star> h) \<star> TTfgh_TfTgh.the_\<theta>) = (f \<star> g \<star> h \<star> TTfgh_TfTgh.the_\<theta>) \<cdot> \<a>[f, g, h \<star> ((tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine] \<cdot> \<a>[f \<star> g, h, ((tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine] \<cdot> (\<a>\<^sup>-\<^sup>1[f, g, h] \<star> ((tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine)
(?h \<cdot> ?g) \<cdot> ?f = ?h \<cdot> ?g \<cdot> ?f
goal (1 subgoal):
1. (\<a>[f, g, h \<star> tab\<^sub>0 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0] \<cdot> \<a>[f \<star> g, h, tab\<^sub>0 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0] \<cdot> (\<a>\<^sup>-\<^sup>1[f, g, h] \<star> tab\<^sub>0 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0) \<cdot> ((f \<star> g \<star> h) \<star> TTfgh_TfTgh.the_\<theta>)) \<cdot> \<a>[f \<star> g \<star> h, (tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine] \<cdot> (\<a>\<^sup>-\<^sup>1[f, g \<star> h, (tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0] \<star> TTfgh_TfTgh.chine) \<cdot> \<a>\<^sup>-\<^sup>1[f, (g \<star> h) \<star> (tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine] \<cdot> (f \<star> \<a>[g \<star> h, tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0, f\<^sub>0gh\<^sub>1.p\<^sub>0] \<star> TTfgh_TfTgh.chine) \<cdot> (f \<star> (\<a>\<^sup>-\<^sup>1[g, h, tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (f \<star> ((g \<star> \<a>[h, tab\<^sub>0 h, g\<^sub>0h\<^sub>1.p\<^sub>0]) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (f \<star> (((g \<star> h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (((g \<star> g\<^sub>0h\<^sub>1.\<phi>) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((\<a>[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (((g.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>1) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (f\<^sub>0gh\<^sub>1.\<phi> \<star> TTfgh_TfTgh.chine) \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>0 f, f\<^sub>0gh\<^sub>1.p\<^sub>1, TTfgh_TfTgh.chine]) \<cdot> \<a>[f, tab\<^sub>0 f, f\<^sub>0gh\<^sub>1.p\<^sub>1 \<star> TTfgh_TfTgh.chine] \<cdot> (f.tab \<star> f\<^sub>0gh\<^sub>1.p\<^sub>1 \<star> TTfgh_TfTgh.chine) \<cdot> \<a>[tab\<^sub>1 f, f\<^sub>0gh\<^sub>1.p\<^sub>1, TTfgh_TfTgh.chine] \<cdot> TTfgh_TfTgh.the_\<nu> \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>1 f, f\<^sub>0g\<^sub>1.p\<^sub>1, fg\<^sub>0h\<^sub>1.p\<^sub>1] = (f \<star> g \<star> h \<star> TTfgh_TfTgh.the_\<theta>) \<cdot> (\<a>[f, g, h \<star> ((tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine] \<cdot> \<a>[f \<star> g, h, ((tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine] \<cdot> (\<a>\<^sup>-\<^sup>1[f, g, h] \<star> ((tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> \<a>[f \<star> g \<star> h, (tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine] \<cdot> (\<a>\<^sup>-\<^sup>1[f, g \<star> h, (tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0] \<star> TTfgh_TfTgh.chine) \<cdot> \<a>\<^sup>-\<^sup>1[f, (g \<star> h) \<star> (tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine] \<cdot> (f \<star> \<a>[g \<star> h, tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0, f\<^sub>0gh\<^sub>1.p\<^sub>0] \<star> TTfgh_TfTgh.chine) \<cdot> (f \<star> (\<a>\<^sup>-\<^sup>1[g, h, tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (f \<star> ((g \<star> \<a>[h, tab\<^sub>0 h, g\<^sub>0h\<^sub>1.p\<^sub>0]) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine)) \<cdot> (f \<star> (((g \<star> h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (((g \<star> g\<^sub>0h\<^sub>1.\<phi>) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((\<a>[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (((g.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>1) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (f\<^sub>0gh\<^sub>1.\<phi> \<star> TTfgh_TfTgh.chine) \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>0 f, f\<^sub>0gh\<^sub>1.p\<^sub>1, TTfgh_TfTgh.chine]) \<cdot> \<a>[f, tab\<^sub>0 f, f\<^sub>0gh\<^sub>1.p\<^sub>1 \<star> TTfgh_TfTgh.chine] \<cdot> (f.tab \<star> f\<^sub>0gh\<^sub>1.p\<^sub>1 \<star> TTfgh_TfTgh.chine) \<cdot> \<a>[tab\<^sub>1 f, f\<^sub>0gh\<^sub>1.p\<^sub>1, TTfgh_TfTgh.chine] \<cdot> TTfgh_TfTgh.the_\<nu> \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>1 f, f\<^sub>0g\<^sub>1.p\<^sub>1, fg\<^sub>0h\<^sub>1.p\<^sub>1]
[PROOF STEP]
by presburger
[PROOF STATE]
proof (state)
this:
(\<a>[f, g, h \<star> tab\<^sub>0 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0] \<cdot> \<a>[f \<star> g, h, tab\<^sub>0 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0] \<cdot> (\<a>\<^sup>-\<^sup>1[f, g, h] \<star> tab\<^sub>0 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0) \<cdot> ((f \<star> g \<star> h) \<star> TTfgh_TfTgh.the_\<theta>)) \<cdot> \<a>[f \<star> g \<star> h, (tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine] \<cdot> (\<a>\<^sup>-\<^sup>1[f, g \<star> h, (tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0] \<star> TTfgh_TfTgh.chine) \<cdot> \<a>\<^sup>-\<^sup>1[f, (g \<star> h) \<star> (tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine] \<cdot> (f \<star> \<a>[g \<star> h, tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0, f\<^sub>0gh\<^sub>1.p\<^sub>0] \<star> TTfgh_TfTgh.chine) \<cdot> (f \<star> (\<a>\<^sup>-\<^sup>1[g, h, tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (f \<star> ((g \<star> \<a>[h, tab\<^sub>0 h, g\<^sub>0h\<^sub>1.p\<^sub>0]) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (f \<star> (((g \<star> h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (((g \<star> g\<^sub>0h\<^sub>1.\<phi>) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((\<a>[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (((g.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>1) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (f\<^sub>0gh\<^sub>1.\<phi> \<star> TTfgh_TfTgh.chine) \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>0 f, f\<^sub>0gh\<^sub>1.p\<^sub>1, TTfgh_TfTgh.chine]) \<cdot> \<a>[f, tab\<^sub>0 f, f\<^sub>0gh\<^sub>1.p\<^sub>1 \<star> TTfgh_TfTgh.chine] \<cdot> (f.tab \<star> f\<^sub>0gh\<^sub>1.p\<^sub>1 \<star> TTfgh_TfTgh.chine) \<cdot> \<a>[tab\<^sub>1 f, f\<^sub>0gh\<^sub>1.p\<^sub>1, TTfgh_TfTgh.chine] \<cdot> TTfgh_TfTgh.the_\<nu> \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>1 f, f\<^sub>0g\<^sub>1.p\<^sub>1, fg\<^sub>0h\<^sub>1.p\<^sub>1] = (f \<star> g \<star> h \<star> TTfgh_TfTgh.the_\<theta>) \<cdot> (\<a>[f, g, h \<star> ((tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine] \<cdot> \<a>[f \<star> g, h, ((tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine] \<cdot> (\<a>\<^sup>-\<^sup>1[f, g, h] \<star> ((tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> \<a>[f \<star> g \<star> h, (tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine] \<cdot> (\<a>\<^sup>-\<^sup>1[f, g \<star> h, (tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0] \<star> TTfgh_TfTgh.chine) \<cdot> \<a>\<^sup>-\<^sup>1[f, (g \<star> h) \<star> (tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine] \<cdot> (f \<star> \<a>[g \<star> h, tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0, f\<^sub>0gh\<^sub>1.p\<^sub>0] \<star> TTfgh_TfTgh.chine) \<cdot> (f \<star> (\<a>\<^sup>-\<^sup>1[g, h, tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (f \<star> ((g \<star> \<a>[h, tab\<^sub>0 h, g\<^sub>0h\<^sub>1.p\<^sub>0]) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine)) \<cdot> (f \<star> (((g \<star> h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (((g \<star> g\<^sub>0h\<^sub>1.\<phi>) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((\<a>[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (((g.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>1) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (f\<^sub>0gh\<^sub>1.\<phi> \<star> TTfgh_TfTgh.chine) \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>0 f, f\<^sub>0gh\<^sub>1.p\<^sub>1, TTfgh_TfTgh.chine]) \<cdot> \<a>[f, tab\<^sub>0 f, f\<^sub>0gh\<^sub>1.p\<^sub>1 \<star> TTfgh_TfTgh.chine] \<cdot> (f.tab \<star> f\<^sub>0gh\<^sub>1.p\<^sub>1 \<star> TTfgh_TfTgh.chine) \<cdot> \<a>[tab\<^sub>1 f, f\<^sub>0gh\<^sub>1.p\<^sub>1, TTfgh_TfTgh.chine] \<cdot> TTfgh_TfTgh.the_\<nu> \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>1 f, f\<^sub>0g\<^sub>1.p\<^sub>1, fg\<^sub>0h\<^sub>1.p\<^sub>1]
goal:
No subgoals!
[PROOF STEP]
qed
[PROOF STATE]
proof (state)
this:
(\<a>[f, g, h \<star> tab\<^sub>0 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0] \<cdot> \<a>[f \<star> g, h, tab\<^sub>0 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0] \<cdot> (\<a>\<^sup>-\<^sup>1[f, g, h] \<star> tab\<^sub>0 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0) \<cdot> ((f \<star> g \<star> h) \<star> TTfgh_TfTgh.the_\<theta>)) \<cdot> \<a>[f \<star> g \<star> h, (tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine] \<cdot> (\<a>\<^sup>-\<^sup>1[f, g \<star> h, (tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0] \<star> TTfgh_TfTgh.chine) \<cdot> \<a>\<^sup>-\<^sup>1[f, (g \<star> h) \<star> (tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine] \<cdot> (f \<star> \<a>[g \<star> h, tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0, f\<^sub>0gh\<^sub>1.p\<^sub>0] \<star> TTfgh_TfTgh.chine) \<cdot> (f \<star> (\<a>\<^sup>-\<^sup>1[g, h, tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (f \<star> ((g \<star> \<a>[h, tab\<^sub>0 h, g\<^sub>0h\<^sub>1.p\<^sub>0]) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (f \<star> (((g \<star> h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (((g \<star> g\<^sub>0h\<^sub>1.\<phi>) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((\<a>[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (((g.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>1) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (f\<^sub>0gh\<^sub>1.\<phi> \<star> TTfgh_TfTgh.chine) \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>0 f, f\<^sub>0gh\<^sub>1.p\<^sub>1, TTfgh_TfTgh.chine]) \<cdot> \<a>[f, tab\<^sub>0 f, f\<^sub>0gh\<^sub>1.p\<^sub>1 \<star> TTfgh_TfTgh.chine] \<cdot> (f.tab \<star> f\<^sub>0gh\<^sub>1.p\<^sub>1 \<star> TTfgh_TfTgh.chine) \<cdot> \<a>[tab\<^sub>1 f, f\<^sub>0gh\<^sub>1.p\<^sub>1, TTfgh_TfTgh.chine] \<cdot> TTfgh_TfTgh.the_\<nu> \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>1 f, f\<^sub>0g\<^sub>1.p\<^sub>1, fg\<^sub>0h\<^sub>1.p\<^sub>1] = (f \<star> g \<star> h \<star> TTfgh_TfTgh.the_\<theta>) \<cdot> (\<a>[f, g, h \<star> ((tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine] \<cdot> \<a>[f \<star> g, h, ((tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine] \<cdot> (\<a>\<^sup>-\<^sup>1[f, g, h] \<star> ((tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> \<a>[f \<star> g \<star> h, (tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine] \<cdot> (\<a>\<^sup>-\<^sup>1[f, g \<star> h, (tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0] \<star> TTfgh_TfTgh.chine) \<cdot> \<a>\<^sup>-\<^sup>1[f, (g \<star> h) \<star> (tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine] \<cdot> (f \<star> \<a>[g \<star> h, tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0, f\<^sub>0gh\<^sub>1.p\<^sub>0] \<star> TTfgh_TfTgh.chine) \<cdot> (f \<star> (\<a>\<^sup>-\<^sup>1[g, h, tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (f \<star> ((g \<star> \<a>[h, tab\<^sub>0 h, g\<^sub>0h\<^sub>1.p\<^sub>0]) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine)) \<cdot> (f \<star> (((g \<star> h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (((g \<star> g\<^sub>0h\<^sub>1.\<phi>) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((\<a>[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (((g.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>1) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (f\<^sub>0gh\<^sub>1.\<phi> \<star> TTfgh_TfTgh.chine) \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>0 f, f\<^sub>0gh\<^sub>1.p\<^sub>1, TTfgh_TfTgh.chine]) \<cdot> \<a>[f, tab\<^sub>0 f, f\<^sub>0gh\<^sub>1.p\<^sub>1 \<star> TTfgh_TfTgh.chine] \<cdot> (f.tab \<star> f\<^sub>0gh\<^sub>1.p\<^sub>1 \<star> TTfgh_TfTgh.chine) \<cdot> \<a>[tab\<^sub>1 f, f\<^sub>0gh\<^sub>1.p\<^sub>1, TTfgh_TfTgh.chine] \<cdot> TTfgh_TfTgh.the_\<nu> \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>1 f, f\<^sub>0g\<^sub>1.p\<^sub>1, fg\<^sub>0h\<^sub>1.p\<^sub>1]
goal (1 subgoal):
1. (f \<star> \<theta>\<^sub>f) \<cdot> \<a>[f, tab\<^sub>0 f, w\<^sub>f] \<cdot> (f.tab \<star> w\<^sub>f) = ((f \<star> \<theta>\<^sub>f') \<cdot> \<a>[f, tab\<^sub>0 f, w\<^sub>f'] \<cdot> (f.tab \<star> w\<^sub>f')) \<cdot> \<beta>\<^sub>f
[PROOF STEP]
also
[PROOF STATE]
proof (state)
this:
(\<a>[f, g, h \<star> tab\<^sub>0 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0] \<cdot> \<a>[f \<star> g, h, tab\<^sub>0 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0] \<cdot> (\<a>\<^sup>-\<^sup>1[f, g, h] \<star> tab\<^sub>0 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0) \<cdot> ((f \<star> g \<star> h) \<star> TTfgh_TfTgh.the_\<theta>)) \<cdot> \<a>[f \<star> g \<star> h, (tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine] \<cdot> (\<a>\<^sup>-\<^sup>1[f, g \<star> h, (tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0] \<star> TTfgh_TfTgh.chine) \<cdot> \<a>\<^sup>-\<^sup>1[f, (g \<star> h) \<star> (tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine] \<cdot> (f \<star> \<a>[g \<star> h, tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0, f\<^sub>0gh\<^sub>1.p\<^sub>0] \<star> TTfgh_TfTgh.chine) \<cdot> (f \<star> (\<a>\<^sup>-\<^sup>1[g, h, tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (f \<star> ((g \<star> \<a>[h, tab\<^sub>0 h, g\<^sub>0h\<^sub>1.p\<^sub>0]) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (f \<star> (((g \<star> h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (((g \<star> g\<^sub>0h\<^sub>1.\<phi>) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((\<a>[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (((g.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>1) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (f\<^sub>0gh\<^sub>1.\<phi> \<star> TTfgh_TfTgh.chine) \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>0 f, f\<^sub>0gh\<^sub>1.p\<^sub>1, TTfgh_TfTgh.chine]) \<cdot> \<a>[f, tab\<^sub>0 f, f\<^sub>0gh\<^sub>1.p\<^sub>1 \<star> TTfgh_TfTgh.chine] \<cdot> (f.tab \<star> f\<^sub>0gh\<^sub>1.p\<^sub>1 \<star> TTfgh_TfTgh.chine) \<cdot> \<a>[tab\<^sub>1 f, f\<^sub>0gh\<^sub>1.p\<^sub>1, TTfgh_TfTgh.chine] \<cdot> TTfgh_TfTgh.the_\<nu> \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>1 f, f\<^sub>0g\<^sub>1.p\<^sub>1, fg\<^sub>0h\<^sub>1.p\<^sub>1] = (f \<star> g \<star> h \<star> TTfgh_TfTgh.the_\<theta>) \<cdot> (\<a>[f, g, h \<star> ((tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine] \<cdot> \<a>[f \<star> g, h, ((tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine] \<cdot> (\<a>\<^sup>-\<^sup>1[f, g, h] \<star> ((tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> \<a>[f \<star> g \<star> h, (tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine] \<cdot> (\<a>\<^sup>-\<^sup>1[f, g \<star> h, (tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0] \<star> TTfgh_TfTgh.chine) \<cdot> \<a>\<^sup>-\<^sup>1[f, (g \<star> h) \<star> (tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine] \<cdot> (f \<star> \<a>[g \<star> h, tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0, f\<^sub>0gh\<^sub>1.p\<^sub>0] \<star> TTfgh_TfTgh.chine) \<cdot> (f \<star> (\<a>\<^sup>-\<^sup>1[g, h, tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (f \<star> ((g \<star> \<a>[h, tab\<^sub>0 h, g\<^sub>0h\<^sub>1.p\<^sub>0]) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine)) \<cdot> (f \<star> (((g \<star> h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (((g \<star> g\<^sub>0h\<^sub>1.\<phi>) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((\<a>[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (((g.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>1) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (f\<^sub>0gh\<^sub>1.\<phi> \<star> TTfgh_TfTgh.chine) \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>0 f, f\<^sub>0gh\<^sub>1.p\<^sub>1, TTfgh_TfTgh.chine]) \<cdot> \<a>[f, tab\<^sub>0 f, f\<^sub>0gh\<^sub>1.p\<^sub>1 \<star> TTfgh_TfTgh.chine] \<cdot> (f.tab \<star> f\<^sub>0gh\<^sub>1.p\<^sub>1 \<star> TTfgh_TfTgh.chine) \<cdot> \<a>[tab\<^sub>1 f, f\<^sub>0gh\<^sub>1.p\<^sub>1, TTfgh_TfTgh.chine] \<cdot> TTfgh_TfTgh.the_\<nu> \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>1 f, f\<^sub>0g\<^sub>1.p\<^sub>1, fg\<^sub>0h\<^sub>1.p\<^sub>1]
goal (1 subgoal):
1. (f \<star> \<theta>\<^sub>f) \<cdot> \<a>[f, tab\<^sub>0 f, w\<^sub>f] \<cdot> (f.tab \<star> w\<^sub>f) = ((f \<star> \<theta>\<^sub>f') \<cdot> \<a>[f, tab\<^sub>0 f, w\<^sub>f'] \<cdot> (f.tab \<star> w\<^sub>f')) \<cdot> \<beta>\<^sub>f
[PROOF STEP]
have "... =
((f \<star> g \<star> h \<star> TTfgh_TfTgh.the_\<theta>) \<cdot>
(f \<star> can (\<^bold>\<langle>g\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> ((\<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>Tgh.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TfTgh.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>)
(((\<^bold>\<langle>g\<^bold>\<rangle> \<^bold>\<star> (\<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>Tgh.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TfTgh.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>)) \<cdot>
(f \<star> (((g \<star> h.tab \<star> Tgh.p\<^sub>0) \<star> TfTgh.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot>
(((g \<star> g\<^sub>0h\<^sub>1.\<phi>) \<star> TfTgh.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot>
((\<a>[g, tab\<^sub>0 g, Tgh.p\<^sub>1] \<star> TfTgh.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot>
(((g.tab \<star> Tgh.p\<^sub>1) \<star> TfTgh.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot>
(f\<^sub>0gh\<^sub>1.\<phi> \<star> TTfgh_TfTgh.chine) \<cdot>
\<a>\<^sup>-\<^sup>1[tab\<^sub>0 f, TfTgh.p\<^sub>1, TTfgh_TfTgh.chine])) \<cdot>
\<a>[f, tab\<^sub>0 f, TfTgh.p\<^sub>1 \<star> TTfgh_TfTgh.chine] \<cdot>
(f.tab \<star> TfTgh.p\<^sub>1 \<star> TTfgh_TfTgh.chine) \<cdot>
\<a>[tab\<^sub>1 f, TfTgh.p\<^sub>1, TTfgh_TfTgh.chine] \<cdot>
TTfgh_TfTgh.the_\<nu> \<cdot>
\<a>\<^sup>-\<^sup>1[tab\<^sub>1 f, Tfg.p\<^sub>1, TTfgh.p\<^sub>1]"
[PROOF STATE]
proof (prove)
goal (1 subgoal):
1. (f \<star> g \<star> h \<star> TTfgh_TfTgh.the_\<theta>) \<cdot> (\<a>[f, g, h \<star> ((tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine] \<cdot> \<a>[f \<star> g, h, ((tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine] \<cdot> (\<a>\<^sup>-\<^sup>1[f, g, h] \<star> ((tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> \<a>[f \<star> g \<star> h, (tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine] \<cdot> (\<a>\<^sup>-\<^sup>1[f, g \<star> h, (tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0] \<star> TTfgh_TfTgh.chine) \<cdot> \<a>\<^sup>-\<^sup>1[f, (g \<star> h) \<star> (tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine] \<cdot> (f \<star> \<a>[g \<star> h, tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0, f\<^sub>0gh\<^sub>1.p\<^sub>0] \<star> TTfgh_TfTgh.chine) \<cdot> (f \<star> (\<a>\<^sup>-\<^sup>1[g, h, tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (f \<star> ((g \<star> \<a>[h, tab\<^sub>0 h, g\<^sub>0h\<^sub>1.p\<^sub>0]) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine)) \<cdot> (f \<star> (((g \<star> h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (((g \<star> g\<^sub>0h\<^sub>1.\<phi>) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((\<a>[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (((g.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>1) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (f\<^sub>0gh\<^sub>1.\<phi> \<star> TTfgh_TfTgh.chine) \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>0 f, f\<^sub>0gh\<^sub>1.p\<^sub>1, TTfgh_TfTgh.chine]) \<cdot> \<a>[f, tab\<^sub>0 f, f\<^sub>0gh\<^sub>1.p\<^sub>1 \<star> TTfgh_TfTgh.chine] \<cdot> (f.tab \<star> f\<^sub>0gh\<^sub>1.p\<^sub>1 \<star> TTfgh_TfTgh.chine) \<cdot> \<a>[tab\<^sub>1 f, f\<^sub>0gh\<^sub>1.p\<^sub>1, TTfgh_TfTgh.chine] \<cdot> TTfgh_TfTgh.the_\<nu> \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>1 f, f\<^sub>0g\<^sub>1.p\<^sub>1, fg\<^sub>0h\<^sub>1.p\<^sub>1] = ((f \<star> g \<star> h \<star> TTfgh_TfTgh.the_\<theta>) \<cdot> (f \<star> can (\<^bold>\<langle>g\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> ((\<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) (((\<^bold>\<langle>g\<^bold>\<rangle> \<^bold>\<star> (\<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>)) \<cdot> (f \<star> (((g \<star> h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (((g \<star> g\<^sub>0h\<^sub>1.\<phi>) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((\<a>[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (((g.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>1) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (f\<^sub>0gh\<^sub>1.\<phi> \<star> TTfgh_TfTgh.chine) \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>0 f, f\<^sub>0gh\<^sub>1.p\<^sub>1, TTfgh_TfTgh.chine])) \<cdot> \<a>[f, tab\<^sub>0 f, f\<^sub>0gh\<^sub>1.p\<^sub>1 \<star> TTfgh_TfTgh.chine] \<cdot> (f.tab \<star> f\<^sub>0gh\<^sub>1.p\<^sub>1 \<star> TTfgh_TfTgh.chine) \<cdot> \<a>[tab\<^sub>1 f, f\<^sub>0gh\<^sub>1.p\<^sub>1, TTfgh_TfTgh.chine] \<cdot> TTfgh_TfTgh.the_\<nu> \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>1 f, f\<^sub>0g\<^sub>1.p\<^sub>1, fg\<^sub>0h\<^sub>1.p\<^sub>1]
[PROOF STEP]
proof -
[PROOF STATE]
proof (state)
goal (1 subgoal):
1. (f \<star> g \<star> h \<star> TTfgh_TfTgh.the_\<theta>) \<cdot> (\<a>[f, g, h \<star> ((tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine] \<cdot> \<a>[f \<star> g, h, ((tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine] \<cdot> (\<a>\<^sup>-\<^sup>1[f, g, h] \<star> ((tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> \<a>[f \<star> g \<star> h, (tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine] \<cdot> (\<a>\<^sup>-\<^sup>1[f, g \<star> h, (tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0] \<star> TTfgh_TfTgh.chine) \<cdot> \<a>\<^sup>-\<^sup>1[f, (g \<star> h) \<star> (tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine] \<cdot> (f \<star> \<a>[g \<star> h, tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0, f\<^sub>0gh\<^sub>1.p\<^sub>0] \<star> TTfgh_TfTgh.chine) \<cdot> (f \<star> (\<a>\<^sup>-\<^sup>1[g, h, tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (f \<star> ((g \<star> \<a>[h, tab\<^sub>0 h, g\<^sub>0h\<^sub>1.p\<^sub>0]) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine)) \<cdot> (f \<star> (((g \<star> h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (((g \<star> g\<^sub>0h\<^sub>1.\<phi>) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((\<a>[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (((g.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>1) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (f\<^sub>0gh\<^sub>1.\<phi> \<star> TTfgh_TfTgh.chine) \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>0 f, f\<^sub>0gh\<^sub>1.p\<^sub>1, TTfgh_TfTgh.chine]) \<cdot> \<a>[f, tab\<^sub>0 f, f\<^sub>0gh\<^sub>1.p\<^sub>1 \<star> TTfgh_TfTgh.chine] \<cdot> (f.tab \<star> f\<^sub>0gh\<^sub>1.p\<^sub>1 \<star> TTfgh_TfTgh.chine) \<cdot> \<a>[tab\<^sub>1 f, f\<^sub>0gh\<^sub>1.p\<^sub>1, TTfgh_TfTgh.chine] \<cdot> TTfgh_TfTgh.the_\<nu> \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>1 f, f\<^sub>0g\<^sub>1.p\<^sub>1, fg\<^sub>0h\<^sub>1.p\<^sub>1] = ((f \<star> g \<star> h \<star> TTfgh_TfTgh.the_\<theta>) \<cdot> (f \<star> can (\<^bold>\<langle>g\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> ((\<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) (((\<^bold>\<langle>g\<^bold>\<rangle> \<^bold>\<star> (\<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>)) \<cdot> (f \<star> (((g \<star> h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (((g \<star> g\<^sub>0h\<^sub>1.\<phi>) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((\<a>[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (((g.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>1) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (f\<^sub>0gh\<^sub>1.\<phi> \<star> TTfgh_TfTgh.chine) \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>0 f, f\<^sub>0gh\<^sub>1.p\<^sub>1, TTfgh_TfTgh.chine])) \<cdot> \<a>[f, tab\<^sub>0 f, f\<^sub>0gh\<^sub>1.p\<^sub>1 \<star> TTfgh_TfTgh.chine] \<cdot> (f.tab \<star> f\<^sub>0gh\<^sub>1.p\<^sub>1 \<star> TTfgh_TfTgh.chine) \<cdot> \<a>[tab\<^sub>1 f, f\<^sub>0gh\<^sub>1.p\<^sub>1, TTfgh_TfTgh.chine] \<cdot> TTfgh_TfTgh.the_\<nu> \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>1 f, f\<^sub>0g\<^sub>1.p\<^sub>1, fg\<^sub>0h\<^sub>1.p\<^sub>1]
[PROOF STEP]
have "\<a>[f, g, h \<star> ((tab\<^sub>0 h \<star> Tgh.p\<^sub>0) \<star> TfTgh.p\<^sub>0) \<star> TTfgh_TfTgh.chine] \<cdot>
\<a>[f \<star> g, h, ((tab\<^sub>0 h \<star> Tgh.p\<^sub>0) \<star> TfTgh.p\<^sub>0) \<star> TTfgh_TfTgh.chine] \<cdot>
(\<a>\<^sup>-\<^sup>1[f, g, h] \<star> ((tab\<^sub>0 h \<star> Tgh.p\<^sub>0) \<star> TfTgh.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot>
\<a>[f \<star> g \<star> h, (tab\<^sub>0 h \<star> Tgh.p\<^sub>0) \<star> TfTgh.p\<^sub>0, TTfgh_TfTgh.chine] \<cdot>
(\<a>\<^sup>-\<^sup>1[f, g \<star> h, (tab\<^sub>0 h \<star> Tgh.p\<^sub>0) \<star> TfTgh.p\<^sub>0] \<star> TTfgh_TfTgh.chine) \<cdot>
\<a>\<^sup>-\<^sup>1[f, (g \<star> h) \<star> (tab\<^sub>0 h \<star> Tgh.p\<^sub>0) \<star> TfTgh.p\<^sub>0, TTfgh_TfTgh.chine] \<cdot>
(f \<star> \<a>[g \<star> h, tab\<^sub>0 h \<star> Tgh.p\<^sub>0, TfTgh.p\<^sub>0] \<star> TTfgh_TfTgh.chine) \<cdot>
(f \<star> (\<a>\<^sup>-\<^sup>1[g, h, tab\<^sub>0 h \<star> Tgh.p\<^sub>0] \<star> TfTgh.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot>
(f \<star> ((g \<star> \<a>[h, tab\<^sub>0 h, Tgh.p\<^sub>0]) \<star> TfTgh.p\<^sub>0) \<star> TTfgh_TfTgh.chine) =
f \<star> can
(\<^bold>\<langle>g\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> ((\<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>Tgh.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TfTgh.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>)
(((\<^bold>\<langle>g\<^bold>\<rangle> \<^bold>\<star> (\<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>Tgh.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TfTgh.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>)"
[PROOF STATE]
proof (prove)
goal (1 subgoal):
1. \<a>[f, g, h \<star> ((tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine] \<cdot> \<a>[f \<star> g, h, ((tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine] \<cdot> (\<a>\<^sup>-\<^sup>1[f, g, h] \<star> ((tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> \<a>[f \<star> g \<star> h, (tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine] \<cdot> (\<a>\<^sup>-\<^sup>1[f, g \<star> h, (tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0] \<star> TTfgh_TfTgh.chine) \<cdot> \<a>\<^sup>-\<^sup>1[f, (g \<star> h) \<star> (tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine] \<cdot> (f \<star> \<a>[g \<star> h, tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0, f\<^sub>0gh\<^sub>1.p\<^sub>0] \<star> TTfgh_TfTgh.chine) \<cdot> (f \<star> (\<a>\<^sup>-\<^sup>1[g, h, tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (f \<star> ((g \<star> \<a>[h, tab\<^sub>0 h, g\<^sub>0h\<^sub>1.p\<^sub>0]) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) = f \<star> can (\<^bold>\<langle>g\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> ((\<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) (((\<^bold>\<langle>g\<^bold>\<rangle> \<^bold>\<star> (\<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>)
[PROOF STEP]
proof -
[PROOF STATE]
proof (state)
goal (1 subgoal):
1. \<a>[f, g, h \<star> ((tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine] \<cdot> \<a>[f \<star> g, h, ((tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine] \<cdot> (\<a>\<^sup>-\<^sup>1[f, g, h] \<star> ((tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> \<a>[f \<star> g \<star> h, (tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine] \<cdot> (\<a>\<^sup>-\<^sup>1[f, g \<star> h, (tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0] \<star> TTfgh_TfTgh.chine) \<cdot> \<a>\<^sup>-\<^sup>1[f, (g \<star> h) \<star> (tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine] \<cdot> (f \<star> \<a>[g \<star> h, tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0, f\<^sub>0gh\<^sub>1.p\<^sub>0] \<star> TTfgh_TfTgh.chine) \<cdot> (f \<star> (\<a>\<^sup>-\<^sup>1[g, h, tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (f \<star> ((g \<star> \<a>[h, tab\<^sub>0 h, g\<^sub>0h\<^sub>1.p\<^sub>0]) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) = f \<star> can (\<^bold>\<langle>g\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> ((\<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) (((\<^bold>\<langle>g\<^bold>\<rangle> \<^bold>\<star> (\<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>)
[PROOF STEP]
have "\<a>[f, g, h \<star> ((tab\<^sub>0 h \<star> Tgh.p\<^sub>0) \<star> TfTgh.p\<^sub>0) \<star> TTfgh_TfTgh.chine] \<cdot>
\<a>[f \<star> g, h, ((tab\<^sub>0 h \<star> Tgh.p\<^sub>0) \<star> TfTgh.p\<^sub>0) \<star> TTfgh_TfTgh.chine] \<cdot>
(\<a>\<^sup>-\<^sup>1[f, g, h] \<star> ((tab\<^sub>0 h \<star> Tgh.p\<^sub>0) \<star> TfTgh.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot>
\<a>[f \<star> g \<star> h, (tab\<^sub>0 h \<star> Tgh.p\<^sub>0) \<star> TfTgh.p\<^sub>0, TTfgh_TfTgh.chine] \<cdot>
(\<a>\<^sup>-\<^sup>1[f, g \<star> h, (tab\<^sub>0 h \<star> Tgh.p\<^sub>0) \<star> TfTgh.p\<^sub>0] \<star> TTfgh_TfTgh.chine) \<cdot>
\<a>\<^sup>-\<^sup>1[f, (g \<star> h) \<star> (tab\<^sub>0 h \<star> Tgh.p\<^sub>0) \<star> TfTgh.p\<^sub>0, TTfgh_TfTgh.chine] \<cdot>
(f \<star> \<a>[g \<star> h, tab\<^sub>0 h \<star> Tgh.p\<^sub>0, TfTgh.p\<^sub>0] \<star> TTfgh_TfTgh.chine) \<cdot>
(f \<star> (\<a>\<^sup>-\<^sup>1[g, h, tab\<^sub>0 h \<star> Tgh.p\<^sub>0] \<star> TfTgh.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot>
(f \<star> ((g \<star> \<a>[h, tab\<^sub>0 h, Tgh.p\<^sub>0]) \<star> TfTgh.p\<^sub>0) \<star> TTfgh_TfTgh.chine) =
\<lbrace>\<^bold>\<a>\<^bold>[\<^bold>\<langle>f\<^bold>\<rangle>, \<^bold>\<langle>g\<^bold>\<rangle>,
\<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> ((\<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>Tgh.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TfTgh.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>\<^bold>] \<^bold>\<cdot>
\<^bold>\<a>\<^bold>[\<^bold>\<langle>f\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>g\<^bold>\<rangle>, \<^bold>\<langle>h\<^bold>\<rangle>,
((\<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>Tgh.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TfTgh.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>\<^bold>] \<^bold>\<cdot>
(\<^bold>\<a>\<^sup>-\<^sup>1\<^bold>[\<^bold>\<langle>f\<^bold>\<rangle>, \<^bold>\<langle>g\<^bold>\<rangle>, \<^bold>\<langle>h\<^bold>\<rangle>\<^bold>]
\<^bold>\<star> ((\<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>Tgh.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TfTgh.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) \<^bold>\<cdot>
\<^bold>\<a>\<^bold>[\<^bold>\<langle>f\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>g\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>h\<^bold>\<rangle>, (\<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>Tgh.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TfTgh.p\<^sub>0\<^bold>\<rangle>,
\<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>\<^bold>] \<^bold>\<cdot>
(\<^bold>\<a>\<^sup>-\<^sup>1\<^bold>[\<^bold>\<langle>f\<^bold>\<rangle>, \<^bold>\<langle>g\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>h\<^bold>\<rangle>,
(\<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>Tgh.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TfTgh.p\<^sub>0\<^bold>\<rangle>\<^bold>] \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) \<^bold>\<cdot>
\<^bold>\<a>\<^sup>-\<^sup>1\<^bold>[\<^bold>\<langle>f\<^bold>\<rangle>, (\<^bold>\<langle>g\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>h\<^bold>\<rangle>) \<^bold>\<star> (\<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>Tgh.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TfTgh.p\<^sub>0\<^bold>\<rangle>,
\<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>\<^bold>] \<^bold>\<cdot>
(\<^bold>\<langle>f\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<a>\<^bold>[\<^bold>\<langle>g\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>h\<^bold>\<rangle>, \<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>Tgh.p\<^sub>0\<^bold>\<rangle>, \<^bold>\<langle>TfTgh.p\<^sub>0\<^bold>\<rangle>\<^bold>] \<^bold>\<star>
\<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) \<^bold>\<cdot>
(\<^bold>\<langle>f\<^bold>\<rangle> \<^bold>\<star> (\<^bold>\<a>\<^sup>-\<^sup>1\<^bold>[\<^bold>\<langle>g\<^bold>\<rangle>, \<^bold>\<langle>h\<^bold>\<rangle>, \<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>Tgh.p\<^sub>0\<^bold>\<rangle>\<^bold>] \<^bold>\<star> \<^bold>\<langle>TfTgh.p\<^sub>0\<^bold>\<rangle>)
\<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) \<^bold>\<cdot>
(\<^bold>\<langle>f\<^bold>\<rangle> \<^bold>\<star> ((\<^bold>\<langle>g\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<a>\<^bold>[\<^bold>\<langle>h\<^bold>\<rangle>, \<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle>, \<^bold>\<langle>Tgh.p\<^sub>0\<^bold>\<rangle>\<^bold>]) \<^bold>\<star> \<^bold>\<langle>TfTgh.p\<^sub>0\<^bold>\<rangle>)
\<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>)\<rbrace>"
[PROOF STATE]
proof (prove)
goal (1 subgoal):
1. \<a>[f, g, h \<star> ((tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine] \<cdot> \<a>[f \<star> g, h, ((tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine] \<cdot> (\<a>\<^sup>-\<^sup>1[f, g, h] \<star> ((tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> \<a>[f \<star> g \<star> h, (tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine] \<cdot> (\<a>\<^sup>-\<^sup>1[f, g \<star> h, (tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0] \<star> TTfgh_TfTgh.chine) \<cdot> \<a>\<^sup>-\<^sup>1[f, (g \<star> h) \<star> (tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine] \<cdot> (f \<star> \<a>[g \<star> h, tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0, f\<^sub>0gh\<^sub>1.p\<^sub>0] \<star> TTfgh_TfTgh.chine) \<cdot> (f \<star> (\<a>\<^sup>-\<^sup>1[g, h, tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (f \<star> ((g \<star> \<a>[h, tab\<^sub>0 h, g\<^sub>0h\<^sub>1.p\<^sub>0]) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) = \<lbrace>\<^bold>\<a>\<^bold>[\<^bold>\<langle>f\<^bold>\<rangle>, \<^bold>\<langle>g\<^bold>\<rangle>, \<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> ((\<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>\<^bold>] \<^bold>\<cdot> \<^bold>\<a>\<^bold>[\<^bold>\<langle>f\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>g\<^bold>\<rangle>, \<^bold>\<langle>h\<^bold>\<rangle>, ((\<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>\<^bold>] \<^bold>\<cdot> (\<^bold>\<a>\<^sup>-\<^sup>1\<^bold>[\<^bold>\<langle>f\<^bold>\<rangle>, \<^bold>\<langle>g\<^bold>\<rangle>, \<^bold>\<langle>h\<^bold>\<rangle>\<^bold>] \<^bold>\<star> ((\<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) \<^bold>\<cdot> \<^bold>\<a>\<^bold>[\<^bold>\<langle>f\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>g\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>h\<^bold>\<rangle>, (\<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>, \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>\<^bold>] \<^bold>\<cdot> (\<^bold>\<a>\<^sup>-\<^sup>1\<^bold>[\<^bold>\<langle>f\<^bold>\<rangle>, \<^bold>\<langle>g\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>h\<^bold>\<rangle>, (\<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>\<^bold>] \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) \<^bold>\<cdot> \<^bold>\<a>\<^sup>-\<^sup>1\<^bold>[\<^bold>\<langle>f\<^bold>\<rangle>, (\<^bold>\<langle>g\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>h\<^bold>\<rangle>) \<^bold>\<star> (\<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>, \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>\<^bold>] \<^bold>\<cdot> (\<^bold>\<langle>f\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<a>\<^bold>[\<^bold>\<langle>g\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>h\<^bold>\<rangle>, \<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>, \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>\<^bold>] \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) \<^bold>\<cdot> (\<^bold>\<langle>f\<^bold>\<rangle> \<^bold>\<star> (\<^bold>\<a>\<^sup>-\<^sup>1\<^bold>[\<^bold>\<langle>g\<^bold>\<rangle>, \<^bold>\<langle>h\<^bold>\<rangle>, \<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>\<^bold>] \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) \<^bold>\<cdot> (\<^bold>\<langle>f\<^bold>\<rangle> \<^bold>\<star> ((\<^bold>\<langle>g\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<a>\<^bold>[\<^bold>\<langle>h\<^bold>\<rangle>, \<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle>, \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>\<^bold>]) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>)\<rbrace>
[PROOF STEP]
using \<a>'_def \<alpha>_def
[PROOF STATE]
proof (prove)
using this:
\<a>\<^sup>-\<^sup>1[?\<mu>, ?\<nu>, ?\<tau>] \<equiv> \<alpha>' (?\<mu>, ?\<nu>, ?\<tau>)
\<alpha> ?\<mu>\<nu>\<tau> \<equiv> \<a>[fst ?\<mu>\<nu>\<tau>, fst (snd ?\<mu>\<nu>\<tau>), snd (snd ?\<mu>\<nu>\<tau>)]
goal (1 subgoal):
1. \<a>[f, g, h \<star> ((tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine] \<cdot> \<a>[f \<star> g, h, ((tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine] \<cdot> (\<a>\<^sup>-\<^sup>1[f, g, h] \<star> ((tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> \<a>[f \<star> g \<star> h, (tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine] \<cdot> (\<a>\<^sup>-\<^sup>1[f, g \<star> h, (tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0] \<star> TTfgh_TfTgh.chine) \<cdot> \<a>\<^sup>-\<^sup>1[f, (g \<star> h) \<star> (tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine] \<cdot> (f \<star> \<a>[g \<star> h, tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0, f\<^sub>0gh\<^sub>1.p\<^sub>0] \<star> TTfgh_TfTgh.chine) \<cdot> (f \<star> (\<a>\<^sup>-\<^sup>1[g, h, tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (f \<star> ((g \<star> \<a>[h, tab\<^sub>0 h, g\<^sub>0h\<^sub>1.p\<^sub>0]) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) = \<lbrace>\<^bold>\<a>\<^bold>[\<^bold>\<langle>f\<^bold>\<rangle>, \<^bold>\<langle>g\<^bold>\<rangle>, \<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> ((\<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>\<^bold>] \<^bold>\<cdot> \<^bold>\<a>\<^bold>[\<^bold>\<langle>f\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>g\<^bold>\<rangle>, \<^bold>\<langle>h\<^bold>\<rangle>, ((\<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>\<^bold>] \<^bold>\<cdot> (\<^bold>\<a>\<^sup>-\<^sup>1\<^bold>[\<^bold>\<langle>f\<^bold>\<rangle>, \<^bold>\<langle>g\<^bold>\<rangle>, \<^bold>\<langle>h\<^bold>\<rangle>\<^bold>] \<^bold>\<star> ((\<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) \<^bold>\<cdot> \<^bold>\<a>\<^bold>[\<^bold>\<langle>f\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>g\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>h\<^bold>\<rangle>, (\<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>, \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>\<^bold>] \<^bold>\<cdot> (\<^bold>\<a>\<^sup>-\<^sup>1\<^bold>[\<^bold>\<langle>f\<^bold>\<rangle>, \<^bold>\<langle>g\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>h\<^bold>\<rangle>, (\<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>\<^bold>] \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) \<^bold>\<cdot> \<^bold>\<a>\<^sup>-\<^sup>1\<^bold>[\<^bold>\<langle>f\<^bold>\<rangle>, (\<^bold>\<langle>g\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>h\<^bold>\<rangle>) \<^bold>\<star> (\<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>, \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>\<^bold>] \<^bold>\<cdot> (\<^bold>\<langle>f\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<a>\<^bold>[\<^bold>\<langle>g\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>h\<^bold>\<rangle>, \<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>, \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>\<^bold>] \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) \<^bold>\<cdot> (\<^bold>\<langle>f\<^bold>\<rangle> \<^bold>\<star> (\<^bold>\<a>\<^sup>-\<^sup>1\<^bold>[\<^bold>\<langle>g\<^bold>\<rangle>, \<^bold>\<langle>h\<^bold>\<rangle>, \<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>\<^bold>] \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) \<^bold>\<cdot> (\<^bold>\<langle>f\<^bold>\<rangle> \<^bold>\<star> ((\<^bold>\<langle>g\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<a>\<^bold>[\<^bold>\<langle>h\<^bold>\<rangle>, \<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle>, \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>\<^bold>]) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>)\<rbrace>
[PROOF STEP]
by simp
[PROOF STATE]
proof (state)
this:
\<a>[f, g, h \<star> ((tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine] \<cdot> \<a>[f \<star> g, h, ((tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine] \<cdot> (\<a>\<^sup>-\<^sup>1[f, g, h] \<star> ((tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> \<a>[f \<star> g \<star> h, (tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine] \<cdot> (\<a>\<^sup>-\<^sup>1[f, g \<star> h, (tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0] \<star> TTfgh_TfTgh.chine) \<cdot> \<a>\<^sup>-\<^sup>1[f, (g \<star> h) \<star> (tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine] \<cdot> (f \<star> \<a>[g \<star> h, tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0, f\<^sub>0gh\<^sub>1.p\<^sub>0] \<star> TTfgh_TfTgh.chine) \<cdot> (f \<star> (\<a>\<^sup>-\<^sup>1[g, h, tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (f \<star> ((g \<star> \<a>[h, tab\<^sub>0 h, g\<^sub>0h\<^sub>1.p\<^sub>0]) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) = \<lbrace>\<^bold>\<a>\<^bold>[\<^bold>\<langle>f\<^bold>\<rangle>, \<^bold>\<langle>g\<^bold>\<rangle>, \<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> ((\<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>\<^bold>] \<^bold>\<cdot> \<^bold>\<a>\<^bold>[\<^bold>\<langle>f\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>g\<^bold>\<rangle>, \<^bold>\<langle>h\<^bold>\<rangle>, ((\<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>\<^bold>] \<^bold>\<cdot> (\<^bold>\<a>\<^sup>-\<^sup>1\<^bold>[\<^bold>\<langle>f\<^bold>\<rangle>, \<^bold>\<langle>g\<^bold>\<rangle>, \<^bold>\<langle>h\<^bold>\<rangle>\<^bold>] \<^bold>\<star> ((\<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) \<^bold>\<cdot> \<^bold>\<a>\<^bold>[\<^bold>\<langle>f\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>g\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>h\<^bold>\<rangle>, (\<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>, \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>\<^bold>] \<^bold>\<cdot> (\<^bold>\<a>\<^sup>-\<^sup>1\<^bold>[\<^bold>\<langle>f\<^bold>\<rangle>, \<^bold>\<langle>g\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>h\<^bold>\<rangle>, (\<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>\<^bold>] \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) \<^bold>\<cdot> \<^bold>\<a>\<^sup>-\<^sup>1\<^bold>[\<^bold>\<langle>f\<^bold>\<rangle>, (\<^bold>\<langle>g\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>h\<^bold>\<rangle>) \<^bold>\<star> (\<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>, \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>\<^bold>] \<^bold>\<cdot> (\<^bold>\<langle>f\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<a>\<^bold>[\<^bold>\<langle>g\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>h\<^bold>\<rangle>, \<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>, \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>\<^bold>] \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) \<^bold>\<cdot> (\<^bold>\<langle>f\<^bold>\<rangle> \<^bold>\<star> (\<^bold>\<a>\<^sup>-\<^sup>1\<^bold>[\<^bold>\<langle>g\<^bold>\<rangle>, \<^bold>\<langle>h\<^bold>\<rangle>, \<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>\<^bold>] \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) \<^bold>\<cdot> (\<^bold>\<langle>f\<^bold>\<rangle> \<^bold>\<star> ((\<^bold>\<langle>g\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<a>\<^bold>[\<^bold>\<langle>h\<^bold>\<rangle>, \<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle>, \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>\<^bold>]) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>)\<rbrace>
goal (1 subgoal):
1. \<a>[f, g, h \<star> ((tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine] \<cdot> \<a>[f \<star> g, h, ((tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine] \<cdot> (\<a>\<^sup>-\<^sup>1[f, g, h] \<star> ((tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> \<a>[f \<star> g \<star> h, (tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine] \<cdot> (\<a>\<^sup>-\<^sup>1[f, g \<star> h, (tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0] \<star> TTfgh_TfTgh.chine) \<cdot> \<a>\<^sup>-\<^sup>1[f, (g \<star> h) \<star> (tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine] \<cdot> (f \<star> \<a>[g \<star> h, tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0, f\<^sub>0gh\<^sub>1.p\<^sub>0] \<star> TTfgh_TfTgh.chine) \<cdot> (f \<star> (\<a>\<^sup>-\<^sup>1[g, h, tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (f \<star> ((g \<star> \<a>[h, tab\<^sub>0 h, g\<^sub>0h\<^sub>1.p\<^sub>0]) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) = f \<star> can (\<^bold>\<langle>g\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> ((\<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) (((\<^bold>\<langle>g\<^bold>\<rangle> \<^bold>\<star> (\<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>)
[PROOF STEP]
also
[PROOF STATE]
proof (state)
this:
\<a>[f, g, h \<star> ((tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine] \<cdot> \<a>[f \<star> g, h, ((tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine] \<cdot> (\<a>\<^sup>-\<^sup>1[f, g, h] \<star> ((tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> \<a>[f \<star> g \<star> h, (tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine] \<cdot> (\<a>\<^sup>-\<^sup>1[f, g \<star> h, (tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0] \<star> TTfgh_TfTgh.chine) \<cdot> \<a>\<^sup>-\<^sup>1[f, (g \<star> h) \<star> (tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine] \<cdot> (f \<star> \<a>[g \<star> h, tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0, f\<^sub>0gh\<^sub>1.p\<^sub>0] \<star> TTfgh_TfTgh.chine) \<cdot> (f \<star> (\<a>\<^sup>-\<^sup>1[g, h, tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (f \<star> ((g \<star> \<a>[h, tab\<^sub>0 h, g\<^sub>0h\<^sub>1.p\<^sub>0]) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) = \<lbrace>\<^bold>\<a>\<^bold>[\<^bold>\<langle>f\<^bold>\<rangle>, \<^bold>\<langle>g\<^bold>\<rangle>, \<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> ((\<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>\<^bold>] \<^bold>\<cdot> \<^bold>\<a>\<^bold>[\<^bold>\<langle>f\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>g\<^bold>\<rangle>, \<^bold>\<langle>h\<^bold>\<rangle>, ((\<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>\<^bold>] \<^bold>\<cdot> (\<^bold>\<a>\<^sup>-\<^sup>1\<^bold>[\<^bold>\<langle>f\<^bold>\<rangle>, \<^bold>\<langle>g\<^bold>\<rangle>, \<^bold>\<langle>h\<^bold>\<rangle>\<^bold>] \<^bold>\<star> ((\<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) \<^bold>\<cdot> \<^bold>\<a>\<^bold>[\<^bold>\<langle>f\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>g\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>h\<^bold>\<rangle>, (\<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>, \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>\<^bold>] \<^bold>\<cdot> (\<^bold>\<a>\<^sup>-\<^sup>1\<^bold>[\<^bold>\<langle>f\<^bold>\<rangle>, \<^bold>\<langle>g\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>h\<^bold>\<rangle>, (\<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>\<^bold>] \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) \<^bold>\<cdot> \<^bold>\<a>\<^sup>-\<^sup>1\<^bold>[\<^bold>\<langle>f\<^bold>\<rangle>, (\<^bold>\<langle>g\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>h\<^bold>\<rangle>) \<^bold>\<star> (\<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>, \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>\<^bold>] \<^bold>\<cdot> (\<^bold>\<langle>f\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<a>\<^bold>[\<^bold>\<langle>g\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>h\<^bold>\<rangle>, \<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>, \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>\<^bold>] \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) \<^bold>\<cdot> (\<^bold>\<langle>f\<^bold>\<rangle> \<^bold>\<star> (\<^bold>\<a>\<^sup>-\<^sup>1\<^bold>[\<^bold>\<langle>g\<^bold>\<rangle>, \<^bold>\<langle>h\<^bold>\<rangle>, \<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>\<^bold>] \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) \<^bold>\<cdot> (\<^bold>\<langle>f\<^bold>\<rangle> \<^bold>\<star> ((\<^bold>\<langle>g\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<a>\<^bold>[\<^bold>\<langle>h\<^bold>\<rangle>, \<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle>, \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>\<^bold>]) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>)\<rbrace>
goal (1 subgoal):
1. \<a>[f, g, h \<star> ((tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine] \<cdot> \<a>[f \<star> g, h, ((tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine] \<cdot> (\<a>\<^sup>-\<^sup>1[f, g, h] \<star> ((tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> \<a>[f \<star> g \<star> h, (tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine] \<cdot> (\<a>\<^sup>-\<^sup>1[f, g \<star> h, (tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0] \<star> TTfgh_TfTgh.chine) \<cdot> \<a>\<^sup>-\<^sup>1[f, (g \<star> h) \<star> (tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine] \<cdot> (f \<star> \<a>[g \<star> h, tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0, f\<^sub>0gh\<^sub>1.p\<^sub>0] \<star> TTfgh_TfTgh.chine) \<cdot> (f \<star> (\<a>\<^sup>-\<^sup>1[g, h, tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (f \<star> ((g \<star> \<a>[h, tab\<^sub>0 h, g\<^sub>0h\<^sub>1.p\<^sub>0]) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) = f \<star> can (\<^bold>\<langle>g\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> ((\<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) (((\<^bold>\<langle>g\<^bold>\<rangle> \<^bold>\<star> (\<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>)
[PROOF STEP]
have "... =
can (\<^bold>\<langle>f\<^bold>\<rangle> \<^bold>\<star> (\<^bold>\<langle>g\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> ((\<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>Tgh.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TfTgh.p\<^sub>0\<^bold>\<rangle>)
\<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>))
(\<^bold>\<langle>f\<^bold>\<rangle> \<^bold>\<star> (((\<^bold>\<langle>g\<^bold>\<rangle> \<^bold>\<star> (\<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>Tgh.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TfTgh.p\<^sub>0\<^bold>\<rangle>)
\<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>))"
[PROOF STATE]
proof (prove)
goal (1 subgoal):
1. \<lbrace>\<^bold>\<a>\<^bold>[\<^bold>\<langle>f\<^bold>\<rangle>, \<^bold>\<langle>g\<^bold>\<rangle>, \<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> ((\<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>\<^bold>] \<^bold>\<cdot> \<^bold>\<a>\<^bold>[\<^bold>\<langle>f\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>g\<^bold>\<rangle>, \<^bold>\<langle>h\<^bold>\<rangle>, ((\<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>\<^bold>] \<^bold>\<cdot> (\<^bold>\<a>\<^sup>-\<^sup>1\<^bold>[\<^bold>\<langle>f\<^bold>\<rangle>, \<^bold>\<langle>g\<^bold>\<rangle>, \<^bold>\<langle>h\<^bold>\<rangle>\<^bold>] \<^bold>\<star> ((\<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) \<^bold>\<cdot> \<^bold>\<a>\<^bold>[\<^bold>\<langle>f\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>g\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>h\<^bold>\<rangle>, (\<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>, \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>\<^bold>] \<^bold>\<cdot> (\<^bold>\<a>\<^sup>-\<^sup>1\<^bold>[\<^bold>\<langle>f\<^bold>\<rangle>, \<^bold>\<langle>g\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>h\<^bold>\<rangle>, (\<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>\<^bold>] \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) \<^bold>\<cdot> \<^bold>\<a>\<^sup>-\<^sup>1\<^bold>[\<^bold>\<langle>f\<^bold>\<rangle>, (\<^bold>\<langle>g\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>h\<^bold>\<rangle>) \<^bold>\<star> (\<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>, \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>\<^bold>] \<^bold>\<cdot> (\<^bold>\<langle>f\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<a>\<^bold>[\<^bold>\<langle>g\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>h\<^bold>\<rangle>, \<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>, \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>\<^bold>] \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) \<^bold>\<cdot> (\<^bold>\<langle>f\<^bold>\<rangle> \<^bold>\<star> (\<^bold>\<a>\<^sup>-\<^sup>1\<^bold>[\<^bold>\<langle>g\<^bold>\<rangle>, \<^bold>\<langle>h\<^bold>\<rangle>, \<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>\<^bold>] \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) \<^bold>\<cdot> (\<^bold>\<langle>f\<^bold>\<rangle> \<^bold>\<star> ((\<^bold>\<langle>g\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<a>\<^bold>[\<^bold>\<langle>h\<^bold>\<rangle>, \<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle>, \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>\<^bold>]) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>)\<rbrace> = can (\<^bold>\<langle>f\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>g\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> ((\<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) (\<^bold>\<langle>f\<^bold>\<rangle> \<^bold>\<star> ((\<^bold>\<langle>g\<^bold>\<rangle> \<^bold>\<star> (\<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>)
[PROOF STEP]
using fg gh
[PROOF STATE]
proof (prove)
using this:
src f = trg g
src g = trg h
goal (1 subgoal):
1. \<lbrace>\<^bold>\<a>\<^bold>[\<^bold>\<langle>f\<^bold>\<rangle>, \<^bold>\<langle>g\<^bold>\<rangle>, \<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> ((\<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>\<^bold>] \<^bold>\<cdot> \<^bold>\<a>\<^bold>[\<^bold>\<langle>f\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>g\<^bold>\<rangle>, \<^bold>\<langle>h\<^bold>\<rangle>, ((\<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>\<^bold>] \<^bold>\<cdot> (\<^bold>\<a>\<^sup>-\<^sup>1\<^bold>[\<^bold>\<langle>f\<^bold>\<rangle>, \<^bold>\<langle>g\<^bold>\<rangle>, \<^bold>\<langle>h\<^bold>\<rangle>\<^bold>] \<^bold>\<star> ((\<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) \<^bold>\<cdot> \<^bold>\<a>\<^bold>[\<^bold>\<langle>f\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>g\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>h\<^bold>\<rangle>, (\<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>, \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>\<^bold>] \<^bold>\<cdot> (\<^bold>\<a>\<^sup>-\<^sup>1\<^bold>[\<^bold>\<langle>f\<^bold>\<rangle>, \<^bold>\<langle>g\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>h\<^bold>\<rangle>, (\<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>\<^bold>] \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) \<^bold>\<cdot> \<^bold>\<a>\<^sup>-\<^sup>1\<^bold>[\<^bold>\<langle>f\<^bold>\<rangle>, (\<^bold>\<langle>g\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>h\<^bold>\<rangle>) \<^bold>\<star> (\<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>, \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>\<^bold>] \<^bold>\<cdot> (\<^bold>\<langle>f\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<a>\<^bold>[\<^bold>\<langle>g\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>h\<^bold>\<rangle>, \<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>, \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>\<^bold>] \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) \<^bold>\<cdot> (\<^bold>\<langle>f\<^bold>\<rangle> \<^bold>\<star> (\<^bold>\<a>\<^sup>-\<^sup>1\<^bold>[\<^bold>\<langle>g\<^bold>\<rangle>, \<^bold>\<langle>h\<^bold>\<rangle>, \<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>\<^bold>] \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) \<^bold>\<cdot> (\<^bold>\<langle>f\<^bold>\<rangle> \<^bold>\<star> ((\<^bold>\<langle>g\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<a>\<^bold>[\<^bold>\<langle>h\<^bold>\<rangle>, \<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle>, \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>\<^bold>]) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>)\<rbrace> = can (\<^bold>\<langle>f\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>g\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> ((\<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) (\<^bold>\<langle>f\<^bold>\<rangle> \<^bold>\<star> ((\<^bold>\<langle>g\<^bold>\<rangle> \<^bold>\<star> (\<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>)
[PROOF STEP]
apply (unfold can_def)
[PROOF STATE]
proof (prove)
goal (1 subgoal):
1. \<lbrakk>src f = trg g; src g = trg h\<rbrakk> \<Longrightarrow> \<lbrace>\<^bold>\<a>\<^bold>[\<^bold>\<langle>f\<^bold>\<rangle>, \<^bold>\<langle>g\<^bold>\<rangle>, \<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> ((\<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>\<^bold>] \<^bold>\<cdot> \<^bold>\<a>\<^bold>[\<^bold>\<langle>f\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>g\<^bold>\<rangle>, \<^bold>\<langle>h\<^bold>\<rangle>, ((\<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>\<^bold>] \<^bold>\<cdot> (\<^bold>\<a>\<^sup>-\<^sup>1\<^bold>[\<^bold>\<langle>f\<^bold>\<rangle>, \<^bold>\<langle>g\<^bold>\<rangle>, \<^bold>\<langle>h\<^bold>\<rangle>\<^bold>] \<^bold>\<star> ((\<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) \<^bold>\<cdot> \<^bold>\<a>\<^bold>[\<^bold>\<langle>f\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>g\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>h\<^bold>\<rangle>, (\<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>, \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>\<^bold>] \<^bold>\<cdot> (\<^bold>\<a>\<^sup>-\<^sup>1\<^bold>[\<^bold>\<langle>f\<^bold>\<rangle>, \<^bold>\<langle>g\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>h\<^bold>\<rangle>, (\<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>\<^bold>] \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) \<^bold>\<cdot> \<^bold>\<a>\<^sup>-\<^sup>1\<^bold>[\<^bold>\<langle>f\<^bold>\<rangle>, (\<^bold>\<langle>g\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>h\<^bold>\<rangle>) \<^bold>\<star> (\<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>, \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>\<^bold>] \<^bold>\<cdot> (\<^bold>\<langle>f\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<a>\<^bold>[\<^bold>\<langle>g\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>h\<^bold>\<rangle>, \<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>, \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>\<^bold>] \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) \<^bold>\<cdot> (\<^bold>\<langle>f\<^bold>\<rangle> \<^bold>\<star> (\<^bold>\<a>\<^sup>-\<^sup>1\<^bold>[\<^bold>\<langle>g\<^bold>\<rangle>, \<^bold>\<langle>h\<^bold>\<rangle>, \<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>\<^bold>] \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) \<^bold>\<cdot> (\<^bold>\<langle>f\<^bold>\<rangle> \<^bold>\<star> ((\<^bold>\<langle>g\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<a>\<^bold>[\<^bold>\<langle>h\<^bold>\<rangle>, \<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle>, \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>\<^bold>]) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>)\<rbrace> = \<lbrace>E.Inv ((\<^bold>\<langle>f\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>g\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> ((\<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>)\<^bold>\<down>) \<^bold>\<cdot> (\<^bold>\<langle>f\<^bold>\<rangle> \<^bold>\<star> ((\<^bold>\<langle>g\<^bold>\<rangle> \<^bold>\<star> (\<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>)\<^bold>\<down>\<rbrace>
[PROOF STEP]
apply (intro E.eval_eqI)
[PROOF STATE]
proof (prove)
goal (2 subgoals):
1. \<lbrakk>src f = trg g; src g = trg h\<rbrakk> \<Longrightarrow> E.VPar (\<^bold>\<a>\<^bold>[\<^bold>\<langle>f\<^bold>\<rangle>, \<^bold>\<langle>g\<^bold>\<rangle>, \<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> ((\<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>\<^bold>] \<^bold>\<cdot> \<^bold>\<a>\<^bold>[\<^bold>\<langle>f\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>g\<^bold>\<rangle>, \<^bold>\<langle>h\<^bold>\<rangle>, ((\<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>\<^bold>] \<^bold>\<cdot> (\<^bold>\<a>\<^sup>-\<^sup>1\<^bold>[\<^bold>\<langle>f\<^bold>\<rangle>, \<^bold>\<langle>g\<^bold>\<rangle>, \<^bold>\<langle>h\<^bold>\<rangle>\<^bold>] \<^bold>\<star> ((\<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) \<^bold>\<cdot> \<^bold>\<a>\<^bold>[\<^bold>\<langle>f\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>g\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>h\<^bold>\<rangle>, (\<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>, \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>\<^bold>] \<^bold>\<cdot> (\<^bold>\<a>\<^sup>-\<^sup>1\<^bold>[\<^bold>\<langle>f\<^bold>\<rangle>, \<^bold>\<langle>g\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>h\<^bold>\<rangle>, (\<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>\<^bold>] \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) \<^bold>\<cdot> \<^bold>\<a>\<^sup>-\<^sup>1\<^bold>[\<^bold>\<langle>f\<^bold>\<rangle>, (\<^bold>\<langle>g\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>h\<^bold>\<rangle>) \<^bold>\<star> (\<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>, \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>\<^bold>] \<^bold>\<cdot> (\<^bold>\<langle>f\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<a>\<^bold>[\<^bold>\<langle>g\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>h\<^bold>\<rangle>, \<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>, \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>\<^bold>] \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) \<^bold>\<cdot> (\<^bold>\<langle>f\<^bold>\<rangle> \<^bold>\<star> (\<^bold>\<a>\<^sup>-\<^sup>1\<^bold>[\<^bold>\<langle>g\<^bold>\<rangle>, \<^bold>\<langle>h\<^bold>\<rangle>, \<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>\<^bold>] \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) \<^bold>\<cdot> (\<^bold>\<langle>f\<^bold>\<rangle> \<^bold>\<star> ((\<^bold>\<langle>g\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<a>\<^bold>[\<^bold>\<langle>h\<^bold>\<rangle>, \<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle>, \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>\<^bold>]) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>)) (E.Inv ((\<^bold>\<langle>f\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>g\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> ((\<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>)\<^bold>\<down>) \<^bold>\<cdot> (\<^bold>\<langle>f\<^bold>\<rangle> \<^bold>\<star> ((\<^bold>\<langle>g\<^bold>\<rangle> \<^bold>\<star> (\<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>)\<^bold>\<down>)
2. \<lbrakk>src f = trg g; src g = trg h\<rbrakk> \<Longrightarrow> \<^bold>\<lfloor>\<^bold>\<a>\<^bold>[\<^bold>\<langle>f\<^bold>\<rangle>, \<^bold>\<langle>g\<^bold>\<rangle>, \<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> ((\<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>\<^bold>] \<^bold>\<cdot> \<^bold>\<a>\<^bold>[\<^bold>\<langle>f\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>g\<^bold>\<rangle>, \<^bold>\<langle>h\<^bold>\<rangle>, ((\<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>\<^bold>] \<^bold>\<cdot> (\<^bold>\<a>\<^sup>-\<^sup>1\<^bold>[\<^bold>\<langle>f\<^bold>\<rangle>, \<^bold>\<langle>g\<^bold>\<rangle>, \<^bold>\<langle>h\<^bold>\<rangle>\<^bold>] \<^bold>\<star> ((\<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) \<^bold>\<cdot> \<^bold>\<a>\<^bold>[\<^bold>\<langle>f\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>g\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>h\<^bold>\<rangle>, (\<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>, \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>\<^bold>] \<^bold>\<cdot> (\<^bold>\<a>\<^sup>-\<^sup>1\<^bold>[\<^bold>\<langle>f\<^bold>\<rangle>, \<^bold>\<langle>g\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>h\<^bold>\<rangle>, (\<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>\<^bold>] \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) \<^bold>\<cdot> \<^bold>\<a>\<^sup>-\<^sup>1\<^bold>[\<^bold>\<langle>f\<^bold>\<rangle>, (\<^bold>\<langle>g\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>h\<^bold>\<rangle>) \<^bold>\<star> (\<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>, \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>\<^bold>] \<^bold>\<cdot> (\<^bold>\<langle>f\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<a>\<^bold>[\<^bold>\<langle>g\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>h\<^bold>\<rangle>, \<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>, \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>\<^bold>] \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) \<^bold>\<cdot> (\<^bold>\<langle>f\<^bold>\<rangle> \<^bold>\<star> (\<^bold>\<a>\<^sup>-\<^sup>1\<^bold>[\<^bold>\<langle>g\<^bold>\<rangle>, \<^bold>\<langle>h\<^bold>\<rangle>, \<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>\<^bold>] \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) \<^bold>\<cdot> (\<^bold>\<langle>f\<^bold>\<rangle> \<^bold>\<star> ((\<^bold>\<langle>g\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<a>\<^bold>[\<^bold>\<langle>h\<^bold>\<rangle>, \<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle>, \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>\<^bold>]) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>)\<^bold>\<rfloor> = \<^bold>\<lfloor>E.Inv ((\<^bold>\<langle>f\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>g\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> ((\<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>)\<^bold>\<down>) \<^bold>\<cdot> (\<^bold>\<langle>f\<^bold>\<rangle> \<^bold>\<star> ((\<^bold>\<langle>g\<^bold>\<rangle> \<^bold>\<star> (\<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>)\<^bold>\<down>\<^bold>\<rfloor>
[PROOF STEP]
by simp_all
[PROOF STATE]
proof (state)
this:
\<lbrace>\<^bold>\<a>\<^bold>[\<^bold>\<langle>f\<^bold>\<rangle>, \<^bold>\<langle>g\<^bold>\<rangle>, \<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> ((\<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>\<^bold>] \<^bold>\<cdot> \<^bold>\<a>\<^bold>[\<^bold>\<langle>f\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>g\<^bold>\<rangle>, \<^bold>\<langle>h\<^bold>\<rangle>, ((\<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>\<^bold>] \<^bold>\<cdot> (\<^bold>\<a>\<^sup>-\<^sup>1\<^bold>[\<^bold>\<langle>f\<^bold>\<rangle>, \<^bold>\<langle>g\<^bold>\<rangle>, \<^bold>\<langle>h\<^bold>\<rangle>\<^bold>] \<^bold>\<star> ((\<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) \<^bold>\<cdot> \<^bold>\<a>\<^bold>[\<^bold>\<langle>f\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>g\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>h\<^bold>\<rangle>, (\<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>, \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>\<^bold>] \<^bold>\<cdot> (\<^bold>\<a>\<^sup>-\<^sup>1\<^bold>[\<^bold>\<langle>f\<^bold>\<rangle>, \<^bold>\<langle>g\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>h\<^bold>\<rangle>, (\<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>\<^bold>] \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) \<^bold>\<cdot> \<^bold>\<a>\<^sup>-\<^sup>1\<^bold>[\<^bold>\<langle>f\<^bold>\<rangle>, (\<^bold>\<langle>g\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>h\<^bold>\<rangle>) \<^bold>\<star> (\<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>, \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>\<^bold>] \<^bold>\<cdot> (\<^bold>\<langle>f\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<a>\<^bold>[\<^bold>\<langle>g\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>h\<^bold>\<rangle>, \<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>, \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>\<^bold>] \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) \<^bold>\<cdot> (\<^bold>\<langle>f\<^bold>\<rangle> \<^bold>\<star> (\<^bold>\<a>\<^sup>-\<^sup>1\<^bold>[\<^bold>\<langle>g\<^bold>\<rangle>, \<^bold>\<langle>h\<^bold>\<rangle>, \<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>\<^bold>] \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) \<^bold>\<cdot> (\<^bold>\<langle>f\<^bold>\<rangle> \<^bold>\<star> ((\<^bold>\<langle>g\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<a>\<^bold>[\<^bold>\<langle>h\<^bold>\<rangle>, \<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle>, \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>\<^bold>]) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>)\<rbrace> = can (\<^bold>\<langle>f\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>g\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> ((\<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) (\<^bold>\<langle>f\<^bold>\<rangle> \<^bold>\<star> ((\<^bold>\<langle>g\<^bold>\<rangle> \<^bold>\<star> (\<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>)
goal (1 subgoal):
1. \<a>[f, g, h \<star> ((tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine] \<cdot> \<a>[f \<star> g, h, ((tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine] \<cdot> (\<a>\<^sup>-\<^sup>1[f, g, h] \<star> ((tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> \<a>[f \<star> g \<star> h, (tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine] \<cdot> (\<a>\<^sup>-\<^sup>1[f, g \<star> h, (tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0] \<star> TTfgh_TfTgh.chine) \<cdot> \<a>\<^sup>-\<^sup>1[f, (g \<star> h) \<star> (tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine] \<cdot> (f \<star> \<a>[g \<star> h, tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0, f\<^sub>0gh\<^sub>1.p\<^sub>0] \<star> TTfgh_TfTgh.chine) \<cdot> (f \<star> (\<a>\<^sup>-\<^sup>1[g, h, tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (f \<star> ((g \<star> \<a>[h, tab\<^sub>0 h, g\<^sub>0h\<^sub>1.p\<^sub>0]) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) = f \<star> can (\<^bold>\<langle>g\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> ((\<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) (((\<^bold>\<langle>g\<^bold>\<rangle> \<^bold>\<star> (\<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>)
[PROOF STEP]
also
[PROOF STATE]
proof (state)
this:
\<lbrace>\<^bold>\<a>\<^bold>[\<^bold>\<langle>f\<^bold>\<rangle>, \<^bold>\<langle>g\<^bold>\<rangle>, \<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> ((\<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>\<^bold>] \<^bold>\<cdot> \<^bold>\<a>\<^bold>[\<^bold>\<langle>f\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>g\<^bold>\<rangle>, \<^bold>\<langle>h\<^bold>\<rangle>, ((\<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>\<^bold>] \<^bold>\<cdot> (\<^bold>\<a>\<^sup>-\<^sup>1\<^bold>[\<^bold>\<langle>f\<^bold>\<rangle>, \<^bold>\<langle>g\<^bold>\<rangle>, \<^bold>\<langle>h\<^bold>\<rangle>\<^bold>] \<^bold>\<star> ((\<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) \<^bold>\<cdot> \<^bold>\<a>\<^bold>[\<^bold>\<langle>f\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>g\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>h\<^bold>\<rangle>, (\<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>, \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>\<^bold>] \<^bold>\<cdot> (\<^bold>\<a>\<^sup>-\<^sup>1\<^bold>[\<^bold>\<langle>f\<^bold>\<rangle>, \<^bold>\<langle>g\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>h\<^bold>\<rangle>, (\<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>\<^bold>] \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) \<^bold>\<cdot> \<^bold>\<a>\<^sup>-\<^sup>1\<^bold>[\<^bold>\<langle>f\<^bold>\<rangle>, (\<^bold>\<langle>g\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>h\<^bold>\<rangle>) \<^bold>\<star> (\<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>, \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>\<^bold>] \<^bold>\<cdot> (\<^bold>\<langle>f\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<a>\<^bold>[\<^bold>\<langle>g\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>h\<^bold>\<rangle>, \<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>, \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>\<^bold>] \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) \<^bold>\<cdot> (\<^bold>\<langle>f\<^bold>\<rangle> \<^bold>\<star> (\<^bold>\<a>\<^sup>-\<^sup>1\<^bold>[\<^bold>\<langle>g\<^bold>\<rangle>, \<^bold>\<langle>h\<^bold>\<rangle>, \<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>\<^bold>] \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) \<^bold>\<cdot> (\<^bold>\<langle>f\<^bold>\<rangle> \<^bold>\<star> ((\<^bold>\<langle>g\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<a>\<^bold>[\<^bold>\<langle>h\<^bold>\<rangle>, \<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle>, \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>\<^bold>]) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>)\<rbrace> = can (\<^bold>\<langle>f\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>g\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> ((\<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) (\<^bold>\<langle>f\<^bold>\<rangle> \<^bold>\<star> ((\<^bold>\<langle>g\<^bold>\<rangle> \<^bold>\<star> (\<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>)
goal (1 subgoal):
1. \<a>[f, g, h \<star> ((tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine] \<cdot> \<a>[f \<star> g, h, ((tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine] \<cdot> (\<a>\<^sup>-\<^sup>1[f, g, h] \<star> ((tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> \<a>[f \<star> g \<star> h, (tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine] \<cdot> (\<a>\<^sup>-\<^sup>1[f, g \<star> h, (tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0] \<star> TTfgh_TfTgh.chine) \<cdot> \<a>\<^sup>-\<^sup>1[f, (g \<star> h) \<star> (tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine] \<cdot> (f \<star> \<a>[g \<star> h, tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0, f\<^sub>0gh\<^sub>1.p\<^sub>0] \<star> TTfgh_TfTgh.chine) \<cdot> (f \<star> (\<a>\<^sup>-\<^sup>1[g, h, tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (f \<star> ((g \<star> \<a>[h, tab\<^sub>0 h, g\<^sub>0h\<^sub>1.p\<^sub>0]) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) = f \<star> can (\<^bold>\<langle>g\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> ((\<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) (((\<^bold>\<langle>g\<^bold>\<rangle> \<^bold>\<star> (\<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>)
[PROOF STEP]
have "... =
f \<star> can (\<^bold>\<langle>g\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> ((\<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>Tgh.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TfTgh.p\<^sub>0\<^bold>\<rangle>)
\<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>)
(((\<^bold>\<langle>g\<^bold>\<rangle> \<^bold>\<star> (\<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>Tgh.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TfTgh.p\<^sub>0\<^bold>\<rangle>)
\<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>)"
[PROOF STATE]
proof (prove)
goal (1 subgoal):
1. can (\<^bold>\<langle>f\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>g\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> ((\<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) (\<^bold>\<langle>f\<^bold>\<rangle> \<^bold>\<star> ((\<^bold>\<langle>g\<^bold>\<rangle> \<^bold>\<star> (\<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) = f \<star> can (\<^bold>\<langle>g\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> ((\<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) (((\<^bold>\<langle>g\<^bold>\<rangle> \<^bold>\<star> (\<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>)
[PROOF STEP]
using fg gh whisker_can_left_0
[PROOF STATE]
proof (prove)
using this:
src f = trg g
src g = trg h
\<lbrakk>E.Ide ?t; E.Ide ?u; \<^bold>\<lfloor>?t\<^bold>\<rfloor> = \<^bold>\<lfloor>?u\<^bold>\<rfloor>; ide ?g; E.Trg ?t = \<^bold>\<langle>src ?g\<^bold>\<rangle>\<^sub>0\<rbrakk> \<Longrightarrow> ?g \<star> can ?u ?t = can (\<^bold>\<langle>?g\<^bold>\<rangle> \<^bold>\<star> ?u) (\<^bold>\<langle>?g\<^bold>\<rangle> \<^bold>\<star> ?t)
goal (1 subgoal):
1. can (\<^bold>\<langle>f\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>g\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> ((\<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) (\<^bold>\<langle>f\<^bold>\<rangle> \<^bold>\<star> ((\<^bold>\<langle>g\<^bold>\<rangle> \<^bold>\<star> (\<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) = f \<star> can (\<^bold>\<langle>g\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> ((\<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) (((\<^bold>\<langle>g\<^bold>\<rangle> \<^bold>\<star> (\<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>)
[PROOF STEP]
by simp
[PROOF STATE]
proof (state)
this:
can (\<^bold>\<langle>f\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>g\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> ((\<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) (\<^bold>\<langle>f\<^bold>\<rangle> \<^bold>\<star> ((\<^bold>\<langle>g\<^bold>\<rangle> \<^bold>\<star> (\<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) = f \<star> can (\<^bold>\<langle>g\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> ((\<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) (((\<^bold>\<langle>g\<^bold>\<rangle> \<^bold>\<star> (\<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>)
goal (1 subgoal):
1. \<a>[f, g, h \<star> ((tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine] \<cdot> \<a>[f \<star> g, h, ((tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine] \<cdot> (\<a>\<^sup>-\<^sup>1[f, g, h] \<star> ((tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> \<a>[f \<star> g \<star> h, (tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine] \<cdot> (\<a>\<^sup>-\<^sup>1[f, g \<star> h, (tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0] \<star> TTfgh_TfTgh.chine) \<cdot> \<a>\<^sup>-\<^sup>1[f, (g \<star> h) \<star> (tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine] \<cdot> (f \<star> \<a>[g \<star> h, tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0, f\<^sub>0gh\<^sub>1.p\<^sub>0] \<star> TTfgh_TfTgh.chine) \<cdot> (f \<star> (\<a>\<^sup>-\<^sup>1[g, h, tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (f \<star> ((g \<star> \<a>[h, tab\<^sub>0 h, g\<^sub>0h\<^sub>1.p\<^sub>0]) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) = f \<star> can (\<^bold>\<langle>g\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> ((\<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) (((\<^bold>\<langle>g\<^bold>\<rangle> \<^bold>\<star> (\<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>)
[PROOF STEP]
finally
[PROOF STATE]
proof (chain)
picking this:
\<a>[f, g, h \<star> ((tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine] \<cdot> \<a>[f \<star> g, h, ((tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine] \<cdot> (\<a>\<^sup>-\<^sup>1[f, g, h] \<star> ((tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> \<a>[f \<star> g \<star> h, (tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine] \<cdot> (\<a>\<^sup>-\<^sup>1[f, g \<star> h, (tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0] \<star> TTfgh_TfTgh.chine) \<cdot> \<a>\<^sup>-\<^sup>1[f, (g \<star> h) \<star> (tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine] \<cdot> (f \<star> \<a>[g \<star> h, tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0, f\<^sub>0gh\<^sub>1.p\<^sub>0] \<star> TTfgh_TfTgh.chine) \<cdot> (f \<star> (\<a>\<^sup>-\<^sup>1[g, h, tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (f \<star> ((g \<star> \<a>[h, tab\<^sub>0 h, g\<^sub>0h\<^sub>1.p\<^sub>0]) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) = f \<star> can (\<^bold>\<langle>g\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> ((\<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) (((\<^bold>\<langle>g\<^bold>\<rangle> \<^bold>\<star> (\<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>)
[PROOF STEP]
show ?thesis
[PROOF STATE]
proof (prove)
using this:
\<a>[f, g, h \<star> ((tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine] \<cdot> \<a>[f \<star> g, h, ((tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine] \<cdot> (\<a>\<^sup>-\<^sup>1[f, g, h] \<star> ((tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> \<a>[f \<star> g \<star> h, (tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine] \<cdot> (\<a>\<^sup>-\<^sup>1[f, g \<star> h, (tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0] \<star> TTfgh_TfTgh.chine) \<cdot> \<a>\<^sup>-\<^sup>1[f, (g \<star> h) \<star> (tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine] \<cdot> (f \<star> \<a>[g \<star> h, tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0, f\<^sub>0gh\<^sub>1.p\<^sub>0] \<star> TTfgh_TfTgh.chine) \<cdot> (f \<star> (\<a>\<^sup>-\<^sup>1[g, h, tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (f \<star> ((g \<star> \<a>[h, tab\<^sub>0 h, g\<^sub>0h\<^sub>1.p\<^sub>0]) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) = f \<star> can (\<^bold>\<langle>g\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> ((\<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) (((\<^bold>\<langle>g\<^bold>\<rangle> \<^bold>\<star> (\<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>)
goal (1 subgoal):
1. \<a>[f, g, h \<star> ((tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine] \<cdot> \<a>[f \<star> g, h, ((tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine] \<cdot> (\<a>\<^sup>-\<^sup>1[f, g, h] \<star> ((tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> \<a>[f \<star> g \<star> h, (tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine] \<cdot> (\<a>\<^sup>-\<^sup>1[f, g \<star> h, (tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0] \<star> TTfgh_TfTgh.chine) \<cdot> \<a>\<^sup>-\<^sup>1[f, (g \<star> h) \<star> (tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine] \<cdot> (f \<star> \<a>[g \<star> h, tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0, f\<^sub>0gh\<^sub>1.p\<^sub>0] \<star> TTfgh_TfTgh.chine) \<cdot> (f \<star> (\<a>\<^sup>-\<^sup>1[g, h, tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (f \<star> ((g \<star> \<a>[h, tab\<^sub>0 h, g\<^sub>0h\<^sub>1.p\<^sub>0]) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) = f \<star> can (\<^bold>\<langle>g\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> ((\<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) (((\<^bold>\<langle>g\<^bold>\<rangle> \<^bold>\<star> (\<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>)
[PROOF STEP]
by blast
[PROOF STATE]
proof (state)
this:
\<a>[f, g, h \<star> ((tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine] \<cdot> \<a>[f \<star> g, h, ((tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine] \<cdot> (\<a>\<^sup>-\<^sup>1[f, g, h] \<star> ((tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> \<a>[f \<star> g \<star> h, (tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine] \<cdot> (\<a>\<^sup>-\<^sup>1[f, g \<star> h, (tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0] \<star> TTfgh_TfTgh.chine) \<cdot> \<a>\<^sup>-\<^sup>1[f, (g \<star> h) \<star> (tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine] \<cdot> (f \<star> \<a>[g \<star> h, tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0, f\<^sub>0gh\<^sub>1.p\<^sub>0] \<star> TTfgh_TfTgh.chine) \<cdot> (f \<star> (\<a>\<^sup>-\<^sup>1[g, h, tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (f \<star> ((g \<star> \<a>[h, tab\<^sub>0 h, g\<^sub>0h\<^sub>1.p\<^sub>0]) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) = f \<star> can (\<^bold>\<langle>g\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> ((\<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) (((\<^bold>\<langle>g\<^bold>\<rangle> \<^bold>\<star> (\<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>)
goal:
No subgoals!
[PROOF STEP]
qed
[PROOF STATE]
proof (state)
this:
\<a>[f, g, h \<star> ((tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine] \<cdot> \<a>[f \<star> g, h, ((tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine] \<cdot> (\<a>\<^sup>-\<^sup>1[f, g, h] \<star> ((tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> \<a>[f \<star> g \<star> h, (tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine] \<cdot> (\<a>\<^sup>-\<^sup>1[f, g \<star> h, (tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0] \<star> TTfgh_TfTgh.chine) \<cdot> \<a>\<^sup>-\<^sup>1[f, (g \<star> h) \<star> (tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine] \<cdot> (f \<star> \<a>[g \<star> h, tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0, f\<^sub>0gh\<^sub>1.p\<^sub>0] \<star> TTfgh_TfTgh.chine) \<cdot> (f \<star> (\<a>\<^sup>-\<^sup>1[g, h, tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (f \<star> ((g \<star> \<a>[h, tab\<^sub>0 h, g\<^sub>0h\<^sub>1.p\<^sub>0]) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) = f \<star> can (\<^bold>\<langle>g\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> ((\<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) (((\<^bold>\<langle>g\<^bold>\<rangle> \<^bold>\<star> (\<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>)
goal (1 subgoal):
1. (f \<star> g \<star> h \<star> TTfgh_TfTgh.the_\<theta>) \<cdot> (\<a>[f, g, h \<star> ((tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine] \<cdot> \<a>[f \<star> g, h, ((tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine] \<cdot> (\<a>\<^sup>-\<^sup>1[f, g, h] \<star> ((tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> \<a>[f \<star> g \<star> h, (tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine] \<cdot> (\<a>\<^sup>-\<^sup>1[f, g \<star> h, (tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0] \<star> TTfgh_TfTgh.chine) \<cdot> \<a>\<^sup>-\<^sup>1[f, (g \<star> h) \<star> (tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine] \<cdot> (f \<star> \<a>[g \<star> h, tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0, f\<^sub>0gh\<^sub>1.p\<^sub>0] \<star> TTfgh_TfTgh.chine) \<cdot> (f \<star> (\<a>\<^sup>-\<^sup>1[g, h, tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (f \<star> ((g \<star> \<a>[h, tab\<^sub>0 h, g\<^sub>0h\<^sub>1.p\<^sub>0]) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine)) \<cdot> (f \<star> (((g \<star> h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (((g \<star> g\<^sub>0h\<^sub>1.\<phi>) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((\<a>[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (((g.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>1) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (f\<^sub>0gh\<^sub>1.\<phi> \<star> TTfgh_TfTgh.chine) \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>0 f, f\<^sub>0gh\<^sub>1.p\<^sub>1, TTfgh_TfTgh.chine]) \<cdot> \<a>[f, tab\<^sub>0 f, f\<^sub>0gh\<^sub>1.p\<^sub>1 \<star> TTfgh_TfTgh.chine] \<cdot> (f.tab \<star> f\<^sub>0gh\<^sub>1.p\<^sub>1 \<star> TTfgh_TfTgh.chine) \<cdot> \<a>[tab\<^sub>1 f, f\<^sub>0gh\<^sub>1.p\<^sub>1, TTfgh_TfTgh.chine] \<cdot> TTfgh_TfTgh.the_\<nu> \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>1 f, f\<^sub>0g\<^sub>1.p\<^sub>1, fg\<^sub>0h\<^sub>1.p\<^sub>1] = ((f \<star> g \<star> h \<star> TTfgh_TfTgh.the_\<theta>) \<cdot> (f \<star> can (\<^bold>\<langle>g\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> ((\<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) (((\<^bold>\<langle>g\<^bold>\<rangle> \<^bold>\<star> (\<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>)) \<cdot> (f \<star> (((g \<star> h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (((g \<star> g\<^sub>0h\<^sub>1.\<phi>) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((\<a>[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (((g.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>1) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (f\<^sub>0gh\<^sub>1.\<phi> \<star> TTfgh_TfTgh.chine) \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>0 f, f\<^sub>0gh\<^sub>1.p\<^sub>1, TTfgh_TfTgh.chine])) \<cdot> \<a>[f, tab\<^sub>0 f, f\<^sub>0gh\<^sub>1.p\<^sub>1 \<star> TTfgh_TfTgh.chine] \<cdot> (f.tab \<star> f\<^sub>0gh\<^sub>1.p\<^sub>1 \<star> TTfgh_TfTgh.chine) \<cdot> \<a>[tab\<^sub>1 f, f\<^sub>0gh\<^sub>1.p\<^sub>1, TTfgh_TfTgh.chine] \<cdot> TTfgh_TfTgh.the_\<nu> \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>1 f, f\<^sub>0g\<^sub>1.p\<^sub>1, fg\<^sub>0h\<^sub>1.p\<^sub>1]
[PROOF STEP]
thus ?thesis
[PROOF STATE]
proof (prove)
using this:
\<a>[f, g, h \<star> ((tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine] \<cdot> \<a>[f \<star> g, h, ((tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine] \<cdot> (\<a>\<^sup>-\<^sup>1[f, g, h] \<star> ((tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> \<a>[f \<star> g \<star> h, (tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine] \<cdot> (\<a>\<^sup>-\<^sup>1[f, g \<star> h, (tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0] \<star> TTfgh_TfTgh.chine) \<cdot> \<a>\<^sup>-\<^sup>1[f, (g \<star> h) \<star> (tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine] \<cdot> (f \<star> \<a>[g \<star> h, tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0, f\<^sub>0gh\<^sub>1.p\<^sub>0] \<star> TTfgh_TfTgh.chine) \<cdot> (f \<star> (\<a>\<^sup>-\<^sup>1[g, h, tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (f \<star> ((g \<star> \<a>[h, tab\<^sub>0 h, g\<^sub>0h\<^sub>1.p\<^sub>0]) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) = f \<star> can (\<^bold>\<langle>g\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> ((\<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) (((\<^bold>\<langle>g\<^bold>\<rangle> \<^bold>\<star> (\<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>)
goal (1 subgoal):
1. (f \<star> g \<star> h \<star> TTfgh_TfTgh.the_\<theta>) \<cdot> (\<a>[f, g, h \<star> ((tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine] \<cdot> \<a>[f \<star> g, h, ((tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine] \<cdot> (\<a>\<^sup>-\<^sup>1[f, g, h] \<star> ((tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> \<a>[f \<star> g \<star> h, (tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine] \<cdot> (\<a>\<^sup>-\<^sup>1[f, g \<star> h, (tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0] \<star> TTfgh_TfTgh.chine) \<cdot> \<a>\<^sup>-\<^sup>1[f, (g \<star> h) \<star> (tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine] \<cdot> (f \<star> \<a>[g \<star> h, tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0, f\<^sub>0gh\<^sub>1.p\<^sub>0] \<star> TTfgh_TfTgh.chine) \<cdot> (f \<star> (\<a>\<^sup>-\<^sup>1[g, h, tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (f \<star> ((g \<star> \<a>[h, tab\<^sub>0 h, g\<^sub>0h\<^sub>1.p\<^sub>0]) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine)) \<cdot> (f \<star> (((g \<star> h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (((g \<star> g\<^sub>0h\<^sub>1.\<phi>) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((\<a>[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (((g.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>1) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (f\<^sub>0gh\<^sub>1.\<phi> \<star> TTfgh_TfTgh.chine) \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>0 f, f\<^sub>0gh\<^sub>1.p\<^sub>1, TTfgh_TfTgh.chine]) \<cdot> \<a>[f, tab\<^sub>0 f, f\<^sub>0gh\<^sub>1.p\<^sub>1 \<star> TTfgh_TfTgh.chine] \<cdot> (f.tab \<star> f\<^sub>0gh\<^sub>1.p\<^sub>1 \<star> TTfgh_TfTgh.chine) \<cdot> \<a>[tab\<^sub>1 f, f\<^sub>0gh\<^sub>1.p\<^sub>1, TTfgh_TfTgh.chine] \<cdot> TTfgh_TfTgh.the_\<nu> \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>1 f, f\<^sub>0g\<^sub>1.p\<^sub>1, fg\<^sub>0h\<^sub>1.p\<^sub>1] = ((f \<star> g \<star> h \<star> TTfgh_TfTgh.the_\<theta>) \<cdot> (f \<star> can (\<^bold>\<langle>g\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> ((\<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) (((\<^bold>\<langle>g\<^bold>\<rangle> \<^bold>\<star> (\<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>)) \<cdot> (f \<star> (((g \<star> h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (((g \<star> g\<^sub>0h\<^sub>1.\<phi>) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((\<a>[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (((g.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>1) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (f\<^sub>0gh\<^sub>1.\<phi> \<star> TTfgh_TfTgh.chine) \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>0 f, f\<^sub>0gh\<^sub>1.p\<^sub>1, TTfgh_TfTgh.chine])) \<cdot> \<a>[f, tab\<^sub>0 f, f\<^sub>0gh\<^sub>1.p\<^sub>1 \<star> TTfgh_TfTgh.chine] \<cdot> (f.tab \<star> f\<^sub>0gh\<^sub>1.p\<^sub>1 \<star> TTfgh_TfTgh.chine) \<cdot> \<a>[tab\<^sub>1 f, f\<^sub>0gh\<^sub>1.p\<^sub>1, TTfgh_TfTgh.chine] \<cdot> TTfgh_TfTgh.the_\<nu> \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>1 f, f\<^sub>0g\<^sub>1.p\<^sub>1, fg\<^sub>0h\<^sub>1.p\<^sub>1]
[PROOF STEP]
using comp_assoc
[PROOF STATE]
proof (prove)
using this:
\<a>[f, g, h \<star> ((tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine] \<cdot> \<a>[f \<star> g, h, ((tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine] \<cdot> (\<a>\<^sup>-\<^sup>1[f, g, h] \<star> ((tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> \<a>[f \<star> g \<star> h, (tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine] \<cdot> (\<a>\<^sup>-\<^sup>1[f, g \<star> h, (tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0] \<star> TTfgh_TfTgh.chine) \<cdot> \<a>\<^sup>-\<^sup>1[f, (g \<star> h) \<star> (tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine] \<cdot> (f \<star> \<a>[g \<star> h, tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0, f\<^sub>0gh\<^sub>1.p\<^sub>0] \<star> TTfgh_TfTgh.chine) \<cdot> (f \<star> (\<a>\<^sup>-\<^sup>1[g, h, tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (f \<star> ((g \<star> \<a>[h, tab\<^sub>0 h, g\<^sub>0h\<^sub>1.p\<^sub>0]) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) = f \<star> can (\<^bold>\<langle>g\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> ((\<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) (((\<^bold>\<langle>g\<^bold>\<rangle> \<^bold>\<star> (\<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>)
(?h \<cdot> ?g) \<cdot> ?f = ?h \<cdot> ?g \<cdot> ?f
goal (1 subgoal):
1. (f \<star> g \<star> h \<star> TTfgh_TfTgh.the_\<theta>) \<cdot> (\<a>[f, g, h \<star> ((tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine] \<cdot> \<a>[f \<star> g, h, ((tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine] \<cdot> (\<a>\<^sup>-\<^sup>1[f, g, h] \<star> ((tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> \<a>[f \<star> g \<star> h, (tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine] \<cdot> (\<a>\<^sup>-\<^sup>1[f, g \<star> h, (tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0] \<star> TTfgh_TfTgh.chine) \<cdot> \<a>\<^sup>-\<^sup>1[f, (g \<star> h) \<star> (tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine] \<cdot> (f \<star> \<a>[g \<star> h, tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0, f\<^sub>0gh\<^sub>1.p\<^sub>0] \<star> TTfgh_TfTgh.chine) \<cdot> (f \<star> (\<a>\<^sup>-\<^sup>1[g, h, tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (f \<star> ((g \<star> \<a>[h, tab\<^sub>0 h, g\<^sub>0h\<^sub>1.p\<^sub>0]) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine)) \<cdot> (f \<star> (((g \<star> h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (((g \<star> g\<^sub>0h\<^sub>1.\<phi>) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((\<a>[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (((g.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>1) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (f\<^sub>0gh\<^sub>1.\<phi> \<star> TTfgh_TfTgh.chine) \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>0 f, f\<^sub>0gh\<^sub>1.p\<^sub>1, TTfgh_TfTgh.chine]) \<cdot> \<a>[f, tab\<^sub>0 f, f\<^sub>0gh\<^sub>1.p\<^sub>1 \<star> TTfgh_TfTgh.chine] \<cdot> (f.tab \<star> f\<^sub>0gh\<^sub>1.p\<^sub>1 \<star> TTfgh_TfTgh.chine) \<cdot> \<a>[tab\<^sub>1 f, f\<^sub>0gh\<^sub>1.p\<^sub>1, TTfgh_TfTgh.chine] \<cdot> TTfgh_TfTgh.the_\<nu> \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>1 f, f\<^sub>0g\<^sub>1.p\<^sub>1, fg\<^sub>0h\<^sub>1.p\<^sub>1] = ((f \<star> g \<star> h \<star> TTfgh_TfTgh.the_\<theta>) \<cdot> (f \<star> can (\<^bold>\<langle>g\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> ((\<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) (((\<^bold>\<langle>g\<^bold>\<rangle> \<^bold>\<star> (\<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>)) \<cdot> (f \<star> (((g \<star> h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (((g \<star> g\<^sub>0h\<^sub>1.\<phi>) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((\<a>[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (((g.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>1) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (f\<^sub>0gh\<^sub>1.\<phi> \<star> TTfgh_TfTgh.chine) \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>0 f, f\<^sub>0gh\<^sub>1.p\<^sub>1, TTfgh_TfTgh.chine])) \<cdot> \<a>[f, tab\<^sub>0 f, f\<^sub>0gh\<^sub>1.p\<^sub>1 \<star> TTfgh_TfTgh.chine] \<cdot> (f.tab \<star> f\<^sub>0gh\<^sub>1.p\<^sub>1 \<star> TTfgh_TfTgh.chine) \<cdot> \<a>[tab\<^sub>1 f, f\<^sub>0gh\<^sub>1.p\<^sub>1, TTfgh_TfTgh.chine] \<cdot> TTfgh_TfTgh.the_\<nu> \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>1 f, f\<^sub>0g\<^sub>1.p\<^sub>1, fg\<^sub>0h\<^sub>1.p\<^sub>1]
[PROOF STEP]
by presburger
[PROOF STATE]
proof (state)
this:
(f \<star> g \<star> h \<star> TTfgh_TfTgh.the_\<theta>) \<cdot> (\<a>[f, g, h \<star> ((tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine] \<cdot> \<a>[f \<star> g, h, ((tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine] \<cdot> (\<a>\<^sup>-\<^sup>1[f, g, h] \<star> ((tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> \<a>[f \<star> g \<star> h, (tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine] \<cdot> (\<a>\<^sup>-\<^sup>1[f, g \<star> h, (tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0] \<star> TTfgh_TfTgh.chine) \<cdot> \<a>\<^sup>-\<^sup>1[f, (g \<star> h) \<star> (tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine] \<cdot> (f \<star> \<a>[g \<star> h, tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0, f\<^sub>0gh\<^sub>1.p\<^sub>0] \<star> TTfgh_TfTgh.chine) \<cdot> (f \<star> (\<a>\<^sup>-\<^sup>1[g, h, tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (f \<star> ((g \<star> \<a>[h, tab\<^sub>0 h, g\<^sub>0h\<^sub>1.p\<^sub>0]) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine)) \<cdot> (f \<star> (((g \<star> h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (((g \<star> g\<^sub>0h\<^sub>1.\<phi>) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((\<a>[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (((g.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>1) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (f\<^sub>0gh\<^sub>1.\<phi> \<star> TTfgh_TfTgh.chine) \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>0 f, f\<^sub>0gh\<^sub>1.p\<^sub>1, TTfgh_TfTgh.chine]) \<cdot> \<a>[f, tab\<^sub>0 f, f\<^sub>0gh\<^sub>1.p\<^sub>1 \<star> TTfgh_TfTgh.chine] \<cdot> (f.tab \<star> f\<^sub>0gh\<^sub>1.p\<^sub>1 \<star> TTfgh_TfTgh.chine) \<cdot> \<a>[tab\<^sub>1 f, f\<^sub>0gh\<^sub>1.p\<^sub>1, TTfgh_TfTgh.chine] \<cdot> TTfgh_TfTgh.the_\<nu> \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>1 f, f\<^sub>0g\<^sub>1.p\<^sub>1, fg\<^sub>0h\<^sub>1.p\<^sub>1] = ((f \<star> g \<star> h \<star> TTfgh_TfTgh.the_\<theta>) \<cdot> (f \<star> can (\<^bold>\<langle>g\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> ((\<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) (((\<^bold>\<langle>g\<^bold>\<rangle> \<^bold>\<star> (\<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>)) \<cdot> (f \<star> (((g \<star> h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (((g \<star> g\<^sub>0h\<^sub>1.\<phi>) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((\<a>[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (((g.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>1) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (f\<^sub>0gh\<^sub>1.\<phi> \<star> TTfgh_TfTgh.chine) \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>0 f, f\<^sub>0gh\<^sub>1.p\<^sub>1, TTfgh_TfTgh.chine])) \<cdot> \<a>[f, tab\<^sub>0 f, f\<^sub>0gh\<^sub>1.p\<^sub>1 \<star> TTfgh_TfTgh.chine] \<cdot> (f.tab \<star> f\<^sub>0gh\<^sub>1.p\<^sub>1 \<star> TTfgh_TfTgh.chine) \<cdot> \<a>[tab\<^sub>1 f, f\<^sub>0gh\<^sub>1.p\<^sub>1, TTfgh_TfTgh.chine] \<cdot> TTfgh_TfTgh.the_\<nu> \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>1 f, f\<^sub>0g\<^sub>1.p\<^sub>1, fg\<^sub>0h\<^sub>1.p\<^sub>1]
goal:
No subgoals!
[PROOF STEP]
qed
[PROOF STATE]
proof (state)
this:
(f \<star> g \<star> h \<star> TTfgh_TfTgh.the_\<theta>) \<cdot> (\<a>[f, g, h \<star> ((tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine] \<cdot> \<a>[f \<star> g, h, ((tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine] \<cdot> (\<a>\<^sup>-\<^sup>1[f, g, h] \<star> ((tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> \<a>[f \<star> g \<star> h, (tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine] \<cdot> (\<a>\<^sup>-\<^sup>1[f, g \<star> h, (tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0] \<star> TTfgh_TfTgh.chine) \<cdot> \<a>\<^sup>-\<^sup>1[f, (g \<star> h) \<star> (tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine] \<cdot> (f \<star> \<a>[g \<star> h, tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0, f\<^sub>0gh\<^sub>1.p\<^sub>0] \<star> TTfgh_TfTgh.chine) \<cdot> (f \<star> (\<a>\<^sup>-\<^sup>1[g, h, tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (f \<star> ((g \<star> \<a>[h, tab\<^sub>0 h, g\<^sub>0h\<^sub>1.p\<^sub>0]) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine)) \<cdot> (f \<star> (((g \<star> h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (((g \<star> g\<^sub>0h\<^sub>1.\<phi>) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((\<a>[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (((g.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>1) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (f\<^sub>0gh\<^sub>1.\<phi> \<star> TTfgh_TfTgh.chine) \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>0 f, f\<^sub>0gh\<^sub>1.p\<^sub>1, TTfgh_TfTgh.chine]) \<cdot> \<a>[f, tab\<^sub>0 f, f\<^sub>0gh\<^sub>1.p\<^sub>1 \<star> TTfgh_TfTgh.chine] \<cdot> (f.tab \<star> f\<^sub>0gh\<^sub>1.p\<^sub>1 \<star> TTfgh_TfTgh.chine) \<cdot> \<a>[tab\<^sub>1 f, f\<^sub>0gh\<^sub>1.p\<^sub>1, TTfgh_TfTgh.chine] \<cdot> TTfgh_TfTgh.the_\<nu> \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>1 f, f\<^sub>0g\<^sub>1.p\<^sub>1, fg\<^sub>0h\<^sub>1.p\<^sub>1] = ((f \<star> g \<star> h \<star> TTfgh_TfTgh.the_\<theta>) \<cdot> (f \<star> can (\<^bold>\<langle>g\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> ((\<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) (((\<^bold>\<langle>g\<^bold>\<rangle> \<^bold>\<star> (\<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>)) \<cdot> (f \<star> (((g \<star> h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (((g \<star> g\<^sub>0h\<^sub>1.\<phi>) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((\<a>[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (((g.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>1) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (f\<^sub>0gh\<^sub>1.\<phi> \<star> TTfgh_TfTgh.chine) \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>0 f, f\<^sub>0gh\<^sub>1.p\<^sub>1, TTfgh_TfTgh.chine])) \<cdot> \<a>[f, tab\<^sub>0 f, f\<^sub>0gh\<^sub>1.p\<^sub>1 \<star> TTfgh_TfTgh.chine] \<cdot> (f.tab \<star> f\<^sub>0gh\<^sub>1.p\<^sub>1 \<star> TTfgh_TfTgh.chine) \<cdot> \<a>[tab\<^sub>1 f, f\<^sub>0gh\<^sub>1.p\<^sub>1, TTfgh_TfTgh.chine] \<cdot> TTfgh_TfTgh.the_\<nu> \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>1 f, f\<^sub>0g\<^sub>1.p\<^sub>1, fg\<^sub>0h\<^sub>1.p\<^sub>1]
goal (1 subgoal):
1. (f \<star> \<theta>\<^sub>f) \<cdot> \<a>[f, tab\<^sub>0 f, w\<^sub>f] \<cdot> (f.tab \<star> w\<^sub>f) = ((f \<star> \<theta>\<^sub>f') \<cdot> \<a>[f, tab\<^sub>0 f, w\<^sub>f'] \<cdot> (f.tab \<star> w\<^sub>f')) \<cdot> \<beta>\<^sub>f
[PROOF STEP]
also
[PROOF STATE]
proof (state)
this:
(f \<star> g \<star> h \<star> TTfgh_TfTgh.the_\<theta>) \<cdot> (\<a>[f, g, h \<star> ((tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine] \<cdot> \<a>[f \<star> g, h, ((tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine] \<cdot> (\<a>\<^sup>-\<^sup>1[f, g, h] \<star> ((tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> \<a>[f \<star> g \<star> h, (tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine] \<cdot> (\<a>\<^sup>-\<^sup>1[f, g \<star> h, (tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0] \<star> TTfgh_TfTgh.chine) \<cdot> \<a>\<^sup>-\<^sup>1[f, (g \<star> h) \<star> (tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine] \<cdot> (f \<star> \<a>[g \<star> h, tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0, f\<^sub>0gh\<^sub>1.p\<^sub>0] \<star> TTfgh_TfTgh.chine) \<cdot> (f \<star> (\<a>\<^sup>-\<^sup>1[g, h, tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (f \<star> ((g \<star> \<a>[h, tab\<^sub>0 h, g\<^sub>0h\<^sub>1.p\<^sub>0]) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine)) \<cdot> (f \<star> (((g \<star> h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (((g \<star> g\<^sub>0h\<^sub>1.\<phi>) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((\<a>[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (((g.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>1) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (f\<^sub>0gh\<^sub>1.\<phi> \<star> TTfgh_TfTgh.chine) \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>0 f, f\<^sub>0gh\<^sub>1.p\<^sub>1, TTfgh_TfTgh.chine]) \<cdot> \<a>[f, tab\<^sub>0 f, f\<^sub>0gh\<^sub>1.p\<^sub>1 \<star> TTfgh_TfTgh.chine] \<cdot> (f.tab \<star> f\<^sub>0gh\<^sub>1.p\<^sub>1 \<star> TTfgh_TfTgh.chine) \<cdot> \<a>[tab\<^sub>1 f, f\<^sub>0gh\<^sub>1.p\<^sub>1, TTfgh_TfTgh.chine] \<cdot> TTfgh_TfTgh.the_\<nu> \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>1 f, f\<^sub>0g\<^sub>1.p\<^sub>1, fg\<^sub>0h\<^sub>1.p\<^sub>1] = ((f \<star> g \<star> h \<star> TTfgh_TfTgh.the_\<theta>) \<cdot> (f \<star> can (\<^bold>\<langle>g\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> ((\<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) (((\<^bold>\<langle>g\<^bold>\<rangle> \<^bold>\<star> (\<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>)) \<cdot> (f \<star> (((g \<star> h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (((g \<star> g\<^sub>0h\<^sub>1.\<phi>) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((\<a>[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (((g.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>1) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (f\<^sub>0gh\<^sub>1.\<phi> \<star> TTfgh_TfTgh.chine) \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>0 f, f\<^sub>0gh\<^sub>1.p\<^sub>1, TTfgh_TfTgh.chine])) \<cdot> \<a>[f, tab\<^sub>0 f, f\<^sub>0gh\<^sub>1.p\<^sub>1 \<star> TTfgh_TfTgh.chine] \<cdot> (f.tab \<star> f\<^sub>0gh\<^sub>1.p\<^sub>1 \<star> TTfgh_TfTgh.chine) \<cdot> \<a>[tab\<^sub>1 f, f\<^sub>0gh\<^sub>1.p\<^sub>1, TTfgh_TfTgh.chine] \<cdot> TTfgh_TfTgh.the_\<nu> \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>1 f, f\<^sub>0g\<^sub>1.p\<^sub>1, fg\<^sub>0h\<^sub>1.p\<^sub>1]
goal (1 subgoal):
1. (f \<star> \<theta>\<^sub>f) \<cdot> \<a>[f, tab\<^sub>0 f, w\<^sub>f] \<cdot> (f.tab \<star> w\<^sub>f) = ((f \<star> \<theta>\<^sub>f') \<cdot> \<a>[f, tab\<^sub>0 f, w\<^sub>f'] \<cdot> (f.tab \<star> w\<^sub>f')) \<cdot> \<beta>\<^sub>f
[PROOF STEP]
have "... =
(f \<star> (g \<star> h \<star> TTfgh_TfTgh.the_\<theta>) \<cdot>
can (\<^bold>\<langle>g\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> ((\<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>Tgh.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TfTgh.p\<^sub>0\<^bold>\<rangle>)
\<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>)
(((\<^bold>\<langle>g\<^bold>\<rangle> \<^bold>\<star> (\<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>Tgh.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TfTgh.p\<^sub>0\<^bold>\<rangle>)
\<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) \<cdot>
(((g \<star> h.tab \<star> Tgh.p\<^sub>0) \<star> TfTgh.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot>
(((g \<star> g\<^sub>0h\<^sub>1.\<phi>) \<star> TfTgh.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot>
((\<a>[g, tab\<^sub>0 g, Tgh.p\<^sub>1] \<star> TfTgh.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot>
(((g.tab \<star> Tgh.p\<^sub>1) \<star> TfTgh.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot>
(f\<^sub>0gh\<^sub>1.\<phi> \<star> TTfgh_TfTgh.chine) \<cdot>
\<a>\<^sup>-\<^sup>1[tab\<^sub>0 f, TfTgh.p\<^sub>1, TTfgh_TfTgh.chine]) \<cdot>
\<a>[f, tab\<^sub>0 f, TfTgh.p\<^sub>1 \<star> TTfgh_TfTgh.chine] \<cdot>
(f.tab \<star> TfTgh.p\<^sub>1 \<star> TTfgh_TfTgh.chine) \<cdot>
\<a>[tab\<^sub>1 f, TfTgh.p\<^sub>1, TTfgh_TfTgh.chine] \<cdot>
TTfgh_TfTgh.the_\<nu> \<cdot>
\<a>\<^sup>-\<^sup>1[tab\<^sub>1 f, Tfg.p\<^sub>1, TTfgh.p\<^sub>1]"
[PROOF STATE]
proof (prove)
goal (1 subgoal):
1. ((f \<star> g \<star> h \<star> TTfgh_TfTgh.the_\<theta>) \<cdot> (f \<star> can (\<^bold>\<langle>g\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> ((\<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) (((\<^bold>\<langle>g\<^bold>\<rangle> \<^bold>\<star> (\<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>)) \<cdot> (f \<star> (((g \<star> h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (((g \<star> g\<^sub>0h\<^sub>1.\<phi>) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((\<a>[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (((g.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>1) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (f\<^sub>0gh\<^sub>1.\<phi> \<star> TTfgh_TfTgh.chine) \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>0 f, f\<^sub>0gh\<^sub>1.p\<^sub>1, TTfgh_TfTgh.chine])) \<cdot> \<a>[f, tab\<^sub>0 f, f\<^sub>0gh\<^sub>1.p\<^sub>1 \<star> TTfgh_TfTgh.chine] \<cdot> (f.tab \<star> f\<^sub>0gh\<^sub>1.p\<^sub>1 \<star> TTfgh_TfTgh.chine) \<cdot> \<a>[tab\<^sub>1 f, f\<^sub>0gh\<^sub>1.p\<^sub>1, TTfgh_TfTgh.chine] \<cdot> TTfgh_TfTgh.the_\<nu> \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>1 f, f\<^sub>0g\<^sub>1.p\<^sub>1, fg\<^sub>0h\<^sub>1.p\<^sub>1] = (f \<star> (g \<star> h \<star> TTfgh_TfTgh.the_\<theta>) \<cdot> can (\<^bold>\<langle>g\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> ((\<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) (((\<^bold>\<langle>g\<^bold>\<rangle> \<^bold>\<star> (\<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) \<cdot> (((g \<star> h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (((g \<star> g\<^sub>0h\<^sub>1.\<phi>) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((\<a>[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (((g.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>1) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (f\<^sub>0gh\<^sub>1.\<phi> \<star> TTfgh_TfTgh.chine) \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>0 f, f\<^sub>0gh\<^sub>1.p\<^sub>1, TTfgh_TfTgh.chine]) \<cdot> \<a>[f, tab\<^sub>0 f, f\<^sub>0gh\<^sub>1.p\<^sub>1 \<star> TTfgh_TfTgh.chine] \<cdot> (f.tab \<star> f\<^sub>0gh\<^sub>1.p\<^sub>1 \<star> TTfgh_TfTgh.chine) \<cdot> \<a>[tab\<^sub>1 f, f\<^sub>0gh\<^sub>1.p\<^sub>1, TTfgh_TfTgh.chine] \<cdot> TTfgh_TfTgh.the_\<nu> \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>1 f, f\<^sub>0g\<^sub>1.p\<^sub>1, fg\<^sub>0h\<^sub>1.p\<^sub>1]
[PROOF STEP]
proof -
[PROOF STATE]
proof (state)
goal (1 subgoal):
1. ((f \<star> g \<star> h \<star> TTfgh_TfTgh.the_\<theta>) \<cdot> (f \<star> can (\<^bold>\<langle>g\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> ((\<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) (((\<^bold>\<langle>g\<^bold>\<rangle> \<^bold>\<star> (\<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>)) \<cdot> (f \<star> (((g \<star> h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (((g \<star> g\<^sub>0h\<^sub>1.\<phi>) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((\<a>[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (((g.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>1) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (f\<^sub>0gh\<^sub>1.\<phi> \<star> TTfgh_TfTgh.chine) \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>0 f, f\<^sub>0gh\<^sub>1.p\<^sub>1, TTfgh_TfTgh.chine])) \<cdot> \<a>[f, tab\<^sub>0 f, f\<^sub>0gh\<^sub>1.p\<^sub>1 \<star> TTfgh_TfTgh.chine] \<cdot> (f.tab \<star> f\<^sub>0gh\<^sub>1.p\<^sub>1 \<star> TTfgh_TfTgh.chine) \<cdot> \<a>[tab\<^sub>1 f, f\<^sub>0gh\<^sub>1.p\<^sub>1, TTfgh_TfTgh.chine] \<cdot> TTfgh_TfTgh.the_\<nu> \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>1 f, f\<^sub>0g\<^sub>1.p\<^sub>1, fg\<^sub>0h\<^sub>1.p\<^sub>1] = (f \<star> (g \<star> h \<star> TTfgh_TfTgh.the_\<theta>) \<cdot> can (\<^bold>\<langle>g\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> ((\<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) (((\<^bold>\<langle>g\<^bold>\<rangle> \<^bold>\<star> (\<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) \<cdot> (((g \<star> h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (((g \<star> g\<^sub>0h\<^sub>1.\<phi>) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((\<a>[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (((g.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>1) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (f\<^sub>0gh\<^sub>1.\<phi> \<star> TTfgh_TfTgh.chine) \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>0 f, f\<^sub>0gh\<^sub>1.p\<^sub>1, TTfgh_TfTgh.chine]) \<cdot> \<a>[f, tab\<^sub>0 f, f\<^sub>0gh\<^sub>1.p\<^sub>1 \<star> TTfgh_TfTgh.chine] \<cdot> (f.tab \<star> f\<^sub>0gh\<^sub>1.p\<^sub>1 \<star> TTfgh_TfTgh.chine) \<cdot> \<a>[tab\<^sub>1 f, f\<^sub>0gh\<^sub>1.p\<^sub>1, TTfgh_TfTgh.chine] \<cdot> TTfgh_TfTgh.the_\<nu> \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>1 f, f\<^sub>0g\<^sub>1.p\<^sub>1, fg\<^sub>0h\<^sub>1.p\<^sub>1]
[PROOF STEP]
have "((f \<star> g \<star> h \<star> TTfgh_TfTgh.the_\<theta>) \<cdot>
(f \<star> can (\<^bold>\<langle>g\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> ((\<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>Tgh.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TfTgh.p\<^sub>0\<^bold>\<rangle>)
\<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>)
(((\<^bold>\<langle>g\<^bold>\<rangle> \<^bold>\<star> (\<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>Tgh.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TfTgh.p\<^sub>0\<^bold>\<rangle>)
\<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>)) \<cdot>
(f \<star> (((g \<star> h.tab \<star> Tgh.p\<^sub>0) \<star> TfTgh.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot>
(((g \<star> g\<^sub>0h\<^sub>1.\<phi>) \<star> TfTgh.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot>
((\<a>[g, tab\<^sub>0 g, Tgh.p\<^sub>1] \<star> TfTgh.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot>
(((g.tab \<star> Tgh.p\<^sub>1) \<star> TfTgh.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot>
(f\<^sub>0gh\<^sub>1.\<phi> \<star> TTfgh_TfTgh.chine) \<cdot>
\<a>\<^sup>-\<^sup>1[tab\<^sub>0 f, TfTgh.p\<^sub>1, TTfgh_TfTgh.chine])) =
f \<star> (g \<star> h \<star> TTfgh_TfTgh.the_\<theta>) \<cdot>
can (\<^bold>\<langle>g\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> ((\<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>Tgh.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TfTgh.p\<^sub>0\<^bold>\<rangle>)
\<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>)
(((\<^bold>\<langle>g\<^bold>\<rangle> \<^bold>\<star> (\<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>Tgh.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TfTgh.p\<^sub>0\<^bold>\<rangle>)
\<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) \<cdot>
(((g \<star> h.tab \<star> Tgh.p\<^sub>0) \<star> TfTgh.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot>
(((g \<star> g\<^sub>0h\<^sub>1.\<phi>) \<star> TfTgh.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot>
((\<a>[g, tab\<^sub>0 g, Tgh.p\<^sub>1] \<star> TfTgh.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot>
(((g.tab \<star> Tgh.p\<^sub>1) \<star> TfTgh.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot>
(f\<^sub>0gh\<^sub>1.\<phi> \<star> TTfgh_TfTgh.chine) \<cdot>
\<a>\<^sup>-\<^sup>1[tab\<^sub>0 f, TfTgh.p\<^sub>1, TTfgh_TfTgh.chine]"
[PROOF STATE]
proof (prove)
goal (1 subgoal):
1. (f \<star> g \<star> h \<star> TTfgh_TfTgh.the_\<theta>) \<cdot> (f \<star> can (\<^bold>\<langle>g\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> ((\<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) (((\<^bold>\<langle>g\<^bold>\<rangle> \<^bold>\<star> (\<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>)) \<cdot> (f \<star> (((g \<star> h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (((g \<star> g\<^sub>0h\<^sub>1.\<phi>) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((\<a>[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (((g.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>1) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (f\<^sub>0gh\<^sub>1.\<phi> \<star> TTfgh_TfTgh.chine) \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>0 f, f\<^sub>0gh\<^sub>1.p\<^sub>1, TTfgh_TfTgh.chine]) = f \<star> (g \<star> h \<star> TTfgh_TfTgh.the_\<theta>) \<cdot> can (\<^bold>\<langle>g\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> ((\<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) (((\<^bold>\<langle>g\<^bold>\<rangle> \<^bold>\<star> (\<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) \<cdot> (((g \<star> h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (((g \<star> g\<^sub>0h\<^sub>1.\<phi>) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((\<a>[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (((g.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>1) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (f\<^sub>0gh\<^sub>1.\<phi> \<star> TTfgh_TfTgh.chine) \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>0 f, f\<^sub>0gh\<^sub>1.p\<^sub>1, TTfgh_TfTgh.chine]
[PROOF STEP]
proof -
[PROOF STATE]
proof (state)
goal (1 subgoal):
1. (f \<star> g \<star> h \<star> TTfgh_TfTgh.the_\<theta>) \<cdot> (f \<star> can (\<^bold>\<langle>g\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> ((\<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) (((\<^bold>\<langle>g\<^bold>\<rangle> \<^bold>\<star> (\<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>)) \<cdot> (f \<star> (((g \<star> h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (((g \<star> g\<^sub>0h\<^sub>1.\<phi>) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((\<a>[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (((g.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>1) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (f\<^sub>0gh\<^sub>1.\<phi> \<star> TTfgh_TfTgh.chine) \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>0 f, f\<^sub>0gh\<^sub>1.p\<^sub>1, TTfgh_TfTgh.chine]) = f \<star> (g \<star> h \<star> TTfgh_TfTgh.the_\<theta>) \<cdot> can (\<^bold>\<langle>g\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> ((\<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) (((\<^bold>\<langle>g\<^bold>\<rangle> \<^bold>\<star> (\<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) \<cdot> (((g \<star> h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (((g \<star> g\<^sub>0h\<^sub>1.\<phi>) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((\<a>[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (((g.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>1) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (f\<^sub>0gh\<^sub>1.\<phi> \<star> TTfgh_TfTgh.chine) \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>0 f, f\<^sub>0gh\<^sub>1.p\<^sub>1, TTfgh_TfTgh.chine]
[PROOF STEP]
have 1: "arr ((g \<star> h \<star> TTfgh_TfTgh.the_\<theta>) \<cdot>
can (\<^bold>\<langle>g\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> ((\<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>Tgh.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TfTgh.p\<^sub>0\<^bold>\<rangle>)
\<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>)
(((\<^bold>\<langle>g\<^bold>\<rangle> \<^bold>\<star> (\<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>Tgh.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TfTgh.p\<^sub>0\<^bold>\<rangle>)
\<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) \<cdot>
(((g \<star> h.tab \<star> Tgh.p\<^sub>0) \<star> TfTgh.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot>
(((g \<star> g\<^sub>0h\<^sub>1.\<phi>) \<star> TfTgh.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot>
((\<a>[g, tab\<^sub>0 g, Tgh.p\<^sub>1] \<star> TfTgh.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot>
(((g.tab \<star> Tgh.p\<^sub>1) \<star> TfTgh.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot>
(f\<^sub>0gh\<^sub>1.\<phi> \<star> TTfgh_TfTgh.chine) \<cdot>
\<a>\<^sup>-\<^sup>1[tab\<^sub>0 f, TfTgh.p\<^sub>1, TTfgh_TfTgh.chine])"
[PROOF STATE]
proof (prove)
goal (1 subgoal):
1. seq (g \<star> h \<star> TTfgh_TfTgh.the_\<theta>) (can (\<^bold>\<langle>g\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> ((\<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) (((\<^bold>\<langle>g\<^bold>\<rangle> \<^bold>\<star> (\<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) \<cdot> (((g \<star> h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (((g \<star> g\<^sub>0h\<^sub>1.\<phi>) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((\<a>[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (((g.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>1) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (f\<^sub>0gh\<^sub>1.\<phi> \<star> TTfgh_TfTgh.chine) \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>0 f, f\<^sub>0gh\<^sub>1.p\<^sub>1, TTfgh_TfTgh.chine])
[PROOF STEP]
using fg gh
[PROOF STATE]
proof (prove)
using this:
src f = trg g
src g = trg h
goal (1 subgoal):
1. seq (g \<star> h \<star> TTfgh_TfTgh.the_\<theta>) (can (\<^bold>\<langle>g\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> ((\<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) (((\<^bold>\<langle>g\<^bold>\<rangle> \<^bold>\<star> (\<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) \<cdot> (((g \<star> h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (((g \<star> g\<^sub>0h\<^sub>1.\<phi>) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((\<a>[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (((g.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>1) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (f\<^sub>0gh\<^sub>1.\<phi> \<star> TTfgh_TfTgh.chine) \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>0 f, f\<^sub>0gh\<^sub>1.p\<^sub>1, TTfgh_TfTgh.chine])
[PROOF STEP]
apply (intro seqI' comp_in_homI)
[PROOF STATE]
proof (prove)
goal (8 subgoals):
1. \<lbrakk>src f = trg g; src g = trg h\<rbrakk> \<Longrightarrow> \<guillemotleft>\<a>\<^sup>-\<^sup>1[tab\<^sub>0 f, f\<^sub>0gh\<^sub>1.p\<^sub>1, TTfgh_TfTgh.chine] : ?a2 \<rightarrow> ?b8\<guillemotright>
2. \<lbrakk>src f = trg g; src g = trg h\<rbrakk> \<Longrightarrow> \<guillemotleft>f\<^sub>0gh\<^sub>1.\<phi> \<star> TTfgh_TfTgh.chine : ?b8 \<rightarrow> ?b7\<guillemotright>
3. \<lbrakk>src f = trg g; src g = trg h\<rbrakk> \<Longrightarrow> \<guillemotleft>((g.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>1) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine : ?b7 \<rightarrow> ?b6\<guillemotright>
4. \<lbrakk>src f = trg g; src g = trg h\<rbrakk> \<Longrightarrow> \<guillemotleft>(\<a>[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine : ?b6 \<rightarrow> ?b5\<guillemotright>
5. \<lbrakk>src f = trg g; src g = trg h\<rbrakk> \<Longrightarrow> \<guillemotleft>((g \<star> g\<^sub>0h\<^sub>1.\<phi>) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine : ?b5 \<rightarrow> ?b4\<guillemotright>
6. \<lbrakk>src f = trg g; src g = trg h\<rbrakk> \<Longrightarrow> \<guillemotleft>((g \<star> h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine : ?b4 \<rightarrow> ?b3\<guillemotright>
7. \<lbrakk>src f = trg g; src g = trg h\<rbrakk> \<Longrightarrow> \<guillemotleft>can (\<^bold>\<langle>g\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> ((\<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) (((\<^bold>\<langle>g\<^bold>\<rangle> \<^bold>\<star> (\<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) : ?b3 \<rightarrow> ?b2\<guillemotright>
8. \<lbrakk>src f = trg g; src g = trg h\<rbrakk> \<Longrightarrow> \<guillemotleft>g \<star> h \<star> TTfgh_TfTgh.the_\<theta> : ?b2 \<rightarrow> ?c2\<guillemotright>
[PROOF STEP]
by auto
[PROOF STATE]
proof (state)
this:
seq (g \<star> h \<star> TTfgh_TfTgh.the_\<theta>) (can (\<^bold>\<langle>g\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> ((\<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) (((\<^bold>\<langle>g\<^bold>\<rangle> \<^bold>\<star> (\<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) \<cdot> (((g \<star> h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (((g \<star> g\<^sub>0h\<^sub>1.\<phi>) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((\<a>[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (((g.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>1) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (f\<^sub>0gh\<^sub>1.\<phi> \<star> TTfgh_TfTgh.chine) \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>0 f, f\<^sub>0gh\<^sub>1.p\<^sub>1, TTfgh_TfTgh.chine])
goal (1 subgoal):
1. (f \<star> g \<star> h \<star> TTfgh_TfTgh.the_\<theta>) \<cdot> (f \<star> can (\<^bold>\<langle>g\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> ((\<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) (((\<^bold>\<langle>g\<^bold>\<rangle> \<^bold>\<star> (\<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>)) \<cdot> (f \<star> (((g \<star> h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (((g \<star> g\<^sub>0h\<^sub>1.\<phi>) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((\<a>[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (((g.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>1) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (f\<^sub>0gh\<^sub>1.\<phi> \<star> TTfgh_TfTgh.chine) \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>0 f, f\<^sub>0gh\<^sub>1.p\<^sub>1, TTfgh_TfTgh.chine]) = f \<star> (g \<star> h \<star> TTfgh_TfTgh.the_\<theta>) \<cdot> can (\<^bold>\<langle>g\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> ((\<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) (((\<^bold>\<langle>g\<^bold>\<rangle> \<^bold>\<star> (\<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) \<cdot> (((g \<star> h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (((g \<star> g\<^sub>0h\<^sub>1.\<phi>) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((\<a>[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (((g.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>1) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (f\<^sub>0gh\<^sub>1.\<phi> \<star> TTfgh_TfTgh.chine) \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>0 f, f\<^sub>0gh\<^sub>1.p\<^sub>1, TTfgh_TfTgh.chine]
[PROOF STEP]
moreover
[PROOF STATE]
proof (state)
this:
seq (g \<star> h \<star> TTfgh_TfTgh.the_\<theta>) (can (\<^bold>\<langle>g\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> ((\<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) (((\<^bold>\<langle>g\<^bold>\<rangle> \<^bold>\<star> (\<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) \<cdot> (((g \<star> h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (((g \<star> g\<^sub>0h\<^sub>1.\<phi>) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((\<a>[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (((g.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>1) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (f\<^sub>0gh\<^sub>1.\<phi> \<star> TTfgh_TfTgh.chine) \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>0 f, f\<^sub>0gh\<^sub>1.p\<^sub>1, TTfgh_TfTgh.chine])
goal (1 subgoal):
1. (f \<star> g \<star> h \<star> TTfgh_TfTgh.the_\<theta>) \<cdot> (f \<star> can (\<^bold>\<langle>g\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> ((\<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) (((\<^bold>\<langle>g\<^bold>\<rangle> \<^bold>\<star> (\<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>)) \<cdot> (f \<star> (((g \<star> h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (((g \<star> g\<^sub>0h\<^sub>1.\<phi>) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((\<a>[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (((g.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>1) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (f\<^sub>0gh\<^sub>1.\<phi> \<star> TTfgh_TfTgh.chine) \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>0 f, f\<^sub>0gh\<^sub>1.p\<^sub>1, TTfgh_TfTgh.chine]) = f \<star> (g \<star> h \<star> TTfgh_TfTgh.the_\<theta>) \<cdot> can (\<^bold>\<langle>g\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> ((\<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) (((\<^bold>\<langle>g\<^bold>\<rangle> \<^bold>\<star> (\<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) \<cdot> (((g \<star> h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (((g \<star> g\<^sub>0h\<^sub>1.\<phi>) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((\<a>[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (((g.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>1) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (f\<^sub>0gh\<^sub>1.\<phi> \<star> TTfgh_TfTgh.chine) \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>0 f, f\<^sub>0gh\<^sub>1.p\<^sub>1, TTfgh_TfTgh.chine]
[PROOF STEP]
have 2: "arr (can (\<^bold>\<langle>g\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> ((\<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>Tgh.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TfTgh.p\<^sub>0\<^bold>\<rangle>)
\<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>)
(((\<^bold>\<langle>g\<^bold>\<rangle> \<^bold>\<star> (\<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>Tgh.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TfTgh.p\<^sub>0\<^bold>\<rangle>)
\<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) \<cdot>
(((g \<star> h.tab \<star> Tgh.p\<^sub>0) \<star> TfTgh.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot>
(((g \<star> g\<^sub>0h\<^sub>1.\<phi>) \<star> TfTgh.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot>
((\<a>[g, tab\<^sub>0 g, Tgh.p\<^sub>1] \<star> TfTgh.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot>
(((g.tab \<star> Tgh.p\<^sub>1) \<star> TfTgh.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot>
(f\<^sub>0gh\<^sub>1.\<phi> \<star> TTfgh_TfTgh.chine) \<cdot>
\<a>\<^sup>-\<^sup>1[tab\<^sub>0 f, TfTgh.p\<^sub>1, TTfgh_TfTgh.chine])"
[PROOF STATE]
proof (prove)
goal (1 subgoal):
1. seq (can (\<^bold>\<langle>g\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> ((\<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) (((\<^bold>\<langle>g\<^bold>\<rangle> \<^bold>\<star> (\<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>)) ((((g \<star> h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (((g \<star> g\<^sub>0h\<^sub>1.\<phi>) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((\<a>[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (((g.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>1) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (f\<^sub>0gh\<^sub>1.\<phi> \<star> TTfgh_TfTgh.chine) \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>0 f, f\<^sub>0gh\<^sub>1.p\<^sub>1, TTfgh_TfTgh.chine])
[PROOF STEP]
using calculation
[PROOF STATE]
proof (prove)
using this:
seq (g \<star> h \<star> TTfgh_TfTgh.the_\<theta>) (can (\<^bold>\<langle>g\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> ((\<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) (((\<^bold>\<langle>g\<^bold>\<rangle> \<^bold>\<star> (\<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) \<cdot> (((g \<star> h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (((g \<star> g\<^sub>0h\<^sub>1.\<phi>) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((\<a>[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (((g.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>1) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (f\<^sub>0gh\<^sub>1.\<phi> \<star> TTfgh_TfTgh.chine) \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>0 f, f\<^sub>0gh\<^sub>1.p\<^sub>1, TTfgh_TfTgh.chine])
goal (1 subgoal):
1. seq (can (\<^bold>\<langle>g\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> ((\<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) (((\<^bold>\<langle>g\<^bold>\<rangle> \<^bold>\<star> (\<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>)) ((((g \<star> h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (((g \<star> g\<^sub>0h\<^sub>1.\<phi>) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((\<a>[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (((g.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>1) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (f\<^sub>0gh\<^sub>1.\<phi> \<star> TTfgh_TfTgh.chine) \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>0 f, f\<^sub>0gh\<^sub>1.p\<^sub>1, TTfgh_TfTgh.chine])
[PROOF STEP]
by blast
[PROOF STATE]
proof (state)
this:
seq (can (\<^bold>\<langle>g\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> ((\<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) (((\<^bold>\<langle>g\<^bold>\<rangle> \<^bold>\<star> (\<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>)) ((((g \<star> h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (((g \<star> g\<^sub>0h\<^sub>1.\<phi>) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((\<a>[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (((g.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>1) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (f\<^sub>0gh\<^sub>1.\<phi> \<star> TTfgh_TfTgh.chine) \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>0 f, f\<^sub>0gh\<^sub>1.p\<^sub>1, TTfgh_TfTgh.chine])
goal (1 subgoal):
1. (f \<star> g \<star> h \<star> TTfgh_TfTgh.the_\<theta>) \<cdot> (f \<star> can (\<^bold>\<langle>g\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> ((\<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) (((\<^bold>\<langle>g\<^bold>\<rangle> \<^bold>\<star> (\<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>)) \<cdot> (f \<star> (((g \<star> h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (((g \<star> g\<^sub>0h\<^sub>1.\<phi>) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((\<a>[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (((g.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>1) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (f\<^sub>0gh\<^sub>1.\<phi> \<star> TTfgh_TfTgh.chine) \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>0 f, f\<^sub>0gh\<^sub>1.p\<^sub>1, TTfgh_TfTgh.chine]) = f \<star> (g \<star> h \<star> TTfgh_TfTgh.the_\<theta>) \<cdot> can (\<^bold>\<langle>g\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> ((\<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) (((\<^bold>\<langle>g\<^bold>\<rangle> \<^bold>\<star> (\<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) \<cdot> (((g \<star> h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (((g \<star> g\<^sub>0h\<^sub>1.\<phi>) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((\<a>[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (((g.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>1) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (f\<^sub>0gh\<^sub>1.\<phi> \<star> TTfgh_TfTgh.chine) \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>0 f, f\<^sub>0gh\<^sub>1.p\<^sub>1, TTfgh_TfTgh.chine]
[PROOF STEP]
ultimately
[PROOF STATE]
proof (chain)
picking this:
seq (g \<star> h \<star> TTfgh_TfTgh.the_\<theta>) (can (\<^bold>\<langle>g\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> ((\<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) (((\<^bold>\<langle>g\<^bold>\<rangle> \<^bold>\<star> (\<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) \<cdot> (((g \<star> h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (((g \<star> g\<^sub>0h\<^sub>1.\<phi>) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((\<a>[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (((g.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>1) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (f\<^sub>0gh\<^sub>1.\<phi> \<star> TTfgh_TfTgh.chine) \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>0 f, f\<^sub>0gh\<^sub>1.p\<^sub>1, TTfgh_TfTgh.chine])
seq (can (\<^bold>\<langle>g\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> ((\<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) (((\<^bold>\<langle>g\<^bold>\<rangle> \<^bold>\<star> (\<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>)) ((((g \<star> h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (((g \<star> g\<^sub>0h\<^sub>1.\<phi>) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((\<a>[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (((g.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>1) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (f\<^sub>0gh\<^sub>1.\<phi> \<star> TTfgh_TfTgh.chine) \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>0 f, f\<^sub>0gh\<^sub>1.p\<^sub>1, TTfgh_TfTgh.chine])
[PROOF STEP]
show ?thesis
[PROOF STATE]
proof (prove)
using this:
seq (g \<star> h \<star> TTfgh_TfTgh.the_\<theta>) (can (\<^bold>\<langle>g\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> ((\<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) (((\<^bold>\<langle>g\<^bold>\<rangle> \<^bold>\<star> (\<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) \<cdot> (((g \<star> h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (((g \<star> g\<^sub>0h\<^sub>1.\<phi>) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((\<a>[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (((g.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>1) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (f\<^sub>0gh\<^sub>1.\<phi> \<star> TTfgh_TfTgh.chine) \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>0 f, f\<^sub>0gh\<^sub>1.p\<^sub>1, TTfgh_TfTgh.chine])
seq (can (\<^bold>\<langle>g\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> ((\<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) (((\<^bold>\<langle>g\<^bold>\<rangle> \<^bold>\<star> (\<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>)) ((((g \<star> h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (((g \<star> g\<^sub>0h\<^sub>1.\<phi>) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((\<a>[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (((g.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>1) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (f\<^sub>0gh\<^sub>1.\<phi> \<star> TTfgh_TfTgh.chine) \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>0 f, f\<^sub>0gh\<^sub>1.p\<^sub>1, TTfgh_TfTgh.chine])
goal (1 subgoal):
1. (f \<star> g \<star> h \<star> TTfgh_TfTgh.the_\<theta>) \<cdot> (f \<star> can (\<^bold>\<langle>g\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> ((\<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) (((\<^bold>\<langle>g\<^bold>\<rangle> \<^bold>\<star> (\<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>)) \<cdot> (f \<star> (((g \<star> h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (((g \<star> g\<^sub>0h\<^sub>1.\<phi>) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((\<a>[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (((g.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>1) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (f\<^sub>0gh\<^sub>1.\<phi> \<star> TTfgh_TfTgh.chine) \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>0 f, f\<^sub>0gh\<^sub>1.p\<^sub>1, TTfgh_TfTgh.chine]) = f \<star> (g \<star> h \<star> TTfgh_TfTgh.the_\<theta>) \<cdot> can (\<^bold>\<langle>g\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> ((\<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) (((\<^bold>\<langle>g\<^bold>\<rangle> \<^bold>\<star> (\<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) \<cdot> (((g \<star> h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (((g \<star> g\<^sub>0h\<^sub>1.\<phi>) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((\<a>[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (((g.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>1) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (f\<^sub>0gh\<^sub>1.\<phi> \<star> TTfgh_TfTgh.chine) \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>0 f, f\<^sub>0gh\<^sub>1.p\<^sub>1, TTfgh_TfTgh.chine]
[PROOF STEP]
using whisker_left f.ide_base
[PROOF STATE]
proof (prove)
using this:
seq (g \<star> h \<star> TTfgh_TfTgh.the_\<theta>) (can (\<^bold>\<langle>g\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> ((\<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) (((\<^bold>\<langle>g\<^bold>\<rangle> \<^bold>\<star> (\<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) \<cdot> (((g \<star> h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (((g \<star> g\<^sub>0h\<^sub>1.\<phi>) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((\<a>[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (((g.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>1) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (f\<^sub>0gh\<^sub>1.\<phi> \<star> TTfgh_TfTgh.chine) \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>0 f, f\<^sub>0gh\<^sub>1.p\<^sub>1, TTfgh_TfTgh.chine])
seq (can (\<^bold>\<langle>g\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> ((\<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) (((\<^bold>\<langle>g\<^bold>\<rangle> \<^bold>\<star> (\<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>)) ((((g \<star> h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (((g \<star> g\<^sub>0h\<^sub>1.\<phi>) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((\<a>[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (((g.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>1) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (f\<^sub>0gh\<^sub>1.\<phi> \<star> TTfgh_TfTgh.chine) \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>0 f, f\<^sub>0gh\<^sub>1.p\<^sub>1, TTfgh_TfTgh.chine])
\<lbrakk>ide ?f; seq ?\<nu> ?\<mu>\<rbrakk> \<Longrightarrow> ?f \<star> ?\<nu> \<cdot> ?\<mu> = (?f \<star> ?\<nu>) \<cdot> (?f \<star> ?\<mu>)
ide f
goal (1 subgoal):
1. (f \<star> g \<star> h \<star> TTfgh_TfTgh.the_\<theta>) \<cdot> (f \<star> can (\<^bold>\<langle>g\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> ((\<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) (((\<^bold>\<langle>g\<^bold>\<rangle> \<^bold>\<star> (\<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>)) \<cdot> (f \<star> (((g \<star> h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (((g \<star> g\<^sub>0h\<^sub>1.\<phi>) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((\<a>[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (((g.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>1) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (f\<^sub>0gh\<^sub>1.\<phi> \<star> TTfgh_TfTgh.chine) \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>0 f, f\<^sub>0gh\<^sub>1.p\<^sub>1, TTfgh_TfTgh.chine]) = f \<star> (g \<star> h \<star> TTfgh_TfTgh.the_\<theta>) \<cdot> can (\<^bold>\<langle>g\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> ((\<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) (((\<^bold>\<langle>g\<^bold>\<rangle> \<^bold>\<star> (\<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) \<cdot> (((g \<star> h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (((g \<star> g\<^sub>0h\<^sub>1.\<phi>) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((\<a>[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (((g.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>1) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (f\<^sub>0gh\<^sub>1.\<phi> \<star> TTfgh_TfTgh.chine) \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>0 f, f\<^sub>0gh\<^sub>1.p\<^sub>1, TTfgh_TfTgh.chine]
[PROOF STEP]
by presburger
[PROOF STATE]
proof (state)
this:
(f \<star> g \<star> h \<star> TTfgh_TfTgh.the_\<theta>) \<cdot> (f \<star> can (\<^bold>\<langle>g\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> ((\<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) (((\<^bold>\<langle>g\<^bold>\<rangle> \<^bold>\<star> (\<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>)) \<cdot> (f \<star> (((g \<star> h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (((g \<star> g\<^sub>0h\<^sub>1.\<phi>) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((\<a>[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (((g.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>1) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (f\<^sub>0gh\<^sub>1.\<phi> \<star> TTfgh_TfTgh.chine) \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>0 f, f\<^sub>0gh\<^sub>1.p\<^sub>1, TTfgh_TfTgh.chine]) = f \<star> (g \<star> h \<star> TTfgh_TfTgh.the_\<theta>) \<cdot> can (\<^bold>\<langle>g\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> ((\<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) (((\<^bold>\<langle>g\<^bold>\<rangle> \<^bold>\<star> (\<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) \<cdot> (((g \<star> h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (((g \<star> g\<^sub>0h\<^sub>1.\<phi>) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((\<a>[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (((g.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>1) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (f\<^sub>0gh\<^sub>1.\<phi> \<star> TTfgh_TfTgh.chine) \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>0 f, f\<^sub>0gh\<^sub>1.p\<^sub>1, TTfgh_TfTgh.chine]
goal:
No subgoals!
[PROOF STEP]
qed
[PROOF STATE]
proof (state)
this:
(f \<star> g \<star> h \<star> TTfgh_TfTgh.the_\<theta>) \<cdot> (f \<star> can (\<^bold>\<langle>g\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> ((\<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) (((\<^bold>\<langle>g\<^bold>\<rangle> \<^bold>\<star> (\<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>)) \<cdot> (f \<star> (((g \<star> h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (((g \<star> g\<^sub>0h\<^sub>1.\<phi>) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((\<a>[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (((g.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>1) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (f\<^sub>0gh\<^sub>1.\<phi> \<star> TTfgh_TfTgh.chine) \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>0 f, f\<^sub>0gh\<^sub>1.p\<^sub>1, TTfgh_TfTgh.chine]) = f \<star> (g \<star> h \<star> TTfgh_TfTgh.the_\<theta>) \<cdot> can (\<^bold>\<langle>g\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> ((\<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) (((\<^bold>\<langle>g\<^bold>\<rangle> \<^bold>\<star> (\<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) \<cdot> (((g \<star> h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (((g \<star> g\<^sub>0h\<^sub>1.\<phi>) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((\<a>[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (((g.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>1) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (f\<^sub>0gh\<^sub>1.\<phi> \<star> TTfgh_TfTgh.chine) \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>0 f, f\<^sub>0gh\<^sub>1.p\<^sub>1, TTfgh_TfTgh.chine]
goal (1 subgoal):
1. ((f \<star> g \<star> h \<star> TTfgh_TfTgh.the_\<theta>) \<cdot> (f \<star> can (\<^bold>\<langle>g\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> ((\<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) (((\<^bold>\<langle>g\<^bold>\<rangle> \<^bold>\<star> (\<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>)) \<cdot> (f \<star> (((g \<star> h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (((g \<star> g\<^sub>0h\<^sub>1.\<phi>) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((\<a>[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (((g.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>1) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (f\<^sub>0gh\<^sub>1.\<phi> \<star> TTfgh_TfTgh.chine) \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>0 f, f\<^sub>0gh\<^sub>1.p\<^sub>1, TTfgh_TfTgh.chine])) \<cdot> \<a>[f, tab\<^sub>0 f, f\<^sub>0gh\<^sub>1.p\<^sub>1 \<star> TTfgh_TfTgh.chine] \<cdot> (f.tab \<star> f\<^sub>0gh\<^sub>1.p\<^sub>1 \<star> TTfgh_TfTgh.chine) \<cdot> \<a>[tab\<^sub>1 f, f\<^sub>0gh\<^sub>1.p\<^sub>1, TTfgh_TfTgh.chine] \<cdot> TTfgh_TfTgh.the_\<nu> \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>1 f, f\<^sub>0g\<^sub>1.p\<^sub>1, fg\<^sub>0h\<^sub>1.p\<^sub>1] = (f \<star> (g \<star> h \<star> TTfgh_TfTgh.the_\<theta>) \<cdot> can (\<^bold>\<langle>g\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> ((\<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) (((\<^bold>\<langle>g\<^bold>\<rangle> \<^bold>\<star> (\<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) \<cdot> (((g \<star> h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (((g \<star> g\<^sub>0h\<^sub>1.\<phi>) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((\<a>[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (((g.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>1) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (f\<^sub>0gh\<^sub>1.\<phi> \<star> TTfgh_TfTgh.chine) \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>0 f, f\<^sub>0gh\<^sub>1.p\<^sub>1, TTfgh_TfTgh.chine]) \<cdot> \<a>[f, tab\<^sub>0 f, f\<^sub>0gh\<^sub>1.p\<^sub>1 \<star> TTfgh_TfTgh.chine] \<cdot> (f.tab \<star> f\<^sub>0gh\<^sub>1.p\<^sub>1 \<star> TTfgh_TfTgh.chine) \<cdot> \<a>[tab\<^sub>1 f, f\<^sub>0gh\<^sub>1.p\<^sub>1, TTfgh_TfTgh.chine] \<cdot> TTfgh_TfTgh.the_\<nu> \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>1 f, f\<^sub>0g\<^sub>1.p\<^sub>1, fg\<^sub>0h\<^sub>1.p\<^sub>1]
[PROOF STEP]
thus ?thesis
[PROOF STATE]
proof (prove)
using this:
(f \<star> g \<star> h \<star> TTfgh_TfTgh.the_\<theta>) \<cdot> (f \<star> can (\<^bold>\<langle>g\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> ((\<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) (((\<^bold>\<langle>g\<^bold>\<rangle> \<^bold>\<star> (\<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>)) \<cdot> (f \<star> (((g \<star> h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (((g \<star> g\<^sub>0h\<^sub>1.\<phi>) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((\<a>[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (((g.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>1) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (f\<^sub>0gh\<^sub>1.\<phi> \<star> TTfgh_TfTgh.chine) \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>0 f, f\<^sub>0gh\<^sub>1.p\<^sub>1, TTfgh_TfTgh.chine]) = f \<star> (g \<star> h \<star> TTfgh_TfTgh.the_\<theta>) \<cdot> can (\<^bold>\<langle>g\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> ((\<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) (((\<^bold>\<langle>g\<^bold>\<rangle> \<^bold>\<star> (\<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) \<cdot> (((g \<star> h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (((g \<star> g\<^sub>0h\<^sub>1.\<phi>) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((\<a>[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (((g.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>1) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (f\<^sub>0gh\<^sub>1.\<phi> \<star> TTfgh_TfTgh.chine) \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>0 f, f\<^sub>0gh\<^sub>1.p\<^sub>1, TTfgh_TfTgh.chine]
goal (1 subgoal):
1. ((f \<star> g \<star> h \<star> TTfgh_TfTgh.the_\<theta>) \<cdot> (f \<star> can (\<^bold>\<langle>g\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> ((\<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) (((\<^bold>\<langle>g\<^bold>\<rangle> \<^bold>\<star> (\<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>)) \<cdot> (f \<star> (((g \<star> h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (((g \<star> g\<^sub>0h\<^sub>1.\<phi>) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((\<a>[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (((g.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>1) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (f\<^sub>0gh\<^sub>1.\<phi> \<star> TTfgh_TfTgh.chine) \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>0 f, f\<^sub>0gh\<^sub>1.p\<^sub>1, TTfgh_TfTgh.chine])) \<cdot> \<a>[f, tab\<^sub>0 f, f\<^sub>0gh\<^sub>1.p\<^sub>1 \<star> TTfgh_TfTgh.chine] \<cdot> (f.tab \<star> f\<^sub>0gh\<^sub>1.p\<^sub>1 \<star> TTfgh_TfTgh.chine) \<cdot> \<a>[tab\<^sub>1 f, f\<^sub>0gh\<^sub>1.p\<^sub>1, TTfgh_TfTgh.chine] \<cdot> TTfgh_TfTgh.the_\<nu> \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>1 f, f\<^sub>0g\<^sub>1.p\<^sub>1, fg\<^sub>0h\<^sub>1.p\<^sub>1] = (f \<star> (g \<star> h \<star> TTfgh_TfTgh.the_\<theta>) \<cdot> can (\<^bold>\<langle>g\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> ((\<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) (((\<^bold>\<langle>g\<^bold>\<rangle> \<^bold>\<star> (\<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) \<cdot> (((g \<star> h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (((g \<star> g\<^sub>0h\<^sub>1.\<phi>) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((\<a>[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (((g.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>1) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (f\<^sub>0gh\<^sub>1.\<phi> \<star> TTfgh_TfTgh.chine) \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>0 f, f\<^sub>0gh\<^sub>1.p\<^sub>1, TTfgh_TfTgh.chine]) \<cdot> \<a>[f, tab\<^sub>0 f, f\<^sub>0gh\<^sub>1.p\<^sub>1 \<star> TTfgh_TfTgh.chine] \<cdot> (f.tab \<star> f\<^sub>0gh\<^sub>1.p\<^sub>1 \<star> TTfgh_TfTgh.chine) \<cdot> \<a>[tab\<^sub>1 f, f\<^sub>0gh\<^sub>1.p\<^sub>1, TTfgh_TfTgh.chine] \<cdot> TTfgh_TfTgh.the_\<nu> \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>1 f, f\<^sub>0g\<^sub>1.p\<^sub>1, fg\<^sub>0h\<^sub>1.p\<^sub>1]
[PROOF STEP]
using comp_assoc
[PROOF STATE]
proof (prove)
using this:
(f \<star> g \<star> h \<star> TTfgh_TfTgh.the_\<theta>) \<cdot> (f \<star> can (\<^bold>\<langle>g\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> ((\<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) (((\<^bold>\<langle>g\<^bold>\<rangle> \<^bold>\<star> (\<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>)) \<cdot> (f \<star> (((g \<star> h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (((g \<star> g\<^sub>0h\<^sub>1.\<phi>) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((\<a>[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (((g.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>1) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (f\<^sub>0gh\<^sub>1.\<phi> \<star> TTfgh_TfTgh.chine) \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>0 f, f\<^sub>0gh\<^sub>1.p\<^sub>1, TTfgh_TfTgh.chine]) = f \<star> (g \<star> h \<star> TTfgh_TfTgh.the_\<theta>) \<cdot> can (\<^bold>\<langle>g\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> ((\<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) (((\<^bold>\<langle>g\<^bold>\<rangle> \<^bold>\<star> (\<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) \<cdot> (((g \<star> h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (((g \<star> g\<^sub>0h\<^sub>1.\<phi>) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((\<a>[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (((g.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>1) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (f\<^sub>0gh\<^sub>1.\<phi> \<star> TTfgh_TfTgh.chine) \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>0 f, f\<^sub>0gh\<^sub>1.p\<^sub>1, TTfgh_TfTgh.chine]
(?h \<cdot> ?g) \<cdot> ?f = ?h \<cdot> ?g \<cdot> ?f
goal (1 subgoal):
1. ((f \<star> g \<star> h \<star> TTfgh_TfTgh.the_\<theta>) \<cdot> (f \<star> can (\<^bold>\<langle>g\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> ((\<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) (((\<^bold>\<langle>g\<^bold>\<rangle> \<^bold>\<star> (\<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>)) \<cdot> (f \<star> (((g \<star> h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (((g \<star> g\<^sub>0h\<^sub>1.\<phi>) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((\<a>[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (((g.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>1) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (f\<^sub>0gh\<^sub>1.\<phi> \<star> TTfgh_TfTgh.chine) \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>0 f, f\<^sub>0gh\<^sub>1.p\<^sub>1, TTfgh_TfTgh.chine])) \<cdot> \<a>[f, tab\<^sub>0 f, f\<^sub>0gh\<^sub>1.p\<^sub>1 \<star> TTfgh_TfTgh.chine] \<cdot> (f.tab \<star> f\<^sub>0gh\<^sub>1.p\<^sub>1 \<star> TTfgh_TfTgh.chine) \<cdot> \<a>[tab\<^sub>1 f, f\<^sub>0gh\<^sub>1.p\<^sub>1, TTfgh_TfTgh.chine] \<cdot> TTfgh_TfTgh.the_\<nu> \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>1 f, f\<^sub>0g\<^sub>1.p\<^sub>1, fg\<^sub>0h\<^sub>1.p\<^sub>1] = (f \<star> (g \<star> h \<star> TTfgh_TfTgh.the_\<theta>) \<cdot> can (\<^bold>\<langle>g\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> ((\<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) (((\<^bold>\<langle>g\<^bold>\<rangle> \<^bold>\<star> (\<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) \<cdot> (((g \<star> h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (((g \<star> g\<^sub>0h\<^sub>1.\<phi>) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((\<a>[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (((g.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>1) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (f\<^sub>0gh\<^sub>1.\<phi> \<star> TTfgh_TfTgh.chine) \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>0 f, f\<^sub>0gh\<^sub>1.p\<^sub>1, TTfgh_TfTgh.chine]) \<cdot> \<a>[f, tab\<^sub>0 f, f\<^sub>0gh\<^sub>1.p\<^sub>1 \<star> TTfgh_TfTgh.chine] \<cdot> (f.tab \<star> f\<^sub>0gh\<^sub>1.p\<^sub>1 \<star> TTfgh_TfTgh.chine) \<cdot> \<a>[tab\<^sub>1 f, f\<^sub>0gh\<^sub>1.p\<^sub>1, TTfgh_TfTgh.chine] \<cdot> TTfgh_TfTgh.the_\<nu> \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>1 f, f\<^sub>0g\<^sub>1.p\<^sub>1, fg\<^sub>0h\<^sub>1.p\<^sub>1]
[PROOF STEP]
by presburger
[PROOF STATE]
proof (state)
this:
((f \<star> g \<star> h \<star> TTfgh_TfTgh.the_\<theta>) \<cdot> (f \<star> can (\<^bold>\<langle>g\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> ((\<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) (((\<^bold>\<langle>g\<^bold>\<rangle> \<^bold>\<star> (\<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>)) \<cdot> (f \<star> (((g \<star> h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (((g \<star> g\<^sub>0h\<^sub>1.\<phi>) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((\<a>[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (((g.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>1) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (f\<^sub>0gh\<^sub>1.\<phi> \<star> TTfgh_TfTgh.chine) \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>0 f, f\<^sub>0gh\<^sub>1.p\<^sub>1, TTfgh_TfTgh.chine])) \<cdot> \<a>[f, tab\<^sub>0 f, f\<^sub>0gh\<^sub>1.p\<^sub>1 \<star> TTfgh_TfTgh.chine] \<cdot> (f.tab \<star> f\<^sub>0gh\<^sub>1.p\<^sub>1 \<star> TTfgh_TfTgh.chine) \<cdot> \<a>[tab\<^sub>1 f, f\<^sub>0gh\<^sub>1.p\<^sub>1, TTfgh_TfTgh.chine] \<cdot> TTfgh_TfTgh.the_\<nu> \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>1 f, f\<^sub>0g\<^sub>1.p\<^sub>1, fg\<^sub>0h\<^sub>1.p\<^sub>1] = (f \<star> (g \<star> h \<star> TTfgh_TfTgh.the_\<theta>) \<cdot> can (\<^bold>\<langle>g\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> ((\<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) (((\<^bold>\<langle>g\<^bold>\<rangle> \<^bold>\<star> (\<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) \<cdot> (((g \<star> h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (((g \<star> g\<^sub>0h\<^sub>1.\<phi>) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((\<a>[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (((g.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>1) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (f\<^sub>0gh\<^sub>1.\<phi> \<star> TTfgh_TfTgh.chine) \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>0 f, f\<^sub>0gh\<^sub>1.p\<^sub>1, TTfgh_TfTgh.chine]) \<cdot> \<a>[f, tab\<^sub>0 f, f\<^sub>0gh\<^sub>1.p\<^sub>1 \<star> TTfgh_TfTgh.chine] \<cdot> (f.tab \<star> f\<^sub>0gh\<^sub>1.p\<^sub>1 \<star> TTfgh_TfTgh.chine) \<cdot> \<a>[tab\<^sub>1 f, f\<^sub>0gh\<^sub>1.p\<^sub>1, TTfgh_TfTgh.chine] \<cdot> TTfgh_TfTgh.the_\<nu> \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>1 f, f\<^sub>0g\<^sub>1.p\<^sub>1, fg\<^sub>0h\<^sub>1.p\<^sub>1]
goal:
No subgoals!
[PROOF STEP]
qed
[PROOF STATE]
proof (state)
this:
((f \<star> g \<star> h \<star> TTfgh_TfTgh.the_\<theta>) \<cdot> (f \<star> can (\<^bold>\<langle>g\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> ((\<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) (((\<^bold>\<langle>g\<^bold>\<rangle> \<^bold>\<star> (\<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>)) \<cdot> (f \<star> (((g \<star> h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (((g \<star> g\<^sub>0h\<^sub>1.\<phi>) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((\<a>[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (((g.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>1) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (f\<^sub>0gh\<^sub>1.\<phi> \<star> TTfgh_TfTgh.chine) \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>0 f, f\<^sub>0gh\<^sub>1.p\<^sub>1, TTfgh_TfTgh.chine])) \<cdot> \<a>[f, tab\<^sub>0 f, f\<^sub>0gh\<^sub>1.p\<^sub>1 \<star> TTfgh_TfTgh.chine] \<cdot> (f.tab \<star> f\<^sub>0gh\<^sub>1.p\<^sub>1 \<star> TTfgh_TfTgh.chine) \<cdot> \<a>[tab\<^sub>1 f, f\<^sub>0gh\<^sub>1.p\<^sub>1, TTfgh_TfTgh.chine] \<cdot> TTfgh_TfTgh.the_\<nu> \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>1 f, f\<^sub>0g\<^sub>1.p\<^sub>1, fg\<^sub>0h\<^sub>1.p\<^sub>1] = (f \<star> (g \<star> h \<star> TTfgh_TfTgh.the_\<theta>) \<cdot> can (\<^bold>\<langle>g\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> ((\<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) (((\<^bold>\<langle>g\<^bold>\<rangle> \<^bold>\<star> (\<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) \<cdot> (((g \<star> h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (((g \<star> g\<^sub>0h\<^sub>1.\<phi>) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((\<a>[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (((g.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>1) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (f\<^sub>0gh\<^sub>1.\<phi> \<star> TTfgh_TfTgh.chine) \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>0 f, f\<^sub>0gh\<^sub>1.p\<^sub>1, TTfgh_TfTgh.chine]) \<cdot> \<a>[f, tab\<^sub>0 f, f\<^sub>0gh\<^sub>1.p\<^sub>1 \<star> TTfgh_TfTgh.chine] \<cdot> (f.tab \<star> f\<^sub>0gh\<^sub>1.p\<^sub>1 \<star> TTfgh_TfTgh.chine) \<cdot> \<a>[tab\<^sub>1 f, f\<^sub>0gh\<^sub>1.p\<^sub>1, TTfgh_TfTgh.chine] \<cdot> TTfgh_TfTgh.the_\<nu> \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>1 f, f\<^sub>0g\<^sub>1.p\<^sub>1, fg\<^sub>0h\<^sub>1.p\<^sub>1]
goal (1 subgoal):
1. (f \<star> \<theta>\<^sub>f) \<cdot> \<a>[f, tab\<^sub>0 f, w\<^sub>f] \<cdot> (f.tab \<star> w\<^sub>f) = ((f \<star> \<theta>\<^sub>f') \<cdot> \<a>[f, tab\<^sub>0 f, w\<^sub>f'] \<cdot> (f.tab \<star> w\<^sub>f')) \<cdot> \<beta>\<^sub>f
[PROOF STEP]
also
[PROOF STATE]
proof (state)
this:
((f \<star> g \<star> h \<star> TTfgh_TfTgh.the_\<theta>) \<cdot> (f \<star> can (\<^bold>\<langle>g\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> ((\<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) (((\<^bold>\<langle>g\<^bold>\<rangle> \<^bold>\<star> (\<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>)) \<cdot> (f \<star> (((g \<star> h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (((g \<star> g\<^sub>0h\<^sub>1.\<phi>) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((\<a>[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (((g.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>1) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (f\<^sub>0gh\<^sub>1.\<phi> \<star> TTfgh_TfTgh.chine) \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>0 f, f\<^sub>0gh\<^sub>1.p\<^sub>1, TTfgh_TfTgh.chine])) \<cdot> \<a>[f, tab\<^sub>0 f, f\<^sub>0gh\<^sub>1.p\<^sub>1 \<star> TTfgh_TfTgh.chine] \<cdot> (f.tab \<star> f\<^sub>0gh\<^sub>1.p\<^sub>1 \<star> TTfgh_TfTgh.chine) \<cdot> \<a>[tab\<^sub>1 f, f\<^sub>0gh\<^sub>1.p\<^sub>1, TTfgh_TfTgh.chine] \<cdot> TTfgh_TfTgh.the_\<nu> \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>1 f, f\<^sub>0g\<^sub>1.p\<^sub>1, fg\<^sub>0h\<^sub>1.p\<^sub>1] = (f \<star> (g \<star> h \<star> TTfgh_TfTgh.the_\<theta>) \<cdot> can (\<^bold>\<langle>g\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> ((\<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) (((\<^bold>\<langle>g\<^bold>\<rangle> \<^bold>\<star> (\<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) \<cdot> (((g \<star> h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (((g \<star> g\<^sub>0h\<^sub>1.\<phi>) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((\<a>[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (((g.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>1) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (f\<^sub>0gh\<^sub>1.\<phi> \<star> TTfgh_TfTgh.chine) \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>0 f, f\<^sub>0gh\<^sub>1.p\<^sub>1, TTfgh_TfTgh.chine]) \<cdot> \<a>[f, tab\<^sub>0 f, f\<^sub>0gh\<^sub>1.p\<^sub>1 \<star> TTfgh_TfTgh.chine] \<cdot> (f.tab \<star> f\<^sub>0gh\<^sub>1.p\<^sub>1 \<star> TTfgh_TfTgh.chine) \<cdot> \<a>[tab\<^sub>1 f, f\<^sub>0gh\<^sub>1.p\<^sub>1, TTfgh_TfTgh.chine] \<cdot> TTfgh_TfTgh.the_\<nu> \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>1 f, f\<^sub>0g\<^sub>1.p\<^sub>1, fg\<^sub>0h\<^sub>1.p\<^sub>1]
goal (1 subgoal):
1. (f \<star> \<theta>\<^sub>f) \<cdot> \<a>[f, tab\<^sub>0 f, w\<^sub>f] \<cdot> (f.tab \<star> w\<^sub>f) = ((f \<star> \<theta>\<^sub>f') \<cdot> \<a>[f, tab\<^sub>0 f, w\<^sub>f'] \<cdot> (f.tab \<star> w\<^sub>f')) \<cdot> \<beta>\<^sub>f
[PROOF STEP]
have "... = f.composite_cell w\<^sub>f' \<theta>\<^sub>f' \<cdot> \<beta>\<^sub>f"
[PROOF STATE]
proof (prove)
goal (1 subgoal):
1. (f \<star> (g \<star> h \<star> TTfgh_TfTgh.the_\<theta>) \<cdot> can (\<^bold>\<langle>g\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> ((\<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) (((\<^bold>\<langle>g\<^bold>\<rangle> \<^bold>\<star> (\<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) \<cdot> (((g \<star> h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (((g \<star> g\<^sub>0h\<^sub>1.\<phi>) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((\<a>[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (((g.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>1) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (f\<^sub>0gh\<^sub>1.\<phi> \<star> TTfgh_TfTgh.chine) \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>0 f, f\<^sub>0gh\<^sub>1.p\<^sub>1, TTfgh_TfTgh.chine]) \<cdot> \<a>[f, tab\<^sub>0 f, f\<^sub>0gh\<^sub>1.p\<^sub>1 \<star> TTfgh_TfTgh.chine] \<cdot> (f.tab \<star> f\<^sub>0gh\<^sub>1.p\<^sub>1 \<star> TTfgh_TfTgh.chine) \<cdot> \<a>[tab\<^sub>1 f, f\<^sub>0gh\<^sub>1.p\<^sub>1, TTfgh_TfTgh.chine] \<cdot> TTfgh_TfTgh.the_\<nu> \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>1 f, f\<^sub>0g\<^sub>1.p\<^sub>1, fg\<^sub>0h\<^sub>1.p\<^sub>1] = ((f \<star> \<theta>\<^sub>f') \<cdot> \<a>[f, tab\<^sub>0 f, w\<^sub>f'] \<cdot> (f.tab \<star> w\<^sub>f')) \<cdot> \<beta>\<^sub>f
[PROOF STEP]
unfolding w\<^sub>f'_def \<theta>\<^sub>f'_def \<beta>\<^sub>f_def
[PROOF STATE]
proof (prove)
goal (1 subgoal):
1. (f \<star> (g \<star> h \<star> TTfgh_TfTgh.the_\<theta>) \<cdot> can (\<^bold>\<langle>g\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> ((\<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) (((\<^bold>\<langle>g\<^bold>\<rangle> \<^bold>\<star> (\<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) \<cdot> (((g \<star> h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (((g \<star> g\<^sub>0h\<^sub>1.\<phi>) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((\<a>[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (((g.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>1) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (f\<^sub>0gh\<^sub>1.\<phi> \<star> TTfgh_TfTgh.chine) \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>0 f, f\<^sub>0gh\<^sub>1.p\<^sub>1, TTfgh_TfTgh.chine]) \<cdot> \<a>[f, tab\<^sub>0 f, f\<^sub>0gh\<^sub>1.p\<^sub>1 \<star> TTfgh_TfTgh.chine] \<cdot> (f.tab \<star> f\<^sub>0gh\<^sub>1.p\<^sub>1 \<star> TTfgh_TfTgh.chine) \<cdot> \<a>[tab\<^sub>1 f, f\<^sub>0gh\<^sub>1.p\<^sub>1, TTfgh_TfTgh.chine] \<cdot> TTfgh_TfTgh.the_\<nu> \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>1 f, f\<^sub>0g\<^sub>1.p\<^sub>1, fg\<^sub>0h\<^sub>1.p\<^sub>1] = ((f \<star> (g \<star> h \<star> TTfgh_TfTgh.the_\<theta>) \<cdot> can (\<^bold>\<langle>g\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> ((\<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) (((\<^bold>\<langle>g\<^bold>\<rangle> \<^bold>\<star> (\<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) \<cdot> (((g \<star> h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (((g \<star> g\<^sub>0h\<^sub>1.\<phi>) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((\<a>[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (((g.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>1) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (f\<^sub>0gh\<^sub>1.\<phi> \<star> TTfgh_TfTgh.chine) \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>0 f, f\<^sub>0gh\<^sub>1.p\<^sub>1, TTfgh_TfTgh.chine]) \<cdot> \<a>[f, tab\<^sub>0 f, f\<^sub>0gh\<^sub>1.p\<^sub>1 \<star> TTfgh_TfTgh.chine] \<cdot> (f.tab \<star> f\<^sub>0gh\<^sub>1.p\<^sub>1 \<star> TTfgh_TfTgh.chine)) \<cdot> \<a>[tab\<^sub>1 f, f\<^sub>0gh\<^sub>1.p\<^sub>1, TTfgh_TfTgh.chine] \<cdot> TTfgh_TfTgh.the_\<nu> \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>1 f, f\<^sub>0g\<^sub>1.p\<^sub>1, fg\<^sub>0h\<^sub>1.p\<^sub>1]
[PROOF STEP]
using comp_assoc
[PROOF STATE]
proof (prove)
using this:
(?h \<cdot> ?g) \<cdot> ?f = ?h \<cdot> ?g \<cdot> ?f
goal (1 subgoal):
1. (f \<star> (g \<star> h \<star> TTfgh_TfTgh.the_\<theta>) \<cdot> can (\<^bold>\<langle>g\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> ((\<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) (((\<^bold>\<langle>g\<^bold>\<rangle> \<^bold>\<star> (\<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) \<cdot> (((g \<star> h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (((g \<star> g\<^sub>0h\<^sub>1.\<phi>) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((\<a>[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (((g.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>1) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (f\<^sub>0gh\<^sub>1.\<phi> \<star> TTfgh_TfTgh.chine) \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>0 f, f\<^sub>0gh\<^sub>1.p\<^sub>1, TTfgh_TfTgh.chine]) \<cdot> \<a>[f, tab\<^sub>0 f, f\<^sub>0gh\<^sub>1.p\<^sub>1 \<star> TTfgh_TfTgh.chine] \<cdot> (f.tab \<star> f\<^sub>0gh\<^sub>1.p\<^sub>1 \<star> TTfgh_TfTgh.chine) \<cdot> \<a>[tab\<^sub>1 f, f\<^sub>0gh\<^sub>1.p\<^sub>1, TTfgh_TfTgh.chine] \<cdot> TTfgh_TfTgh.the_\<nu> \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>1 f, f\<^sub>0g\<^sub>1.p\<^sub>1, fg\<^sub>0h\<^sub>1.p\<^sub>1] = ((f \<star> (g \<star> h \<star> TTfgh_TfTgh.the_\<theta>) \<cdot> can (\<^bold>\<langle>g\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> ((\<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) (((\<^bold>\<langle>g\<^bold>\<rangle> \<^bold>\<star> (\<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) \<cdot> (((g \<star> h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (((g \<star> g\<^sub>0h\<^sub>1.\<phi>) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((\<a>[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (((g.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>1) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (f\<^sub>0gh\<^sub>1.\<phi> \<star> TTfgh_TfTgh.chine) \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>0 f, f\<^sub>0gh\<^sub>1.p\<^sub>1, TTfgh_TfTgh.chine]) \<cdot> \<a>[f, tab\<^sub>0 f, f\<^sub>0gh\<^sub>1.p\<^sub>1 \<star> TTfgh_TfTgh.chine] \<cdot> (f.tab \<star> f\<^sub>0gh\<^sub>1.p\<^sub>1 \<star> TTfgh_TfTgh.chine)) \<cdot> \<a>[tab\<^sub>1 f, f\<^sub>0gh\<^sub>1.p\<^sub>1, TTfgh_TfTgh.chine] \<cdot> TTfgh_TfTgh.the_\<nu> \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>1 f, f\<^sub>0g\<^sub>1.p\<^sub>1, fg\<^sub>0h\<^sub>1.p\<^sub>1]
[PROOF STEP]
by presburger
[PROOF STATE]
proof (state)
this:
(f \<star> (g \<star> h \<star> TTfgh_TfTgh.the_\<theta>) \<cdot> can (\<^bold>\<langle>g\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> ((\<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) (((\<^bold>\<langle>g\<^bold>\<rangle> \<^bold>\<star> (\<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) \<cdot> (((g \<star> h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (((g \<star> g\<^sub>0h\<^sub>1.\<phi>) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((\<a>[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (((g.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>1) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (f\<^sub>0gh\<^sub>1.\<phi> \<star> TTfgh_TfTgh.chine) \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>0 f, f\<^sub>0gh\<^sub>1.p\<^sub>1, TTfgh_TfTgh.chine]) \<cdot> \<a>[f, tab\<^sub>0 f, f\<^sub>0gh\<^sub>1.p\<^sub>1 \<star> TTfgh_TfTgh.chine] \<cdot> (f.tab \<star> f\<^sub>0gh\<^sub>1.p\<^sub>1 \<star> TTfgh_TfTgh.chine) \<cdot> \<a>[tab\<^sub>1 f, f\<^sub>0gh\<^sub>1.p\<^sub>1, TTfgh_TfTgh.chine] \<cdot> TTfgh_TfTgh.the_\<nu> \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>1 f, f\<^sub>0g\<^sub>1.p\<^sub>1, fg\<^sub>0h\<^sub>1.p\<^sub>1] = ((f \<star> \<theta>\<^sub>f') \<cdot> \<a>[f, tab\<^sub>0 f, w\<^sub>f'] \<cdot> (f.tab \<star> w\<^sub>f')) \<cdot> \<beta>\<^sub>f
goal (1 subgoal):
1. (f \<star> \<theta>\<^sub>f) \<cdot> \<a>[f, tab\<^sub>0 f, w\<^sub>f] \<cdot> (f.tab \<star> w\<^sub>f) = ((f \<star> \<theta>\<^sub>f') \<cdot> \<a>[f, tab\<^sub>0 f, w\<^sub>f'] \<cdot> (f.tab \<star> w\<^sub>f')) \<cdot> \<beta>\<^sub>f
[PROOF STEP]
finally
[PROOF STATE]
proof (chain)
picking this:
(f \<star> \<theta>\<^sub>f) \<cdot> \<a>[f, tab\<^sub>0 f, w\<^sub>f] \<cdot> (f.tab \<star> w\<^sub>f) = ((f \<star> \<theta>\<^sub>f') \<cdot> \<a>[f, tab\<^sub>0 f, w\<^sub>f'] \<cdot> (f.tab \<star> w\<^sub>f')) \<cdot> \<beta>\<^sub>f
[PROOF STEP]
show ?thesis
[PROOF STATE]
proof (prove)
using this:
(f \<star> \<theta>\<^sub>f) \<cdot> \<a>[f, tab\<^sub>0 f, w\<^sub>f] \<cdot> (f.tab \<star> w\<^sub>f) = ((f \<star> \<theta>\<^sub>f') \<cdot> \<a>[f, tab\<^sub>0 f, w\<^sub>f'] \<cdot> (f.tab \<star> w\<^sub>f')) \<cdot> \<beta>\<^sub>f
goal (1 subgoal):
1. (f \<star> \<theta>\<^sub>f) \<cdot> \<a>[f, tab\<^sub>0 f, w\<^sub>f] \<cdot> (f.tab \<star> w\<^sub>f) = ((f \<star> \<theta>\<^sub>f') \<cdot> \<a>[f, tab\<^sub>0 f, w\<^sub>f'] \<cdot> (f.tab \<star> w\<^sub>f')) \<cdot> \<beta>\<^sub>f
[PROOF STEP]
by blast
[PROOF STATE]
proof (state)
this:
(f \<star> \<theta>\<^sub>f) \<cdot> \<a>[f, tab\<^sub>0 f, w\<^sub>f] \<cdot> (f.tab \<star> w\<^sub>f) = ((f \<star> \<theta>\<^sub>f') \<cdot> \<a>[f, tab\<^sub>0 f, w\<^sub>f'] \<cdot> (f.tab \<star> w\<^sub>f')) \<cdot> \<beta>\<^sub>f
goal:
No subgoals!
[PROOF STEP]
qed
[PROOF STATE]
proof (state)
this:
(f \<star> \<theta>\<^sub>f) \<cdot> \<a>[f, tab\<^sub>0 f, w\<^sub>f] \<cdot> (f.tab \<star> w\<^sub>f) = ((f \<star> \<theta>\<^sub>f') \<cdot> \<a>[f, tab\<^sub>0 f, w\<^sub>f'] \<cdot> (f.tab \<star> w\<^sub>f')) \<cdot> \<beta>\<^sub>f
goal (1 subgoal):
1. \<exists>!\<gamma>. \<guillemotleft>\<gamma> : w\<^sub>f \<rightarrow> w\<^sub>f'\<guillemotright> \<and> \<beta>\<^sub>f = tab\<^sub>1 f \<star> \<gamma> \<and> \<theta>\<^sub>f = \<theta>\<^sub>f' \<cdot> (tab\<^sub>0 f \<star> \<gamma>)
[PROOF STEP]
show ?thesis
[PROOF STATE]
proof (prove)
goal (1 subgoal):
1. \<exists>!\<gamma>. \<guillemotleft>\<gamma> : w\<^sub>f \<rightarrow> w\<^sub>f'\<guillemotright> \<and> \<beta>\<^sub>f = tab\<^sub>1 f \<star> \<gamma> \<and> \<theta>\<^sub>f = \<theta>\<^sub>f' \<cdot> (tab\<^sub>0 f \<star> \<gamma>)
[PROOF STEP]
using w\<^sub>f w\<^sub>f' \<theta>\<^sub>f \<theta>\<^sub>f' \<beta>\<^sub>f f.T2 [of w\<^sub>f w\<^sub>f' \<theta>\<^sub>f u\<^sub>f \<theta>\<^sub>f' \<beta>\<^sub>f] eq\<^sub>f
[PROOF STATE]
proof (prove)
using this:
ide w\<^sub>f
ide w\<^sub>f'
\<guillemotleft>\<theta>\<^sub>f : tab\<^sub>0 f \<star> w\<^sub>f \<rightarrow> u\<^sub>f\<guillemotright>
\<guillemotleft>\<theta>\<^sub>f' : tab\<^sub>0 f \<star> w\<^sub>f' \<rightarrow> u\<^sub>f\<guillemotright>
\<guillemotleft>\<beta>\<^sub>f : tab\<^sub>1 f \<star> w\<^sub>f \<rightarrow> tab\<^sub>1 f \<star> w\<^sub>f'\<guillemotright>
\<lbrakk>ide w\<^sub>f; ide w\<^sub>f'; \<guillemotleft>\<theta>\<^sub>f : tab\<^sub>0 f \<star> w\<^sub>f \<rightarrow> u\<^sub>f\<guillemotright>; \<guillemotleft>\<theta>\<^sub>f' : tab\<^sub>0 f \<star> w\<^sub>f' \<rightarrow> u\<^sub>f\<guillemotright>; \<guillemotleft>\<beta>\<^sub>f : tab\<^sub>1 f \<star> w\<^sub>f \<rightarrow> tab\<^sub>1 f \<star> w\<^sub>f'\<guillemotright>; (f \<star> \<theta>\<^sub>f) \<cdot> \<a>[f, tab\<^sub>0 f, w\<^sub>f] \<cdot> (f.tab \<star> w\<^sub>f) = ((f \<star> \<theta>\<^sub>f') \<cdot> \<a>[f, tab\<^sub>0 f, w\<^sub>f'] \<cdot> (f.tab \<star> w\<^sub>f')) \<cdot> \<beta>\<^sub>f\<rbrakk> \<Longrightarrow> \<exists>!\<gamma>. \<guillemotleft>\<gamma> : w\<^sub>f \<rightarrow> w\<^sub>f'\<guillemotright> \<and> \<beta>\<^sub>f = tab\<^sub>1 f \<star> \<gamma> \<and> \<theta>\<^sub>f = \<theta>\<^sub>f' \<cdot> (tab\<^sub>0 f \<star> \<gamma>)
(f \<star> \<theta>\<^sub>f) \<cdot> \<a>[f, tab\<^sub>0 f, w\<^sub>f] \<cdot> (f.tab \<star> w\<^sub>f) = ((f \<star> \<theta>\<^sub>f') \<cdot> \<a>[f, tab\<^sub>0 f, w\<^sub>f'] \<cdot> (f.tab \<star> w\<^sub>f')) \<cdot> \<beta>\<^sub>f
goal (1 subgoal):
1. \<exists>!\<gamma>. \<guillemotleft>\<gamma> : w\<^sub>f \<rightarrow> w\<^sub>f'\<guillemotright> \<and> \<beta>\<^sub>f = tab\<^sub>1 f \<star> \<gamma> \<and> \<theta>\<^sub>f = \<theta>\<^sub>f' \<cdot> (tab\<^sub>0 f \<star> \<gamma>)
[PROOF STEP]
by fast
[PROOF STATE]
proof (state)
this:
\<exists>!\<gamma>. \<guillemotleft>\<gamma> : w\<^sub>f \<rightarrow> w\<^sub>f'\<guillemotright> \<and> \<beta>\<^sub>f = tab\<^sub>1 f \<star> \<gamma> \<and> \<theta>\<^sub>f = \<theta>\<^sub>f' \<cdot> (tab\<^sub>0 f \<star> \<gamma>)
goal:
No subgoals!
[PROOF STEP]
qed
[PROOF STATE]
proof (state)
this:
\<exists>!\<gamma>. \<guillemotleft>\<gamma> : w\<^sub>f \<rightarrow> w\<^sub>f'\<guillemotright> \<and> \<beta>\<^sub>f = tab\<^sub>1 f \<star> \<gamma> \<and> \<theta>\<^sub>f = \<theta>\<^sub>f' \<cdot> (tab\<^sub>0 f \<star> \<gamma>)
goal (3 subgoals):
1. \<lbrakk>\<lbrakk>f\<^sub>0gh\<^sub>1.p\<^sub>1 \<star> TTfgh_TfTgh.chine\<rbrakk>\<rbrakk> = \<lbrakk>\<lbrakk>f\<^sub>0g\<^sub>1.p\<^sub>1 \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1\<rbrakk>\<rbrakk>
2. \<lbrakk>\<lbrakk>g\<^sub>0h\<^sub>1.p\<^sub>1 \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine\<rbrakk>\<rbrakk> = \<lbrakk>\<lbrakk>f\<^sub>0g\<^sub>1.p\<^sub>0 \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1\<rbrakk>\<rbrakk>
3. \<lbrakk>\<lbrakk>g\<^sub>0h\<^sub>1.p\<^sub>0 \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine\<rbrakk>\<rbrakk> = \<lbrakk>\<lbrakk>fg\<^sub>0h\<^sub>1.p\<^sub>0\<rbrakk>\<rbrakk>
[PROOF STEP]
obtain \<gamma>\<^sub>f where \<gamma>\<^sub>f: "\<guillemotleft>\<gamma>\<^sub>f : w\<^sub>f \<Rightarrow> w\<^sub>f'\<guillemotright> \<and> \<beta>\<^sub>f = tab\<^sub>1 f \<star> \<gamma>\<^sub>f \<and> \<theta>\<^sub>f = \<theta>\<^sub>f' \<cdot> (tab\<^sub>0 f \<star> \<gamma>\<^sub>f)"
[PROOF STATE]
proof (prove)
goal (1 subgoal):
1. (\<And>\<gamma>\<^sub>f. \<guillemotleft>\<gamma>\<^sub>f : w\<^sub>f \<rightarrow> w\<^sub>f'\<guillemotright> \<and> \<beta>\<^sub>f = tab\<^sub>1 f \<star> \<gamma>\<^sub>f \<and> \<theta>\<^sub>f = \<theta>\<^sub>f' \<cdot> (tab\<^sub>0 f \<star> \<gamma>\<^sub>f) \<Longrightarrow> thesis) \<Longrightarrow> thesis
[PROOF STEP]
using 5
[PROOF STATE]
proof (prove)
using this:
\<exists>!\<gamma>. \<guillemotleft>\<gamma> : w\<^sub>f \<rightarrow> w\<^sub>f'\<guillemotright> \<and> \<beta>\<^sub>f = tab\<^sub>1 f \<star> \<gamma> \<and> \<theta>\<^sub>f = \<theta>\<^sub>f' \<cdot> (tab\<^sub>0 f \<star> \<gamma>)
goal (1 subgoal):
1. (\<And>\<gamma>\<^sub>f. \<guillemotleft>\<gamma>\<^sub>f : w\<^sub>f \<rightarrow> w\<^sub>f'\<guillemotright> \<and> \<beta>\<^sub>f = tab\<^sub>1 f \<star> \<gamma>\<^sub>f \<and> \<theta>\<^sub>f = \<theta>\<^sub>f' \<cdot> (tab\<^sub>0 f \<star> \<gamma>\<^sub>f) \<Longrightarrow> thesis) \<Longrightarrow> thesis
[PROOF STEP]
by auto
[PROOF STATE]
proof (state)
this:
\<guillemotleft>\<gamma>\<^sub>f : w\<^sub>f \<rightarrow> w\<^sub>f'\<guillemotright> \<and> \<beta>\<^sub>f = tab\<^sub>1 f \<star> \<gamma>\<^sub>f \<and> \<theta>\<^sub>f = \<theta>\<^sub>f' \<cdot> (tab\<^sub>0 f \<star> \<gamma>\<^sub>f)
goal (3 subgoals):
1. \<lbrakk>\<lbrakk>f\<^sub>0gh\<^sub>1.p\<^sub>1 \<star> TTfgh_TfTgh.chine\<rbrakk>\<rbrakk> = \<lbrakk>\<lbrakk>f\<^sub>0g\<^sub>1.p\<^sub>1 \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1\<rbrakk>\<rbrakk>
2. \<lbrakk>\<lbrakk>g\<^sub>0h\<^sub>1.p\<^sub>1 \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine\<rbrakk>\<rbrakk> = \<lbrakk>\<lbrakk>f\<^sub>0g\<^sub>1.p\<^sub>0 \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1\<rbrakk>\<rbrakk>
3. \<lbrakk>\<lbrakk>g\<^sub>0h\<^sub>1.p\<^sub>0 \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine\<rbrakk>\<rbrakk> = \<lbrakk>\<lbrakk>fg\<^sub>0h\<^sub>1.p\<^sub>0\<rbrakk>\<rbrakk>
[PROOF STEP]
show "\<lbrakk>\<lbrakk>TfTgh.p\<^sub>1 \<star> TTfgh_TfTgh.chine\<rbrakk>\<rbrakk> = \<lbrakk>\<lbrakk>Tfg.p\<^sub>1 \<star> TTfgh.p\<^sub>1\<rbrakk>\<rbrakk>"
[PROOF STATE]
proof (prove)
goal (1 subgoal):
1. \<lbrakk>\<lbrakk>f\<^sub>0gh\<^sub>1.p\<^sub>1 \<star> TTfgh_TfTgh.chine\<rbrakk>\<rbrakk> = \<lbrakk>\<lbrakk>f\<^sub>0g\<^sub>1.p\<^sub>1 \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1\<rbrakk>\<rbrakk>
[PROOF STEP]
proof -
[PROOF STATE]
proof (state)
goal (1 subgoal):
1. \<lbrakk>\<lbrakk>f\<^sub>0gh\<^sub>1.p\<^sub>1 \<star> TTfgh_TfTgh.chine\<rbrakk>\<rbrakk> = \<lbrakk>\<lbrakk>f\<^sub>0g\<^sub>1.p\<^sub>1 \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1\<rbrakk>\<rbrakk>
[PROOF STEP]
have "iso \<gamma>\<^sub>f"
[PROOF STATE]
proof (prove)
goal (1 subgoal):
1. local.iso \<gamma>\<^sub>f
[PROOF STEP]
using \<gamma>\<^sub>f BS3 w\<^sub>f_is_map w\<^sub>f'_is_map
[PROOF STATE]
proof (prove)
using this:
\<guillemotleft>\<gamma>\<^sub>f : w\<^sub>f \<rightarrow> w\<^sub>f'\<guillemotright> \<and> \<beta>\<^sub>f = tab\<^sub>1 f \<star> \<gamma>\<^sub>f \<and> \<theta>\<^sub>f = \<theta>\<^sub>f' \<cdot> (tab\<^sub>0 f \<star> \<gamma>\<^sub>f)
\<lbrakk>is_left_adjoint ?f; is_left_adjoint ?f'; \<guillemotleft>?\<mu> : ?f \<rightarrow> ?f'\<guillemotright>; \<guillemotleft>?\<mu>' : ?f \<rightarrow> ?f'\<guillemotright>\<rbrakk> \<Longrightarrow> local.iso ?\<mu> \<and> local.iso ?\<mu>' \<and> ?\<mu> = ?\<mu>'
is_left_adjoint w\<^sub>f
is_left_adjoint w\<^sub>f'
goal (1 subgoal):
1. local.iso \<gamma>\<^sub>f
[PROOF STEP]
by blast
[PROOF STATE]
proof (state)
this:
local.iso \<gamma>\<^sub>f
goal (1 subgoal):
1. \<lbrakk>\<lbrakk>f\<^sub>0gh\<^sub>1.p\<^sub>1 \<star> TTfgh_TfTgh.chine\<rbrakk>\<rbrakk> = \<lbrakk>\<lbrakk>f\<^sub>0g\<^sub>1.p\<^sub>1 \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1\<rbrakk>\<rbrakk>
[PROOF STEP]
hence "isomorphic w\<^sub>f w\<^sub>f'"
[PROOF STATE]
proof (prove)
using this:
local.iso \<gamma>\<^sub>f
goal (1 subgoal):
1. w\<^sub>f \<cong> w\<^sub>f'
[PROOF STEP]
using \<gamma>\<^sub>f isomorphic_def isomorphic_symmetric
[PROOF STATE]
proof (prove)
using this:
local.iso \<gamma>\<^sub>f
\<guillemotleft>\<gamma>\<^sub>f : w\<^sub>f \<rightarrow> w\<^sub>f'\<guillemotright> \<and> \<beta>\<^sub>f = tab\<^sub>1 f \<star> \<gamma>\<^sub>f \<and> \<theta>\<^sub>f = \<theta>\<^sub>f' \<cdot> (tab\<^sub>0 f \<star> \<gamma>\<^sub>f)
(?a \<cong> ?a') = (\<exists>f. \<guillemotleft>f : ?a \<rightarrow> ?a'\<guillemotright> \<and> local.iso f)
?f \<cong> ?g \<Longrightarrow> ?g \<cong> ?f
goal (1 subgoal):
1. w\<^sub>f \<cong> w\<^sub>f'
[PROOF STEP]
by auto
[PROOF STATE]
proof (state)
this:
w\<^sub>f \<cong> w\<^sub>f'
goal (1 subgoal):
1. \<lbrakk>\<lbrakk>f\<^sub>0gh\<^sub>1.p\<^sub>1 \<star> TTfgh_TfTgh.chine\<rbrakk>\<rbrakk> = \<lbrakk>\<lbrakk>f\<^sub>0g\<^sub>1.p\<^sub>1 \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1\<rbrakk>\<rbrakk>
[PROOF STEP]
thus ?thesis
[PROOF STATE]
proof (prove)
using this:
w\<^sub>f \<cong> w\<^sub>f'
goal (1 subgoal):
1. \<lbrakk>\<lbrakk>f\<^sub>0gh\<^sub>1.p\<^sub>1 \<star> TTfgh_TfTgh.chine\<rbrakk>\<rbrakk> = \<lbrakk>\<lbrakk>f\<^sub>0g\<^sub>1.p\<^sub>1 \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1\<rbrakk>\<rbrakk>
[PROOF STEP]
using w\<^sub>f w\<^sub>f_def w\<^sub>f'_def Maps.CLS_eqI isomorphic_symmetric
[PROOF STATE]
proof (prove)
using this:
w\<^sub>f \<cong> w\<^sub>f'
ide w\<^sub>f
w\<^sub>f = f\<^sub>0g\<^sub>1.p\<^sub>1 \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1
w\<^sub>f' = f\<^sub>0gh\<^sub>1.p\<^sub>1 \<star> TTfgh_TfTgh.chine
ide ?f \<Longrightarrow> (\<lbrakk>\<lbrakk>?f\<rbrakk>\<rbrakk> = \<lbrakk>\<lbrakk>?g\<rbrakk>\<rbrakk>) = (?f \<cong> ?g)
?f \<cong> ?g \<Longrightarrow> ?g \<cong> ?f
goal (1 subgoal):
1. \<lbrakk>\<lbrakk>f\<^sub>0gh\<^sub>1.p\<^sub>1 \<star> TTfgh_TfTgh.chine\<rbrakk>\<rbrakk> = \<lbrakk>\<lbrakk>f\<^sub>0g\<^sub>1.p\<^sub>1 \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1\<rbrakk>\<rbrakk>
[PROOF STEP]
by auto
[PROOF STATE]
proof (state)
this:
\<lbrakk>\<lbrakk>f\<^sub>0gh\<^sub>1.p\<^sub>1 \<star> TTfgh_TfTgh.chine\<rbrakk>\<rbrakk> = \<lbrakk>\<lbrakk>f\<^sub>0g\<^sub>1.p\<^sub>1 \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1\<rbrakk>\<rbrakk>
goal:
No subgoals!
[PROOF STEP]
qed
[PROOF STATE]
proof (state)
this:
\<lbrakk>\<lbrakk>f\<^sub>0gh\<^sub>1.p\<^sub>1 \<star> TTfgh_TfTgh.chine\<rbrakk>\<rbrakk> = \<lbrakk>\<lbrakk>f\<^sub>0g\<^sub>1.p\<^sub>1 \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1\<rbrakk>\<rbrakk>
goal (2 subgoals):
1. \<lbrakk>\<lbrakk>g\<^sub>0h\<^sub>1.p\<^sub>1 \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine\<rbrakk>\<rbrakk> = \<lbrakk>\<lbrakk>f\<^sub>0g\<^sub>1.p\<^sub>0 \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1\<rbrakk>\<rbrakk>
2. \<lbrakk>\<lbrakk>g\<^sub>0h\<^sub>1.p\<^sub>0 \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine\<rbrakk>\<rbrakk> = \<lbrakk>\<lbrakk>fg\<^sub>0h\<^sub>1.p\<^sub>0\<rbrakk>\<rbrakk>
[PROOF STEP]
define u\<^sub>g where "u\<^sub>g = h \<star> tab\<^sub>0 h \<star> TTfgh.p\<^sub>0"
[PROOF STATE]
proof (state)
this:
u\<^sub>g = h \<star> tab\<^sub>0 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0
goal (2 subgoals):
1. \<lbrakk>\<lbrakk>g\<^sub>0h\<^sub>1.p\<^sub>1 \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine\<rbrakk>\<rbrakk> = \<lbrakk>\<lbrakk>f\<^sub>0g\<^sub>1.p\<^sub>0 \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1\<rbrakk>\<rbrakk>
2. \<lbrakk>\<lbrakk>g\<^sub>0h\<^sub>1.p\<^sub>0 \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine\<rbrakk>\<rbrakk> = \<lbrakk>\<lbrakk>fg\<^sub>0h\<^sub>1.p\<^sub>0\<rbrakk>\<rbrakk>
[PROOF STEP]
define w\<^sub>g where "w\<^sub>g = Tfg.p\<^sub>0 \<star> TTfgh.p\<^sub>1"
[PROOF STATE]
proof (state)
this:
w\<^sub>g = f\<^sub>0g\<^sub>1.p\<^sub>0 \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1
goal (2 subgoals):
1. \<lbrakk>\<lbrakk>g\<^sub>0h\<^sub>1.p\<^sub>1 \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine\<rbrakk>\<rbrakk> = \<lbrakk>\<lbrakk>f\<^sub>0g\<^sub>1.p\<^sub>0 \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1\<rbrakk>\<rbrakk>
2. \<lbrakk>\<lbrakk>g\<^sub>0h\<^sub>1.p\<^sub>0 \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine\<rbrakk>\<rbrakk> = \<lbrakk>\<lbrakk>fg\<^sub>0h\<^sub>1.p\<^sub>0\<rbrakk>\<rbrakk>
[PROOF STEP]
define w\<^sub>g' where "w\<^sub>g' = Tgh.p\<^sub>1 \<star> TfTgh.p\<^sub>0 \<star> TTfgh_TfTgh.chine"
[PROOF STATE]
proof (state)
this:
w\<^sub>g' = g\<^sub>0h\<^sub>1.p\<^sub>1 \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine
goal (2 subgoals):
1. \<lbrakk>\<lbrakk>g\<^sub>0h\<^sub>1.p\<^sub>1 \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine\<rbrakk>\<rbrakk> = \<lbrakk>\<lbrakk>f\<^sub>0g\<^sub>1.p\<^sub>0 \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1\<rbrakk>\<rbrakk>
2. \<lbrakk>\<lbrakk>g\<^sub>0h\<^sub>1.p\<^sub>0 \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine\<rbrakk>\<rbrakk> = \<lbrakk>\<lbrakk>fg\<^sub>0h\<^sub>1.p\<^sub>0\<rbrakk>\<rbrakk>
[PROOF STEP]
define \<theta>\<^sub>g
where "\<theta>\<^sub>g = \<a>[h, tab\<^sub>0 h, TTfgh.p\<^sub>0] \<cdot> (h.tab \<star> TTfgh.p\<^sub>0) \<cdot> fg\<^sub>0h\<^sub>1.\<phi> \<cdot>
\<a>\<^sup>-\<^sup>1[tab\<^sub>0 g, Tfg.p\<^sub>0, TTfgh.p\<^sub>1]"
[PROOF STATE]
proof (state)
this:
\<theta>\<^sub>g = \<a>[h, tab\<^sub>0 h, fg\<^sub>0h\<^sub>1.p\<^sub>0] \<cdot> (h.tab \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0) \<cdot> fg\<^sub>0h\<^sub>1.\<phi> \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>0 g, f\<^sub>0g\<^sub>1.p\<^sub>0, fg\<^sub>0h\<^sub>1.p\<^sub>1]
goal (2 subgoals):
1. \<lbrakk>\<lbrakk>g\<^sub>0h\<^sub>1.p\<^sub>1 \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine\<rbrakk>\<rbrakk> = \<lbrakk>\<lbrakk>f\<^sub>0g\<^sub>1.p\<^sub>0 \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1\<rbrakk>\<rbrakk>
2. \<lbrakk>\<lbrakk>g\<^sub>0h\<^sub>1.p\<^sub>0 \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine\<rbrakk>\<rbrakk> = \<lbrakk>\<lbrakk>fg\<^sub>0h\<^sub>1.p\<^sub>0\<rbrakk>\<rbrakk>
[PROOF STEP]
define \<theta>\<^sub>g'
where "\<theta>\<^sub>g' = (h \<star> TTfgh_TfTgh.the_\<theta>) \<cdot>
can (\<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> ((\<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>Tgh.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TfTgh.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>)
(((\<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>Tgh.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TfTgh.p\<^sub>0\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) \<cdot>
((h.tab \<star> Tgh.p\<^sub>0) \<star> TfTgh.p\<^sub>0 \<star> TTfgh_TfTgh.chine) \<cdot>
(g\<^sub>0h\<^sub>1.\<phi> \<star> TfTgh.p\<^sub>0 \<star> TTfgh_TfTgh.chine) \<cdot>
\<a>\<^sup>-\<^sup>1[tab\<^sub>0 g, Tgh.p\<^sub>1, TfTgh.p\<^sub>0 \<star> TTfgh_TfTgh.chine]"
[PROOF STATE]
proof (state)
this:
\<theta>\<^sub>g' = (h \<star> TTfgh_TfTgh.the_\<theta>) \<cdot> can (\<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> ((\<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) (((\<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) \<cdot> ((h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine) \<cdot> (g\<^sub>0h\<^sub>1.\<phi> \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine) \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1, f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine]
goal (2 subgoals):
1. \<lbrakk>\<lbrakk>g\<^sub>0h\<^sub>1.p\<^sub>1 \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine\<rbrakk>\<rbrakk> = \<lbrakk>\<lbrakk>f\<^sub>0g\<^sub>1.p\<^sub>0 \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1\<rbrakk>\<rbrakk>
2. \<lbrakk>\<lbrakk>g\<^sub>0h\<^sub>1.p\<^sub>0 \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine\<rbrakk>\<rbrakk> = \<lbrakk>\<lbrakk>fg\<^sub>0h\<^sub>1.p\<^sub>0\<rbrakk>\<rbrakk>
[PROOF STEP]
define \<beta>\<^sub>g
where "\<beta>\<^sub>g = \<a>[tab\<^sub>1 g, Tgh.p\<^sub>1, TfTgh.p\<^sub>0 \<star> TTfgh_TfTgh.chine] \<cdot>
\<a>[tab\<^sub>1 g \<star> Tgh.p\<^sub>1, TfTgh.p\<^sub>0, TTfgh_TfTgh.chine] \<cdot>
(f\<^sub>0gh\<^sub>1.\<phi> \<star> TTfgh_TfTgh.chine) \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>0 f, TfTgh.p\<^sub>1, TTfgh_TfTgh.chine] \<cdot>
(tab\<^sub>0 f \<star> \<gamma>\<^sub>f) \<cdot> \<a>[tab\<^sub>0 f, Tfg.p\<^sub>1, TTfgh.p\<^sub>1] \<cdot> (inv f\<^sub>0g\<^sub>1.\<phi> \<star> TTfgh.p\<^sub>1) \<cdot>
\<a>\<^sup>-\<^sup>1[tab\<^sub>1 g, Tfg.p\<^sub>0, TTfgh.p\<^sub>1]"
[PROOF STATE]
proof (state)
this:
\<beta>\<^sub>g = \<a>[tab\<^sub>1 g, g\<^sub>0h\<^sub>1.p\<^sub>1, f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine] \<cdot> \<a>[tab\<^sub>1 g \<star> g\<^sub>0h\<^sub>1.p\<^sub>1, f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine] \<cdot> (f\<^sub>0gh\<^sub>1.\<phi> \<star> TTfgh_TfTgh.chine) \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>0 f, f\<^sub>0gh\<^sub>1.p\<^sub>1, TTfgh_TfTgh.chine] \<cdot> (tab\<^sub>0 f \<star> \<gamma>\<^sub>f) \<cdot> \<a>[tab\<^sub>0 f, f\<^sub>0g\<^sub>1.p\<^sub>1, fg\<^sub>0h\<^sub>1.p\<^sub>1] \<cdot> (local.inv f\<^sub>0g\<^sub>1.\<phi> \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>1 g, f\<^sub>0g\<^sub>1.p\<^sub>0, fg\<^sub>0h\<^sub>1.p\<^sub>1]
goal (2 subgoals):
1. \<lbrakk>\<lbrakk>g\<^sub>0h\<^sub>1.p\<^sub>1 \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine\<rbrakk>\<rbrakk> = \<lbrakk>\<lbrakk>f\<^sub>0g\<^sub>1.p\<^sub>0 \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1\<rbrakk>\<rbrakk>
2. \<lbrakk>\<lbrakk>g\<^sub>0h\<^sub>1.p\<^sub>0 \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine\<rbrakk>\<rbrakk> = \<lbrakk>\<lbrakk>fg\<^sub>0h\<^sub>1.p\<^sub>0\<rbrakk>\<rbrakk>
[PROOF STEP]
have u\<^sub>g: "ide u\<^sub>g"
[PROOF STATE]
proof (prove)
goal (1 subgoal):
1. ide u\<^sub>g
[PROOF STEP]
unfolding u\<^sub>g_def
[PROOF STATE]
proof (prove)
goal (1 subgoal):
1. ide (h \<star> tab\<^sub>0 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0)
[PROOF STEP]
by simp
[PROOF STATE]
proof (state)
this:
ide u\<^sub>g
goal (2 subgoals):
1. \<lbrakk>\<lbrakk>g\<^sub>0h\<^sub>1.p\<^sub>1 \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine\<rbrakk>\<rbrakk> = \<lbrakk>\<lbrakk>f\<^sub>0g\<^sub>1.p\<^sub>0 \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1\<rbrakk>\<rbrakk>
2. \<lbrakk>\<lbrakk>g\<^sub>0h\<^sub>1.p\<^sub>0 \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine\<rbrakk>\<rbrakk> = \<lbrakk>\<lbrakk>fg\<^sub>0h\<^sub>1.p\<^sub>0\<rbrakk>\<rbrakk>
[PROOF STEP]
have w\<^sub>g: "ide w\<^sub>g"
[PROOF STATE]
proof (prove)
goal (1 subgoal):
1. ide w\<^sub>g
[PROOF STEP]
unfolding w\<^sub>g_def
[PROOF STATE]
proof (prove)
goal (1 subgoal):
1. ide (f\<^sub>0g\<^sub>1.p\<^sub>0 \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1)
[PROOF STEP]
using fg\<^sub>0h\<^sub>1.p\<^sub>1_simps
[PROOF STATE]
proof (prove)
using this:
trg fg\<^sub>0h\<^sub>1.p\<^sub>1 = src (tab\<^sub>0 g \<star> f\<^sub>0g\<^sub>1.p\<^sub>0)
goal (1 subgoal):
1. ide (f\<^sub>0g\<^sub>1.p\<^sub>0 \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1)
[PROOF STEP]
by auto
[PROOF STATE]
proof (state)
this:
ide w\<^sub>g
goal (2 subgoals):
1. \<lbrakk>\<lbrakk>g\<^sub>0h\<^sub>1.p\<^sub>1 \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine\<rbrakk>\<rbrakk> = \<lbrakk>\<lbrakk>f\<^sub>0g\<^sub>1.p\<^sub>0 \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1\<rbrakk>\<rbrakk>
2. \<lbrakk>\<lbrakk>g\<^sub>0h\<^sub>1.p\<^sub>0 \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine\<rbrakk>\<rbrakk> = \<lbrakk>\<lbrakk>fg\<^sub>0h\<^sub>1.p\<^sub>0\<rbrakk>\<rbrakk>
[PROOF STEP]
have w\<^sub>g_is_map: "is_left_adjoint w\<^sub>g"
[PROOF STATE]
proof (prove)
goal (1 subgoal):
1. is_left_adjoint w\<^sub>g
[PROOF STEP]
unfolding w\<^sub>g_def
[PROOF STATE]
proof (prove)
goal (1 subgoal):
1. is_left_adjoint (f\<^sub>0g\<^sub>1.p\<^sub>0 \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1)
[PROOF STEP]
using fg\<^sub>0h\<^sub>1.p\<^sub>1_simps left_adjoints_compose
[PROOF STATE]
proof (prove)
using this:
trg fg\<^sub>0h\<^sub>1.p\<^sub>1 = src (tab\<^sub>0 g \<star> f\<^sub>0g\<^sub>1.p\<^sub>0)
\<lbrakk>is_left_adjoint ?f; is_left_adjoint ?f'; src ?f = trg ?f'\<rbrakk> \<Longrightarrow> is_left_adjoint (?f \<star> ?f')
goal (1 subgoal):
1. is_left_adjoint (f\<^sub>0g\<^sub>1.p\<^sub>0 \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1)
[PROOF STEP]
by simp
[PROOF STATE]
proof (state)
this:
is_left_adjoint w\<^sub>g
goal (2 subgoals):
1. \<lbrakk>\<lbrakk>g\<^sub>0h\<^sub>1.p\<^sub>1 \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine\<rbrakk>\<rbrakk> = \<lbrakk>\<lbrakk>f\<^sub>0g\<^sub>1.p\<^sub>0 \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1\<rbrakk>\<rbrakk>
2. \<lbrakk>\<lbrakk>g\<^sub>0h\<^sub>1.p\<^sub>0 \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine\<rbrakk>\<rbrakk> = \<lbrakk>\<lbrakk>fg\<^sub>0h\<^sub>1.p\<^sub>0\<rbrakk>\<rbrakk>
[PROOF STEP]
have w\<^sub>g': "ide w\<^sub>g'"
[PROOF STATE]
proof (prove)
goal (1 subgoal):
1. ide w\<^sub>g'
[PROOF STEP]
unfolding w\<^sub>g'_def
[PROOF STATE]
proof (prove)
goal (1 subgoal):
1. ide (g\<^sub>0h\<^sub>1.p\<^sub>1 \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine)
[PROOF STEP]
by simp
[PROOF STATE]
proof (state)
this:
ide w\<^sub>g'
goal (2 subgoals):
1. \<lbrakk>\<lbrakk>g\<^sub>0h\<^sub>1.p\<^sub>1 \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine\<rbrakk>\<rbrakk> = \<lbrakk>\<lbrakk>f\<^sub>0g\<^sub>1.p\<^sub>0 \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1\<rbrakk>\<rbrakk>
2. \<lbrakk>\<lbrakk>g\<^sub>0h\<^sub>1.p\<^sub>0 \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine\<rbrakk>\<rbrakk> = \<lbrakk>\<lbrakk>fg\<^sub>0h\<^sub>1.p\<^sub>0\<rbrakk>\<rbrakk>
[PROOF STEP]
have w\<^sub>g'_is_map: "is_left_adjoint w\<^sub>g'"
[PROOF STATE]
proof (prove)
goal (1 subgoal):
1. is_left_adjoint w\<^sub>g'
[PROOF STEP]
unfolding w\<^sub>g'_def
[PROOF STATE]
proof (prove)
goal (1 subgoal):
1. is_left_adjoint (g\<^sub>0h\<^sub>1.p\<^sub>1 \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine)
[PROOF STEP]
using TTfgh_TfTgh.is_map left_adjoints_compose
[PROOF STATE]
proof (prove)
using this:
is_left_adjoint TTfgh_TfTgh.chine
\<lbrakk>is_left_adjoint ?f; is_left_adjoint ?f'; src ?f = trg ?f'\<rbrakk> \<Longrightarrow> is_left_adjoint (?f \<star> ?f')
goal (1 subgoal):
1. is_left_adjoint (g\<^sub>0h\<^sub>1.p\<^sub>1 \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine)
[PROOF STEP]
by simp
[PROOF STATE]
proof (state)
this:
is_left_adjoint w\<^sub>g'
goal (2 subgoals):
1. \<lbrakk>\<lbrakk>g\<^sub>0h\<^sub>1.p\<^sub>1 \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine\<rbrakk>\<rbrakk> = \<lbrakk>\<lbrakk>f\<^sub>0g\<^sub>1.p\<^sub>0 \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1\<rbrakk>\<rbrakk>
2. \<lbrakk>\<lbrakk>g\<^sub>0h\<^sub>1.p\<^sub>0 \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine\<rbrakk>\<rbrakk> = \<lbrakk>\<lbrakk>fg\<^sub>0h\<^sub>1.p\<^sub>0\<rbrakk>\<rbrakk>
[PROOF STEP]
have \<theta>\<^sub>g: "\<guillemotleft>\<theta>\<^sub>g : tab\<^sub>0 g \<star> w\<^sub>g \<Rightarrow> u\<^sub>g\<guillemotright>"
[PROOF STATE]
proof (prove)
goal (1 subgoal):
1. \<guillemotleft>\<theta>\<^sub>g : tab\<^sub>0 g \<star> w\<^sub>g \<rightarrow> u\<^sub>g\<guillemotright>
[PROOF STEP]
using w\<^sub>g_def u\<^sub>g_def \<theta>\<^sub>g_def fg\<^sub>0h\<^sub>1.p\<^sub>1_simps fg\<^sub>0h\<^sub>1.\<phi>_in_hom
[PROOF STATE]
proof (prove)
using this:
w\<^sub>g = f\<^sub>0g\<^sub>1.p\<^sub>0 \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1
u\<^sub>g = h \<star> tab\<^sub>0 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0
\<theta>\<^sub>g = \<a>[h, tab\<^sub>0 h, fg\<^sub>0h\<^sub>1.p\<^sub>0] \<cdot> (h.tab \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0) \<cdot> fg\<^sub>0h\<^sub>1.\<phi> \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>0 g, f\<^sub>0g\<^sub>1.p\<^sub>0, fg\<^sub>0h\<^sub>1.p\<^sub>1]
trg fg\<^sub>0h\<^sub>1.p\<^sub>1 = src (tab\<^sub>0 g \<star> f\<^sub>0g\<^sub>1.p\<^sub>0)
\<guillemotleft>fg\<^sub>0h\<^sub>1.\<phi> : src fg\<^sub>0h\<^sub>1.p\<^sub>0 \<rightarrow> trg (tab\<^sub>1 h)\<guillemotright>
\<guillemotleft>fg\<^sub>0h\<^sub>1.\<phi> : (tab\<^sub>0 g \<star> f\<^sub>0g\<^sub>1.p\<^sub>0) \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1 \<rightarrow> tab\<^sub>1 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0\<guillemotright>
goal (1 subgoal):
1. \<guillemotleft>\<theta>\<^sub>g : tab\<^sub>0 g \<star> w\<^sub>g \<rightarrow> u\<^sub>g\<guillemotright>
[PROOF STEP]
by auto
[PROOF STATE]
proof (state)
this:
\<guillemotleft>\<theta>\<^sub>g : tab\<^sub>0 g \<star> w\<^sub>g \<rightarrow> u\<^sub>g\<guillemotright>
goal (2 subgoals):
1. \<lbrakk>\<lbrakk>g\<^sub>0h\<^sub>1.p\<^sub>1 \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine\<rbrakk>\<rbrakk> = \<lbrakk>\<lbrakk>f\<^sub>0g\<^sub>1.p\<^sub>0 \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1\<rbrakk>\<rbrakk>
2. \<lbrakk>\<lbrakk>g\<^sub>0h\<^sub>1.p\<^sub>0 \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine\<rbrakk>\<rbrakk> = \<lbrakk>\<lbrakk>fg\<^sub>0h\<^sub>1.p\<^sub>0\<rbrakk>\<rbrakk>
[PROOF STEP]
have \<theta>\<^sub>g': "\<guillemotleft>\<theta>\<^sub>g' : tab\<^sub>0 g \<star> w\<^sub>g' \<Rightarrow> u\<^sub>g\<guillemotright>"
[PROOF STATE]
proof (prove)
goal (1 subgoal):
1. \<guillemotleft>\<theta>\<^sub>g' : tab\<^sub>0 g \<star> w\<^sub>g' \<rightarrow> u\<^sub>g\<guillemotright>
[PROOF STEP]
unfolding w\<^sub>g'_def u\<^sub>g_def \<theta>\<^sub>g'_def
[PROOF STATE]
proof (prove)
goal (1 subgoal):
1. \<guillemotleft>(h \<star> TTfgh_TfTgh.the_\<theta>) \<cdot> can (\<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> ((\<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) (((\<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) \<cdot> ((h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine) \<cdot> (g\<^sub>0h\<^sub>1.\<phi> \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine) \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1, f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine] : tab\<^sub>0 g \<star> g\<^sub>0h\<^sub>1.p\<^sub>1 \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine \<rightarrow> h \<star> tab\<^sub>0 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0\<guillemotright>
[PROOF STEP]
by (intro comp_in_homI) auto
[PROOF STATE]
proof (state)
this:
\<guillemotleft>\<theta>\<^sub>g' : tab\<^sub>0 g \<star> w\<^sub>g' \<rightarrow> u\<^sub>g\<guillemotright>
goal (2 subgoals):
1. \<lbrakk>\<lbrakk>g\<^sub>0h\<^sub>1.p\<^sub>1 \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine\<rbrakk>\<rbrakk> = \<lbrakk>\<lbrakk>f\<^sub>0g\<^sub>1.p\<^sub>0 \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1\<rbrakk>\<rbrakk>
2. \<lbrakk>\<lbrakk>g\<^sub>0h\<^sub>1.p\<^sub>0 \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine\<rbrakk>\<rbrakk> = \<lbrakk>\<lbrakk>fg\<^sub>0h\<^sub>1.p\<^sub>0\<rbrakk>\<rbrakk>
[PROOF STEP]
have w\<^sub>g_in_hhom: "in_hhom w\<^sub>g (src u\<^sub>g) (src (tab\<^sub>0 g))"
[PROOF STATE]
proof (prove)
goal (1 subgoal):
1. \<guillemotleft>w\<^sub>g : src u\<^sub>g \<rightarrow> src (tab\<^sub>0 g)\<guillemotright>
[PROOF STEP]
unfolding w\<^sub>g_def u\<^sub>g_def
[PROOF STATE]
proof (prove)
goal (1 subgoal):
1. \<guillemotleft>f\<^sub>0g\<^sub>1.p\<^sub>0 \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1 : src (h \<star> tab\<^sub>0 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0) \<rightarrow> src (tab\<^sub>0 g)\<guillemotright>
[PROOF STEP]
by auto
[PROOF STATE]
proof (state)
this:
\<guillemotleft>w\<^sub>g : src u\<^sub>g \<rightarrow> src (tab\<^sub>0 g)\<guillemotright>
goal (2 subgoals):
1. \<lbrakk>\<lbrakk>g\<^sub>0h\<^sub>1.p\<^sub>1 \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine\<rbrakk>\<rbrakk> = \<lbrakk>\<lbrakk>f\<^sub>0g\<^sub>1.p\<^sub>0 \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1\<rbrakk>\<rbrakk>
2. \<lbrakk>\<lbrakk>g\<^sub>0h\<^sub>1.p\<^sub>0 \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine\<rbrakk>\<rbrakk> = \<lbrakk>\<lbrakk>fg\<^sub>0h\<^sub>1.p\<^sub>0\<rbrakk>\<rbrakk>
[PROOF STEP]
have w\<^sub>g'_in_hhom: "in_hhom w\<^sub>g' (src u\<^sub>g) (src (tab\<^sub>0 g))"
[PROOF STATE]
proof (prove)
goal (1 subgoal):
1. \<guillemotleft>w\<^sub>g' : src u\<^sub>g \<rightarrow> src (tab\<^sub>0 g)\<guillemotright>
[PROOF STEP]
unfolding w\<^sub>g'_def u\<^sub>g_def
[PROOF STATE]
proof (prove)
goal (1 subgoal):
1. \<guillemotleft>g\<^sub>0h\<^sub>1.p\<^sub>1 \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine : src (h \<star> tab\<^sub>0 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0) \<rightarrow> src (tab\<^sub>0 g)\<guillemotright>
[PROOF STEP]
by auto
[PROOF STATE]
proof (state)
this:
\<guillemotleft>w\<^sub>g' : src u\<^sub>g \<rightarrow> src (tab\<^sub>0 g)\<guillemotright>
goal (2 subgoals):
1. \<lbrakk>\<lbrakk>g\<^sub>0h\<^sub>1.p\<^sub>1 \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine\<rbrakk>\<rbrakk> = \<lbrakk>\<lbrakk>f\<^sub>0g\<^sub>1.p\<^sub>0 \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1\<rbrakk>\<rbrakk>
2. \<lbrakk>\<lbrakk>g\<^sub>0h\<^sub>1.p\<^sub>0 \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine\<rbrakk>\<rbrakk> = \<lbrakk>\<lbrakk>fg\<^sub>0h\<^sub>1.p\<^sub>0\<rbrakk>\<rbrakk>
[PROOF STEP]
have \<beta>\<^sub>g: "\<guillemotleft>\<beta>\<^sub>g : tab\<^sub>1 g \<star> w\<^sub>g \<Rightarrow> tab\<^sub>1 g \<star> w\<^sub>g'\<guillemotright>"
[PROOF STATE]
proof (prove)
goal (1 subgoal):
1. \<guillemotleft>\<beta>\<^sub>g : tab\<^sub>1 g \<star> w\<^sub>g \<rightarrow> tab\<^sub>1 g \<star> w\<^sub>g'\<guillemotright>
[PROOF STEP]
proof (unfold \<beta>\<^sub>g_def w\<^sub>g_def, intro comp_in_homI)
[PROOF STATE]
proof (state)
goal (8 subgoals):
1. \<guillemotleft>\<a>\<^sup>-\<^sup>1[tab\<^sub>1 g, f\<^sub>0g\<^sub>1.p\<^sub>0, fg\<^sub>0h\<^sub>1.p\<^sub>1] : tab\<^sub>1 g \<star> f\<^sub>0g\<^sub>1.p\<^sub>0 \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1 \<rightarrow> ?b6\<guillemotright>
2. \<guillemotleft>local.inv f\<^sub>0g\<^sub>1.\<phi> \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1 : ?b6 \<rightarrow> ?b5\<guillemotright>
3. \<guillemotleft>\<a>[tab\<^sub>0 f, f\<^sub>0g\<^sub>1.p\<^sub>1, fg\<^sub>0h\<^sub>1.p\<^sub>1] : ?b5 \<rightarrow> ?b4\<guillemotright>
4. \<guillemotleft>tab\<^sub>0 f \<star> \<gamma>\<^sub>f : ?b4 \<rightarrow> ?b3\<guillemotright>
5. \<guillemotleft>\<a>\<^sup>-\<^sup>1[tab\<^sub>0 f, f\<^sub>0gh\<^sub>1.p\<^sub>1, TTfgh_TfTgh.chine] : ?b3 \<rightarrow> ?b2\<guillemotright>
6. \<guillemotleft>f\<^sub>0gh\<^sub>1.\<phi> \<star> TTfgh_TfTgh.chine : ?b2 \<rightarrow> ?b1\<guillemotright>
7. \<guillemotleft>\<a>[tab\<^sub>1 g \<star> g\<^sub>0h\<^sub>1.p\<^sub>1, f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine] : ?b1 \<rightarrow> ?b\<guillemotright>
8. \<guillemotleft>\<a>[tab\<^sub>1 g, g\<^sub>0h\<^sub>1.p\<^sub>1, f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine] : ?b \<rightarrow> tab\<^sub>1 g \<star> w\<^sub>g'\<guillemotright>
[PROOF STEP]
show "\<guillemotleft>\<a>\<^sup>-\<^sup>1[tab\<^sub>1 g, Tfg.p\<^sub>0, TTfgh.p\<^sub>1] :
tab\<^sub>1 g \<star> Tfg.p\<^sub>0 \<star> TTfgh.p\<^sub>1 \<Rightarrow> (tab\<^sub>1 g \<star> Tfg.p\<^sub>0) \<star> TTfgh.p\<^sub>1\<guillemotright>"
[PROOF STATE]
proof (prove)
goal (1 subgoal):
1. \<guillemotleft>\<a>\<^sup>-\<^sup>1[tab\<^sub>1 g, f\<^sub>0g\<^sub>1.p\<^sub>0, fg\<^sub>0h\<^sub>1.p\<^sub>1] : tab\<^sub>1 g \<star> f\<^sub>0g\<^sub>1.p\<^sub>0 \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1 \<rightarrow> (tab\<^sub>1 g \<star> f\<^sub>0g\<^sub>1.p\<^sub>0) \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1\<guillemotright>
[PROOF STEP]
using fg\<^sub>0h\<^sub>1.p\<^sub>1_simps
[PROOF STATE]
proof (prove)
using this:
trg fg\<^sub>0h\<^sub>1.p\<^sub>1 = src (tab\<^sub>0 g \<star> f\<^sub>0g\<^sub>1.p\<^sub>0)
goal (1 subgoal):
1. \<guillemotleft>\<a>\<^sup>-\<^sup>1[tab\<^sub>1 g, f\<^sub>0g\<^sub>1.p\<^sub>0, fg\<^sub>0h\<^sub>1.p\<^sub>1] : tab\<^sub>1 g \<star> f\<^sub>0g\<^sub>1.p\<^sub>0 \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1 \<rightarrow> (tab\<^sub>1 g \<star> f\<^sub>0g\<^sub>1.p\<^sub>0) \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1\<guillemotright>
[PROOF STEP]
by auto
[PROOF STATE]
proof (state)
this:
\<guillemotleft>\<a>\<^sup>-\<^sup>1[tab\<^sub>1 g, f\<^sub>0g\<^sub>1.p\<^sub>0, fg\<^sub>0h\<^sub>1.p\<^sub>1] : tab\<^sub>1 g \<star> f\<^sub>0g\<^sub>1.p\<^sub>0 \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1 \<rightarrow> (tab\<^sub>1 g \<star> f\<^sub>0g\<^sub>1.p\<^sub>0) \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1\<guillemotright>
goal (7 subgoals):
1. \<guillemotleft>local.inv f\<^sub>0g\<^sub>1.\<phi> \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1 : (tab\<^sub>1 g \<star> f\<^sub>0g\<^sub>1.p\<^sub>0) \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1 \<rightarrow> ?b5\<guillemotright>
2. \<guillemotleft>\<a>[tab\<^sub>0 f, f\<^sub>0g\<^sub>1.p\<^sub>1, fg\<^sub>0h\<^sub>1.p\<^sub>1] : ?b5 \<rightarrow> ?b4\<guillemotright>
3. \<guillemotleft>tab\<^sub>0 f \<star> \<gamma>\<^sub>f : ?b4 \<rightarrow> ?b3\<guillemotright>
4. \<guillemotleft>\<a>\<^sup>-\<^sup>1[tab\<^sub>0 f, f\<^sub>0gh\<^sub>1.p\<^sub>1, TTfgh_TfTgh.chine] : ?b3 \<rightarrow> ?b2\<guillemotright>
5. \<guillemotleft>f\<^sub>0gh\<^sub>1.\<phi> \<star> TTfgh_TfTgh.chine : ?b2 \<rightarrow> ?b1\<guillemotright>
6. \<guillemotleft>\<a>[tab\<^sub>1 g \<star> g\<^sub>0h\<^sub>1.p\<^sub>1, f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine] : ?b1 \<rightarrow> ?b\<guillemotright>
7. \<guillemotleft>\<a>[tab\<^sub>1 g, g\<^sub>0h\<^sub>1.p\<^sub>1, f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine] : ?b \<rightarrow> tab\<^sub>1 g \<star> w\<^sub>g'\<guillemotright>
[PROOF STEP]
show "\<guillemotleft>inv f\<^sub>0g\<^sub>1.\<phi> \<star> TTfgh.p\<^sub>1 :
(tab\<^sub>1 g \<star> Tfg.p\<^sub>0) \<star> TTfgh.p\<^sub>1 \<Rightarrow> (tab\<^sub>0 f \<star> Tfg.p\<^sub>1) \<star> TTfgh.p\<^sub>1\<guillemotright>"
[PROOF STATE]
proof (prove)
goal (1 subgoal):
1. \<guillemotleft>local.inv f\<^sub>0g\<^sub>1.\<phi> \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1 : (tab\<^sub>1 g \<star> f\<^sub>0g\<^sub>1.p\<^sub>0) \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1 \<rightarrow> (tab\<^sub>0 f \<star> f\<^sub>0g\<^sub>1.p\<^sub>1) \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1\<guillemotright>
[PROOF STEP]
using fg\<^sub>0h\<^sub>1.p\<^sub>1_simps f\<^sub>0g\<^sub>1.\<phi>_in_hom f\<^sub>0g\<^sub>1.\<phi>_uniqueness(2)
[PROOF STATE]
proof (prove)
using this:
trg fg\<^sub>0h\<^sub>1.p\<^sub>1 = src (tab\<^sub>0 g \<star> f\<^sub>0g\<^sub>1.p\<^sub>0)
\<guillemotleft>f\<^sub>0g\<^sub>1.\<phi> : src f\<^sub>0g\<^sub>1.p\<^sub>0 \<rightarrow> trg (tab\<^sub>1 g)\<guillemotright>
\<guillemotleft>f\<^sub>0g\<^sub>1.\<phi> : tab\<^sub>0 f \<star> f\<^sub>0g\<^sub>1.p\<^sub>1 \<rightarrow> tab\<^sub>1 g \<star> f\<^sub>0g\<^sub>1.p\<^sub>0\<guillemotright>
local.iso f\<^sub>0g\<^sub>1.\<phi>
goal (1 subgoal):
1. \<guillemotleft>local.inv f\<^sub>0g\<^sub>1.\<phi> \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1 : (tab\<^sub>1 g \<star> f\<^sub>0g\<^sub>1.p\<^sub>0) \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1 \<rightarrow> (tab\<^sub>0 f \<star> f\<^sub>0g\<^sub>1.p\<^sub>1) \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1\<guillemotright>
[PROOF STEP]
by (intro hcomp_in_vhom) auto
[PROOF STATE]
proof (state)
this:
\<guillemotleft>local.inv f\<^sub>0g\<^sub>1.\<phi> \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1 : (tab\<^sub>1 g \<star> f\<^sub>0g\<^sub>1.p\<^sub>0) \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1 \<rightarrow> (tab\<^sub>0 f \<star> f\<^sub>0g\<^sub>1.p\<^sub>1) \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1\<guillemotright>
goal (6 subgoals):
1. \<guillemotleft>\<a>[tab\<^sub>0 f, f\<^sub>0g\<^sub>1.p\<^sub>1, fg\<^sub>0h\<^sub>1.p\<^sub>1] : (tab\<^sub>0 f \<star> f\<^sub>0g\<^sub>1.p\<^sub>1) \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1 \<rightarrow> ?b4\<guillemotright>
2. \<guillemotleft>tab\<^sub>0 f \<star> \<gamma>\<^sub>f : ?b4 \<rightarrow> ?b3\<guillemotright>
3. \<guillemotleft>\<a>\<^sup>-\<^sup>1[tab\<^sub>0 f, f\<^sub>0gh\<^sub>1.p\<^sub>1, TTfgh_TfTgh.chine] : ?b3 \<rightarrow> ?b2\<guillemotright>
4. \<guillemotleft>f\<^sub>0gh\<^sub>1.\<phi> \<star> TTfgh_TfTgh.chine : ?b2 \<rightarrow> ?b1\<guillemotright>
5. \<guillemotleft>\<a>[tab\<^sub>1 g \<star> g\<^sub>0h\<^sub>1.p\<^sub>1, f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine] : ?b1 \<rightarrow> ?b\<guillemotright>
6. \<guillemotleft>\<a>[tab\<^sub>1 g, g\<^sub>0h\<^sub>1.p\<^sub>1, f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine] : ?b \<rightarrow> tab\<^sub>1 g \<star> w\<^sub>g'\<guillemotright>
[PROOF STEP]
show "\<guillemotleft>\<a>[tab\<^sub>0 f, Tfg.p\<^sub>1, TTfgh.p\<^sub>1] :
(tab\<^sub>0 f \<star> Tfg.p\<^sub>1) \<star> TTfgh.p\<^sub>1 \<Rightarrow> tab\<^sub>0 f \<star> Tfg.p\<^sub>1 \<star> TTfgh.p\<^sub>1\<guillemotright>"
[PROOF STATE]
proof (prove)
goal (1 subgoal):
1. \<guillemotleft>\<a>[tab\<^sub>0 f, f\<^sub>0g\<^sub>1.p\<^sub>1, fg\<^sub>0h\<^sub>1.p\<^sub>1] : (tab\<^sub>0 f \<star> f\<^sub>0g\<^sub>1.p\<^sub>1) \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1 \<rightarrow> tab\<^sub>0 f \<star> f\<^sub>0g\<^sub>1.p\<^sub>1 \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1\<guillemotright>
[PROOF STEP]
using fg\<^sub>0h\<^sub>1.p\<^sub>1_simps \<gamma>\<^sub>f w\<^sub>f_def w\<^sub>f'_def
[PROOF STATE]
proof (prove)
using this:
trg fg\<^sub>0h\<^sub>1.p\<^sub>1 = src (tab\<^sub>0 g \<star> f\<^sub>0g\<^sub>1.p\<^sub>0)
\<guillemotleft>\<gamma>\<^sub>f : w\<^sub>f \<rightarrow> w\<^sub>f'\<guillemotright> \<and> \<beta>\<^sub>f = tab\<^sub>1 f \<star> \<gamma>\<^sub>f \<and> \<theta>\<^sub>f = \<theta>\<^sub>f' \<cdot> (tab\<^sub>0 f \<star> \<gamma>\<^sub>f)
w\<^sub>f = f\<^sub>0g\<^sub>1.p\<^sub>1 \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1
w\<^sub>f' = f\<^sub>0gh\<^sub>1.p\<^sub>1 \<star> TTfgh_TfTgh.chine
goal (1 subgoal):
1. \<guillemotleft>\<a>[tab\<^sub>0 f, f\<^sub>0g\<^sub>1.p\<^sub>1, fg\<^sub>0h\<^sub>1.p\<^sub>1] : (tab\<^sub>0 f \<star> f\<^sub>0g\<^sub>1.p\<^sub>1) \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1 \<rightarrow> tab\<^sub>0 f \<star> f\<^sub>0g\<^sub>1.p\<^sub>1 \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1\<guillemotright>
[PROOF STEP]
by auto
[PROOF STATE]
proof (state)
this:
\<guillemotleft>\<a>[tab\<^sub>0 f, f\<^sub>0g\<^sub>1.p\<^sub>1, fg\<^sub>0h\<^sub>1.p\<^sub>1] : (tab\<^sub>0 f \<star> f\<^sub>0g\<^sub>1.p\<^sub>1) \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1 \<rightarrow> tab\<^sub>0 f \<star> f\<^sub>0g\<^sub>1.p\<^sub>1 \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1\<guillemotright>
goal (5 subgoals):
1. \<guillemotleft>tab\<^sub>0 f \<star> \<gamma>\<^sub>f : tab\<^sub>0 f \<star> f\<^sub>0g\<^sub>1.p\<^sub>1 \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1 \<rightarrow> ?b3\<guillemotright>
2. \<guillemotleft>\<a>\<^sup>-\<^sup>1[tab\<^sub>0 f, f\<^sub>0gh\<^sub>1.p\<^sub>1, TTfgh_TfTgh.chine] : ?b3 \<rightarrow> ?b2\<guillemotright>
3. \<guillemotleft>f\<^sub>0gh\<^sub>1.\<phi> \<star> TTfgh_TfTgh.chine : ?b2 \<rightarrow> ?b1\<guillemotright>
4. \<guillemotleft>\<a>[tab\<^sub>1 g \<star> g\<^sub>0h\<^sub>1.p\<^sub>1, f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine] : ?b1 \<rightarrow> ?b\<guillemotright>
5. \<guillemotleft>\<a>[tab\<^sub>1 g, g\<^sub>0h\<^sub>1.p\<^sub>1, f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine] : ?b \<rightarrow> tab\<^sub>1 g \<star> w\<^sub>g'\<guillemotright>
[PROOF STEP]
show "\<guillemotleft>tab\<^sub>0 f \<star> \<gamma>\<^sub>f : tab\<^sub>0 f \<star> Tfg.p\<^sub>1 \<star> TTfgh.p\<^sub>1 \<Rightarrow> tab\<^sub>0 f \<star> TfTgh.p\<^sub>1 \<star> TTfgh_TfTgh.chine\<guillemotright>"
[PROOF STATE]
proof (prove)
goal (1 subgoal):
1. \<guillemotleft>tab\<^sub>0 f \<star> \<gamma>\<^sub>f : tab\<^sub>0 f \<star> f\<^sub>0g\<^sub>1.p\<^sub>1 \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1 \<rightarrow> tab\<^sub>0 f \<star> f\<^sub>0gh\<^sub>1.p\<^sub>1 \<star> TTfgh_TfTgh.chine\<guillemotright>
[PROOF STEP]
using fg\<^sub>0h\<^sub>1.p\<^sub>1_simps \<gamma>\<^sub>f w\<^sub>f_def w\<^sub>f'_def
[PROOF STATE]
proof (prove)
using this:
trg fg\<^sub>0h\<^sub>1.p\<^sub>1 = src (tab\<^sub>0 g \<star> f\<^sub>0g\<^sub>1.p\<^sub>0)
\<guillemotleft>\<gamma>\<^sub>f : w\<^sub>f \<rightarrow> w\<^sub>f'\<guillemotright> \<and> \<beta>\<^sub>f = tab\<^sub>1 f \<star> \<gamma>\<^sub>f \<and> \<theta>\<^sub>f = \<theta>\<^sub>f' \<cdot> (tab\<^sub>0 f \<star> \<gamma>\<^sub>f)
w\<^sub>f = f\<^sub>0g\<^sub>1.p\<^sub>1 \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1
w\<^sub>f' = f\<^sub>0gh\<^sub>1.p\<^sub>1 \<star> TTfgh_TfTgh.chine
goal (1 subgoal):
1. \<guillemotleft>tab\<^sub>0 f \<star> \<gamma>\<^sub>f : tab\<^sub>0 f \<star> f\<^sub>0g\<^sub>1.p\<^sub>1 \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1 \<rightarrow> tab\<^sub>0 f \<star> f\<^sub>0gh\<^sub>1.p\<^sub>1 \<star> TTfgh_TfTgh.chine\<guillemotright>
[PROOF STEP]
by auto
[PROOF STATE]
proof (state)
this:
\<guillemotleft>tab\<^sub>0 f \<star> \<gamma>\<^sub>f : tab\<^sub>0 f \<star> f\<^sub>0g\<^sub>1.p\<^sub>1 \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1 \<rightarrow> tab\<^sub>0 f \<star> f\<^sub>0gh\<^sub>1.p\<^sub>1 \<star> TTfgh_TfTgh.chine\<guillemotright>
goal (4 subgoals):
1. \<guillemotleft>\<a>\<^sup>-\<^sup>1[tab\<^sub>0 f, f\<^sub>0gh\<^sub>1.p\<^sub>1, TTfgh_TfTgh.chine] : tab\<^sub>0 f \<star> f\<^sub>0gh\<^sub>1.p\<^sub>1 \<star> TTfgh_TfTgh.chine \<rightarrow> ?b2\<guillemotright>
2. \<guillemotleft>f\<^sub>0gh\<^sub>1.\<phi> \<star> TTfgh_TfTgh.chine : ?b2 \<rightarrow> ?b1\<guillemotright>
3. \<guillemotleft>\<a>[tab\<^sub>1 g \<star> g\<^sub>0h\<^sub>1.p\<^sub>1, f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine] : ?b1 \<rightarrow> ?b\<guillemotright>
4. \<guillemotleft>\<a>[tab\<^sub>1 g, g\<^sub>0h\<^sub>1.p\<^sub>1, f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine] : ?b \<rightarrow> tab\<^sub>1 g \<star> w\<^sub>g'\<guillemotright>
[PROOF STEP]
show "\<guillemotleft>\<a>\<^sup>-\<^sup>1[tab\<^sub>0 f, TfTgh.p\<^sub>1, TTfgh_TfTgh.chine] :
tab\<^sub>0 f \<star> TfTgh.p\<^sub>1 \<star> TTfgh_TfTgh.chine \<Rightarrow> (tab\<^sub>0 f \<star> TfTgh.p\<^sub>1) \<star> TTfgh_TfTgh.chine\<guillemotright>"
[PROOF STATE]
proof (prove)
goal (1 subgoal):
1. \<guillemotleft>\<a>\<^sup>-\<^sup>1[tab\<^sub>0 f, f\<^sub>0gh\<^sub>1.p\<^sub>1, TTfgh_TfTgh.chine] : tab\<^sub>0 f \<star> f\<^sub>0gh\<^sub>1.p\<^sub>1 \<star> TTfgh_TfTgh.chine \<rightarrow> (tab\<^sub>0 f \<star> f\<^sub>0gh\<^sub>1.p\<^sub>1) \<star> TTfgh_TfTgh.chine\<guillemotright>
[PROOF STEP]
by auto
[PROOF STATE]
proof (state)
this:
\<guillemotleft>\<a>\<^sup>-\<^sup>1[tab\<^sub>0 f, f\<^sub>0gh\<^sub>1.p\<^sub>1, TTfgh_TfTgh.chine] : tab\<^sub>0 f \<star> f\<^sub>0gh\<^sub>1.p\<^sub>1 \<star> TTfgh_TfTgh.chine \<rightarrow> (tab\<^sub>0 f \<star> f\<^sub>0gh\<^sub>1.p\<^sub>1) \<star> TTfgh_TfTgh.chine\<guillemotright>
goal (3 subgoals):
1. \<guillemotleft>f\<^sub>0gh\<^sub>1.\<phi> \<star> TTfgh_TfTgh.chine : (tab\<^sub>0 f \<star> f\<^sub>0gh\<^sub>1.p\<^sub>1) \<star> TTfgh_TfTgh.chine \<rightarrow> ?b1\<guillemotright>
2. \<guillemotleft>\<a>[tab\<^sub>1 g \<star> g\<^sub>0h\<^sub>1.p\<^sub>1, f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine] : ?b1 \<rightarrow> ?b\<guillemotright>
3. \<guillemotleft>\<a>[tab\<^sub>1 g, g\<^sub>0h\<^sub>1.p\<^sub>1, f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine] : ?b \<rightarrow> tab\<^sub>1 g \<star> w\<^sub>g'\<guillemotright>
[PROOF STEP]
show "\<guillemotleft>f\<^sub>0gh\<^sub>1.\<phi> \<star> TTfgh_TfTgh.chine :
(tab\<^sub>0 f \<star> TfTgh.p\<^sub>1) \<star> TTfgh_TfTgh.chine
\<Rightarrow> ((tab\<^sub>1 g \<star> Tgh.p\<^sub>1) \<star> TfTgh.p\<^sub>0) \<star> TTfgh_TfTgh.chine\<guillemotright>"
[PROOF STATE]
proof (prove)
goal (1 subgoal):
1. \<guillemotleft>f\<^sub>0gh\<^sub>1.\<phi> \<star> TTfgh_TfTgh.chine : (tab\<^sub>0 f \<star> f\<^sub>0gh\<^sub>1.p\<^sub>1) \<star> TTfgh_TfTgh.chine \<rightarrow> ((tab\<^sub>1 g \<star> g\<^sub>0h\<^sub>1.p\<^sub>1) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine\<guillemotright>
[PROOF STEP]
using f\<^sub>0gh\<^sub>1.\<phi>_in_hom
[PROOF STATE]
proof (prove)
using this:
\<guillemotleft>f\<^sub>0gh\<^sub>1.\<phi> : src f\<^sub>0gh\<^sub>1.p\<^sub>0 \<rightarrow> trg (tab\<^sub>1 g \<star> g\<^sub>0h\<^sub>1.p\<^sub>1)\<guillemotright>
\<guillemotleft>f\<^sub>0gh\<^sub>1.\<phi> : tab\<^sub>0 f \<star> f\<^sub>0gh\<^sub>1.p\<^sub>1 \<rightarrow> (tab\<^sub>1 g \<star> g\<^sub>0h\<^sub>1.p\<^sub>1) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0\<guillemotright>
goal (1 subgoal):
1. \<guillemotleft>f\<^sub>0gh\<^sub>1.\<phi> \<star> TTfgh_TfTgh.chine : (tab\<^sub>0 f \<star> f\<^sub>0gh\<^sub>1.p\<^sub>1) \<star> TTfgh_TfTgh.chine \<rightarrow> ((tab\<^sub>1 g \<star> g\<^sub>0h\<^sub>1.p\<^sub>1) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine\<guillemotright>
[PROOF STEP]
by (intro hcomp_in_vhom) auto
[PROOF STATE]
proof (state)
this:
\<guillemotleft>f\<^sub>0gh\<^sub>1.\<phi> \<star> TTfgh_TfTgh.chine : (tab\<^sub>0 f \<star> f\<^sub>0gh\<^sub>1.p\<^sub>1) \<star> TTfgh_TfTgh.chine \<rightarrow> ((tab\<^sub>1 g \<star> g\<^sub>0h\<^sub>1.p\<^sub>1) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine\<guillemotright>
goal (2 subgoals):
1. \<guillemotleft>\<a>[tab\<^sub>1 g \<star> g\<^sub>0h\<^sub>1.p\<^sub>1, f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine] : ((tab\<^sub>1 g \<star> g\<^sub>0h\<^sub>1.p\<^sub>1) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine \<rightarrow> ?b\<guillemotright>
2. \<guillemotleft>\<a>[tab\<^sub>1 g, g\<^sub>0h\<^sub>1.p\<^sub>1, f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine] : ?b \<rightarrow> tab\<^sub>1 g \<star> w\<^sub>g'\<guillemotright>
[PROOF STEP]
show "\<guillemotleft>\<a>[tab\<^sub>1 g \<star> Tgh.p\<^sub>1, TfTgh.p\<^sub>0, TTfgh_TfTgh.chine] :
((tab\<^sub>1 g \<star> Tgh.p\<^sub>1) \<star> TfTgh.p\<^sub>0) \<star> TTfgh_TfTgh.chine
\<Rightarrow> (tab\<^sub>1 g \<star> Tgh.p\<^sub>1) \<star> TfTgh.p\<^sub>0 \<star> TTfgh_TfTgh.chine\<guillemotright>"
[PROOF STATE]
proof (prove)
goal (1 subgoal):
1. \<guillemotleft>\<a>[tab\<^sub>1 g \<star> g\<^sub>0h\<^sub>1.p\<^sub>1, f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine] : ((tab\<^sub>1 g \<star> g\<^sub>0h\<^sub>1.p\<^sub>1) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine \<rightarrow> (tab\<^sub>1 g \<star> g\<^sub>0h\<^sub>1.p\<^sub>1) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine\<guillemotright>
[PROOF STEP]
by auto
[PROOF STATE]
proof (state)
this:
\<guillemotleft>\<a>[tab\<^sub>1 g \<star> g\<^sub>0h\<^sub>1.p\<^sub>1, f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine] : ((tab\<^sub>1 g \<star> g\<^sub>0h\<^sub>1.p\<^sub>1) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine \<rightarrow> (tab\<^sub>1 g \<star> g\<^sub>0h\<^sub>1.p\<^sub>1) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine\<guillemotright>
goal (1 subgoal):
1. \<guillemotleft>\<a>[tab\<^sub>1 g, g\<^sub>0h\<^sub>1.p\<^sub>1, f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine] : (tab\<^sub>1 g \<star> g\<^sub>0h\<^sub>1.p\<^sub>1) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine \<rightarrow> tab\<^sub>1 g \<star> w\<^sub>g'\<guillemotright>
[PROOF STEP]
show "\<guillemotleft>\<a>[tab\<^sub>1 g, Tgh.p\<^sub>1, TfTgh.p\<^sub>0 \<star> TTfgh_TfTgh.chine] :
(tab\<^sub>1 g \<star> Tgh.p\<^sub>1) \<star> TfTgh.p\<^sub>0 \<star> TTfgh_TfTgh.chine \<Rightarrow> tab\<^sub>1 g \<star> w\<^sub>g'\<guillemotright>"
[PROOF STATE]
proof (prove)
goal (1 subgoal):
1. \<guillemotleft>\<a>[tab\<^sub>1 g, g\<^sub>0h\<^sub>1.p\<^sub>1, f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine] : (tab\<^sub>1 g \<star> g\<^sub>0h\<^sub>1.p\<^sub>1) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine \<rightarrow> tab\<^sub>1 g \<star> w\<^sub>g'\<guillemotright>
[PROOF STEP]
using w\<^sub>g'_def
[PROOF STATE]
proof (prove)
using this:
w\<^sub>g' = g\<^sub>0h\<^sub>1.p\<^sub>1 \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine
goal (1 subgoal):
1. \<guillemotleft>\<a>[tab\<^sub>1 g, g\<^sub>0h\<^sub>1.p\<^sub>1, f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine] : (tab\<^sub>1 g \<star> g\<^sub>0h\<^sub>1.p\<^sub>1) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine \<rightarrow> tab\<^sub>1 g \<star> w\<^sub>g'\<guillemotright>
[PROOF STEP]
by auto
[PROOF STATE]
proof (state)
this:
\<guillemotleft>\<a>[tab\<^sub>1 g, g\<^sub>0h\<^sub>1.p\<^sub>1, f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine] : (tab\<^sub>1 g \<star> g\<^sub>0h\<^sub>1.p\<^sub>1) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine \<rightarrow> tab\<^sub>1 g \<star> w\<^sub>g'\<guillemotright>
goal:
No subgoals!
[PROOF STEP]
qed
[PROOF STATE]
proof (state)
this:
\<guillemotleft>\<beta>\<^sub>g : tab\<^sub>1 g \<star> w\<^sub>g \<rightarrow> tab\<^sub>1 g \<star> w\<^sub>g'\<guillemotright>
goal (2 subgoals):
1. \<lbrakk>\<lbrakk>g\<^sub>0h\<^sub>1.p\<^sub>1 \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine\<rbrakk>\<rbrakk> = \<lbrakk>\<lbrakk>f\<^sub>0g\<^sub>1.p\<^sub>0 \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1\<rbrakk>\<rbrakk>
2. \<lbrakk>\<lbrakk>g\<^sub>0h\<^sub>1.p\<^sub>0 \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine\<rbrakk>\<rbrakk> = \<lbrakk>\<lbrakk>fg\<^sub>0h\<^sub>1.p\<^sub>0\<rbrakk>\<rbrakk>
[PROOF STEP]
have eq\<^sub>g: "g.composite_cell w\<^sub>g \<theta>\<^sub>g = g.composite_cell w\<^sub>g' \<theta>\<^sub>g' \<cdot> \<beta>\<^sub>g"
[PROOF STATE]
proof (prove)
goal (1 subgoal):
1. (g \<star> \<theta>\<^sub>g) \<cdot> \<a>[g, tab\<^sub>0 g, w\<^sub>g] \<cdot> (g.tab \<star> w\<^sub>g) = ((g \<star> \<theta>\<^sub>g') \<cdot> \<a>[g, tab\<^sub>0 g, w\<^sub>g'] \<cdot> (g.tab \<star> w\<^sub>g')) \<cdot> \<beta>\<^sub>g
[PROOF STEP]
proof -
[PROOF STATE]
proof (state)
goal (1 subgoal):
1. (g \<star> \<theta>\<^sub>g) \<cdot> \<a>[g, tab\<^sub>0 g, w\<^sub>g] \<cdot> (g.tab \<star> w\<^sub>g) = ((g \<star> \<theta>\<^sub>g') \<cdot> \<a>[g, tab\<^sub>0 g, w\<^sub>g'] \<cdot> (g.tab \<star> w\<^sub>g')) \<cdot> \<beta>\<^sub>g
[PROOF STEP]
have "g.composite_cell w\<^sub>g \<theta>\<^sub>g =
(g \<star> \<a>[h, tab\<^sub>0 h, TTfgh.p\<^sub>0] \<cdot>
(h.tab \<star> TTfgh.p\<^sub>0) \<cdot>
fg\<^sub>0h\<^sub>1.\<phi> \<cdot>
\<a>\<^sup>-\<^sup>1[tab\<^sub>0 g, Tfg.p\<^sub>0, TTfgh.p\<^sub>1]) \<cdot>
\<a>[g, tab\<^sub>0 g, Tfg.p\<^sub>0 \<star> TTfgh.p\<^sub>1] \<cdot>
(g.tab \<star> Tfg.p\<^sub>0 \<star> TTfgh.p\<^sub>1)"
[PROOF STATE]
proof (prove)
goal (1 subgoal):
1. (g \<star> \<theta>\<^sub>g) \<cdot> \<a>[g, tab\<^sub>0 g, w\<^sub>g] \<cdot> (g.tab \<star> w\<^sub>g) = (g \<star> \<a>[h, tab\<^sub>0 h, fg\<^sub>0h\<^sub>1.p\<^sub>0] \<cdot> (h.tab \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0) \<cdot> fg\<^sub>0h\<^sub>1.\<phi> \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>0 g, f\<^sub>0g\<^sub>1.p\<^sub>0, fg\<^sub>0h\<^sub>1.p\<^sub>1]) \<cdot> \<a>[g, tab\<^sub>0 g, f\<^sub>0g\<^sub>1.p\<^sub>0 \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1] \<cdot> (g.tab \<star> f\<^sub>0g\<^sub>1.p\<^sub>0 \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1)
[PROOF STEP]
unfolding w\<^sub>g_def \<theta>\<^sub>g_def
[PROOF STATE]
proof (prove)
goal (1 subgoal):
1. (g \<star> \<a>[h, tab\<^sub>0 h, fg\<^sub>0h\<^sub>1.p\<^sub>0] \<cdot> (h.tab \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0) \<cdot> fg\<^sub>0h\<^sub>1.\<phi> \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>0 g, f\<^sub>0g\<^sub>1.p\<^sub>0, fg\<^sub>0h\<^sub>1.p\<^sub>1]) \<cdot> \<a>[g, tab\<^sub>0 g, f\<^sub>0g\<^sub>1.p\<^sub>0 \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1] \<cdot> (g.tab \<star> f\<^sub>0g\<^sub>1.p\<^sub>0 \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) = (g \<star> \<a>[h, tab\<^sub>0 h, fg\<^sub>0h\<^sub>1.p\<^sub>0] \<cdot> (h.tab \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0) \<cdot> fg\<^sub>0h\<^sub>1.\<phi> \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>0 g, f\<^sub>0g\<^sub>1.p\<^sub>0, fg\<^sub>0h\<^sub>1.p\<^sub>1]) \<cdot> \<a>[g, tab\<^sub>0 g, f\<^sub>0g\<^sub>1.p\<^sub>0 \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1] \<cdot> (g.tab \<star> f\<^sub>0g\<^sub>1.p\<^sub>0 \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1)
[PROOF STEP]
by simp
[PROOF STATE]
proof (state)
this:
(g \<star> \<theta>\<^sub>g) \<cdot> \<a>[g, tab\<^sub>0 g, w\<^sub>g] \<cdot> (g.tab \<star> w\<^sub>g) = (g \<star> \<a>[h, tab\<^sub>0 h, fg\<^sub>0h\<^sub>1.p\<^sub>0] \<cdot> (h.tab \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0) \<cdot> fg\<^sub>0h\<^sub>1.\<phi> \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>0 g, f\<^sub>0g\<^sub>1.p\<^sub>0, fg\<^sub>0h\<^sub>1.p\<^sub>1]) \<cdot> \<a>[g, tab\<^sub>0 g, f\<^sub>0g\<^sub>1.p\<^sub>0 \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1] \<cdot> (g.tab \<star> f\<^sub>0g\<^sub>1.p\<^sub>0 \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1)
goal (1 subgoal):
1. (g \<star> \<theta>\<^sub>g) \<cdot> \<a>[g, tab\<^sub>0 g, w\<^sub>g] \<cdot> (g.tab \<star> w\<^sub>g) = ((g \<star> \<theta>\<^sub>g') \<cdot> \<a>[g, tab\<^sub>0 g, w\<^sub>g'] \<cdot> (g.tab \<star> w\<^sub>g')) \<cdot> \<beta>\<^sub>g
[PROOF STEP]
also
[PROOF STATE]
proof (state)
this:
(g \<star> \<theta>\<^sub>g) \<cdot> \<a>[g, tab\<^sub>0 g, w\<^sub>g] \<cdot> (g.tab \<star> w\<^sub>g) = (g \<star> \<a>[h, tab\<^sub>0 h, fg\<^sub>0h\<^sub>1.p\<^sub>0] \<cdot> (h.tab \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0) \<cdot> fg\<^sub>0h\<^sub>1.\<phi> \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>0 g, f\<^sub>0g\<^sub>1.p\<^sub>0, fg\<^sub>0h\<^sub>1.p\<^sub>1]) \<cdot> \<a>[g, tab\<^sub>0 g, f\<^sub>0g\<^sub>1.p\<^sub>0 \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1] \<cdot> (g.tab \<star> f\<^sub>0g\<^sub>1.p\<^sub>0 \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1)
goal (1 subgoal):
1. (g \<star> \<theta>\<^sub>g) \<cdot> \<a>[g, tab\<^sub>0 g, w\<^sub>g] \<cdot> (g.tab \<star> w\<^sub>g) = ((g \<star> \<theta>\<^sub>g') \<cdot> \<a>[g, tab\<^sub>0 g, w\<^sub>g'] \<cdot> (g.tab \<star> w\<^sub>g')) \<cdot> \<beta>\<^sub>g
[PROOF STEP]
have "... =
(g \<star> \<a>[h, tab\<^sub>0 h, TTfgh.p\<^sub>0]) \<cdot>
(g \<star> h.tab \<star> TTfgh.p\<^sub>0) \<cdot>
(g \<star> fg\<^sub>0h\<^sub>1.\<phi>) \<cdot>
((g \<star> \<a>\<^sup>-\<^sup>1[tab\<^sub>0 g, Tfg.p\<^sub>0, TTfgh.p\<^sub>1]) \<cdot>
\<a>[g, tab\<^sub>0 g, Tfg.p\<^sub>0 \<star> TTfgh.p\<^sub>1]) \<cdot>
(g.tab \<star> Tfg.p\<^sub>0 \<star> TTfgh.p\<^sub>1)"
[PROOF STATE]
proof (prove)
goal (1 subgoal):
1. (g \<star> \<a>[h, tab\<^sub>0 h, fg\<^sub>0h\<^sub>1.p\<^sub>0] \<cdot> (h.tab \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0) \<cdot> fg\<^sub>0h\<^sub>1.\<phi> \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>0 g, f\<^sub>0g\<^sub>1.p\<^sub>0, fg\<^sub>0h\<^sub>1.p\<^sub>1]) \<cdot> \<a>[g, tab\<^sub>0 g, f\<^sub>0g\<^sub>1.p\<^sub>0 \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1] \<cdot> (g.tab \<star> f\<^sub>0g\<^sub>1.p\<^sub>0 \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) = (g \<star> \<a>[h, tab\<^sub>0 h, fg\<^sub>0h\<^sub>1.p\<^sub>0]) \<cdot> (g \<star> h.tab \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0) \<cdot> (g \<star> fg\<^sub>0h\<^sub>1.\<phi>) \<cdot> ((g \<star> \<a>\<^sup>-\<^sup>1[tab\<^sub>0 g, f\<^sub>0g\<^sub>1.p\<^sub>0, fg\<^sub>0h\<^sub>1.p\<^sub>1]) \<cdot> \<a>[g, tab\<^sub>0 g, f\<^sub>0g\<^sub>1.p\<^sub>0 \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1]) \<cdot> (g.tab \<star> f\<^sub>0g\<^sub>1.p\<^sub>0 \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1)
[PROOF STEP]
using fg gh f\<^sub>0g\<^sub>1.p\<^sub>0_simps fg\<^sub>0h\<^sub>1.p\<^sub>0_simps fg\<^sub>0h\<^sub>1.p\<^sub>1_simps whisker_left
comp_assoc
[PROOF STATE]
proof (prove)
using this:
src f = trg g
src g = trg h
trg f\<^sub>0g\<^sub>1.p\<^sub>0 = src (tab\<^sub>1 g)
trg fg\<^sub>0h\<^sub>1.p\<^sub>0 = src (tab\<^sub>1 h)
trg fg\<^sub>0h\<^sub>1.p\<^sub>1 = src (tab\<^sub>0 g \<star> f\<^sub>0g\<^sub>1.p\<^sub>0)
\<lbrakk>ide ?f; seq ?\<nu> ?\<mu>\<rbrakk> \<Longrightarrow> ?f \<star> ?\<nu> \<cdot> ?\<mu> = (?f \<star> ?\<nu>) \<cdot> (?f \<star> ?\<mu>)
(?h \<cdot> ?g) \<cdot> ?f = ?h \<cdot> ?g \<cdot> ?f
goal (1 subgoal):
1. (g \<star> \<a>[h, tab\<^sub>0 h, fg\<^sub>0h\<^sub>1.p\<^sub>0] \<cdot> (h.tab \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0) \<cdot> fg\<^sub>0h\<^sub>1.\<phi> \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>0 g, f\<^sub>0g\<^sub>1.p\<^sub>0, fg\<^sub>0h\<^sub>1.p\<^sub>1]) \<cdot> \<a>[g, tab\<^sub>0 g, f\<^sub>0g\<^sub>1.p\<^sub>0 \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1] \<cdot> (g.tab \<star> f\<^sub>0g\<^sub>1.p\<^sub>0 \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) = (g \<star> \<a>[h, tab\<^sub>0 h, fg\<^sub>0h\<^sub>1.p\<^sub>0]) \<cdot> (g \<star> h.tab \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0) \<cdot> (g \<star> fg\<^sub>0h\<^sub>1.\<phi>) \<cdot> ((g \<star> \<a>\<^sup>-\<^sup>1[tab\<^sub>0 g, f\<^sub>0g\<^sub>1.p\<^sub>0, fg\<^sub>0h\<^sub>1.p\<^sub>1]) \<cdot> \<a>[g, tab\<^sub>0 g, f\<^sub>0g\<^sub>1.p\<^sub>0 \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1]) \<cdot> (g.tab \<star> f\<^sub>0g\<^sub>1.p\<^sub>0 \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1)
[PROOF STEP]
by simp
[PROOF STATE]
proof (state)
this:
(g \<star> \<a>[h, tab\<^sub>0 h, fg\<^sub>0h\<^sub>1.p\<^sub>0] \<cdot> (h.tab \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0) \<cdot> fg\<^sub>0h\<^sub>1.\<phi> \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>0 g, f\<^sub>0g\<^sub>1.p\<^sub>0, fg\<^sub>0h\<^sub>1.p\<^sub>1]) \<cdot> \<a>[g, tab\<^sub>0 g, f\<^sub>0g\<^sub>1.p\<^sub>0 \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1] \<cdot> (g.tab \<star> f\<^sub>0g\<^sub>1.p\<^sub>0 \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) = (g \<star> \<a>[h, tab\<^sub>0 h, fg\<^sub>0h\<^sub>1.p\<^sub>0]) \<cdot> (g \<star> h.tab \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0) \<cdot> (g \<star> fg\<^sub>0h\<^sub>1.\<phi>) \<cdot> ((g \<star> \<a>\<^sup>-\<^sup>1[tab\<^sub>0 g, f\<^sub>0g\<^sub>1.p\<^sub>0, fg\<^sub>0h\<^sub>1.p\<^sub>1]) \<cdot> \<a>[g, tab\<^sub>0 g, f\<^sub>0g\<^sub>1.p\<^sub>0 \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1]) \<cdot> (g.tab \<star> f\<^sub>0g\<^sub>1.p\<^sub>0 \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1)
goal (1 subgoal):
1. (g \<star> \<theta>\<^sub>g) \<cdot> \<a>[g, tab\<^sub>0 g, w\<^sub>g] \<cdot> (g.tab \<star> w\<^sub>g) = ((g \<star> \<theta>\<^sub>g') \<cdot> \<a>[g, tab\<^sub>0 g, w\<^sub>g'] \<cdot> (g.tab \<star> w\<^sub>g')) \<cdot> \<beta>\<^sub>g
[PROOF STEP]
also
[PROOF STATE]
proof (state)
this:
(g \<star> \<a>[h, tab\<^sub>0 h, fg\<^sub>0h\<^sub>1.p\<^sub>0] \<cdot> (h.tab \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0) \<cdot> fg\<^sub>0h\<^sub>1.\<phi> \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>0 g, f\<^sub>0g\<^sub>1.p\<^sub>0, fg\<^sub>0h\<^sub>1.p\<^sub>1]) \<cdot> \<a>[g, tab\<^sub>0 g, f\<^sub>0g\<^sub>1.p\<^sub>0 \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1] \<cdot> (g.tab \<star> f\<^sub>0g\<^sub>1.p\<^sub>0 \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) = (g \<star> \<a>[h, tab\<^sub>0 h, fg\<^sub>0h\<^sub>1.p\<^sub>0]) \<cdot> (g \<star> h.tab \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0) \<cdot> (g \<star> fg\<^sub>0h\<^sub>1.\<phi>) \<cdot> ((g \<star> \<a>\<^sup>-\<^sup>1[tab\<^sub>0 g, f\<^sub>0g\<^sub>1.p\<^sub>0, fg\<^sub>0h\<^sub>1.p\<^sub>1]) \<cdot> \<a>[g, tab\<^sub>0 g, f\<^sub>0g\<^sub>1.p\<^sub>0 \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1]) \<cdot> (g.tab \<star> f\<^sub>0g\<^sub>1.p\<^sub>0 \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1)
goal (1 subgoal):
1. (g \<star> \<theta>\<^sub>g) \<cdot> \<a>[g, tab\<^sub>0 g, w\<^sub>g] \<cdot> (g.tab \<star> w\<^sub>g) = ((g \<star> \<theta>\<^sub>g') \<cdot> \<a>[g, tab\<^sub>0 g, w\<^sub>g'] \<cdot> (g.tab \<star> w\<^sub>g')) \<cdot> \<beta>\<^sub>g
[PROOF STEP]
have "... =
(g \<star> \<a>[h, tab\<^sub>0 h, TTfgh.p\<^sub>0]) \<cdot>
(g \<star> h.tab \<star> TTfgh.p\<^sub>0) \<cdot>
(g \<star> fg\<^sub>0h\<^sub>1.\<phi>) \<cdot>
(\<a>[g, tab\<^sub>0 g \<star> Tfg.p\<^sub>0, TTfgh.p\<^sub>1] \<cdot>
(\<a>\<^sup>-\<^sup>1[g, tab\<^sub>0 g \<star> Tfg.p\<^sub>0, TTfgh.p\<^sub>1] \<cdot>
(g \<star> \<a>\<^sup>-\<^sup>1[tab\<^sub>0 g, Tfg.p\<^sub>0, TTfgh.p\<^sub>1]))) \<cdot>
\<a>[g, tab\<^sub>0 g, Tfg.p\<^sub>0 \<star> TTfgh.p\<^sub>1] \<cdot>
(g.tab \<star> Tfg.p\<^sub>0 \<star> TTfgh.p\<^sub>1)"
[PROOF STATE]
proof (prove)
goal (1 subgoal):
1. (g \<star> \<a>[h, tab\<^sub>0 h, fg\<^sub>0h\<^sub>1.p\<^sub>0]) \<cdot> (g \<star> h.tab \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0) \<cdot> (g \<star> fg\<^sub>0h\<^sub>1.\<phi>) \<cdot> ((g \<star> \<a>\<^sup>-\<^sup>1[tab\<^sub>0 g, f\<^sub>0g\<^sub>1.p\<^sub>0, fg\<^sub>0h\<^sub>1.p\<^sub>1]) \<cdot> \<a>[g, tab\<^sub>0 g, f\<^sub>0g\<^sub>1.p\<^sub>0 \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1]) \<cdot> (g.tab \<star> f\<^sub>0g\<^sub>1.p\<^sub>0 \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) = (g \<star> \<a>[h, tab\<^sub>0 h, fg\<^sub>0h\<^sub>1.p\<^sub>0]) \<cdot> (g \<star> h.tab \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0) \<cdot> (g \<star> fg\<^sub>0h\<^sub>1.\<phi>) \<cdot> (\<a>[g, tab\<^sub>0 g \<star> f\<^sub>0g\<^sub>1.p\<^sub>0, fg\<^sub>0h\<^sub>1.p\<^sub>1] \<cdot> \<a>\<^sup>-\<^sup>1[g, tab\<^sub>0 g \<star> f\<^sub>0g\<^sub>1.p\<^sub>0, fg\<^sub>0h\<^sub>1.p\<^sub>1] \<cdot> (g \<star> \<a>\<^sup>-\<^sup>1[tab\<^sub>0 g, f\<^sub>0g\<^sub>1.p\<^sub>0, fg\<^sub>0h\<^sub>1.p\<^sub>1])) \<cdot> \<a>[g, tab\<^sub>0 g, f\<^sub>0g\<^sub>1.p\<^sub>0 \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1] \<cdot> (g.tab \<star> f\<^sub>0g\<^sub>1.p\<^sub>0 \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1)
[PROOF STEP]
proof -
[PROOF STATE]
proof (state)
goal (1 subgoal):
1. (g \<star> \<a>[h, tab\<^sub>0 h, fg\<^sub>0h\<^sub>1.p\<^sub>0]) \<cdot> (g \<star> h.tab \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0) \<cdot> (g \<star> fg\<^sub>0h\<^sub>1.\<phi>) \<cdot> ((g \<star> \<a>\<^sup>-\<^sup>1[tab\<^sub>0 g, f\<^sub>0g\<^sub>1.p\<^sub>0, fg\<^sub>0h\<^sub>1.p\<^sub>1]) \<cdot> \<a>[g, tab\<^sub>0 g, f\<^sub>0g\<^sub>1.p\<^sub>0 \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1]) \<cdot> (g.tab \<star> f\<^sub>0g\<^sub>1.p\<^sub>0 \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) = (g \<star> \<a>[h, tab\<^sub>0 h, fg\<^sub>0h\<^sub>1.p\<^sub>0]) \<cdot> (g \<star> h.tab \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0) \<cdot> (g \<star> fg\<^sub>0h\<^sub>1.\<phi>) \<cdot> (\<a>[g, tab\<^sub>0 g \<star> f\<^sub>0g\<^sub>1.p\<^sub>0, fg\<^sub>0h\<^sub>1.p\<^sub>1] \<cdot> \<a>\<^sup>-\<^sup>1[g, tab\<^sub>0 g \<star> f\<^sub>0g\<^sub>1.p\<^sub>0, fg\<^sub>0h\<^sub>1.p\<^sub>1] \<cdot> (g \<star> \<a>\<^sup>-\<^sup>1[tab\<^sub>0 g, f\<^sub>0g\<^sub>1.p\<^sub>0, fg\<^sub>0h\<^sub>1.p\<^sub>1])) \<cdot> \<a>[g, tab\<^sub>0 g, f\<^sub>0g\<^sub>1.p\<^sub>0 \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1] \<cdot> (g.tab \<star> f\<^sub>0g\<^sub>1.p\<^sub>0 \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1)
[PROOF STEP]
have "(\<a>[g, tab\<^sub>0 g \<star> Tfg.p\<^sub>0, TTfgh.p\<^sub>1] \<cdot>
\<a>\<^sup>-\<^sup>1[g, tab\<^sub>0 g \<star> Tfg.p\<^sub>0, TTfgh.p\<^sub>1]) \<cdot>
(g \<star> \<a>\<^sup>-\<^sup>1[tab\<^sub>0 g, Tfg.p\<^sub>0, TTfgh.p\<^sub>1]) =
g \<star> \<a>\<^sup>-\<^sup>1[tab\<^sub>0 g, Tfg.p\<^sub>0, TTfgh.p\<^sub>1]"
[PROOF STATE]
proof (prove)
goal (1 subgoal):
1. (\<a>[g, tab\<^sub>0 g \<star> f\<^sub>0g\<^sub>1.p\<^sub>0, fg\<^sub>0h\<^sub>1.p\<^sub>1] \<cdot> \<a>\<^sup>-\<^sup>1[g, tab\<^sub>0 g \<star> f\<^sub>0g\<^sub>1.p\<^sub>0, fg\<^sub>0h\<^sub>1.p\<^sub>1]) \<cdot> (g \<star> \<a>\<^sup>-\<^sup>1[tab\<^sub>0 g, f\<^sub>0g\<^sub>1.p\<^sub>0, fg\<^sub>0h\<^sub>1.p\<^sub>1]) = g \<star> \<a>\<^sup>-\<^sup>1[tab\<^sub>0 g, f\<^sub>0g\<^sub>1.p\<^sub>0, fg\<^sub>0h\<^sub>1.p\<^sub>1]
[PROOF STEP]
using fg\<^sub>0h\<^sub>1.p\<^sub>1_simps comp_cod_arr comp_assoc_assoc'
[PROOF STATE]
proof (prove)
using this:
trg fg\<^sub>0h\<^sub>1.p\<^sub>1 = src (tab\<^sub>0 g \<star> f\<^sub>0g\<^sub>1.p\<^sub>0)
\<lbrakk>arr ?f; cod ?f = ?b\<rbrakk> \<Longrightarrow> ?b \<cdot> ?f = ?f
\<lbrakk>ide ?f; ide ?g; ide ?h; src ?f = trg ?g; src ?g = trg ?h\<rbrakk> \<Longrightarrow> \<a>[?f, ?g, ?h] \<cdot> \<a>\<^sup>-\<^sup>1[?f, ?g, ?h] = ?f \<star> ?g \<star> ?h
\<lbrakk>ide ?f; ide ?g; ide ?h; src ?f = trg ?g; src ?g = trg ?h\<rbrakk> \<Longrightarrow> \<a>\<^sup>-\<^sup>1[?f, ?g, ?h] \<cdot> \<a>[?f, ?g, ?h] = (?f \<star> ?g) \<star> ?h
goal (1 subgoal):
1. (\<a>[g, tab\<^sub>0 g \<star> f\<^sub>0g\<^sub>1.p\<^sub>0, fg\<^sub>0h\<^sub>1.p\<^sub>1] \<cdot> \<a>\<^sup>-\<^sup>1[g, tab\<^sub>0 g \<star> f\<^sub>0g\<^sub>1.p\<^sub>0, fg\<^sub>0h\<^sub>1.p\<^sub>1]) \<cdot> (g \<star> \<a>\<^sup>-\<^sup>1[tab\<^sub>0 g, f\<^sub>0g\<^sub>1.p\<^sub>0, fg\<^sub>0h\<^sub>1.p\<^sub>1]) = g \<star> \<a>\<^sup>-\<^sup>1[tab\<^sub>0 g, f\<^sub>0g\<^sub>1.p\<^sub>0, fg\<^sub>0h\<^sub>1.p\<^sub>1]
[PROOF STEP]
by simp
[PROOF STATE]
proof (state)
this:
(\<a>[g, tab\<^sub>0 g \<star> f\<^sub>0g\<^sub>1.p\<^sub>0, fg\<^sub>0h\<^sub>1.p\<^sub>1] \<cdot> \<a>\<^sup>-\<^sup>1[g, tab\<^sub>0 g \<star> f\<^sub>0g\<^sub>1.p\<^sub>0, fg\<^sub>0h\<^sub>1.p\<^sub>1]) \<cdot> (g \<star> \<a>\<^sup>-\<^sup>1[tab\<^sub>0 g, f\<^sub>0g\<^sub>1.p\<^sub>0, fg\<^sub>0h\<^sub>1.p\<^sub>1]) = g \<star> \<a>\<^sup>-\<^sup>1[tab\<^sub>0 g, f\<^sub>0g\<^sub>1.p\<^sub>0, fg\<^sub>0h\<^sub>1.p\<^sub>1]
goal (1 subgoal):
1. (g \<star> \<a>[h, tab\<^sub>0 h, fg\<^sub>0h\<^sub>1.p\<^sub>0]) \<cdot> (g \<star> h.tab \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0) \<cdot> (g \<star> fg\<^sub>0h\<^sub>1.\<phi>) \<cdot> ((g \<star> \<a>\<^sup>-\<^sup>1[tab\<^sub>0 g, f\<^sub>0g\<^sub>1.p\<^sub>0, fg\<^sub>0h\<^sub>1.p\<^sub>1]) \<cdot> \<a>[g, tab\<^sub>0 g, f\<^sub>0g\<^sub>1.p\<^sub>0 \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1]) \<cdot> (g.tab \<star> f\<^sub>0g\<^sub>1.p\<^sub>0 \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) = (g \<star> \<a>[h, tab\<^sub>0 h, fg\<^sub>0h\<^sub>1.p\<^sub>0]) \<cdot> (g \<star> h.tab \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0) \<cdot> (g \<star> fg\<^sub>0h\<^sub>1.\<phi>) \<cdot> (\<a>[g, tab\<^sub>0 g \<star> f\<^sub>0g\<^sub>1.p\<^sub>0, fg\<^sub>0h\<^sub>1.p\<^sub>1] \<cdot> \<a>\<^sup>-\<^sup>1[g, tab\<^sub>0 g \<star> f\<^sub>0g\<^sub>1.p\<^sub>0, fg\<^sub>0h\<^sub>1.p\<^sub>1] \<cdot> (g \<star> \<a>\<^sup>-\<^sup>1[tab\<^sub>0 g, f\<^sub>0g\<^sub>1.p\<^sub>0, fg\<^sub>0h\<^sub>1.p\<^sub>1])) \<cdot> \<a>[g, tab\<^sub>0 g, f\<^sub>0g\<^sub>1.p\<^sub>0 \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1] \<cdot> (g.tab \<star> f\<^sub>0g\<^sub>1.p\<^sub>0 \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1)
[PROOF STEP]
thus ?thesis
[PROOF STATE]
proof (prove)
using this:
(\<a>[g, tab\<^sub>0 g \<star> f\<^sub>0g\<^sub>1.p\<^sub>0, fg\<^sub>0h\<^sub>1.p\<^sub>1] \<cdot> \<a>\<^sup>-\<^sup>1[g, tab\<^sub>0 g \<star> f\<^sub>0g\<^sub>1.p\<^sub>0, fg\<^sub>0h\<^sub>1.p\<^sub>1]) \<cdot> (g \<star> \<a>\<^sup>-\<^sup>1[tab\<^sub>0 g, f\<^sub>0g\<^sub>1.p\<^sub>0, fg\<^sub>0h\<^sub>1.p\<^sub>1]) = g \<star> \<a>\<^sup>-\<^sup>1[tab\<^sub>0 g, f\<^sub>0g\<^sub>1.p\<^sub>0, fg\<^sub>0h\<^sub>1.p\<^sub>1]
goal (1 subgoal):
1. (g \<star> \<a>[h, tab\<^sub>0 h, fg\<^sub>0h\<^sub>1.p\<^sub>0]) \<cdot> (g \<star> h.tab \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0) \<cdot> (g \<star> fg\<^sub>0h\<^sub>1.\<phi>) \<cdot> ((g \<star> \<a>\<^sup>-\<^sup>1[tab\<^sub>0 g, f\<^sub>0g\<^sub>1.p\<^sub>0, fg\<^sub>0h\<^sub>1.p\<^sub>1]) \<cdot> \<a>[g, tab\<^sub>0 g, f\<^sub>0g\<^sub>1.p\<^sub>0 \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1]) \<cdot> (g.tab \<star> f\<^sub>0g\<^sub>1.p\<^sub>0 \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) = (g \<star> \<a>[h, tab\<^sub>0 h, fg\<^sub>0h\<^sub>1.p\<^sub>0]) \<cdot> (g \<star> h.tab \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0) \<cdot> (g \<star> fg\<^sub>0h\<^sub>1.\<phi>) \<cdot> (\<a>[g, tab\<^sub>0 g \<star> f\<^sub>0g\<^sub>1.p\<^sub>0, fg\<^sub>0h\<^sub>1.p\<^sub>1] \<cdot> \<a>\<^sup>-\<^sup>1[g, tab\<^sub>0 g \<star> f\<^sub>0g\<^sub>1.p\<^sub>0, fg\<^sub>0h\<^sub>1.p\<^sub>1] \<cdot> (g \<star> \<a>\<^sup>-\<^sup>1[tab\<^sub>0 g, f\<^sub>0g\<^sub>1.p\<^sub>0, fg\<^sub>0h\<^sub>1.p\<^sub>1])) \<cdot> \<a>[g, tab\<^sub>0 g, f\<^sub>0g\<^sub>1.p\<^sub>0 \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1] \<cdot> (g.tab \<star> f\<^sub>0g\<^sub>1.p\<^sub>0 \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1)
[PROOF STEP]
by (simp add: comp_assoc)
[PROOF STATE]
proof (state)
this:
(g \<star> \<a>[h, tab\<^sub>0 h, fg\<^sub>0h\<^sub>1.p\<^sub>0]) \<cdot> (g \<star> h.tab \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0) \<cdot> (g \<star> fg\<^sub>0h\<^sub>1.\<phi>) \<cdot> ((g \<star> \<a>\<^sup>-\<^sup>1[tab\<^sub>0 g, f\<^sub>0g\<^sub>1.p\<^sub>0, fg\<^sub>0h\<^sub>1.p\<^sub>1]) \<cdot> \<a>[g, tab\<^sub>0 g, f\<^sub>0g\<^sub>1.p\<^sub>0 \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1]) \<cdot> (g.tab \<star> f\<^sub>0g\<^sub>1.p\<^sub>0 \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) = (g \<star> \<a>[h, tab\<^sub>0 h, fg\<^sub>0h\<^sub>1.p\<^sub>0]) \<cdot> (g \<star> h.tab \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0) \<cdot> (g \<star> fg\<^sub>0h\<^sub>1.\<phi>) \<cdot> (\<a>[g, tab\<^sub>0 g \<star> f\<^sub>0g\<^sub>1.p\<^sub>0, fg\<^sub>0h\<^sub>1.p\<^sub>1] \<cdot> \<a>\<^sup>-\<^sup>1[g, tab\<^sub>0 g \<star> f\<^sub>0g\<^sub>1.p\<^sub>0, fg\<^sub>0h\<^sub>1.p\<^sub>1] \<cdot> (g \<star> \<a>\<^sup>-\<^sup>1[tab\<^sub>0 g, f\<^sub>0g\<^sub>1.p\<^sub>0, fg\<^sub>0h\<^sub>1.p\<^sub>1])) \<cdot> \<a>[g, tab\<^sub>0 g, f\<^sub>0g\<^sub>1.p\<^sub>0 \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1] \<cdot> (g.tab \<star> f\<^sub>0g\<^sub>1.p\<^sub>0 \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1)
goal:
No subgoals!
[PROOF STEP]
qed
[PROOF STATE]
proof (state)
this:
(g \<star> \<a>[h, tab\<^sub>0 h, fg\<^sub>0h\<^sub>1.p\<^sub>0]) \<cdot> (g \<star> h.tab \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0) \<cdot> (g \<star> fg\<^sub>0h\<^sub>1.\<phi>) \<cdot> ((g \<star> \<a>\<^sup>-\<^sup>1[tab\<^sub>0 g, f\<^sub>0g\<^sub>1.p\<^sub>0, fg\<^sub>0h\<^sub>1.p\<^sub>1]) \<cdot> \<a>[g, tab\<^sub>0 g, f\<^sub>0g\<^sub>1.p\<^sub>0 \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1]) \<cdot> (g.tab \<star> f\<^sub>0g\<^sub>1.p\<^sub>0 \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) = (g \<star> \<a>[h, tab\<^sub>0 h, fg\<^sub>0h\<^sub>1.p\<^sub>0]) \<cdot> (g \<star> h.tab \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0) \<cdot> (g \<star> fg\<^sub>0h\<^sub>1.\<phi>) \<cdot> (\<a>[g, tab\<^sub>0 g \<star> f\<^sub>0g\<^sub>1.p\<^sub>0, fg\<^sub>0h\<^sub>1.p\<^sub>1] \<cdot> \<a>\<^sup>-\<^sup>1[g, tab\<^sub>0 g \<star> f\<^sub>0g\<^sub>1.p\<^sub>0, fg\<^sub>0h\<^sub>1.p\<^sub>1] \<cdot> (g \<star> \<a>\<^sup>-\<^sup>1[tab\<^sub>0 g, f\<^sub>0g\<^sub>1.p\<^sub>0, fg\<^sub>0h\<^sub>1.p\<^sub>1])) \<cdot> \<a>[g, tab\<^sub>0 g, f\<^sub>0g\<^sub>1.p\<^sub>0 \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1] \<cdot> (g.tab \<star> f\<^sub>0g\<^sub>1.p\<^sub>0 \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1)
goal (1 subgoal):
1. (g \<star> \<theta>\<^sub>g) \<cdot> \<a>[g, tab\<^sub>0 g, w\<^sub>g] \<cdot> (g.tab \<star> w\<^sub>g) = ((g \<star> \<theta>\<^sub>g') \<cdot> \<a>[g, tab\<^sub>0 g, w\<^sub>g'] \<cdot> (g.tab \<star> w\<^sub>g')) \<cdot> \<beta>\<^sub>g
[PROOF STEP]
also
[PROOF STATE]
proof (state)
this:
(g \<star> \<a>[h, tab\<^sub>0 h, fg\<^sub>0h\<^sub>1.p\<^sub>0]) \<cdot> (g \<star> h.tab \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0) \<cdot> (g \<star> fg\<^sub>0h\<^sub>1.\<phi>) \<cdot> ((g \<star> \<a>\<^sup>-\<^sup>1[tab\<^sub>0 g, f\<^sub>0g\<^sub>1.p\<^sub>0, fg\<^sub>0h\<^sub>1.p\<^sub>1]) \<cdot> \<a>[g, tab\<^sub>0 g, f\<^sub>0g\<^sub>1.p\<^sub>0 \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1]) \<cdot> (g.tab \<star> f\<^sub>0g\<^sub>1.p\<^sub>0 \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) = (g \<star> \<a>[h, tab\<^sub>0 h, fg\<^sub>0h\<^sub>1.p\<^sub>0]) \<cdot> (g \<star> h.tab \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0) \<cdot> (g \<star> fg\<^sub>0h\<^sub>1.\<phi>) \<cdot> (\<a>[g, tab\<^sub>0 g \<star> f\<^sub>0g\<^sub>1.p\<^sub>0, fg\<^sub>0h\<^sub>1.p\<^sub>1] \<cdot> \<a>\<^sup>-\<^sup>1[g, tab\<^sub>0 g \<star> f\<^sub>0g\<^sub>1.p\<^sub>0, fg\<^sub>0h\<^sub>1.p\<^sub>1] \<cdot> (g \<star> \<a>\<^sup>-\<^sup>1[tab\<^sub>0 g, f\<^sub>0g\<^sub>1.p\<^sub>0, fg\<^sub>0h\<^sub>1.p\<^sub>1])) \<cdot> \<a>[g, tab\<^sub>0 g, f\<^sub>0g\<^sub>1.p\<^sub>0 \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1] \<cdot> (g.tab \<star> f\<^sub>0g\<^sub>1.p\<^sub>0 \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1)
goal (1 subgoal):
1. (g \<star> \<theta>\<^sub>g) \<cdot> \<a>[g, tab\<^sub>0 g, w\<^sub>g] \<cdot> (g.tab \<star> w\<^sub>g) = ((g \<star> \<theta>\<^sub>g') \<cdot> \<a>[g, tab\<^sub>0 g, w\<^sub>g'] \<cdot> (g.tab \<star> w\<^sub>g')) \<cdot> \<beta>\<^sub>g
[PROOF STEP]
have "... =
(g \<star> \<a>[h, tab\<^sub>0 h, TTfgh.p\<^sub>0]) \<cdot>
(g \<star> h.tab \<star> TTfgh.p\<^sub>0) \<cdot>
(g \<star> fg\<^sub>0h\<^sub>1.\<phi>) \<cdot>
\<a>[g, tab\<^sub>0 g \<star> Tfg.p\<^sub>0, TTfgh.p\<^sub>1] \<cdot>
(\<a>\<^sup>-\<^sup>1[g, tab\<^sub>0 g \<star> Tfg.p\<^sub>0, TTfgh.p\<^sub>1] \<cdot>
(g \<star> \<a>\<^sup>-\<^sup>1[tab\<^sub>0 g, Tfg.p\<^sub>0, TTfgh.p\<^sub>1]) \<cdot>
\<a>[g, tab\<^sub>0 g, Tfg.p\<^sub>0 \<star> TTfgh.p\<^sub>1]) \<cdot>
(g.tab \<star> Tfg.p\<^sub>0 \<star> TTfgh.p\<^sub>1)"
[PROOF STATE]
proof (prove)
goal (1 subgoal):
1. (g \<star> \<a>[h, tab\<^sub>0 h, fg\<^sub>0h\<^sub>1.p\<^sub>0]) \<cdot> (g \<star> h.tab \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0) \<cdot> (g \<star> fg\<^sub>0h\<^sub>1.\<phi>) \<cdot> (\<a>[g, tab\<^sub>0 g \<star> f\<^sub>0g\<^sub>1.p\<^sub>0, fg\<^sub>0h\<^sub>1.p\<^sub>1] \<cdot> \<a>\<^sup>-\<^sup>1[g, tab\<^sub>0 g \<star> f\<^sub>0g\<^sub>1.p\<^sub>0, fg\<^sub>0h\<^sub>1.p\<^sub>1] \<cdot> (g \<star> \<a>\<^sup>-\<^sup>1[tab\<^sub>0 g, f\<^sub>0g\<^sub>1.p\<^sub>0, fg\<^sub>0h\<^sub>1.p\<^sub>1])) \<cdot> \<a>[g, tab\<^sub>0 g, f\<^sub>0g\<^sub>1.p\<^sub>0 \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1] \<cdot> (g.tab \<star> f\<^sub>0g\<^sub>1.p\<^sub>0 \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) = (g \<star> \<a>[h, tab\<^sub>0 h, fg\<^sub>0h\<^sub>1.p\<^sub>0]) \<cdot> (g \<star> h.tab \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0) \<cdot> (g \<star> fg\<^sub>0h\<^sub>1.\<phi>) \<cdot> \<a>[g, tab\<^sub>0 g \<star> f\<^sub>0g\<^sub>1.p\<^sub>0, fg\<^sub>0h\<^sub>1.p\<^sub>1] \<cdot> (\<a>\<^sup>-\<^sup>1[g, tab\<^sub>0 g \<star> f\<^sub>0g\<^sub>1.p\<^sub>0, fg\<^sub>0h\<^sub>1.p\<^sub>1] \<cdot> (g \<star> \<a>\<^sup>-\<^sup>1[tab\<^sub>0 g, f\<^sub>0g\<^sub>1.p\<^sub>0, fg\<^sub>0h\<^sub>1.p\<^sub>1]) \<cdot> \<a>[g, tab\<^sub>0 g, f\<^sub>0g\<^sub>1.p\<^sub>0 \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1]) \<cdot> (g.tab \<star> f\<^sub>0g\<^sub>1.p\<^sub>0 \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1)
[PROOF STEP]
using comp_assoc
[PROOF STATE]
proof (prove)
using this:
(?h \<cdot> ?g) \<cdot> ?f = ?h \<cdot> ?g \<cdot> ?f
goal (1 subgoal):
1. (g \<star> \<a>[h, tab\<^sub>0 h, fg\<^sub>0h\<^sub>1.p\<^sub>0]) \<cdot> (g \<star> h.tab \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0) \<cdot> (g \<star> fg\<^sub>0h\<^sub>1.\<phi>) \<cdot> (\<a>[g, tab\<^sub>0 g \<star> f\<^sub>0g\<^sub>1.p\<^sub>0, fg\<^sub>0h\<^sub>1.p\<^sub>1] \<cdot> \<a>\<^sup>-\<^sup>1[g, tab\<^sub>0 g \<star> f\<^sub>0g\<^sub>1.p\<^sub>0, fg\<^sub>0h\<^sub>1.p\<^sub>1] \<cdot> (g \<star> \<a>\<^sup>-\<^sup>1[tab\<^sub>0 g, f\<^sub>0g\<^sub>1.p\<^sub>0, fg\<^sub>0h\<^sub>1.p\<^sub>1])) \<cdot> \<a>[g, tab\<^sub>0 g, f\<^sub>0g\<^sub>1.p\<^sub>0 \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1] \<cdot> (g.tab \<star> f\<^sub>0g\<^sub>1.p\<^sub>0 \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) = (g \<star> \<a>[h, tab\<^sub>0 h, fg\<^sub>0h\<^sub>1.p\<^sub>0]) \<cdot> (g \<star> h.tab \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0) \<cdot> (g \<star> fg\<^sub>0h\<^sub>1.\<phi>) \<cdot> \<a>[g, tab\<^sub>0 g \<star> f\<^sub>0g\<^sub>1.p\<^sub>0, fg\<^sub>0h\<^sub>1.p\<^sub>1] \<cdot> (\<a>\<^sup>-\<^sup>1[g, tab\<^sub>0 g \<star> f\<^sub>0g\<^sub>1.p\<^sub>0, fg\<^sub>0h\<^sub>1.p\<^sub>1] \<cdot> (g \<star> \<a>\<^sup>-\<^sup>1[tab\<^sub>0 g, f\<^sub>0g\<^sub>1.p\<^sub>0, fg\<^sub>0h\<^sub>1.p\<^sub>1]) \<cdot> \<a>[g, tab\<^sub>0 g, f\<^sub>0g\<^sub>1.p\<^sub>0 \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1]) \<cdot> (g.tab \<star> f\<^sub>0g\<^sub>1.p\<^sub>0 \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1)
[PROOF STEP]
by presburger
[PROOF STATE]
proof (state)
this:
(g \<star> \<a>[h, tab\<^sub>0 h, fg\<^sub>0h\<^sub>1.p\<^sub>0]) \<cdot> (g \<star> h.tab \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0) \<cdot> (g \<star> fg\<^sub>0h\<^sub>1.\<phi>) \<cdot> (\<a>[g, tab\<^sub>0 g \<star> f\<^sub>0g\<^sub>1.p\<^sub>0, fg\<^sub>0h\<^sub>1.p\<^sub>1] \<cdot> \<a>\<^sup>-\<^sup>1[g, tab\<^sub>0 g \<star> f\<^sub>0g\<^sub>1.p\<^sub>0, fg\<^sub>0h\<^sub>1.p\<^sub>1] \<cdot> (g \<star> \<a>\<^sup>-\<^sup>1[tab\<^sub>0 g, f\<^sub>0g\<^sub>1.p\<^sub>0, fg\<^sub>0h\<^sub>1.p\<^sub>1])) \<cdot> \<a>[g, tab\<^sub>0 g, f\<^sub>0g\<^sub>1.p\<^sub>0 \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1] \<cdot> (g.tab \<star> f\<^sub>0g\<^sub>1.p\<^sub>0 \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) = (g \<star> \<a>[h, tab\<^sub>0 h, fg\<^sub>0h\<^sub>1.p\<^sub>0]) \<cdot> (g \<star> h.tab \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0) \<cdot> (g \<star> fg\<^sub>0h\<^sub>1.\<phi>) \<cdot> \<a>[g, tab\<^sub>0 g \<star> f\<^sub>0g\<^sub>1.p\<^sub>0, fg\<^sub>0h\<^sub>1.p\<^sub>1] \<cdot> (\<a>\<^sup>-\<^sup>1[g, tab\<^sub>0 g \<star> f\<^sub>0g\<^sub>1.p\<^sub>0, fg\<^sub>0h\<^sub>1.p\<^sub>1] \<cdot> (g \<star> \<a>\<^sup>-\<^sup>1[tab\<^sub>0 g, f\<^sub>0g\<^sub>1.p\<^sub>0, fg\<^sub>0h\<^sub>1.p\<^sub>1]) \<cdot> \<a>[g, tab\<^sub>0 g, f\<^sub>0g\<^sub>1.p\<^sub>0 \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1]) \<cdot> (g.tab \<star> f\<^sub>0g\<^sub>1.p\<^sub>0 \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1)
goal (1 subgoal):
1. (g \<star> \<theta>\<^sub>g) \<cdot> \<a>[g, tab\<^sub>0 g, w\<^sub>g] \<cdot> (g.tab \<star> w\<^sub>g) = ((g \<star> \<theta>\<^sub>g') \<cdot> \<a>[g, tab\<^sub>0 g, w\<^sub>g'] \<cdot> (g.tab \<star> w\<^sub>g')) \<cdot> \<beta>\<^sub>g
[PROOF STEP]
also
[PROOF STATE]
proof (state)
this:
(g \<star> \<a>[h, tab\<^sub>0 h, fg\<^sub>0h\<^sub>1.p\<^sub>0]) \<cdot> (g \<star> h.tab \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0) \<cdot> (g \<star> fg\<^sub>0h\<^sub>1.\<phi>) \<cdot> (\<a>[g, tab\<^sub>0 g \<star> f\<^sub>0g\<^sub>1.p\<^sub>0, fg\<^sub>0h\<^sub>1.p\<^sub>1] \<cdot> \<a>\<^sup>-\<^sup>1[g, tab\<^sub>0 g \<star> f\<^sub>0g\<^sub>1.p\<^sub>0, fg\<^sub>0h\<^sub>1.p\<^sub>1] \<cdot> (g \<star> \<a>\<^sup>-\<^sup>1[tab\<^sub>0 g, f\<^sub>0g\<^sub>1.p\<^sub>0, fg\<^sub>0h\<^sub>1.p\<^sub>1])) \<cdot> \<a>[g, tab\<^sub>0 g, f\<^sub>0g\<^sub>1.p\<^sub>0 \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1] \<cdot> (g.tab \<star> f\<^sub>0g\<^sub>1.p\<^sub>0 \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) = (g \<star> \<a>[h, tab\<^sub>0 h, fg\<^sub>0h\<^sub>1.p\<^sub>0]) \<cdot> (g \<star> h.tab \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0) \<cdot> (g \<star> fg\<^sub>0h\<^sub>1.\<phi>) \<cdot> \<a>[g, tab\<^sub>0 g \<star> f\<^sub>0g\<^sub>1.p\<^sub>0, fg\<^sub>0h\<^sub>1.p\<^sub>1] \<cdot> (\<a>\<^sup>-\<^sup>1[g, tab\<^sub>0 g \<star> f\<^sub>0g\<^sub>1.p\<^sub>0, fg\<^sub>0h\<^sub>1.p\<^sub>1] \<cdot> (g \<star> \<a>\<^sup>-\<^sup>1[tab\<^sub>0 g, f\<^sub>0g\<^sub>1.p\<^sub>0, fg\<^sub>0h\<^sub>1.p\<^sub>1]) \<cdot> \<a>[g, tab\<^sub>0 g, f\<^sub>0g\<^sub>1.p\<^sub>0 \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1]) \<cdot> (g.tab \<star> f\<^sub>0g\<^sub>1.p\<^sub>0 \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1)
goal (1 subgoal):
1. (g \<star> \<theta>\<^sub>g) \<cdot> \<a>[g, tab\<^sub>0 g, w\<^sub>g] \<cdot> (g.tab \<star> w\<^sub>g) = ((g \<star> \<theta>\<^sub>g') \<cdot> \<a>[g, tab\<^sub>0 g, w\<^sub>g'] \<cdot> (g.tab \<star> w\<^sub>g')) \<cdot> \<beta>\<^sub>g
[PROOF STEP]
have "... =
(g \<star> \<a>[h, tab\<^sub>0 h, TTfgh.p\<^sub>0]) \<cdot>
(g \<star> h.tab \<star> TTfgh.p\<^sub>0) \<cdot>
(g \<star> fg\<^sub>0h\<^sub>1.\<phi>) \<cdot>
\<a>[g, tab\<^sub>0 g \<star> Tfg.p\<^sub>0, TTfgh.p\<^sub>1] \<cdot>
(\<a>[g, tab\<^sub>0 g, Tfg.p\<^sub>0] \<star> TTfgh.p\<^sub>1) \<cdot>
(\<a>\<^sup>-\<^sup>1[g \<star> tab\<^sub>0 g, Tfg.p\<^sub>0, TTfgh.p\<^sub>1] \<cdot>
(g.tab \<star> Tfg.p\<^sub>0 \<star> TTfgh.p\<^sub>1))"
[PROOF STATE]
proof (prove)
goal (1 subgoal):
1. (g \<star> \<a>[h, tab\<^sub>0 h, fg\<^sub>0h\<^sub>1.p\<^sub>0]) \<cdot> (g \<star> h.tab \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0) \<cdot> (g \<star> fg\<^sub>0h\<^sub>1.\<phi>) \<cdot> \<a>[g, tab\<^sub>0 g \<star> f\<^sub>0g\<^sub>1.p\<^sub>0, fg\<^sub>0h\<^sub>1.p\<^sub>1] \<cdot> (\<a>\<^sup>-\<^sup>1[g, tab\<^sub>0 g \<star> f\<^sub>0g\<^sub>1.p\<^sub>0, fg\<^sub>0h\<^sub>1.p\<^sub>1] \<cdot> (g \<star> \<a>\<^sup>-\<^sup>1[tab\<^sub>0 g, f\<^sub>0g\<^sub>1.p\<^sub>0, fg\<^sub>0h\<^sub>1.p\<^sub>1]) \<cdot> \<a>[g, tab\<^sub>0 g, f\<^sub>0g\<^sub>1.p\<^sub>0 \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1]) \<cdot> (g.tab \<star> f\<^sub>0g\<^sub>1.p\<^sub>0 \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) = (g \<star> \<a>[h, tab\<^sub>0 h, fg\<^sub>0h\<^sub>1.p\<^sub>0]) \<cdot> (g \<star> h.tab \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0) \<cdot> (g \<star> fg\<^sub>0h\<^sub>1.\<phi>) \<cdot> \<a>[g, tab\<^sub>0 g \<star> f\<^sub>0g\<^sub>1.p\<^sub>0, fg\<^sub>0h\<^sub>1.p\<^sub>1] \<cdot> (\<a>[g, tab\<^sub>0 g, f\<^sub>0g\<^sub>1.p\<^sub>0] \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> \<a>\<^sup>-\<^sup>1[g \<star> tab\<^sub>0 g, f\<^sub>0g\<^sub>1.p\<^sub>0, fg\<^sub>0h\<^sub>1.p\<^sub>1] \<cdot> (g.tab \<star> f\<^sub>0g\<^sub>1.p\<^sub>0 \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1)
[PROOF STEP]
using fg gh f\<^sub>0g\<^sub>1.p\<^sub>0_simps fg\<^sub>0h\<^sub>1.p\<^sub>0_simps fg\<^sub>0h\<^sub>1.p\<^sub>1_simps fg\<^sub>0h\<^sub>1.p\<^sub>1_simps comp_assoc pentagon'
invert_opposite_sides_of_square
[of "\<a>\<^sup>-\<^sup>1[g, tab\<^sub>0 g, Tfg.p\<^sub>0] \<star> TTfgh.p\<^sub>1"
"(\<a>\<^sup>-\<^sup>1[g, tab\<^sub>0 g \<star> Tfg.p\<^sub>0, TTfgh.p\<^sub>1]) \<cdot> (g \<star> \<a>\<^sup>-\<^sup>1[tab\<^sub>0 g, Tfg.p\<^sub>0, TTfgh.p\<^sub>1])"
"\<a>\<^sup>-\<^sup>1[g \<star> tab\<^sub>0 g, Tfg.p\<^sub>0, TTfgh.p\<^sub>1]" "\<a>\<^sup>-\<^sup>1[g, tab\<^sub>0 g, Tfg.p\<^sub>0 \<star> TTfgh.p\<^sub>1]"]
[PROOF STATE]
proof (prove)
using this:
src f = trg g
src g = trg h
trg f\<^sub>0g\<^sub>1.p\<^sub>0 = src (tab\<^sub>1 g)
trg fg\<^sub>0h\<^sub>1.p\<^sub>0 = src (tab\<^sub>1 h)
trg fg\<^sub>0h\<^sub>1.p\<^sub>1 = src (tab\<^sub>0 g \<star> f\<^sub>0g\<^sub>1.p\<^sub>0)
trg fg\<^sub>0h\<^sub>1.p\<^sub>1 = src (tab\<^sub>0 g \<star> f\<^sub>0g\<^sub>1.p\<^sub>0)
(?h \<cdot> ?g) \<cdot> ?f = ?h \<cdot> ?g \<cdot> ?f
\<lbrakk>ide ?f; ide ?g; ide ?h; ide ?k; src ?f = trg ?g; src ?g = trg ?h; src ?h = trg ?k\<rbrakk> \<Longrightarrow> ((\<a>\<^sup>-\<^sup>1[?f, ?g, ?h] \<star> ?k) \<cdot> \<a>\<^sup>-\<^sup>1[?f, ?g \<star> ?h, ?k]) \<cdot> (?f \<star> \<a>\<^sup>-\<^sup>1[?g, ?h, ?k]) = \<a>\<^sup>-\<^sup>1[?f \<star> ?g, ?h, ?k] \<cdot> \<a>\<^sup>-\<^sup>1[?f, ?g, ?h \<star> ?k]
\<lbrakk>seq (\<a>\<^sup>-\<^sup>1[g, tab\<^sub>0 g, f\<^sub>0g\<^sub>1.p\<^sub>0] \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) (\<a>\<^sup>-\<^sup>1[g, tab\<^sub>0 g \<star> f\<^sub>0g\<^sub>1.p\<^sub>0, fg\<^sub>0h\<^sub>1.p\<^sub>1] \<cdot> (g \<star> \<a>\<^sup>-\<^sup>1[tab\<^sub>0 g, f\<^sub>0g\<^sub>1.p\<^sub>0, fg\<^sub>0h\<^sub>1.p\<^sub>1])); (\<a>\<^sup>-\<^sup>1[g, tab\<^sub>0 g, f\<^sub>0g\<^sub>1.p\<^sub>0] \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> \<a>\<^sup>-\<^sup>1[g, tab\<^sub>0 g \<star> f\<^sub>0g\<^sub>1.p\<^sub>0, fg\<^sub>0h\<^sub>1.p\<^sub>1] \<cdot> (g \<star> \<a>\<^sup>-\<^sup>1[tab\<^sub>0 g, f\<^sub>0g\<^sub>1.p\<^sub>0, fg\<^sub>0h\<^sub>1.p\<^sub>1]) = \<a>\<^sup>-\<^sup>1[g \<star> tab\<^sub>0 g, f\<^sub>0g\<^sub>1.p\<^sub>0, fg\<^sub>0h\<^sub>1.p\<^sub>1] \<cdot> \<a>\<^sup>-\<^sup>1[g, tab\<^sub>0 g, f\<^sub>0g\<^sub>1.p\<^sub>0 \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1]; local.iso (\<a>\<^sup>-\<^sup>1[g, tab\<^sub>0 g, f\<^sub>0g\<^sub>1.p\<^sub>0] \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1); local.iso \<a>\<^sup>-\<^sup>1[g, tab\<^sub>0 g, f\<^sub>0g\<^sub>1.p\<^sub>0 \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1]\<rbrakk> \<Longrightarrow> seq (\<a>\<^sup>-\<^sup>1[g, tab\<^sub>0 g \<star> f\<^sub>0g\<^sub>1.p\<^sub>0, fg\<^sub>0h\<^sub>1.p\<^sub>1] \<cdot> (g \<star> \<a>\<^sup>-\<^sup>1[tab\<^sub>0 g, f\<^sub>0g\<^sub>1.p\<^sub>0, fg\<^sub>0h\<^sub>1.p\<^sub>1])) (local.inv \<a>\<^sup>-\<^sup>1[g, tab\<^sub>0 g, f\<^sub>0g\<^sub>1.p\<^sub>0 \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1]) \<and> seq (local.inv (\<a>\<^sup>-\<^sup>1[g, tab\<^sub>0 g, f\<^sub>0g\<^sub>1.p\<^sub>0] \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1)) \<a>\<^sup>-\<^sup>1[g \<star> tab\<^sub>0 g, f\<^sub>0g\<^sub>1.p\<^sub>0, fg\<^sub>0h\<^sub>1.p\<^sub>1] \<and> (\<a>\<^sup>-\<^sup>1[g, tab\<^sub>0 g \<star> f\<^sub>0g\<^sub>1.p\<^sub>0, fg\<^sub>0h\<^sub>1.p\<^sub>1] \<cdot> (g \<star> \<a>\<^sup>-\<^sup>1[tab\<^sub>0 g, f\<^sub>0g\<^sub>1.p\<^sub>0, fg\<^sub>0h\<^sub>1.p\<^sub>1])) \<cdot> local.inv \<a>\<^sup>-\<^sup>1[g, tab\<^sub>0 g, f\<^sub>0g\<^sub>1.p\<^sub>0 \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1] = local.inv (\<a>\<^sup>-\<^sup>1[g, tab\<^sub>0 g, f\<^sub>0g\<^sub>1.p\<^sub>0] \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> \<a>\<^sup>-\<^sup>1[g \<star> tab\<^sub>0 g, f\<^sub>0g\<^sub>1.p\<^sub>0, fg\<^sub>0h\<^sub>1.p\<^sub>1]
goal (1 subgoal):
1. (g \<star> \<a>[h, tab\<^sub>0 h, fg\<^sub>0h\<^sub>1.p\<^sub>0]) \<cdot> (g \<star> h.tab \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0) \<cdot> (g \<star> fg\<^sub>0h\<^sub>1.\<phi>) \<cdot> \<a>[g, tab\<^sub>0 g \<star> f\<^sub>0g\<^sub>1.p\<^sub>0, fg\<^sub>0h\<^sub>1.p\<^sub>1] \<cdot> (\<a>\<^sup>-\<^sup>1[g, tab\<^sub>0 g \<star> f\<^sub>0g\<^sub>1.p\<^sub>0, fg\<^sub>0h\<^sub>1.p\<^sub>1] \<cdot> (g \<star> \<a>\<^sup>-\<^sup>1[tab\<^sub>0 g, f\<^sub>0g\<^sub>1.p\<^sub>0, fg\<^sub>0h\<^sub>1.p\<^sub>1]) \<cdot> \<a>[g, tab\<^sub>0 g, f\<^sub>0g\<^sub>1.p\<^sub>0 \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1]) \<cdot> (g.tab \<star> f\<^sub>0g\<^sub>1.p\<^sub>0 \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) = (g \<star> \<a>[h, tab\<^sub>0 h, fg\<^sub>0h\<^sub>1.p\<^sub>0]) \<cdot> (g \<star> h.tab \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0) \<cdot> (g \<star> fg\<^sub>0h\<^sub>1.\<phi>) \<cdot> \<a>[g, tab\<^sub>0 g \<star> f\<^sub>0g\<^sub>1.p\<^sub>0, fg\<^sub>0h\<^sub>1.p\<^sub>1] \<cdot> (\<a>[g, tab\<^sub>0 g, f\<^sub>0g\<^sub>1.p\<^sub>0] \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> \<a>\<^sup>-\<^sup>1[g \<star> tab\<^sub>0 g, f\<^sub>0g\<^sub>1.p\<^sub>0, fg\<^sub>0h\<^sub>1.p\<^sub>1] \<cdot> (g.tab \<star> f\<^sub>0g\<^sub>1.p\<^sub>0 \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1)
[PROOF STEP]
by simp
[PROOF STATE]
proof (state)
this:
(g \<star> \<a>[h, tab\<^sub>0 h, fg\<^sub>0h\<^sub>1.p\<^sub>0]) \<cdot> (g \<star> h.tab \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0) \<cdot> (g \<star> fg\<^sub>0h\<^sub>1.\<phi>) \<cdot> \<a>[g, tab\<^sub>0 g \<star> f\<^sub>0g\<^sub>1.p\<^sub>0, fg\<^sub>0h\<^sub>1.p\<^sub>1] \<cdot> (\<a>\<^sup>-\<^sup>1[g, tab\<^sub>0 g \<star> f\<^sub>0g\<^sub>1.p\<^sub>0, fg\<^sub>0h\<^sub>1.p\<^sub>1] \<cdot> (g \<star> \<a>\<^sup>-\<^sup>1[tab\<^sub>0 g, f\<^sub>0g\<^sub>1.p\<^sub>0, fg\<^sub>0h\<^sub>1.p\<^sub>1]) \<cdot> \<a>[g, tab\<^sub>0 g, f\<^sub>0g\<^sub>1.p\<^sub>0 \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1]) \<cdot> (g.tab \<star> f\<^sub>0g\<^sub>1.p\<^sub>0 \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) = (g \<star> \<a>[h, tab\<^sub>0 h, fg\<^sub>0h\<^sub>1.p\<^sub>0]) \<cdot> (g \<star> h.tab \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0) \<cdot> (g \<star> fg\<^sub>0h\<^sub>1.\<phi>) \<cdot> \<a>[g, tab\<^sub>0 g \<star> f\<^sub>0g\<^sub>1.p\<^sub>0, fg\<^sub>0h\<^sub>1.p\<^sub>1] \<cdot> (\<a>[g, tab\<^sub>0 g, f\<^sub>0g\<^sub>1.p\<^sub>0] \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> \<a>\<^sup>-\<^sup>1[g \<star> tab\<^sub>0 g, f\<^sub>0g\<^sub>1.p\<^sub>0, fg\<^sub>0h\<^sub>1.p\<^sub>1] \<cdot> (g.tab \<star> f\<^sub>0g\<^sub>1.p\<^sub>0 \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1)
goal (1 subgoal):
1. (g \<star> \<theta>\<^sub>g) \<cdot> \<a>[g, tab\<^sub>0 g, w\<^sub>g] \<cdot> (g.tab \<star> w\<^sub>g) = ((g \<star> \<theta>\<^sub>g') \<cdot> \<a>[g, tab\<^sub>0 g, w\<^sub>g'] \<cdot> (g.tab \<star> w\<^sub>g')) \<cdot> \<beta>\<^sub>g
[PROOF STEP]
also
[PROOF STATE]
proof (state)
this:
(g \<star> \<a>[h, tab\<^sub>0 h, fg\<^sub>0h\<^sub>1.p\<^sub>0]) \<cdot> (g \<star> h.tab \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0) \<cdot> (g \<star> fg\<^sub>0h\<^sub>1.\<phi>) \<cdot> \<a>[g, tab\<^sub>0 g \<star> f\<^sub>0g\<^sub>1.p\<^sub>0, fg\<^sub>0h\<^sub>1.p\<^sub>1] \<cdot> (\<a>\<^sup>-\<^sup>1[g, tab\<^sub>0 g \<star> f\<^sub>0g\<^sub>1.p\<^sub>0, fg\<^sub>0h\<^sub>1.p\<^sub>1] \<cdot> (g \<star> \<a>\<^sup>-\<^sup>1[tab\<^sub>0 g, f\<^sub>0g\<^sub>1.p\<^sub>0, fg\<^sub>0h\<^sub>1.p\<^sub>1]) \<cdot> \<a>[g, tab\<^sub>0 g, f\<^sub>0g\<^sub>1.p\<^sub>0 \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1]) \<cdot> (g.tab \<star> f\<^sub>0g\<^sub>1.p\<^sub>0 \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) = (g \<star> \<a>[h, tab\<^sub>0 h, fg\<^sub>0h\<^sub>1.p\<^sub>0]) \<cdot> (g \<star> h.tab \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0) \<cdot> (g \<star> fg\<^sub>0h\<^sub>1.\<phi>) \<cdot> \<a>[g, tab\<^sub>0 g \<star> f\<^sub>0g\<^sub>1.p\<^sub>0, fg\<^sub>0h\<^sub>1.p\<^sub>1] \<cdot> (\<a>[g, tab\<^sub>0 g, f\<^sub>0g\<^sub>1.p\<^sub>0] \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> \<a>\<^sup>-\<^sup>1[g \<star> tab\<^sub>0 g, f\<^sub>0g\<^sub>1.p\<^sub>0, fg\<^sub>0h\<^sub>1.p\<^sub>1] \<cdot> (g.tab \<star> f\<^sub>0g\<^sub>1.p\<^sub>0 \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1)
goal (1 subgoal):
1. (g \<star> \<theta>\<^sub>g) \<cdot> \<a>[g, tab\<^sub>0 g, w\<^sub>g] \<cdot> (g.tab \<star> w\<^sub>g) = ((g \<star> \<theta>\<^sub>g') \<cdot> \<a>[g, tab\<^sub>0 g, w\<^sub>g'] \<cdot> (g.tab \<star> w\<^sub>g')) \<cdot> \<beta>\<^sub>g
[PROOF STEP]
have "... =
(g \<star> \<a>[h, tab\<^sub>0 h, TTfgh.p\<^sub>0]) \<cdot>
(g \<star> h.tab \<star> TTfgh.p\<^sub>0) \<cdot>
(g \<star> fg\<^sub>0h\<^sub>1.\<phi>) \<cdot>
\<a>[g, tab\<^sub>0 g \<star> Tfg.p\<^sub>0, TTfgh.p\<^sub>1] \<cdot>
(\<a>[g, tab\<^sub>0 g, Tfg.p\<^sub>0] \<star> TTfgh.p\<^sub>1) \<cdot>
((g.tab \<star> Tfg.p\<^sub>0) \<star> TTfgh.p\<^sub>1) \<cdot>
\<a>\<^sup>-\<^sup>1[tab\<^sub>1 g, Tfg.p\<^sub>0, TTfgh.p\<^sub>1]"
[PROOF STATE]
proof (prove)
goal (1 subgoal):
1. (g \<star> \<a>[h, tab\<^sub>0 h, fg\<^sub>0h\<^sub>1.p\<^sub>0]) \<cdot> (g \<star> h.tab \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0) \<cdot> (g \<star> fg\<^sub>0h\<^sub>1.\<phi>) \<cdot> \<a>[g, tab\<^sub>0 g \<star> f\<^sub>0g\<^sub>1.p\<^sub>0, fg\<^sub>0h\<^sub>1.p\<^sub>1] \<cdot> (\<a>[g, tab\<^sub>0 g, f\<^sub>0g\<^sub>1.p\<^sub>0] \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> \<a>\<^sup>-\<^sup>1[g \<star> tab\<^sub>0 g, f\<^sub>0g\<^sub>1.p\<^sub>0, fg\<^sub>0h\<^sub>1.p\<^sub>1] \<cdot> (g.tab \<star> f\<^sub>0g\<^sub>1.p\<^sub>0 \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) = (g \<star> \<a>[h, tab\<^sub>0 h, fg\<^sub>0h\<^sub>1.p\<^sub>0]) \<cdot> (g \<star> h.tab \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0) \<cdot> (g \<star> fg\<^sub>0h\<^sub>1.\<phi>) \<cdot> \<a>[g, tab\<^sub>0 g \<star> f\<^sub>0g\<^sub>1.p\<^sub>0, fg\<^sub>0h\<^sub>1.p\<^sub>1] \<cdot> (\<a>[g, tab\<^sub>0 g, f\<^sub>0g\<^sub>1.p\<^sub>0] \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> ((g.tab \<star> f\<^sub>0g\<^sub>1.p\<^sub>0) \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>1 g, f\<^sub>0g\<^sub>1.p\<^sub>0, fg\<^sub>0h\<^sub>1.p\<^sub>1]
[PROOF STEP]
using fg\<^sub>0h\<^sub>1.p\<^sub>1_simps assoc'_naturality [of g.tab Tfg.p\<^sub>0 TTfgh.p\<^sub>1]
[PROOF STATE]
proof (prove)
using this:
trg fg\<^sub>0h\<^sub>1.p\<^sub>1 = src (tab\<^sub>0 g \<star> f\<^sub>0g\<^sub>1.p\<^sub>0)
\<lbrakk>arr g.tab; arr f\<^sub>0g\<^sub>1.p\<^sub>0; arr fg\<^sub>0h\<^sub>1.p\<^sub>1; src g.tab = trg f\<^sub>0g\<^sub>1.p\<^sub>0; src f\<^sub>0g\<^sub>1.p\<^sub>0 = trg fg\<^sub>0h\<^sub>1.p\<^sub>1\<rbrakk> \<Longrightarrow> \<a>\<^sup>-\<^sup>1[cod g.tab, cod f\<^sub>0g\<^sub>1.p\<^sub>0, cod fg\<^sub>0h\<^sub>1.p\<^sub>1] \<cdot> (g.tab \<star> f\<^sub>0g\<^sub>1.p\<^sub>0 \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) = ((g.tab \<star> f\<^sub>0g\<^sub>1.p\<^sub>0) \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> \<a>\<^sup>-\<^sup>1[local.dom g.tab, local.dom f\<^sub>0g\<^sub>1.p\<^sub>0, local.dom fg\<^sub>0h\<^sub>1.p\<^sub>1]
goal (1 subgoal):
1. (g \<star> \<a>[h, tab\<^sub>0 h, fg\<^sub>0h\<^sub>1.p\<^sub>0]) \<cdot> (g \<star> h.tab \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0) \<cdot> (g \<star> fg\<^sub>0h\<^sub>1.\<phi>) \<cdot> \<a>[g, tab\<^sub>0 g \<star> f\<^sub>0g\<^sub>1.p\<^sub>0, fg\<^sub>0h\<^sub>1.p\<^sub>1] \<cdot> (\<a>[g, tab\<^sub>0 g, f\<^sub>0g\<^sub>1.p\<^sub>0] \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> \<a>\<^sup>-\<^sup>1[g \<star> tab\<^sub>0 g, f\<^sub>0g\<^sub>1.p\<^sub>0, fg\<^sub>0h\<^sub>1.p\<^sub>1] \<cdot> (g.tab \<star> f\<^sub>0g\<^sub>1.p\<^sub>0 \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) = (g \<star> \<a>[h, tab\<^sub>0 h, fg\<^sub>0h\<^sub>1.p\<^sub>0]) \<cdot> (g \<star> h.tab \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0) \<cdot> (g \<star> fg\<^sub>0h\<^sub>1.\<phi>) \<cdot> \<a>[g, tab\<^sub>0 g \<star> f\<^sub>0g\<^sub>1.p\<^sub>0, fg\<^sub>0h\<^sub>1.p\<^sub>1] \<cdot> (\<a>[g, tab\<^sub>0 g, f\<^sub>0g\<^sub>1.p\<^sub>0] \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> ((g.tab \<star> f\<^sub>0g\<^sub>1.p\<^sub>0) \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>1 g, f\<^sub>0g\<^sub>1.p\<^sub>0, fg\<^sub>0h\<^sub>1.p\<^sub>1]
[PROOF STEP]
by simp
[PROOF STATE]
proof (state)
this:
(g \<star> \<a>[h, tab\<^sub>0 h, fg\<^sub>0h\<^sub>1.p\<^sub>0]) \<cdot> (g \<star> h.tab \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0) \<cdot> (g \<star> fg\<^sub>0h\<^sub>1.\<phi>) \<cdot> \<a>[g, tab\<^sub>0 g \<star> f\<^sub>0g\<^sub>1.p\<^sub>0, fg\<^sub>0h\<^sub>1.p\<^sub>1] \<cdot> (\<a>[g, tab\<^sub>0 g, f\<^sub>0g\<^sub>1.p\<^sub>0] \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> \<a>\<^sup>-\<^sup>1[g \<star> tab\<^sub>0 g, f\<^sub>0g\<^sub>1.p\<^sub>0, fg\<^sub>0h\<^sub>1.p\<^sub>1] \<cdot> (g.tab \<star> f\<^sub>0g\<^sub>1.p\<^sub>0 \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) = (g \<star> \<a>[h, tab\<^sub>0 h, fg\<^sub>0h\<^sub>1.p\<^sub>0]) \<cdot> (g \<star> h.tab \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0) \<cdot> (g \<star> fg\<^sub>0h\<^sub>1.\<phi>) \<cdot> \<a>[g, tab\<^sub>0 g \<star> f\<^sub>0g\<^sub>1.p\<^sub>0, fg\<^sub>0h\<^sub>1.p\<^sub>1] \<cdot> (\<a>[g, tab\<^sub>0 g, f\<^sub>0g\<^sub>1.p\<^sub>0] \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> ((g.tab \<star> f\<^sub>0g\<^sub>1.p\<^sub>0) \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>1 g, f\<^sub>0g\<^sub>1.p\<^sub>0, fg\<^sub>0h\<^sub>1.p\<^sub>1]
goal (1 subgoal):
1. (g \<star> \<theta>\<^sub>g) \<cdot> \<a>[g, tab\<^sub>0 g, w\<^sub>g] \<cdot> (g.tab \<star> w\<^sub>g) = ((g \<star> \<theta>\<^sub>g') \<cdot> \<a>[g, tab\<^sub>0 g, w\<^sub>g'] \<cdot> (g.tab \<star> w\<^sub>g')) \<cdot> \<beta>\<^sub>g
[PROOF STEP]
also
[PROOF STATE]
proof (state)
this:
(g \<star> \<a>[h, tab\<^sub>0 h, fg\<^sub>0h\<^sub>1.p\<^sub>0]) \<cdot> (g \<star> h.tab \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0) \<cdot> (g \<star> fg\<^sub>0h\<^sub>1.\<phi>) \<cdot> \<a>[g, tab\<^sub>0 g \<star> f\<^sub>0g\<^sub>1.p\<^sub>0, fg\<^sub>0h\<^sub>1.p\<^sub>1] \<cdot> (\<a>[g, tab\<^sub>0 g, f\<^sub>0g\<^sub>1.p\<^sub>0] \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> \<a>\<^sup>-\<^sup>1[g \<star> tab\<^sub>0 g, f\<^sub>0g\<^sub>1.p\<^sub>0, fg\<^sub>0h\<^sub>1.p\<^sub>1] \<cdot> (g.tab \<star> f\<^sub>0g\<^sub>1.p\<^sub>0 \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) = (g \<star> \<a>[h, tab\<^sub>0 h, fg\<^sub>0h\<^sub>1.p\<^sub>0]) \<cdot> (g \<star> h.tab \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0) \<cdot> (g \<star> fg\<^sub>0h\<^sub>1.\<phi>) \<cdot> \<a>[g, tab\<^sub>0 g \<star> f\<^sub>0g\<^sub>1.p\<^sub>0, fg\<^sub>0h\<^sub>1.p\<^sub>1] \<cdot> (\<a>[g, tab\<^sub>0 g, f\<^sub>0g\<^sub>1.p\<^sub>0] \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> ((g.tab \<star> f\<^sub>0g\<^sub>1.p\<^sub>0) \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>1 g, f\<^sub>0g\<^sub>1.p\<^sub>0, fg\<^sub>0h\<^sub>1.p\<^sub>1]
goal (1 subgoal):
1. (g \<star> \<theta>\<^sub>g) \<cdot> \<a>[g, tab\<^sub>0 g, w\<^sub>g] \<cdot> (g.tab \<star> w\<^sub>g) = ((g \<star> \<theta>\<^sub>g') \<cdot> \<a>[g, tab\<^sub>0 g, w\<^sub>g'] \<cdot> (g.tab \<star> w\<^sub>g')) \<cdot> \<beta>\<^sub>g
[PROOF STEP]
have "... =
(g \<star> \<a>[h, tab\<^sub>0 h, TTfgh.p\<^sub>0]) \<cdot>
(g \<star> h.tab \<star> TTfgh.p\<^sub>0) \<cdot>
(g \<star> fg\<^sub>0h\<^sub>1.\<phi>) \<cdot>
\<a>[g, tab\<^sub>0 g \<star> Tfg.p\<^sub>0, TTfgh.p\<^sub>1] \<cdot>
(\<a>[g, tab\<^sub>0 g, Tfg.p\<^sub>0] \<star> TTfgh.p\<^sub>1) \<cdot>
((g.tab \<star> Tfg.p\<^sub>0) \<star> TTfgh.p\<^sub>1) \<cdot>
(f\<^sub>0g\<^sub>1.\<phi> \<star> TTfgh.p\<^sub>1) \<cdot>
\<a>\<^sup>-\<^sup>1[tab\<^sub>0 f, Tfg.p\<^sub>1, TTfgh.p\<^sub>1] \<cdot>
\<a>[tab\<^sub>0 f, Tfg.p\<^sub>1, TTfgh.p\<^sub>1] \<cdot>
(inv f\<^sub>0g\<^sub>1.\<phi> \<star> TTfgh.p\<^sub>1) \<cdot>
\<a>\<^sup>-\<^sup>1[tab\<^sub>1 g, Tfg.p\<^sub>0, TTfgh.p\<^sub>1]"
[PROOF STATE]
proof (prove)
goal (1 subgoal):
1. (g \<star> \<a>[h, tab\<^sub>0 h, fg\<^sub>0h\<^sub>1.p\<^sub>0]) \<cdot> (g \<star> h.tab \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0) \<cdot> (g \<star> fg\<^sub>0h\<^sub>1.\<phi>) \<cdot> \<a>[g, tab\<^sub>0 g \<star> f\<^sub>0g\<^sub>1.p\<^sub>0, fg\<^sub>0h\<^sub>1.p\<^sub>1] \<cdot> (\<a>[g, tab\<^sub>0 g, f\<^sub>0g\<^sub>1.p\<^sub>0] \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> ((g.tab \<star> f\<^sub>0g\<^sub>1.p\<^sub>0) \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>1 g, f\<^sub>0g\<^sub>1.p\<^sub>0, fg\<^sub>0h\<^sub>1.p\<^sub>1] = (g \<star> \<a>[h, tab\<^sub>0 h, fg\<^sub>0h\<^sub>1.p\<^sub>0]) \<cdot> (g \<star> h.tab \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0) \<cdot> (g \<star> fg\<^sub>0h\<^sub>1.\<phi>) \<cdot> \<a>[g, tab\<^sub>0 g \<star> f\<^sub>0g\<^sub>1.p\<^sub>0, fg\<^sub>0h\<^sub>1.p\<^sub>1] \<cdot> (\<a>[g, tab\<^sub>0 g, f\<^sub>0g\<^sub>1.p\<^sub>0] \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> ((g.tab \<star> f\<^sub>0g\<^sub>1.p\<^sub>0) \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> (f\<^sub>0g\<^sub>1.\<phi> \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>0 f, f\<^sub>0g\<^sub>1.p\<^sub>1, fg\<^sub>0h\<^sub>1.p\<^sub>1] \<cdot> \<a>[tab\<^sub>0 f, f\<^sub>0g\<^sub>1.p\<^sub>1, fg\<^sub>0h\<^sub>1.p\<^sub>1] \<cdot> (local.inv f\<^sub>0g\<^sub>1.\<phi> \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>1 g, f\<^sub>0g\<^sub>1.p\<^sub>0, fg\<^sub>0h\<^sub>1.p\<^sub>1]
[PROOF STEP]
proof -
[PROOF STATE]
proof (state)
goal (1 subgoal):
1. (g \<star> \<a>[h, tab\<^sub>0 h, fg\<^sub>0h\<^sub>1.p\<^sub>0]) \<cdot> (g \<star> h.tab \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0) \<cdot> (g \<star> fg\<^sub>0h\<^sub>1.\<phi>) \<cdot> \<a>[g, tab\<^sub>0 g \<star> f\<^sub>0g\<^sub>1.p\<^sub>0, fg\<^sub>0h\<^sub>1.p\<^sub>1] \<cdot> (\<a>[g, tab\<^sub>0 g, f\<^sub>0g\<^sub>1.p\<^sub>0] \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> ((g.tab \<star> f\<^sub>0g\<^sub>1.p\<^sub>0) \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>1 g, f\<^sub>0g\<^sub>1.p\<^sub>0, fg\<^sub>0h\<^sub>1.p\<^sub>1] = (g \<star> \<a>[h, tab\<^sub>0 h, fg\<^sub>0h\<^sub>1.p\<^sub>0]) \<cdot> (g \<star> h.tab \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0) \<cdot> (g \<star> fg\<^sub>0h\<^sub>1.\<phi>) \<cdot> \<a>[g, tab\<^sub>0 g \<star> f\<^sub>0g\<^sub>1.p\<^sub>0, fg\<^sub>0h\<^sub>1.p\<^sub>1] \<cdot> (\<a>[g, tab\<^sub>0 g, f\<^sub>0g\<^sub>1.p\<^sub>0] \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> ((g.tab \<star> f\<^sub>0g\<^sub>1.p\<^sub>0) \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> (f\<^sub>0g\<^sub>1.\<phi> \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>0 f, f\<^sub>0g\<^sub>1.p\<^sub>1, fg\<^sub>0h\<^sub>1.p\<^sub>1] \<cdot> \<a>[tab\<^sub>0 f, f\<^sub>0g\<^sub>1.p\<^sub>1, fg\<^sub>0h\<^sub>1.p\<^sub>1] \<cdot> (local.inv f\<^sub>0g\<^sub>1.\<phi> \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>1 g, f\<^sub>0g\<^sub>1.p\<^sub>0, fg\<^sub>0h\<^sub>1.p\<^sub>1]
[PROOF STEP]
have "(f\<^sub>0g\<^sub>1.\<phi> \<star> TTfgh.p\<^sub>1) \<cdot>
\<a>\<^sup>-\<^sup>1[tab\<^sub>0 f, Tfg.p\<^sub>1, TTfgh.p\<^sub>1] \<cdot>
\<a>[tab\<^sub>0 f, Tfg.p\<^sub>1, TTfgh.p\<^sub>1] \<cdot>
(inv f\<^sub>0g\<^sub>1.\<phi> \<star> TTfgh.p\<^sub>1) \<cdot>
\<a>\<^sup>-\<^sup>1[tab\<^sub>1 g, Tfg.p\<^sub>0, TTfgh.p\<^sub>1] =
((f\<^sub>0g\<^sub>1.\<phi> \<star> TTfgh.p\<^sub>1) \<cdot>
(\<a>\<^sup>-\<^sup>1[tab\<^sub>0 f, Tfg.p\<^sub>1, TTfgh.p\<^sub>1] \<cdot>
\<a>[tab\<^sub>0 f, Tfg.p\<^sub>1, TTfgh.p\<^sub>1]) \<cdot>
(inv f\<^sub>0g\<^sub>1.\<phi> \<star> TTfgh.p\<^sub>1)) \<cdot>
\<a>\<^sup>-\<^sup>1[tab\<^sub>1 g, Tfg.p\<^sub>0, TTfgh.p\<^sub>1]"
[PROOF STATE]
proof (prove)
goal (1 subgoal):
1. (f\<^sub>0g\<^sub>1.\<phi> \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>0 f, f\<^sub>0g\<^sub>1.p\<^sub>1, fg\<^sub>0h\<^sub>1.p\<^sub>1] \<cdot> \<a>[tab\<^sub>0 f, f\<^sub>0g\<^sub>1.p\<^sub>1, fg\<^sub>0h\<^sub>1.p\<^sub>1] \<cdot> (local.inv f\<^sub>0g\<^sub>1.\<phi> \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>1 g, f\<^sub>0g\<^sub>1.p\<^sub>0, fg\<^sub>0h\<^sub>1.p\<^sub>1] = ((f\<^sub>0g\<^sub>1.\<phi> \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> (\<a>\<^sup>-\<^sup>1[tab\<^sub>0 f, f\<^sub>0g\<^sub>1.p\<^sub>1, fg\<^sub>0h\<^sub>1.p\<^sub>1] \<cdot> \<a>[tab\<^sub>0 f, f\<^sub>0g\<^sub>1.p\<^sub>1, fg\<^sub>0h\<^sub>1.p\<^sub>1]) \<cdot> (local.inv f\<^sub>0g\<^sub>1.\<phi> \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1)) \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>1 g, f\<^sub>0g\<^sub>1.p\<^sub>0, fg\<^sub>0h\<^sub>1.p\<^sub>1]
[PROOF STEP]
using comp_assoc
[PROOF STATE]
proof (prove)
using this:
(?h \<cdot> ?g) \<cdot> ?f = ?h \<cdot> ?g \<cdot> ?f
goal (1 subgoal):
1. (f\<^sub>0g\<^sub>1.\<phi> \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>0 f, f\<^sub>0g\<^sub>1.p\<^sub>1, fg\<^sub>0h\<^sub>1.p\<^sub>1] \<cdot> \<a>[tab\<^sub>0 f, f\<^sub>0g\<^sub>1.p\<^sub>1, fg\<^sub>0h\<^sub>1.p\<^sub>1] \<cdot> (local.inv f\<^sub>0g\<^sub>1.\<phi> \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>1 g, f\<^sub>0g\<^sub>1.p\<^sub>0, fg\<^sub>0h\<^sub>1.p\<^sub>1] = ((f\<^sub>0g\<^sub>1.\<phi> \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> (\<a>\<^sup>-\<^sup>1[tab\<^sub>0 f, f\<^sub>0g\<^sub>1.p\<^sub>1, fg\<^sub>0h\<^sub>1.p\<^sub>1] \<cdot> \<a>[tab\<^sub>0 f, f\<^sub>0g\<^sub>1.p\<^sub>1, fg\<^sub>0h\<^sub>1.p\<^sub>1]) \<cdot> (local.inv f\<^sub>0g\<^sub>1.\<phi> \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1)) \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>1 g, f\<^sub>0g\<^sub>1.p\<^sub>0, fg\<^sub>0h\<^sub>1.p\<^sub>1]
[PROOF STEP]
by presburger
[PROOF STATE]
proof (state)
this:
(f\<^sub>0g\<^sub>1.\<phi> \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>0 f, f\<^sub>0g\<^sub>1.p\<^sub>1, fg\<^sub>0h\<^sub>1.p\<^sub>1] \<cdot> \<a>[tab\<^sub>0 f, f\<^sub>0g\<^sub>1.p\<^sub>1, fg\<^sub>0h\<^sub>1.p\<^sub>1] \<cdot> (local.inv f\<^sub>0g\<^sub>1.\<phi> \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>1 g, f\<^sub>0g\<^sub>1.p\<^sub>0, fg\<^sub>0h\<^sub>1.p\<^sub>1] = ((f\<^sub>0g\<^sub>1.\<phi> \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> (\<a>\<^sup>-\<^sup>1[tab\<^sub>0 f, f\<^sub>0g\<^sub>1.p\<^sub>1, fg\<^sub>0h\<^sub>1.p\<^sub>1] \<cdot> \<a>[tab\<^sub>0 f, f\<^sub>0g\<^sub>1.p\<^sub>1, fg\<^sub>0h\<^sub>1.p\<^sub>1]) \<cdot> (local.inv f\<^sub>0g\<^sub>1.\<phi> \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1)) \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>1 g, f\<^sub>0g\<^sub>1.p\<^sub>0, fg\<^sub>0h\<^sub>1.p\<^sub>1]
goal (1 subgoal):
1. (g \<star> \<a>[h, tab\<^sub>0 h, fg\<^sub>0h\<^sub>1.p\<^sub>0]) \<cdot> (g \<star> h.tab \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0) \<cdot> (g \<star> fg\<^sub>0h\<^sub>1.\<phi>) \<cdot> \<a>[g, tab\<^sub>0 g \<star> f\<^sub>0g\<^sub>1.p\<^sub>0, fg\<^sub>0h\<^sub>1.p\<^sub>1] \<cdot> (\<a>[g, tab\<^sub>0 g, f\<^sub>0g\<^sub>1.p\<^sub>0] \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> ((g.tab \<star> f\<^sub>0g\<^sub>1.p\<^sub>0) \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>1 g, f\<^sub>0g\<^sub>1.p\<^sub>0, fg\<^sub>0h\<^sub>1.p\<^sub>1] = (g \<star> \<a>[h, tab\<^sub>0 h, fg\<^sub>0h\<^sub>1.p\<^sub>0]) \<cdot> (g \<star> h.tab \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0) \<cdot> (g \<star> fg\<^sub>0h\<^sub>1.\<phi>) \<cdot> \<a>[g, tab\<^sub>0 g \<star> f\<^sub>0g\<^sub>1.p\<^sub>0, fg\<^sub>0h\<^sub>1.p\<^sub>1] \<cdot> (\<a>[g, tab\<^sub>0 g, f\<^sub>0g\<^sub>1.p\<^sub>0] \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> ((g.tab \<star> f\<^sub>0g\<^sub>1.p\<^sub>0) \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> (f\<^sub>0g\<^sub>1.\<phi> \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>0 f, f\<^sub>0g\<^sub>1.p\<^sub>1, fg\<^sub>0h\<^sub>1.p\<^sub>1] \<cdot> \<a>[tab\<^sub>0 f, f\<^sub>0g\<^sub>1.p\<^sub>1, fg\<^sub>0h\<^sub>1.p\<^sub>1] \<cdot> (local.inv f\<^sub>0g\<^sub>1.\<phi> \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>1 g, f\<^sub>0g\<^sub>1.p\<^sub>0, fg\<^sub>0h\<^sub>1.p\<^sub>1]
[PROOF STEP]
also
[PROOF STATE]
proof (state)
this:
(f\<^sub>0g\<^sub>1.\<phi> \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>0 f, f\<^sub>0g\<^sub>1.p\<^sub>1, fg\<^sub>0h\<^sub>1.p\<^sub>1] \<cdot> \<a>[tab\<^sub>0 f, f\<^sub>0g\<^sub>1.p\<^sub>1, fg\<^sub>0h\<^sub>1.p\<^sub>1] \<cdot> (local.inv f\<^sub>0g\<^sub>1.\<phi> \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>1 g, f\<^sub>0g\<^sub>1.p\<^sub>0, fg\<^sub>0h\<^sub>1.p\<^sub>1] = ((f\<^sub>0g\<^sub>1.\<phi> \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> (\<a>\<^sup>-\<^sup>1[tab\<^sub>0 f, f\<^sub>0g\<^sub>1.p\<^sub>1, fg\<^sub>0h\<^sub>1.p\<^sub>1] \<cdot> \<a>[tab\<^sub>0 f, f\<^sub>0g\<^sub>1.p\<^sub>1, fg\<^sub>0h\<^sub>1.p\<^sub>1]) \<cdot> (local.inv f\<^sub>0g\<^sub>1.\<phi> \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1)) \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>1 g, f\<^sub>0g\<^sub>1.p\<^sub>0, fg\<^sub>0h\<^sub>1.p\<^sub>1]
goal (1 subgoal):
1. (g \<star> \<a>[h, tab\<^sub>0 h, fg\<^sub>0h\<^sub>1.p\<^sub>0]) \<cdot> (g \<star> h.tab \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0) \<cdot> (g \<star> fg\<^sub>0h\<^sub>1.\<phi>) \<cdot> \<a>[g, tab\<^sub>0 g \<star> f\<^sub>0g\<^sub>1.p\<^sub>0, fg\<^sub>0h\<^sub>1.p\<^sub>1] \<cdot> (\<a>[g, tab\<^sub>0 g, f\<^sub>0g\<^sub>1.p\<^sub>0] \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> ((g.tab \<star> f\<^sub>0g\<^sub>1.p\<^sub>0) \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>1 g, f\<^sub>0g\<^sub>1.p\<^sub>0, fg\<^sub>0h\<^sub>1.p\<^sub>1] = (g \<star> \<a>[h, tab\<^sub>0 h, fg\<^sub>0h\<^sub>1.p\<^sub>0]) \<cdot> (g \<star> h.tab \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0) \<cdot> (g \<star> fg\<^sub>0h\<^sub>1.\<phi>) \<cdot> \<a>[g, tab\<^sub>0 g \<star> f\<^sub>0g\<^sub>1.p\<^sub>0, fg\<^sub>0h\<^sub>1.p\<^sub>1] \<cdot> (\<a>[g, tab\<^sub>0 g, f\<^sub>0g\<^sub>1.p\<^sub>0] \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> ((g.tab \<star> f\<^sub>0g\<^sub>1.p\<^sub>0) \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> (f\<^sub>0g\<^sub>1.\<phi> \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>0 f, f\<^sub>0g\<^sub>1.p\<^sub>1, fg\<^sub>0h\<^sub>1.p\<^sub>1] \<cdot> \<a>[tab\<^sub>0 f, f\<^sub>0g\<^sub>1.p\<^sub>1, fg\<^sub>0h\<^sub>1.p\<^sub>1] \<cdot> (local.inv f\<^sub>0g\<^sub>1.\<phi> \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>1 g, f\<^sub>0g\<^sub>1.p\<^sub>0, fg\<^sub>0h\<^sub>1.p\<^sub>1]
[PROOF STEP]
have "... = ((f\<^sub>0g\<^sub>1.\<phi> \<star> TTfgh.p\<^sub>1) \<cdot>
((tab\<^sub>0 f \<star> Tfg.p\<^sub>1) \<star> TTfgh.p\<^sub>1) \<cdot>
(inv f\<^sub>0g\<^sub>1.\<phi> \<star> TTfgh.p\<^sub>1)) \<cdot>
\<a>\<^sup>-\<^sup>1[tab\<^sub>1 g, Tfg.p\<^sub>0, TTfgh.p\<^sub>1]"
[PROOF STATE]
proof (prove)
goal (1 subgoal):
1. ((f\<^sub>0g\<^sub>1.\<phi> \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> (\<a>\<^sup>-\<^sup>1[tab\<^sub>0 f, f\<^sub>0g\<^sub>1.p\<^sub>1, fg\<^sub>0h\<^sub>1.p\<^sub>1] \<cdot> \<a>[tab\<^sub>0 f, f\<^sub>0g\<^sub>1.p\<^sub>1, fg\<^sub>0h\<^sub>1.p\<^sub>1]) \<cdot> (local.inv f\<^sub>0g\<^sub>1.\<phi> \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1)) \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>1 g, f\<^sub>0g\<^sub>1.p\<^sub>0, fg\<^sub>0h\<^sub>1.p\<^sub>1] = ((f\<^sub>0g\<^sub>1.\<phi> \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> ((tab\<^sub>0 f \<star> f\<^sub>0g\<^sub>1.p\<^sub>1) \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> (local.inv f\<^sub>0g\<^sub>1.\<phi> \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1)) \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>1 g, f\<^sub>0g\<^sub>1.p\<^sub>0, fg\<^sub>0h\<^sub>1.p\<^sub>1]
[PROOF STEP]
using fg\<^sub>0h\<^sub>1.p\<^sub>1_simps whisker_right comp_assoc_assoc'
[PROOF STATE]
proof (prove)
using this:
trg fg\<^sub>0h\<^sub>1.p\<^sub>1 = src (tab\<^sub>0 g \<star> f\<^sub>0g\<^sub>1.p\<^sub>0)
\<lbrakk>ide ?f; seq ?\<nu> ?\<mu>\<rbrakk> \<Longrightarrow> ?\<nu> \<cdot> ?\<mu> \<star> ?f = (?\<nu> \<star> ?f) \<cdot> (?\<mu> \<star> ?f)
\<lbrakk>ide ?f; ide ?g; ide ?h; src ?f = trg ?g; src ?g = trg ?h\<rbrakk> \<Longrightarrow> \<a>[?f, ?g, ?h] \<cdot> \<a>\<^sup>-\<^sup>1[?f, ?g, ?h] = ?f \<star> ?g \<star> ?h
\<lbrakk>ide ?f; ide ?g; ide ?h; src ?f = trg ?g; src ?g = trg ?h\<rbrakk> \<Longrightarrow> \<a>\<^sup>-\<^sup>1[?f, ?g, ?h] \<cdot> \<a>[?f, ?g, ?h] = (?f \<star> ?g) \<star> ?h
goal (1 subgoal):
1. ((f\<^sub>0g\<^sub>1.\<phi> \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> (\<a>\<^sup>-\<^sup>1[tab\<^sub>0 f, f\<^sub>0g\<^sub>1.p\<^sub>1, fg\<^sub>0h\<^sub>1.p\<^sub>1] \<cdot> \<a>[tab\<^sub>0 f, f\<^sub>0g\<^sub>1.p\<^sub>1, fg\<^sub>0h\<^sub>1.p\<^sub>1]) \<cdot> (local.inv f\<^sub>0g\<^sub>1.\<phi> \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1)) \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>1 g, f\<^sub>0g\<^sub>1.p\<^sub>0, fg\<^sub>0h\<^sub>1.p\<^sub>1] = ((f\<^sub>0g\<^sub>1.\<phi> \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> ((tab\<^sub>0 f \<star> f\<^sub>0g\<^sub>1.p\<^sub>1) \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> (local.inv f\<^sub>0g\<^sub>1.\<phi> \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1)) \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>1 g, f\<^sub>0g\<^sub>1.p\<^sub>0, fg\<^sub>0h\<^sub>1.p\<^sub>1]
[PROOF STEP]
by simp
[PROOF STATE]
proof (state)
this:
((f\<^sub>0g\<^sub>1.\<phi> \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> (\<a>\<^sup>-\<^sup>1[tab\<^sub>0 f, f\<^sub>0g\<^sub>1.p\<^sub>1, fg\<^sub>0h\<^sub>1.p\<^sub>1] \<cdot> \<a>[tab\<^sub>0 f, f\<^sub>0g\<^sub>1.p\<^sub>1, fg\<^sub>0h\<^sub>1.p\<^sub>1]) \<cdot> (local.inv f\<^sub>0g\<^sub>1.\<phi> \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1)) \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>1 g, f\<^sub>0g\<^sub>1.p\<^sub>0, fg\<^sub>0h\<^sub>1.p\<^sub>1] = ((f\<^sub>0g\<^sub>1.\<phi> \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> ((tab\<^sub>0 f \<star> f\<^sub>0g\<^sub>1.p\<^sub>1) \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> (local.inv f\<^sub>0g\<^sub>1.\<phi> \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1)) \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>1 g, f\<^sub>0g\<^sub>1.p\<^sub>0, fg\<^sub>0h\<^sub>1.p\<^sub>1]
goal (1 subgoal):
1. (g \<star> \<a>[h, tab\<^sub>0 h, fg\<^sub>0h\<^sub>1.p\<^sub>0]) \<cdot> (g \<star> h.tab \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0) \<cdot> (g \<star> fg\<^sub>0h\<^sub>1.\<phi>) \<cdot> \<a>[g, tab\<^sub>0 g \<star> f\<^sub>0g\<^sub>1.p\<^sub>0, fg\<^sub>0h\<^sub>1.p\<^sub>1] \<cdot> (\<a>[g, tab\<^sub>0 g, f\<^sub>0g\<^sub>1.p\<^sub>0] \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> ((g.tab \<star> f\<^sub>0g\<^sub>1.p\<^sub>0) \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>1 g, f\<^sub>0g\<^sub>1.p\<^sub>0, fg\<^sub>0h\<^sub>1.p\<^sub>1] = (g \<star> \<a>[h, tab\<^sub>0 h, fg\<^sub>0h\<^sub>1.p\<^sub>0]) \<cdot> (g \<star> h.tab \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0) \<cdot> (g \<star> fg\<^sub>0h\<^sub>1.\<phi>) \<cdot> \<a>[g, tab\<^sub>0 g \<star> f\<^sub>0g\<^sub>1.p\<^sub>0, fg\<^sub>0h\<^sub>1.p\<^sub>1] \<cdot> (\<a>[g, tab\<^sub>0 g, f\<^sub>0g\<^sub>1.p\<^sub>0] \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> ((g.tab \<star> f\<^sub>0g\<^sub>1.p\<^sub>0) \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> (f\<^sub>0g\<^sub>1.\<phi> \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>0 f, f\<^sub>0g\<^sub>1.p\<^sub>1, fg\<^sub>0h\<^sub>1.p\<^sub>1] \<cdot> \<a>[tab\<^sub>0 f, f\<^sub>0g\<^sub>1.p\<^sub>1, fg\<^sub>0h\<^sub>1.p\<^sub>1] \<cdot> (local.inv f\<^sub>0g\<^sub>1.\<phi> \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>1 g, f\<^sub>0g\<^sub>1.p\<^sub>0, fg\<^sub>0h\<^sub>1.p\<^sub>1]
[PROOF STEP]
also
[PROOF STATE]
proof (state)
this:
((f\<^sub>0g\<^sub>1.\<phi> \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> (\<a>\<^sup>-\<^sup>1[tab\<^sub>0 f, f\<^sub>0g\<^sub>1.p\<^sub>1, fg\<^sub>0h\<^sub>1.p\<^sub>1] \<cdot> \<a>[tab\<^sub>0 f, f\<^sub>0g\<^sub>1.p\<^sub>1, fg\<^sub>0h\<^sub>1.p\<^sub>1]) \<cdot> (local.inv f\<^sub>0g\<^sub>1.\<phi> \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1)) \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>1 g, f\<^sub>0g\<^sub>1.p\<^sub>0, fg\<^sub>0h\<^sub>1.p\<^sub>1] = ((f\<^sub>0g\<^sub>1.\<phi> \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> ((tab\<^sub>0 f \<star> f\<^sub>0g\<^sub>1.p\<^sub>1) \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> (local.inv f\<^sub>0g\<^sub>1.\<phi> \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1)) \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>1 g, f\<^sub>0g\<^sub>1.p\<^sub>0, fg\<^sub>0h\<^sub>1.p\<^sub>1]
goal (1 subgoal):
1. (g \<star> \<a>[h, tab\<^sub>0 h, fg\<^sub>0h\<^sub>1.p\<^sub>0]) \<cdot> (g \<star> h.tab \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0) \<cdot> (g \<star> fg\<^sub>0h\<^sub>1.\<phi>) \<cdot> \<a>[g, tab\<^sub>0 g \<star> f\<^sub>0g\<^sub>1.p\<^sub>0, fg\<^sub>0h\<^sub>1.p\<^sub>1] \<cdot> (\<a>[g, tab\<^sub>0 g, f\<^sub>0g\<^sub>1.p\<^sub>0] \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> ((g.tab \<star> f\<^sub>0g\<^sub>1.p\<^sub>0) \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>1 g, f\<^sub>0g\<^sub>1.p\<^sub>0, fg\<^sub>0h\<^sub>1.p\<^sub>1] = (g \<star> \<a>[h, tab\<^sub>0 h, fg\<^sub>0h\<^sub>1.p\<^sub>0]) \<cdot> (g \<star> h.tab \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0) \<cdot> (g \<star> fg\<^sub>0h\<^sub>1.\<phi>) \<cdot> \<a>[g, tab\<^sub>0 g \<star> f\<^sub>0g\<^sub>1.p\<^sub>0, fg\<^sub>0h\<^sub>1.p\<^sub>1] \<cdot> (\<a>[g, tab\<^sub>0 g, f\<^sub>0g\<^sub>1.p\<^sub>0] \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> ((g.tab \<star> f\<^sub>0g\<^sub>1.p\<^sub>0) \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> (f\<^sub>0g\<^sub>1.\<phi> \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>0 f, f\<^sub>0g\<^sub>1.p\<^sub>1, fg\<^sub>0h\<^sub>1.p\<^sub>1] \<cdot> \<a>[tab\<^sub>0 f, f\<^sub>0g\<^sub>1.p\<^sub>1, fg\<^sub>0h\<^sub>1.p\<^sub>1] \<cdot> (local.inv f\<^sub>0g\<^sub>1.\<phi> \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>1 g, f\<^sub>0g\<^sub>1.p\<^sub>0, fg\<^sub>0h\<^sub>1.p\<^sub>1]
[PROOF STEP]
have "... = ((f\<^sub>0g\<^sub>1.\<phi> \<star> TTfgh.p\<^sub>1) \<cdot>
(inv f\<^sub>0g\<^sub>1.\<phi> \<star> TTfgh.p\<^sub>1)) \<cdot>
\<a>\<^sup>-\<^sup>1[tab\<^sub>1 g, Tfg.p\<^sub>0, TTfgh.p\<^sub>1]"
[PROOF STATE]
proof (prove)
goal (1 subgoal):
1. ((f\<^sub>0g\<^sub>1.\<phi> \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> ((tab\<^sub>0 f \<star> f\<^sub>0g\<^sub>1.p\<^sub>1) \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> (local.inv f\<^sub>0g\<^sub>1.\<phi> \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1)) \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>1 g, f\<^sub>0g\<^sub>1.p\<^sub>0, fg\<^sub>0h\<^sub>1.p\<^sub>1] = ((f\<^sub>0g\<^sub>1.\<phi> \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> (local.inv f\<^sub>0g\<^sub>1.\<phi> \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1)) \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>1 g, f\<^sub>0g\<^sub>1.p\<^sub>0, fg\<^sub>0h\<^sub>1.p\<^sub>1]
[PROOF STEP]
using fg\<^sub>0h\<^sub>1.p\<^sub>1_simps f\<^sub>0g\<^sub>1.\<phi>_uniqueness comp_cod_arr
[PROOF STATE]
proof (prove)
using this:
trg fg\<^sub>0h\<^sub>1.p\<^sub>1 = src (tab\<^sub>0 g \<star> f\<^sub>0g\<^sub>1.p\<^sub>0)
\<guillemotleft>?\<mu> : tab\<^sub>0 f \<star> f\<^sub>0g\<^sub>1.p\<^sub>1 \<rightarrow> tab\<^sub>1 g \<star> f\<^sub>0g\<^sub>1.p\<^sub>0\<guillemotright> \<Longrightarrow> ?\<mu> = f\<^sub>0g\<^sub>1.\<phi>
local.iso f\<^sub>0g\<^sub>1.\<phi>
\<lbrakk>arr ?f; cod ?f = ?b\<rbrakk> \<Longrightarrow> ?b \<cdot> ?f = ?f
goal (1 subgoal):
1. ((f\<^sub>0g\<^sub>1.\<phi> \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> ((tab\<^sub>0 f \<star> f\<^sub>0g\<^sub>1.p\<^sub>1) \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> (local.inv f\<^sub>0g\<^sub>1.\<phi> \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1)) \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>1 g, f\<^sub>0g\<^sub>1.p\<^sub>0, fg\<^sub>0h\<^sub>1.p\<^sub>1] = ((f\<^sub>0g\<^sub>1.\<phi> \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> (local.inv f\<^sub>0g\<^sub>1.\<phi> \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1)) \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>1 g, f\<^sub>0g\<^sub>1.p\<^sub>0, fg\<^sub>0h\<^sub>1.p\<^sub>1]
[PROOF STEP]
by simp
[PROOF STATE]
proof (state)
this:
((f\<^sub>0g\<^sub>1.\<phi> \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> ((tab\<^sub>0 f \<star> f\<^sub>0g\<^sub>1.p\<^sub>1) \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> (local.inv f\<^sub>0g\<^sub>1.\<phi> \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1)) \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>1 g, f\<^sub>0g\<^sub>1.p\<^sub>0, fg\<^sub>0h\<^sub>1.p\<^sub>1] = ((f\<^sub>0g\<^sub>1.\<phi> \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> (local.inv f\<^sub>0g\<^sub>1.\<phi> \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1)) \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>1 g, f\<^sub>0g\<^sub>1.p\<^sub>0, fg\<^sub>0h\<^sub>1.p\<^sub>1]
goal (1 subgoal):
1. (g \<star> \<a>[h, tab\<^sub>0 h, fg\<^sub>0h\<^sub>1.p\<^sub>0]) \<cdot> (g \<star> h.tab \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0) \<cdot> (g \<star> fg\<^sub>0h\<^sub>1.\<phi>) \<cdot> \<a>[g, tab\<^sub>0 g \<star> f\<^sub>0g\<^sub>1.p\<^sub>0, fg\<^sub>0h\<^sub>1.p\<^sub>1] \<cdot> (\<a>[g, tab\<^sub>0 g, f\<^sub>0g\<^sub>1.p\<^sub>0] \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> ((g.tab \<star> f\<^sub>0g\<^sub>1.p\<^sub>0) \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>1 g, f\<^sub>0g\<^sub>1.p\<^sub>0, fg\<^sub>0h\<^sub>1.p\<^sub>1] = (g \<star> \<a>[h, tab\<^sub>0 h, fg\<^sub>0h\<^sub>1.p\<^sub>0]) \<cdot> (g \<star> h.tab \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0) \<cdot> (g \<star> fg\<^sub>0h\<^sub>1.\<phi>) \<cdot> \<a>[g, tab\<^sub>0 g \<star> f\<^sub>0g\<^sub>1.p\<^sub>0, fg\<^sub>0h\<^sub>1.p\<^sub>1] \<cdot> (\<a>[g, tab\<^sub>0 g, f\<^sub>0g\<^sub>1.p\<^sub>0] \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> ((g.tab \<star> f\<^sub>0g\<^sub>1.p\<^sub>0) \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> (f\<^sub>0g\<^sub>1.\<phi> \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>0 f, f\<^sub>0g\<^sub>1.p\<^sub>1, fg\<^sub>0h\<^sub>1.p\<^sub>1] \<cdot> \<a>[tab\<^sub>0 f, f\<^sub>0g\<^sub>1.p\<^sub>1, fg\<^sub>0h\<^sub>1.p\<^sub>1] \<cdot> (local.inv f\<^sub>0g\<^sub>1.\<phi> \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>1 g, f\<^sub>0g\<^sub>1.p\<^sub>0, fg\<^sub>0h\<^sub>1.p\<^sub>1]
[PROOF STEP]
also
[PROOF STATE]
proof (state)
this:
((f\<^sub>0g\<^sub>1.\<phi> \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> ((tab\<^sub>0 f \<star> f\<^sub>0g\<^sub>1.p\<^sub>1) \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> (local.inv f\<^sub>0g\<^sub>1.\<phi> \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1)) \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>1 g, f\<^sub>0g\<^sub>1.p\<^sub>0, fg\<^sub>0h\<^sub>1.p\<^sub>1] = ((f\<^sub>0g\<^sub>1.\<phi> \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> (local.inv f\<^sub>0g\<^sub>1.\<phi> \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1)) \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>1 g, f\<^sub>0g\<^sub>1.p\<^sub>0, fg\<^sub>0h\<^sub>1.p\<^sub>1]
goal (1 subgoal):
1. (g \<star> \<a>[h, tab\<^sub>0 h, fg\<^sub>0h\<^sub>1.p\<^sub>0]) \<cdot> (g \<star> h.tab \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0) \<cdot> (g \<star> fg\<^sub>0h\<^sub>1.\<phi>) \<cdot> \<a>[g, tab\<^sub>0 g \<star> f\<^sub>0g\<^sub>1.p\<^sub>0, fg\<^sub>0h\<^sub>1.p\<^sub>1] \<cdot> (\<a>[g, tab\<^sub>0 g, f\<^sub>0g\<^sub>1.p\<^sub>0] \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> ((g.tab \<star> f\<^sub>0g\<^sub>1.p\<^sub>0) \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>1 g, f\<^sub>0g\<^sub>1.p\<^sub>0, fg\<^sub>0h\<^sub>1.p\<^sub>1] = (g \<star> \<a>[h, tab\<^sub>0 h, fg\<^sub>0h\<^sub>1.p\<^sub>0]) \<cdot> (g \<star> h.tab \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0) \<cdot> (g \<star> fg\<^sub>0h\<^sub>1.\<phi>) \<cdot> \<a>[g, tab\<^sub>0 g \<star> f\<^sub>0g\<^sub>1.p\<^sub>0, fg\<^sub>0h\<^sub>1.p\<^sub>1] \<cdot> (\<a>[g, tab\<^sub>0 g, f\<^sub>0g\<^sub>1.p\<^sub>0] \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> ((g.tab \<star> f\<^sub>0g\<^sub>1.p\<^sub>0) \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> (f\<^sub>0g\<^sub>1.\<phi> \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>0 f, f\<^sub>0g\<^sub>1.p\<^sub>1, fg\<^sub>0h\<^sub>1.p\<^sub>1] \<cdot> \<a>[tab\<^sub>0 f, f\<^sub>0g\<^sub>1.p\<^sub>1, fg\<^sub>0h\<^sub>1.p\<^sub>1] \<cdot> (local.inv f\<^sub>0g\<^sub>1.\<phi> \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>1 g, f\<^sub>0g\<^sub>1.p\<^sub>0, fg\<^sub>0h\<^sub>1.p\<^sub>1]
[PROOF STEP]
have "... = ((tab\<^sub>1 g \<star> Tfg.p\<^sub>0) \<star> TTfgh.p\<^sub>1) \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>1 g, Tfg.p\<^sub>0, TTfgh.p\<^sub>1]"
[PROOF STATE]
proof (prove)
goal (1 subgoal):
1. ((f\<^sub>0g\<^sub>1.\<phi> \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> (local.inv f\<^sub>0g\<^sub>1.\<phi> \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1)) \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>1 g, f\<^sub>0g\<^sub>1.p\<^sub>0, fg\<^sub>0h\<^sub>1.p\<^sub>1] = ((tab\<^sub>1 g \<star> f\<^sub>0g\<^sub>1.p\<^sub>0) \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>1 g, f\<^sub>0g\<^sub>1.p\<^sub>0, fg\<^sub>0h\<^sub>1.p\<^sub>1]
[PROOF STEP]
proof -
[PROOF STATE]
proof (state)
goal (1 subgoal):
1. ((f\<^sub>0g\<^sub>1.\<phi> \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> (local.inv f\<^sub>0g\<^sub>1.\<phi> \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1)) \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>1 g, f\<^sub>0g\<^sub>1.p\<^sub>0, fg\<^sub>0h\<^sub>1.p\<^sub>1] = ((tab\<^sub>1 g \<star> f\<^sub>0g\<^sub>1.p\<^sub>0) \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>1 g, f\<^sub>0g\<^sub>1.p\<^sub>0, fg\<^sub>0h\<^sub>1.p\<^sub>1]
[PROOF STEP]
have "(f\<^sub>0g\<^sub>1.\<phi> \<star> TTfgh.p\<^sub>1) \<cdot> (inv f\<^sub>0g\<^sub>1.\<phi> \<star> TTfgh.p\<^sub>1) =
f\<^sub>0g\<^sub>1.\<phi> \<cdot> inv f\<^sub>0g\<^sub>1.\<phi> \<star> TTfgh.p\<^sub>1"
[PROOF STATE]
proof (prove)
goal (1 subgoal):
1. (f\<^sub>0g\<^sub>1.\<phi> \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> (local.inv f\<^sub>0g\<^sub>1.\<phi> \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) = f\<^sub>0g\<^sub>1.\<phi> \<cdot> local.inv f\<^sub>0g\<^sub>1.\<phi> \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1
[PROOF STEP]
using f\<^sub>0g\<^sub>1.\<phi>_uniqueness whisker_right
[PROOF STATE]
proof (prove)
using this:
\<guillemotleft>?\<mu> : tab\<^sub>0 f \<star> f\<^sub>0g\<^sub>1.p\<^sub>1 \<rightarrow> tab\<^sub>1 g \<star> f\<^sub>0g\<^sub>1.p\<^sub>0\<guillemotright> \<Longrightarrow> ?\<mu> = f\<^sub>0g\<^sub>1.\<phi>
local.iso f\<^sub>0g\<^sub>1.\<phi>
\<lbrakk>ide ?f; seq ?\<nu> ?\<mu>\<rbrakk> \<Longrightarrow> ?\<nu> \<cdot> ?\<mu> \<star> ?f = (?\<nu> \<star> ?f) \<cdot> (?\<mu> \<star> ?f)
goal (1 subgoal):
1. (f\<^sub>0g\<^sub>1.\<phi> \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> (local.inv f\<^sub>0g\<^sub>1.\<phi> \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) = f\<^sub>0g\<^sub>1.\<phi> \<cdot> local.inv f\<^sub>0g\<^sub>1.\<phi> \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1
[PROOF STEP]
by simp
[PROOF STATE]
proof (state)
this:
(f\<^sub>0g\<^sub>1.\<phi> \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> (local.inv f\<^sub>0g\<^sub>1.\<phi> \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) = f\<^sub>0g\<^sub>1.\<phi> \<cdot> local.inv f\<^sub>0g\<^sub>1.\<phi> \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1
goal (1 subgoal):
1. ((f\<^sub>0g\<^sub>1.\<phi> \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> (local.inv f\<^sub>0g\<^sub>1.\<phi> \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1)) \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>1 g, f\<^sub>0g\<^sub>1.p\<^sub>0, fg\<^sub>0h\<^sub>1.p\<^sub>1] = ((tab\<^sub>1 g \<star> f\<^sub>0g\<^sub>1.p\<^sub>0) \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>1 g, f\<^sub>0g\<^sub>1.p\<^sub>0, fg\<^sub>0h\<^sub>1.p\<^sub>1]
[PROOF STEP]
also
[PROOF STATE]
proof (state)
this:
(f\<^sub>0g\<^sub>1.\<phi> \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> (local.inv f\<^sub>0g\<^sub>1.\<phi> \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) = f\<^sub>0g\<^sub>1.\<phi> \<cdot> local.inv f\<^sub>0g\<^sub>1.\<phi> \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1
goal (1 subgoal):
1. ((f\<^sub>0g\<^sub>1.\<phi> \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> (local.inv f\<^sub>0g\<^sub>1.\<phi> \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1)) \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>1 g, f\<^sub>0g\<^sub>1.p\<^sub>0, fg\<^sub>0h\<^sub>1.p\<^sub>1] = ((tab\<^sub>1 g \<star> f\<^sub>0g\<^sub>1.p\<^sub>0) \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>1 g, f\<^sub>0g\<^sub>1.p\<^sub>0, fg\<^sub>0h\<^sub>1.p\<^sub>1]
[PROOF STEP]
have "... = (tab\<^sub>1 g \<star> Tfg.p\<^sub>0) \<star> TTfgh.p\<^sub>1"
[PROOF STATE]
proof (prove)
goal (1 subgoal):
1. f\<^sub>0g\<^sub>1.\<phi> \<cdot> local.inv f\<^sub>0g\<^sub>1.\<phi> \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1 = (tab\<^sub>1 g \<star> f\<^sub>0g\<^sub>1.p\<^sub>0) \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1
[PROOF STEP]
using f\<^sub>0g\<^sub>1.\<phi>_uniqueness comp_arr_inv'
[PROOF STATE]
proof (prove)
using this:
\<guillemotleft>?\<mu> : tab\<^sub>0 f \<star> f\<^sub>0g\<^sub>1.p\<^sub>1 \<rightarrow> tab\<^sub>1 g \<star> f\<^sub>0g\<^sub>1.p\<^sub>0\<guillemotright> \<Longrightarrow> ?\<mu> = f\<^sub>0g\<^sub>1.\<phi>
local.iso f\<^sub>0g\<^sub>1.\<phi>
local.iso ?f \<Longrightarrow> ?f \<cdot> local.inv ?f = cod ?f
goal (1 subgoal):
1. f\<^sub>0g\<^sub>1.\<phi> \<cdot> local.inv f\<^sub>0g\<^sub>1.\<phi> \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1 = (tab\<^sub>1 g \<star> f\<^sub>0g\<^sub>1.p\<^sub>0) \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1
[PROOF STEP]
by simp
[PROOF STATE]
proof (state)
this:
f\<^sub>0g\<^sub>1.\<phi> \<cdot> local.inv f\<^sub>0g\<^sub>1.\<phi> \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1 = (tab\<^sub>1 g \<star> f\<^sub>0g\<^sub>1.p\<^sub>0) \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1
goal (1 subgoal):
1. ((f\<^sub>0g\<^sub>1.\<phi> \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> (local.inv f\<^sub>0g\<^sub>1.\<phi> \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1)) \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>1 g, f\<^sub>0g\<^sub>1.p\<^sub>0, fg\<^sub>0h\<^sub>1.p\<^sub>1] = ((tab\<^sub>1 g \<star> f\<^sub>0g\<^sub>1.p\<^sub>0) \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>1 g, f\<^sub>0g\<^sub>1.p\<^sub>0, fg\<^sub>0h\<^sub>1.p\<^sub>1]
[PROOF STEP]
finally
[PROOF STATE]
proof (chain)
picking this:
(f\<^sub>0g\<^sub>1.\<phi> \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> (local.inv f\<^sub>0g\<^sub>1.\<phi> \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) = (tab\<^sub>1 g \<star> f\<^sub>0g\<^sub>1.p\<^sub>0) \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1
[PROOF STEP]
show ?thesis
[PROOF STATE]
proof (prove)
using this:
(f\<^sub>0g\<^sub>1.\<phi> \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> (local.inv f\<^sub>0g\<^sub>1.\<phi> \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) = (tab\<^sub>1 g \<star> f\<^sub>0g\<^sub>1.p\<^sub>0) \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1
goal (1 subgoal):
1. ((f\<^sub>0g\<^sub>1.\<phi> \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> (local.inv f\<^sub>0g\<^sub>1.\<phi> \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1)) \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>1 g, f\<^sub>0g\<^sub>1.p\<^sub>0, fg\<^sub>0h\<^sub>1.p\<^sub>1] = ((tab\<^sub>1 g \<star> f\<^sub>0g\<^sub>1.p\<^sub>0) \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>1 g, f\<^sub>0g\<^sub>1.p\<^sub>0, fg\<^sub>0h\<^sub>1.p\<^sub>1]
[PROOF STEP]
by simp
[PROOF STATE]
proof (state)
this:
((f\<^sub>0g\<^sub>1.\<phi> \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> (local.inv f\<^sub>0g\<^sub>1.\<phi> \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1)) \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>1 g, f\<^sub>0g\<^sub>1.p\<^sub>0, fg\<^sub>0h\<^sub>1.p\<^sub>1] = ((tab\<^sub>1 g \<star> f\<^sub>0g\<^sub>1.p\<^sub>0) \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>1 g, f\<^sub>0g\<^sub>1.p\<^sub>0, fg\<^sub>0h\<^sub>1.p\<^sub>1]
goal:
No subgoals!
[PROOF STEP]
qed
[PROOF STATE]
proof (state)
this:
((f\<^sub>0g\<^sub>1.\<phi> \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> (local.inv f\<^sub>0g\<^sub>1.\<phi> \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1)) \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>1 g, f\<^sub>0g\<^sub>1.p\<^sub>0, fg\<^sub>0h\<^sub>1.p\<^sub>1] = ((tab\<^sub>1 g \<star> f\<^sub>0g\<^sub>1.p\<^sub>0) \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>1 g, f\<^sub>0g\<^sub>1.p\<^sub>0, fg\<^sub>0h\<^sub>1.p\<^sub>1]
goal (1 subgoal):
1. (g \<star> \<a>[h, tab\<^sub>0 h, fg\<^sub>0h\<^sub>1.p\<^sub>0]) \<cdot> (g \<star> h.tab \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0) \<cdot> (g \<star> fg\<^sub>0h\<^sub>1.\<phi>) \<cdot> \<a>[g, tab\<^sub>0 g \<star> f\<^sub>0g\<^sub>1.p\<^sub>0, fg\<^sub>0h\<^sub>1.p\<^sub>1] \<cdot> (\<a>[g, tab\<^sub>0 g, f\<^sub>0g\<^sub>1.p\<^sub>0] \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> ((g.tab \<star> f\<^sub>0g\<^sub>1.p\<^sub>0) \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>1 g, f\<^sub>0g\<^sub>1.p\<^sub>0, fg\<^sub>0h\<^sub>1.p\<^sub>1] = (g \<star> \<a>[h, tab\<^sub>0 h, fg\<^sub>0h\<^sub>1.p\<^sub>0]) \<cdot> (g \<star> h.tab \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0) \<cdot> (g \<star> fg\<^sub>0h\<^sub>1.\<phi>) \<cdot> \<a>[g, tab\<^sub>0 g \<star> f\<^sub>0g\<^sub>1.p\<^sub>0, fg\<^sub>0h\<^sub>1.p\<^sub>1] \<cdot> (\<a>[g, tab\<^sub>0 g, f\<^sub>0g\<^sub>1.p\<^sub>0] \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> ((g.tab \<star> f\<^sub>0g\<^sub>1.p\<^sub>0) \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> (f\<^sub>0g\<^sub>1.\<phi> \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>0 f, f\<^sub>0g\<^sub>1.p\<^sub>1, fg\<^sub>0h\<^sub>1.p\<^sub>1] \<cdot> \<a>[tab\<^sub>0 f, f\<^sub>0g\<^sub>1.p\<^sub>1, fg\<^sub>0h\<^sub>1.p\<^sub>1] \<cdot> (local.inv f\<^sub>0g\<^sub>1.\<phi> \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>1 g, f\<^sub>0g\<^sub>1.p\<^sub>0, fg\<^sub>0h\<^sub>1.p\<^sub>1]
[PROOF STEP]
also
[PROOF STATE]
proof (state)
this:
((f\<^sub>0g\<^sub>1.\<phi> \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> (local.inv f\<^sub>0g\<^sub>1.\<phi> \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1)) \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>1 g, f\<^sub>0g\<^sub>1.p\<^sub>0, fg\<^sub>0h\<^sub>1.p\<^sub>1] = ((tab\<^sub>1 g \<star> f\<^sub>0g\<^sub>1.p\<^sub>0) \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>1 g, f\<^sub>0g\<^sub>1.p\<^sub>0, fg\<^sub>0h\<^sub>1.p\<^sub>1]
goal (1 subgoal):
1. (g \<star> \<a>[h, tab\<^sub>0 h, fg\<^sub>0h\<^sub>1.p\<^sub>0]) \<cdot> (g \<star> h.tab \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0) \<cdot> (g \<star> fg\<^sub>0h\<^sub>1.\<phi>) \<cdot> \<a>[g, tab\<^sub>0 g \<star> f\<^sub>0g\<^sub>1.p\<^sub>0, fg\<^sub>0h\<^sub>1.p\<^sub>1] \<cdot> (\<a>[g, tab\<^sub>0 g, f\<^sub>0g\<^sub>1.p\<^sub>0] \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> ((g.tab \<star> f\<^sub>0g\<^sub>1.p\<^sub>0) \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>1 g, f\<^sub>0g\<^sub>1.p\<^sub>0, fg\<^sub>0h\<^sub>1.p\<^sub>1] = (g \<star> \<a>[h, tab\<^sub>0 h, fg\<^sub>0h\<^sub>1.p\<^sub>0]) \<cdot> (g \<star> h.tab \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0) \<cdot> (g \<star> fg\<^sub>0h\<^sub>1.\<phi>) \<cdot> \<a>[g, tab\<^sub>0 g \<star> f\<^sub>0g\<^sub>1.p\<^sub>0, fg\<^sub>0h\<^sub>1.p\<^sub>1] \<cdot> (\<a>[g, tab\<^sub>0 g, f\<^sub>0g\<^sub>1.p\<^sub>0] \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> ((g.tab \<star> f\<^sub>0g\<^sub>1.p\<^sub>0) \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> (f\<^sub>0g\<^sub>1.\<phi> \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>0 f, f\<^sub>0g\<^sub>1.p\<^sub>1, fg\<^sub>0h\<^sub>1.p\<^sub>1] \<cdot> \<a>[tab\<^sub>0 f, f\<^sub>0g\<^sub>1.p\<^sub>1, fg\<^sub>0h\<^sub>1.p\<^sub>1] \<cdot> (local.inv f\<^sub>0g\<^sub>1.\<phi> \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>1 g, f\<^sub>0g\<^sub>1.p\<^sub>0, fg\<^sub>0h\<^sub>1.p\<^sub>1]
[PROOF STEP]
have "... = \<a>\<^sup>-\<^sup>1[tab\<^sub>1 g, Tfg.p\<^sub>0, TTfgh.p\<^sub>1]"
[PROOF STATE]
proof (prove)
goal (1 subgoal):
1. ((tab\<^sub>1 g \<star> f\<^sub>0g\<^sub>1.p\<^sub>0) \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>1 g, f\<^sub>0g\<^sub>1.p\<^sub>0, fg\<^sub>0h\<^sub>1.p\<^sub>1] = \<a>\<^sup>-\<^sup>1[tab\<^sub>1 g, f\<^sub>0g\<^sub>1.p\<^sub>0, fg\<^sub>0h\<^sub>1.p\<^sub>1]
[PROOF STEP]
using fg\<^sub>0h\<^sub>1.p\<^sub>1_simps comp_cod_arr
[PROOF STATE]
proof (prove)
using this:
trg fg\<^sub>0h\<^sub>1.p\<^sub>1 = src (tab\<^sub>0 g \<star> f\<^sub>0g\<^sub>1.p\<^sub>0)
\<lbrakk>arr ?f; cod ?f = ?b\<rbrakk> \<Longrightarrow> ?b \<cdot> ?f = ?f
goal (1 subgoal):
1. ((tab\<^sub>1 g \<star> f\<^sub>0g\<^sub>1.p\<^sub>0) \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>1 g, f\<^sub>0g\<^sub>1.p\<^sub>0, fg\<^sub>0h\<^sub>1.p\<^sub>1] = \<a>\<^sup>-\<^sup>1[tab\<^sub>1 g, f\<^sub>0g\<^sub>1.p\<^sub>0, fg\<^sub>0h\<^sub>1.p\<^sub>1]
[PROOF STEP]
by simp
[PROOF STATE]
proof (state)
this:
((tab\<^sub>1 g \<star> f\<^sub>0g\<^sub>1.p\<^sub>0) \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>1 g, f\<^sub>0g\<^sub>1.p\<^sub>0, fg\<^sub>0h\<^sub>1.p\<^sub>1] = \<a>\<^sup>-\<^sup>1[tab\<^sub>1 g, f\<^sub>0g\<^sub>1.p\<^sub>0, fg\<^sub>0h\<^sub>1.p\<^sub>1]
goal (1 subgoal):
1. (g \<star> \<a>[h, tab\<^sub>0 h, fg\<^sub>0h\<^sub>1.p\<^sub>0]) \<cdot> (g \<star> h.tab \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0) \<cdot> (g \<star> fg\<^sub>0h\<^sub>1.\<phi>) \<cdot> \<a>[g, tab\<^sub>0 g \<star> f\<^sub>0g\<^sub>1.p\<^sub>0, fg\<^sub>0h\<^sub>1.p\<^sub>1] \<cdot> (\<a>[g, tab\<^sub>0 g, f\<^sub>0g\<^sub>1.p\<^sub>0] \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> ((g.tab \<star> f\<^sub>0g\<^sub>1.p\<^sub>0) \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>1 g, f\<^sub>0g\<^sub>1.p\<^sub>0, fg\<^sub>0h\<^sub>1.p\<^sub>1] = (g \<star> \<a>[h, tab\<^sub>0 h, fg\<^sub>0h\<^sub>1.p\<^sub>0]) \<cdot> (g \<star> h.tab \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0) \<cdot> (g \<star> fg\<^sub>0h\<^sub>1.\<phi>) \<cdot> \<a>[g, tab\<^sub>0 g \<star> f\<^sub>0g\<^sub>1.p\<^sub>0, fg\<^sub>0h\<^sub>1.p\<^sub>1] \<cdot> (\<a>[g, tab\<^sub>0 g, f\<^sub>0g\<^sub>1.p\<^sub>0] \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> ((g.tab \<star> f\<^sub>0g\<^sub>1.p\<^sub>0) \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> (f\<^sub>0g\<^sub>1.\<phi> \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>0 f, f\<^sub>0g\<^sub>1.p\<^sub>1, fg\<^sub>0h\<^sub>1.p\<^sub>1] \<cdot> \<a>[tab\<^sub>0 f, f\<^sub>0g\<^sub>1.p\<^sub>1, fg\<^sub>0h\<^sub>1.p\<^sub>1] \<cdot> (local.inv f\<^sub>0g\<^sub>1.\<phi> \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>1 g, f\<^sub>0g\<^sub>1.p\<^sub>0, fg\<^sub>0h\<^sub>1.p\<^sub>1]
[PROOF STEP]
finally
[PROOF STATE]
proof (chain)
picking this:
(f\<^sub>0g\<^sub>1.\<phi> \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>0 f, f\<^sub>0g\<^sub>1.p\<^sub>1, fg\<^sub>0h\<^sub>1.p\<^sub>1] \<cdot> \<a>[tab\<^sub>0 f, f\<^sub>0g\<^sub>1.p\<^sub>1, fg\<^sub>0h\<^sub>1.p\<^sub>1] \<cdot> (local.inv f\<^sub>0g\<^sub>1.\<phi> \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>1 g, f\<^sub>0g\<^sub>1.p\<^sub>0, fg\<^sub>0h\<^sub>1.p\<^sub>1] = \<a>\<^sup>-\<^sup>1[tab\<^sub>1 g, f\<^sub>0g\<^sub>1.p\<^sub>0, fg\<^sub>0h\<^sub>1.p\<^sub>1]
[PROOF STEP]
have "(f\<^sub>0g\<^sub>1.\<phi> \<star> TTfgh.p\<^sub>1) \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>0 f, Tfg.p\<^sub>1, TTfgh.p\<^sub>1] \<cdot>
\<a>[tab\<^sub>0 f, Tfg.p\<^sub>1, TTfgh.p\<^sub>1] \<cdot> (inv f\<^sub>0g\<^sub>1.\<phi> \<star> TTfgh.p\<^sub>1) \<cdot>
\<a>\<^sup>-\<^sup>1[tab\<^sub>1 g, Tfg.p\<^sub>0, TTfgh.p\<^sub>1] = \<a>\<^sup>-\<^sup>1[tab\<^sub>1 g, Tfg.p\<^sub>0, TTfgh.p\<^sub>1]"
[PROOF STATE]
proof (prove)
using this:
(f\<^sub>0g\<^sub>1.\<phi> \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>0 f, f\<^sub>0g\<^sub>1.p\<^sub>1, fg\<^sub>0h\<^sub>1.p\<^sub>1] \<cdot> \<a>[tab\<^sub>0 f, f\<^sub>0g\<^sub>1.p\<^sub>1, fg\<^sub>0h\<^sub>1.p\<^sub>1] \<cdot> (local.inv f\<^sub>0g\<^sub>1.\<phi> \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>1 g, f\<^sub>0g\<^sub>1.p\<^sub>0, fg\<^sub>0h\<^sub>1.p\<^sub>1] = \<a>\<^sup>-\<^sup>1[tab\<^sub>1 g, f\<^sub>0g\<^sub>1.p\<^sub>0, fg\<^sub>0h\<^sub>1.p\<^sub>1]
goal (1 subgoal):
1. (f\<^sub>0g\<^sub>1.\<phi> \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>0 f, f\<^sub>0g\<^sub>1.p\<^sub>1, fg\<^sub>0h\<^sub>1.p\<^sub>1] \<cdot> \<a>[tab\<^sub>0 f, f\<^sub>0g\<^sub>1.p\<^sub>1, fg\<^sub>0h\<^sub>1.p\<^sub>1] \<cdot> (local.inv f\<^sub>0g\<^sub>1.\<phi> \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>1 g, f\<^sub>0g\<^sub>1.p\<^sub>0, fg\<^sub>0h\<^sub>1.p\<^sub>1] = \<a>\<^sup>-\<^sup>1[tab\<^sub>1 g, f\<^sub>0g\<^sub>1.p\<^sub>0, fg\<^sub>0h\<^sub>1.p\<^sub>1]
[PROOF STEP]
by simp
[PROOF STATE]
proof (state)
this:
(f\<^sub>0g\<^sub>1.\<phi> \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>0 f, f\<^sub>0g\<^sub>1.p\<^sub>1, fg\<^sub>0h\<^sub>1.p\<^sub>1] \<cdot> \<a>[tab\<^sub>0 f, f\<^sub>0g\<^sub>1.p\<^sub>1, fg\<^sub>0h\<^sub>1.p\<^sub>1] \<cdot> (local.inv f\<^sub>0g\<^sub>1.\<phi> \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>1 g, f\<^sub>0g\<^sub>1.p\<^sub>0, fg\<^sub>0h\<^sub>1.p\<^sub>1] = \<a>\<^sup>-\<^sup>1[tab\<^sub>1 g, f\<^sub>0g\<^sub>1.p\<^sub>0, fg\<^sub>0h\<^sub>1.p\<^sub>1]
goal (1 subgoal):
1. (g \<star> \<a>[h, tab\<^sub>0 h, fg\<^sub>0h\<^sub>1.p\<^sub>0]) \<cdot> (g \<star> h.tab \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0) \<cdot> (g \<star> fg\<^sub>0h\<^sub>1.\<phi>) \<cdot> \<a>[g, tab\<^sub>0 g \<star> f\<^sub>0g\<^sub>1.p\<^sub>0, fg\<^sub>0h\<^sub>1.p\<^sub>1] \<cdot> (\<a>[g, tab\<^sub>0 g, f\<^sub>0g\<^sub>1.p\<^sub>0] \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> ((g.tab \<star> f\<^sub>0g\<^sub>1.p\<^sub>0) \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>1 g, f\<^sub>0g\<^sub>1.p\<^sub>0, fg\<^sub>0h\<^sub>1.p\<^sub>1] = (g \<star> \<a>[h, tab\<^sub>0 h, fg\<^sub>0h\<^sub>1.p\<^sub>0]) \<cdot> (g \<star> h.tab \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0) \<cdot> (g \<star> fg\<^sub>0h\<^sub>1.\<phi>) \<cdot> \<a>[g, tab\<^sub>0 g \<star> f\<^sub>0g\<^sub>1.p\<^sub>0, fg\<^sub>0h\<^sub>1.p\<^sub>1] \<cdot> (\<a>[g, tab\<^sub>0 g, f\<^sub>0g\<^sub>1.p\<^sub>0] \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> ((g.tab \<star> f\<^sub>0g\<^sub>1.p\<^sub>0) \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> (f\<^sub>0g\<^sub>1.\<phi> \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>0 f, f\<^sub>0g\<^sub>1.p\<^sub>1, fg\<^sub>0h\<^sub>1.p\<^sub>1] \<cdot> \<a>[tab\<^sub>0 f, f\<^sub>0g\<^sub>1.p\<^sub>1, fg\<^sub>0h\<^sub>1.p\<^sub>1] \<cdot> (local.inv f\<^sub>0g\<^sub>1.\<phi> \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>1 g, f\<^sub>0g\<^sub>1.p\<^sub>0, fg\<^sub>0h\<^sub>1.p\<^sub>1]
[PROOF STEP]
thus ?thesis
[PROOF STATE]
proof (prove)
using this:
(f\<^sub>0g\<^sub>1.\<phi> \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>0 f, f\<^sub>0g\<^sub>1.p\<^sub>1, fg\<^sub>0h\<^sub>1.p\<^sub>1] \<cdot> \<a>[tab\<^sub>0 f, f\<^sub>0g\<^sub>1.p\<^sub>1, fg\<^sub>0h\<^sub>1.p\<^sub>1] \<cdot> (local.inv f\<^sub>0g\<^sub>1.\<phi> \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>1 g, f\<^sub>0g\<^sub>1.p\<^sub>0, fg\<^sub>0h\<^sub>1.p\<^sub>1] = \<a>\<^sup>-\<^sup>1[tab\<^sub>1 g, f\<^sub>0g\<^sub>1.p\<^sub>0, fg\<^sub>0h\<^sub>1.p\<^sub>1]
goal (1 subgoal):
1. (g \<star> \<a>[h, tab\<^sub>0 h, fg\<^sub>0h\<^sub>1.p\<^sub>0]) \<cdot> (g \<star> h.tab \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0) \<cdot> (g \<star> fg\<^sub>0h\<^sub>1.\<phi>) \<cdot> \<a>[g, tab\<^sub>0 g \<star> f\<^sub>0g\<^sub>1.p\<^sub>0, fg\<^sub>0h\<^sub>1.p\<^sub>1] \<cdot> (\<a>[g, tab\<^sub>0 g, f\<^sub>0g\<^sub>1.p\<^sub>0] \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> ((g.tab \<star> f\<^sub>0g\<^sub>1.p\<^sub>0) \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>1 g, f\<^sub>0g\<^sub>1.p\<^sub>0, fg\<^sub>0h\<^sub>1.p\<^sub>1] = (g \<star> \<a>[h, tab\<^sub>0 h, fg\<^sub>0h\<^sub>1.p\<^sub>0]) \<cdot> (g \<star> h.tab \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0) \<cdot> (g \<star> fg\<^sub>0h\<^sub>1.\<phi>) \<cdot> \<a>[g, tab\<^sub>0 g \<star> f\<^sub>0g\<^sub>1.p\<^sub>0, fg\<^sub>0h\<^sub>1.p\<^sub>1] \<cdot> (\<a>[g, tab\<^sub>0 g, f\<^sub>0g\<^sub>1.p\<^sub>0] \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> ((g.tab \<star> f\<^sub>0g\<^sub>1.p\<^sub>0) \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> (f\<^sub>0g\<^sub>1.\<phi> \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>0 f, f\<^sub>0g\<^sub>1.p\<^sub>1, fg\<^sub>0h\<^sub>1.p\<^sub>1] \<cdot> \<a>[tab\<^sub>0 f, f\<^sub>0g\<^sub>1.p\<^sub>1, fg\<^sub>0h\<^sub>1.p\<^sub>1] \<cdot> (local.inv f\<^sub>0g\<^sub>1.\<phi> \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>1 g, f\<^sub>0g\<^sub>1.p\<^sub>0, fg\<^sub>0h\<^sub>1.p\<^sub>1]
[PROOF STEP]
using comp_assoc
[PROOF STATE]
proof (prove)
using this:
(f\<^sub>0g\<^sub>1.\<phi> \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>0 f, f\<^sub>0g\<^sub>1.p\<^sub>1, fg\<^sub>0h\<^sub>1.p\<^sub>1] \<cdot> \<a>[tab\<^sub>0 f, f\<^sub>0g\<^sub>1.p\<^sub>1, fg\<^sub>0h\<^sub>1.p\<^sub>1] \<cdot> (local.inv f\<^sub>0g\<^sub>1.\<phi> \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>1 g, f\<^sub>0g\<^sub>1.p\<^sub>0, fg\<^sub>0h\<^sub>1.p\<^sub>1] = \<a>\<^sup>-\<^sup>1[tab\<^sub>1 g, f\<^sub>0g\<^sub>1.p\<^sub>0, fg\<^sub>0h\<^sub>1.p\<^sub>1]
(?h \<cdot> ?g) \<cdot> ?f = ?h \<cdot> ?g \<cdot> ?f
goal (1 subgoal):
1. (g \<star> \<a>[h, tab\<^sub>0 h, fg\<^sub>0h\<^sub>1.p\<^sub>0]) \<cdot> (g \<star> h.tab \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0) \<cdot> (g \<star> fg\<^sub>0h\<^sub>1.\<phi>) \<cdot> \<a>[g, tab\<^sub>0 g \<star> f\<^sub>0g\<^sub>1.p\<^sub>0, fg\<^sub>0h\<^sub>1.p\<^sub>1] \<cdot> (\<a>[g, tab\<^sub>0 g, f\<^sub>0g\<^sub>1.p\<^sub>0] \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> ((g.tab \<star> f\<^sub>0g\<^sub>1.p\<^sub>0) \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>1 g, f\<^sub>0g\<^sub>1.p\<^sub>0, fg\<^sub>0h\<^sub>1.p\<^sub>1] = (g \<star> \<a>[h, tab\<^sub>0 h, fg\<^sub>0h\<^sub>1.p\<^sub>0]) \<cdot> (g \<star> h.tab \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0) \<cdot> (g \<star> fg\<^sub>0h\<^sub>1.\<phi>) \<cdot> \<a>[g, tab\<^sub>0 g \<star> f\<^sub>0g\<^sub>1.p\<^sub>0, fg\<^sub>0h\<^sub>1.p\<^sub>1] \<cdot> (\<a>[g, tab\<^sub>0 g, f\<^sub>0g\<^sub>1.p\<^sub>0] \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> ((g.tab \<star> f\<^sub>0g\<^sub>1.p\<^sub>0) \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> (f\<^sub>0g\<^sub>1.\<phi> \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>0 f, f\<^sub>0g\<^sub>1.p\<^sub>1, fg\<^sub>0h\<^sub>1.p\<^sub>1] \<cdot> \<a>[tab\<^sub>0 f, f\<^sub>0g\<^sub>1.p\<^sub>1, fg\<^sub>0h\<^sub>1.p\<^sub>1] \<cdot> (local.inv f\<^sub>0g\<^sub>1.\<phi> \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>1 g, f\<^sub>0g\<^sub>1.p\<^sub>0, fg\<^sub>0h\<^sub>1.p\<^sub>1]
[PROOF STEP]
by presburger
[PROOF STATE]
proof (state)
this:
(g \<star> \<a>[h, tab\<^sub>0 h, fg\<^sub>0h\<^sub>1.p\<^sub>0]) \<cdot> (g \<star> h.tab \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0) \<cdot> (g \<star> fg\<^sub>0h\<^sub>1.\<phi>) \<cdot> \<a>[g, tab\<^sub>0 g \<star> f\<^sub>0g\<^sub>1.p\<^sub>0, fg\<^sub>0h\<^sub>1.p\<^sub>1] \<cdot> (\<a>[g, tab\<^sub>0 g, f\<^sub>0g\<^sub>1.p\<^sub>0] \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> ((g.tab \<star> f\<^sub>0g\<^sub>1.p\<^sub>0) \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>1 g, f\<^sub>0g\<^sub>1.p\<^sub>0, fg\<^sub>0h\<^sub>1.p\<^sub>1] = (g \<star> \<a>[h, tab\<^sub>0 h, fg\<^sub>0h\<^sub>1.p\<^sub>0]) \<cdot> (g \<star> h.tab \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0) \<cdot> (g \<star> fg\<^sub>0h\<^sub>1.\<phi>) \<cdot> \<a>[g, tab\<^sub>0 g \<star> f\<^sub>0g\<^sub>1.p\<^sub>0, fg\<^sub>0h\<^sub>1.p\<^sub>1] \<cdot> (\<a>[g, tab\<^sub>0 g, f\<^sub>0g\<^sub>1.p\<^sub>0] \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> ((g.tab \<star> f\<^sub>0g\<^sub>1.p\<^sub>0) \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> (f\<^sub>0g\<^sub>1.\<phi> \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>0 f, f\<^sub>0g\<^sub>1.p\<^sub>1, fg\<^sub>0h\<^sub>1.p\<^sub>1] \<cdot> \<a>[tab\<^sub>0 f, f\<^sub>0g\<^sub>1.p\<^sub>1, fg\<^sub>0h\<^sub>1.p\<^sub>1] \<cdot> (local.inv f\<^sub>0g\<^sub>1.\<phi> \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>1 g, f\<^sub>0g\<^sub>1.p\<^sub>0, fg\<^sub>0h\<^sub>1.p\<^sub>1]
goal:
No subgoals!
[PROOF STEP]
qed
[PROOF STATE]
proof (state)
this:
(g \<star> \<a>[h, tab\<^sub>0 h, fg\<^sub>0h\<^sub>1.p\<^sub>0]) \<cdot> (g \<star> h.tab \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0) \<cdot> (g \<star> fg\<^sub>0h\<^sub>1.\<phi>) \<cdot> \<a>[g, tab\<^sub>0 g \<star> f\<^sub>0g\<^sub>1.p\<^sub>0, fg\<^sub>0h\<^sub>1.p\<^sub>1] \<cdot> (\<a>[g, tab\<^sub>0 g, f\<^sub>0g\<^sub>1.p\<^sub>0] \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> ((g.tab \<star> f\<^sub>0g\<^sub>1.p\<^sub>0) \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>1 g, f\<^sub>0g\<^sub>1.p\<^sub>0, fg\<^sub>0h\<^sub>1.p\<^sub>1] = (g \<star> \<a>[h, tab\<^sub>0 h, fg\<^sub>0h\<^sub>1.p\<^sub>0]) \<cdot> (g \<star> h.tab \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0) \<cdot> (g \<star> fg\<^sub>0h\<^sub>1.\<phi>) \<cdot> \<a>[g, tab\<^sub>0 g \<star> f\<^sub>0g\<^sub>1.p\<^sub>0, fg\<^sub>0h\<^sub>1.p\<^sub>1] \<cdot> (\<a>[g, tab\<^sub>0 g, f\<^sub>0g\<^sub>1.p\<^sub>0] \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> ((g.tab \<star> f\<^sub>0g\<^sub>1.p\<^sub>0) \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> (f\<^sub>0g\<^sub>1.\<phi> \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>0 f, f\<^sub>0g\<^sub>1.p\<^sub>1, fg\<^sub>0h\<^sub>1.p\<^sub>1] \<cdot> \<a>[tab\<^sub>0 f, f\<^sub>0g\<^sub>1.p\<^sub>1, fg\<^sub>0h\<^sub>1.p\<^sub>1] \<cdot> (local.inv f\<^sub>0g\<^sub>1.\<phi> \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>1 g, f\<^sub>0g\<^sub>1.p\<^sub>0, fg\<^sub>0h\<^sub>1.p\<^sub>1]
goal (1 subgoal):
1. (g \<star> \<theta>\<^sub>g) \<cdot> \<a>[g, tab\<^sub>0 g, w\<^sub>g] \<cdot> (g.tab \<star> w\<^sub>g) = ((g \<star> \<theta>\<^sub>g') \<cdot> \<a>[g, tab\<^sub>0 g, w\<^sub>g'] \<cdot> (g.tab \<star> w\<^sub>g')) \<cdot> \<beta>\<^sub>g
[PROOF STEP]
also
[PROOF STATE]
proof (state)
this:
(g \<star> \<a>[h, tab\<^sub>0 h, fg\<^sub>0h\<^sub>1.p\<^sub>0]) \<cdot> (g \<star> h.tab \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0) \<cdot> (g \<star> fg\<^sub>0h\<^sub>1.\<phi>) \<cdot> \<a>[g, tab\<^sub>0 g \<star> f\<^sub>0g\<^sub>1.p\<^sub>0, fg\<^sub>0h\<^sub>1.p\<^sub>1] \<cdot> (\<a>[g, tab\<^sub>0 g, f\<^sub>0g\<^sub>1.p\<^sub>0] \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> ((g.tab \<star> f\<^sub>0g\<^sub>1.p\<^sub>0) \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>1 g, f\<^sub>0g\<^sub>1.p\<^sub>0, fg\<^sub>0h\<^sub>1.p\<^sub>1] = (g \<star> \<a>[h, tab\<^sub>0 h, fg\<^sub>0h\<^sub>1.p\<^sub>0]) \<cdot> (g \<star> h.tab \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0) \<cdot> (g \<star> fg\<^sub>0h\<^sub>1.\<phi>) \<cdot> \<a>[g, tab\<^sub>0 g \<star> f\<^sub>0g\<^sub>1.p\<^sub>0, fg\<^sub>0h\<^sub>1.p\<^sub>1] \<cdot> (\<a>[g, tab\<^sub>0 g, f\<^sub>0g\<^sub>1.p\<^sub>0] \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> ((g.tab \<star> f\<^sub>0g\<^sub>1.p\<^sub>0) \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> (f\<^sub>0g\<^sub>1.\<phi> \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>0 f, f\<^sub>0g\<^sub>1.p\<^sub>1, fg\<^sub>0h\<^sub>1.p\<^sub>1] \<cdot> \<a>[tab\<^sub>0 f, f\<^sub>0g\<^sub>1.p\<^sub>1, fg\<^sub>0h\<^sub>1.p\<^sub>1] \<cdot> (local.inv f\<^sub>0g\<^sub>1.\<phi> \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>1 g, f\<^sub>0g\<^sub>1.p\<^sub>0, fg\<^sub>0h\<^sub>1.p\<^sub>1]
goal (1 subgoal):
1. (g \<star> \<theta>\<^sub>g) \<cdot> \<a>[g, tab\<^sub>0 g, w\<^sub>g] \<cdot> (g.tab \<star> w\<^sub>g) = ((g \<star> \<theta>\<^sub>g') \<cdot> \<a>[g, tab\<^sub>0 g, w\<^sub>g'] \<cdot> (g.tab \<star> w\<^sub>g')) \<cdot> \<beta>\<^sub>g
[PROOF STEP]
have "... = \<theta>\<^sub>f \<cdot>
\<a>[tab\<^sub>0 f, Tfg.p\<^sub>1, TTfgh.p\<^sub>1] \<cdot>
(inv f\<^sub>0g\<^sub>1.\<phi> \<star> TTfgh.p\<^sub>1) \<cdot>
\<a>\<^sup>-\<^sup>1[tab\<^sub>1 g, Tfg.p\<^sub>0, TTfgh.p\<^sub>1]"
[PROOF STATE]
proof (prove)
goal (1 subgoal):
1. (g \<star> \<a>[h, tab\<^sub>0 h, fg\<^sub>0h\<^sub>1.p\<^sub>0]) \<cdot> (g \<star> h.tab \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0) \<cdot> (g \<star> fg\<^sub>0h\<^sub>1.\<phi>) \<cdot> \<a>[g, tab\<^sub>0 g \<star> f\<^sub>0g\<^sub>1.p\<^sub>0, fg\<^sub>0h\<^sub>1.p\<^sub>1] \<cdot> (\<a>[g, tab\<^sub>0 g, f\<^sub>0g\<^sub>1.p\<^sub>0] \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> ((g.tab \<star> f\<^sub>0g\<^sub>1.p\<^sub>0) \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> (f\<^sub>0g\<^sub>1.\<phi> \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>0 f, f\<^sub>0g\<^sub>1.p\<^sub>1, fg\<^sub>0h\<^sub>1.p\<^sub>1] \<cdot> \<a>[tab\<^sub>0 f, f\<^sub>0g\<^sub>1.p\<^sub>1, fg\<^sub>0h\<^sub>1.p\<^sub>1] \<cdot> (local.inv f\<^sub>0g\<^sub>1.\<phi> \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>1 g, f\<^sub>0g\<^sub>1.p\<^sub>0, fg\<^sub>0h\<^sub>1.p\<^sub>1] = \<theta>\<^sub>f \<cdot> \<a>[tab\<^sub>0 f, f\<^sub>0g\<^sub>1.p\<^sub>1, fg\<^sub>0h\<^sub>1.p\<^sub>1] \<cdot> (local.inv f\<^sub>0g\<^sub>1.\<phi> \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>1 g, f\<^sub>0g\<^sub>1.p\<^sub>0, fg\<^sub>0h\<^sub>1.p\<^sub>1]
[PROOF STEP]
unfolding \<theta>\<^sub>f_def
[PROOF STATE]
proof (prove)
goal (1 subgoal):
1. (g \<star> \<a>[h, tab\<^sub>0 h, fg\<^sub>0h\<^sub>1.p\<^sub>0]) \<cdot> (g \<star> h.tab \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0) \<cdot> (g \<star> fg\<^sub>0h\<^sub>1.\<phi>) \<cdot> \<a>[g, tab\<^sub>0 g \<star> f\<^sub>0g\<^sub>1.p\<^sub>0, fg\<^sub>0h\<^sub>1.p\<^sub>1] \<cdot> (\<a>[g, tab\<^sub>0 g, f\<^sub>0g\<^sub>1.p\<^sub>0] \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> ((g.tab \<star> f\<^sub>0g\<^sub>1.p\<^sub>0) \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> (f\<^sub>0g\<^sub>1.\<phi> \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>0 f, f\<^sub>0g\<^sub>1.p\<^sub>1, fg\<^sub>0h\<^sub>1.p\<^sub>1] \<cdot> \<a>[tab\<^sub>0 f, f\<^sub>0g\<^sub>1.p\<^sub>1, fg\<^sub>0h\<^sub>1.p\<^sub>1] \<cdot> (local.inv f\<^sub>0g\<^sub>1.\<phi> \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>1 g, f\<^sub>0g\<^sub>1.p\<^sub>0, fg\<^sub>0h\<^sub>1.p\<^sub>1] = ((g \<star> \<a>[h, tab\<^sub>0 h, fg\<^sub>0h\<^sub>1.p\<^sub>0]) \<cdot> (g \<star> h.tab \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0) \<cdot> (g \<star> fg\<^sub>0h\<^sub>1.\<phi>) \<cdot> \<a>[g, tab\<^sub>0 g \<star> f\<^sub>0g\<^sub>1.p\<^sub>0, fg\<^sub>0h\<^sub>1.p\<^sub>1] \<cdot> (\<a>[g, tab\<^sub>0 g, f\<^sub>0g\<^sub>1.p\<^sub>0] \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> ((g.tab \<star> f\<^sub>0g\<^sub>1.p\<^sub>0) \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> (f\<^sub>0g\<^sub>1.\<phi> \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>0 f, f\<^sub>0g\<^sub>1.p\<^sub>1, fg\<^sub>0h\<^sub>1.p\<^sub>1]) \<cdot> \<a>[tab\<^sub>0 f, f\<^sub>0g\<^sub>1.p\<^sub>1, fg\<^sub>0h\<^sub>1.p\<^sub>1] \<cdot> (local.inv f\<^sub>0g\<^sub>1.\<phi> \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>1 g, f\<^sub>0g\<^sub>1.p\<^sub>0, fg\<^sub>0h\<^sub>1.p\<^sub>1]
[PROOF STEP]
using comp_assoc
[PROOF STATE]
proof (prove)
using this:
(?h \<cdot> ?g) \<cdot> ?f = ?h \<cdot> ?g \<cdot> ?f
goal (1 subgoal):
1. (g \<star> \<a>[h, tab\<^sub>0 h, fg\<^sub>0h\<^sub>1.p\<^sub>0]) \<cdot> (g \<star> h.tab \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0) \<cdot> (g \<star> fg\<^sub>0h\<^sub>1.\<phi>) \<cdot> \<a>[g, tab\<^sub>0 g \<star> f\<^sub>0g\<^sub>1.p\<^sub>0, fg\<^sub>0h\<^sub>1.p\<^sub>1] \<cdot> (\<a>[g, tab\<^sub>0 g, f\<^sub>0g\<^sub>1.p\<^sub>0] \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> ((g.tab \<star> f\<^sub>0g\<^sub>1.p\<^sub>0) \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> (f\<^sub>0g\<^sub>1.\<phi> \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>0 f, f\<^sub>0g\<^sub>1.p\<^sub>1, fg\<^sub>0h\<^sub>1.p\<^sub>1] \<cdot> \<a>[tab\<^sub>0 f, f\<^sub>0g\<^sub>1.p\<^sub>1, fg\<^sub>0h\<^sub>1.p\<^sub>1] \<cdot> (local.inv f\<^sub>0g\<^sub>1.\<phi> \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>1 g, f\<^sub>0g\<^sub>1.p\<^sub>0, fg\<^sub>0h\<^sub>1.p\<^sub>1] = ((g \<star> \<a>[h, tab\<^sub>0 h, fg\<^sub>0h\<^sub>1.p\<^sub>0]) \<cdot> (g \<star> h.tab \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0) \<cdot> (g \<star> fg\<^sub>0h\<^sub>1.\<phi>) \<cdot> \<a>[g, tab\<^sub>0 g \<star> f\<^sub>0g\<^sub>1.p\<^sub>0, fg\<^sub>0h\<^sub>1.p\<^sub>1] \<cdot> (\<a>[g, tab\<^sub>0 g, f\<^sub>0g\<^sub>1.p\<^sub>0] \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> ((g.tab \<star> f\<^sub>0g\<^sub>1.p\<^sub>0) \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> (f\<^sub>0g\<^sub>1.\<phi> \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>0 f, f\<^sub>0g\<^sub>1.p\<^sub>1, fg\<^sub>0h\<^sub>1.p\<^sub>1]) \<cdot> \<a>[tab\<^sub>0 f, f\<^sub>0g\<^sub>1.p\<^sub>1, fg\<^sub>0h\<^sub>1.p\<^sub>1] \<cdot> (local.inv f\<^sub>0g\<^sub>1.\<phi> \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>1 g, f\<^sub>0g\<^sub>1.p\<^sub>0, fg\<^sub>0h\<^sub>1.p\<^sub>1]
[PROOF STEP]
by presburger
[PROOF STATE]
proof (state)
this:
(g \<star> \<a>[h, tab\<^sub>0 h, fg\<^sub>0h\<^sub>1.p\<^sub>0]) \<cdot> (g \<star> h.tab \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0) \<cdot> (g \<star> fg\<^sub>0h\<^sub>1.\<phi>) \<cdot> \<a>[g, tab\<^sub>0 g \<star> f\<^sub>0g\<^sub>1.p\<^sub>0, fg\<^sub>0h\<^sub>1.p\<^sub>1] \<cdot> (\<a>[g, tab\<^sub>0 g, f\<^sub>0g\<^sub>1.p\<^sub>0] \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> ((g.tab \<star> f\<^sub>0g\<^sub>1.p\<^sub>0) \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> (f\<^sub>0g\<^sub>1.\<phi> \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>0 f, f\<^sub>0g\<^sub>1.p\<^sub>1, fg\<^sub>0h\<^sub>1.p\<^sub>1] \<cdot> \<a>[tab\<^sub>0 f, f\<^sub>0g\<^sub>1.p\<^sub>1, fg\<^sub>0h\<^sub>1.p\<^sub>1] \<cdot> (local.inv f\<^sub>0g\<^sub>1.\<phi> \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>1 g, f\<^sub>0g\<^sub>1.p\<^sub>0, fg\<^sub>0h\<^sub>1.p\<^sub>1] = \<theta>\<^sub>f \<cdot> \<a>[tab\<^sub>0 f, f\<^sub>0g\<^sub>1.p\<^sub>1, fg\<^sub>0h\<^sub>1.p\<^sub>1] \<cdot> (local.inv f\<^sub>0g\<^sub>1.\<phi> \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>1 g, f\<^sub>0g\<^sub>1.p\<^sub>0, fg\<^sub>0h\<^sub>1.p\<^sub>1]
goal (1 subgoal):
1. (g \<star> \<theta>\<^sub>g) \<cdot> \<a>[g, tab\<^sub>0 g, w\<^sub>g] \<cdot> (g.tab \<star> w\<^sub>g) = ((g \<star> \<theta>\<^sub>g') \<cdot> \<a>[g, tab\<^sub>0 g, w\<^sub>g'] \<cdot> (g.tab \<star> w\<^sub>g')) \<cdot> \<beta>\<^sub>g
[PROOF STEP]
also
[PROOF STATE]
proof (state)
this:
(g \<star> \<a>[h, tab\<^sub>0 h, fg\<^sub>0h\<^sub>1.p\<^sub>0]) \<cdot> (g \<star> h.tab \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0) \<cdot> (g \<star> fg\<^sub>0h\<^sub>1.\<phi>) \<cdot> \<a>[g, tab\<^sub>0 g \<star> f\<^sub>0g\<^sub>1.p\<^sub>0, fg\<^sub>0h\<^sub>1.p\<^sub>1] \<cdot> (\<a>[g, tab\<^sub>0 g, f\<^sub>0g\<^sub>1.p\<^sub>0] \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> ((g.tab \<star> f\<^sub>0g\<^sub>1.p\<^sub>0) \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> (f\<^sub>0g\<^sub>1.\<phi> \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>0 f, f\<^sub>0g\<^sub>1.p\<^sub>1, fg\<^sub>0h\<^sub>1.p\<^sub>1] \<cdot> \<a>[tab\<^sub>0 f, f\<^sub>0g\<^sub>1.p\<^sub>1, fg\<^sub>0h\<^sub>1.p\<^sub>1] \<cdot> (local.inv f\<^sub>0g\<^sub>1.\<phi> \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>1 g, f\<^sub>0g\<^sub>1.p\<^sub>0, fg\<^sub>0h\<^sub>1.p\<^sub>1] = \<theta>\<^sub>f \<cdot> \<a>[tab\<^sub>0 f, f\<^sub>0g\<^sub>1.p\<^sub>1, fg\<^sub>0h\<^sub>1.p\<^sub>1] \<cdot> (local.inv f\<^sub>0g\<^sub>1.\<phi> \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>1 g, f\<^sub>0g\<^sub>1.p\<^sub>0, fg\<^sub>0h\<^sub>1.p\<^sub>1]
goal (1 subgoal):
1. (g \<star> \<theta>\<^sub>g) \<cdot> \<a>[g, tab\<^sub>0 g, w\<^sub>g] \<cdot> (g.tab \<star> w\<^sub>g) = ((g \<star> \<theta>\<^sub>g') \<cdot> \<a>[g, tab\<^sub>0 g, w\<^sub>g'] \<cdot> (g.tab \<star> w\<^sub>g')) \<cdot> \<beta>\<^sub>g
[PROOF STEP]
have "... = \<theta>\<^sub>f' \<cdot> (tab\<^sub>0 f \<star> \<gamma>\<^sub>f) \<cdot>
\<a>[tab\<^sub>0 f, Tfg.p\<^sub>1, TTfgh.p\<^sub>1] \<cdot>
(inv f\<^sub>0g\<^sub>1.\<phi> \<star> TTfgh.p\<^sub>1) \<cdot>
\<a>\<^sup>-\<^sup>1[tab\<^sub>1 g, Tfg.p\<^sub>0, TTfgh.p\<^sub>1]"
[PROOF STATE]
proof (prove)
goal (1 subgoal):
1. \<theta>\<^sub>f \<cdot> \<a>[tab\<^sub>0 f, f\<^sub>0g\<^sub>1.p\<^sub>1, fg\<^sub>0h\<^sub>1.p\<^sub>1] \<cdot> (local.inv f\<^sub>0g\<^sub>1.\<phi> \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>1 g, f\<^sub>0g\<^sub>1.p\<^sub>0, fg\<^sub>0h\<^sub>1.p\<^sub>1] = \<theta>\<^sub>f' \<cdot> (tab\<^sub>0 f \<star> \<gamma>\<^sub>f) \<cdot> \<a>[tab\<^sub>0 f, f\<^sub>0g\<^sub>1.p\<^sub>1, fg\<^sub>0h\<^sub>1.p\<^sub>1] \<cdot> (local.inv f\<^sub>0g\<^sub>1.\<phi> \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>1 g, f\<^sub>0g\<^sub>1.p\<^sub>0, fg\<^sub>0h\<^sub>1.p\<^sub>1]
[PROOF STEP]
using \<gamma>\<^sub>f comp_assoc
[PROOF STATE]
proof (prove)
using this:
\<guillemotleft>\<gamma>\<^sub>f : w\<^sub>f \<rightarrow> w\<^sub>f'\<guillemotright> \<and> \<beta>\<^sub>f = tab\<^sub>1 f \<star> \<gamma>\<^sub>f \<and> \<theta>\<^sub>f = \<theta>\<^sub>f' \<cdot> (tab\<^sub>0 f \<star> \<gamma>\<^sub>f)
(?h \<cdot> ?g) \<cdot> ?f = ?h \<cdot> ?g \<cdot> ?f
goal (1 subgoal):
1. \<theta>\<^sub>f \<cdot> \<a>[tab\<^sub>0 f, f\<^sub>0g\<^sub>1.p\<^sub>1, fg\<^sub>0h\<^sub>1.p\<^sub>1] \<cdot> (local.inv f\<^sub>0g\<^sub>1.\<phi> \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>1 g, f\<^sub>0g\<^sub>1.p\<^sub>0, fg\<^sub>0h\<^sub>1.p\<^sub>1] = \<theta>\<^sub>f' \<cdot> (tab\<^sub>0 f \<star> \<gamma>\<^sub>f) \<cdot> \<a>[tab\<^sub>0 f, f\<^sub>0g\<^sub>1.p\<^sub>1, fg\<^sub>0h\<^sub>1.p\<^sub>1] \<cdot> (local.inv f\<^sub>0g\<^sub>1.\<phi> \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>1 g, f\<^sub>0g\<^sub>1.p\<^sub>0, fg\<^sub>0h\<^sub>1.p\<^sub>1]
[PROOF STEP]
by simp
[PROOF STATE]
proof (state)
this:
\<theta>\<^sub>f \<cdot> \<a>[tab\<^sub>0 f, f\<^sub>0g\<^sub>1.p\<^sub>1, fg\<^sub>0h\<^sub>1.p\<^sub>1] \<cdot> (local.inv f\<^sub>0g\<^sub>1.\<phi> \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>1 g, f\<^sub>0g\<^sub>1.p\<^sub>0, fg\<^sub>0h\<^sub>1.p\<^sub>1] = \<theta>\<^sub>f' \<cdot> (tab\<^sub>0 f \<star> \<gamma>\<^sub>f) \<cdot> \<a>[tab\<^sub>0 f, f\<^sub>0g\<^sub>1.p\<^sub>1, fg\<^sub>0h\<^sub>1.p\<^sub>1] \<cdot> (local.inv f\<^sub>0g\<^sub>1.\<phi> \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>1 g, f\<^sub>0g\<^sub>1.p\<^sub>0, fg\<^sub>0h\<^sub>1.p\<^sub>1]
goal (1 subgoal):
1. (g \<star> \<theta>\<^sub>g) \<cdot> \<a>[g, tab\<^sub>0 g, w\<^sub>g] \<cdot> (g.tab \<star> w\<^sub>g) = ((g \<star> \<theta>\<^sub>g') \<cdot> \<a>[g, tab\<^sub>0 g, w\<^sub>g'] \<cdot> (g.tab \<star> w\<^sub>g')) \<cdot> \<beta>\<^sub>g
[PROOF STEP]
also
[PROOF STATE]
proof (state)
this:
\<theta>\<^sub>f \<cdot> \<a>[tab\<^sub>0 f, f\<^sub>0g\<^sub>1.p\<^sub>1, fg\<^sub>0h\<^sub>1.p\<^sub>1] \<cdot> (local.inv f\<^sub>0g\<^sub>1.\<phi> \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>1 g, f\<^sub>0g\<^sub>1.p\<^sub>0, fg\<^sub>0h\<^sub>1.p\<^sub>1] = \<theta>\<^sub>f' \<cdot> (tab\<^sub>0 f \<star> \<gamma>\<^sub>f) \<cdot> \<a>[tab\<^sub>0 f, f\<^sub>0g\<^sub>1.p\<^sub>1, fg\<^sub>0h\<^sub>1.p\<^sub>1] \<cdot> (local.inv f\<^sub>0g\<^sub>1.\<phi> \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>1 g, f\<^sub>0g\<^sub>1.p\<^sub>0, fg\<^sub>0h\<^sub>1.p\<^sub>1]
goal (1 subgoal):
1. (g \<star> \<theta>\<^sub>g) \<cdot> \<a>[g, tab\<^sub>0 g, w\<^sub>g] \<cdot> (g.tab \<star> w\<^sub>g) = ((g \<star> \<theta>\<^sub>g') \<cdot> \<a>[g, tab\<^sub>0 g, w\<^sub>g'] \<cdot> (g.tab \<star> w\<^sub>g')) \<cdot> \<beta>\<^sub>g
[PROOF STEP]
have "... = (g \<star> h \<star> TTfgh_TfTgh.the_\<theta>) \<cdot>
can (\<^bold>\<langle>g\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> ((\<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>Tgh.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TfTgh.p\<^sub>0\<^bold>\<rangle>)
\<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>)
(((\<^bold>\<langle>g\<^bold>\<rangle> \<^bold>\<star> (\<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>Tgh.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TfTgh.p\<^sub>0\<^bold>\<rangle>)
\<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) \<cdot>
(((g \<star> h.tab \<star> Tgh.p\<^sub>0) \<star> TfTgh.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot>
(((g \<star> g\<^sub>0h\<^sub>1.\<phi>) \<star> TfTgh.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot>
((\<a>[g, tab\<^sub>0 g, Tgh.p\<^sub>1] \<star> TfTgh.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot>
(((g.tab \<star> Tgh.p\<^sub>1) \<star> TfTgh.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot>
(f\<^sub>0gh\<^sub>1.\<phi> \<star> TTfgh_TfTgh.chine) \<cdot>
\<a>\<^sup>-\<^sup>1[tab\<^sub>0 f, TfTgh.p\<^sub>1, TTfgh_TfTgh.chine] \<cdot>
(tab\<^sub>0 f \<star> \<gamma>\<^sub>f) \<cdot>
\<a>[tab\<^sub>0 f, Tfg.p\<^sub>1, TTfgh.p\<^sub>1] \<cdot>
(inv f\<^sub>0g\<^sub>1.\<phi> \<star> TTfgh.p\<^sub>1) \<cdot>
\<a>\<^sup>-\<^sup>1[tab\<^sub>1 g, Tfg.p\<^sub>0, TTfgh.p\<^sub>1]"
[PROOF STATE]
proof (prove)
goal (1 subgoal):
1. \<theta>\<^sub>f' \<cdot> (tab\<^sub>0 f \<star> \<gamma>\<^sub>f) \<cdot> \<a>[tab\<^sub>0 f, f\<^sub>0g\<^sub>1.p\<^sub>1, fg\<^sub>0h\<^sub>1.p\<^sub>1] \<cdot> (local.inv f\<^sub>0g\<^sub>1.\<phi> \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>1 g, f\<^sub>0g\<^sub>1.p\<^sub>0, fg\<^sub>0h\<^sub>1.p\<^sub>1] = (g \<star> h \<star> TTfgh_TfTgh.the_\<theta>) \<cdot> can (\<^bold>\<langle>g\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> ((\<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) (((\<^bold>\<langle>g\<^bold>\<rangle> \<^bold>\<star> (\<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) \<cdot> (((g \<star> h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (((g \<star> g\<^sub>0h\<^sub>1.\<phi>) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((\<a>[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (((g.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>1) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (f\<^sub>0gh\<^sub>1.\<phi> \<star> TTfgh_TfTgh.chine) \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>0 f, f\<^sub>0gh\<^sub>1.p\<^sub>1, TTfgh_TfTgh.chine] \<cdot> (tab\<^sub>0 f \<star> \<gamma>\<^sub>f) \<cdot> \<a>[tab\<^sub>0 f, f\<^sub>0g\<^sub>1.p\<^sub>1, fg\<^sub>0h\<^sub>1.p\<^sub>1] \<cdot> (local.inv f\<^sub>0g\<^sub>1.\<phi> \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>1 g, f\<^sub>0g\<^sub>1.p\<^sub>0, fg\<^sub>0h\<^sub>1.p\<^sub>1]
[PROOF STEP]
unfolding \<theta>\<^sub>f'_def
[PROOF STATE]
proof (prove)
goal (1 subgoal):
1. ((g \<star> h \<star> TTfgh_TfTgh.the_\<theta>) \<cdot> can (\<^bold>\<langle>g\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> ((\<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) (((\<^bold>\<langle>g\<^bold>\<rangle> \<^bold>\<star> (\<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) \<cdot> (((g \<star> h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (((g \<star> g\<^sub>0h\<^sub>1.\<phi>) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((\<a>[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (((g.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>1) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (f\<^sub>0gh\<^sub>1.\<phi> \<star> TTfgh_TfTgh.chine) \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>0 f, f\<^sub>0gh\<^sub>1.p\<^sub>1, TTfgh_TfTgh.chine]) \<cdot> (tab\<^sub>0 f \<star> \<gamma>\<^sub>f) \<cdot> \<a>[tab\<^sub>0 f, f\<^sub>0g\<^sub>1.p\<^sub>1, fg\<^sub>0h\<^sub>1.p\<^sub>1] \<cdot> (local.inv f\<^sub>0g\<^sub>1.\<phi> \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>1 g, f\<^sub>0g\<^sub>1.p\<^sub>0, fg\<^sub>0h\<^sub>1.p\<^sub>1] = (g \<star> h \<star> TTfgh_TfTgh.the_\<theta>) \<cdot> can (\<^bold>\<langle>g\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> ((\<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) (((\<^bold>\<langle>g\<^bold>\<rangle> \<^bold>\<star> (\<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) \<cdot> (((g \<star> h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (((g \<star> g\<^sub>0h\<^sub>1.\<phi>) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((\<a>[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (((g.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>1) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (f\<^sub>0gh\<^sub>1.\<phi> \<star> TTfgh_TfTgh.chine) \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>0 f, f\<^sub>0gh\<^sub>1.p\<^sub>1, TTfgh_TfTgh.chine] \<cdot> (tab\<^sub>0 f \<star> \<gamma>\<^sub>f) \<cdot> \<a>[tab\<^sub>0 f, f\<^sub>0g\<^sub>1.p\<^sub>1, fg\<^sub>0h\<^sub>1.p\<^sub>1] \<cdot> (local.inv f\<^sub>0g\<^sub>1.\<phi> \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>1 g, f\<^sub>0g\<^sub>1.p\<^sub>0, fg\<^sub>0h\<^sub>1.p\<^sub>1]
[PROOF STEP]
using comp_assoc
[PROOF STATE]
proof (prove)
using this:
(?h \<cdot> ?g) \<cdot> ?f = ?h \<cdot> ?g \<cdot> ?f
goal (1 subgoal):
1. ((g \<star> h \<star> TTfgh_TfTgh.the_\<theta>) \<cdot> can (\<^bold>\<langle>g\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> ((\<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) (((\<^bold>\<langle>g\<^bold>\<rangle> \<^bold>\<star> (\<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) \<cdot> (((g \<star> h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (((g \<star> g\<^sub>0h\<^sub>1.\<phi>) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((\<a>[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (((g.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>1) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (f\<^sub>0gh\<^sub>1.\<phi> \<star> TTfgh_TfTgh.chine) \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>0 f, f\<^sub>0gh\<^sub>1.p\<^sub>1, TTfgh_TfTgh.chine]) \<cdot> (tab\<^sub>0 f \<star> \<gamma>\<^sub>f) \<cdot> \<a>[tab\<^sub>0 f, f\<^sub>0g\<^sub>1.p\<^sub>1, fg\<^sub>0h\<^sub>1.p\<^sub>1] \<cdot> (local.inv f\<^sub>0g\<^sub>1.\<phi> \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>1 g, f\<^sub>0g\<^sub>1.p\<^sub>0, fg\<^sub>0h\<^sub>1.p\<^sub>1] = (g \<star> h \<star> TTfgh_TfTgh.the_\<theta>) \<cdot> can (\<^bold>\<langle>g\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> ((\<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) (((\<^bold>\<langle>g\<^bold>\<rangle> \<^bold>\<star> (\<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) \<cdot> (((g \<star> h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (((g \<star> g\<^sub>0h\<^sub>1.\<phi>) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((\<a>[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (((g.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>1) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (f\<^sub>0gh\<^sub>1.\<phi> \<star> TTfgh_TfTgh.chine) \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>0 f, f\<^sub>0gh\<^sub>1.p\<^sub>1, TTfgh_TfTgh.chine] \<cdot> (tab\<^sub>0 f \<star> \<gamma>\<^sub>f) \<cdot> \<a>[tab\<^sub>0 f, f\<^sub>0g\<^sub>1.p\<^sub>1, fg\<^sub>0h\<^sub>1.p\<^sub>1] \<cdot> (local.inv f\<^sub>0g\<^sub>1.\<phi> \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>1 g, f\<^sub>0g\<^sub>1.p\<^sub>0, fg\<^sub>0h\<^sub>1.p\<^sub>1]
[PROOF STEP]
by presburger
[PROOF STATE]
proof (state)
this:
\<theta>\<^sub>f' \<cdot> (tab\<^sub>0 f \<star> \<gamma>\<^sub>f) \<cdot> \<a>[tab\<^sub>0 f, f\<^sub>0g\<^sub>1.p\<^sub>1, fg\<^sub>0h\<^sub>1.p\<^sub>1] \<cdot> (local.inv f\<^sub>0g\<^sub>1.\<phi> \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>1 g, f\<^sub>0g\<^sub>1.p\<^sub>0, fg\<^sub>0h\<^sub>1.p\<^sub>1] = (g \<star> h \<star> TTfgh_TfTgh.the_\<theta>) \<cdot> can (\<^bold>\<langle>g\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> ((\<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) (((\<^bold>\<langle>g\<^bold>\<rangle> \<^bold>\<star> (\<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) \<cdot> (((g \<star> h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (((g \<star> g\<^sub>0h\<^sub>1.\<phi>) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((\<a>[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (((g.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>1) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (f\<^sub>0gh\<^sub>1.\<phi> \<star> TTfgh_TfTgh.chine) \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>0 f, f\<^sub>0gh\<^sub>1.p\<^sub>1, TTfgh_TfTgh.chine] \<cdot> (tab\<^sub>0 f \<star> \<gamma>\<^sub>f) \<cdot> \<a>[tab\<^sub>0 f, f\<^sub>0g\<^sub>1.p\<^sub>1, fg\<^sub>0h\<^sub>1.p\<^sub>1] \<cdot> (local.inv f\<^sub>0g\<^sub>1.\<phi> \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>1 g, f\<^sub>0g\<^sub>1.p\<^sub>0, fg\<^sub>0h\<^sub>1.p\<^sub>1]
goal (1 subgoal):
1. (g \<star> \<theta>\<^sub>g) \<cdot> \<a>[g, tab\<^sub>0 g, w\<^sub>g] \<cdot> (g.tab \<star> w\<^sub>g) = ((g \<star> \<theta>\<^sub>g') \<cdot> \<a>[g, tab\<^sub>0 g, w\<^sub>g'] \<cdot> (g.tab \<star> w\<^sub>g')) \<cdot> \<beta>\<^sub>g
[PROOF STEP]
also
[PROOF STATE]
proof (state)
this:
\<theta>\<^sub>f' \<cdot> (tab\<^sub>0 f \<star> \<gamma>\<^sub>f) \<cdot> \<a>[tab\<^sub>0 f, f\<^sub>0g\<^sub>1.p\<^sub>1, fg\<^sub>0h\<^sub>1.p\<^sub>1] \<cdot> (local.inv f\<^sub>0g\<^sub>1.\<phi> \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>1 g, f\<^sub>0g\<^sub>1.p\<^sub>0, fg\<^sub>0h\<^sub>1.p\<^sub>1] = (g \<star> h \<star> TTfgh_TfTgh.the_\<theta>) \<cdot> can (\<^bold>\<langle>g\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> ((\<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) (((\<^bold>\<langle>g\<^bold>\<rangle> \<^bold>\<star> (\<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) \<cdot> (((g \<star> h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (((g \<star> g\<^sub>0h\<^sub>1.\<phi>) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((\<a>[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (((g.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>1) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (f\<^sub>0gh\<^sub>1.\<phi> \<star> TTfgh_TfTgh.chine) \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>0 f, f\<^sub>0gh\<^sub>1.p\<^sub>1, TTfgh_TfTgh.chine] \<cdot> (tab\<^sub>0 f \<star> \<gamma>\<^sub>f) \<cdot> \<a>[tab\<^sub>0 f, f\<^sub>0g\<^sub>1.p\<^sub>1, fg\<^sub>0h\<^sub>1.p\<^sub>1] \<cdot> (local.inv f\<^sub>0g\<^sub>1.\<phi> \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>1 g, f\<^sub>0g\<^sub>1.p\<^sub>0, fg\<^sub>0h\<^sub>1.p\<^sub>1]
goal (1 subgoal):
1. (g \<star> \<theta>\<^sub>g) \<cdot> \<a>[g, tab\<^sub>0 g, w\<^sub>g] \<cdot> (g.tab \<star> w\<^sub>g) = ((g \<star> \<theta>\<^sub>g') \<cdot> \<a>[g, tab\<^sub>0 g, w\<^sub>g'] \<cdot> (g.tab \<star> w\<^sub>g')) \<cdot> \<beta>\<^sub>g
[PROOF STEP]
have "... = (g \<star> h \<star> TTfgh_TfTgh.the_\<theta>) \<cdot>
can (\<^bold>\<langle>g\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> ((\<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>Tgh.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TfTgh.p\<^sub>0\<^bold>\<rangle>)
\<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>)
(((\<^bold>\<langle>g\<^bold>\<rangle> \<^bold>\<star> (\<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>Tgh.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TfTgh.p\<^sub>0\<^bold>\<rangle>)
\<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) \<cdot>
(((g \<star> h.tab \<star> Tgh.p\<^sub>0) \<star> TfTgh.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot>
(((g \<star> g\<^sub>0h\<^sub>1.\<phi>) \<star> TfTgh.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot>
((\<a>[g, tab\<^sub>0 g, Tgh.p\<^sub>1] \<star> TfTgh.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot>
(((g.tab \<star> Tgh.p\<^sub>1) \<star> TfTgh.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot>
(\<a>\<^sup>-\<^sup>1[tab\<^sub>1 g \<star> Tgh.p\<^sub>1, TfTgh.p\<^sub>0, TTfgh_TfTgh.chine] \<cdot>
(\<a>\<^sup>-\<^sup>1[tab\<^sub>1 g, Tgh.p\<^sub>1, TfTgh.p\<^sub>0 \<star> TTfgh_TfTgh.chine] \<cdot>
\<a>[tab\<^sub>1 g, Tgh.p\<^sub>1, TfTgh.p\<^sub>0 \<star> TTfgh_TfTgh.chine]) \<cdot>
\<a>[tab\<^sub>1 g \<star> Tgh.p\<^sub>1, TfTgh.p\<^sub>0, TTfgh_TfTgh.chine] \<cdot>
(f\<^sub>0gh\<^sub>1.\<phi> \<star> TTfgh_TfTgh.chine)) \<cdot>
\<a>\<^sup>-\<^sup>1[tab\<^sub>0 f, TfTgh.p\<^sub>1, TTfgh_TfTgh.chine] \<cdot>
(tab\<^sub>0 f \<star> \<gamma>\<^sub>f) \<cdot>
\<a>[tab\<^sub>0 f, Tfg.p\<^sub>1, TTfgh.p\<^sub>1] \<cdot>
(inv f\<^sub>0g\<^sub>1.\<phi> \<star> TTfgh.p\<^sub>1) \<cdot>
\<a>\<^sup>-\<^sup>1[tab\<^sub>1 g, Tfg.p\<^sub>0, TTfgh.p\<^sub>1]"
[PROOF STATE]
proof (prove)
goal (1 subgoal):
1. (g \<star> h \<star> TTfgh_TfTgh.the_\<theta>) \<cdot> can (\<^bold>\<langle>g\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> ((\<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) (((\<^bold>\<langle>g\<^bold>\<rangle> \<^bold>\<star> (\<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) \<cdot> (((g \<star> h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (((g \<star> g\<^sub>0h\<^sub>1.\<phi>) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((\<a>[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (((g.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>1) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (f\<^sub>0gh\<^sub>1.\<phi> \<star> TTfgh_TfTgh.chine) \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>0 f, f\<^sub>0gh\<^sub>1.p\<^sub>1, TTfgh_TfTgh.chine] \<cdot> (tab\<^sub>0 f \<star> \<gamma>\<^sub>f) \<cdot> \<a>[tab\<^sub>0 f, f\<^sub>0g\<^sub>1.p\<^sub>1, fg\<^sub>0h\<^sub>1.p\<^sub>1] \<cdot> (local.inv f\<^sub>0g\<^sub>1.\<phi> \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>1 g, f\<^sub>0g\<^sub>1.p\<^sub>0, fg\<^sub>0h\<^sub>1.p\<^sub>1] = (g \<star> h \<star> TTfgh_TfTgh.the_\<theta>) \<cdot> can (\<^bold>\<langle>g\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> ((\<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) (((\<^bold>\<langle>g\<^bold>\<rangle> \<^bold>\<star> (\<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) \<cdot> (((g \<star> h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (((g \<star> g\<^sub>0h\<^sub>1.\<phi>) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((\<a>[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (((g.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>1) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (\<a>\<^sup>-\<^sup>1[tab\<^sub>1 g \<star> g\<^sub>0h\<^sub>1.p\<^sub>1, f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine] \<cdot> (\<a>\<^sup>-\<^sup>1[tab\<^sub>1 g, g\<^sub>0h\<^sub>1.p\<^sub>1, f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine] \<cdot> \<a>[tab\<^sub>1 g, g\<^sub>0h\<^sub>1.p\<^sub>1, f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine]) \<cdot> \<a>[tab\<^sub>1 g \<star> g\<^sub>0h\<^sub>1.p\<^sub>1, f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine] \<cdot> (f\<^sub>0gh\<^sub>1.\<phi> \<star> TTfgh_TfTgh.chine)) \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>0 f, f\<^sub>0gh\<^sub>1.p\<^sub>1, TTfgh_TfTgh.chine] \<cdot> (tab\<^sub>0 f \<star> \<gamma>\<^sub>f) \<cdot> \<a>[tab\<^sub>0 f, f\<^sub>0g\<^sub>1.p\<^sub>1, fg\<^sub>0h\<^sub>1.p\<^sub>1] \<cdot> (local.inv f\<^sub>0g\<^sub>1.\<phi> \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>1 g, f\<^sub>0g\<^sub>1.p\<^sub>0, fg\<^sub>0h\<^sub>1.p\<^sub>1]
[PROOF STEP]
proof -
[PROOF STATE]
proof (state)
goal (1 subgoal):
1. (g \<star> h \<star> TTfgh_TfTgh.the_\<theta>) \<cdot> can (\<^bold>\<langle>g\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> ((\<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) (((\<^bold>\<langle>g\<^bold>\<rangle> \<^bold>\<star> (\<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) \<cdot> (((g \<star> h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (((g \<star> g\<^sub>0h\<^sub>1.\<phi>) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((\<a>[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (((g.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>1) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (f\<^sub>0gh\<^sub>1.\<phi> \<star> TTfgh_TfTgh.chine) \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>0 f, f\<^sub>0gh\<^sub>1.p\<^sub>1, TTfgh_TfTgh.chine] \<cdot> (tab\<^sub>0 f \<star> \<gamma>\<^sub>f) \<cdot> \<a>[tab\<^sub>0 f, f\<^sub>0g\<^sub>1.p\<^sub>1, fg\<^sub>0h\<^sub>1.p\<^sub>1] \<cdot> (local.inv f\<^sub>0g\<^sub>1.\<phi> \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>1 g, f\<^sub>0g\<^sub>1.p\<^sub>0, fg\<^sub>0h\<^sub>1.p\<^sub>1] = (g \<star> h \<star> TTfgh_TfTgh.the_\<theta>) \<cdot> can (\<^bold>\<langle>g\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> ((\<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) (((\<^bold>\<langle>g\<^bold>\<rangle> \<^bold>\<star> (\<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) \<cdot> (((g \<star> h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (((g \<star> g\<^sub>0h\<^sub>1.\<phi>) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((\<a>[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (((g.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>1) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (\<a>\<^sup>-\<^sup>1[tab\<^sub>1 g \<star> g\<^sub>0h\<^sub>1.p\<^sub>1, f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine] \<cdot> (\<a>\<^sup>-\<^sup>1[tab\<^sub>1 g, g\<^sub>0h\<^sub>1.p\<^sub>1, f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine] \<cdot> \<a>[tab\<^sub>1 g, g\<^sub>0h\<^sub>1.p\<^sub>1, f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine]) \<cdot> \<a>[tab\<^sub>1 g \<star> g\<^sub>0h\<^sub>1.p\<^sub>1, f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine] \<cdot> (f\<^sub>0gh\<^sub>1.\<phi> \<star> TTfgh_TfTgh.chine)) \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>0 f, f\<^sub>0gh\<^sub>1.p\<^sub>1, TTfgh_TfTgh.chine] \<cdot> (tab\<^sub>0 f \<star> \<gamma>\<^sub>f) \<cdot> \<a>[tab\<^sub>0 f, f\<^sub>0g\<^sub>1.p\<^sub>1, fg\<^sub>0h\<^sub>1.p\<^sub>1] \<cdot> (local.inv f\<^sub>0g\<^sub>1.\<phi> \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>1 g, f\<^sub>0g\<^sub>1.p\<^sub>0, fg\<^sub>0h\<^sub>1.p\<^sub>1]
[PROOF STEP]
have "(\<a>\<^sup>-\<^sup>1[tab\<^sub>1 g \<star> Tgh.p\<^sub>1, TfTgh.p\<^sub>0, TTfgh_TfTgh.chine] \<cdot>
(\<a>\<^sup>-\<^sup>1[tab\<^sub>1 g, Tgh.p\<^sub>1, TfTgh.p\<^sub>0 \<star> TTfgh_TfTgh.chine] \<cdot>
\<a>[tab\<^sub>1 g, Tgh.p\<^sub>1, TfTgh.p\<^sub>0 \<star> TTfgh_TfTgh.chine]) \<cdot>
\<a>[tab\<^sub>1 g \<star> Tgh.p\<^sub>1, TfTgh.p\<^sub>0, TTfgh_TfTgh.chine]) \<cdot>
(f\<^sub>0gh\<^sub>1.\<phi> \<star> TTfgh_TfTgh.chine) =
f\<^sub>0gh\<^sub>1.\<phi> \<star> TTfgh_TfTgh.chine"
[PROOF STATE]
proof (prove)
goal (1 subgoal):
1. (\<a>\<^sup>-\<^sup>1[tab\<^sub>1 g \<star> g\<^sub>0h\<^sub>1.p\<^sub>1, f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine] \<cdot> (\<a>\<^sup>-\<^sup>1[tab\<^sub>1 g, g\<^sub>0h\<^sub>1.p\<^sub>1, f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine] \<cdot> \<a>[tab\<^sub>1 g, g\<^sub>0h\<^sub>1.p\<^sub>1, f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine]) \<cdot> \<a>[tab\<^sub>1 g \<star> g\<^sub>0h\<^sub>1.p\<^sub>1, f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine]) \<cdot> (f\<^sub>0gh\<^sub>1.\<phi> \<star> TTfgh_TfTgh.chine) = f\<^sub>0gh\<^sub>1.\<phi> \<star> TTfgh_TfTgh.chine
[PROOF STEP]
using f\<^sub>0gh\<^sub>1.p\<^sub>0_simps comp_cod_arr comp_arr_dom comp_assoc_assoc'
[PROOF STATE]
proof (prove)
using this:
trg f\<^sub>0gh\<^sub>1.p\<^sub>0 = src (tab\<^sub>1 g \<star> g\<^sub>0h\<^sub>1.p\<^sub>1)
\<lbrakk>arr ?f; cod ?f = ?b\<rbrakk> \<Longrightarrow> ?b \<cdot> ?f = ?f
\<lbrakk>arr ?f; local.dom ?f = ?a\<rbrakk> \<Longrightarrow> ?f \<cdot> ?a = ?f
\<lbrakk>ide ?f; ide ?g; ide ?h; src ?f = trg ?g; src ?g = trg ?h\<rbrakk> \<Longrightarrow> \<a>[?f, ?g, ?h] \<cdot> \<a>\<^sup>-\<^sup>1[?f, ?g, ?h] = ?f \<star> ?g \<star> ?h
\<lbrakk>ide ?f; ide ?g; ide ?h; src ?f = trg ?g; src ?g = trg ?h\<rbrakk> \<Longrightarrow> \<a>\<^sup>-\<^sup>1[?f, ?g, ?h] \<cdot> \<a>[?f, ?g, ?h] = (?f \<star> ?g) \<star> ?h
goal (1 subgoal):
1. (\<a>\<^sup>-\<^sup>1[tab\<^sub>1 g \<star> g\<^sub>0h\<^sub>1.p\<^sub>1, f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine] \<cdot> (\<a>\<^sup>-\<^sup>1[tab\<^sub>1 g, g\<^sub>0h\<^sub>1.p\<^sub>1, f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine] \<cdot> \<a>[tab\<^sub>1 g, g\<^sub>0h\<^sub>1.p\<^sub>1, f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine]) \<cdot> \<a>[tab\<^sub>1 g \<star> g\<^sub>0h\<^sub>1.p\<^sub>1, f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine]) \<cdot> (f\<^sub>0gh\<^sub>1.\<phi> \<star> TTfgh_TfTgh.chine) = f\<^sub>0gh\<^sub>1.\<phi> \<star> TTfgh_TfTgh.chine
[PROOF STEP]
by simp
[PROOF STATE]
proof (state)
this:
(\<a>\<^sup>-\<^sup>1[tab\<^sub>1 g \<star> g\<^sub>0h\<^sub>1.p\<^sub>1, f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine] \<cdot> (\<a>\<^sup>-\<^sup>1[tab\<^sub>1 g, g\<^sub>0h\<^sub>1.p\<^sub>1, f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine] \<cdot> \<a>[tab\<^sub>1 g, g\<^sub>0h\<^sub>1.p\<^sub>1, f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine]) \<cdot> \<a>[tab\<^sub>1 g \<star> g\<^sub>0h\<^sub>1.p\<^sub>1, f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine]) \<cdot> (f\<^sub>0gh\<^sub>1.\<phi> \<star> TTfgh_TfTgh.chine) = f\<^sub>0gh\<^sub>1.\<phi> \<star> TTfgh_TfTgh.chine
goal (1 subgoal):
1. (g \<star> h \<star> TTfgh_TfTgh.the_\<theta>) \<cdot> can (\<^bold>\<langle>g\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> ((\<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) (((\<^bold>\<langle>g\<^bold>\<rangle> \<^bold>\<star> (\<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) \<cdot> (((g \<star> h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (((g \<star> g\<^sub>0h\<^sub>1.\<phi>) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((\<a>[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (((g.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>1) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (f\<^sub>0gh\<^sub>1.\<phi> \<star> TTfgh_TfTgh.chine) \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>0 f, f\<^sub>0gh\<^sub>1.p\<^sub>1, TTfgh_TfTgh.chine] \<cdot> (tab\<^sub>0 f \<star> \<gamma>\<^sub>f) \<cdot> \<a>[tab\<^sub>0 f, f\<^sub>0g\<^sub>1.p\<^sub>1, fg\<^sub>0h\<^sub>1.p\<^sub>1] \<cdot> (local.inv f\<^sub>0g\<^sub>1.\<phi> \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>1 g, f\<^sub>0g\<^sub>1.p\<^sub>0, fg\<^sub>0h\<^sub>1.p\<^sub>1] = (g \<star> h \<star> TTfgh_TfTgh.the_\<theta>) \<cdot> can (\<^bold>\<langle>g\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> ((\<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) (((\<^bold>\<langle>g\<^bold>\<rangle> \<^bold>\<star> (\<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) \<cdot> (((g \<star> h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (((g \<star> g\<^sub>0h\<^sub>1.\<phi>) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((\<a>[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (((g.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>1) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (\<a>\<^sup>-\<^sup>1[tab\<^sub>1 g \<star> g\<^sub>0h\<^sub>1.p\<^sub>1, f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine] \<cdot> (\<a>\<^sup>-\<^sup>1[tab\<^sub>1 g, g\<^sub>0h\<^sub>1.p\<^sub>1, f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine] \<cdot> \<a>[tab\<^sub>1 g, g\<^sub>0h\<^sub>1.p\<^sub>1, f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine]) \<cdot> \<a>[tab\<^sub>1 g \<star> g\<^sub>0h\<^sub>1.p\<^sub>1, f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine] \<cdot> (f\<^sub>0gh\<^sub>1.\<phi> \<star> TTfgh_TfTgh.chine)) \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>0 f, f\<^sub>0gh\<^sub>1.p\<^sub>1, TTfgh_TfTgh.chine] \<cdot> (tab\<^sub>0 f \<star> \<gamma>\<^sub>f) \<cdot> \<a>[tab\<^sub>0 f, f\<^sub>0g\<^sub>1.p\<^sub>1, fg\<^sub>0h\<^sub>1.p\<^sub>1] \<cdot> (local.inv f\<^sub>0g\<^sub>1.\<phi> \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>1 g, f\<^sub>0g\<^sub>1.p\<^sub>0, fg\<^sub>0h\<^sub>1.p\<^sub>1]
[PROOF STEP]
thus ?thesis
[PROOF STATE]
proof (prove)
using this:
(\<a>\<^sup>-\<^sup>1[tab\<^sub>1 g \<star> g\<^sub>0h\<^sub>1.p\<^sub>1, f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine] \<cdot> (\<a>\<^sup>-\<^sup>1[tab\<^sub>1 g, g\<^sub>0h\<^sub>1.p\<^sub>1, f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine] \<cdot> \<a>[tab\<^sub>1 g, g\<^sub>0h\<^sub>1.p\<^sub>1, f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine]) \<cdot> \<a>[tab\<^sub>1 g \<star> g\<^sub>0h\<^sub>1.p\<^sub>1, f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine]) \<cdot> (f\<^sub>0gh\<^sub>1.\<phi> \<star> TTfgh_TfTgh.chine) = f\<^sub>0gh\<^sub>1.\<phi> \<star> TTfgh_TfTgh.chine
goal (1 subgoal):
1. (g \<star> h \<star> TTfgh_TfTgh.the_\<theta>) \<cdot> can (\<^bold>\<langle>g\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> ((\<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) (((\<^bold>\<langle>g\<^bold>\<rangle> \<^bold>\<star> (\<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) \<cdot> (((g \<star> h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (((g \<star> g\<^sub>0h\<^sub>1.\<phi>) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((\<a>[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (((g.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>1) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (f\<^sub>0gh\<^sub>1.\<phi> \<star> TTfgh_TfTgh.chine) \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>0 f, f\<^sub>0gh\<^sub>1.p\<^sub>1, TTfgh_TfTgh.chine] \<cdot> (tab\<^sub>0 f \<star> \<gamma>\<^sub>f) \<cdot> \<a>[tab\<^sub>0 f, f\<^sub>0g\<^sub>1.p\<^sub>1, fg\<^sub>0h\<^sub>1.p\<^sub>1] \<cdot> (local.inv f\<^sub>0g\<^sub>1.\<phi> \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>1 g, f\<^sub>0g\<^sub>1.p\<^sub>0, fg\<^sub>0h\<^sub>1.p\<^sub>1] = (g \<star> h \<star> TTfgh_TfTgh.the_\<theta>) \<cdot> can (\<^bold>\<langle>g\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> ((\<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) (((\<^bold>\<langle>g\<^bold>\<rangle> \<^bold>\<star> (\<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) \<cdot> (((g \<star> h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (((g \<star> g\<^sub>0h\<^sub>1.\<phi>) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((\<a>[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (((g.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>1) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (\<a>\<^sup>-\<^sup>1[tab\<^sub>1 g \<star> g\<^sub>0h\<^sub>1.p\<^sub>1, f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine] \<cdot> (\<a>\<^sup>-\<^sup>1[tab\<^sub>1 g, g\<^sub>0h\<^sub>1.p\<^sub>1, f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine] \<cdot> \<a>[tab\<^sub>1 g, g\<^sub>0h\<^sub>1.p\<^sub>1, f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine]) \<cdot> \<a>[tab\<^sub>1 g \<star> g\<^sub>0h\<^sub>1.p\<^sub>1, f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine] \<cdot> (f\<^sub>0gh\<^sub>1.\<phi> \<star> TTfgh_TfTgh.chine)) \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>0 f, f\<^sub>0gh\<^sub>1.p\<^sub>1, TTfgh_TfTgh.chine] \<cdot> (tab\<^sub>0 f \<star> \<gamma>\<^sub>f) \<cdot> \<a>[tab\<^sub>0 f, f\<^sub>0g\<^sub>1.p\<^sub>1, fg\<^sub>0h\<^sub>1.p\<^sub>1] \<cdot> (local.inv f\<^sub>0g\<^sub>1.\<phi> \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>1 g, f\<^sub>0g\<^sub>1.p\<^sub>0, fg\<^sub>0h\<^sub>1.p\<^sub>1]
[PROOF STEP]
using comp_assoc
[PROOF STATE]
proof (prove)
using this:
(\<a>\<^sup>-\<^sup>1[tab\<^sub>1 g \<star> g\<^sub>0h\<^sub>1.p\<^sub>1, f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine] \<cdot> (\<a>\<^sup>-\<^sup>1[tab\<^sub>1 g, g\<^sub>0h\<^sub>1.p\<^sub>1, f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine] \<cdot> \<a>[tab\<^sub>1 g, g\<^sub>0h\<^sub>1.p\<^sub>1, f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine]) \<cdot> \<a>[tab\<^sub>1 g \<star> g\<^sub>0h\<^sub>1.p\<^sub>1, f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine]) \<cdot> (f\<^sub>0gh\<^sub>1.\<phi> \<star> TTfgh_TfTgh.chine) = f\<^sub>0gh\<^sub>1.\<phi> \<star> TTfgh_TfTgh.chine
(?h \<cdot> ?g) \<cdot> ?f = ?h \<cdot> ?g \<cdot> ?f
goal (1 subgoal):
1. (g \<star> h \<star> TTfgh_TfTgh.the_\<theta>) \<cdot> can (\<^bold>\<langle>g\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> ((\<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) (((\<^bold>\<langle>g\<^bold>\<rangle> \<^bold>\<star> (\<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) \<cdot> (((g \<star> h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (((g \<star> g\<^sub>0h\<^sub>1.\<phi>) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((\<a>[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (((g.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>1) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (f\<^sub>0gh\<^sub>1.\<phi> \<star> TTfgh_TfTgh.chine) \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>0 f, f\<^sub>0gh\<^sub>1.p\<^sub>1, TTfgh_TfTgh.chine] \<cdot> (tab\<^sub>0 f \<star> \<gamma>\<^sub>f) \<cdot> \<a>[tab\<^sub>0 f, f\<^sub>0g\<^sub>1.p\<^sub>1, fg\<^sub>0h\<^sub>1.p\<^sub>1] \<cdot> (local.inv f\<^sub>0g\<^sub>1.\<phi> \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>1 g, f\<^sub>0g\<^sub>1.p\<^sub>0, fg\<^sub>0h\<^sub>1.p\<^sub>1] = (g \<star> h \<star> TTfgh_TfTgh.the_\<theta>) \<cdot> can (\<^bold>\<langle>g\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> ((\<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) (((\<^bold>\<langle>g\<^bold>\<rangle> \<^bold>\<star> (\<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) \<cdot> (((g \<star> h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (((g \<star> g\<^sub>0h\<^sub>1.\<phi>) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((\<a>[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (((g.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>1) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (\<a>\<^sup>-\<^sup>1[tab\<^sub>1 g \<star> g\<^sub>0h\<^sub>1.p\<^sub>1, f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine] \<cdot> (\<a>\<^sup>-\<^sup>1[tab\<^sub>1 g, g\<^sub>0h\<^sub>1.p\<^sub>1, f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine] \<cdot> \<a>[tab\<^sub>1 g, g\<^sub>0h\<^sub>1.p\<^sub>1, f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine]) \<cdot> \<a>[tab\<^sub>1 g \<star> g\<^sub>0h\<^sub>1.p\<^sub>1, f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine] \<cdot> (f\<^sub>0gh\<^sub>1.\<phi> \<star> TTfgh_TfTgh.chine)) \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>0 f, f\<^sub>0gh\<^sub>1.p\<^sub>1, TTfgh_TfTgh.chine] \<cdot> (tab\<^sub>0 f \<star> \<gamma>\<^sub>f) \<cdot> \<a>[tab\<^sub>0 f, f\<^sub>0g\<^sub>1.p\<^sub>1, fg\<^sub>0h\<^sub>1.p\<^sub>1] \<cdot> (local.inv f\<^sub>0g\<^sub>1.\<phi> \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>1 g, f\<^sub>0g\<^sub>1.p\<^sub>0, fg\<^sub>0h\<^sub>1.p\<^sub>1]
[PROOF STEP]
by fastforce
[PROOF STATE]
proof (state)
this:
(g \<star> h \<star> TTfgh_TfTgh.the_\<theta>) \<cdot> can (\<^bold>\<langle>g\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> ((\<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) (((\<^bold>\<langle>g\<^bold>\<rangle> \<^bold>\<star> (\<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) \<cdot> (((g \<star> h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (((g \<star> g\<^sub>0h\<^sub>1.\<phi>) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((\<a>[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (((g.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>1) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (f\<^sub>0gh\<^sub>1.\<phi> \<star> TTfgh_TfTgh.chine) \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>0 f, f\<^sub>0gh\<^sub>1.p\<^sub>1, TTfgh_TfTgh.chine] \<cdot> (tab\<^sub>0 f \<star> \<gamma>\<^sub>f) \<cdot> \<a>[tab\<^sub>0 f, f\<^sub>0g\<^sub>1.p\<^sub>1, fg\<^sub>0h\<^sub>1.p\<^sub>1] \<cdot> (local.inv f\<^sub>0g\<^sub>1.\<phi> \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>1 g, f\<^sub>0g\<^sub>1.p\<^sub>0, fg\<^sub>0h\<^sub>1.p\<^sub>1] = (g \<star> h \<star> TTfgh_TfTgh.the_\<theta>) \<cdot> can (\<^bold>\<langle>g\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> ((\<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) (((\<^bold>\<langle>g\<^bold>\<rangle> \<^bold>\<star> (\<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) \<cdot> (((g \<star> h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (((g \<star> g\<^sub>0h\<^sub>1.\<phi>) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((\<a>[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (((g.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>1) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (\<a>\<^sup>-\<^sup>1[tab\<^sub>1 g \<star> g\<^sub>0h\<^sub>1.p\<^sub>1, f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine] \<cdot> (\<a>\<^sup>-\<^sup>1[tab\<^sub>1 g, g\<^sub>0h\<^sub>1.p\<^sub>1, f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine] \<cdot> \<a>[tab\<^sub>1 g, g\<^sub>0h\<^sub>1.p\<^sub>1, f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine]) \<cdot> \<a>[tab\<^sub>1 g \<star> g\<^sub>0h\<^sub>1.p\<^sub>1, f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine] \<cdot> (f\<^sub>0gh\<^sub>1.\<phi> \<star> TTfgh_TfTgh.chine)) \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>0 f, f\<^sub>0gh\<^sub>1.p\<^sub>1, TTfgh_TfTgh.chine] \<cdot> (tab\<^sub>0 f \<star> \<gamma>\<^sub>f) \<cdot> \<a>[tab\<^sub>0 f, f\<^sub>0g\<^sub>1.p\<^sub>1, fg\<^sub>0h\<^sub>1.p\<^sub>1] \<cdot> (local.inv f\<^sub>0g\<^sub>1.\<phi> \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>1 g, f\<^sub>0g\<^sub>1.p\<^sub>0, fg\<^sub>0h\<^sub>1.p\<^sub>1]
goal:
No subgoals!
[PROOF STEP]
qed
[PROOF STATE]
proof (state)
this:
(g \<star> h \<star> TTfgh_TfTgh.the_\<theta>) \<cdot> can (\<^bold>\<langle>g\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> ((\<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) (((\<^bold>\<langle>g\<^bold>\<rangle> \<^bold>\<star> (\<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) \<cdot> (((g \<star> h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (((g \<star> g\<^sub>0h\<^sub>1.\<phi>) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((\<a>[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (((g.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>1) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (f\<^sub>0gh\<^sub>1.\<phi> \<star> TTfgh_TfTgh.chine) \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>0 f, f\<^sub>0gh\<^sub>1.p\<^sub>1, TTfgh_TfTgh.chine] \<cdot> (tab\<^sub>0 f \<star> \<gamma>\<^sub>f) \<cdot> \<a>[tab\<^sub>0 f, f\<^sub>0g\<^sub>1.p\<^sub>1, fg\<^sub>0h\<^sub>1.p\<^sub>1] \<cdot> (local.inv f\<^sub>0g\<^sub>1.\<phi> \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>1 g, f\<^sub>0g\<^sub>1.p\<^sub>0, fg\<^sub>0h\<^sub>1.p\<^sub>1] = (g \<star> h \<star> TTfgh_TfTgh.the_\<theta>) \<cdot> can (\<^bold>\<langle>g\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> ((\<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) (((\<^bold>\<langle>g\<^bold>\<rangle> \<^bold>\<star> (\<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) \<cdot> (((g \<star> h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (((g \<star> g\<^sub>0h\<^sub>1.\<phi>) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((\<a>[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (((g.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>1) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (\<a>\<^sup>-\<^sup>1[tab\<^sub>1 g \<star> g\<^sub>0h\<^sub>1.p\<^sub>1, f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine] \<cdot> (\<a>\<^sup>-\<^sup>1[tab\<^sub>1 g, g\<^sub>0h\<^sub>1.p\<^sub>1, f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine] \<cdot> \<a>[tab\<^sub>1 g, g\<^sub>0h\<^sub>1.p\<^sub>1, f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine]) \<cdot> \<a>[tab\<^sub>1 g \<star> g\<^sub>0h\<^sub>1.p\<^sub>1, f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine] \<cdot> (f\<^sub>0gh\<^sub>1.\<phi> \<star> TTfgh_TfTgh.chine)) \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>0 f, f\<^sub>0gh\<^sub>1.p\<^sub>1, TTfgh_TfTgh.chine] \<cdot> (tab\<^sub>0 f \<star> \<gamma>\<^sub>f) \<cdot> \<a>[tab\<^sub>0 f, f\<^sub>0g\<^sub>1.p\<^sub>1, fg\<^sub>0h\<^sub>1.p\<^sub>1] \<cdot> (local.inv f\<^sub>0g\<^sub>1.\<phi> \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>1 g, f\<^sub>0g\<^sub>1.p\<^sub>0, fg\<^sub>0h\<^sub>1.p\<^sub>1]
goal (1 subgoal):
1. (g \<star> \<theta>\<^sub>g) \<cdot> \<a>[g, tab\<^sub>0 g, w\<^sub>g] \<cdot> (g.tab \<star> w\<^sub>g) = ((g \<star> \<theta>\<^sub>g') \<cdot> \<a>[g, tab\<^sub>0 g, w\<^sub>g'] \<cdot> (g.tab \<star> w\<^sub>g')) \<cdot> \<beta>\<^sub>g
[PROOF STEP]
also
[PROOF STATE]
proof (state)
this:
(g \<star> h \<star> TTfgh_TfTgh.the_\<theta>) \<cdot> can (\<^bold>\<langle>g\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> ((\<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) (((\<^bold>\<langle>g\<^bold>\<rangle> \<^bold>\<star> (\<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) \<cdot> (((g \<star> h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (((g \<star> g\<^sub>0h\<^sub>1.\<phi>) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((\<a>[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (((g.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>1) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (f\<^sub>0gh\<^sub>1.\<phi> \<star> TTfgh_TfTgh.chine) \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>0 f, f\<^sub>0gh\<^sub>1.p\<^sub>1, TTfgh_TfTgh.chine] \<cdot> (tab\<^sub>0 f \<star> \<gamma>\<^sub>f) \<cdot> \<a>[tab\<^sub>0 f, f\<^sub>0g\<^sub>1.p\<^sub>1, fg\<^sub>0h\<^sub>1.p\<^sub>1] \<cdot> (local.inv f\<^sub>0g\<^sub>1.\<phi> \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>1 g, f\<^sub>0g\<^sub>1.p\<^sub>0, fg\<^sub>0h\<^sub>1.p\<^sub>1] = (g \<star> h \<star> TTfgh_TfTgh.the_\<theta>) \<cdot> can (\<^bold>\<langle>g\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> ((\<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) (((\<^bold>\<langle>g\<^bold>\<rangle> \<^bold>\<star> (\<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) \<cdot> (((g \<star> h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (((g \<star> g\<^sub>0h\<^sub>1.\<phi>) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((\<a>[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (((g.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>1) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (\<a>\<^sup>-\<^sup>1[tab\<^sub>1 g \<star> g\<^sub>0h\<^sub>1.p\<^sub>1, f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine] \<cdot> (\<a>\<^sup>-\<^sup>1[tab\<^sub>1 g, g\<^sub>0h\<^sub>1.p\<^sub>1, f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine] \<cdot> \<a>[tab\<^sub>1 g, g\<^sub>0h\<^sub>1.p\<^sub>1, f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine]) \<cdot> \<a>[tab\<^sub>1 g \<star> g\<^sub>0h\<^sub>1.p\<^sub>1, f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine] \<cdot> (f\<^sub>0gh\<^sub>1.\<phi> \<star> TTfgh_TfTgh.chine)) \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>0 f, f\<^sub>0gh\<^sub>1.p\<^sub>1, TTfgh_TfTgh.chine] \<cdot> (tab\<^sub>0 f \<star> \<gamma>\<^sub>f) \<cdot> \<a>[tab\<^sub>0 f, f\<^sub>0g\<^sub>1.p\<^sub>1, fg\<^sub>0h\<^sub>1.p\<^sub>1] \<cdot> (local.inv f\<^sub>0g\<^sub>1.\<phi> \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>1 g, f\<^sub>0g\<^sub>1.p\<^sub>0, fg\<^sub>0h\<^sub>1.p\<^sub>1]
goal (1 subgoal):
1. (g \<star> \<theta>\<^sub>g) \<cdot> \<a>[g, tab\<^sub>0 g, w\<^sub>g] \<cdot> (g.tab \<star> w\<^sub>g) = ((g \<star> \<theta>\<^sub>g') \<cdot> \<a>[g, tab\<^sub>0 g, w\<^sub>g'] \<cdot> (g.tab \<star> w\<^sub>g')) \<cdot> \<beta>\<^sub>g
[PROOF STEP]
have "... = (g \<star> h \<star> TTfgh_TfTgh.the_\<theta>) \<cdot>
can (\<^bold>\<langle>g\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> ((\<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>Tgh.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TfTgh.p\<^sub>0\<^bold>\<rangle>)
\<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>)
(((\<^bold>\<langle>g\<^bold>\<rangle> \<^bold>\<star> (\<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>Tgh.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TfTgh.p\<^sub>0\<^bold>\<rangle>)
\<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) \<cdot>
(((g \<star> h.tab \<star> Tgh.p\<^sub>0) \<star> TfTgh.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot>
(((g \<star> g\<^sub>0h\<^sub>1.\<phi>) \<star> TfTgh.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot>
((\<a>[g, tab\<^sub>0 g, Tgh.p\<^sub>1] \<star> TfTgh.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot>
((((g.tab \<star> Tgh.p\<^sub>1) \<star> TfTgh.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot>
\<a>\<^sup>-\<^sup>1[tab\<^sub>1 g \<star> Tgh.p\<^sub>1, TfTgh.p\<^sub>0, TTfgh_TfTgh.chine]) \<cdot>
\<a>\<^sup>-\<^sup>1[tab\<^sub>1 g, Tgh.p\<^sub>1, TfTgh.p\<^sub>0 \<star> TTfgh_TfTgh.chine] \<cdot>
\<beta>\<^sub>g"
[PROOF STATE]
proof (prove)
goal (1 subgoal):
1. (g \<star> h \<star> TTfgh_TfTgh.the_\<theta>) \<cdot> can (\<^bold>\<langle>g\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> ((\<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) (((\<^bold>\<langle>g\<^bold>\<rangle> \<^bold>\<star> (\<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) \<cdot> (((g \<star> h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (((g \<star> g\<^sub>0h\<^sub>1.\<phi>) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((\<a>[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (((g.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>1) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (\<a>\<^sup>-\<^sup>1[tab\<^sub>1 g \<star> g\<^sub>0h\<^sub>1.p\<^sub>1, f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine] \<cdot> (\<a>\<^sup>-\<^sup>1[tab\<^sub>1 g, g\<^sub>0h\<^sub>1.p\<^sub>1, f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine] \<cdot> \<a>[tab\<^sub>1 g, g\<^sub>0h\<^sub>1.p\<^sub>1, f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine]) \<cdot> \<a>[tab\<^sub>1 g \<star> g\<^sub>0h\<^sub>1.p\<^sub>1, f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine] \<cdot> (f\<^sub>0gh\<^sub>1.\<phi> \<star> TTfgh_TfTgh.chine)) \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>0 f, f\<^sub>0gh\<^sub>1.p\<^sub>1, TTfgh_TfTgh.chine] \<cdot> (tab\<^sub>0 f \<star> \<gamma>\<^sub>f) \<cdot> \<a>[tab\<^sub>0 f, f\<^sub>0g\<^sub>1.p\<^sub>1, fg\<^sub>0h\<^sub>1.p\<^sub>1] \<cdot> (local.inv f\<^sub>0g\<^sub>1.\<phi> \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>1 g, f\<^sub>0g\<^sub>1.p\<^sub>0, fg\<^sub>0h\<^sub>1.p\<^sub>1] = (g \<star> h \<star> TTfgh_TfTgh.the_\<theta>) \<cdot> can (\<^bold>\<langle>g\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> ((\<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) (((\<^bold>\<langle>g\<^bold>\<rangle> \<^bold>\<star> (\<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) \<cdot> (((g \<star> h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (((g \<star> g\<^sub>0h\<^sub>1.\<phi>) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((\<a>[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((((g.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>1) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>1 g \<star> g\<^sub>0h\<^sub>1.p\<^sub>1, f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine]) \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>1 g, g\<^sub>0h\<^sub>1.p\<^sub>1, f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine] \<cdot> \<beta>\<^sub>g
[PROOF STEP]
unfolding \<beta>\<^sub>g_def
[PROOF STATE]
proof (prove)
goal (1 subgoal):
1. (g \<star> h \<star> TTfgh_TfTgh.the_\<theta>) \<cdot> can (\<^bold>\<langle>g\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> ((\<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) (((\<^bold>\<langle>g\<^bold>\<rangle> \<^bold>\<star> (\<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) \<cdot> (((g \<star> h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (((g \<star> g\<^sub>0h\<^sub>1.\<phi>) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((\<a>[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (((g.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>1) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (\<a>\<^sup>-\<^sup>1[tab\<^sub>1 g \<star> g\<^sub>0h\<^sub>1.p\<^sub>1, f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine] \<cdot> (\<a>\<^sup>-\<^sup>1[tab\<^sub>1 g, g\<^sub>0h\<^sub>1.p\<^sub>1, f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine] \<cdot> \<a>[tab\<^sub>1 g, g\<^sub>0h\<^sub>1.p\<^sub>1, f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine]) \<cdot> \<a>[tab\<^sub>1 g \<star> g\<^sub>0h\<^sub>1.p\<^sub>1, f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine] \<cdot> (f\<^sub>0gh\<^sub>1.\<phi> \<star> TTfgh_TfTgh.chine)) \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>0 f, f\<^sub>0gh\<^sub>1.p\<^sub>1, TTfgh_TfTgh.chine] \<cdot> (tab\<^sub>0 f \<star> \<gamma>\<^sub>f) \<cdot> \<a>[tab\<^sub>0 f, f\<^sub>0g\<^sub>1.p\<^sub>1, fg\<^sub>0h\<^sub>1.p\<^sub>1] \<cdot> (local.inv f\<^sub>0g\<^sub>1.\<phi> \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>1 g, f\<^sub>0g\<^sub>1.p\<^sub>0, fg\<^sub>0h\<^sub>1.p\<^sub>1] = (g \<star> h \<star> TTfgh_TfTgh.the_\<theta>) \<cdot> can (\<^bold>\<langle>g\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> ((\<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) (((\<^bold>\<langle>g\<^bold>\<rangle> \<^bold>\<star> (\<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) \<cdot> (((g \<star> h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (((g \<star> g\<^sub>0h\<^sub>1.\<phi>) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((\<a>[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((((g.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>1) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>1 g \<star> g\<^sub>0h\<^sub>1.p\<^sub>1, f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine]) \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>1 g, g\<^sub>0h\<^sub>1.p\<^sub>1, f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine] \<cdot> \<a>[tab\<^sub>1 g, g\<^sub>0h\<^sub>1.p\<^sub>1, f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine] \<cdot> \<a>[tab\<^sub>1 g \<star> g\<^sub>0h\<^sub>1.p\<^sub>1, f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine] \<cdot> (f\<^sub>0gh\<^sub>1.\<phi> \<star> TTfgh_TfTgh.chine) \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>0 f, f\<^sub>0gh\<^sub>1.p\<^sub>1, TTfgh_TfTgh.chine] \<cdot> (tab\<^sub>0 f \<star> \<gamma>\<^sub>f) \<cdot> \<a>[tab\<^sub>0 f, f\<^sub>0g\<^sub>1.p\<^sub>1, fg\<^sub>0h\<^sub>1.p\<^sub>1] \<cdot> (local.inv f\<^sub>0g\<^sub>1.\<phi> \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>1 g, f\<^sub>0g\<^sub>1.p\<^sub>0, fg\<^sub>0h\<^sub>1.p\<^sub>1]
[PROOF STEP]
using comp_assoc
[PROOF STATE]
proof (prove)
using this:
(?h \<cdot> ?g) \<cdot> ?f = ?h \<cdot> ?g \<cdot> ?f
goal (1 subgoal):
1. (g \<star> h \<star> TTfgh_TfTgh.the_\<theta>) \<cdot> can (\<^bold>\<langle>g\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> ((\<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) (((\<^bold>\<langle>g\<^bold>\<rangle> \<^bold>\<star> (\<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) \<cdot> (((g \<star> h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (((g \<star> g\<^sub>0h\<^sub>1.\<phi>) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((\<a>[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (((g.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>1) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (\<a>\<^sup>-\<^sup>1[tab\<^sub>1 g \<star> g\<^sub>0h\<^sub>1.p\<^sub>1, f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine] \<cdot> (\<a>\<^sup>-\<^sup>1[tab\<^sub>1 g, g\<^sub>0h\<^sub>1.p\<^sub>1, f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine] \<cdot> \<a>[tab\<^sub>1 g, g\<^sub>0h\<^sub>1.p\<^sub>1, f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine]) \<cdot> \<a>[tab\<^sub>1 g \<star> g\<^sub>0h\<^sub>1.p\<^sub>1, f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine] \<cdot> (f\<^sub>0gh\<^sub>1.\<phi> \<star> TTfgh_TfTgh.chine)) \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>0 f, f\<^sub>0gh\<^sub>1.p\<^sub>1, TTfgh_TfTgh.chine] \<cdot> (tab\<^sub>0 f \<star> \<gamma>\<^sub>f) \<cdot> \<a>[tab\<^sub>0 f, f\<^sub>0g\<^sub>1.p\<^sub>1, fg\<^sub>0h\<^sub>1.p\<^sub>1] \<cdot> (local.inv f\<^sub>0g\<^sub>1.\<phi> \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>1 g, f\<^sub>0g\<^sub>1.p\<^sub>0, fg\<^sub>0h\<^sub>1.p\<^sub>1] = (g \<star> h \<star> TTfgh_TfTgh.the_\<theta>) \<cdot> can (\<^bold>\<langle>g\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> ((\<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) (((\<^bold>\<langle>g\<^bold>\<rangle> \<^bold>\<star> (\<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) \<cdot> (((g \<star> h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (((g \<star> g\<^sub>0h\<^sub>1.\<phi>) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((\<a>[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((((g.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>1) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>1 g \<star> g\<^sub>0h\<^sub>1.p\<^sub>1, f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine]) \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>1 g, g\<^sub>0h\<^sub>1.p\<^sub>1, f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine] \<cdot> \<a>[tab\<^sub>1 g, g\<^sub>0h\<^sub>1.p\<^sub>1, f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine] \<cdot> \<a>[tab\<^sub>1 g \<star> g\<^sub>0h\<^sub>1.p\<^sub>1, f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine] \<cdot> (f\<^sub>0gh\<^sub>1.\<phi> \<star> TTfgh_TfTgh.chine) \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>0 f, f\<^sub>0gh\<^sub>1.p\<^sub>1, TTfgh_TfTgh.chine] \<cdot> (tab\<^sub>0 f \<star> \<gamma>\<^sub>f) \<cdot> \<a>[tab\<^sub>0 f, f\<^sub>0g\<^sub>1.p\<^sub>1, fg\<^sub>0h\<^sub>1.p\<^sub>1] \<cdot> (local.inv f\<^sub>0g\<^sub>1.\<phi> \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>1 g, f\<^sub>0g\<^sub>1.p\<^sub>0, fg\<^sub>0h\<^sub>1.p\<^sub>1]
[PROOF STEP]
by presburger
[PROOF STATE]
proof (state)
this:
(g \<star> h \<star> TTfgh_TfTgh.the_\<theta>) \<cdot> can (\<^bold>\<langle>g\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> ((\<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) (((\<^bold>\<langle>g\<^bold>\<rangle> \<^bold>\<star> (\<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) \<cdot> (((g \<star> h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (((g \<star> g\<^sub>0h\<^sub>1.\<phi>) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((\<a>[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (((g.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>1) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (\<a>\<^sup>-\<^sup>1[tab\<^sub>1 g \<star> g\<^sub>0h\<^sub>1.p\<^sub>1, f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine] \<cdot> (\<a>\<^sup>-\<^sup>1[tab\<^sub>1 g, g\<^sub>0h\<^sub>1.p\<^sub>1, f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine] \<cdot> \<a>[tab\<^sub>1 g, g\<^sub>0h\<^sub>1.p\<^sub>1, f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine]) \<cdot> \<a>[tab\<^sub>1 g \<star> g\<^sub>0h\<^sub>1.p\<^sub>1, f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine] \<cdot> (f\<^sub>0gh\<^sub>1.\<phi> \<star> TTfgh_TfTgh.chine)) \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>0 f, f\<^sub>0gh\<^sub>1.p\<^sub>1, TTfgh_TfTgh.chine] \<cdot> (tab\<^sub>0 f \<star> \<gamma>\<^sub>f) \<cdot> \<a>[tab\<^sub>0 f, f\<^sub>0g\<^sub>1.p\<^sub>1, fg\<^sub>0h\<^sub>1.p\<^sub>1] \<cdot> (local.inv f\<^sub>0g\<^sub>1.\<phi> \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>1 g, f\<^sub>0g\<^sub>1.p\<^sub>0, fg\<^sub>0h\<^sub>1.p\<^sub>1] = (g \<star> h \<star> TTfgh_TfTgh.the_\<theta>) \<cdot> can (\<^bold>\<langle>g\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> ((\<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) (((\<^bold>\<langle>g\<^bold>\<rangle> \<^bold>\<star> (\<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) \<cdot> (((g \<star> h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (((g \<star> g\<^sub>0h\<^sub>1.\<phi>) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((\<a>[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((((g.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>1) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>1 g \<star> g\<^sub>0h\<^sub>1.p\<^sub>1, f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine]) \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>1 g, g\<^sub>0h\<^sub>1.p\<^sub>1, f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine] \<cdot> \<beta>\<^sub>g
goal (1 subgoal):
1. (g \<star> \<theta>\<^sub>g) \<cdot> \<a>[g, tab\<^sub>0 g, w\<^sub>g] \<cdot> (g.tab \<star> w\<^sub>g) = ((g \<star> \<theta>\<^sub>g') \<cdot> \<a>[g, tab\<^sub>0 g, w\<^sub>g'] \<cdot> (g.tab \<star> w\<^sub>g')) \<cdot> \<beta>\<^sub>g
[PROOF STEP]
also
[PROOF STATE]
proof (state)
this:
(g \<star> h \<star> TTfgh_TfTgh.the_\<theta>) \<cdot> can (\<^bold>\<langle>g\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> ((\<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) (((\<^bold>\<langle>g\<^bold>\<rangle> \<^bold>\<star> (\<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) \<cdot> (((g \<star> h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (((g \<star> g\<^sub>0h\<^sub>1.\<phi>) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((\<a>[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (((g.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>1) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (\<a>\<^sup>-\<^sup>1[tab\<^sub>1 g \<star> g\<^sub>0h\<^sub>1.p\<^sub>1, f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine] \<cdot> (\<a>\<^sup>-\<^sup>1[tab\<^sub>1 g, g\<^sub>0h\<^sub>1.p\<^sub>1, f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine] \<cdot> \<a>[tab\<^sub>1 g, g\<^sub>0h\<^sub>1.p\<^sub>1, f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine]) \<cdot> \<a>[tab\<^sub>1 g \<star> g\<^sub>0h\<^sub>1.p\<^sub>1, f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine] \<cdot> (f\<^sub>0gh\<^sub>1.\<phi> \<star> TTfgh_TfTgh.chine)) \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>0 f, f\<^sub>0gh\<^sub>1.p\<^sub>1, TTfgh_TfTgh.chine] \<cdot> (tab\<^sub>0 f \<star> \<gamma>\<^sub>f) \<cdot> \<a>[tab\<^sub>0 f, f\<^sub>0g\<^sub>1.p\<^sub>1, fg\<^sub>0h\<^sub>1.p\<^sub>1] \<cdot> (local.inv f\<^sub>0g\<^sub>1.\<phi> \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>1 g, f\<^sub>0g\<^sub>1.p\<^sub>0, fg\<^sub>0h\<^sub>1.p\<^sub>1] = (g \<star> h \<star> TTfgh_TfTgh.the_\<theta>) \<cdot> can (\<^bold>\<langle>g\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> ((\<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) (((\<^bold>\<langle>g\<^bold>\<rangle> \<^bold>\<star> (\<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) \<cdot> (((g \<star> h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (((g \<star> g\<^sub>0h\<^sub>1.\<phi>) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((\<a>[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((((g.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>1) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>1 g \<star> g\<^sub>0h\<^sub>1.p\<^sub>1, f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine]) \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>1 g, g\<^sub>0h\<^sub>1.p\<^sub>1, f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine] \<cdot> \<beta>\<^sub>g
goal (1 subgoal):
1. (g \<star> \<theta>\<^sub>g) \<cdot> \<a>[g, tab\<^sub>0 g, w\<^sub>g] \<cdot> (g.tab \<star> w\<^sub>g) = ((g \<star> \<theta>\<^sub>g') \<cdot> \<a>[g, tab\<^sub>0 g, w\<^sub>g'] \<cdot> (g.tab \<star> w\<^sub>g')) \<cdot> \<beta>\<^sub>g
[PROOF STEP]
have "... = (g \<star> h \<star> TTfgh_TfTgh.the_\<theta>) \<cdot>
can (\<^bold>\<langle>g\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> ((\<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>Tgh.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TfTgh.p\<^sub>0\<^bold>\<rangle>)
\<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>)
(((\<^bold>\<langle>g\<^bold>\<rangle> \<^bold>\<star> (\<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>Tgh.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TfTgh.p\<^sub>0\<^bold>\<rangle>)
\<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) \<cdot>
(((g \<star> h.tab \<star> Tgh.p\<^sub>0) \<star> TfTgh.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot>
(((g \<star> g\<^sub>0h\<^sub>1.\<phi>) \<star> TfTgh.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot>
((\<a>[g, tab\<^sub>0 g, Tgh.p\<^sub>1] \<star> TfTgh.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot>
\<a>\<^sup>-\<^sup>1[(g \<star> tab\<^sub>0 g) \<star> Tgh.p\<^sub>1, TfTgh.p\<^sub>0, TTfgh_TfTgh.chine] \<cdot>
(((g.tab \<star> Tgh.p\<^sub>1) \<star> TfTgh.p\<^sub>0 \<star> TTfgh_TfTgh.chine) \<cdot>
\<a>\<^sup>-\<^sup>1[tab\<^sub>1 g, Tgh.p\<^sub>1, TfTgh.p\<^sub>0 \<star> TTfgh_TfTgh.chine]) \<cdot>
\<beta>\<^sub>g"
[PROOF STATE]
proof (prove)
goal (1 subgoal):
1. (g \<star> h \<star> TTfgh_TfTgh.the_\<theta>) \<cdot> can (\<^bold>\<langle>g\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> ((\<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) (((\<^bold>\<langle>g\<^bold>\<rangle> \<^bold>\<star> (\<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) \<cdot> (((g \<star> h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (((g \<star> g\<^sub>0h\<^sub>1.\<phi>) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((\<a>[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((((g.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>1) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>1 g \<star> g\<^sub>0h\<^sub>1.p\<^sub>1, f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine]) \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>1 g, g\<^sub>0h\<^sub>1.p\<^sub>1, f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine] \<cdot> \<beta>\<^sub>g = (g \<star> h \<star> TTfgh_TfTgh.the_\<theta>) \<cdot> can (\<^bold>\<langle>g\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> ((\<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) (((\<^bold>\<langle>g\<^bold>\<rangle> \<^bold>\<star> (\<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) \<cdot> (((g \<star> h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (((g \<star> g\<^sub>0h\<^sub>1.\<phi>) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((\<a>[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> \<a>\<^sup>-\<^sup>1[(g \<star> tab\<^sub>0 g) \<star> g\<^sub>0h\<^sub>1.p\<^sub>1, f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine] \<cdot> (((g.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>1) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine) \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>1 g, g\<^sub>0h\<^sub>1.p\<^sub>1, f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine]) \<cdot> \<beta>\<^sub>g
[PROOF STEP]
proof -
[PROOF STATE]
proof (state)
goal (1 subgoal):
1. (g \<star> h \<star> TTfgh_TfTgh.the_\<theta>) \<cdot> can (\<^bold>\<langle>g\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> ((\<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) (((\<^bold>\<langle>g\<^bold>\<rangle> \<^bold>\<star> (\<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) \<cdot> (((g \<star> h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (((g \<star> g\<^sub>0h\<^sub>1.\<phi>) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((\<a>[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((((g.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>1) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>1 g \<star> g\<^sub>0h\<^sub>1.p\<^sub>1, f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine]) \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>1 g, g\<^sub>0h\<^sub>1.p\<^sub>1, f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine] \<cdot> \<beta>\<^sub>g = (g \<star> h \<star> TTfgh_TfTgh.the_\<theta>) \<cdot> can (\<^bold>\<langle>g\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> ((\<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) (((\<^bold>\<langle>g\<^bold>\<rangle> \<^bold>\<star> (\<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) \<cdot> (((g \<star> h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (((g \<star> g\<^sub>0h\<^sub>1.\<phi>) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((\<a>[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> \<a>\<^sup>-\<^sup>1[(g \<star> tab\<^sub>0 g) \<star> g\<^sub>0h\<^sub>1.p\<^sub>1, f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine] \<cdot> (((g.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>1) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine) \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>1 g, g\<^sub>0h\<^sub>1.p\<^sub>1, f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine]) \<cdot> \<beta>\<^sub>g
[PROOF STEP]
have "(((g.tab \<star> Tgh.p\<^sub>1) \<star> TfTgh.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot>
\<a>\<^sup>-\<^sup>1[tab\<^sub>1 g \<star> Tgh.p\<^sub>1, TfTgh.p\<^sub>0, TTfgh_TfTgh.chine] =
\<a>\<^sup>-\<^sup>1[(g \<star> tab\<^sub>0 g) \<star> Tgh.p\<^sub>1, TfTgh.p\<^sub>0, TTfgh_TfTgh.chine] \<cdot>
((g.tab \<star> Tgh.p\<^sub>1) \<star> TfTgh.p\<^sub>0 \<star> TTfgh_TfTgh.chine)"
[PROOF STATE]
proof (prove)
goal (1 subgoal):
1. (((g.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>1) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>1 g \<star> g\<^sub>0h\<^sub>1.p\<^sub>1, f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine] = \<a>\<^sup>-\<^sup>1[(g \<star> tab\<^sub>0 g) \<star> g\<^sub>0h\<^sub>1.p\<^sub>1, f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine] \<cdot> ((g.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>1) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine)
[PROOF STEP]
using f\<^sub>0gh\<^sub>1.p\<^sub>0_simps
assoc'_naturality [of "(g.tab \<star> Tgh.p\<^sub>1)" TfTgh.p\<^sub>0 TTfgh_TfTgh.chine]
[PROOF STATE]
proof (prove)
using this:
trg f\<^sub>0gh\<^sub>1.p\<^sub>0 = src (tab\<^sub>1 g \<star> g\<^sub>0h\<^sub>1.p\<^sub>1)
\<lbrakk>hseq g.tab g\<^sub>0h\<^sub>1.p\<^sub>1; arr f\<^sub>0gh\<^sub>1.p\<^sub>0; arr TTfgh_TfTgh.chine; src (g.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>1) = trg f\<^sub>0gh\<^sub>1.p\<^sub>0; src f\<^sub>0gh\<^sub>1.p\<^sub>0 = trg TTfgh_TfTgh.chine\<rbrakk> \<Longrightarrow> \<a>\<^sup>-\<^sup>1[cod (g.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>1), cod f\<^sub>0gh\<^sub>1.p\<^sub>0, cod TTfgh_TfTgh.chine] \<cdot> ((g.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>1) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine) = (((g.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>1) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> \<a>\<^sup>-\<^sup>1[local.dom (g.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>1), local.dom f\<^sub>0gh\<^sub>1.p\<^sub>0, local.dom TTfgh_TfTgh.chine]
goal (1 subgoal):
1. (((g.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>1) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>1 g \<star> g\<^sub>0h\<^sub>1.p\<^sub>1, f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine] = \<a>\<^sup>-\<^sup>1[(g \<star> tab\<^sub>0 g) \<star> g\<^sub>0h\<^sub>1.p\<^sub>1, f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine] \<cdot> ((g.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>1) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine)
[PROOF STEP]
by simp
[PROOF STATE]
proof (state)
this:
(((g.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>1) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>1 g \<star> g\<^sub>0h\<^sub>1.p\<^sub>1, f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine] = \<a>\<^sup>-\<^sup>1[(g \<star> tab\<^sub>0 g) \<star> g\<^sub>0h\<^sub>1.p\<^sub>1, f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine] \<cdot> ((g.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>1) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine)
goal (1 subgoal):
1. (g \<star> h \<star> TTfgh_TfTgh.the_\<theta>) \<cdot> can (\<^bold>\<langle>g\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> ((\<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) (((\<^bold>\<langle>g\<^bold>\<rangle> \<^bold>\<star> (\<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) \<cdot> (((g \<star> h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (((g \<star> g\<^sub>0h\<^sub>1.\<phi>) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((\<a>[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((((g.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>1) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>1 g \<star> g\<^sub>0h\<^sub>1.p\<^sub>1, f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine]) \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>1 g, g\<^sub>0h\<^sub>1.p\<^sub>1, f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine] \<cdot> \<beta>\<^sub>g = (g \<star> h \<star> TTfgh_TfTgh.the_\<theta>) \<cdot> can (\<^bold>\<langle>g\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> ((\<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) (((\<^bold>\<langle>g\<^bold>\<rangle> \<^bold>\<star> (\<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) \<cdot> (((g \<star> h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (((g \<star> g\<^sub>0h\<^sub>1.\<phi>) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((\<a>[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> \<a>\<^sup>-\<^sup>1[(g \<star> tab\<^sub>0 g) \<star> g\<^sub>0h\<^sub>1.p\<^sub>1, f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine] \<cdot> (((g.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>1) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine) \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>1 g, g\<^sub>0h\<^sub>1.p\<^sub>1, f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine]) \<cdot> \<beta>\<^sub>g
[PROOF STEP]
thus ?thesis
[PROOF STATE]
proof (prove)
using this:
(((g.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>1) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>1 g \<star> g\<^sub>0h\<^sub>1.p\<^sub>1, f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine] = \<a>\<^sup>-\<^sup>1[(g \<star> tab\<^sub>0 g) \<star> g\<^sub>0h\<^sub>1.p\<^sub>1, f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine] \<cdot> ((g.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>1) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine)
goal (1 subgoal):
1. (g \<star> h \<star> TTfgh_TfTgh.the_\<theta>) \<cdot> can (\<^bold>\<langle>g\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> ((\<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) (((\<^bold>\<langle>g\<^bold>\<rangle> \<^bold>\<star> (\<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) \<cdot> (((g \<star> h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (((g \<star> g\<^sub>0h\<^sub>1.\<phi>) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((\<a>[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((((g.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>1) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>1 g \<star> g\<^sub>0h\<^sub>1.p\<^sub>1, f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine]) \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>1 g, g\<^sub>0h\<^sub>1.p\<^sub>1, f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine] \<cdot> \<beta>\<^sub>g = (g \<star> h \<star> TTfgh_TfTgh.the_\<theta>) \<cdot> can (\<^bold>\<langle>g\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> ((\<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) (((\<^bold>\<langle>g\<^bold>\<rangle> \<^bold>\<star> (\<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) \<cdot> (((g \<star> h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (((g \<star> g\<^sub>0h\<^sub>1.\<phi>) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((\<a>[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> \<a>\<^sup>-\<^sup>1[(g \<star> tab\<^sub>0 g) \<star> g\<^sub>0h\<^sub>1.p\<^sub>1, f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine] \<cdot> (((g.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>1) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine) \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>1 g, g\<^sub>0h\<^sub>1.p\<^sub>1, f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine]) \<cdot> \<beta>\<^sub>g
[PROOF STEP]
using comp_assoc
[PROOF STATE]
proof (prove)
using this:
(((g.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>1) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>1 g \<star> g\<^sub>0h\<^sub>1.p\<^sub>1, f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine] = \<a>\<^sup>-\<^sup>1[(g \<star> tab\<^sub>0 g) \<star> g\<^sub>0h\<^sub>1.p\<^sub>1, f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine] \<cdot> ((g.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>1) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine)
(?h \<cdot> ?g) \<cdot> ?f = ?h \<cdot> ?g \<cdot> ?f
goal (1 subgoal):
1. (g \<star> h \<star> TTfgh_TfTgh.the_\<theta>) \<cdot> can (\<^bold>\<langle>g\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> ((\<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) (((\<^bold>\<langle>g\<^bold>\<rangle> \<^bold>\<star> (\<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) \<cdot> (((g \<star> h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (((g \<star> g\<^sub>0h\<^sub>1.\<phi>) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((\<a>[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((((g.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>1) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>1 g \<star> g\<^sub>0h\<^sub>1.p\<^sub>1, f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine]) \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>1 g, g\<^sub>0h\<^sub>1.p\<^sub>1, f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine] \<cdot> \<beta>\<^sub>g = (g \<star> h \<star> TTfgh_TfTgh.the_\<theta>) \<cdot> can (\<^bold>\<langle>g\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> ((\<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) (((\<^bold>\<langle>g\<^bold>\<rangle> \<^bold>\<star> (\<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) \<cdot> (((g \<star> h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (((g \<star> g\<^sub>0h\<^sub>1.\<phi>) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((\<a>[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> \<a>\<^sup>-\<^sup>1[(g \<star> tab\<^sub>0 g) \<star> g\<^sub>0h\<^sub>1.p\<^sub>1, f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine] \<cdot> (((g.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>1) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine) \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>1 g, g\<^sub>0h\<^sub>1.p\<^sub>1, f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine]) \<cdot> \<beta>\<^sub>g
[PROOF STEP]
by presburger
[PROOF STATE]
proof (state)
this:
(g \<star> h \<star> TTfgh_TfTgh.the_\<theta>) \<cdot> can (\<^bold>\<langle>g\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> ((\<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) (((\<^bold>\<langle>g\<^bold>\<rangle> \<^bold>\<star> (\<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) \<cdot> (((g \<star> h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (((g \<star> g\<^sub>0h\<^sub>1.\<phi>) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((\<a>[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((((g.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>1) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>1 g \<star> g\<^sub>0h\<^sub>1.p\<^sub>1, f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine]) \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>1 g, g\<^sub>0h\<^sub>1.p\<^sub>1, f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine] \<cdot> \<beta>\<^sub>g = (g \<star> h \<star> TTfgh_TfTgh.the_\<theta>) \<cdot> can (\<^bold>\<langle>g\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> ((\<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) (((\<^bold>\<langle>g\<^bold>\<rangle> \<^bold>\<star> (\<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) \<cdot> (((g \<star> h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (((g \<star> g\<^sub>0h\<^sub>1.\<phi>) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((\<a>[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> \<a>\<^sup>-\<^sup>1[(g \<star> tab\<^sub>0 g) \<star> g\<^sub>0h\<^sub>1.p\<^sub>1, f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine] \<cdot> (((g.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>1) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine) \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>1 g, g\<^sub>0h\<^sub>1.p\<^sub>1, f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine]) \<cdot> \<beta>\<^sub>g
goal:
No subgoals!
[PROOF STEP]
qed
[PROOF STATE]
proof (state)
this:
(g \<star> h \<star> TTfgh_TfTgh.the_\<theta>) \<cdot> can (\<^bold>\<langle>g\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> ((\<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) (((\<^bold>\<langle>g\<^bold>\<rangle> \<^bold>\<star> (\<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) \<cdot> (((g \<star> h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (((g \<star> g\<^sub>0h\<^sub>1.\<phi>) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((\<a>[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((((g.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>1) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>1 g \<star> g\<^sub>0h\<^sub>1.p\<^sub>1, f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine]) \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>1 g, g\<^sub>0h\<^sub>1.p\<^sub>1, f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine] \<cdot> \<beta>\<^sub>g = (g \<star> h \<star> TTfgh_TfTgh.the_\<theta>) \<cdot> can (\<^bold>\<langle>g\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> ((\<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) (((\<^bold>\<langle>g\<^bold>\<rangle> \<^bold>\<star> (\<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) \<cdot> (((g \<star> h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (((g \<star> g\<^sub>0h\<^sub>1.\<phi>) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((\<a>[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> \<a>\<^sup>-\<^sup>1[(g \<star> tab\<^sub>0 g) \<star> g\<^sub>0h\<^sub>1.p\<^sub>1, f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine] \<cdot> (((g.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>1) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine) \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>1 g, g\<^sub>0h\<^sub>1.p\<^sub>1, f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine]) \<cdot> \<beta>\<^sub>g
goal (1 subgoal):
1. (g \<star> \<theta>\<^sub>g) \<cdot> \<a>[g, tab\<^sub>0 g, w\<^sub>g] \<cdot> (g.tab \<star> w\<^sub>g) = ((g \<star> \<theta>\<^sub>g') \<cdot> \<a>[g, tab\<^sub>0 g, w\<^sub>g'] \<cdot> (g.tab \<star> w\<^sub>g')) \<cdot> \<beta>\<^sub>g
[PROOF STEP]
also
[PROOF STATE]
proof (state)
this:
(g \<star> h \<star> TTfgh_TfTgh.the_\<theta>) \<cdot> can (\<^bold>\<langle>g\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> ((\<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) (((\<^bold>\<langle>g\<^bold>\<rangle> \<^bold>\<star> (\<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) \<cdot> (((g \<star> h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (((g \<star> g\<^sub>0h\<^sub>1.\<phi>) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((\<a>[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((((g.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>1) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>1 g \<star> g\<^sub>0h\<^sub>1.p\<^sub>1, f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine]) \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>1 g, g\<^sub>0h\<^sub>1.p\<^sub>1, f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine] \<cdot> \<beta>\<^sub>g = (g \<star> h \<star> TTfgh_TfTgh.the_\<theta>) \<cdot> can (\<^bold>\<langle>g\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> ((\<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) (((\<^bold>\<langle>g\<^bold>\<rangle> \<^bold>\<star> (\<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) \<cdot> (((g \<star> h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (((g \<star> g\<^sub>0h\<^sub>1.\<phi>) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((\<a>[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> \<a>\<^sup>-\<^sup>1[(g \<star> tab\<^sub>0 g) \<star> g\<^sub>0h\<^sub>1.p\<^sub>1, f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine] \<cdot> (((g.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>1) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine) \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>1 g, g\<^sub>0h\<^sub>1.p\<^sub>1, f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine]) \<cdot> \<beta>\<^sub>g
goal (1 subgoal):
1. (g \<star> \<theta>\<^sub>g) \<cdot> \<a>[g, tab\<^sub>0 g, w\<^sub>g] \<cdot> (g.tab \<star> w\<^sub>g) = ((g \<star> \<theta>\<^sub>g') \<cdot> \<a>[g, tab\<^sub>0 g, w\<^sub>g'] \<cdot> (g.tab \<star> w\<^sub>g')) \<cdot> \<beta>\<^sub>g
[PROOF STEP]
have "... = (g \<star> h \<star> TTfgh_TfTgh.the_\<theta>) \<cdot>
can (\<^bold>\<langle>g\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> ((\<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>Tgh.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TfTgh.p\<^sub>0\<^bold>\<rangle>)
\<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>)
(((\<^bold>\<langle>g\<^bold>\<rangle> \<^bold>\<star> (\<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>Tgh.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TfTgh.p\<^sub>0\<^bold>\<rangle>)
\<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) \<cdot>
(((g \<star> h.tab \<star> Tgh.p\<^sub>0) \<star> TfTgh.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot>
(((g \<star> g\<^sub>0h\<^sub>1.\<phi>) \<star> TfTgh.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot>
((\<a>[g, tab\<^sub>0 g, Tgh.p\<^sub>1] \<star> TfTgh.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot>
\<a>\<^sup>-\<^sup>1[(g \<star> tab\<^sub>0 g) \<star> Tgh.p\<^sub>1, TfTgh.p\<^sub>0, TTfgh_TfTgh.chine] \<cdot>
\<a>\<^sup>-\<^sup>1[g \<star> tab\<^sub>0 g, Tgh.p\<^sub>1, TfTgh.p\<^sub>0 \<star> TTfgh_TfTgh.chine] \<cdot>
(g.tab \<star> Tgh.p\<^sub>1 \<star> TfTgh.p\<^sub>0 \<star> TTfgh_TfTgh.chine) \<cdot>
\<beta>\<^sub>g"
[PROOF STATE]
proof (prove)
goal (1 subgoal):
1. (g \<star> h \<star> TTfgh_TfTgh.the_\<theta>) \<cdot> can (\<^bold>\<langle>g\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> ((\<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) (((\<^bold>\<langle>g\<^bold>\<rangle> \<^bold>\<star> (\<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) \<cdot> (((g \<star> h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (((g \<star> g\<^sub>0h\<^sub>1.\<phi>) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((\<a>[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> \<a>\<^sup>-\<^sup>1[(g \<star> tab\<^sub>0 g) \<star> g\<^sub>0h\<^sub>1.p\<^sub>1, f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine] \<cdot> (((g.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>1) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine) \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>1 g, g\<^sub>0h\<^sub>1.p\<^sub>1, f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine]) \<cdot> \<beta>\<^sub>g = (g \<star> h \<star> TTfgh_TfTgh.the_\<theta>) \<cdot> can (\<^bold>\<langle>g\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> ((\<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) (((\<^bold>\<langle>g\<^bold>\<rangle> \<^bold>\<star> (\<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) \<cdot> (((g \<star> h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (((g \<star> g\<^sub>0h\<^sub>1.\<phi>) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((\<a>[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> \<a>\<^sup>-\<^sup>1[(g \<star> tab\<^sub>0 g) \<star> g\<^sub>0h\<^sub>1.p\<^sub>1, f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine] \<cdot> \<a>\<^sup>-\<^sup>1[g \<star> tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1, f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine] \<cdot> (g.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>1 \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine) \<cdot> \<beta>\<^sub>g
[PROOF STEP]
proof -
[PROOF STATE]
proof (state)
goal (1 subgoal):
1. (g \<star> h \<star> TTfgh_TfTgh.the_\<theta>) \<cdot> can (\<^bold>\<langle>g\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> ((\<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) (((\<^bold>\<langle>g\<^bold>\<rangle> \<^bold>\<star> (\<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) \<cdot> (((g \<star> h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (((g \<star> g\<^sub>0h\<^sub>1.\<phi>) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((\<a>[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> \<a>\<^sup>-\<^sup>1[(g \<star> tab\<^sub>0 g) \<star> g\<^sub>0h\<^sub>1.p\<^sub>1, f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine] \<cdot> (((g.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>1) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine) \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>1 g, g\<^sub>0h\<^sub>1.p\<^sub>1, f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine]) \<cdot> \<beta>\<^sub>g = (g \<star> h \<star> TTfgh_TfTgh.the_\<theta>) \<cdot> can (\<^bold>\<langle>g\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> ((\<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) (((\<^bold>\<langle>g\<^bold>\<rangle> \<^bold>\<star> (\<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) \<cdot> (((g \<star> h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (((g \<star> g\<^sub>0h\<^sub>1.\<phi>) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((\<a>[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> \<a>\<^sup>-\<^sup>1[(g \<star> tab\<^sub>0 g) \<star> g\<^sub>0h\<^sub>1.p\<^sub>1, f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine] \<cdot> \<a>\<^sup>-\<^sup>1[g \<star> tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1, f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine] \<cdot> (g.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>1 \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine) \<cdot> \<beta>\<^sub>g
[PROOF STEP]
have "((g.tab \<star> Tgh.p\<^sub>1) \<star> TfTgh.p\<^sub>0 \<star> TTfgh_TfTgh.chine) \<cdot>
\<a>\<^sup>-\<^sup>1[tab\<^sub>1 g, Tgh.p\<^sub>1, TfTgh.p\<^sub>0 \<star> TTfgh_TfTgh.chine] =
\<a>\<^sup>-\<^sup>1[g \<star> tab\<^sub>0 g, Tgh.p\<^sub>1, TfTgh.p\<^sub>0 \<star> TTfgh_TfTgh.chine] \<cdot>
(g.tab \<star> Tgh.p\<^sub>1 \<star> TfTgh.p\<^sub>0 \<star> TTfgh_TfTgh.chine)"
[PROOF STATE]
proof (prove)
goal (1 subgoal):
1. ((g.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>1) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine) \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>1 g, g\<^sub>0h\<^sub>1.p\<^sub>1, f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine] = \<a>\<^sup>-\<^sup>1[g \<star> tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1, f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine] \<cdot> (g.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>1 \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine)
[PROOF STEP]
using g\<^sub>0h\<^sub>1.p\<^sub>1_simps f\<^sub>0gh\<^sub>1.p\<^sub>0_simps
assoc'_naturality [of g.tab Tgh.p\<^sub>1 "TfTgh.p\<^sub>0 \<star> TTfgh_TfTgh.chine"]
[PROOF STATE]
proof (prove)
using this:
trg g\<^sub>0h\<^sub>1.p\<^sub>1 = src (tab\<^sub>0 g)
trg f\<^sub>0gh\<^sub>1.p\<^sub>0 = src (tab\<^sub>1 g \<star> g\<^sub>0h\<^sub>1.p\<^sub>1)
\<lbrakk>arr g.tab; arr g\<^sub>0h\<^sub>1.p\<^sub>1; hseq f\<^sub>0gh\<^sub>1.p\<^sub>0 TTfgh_TfTgh.chine; src g.tab = trg g\<^sub>0h\<^sub>1.p\<^sub>1; src g\<^sub>0h\<^sub>1.p\<^sub>1 = trg (f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine)\<rbrakk> \<Longrightarrow> \<a>\<^sup>-\<^sup>1[cod g.tab, cod g\<^sub>0h\<^sub>1.p\<^sub>1, cod (f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine)] \<cdot> (g.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>1 \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine) = ((g.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>1) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine) \<cdot> \<a>\<^sup>-\<^sup>1[local.dom g.tab, local.dom g\<^sub>0h\<^sub>1.p\<^sub>1, local.dom (f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine)]
goal (1 subgoal):
1. ((g.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>1) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine) \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>1 g, g\<^sub>0h\<^sub>1.p\<^sub>1, f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine] = \<a>\<^sup>-\<^sup>1[g \<star> tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1, f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine] \<cdot> (g.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>1 \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine)
[PROOF STEP]
by simp
[PROOF STATE]
proof (state)
this:
((g.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>1) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine) \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>1 g, g\<^sub>0h\<^sub>1.p\<^sub>1, f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine] = \<a>\<^sup>-\<^sup>1[g \<star> tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1, f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine] \<cdot> (g.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>1 \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine)
goal (1 subgoal):
1. (g \<star> h \<star> TTfgh_TfTgh.the_\<theta>) \<cdot> can (\<^bold>\<langle>g\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> ((\<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) (((\<^bold>\<langle>g\<^bold>\<rangle> \<^bold>\<star> (\<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) \<cdot> (((g \<star> h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (((g \<star> g\<^sub>0h\<^sub>1.\<phi>) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((\<a>[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> \<a>\<^sup>-\<^sup>1[(g \<star> tab\<^sub>0 g) \<star> g\<^sub>0h\<^sub>1.p\<^sub>1, f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine] \<cdot> (((g.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>1) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine) \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>1 g, g\<^sub>0h\<^sub>1.p\<^sub>1, f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine]) \<cdot> \<beta>\<^sub>g = (g \<star> h \<star> TTfgh_TfTgh.the_\<theta>) \<cdot> can (\<^bold>\<langle>g\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> ((\<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) (((\<^bold>\<langle>g\<^bold>\<rangle> \<^bold>\<star> (\<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) \<cdot> (((g \<star> h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (((g \<star> g\<^sub>0h\<^sub>1.\<phi>) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((\<a>[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> \<a>\<^sup>-\<^sup>1[(g \<star> tab\<^sub>0 g) \<star> g\<^sub>0h\<^sub>1.p\<^sub>1, f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine] \<cdot> \<a>\<^sup>-\<^sup>1[g \<star> tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1, f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine] \<cdot> (g.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>1 \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine) \<cdot> \<beta>\<^sub>g
[PROOF STEP]
thus ?thesis
[PROOF STATE]
proof (prove)
using this:
((g.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>1) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine) \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>1 g, g\<^sub>0h\<^sub>1.p\<^sub>1, f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine] = \<a>\<^sup>-\<^sup>1[g \<star> tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1, f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine] \<cdot> (g.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>1 \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine)
goal (1 subgoal):
1. (g \<star> h \<star> TTfgh_TfTgh.the_\<theta>) \<cdot> can (\<^bold>\<langle>g\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> ((\<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) (((\<^bold>\<langle>g\<^bold>\<rangle> \<^bold>\<star> (\<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) \<cdot> (((g \<star> h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (((g \<star> g\<^sub>0h\<^sub>1.\<phi>) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((\<a>[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> \<a>\<^sup>-\<^sup>1[(g \<star> tab\<^sub>0 g) \<star> g\<^sub>0h\<^sub>1.p\<^sub>1, f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine] \<cdot> (((g.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>1) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine) \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>1 g, g\<^sub>0h\<^sub>1.p\<^sub>1, f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine]) \<cdot> \<beta>\<^sub>g = (g \<star> h \<star> TTfgh_TfTgh.the_\<theta>) \<cdot> can (\<^bold>\<langle>g\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> ((\<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) (((\<^bold>\<langle>g\<^bold>\<rangle> \<^bold>\<star> (\<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) \<cdot> (((g \<star> h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (((g \<star> g\<^sub>0h\<^sub>1.\<phi>) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((\<a>[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> \<a>\<^sup>-\<^sup>1[(g \<star> tab\<^sub>0 g) \<star> g\<^sub>0h\<^sub>1.p\<^sub>1, f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine] \<cdot> \<a>\<^sup>-\<^sup>1[g \<star> tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1, f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine] \<cdot> (g.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>1 \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine) \<cdot> \<beta>\<^sub>g
[PROOF STEP]
using comp_assoc
[PROOF STATE]
proof (prove)
using this:
((g.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>1) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine) \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>1 g, g\<^sub>0h\<^sub>1.p\<^sub>1, f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine] = \<a>\<^sup>-\<^sup>1[g \<star> tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1, f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine] \<cdot> (g.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>1 \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine)
(?h \<cdot> ?g) \<cdot> ?f = ?h \<cdot> ?g \<cdot> ?f
goal (1 subgoal):
1. (g \<star> h \<star> TTfgh_TfTgh.the_\<theta>) \<cdot> can (\<^bold>\<langle>g\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> ((\<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) (((\<^bold>\<langle>g\<^bold>\<rangle> \<^bold>\<star> (\<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) \<cdot> (((g \<star> h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (((g \<star> g\<^sub>0h\<^sub>1.\<phi>) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((\<a>[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> \<a>\<^sup>-\<^sup>1[(g \<star> tab\<^sub>0 g) \<star> g\<^sub>0h\<^sub>1.p\<^sub>1, f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine] \<cdot> (((g.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>1) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine) \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>1 g, g\<^sub>0h\<^sub>1.p\<^sub>1, f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine]) \<cdot> \<beta>\<^sub>g = (g \<star> h \<star> TTfgh_TfTgh.the_\<theta>) \<cdot> can (\<^bold>\<langle>g\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> ((\<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) (((\<^bold>\<langle>g\<^bold>\<rangle> \<^bold>\<star> (\<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) \<cdot> (((g \<star> h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (((g \<star> g\<^sub>0h\<^sub>1.\<phi>) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((\<a>[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> \<a>\<^sup>-\<^sup>1[(g \<star> tab\<^sub>0 g) \<star> g\<^sub>0h\<^sub>1.p\<^sub>1, f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine] \<cdot> \<a>\<^sup>-\<^sup>1[g \<star> tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1, f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine] \<cdot> (g.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>1 \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine) \<cdot> \<beta>\<^sub>g
[PROOF STEP]
by presburger
[PROOF STATE]
proof (state)
this:
(g \<star> h \<star> TTfgh_TfTgh.the_\<theta>) \<cdot> can (\<^bold>\<langle>g\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> ((\<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) (((\<^bold>\<langle>g\<^bold>\<rangle> \<^bold>\<star> (\<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) \<cdot> (((g \<star> h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (((g \<star> g\<^sub>0h\<^sub>1.\<phi>) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((\<a>[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> \<a>\<^sup>-\<^sup>1[(g \<star> tab\<^sub>0 g) \<star> g\<^sub>0h\<^sub>1.p\<^sub>1, f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine] \<cdot> (((g.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>1) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine) \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>1 g, g\<^sub>0h\<^sub>1.p\<^sub>1, f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine]) \<cdot> \<beta>\<^sub>g = (g \<star> h \<star> TTfgh_TfTgh.the_\<theta>) \<cdot> can (\<^bold>\<langle>g\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> ((\<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) (((\<^bold>\<langle>g\<^bold>\<rangle> \<^bold>\<star> (\<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) \<cdot> (((g \<star> h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (((g \<star> g\<^sub>0h\<^sub>1.\<phi>) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((\<a>[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> \<a>\<^sup>-\<^sup>1[(g \<star> tab\<^sub>0 g) \<star> g\<^sub>0h\<^sub>1.p\<^sub>1, f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine] \<cdot> \<a>\<^sup>-\<^sup>1[g \<star> tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1, f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine] \<cdot> (g.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>1 \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine) \<cdot> \<beta>\<^sub>g
goal:
No subgoals!
[PROOF STEP]
qed
[PROOF STATE]
proof (state)
this:
(g \<star> h \<star> TTfgh_TfTgh.the_\<theta>) \<cdot> can (\<^bold>\<langle>g\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> ((\<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) (((\<^bold>\<langle>g\<^bold>\<rangle> \<^bold>\<star> (\<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) \<cdot> (((g \<star> h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (((g \<star> g\<^sub>0h\<^sub>1.\<phi>) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((\<a>[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> \<a>\<^sup>-\<^sup>1[(g \<star> tab\<^sub>0 g) \<star> g\<^sub>0h\<^sub>1.p\<^sub>1, f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine] \<cdot> (((g.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>1) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine) \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>1 g, g\<^sub>0h\<^sub>1.p\<^sub>1, f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine]) \<cdot> \<beta>\<^sub>g = (g \<star> h \<star> TTfgh_TfTgh.the_\<theta>) \<cdot> can (\<^bold>\<langle>g\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> ((\<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) (((\<^bold>\<langle>g\<^bold>\<rangle> \<^bold>\<star> (\<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) \<cdot> (((g \<star> h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (((g \<star> g\<^sub>0h\<^sub>1.\<phi>) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((\<a>[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> \<a>\<^sup>-\<^sup>1[(g \<star> tab\<^sub>0 g) \<star> g\<^sub>0h\<^sub>1.p\<^sub>1, f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine] \<cdot> \<a>\<^sup>-\<^sup>1[g \<star> tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1, f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine] \<cdot> (g.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>1 \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine) \<cdot> \<beta>\<^sub>g
goal (1 subgoal):
1. (g \<star> \<theta>\<^sub>g) \<cdot> \<a>[g, tab\<^sub>0 g, w\<^sub>g] \<cdot> (g.tab \<star> w\<^sub>g) = ((g \<star> \<theta>\<^sub>g') \<cdot> \<a>[g, tab\<^sub>0 g, w\<^sub>g'] \<cdot> (g.tab \<star> w\<^sub>g')) \<cdot> \<beta>\<^sub>g
[PROOF STEP]
also
[PROOF STATE]
proof (state)
this:
(g \<star> h \<star> TTfgh_TfTgh.the_\<theta>) \<cdot> can (\<^bold>\<langle>g\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> ((\<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) (((\<^bold>\<langle>g\<^bold>\<rangle> \<^bold>\<star> (\<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) \<cdot> (((g \<star> h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (((g \<star> g\<^sub>0h\<^sub>1.\<phi>) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((\<a>[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> \<a>\<^sup>-\<^sup>1[(g \<star> tab\<^sub>0 g) \<star> g\<^sub>0h\<^sub>1.p\<^sub>1, f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine] \<cdot> (((g.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>1) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine) \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>1 g, g\<^sub>0h\<^sub>1.p\<^sub>1, f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine]) \<cdot> \<beta>\<^sub>g = (g \<star> h \<star> TTfgh_TfTgh.the_\<theta>) \<cdot> can (\<^bold>\<langle>g\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> ((\<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) (((\<^bold>\<langle>g\<^bold>\<rangle> \<^bold>\<star> (\<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) \<cdot> (((g \<star> h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (((g \<star> g\<^sub>0h\<^sub>1.\<phi>) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((\<a>[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> \<a>\<^sup>-\<^sup>1[(g \<star> tab\<^sub>0 g) \<star> g\<^sub>0h\<^sub>1.p\<^sub>1, f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine] \<cdot> \<a>\<^sup>-\<^sup>1[g \<star> tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1, f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine] \<cdot> (g.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>1 \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine) \<cdot> \<beta>\<^sub>g
goal (1 subgoal):
1. (g \<star> \<theta>\<^sub>g) \<cdot> \<a>[g, tab\<^sub>0 g, w\<^sub>g] \<cdot> (g.tab \<star> w\<^sub>g) = ((g \<star> \<theta>\<^sub>g') \<cdot> \<a>[g, tab\<^sub>0 g, w\<^sub>g'] \<cdot> (g.tab \<star> w\<^sub>g')) \<cdot> \<beta>\<^sub>g
[PROOF STEP]
have "... = (g \<star> h \<star> TTfgh_TfTgh.the_\<theta>) \<cdot>
can (\<^bold>\<langle>g\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> ((\<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>Tgh.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TfTgh.p\<^sub>0\<^bold>\<rangle>)
\<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>)
(((\<^bold>\<langle>g\<^bold>\<rangle> \<^bold>\<star> (\<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>Tgh.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TfTgh.p\<^sub>0\<^bold>\<rangle>)
\<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) \<cdot>
(((g \<star> h.tab \<star> Tgh.p\<^sub>0) \<star> TfTgh.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot>
(((g \<star> g\<^sub>0h\<^sub>1.\<phi>) \<star> TfTgh.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot>
((\<a>[g, tab\<^sub>0 g, Tgh.p\<^sub>1] \<star> TfTgh.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot>
\<a>\<^sup>-\<^sup>1[(g \<star> tab\<^sub>0 g) \<star> Tgh.p\<^sub>1, TfTgh.p\<^sub>0, TTfgh_TfTgh.chine] \<cdot>
\<a>\<^sup>-\<^sup>1[g \<star> tab\<^sub>0 g, Tgh.p\<^sub>1, TfTgh.p\<^sub>0 \<star> TTfgh_TfTgh.chine] \<cdot>
((\<a>\<^sup>-\<^sup>1[g, tab\<^sub>0 g, Tgh.p\<^sub>1 \<star> TfTgh.p\<^sub>0 \<star> TTfgh_TfTgh.chine] \<cdot>
\<a>[g, tab\<^sub>0 g, Tgh.p\<^sub>1 \<star> TfTgh.p\<^sub>0 \<star> TTfgh_TfTgh.chine]) \<cdot>
(g.tab \<star> Tgh.p\<^sub>1 \<star> TfTgh.p\<^sub>0 \<star> TTfgh_TfTgh.chine)) \<cdot>
\<beta>\<^sub>g"
[PROOF STATE]
proof (prove)
goal (1 subgoal):
1. (g \<star> h \<star> TTfgh_TfTgh.the_\<theta>) \<cdot> can (\<^bold>\<langle>g\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> ((\<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) (((\<^bold>\<langle>g\<^bold>\<rangle> \<^bold>\<star> (\<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) \<cdot> (((g \<star> h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (((g \<star> g\<^sub>0h\<^sub>1.\<phi>) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((\<a>[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> \<a>\<^sup>-\<^sup>1[(g \<star> tab\<^sub>0 g) \<star> g\<^sub>0h\<^sub>1.p\<^sub>1, f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine] \<cdot> \<a>\<^sup>-\<^sup>1[g \<star> tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1, f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine] \<cdot> (g.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>1 \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine) \<cdot> \<beta>\<^sub>g = (g \<star> h \<star> TTfgh_TfTgh.the_\<theta>) \<cdot> can (\<^bold>\<langle>g\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> ((\<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) (((\<^bold>\<langle>g\<^bold>\<rangle> \<^bold>\<star> (\<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) \<cdot> (((g \<star> h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (((g \<star> g\<^sub>0h\<^sub>1.\<phi>) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((\<a>[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> \<a>\<^sup>-\<^sup>1[(g \<star> tab\<^sub>0 g) \<star> g\<^sub>0h\<^sub>1.p\<^sub>1, f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine] \<cdot> \<a>\<^sup>-\<^sup>1[g \<star> tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1, f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine] \<cdot> ((\<a>\<^sup>-\<^sup>1[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1 \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine] \<cdot> \<a>[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1 \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine]) \<cdot> (g.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>1 \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine)) \<cdot> \<beta>\<^sub>g
[PROOF STEP]
proof -
[PROOF STATE]
proof (state)
goal (1 subgoal):
1. (g \<star> h \<star> TTfgh_TfTgh.the_\<theta>) \<cdot> can (\<^bold>\<langle>g\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> ((\<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) (((\<^bold>\<langle>g\<^bold>\<rangle> \<^bold>\<star> (\<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) \<cdot> (((g \<star> h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (((g \<star> g\<^sub>0h\<^sub>1.\<phi>) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((\<a>[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> \<a>\<^sup>-\<^sup>1[(g \<star> tab\<^sub>0 g) \<star> g\<^sub>0h\<^sub>1.p\<^sub>1, f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine] \<cdot> \<a>\<^sup>-\<^sup>1[g \<star> tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1, f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine] \<cdot> (g.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>1 \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine) \<cdot> \<beta>\<^sub>g = (g \<star> h \<star> TTfgh_TfTgh.the_\<theta>) \<cdot> can (\<^bold>\<langle>g\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> ((\<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) (((\<^bold>\<langle>g\<^bold>\<rangle> \<^bold>\<star> (\<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) \<cdot> (((g \<star> h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (((g \<star> g\<^sub>0h\<^sub>1.\<phi>) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((\<a>[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> \<a>\<^sup>-\<^sup>1[(g \<star> tab\<^sub>0 g) \<star> g\<^sub>0h\<^sub>1.p\<^sub>1, f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine] \<cdot> \<a>\<^sup>-\<^sup>1[g \<star> tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1, f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine] \<cdot> ((\<a>\<^sup>-\<^sup>1[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1 \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine] \<cdot> \<a>[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1 \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine]) \<cdot> (g.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>1 \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine)) \<cdot> \<beta>\<^sub>g
[PROOF STEP]
have "(\<a>\<^sup>-\<^sup>1[g, tab\<^sub>0 g, Tgh.p\<^sub>1 \<star> TfTgh.p\<^sub>0 \<star> TTfgh_TfTgh.chine] \<cdot>
\<a>[g, tab\<^sub>0 g, Tgh.p\<^sub>1 \<star> TfTgh.p\<^sub>0 \<star> TTfgh_TfTgh.chine]) \<cdot>
(g.tab \<star> Tgh.p\<^sub>1 \<star> TfTgh.p\<^sub>0 \<star> TTfgh_TfTgh.chine) =
g.tab \<star> Tgh.p\<^sub>1 \<star> TfTgh.p\<^sub>0 \<star> TTfgh_TfTgh.chine"
[PROOF STATE]
proof (prove)
goal (1 subgoal):
1. (\<a>\<^sup>-\<^sup>1[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1 \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine] \<cdot> \<a>[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1 \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine]) \<cdot> (g.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>1 \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine) = g.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>1 \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine
[PROOF STEP]
using comp_cod_arr comp_assoc_assoc'
[PROOF STATE]
proof (prove)
using this:
\<lbrakk>arr ?f; cod ?f = ?b\<rbrakk> \<Longrightarrow> ?b \<cdot> ?f = ?f
\<lbrakk>ide ?f; ide ?g; ide ?h; src ?f = trg ?g; src ?g = trg ?h\<rbrakk> \<Longrightarrow> \<a>[?f, ?g, ?h] \<cdot> \<a>\<^sup>-\<^sup>1[?f, ?g, ?h] = ?f \<star> ?g \<star> ?h
\<lbrakk>ide ?f; ide ?g; ide ?h; src ?f = trg ?g; src ?g = trg ?h\<rbrakk> \<Longrightarrow> \<a>\<^sup>-\<^sup>1[?f, ?g, ?h] \<cdot> \<a>[?f, ?g, ?h] = (?f \<star> ?g) \<star> ?h
goal (1 subgoal):
1. (\<a>\<^sup>-\<^sup>1[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1 \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine] \<cdot> \<a>[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1 \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine]) \<cdot> (g.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>1 \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine) = g.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>1 \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine
[PROOF STEP]
by simp
[PROOF STATE]
proof (state)
this:
(\<a>\<^sup>-\<^sup>1[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1 \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine] \<cdot> \<a>[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1 \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine]) \<cdot> (g.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>1 \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine) = g.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>1 \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine
goal (1 subgoal):
1. (g \<star> h \<star> TTfgh_TfTgh.the_\<theta>) \<cdot> can (\<^bold>\<langle>g\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> ((\<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) (((\<^bold>\<langle>g\<^bold>\<rangle> \<^bold>\<star> (\<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) \<cdot> (((g \<star> h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (((g \<star> g\<^sub>0h\<^sub>1.\<phi>) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((\<a>[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> \<a>\<^sup>-\<^sup>1[(g \<star> tab\<^sub>0 g) \<star> g\<^sub>0h\<^sub>1.p\<^sub>1, f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine] \<cdot> \<a>\<^sup>-\<^sup>1[g \<star> tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1, f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine] \<cdot> (g.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>1 \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine) \<cdot> \<beta>\<^sub>g = (g \<star> h \<star> TTfgh_TfTgh.the_\<theta>) \<cdot> can (\<^bold>\<langle>g\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> ((\<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) (((\<^bold>\<langle>g\<^bold>\<rangle> \<^bold>\<star> (\<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) \<cdot> (((g \<star> h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (((g \<star> g\<^sub>0h\<^sub>1.\<phi>) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((\<a>[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> \<a>\<^sup>-\<^sup>1[(g \<star> tab\<^sub>0 g) \<star> g\<^sub>0h\<^sub>1.p\<^sub>1, f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine] \<cdot> \<a>\<^sup>-\<^sup>1[g \<star> tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1, f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine] \<cdot> ((\<a>\<^sup>-\<^sup>1[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1 \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine] \<cdot> \<a>[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1 \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine]) \<cdot> (g.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>1 \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine)) \<cdot> \<beta>\<^sub>g
[PROOF STEP]
thus ?thesis
[PROOF STATE]
proof (prove)
using this:
(\<a>\<^sup>-\<^sup>1[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1 \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine] \<cdot> \<a>[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1 \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine]) \<cdot> (g.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>1 \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine) = g.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>1 \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine
goal (1 subgoal):
1. (g \<star> h \<star> TTfgh_TfTgh.the_\<theta>) \<cdot> can (\<^bold>\<langle>g\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> ((\<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) (((\<^bold>\<langle>g\<^bold>\<rangle> \<^bold>\<star> (\<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) \<cdot> (((g \<star> h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (((g \<star> g\<^sub>0h\<^sub>1.\<phi>) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((\<a>[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> \<a>\<^sup>-\<^sup>1[(g \<star> tab\<^sub>0 g) \<star> g\<^sub>0h\<^sub>1.p\<^sub>1, f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine] \<cdot> \<a>\<^sup>-\<^sup>1[g \<star> tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1, f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine] \<cdot> (g.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>1 \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine) \<cdot> \<beta>\<^sub>g = (g \<star> h \<star> TTfgh_TfTgh.the_\<theta>) \<cdot> can (\<^bold>\<langle>g\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> ((\<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) (((\<^bold>\<langle>g\<^bold>\<rangle> \<^bold>\<star> (\<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) \<cdot> (((g \<star> h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (((g \<star> g\<^sub>0h\<^sub>1.\<phi>) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((\<a>[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> \<a>\<^sup>-\<^sup>1[(g \<star> tab\<^sub>0 g) \<star> g\<^sub>0h\<^sub>1.p\<^sub>1, f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine] \<cdot> \<a>\<^sup>-\<^sup>1[g \<star> tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1, f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine] \<cdot> ((\<a>\<^sup>-\<^sup>1[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1 \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine] \<cdot> \<a>[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1 \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine]) \<cdot> (g.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>1 \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine)) \<cdot> \<beta>\<^sub>g
[PROOF STEP]
using comp_assoc g\<^sub>0h\<^sub>1.\<phi>_in_hom
[PROOF STATE]
proof (prove)
using this:
(\<a>\<^sup>-\<^sup>1[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1 \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine] \<cdot> \<a>[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1 \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine]) \<cdot> (g.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>1 \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine) = g.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>1 \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine
(?h \<cdot> ?g) \<cdot> ?f = ?h \<cdot> ?g \<cdot> ?f
\<guillemotleft>g\<^sub>0h\<^sub>1.\<phi> : src g\<^sub>0h\<^sub>1.p\<^sub>0 \<rightarrow> trg (tab\<^sub>1 h)\<guillemotright>
\<guillemotleft>g\<^sub>0h\<^sub>1.\<phi> : tab\<^sub>0 g \<star> g\<^sub>0h\<^sub>1.p\<^sub>1 \<rightarrow> tab\<^sub>1 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0\<guillemotright>
goal (1 subgoal):
1. (g \<star> h \<star> TTfgh_TfTgh.the_\<theta>) \<cdot> can (\<^bold>\<langle>g\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> ((\<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) (((\<^bold>\<langle>g\<^bold>\<rangle> \<^bold>\<star> (\<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) \<cdot> (((g \<star> h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (((g \<star> g\<^sub>0h\<^sub>1.\<phi>) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((\<a>[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> \<a>\<^sup>-\<^sup>1[(g \<star> tab\<^sub>0 g) \<star> g\<^sub>0h\<^sub>1.p\<^sub>1, f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine] \<cdot> \<a>\<^sup>-\<^sup>1[g \<star> tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1, f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine] \<cdot> (g.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>1 \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine) \<cdot> \<beta>\<^sub>g = (g \<star> h \<star> TTfgh_TfTgh.the_\<theta>) \<cdot> can (\<^bold>\<langle>g\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> ((\<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) (((\<^bold>\<langle>g\<^bold>\<rangle> \<^bold>\<star> (\<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) \<cdot> (((g \<star> h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (((g \<star> g\<^sub>0h\<^sub>1.\<phi>) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((\<a>[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> \<a>\<^sup>-\<^sup>1[(g \<star> tab\<^sub>0 g) \<star> g\<^sub>0h\<^sub>1.p\<^sub>1, f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine] \<cdot> \<a>\<^sup>-\<^sup>1[g \<star> tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1, f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine] \<cdot> ((\<a>\<^sup>-\<^sup>1[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1 \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine] \<cdot> \<a>[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1 \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine]) \<cdot> (g.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>1 \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine)) \<cdot> \<beta>\<^sub>g
[PROOF STEP]
by simp
[PROOF STATE]
proof (state)
this:
(g \<star> h \<star> TTfgh_TfTgh.the_\<theta>) \<cdot> can (\<^bold>\<langle>g\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> ((\<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) (((\<^bold>\<langle>g\<^bold>\<rangle> \<^bold>\<star> (\<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) \<cdot> (((g \<star> h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (((g \<star> g\<^sub>0h\<^sub>1.\<phi>) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((\<a>[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> \<a>\<^sup>-\<^sup>1[(g \<star> tab\<^sub>0 g) \<star> g\<^sub>0h\<^sub>1.p\<^sub>1, f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine] \<cdot> \<a>\<^sup>-\<^sup>1[g \<star> tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1, f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine] \<cdot> (g.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>1 \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine) \<cdot> \<beta>\<^sub>g = (g \<star> h \<star> TTfgh_TfTgh.the_\<theta>) \<cdot> can (\<^bold>\<langle>g\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> ((\<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) (((\<^bold>\<langle>g\<^bold>\<rangle> \<^bold>\<star> (\<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) \<cdot> (((g \<star> h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (((g \<star> g\<^sub>0h\<^sub>1.\<phi>) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((\<a>[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> \<a>\<^sup>-\<^sup>1[(g \<star> tab\<^sub>0 g) \<star> g\<^sub>0h\<^sub>1.p\<^sub>1, f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine] \<cdot> \<a>\<^sup>-\<^sup>1[g \<star> tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1, f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine] \<cdot> ((\<a>\<^sup>-\<^sup>1[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1 \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine] \<cdot> \<a>[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1 \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine]) \<cdot> (g.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>1 \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine)) \<cdot> \<beta>\<^sub>g
goal:
No subgoals!
[PROOF STEP]
qed
[PROOF STATE]
proof (state)
this:
(g \<star> h \<star> TTfgh_TfTgh.the_\<theta>) \<cdot> can (\<^bold>\<langle>g\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> ((\<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) (((\<^bold>\<langle>g\<^bold>\<rangle> \<^bold>\<star> (\<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) \<cdot> (((g \<star> h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (((g \<star> g\<^sub>0h\<^sub>1.\<phi>) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((\<a>[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> \<a>\<^sup>-\<^sup>1[(g \<star> tab\<^sub>0 g) \<star> g\<^sub>0h\<^sub>1.p\<^sub>1, f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine] \<cdot> \<a>\<^sup>-\<^sup>1[g \<star> tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1, f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine] \<cdot> (g.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>1 \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine) \<cdot> \<beta>\<^sub>g = (g \<star> h \<star> TTfgh_TfTgh.the_\<theta>) \<cdot> can (\<^bold>\<langle>g\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> ((\<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) (((\<^bold>\<langle>g\<^bold>\<rangle> \<^bold>\<star> (\<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) \<cdot> (((g \<star> h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (((g \<star> g\<^sub>0h\<^sub>1.\<phi>) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((\<a>[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> \<a>\<^sup>-\<^sup>1[(g \<star> tab\<^sub>0 g) \<star> g\<^sub>0h\<^sub>1.p\<^sub>1, f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine] \<cdot> \<a>\<^sup>-\<^sup>1[g \<star> tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1, f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine] \<cdot> ((\<a>\<^sup>-\<^sup>1[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1 \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine] \<cdot> \<a>[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1 \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine]) \<cdot> (g.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>1 \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine)) \<cdot> \<beta>\<^sub>g
goal (1 subgoal):
1. (g \<star> \<theta>\<^sub>g) \<cdot> \<a>[g, tab\<^sub>0 g, w\<^sub>g] \<cdot> (g.tab \<star> w\<^sub>g) = ((g \<star> \<theta>\<^sub>g') \<cdot> \<a>[g, tab\<^sub>0 g, w\<^sub>g'] \<cdot> (g.tab \<star> w\<^sub>g')) \<cdot> \<beta>\<^sub>g
[PROOF STEP]
also
[PROOF STATE]
proof (state)
this:
(g \<star> h \<star> TTfgh_TfTgh.the_\<theta>) \<cdot> can (\<^bold>\<langle>g\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> ((\<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) (((\<^bold>\<langle>g\<^bold>\<rangle> \<^bold>\<star> (\<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) \<cdot> (((g \<star> h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (((g \<star> g\<^sub>0h\<^sub>1.\<phi>) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((\<a>[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> \<a>\<^sup>-\<^sup>1[(g \<star> tab\<^sub>0 g) \<star> g\<^sub>0h\<^sub>1.p\<^sub>1, f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine] \<cdot> \<a>\<^sup>-\<^sup>1[g \<star> tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1, f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine] \<cdot> (g.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>1 \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine) \<cdot> \<beta>\<^sub>g = (g \<star> h \<star> TTfgh_TfTgh.the_\<theta>) \<cdot> can (\<^bold>\<langle>g\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> ((\<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) (((\<^bold>\<langle>g\<^bold>\<rangle> \<^bold>\<star> (\<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) \<cdot> (((g \<star> h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (((g \<star> g\<^sub>0h\<^sub>1.\<phi>) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((\<a>[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> \<a>\<^sup>-\<^sup>1[(g \<star> tab\<^sub>0 g) \<star> g\<^sub>0h\<^sub>1.p\<^sub>1, f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine] \<cdot> \<a>\<^sup>-\<^sup>1[g \<star> tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1, f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine] \<cdot> ((\<a>\<^sup>-\<^sup>1[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1 \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine] \<cdot> \<a>[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1 \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine]) \<cdot> (g.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>1 \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine)) \<cdot> \<beta>\<^sub>g
goal (1 subgoal):
1. (g \<star> \<theta>\<^sub>g) \<cdot> \<a>[g, tab\<^sub>0 g, w\<^sub>g] \<cdot> (g.tab \<star> w\<^sub>g) = ((g \<star> \<theta>\<^sub>g') \<cdot> \<a>[g, tab\<^sub>0 g, w\<^sub>g'] \<cdot> (g.tab \<star> w\<^sub>g')) \<cdot> \<beta>\<^sub>g
[PROOF STEP]
have "... = (g \<star> h \<star> TTfgh_TfTgh.the_\<theta>) \<cdot>
can (\<^bold>\<langle>g\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> ((\<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>Tgh.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TfTgh.p\<^sub>0\<^bold>\<rangle>)
\<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>)
(((\<^bold>\<langle>g\<^bold>\<rangle> \<^bold>\<star> (\<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>Tgh.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TfTgh.p\<^sub>0\<^bold>\<rangle>)
\<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) \<cdot>
(((g \<star> h.tab \<star> Tgh.p\<^sub>0) \<star> TfTgh.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot>
(((g \<star> g\<^sub>0h\<^sub>1.\<phi>) \<star> TfTgh.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot>
((\<a>\<^sup>-\<^sup>1[g \<star> tab\<^sub>0 g \<star> Tgh.p\<^sub>1, TfTgh.p\<^sub>0, TTfgh_TfTgh.chine] \<cdot>
(\<a>\<^sup>-\<^sup>1[g, tab\<^sub>0 g \<star> Tgh.p\<^sub>1, TfTgh.p\<^sub>0 \<star> TTfgh_TfTgh.chine] \<cdot>
\<a>[g, tab\<^sub>0 g \<star> Tgh.p\<^sub>1, TfTgh.p\<^sub>0 \<star> TTfgh_TfTgh.chine]) \<cdot>
\<a>[g \<star> tab\<^sub>0 g \<star> Tgh.p\<^sub>1, TfTgh.p\<^sub>0, TTfgh_TfTgh.chine]) \<cdot>
((\<a>[g, tab\<^sub>0 g, Tgh.p\<^sub>1] \<star> TfTgh.p\<^sub>0) \<star> TTfgh_TfTgh.chine)) \<cdot>
\<a>\<^sup>-\<^sup>1[(g \<star> tab\<^sub>0 g) \<star> Tgh.p\<^sub>1, TfTgh.p\<^sub>0, TTfgh_TfTgh.chine] \<cdot>
\<a>\<^sup>-\<^sup>1[g \<star> tab\<^sub>0 g, Tgh.p\<^sub>1, TfTgh.p\<^sub>0 \<star> TTfgh_TfTgh.chine] \<cdot>
((\<a>\<^sup>-\<^sup>1[g, tab\<^sub>0 g, Tgh.p\<^sub>1 \<star> TfTgh.p\<^sub>0 \<star> TTfgh_TfTgh.chine] \<cdot>
\<a>[g, tab\<^sub>0 g, Tgh.p\<^sub>1 \<star> TfTgh.p\<^sub>0 \<star> TTfgh_TfTgh.chine]) \<cdot>
(g.tab \<star> Tgh.p\<^sub>1 \<star> TfTgh.p\<^sub>0 \<star> TTfgh_TfTgh.chine)) \<cdot>
\<beta>\<^sub>g"
[PROOF STATE]
proof (prove)
goal (1 subgoal):
1. (g \<star> h \<star> TTfgh_TfTgh.the_\<theta>) \<cdot> can (\<^bold>\<langle>g\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> ((\<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) (((\<^bold>\<langle>g\<^bold>\<rangle> \<^bold>\<star> (\<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) \<cdot> (((g \<star> h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (((g \<star> g\<^sub>0h\<^sub>1.\<phi>) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((\<a>[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> \<a>\<^sup>-\<^sup>1[(g \<star> tab\<^sub>0 g) \<star> g\<^sub>0h\<^sub>1.p\<^sub>1, f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine] \<cdot> \<a>\<^sup>-\<^sup>1[g \<star> tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1, f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine] \<cdot> ((\<a>\<^sup>-\<^sup>1[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1 \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine] \<cdot> \<a>[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1 \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine]) \<cdot> (g.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>1 \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine)) \<cdot> \<beta>\<^sub>g = (g \<star> h \<star> TTfgh_TfTgh.the_\<theta>) \<cdot> can (\<^bold>\<langle>g\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> ((\<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) (((\<^bold>\<langle>g\<^bold>\<rangle> \<^bold>\<star> (\<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) \<cdot> (((g \<star> h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (((g \<star> g\<^sub>0h\<^sub>1.\<phi>) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((\<a>\<^sup>-\<^sup>1[g \<star> tab\<^sub>0 g \<star> g\<^sub>0h\<^sub>1.p\<^sub>1, f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine] \<cdot> (\<a>\<^sup>-\<^sup>1[g, tab\<^sub>0 g \<star> g\<^sub>0h\<^sub>1.p\<^sub>1, f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine] \<cdot> \<a>[g, tab\<^sub>0 g \<star> g\<^sub>0h\<^sub>1.p\<^sub>1, f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine]) \<cdot> \<a>[g \<star> tab\<^sub>0 g \<star> g\<^sub>0h\<^sub>1.p\<^sub>1, f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine]) \<cdot> ((\<a>[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine)) \<cdot> \<a>\<^sup>-\<^sup>1[(g \<star> tab\<^sub>0 g) \<star> g\<^sub>0h\<^sub>1.p\<^sub>1, f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine] \<cdot> \<a>\<^sup>-\<^sup>1[g \<star> tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1, f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine] \<cdot> ((\<a>\<^sup>-\<^sup>1[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1 \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine] \<cdot> \<a>[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1 \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine]) \<cdot> (g.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>1 \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine)) \<cdot> \<beta>\<^sub>g
[PROOF STEP]
proof -
[PROOF STATE]
proof (state)
goal (1 subgoal):
1. (g \<star> h \<star> TTfgh_TfTgh.the_\<theta>) \<cdot> can (\<^bold>\<langle>g\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> ((\<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) (((\<^bold>\<langle>g\<^bold>\<rangle> \<^bold>\<star> (\<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) \<cdot> (((g \<star> h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (((g \<star> g\<^sub>0h\<^sub>1.\<phi>) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((\<a>[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> \<a>\<^sup>-\<^sup>1[(g \<star> tab\<^sub>0 g) \<star> g\<^sub>0h\<^sub>1.p\<^sub>1, f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine] \<cdot> \<a>\<^sup>-\<^sup>1[g \<star> tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1, f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine] \<cdot> ((\<a>\<^sup>-\<^sup>1[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1 \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine] \<cdot> \<a>[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1 \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine]) \<cdot> (g.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>1 \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine)) \<cdot> \<beta>\<^sub>g = (g \<star> h \<star> TTfgh_TfTgh.the_\<theta>) \<cdot> can (\<^bold>\<langle>g\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> ((\<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) (((\<^bold>\<langle>g\<^bold>\<rangle> \<^bold>\<star> (\<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) \<cdot> (((g \<star> h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (((g \<star> g\<^sub>0h\<^sub>1.\<phi>) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((\<a>\<^sup>-\<^sup>1[g \<star> tab\<^sub>0 g \<star> g\<^sub>0h\<^sub>1.p\<^sub>1, f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine] \<cdot> (\<a>\<^sup>-\<^sup>1[g, tab\<^sub>0 g \<star> g\<^sub>0h\<^sub>1.p\<^sub>1, f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine] \<cdot> \<a>[g, tab\<^sub>0 g \<star> g\<^sub>0h\<^sub>1.p\<^sub>1, f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine]) \<cdot> \<a>[g \<star> tab\<^sub>0 g \<star> g\<^sub>0h\<^sub>1.p\<^sub>1, f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine]) \<cdot> ((\<a>[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine)) \<cdot> \<a>\<^sup>-\<^sup>1[(g \<star> tab\<^sub>0 g) \<star> g\<^sub>0h\<^sub>1.p\<^sub>1, f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine] \<cdot> \<a>\<^sup>-\<^sup>1[g \<star> tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1, f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine] \<cdot> ((\<a>\<^sup>-\<^sup>1[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1 \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine] \<cdot> \<a>[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1 \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine]) \<cdot> (g.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>1 \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine)) \<cdot> \<beta>\<^sub>g
[PROOF STEP]
have "(\<a>\<^sup>-\<^sup>1[g \<star> tab\<^sub>0 g \<star> Tgh.p\<^sub>1, TfTgh.p\<^sub>0, TTfgh_TfTgh.chine] \<cdot>
(\<a>\<^sup>-\<^sup>1[g, tab\<^sub>0 g \<star> Tgh.p\<^sub>1, TfTgh.p\<^sub>0 \<star> TTfgh_TfTgh.chine] \<cdot>
\<a>[g, tab\<^sub>0 g \<star> Tgh.p\<^sub>1, TfTgh.p\<^sub>0 \<star> TTfgh_TfTgh.chine]) \<cdot>
\<a>[g \<star> tab\<^sub>0 g \<star> Tgh.p\<^sub>1, TfTgh.p\<^sub>0, TTfgh_TfTgh.chine]) \<cdot>
((\<a>[g, tab\<^sub>0 g, Tgh.p\<^sub>1] \<star> TfTgh.p\<^sub>0) \<star> TTfgh_TfTgh.chine) =
(\<a>\<^sup>-\<^sup>1[g \<star> tab\<^sub>0 g \<star> Tgh.p\<^sub>1, TfTgh.p\<^sub>0, TTfgh_TfTgh.chine] \<cdot>
((g \<star> (tab\<^sub>0 g \<star> Tgh.p\<^sub>1)) \<star> TfTgh.p\<^sub>0 \<star> TTfgh_TfTgh.chine) \<cdot>
\<a>[g \<star> tab\<^sub>0 g \<star> Tgh.p\<^sub>1, TfTgh.p\<^sub>0, TTfgh_TfTgh.chine]) \<cdot>
((\<a>[g, tab\<^sub>0 g, Tgh.p\<^sub>1] \<star> TfTgh.p\<^sub>0) \<star> TTfgh_TfTgh.chine)"
[PROOF STATE]
proof (prove)
goal (1 subgoal):
1. (\<a>\<^sup>-\<^sup>1[g \<star> tab\<^sub>0 g \<star> g\<^sub>0h\<^sub>1.p\<^sub>1, f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine] \<cdot> (\<a>\<^sup>-\<^sup>1[g, tab\<^sub>0 g \<star> g\<^sub>0h\<^sub>1.p\<^sub>1, f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine] \<cdot> \<a>[g, tab\<^sub>0 g \<star> g\<^sub>0h\<^sub>1.p\<^sub>1, f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine]) \<cdot> \<a>[g \<star> tab\<^sub>0 g \<star> g\<^sub>0h\<^sub>1.p\<^sub>1, f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine]) \<cdot> ((\<a>[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) = (\<a>\<^sup>-\<^sup>1[g \<star> tab\<^sub>0 g \<star> g\<^sub>0h\<^sub>1.p\<^sub>1, f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine] \<cdot> ((g \<star> tab\<^sub>0 g \<star> g\<^sub>0h\<^sub>1.p\<^sub>1) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine) \<cdot> \<a>[g \<star> tab\<^sub>0 g \<star> g\<^sub>0h\<^sub>1.p\<^sub>1, f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine]) \<cdot> ((\<a>[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine)
[PROOF STEP]
using g\<^sub>0h\<^sub>1.p\<^sub>1_simps f\<^sub>0gh\<^sub>1.p\<^sub>0_simps comp_assoc comp_assoc_assoc'
[PROOF STATE]
proof (prove)
using this:
trg g\<^sub>0h\<^sub>1.p\<^sub>1 = src (tab\<^sub>0 g)
trg f\<^sub>0gh\<^sub>1.p\<^sub>0 = src (tab\<^sub>1 g \<star> g\<^sub>0h\<^sub>1.p\<^sub>1)
(?h \<cdot> ?g) \<cdot> ?f = ?h \<cdot> ?g \<cdot> ?f
\<lbrakk>ide ?f; ide ?g; ide ?h; src ?f = trg ?g; src ?g = trg ?h\<rbrakk> \<Longrightarrow> \<a>[?f, ?g, ?h] \<cdot> \<a>\<^sup>-\<^sup>1[?f, ?g, ?h] = ?f \<star> ?g \<star> ?h
\<lbrakk>ide ?f; ide ?g; ide ?h; src ?f = trg ?g; src ?g = trg ?h\<rbrakk> \<Longrightarrow> \<a>\<^sup>-\<^sup>1[?f, ?g, ?h] \<cdot> \<a>[?f, ?g, ?h] = (?f \<star> ?g) \<star> ?h
goal (1 subgoal):
1. (\<a>\<^sup>-\<^sup>1[g \<star> tab\<^sub>0 g \<star> g\<^sub>0h\<^sub>1.p\<^sub>1, f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine] \<cdot> (\<a>\<^sup>-\<^sup>1[g, tab\<^sub>0 g \<star> g\<^sub>0h\<^sub>1.p\<^sub>1, f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine] \<cdot> \<a>[g, tab\<^sub>0 g \<star> g\<^sub>0h\<^sub>1.p\<^sub>1, f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine]) \<cdot> \<a>[g \<star> tab\<^sub>0 g \<star> g\<^sub>0h\<^sub>1.p\<^sub>1, f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine]) \<cdot> ((\<a>[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) = (\<a>\<^sup>-\<^sup>1[g \<star> tab\<^sub>0 g \<star> g\<^sub>0h\<^sub>1.p\<^sub>1, f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine] \<cdot> ((g \<star> tab\<^sub>0 g \<star> g\<^sub>0h\<^sub>1.p\<^sub>1) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine) \<cdot> \<a>[g \<star> tab\<^sub>0 g \<star> g\<^sub>0h\<^sub>1.p\<^sub>1, f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine]) \<cdot> ((\<a>[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine)
[PROOF STEP]
by simp
[PROOF STATE]
proof (state)
this:
(\<a>\<^sup>-\<^sup>1[g \<star> tab\<^sub>0 g \<star> g\<^sub>0h\<^sub>1.p\<^sub>1, f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine] \<cdot> (\<a>\<^sup>-\<^sup>1[g, tab\<^sub>0 g \<star> g\<^sub>0h\<^sub>1.p\<^sub>1, f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine] \<cdot> \<a>[g, tab\<^sub>0 g \<star> g\<^sub>0h\<^sub>1.p\<^sub>1, f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine]) \<cdot> \<a>[g \<star> tab\<^sub>0 g \<star> g\<^sub>0h\<^sub>1.p\<^sub>1, f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine]) \<cdot> ((\<a>[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) = (\<a>\<^sup>-\<^sup>1[g \<star> tab\<^sub>0 g \<star> g\<^sub>0h\<^sub>1.p\<^sub>1, f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine] \<cdot> ((g \<star> tab\<^sub>0 g \<star> g\<^sub>0h\<^sub>1.p\<^sub>1) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine) \<cdot> \<a>[g \<star> tab\<^sub>0 g \<star> g\<^sub>0h\<^sub>1.p\<^sub>1, f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine]) \<cdot> ((\<a>[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine)
goal (1 subgoal):
1. (g \<star> h \<star> TTfgh_TfTgh.the_\<theta>) \<cdot> can (\<^bold>\<langle>g\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> ((\<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) (((\<^bold>\<langle>g\<^bold>\<rangle> \<^bold>\<star> (\<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) \<cdot> (((g \<star> h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (((g \<star> g\<^sub>0h\<^sub>1.\<phi>) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((\<a>[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> \<a>\<^sup>-\<^sup>1[(g \<star> tab\<^sub>0 g) \<star> g\<^sub>0h\<^sub>1.p\<^sub>1, f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine] \<cdot> \<a>\<^sup>-\<^sup>1[g \<star> tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1, f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine] \<cdot> ((\<a>\<^sup>-\<^sup>1[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1 \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine] \<cdot> \<a>[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1 \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine]) \<cdot> (g.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>1 \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine)) \<cdot> \<beta>\<^sub>g = (g \<star> h \<star> TTfgh_TfTgh.the_\<theta>) \<cdot> can (\<^bold>\<langle>g\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> ((\<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) (((\<^bold>\<langle>g\<^bold>\<rangle> \<^bold>\<star> (\<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) \<cdot> (((g \<star> h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (((g \<star> g\<^sub>0h\<^sub>1.\<phi>) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((\<a>\<^sup>-\<^sup>1[g \<star> tab\<^sub>0 g \<star> g\<^sub>0h\<^sub>1.p\<^sub>1, f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine] \<cdot> (\<a>\<^sup>-\<^sup>1[g, tab\<^sub>0 g \<star> g\<^sub>0h\<^sub>1.p\<^sub>1, f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine] \<cdot> \<a>[g, tab\<^sub>0 g \<star> g\<^sub>0h\<^sub>1.p\<^sub>1, f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine]) \<cdot> \<a>[g \<star> tab\<^sub>0 g \<star> g\<^sub>0h\<^sub>1.p\<^sub>1, f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine]) \<cdot> ((\<a>[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine)) \<cdot> \<a>\<^sup>-\<^sup>1[(g \<star> tab\<^sub>0 g) \<star> g\<^sub>0h\<^sub>1.p\<^sub>1, f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine] \<cdot> \<a>\<^sup>-\<^sup>1[g \<star> tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1, f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine] \<cdot> ((\<a>\<^sup>-\<^sup>1[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1 \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine] \<cdot> \<a>[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1 \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine]) \<cdot> (g.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>1 \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine)) \<cdot> \<beta>\<^sub>g
[PROOF STEP]
also
[PROOF STATE]
proof (state)
this:
(\<a>\<^sup>-\<^sup>1[g \<star> tab\<^sub>0 g \<star> g\<^sub>0h\<^sub>1.p\<^sub>1, f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine] \<cdot> (\<a>\<^sup>-\<^sup>1[g, tab\<^sub>0 g \<star> g\<^sub>0h\<^sub>1.p\<^sub>1, f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine] \<cdot> \<a>[g, tab\<^sub>0 g \<star> g\<^sub>0h\<^sub>1.p\<^sub>1, f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine]) \<cdot> \<a>[g \<star> tab\<^sub>0 g \<star> g\<^sub>0h\<^sub>1.p\<^sub>1, f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine]) \<cdot> ((\<a>[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) = (\<a>\<^sup>-\<^sup>1[g \<star> tab\<^sub>0 g \<star> g\<^sub>0h\<^sub>1.p\<^sub>1, f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine] \<cdot> ((g \<star> tab\<^sub>0 g \<star> g\<^sub>0h\<^sub>1.p\<^sub>1) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine) \<cdot> \<a>[g \<star> tab\<^sub>0 g \<star> g\<^sub>0h\<^sub>1.p\<^sub>1, f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine]) \<cdot> ((\<a>[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine)
goal (1 subgoal):
1. (g \<star> h \<star> TTfgh_TfTgh.the_\<theta>) \<cdot> can (\<^bold>\<langle>g\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> ((\<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) (((\<^bold>\<langle>g\<^bold>\<rangle> \<^bold>\<star> (\<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) \<cdot> (((g \<star> h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (((g \<star> g\<^sub>0h\<^sub>1.\<phi>) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((\<a>[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> \<a>\<^sup>-\<^sup>1[(g \<star> tab\<^sub>0 g) \<star> g\<^sub>0h\<^sub>1.p\<^sub>1, f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine] \<cdot> \<a>\<^sup>-\<^sup>1[g \<star> tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1, f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine] \<cdot> ((\<a>\<^sup>-\<^sup>1[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1 \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine] \<cdot> \<a>[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1 \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine]) \<cdot> (g.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>1 \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine)) \<cdot> \<beta>\<^sub>g = (g \<star> h \<star> TTfgh_TfTgh.the_\<theta>) \<cdot> can (\<^bold>\<langle>g\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> ((\<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) (((\<^bold>\<langle>g\<^bold>\<rangle> \<^bold>\<star> (\<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) \<cdot> (((g \<star> h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (((g \<star> g\<^sub>0h\<^sub>1.\<phi>) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((\<a>\<^sup>-\<^sup>1[g \<star> tab\<^sub>0 g \<star> g\<^sub>0h\<^sub>1.p\<^sub>1, f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine] \<cdot> (\<a>\<^sup>-\<^sup>1[g, tab\<^sub>0 g \<star> g\<^sub>0h\<^sub>1.p\<^sub>1, f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine] \<cdot> \<a>[g, tab\<^sub>0 g \<star> g\<^sub>0h\<^sub>1.p\<^sub>1, f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine]) \<cdot> \<a>[g \<star> tab\<^sub>0 g \<star> g\<^sub>0h\<^sub>1.p\<^sub>1, f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine]) \<cdot> ((\<a>[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine)) \<cdot> \<a>\<^sup>-\<^sup>1[(g \<star> tab\<^sub>0 g) \<star> g\<^sub>0h\<^sub>1.p\<^sub>1, f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine] \<cdot> \<a>\<^sup>-\<^sup>1[g \<star> tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1, f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine] \<cdot> ((\<a>\<^sup>-\<^sup>1[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1 \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine] \<cdot> \<a>[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1 \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine]) \<cdot> (g.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>1 \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine)) \<cdot> \<beta>\<^sub>g
[PROOF STEP]
have "... = (\<a>\<^sup>-\<^sup>1[g \<star> tab\<^sub>0 g \<star> Tgh.p\<^sub>1, TfTgh.p\<^sub>0, TTfgh_TfTgh.chine] \<cdot>
\<a>[g \<star> tab\<^sub>0 g \<star> Tgh.p\<^sub>1, TfTgh.p\<^sub>0, TTfgh_TfTgh.chine]) \<cdot>
((\<a>[g, tab\<^sub>0 g, Tgh.p\<^sub>1] \<star> TfTgh.p\<^sub>0) \<star> TTfgh_TfTgh.chine)"
[PROOF STATE]
proof (prove)
goal (1 subgoal):
1. (\<a>\<^sup>-\<^sup>1[g \<star> tab\<^sub>0 g \<star> g\<^sub>0h\<^sub>1.p\<^sub>1, f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine] \<cdot> ((g \<star> tab\<^sub>0 g \<star> g\<^sub>0h\<^sub>1.p\<^sub>1) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine) \<cdot> \<a>[g \<star> tab\<^sub>0 g \<star> g\<^sub>0h\<^sub>1.p\<^sub>1, f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine]) \<cdot> ((\<a>[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) = (\<a>\<^sup>-\<^sup>1[g \<star> tab\<^sub>0 g \<star> g\<^sub>0h\<^sub>1.p\<^sub>1, f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine] \<cdot> \<a>[g \<star> tab\<^sub>0 g \<star> g\<^sub>0h\<^sub>1.p\<^sub>1, f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine]) \<cdot> ((\<a>[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine)
[PROOF STEP]
using g\<^sub>0h\<^sub>1.p\<^sub>1_simps f\<^sub>0gh\<^sub>1.p\<^sub>0_simps comp_cod_arr comp_assoc_assoc'
[PROOF STATE]
proof (prove)
using this:
trg g\<^sub>0h\<^sub>1.p\<^sub>1 = src (tab\<^sub>0 g)
trg f\<^sub>0gh\<^sub>1.p\<^sub>0 = src (tab\<^sub>1 g \<star> g\<^sub>0h\<^sub>1.p\<^sub>1)
\<lbrakk>arr ?f; cod ?f = ?b\<rbrakk> \<Longrightarrow> ?b \<cdot> ?f = ?f
\<lbrakk>ide ?f; ide ?g; ide ?h; src ?f = trg ?g; src ?g = trg ?h\<rbrakk> \<Longrightarrow> \<a>[?f, ?g, ?h] \<cdot> \<a>\<^sup>-\<^sup>1[?f, ?g, ?h] = ?f \<star> ?g \<star> ?h
\<lbrakk>ide ?f; ide ?g; ide ?h; src ?f = trg ?g; src ?g = trg ?h\<rbrakk> \<Longrightarrow> \<a>\<^sup>-\<^sup>1[?f, ?g, ?h] \<cdot> \<a>[?f, ?g, ?h] = (?f \<star> ?g) \<star> ?h
goal (1 subgoal):
1. (\<a>\<^sup>-\<^sup>1[g \<star> tab\<^sub>0 g \<star> g\<^sub>0h\<^sub>1.p\<^sub>1, f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine] \<cdot> ((g \<star> tab\<^sub>0 g \<star> g\<^sub>0h\<^sub>1.p\<^sub>1) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine) \<cdot> \<a>[g \<star> tab\<^sub>0 g \<star> g\<^sub>0h\<^sub>1.p\<^sub>1, f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine]) \<cdot> ((\<a>[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) = (\<a>\<^sup>-\<^sup>1[g \<star> tab\<^sub>0 g \<star> g\<^sub>0h\<^sub>1.p\<^sub>1, f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine] \<cdot> \<a>[g \<star> tab\<^sub>0 g \<star> g\<^sub>0h\<^sub>1.p\<^sub>1, f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine]) \<cdot> ((\<a>[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine)
[PROOF STEP]
by simp
[PROOF STATE]
proof (state)
this:
(\<a>\<^sup>-\<^sup>1[g \<star> tab\<^sub>0 g \<star> g\<^sub>0h\<^sub>1.p\<^sub>1, f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine] \<cdot> ((g \<star> tab\<^sub>0 g \<star> g\<^sub>0h\<^sub>1.p\<^sub>1) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine) \<cdot> \<a>[g \<star> tab\<^sub>0 g \<star> g\<^sub>0h\<^sub>1.p\<^sub>1, f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine]) \<cdot> ((\<a>[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) = (\<a>\<^sup>-\<^sup>1[g \<star> tab\<^sub>0 g \<star> g\<^sub>0h\<^sub>1.p\<^sub>1, f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine] \<cdot> \<a>[g \<star> tab\<^sub>0 g \<star> g\<^sub>0h\<^sub>1.p\<^sub>1, f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine]) \<cdot> ((\<a>[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine)
goal (1 subgoal):
1. (g \<star> h \<star> TTfgh_TfTgh.the_\<theta>) \<cdot> can (\<^bold>\<langle>g\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> ((\<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) (((\<^bold>\<langle>g\<^bold>\<rangle> \<^bold>\<star> (\<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) \<cdot> (((g \<star> h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (((g \<star> g\<^sub>0h\<^sub>1.\<phi>) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((\<a>[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> \<a>\<^sup>-\<^sup>1[(g \<star> tab\<^sub>0 g) \<star> g\<^sub>0h\<^sub>1.p\<^sub>1, f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine] \<cdot> \<a>\<^sup>-\<^sup>1[g \<star> tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1, f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine] \<cdot> ((\<a>\<^sup>-\<^sup>1[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1 \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine] \<cdot> \<a>[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1 \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine]) \<cdot> (g.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>1 \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine)) \<cdot> \<beta>\<^sub>g = (g \<star> h \<star> TTfgh_TfTgh.the_\<theta>) \<cdot> can (\<^bold>\<langle>g\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> ((\<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) (((\<^bold>\<langle>g\<^bold>\<rangle> \<^bold>\<star> (\<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) \<cdot> (((g \<star> h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (((g \<star> g\<^sub>0h\<^sub>1.\<phi>) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((\<a>\<^sup>-\<^sup>1[g \<star> tab\<^sub>0 g \<star> g\<^sub>0h\<^sub>1.p\<^sub>1, f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine] \<cdot> (\<a>\<^sup>-\<^sup>1[g, tab\<^sub>0 g \<star> g\<^sub>0h\<^sub>1.p\<^sub>1, f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine] \<cdot> \<a>[g, tab\<^sub>0 g \<star> g\<^sub>0h\<^sub>1.p\<^sub>1, f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine]) \<cdot> \<a>[g \<star> tab\<^sub>0 g \<star> g\<^sub>0h\<^sub>1.p\<^sub>1, f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine]) \<cdot> ((\<a>[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine)) \<cdot> \<a>\<^sup>-\<^sup>1[(g \<star> tab\<^sub>0 g) \<star> g\<^sub>0h\<^sub>1.p\<^sub>1, f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine] \<cdot> \<a>\<^sup>-\<^sup>1[g \<star> tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1, f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine] \<cdot> ((\<a>\<^sup>-\<^sup>1[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1 \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine] \<cdot> \<a>[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1 \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine]) \<cdot> (g.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>1 \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine)) \<cdot> \<beta>\<^sub>g
[PROOF STEP]
also
[PROOF STATE]
proof (state)
this:
(\<a>\<^sup>-\<^sup>1[g \<star> tab\<^sub>0 g \<star> g\<^sub>0h\<^sub>1.p\<^sub>1, f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine] \<cdot> ((g \<star> tab\<^sub>0 g \<star> g\<^sub>0h\<^sub>1.p\<^sub>1) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine) \<cdot> \<a>[g \<star> tab\<^sub>0 g \<star> g\<^sub>0h\<^sub>1.p\<^sub>1, f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine]) \<cdot> ((\<a>[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) = (\<a>\<^sup>-\<^sup>1[g \<star> tab\<^sub>0 g \<star> g\<^sub>0h\<^sub>1.p\<^sub>1, f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine] \<cdot> \<a>[g \<star> tab\<^sub>0 g \<star> g\<^sub>0h\<^sub>1.p\<^sub>1, f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine]) \<cdot> ((\<a>[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine)
goal (1 subgoal):
1. (g \<star> h \<star> TTfgh_TfTgh.the_\<theta>) \<cdot> can (\<^bold>\<langle>g\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> ((\<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) (((\<^bold>\<langle>g\<^bold>\<rangle> \<^bold>\<star> (\<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) \<cdot> (((g \<star> h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (((g \<star> g\<^sub>0h\<^sub>1.\<phi>) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((\<a>[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> \<a>\<^sup>-\<^sup>1[(g \<star> tab\<^sub>0 g) \<star> g\<^sub>0h\<^sub>1.p\<^sub>1, f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine] \<cdot> \<a>\<^sup>-\<^sup>1[g \<star> tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1, f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine] \<cdot> ((\<a>\<^sup>-\<^sup>1[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1 \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine] \<cdot> \<a>[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1 \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine]) \<cdot> (g.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>1 \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine)) \<cdot> \<beta>\<^sub>g = (g \<star> h \<star> TTfgh_TfTgh.the_\<theta>) \<cdot> can (\<^bold>\<langle>g\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> ((\<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) (((\<^bold>\<langle>g\<^bold>\<rangle> \<^bold>\<star> (\<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) \<cdot> (((g \<star> h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (((g \<star> g\<^sub>0h\<^sub>1.\<phi>) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((\<a>\<^sup>-\<^sup>1[g \<star> tab\<^sub>0 g \<star> g\<^sub>0h\<^sub>1.p\<^sub>1, f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine] \<cdot> (\<a>\<^sup>-\<^sup>1[g, tab\<^sub>0 g \<star> g\<^sub>0h\<^sub>1.p\<^sub>1, f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine] \<cdot> \<a>[g, tab\<^sub>0 g \<star> g\<^sub>0h\<^sub>1.p\<^sub>1, f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine]) \<cdot> \<a>[g \<star> tab\<^sub>0 g \<star> g\<^sub>0h\<^sub>1.p\<^sub>1, f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine]) \<cdot> ((\<a>[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine)) \<cdot> \<a>\<^sup>-\<^sup>1[(g \<star> tab\<^sub>0 g) \<star> g\<^sub>0h\<^sub>1.p\<^sub>1, f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine] \<cdot> \<a>\<^sup>-\<^sup>1[g \<star> tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1, f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine] \<cdot> ((\<a>\<^sup>-\<^sup>1[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1 \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine] \<cdot> \<a>[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1 \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine]) \<cdot> (g.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>1 \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine)) \<cdot> \<beta>\<^sub>g
[PROOF STEP]
have "... = (((g \<star> (tab\<^sub>0 g \<star> Tgh.p\<^sub>1)) \<star> TfTgh.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot>
((\<a>[g, tab\<^sub>0 g, Tgh.p\<^sub>1] \<star> TfTgh.p\<^sub>0) \<star> TTfgh_TfTgh.chine)"
[PROOF STATE]
proof (prove)
goal (1 subgoal):
1. (\<a>\<^sup>-\<^sup>1[g \<star> tab\<^sub>0 g \<star> g\<^sub>0h\<^sub>1.p\<^sub>1, f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine] \<cdot> \<a>[g \<star> tab\<^sub>0 g \<star> g\<^sub>0h\<^sub>1.p\<^sub>1, f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine]) \<cdot> ((\<a>[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) = (((g \<star> tab\<^sub>0 g \<star> g\<^sub>0h\<^sub>1.p\<^sub>1) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((\<a>[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine)
[PROOF STEP]
using g\<^sub>0h\<^sub>1.p\<^sub>1_simps f\<^sub>0gh\<^sub>1.p\<^sub>0_simps whisker_right comp_assoc_assoc'
[PROOF STATE]
proof (prove)
using this:
trg g\<^sub>0h\<^sub>1.p\<^sub>1 = src (tab\<^sub>0 g)
trg f\<^sub>0gh\<^sub>1.p\<^sub>0 = src (tab\<^sub>1 g \<star> g\<^sub>0h\<^sub>1.p\<^sub>1)
\<lbrakk>ide ?f; seq ?\<nu> ?\<mu>\<rbrakk> \<Longrightarrow> ?\<nu> \<cdot> ?\<mu> \<star> ?f = (?\<nu> \<star> ?f) \<cdot> (?\<mu> \<star> ?f)
\<lbrakk>ide ?f; ide ?g; ide ?h; src ?f = trg ?g; src ?g = trg ?h\<rbrakk> \<Longrightarrow> \<a>[?f, ?g, ?h] \<cdot> \<a>\<^sup>-\<^sup>1[?f, ?g, ?h] = ?f \<star> ?g \<star> ?h
\<lbrakk>ide ?f; ide ?g; ide ?h; src ?f = trg ?g; src ?g = trg ?h\<rbrakk> \<Longrightarrow> \<a>\<^sup>-\<^sup>1[?f, ?g, ?h] \<cdot> \<a>[?f, ?g, ?h] = (?f \<star> ?g) \<star> ?h
goal (1 subgoal):
1. (\<a>\<^sup>-\<^sup>1[g \<star> tab\<^sub>0 g \<star> g\<^sub>0h\<^sub>1.p\<^sub>1, f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine] \<cdot> \<a>[g \<star> tab\<^sub>0 g \<star> g\<^sub>0h\<^sub>1.p\<^sub>1, f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine]) \<cdot> ((\<a>[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) = (((g \<star> tab\<^sub>0 g \<star> g\<^sub>0h\<^sub>1.p\<^sub>1) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((\<a>[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine)
[PROOF STEP]
by simp
[PROOF STATE]
proof (state)
this:
(\<a>\<^sup>-\<^sup>1[g \<star> tab\<^sub>0 g \<star> g\<^sub>0h\<^sub>1.p\<^sub>1, f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine] \<cdot> \<a>[g \<star> tab\<^sub>0 g \<star> g\<^sub>0h\<^sub>1.p\<^sub>1, f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine]) \<cdot> ((\<a>[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) = (((g \<star> tab\<^sub>0 g \<star> g\<^sub>0h\<^sub>1.p\<^sub>1) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((\<a>[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine)
goal (1 subgoal):
1. (g \<star> h \<star> TTfgh_TfTgh.the_\<theta>) \<cdot> can (\<^bold>\<langle>g\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> ((\<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) (((\<^bold>\<langle>g\<^bold>\<rangle> \<^bold>\<star> (\<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) \<cdot> (((g \<star> h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (((g \<star> g\<^sub>0h\<^sub>1.\<phi>) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((\<a>[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> \<a>\<^sup>-\<^sup>1[(g \<star> tab\<^sub>0 g) \<star> g\<^sub>0h\<^sub>1.p\<^sub>1, f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine] \<cdot> \<a>\<^sup>-\<^sup>1[g \<star> tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1, f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine] \<cdot> ((\<a>\<^sup>-\<^sup>1[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1 \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine] \<cdot> \<a>[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1 \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine]) \<cdot> (g.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>1 \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine)) \<cdot> \<beta>\<^sub>g = (g \<star> h \<star> TTfgh_TfTgh.the_\<theta>) \<cdot> can (\<^bold>\<langle>g\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> ((\<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) (((\<^bold>\<langle>g\<^bold>\<rangle> \<^bold>\<star> (\<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) \<cdot> (((g \<star> h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (((g \<star> g\<^sub>0h\<^sub>1.\<phi>) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((\<a>\<^sup>-\<^sup>1[g \<star> tab\<^sub>0 g \<star> g\<^sub>0h\<^sub>1.p\<^sub>1, f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine] \<cdot> (\<a>\<^sup>-\<^sup>1[g, tab\<^sub>0 g \<star> g\<^sub>0h\<^sub>1.p\<^sub>1, f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine] \<cdot> \<a>[g, tab\<^sub>0 g \<star> g\<^sub>0h\<^sub>1.p\<^sub>1, f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine]) \<cdot> \<a>[g \<star> tab\<^sub>0 g \<star> g\<^sub>0h\<^sub>1.p\<^sub>1, f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine]) \<cdot> ((\<a>[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine)) \<cdot> \<a>\<^sup>-\<^sup>1[(g \<star> tab\<^sub>0 g) \<star> g\<^sub>0h\<^sub>1.p\<^sub>1, f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine] \<cdot> \<a>\<^sup>-\<^sup>1[g \<star> tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1, f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine] \<cdot> ((\<a>\<^sup>-\<^sup>1[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1 \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine] \<cdot> \<a>[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1 \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine]) \<cdot> (g.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>1 \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine)) \<cdot> \<beta>\<^sub>g
[PROOF STEP]
also
[PROOF STATE]
proof (state)
this:
(\<a>\<^sup>-\<^sup>1[g \<star> tab\<^sub>0 g \<star> g\<^sub>0h\<^sub>1.p\<^sub>1, f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine] \<cdot> \<a>[g \<star> tab\<^sub>0 g \<star> g\<^sub>0h\<^sub>1.p\<^sub>1, f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine]) \<cdot> ((\<a>[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) = (((g \<star> tab\<^sub>0 g \<star> g\<^sub>0h\<^sub>1.p\<^sub>1) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((\<a>[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine)
goal (1 subgoal):
1. (g \<star> h \<star> TTfgh_TfTgh.the_\<theta>) \<cdot> can (\<^bold>\<langle>g\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> ((\<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) (((\<^bold>\<langle>g\<^bold>\<rangle> \<^bold>\<star> (\<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) \<cdot> (((g \<star> h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (((g \<star> g\<^sub>0h\<^sub>1.\<phi>) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((\<a>[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> \<a>\<^sup>-\<^sup>1[(g \<star> tab\<^sub>0 g) \<star> g\<^sub>0h\<^sub>1.p\<^sub>1, f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine] \<cdot> \<a>\<^sup>-\<^sup>1[g \<star> tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1, f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine] \<cdot> ((\<a>\<^sup>-\<^sup>1[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1 \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine] \<cdot> \<a>[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1 \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine]) \<cdot> (g.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>1 \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine)) \<cdot> \<beta>\<^sub>g = (g \<star> h \<star> TTfgh_TfTgh.the_\<theta>) \<cdot> can (\<^bold>\<langle>g\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> ((\<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) (((\<^bold>\<langle>g\<^bold>\<rangle> \<^bold>\<star> (\<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) \<cdot> (((g \<star> h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (((g \<star> g\<^sub>0h\<^sub>1.\<phi>) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((\<a>\<^sup>-\<^sup>1[g \<star> tab\<^sub>0 g \<star> g\<^sub>0h\<^sub>1.p\<^sub>1, f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine] \<cdot> (\<a>\<^sup>-\<^sup>1[g, tab\<^sub>0 g \<star> g\<^sub>0h\<^sub>1.p\<^sub>1, f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine] \<cdot> \<a>[g, tab\<^sub>0 g \<star> g\<^sub>0h\<^sub>1.p\<^sub>1, f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine]) \<cdot> \<a>[g \<star> tab\<^sub>0 g \<star> g\<^sub>0h\<^sub>1.p\<^sub>1, f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine]) \<cdot> ((\<a>[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine)) \<cdot> \<a>\<^sup>-\<^sup>1[(g \<star> tab\<^sub>0 g) \<star> g\<^sub>0h\<^sub>1.p\<^sub>1, f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine] \<cdot> \<a>\<^sup>-\<^sup>1[g \<star> tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1, f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine] \<cdot> ((\<a>\<^sup>-\<^sup>1[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1 \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine] \<cdot> \<a>[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1 \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine]) \<cdot> (g.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>1 \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine)) \<cdot> \<beta>\<^sub>g
[PROOF STEP]
have "... = (\<a>[g, tab\<^sub>0 g, Tgh.p\<^sub>1] \<star> TfTgh.p\<^sub>0) \<star> TTfgh_TfTgh.chine"
[PROOF STATE]
proof (prove)
goal (1 subgoal):
1. (((g \<star> tab\<^sub>0 g \<star> g\<^sub>0h\<^sub>1.p\<^sub>1) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((\<a>[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) = (\<a>[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine
[PROOF STEP]
using g\<^sub>0h\<^sub>1.p\<^sub>1_simps f\<^sub>0gh\<^sub>1.p\<^sub>0_simps comp_cod_arr
[PROOF STATE]
proof (prove)
using this:
trg g\<^sub>0h\<^sub>1.p\<^sub>1 = src (tab\<^sub>0 g)
trg f\<^sub>0gh\<^sub>1.p\<^sub>0 = src (tab\<^sub>1 g \<star> g\<^sub>0h\<^sub>1.p\<^sub>1)
\<lbrakk>arr ?f; cod ?f = ?b\<rbrakk> \<Longrightarrow> ?b \<cdot> ?f = ?f
goal (1 subgoal):
1. (((g \<star> tab\<^sub>0 g \<star> g\<^sub>0h\<^sub>1.p\<^sub>1) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((\<a>[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) = (\<a>[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine
[PROOF STEP]
by simp
[PROOF STATE]
proof (state)
this:
(((g \<star> tab\<^sub>0 g \<star> g\<^sub>0h\<^sub>1.p\<^sub>1) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((\<a>[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) = (\<a>[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine
goal (1 subgoal):
1. (g \<star> h \<star> TTfgh_TfTgh.the_\<theta>) \<cdot> can (\<^bold>\<langle>g\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> ((\<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) (((\<^bold>\<langle>g\<^bold>\<rangle> \<^bold>\<star> (\<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) \<cdot> (((g \<star> h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (((g \<star> g\<^sub>0h\<^sub>1.\<phi>) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((\<a>[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> \<a>\<^sup>-\<^sup>1[(g \<star> tab\<^sub>0 g) \<star> g\<^sub>0h\<^sub>1.p\<^sub>1, f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine] \<cdot> \<a>\<^sup>-\<^sup>1[g \<star> tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1, f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine] \<cdot> ((\<a>\<^sup>-\<^sup>1[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1 \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine] \<cdot> \<a>[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1 \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine]) \<cdot> (g.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>1 \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine)) \<cdot> \<beta>\<^sub>g = (g \<star> h \<star> TTfgh_TfTgh.the_\<theta>) \<cdot> can (\<^bold>\<langle>g\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> ((\<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) (((\<^bold>\<langle>g\<^bold>\<rangle> \<^bold>\<star> (\<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) \<cdot> (((g \<star> h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (((g \<star> g\<^sub>0h\<^sub>1.\<phi>) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((\<a>\<^sup>-\<^sup>1[g \<star> tab\<^sub>0 g \<star> g\<^sub>0h\<^sub>1.p\<^sub>1, f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine] \<cdot> (\<a>\<^sup>-\<^sup>1[g, tab\<^sub>0 g \<star> g\<^sub>0h\<^sub>1.p\<^sub>1, f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine] \<cdot> \<a>[g, tab\<^sub>0 g \<star> g\<^sub>0h\<^sub>1.p\<^sub>1, f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine]) \<cdot> \<a>[g \<star> tab\<^sub>0 g \<star> g\<^sub>0h\<^sub>1.p\<^sub>1, f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine]) \<cdot> ((\<a>[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine)) \<cdot> \<a>\<^sup>-\<^sup>1[(g \<star> tab\<^sub>0 g) \<star> g\<^sub>0h\<^sub>1.p\<^sub>1, f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine] \<cdot> \<a>\<^sup>-\<^sup>1[g \<star> tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1, f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine] \<cdot> ((\<a>\<^sup>-\<^sup>1[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1 \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine] \<cdot> \<a>[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1 \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine]) \<cdot> (g.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>1 \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine)) \<cdot> \<beta>\<^sub>g
[PROOF STEP]
finally
[PROOF STATE]
proof (chain)
picking this:
(\<a>\<^sup>-\<^sup>1[g \<star> tab\<^sub>0 g \<star> g\<^sub>0h\<^sub>1.p\<^sub>1, f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine] \<cdot> (\<a>\<^sup>-\<^sup>1[g, tab\<^sub>0 g \<star> g\<^sub>0h\<^sub>1.p\<^sub>1, f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine] \<cdot> \<a>[g, tab\<^sub>0 g \<star> g\<^sub>0h\<^sub>1.p\<^sub>1, f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine]) \<cdot> \<a>[g \<star> tab\<^sub>0 g \<star> g\<^sub>0h\<^sub>1.p\<^sub>1, f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine]) \<cdot> ((\<a>[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) = (\<a>[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine
[PROOF STEP]
show ?thesis
[PROOF STATE]
proof (prove)
using this:
(\<a>\<^sup>-\<^sup>1[g \<star> tab\<^sub>0 g \<star> g\<^sub>0h\<^sub>1.p\<^sub>1, f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine] \<cdot> (\<a>\<^sup>-\<^sup>1[g, tab\<^sub>0 g \<star> g\<^sub>0h\<^sub>1.p\<^sub>1, f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine] \<cdot> \<a>[g, tab\<^sub>0 g \<star> g\<^sub>0h\<^sub>1.p\<^sub>1, f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine]) \<cdot> \<a>[g \<star> tab\<^sub>0 g \<star> g\<^sub>0h\<^sub>1.p\<^sub>1, f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine]) \<cdot> ((\<a>[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) = (\<a>[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine
goal (1 subgoal):
1. (g \<star> h \<star> TTfgh_TfTgh.the_\<theta>) \<cdot> can (\<^bold>\<langle>g\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> ((\<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) (((\<^bold>\<langle>g\<^bold>\<rangle> \<^bold>\<star> (\<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) \<cdot> (((g \<star> h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (((g \<star> g\<^sub>0h\<^sub>1.\<phi>) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((\<a>[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> \<a>\<^sup>-\<^sup>1[(g \<star> tab\<^sub>0 g) \<star> g\<^sub>0h\<^sub>1.p\<^sub>1, f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine] \<cdot> \<a>\<^sup>-\<^sup>1[g \<star> tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1, f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine] \<cdot> ((\<a>\<^sup>-\<^sup>1[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1 \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine] \<cdot> \<a>[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1 \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine]) \<cdot> (g.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>1 \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine)) \<cdot> \<beta>\<^sub>g = (g \<star> h \<star> TTfgh_TfTgh.the_\<theta>) \<cdot> can (\<^bold>\<langle>g\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> ((\<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) (((\<^bold>\<langle>g\<^bold>\<rangle> \<^bold>\<star> (\<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) \<cdot> (((g \<star> h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (((g \<star> g\<^sub>0h\<^sub>1.\<phi>) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((\<a>\<^sup>-\<^sup>1[g \<star> tab\<^sub>0 g \<star> g\<^sub>0h\<^sub>1.p\<^sub>1, f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine] \<cdot> (\<a>\<^sup>-\<^sup>1[g, tab\<^sub>0 g \<star> g\<^sub>0h\<^sub>1.p\<^sub>1, f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine] \<cdot> \<a>[g, tab\<^sub>0 g \<star> g\<^sub>0h\<^sub>1.p\<^sub>1, f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine]) \<cdot> \<a>[g \<star> tab\<^sub>0 g \<star> g\<^sub>0h\<^sub>1.p\<^sub>1, f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine]) \<cdot> ((\<a>[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine)) \<cdot> \<a>\<^sup>-\<^sup>1[(g \<star> tab\<^sub>0 g) \<star> g\<^sub>0h\<^sub>1.p\<^sub>1, f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine] \<cdot> \<a>\<^sup>-\<^sup>1[g \<star> tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1, f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine] \<cdot> ((\<a>\<^sup>-\<^sup>1[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1 \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine] \<cdot> \<a>[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1 \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine]) \<cdot> (g.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>1 \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine)) \<cdot> \<beta>\<^sub>g
[PROOF STEP]
by presburger
[PROOF STATE]
proof (state)
this:
(g \<star> h \<star> TTfgh_TfTgh.the_\<theta>) \<cdot> can (\<^bold>\<langle>g\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> ((\<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) (((\<^bold>\<langle>g\<^bold>\<rangle> \<^bold>\<star> (\<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) \<cdot> (((g \<star> h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (((g \<star> g\<^sub>0h\<^sub>1.\<phi>) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((\<a>[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> \<a>\<^sup>-\<^sup>1[(g \<star> tab\<^sub>0 g) \<star> g\<^sub>0h\<^sub>1.p\<^sub>1, f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine] \<cdot> \<a>\<^sup>-\<^sup>1[g \<star> tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1, f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine] \<cdot> ((\<a>\<^sup>-\<^sup>1[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1 \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine] \<cdot> \<a>[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1 \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine]) \<cdot> (g.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>1 \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine)) \<cdot> \<beta>\<^sub>g = (g \<star> h \<star> TTfgh_TfTgh.the_\<theta>) \<cdot> can (\<^bold>\<langle>g\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> ((\<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) (((\<^bold>\<langle>g\<^bold>\<rangle> \<^bold>\<star> (\<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) \<cdot> (((g \<star> h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (((g \<star> g\<^sub>0h\<^sub>1.\<phi>) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((\<a>\<^sup>-\<^sup>1[g \<star> tab\<^sub>0 g \<star> g\<^sub>0h\<^sub>1.p\<^sub>1, f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine] \<cdot> (\<a>\<^sup>-\<^sup>1[g, tab\<^sub>0 g \<star> g\<^sub>0h\<^sub>1.p\<^sub>1, f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine] \<cdot> \<a>[g, tab\<^sub>0 g \<star> g\<^sub>0h\<^sub>1.p\<^sub>1, f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine]) \<cdot> \<a>[g \<star> tab\<^sub>0 g \<star> g\<^sub>0h\<^sub>1.p\<^sub>1, f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine]) \<cdot> ((\<a>[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine)) \<cdot> \<a>\<^sup>-\<^sup>1[(g \<star> tab\<^sub>0 g) \<star> g\<^sub>0h\<^sub>1.p\<^sub>1, f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine] \<cdot> \<a>\<^sup>-\<^sup>1[g \<star> tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1, f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine] \<cdot> ((\<a>\<^sup>-\<^sup>1[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1 \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine] \<cdot> \<a>[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1 \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine]) \<cdot> (g.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>1 \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine)) \<cdot> \<beta>\<^sub>g
goal:
No subgoals!
[PROOF STEP]
qed
[PROOF STATE]
proof (state)
this:
(g \<star> h \<star> TTfgh_TfTgh.the_\<theta>) \<cdot> can (\<^bold>\<langle>g\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> ((\<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) (((\<^bold>\<langle>g\<^bold>\<rangle> \<^bold>\<star> (\<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) \<cdot> (((g \<star> h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (((g \<star> g\<^sub>0h\<^sub>1.\<phi>) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((\<a>[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> \<a>\<^sup>-\<^sup>1[(g \<star> tab\<^sub>0 g) \<star> g\<^sub>0h\<^sub>1.p\<^sub>1, f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine] \<cdot> \<a>\<^sup>-\<^sup>1[g \<star> tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1, f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine] \<cdot> ((\<a>\<^sup>-\<^sup>1[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1 \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine] \<cdot> \<a>[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1 \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine]) \<cdot> (g.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>1 \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine)) \<cdot> \<beta>\<^sub>g = (g \<star> h \<star> TTfgh_TfTgh.the_\<theta>) \<cdot> can (\<^bold>\<langle>g\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> ((\<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) (((\<^bold>\<langle>g\<^bold>\<rangle> \<^bold>\<star> (\<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) \<cdot> (((g \<star> h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (((g \<star> g\<^sub>0h\<^sub>1.\<phi>) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((\<a>\<^sup>-\<^sup>1[g \<star> tab\<^sub>0 g \<star> g\<^sub>0h\<^sub>1.p\<^sub>1, f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine] \<cdot> (\<a>\<^sup>-\<^sup>1[g, tab\<^sub>0 g \<star> g\<^sub>0h\<^sub>1.p\<^sub>1, f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine] \<cdot> \<a>[g, tab\<^sub>0 g \<star> g\<^sub>0h\<^sub>1.p\<^sub>1, f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine]) \<cdot> \<a>[g \<star> tab\<^sub>0 g \<star> g\<^sub>0h\<^sub>1.p\<^sub>1, f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine]) \<cdot> ((\<a>[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine)) \<cdot> \<a>\<^sup>-\<^sup>1[(g \<star> tab\<^sub>0 g) \<star> g\<^sub>0h\<^sub>1.p\<^sub>1, f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine] \<cdot> \<a>\<^sup>-\<^sup>1[g \<star> tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1, f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine] \<cdot> ((\<a>\<^sup>-\<^sup>1[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1 \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine] \<cdot> \<a>[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1 \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine]) \<cdot> (g.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>1 \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine)) \<cdot> \<beta>\<^sub>g
goal (1 subgoal):
1. (g \<star> \<theta>\<^sub>g) \<cdot> \<a>[g, tab\<^sub>0 g, w\<^sub>g] \<cdot> (g.tab \<star> w\<^sub>g) = ((g \<star> \<theta>\<^sub>g') \<cdot> \<a>[g, tab\<^sub>0 g, w\<^sub>g'] \<cdot> (g.tab \<star> w\<^sub>g')) \<cdot> \<beta>\<^sub>g
[PROOF STEP]
also
[PROOF STATE]
proof (state)
this:
(g \<star> h \<star> TTfgh_TfTgh.the_\<theta>) \<cdot> can (\<^bold>\<langle>g\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> ((\<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) (((\<^bold>\<langle>g\<^bold>\<rangle> \<^bold>\<star> (\<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) \<cdot> (((g \<star> h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (((g \<star> g\<^sub>0h\<^sub>1.\<phi>) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((\<a>[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> \<a>\<^sup>-\<^sup>1[(g \<star> tab\<^sub>0 g) \<star> g\<^sub>0h\<^sub>1.p\<^sub>1, f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine] \<cdot> \<a>\<^sup>-\<^sup>1[g \<star> tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1, f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine] \<cdot> ((\<a>\<^sup>-\<^sup>1[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1 \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine] \<cdot> \<a>[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1 \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine]) \<cdot> (g.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>1 \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine)) \<cdot> \<beta>\<^sub>g = (g \<star> h \<star> TTfgh_TfTgh.the_\<theta>) \<cdot> can (\<^bold>\<langle>g\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> ((\<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) (((\<^bold>\<langle>g\<^bold>\<rangle> \<^bold>\<star> (\<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) \<cdot> (((g \<star> h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (((g \<star> g\<^sub>0h\<^sub>1.\<phi>) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((\<a>\<^sup>-\<^sup>1[g \<star> tab\<^sub>0 g \<star> g\<^sub>0h\<^sub>1.p\<^sub>1, f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine] \<cdot> (\<a>\<^sup>-\<^sup>1[g, tab\<^sub>0 g \<star> g\<^sub>0h\<^sub>1.p\<^sub>1, f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine] \<cdot> \<a>[g, tab\<^sub>0 g \<star> g\<^sub>0h\<^sub>1.p\<^sub>1, f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine]) \<cdot> \<a>[g \<star> tab\<^sub>0 g \<star> g\<^sub>0h\<^sub>1.p\<^sub>1, f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine]) \<cdot> ((\<a>[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine)) \<cdot> \<a>\<^sup>-\<^sup>1[(g \<star> tab\<^sub>0 g) \<star> g\<^sub>0h\<^sub>1.p\<^sub>1, f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine] \<cdot> \<a>\<^sup>-\<^sup>1[g \<star> tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1, f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine] \<cdot> ((\<a>\<^sup>-\<^sup>1[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1 \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine] \<cdot> \<a>[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1 \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine]) \<cdot> (g.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>1 \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine)) \<cdot> \<beta>\<^sub>g
goal (1 subgoal):
1. (g \<star> \<theta>\<^sub>g) \<cdot> \<a>[g, tab\<^sub>0 g, w\<^sub>g] \<cdot> (g.tab \<star> w\<^sub>g) = ((g \<star> \<theta>\<^sub>g') \<cdot> \<a>[g, tab\<^sub>0 g, w\<^sub>g'] \<cdot> (g.tab \<star> w\<^sub>g')) \<cdot> \<beta>\<^sub>g
[PROOF STEP]
have "... = (g \<star> h \<star> TTfgh_TfTgh.the_\<theta>) \<cdot>
can (\<^bold>\<langle>g\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> ((\<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>Tgh.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TfTgh.p\<^sub>0\<^bold>\<rangle>)
\<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>)
(((\<^bold>\<langle>g\<^bold>\<rangle> \<^bold>\<star> (\<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>Tgh.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TfTgh.p\<^sub>0\<^bold>\<rangle>)
\<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) \<cdot>
((((g \<star> h.tab \<star> Tgh.p\<^sub>0) \<star> TfTgh.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot>
(((g \<star> g\<^sub>0h\<^sub>1.\<phi>) \<star> TfTgh.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot>
\<a>\<^sup>-\<^sup>1[g \<star> tab\<^sub>0 g \<star> Tgh.p\<^sub>1, TfTgh.p\<^sub>0, TTfgh_TfTgh.chine] \<cdot>
\<a>\<^sup>-\<^sup>1[g, tab\<^sub>0 g \<star> Tgh.p\<^sub>1, TfTgh.p\<^sub>0 \<star> TTfgh_TfTgh.chine]) \<cdot>
\<a>[g, tab\<^sub>0 g \<star> Tgh.p\<^sub>1, TfTgh.p\<^sub>0 \<star> TTfgh_TfTgh.chine] \<cdot>
\<a>[g \<star> tab\<^sub>0 g \<star> Tgh.p\<^sub>1, TfTgh.p\<^sub>0, TTfgh_TfTgh.chine] \<cdot>
((\<a>[g, tab\<^sub>0 g, Tgh.p\<^sub>1] \<star> TfTgh.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot>
\<a>\<^sup>-\<^sup>1[(g \<star> tab\<^sub>0 g) \<star> Tgh.p\<^sub>1, TfTgh.p\<^sub>0, TTfgh_TfTgh.chine] \<cdot>
\<a>\<^sup>-\<^sup>1[g \<star> tab\<^sub>0 g, Tgh.p\<^sub>1, TfTgh.p\<^sub>0 \<star> TTfgh_TfTgh.chine] \<cdot>
(\<a>\<^sup>-\<^sup>1[g, tab\<^sub>0 g, Tgh.p\<^sub>1 \<star> TfTgh.p\<^sub>0 \<star> TTfgh_TfTgh.chine] \<cdot>
\<a>[g, tab\<^sub>0 g, Tgh.p\<^sub>1 \<star> TfTgh.p\<^sub>0 \<star> TTfgh_TfTgh.chine] \<cdot>
(g.tab \<star> Tgh.p\<^sub>1 \<star> TfTgh.p\<^sub>0 \<star> TTfgh_TfTgh.chine)) \<cdot>
\<beta>\<^sub>g"
[PROOF STATE]
proof (prove)
goal (1 subgoal):
1. (g \<star> h \<star> TTfgh_TfTgh.the_\<theta>) \<cdot> can (\<^bold>\<langle>g\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> ((\<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) (((\<^bold>\<langle>g\<^bold>\<rangle> \<^bold>\<star> (\<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) \<cdot> (((g \<star> h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (((g \<star> g\<^sub>0h\<^sub>1.\<phi>) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((\<a>\<^sup>-\<^sup>1[g \<star> tab\<^sub>0 g \<star> g\<^sub>0h\<^sub>1.p\<^sub>1, f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine] \<cdot> (\<a>\<^sup>-\<^sup>1[g, tab\<^sub>0 g \<star> g\<^sub>0h\<^sub>1.p\<^sub>1, f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine] \<cdot> \<a>[g, tab\<^sub>0 g \<star> g\<^sub>0h\<^sub>1.p\<^sub>1, f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine]) \<cdot> \<a>[g \<star> tab\<^sub>0 g \<star> g\<^sub>0h\<^sub>1.p\<^sub>1, f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine]) \<cdot> ((\<a>[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine)) \<cdot> \<a>\<^sup>-\<^sup>1[(g \<star> tab\<^sub>0 g) \<star> g\<^sub>0h\<^sub>1.p\<^sub>1, f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine] \<cdot> \<a>\<^sup>-\<^sup>1[g \<star> tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1, f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine] \<cdot> ((\<a>\<^sup>-\<^sup>1[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1 \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine] \<cdot> \<a>[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1 \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine]) \<cdot> (g.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>1 \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine)) \<cdot> \<beta>\<^sub>g = (g \<star> h \<star> TTfgh_TfTgh.the_\<theta>) \<cdot> can (\<^bold>\<langle>g\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> ((\<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) (((\<^bold>\<langle>g\<^bold>\<rangle> \<^bold>\<star> (\<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) \<cdot> ((((g \<star> h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (((g \<star> g\<^sub>0h\<^sub>1.\<phi>) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> \<a>\<^sup>-\<^sup>1[g \<star> tab\<^sub>0 g \<star> g\<^sub>0h\<^sub>1.p\<^sub>1, f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine] \<cdot> \<a>\<^sup>-\<^sup>1[g, tab\<^sub>0 g \<star> g\<^sub>0h\<^sub>1.p\<^sub>1, f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine]) \<cdot> \<a>[g, tab\<^sub>0 g \<star> g\<^sub>0h\<^sub>1.p\<^sub>1, f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine] \<cdot> \<a>[g \<star> tab\<^sub>0 g \<star> g\<^sub>0h\<^sub>1.p\<^sub>1, f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine] \<cdot> ((\<a>[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> \<a>\<^sup>-\<^sup>1[(g \<star> tab\<^sub>0 g) \<star> g\<^sub>0h\<^sub>1.p\<^sub>1, f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine] \<cdot> \<a>\<^sup>-\<^sup>1[g \<star> tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1, f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine] \<cdot> (\<a>\<^sup>-\<^sup>1[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1 \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine] \<cdot> \<a>[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1 \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine] \<cdot> (g.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>1 \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine)) \<cdot> \<beta>\<^sub>g
[PROOF STEP]
using comp_assoc
[PROOF STATE]
proof (prove)
using this:
(?h \<cdot> ?g) \<cdot> ?f = ?h \<cdot> ?g \<cdot> ?f
goal (1 subgoal):
1. (g \<star> h \<star> TTfgh_TfTgh.the_\<theta>) \<cdot> can (\<^bold>\<langle>g\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> ((\<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) (((\<^bold>\<langle>g\<^bold>\<rangle> \<^bold>\<star> (\<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) \<cdot> (((g \<star> h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (((g \<star> g\<^sub>0h\<^sub>1.\<phi>) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((\<a>\<^sup>-\<^sup>1[g \<star> tab\<^sub>0 g \<star> g\<^sub>0h\<^sub>1.p\<^sub>1, f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine] \<cdot> (\<a>\<^sup>-\<^sup>1[g, tab\<^sub>0 g \<star> g\<^sub>0h\<^sub>1.p\<^sub>1, f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine] \<cdot> \<a>[g, tab\<^sub>0 g \<star> g\<^sub>0h\<^sub>1.p\<^sub>1, f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine]) \<cdot> \<a>[g \<star> tab\<^sub>0 g \<star> g\<^sub>0h\<^sub>1.p\<^sub>1, f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine]) \<cdot> ((\<a>[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine)) \<cdot> \<a>\<^sup>-\<^sup>1[(g \<star> tab\<^sub>0 g) \<star> g\<^sub>0h\<^sub>1.p\<^sub>1, f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine] \<cdot> \<a>\<^sup>-\<^sup>1[g \<star> tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1, f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine] \<cdot> ((\<a>\<^sup>-\<^sup>1[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1 \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine] \<cdot> \<a>[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1 \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine]) \<cdot> (g.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>1 \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine)) \<cdot> \<beta>\<^sub>g = (g \<star> h \<star> TTfgh_TfTgh.the_\<theta>) \<cdot> can (\<^bold>\<langle>g\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> ((\<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) (((\<^bold>\<langle>g\<^bold>\<rangle> \<^bold>\<star> (\<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) \<cdot> ((((g \<star> h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (((g \<star> g\<^sub>0h\<^sub>1.\<phi>) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> \<a>\<^sup>-\<^sup>1[g \<star> tab\<^sub>0 g \<star> g\<^sub>0h\<^sub>1.p\<^sub>1, f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine] \<cdot> \<a>\<^sup>-\<^sup>1[g, tab\<^sub>0 g \<star> g\<^sub>0h\<^sub>1.p\<^sub>1, f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine]) \<cdot> \<a>[g, tab\<^sub>0 g \<star> g\<^sub>0h\<^sub>1.p\<^sub>1, f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine] \<cdot> \<a>[g \<star> tab\<^sub>0 g \<star> g\<^sub>0h\<^sub>1.p\<^sub>1, f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine] \<cdot> ((\<a>[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> \<a>\<^sup>-\<^sup>1[(g \<star> tab\<^sub>0 g) \<star> g\<^sub>0h\<^sub>1.p\<^sub>1, f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine] \<cdot> \<a>\<^sup>-\<^sup>1[g \<star> tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1, f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine] \<cdot> (\<a>\<^sup>-\<^sup>1[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1 \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine] \<cdot> \<a>[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1 \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine] \<cdot> (g.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>1 \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine)) \<cdot> \<beta>\<^sub>g
[PROOF STEP]
by presburger
[PROOF STATE]
proof (state)
this:
(g \<star> h \<star> TTfgh_TfTgh.the_\<theta>) \<cdot> can (\<^bold>\<langle>g\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> ((\<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) (((\<^bold>\<langle>g\<^bold>\<rangle> \<^bold>\<star> (\<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) \<cdot> (((g \<star> h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (((g \<star> g\<^sub>0h\<^sub>1.\<phi>) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((\<a>\<^sup>-\<^sup>1[g \<star> tab\<^sub>0 g \<star> g\<^sub>0h\<^sub>1.p\<^sub>1, f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine] \<cdot> (\<a>\<^sup>-\<^sup>1[g, tab\<^sub>0 g \<star> g\<^sub>0h\<^sub>1.p\<^sub>1, f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine] \<cdot> \<a>[g, tab\<^sub>0 g \<star> g\<^sub>0h\<^sub>1.p\<^sub>1, f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine]) \<cdot> \<a>[g \<star> tab\<^sub>0 g \<star> g\<^sub>0h\<^sub>1.p\<^sub>1, f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine]) \<cdot> ((\<a>[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine)) \<cdot> \<a>\<^sup>-\<^sup>1[(g \<star> tab\<^sub>0 g) \<star> g\<^sub>0h\<^sub>1.p\<^sub>1, f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine] \<cdot> \<a>\<^sup>-\<^sup>1[g \<star> tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1, f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine] \<cdot> ((\<a>\<^sup>-\<^sup>1[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1 \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine] \<cdot> \<a>[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1 \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine]) \<cdot> (g.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>1 \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine)) \<cdot> \<beta>\<^sub>g = (g \<star> h \<star> TTfgh_TfTgh.the_\<theta>) \<cdot> can (\<^bold>\<langle>g\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> ((\<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) (((\<^bold>\<langle>g\<^bold>\<rangle> \<^bold>\<star> (\<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) \<cdot> ((((g \<star> h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (((g \<star> g\<^sub>0h\<^sub>1.\<phi>) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> \<a>\<^sup>-\<^sup>1[g \<star> tab\<^sub>0 g \<star> g\<^sub>0h\<^sub>1.p\<^sub>1, f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine] \<cdot> \<a>\<^sup>-\<^sup>1[g, tab\<^sub>0 g \<star> g\<^sub>0h\<^sub>1.p\<^sub>1, f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine]) \<cdot> \<a>[g, tab\<^sub>0 g \<star> g\<^sub>0h\<^sub>1.p\<^sub>1, f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine] \<cdot> \<a>[g \<star> tab\<^sub>0 g \<star> g\<^sub>0h\<^sub>1.p\<^sub>1, f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine] \<cdot> ((\<a>[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> \<a>\<^sup>-\<^sup>1[(g \<star> tab\<^sub>0 g) \<star> g\<^sub>0h\<^sub>1.p\<^sub>1, f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine] \<cdot> \<a>\<^sup>-\<^sup>1[g \<star> tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1, f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine] \<cdot> (\<a>\<^sup>-\<^sup>1[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1 \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine] \<cdot> \<a>[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1 \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine] \<cdot> (g.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>1 \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine)) \<cdot> \<beta>\<^sub>g
goal (1 subgoal):
1. (g \<star> \<theta>\<^sub>g) \<cdot> \<a>[g, tab\<^sub>0 g, w\<^sub>g] \<cdot> (g.tab \<star> w\<^sub>g) = ((g \<star> \<theta>\<^sub>g') \<cdot> \<a>[g, tab\<^sub>0 g, w\<^sub>g'] \<cdot> (g.tab \<star> w\<^sub>g')) \<cdot> \<beta>\<^sub>g
[PROOF STEP]
also
[PROOF STATE]
proof (state)
this:
(g \<star> h \<star> TTfgh_TfTgh.the_\<theta>) \<cdot> can (\<^bold>\<langle>g\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> ((\<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) (((\<^bold>\<langle>g\<^bold>\<rangle> \<^bold>\<star> (\<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) \<cdot> (((g \<star> h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (((g \<star> g\<^sub>0h\<^sub>1.\<phi>) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((\<a>\<^sup>-\<^sup>1[g \<star> tab\<^sub>0 g \<star> g\<^sub>0h\<^sub>1.p\<^sub>1, f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine] \<cdot> (\<a>\<^sup>-\<^sup>1[g, tab\<^sub>0 g \<star> g\<^sub>0h\<^sub>1.p\<^sub>1, f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine] \<cdot> \<a>[g, tab\<^sub>0 g \<star> g\<^sub>0h\<^sub>1.p\<^sub>1, f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine]) \<cdot> \<a>[g \<star> tab\<^sub>0 g \<star> g\<^sub>0h\<^sub>1.p\<^sub>1, f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine]) \<cdot> ((\<a>[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine)) \<cdot> \<a>\<^sup>-\<^sup>1[(g \<star> tab\<^sub>0 g) \<star> g\<^sub>0h\<^sub>1.p\<^sub>1, f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine] \<cdot> \<a>\<^sup>-\<^sup>1[g \<star> tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1, f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine] \<cdot> ((\<a>\<^sup>-\<^sup>1[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1 \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine] \<cdot> \<a>[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1 \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine]) \<cdot> (g.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>1 \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine)) \<cdot> \<beta>\<^sub>g = (g \<star> h \<star> TTfgh_TfTgh.the_\<theta>) \<cdot> can (\<^bold>\<langle>g\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> ((\<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) (((\<^bold>\<langle>g\<^bold>\<rangle> \<^bold>\<star> (\<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) \<cdot> ((((g \<star> h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (((g \<star> g\<^sub>0h\<^sub>1.\<phi>) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> \<a>\<^sup>-\<^sup>1[g \<star> tab\<^sub>0 g \<star> g\<^sub>0h\<^sub>1.p\<^sub>1, f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine] \<cdot> \<a>\<^sup>-\<^sup>1[g, tab\<^sub>0 g \<star> g\<^sub>0h\<^sub>1.p\<^sub>1, f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine]) \<cdot> \<a>[g, tab\<^sub>0 g \<star> g\<^sub>0h\<^sub>1.p\<^sub>1, f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine] \<cdot> \<a>[g \<star> tab\<^sub>0 g \<star> g\<^sub>0h\<^sub>1.p\<^sub>1, f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine] \<cdot> ((\<a>[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> \<a>\<^sup>-\<^sup>1[(g \<star> tab\<^sub>0 g) \<star> g\<^sub>0h\<^sub>1.p\<^sub>1, f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine] \<cdot> \<a>\<^sup>-\<^sup>1[g \<star> tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1, f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine] \<cdot> (\<a>\<^sup>-\<^sup>1[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1 \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine] \<cdot> \<a>[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1 \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine] \<cdot> (g.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>1 \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine)) \<cdot> \<beta>\<^sub>g
goal (1 subgoal):
1. (g \<star> \<theta>\<^sub>g) \<cdot> \<a>[g, tab\<^sub>0 g, w\<^sub>g] \<cdot> (g.tab \<star> w\<^sub>g) = ((g \<star> \<theta>\<^sub>g') \<cdot> \<a>[g, tab\<^sub>0 g, w\<^sub>g'] \<cdot> (g.tab \<star> w\<^sub>g')) \<cdot> \<beta>\<^sub>g
[PROOF STEP]
have "... = (g \<star> h \<star> TTfgh_TfTgh.the_\<theta>) \<cdot>
(can (\<^bold>\<langle>g\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> ((\<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>Tgh.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TfTgh.p\<^sub>0\<^bold>\<rangle>)
\<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>)
(((\<^bold>\<langle>g\<^bold>\<rangle> \<^bold>\<star> (\<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>Tgh.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TfTgh.p\<^sub>0\<^bold>\<rangle>)
\<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) \<cdot>
\<a>\<^sup>-\<^sup>1[g \<star> (h \<star> tab\<^sub>0 h) \<star> Tgh.p\<^sub>0, TfTgh.p\<^sub>0, TTfgh_TfTgh.chine] \<cdot>
\<a>\<^sup>-\<^sup>1[g, (h \<star> tab\<^sub>0 h) \<star> Tgh.p\<^sub>0, TfTgh.p\<^sub>0 \<star> TTfgh_TfTgh.chine]) \<cdot>
(g \<star> (h.tab \<star> Tgh.p\<^sub>0) \<star> TfTgh.p\<^sub>0 \<star> TTfgh_TfTgh.chine) \<cdot>
(g \<star> g\<^sub>0h\<^sub>1.\<phi> \<star> TfTgh.p\<^sub>0 \<star> TTfgh_TfTgh.chine) \<cdot>
(\<a>[g, tab\<^sub>0 g \<star> Tgh.p\<^sub>1, TfTgh.p\<^sub>0 \<star> TTfgh_TfTgh.chine] \<cdot>
\<a>[g \<star> tab\<^sub>0 g \<star> Tgh.p\<^sub>1, TfTgh.p\<^sub>0, TTfgh_TfTgh.chine] \<cdot>
((\<a>[g, tab\<^sub>0 g, Tgh.p\<^sub>1] \<star> TfTgh.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot>
\<a>\<^sup>-\<^sup>1[(g \<star> tab\<^sub>0 g) \<star> Tgh.p\<^sub>1, TfTgh.p\<^sub>0, TTfgh_TfTgh.chine] \<cdot>
\<a>\<^sup>-\<^sup>1[g \<star> tab\<^sub>0 g, Tgh.p\<^sub>1, TfTgh.p\<^sub>0 \<star> TTfgh_TfTgh.chine] \<cdot>
\<a>\<^sup>-\<^sup>1[g, tab\<^sub>0 g, Tgh.p\<^sub>1 \<star> TfTgh.p\<^sub>0 \<star> TTfgh_TfTgh.chine]) \<cdot>
\<a>[g, tab\<^sub>0 g, Tgh.p\<^sub>1 \<star> TfTgh.p\<^sub>0 \<star> TTfgh_TfTgh.chine] \<cdot>
(g.tab \<star> Tgh.p\<^sub>1 \<star> TfTgh.p\<^sub>0 \<star> TTfgh_TfTgh.chine) \<cdot>
\<beta>\<^sub>g"
[PROOF STATE]
proof (prove)
goal (1 subgoal):
1. (g \<star> h \<star> TTfgh_TfTgh.the_\<theta>) \<cdot> can (\<^bold>\<langle>g\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> ((\<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) (((\<^bold>\<langle>g\<^bold>\<rangle> \<^bold>\<star> (\<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) \<cdot> ((((g \<star> h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (((g \<star> g\<^sub>0h\<^sub>1.\<phi>) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> \<a>\<^sup>-\<^sup>1[g \<star> tab\<^sub>0 g \<star> g\<^sub>0h\<^sub>1.p\<^sub>1, f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine] \<cdot> \<a>\<^sup>-\<^sup>1[g, tab\<^sub>0 g \<star> g\<^sub>0h\<^sub>1.p\<^sub>1, f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine]) \<cdot> \<a>[g, tab\<^sub>0 g \<star> g\<^sub>0h\<^sub>1.p\<^sub>1, f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine] \<cdot> \<a>[g \<star> tab\<^sub>0 g \<star> g\<^sub>0h\<^sub>1.p\<^sub>1, f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine] \<cdot> ((\<a>[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> \<a>\<^sup>-\<^sup>1[(g \<star> tab\<^sub>0 g) \<star> g\<^sub>0h\<^sub>1.p\<^sub>1, f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine] \<cdot> \<a>\<^sup>-\<^sup>1[g \<star> tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1, f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine] \<cdot> (\<a>\<^sup>-\<^sup>1[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1 \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine] \<cdot> \<a>[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1 \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine] \<cdot> (g.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>1 \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine)) \<cdot> \<beta>\<^sub>g = (g \<star> h \<star> TTfgh_TfTgh.the_\<theta>) \<cdot> (can (\<^bold>\<langle>g\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> ((\<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) (((\<^bold>\<langle>g\<^bold>\<rangle> \<^bold>\<star> (\<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) \<cdot> \<a>\<^sup>-\<^sup>1[g \<star> (h \<star> tab\<^sub>0 h) \<star> g\<^sub>0h\<^sub>1.p\<^sub>0, f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine] \<cdot> \<a>\<^sup>-\<^sup>1[g, (h \<star> tab\<^sub>0 h) \<star> g\<^sub>0h\<^sub>1.p\<^sub>0, f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine]) \<cdot> (g \<star> (h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine) \<cdot> (g \<star> g\<^sub>0h\<^sub>1.\<phi> \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine) \<cdot> (\<a>[g, tab\<^sub>0 g \<star> g\<^sub>0h\<^sub>1.p\<^sub>1, f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine] \<cdot> \<a>[g \<star> tab\<^sub>0 g \<star> g\<^sub>0h\<^sub>1.p\<^sub>1, f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine] \<cdot> ((\<a>[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> \<a>\<^sup>-\<^sup>1[(g \<star> tab\<^sub>0 g) \<star> g\<^sub>0h\<^sub>1.p\<^sub>1, f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine] \<cdot> \<a>\<^sup>-\<^sup>1[g \<star> tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1, f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine] \<cdot> \<a>\<^sup>-\<^sup>1[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1 \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine]) \<cdot> \<a>[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1 \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine] \<cdot> (g.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>1 \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine) \<cdot> \<beta>\<^sub>g
[PROOF STEP]
proof -
[PROOF STATE]
proof (state)
goal (1 subgoal):
1. (g \<star> h \<star> TTfgh_TfTgh.the_\<theta>) \<cdot> can (\<^bold>\<langle>g\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> ((\<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) (((\<^bold>\<langle>g\<^bold>\<rangle> \<^bold>\<star> (\<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) \<cdot> ((((g \<star> h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (((g \<star> g\<^sub>0h\<^sub>1.\<phi>) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> \<a>\<^sup>-\<^sup>1[g \<star> tab\<^sub>0 g \<star> g\<^sub>0h\<^sub>1.p\<^sub>1, f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine] \<cdot> \<a>\<^sup>-\<^sup>1[g, tab\<^sub>0 g \<star> g\<^sub>0h\<^sub>1.p\<^sub>1, f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine]) \<cdot> \<a>[g, tab\<^sub>0 g \<star> g\<^sub>0h\<^sub>1.p\<^sub>1, f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine] \<cdot> \<a>[g \<star> tab\<^sub>0 g \<star> g\<^sub>0h\<^sub>1.p\<^sub>1, f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine] \<cdot> ((\<a>[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> \<a>\<^sup>-\<^sup>1[(g \<star> tab\<^sub>0 g) \<star> g\<^sub>0h\<^sub>1.p\<^sub>1, f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine] \<cdot> \<a>\<^sup>-\<^sup>1[g \<star> tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1, f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine] \<cdot> (\<a>\<^sup>-\<^sup>1[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1 \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine] \<cdot> \<a>[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1 \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine] \<cdot> (g.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>1 \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine)) \<cdot> \<beta>\<^sub>g = (g \<star> h \<star> TTfgh_TfTgh.the_\<theta>) \<cdot> (can (\<^bold>\<langle>g\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> ((\<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) (((\<^bold>\<langle>g\<^bold>\<rangle> \<^bold>\<star> (\<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) \<cdot> \<a>\<^sup>-\<^sup>1[g \<star> (h \<star> tab\<^sub>0 h) \<star> g\<^sub>0h\<^sub>1.p\<^sub>0, f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine] \<cdot> \<a>\<^sup>-\<^sup>1[g, (h \<star> tab\<^sub>0 h) \<star> g\<^sub>0h\<^sub>1.p\<^sub>0, f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine]) \<cdot> (g \<star> (h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine) \<cdot> (g \<star> g\<^sub>0h\<^sub>1.\<phi> \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine) \<cdot> (\<a>[g, tab\<^sub>0 g \<star> g\<^sub>0h\<^sub>1.p\<^sub>1, f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine] \<cdot> \<a>[g \<star> tab\<^sub>0 g \<star> g\<^sub>0h\<^sub>1.p\<^sub>1, f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine] \<cdot> ((\<a>[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> \<a>\<^sup>-\<^sup>1[(g \<star> tab\<^sub>0 g) \<star> g\<^sub>0h\<^sub>1.p\<^sub>1, f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine] \<cdot> \<a>\<^sup>-\<^sup>1[g \<star> tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1, f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine] \<cdot> \<a>\<^sup>-\<^sup>1[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1 \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine]) \<cdot> \<a>[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1 \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine] \<cdot> (g.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>1 \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine) \<cdot> \<beta>\<^sub>g
[PROOF STEP]
have "(((g \<star> h.tab \<star> Tgh.p\<^sub>0) \<star> TfTgh.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot>
(((g \<star> g\<^sub>0h\<^sub>1.\<phi>) \<star> TfTgh.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot>
\<a>\<^sup>-\<^sup>1[g \<star> tab\<^sub>0 g \<star> Tgh.p\<^sub>1, TfTgh.p\<^sub>0, TTfgh_TfTgh.chine] \<cdot>
\<a>\<^sup>-\<^sup>1[g, tab\<^sub>0 g \<star> Tgh.p\<^sub>1, TfTgh.p\<^sub>0 \<star> TTfgh_TfTgh.chine] =
\<a>\<^sup>-\<^sup>1[g \<star> (h \<star> tab\<^sub>0 h) \<star> Tgh.p\<^sub>0, TfTgh.p\<^sub>0, TTfgh_TfTgh.chine] \<cdot>
\<a>\<^sup>-\<^sup>1[g, (h \<star> tab\<^sub>0 h) \<star> Tgh.p\<^sub>0, TfTgh.p\<^sub>0 \<star> TTfgh_TfTgh.chine] \<cdot>
(g \<star> (h.tab \<star> Tgh.p\<^sub>0) \<star> TfTgh.p\<^sub>0 \<star> TTfgh_TfTgh.chine) \<cdot>
(g \<star> g\<^sub>0h\<^sub>1.\<phi> \<star> TfTgh.p\<^sub>0 \<star> TTfgh_TfTgh.chine)"
[PROOF STATE]
proof (prove)
goal (1 subgoal):
1. (((g \<star> h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (((g \<star> g\<^sub>0h\<^sub>1.\<phi>) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> \<a>\<^sup>-\<^sup>1[g \<star> tab\<^sub>0 g \<star> g\<^sub>0h\<^sub>1.p\<^sub>1, f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine] \<cdot> \<a>\<^sup>-\<^sup>1[g, tab\<^sub>0 g \<star> g\<^sub>0h\<^sub>1.p\<^sub>1, f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine] = \<a>\<^sup>-\<^sup>1[g \<star> (h \<star> tab\<^sub>0 h) \<star> g\<^sub>0h\<^sub>1.p\<^sub>0, f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine] \<cdot> \<a>\<^sup>-\<^sup>1[g, (h \<star> tab\<^sub>0 h) \<star> g\<^sub>0h\<^sub>1.p\<^sub>0, f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine] \<cdot> (g \<star> (h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine) \<cdot> (g \<star> g\<^sub>0h\<^sub>1.\<phi> \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine)
[PROOF STEP]
proof -
[PROOF STATE]
proof (state)
goal (1 subgoal):
1. (((g \<star> h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (((g \<star> g\<^sub>0h\<^sub>1.\<phi>) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> \<a>\<^sup>-\<^sup>1[g \<star> tab\<^sub>0 g \<star> g\<^sub>0h\<^sub>1.p\<^sub>1, f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine] \<cdot> \<a>\<^sup>-\<^sup>1[g, tab\<^sub>0 g \<star> g\<^sub>0h\<^sub>1.p\<^sub>1, f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine] = \<a>\<^sup>-\<^sup>1[g \<star> (h \<star> tab\<^sub>0 h) \<star> g\<^sub>0h\<^sub>1.p\<^sub>0, f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine] \<cdot> \<a>\<^sup>-\<^sup>1[g, (h \<star> tab\<^sub>0 h) \<star> g\<^sub>0h\<^sub>1.p\<^sub>0, f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine] \<cdot> (g \<star> (h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine) \<cdot> (g \<star> g\<^sub>0h\<^sub>1.\<phi> \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine)
[PROOF STEP]
have "(((g \<star> h.tab \<star> Tgh.p\<^sub>0) \<star> TfTgh.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot>
(((g \<star> g\<^sub>0h\<^sub>1.\<phi>) \<star> TfTgh.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot>
\<a>\<^sup>-\<^sup>1[g \<star> tab\<^sub>0 g \<star> Tgh.p\<^sub>1, TfTgh.p\<^sub>0, TTfgh_TfTgh.chine] \<cdot>
\<a>\<^sup>-\<^sup>1[g, tab\<^sub>0 g \<star> Tgh.p\<^sub>1, TfTgh.p\<^sub>0 \<star> TTfgh_TfTgh.chine] =
(((g \<star> h.tab \<star> Tgh.p\<^sub>0) \<star> TfTgh.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot>
((((g \<star> g\<^sub>0h\<^sub>1.\<phi>) \<star> TfTgh.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot>
\<a>\<^sup>-\<^sup>1[g \<star> tab\<^sub>0 g \<star> Tgh.p\<^sub>1, TfTgh.p\<^sub>0, TTfgh_TfTgh.chine]) \<cdot>
\<a>\<^sup>-\<^sup>1[g, tab\<^sub>0 g \<star> Tgh.p\<^sub>1, TfTgh.p\<^sub>0 \<star> TTfgh_TfTgh.chine]"
[PROOF STATE]
proof (prove)
goal (1 subgoal):
1. (((g \<star> h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (((g \<star> g\<^sub>0h\<^sub>1.\<phi>) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> \<a>\<^sup>-\<^sup>1[g \<star> tab\<^sub>0 g \<star> g\<^sub>0h\<^sub>1.p\<^sub>1, f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine] \<cdot> \<a>\<^sup>-\<^sup>1[g, tab\<^sub>0 g \<star> g\<^sub>0h\<^sub>1.p\<^sub>1, f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine] = (((g \<star> h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((((g \<star> g\<^sub>0h\<^sub>1.\<phi>) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> \<a>\<^sup>-\<^sup>1[g \<star> tab\<^sub>0 g \<star> g\<^sub>0h\<^sub>1.p\<^sub>1, f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine]) \<cdot> \<a>\<^sup>-\<^sup>1[g, tab\<^sub>0 g \<star> g\<^sub>0h\<^sub>1.p\<^sub>1, f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine]
[PROOF STEP]
using comp_assoc
[PROOF STATE]
proof (prove)
using this:
(?h \<cdot> ?g) \<cdot> ?f = ?h \<cdot> ?g \<cdot> ?f
goal (1 subgoal):
1. (((g \<star> h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (((g \<star> g\<^sub>0h\<^sub>1.\<phi>) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> \<a>\<^sup>-\<^sup>1[g \<star> tab\<^sub>0 g \<star> g\<^sub>0h\<^sub>1.p\<^sub>1, f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine] \<cdot> \<a>\<^sup>-\<^sup>1[g, tab\<^sub>0 g \<star> g\<^sub>0h\<^sub>1.p\<^sub>1, f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine] = (((g \<star> h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((((g \<star> g\<^sub>0h\<^sub>1.\<phi>) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> \<a>\<^sup>-\<^sup>1[g \<star> tab\<^sub>0 g \<star> g\<^sub>0h\<^sub>1.p\<^sub>1, f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine]) \<cdot> \<a>\<^sup>-\<^sup>1[g, tab\<^sub>0 g \<star> g\<^sub>0h\<^sub>1.p\<^sub>1, f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine]
[PROOF STEP]
by presburger
[PROOF STATE]
proof (state)
this:
(((g \<star> h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (((g \<star> g\<^sub>0h\<^sub>1.\<phi>) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> \<a>\<^sup>-\<^sup>1[g \<star> tab\<^sub>0 g \<star> g\<^sub>0h\<^sub>1.p\<^sub>1, f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine] \<cdot> \<a>\<^sup>-\<^sup>1[g, tab\<^sub>0 g \<star> g\<^sub>0h\<^sub>1.p\<^sub>1, f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine] = (((g \<star> h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((((g \<star> g\<^sub>0h\<^sub>1.\<phi>) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> \<a>\<^sup>-\<^sup>1[g \<star> tab\<^sub>0 g \<star> g\<^sub>0h\<^sub>1.p\<^sub>1, f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine]) \<cdot> \<a>\<^sup>-\<^sup>1[g, tab\<^sub>0 g \<star> g\<^sub>0h\<^sub>1.p\<^sub>1, f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine]
goal (1 subgoal):
1. (((g \<star> h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (((g \<star> g\<^sub>0h\<^sub>1.\<phi>) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> \<a>\<^sup>-\<^sup>1[g \<star> tab\<^sub>0 g \<star> g\<^sub>0h\<^sub>1.p\<^sub>1, f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine] \<cdot> \<a>\<^sup>-\<^sup>1[g, tab\<^sub>0 g \<star> g\<^sub>0h\<^sub>1.p\<^sub>1, f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine] = \<a>\<^sup>-\<^sup>1[g \<star> (h \<star> tab\<^sub>0 h) \<star> g\<^sub>0h\<^sub>1.p\<^sub>0, f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine] \<cdot> \<a>\<^sup>-\<^sup>1[g, (h \<star> tab\<^sub>0 h) \<star> g\<^sub>0h\<^sub>1.p\<^sub>0, f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine] \<cdot> (g \<star> (h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine) \<cdot> (g \<star> g\<^sub>0h\<^sub>1.\<phi> \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine)
[PROOF STEP]
also
[PROOF STATE]
proof (state)
this:
(((g \<star> h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (((g \<star> g\<^sub>0h\<^sub>1.\<phi>) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> \<a>\<^sup>-\<^sup>1[g \<star> tab\<^sub>0 g \<star> g\<^sub>0h\<^sub>1.p\<^sub>1, f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine] \<cdot> \<a>\<^sup>-\<^sup>1[g, tab\<^sub>0 g \<star> g\<^sub>0h\<^sub>1.p\<^sub>1, f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine] = (((g \<star> h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((((g \<star> g\<^sub>0h\<^sub>1.\<phi>) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> \<a>\<^sup>-\<^sup>1[g \<star> tab\<^sub>0 g \<star> g\<^sub>0h\<^sub>1.p\<^sub>1, f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine]) \<cdot> \<a>\<^sup>-\<^sup>1[g, tab\<^sub>0 g \<star> g\<^sub>0h\<^sub>1.p\<^sub>1, f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine]
goal (1 subgoal):
1. (((g \<star> h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (((g \<star> g\<^sub>0h\<^sub>1.\<phi>) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> \<a>\<^sup>-\<^sup>1[g \<star> tab\<^sub>0 g \<star> g\<^sub>0h\<^sub>1.p\<^sub>1, f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine] \<cdot> \<a>\<^sup>-\<^sup>1[g, tab\<^sub>0 g \<star> g\<^sub>0h\<^sub>1.p\<^sub>1, f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine] = \<a>\<^sup>-\<^sup>1[g \<star> (h \<star> tab\<^sub>0 h) \<star> g\<^sub>0h\<^sub>1.p\<^sub>0, f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine] \<cdot> \<a>\<^sup>-\<^sup>1[g, (h \<star> tab\<^sub>0 h) \<star> g\<^sub>0h\<^sub>1.p\<^sub>0, f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine] \<cdot> (g \<star> (h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine) \<cdot> (g \<star> g\<^sub>0h\<^sub>1.\<phi> \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine)
[PROOF STEP]
have "... = ((((g \<star> h.tab \<star> Tgh.p\<^sub>0) \<star> TfTgh.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot>
\<a>\<^sup>-\<^sup>1[g \<star> tab\<^sub>1 h \<star> Tgh.p\<^sub>0, TfTgh.p\<^sub>0, TTfgh_TfTgh.chine]) \<cdot>
((g \<star> g\<^sub>0h\<^sub>1.\<phi>) \<star> TfTgh.p\<^sub>0 \<star> TTfgh_TfTgh.chine) \<cdot>
\<a>\<^sup>-\<^sup>1[g, tab\<^sub>0 g \<star> Tgh.p\<^sub>1, TfTgh.p\<^sub>0 \<star> TTfgh_TfTgh.chine]"
[PROOF STATE]
proof (prove)
goal (1 subgoal):
1. (((g \<star> h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((((g \<star> g\<^sub>0h\<^sub>1.\<phi>) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> \<a>\<^sup>-\<^sup>1[g \<star> tab\<^sub>0 g \<star> g\<^sub>0h\<^sub>1.p\<^sub>1, f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine]) \<cdot> \<a>\<^sup>-\<^sup>1[g, tab\<^sub>0 g \<star> g\<^sub>0h\<^sub>1.p\<^sub>1, f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine] = ((((g \<star> h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> \<a>\<^sup>-\<^sup>1[g \<star> tab\<^sub>1 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0, f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine]) \<cdot> ((g \<star> g\<^sub>0h\<^sub>1.\<phi>) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine) \<cdot> \<a>\<^sup>-\<^sup>1[g, tab\<^sub>0 g \<star> g\<^sub>0h\<^sub>1.p\<^sub>1, f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine]
[PROOF STEP]
proof -
[PROOF STATE]
proof (state)
goal (1 subgoal):
1. (((g \<star> h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((((g \<star> g\<^sub>0h\<^sub>1.\<phi>) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> \<a>\<^sup>-\<^sup>1[g \<star> tab\<^sub>0 g \<star> g\<^sub>0h\<^sub>1.p\<^sub>1, f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine]) \<cdot> \<a>\<^sup>-\<^sup>1[g, tab\<^sub>0 g \<star> g\<^sub>0h\<^sub>1.p\<^sub>1, f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine] = ((((g \<star> h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> \<a>\<^sup>-\<^sup>1[g \<star> tab\<^sub>1 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0, f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine]) \<cdot> ((g \<star> g\<^sub>0h\<^sub>1.\<phi>) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine) \<cdot> \<a>\<^sup>-\<^sup>1[g, tab\<^sub>0 g \<star> g\<^sub>0h\<^sub>1.p\<^sub>1, f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine]
[PROOF STEP]
have "(((g \<star> g\<^sub>0h\<^sub>1.\<phi>) \<star> TfTgh.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot>
\<a>\<^sup>-\<^sup>1[g \<star> tab\<^sub>0 g \<star> Tgh.p\<^sub>1, TfTgh.p\<^sub>0, TTfgh_TfTgh.chine] =
\<a>\<^sup>-\<^sup>1[g \<star> tab\<^sub>1 h \<star> Tgh.p\<^sub>0, TfTgh.p\<^sub>0, TTfgh_TfTgh.chine] \<cdot>
((g \<star> g\<^sub>0h\<^sub>1.\<phi>) \<star> TfTgh.p\<^sub>0 \<star> TTfgh_TfTgh.chine)"
[PROOF STATE]
proof (prove)
goal (1 subgoal):
1. (((g \<star> g\<^sub>0h\<^sub>1.\<phi>) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> \<a>\<^sup>-\<^sup>1[g \<star> tab\<^sub>0 g \<star> g\<^sub>0h\<^sub>1.p\<^sub>1, f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine] = \<a>\<^sup>-\<^sup>1[g \<star> tab\<^sub>1 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0, f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine] \<cdot> ((g \<star> g\<^sub>0h\<^sub>1.\<phi>) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine)
[PROOF STEP]
using gh f\<^sub>0gh\<^sub>1.p\<^sub>0_simps
assoc'_naturality [of "g \<star> g\<^sub>0h\<^sub>1.\<phi>" TfTgh.p\<^sub>0 TTfgh_TfTgh.chine]
[PROOF STATE]
proof (prove)
using this:
src g = trg h
trg f\<^sub>0gh\<^sub>1.p\<^sub>0 = src (tab\<^sub>1 g \<star> g\<^sub>0h\<^sub>1.p\<^sub>1)
\<lbrakk>hseq g g\<^sub>0h\<^sub>1.\<phi>; arr f\<^sub>0gh\<^sub>1.p\<^sub>0; arr TTfgh_TfTgh.chine; src (g \<star> g\<^sub>0h\<^sub>1.\<phi>) = trg f\<^sub>0gh\<^sub>1.p\<^sub>0; src f\<^sub>0gh\<^sub>1.p\<^sub>0 = trg TTfgh_TfTgh.chine\<rbrakk> \<Longrightarrow> \<a>\<^sup>-\<^sup>1[cod (g \<star> g\<^sub>0h\<^sub>1.\<phi>), cod f\<^sub>0gh\<^sub>1.p\<^sub>0, cod TTfgh_TfTgh.chine] \<cdot> ((g \<star> g\<^sub>0h\<^sub>1.\<phi>) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine) = (((g \<star> g\<^sub>0h\<^sub>1.\<phi>) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> \<a>\<^sup>-\<^sup>1[local.dom (g \<star> g\<^sub>0h\<^sub>1.\<phi>), local.dom f\<^sub>0gh\<^sub>1.p\<^sub>0, local.dom TTfgh_TfTgh.chine]
goal (1 subgoal):
1. (((g \<star> g\<^sub>0h\<^sub>1.\<phi>) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> \<a>\<^sup>-\<^sup>1[g \<star> tab\<^sub>0 g \<star> g\<^sub>0h\<^sub>1.p\<^sub>1, f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine] = \<a>\<^sup>-\<^sup>1[g \<star> tab\<^sub>1 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0, f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine] \<cdot> ((g \<star> g\<^sub>0h\<^sub>1.\<phi>) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine)
[PROOF STEP]
by simp
[PROOF STATE]
proof (state)
this:
(((g \<star> g\<^sub>0h\<^sub>1.\<phi>) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> \<a>\<^sup>-\<^sup>1[g \<star> tab\<^sub>0 g \<star> g\<^sub>0h\<^sub>1.p\<^sub>1, f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine] = \<a>\<^sup>-\<^sup>1[g \<star> tab\<^sub>1 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0, f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine] \<cdot> ((g \<star> g\<^sub>0h\<^sub>1.\<phi>) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine)
goal (1 subgoal):
1. (((g \<star> h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((((g \<star> g\<^sub>0h\<^sub>1.\<phi>) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> \<a>\<^sup>-\<^sup>1[g \<star> tab\<^sub>0 g \<star> g\<^sub>0h\<^sub>1.p\<^sub>1, f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine]) \<cdot> \<a>\<^sup>-\<^sup>1[g, tab\<^sub>0 g \<star> g\<^sub>0h\<^sub>1.p\<^sub>1, f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine] = ((((g \<star> h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> \<a>\<^sup>-\<^sup>1[g \<star> tab\<^sub>1 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0, f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine]) \<cdot> ((g \<star> g\<^sub>0h\<^sub>1.\<phi>) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine) \<cdot> \<a>\<^sup>-\<^sup>1[g, tab\<^sub>0 g \<star> g\<^sub>0h\<^sub>1.p\<^sub>1, f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine]
[PROOF STEP]
thus ?thesis
[PROOF STATE]
proof (prove)
using this:
(((g \<star> g\<^sub>0h\<^sub>1.\<phi>) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> \<a>\<^sup>-\<^sup>1[g \<star> tab\<^sub>0 g \<star> g\<^sub>0h\<^sub>1.p\<^sub>1, f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine] = \<a>\<^sup>-\<^sup>1[g \<star> tab\<^sub>1 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0, f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine] \<cdot> ((g \<star> g\<^sub>0h\<^sub>1.\<phi>) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine)
goal (1 subgoal):
1. (((g \<star> h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((((g \<star> g\<^sub>0h\<^sub>1.\<phi>) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> \<a>\<^sup>-\<^sup>1[g \<star> tab\<^sub>0 g \<star> g\<^sub>0h\<^sub>1.p\<^sub>1, f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine]) \<cdot> \<a>\<^sup>-\<^sup>1[g, tab\<^sub>0 g \<star> g\<^sub>0h\<^sub>1.p\<^sub>1, f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine] = ((((g \<star> h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> \<a>\<^sup>-\<^sup>1[g \<star> tab\<^sub>1 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0, f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine]) \<cdot> ((g \<star> g\<^sub>0h\<^sub>1.\<phi>) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine) \<cdot> \<a>\<^sup>-\<^sup>1[g, tab\<^sub>0 g \<star> g\<^sub>0h\<^sub>1.p\<^sub>1, f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine]
[PROOF STEP]
using comp_assoc
[PROOF STATE]
proof (prove)
using this:
(((g \<star> g\<^sub>0h\<^sub>1.\<phi>) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> \<a>\<^sup>-\<^sup>1[g \<star> tab\<^sub>0 g \<star> g\<^sub>0h\<^sub>1.p\<^sub>1, f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine] = \<a>\<^sup>-\<^sup>1[g \<star> tab\<^sub>1 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0, f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine] \<cdot> ((g \<star> g\<^sub>0h\<^sub>1.\<phi>) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine)
(?h \<cdot> ?g) \<cdot> ?f = ?h \<cdot> ?g \<cdot> ?f
goal (1 subgoal):
1. (((g \<star> h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((((g \<star> g\<^sub>0h\<^sub>1.\<phi>) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> \<a>\<^sup>-\<^sup>1[g \<star> tab\<^sub>0 g \<star> g\<^sub>0h\<^sub>1.p\<^sub>1, f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine]) \<cdot> \<a>\<^sup>-\<^sup>1[g, tab\<^sub>0 g \<star> g\<^sub>0h\<^sub>1.p\<^sub>1, f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine] = ((((g \<star> h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> \<a>\<^sup>-\<^sup>1[g \<star> tab\<^sub>1 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0, f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine]) \<cdot> ((g \<star> g\<^sub>0h\<^sub>1.\<phi>) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine) \<cdot> \<a>\<^sup>-\<^sup>1[g, tab\<^sub>0 g \<star> g\<^sub>0h\<^sub>1.p\<^sub>1, f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine]
[PROOF STEP]
by presburger
[PROOF STATE]
proof (state)
this:
(((g \<star> h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((((g \<star> g\<^sub>0h\<^sub>1.\<phi>) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> \<a>\<^sup>-\<^sup>1[g \<star> tab\<^sub>0 g \<star> g\<^sub>0h\<^sub>1.p\<^sub>1, f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine]) \<cdot> \<a>\<^sup>-\<^sup>1[g, tab\<^sub>0 g \<star> g\<^sub>0h\<^sub>1.p\<^sub>1, f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine] = ((((g \<star> h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> \<a>\<^sup>-\<^sup>1[g \<star> tab\<^sub>1 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0, f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine]) \<cdot> ((g \<star> g\<^sub>0h\<^sub>1.\<phi>) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine) \<cdot> \<a>\<^sup>-\<^sup>1[g, tab\<^sub>0 g \<star> g\<^sub>0h\<^sub>1.p\<^sub>1, f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine]
goal:
No subgoals!
[PROOF STEP]
qed
[PROOF STATE]
proof (state)
this:
(((g \<star> h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((((g \<star> g\<^sub>0h\<^sub>1.\<phi>) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> \<a>\<^sup>-\<^sup>1[g \<star> tab\<^sub>0 g \<star> g\<^sub>0h\<^sub>1.p\<^sub>1, f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine]) \<cdot> \<a>\<^sup>-\<^sup>1[g, tab\<^sub>0 g \<star> g\<^sub>0h\<^sub>1.p\<^sub>1, f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine] = ((((g \<star> h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> \<a>\<^sup>-\<^sup>1[g \<star> tab\<^sub>1 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0, f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine]) \<cdot> ((g \<star> g\<^sub>0h\<^sub>1.\<phi>) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine) \<cdot> \<a>\<^sup>-\<^sup>1[g, tab\<^sub>0 g \<star> g\<^sub>0h\<^sub>1.p\<^sub>1, f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine]
goal (1 subgoal):
1. (((g \<star> h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (((g \<star> g\<^sub>0h\<^sub>1.\<phi>) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> \<a>\<^sup>-\<^sup>1[g \<star> tab\<^sub>0 g \<star> g\<^sub>0h\<^sub>1.p\<^sub>1, f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine] \<cdot> \<a>\<^sup>-\<^sup>1[g, tab\<^sub>0 g \<star> g\<^sub>0h\<^sub>1.p\<^sub>1, f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine] = \<a>\<^sup>-\<^sup>1[g \<star> (h \<star> tab\<^sub>0 h) \<star> g\<^sub>0h\<^sub>1.p\<^sub>0, f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine] \<cdot> \<a>\<^sup>-\<^sup>1[g, (h \<star> tab\<^sub>0 h) \<star> g\<^sub>0h\<^sub>1.p\<^sub>0, f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine] \<cdot> (g \<star> (h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine) \<cdot> (g \<star> g\<^sub>0h\<^sub>1.\<phi> \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine)
[PROOF STEP]
also
[PROOF STATE]
proof (state)
this:
(((g \<star> h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> ((((g \<star> g\<^sub>0h\<^sub>1.\<phi>) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> \<a>\<^sup>-\<^sup>1[g \<star> tab\<^sub>0 g \<star> g\<^sub>0h\<^sub>1.p\<^sub>1, f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine]) \<cdot> \<a>\<^sup>-\<^sup>1[g, tab\<^sub>0 g \<star> g\<^sub>0h\<^sub>1.p\<^sub>1, f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine] = ((((g \<star> h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> \<a>\<^sup>-\<^sup>1[g \<star> tab\<^sub>1 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0, f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine]) \<cdot> ((g \<star> g\<^sub>0h\<^sub>1.\<phi>) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine) \<cdot> \<a>\<^sup>-\<^sup>1[g, tab\<^sub>0 g \<star> g\<^sub>0h\<^sub>1.p\<^sub>1, f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine]
goal (1 subgoal):
1. (((g \<star> h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (((g \<star> g\<^sub>0h\<^sub>1.\<phi>) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> \<a>\<^sup>-\<^sup>1[g \<star> tab\<^sub>0 g \<star> g\<^sub>0h\<^sub>1.p\<^sub>1, f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine] \<cdot> \<a>\<^sup>-\<^sup>1[g, tab\<^sub>0 g \<star> g\<^sub>0h\<^sub>1.p\<^sub>1, f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine] = \<a>\<^sup>-\<^sup>1[g \<star> (h \<star> tab\<^sub>0 h) \<star> g\<^sub>0h\<^sub>1.p\<^sub>0, f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine] \<cdot> \<a>\<^sup>-\<^sup>1[g, (h \<star> tab\<^sub>0 h) \<star> g\<^sub>0h\<^sub>1.p\<^sub>0, f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine] \<cdot> (g \<star> (h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine) \<cdot> (g \<star> g\<^sub>0h\<^sub>1.\<phi> \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine)
[PROOF STEP]
have "... = \<a>\<^sup>-\<^sup>1[g \<star> (h \<star> tab\<^sub>0 h) \<star> Tgh.p\<^sub>0, TfTgh.p\<^sub>0, TTfgh_TfTgh.chine] \<cdot>
((g \<star> h.tab \<star> Tgh.p\<^sub>0) \<star> TfTgh.p\<^sub>0 \<star> TTfgh_TfTgh.chine) \<cdot>
((g \<star> g\<^sub>0h\<^sub>1.\<phi>) \<star> TfTgh.p\<^sub>0 \<star> TTfgh_TfTgh.chine) \<cdot>
\<a>\<^sup>-\<^sup>1[g, tab\<^sub>0 g \<star> Tgh.p\<^sub>1, TfTgh.p\<^sub>0 \<star> TTfgh_TfTgh.chine]"
[PROOF STATE]
proof (prove)
goal (1 subgoal):
1. ((((g \<star> h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> \<a>\<^sup>-\<^sup>1[g \<star> tab\<^sub>1 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0, f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine]) \<cdot> ((g \<star> g\<^sub>0h\<^sub>1.\<phi>) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine) \<cdot> \<a>\<^sup>-\<^sup>1[g, tab\<^sub>0 g \<star> g\<^sub>0h\<^sub>1.p\<^sub>1, f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine] = \<a>\<^sup>-\<^sup>1[g \<star> (h \<star> tab\<^sub>0 h) \<star> g\<^sub>0h\<^sub>1.p\<^sub>0, f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine] \<cdot> ((g \<star> h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine) \<cdot> ((g \<star> g\<^sub>0h\<^sub>1.\<phi>) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine) \<cdot> \<a>\<^sup>-\<^sup>1[g, tab\<^sub>0 g \<star> g\<^sub>0h\<^sub>1.p\<^sub>1, f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine]
[PROOF STEP]
proof -
[PROOF STATE]
proof (state)
goal (1 subgoal):
1. ((((g \<star> h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> \<a>\<^sup>-\<^sup>1[g \<star> tab\<^sub>1 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0, f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine]) \<cdot> ((g \<star> g\<^sub>0h\<^sub>1.\<phi>) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine) \<cdot> \<a>\<^sup>-\<^sup>1[g, tab\<^sub>0 g \<star> g\<^sub>0h\<^sub>1.p\<^sub>1, f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine] = \<a>\<^sup>-\<^sup>1[g \<star> (h \<star> tab\<^sub>0 h) \<star> g\<^sub>0h\<^sub>1.p\<^sub>0, f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine] \<cdot> ((g \<star> h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine) \<cdot> ((g \<star> g\<^sub>0h\<^sub>1.\<phi>) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine) \<cdot> \<a>\<^sup>-\<^sup>1[g, tab\<^sub>0 g \<star> g\<^sub>0h\<^sub>1.p\<^sub>1, f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine]
[PROOF STEP]
have "(((g \<star> h.tab \<star> Tgh.p\<^sub>0) \<star> TfTgh.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot>
\<a>\<^sup>-\<^sup>1[g \<star> tab\<^sub>1 h \<star> Tgh.p\<^sub>0, TfTgh.p\<^sub>0, TTfgh_TfTgh.chine] =
\<a>\<^sup>-\<^sup>1[g \<star> (h \<star> tab\<^sub>0 h) \<star> Tgh.p\<^sub>0, TfTgh.p\<^sub>0, TTfgh_TfTgh.chine] \<cdot>
((g \<star> h.tab \<star> Tgh.p\<^sub>0) \<star> TfTgh.p\<^sub>0 \<star> TTfgh_TfTgh.chine)"
[PROOF STATE]
proof (prove)
goal (1 subgoal):
1. (((g \<star> h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> \<a>\<^sup>-\<^sup>1[g \<star> tab\<^sub>1 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0, f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine] = \<a>\<^sup>-\<^sup>1[g \<star> (h \<star> tab\<^sub>0 h) \<star> g\<^sub>0h\<^sub>1.p\<^sub>0, f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine] \<cdot> ((g \<star> h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine)
[PROOF STEP]
using gh f\<^sub>0gh\<^sub>1.p\<^sub>0_simps
assoc'_naturality [of "g \<star> h.tab \<star> Tgh.p\<^sub>0" TfTgh.p\<^sub>0 TTfgh_TfTgh.chine]
[PROOF STATE]
proof (prove)
using this:
src g = trg h
trg f\<^sub>0gh\<^sub>1.p\<^sub>0 = src (tab\<^sub>1 g \<star> g\<^sub>0h\<^sub>1.p\<^sub>1)
\<lbrakk>hseq g (h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0); arr f\<^sub>0gh\<^sub>1.p\<^sub>0; arr TTfgh_TfTgh.chine; src (g \<star> h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) = trg f\<^sub>0gh\<^sub>1.p\<^sub>0; src f\<^sub>0gh\<^sub>1.p\<^sub>0 = trg TTfgh_TfTgh.chine\<rbrakk> \<Longrightarrow> \<a>\<^sup>-\<^sup>1[cod (g \<star> h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0), cod f\<^sub>0gh\<^sub>1.p\<^sub>0, cod TTfgh_TfTgh.chine] \<cdot> ((g \<star> h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine) = (((g \<star> h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> \<a>\<^sup>-\<^sup>1[local.dom (g \<star> h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0), local.dom f\<^sub>0gh\<^sub>1.p\<^sub>0, local.dom TTfgh_TfTgh.chine]
goal (1 subgoal):
1. (((g \<star> h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> \<a>\<^sup>-\<^sup>1[g \<star> tab\<^sub>1 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0, f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine] = \<a>\<^sup>-\<^sup>1[g \<star> (h \<star> tab\<^sub>0 h) \<star> g\<^sub>0h\<^sub>1.p\<^sub>0, f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine] \<cdot> ((g \<star> h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine)
[PROOF STEP]
by simp
[PROOF STATE]
proof (state)
this:
(((g \<star> h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> \<a>\<^sup>-\<^sup>1[g \<star> tab\<^sub>1 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0, f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine] = \<a>\<^sup>-\<^sup>1[g \<star> (h \<star> tab\<^sub>0 h) \<star> g\<^sub>0h\<^sub>1.p\<^sub>0, f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine] \<cdot> ((g \<star> h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine)
goal (1 subgoal):
1. ((((g \<star> h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> \<a>\<^sup>-\<^sup>1[g \<star> tab\<^sub>1 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0, f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine]) \<cdot> ((g \<star> g\<^sub>0h\<^sub>1.\<phi>) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine) \<cdot> \<a>\<^sup>-\<^sup>1[g, tab\<^sub>0 g \<star> g\<^sub>0h\<^sub>1.p\<^sub>1, f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine] = \<a>\<^sup>-\<^sup>1[g \<star> (h \<star> tab\<^sub>0 h) \<star> g\<^sub>0h\<^sub>1.p\<^sub>0, f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine] \<cdot> ((g \<star> h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine) \<cdot> ((g \<star> g\<^sub>0h\<^sub>1.\<phi>) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine) \<cdot> \<a>\<^sup>-\<^sup>1[g, tab\<^sub>0 g \<star> g\<^sub>0h\<^sub>1.p\<^sub>1, f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine]
[PROOF STEP]
thus ?thesis
[PROOF STATE]
proof (prove)
using this:
(((g \<star> h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> \<a>\<^sup>-\<^sup>1[g \<star> tab\<^sub>1 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0, f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine] = \<a>\<^sup>-\<^sup>1[g \<star> (h \<star> tab\<^sub>0 h) \<star> g\<^sub>0h\<^sub>1.p\<^sub>0, f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine] \<cdot> ((g \<star> h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine)
goal (1 subgoal):
1. ((((g \<star> h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> \<a>\<^sup>-\<^sup>1[g \<star> tab\<^sub>1 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0, f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine]) \<cdot> ((g \<star> g\<^sub>0h\<^sub>1.\<phi>) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine) \<cdot> \<a>\<^sup>-\<^sup>1[g, tab\<^sub>0 g \<star> g\<^sub>0h\<^sub>1.p\<^sub>1, f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine] = \<a>\<^sup>-\<^sup>1[g \<star> (h \<star> tab\<^sub>0 h) \<star> g\<^sub>0h\<^sub>1.p\<^sub>0, f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine] \<cdot> ((g \<star> h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine) \<cdot> ((g \<star> g\<^sub>0h\<^sub>1.\<phi>) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine) \<cdot> \<a>\<^sup>-\<^sup>1[g, tab\<^sub>0 g \<star> g\<^sub>0h\<^sub>1.p\<^sub>1, f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine]
[PROOF STEP]
using comp_assoc
[PROOF STATE]
proof (prove)
using this:
(((g \<star> h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> \<a>\<^sup>-\<^sup>1[g \<star> tab\<^sub>1 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0, f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine] = \<a>\<^sup>-\<^sup>1[g \<star> (h \<star> tab\<^sub>0 h) \<star> g\<^sub>0h\<^sub>1.p\<^sub>0, f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine] \<cdot> ((g \<star> h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine)
(?h \<cdot> ?g) \<cdot> ?f = ?h \<cdot> ?g \<cdot> ?f
goal (1 subgoal):
1. ((((g \<star> h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> \<a>\<^sup>-\<^sup>1[g \<star> tab\<^sub>1 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0, f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine]) \<cdot> ((g \<star> g\<^sub>0h\<^sub>1.\<phi>) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine) \<cdot> \<a>\<^sup>-\<^sup>1[g, tab\<^sub>0 g \<star> g\<^sub>0h\<^sub>1.p\<^sub>1, f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine] = \<a>\<^sup>-\<^sup>1[g \<star> (h \<star> tab\<^sub>0 h) \<star> g\<^sub>0h\<^sub>1.p\<^sub>0, f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine] \<cdot> ((g \<star> h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine) \<cdot> ((g \<star> g\<^sub>0h\<^sub>1.\<phi>) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine) \<cdot> \<a>\<^sup>-\<^sup>1[g, tab\<^sub>0 g \<star> g\<^sub>0h\<^sub>1.p\<^sub>1, f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine]
[PROOF STEP]
by presburger
[PROOF STATE]
proof (state)
this:
((((g \<star> h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> \<a>\<^sup>-\<^sup>1[g \<star> tab\<^sub>1 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0, f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine]) \<cdot> ((g \<star> g\<^sub>0h\<^sub>1.\<phi>) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine) \<cdot> \<a>\<^sup>-\<^sup>1[g, tab\<^sub>0 g \<star> g\<^sub>0h\<^sub>1.p\<^sub>1, f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine] = \<a>\<^sup>-\<^sup>1[g \<star> (h \<star> tab\<^sub>0 h) \<star> g\<^sub>0h\<^sub>1.p\<^sub>0, f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine] \<cdot> ((g \<star> h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine) \<cdot> ((g \<star> g\<^sub>0h\<^sub>1.\<phi>) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine) \<cdot> \<a>\<^sup>-\<^sup>1[g, tab\<^sub>0 g \<star> g\<^sub>0h\<^sub>1.p\<^sub>1, f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine]
goal:
No subgoals!
[PROOF STEP]
qed
[PROOF STATE]
proof (state)
this:
((((g \<star> h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> \<a>\<^sup>-\<^sup>1[g \<star> tab\<^sub>1 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0, f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine]) \<cdot> ((g \<star> g\<^sub>0h\<^sub>1.\<phi>) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine) \<cdot> \<a>\<^sup>-\<^sup>1[g, tab\<^sub>0 g \<star> g\<^sub>0h\<^sub>1.p\<^sub>1, f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine] = \<a>\<^sup>-\<^sup>1[g \<star> (h \<star> tab\<^sub>0 h) \<star> g\<^sub>0h\<^sub>1.p\<^sub>0, f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine] \<cdot> ((g \<star> h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine) \<cdot> ((g \<star> g\<^sub>0h\<^sub>1.\<phi>) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine) \<cdot> \<a>\<^sup>-\<^sup>1[g, tab\<^sub>0 g \<star> g\<^sub>0h\<^sub>1.p\<^sub>1, f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine]
goal (1 subgoal):
1. (((g \<star> h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (((g \<star> g\<^sub>0h\<^sub>1.\<phi>) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> \<a>\<^sup>-\<^sup>1[g \<star> tab\<^sub>0 g \<star> g\<^sub>0h\<^sub>1.p\<^sub>1, f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine] \<cdot> \<a>\<^sup>-\<^sup>1[g, tab\<^sub>0 g \<star> g\<^sub>0h\<^sub>1.p\<^sub>1, f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine] = \<a>\<^sup>-\<^sup>1[g \<star> (h \<star> tab\<^sub>0 h) \<star> g\<^sub>0h\<^sub>1.p\<^sub>0, f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine] \<cdot> \<a>\<^sup>-\<^sup>1[g, (h \<star> tab\<^sub>0 h) \<star> g\<^sub>0h\<^sub>1.p\<^sub>0, f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine] \<cdot> (g \<star> (h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine) \<cdot> (g \<star> g\<^sub>0h\<^sub>1.\<phi> \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine)
[PROOF STEP]
also
[PROOF STATE]
proof (state)
this:
((((g \<star> h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> \<a>\<^sup>-\<^sup>1[g \<star> tab\<^sub>1 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0, f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine]) \<cdot> ((g \<star> g\<^sub>0h\<^sub>1.\<phi>) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine) \<cdot> \<a>\<^sup>-\<^sup>1[g, tab\<^sub>0 g \<star> g\<^sub>0h\<^sub>1.p\<^sub>1, f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine] = \<a>\<^sup>-\<^sup>1[g \<star> (h \<star> tab\<^sub>0 h) \<star> g\<^sub>0h\<^sub>1.p\<^sub>0, f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine] \<cdot> ((g \<star> h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine) \<cdot> ((g \<star> g\<^sub>0h\<^sub>1.\<phi>) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine) \<cdot> \<a>\<^sup>-\<^sup>1[g, tab\<^sub>0 g \<star> g\<^sub>0h\<^sub>1.p\<^sub>1, f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine]
goal (1 subgoal):
1. (((g \<star> h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (((g \<star> g\<^sub>0h\<^sub>1.\<phi>) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> \<a>\<^sup>-\<^sup>1[g \<star> tab\<^sub>0 g \<star> g\<^sub>0h\<^sub>1.p\<^sub>1, f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine] \<cdot> \<a>\<^sup>-\<^sup>1[g, tab\<^sub>0 g \<star> g\<^sub>0h\<^sub>1.p\<^sub>1, f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine] = \<a>\<^sup>-\<^sup>1[g \<star> (h \<star> tab\<^sub>0 h) \<star> g\<^sub>0h\<^sub>1.p\<^sub>0, f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine] \<cdot> \<a>\<^sup>-\<^sup>1[g, (h \<star> tab\<^sub>0 h) \<star> g\<^sub>0h\<^sub>1.p\<^sub>0, f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine] \<cdot> (g \<star> (h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine) \<cdot> (g \<star> g\<^sub>0h\<^sub>1.\<phi> \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine)
[PROOF STEP]
have "... = \<a>\<^sup>-\<^sup>1[g \<star> (h \<star> tab\<^sub>0 h) \<star> Tgh.p\<^sub>0, TfTgh.p\<^sub>0, TTfgh_TfTgh.chine] \<cdot>
(((g \<star> h.tab \<star> Tgh.p\<^sub>0) \<star> TfTgh.p\<^sub>0 \<star> TTfgh_TfTgh.chine) \<cdot>
\<a>\<^sup>-\<^sup>1[g, tab\<^sub>1 h \<star> Tgh.p\<^sub>0, TfTgh.p\<^sub>0 \<star> TTfgh_TfTgh.chine]) \<cdot>
(g \<star> g\<^sub>0h\<^sub>1.\<phi> \<star> TfTgh.p\<^sub>0 \<star> TTfgh_TfTgh.chine)"
[PROOF STATE]
proof (prove)
goal (1 subgoal):
1. \<a>\<^sup>-\<^sup>1[g \<star> (h \<star> tab\<^sub>0 h) \<star> g\<^sub>0h\<^sub>1.p\<^sub>0, f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine] \<cdot> ((g \<star> h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine) \<cdot> ((g \<star> g\<^sub>0h\<^sub>1.\<phi>) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine) \<cdot> \<a>\<^sup>-\<^sup>1[g, tab\<^sub>0 g \<star> g\<^sub>0h\<^sub>1.p\<^sub>1, f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine] = \<a>\<^sup>-\<^sup>1[g \<star> (h \<star> tab\<^sub>0 h) \<star> g\<^sub>0h\<^sub>1.p\<^sub>0, f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine] \<cdot> (((g \<star> h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine) \<cdot> \<a>\<^sup>-\<^sup>1[g, tab\<^sub>1 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0, f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine]) \<cdot> (g \<star> g\<^sub>0h\<^sub>1.\<phi> \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine)
[PROOF STEP]
proof -
[PROOF STATE]
proof (state)
goal (1 subgoal):
1. \<a>\<^sup>-\<^sup>1[g \<star> (h \<star> tab\<^sub>0 h) \<star> g\<^sub>0h\<^sub>1.p\<^sub>0, f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine] \<cdot> ((g \<star> h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine) \<cdot> ((g \<star> g\<^sub>0h\<^sub>1.\<phi>) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine) \<cdot> \<a>\<^sup>-\<^sup>1[g, tab\<^sub>0 g \<star> g\<^sub>0h\<^sub>1.p\<^sub>1, f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine] = \<a>\<^sup>-\<^sup>1[g \<star> (h \<star> tab\<^sub>0 h) \<star> g\<^sub>0h\<^sub>1.p\<^sub>0, f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine] \<cdot> (((g \<star> h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine) \<cdot> \<a>\<^sup>-\<^sup>1[g, tab\<^sub>1 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0, f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine]) \<cdot> (g \<star> g\<^sub>0h\<^sub>1.\<phi> \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine)
[PROOF STEP]
have "((g \<star> g\<^sub>0h\<^sub>1.\<phi>) \<star> TfTgh.p\<^sub>0 \<star> TTfgh_TfTgh.chine) \<cdot>
\<a>\<^sup>-\<^sup>1[g, tab\<^sub>0 g \<star> Tgh.p\<^sub>1, TfTgh.p\<^sub>0 \<star> TTfgh_TfTgh.chine] =
\<a>\<^sup>-\<^sup>1[g, tab\<^sub>1 h \<star> Tgh.p\<^sub>0, TfTgh.p\<^sub>0 \<star> TTfgh_TfTgh.chine] \<cdot>
(g \<star> g\<^sub>0h\<^sub>1.\<phi> \<star> TfTgh.p\<^sub>0 \<star> TTfgh_TfTgh.chine)"
[PROOF STATE]
proof (prove)
goal (1 subgoal):
1. ((g \<star> g\<^sub>0h\<^sub>1.\<phi>) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine) \<cdot> \<a>\<^sup>-\<^sup>1[g, tab\<^sub>0 g \<star> g\<^sub>0h\<^sub>1.p\<^sub>1, f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine] = \<a>\<^sup>-\<^sup>1[g, tab\<^sub>1 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0, f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine] \<cdot> (g \<star> g\<^sub>0h\<^sub>1.\<phi> \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine)
[PROOF STEP]
using gh f\<^sub>0gh\<^sub>1.p\<^sub>0_simps
assoc'_naturality [of g g\<^sub>0h\<^sub>1.\<phi> "TfTgh.p\<^sub>0 \<star> TTfgh_TfTgh.chine"]
[PROOF STATE]
proof (prove)
using this:
src g = trg h
trg f\<^sub>0gh\<^sub>1.p\<^sub>0 = src (tab\<^sub>1 g \<star> g\<^sub>0h\<^sub>1.p\<^sub>1)
\<lbrakk>arr g; arr g\<^sub>0h\<^sub>1.\<phi>; hseq f\<^sub>0gh\<^sub>1.p\<^sub>0 TTfgh_TfTgh.chine; src g = trg g\<^sub>0h\<^sub>1.\<phi>; src g\<^sub>0h\<^sub>1.\<phi> = trg (f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine)\<rbrakk> \<Longrightarrow> \<a>\<^sup>-\<^sup>1[cod g, cod g\<^sub>0h\<^sub>1.\<phi>, cod (f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine)] \<cdot> (g \<star> g\<^sub>0h\<^sub>1.\<phi> \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine) = ((g \<star> g\<^sub>0h\<^sub>1.\<phi>) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine) \<cdot> \<a>\<^sup>-\<^sup>1[local.dom g, local.dom g\<^sub>0h\<^sub>1.\<phi>, local.dom (f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine)]
goal (1 subgoal):
1. ((g \<star> g\<^sub>0h\<^sub>1.\<phi>) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine) \<cdot> \<a>\<^sup>-\<^sup>1[g, tab\<^sub>0 g \<star> g\<^sub>0h\<^sub>1.p\<^sub>1, f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine] = \<a>\<^sup>-\<^sup>1[g, tab\<^sub>1 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0, f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine] \<cdot> (g \<star> g\<^sub>0h\<^sub>1.\<phi> \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine)
[PROOF STEP]
by simp
[PROOF STATE]
proof (state)
this:
((g \<star> g\<^sub>0h\<^sub>1.\<phi>) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine) \<cdot> \<a>\<^sup>-\<^sup>1[g, tab\<^sub>0 g \<star> g\<^sub>0h\<^sub>1.p\<^sub>1, f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine] = \<a>\<^sup>-\<^sup>1[g, tab\<^sub>1 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0, f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine] \<cdot> (g \<star> g\<^sub>0h\<^sub>1.\<phi> \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine)
goal (1 subgoal):
1. \<a>\<^sup>-\<^sup>1[g \<star> (h \<star> tab\<^sub>0 h) \<star> g\<^sub>0h\<^sub>1.p\<^sub>0, f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine] \<cdot> ((g \<star> h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine) \<cdot> ((g \<star> g\<^sub>0h\<^sub>1.\<phi>) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine) \<cdot> \<a>\<^sup>-\<^sup>1[g, tab\<^sub>0 g \<star> g\<^sub>0h\<^sub>1.p\<^sub>1, f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine] = \<a>\<^sup>-\<^sup>1[g \<star> (h \<star> tab\<^sub>0 h) \<star> g\<^sub>0h\<^sub>1.p\<^sub>0, f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine] \<cdot> (((g \<star> h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine) \<cdot> \<a>\<^sup>-\<^sup>1[g, tab\<^sub>1 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0, f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine]) \<cdot> (g \<star> g\<^sub>0h\<^sub>1.\<phi> \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine)
[PROOF STEP]
thus ?thesis
[PROOF STATE]
proof (prove)
using this:
((g \<star> g\<^sub>0h\<^sub>1.\<phi>) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine) \<cdot> \<a>\<^sup>-\<^sup>1[g, tab\<^sub>0 g \<star> g\<^sub>0h\<^sub>1.p\<^sub>1, f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine] = \<a>\<^sup>-\<^sup>1[g, tab\<^sub>1 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0, f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine] \<cdot> (g \<star> g\<^sub>0h\<^sub>1.\<phi> \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine)
goal (1 subgoal):
1. \<a>\<^sup>-\<^sup>1[g \<star> (h \<star> tab\<^sub>0 h) \<star> g\<^sub>0h\<^sub>1.p\<^sub>0, f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine] \<cdot> ((g \<star> h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine) \<cdot> ((g \<star> g\<^sub>0h\<^sub>1.\<phi>) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine) \<cdot> \<a>\<^sup>-\<^sup>1[g, tab\<^sub>0 g \<star> g\<^sub>0h\<^sub>1.p\<^sub>1, f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine] = \<a>\<^sup>-\<^sup>1[g \<star> (h \<star> tab\<^sub>0 h) \<star> g\<^sub>0h\<^sub>1.p\<^sub>0, f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine] \<cdot> (((g \<star> h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine) \<cdot> \<a>\<^sup>-\<^sup>1[g, tab\<^sub>1 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0, f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine]) \<cdot> (g \<star> g\<^sub>0h\<^sub>1.\<phi> \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine)
[PROOF STEP]
using comp_assoc
[PROOF STATE]
proof (prove)
using this:
((g \<star> g\<^sub>0h\<^sub>1.\<phi>) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine) \<cdot> \<a>\<^sup>-\<^sup>1[g, tab\<^sub>0 g \<star> g\<^sub>0h\<^sub>1.p\<^sub>1, f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine] = \<a>\<^sup>-\<^sup>1[g, tab\<^sub>1 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0, f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine] \<cdot> (g \<star> g\<^sub>0h\<^sub>1.\<phi> \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine)
(?h \<cdot> ?g) \<cdot> ?f = ?h \<cdot> ?g \<cdot> ?f
goal (1 subgoal):
1. \<a>\<^sup>-\<^sup>1[g \<star> (h \<star> tab\<^sub>0 h) \<star> g\<^sub>0h\<^sub>1.p\<^sub>0, f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine] \<cdot> ((g \<star> h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine) \<cdot> ((g \<star> g\<^sub>0h\<^sub>1.\<phi>) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine) \<cdot> \<a>\<^sup>-\<^sup>1[g, tab\<^sub>0 g \<star> g\<^sub>0h\<^sub>1.p\<^sub>1, f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine] = \<a>\<^sup>-\<^sup>1[g \<star> (h \<star> tab\<^sub>0 h) \<star> g\<^sub>0h\<^sub>1.p\<^sub>0, f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine] \<cdot> (((g \<star> h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine) \<cdot> \<a>\<^sup>-\<^sup>1[g, tab\<^sub>1 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0, f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine]) \<cdot> (g \<star> g\<^sub>0h\<^sub>1.\<phi> \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine)
[PROOF STEP]
by presburger
[PROOF STATE]
proof (state)
this:
\<a>\<^sup>-\<^sup>1[g \<star> (h \<star> tab\<^sub>0 h) \<star> g\<^sub>0h\<^sub>1.p\<^sub>0, f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine] \<cdot> ((g \<star> h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine) \<cdot> ((g \<star> g\<^sub>0h\<^sub>1.\<phi>) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine) \<cdot> \<a>\<^sup>-\<^sup>1[g, tab\<^sub>0 g \<star> g\<^sub>0h\<^sub>1.p\<^sub>1, f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine] = \<a>\<^sup>-\<^sup>1[g \<star> (h \<star> tab\<^sub>0 h) \<star> g\<^sub>0h\<^sub>1.p\<^sub>0, f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine] \<cdot> (((g \<star> h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine) \<cdot> \<a>\<^sup>-\<^sup>1[g, tab\<^sub>1 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0, f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine]) \<cdot> (g \<star> g\<^sub>0h\<^sub>1.\<phi> \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine)
goal:
No subgoals!
[PROOF STEP]
qed
[PROOF STATE]
proof (state)
this:
\<a>\<^sup>-\<^sup>1[g \<star> (h \<star> tab\<^sub>0 h) \<star> g\<^sub>0h\<^sub>1.p\<^sub>0, f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine] \<cdot> ((g \<star> h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine) \<cdot> ((g \<star> g\<^sub>0h\<^sub>1.\<phi>) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine) \<cdot> \<a>\<^sup>-\<^sup>1[g, tab\<^sub>0 g \<star> g\<^sub>0h\<^sub>1.p\<^sub>1, f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine] = \<a>\<^sup>-\<^sup>1[g \<star> (h \<star> tab\<^sub>0 h) \<star> g\<^sub>0h\<^sub>1.p\<^sub>0, f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine] \<cdot> (((g \<star> h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine) \<cdot> \<a>\<^sup>-\<^sup>1[g, tab\<^sub>1 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0, f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine]) \<cdot> (g \<star> g\<^sub>0h\<^sub>1.\<phi> \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine)
goal (1 subgoal):
1. (((g \<star> h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (((g \<star> g\<^sub>0h\<^sub>1.\<phi>) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> \<a>\<^sup>-\<^sup>1[g \<star> tab\<^sub>0 g \<star> g\<^sub>0h\<^sub>1.p\<^sub>1, f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine] \<cdot> \<a>\<^sup>-\<^sup>1[g, tab\<^sub>0 g \<star> g\<^sub>0h\<^sub>1.p\<^sub>1, f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine] = \<a>\<^sup>-\<^sup>1[g \<star> (h \<star> tab\<^sub>0 h) \<star> g\<^sub>0h\<^sub>1.p\<^sub>0, f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine] \<cdot> \<a>\<^sup>-\<^sup>1[g, (h \<star> tab\<^sub>0 h) \<star> g\<^sub>0h\<^sub>1.p\<^sub>0, f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine] \<cdot> (g \<star> (h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine) \<cdot> (g \<star> g\<^sub>0h\<^sub>1.\<phi> \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine)
[PROOF STEP]
also
[PROOF STATE]
proof (state)
this:
\<a>\<^sup>-\<^sup>1[g \<star> (h \<star> tab\<^sub>0 h) \<star> g\<^sub>0h\<^sub>1.p\<^sub>0, f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine] \<cdot> ((g \<star> h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine) \<cdot> ((g \<star> g\<^sub>0h\<^sub>1.\<phi>) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine) \<cdot> \<a>\<^sup>-\<^sup>1[g, tab\<^sub>0 g \<star> g\<^sub>0h\<^sub>1.p\<^sub>1, f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine] = \<a>\<^sup>-\<^sup>1[g \<star> (h \<star> tab\<^sub>0 h) \<star> g\<^sub>0h\<^sub>1.p\<^sub>0, f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine] \<cdot> (((g \<star> h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine) \<cdot> \<a>\<^sup>-\<^sup>1[g, tab\<^sub>1 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0, f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine]) \<cdot> (g \<star> g\<^sub>0h\<^sub>1.\<phi> \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine)
goal (1 subgoal):
1. (((g \<star> h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (((g \<star> g\<^sub>0h\<^sub>1.\<phi>) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> \<a>\<^sup>-\<^sup>1[g \<star> tab\<^sub>0 g \<star> g\<^sub>0h\<^sub>1.p\<^sub>1, f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine] \<cdot> \<a>\<^sup>-\<^sup>1[g, tab\<^sub>0 g \<star> g\<^sub>0h\<^sub>1.p\<^sub>1, f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine] = \<a>\<^sup>-\<^sup>1[g \<star> (h \<star> tab\<^sub>0 h) \<star> g\<^sub>0h\<^sub>1.p\<^sub>0, f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine] \<cdot> \<a>\<^sup>-\<^sup>1[g, (h \<star> tab\<^sub>0 h) \<star> g\<^sub>0h\<^sub>1.p\<^sub>0, f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine] \<cdot> (g \<star> (h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine) \<cdot> (g \<star> g\<^sub>0h\<^sub>1.\<phi> \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine)
[PROOF STEP]
have "... = \<a>\<^sup>-\<^sup>1[g \<star> (h \<star> tab\<^sub>0 h) \<star> Tgh.p\<^sub>0, TfTgh.p\<^sub>0, TTfgh_TfTgh.chine] \<cdot>
\<a>\<^sup>-\<^sup>1[g, (h \<star> tab\<^sub>0 h) \<star> Tgh.p\<^sub>0, TfTgh.p\<^sub>0 \<star> TTfgh_TfTgh.chine] \<cdot>
(g \<star> (h.tab \<star> Tgh.p\<^sub>0) \<star> TfTgh.p\<^sub>0 \<star> TTfgh_TfTgh.chine) \<cdot>
(g \<star> g\<^sub>0h\<^sub>1.\<phi> \<star> TfTgh.p\<^sub>0 \<star> TTfgh_TfTgh.chine)"
[PROOF STATE]
proof (prove)
goal (1 subgoal):
1. \<a>\<^sup>-\<^sup>1[g \<star> (h \<star> tab\<^sub>0 h) \<star> g\<^sub>0h\<^sub>1.p\<^sub>0, f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine] \<cdot> (((g \<star> h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine) \<cdot> \<a>\<^sup>-\<^sup>1[g, tab\<^sub>1 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0, f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine]) \<cdot> (g \<star> g\<^sub>0h\<^sub>1.\<phi> \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine) = \<a>\<^sup>-\<^sup>1[g \<star> (h \<star> tab\<^sub>0 h) \<star> g\<^sub>0h\<^sub>1.p\<^sub>0, f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine] \<cdot> \<a>\<^sup>-\<^sup>1[g, (h \<star> tab\<^sub>0 h) \<star> g\<^sub>0h\<^sub>1.p\<^sub>0, f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine] \<cdot> (g \<star> (h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine) \<cdot> (g \<star> g\<^sub>0h\<^sub>1.\<phi> \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine)
[PROOF STEP]
proof -
[PROOF STATE]
proof (state)
goal (1 subgoal):
1. \<a>\<^sup>-\<^sup>1[g \<star> (h \<star> tab\<^sub>0 h) \<star> g\<^sub>0h\<^sub>1.p\<^sub>0, f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine] \<cdot> (((g \<star> h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine) \<cdot> \<a>\<^sup>-\<^sup>1[g, tab\<^sub>1 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0, f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine]) \<cdot> (g \<star> g\<^sub>0h\<^sub>1.\<phi> \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine) = \<a>\<^sup>-\<^sup>1[g \<star> (h \<star> tab\<^sub>0 h) \<star> g\<^sub>0h\<^sub>1.p\<^sub>0, f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine] \<cdot> \<a>\<^sup>-\<^sup>1[g, (h \<star> tab\<^sub>0 h) \<star> g\<^sub>0h\<^sub>1.p\<^sub>0, f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine] \<cdot> (g \<star> (h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine) \<cdot> (g \<star> g\<^sub>0h\<^sub>1.\<phi> \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine)
[PROOF STEP]
have "((g \<star> h.tab \<star> Tgh.p\<^sub>0) \<star> TfTgh.p\<^sub>0 \<star> TTfgh_TfTgh.chine) \<cdot>
\<a>\<^sup>-\<^sup>1[g, tab\<^sub>1 h \<star> Tgh.p\<^sub>0, TfTgh.p\<^sub>0 \<star> TTfgh_TfTgh.chine] =
\<a>\<^sup>-\<^sup>1[g, (h \<star> tab\<^sub>0 h) \<star> Tgh.p\<^sub>0, TfTgh.p\<^sub>0 \<star> TTfgh_TfTgh.chine] \<cdot>
(g \<star> (h.tab \<star> Tgh.p\<^sub>0) \<star> TfTgh.p\<^sub>0 \<star> TTfgh_TfTgh.chine)"
[PROOF STATE]
proof (prove)
goal (1 subgoal):
1. ((g \<star> h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine) \<cdot> \<a>\<^sup>-\<^sup>1[g, tab\<^sub>1 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0, f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine] = \<a>\<^sup>-\<^sup>1[g, (h \<star> tab\<^sub>0 h) \<star> g\<^sub>0h\<^sub>1.p\<^sub>0, f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine] \<cdot> (g \<star> (h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine)
[PROOF STEP]
using gh f\<^sub>0gh\<^sub>1.p\<^sub>0_simps
assoc'_naturality [of g "h.tab \<star> Tgh.p\<^sub>0" "TfTgh.p\<^sub>0 \<star> TTfgh_TfTgh.chine"]
[PROOF STATE]
proof (prove)
using this:
src g = trg h
trg f\<^sub>0gh\<^sub>1.p\<^sub>0 = src (tab\<^sub>1 g \<star> g\<^sub>0h\<^sub>1.p\<^sub>1)
\<lbrakk>arr g; hseq h.tab g\<^sub>0h\<^sub>1.p\<^sub>0; hseq f\<^sub>0gh\<^sub>1.p\<^sub>0 TTfgh_TfTgh.chine; src g = trg (h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0); src (h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) = trg (f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine)\<rbrakk> \<Longrightarrow> \<a>\<^sup>-\<^sup>1[cod g, cod (h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0), cod (f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine)] \<cdot> (g \<star> (h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine) = ((g \<star> h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine) \<cdot> \<a>\<^sup>-\<^sup>1[local.dom g, local.dom (h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0), local.dom (f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine)]
goal (1 subgoal):
1. ((g \<star> h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine) \<cdot> \<a>\<^sup>-\<^sup>1[g, tab\<^sub>1 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0, f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine] = \<a>\<^sup>-\<^sup>1[g, (h \<star> tab\<^sub>0 h) \<star> g\<^sub>0h\<^sub>1.p\<^sub>0, f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine] \<cdot> (g \<star> (h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine)
[PROOF STEP]
by simp
[PROOF STATE]
proof (state)
this:
((g \<star> h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine) \<cdot> \<a>\<^sup>-\<^sup>1[g, tab\<^sub>1 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0, f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine] = \<a>\<^sup>-\<^sup>1[g, (h \<star> tab\<^sub>0 h) \<star> g\<^sub>0h\<^sub>1.p\<^sub>0, f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine] \<cdot> (g \<star> (h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine)
goal (1 subgoal):
1. \<a>\<^sup>-\<^sup>1[g \<star> (h \<star> tab\<^sub>0 h) \<star> g\<^sub>0h\<^sub>1.p\<^sub>0, f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine] \<cdot> (((g \<star> h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine) \<cdot> \<a>\<^sup>-\<^sup>1[g, tab\<^sub>1 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0, f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine]) \<cdot> (g \<star> g\<^sub>0h\<^sub>1.\<phi> \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine) = \<a>\<^sup>-\<^sup>1[g \<star> (h \<star> tab\<^sub>0 h) \<star> g\<^sub>0h\<^sub>1.p\<^sub>0, f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine] \<cdot> \<a>\<^sup>-\<^sup>1[g, (h \<star> tab\<^sub>0 h) \<star> g\<^sub>0h\<^sub>1.p\<^sub>0, f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine] \<cdot> (g \<star> (h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine) \<cdot> (g \<star> g\<^sub>0h\<^sub>1.\<phi> \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine)
[PROOF STEP]
thus ?thesis
[PROOF STATE]
proof (prove)
using this:
((g \<star> h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine) \<cdot> \<a>\<^sup>-\<^sup>1[g, tab\<^sub>1 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0, f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine] = \<a>\<^sup>-\<^sup>1[g, (h \<star> tab\<^sub>0 h) \<star> g\<^sub>0h\<^sub>1.p\<^sub>0, f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine] \<cdot> (g \<star> (h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine)
goal (1 subgoal):
1. \<a>\<^sup>-\<^sup>1[g \<star> (h \<star> tab\<^sub>0 h) \<star> g\<^sub>0h\<^sub>1.p\<^sub>0, f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine] \<cdot> (((g \<star> h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine) \<cdot> \<a>\<^sup>-\<^sup>1[g, tab\<^sub>1 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0, f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine]) \<cdot> (g \<star> g\<^sub>0h\<^sub>1.\<phi> \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine) = \<a>\<^sup>-\<^sup>1[g \<star> (h \<star> tab\<^sub>0 h) \<star> g\<^sub>0h\<^sub>1.p\<^sub>0, f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine] \<cdot> \<a>\<^sup>-\<^sup>1[g, (h \<star> tab\<^sub>0 h) \<star> g\<^sub>0h\<^sub>1.p\<^sub>0, f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine] \<cdot> (g \<star> (h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine) \<cdot> (g \<star> g\<^sub>0h\<^sub>1.\<phi> \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine)
[PROOF STEP]
using comp_assoc
[PROOF STATE]
proof (prove)
using this:
((g \<star> h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine) \<cdot> \<a>\<^sup>-\<^sup>1[g, tab\<^sub>1 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0, f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine] = \<a>\<^sup>-\<^sup>1[g, (h \<star> tab\<^sub>0 h) \<star> g\<^sub>0h\<^sub>1.p\<^sub>0, f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine] \<cdot> (g \<star> (h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine)
(?h \<cdot> ?g) \<cdot> ?f = ?h \<cdot> ?g \<cdot> ?f
goal (1 subgoal):
1. \<a>\<^sup>-\<^sup>1[g \<star> (h \<star> tab\<^sub>0 h) \<star> g\<^sub>0h\<^sub>1.p\<^sub>0, f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine] \<cdot> (((g \<star> h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine) \<cdot> \<a>\<^sup>-\<^sup>1[g, tab\<^sub>1 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0, f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine]) \<cdot> (g \<star> g\<^sub>0h\<^sub>1.\<phi> \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine) = \<a>\<^sup>-\<^sup>1[g \<star> (h \<star> tab\<^sub>0 h) \<star> g\<^sub>0h\<^sub>1.p\<^sub>0, f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine] \<cdot> \<a>\<^sup>-\<^sup>1[g, (h \<star> tab\<^sub>0 h) \<star> g\<^sub>0h\<^sub>1.p\<^sub>0, f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine] \<cdot> (g \<star> (h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine) \<cdot> (g \<star> g\<^sub>0h\<^sub>1.\<phi> \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine)
[PROOF STEP]
by presburger
[PROOF STATE]
proof (state)
this:
\<a>\<^sup>-\<^sup>1[g \<star> (h \<star> tab\<^sub>0 h) \<star> g\<^sub>0h\<^sub>1.p\<^sub>0, f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine] \<cdot> (((g \<star> h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine) \<cdot> \<a>\<^sup>-\<^sup>1[g, tab\<^sub>1 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0, f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine]) \<cdot> (g \<star> g\<^sub>0h\<^sub>1.\<phi> \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine) = \<a>\<^sup>-\<^sup>1[g \<star> (h \<star> tab\<^sub>0 h) \<star> g\<^sub>0h\<^sub>1.p\<^sub>0, f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine] \<cdot> \<a>\<^sup>-\<^sup>1[g, (h \<star> tab\<^sub>0 h) \<star> g\<^sub>0h\<^sub>1.p\<^sub>0, f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine] \<cdot> (g \<star> (h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine) \<cdot> (g \<star> g\<^sub>0h\<^sub>1.\<phi> \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine)
goal:
No subgoals!
[PROOF STEP]
qed
[PROOF STATE]
proof (state)
this:
\<a>\<^sup>-\<^sup>1[g \<star> (h \<star> tab\<^sub>0 h) \<star> g\<^sub>0h\<^sub>1.p\<^sub>0, f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine] \<cdot> (((g \<star> h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine) \<cdot> \<a>\<^sup>-\<^sup>1[g, tab\<^sub>1 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0, f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine]) \<cdot> (g \<star> g\<^sub>0h\<^sub>1.\<phi> \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine) = \<a>\<^sup>-\<^sup>1[g \<star> (h \<star> tab\<^sub>0 h) \<star> g\<^sub>0h\<^sub>1.p\<^sub>0, f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine] \<cdot> \<a>\<^sup>-\<^sup>1[g, (h \<star> tab\<^sub>0 h) \<star> g\<^sub>0h\<^sub>1.p\<^sub>0, f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine] \<cdot> (g \<star> (h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine) \<cdot> (g \<star> g\<^sub>0h\<^sub>1.\<phi> \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine)
goal (1 subgoal):
1. (((g \<star> h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (((g \<star> g\<^sub>0h\<^sub>1.\<phi>) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> \<a>\<^sup>-\<^sup>1[g \<star> tab\<^sub>0 g \<star> g\<^sub>0h\<^sub>1.p\<^sub>1, f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine] \<cdot> \<a>\<^sup>-\<^sup>1[g, tab\<^sub>0 g \<star> g\<^sub>0h\<^sub>1.p\<^sub>1, f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine] = \<a>\<^sup>-\<^sup>1[g \<star> (h \<star> tab\<^sub>0 h) \<star> g\<^sub>0h\<^sub>1.p\<^sub>0, f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine] \<cdot> \<a>\<^sup>-\<^sup>1[g, (h \<star> tab\<^sub>0 h) \<star> g\<^sub>0h\<^sub>1.p\<^sub>0, f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine] \<cdot> (g \<star> (h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine) \<cdot> (g \<star> g\<^sub>0h\<^sub>1.\<phi> \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine)
[PROOF STEP]
finally
[PROOF STATE]
proof (chain)
picking this:
(((g \<star> h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (((g \<star> g\<^sub>0h\<^sub>1.\<phi>) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> \<a>\<^sup>-\<^sup>1[g \<star> tab\<^sub>0 g \<star> g\<^sub>0h\<^sub>1.p\<^sub>1, f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine] \<cdot> \<a>\<^sup>-\<^sup>1[g, tab\<^sub>0 g \<star> g\<^sub>0h\<^sub>1.p\<^sub>1, f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine] = \<a>\<^sup>-\<^sup>1[g \<star> (h \<star> tab\<^sub>0 h) \<star> g\<^sub>0h\<^sub>1.p\<^sub>0, f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine] \<cdot> \<a>\<^sup>-\<^sup>1[g, (h \<star> tab\<^sub>0 h) \<star> g\<^sub>0h\<^sub>1.p\<^sub>0, f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine] \<cdot> (g \<star> (h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine) \<cdot> (g \<star> g\<^sub>0h\<^sub>1.\<phi> \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine)
[PROOF STEP]
show ?thesis
[PROOF STATE]
proof (prove)
using this:
(((g \<star> h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (((g \<star> g\<^sub>0h\<^sub>1.\<phi>) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> \<a>\<^sup>-\<^sup>1[g \<star> tab\<^sub>0 g \<star> g\<^sub>0h\<^sub>1.p\<^sub>1, f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine] \<cdot> \<a>\<^sup>-\<^sup>1[g, tab\<^sub>0 g \<star> g\<^sub>0h\<^sub>1.p\<^sub>1, f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine] = \<a>\<^sup>-\<^sup>1[g \<star> (h \<star> tab\<^sub>0 h) \<star> g\<^sub>0h\<^sub>1.p\<^sub>0, f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine] \<cdot> \<a>\<^sup>-\<^sup>1[g, (h \<star> tab\<^sub>0 h) \<star> g\<^sub>0h\<^sub>1.p\<^sub>0, f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine] \<cdot> (g \<star> (h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine) \<cdot> (g \<star> g\<^sub>0h\<^sub>1.\<phi> \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine)
goal (1 subgoal):
1. (((g \<star> h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (((g \<star> g\<^sub>0h\<^sub>1.\<phi>) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> \<a>\<^sup>-\<^sup>1[g \<star> tab\<^sub>0 g \<star> g\<^sub>0h\<^sub>1.p\<^sub>1, f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine] \<cdot> \<a>\<^sup>-\<^sup>1[g, tab\<^sub>0 g \<star> g\<^sub>0h\<^sub>1.p\<^sub>1, f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine] = \<a>\<^sup>-\<^sup>1[g \<star> (h \<star> tab\<^sub>0 h) \<star> g\<^sub>0h\<^sub>1.p\<^sub>0, f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine] \<cdot> \<a>\<^sup>-\<^sup>1[g, (h \<star> tab\<^sub>0 h) \<star> g\<^sub>0h\<^sub>1.p\<^sub>0, f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine] \<cdot> (g \<star> (h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine) \<cdot> (g \<star> g\<^sub>0h\<^sub>1.\<phi> \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine)
[PROOF STEP]
by simp
[PROOF STATE]
proof (state)
this:
(((g \<star> h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (((g \<star> g\<^sub>0h\<^sub>1.\<phi>) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> \<a>\<^sup>-\<^sup>1[g \<star> tab\<^sub>0 g \<star> g\<^sub>0h\<^sub>1.p\<^sub>1, f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine] \<cdot> \<a>\<^sup>-\<^sup>1[g, tab\<^sub>0 g \<star> g\<^sub>0h\<^sub>1.p\<^sub>1, f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine] = \<a>\<^sup>-\<^sup>1[g \<star> (h \<star> tab\<^sub>0 h) \<star> g\<^sub>0h\<^sub>1.p\<^sub>0, f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine] \<cdot> \<a>\<^sup>-\<^sup>1[g, (h \<star> tab\<^sub>0 h) \<star> g\<^sub>0h\<^sub>1.p\<^sub>0, f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine] \<cdot> (g \<star> (h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine) \<cdot> (g \<star> g\<^sub>0h\<^sub>1.\<phi> \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine)
goal:
No subgoals!
[PROOF STEP]
qed
[PROOF STATE]
proof (state)
this:
(((g \<star> h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (((g \<star> g\<^sub>0h\<^sub>1.\<phi>) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> \<a>\<^sup>-\<^sup>1[g \<star> tab\<^sub>0 g \<star> g\<^sub>0h\<^sub>1.p\<^sub>1, f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine] \<cdot> \<a>\<^sup>-\<^sup>1[g, tab\<^sub>0 g \<star> g\<^sub>0h\<^sub>1.p\<^sub>1, f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine] = \<a>\<^sup>-\<^sup>1[g \<star> (h \<star> tab\<^sub>0 h) \<star> g\<^sub>0h\<^sub>1.p\<^sub>0, f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine] \<cdot> \<a>\<^sup>-\<^sup>1[g, (h \<star> tab\<^sub>0 h) \<star> g\<^sub>0h\<^sub>1.p\<^sub>0, f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine] \<cdot> (g \<star> (h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine) \<cdot> (g \<star> g\<^sub>0h\<^sub>1.\<phi> \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine)
goal (1 subgoal):
1. (g \<star> h \<star> TTfgh_TfTgh.the_\<theta>) \<cdot> can (\<^bold>\<langle>g\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> ((\<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) (((\<^bold>\<langle>g\<^bold>\<rangle> \<^bold>\<star> (\<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) \<cdot> ((((g \<star> h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (((g \<star> g\<^sub>0h\<^sub>1.\<phi>) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> \<a>\<^sup>-\<^sup>1[g \<star> tab\<^sub>0 g \<star> g\<^sub>0h\<^sub>1.p\<^sub>1, f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine] \<cdot> \<a>\<^sup>-\<^sup>1[g, tab\<^sub>0 g \<star> g\<^sub>0h\<^sub>1.p\<^sub>1, f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine]) \<cdot> \<a>[g, tab\<^sub>0 g \<star> g\<^sub>0h\<^sub>1.p\<^sub>1, f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine] \<cdot> \<a>[g \<star> tab\<^sub>0 g \<star> g\<^sub>0h\<^sub>1.p\<^sub>1, f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine] \<cdot> ((\<a>[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> \<a>\<^sup>-\<^sup>1[(g \<star> tab\<^sub>0 g) \<star> g\<^sub>0h\<^sub>1.p\<^sub>1, f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine] \<cdot> \<a>\<^sup>-\<^sup>1[g \<star> tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1, f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine] \<cdot> (\<a>\<^sup>-\<^sup>1[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1 \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine] \<cdot> \<a>[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1 \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine] \<cdot> (g.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>1 \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine)) \<cdot> \<beta>\<^sub>g = (g \<star> h \<star> TTfgh_TfTgh.the_\<theta>) \<cdot> (can (\<^bold>\<langle>g\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> ((\<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) (((\<^bold>\<langle>g\<^bold>\<rangle> \<^bold>\<star> (\<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) \<cdot> \<a>\<^sup>-\<^sup>1[g \<star> (h \<star> tab\<^sub>0 h) \<star> g\<^sub>0h\<^sub>1.p\<^sub>0, f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine] \<cdot> \<a>\<^sup>-\<^sup>1[g, (h \<star> tab\<^sub>0 h) \<star> g\<^sub>0h\<^sub>1.p\<^sub>0, f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine]) \<cdot> (g \<star> (h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine) \<cdot> (g \<star> g\<^sub>0h\<^sub>1.\<phi> \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine) \<cdot> (\<a>[g, tab\<^sub>0 g \<star> g\<^sub>0h\<^sub>1.p\<^sub>1, f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine] \<cdot> \<a>[g \<star> tab\<^sub>0 g \<star> g\<^sub>0h\<^sub>1.p\<^sub>1, f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine] \<cdot> ((\<a>[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> \<a>\<^sup>-\<^sup>1[(g \<star> tab\<^sub>0 g) \<star> g\<^sub>0h\<^sub>1.p\<^sub>1, f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine] \<cdot> \<a>\<^sup>-\<^sup>1[g \<star> tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1, f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine] \<cdot> \<a>\<^sup>-\<^sup>1[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1 \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine]) \<cdot> \<a>[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1 \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine] \<cdot> (g.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>1 \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine) \<cdot> \<beta>\<^sub>g
[PROOF STEP]
thus ?thesis
[PROOF STATE]
proof (prove)
using this:
(((g \<star> h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (((g \<star> g\<^sub>0h\<^sub>1.\<phi>) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> \<a>\<^sup>-\<^sup>1[g \<star> tab\<^sub>0 g \<star> g\<^sub>0h\<^sub>1.p\<^sub>1, f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine] \<cdot> \<a>\<^sup>-\<^sup>1[g, tab\<^sub>0 g \<star> g\<^sub>0h\<^sub>1.p\<^sub>1, f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine] = \<a>\<^sup>-\<^sup>1[g \<star> (h \<star> tab\<^sub>0 h) \<star> g\<^sub>0h\<^sub>1.p\<^sub>0, f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine] \<cdot> \<a>\<^sup>-\<^sup>1[g, (h \<star> tab\<^sub>0 h) \<star> g\<^sub>0h\<^sub>1.p\<^sub>0, f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine] \<cdot> (g \<star> (h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine) \<cdot> (g \<star> g\<^sub>0h\<^sub>1.\<phi> \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine)
goal (1 subgoal):
1. (g \<star> h \<star> TTfgh_TfTgh.the_\<theta>) \<cdot> can (\<^bold>\<langle>g\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> ((\<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) (((\<^bold>\<langle>g\<^bold>\<rangle> \<^bold>\<star> (\<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) \<cdot> ((((g \<star> h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (((g \<star> g\<^sub>0h\<^sub>1.\<phi>) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> \<a>\<^sup>-\<^sup>1[g \<star> tab\<^sub>0 g \<star> g\<^sub>0h\<^sub>1.p\<^sub>1, f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine] \<cdot> \<a>\<^sup>-\<^sup>1[g, tab\<^sub>0 g \<star> g\<^sub>0h\<^sub>1.p\<^sub>1, f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine]) \<cdot> \<a>[g, tab\<^sub>0 g \<star> g\<^sub>0h\<^sub>1.p\<^sub>1, f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine] \<cdot> \<a>[g \<star> tab\<^sub>0 g \<star> g\<^sub>0h\<^sub>1.p\<^sub>1, f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine] \<cdot> ((\<a>[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> \<a>\<^sup>-\<^sup>1[(g \<star> tab\<^sub>0 g) \<star> g\<^sub>0h\<^sub>1.p\<^sub>1, f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine] \<cdot> \<a>\<^sup>-\<^sup>1[g \<star> tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1, f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine] \<cdot> (\<a>\<^sup>-\<^sup>1[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1 \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine] \<cdot> \<a>[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1 \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine] \<cdot> (g.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>1 \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine)) \<cdot> \<beta>\<^sub>g = (g \<star> h \<star> TTfgh_TfTgh.the_\<theta>) \<cdot> (can (\<^bold>\<langle>g\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> ((\<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) (((\<^bold>\<langle>g\<^bold>\<rangle> \<^bold>\<star> (\<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) \<cdot> \<a>\<^sup>-\<^sup>1[g \<star> (h \<star> tab\<^sub>0 h) \<star> g\<^sub>0h\<^sub>1.p\<^sub>0, f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine] \<cdot> \<a>\<^sup>-\<^sup>1[g, (h \<star> tab\<^sub>0 h) \<star> g\<^sub>0h\<^sub>1.p\<^sub>0, f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine]) \<cdot> (g \<star> (h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine) \<cdot> (g \<star> g\<^sub>0h\<^sub>1.\<phi> \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine) \<cdot> (\<a>[g, tab\<^sub>0 g \<star> g\<^sub>0h\<^sub>1.p\<^sub>1, f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine] \<cdot> \<a>[g \<star> tab\<^sub>0 g \<star> g\<^sub>0h\<^sub>1.p\<^sub>1, f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine] \<cdot> ((\<a>[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> \<a>\<^sup>-\<^sup>1[(g \<star> tab\<^sub>0 g) \<star> g\<^sub>0h\<^sub>1.p\<^sub>1, f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine] \<cdot> \<a>\<^sup>-\<^sup>1[g \<star> tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1, f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine] \<cdot> \<a>\<^sup>-\<^sup>1[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1 \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine]) \<cdot> \<a>[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1 \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine] \<cdot> (g.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>1 \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine) \<cdot> \<beta>\<^sub>g
[PROOF STEP]
using comp_assoc
[PROOF STATE]
proof (prove)
using this:
(((g \<star> h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (((g \<star> g\<^sub>0h\<^sub>1.\<phi>) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> \<a>\<^sup>-\<^sup>1[g \<star> tab\<^sub>0 g \<star> g\<^sub>0h\<^sub>1.p\<^sub>1, f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine] \<cdot> \<a>\<^sup>-\<^sup>1[g, tab\<^sub>0 g \<star> g\<^sub>0h\<^sub>1.p\<^sub>1, f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine] = \<a>\<^sup>-\<^sup>1[g \<star> (h \<star> tab\<^sub>0 h) \<star> g\<^sub>0h\<^sub>1.p\<^sub>0, f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine] \<cdot> \<a>\<^sup>-\<^sup>1[g, (h \<star> tab\<^sub>0 h) \<star> g\<^sub>0h\<^sub>1.p\<^sub>0, f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine] \<cdot> (g \<star> (h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine) \<cdot> (g \<star> g\<^sub>0h\<^sub>1.\<phi> \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine)
(?h \<cdot> ?g) \<cdot> ?f = ?h \<cdot> ?g \<cdot> ?f
goal (1 subgoal):
1. (g \<star> h \<star> TTfgh_TfTgh.the_\<theta>) \<cdot> can (\<^bold>\<langle>g\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> ((\<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) (((\<^bold>\<langle>g\<^bold>\<rangle> \<^bold>\<star> (\<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) \<cdot> ((((g \<star> h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (((g \<star> g\<^sub>0h\<^sub>1.\<phi>) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> \<a>\<^sup>-\<^sup>1[g \<star> tab\<^sub>0 g \<star> g\<^sub>0h\<^sub>1.p\<^sub>1, f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine] \<cdot> \<a>\<^sup>-\<^sup>1[g, tab\<^sub>0 g \<star> g\<^sub>0h\<^sub>1.p\<^sub>1, f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine]) \<cdot> \<a>[g, tab\<^sub>0 g \<star> g\<^sub>0h\<^sub>1.p\<^sub>1, f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine] \<cdot> \<a>[g \<star> tab\<^sub>0 g \<star> g\<^sub>0h\<^sub>1.p\<^sub>1, f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine] \<cdot> ((\<a>[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> \<a>\<^sup>-\<^sup>1[(g \<star> tab\<^sub>0 g) \<star> g\<^sub>0h\<^sub>1.p\<^sub>1, f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine] \<cdot> \<a>\<^sup>-\<^sup>1[g \<star> tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1, f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine] \<cdot> (\<a>\<^sup>-\<^sup>1[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1 \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine] \<cdot> \<a>[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1 \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine] \<cdot> (g.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>1 \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine)) \<cdot> \<beta>\<^sub>g = (g \<star> h \<star> TTfgh_TfTgh.the_\<theta>) \<cdot> (can (\<^bold>\<langle>g\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> ((\<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) (((\<^bold>\<langle>g\<^bold>\<rangle> \<^bold>\<star> (\<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) \<cdot> \<a>\<^sup>-\<^sup>1[g \<star> (h \<star> tab\<^sub>0 h) \<star> g\<^sub>0h\<^sub>1.p\<^sub>0, f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine] \<cdot> \<a>\<^sup>-\<^sup>1[g, (h \<star> tab\<^sub>0 h) \<star> g\<^sub>0h\<^sub>1.p\<^sub>0, f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine]) \<cdot> (g \<star> (h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine) \<cdot> (g \<star> g\<^sub>0h\<^sub>1.\<phi> \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine) \<cdot> (\<a>[g, tab\<^sub>0 g \<star> g\<^sub>0h\<^sub>1.p\<^sub>1, f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine] \<cdot> \<a>[g \<star> tab\<^sub>0 g \<star> g\<^sub>0h\<^sub>1.p\<^sub>1, f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine] \<cdot> ((\<a>[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> \<a>\<^sup>-\<^sup>1[(g \<star> tab\<^sub>0 g) \<star> g\<^sub>0h\<^sub>1.p\<^sub>1, f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine] \<cdot> \<a>\<^sup>-\<^sup>1[g \<star> tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1, f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine] \<cdot> \<a>\<^sup>-\<^sup>1[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1 \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine]) \<cdot> \<a>[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1 \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine] \<cdot> (g.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>1 \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine) \<cdot> \<beta>\<^sub>g
[PROOF STEP]
by presburger
[PROOF STATE]
proof (state)
this:
(g \<star> h \<star> TTfgh_TfTgh.the_\<theta>) \<cdot> can (\<^bold>\<langle>g\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> ((\<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) (((\<^bold>\<langle>g\<^bold>\<rangle> \<^bold>\<star> (\<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) \<cdot> ((((g \<star> h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (((g \<star> g\<^sub>0h\<^sub>1.\<phi>) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> \<a>\<^sup>-\<^sup>1[g \<star> tab\<^sub>0 g \<star> g\<^sub>0h\<^sub>1.p\<^sub>1, f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine] \<cdot> \<a>\<^sup>-\<^sup>1[g, tab\<^sub>0 g \<star> g\<^sub>0h\<^sub>1.p\<^sub>1, f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine]) \<cdot> \<a>[g, tab\<^sub>0 g \<star> g\<^sub>0h\<^sub>1.p\<^sub>1, f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine] \<cdot> \<a>[g \<star> tab\<^sub>0 g \<star> g\<^sub>0h\<^sub>1.p\<^sub>1, f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine] \<cdot> ((\<a>[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> \<a>\<^sup>-\<^sup>1[(g \<star> tab\<^sub>0 g) \<star> g\<^sub>0h\<^sub>1.p\<^sub>1, f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine] \<cdot> \<a>\<^sup>-\<^sup>1[g \<star> tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1, f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine] \<cdot> (\<a>\<^sup>-\<^sup>1[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1 \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine] \<cdot> \<a>[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1 \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine] \<cdot> (g.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>1 \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine)) \<cdot> \<beta>\<^sub>g = (g \<star> h \<star> TTfgh_TfTgh.the_\<theta>) \<cdot> (can (\<^bold>\<langle>g\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> ((\<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) (((\<^bold>\<langle>g\<^bold>\<rangle> \<^bold>\<star> (\<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) \<cdot> \<a>\<^sup>-\<^sup>1[g \<star> (h \<star> tab\<^sub>0 h) \<star> g\<^sub>0h\<^sub>1.p\<^sub>0, f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine] \<cdot> \<a>\<^sup>-\<^sup>1[g, (h \<star> tab\<^sub>0 h) \<star> g\<^sub>0h\<^sub>1.p\<^sub>0, f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine]) \<cdot> (g \<star> (h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine) \<cdot> (g \<star> g\<^sub>0h\<^sub>1.\<phi> \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine) \<cdot> (\<a>[g, tab\<^sub>0 g \<star> g\<^sub>0h\<^sub>1.p\<^sub>1, f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine] \<cdot> \<a>[g \<star> tab\<^sub>0 g \<star> g\<^sub>0h\<^sub>1.p\<^sub>1, f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine] \<cdot> ((\<a>[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> \<a>\<^sup>-\<^sup>1[(g \<star> tab\<^sub>0 g) \<star> g\<^sub>0h\<^sub>1.p\<^sub>1, f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine] \<cdot> \<a>\<^sup>-\<^sup>1[g \<star> tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1, f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine] \<cdot> \<a>\<^sup>-\<^sup>1[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1 \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine]) \<cdot> \<a>[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1 \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine] \<cdot> (g.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>1 \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine) \<cdot> \<beta>\<^sub>g
goal:
No subgoals!
[PROOF STEP]
qed
[PROOF STATE]
proof (state)
this:
(g \<star> h \<star> TTfgh_TfTgh.the_\<theta>) \<cdot> can (\<^bold>\<langle>g\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> ((\<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) (((\<^bold>\<langle>g\<^bold>\<rangle> \<^bold>\<star> (\<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) \<cdot> ((((g \<star> h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (((g \<star> g\<^sub>0h\<^sub>1.\<phi>) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> \<a>\<^sup>-\<^sup>1[g \<star> tab\<^sub>0 g \<star> g\<^sub>0h\<^sub>1.p\<^sub>1, f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine] \<cdot> \<a>\<^sup>-\<^sup>1[g, tab\<^sub>0 g \<star> g\<^sub>0h\<^sub>1.p\<^sub>1, f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine]) \<cdot> \<a>[g, tab\<^sub>0 g \<star> g\<^sub>0h\<^sub>1.p\<^sub>1, f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine] \<cdot> \<a>[g \<star> tab\<^sub>0 g \<star> g\<^sub>0h\<^sub>1.p\<^sub>1, f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine] \<cdot> ((\<a>[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> \<a>\<^sup>-\<^sup>1[(g \<star> tab\<^sub>0 g) \<star> g\<^sub>0h\<^sub>1.p\<^sub>1, f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine] \<cdot> \<a>\<^sup>-\<^sup>1[g \<star> tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1, f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine] \<cdot> (\<a>\<^sup>-\<^sup>1[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1 \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine] \<cdot> \<a>[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1 \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine] \<cdot> (g.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>1 \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine)) \<cdot> \<beta>\<^sub>g = (g \<star> h \<star> TTfgh_TfTgh.the_\<theta>) \<cdot> (can (\<^bold>\<langle>g\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> ((\<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) (((\<^bold>\<langle>g\<^bold>\<rangle> \<^bold>\<star> (\<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) \<cdot> \<a>\<^sup>-\<^sup>1[g \<star> (h \<star> tab\<^sub>0 h) \<star> g\<^sub>0h\<^sub>1.p\<^sub>0, f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine] \<cdot> \<a>\<^sup>-\<^sup>1[g, (h \<star> tab\<^sub>0 h) \<star> g\<^sub>0h\<^sub>1.p\<^sub>0, f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine]) \<cdot> (g \<star> (h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine) \<cdot> (g \<star> g\<^sub>0h\<^sub>1.\<phi> \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine) \<cdot> (\<a>[g, tab\<^sub>0 g \<star> g\<^sub>0h\<^sub>1.p\<^sub>1, f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine] \<cdot> \<a>[g \<star> tab\<^sub>0 g \<star> g\<^sub>0h\<^sub>1.p\<^sub>1, f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine] \<cdot> ((\<a>[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> \<a>\<^sup>-\<^sup>1[(g \<star> tab\<^sub>0 g) \<star> g\<^sub>0h\<^sub>1.p\<^sub>1, f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine] \<cdot> \<a>\<^sup>-\<^sup>1[g \<star> tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1, f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine] \<cdot> \<a>\<^sup>-\<^sup>1[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1 \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine]) \<cdot> \<a>[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1 \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine] \<cdot> (g.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>1 \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine) \<cdot> \<beta>\<^sub>g
goal (1 subgoal):
1. (g \<star> \<theta>\<^sub>g) \<cdot> \<a>[g, tab\<^sub>0 g, w\<^sub>g] \<cdot> (g.tab \<star> w\<^sub>g) = ((g \<star> \<theta>\<^sub>g') \<cdot> \<a>[g, tab\<^sub>0 g, w\<^sub>g'] \<cdot> (g.tab \<star> w\<^sub>g')) \<cdot> \<beta>\<^sub>g
[PROOF STEP]
also
[PROOF STATE]
proof (state)
this:
(g \<star> h \<star> TTfgh_TfTgh.the_\<theta>) \<cdot> can (\<^bold>\<langle>g\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> ((\<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) (((\<^bold>\<langle>g\<^bold>\<rangle> \<^bold>\<star> (\<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) \<cdot> ((((g \<star> h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> (((g \<star> g\<^sub>0h\<^sub>1.\<phi>) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> \<a>\<^sup>-\<^sup>1[g \<star> tab\<^sub>0 g \<star> g\<^sub>0h\<^sub>1.p\<^sub>1, f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine] \<cdot> \<a>\<^sup>-\<^sup>1[g, tab\<^sub>0 g \<star> g\<^sub>0h\<^sub>1.p\<^sub>1, f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine]) \<cdot> \<a>[g, tab\<^sub>0 g \<star> g\<^sub>0h\<^sub>1.p\<^sub>1, f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine] \<cdot> \<a>[g \<star> tab\<^sub>0 g \<star> g\<^sub>0h\<^sub>1.p\<^sub>1, f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine] \<cdot> ((\<a>[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> \<a>\<^sup>-\<^sup>1[(g \<star> tab\<^sub>0 g) \<star> g\<^sub>0h\<^sub>1.p\<^sub>1, f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine] \<cdot> \<a>\<^sup>-\<^sup>1[g \<star> tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1, f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine] \<cdot> (\<a>\<^sup>-\<^sup>1[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1 \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine] \<cdot> \<a>[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1 \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine] \<cdot> (g.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>1 \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine)) \<cdot> \<beta>\<^sub>g = (g \<star> h \<star> TTfgh_TfTgh.the_\<theta>) \<cdot> (can (\<^bold>\<langle>g\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> ((\<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) (((\<^bold>\<langle>g\<^bold>\<rangle> \<^bold>\<star> (\<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) \<cdot> \<a>\<^sup>-\<^sup>1[g \<star> (h \<star> tab\<^sub>0 h) \<star> g\<^sub>0h\<^sub>1.p\<^sub>0, f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine] \<cdot> \<a>\<^sup>-\<^sup>1[g, (h \<star> tab\<^sub>0 h) \<star> g\<^sub>0h\<^sub>1.p\<^sub>0, f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine]) \<cdot> (g \<star> (h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine) \<cdot> (g \<star> g\<^sub>0h\<^sub>1.\<phi> \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine) \<cdot> (\<a>[g, tab\<^sub>0 g \<star> g\<^sub>0h\<^sub>1.p\<^sub>1, f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine] \<cdot> \<a>[g \<star> tab\<^sub>0 g \<star> g\<^sub>0h\<^sub>1.p\<^sub>1, f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine] \<cdot> ((\<a>[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> \<a>\<^sup>-\<^sup>1[(g \<star> tab\<^sub>0 g) \<star> g\<^sub>0h\<^sub>1.p\<^sub>1, f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine] \<cdot> \<a>\<^sup>-\<^sup>1[g \<star> tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1, f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine] \<cdot> \<a>\<^sup>-\<^sup>1[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1 \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine]) \<cdot> \<a>[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1 \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine] \<cdot> (g.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>1 \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine) \<cdot> \<beta>\<^sub>g
goal (1 subgoal):
1. (g \<star> \<theta>\<^sub>g) \<cdot> \<a>[g, tab\<^sub>0 g, w\<^sub>g] \<cdot> (g.tab \<star> w\<^sub>g) = ((g \<star> \<theta>\<^sub>g') \<cdot> \<a>[g, tab\<^sub>0 g, w\<^sub>g'] \<cdot> (g.tab \<star> w\<^sub>g')) \<cdot> \<beta>\<^sub>g
[PROOF STEP]
have "... = ((g \<star> h \<star> TTfgh_TfTgh.the_\<theta>) \<cdot>
(g \<star> can (\<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> ((\<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>Tgh.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TfTgh.p\<^sub>0\<^bold>\<rangle>)
\<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>)
(((\<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>Tgh.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TfTgh.p\<^sub>0\<^bold>\<rangle>
\<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>)) \<cdot>
(g \<star> (h.tab \<star> Tgh.p\<^sub>0) \<star> TfTgh.p\<^sub>0 \<star> TTfgh_TfTgh.chine) \<cdot>
(g \<star> g\<^sub>0h\<^sub>1.\<phi> \<star> TfTgh.p\<^sub>0 \<star> TTfgh_TfTgh.chine) \<cdot>
(g \<star> \<a>\<^sup>-\<^sup>1[tab\<^sub>0 g, Tgh.p\<^sub>1, TfTgh.p\<^sub>0 \<star> TTfgh_TfTgh.chine])) \<cdot>
\<a>[g, tab\<^sub>0 g, Tgh.p\<^sub>1 \<star> TfTgh.p\<^sub>0 \<star> TTfgh_TfTgh.chine] \<cdot>
(g.tab \<star> Tgh.p\<^sub>1 \<star> TfTgh.p\<^sub>0 \<star> TTfgh_TfTgh.chine) \<cdot>
\<beta>\<^sub>g"
[PROOF STATE]
proof (prove)
goal (1 subgoal):
1. (g \<star> h \<star> TTfgh_TfTgh.the_\<theta>) \<cdot> (can (\<^bold>\<langle>g\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> ((\<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) (((\<^bold>\<langle>g\<^bold>\<rangle> \<^bold>\<star> (\<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) \<cdot> \<a>\<^sup>-\<^sup>1[g \<star> (h \<star> tab\<^sub>0 h) \<star> g\<^sub>0h\<^sub>1.p\<^sub>0, f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine] \<cdot> \<a>\<^sup>-\<^sup>1[g, (h \<star> tab\<^sub>0 h) \<star> g\<^sub>0h\<^sub>1.p\<^sub>0, f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine]) \<cdot> (g \<star> (h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine) \<cdot> (g \<star> g\<^sub>0h\<^sub>1.\<phi> \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine) \<cdot> (\<a>[g, tab\<^sub>0 g \<star> g\<^sub>0h\<^sub>1.p\<^sub>1, f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine] \<cdot> \<a>[g \<star> tab\<^sub>0 g \<star> g\<^sub>0h\<^sub>1.p\<^sub>1, f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine] \<cdot> ((\<a>[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> \<a>\<^sup>-\<^sup>1[(g \<star> tab\<^sub>0 g) \<star> g\<^sub>0h\<^sub>1.p\<^sub>1, f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine] \<cdot> \<a>\<^sup>-\<^sup>1[g \<star> tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1, f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine] \<cdot> \<a>\<^sup>-\<^sup>1[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1 \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine]) \<cdot> \<a>[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1 \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine] \<cdot> (g.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>1 \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine) \<cdot> \<beta>\<^sub>g = ((g \<star> h \<star> TTfgh_TfTgh.the_\<theta>) \<cdot> (g \<star> can (\<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> ((\<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) (((\<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>)) \<cdot> (g \<star> (h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine) \<cdot> (g \<star> g\<^sub>0h\<^sub>1.\<phi> \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine) \<cdot> (g \<star> \<a>\<^sup>-\<^sup>1[tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1, f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine])) \<cdot> \<a>[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1 \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine] \<cdot> (g.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>1 \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine) \<cdot> \<beta>\<^sub>g
[PROOF STEP]
proof -
[PROOF STATE]
proof (state)
goal (1 subgoal):
1. (g \<star> h \<star> TTfgh_TfTgh.the_\<theta>) \<cdot> (can (\<^bold>\<langle>g\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> ((\<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) (((\<^bold>\<langle>g\<^bold>\<rangle> \<^bold>\<star> (\<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) \<cdot> \<a>\<^sup>-\<^sup>1[g \<star> (h \<star> tab\<^sub>0 h) \<star> g\<^sub>0h\<^sub>1.p\<^sub>0, f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine] \<cdot> \<a>\<^sup>-\<^sup>1[g, (h \<star> tab\<^sub>0 h) \<star> g\<^sub>0h\<^sub>1.p\<^sub>0, f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine]) \<cdot> (g \<star> (h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine) \<cdot> (g \<star> g\<^sub>0h\<^sub>1.\<phi> \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine) \<cdot> (\<a>[g, tab\<^sub>0 g \<star> g\<^sub>0h\<^sub>1.p\<^sub>1, f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine] \<cdot> \<a>[g \<star> tab\<^sub>0 g \<star> g\<^sub>0h\<^sub>1.p\<^sub>1, f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine] \<cdot> ((\<a>[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> \<a>\<^sup>-\<^sup>1[(g \<star> tab\<^sub>0 g) \<star> g\<^sub>0h\<^sub>1.p\<^sub>1, f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine] \<cdot> \<a>\<^sup>-\<^sup>1[g \<star> tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1, f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine] \<cdot> \<a>\<^sup>-\<^sup>1[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1 \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine]) \<cdot> \<a>[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1 \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine] \<cdot> (g.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>1 \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine) \<cdot> \<beta>\<^sub>g = ((g \<star> h \<star> TTfgh_TfTgh.the_\<theta>) \<cdot> (g \<star> can (\<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> ((\<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) (((\<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>)) \<cdot> (g \<star> (h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine) \<cdot> (g \<star> g\<^sub>0h\<^sub>1.\<phi> \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine) \<cdot> (g \<star> \<a>\<^sup>-\<^sup>1[tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1, f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine])) \<cdot> \<a>[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1 \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine] \<cdot> (g.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>1 \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine) \<cdot> \<beta>\<^sub>g
[PROOF STEP]
have "can (\<^bold>\<langle>g\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> ((\<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>Tgh.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TfTgh.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>)
(((\<^bold>\<langle>g\<^bold>\<rangle> \<^bold>\<star> (\<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>Tgh.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TfTgh.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) \<cdot>
\<a>\<^sup>-\<^sup>1[g \<star> (h \<star> tab\<^sub>0 h) \<star> Tgh.p\<^sub>0, TfTgh.p\<^sub>0, TTfgh_TfTgh.chine] \<cdot>
\<a>\<^sup>-\<^sup>1[g, (h \<star> tab\<^sub>0 h) \<star> Tgh.p\<^sub>0, TfTgh.p\<^sub>0 \<star> TTfgh_TfTgh.chine] =
g \<star> can (\<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> ((\<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>Tgh.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TfTgh.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>)
(((\<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>Tgh.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TfTgh.p\<^sub>0\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>)"
[PROOF STATE]
proof (prove)
goal (1 subgoal):
1. can (\<^bold>\<langle>g\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> ((\<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) (((\<^bold>\<langle>g\<^bold>\<rangle> \<^bold>\<star> (\<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) \<cdot> \<a>\<^sup>-\<^sup>1[g \<star> (h \<star> tab\<^sub>0 h) \<star> g\<^sub>0h\<^sub>1.p\<^sub>0, f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine] \<cdot> \<a>\<^sup>-\<^sup>1[g, (h \<star> tab\<^sub>0 h) \<star> g\<^sub>0h\<^sub>1.p\<^sub>0, f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine] = g \<star> can (\<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> ((\<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) (((\<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>)
[PROOF STEP]
proof -
[PROOF STATE]
proof (state)
goal (1 subgoal):
1. can (\<^bold>\<langle>g\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> ((\<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) (((\<^bold>\<langle>g\<^bold>\<rangle> \<^bold>\<star> (\<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) \<cdot> \<a>\<^sup>-\<^sup>1[g \<star> (h \<star> tab\<^sub>0 h) \<star> g\<^sub>0h\<^sub>1.p\<^sub>0, f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine] \<cdot> \<a>\<^sup>-\<^sup>1[g, (h \<star> tab\<^sub>0 h) \<star> g\<^sub>0h\<^sub>1.p\<^sub>0, f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine] = g \<star> can (\<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> ((\<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) (((\<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>)
[PROOF STEP]
have "\<a>\<^sup>-\<^sup>1[g \<star> (h \<star> tab\<^sub>0 h) \<star> Tgh.p\<^sub>0, TfTgh.p\<^sub>0, TTfgh_TfTgh.chine] =
can (((\<^bold>\<langle>g\<^bold>\<rangle> \<^bold>\<star> (\<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>Tgh.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TfTgh.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>)
((\<^bold>\<langle>g\<^bold>\<rangle> \<^bold>\<star> (\<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>Tgh.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TfTgh.p\<^sub>0\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>)"
[PROOF STATE]
proof (prove)
goal (1 subgoal):
1. \<a>\<^sup>-\<^sup>1[g \<star> (h \<star> tab\<^sub>0 h) \<star> g\<^sub>0h\<^sub>1.p\<^sub>0, f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine] = can (((\<^bold>\<langle>g\<^bold>\<rangle> \<^bold>\<star> (\<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) ((\<^bold>\<langle>g\<^bold>\<rangle> \<^bold>\<star> (\<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>)
[PROOF STEP]
proof -
[PROOF STATE]
proof (state)
goal (1 subgoal):
1. \<a>\<^sup>-\<^sup>1[g \<star> (h \<star> tab\<^sub>0 h) \<star> g\<^sub>0h\<^sub>1.p\<^sub>0, f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine] = can (((\<^bold>\<langle>g\<^bold>\<rangle> \<^bold>\<star> (\<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) ((\<^bold>\<langle>g\<^bold>\<rangle> \<^bold>\<star> (\<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>)
[PROOF STEP]
have "\<a>\<^sup>-\<^sup>1[g \<star> (h \<star> tab\<^sub>0 h) \<star> Tgh.p\<^sub>0, TfTgh.p\<^sub>0, TTfgh_TfTgh.chine] =
\<lbrace>\<^bold>\<a>\<^sup>-\<^sup>1\<^bold>[\<^bold>\<langle>g\<^bold>\<rangle> \<^bold>\<star> (\<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>Tgh.p\<^sub>0\<^bold>\<rangle>, \<^bold>\<langle>TfTgh.p\<^sub>0\<^bold>\<rangle>, \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>\<^bold>]\<rbrace>"
[PROOF STATE]
proof (prove)
goal (1 subgoal):
1. \<a>\<^sup>-\<^sup>1[g \<star> (h \<star> tab\<^sub>0 h) \<star> g\<^sub>0h\<^sub>1.p\<^sub>0, f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine] = \<lbrace>\<^bold>\<a>\<^sup>-\<^sup>1\<^bold>[\<^bold>\<langle>g\<^bold>\<rangle> \<^bold>\<star> (\<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>, \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>, \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>\<^bold>]\<rbrace>
[PROOF STEP]
using gh f\<^sub>0gh\<^sub>1.p\<^sub>0_simps canI_associator_0 \<a>'_def \<alpha>_def
[PROOF STATE]
proof (prove)
using this:
src g = trg h
trg f\<^sub>0gh\<^sub>1.p\<^sub>0 = src (tab\<^sub>1 g \<star> g\<^sub>0h\<^sub>1.p\<^sub>1)
\<lbrakk>ide ?f; ide ?g; ide ?h; src ?f = trg ?g; src ?g = trg ?h\<rbrakk> \<Longrightarrow> \<a>[?f, ?g, ?h] = can (\<^bold>\<langle>?f\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>?g\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>?h\<^bold>\<rangle>) ((\<^bold>\<langle>?f\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>?g\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>?h\<^bold>\<rangle>)
\<a>\<^sup>-\<^sup>1[?\<mu>, ?\<nu>, ?\<tau>] \<equiv> \<alpha>' (?\<mu>, ?\<nu>, ?\<tau>)
\<alpha> ?\<mu>\<nu>\<tau> \<equiv> \<a>[fst ?\<mu>\<nu>\<tau>, fst (snd ?\<mu>\<nu>\<tau>), snd (snd ?\<mu>\<nu>\<tau>)]
goal (1 subgoal):
1. \<a>\<^sup>-\<^sup>1[g \<star> (h \<star> tab\<^sub>0 h) \<star> g\<^sub>0h\<^sub>1.p\<^sub>0, f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine] = \<lbrace>\<^bold>\<a>\<^sup>-\<^sup>1\<^bold>[\<^bold>\<langle>g\<^bold>\<rangle> \<^bold>\<star> (\<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>, \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>, \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>\<^bold>]\<rbrace>
[PROOF STEP]
by simp
[PROOF STATE]
proof (state)
this:
\<a>\<^sup>-\<^sup>1[g \<star> (h \<star> tab\<^sub>0 h) \<star> g\<^sub>0h\<^sub>1.p\<^sub>0, f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine] = \<lbrace>\<^bold>\<a>\<^sup>-\<^sup>1\<^bold>[\<^bold>\<langle>g\<^bold>\<rangle> \<^bold>\<star> (\<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>, \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>, \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>\<^bold>]\<rbrace>
goal (1 subgoal):
1. \<a>\<^sup>-\<^sup>1[g \<star> (h \<star> tab\<^sub>0 h) \<star> g\<^sub>0h\<^sub>1.p\<^sub>0, f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine] = can (((\<^bold>\<langle>g\<^bold>\<rangle> \<^bold>\<star> (\<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) ((\<^bold>\<langle>g\<^bold>\<rangle> \<^bold>\<star> (\<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>)
[PROOF STEP]
also
[PROOF STATE]
proof (state)
this:
\<a>\<^sup>-\<^sup>1[g \<star> (h \<star> tab\<^sub>0 h) \<star> g\<^sub>0h\<^sub>1.p\<^sub>0, f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine] = \<lbrace>\<^bold>\<a>\<^sup>-\<^sup>1\<^bold>[\<^bold>\<langle>g\<^bold>\<rangle> \<^bold>\<star> (\<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>, \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>, \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>\<^bold>]\<rbrace>
goal (1 subgoal):
1. \<a>\<^sup>-\<^sup>1[g \<star> (h \<star> tab\<^sub>0 h) \<star> g\<^sub>0h\<^sub>1.p\<^sub>0, f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine] = can (((\<^bold>\<langle>g\<^bold>\<rangle> \<^bold>\<star> (\<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) ((\<^bold>\<langle>g\<^bold>\<rangle> \<^bold>\<star> (\<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>)
[PROOF STEP]
have "... = can (((\<^bold>\<langle>g\<^bold>\<rangle> \<^bold>\<star> (\<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>Tgh.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TfTgh.p\<^sub>0\<^bold>\<rangle>)
\<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>)
((\<^bold>\<langle>g\<^bold>\<rangle> \<^bold>\<star> (\<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>Tgh.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TfTgh.p\<^sub>0\<^bold>\<rangle>
\<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>)"
[PROOF STATE]
proof (prove)
goal (1 subgoal):
1. \<lbrace>\<^bold>\<a>\<^sup>-\<^sup>1\<^bold>[\<^bold>\<langle>g\<^bold>\<rangle> \<^bold>\<star> (\<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>, \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>, \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>\<^bold>]\<rbrace> = can (((\<^bold>\<langle>g\<^bold>\<rangle> \<^bold>\<star> (\<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) ((\<^bold>\<langle>g\<^bold>\<rangle> \<^bold>\<star> (\<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>)
[PROOF STEP]
unfolding can_def
[PROOF STATE]
proof (prove)
goal (1 subgoal):
1. \<lbrace>\<^bold>\<a>\<^sup>-\<^sup>1\<^bold>[\<^bold>\<langle>g\<^bold>\<rangle> \<^bold>\<star> (\<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>, \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>, \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>\<^bold>]\<rbrace> = \<lbrace>E.Inv ((((\<^bold>\<langle>g\<^bold>\<rangle> \<^bold>\<star> (\<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>)\<^bold>\<down>) \<^bold>\<cdot> ((\<^bold>\<langle>g\<^bold>\<rangle> \<^bold>\<star> (\<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>)\<^bold>\<down>\<rbrace>
[PROOF STEP]
using gh
[PROOF STATE]
proof (prove)
using this:
src g = trg h
goal (1 subgoal):
1. \<lbrace>\<^bold>\<a>\<^sup>-\<^sup>1\<^bold>[\<^bold>\<langle>g\<^bold>\<rangle> \<^bold>\<star> (\<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>, \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>, \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>\<^bold>]\<rbrace> = \<lbrace>E.Inv ((((\<^bold>\<langle>g\<^bold>\<rangle> \<^bold>\<star> (\<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>)\<^bold>\<down>) \<^bold>\<cdot> ((\<^bold>\<langle>g\<^bold>\<rangle> \<^bold>\<star> (\<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>)\<^bold>\<down>\<rbrace>
[PROOF STEP]
apply (intro E.eval_eqI)
[PROOF STATE]
proof (prove)
goal (2 subgoals):
1. src g = trg h \<Longrightarrow> E.VPar \<^bold>\<a>\<^sup>-\<^sup>1\<^bold>[\<^bold>\<langle>g\<^bold>\<rangle> \<^bold>\<star> (\<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>, \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>, \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>\<^bold>] (E.Inv ((((\<^bold>\<langle>g\<^bold>\<rangle> \<^bold>\<star> (\<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>)\<^bold>\<down>) \<^bold>\<cdot> ((\<^bold>\<langle>g\<^bold>\<rangle> \<^bold>\<star> (\<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>)\<^bold>\<down>)
2. src g = trg h \<Longrightarrow> \<^bold>\<lfloor>\<^bold>\<a>\<^sup>-\<^sup>1\<^bold>[\<^bold>\<langle>g\<^bold>\<rangle> \<^bold>\<star> (\<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>, \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>, \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>\<^bold>]\<^bold>\<rfloor> = \<^bold>\<lfloor>E.Inv ((((\<^bold>\<langle>g\<^bold>\<rangle> \<^bold>\<star> (\<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>)\<^bold>\<down>) \<^bold>\<cdot> ((\<^bold>\<langle>g\<^bold>\<rangle> \<^bold>\<star> (\<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>)\<^bold>\<down>\<^bold>\<rfloor>
[PROOF STEP]
by simp_all
[PROOF STATE]
proof (state)
this:
\<lbrace>\<^bold>\<a>\<^sup>-\<^sup>1\<^bold>[\<^bold>\<langle>g\<^bold>\<rangle> \<^bold>\<star> (\<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>, \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>, \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>\<^bold>]\<rbrace> = can (((\<^bold>\<langle>g\<^bold>\<rangle> \<^bold>\<star> (\<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) ((\<^bold>\<langle>g\<^bold>\<rangle> \<^bold>\<star> (\<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>)
goal (1 subgoal):
1. \<a>\<^sup>-\<^sup>1[g \<star> (h \<star> tab\<^sub>0 h) \<star> g\<^sub>0h\<^sub>1.p\<^sub>0, f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine] = can (((\<^bold>\<langle>g\<^bold>\<rangle> \<^bold>\<star> (\<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) ((\<^bold>\<langle>g\<^bold>\<rangle> \<^bold>\<star> (\<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>)
[PROOF STEP]
finally
[PROOF STATE]
proof (chain)
picking this:
\<a>\<^sup>-\<^sup>1[g \<star> (h \<star> tab\<^sub>0 h) \<star> g\<^sub>0h\<^sub>1.p\<^sub>0, f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine] = can (((\<^bold>\<langle>g\<^bold>\<rangle> \<^bold>\<star> (\<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) ((\<^bold>\<langle>g\<^bold>\<rangle> \<^bold>\<star> (\<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>)
[PROOF STEP]
show ?thesis
[PROOF STATE]
proof (prove)
using this:
\<a>\<^sup>-\<^sup>1[g \<star> (h \<star> tab\<^sub>0 h) \<star> g\<^sub>0h\<^sub>1.p\<^sub>0, f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine] = can (((\<^bold>\<langle>g\<^bold>\<rangle> \<^bold>\<star> (\<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) ((\<^bold>\<langle>g\<^bold>\<rangle> \<^bold>\<star> (\<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>)
goal (1 subgoal):
1. \<a>\<^sup>-\<^sup>1[g \<star> (h \<star> tab\<^sub>0 h) \<star> g\<^sub>0h\<^sub>1.p\<^sub>0, f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine] = can (((\<^bold>\<langle>g\<^bold>\<rangle> \<^bold>\<star> (\<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) ((\<^bold>\<langle>g\<^bold>\<rangle> \<^bold>\<star> (\<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>)
[PROOF STEP]
by blast
[PROOF STATE]
proof (state)
this:
\<a>\<^sup>-\<^sup>1[g \<star> (h \<star> tab\<^sub>0 h) \<star> g\<^sub>0h\<^sub>1.p\<^sub>0, f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine] = can (((\<^bold>\<langle>g\<^bold>\<rangle> \<^bold>\<star> (\<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) ((\<^bold>\<langle>g\<^bold>\<rangle> \<^bold>\<star> (\<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>)
goal:
No subgoals!
[PROOF STEP]
qed
[PROOF STATE]
proof (state)
this:
\<a>\<^sup>-\<^sup>1[g \<star> (h \<star> tab\<^sub>0 h) \<star> g\<^sub>0h\<^sub>1.p\<^sub>0, f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine] = can (((\<^bold>\<langle>g\<^bold>\<rangle> \<^bold>\<star> (\<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) ((\<^bold>\<langle>g\<^bold>\<rangle> \<^bold>\<star> (\<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>)
goal (1 subgoal):
1. can (\<^bold>\<langle>g\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> ((\<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) (((\<^bold>\<langle>g\<^bold>\<rangle> \<^bold>\<star> (\<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) \<cdot> \<a>\<^sup>-\<^sup>1[g \<star> (h \<star> tab\<^sub>0 h) \<star> g\<^sub>0h\<^sub>1.p\<^sub>0, f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine] \<cdot> \<a>\<^sup>-\<^sup>1[g, (h \<star> tab\<^sub>0 h) \<star> g\<^sub>0h\<^sub>1.p\<^sub>0, f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine] = g \<star> can (\<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> ((\<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) (((\<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>)
[PROOF STEP]
moreover
[PROOF STATE]
proof (state)
this:
\<a>\<^sup>-\<^sup>1[g \<star> (h \<star> tab\<^sub>0 h) \<star> g\<^sub>0h\<^sub>1.p\<^sub>0, f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine] = can (((\<^bold>\<langle>g\<^bold>\<rangle> \<^bold>\<star> (\<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) ((\<^bold>\<langle>g\<^bold>\<rangle> \<^bold>\<star> (\<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>)
goal (1 subgoal):
1. can (\<^bold>\<langle>g\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> ((\<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) (((\<^bold>\<langle>g\<^bold>\<rangle> \<^bold>\<star> (\<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) \<cdot> \<a>\<^sup>-\<^sup>1[g \<star> (h \<star> tab\<^sub>0 h) \<star> g\<^sub>0h\<^sub>1.p\<^sub>0, f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine] \<cdot> \<a>\<^sup>-\<^sup>1[g, (h \<star> tab\<^sub>0 h) \<star> g\<^sub>0h\<^sub>1.p\<^sub>0, f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine] = g \<star> can (\<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> ((\<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) (((\<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>)
[PROOF STEP]
have "\<a>\<^sup>-\<^sup>1[g, (h \<star> tab\<^sub>0 h) \<star> Tgh.p\<^sub>0, TfTgh.p\<^sub>0 \<star> TTfgh_TfTgh.chine] =
can ((\<^bold>\<langle>g\<^bold>\<rangle> \<^bold>\<star> (\<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>Tgh.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TfTgh.p\<^sub>0\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>)
(\<^bold>\<langle>g\<^bold>\<rangle> \<^bold>\<star> ((\<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>Tgh.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TfTgh.p\<^sub>0\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>)"
[PROOF STATE]
proof (prove)
goal (1 subgoal):
1. \<a>\<^sup>-\<^sup>1[g, (h \<star> tab\<^sub>0 h) \<star> g\<^sub>0h\<^sub>1.p\<^sub>0, f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine] = can ((\<^bold>\<langle>g\<^bold>\<rangle> \<^bold>\<star> (\<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) (\<^bold>\<langle>g\<^bold>\<rangle> \<^bold>\<star> ((\<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>)
[PROOF STEP]
proof -
[PROOF STATE]
proof (state)
goal (1 subgoal):
1. \<a>\<^sup>-\<^sup>1[g, (h \<star> tab\<^sub>0 h) \<star> g\<^sub>0h\<^sub>1.p\<^sub>0, f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine] = can ((\<^bold>\<langle>g\<^bold>\<rangle> \<^bold>\<star> (\<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) (\<^bold>\<langle>g\<^bold>\<rangle> \<^bold>\<star> ((\<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>)
[PROOF STEP]
have "\<a>\<^sup>-\<^sup>1[g, (h \<star> tab\<^sub>0 h) \<star> Tgh.p\<^sub>0, TfTgh.p\<^sub>0 \<star> TTfgh_TfTgh.chine] =
\<lbrace>\<^bold>\<a>\<^sup>-\<^sup>1\<^bold>[\<^bold>\<langle>g\<^bold>\<rangle>, (\<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>Tgh.p\<^sub>0\<^bold>\<rangle>, \<^bold>\<langle>TfTgh.p\<^sub>0\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>\<^bold>]\<rbrace>"
[PROOF STATE]
proof (prove)
goal (1 subgoal):
1. \<a>\<^sup>-\<^sup>1[g, (h \<star> tab\<^sub>0 h) \<star> g\<^sub>0h\<^sub>1.p\<^sub>0, f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine] = \<lbrace>\<^bold>\<a>\<^sup>-\<^sup>1\<^bold>[\<^bold>\<langle>g\<^bold>\<rangle>, (\<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>, \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>\<^bold>]\<rbrace>
[PROOF STEP]
using gh f\<^sub>0gh\<^sub>1.p\<^sub>0_simps canI_associator_0 \<a>'_def \<alpha>_def
[PROOF STATE]
proof (prove)
using this:
src g = trg h
trg f\<^sub>0gh\<^sub>1.p\<^sub>0 = src (tab\<^sub>1 g \<star> g\<^sub>0h\<^sub>1.p\<^sub>1)
\<lbrakk>ide ?f; ide ?g; ide ?h; src ?f = trg ?g; src ?g = trg ?h\<rbrakk> \<Longrightarrow> \<a>[?f, ?g, ?h] = can (\<^bold>\<langle>?f\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>?g\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>?h\<^bold>\<rangle>) ((\<^bold>\<langle>?f\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>?g\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>?h\<^bold>\<rangle>)
\<a>\<^sup>-\<^sup>1[?\<mu>, ?\<nu>, ?\<tau>] \<equiv> \<alpha>' (?\<mu>, ?\<nu>, ?\<tau>)
\<alpha> ?\<mu>\<nu>\<tau> \<equiv> \<a>[fst ?\<mu>\<nu>\<tau>, fst (snd ?\<mu>\<nu>\<tau>), snd (snd ?\<mu>\<nu>\<tau>)]
goal (1 subgoal):
1. \<a>\<^sup>-\<^sup>1[g, (h \<star> tab\<^sub>0 h) \<star> g\<^sub>0h\<^sub>1.p\<^sub>0, f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine] = \<lbrace>\<^bold>\<a>\<^sup>-\<^sup>1\<^bold>[\<^bold>\<langle>g\<^bold>\<rangle>, (\<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>, \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>\<^bold>]\<rbrace>
[PROOF STEP]
by simp
[PROOF STATE]
proof (state)
this:
\<a>\<^sup>-\<^sup>1[g, (h \<star> tab\<^sub>0 h) \<star> g\<^sub>0h\<^sub>1.p\<^sub>0, f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine] = \<lbrace>\<^bold>\<a>\<^sup>-\<^sup>1\<^bold>[\<^bold>\<langle>g\<^bold>\<rangle>, (\<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>, \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>\<^bold>]\<rbrace>
goal (1 subgoal):
1. \<a>\<^sup>-\<^sup>1[g, (h \<star> tab\<^sub>0 h) \<star> g\<^sub>0h\<^sub>1.p\<^sub>0, f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine] = can ((\<^bold>\<langle>g\<^bold>\<rangle> \<^bold>\<star> (\<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) (\<^bold>\<langle>g\<^bold>\<rangle> \<^bold>\<star> ((\<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>)
[PROOF STEP]
also
[PROOF STATE]
proof (state)
this:
\<a>\<^sup>-\<^sup>1[g, (h \<star> tab\<^sub>0 h) \<star> g\<^sub>0h\<^sub>1.p\<^sub>0, f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine] = \<lbrace>\<^bold>\<a>\<^sup>-\<^sup>1\<^bold>[\<^bold>\<langle>g\<^bold>\<rangle>, (\<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>, \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>\<^bold>]\<rbrace>
goal (1 subgoal):
1. \<a>\<^sup>-\<^sup>1[g, (h \<star> tab\<^sub>0 h) \<star> g\<^sub>0h\<^sub>1.p\<^sub>0, f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine] = can ((\<^bold>\<langle>g\<^bold>\<rangle> \<^bold>\<star> (\<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) (\<^bold>\<langle>g\<^bold>\<rangle> \<^bold>\<star> ((\<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>)
[PROOF STEP]
have "... = can ((\<^bold>\<langle>g\<^bold>\<rangle> \<^bold>\<star> (\<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>Tgh.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TfTgh.p\<^sub>0\<^bold>\<rangle>
\<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>)
(\<^bold>\<langle>g\<^bold>\<rangle> \<^bold>\<star> ((\<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>Tgh.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TfTgh.p\<^sub>0\<^bold>\<rangle>
\<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>)"
[PROOF STATE]
proof (prove)
goal (1 subgoal):
1. \<lbrace>\<^bold>\<a>\<^sup>-\<^sup>1\<^bold>[\<^bold>\<langle>g\<^bold>\<rangle>, (\<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>, \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>\<^bold>]\<rbrace> = can ((\<^bold>\<langle>g\<^bold>\<rangle> \<^bold>\<star> (\<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) (\<^bold>\<langle>g\<^bold>\<rangle> \<^bold>\<star> ((\<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>)
[PROOF STEP]
unfolding can_def
[PROOF STATE]
proof (prove)
goal (1 subgoal):
1. \<lbrace>\<^bold>\<a>\<^sup>-\<^sup>1\<^bold>[\<^bold>\<langle>g\<^bold>\<rangle>, (\<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>, \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>\<^bold>]\<rbrace> = \<lbrace>E.Inv (((\<^bold>\<langle>g\<^bold>\<rangle> \<^bold>\<star> (\<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>)\<^bold>\<down>) \<^bold>\<cdot> (\<^bold>\<langle>g\<^bold>\<rangle> \<^bold>\<star> ((\<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>)\<^bold>\<down>\<rbrace>
[PROOF STEP]
using gh
[PROOF STATE]
proof (prove)
using this:
src g = trg h
goal (1 subgoal):
1. \<lbrace>\<^bold>\<a>\<^sup>-\<^sup>1\<^bold>[\<^bold>\<langle>g\<^bold>\<rangle>, (\<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>, \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>\<^bold>]\<rbrace> = \<lbrace>E.Inv (((\<^bold>\<langle>g\<^bold>\<rangle> \<^bold>\<star> (\<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>)\<^bold>\<down>) \<^bold>\<cdot> (\<^bold>\<langle>g\<^bold>\<rangle> \<^bold>\<star> ((\<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>)\<^bold>\<down>\<rbrace>
[PROOF STEP]
apply (intro E.eval_eqI)
[PROOF STATE]
proof (prove)
goal (2 subgoals):
1. src g = trg h \<Longrightarrow> E.VPar \<^bold>\<a>\<^sup>-\<^sup>1\<^bold>[\<^bold>\<langle>g\<^bold>\<rangle>, (\<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>, \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>\<^bold>] (E.Inv (((\<^bold>\<langle>g\<^bold>\<rangle> \<^bold>\<star> (\<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>)\<^bold>\<down>) \<^bold>\<cdot> (\<^bold>\<langle>g\<^bold>\<rangle> \<^bold>\<star> ((\<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>)\<^bold>\<down>)
2. src g = trg h \<Longrightarrow> \<^bold>\<lfloor>\<^bold>\<a>\<^sup>-\<^sup>1\<^bold>[\<^bold>\<langle>g\<^bold>\<rangle>, (\<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>, \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>\<^bold>]\<^bold>\<rfloor> = \<^bold>\<lfloor>E.Inv (((\<^bold>\<langle>g\<^bold>\<rangle> \<^bold>\<star> (\<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>)\<^bold>\<down>) \<^bold>\<cdot> (\<^bold>\<langle>g\<^bold>\<rangle> \<^bold>\<star> ((\<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>)\<^bold>\<down>\<^bold>\<rfloor>
[PROOF STEP]
by simp_all
[PROOF STATE]
proof (state)
this:
\<lbrace>\<^bold>\<a>\<^sup>-\<^sup>1\<^bold>[\<^bold>\<langle>g\<^bold>\<rangle>, (\<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>, \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>\<^bold>]\<rbrace> = can ((\<^bold>\<langle>g\<^bold>\<rangle> \<^bold>\<star> (\<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) (\<^bold>\<langle>g\<^bold>\<rangle> \<^bold>\<star> ((\<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>)
goal (1 subgoal):
1. \<a>\<^sup>-\<^sup>1[g, (h \<star> tab\<^sub>0 h) \<star> g\<^sub>0h\<^sub>1.p\<^sub>0, f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine] = can ((\<^bold>\<langle>g\<^bold>\<rangle> \<^bold>\<star> (\<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) (\<^bold>\<langle>g\<^bold>\<rangle> \<^bold>\<star> ((\<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>)
[PROOF STEP]
finally
[PROOF STATE]
proof (chain)
picking this:
\<a>\<^sup>-\<^sup>1[g, (h \<star> tab\<^sub>0 h) \<star> g\<^sub>0h\<^sub>1.p\<^sub>0, f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine] = can ((\<^bold>\<langle>g\<^bold>\<rangle> \<^bold>\<star> (\<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) (\<^bold>\<langle>g\<^bold>\<rangle> \<^bold>\<star> ((\<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>)
[PROOF STEP]
show ?thesis
[PROOF STATE]
proof (prove)
using this:
\<a>\<^sup>-\<^sup>1[g, (h \<star> tab\<^sub>0 h) \<star> g\<^sub>0h\<^sub>1.p\<^sub>0, f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine] = can ((\<^bold>\<langle>g\<^bold>\<rangle> \<^bold>\<star> (\<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) (\<^bold>\<langle>g\<^bold>\<rangle> \<^bold>\<star> ((\<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>)
goal (1 subgoal):
1. \<a>\<^sup>-\<^sup>1[g, (h \<star> tab\<^sub>0 h) \<star> g\<^sub>0h\<^sub>1.p\<^sub>0, f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine] = can ((\<^bold>\<langle>g\<^bold>\<rangle> \<^bold>\<star> (\<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) (\<^bold>\<langle>g\<^bold>\<rangle> \<^bold>\<star> ((\<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>)
[PROOF STEP]
by blast
[PROOF STATE]
proof (state)
this:
\<a>\<^sup>-\<^sup>1[g, (h \<star> tab\<^sub>0 h) \<star> g\<^sub>0h\<^sub>1.p\<^sub>0, f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine] = can ((\<^bold>\<langle>g\<^bold>\<rangle> \<^bold>\<star> (\<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) (\<^bold>\<langle>g\<^bold>\<rangle> \<^bold>\<star> ((\<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>)
goal:
No subgoals!
[PROOF STEP]
qed
[PROOF STATE]
proof (state)
this:
\<a>\<^sup>-\<^sup>1[g, (h \<star> tab\<^sub>0 h) \<star> g\<^sub>0h\<^sub>1.p\<^sub>0, f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine] = can ((\<^bold>\<langle>g\<^bold>\<rangle> \<^bold>\<star> (\<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) (\<^bold>\<langle>g\<^bold>\<rangle> \<^bold>\<star> ((\<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>)
goal (1 subgoal):
1. can (\<^bold>\<langle>g\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> ((\<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) (((\<^bold>\<langle>g\<^bold>\<rangle> \<^bold>\<star> (\<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) \<cdot> \<a>\<^sup>-\<^sup>1[g \<star> (h \<star> tab\<^sub>0 h) \<star> g\<^sub>0h\<^sub>1.p\<^sub>0, f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine] \<cdot> \<a>\<^sup>-\<^sup>1[g, (h \<star> tab\<^sub>0 h) \<star> g\<^sub>0h\<^sub>1.p\<^sub>0, f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine] = g \<star> can (\<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> ((\<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) (((\<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>)
[PROOF STEP]
ultimately
[PROOF STATE]
proof (chain)
picking this:
\<a>\<^sup>-\<^sup>1[g \<star> (h \<star> tab\<^sub>0 h) \<star> g\<^sub>0h\<^sub>1.p\<^sub>0, f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine] = can (((\<^bold>\<langle>g\<^bold>\<rangle> \<^bold>\<star> (\<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) ((\<^bold>\<langle>g\<^bold>\<rangle> \<^bold>\<star> (\<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>)
\<a>\<^sup>-\<^sup>1[g, (h \<star> tab\<^sub>0 h) \<star> g\<^sub>0h\<^sub>1.p\<^sub>0, f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine] = can ((\<^bold>\<langle>g\<^bold>\<rangle> \<^bold>\<star> (\<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) (\<^bold>\<langle>g\<^bold>\<rangle> \<^bold>\<star> ((\<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>)
[PROOF STEP]
show ?thesis
[PROOF STATE]
proof (prove)
using this:
\<a>\<^sup>-\<^sup>1[g \<star> (h \<star> tab\<^sub>0 h) \<star> g\<^sub>0h\<^sub>1.p\<^sub>0, f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine] = can (((\<^bold>\<langle>g\<^bold>\<rangle> \<^bold>\<star> (\<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) ((\<^bold>\<langle>g\<^bold>\<rangle> \<^bold>\<star> (\<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>)
\<a>\<^sup>-\<^sup>1[g, (h \<star> tab\<^sub>0 h) \<star> g\<^sub>0h\<^sub>1.p\<^sub>0, f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine] = can ((\<^bold>\<langle>g\<^bold>\<rangle> \<^bold>\<star> (\<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) (\<^bold>\<langle>g\<^bold>\<rangle> \<^bold>\<star> ((\<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>)
goal (1 subgoal):
1. can (\<^bold>\<langle>g\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> ((\<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) (((\<^bold>\<langle>g\<^bold>\<rangle> \<^bold>\<star> (\<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) \<cdot> \<a>\<^sup>-\<^sup>1[g \<star> (h \<star> tab\<^sub>0 h) \<star> g\<^sub>0h\<^sub>1.p\<^sub>0, f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine] \<cdot> \<a>\<^sup>-\<^sup>1[g, (h \<star> tab\<^sub>0 h) \<star> g\<^sub>0h\<^sub>1.p\<^sub>0, f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine] = g \<star> can (\<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> ((\<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) (((\<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>)
[PROOF STEP]
using gh whisker_can_left_0
[PROOF STATE]
proof (prove)
using this:
\<a>\<^sup>-\<^sup>1[g \<star> (h \<star> tab\<^sub>0 h) \<star> g\<^sub>0h\<^sub>1.p\<^sub>0, f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine] = can (((\<^bold>\<langle>g\<^bold>\<rangle> \<^bold>\<star> (\<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) ((\<^bold>\<langle>g\<^bold>\<rangle> \<^bold>\<star> (\<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>)
\<a>\<^sup>-\<^sup>1[g, (h \<star> tab\<^sub>0 h) \<star> g\<^sub>0h\<^sub>1.p\<^sub>0, f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine] = can ((\<^bold>\<langle>g\<^bold>\<rangle> \<^bold>\<star> (\<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) (\<^bold>\<langle>g\<^bold>\<rangle> \<^bold>\<star> ((\<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>)
src g = trg h
\<lbrakk>E.Ide ?t; E.Ide ?u; \<^bold>\<lfloor>?t\<^bold>\<rfloor> = \<^bold>\<lfloor>?u\<^bold>\<rfloor>; ide ?g; E.Trg ?t = \<^bold>\<langle>src ?g\<^bold>\<rangle>\<^sub>0\<rbrakk> \<Longrightarrow> ?g \<star> can ?u ?t = can (\<^bold>\<langle>?g\<^bold>\<rangle> \<^bold>\<star> ?u) (\<^bold>\<langle>?g\<^bold>\<rangle> \<^bold>\<star> ?t)
goal (1 subgoal):
1. can (\<^bold>\<langle>g\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> ((\<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) (((\<^bold>\<langle>g\<^bold>\<rangle> \<^bold>\<star> (\<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) \<cdot> \<a>\<^sup>-\<^sup>1[g \<star> (h \<star> tab\<^sub>0 h) \<star> g\<^sub>0h\<^sub>1.p\<^sub>0, f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine] \<cdot> \<a>\<^sup>-\<^sup>1[g, (h \<star> tab\<^sub>0 h) \<star> g\<^sub>0h\<^sub>1.p\<^sub>0, f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine] = g \<star> can (\<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> ((\<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) (((\<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>)
[PROOF STEP]
by simp
[PROOF STATE]
proof (state)
this:
can (\<^bold>\<langle>g\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> ((\<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) (((\<^bold>\<langle>g\<^bold>\<rangle> \<^bold>\<star> (\<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) \<cdot> \<a>\<^sup>-\<^sup>1[g \<star> (h \<star> tab\<^sub>0 h) \<star> g\<^sub>0h\<^sub>1.p\<^sub>0, f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine] \<cdot> \<a>\<^sup>-\<^sup>1[g, (h \<star> tab\<^sub>0 h) \<star> g\<^sub>0h\<^sub>1.p\<^sub>0, f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine] = g \<star> can (\<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> ((\<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) (((\<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>)
goal:
No subgoals!
[PROOF STEP]
qed
[PROOF STATE]
proof (state)
this:
can (\<^bold>\<langle>g\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> ((\<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) (((\<^bold>\<langle>g\<^bold>\<rangle> \<^bold>\<star> (\<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) \<cdot> \<a>\<^sup>-\<^sup>1[g \<star> (h \<star> tab\<^sub>0 h) \<star> g\<^sub>0h\<^sub>1.p\<^sub>0, f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine] \<cdot> \<a>\<^sup>-\<^sup>1[g, (h \<star> tab\<^sub>0 h) \<star> g\<^sub>0h\<^sub>1.p\<^sub>0, f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine] = g \<star> can (\<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> ((\<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) (((\<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>)
goal (1 subgoal):
1. (g \<star> h \<star> TTfgh_TfTgh.the_\<theta>) \<cdot> (can (\<^bold>\<langle>g\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> ((\<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) (((\<^bold>\<langle>g\<^bold>\<rangle> \<^bold>\<star> (\<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) \<cdot> \<a>\<^sup>-\<^sup>1[g \<star> (h \<star> tab\<^sub>0 h) \<star> g\<^sub>0h\<^sub>1.p\<^sub>0, f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine] \<cdot> \<a>\<^sup>-\<^sup>1[g, (h \<star> tab\<^sub>0 h) \<star> g\<^sub>0h\<^sub>1.p\<^sub>0, f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine]) \<cdot> (g \<star> (h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine) \<cdot> (g \<star> g\<^sub>0h\<^sub>1.\<phi> \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine) \<cdot> (\<a>[g, tab\<^sub>0 g \<star> g\<^sub>0h\<^sub>1.p\<^sub>1, f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine] \<cdot> \<a>[g \<star> tab\<^sub>0 g \<star> g\<^sub>0h\<^sub>1.p\<^sub>1, f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine] \<cdot> ((\<a>[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> \<a>\<^sup>-\<^sup>1[(g \<star> tab\<^sub>0 g) \<star> g\<^sub>0h\<^sub>1.p\<^sub>1, f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine] \<cdot> \<a>\<^sup>-\<^sup>1[g \<star> tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1, f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine] \<cdot> \<a>\<^sup>-\<^sup>1[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1 \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine]) \<cdot> \<a>[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1 \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine] \<cdot> (g.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>1 \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine) \<cdot> \<beta>\<^sub>g = ((g \<star> h \<star> TTfgh_TfTgh.the_\<theta>) \<cdot> (g \<star> can (\<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> ((\<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) (((\<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>)) \<cdot> (g \<star> (h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine) \<cdot> (g \<star> g\<^sub>0h\<^sub>1.\<phi> \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine) \<cdot> (g \<star> \<a>\<^sup>-\<^sup>1[tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1, f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine])) \<cdot> \<a>[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1 \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine] \<cdot> (g.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>1 \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine) \<cdot> \<beta>\<^sub>g
[PROOF STEP]
moreover
[PROOF STATE]
proof (state)
this:
can (\<^bold>\<langle>g\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> ((\<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) (((\<^bold>\<langle>g\<^bold>\<rangle> \<^bold>\<star> (\<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) \<cdot> \<a>\<^sup>-\<^sup>1[g \<star> (h \<star> tab\<^sub>0 h) \<star> g\<^sub>0h\<^sub>1.p\<^sub>0, f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine] \<cdot> \<a>\<^sup>-\<^sup>1[g, (h \<star> tab\<^sub>0 h) \<star> g\<^sub>0h\<^sub>1.p\<^sub>0, f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine] = g \<star> can (\<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> ((\<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) (((\<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>)
goal (1 subgoal):
1. (g \<star> h \<star> TTfgh_TfTgh.the_\<theta>) \<cdot> (can (\<^bold>\<langle>g\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> ((\<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) (((\<^bold>\<langle>g\<^bold>\<rangle> \<^bold>\<star> (\<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) \<cdot> \<a>\<^sup>-\<^sup>1[g \<star> (h \<star> tab\<^sub>0 h) \<star> g\<^sub>0h\<^sub>1.p\<^sub>0, f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine] \<cdot> \<a>\<^sup>-\<^sup>1[g, (h \<star> tab\<^sub>0 h) \<star> g\<^sub>0h\<^sub>1.p\<^sub>0, f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine]) \<cdot> (g \<star> (h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine) \<cdot> (g \<star> g\<^sub>0h\<^sub>1.\<phi> \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine) \<cdot> (\<a>[g, tab\<^sub>0 g \<star> g\<^sub>0h\<^sub>1.p\<^sub>1, f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine] \<cdot> \<a>[g \<star> tab\<^sub>0 g \<star> g\<^sub>0h\<^sub>1.p\<^sub>1, f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine] \<cdot> ((\<a>[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> \<a>\<^sup>-\<^sup>1[(g \<star> tab\<^sub>0 g) \<star> g\<^sub>0h\<^sub>1.p\<^sub>1, f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine] \<cdot> \<a>\<^sup>-\<^sup>1[g \<star> tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1, f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine] \<cdot> \<a>\<^sup>-\<^sup>1[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1 \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine]) \<cdot> \<a>[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1 \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine] \<cdot> (g.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>1 \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine) \<cdot> \<beta>\<^sub>g = ((g \<star> h \<star> TTfgh_TfTgh.the_\<theta>) \<cdot> (g \<star> can (\<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> ((\<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) (((\<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>)) \<cdot> (g \<star> (h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine) \<cdot> (g \<star> g\<^sub>0h\<^sub>1.\<phi> \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine) \<cdot> (g \<star> \<a>\<^sup>-\<^sup>1[tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1, f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine])) \<cdot> \<a>[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1 \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine] \<cdot> (g.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>1 \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine) \<cdot> \<beta>\<^sub>g
[PROOF STEP]
have "\<a>[g, tab\<^sub>0 g \<star> Tgh.p\<^sub>1, TfTgh.p\<^sub>0 \<star> TTfgh_TfTgh.chine] \<cdot>
\<a>[g \<star> tab\<^sub>0 g \<star> Tgh.p\<^sub>1, TfTgh.p\<^sub>0, TTfgh_TfTgh.chine] \<cdot>
((\<a>[g, tab\<^sub>0 g, Tgh.p\<^sub>1] \<star> TfTgh.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot>
\<a>\<^sup>-\<^sup>1[(g \<star> tab\<^sub>0 g) \<star> Tgh.p\<^sub>1, TfTgh.p\<^sub>0, TTfgh_TfTgh.chine] \<cdot>
\<a>\<^sup>-\<^sup>1[g \<star> tab\<^sub>0 g, Tgh.p\<^sub>1, TfTgh.p\<^sub>0 \<star> TTfgh_TfTgh.chine] \<cdot>
\<a>\<^sup>-\<^sup>1[g, tab\<^sub>0 g, Tgh.p\<^sub>1 \<star> TfTgh.p\<^sub>0 \<star> TTfgh_TfTgh.chine] =
g \<star> \<a>\<^sup>-\<^sup>1[tab\<^sub>0 g, Tgh.p\<^sub>1, TfTgh.p\<^sub>0 \<star> TTfgh_TfTgh.chine]"
[PROOF STATE]
proof (prove)
goal (1 subgoal):
1. \<a>[g, tab\<^sub>0 g \<star> g\<^sub>0h\<^sub>1.p\<^sub>1, f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine] \<cdot> \<a>[g \<star> tab\<^sub>0 g \<star> g\<^sub>0h\<^sub>1.p\<^sub>1, f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine] \<cdot> ((\<a>[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> \<a>\<^sup>-\<^sup>1[(g \<star> tab\<^sub>0 g) \<star> g\<^sub>0h\<^sub>1.p\<^sub>1, f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine] \<cdot> \<a>\<^sup>-\<^sup>1[g \<star> tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1, f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine] \<cdot> \<a>\<^sup>-\<^sup>1[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1 \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine] = g \<star> \<a>\<^sup>-\<^sup>1[tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1, f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine]
[PROOF STEP]
proof -
[PROOF STATE]
proof (state)
goal (1 subgoal):
1. \<a>[g, tab\<^sub>0 g \<star> g\<^sub>0h\<^sub>1.p\<^sub>1, f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine] \<cdot> \<a>[g \<star> tab\<^sub>0 g \<star> g\<^sub>0h\<^sub>1.p\<^sub>1, f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine] \<cdot> ((\<a>[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> \<a>\<^sup>-\<^sup>1[(g \<star> tab\<^sub>0 g) \<star> g\<^sub>0h\<^sub>1.p\<^sub>1, f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine] \<cdot> \<a>\<^sup>-\<^sup>1[g \<star> tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1, f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine] \<cdot> \<a>\<^sup>-\<^sup>1[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1 \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine] = g \<star> \<a>\<^sup>-\<^sup>1[tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1, f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine]
[PROOF STEP]
have "\<a>[g, tab\<^sub>0 g \<star> Tgh.p\<^sub>1, TfTgh.p\<^sub>0 \<star> TTfgh_TfTgh.chine] \<cdot>
\<a>[g \<star> tab\<^sub>0 g \<star> Tgh.p\<^sub>1, TfTgh.p\<^sub>0, TTfgh_TfTgh.chine] \<cdot>
((\<a>[g, tab\<^sub>0 g, Tgh.p\<^sub>1] \<star> TfTgh.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot>
\<a>\<^sup>-\<^sup>1[(g \<star> tab\<^sub>0 g) \<star> Tgh.p\<^sub>1, TfTgh.p\<^sub>0, TTfgh_TfTgh.chine] \<cdot>
\<a>\<^sup>-\<^sup>1[g \<star> tab\<^sub>0 g, Tgh.p\<^sub>1, TfTgh.p\<^sub>0 \<star> TTfgh_TfTgh.chine] \<cdot>
\<a>\<^sup>-\<^sup>1[g, tab\<^sub>0 g, Tgh.p\<^sub>1 \<star> TfTgh.p\<^sub>0 \<star> TTfgh_TfTgh.chine] =
\<lbrace>\<^bold>\<a>\<^bold>[\<^bold>\<langle>g\<^bold>\<rangle>, \<^bold>\<langle>tab\<^sub>0 g\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>Tgh.p\<^sub>1\<^bold>\<rangle>, \<^bold>\<langle>TfTgh.p\<^sub>0\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>\<^bold>] \<^bold>\<cdot>
\<^bold>\<a>\<^bold>[\<^bold>\<langle>g\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>tab\<^sub>0 g\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>Tgh.p\<^sub>1\<^bold>\<rangle>, \<^bold>\<langle>TfTgh.p\<^sub>0\<^bold>\<rangle>, \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>\<^bold>] \<^bold>\<cdot>
((\<^bold>\<a>\<^bold>[\<^bold>\<langle>g\<^bold>\<rangle>, \<^bold>\<langle>tab\<^sub>0 g\<^bold>\<rangle>, \<^bold>\<langle>Tgh.p\<^sub>1\<^bold>\<rangle>\<^bold>] \<^bold>\<star> \<^bold>\<langle>TfTgh.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) \<^bold>\<cdot>
\<^bold>\<a>\<^sup>-\<^sup>1\<^bold>[(\<^bold>\<langle>g\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>tab\<^sub>0 g\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>Tgh.p\<^sub>1\<^bold>\<rangle>, \<^bold>\<langle>TfTgh.p\<^sub>0\<^bold>\<rangle>, \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>\<^bold>] \<^bold>\<cdot>
\<^bold>\<a>\<^sup>-\<^sup>1\<^bold>[\<^bold>\<langle>g\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>tab\<^sub>0 g\<^bold>\<rangle>, \<^bold>\<langle>Tgh.p\<^sub>1\<^bold>\<rangle>, \<^bold>\<langle>TfTgh.p\<^sub>0\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>\<^bold>] \<^bold>\<cdot>
\<^bold>\<a>\<^sup>-\<^sup>1\<^bold>[\<^bold>\<langle>g\<^bold>\<rangle>, \<^bold>\<langle>tab\<^sub>0 g\<^bold>\<rangle>, \<^bold>\<langle>Tgh.p\<^sub>1\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>TfTgh.p\<^sub>0\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>\<^bold>]\<rbrace>"
[PROOF STATE]
proof (prove)
goal (1 subgoal):
1. \<a>[g, tab\<^sub>0 g \<star> g\<^sub>0h\<^sub>1.p\<^sub>1, f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine] \<cdot> \<a>[g \<star> tab\<^sub>0 g \<star> g\<^sub>0h\<^sub>1.p\<^sub>1, f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine] \<cdot> ((\<a>[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> \<a>\<^sup>-\<^sup>1[(g \<star> tab\<^sub>0 g) \<star> g\<^sub>0h\<^sub>1.p\<^sub>1, f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine] \<cdot> \<a>\<^sup>-\<^sup>1[g \<star> tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1, f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine] \<cdot> \<a>\<^sup>-\<^sup>1[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1 \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine] = \<lbrace>\<^bold>\<a>\<^bold>[\<^bold>\<langle>g\<^bold>\<rangle>, \<^bold>\<langle>tab\<^sub>0 g\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>1\<^bold>\<rangle>, \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>\<^bold>] \<^bold>\<cdot> \<^bold>\<a>\<^bold>[\<^bold>\<langle>g\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>tab\<^sub>0 g\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>1\<^bold>\<rangle>, \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>, \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>\<^bold>] \<^bold>\<cdot> ((\<^bold>\<a>\<^bold>[\<^bold>\<langle>g\<^bold>\<rangle>, \<^bold>\<langle>tab\<^sub>0 g\<^bold>\<rangle>, \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>1\<^bold>\<rangle>\<^bold>] \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) \<^bold>\<cdot> \<^bold>\<a>\<^sup>-\<^sup>1\<^bold>[(\<^bold>\<langle>g\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>tab\<^sub>0 g\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>1\<^bold>\<rangle>, \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>, \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>\<^bold>] \<^bold>\<cdot> \<^bold>\<a>\<^sup>-\<^sup>1\<^bold>[\<^bold>\<langle>g\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>tab\<^sub>0 g\<^bold>\<rangle>, \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>1\<^bold>\<rangle>, \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>\<^bold>] \<^bold>\<cdot> \<^bold>\<a>\<^sup>-\<^sup>1\<^bold>[\<^bold>\<langle>g\<^bold>\<rangle>, \<^bold>\<langle>tab\<^sub>0 g\<^bold>\<rangle>, \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>1\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>\<^bold>]\<rbrace>
[PROOF STEP]
using gh g\<^sub>0h\<^sub>1.p\<^sub>1_simps f\<^sub>0gh\<^sub>1.p\<^sub>0_simps \<a>'_def \<alpha>_def
[PROOF STATE]
proof (prove)
using this:
src g = trg h
trg g\<^sub>0h\<^sub>1.p\<^sub>1 = src (tab\<^sub>0 g)
trg f\<^sub>0gh\<^sub>1.p\<^sub>0 = src (tab\<^sub>1 g \<star> g\<^sub>0h\<^sub>1.p\<^sub>1)
\<a>\<^sup>-\<^sup>1[?\<mu>, ?\<nu>, ?\<tau>] \<equiv> \<alpha>' (?\<mu>, ?\<nu>, ?\<tau>)
\<alpha> ?\<mu>\<nu>\<tau> \<equiv> \<a>[fst ?\<mu>\<nu>\<tau>, fst (snd ?\<mu>\<nu>\<tau>), snd (snd ?\<mu>\<nu>\<tau>)]
goal (1 subgoal):
1. \<a>[g, tab\<^sub>0 g \<star> g\<^sub>0h\<^sub>1.p\<^sub>1, f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine] \<cdot> \<a>[g \<star> tab\<^sub>0 g \<star> g\<^sub>0h\<^sub>1.p\<^sub>1, f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine] \<cdot> ((\<a>[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> \<a>\<^sup>-\<^sup>1[(g \<star> tab\<^sub>0 g) \<star> g\<^sub>0h\<^sub>1.p\<^sub>1, f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine] \<cdot> \<a>\<^sup>-\<^sup>1[g \<star> tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1, f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine] \<cdot> \<a>\<^sup>-\<^sup>1[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1 \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine] = \<lbrace>\<^bold>\<a>\<^bold>[\<^bold>\<langle>g\<^bold>\<rangle>, \<^bold>\<langle>tab\<^sub>0 g\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>1\<^bold>\<rangle>, \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>\<^bold>] \<^bold>\<cdot> \<^bold>\<a>\<^bold>[\<^bold>\<langle>g\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>tab\<^sub>0 g\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>1\<^bold>\<rangle>, \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>, \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>\<^bold>] \<^bold>\<cdot> ((\<^bold>\<a>\<^bold>[\<^bold>\<langle>g\<^bold>\<rangle>, \<^bold>\<langle>tab\<^sub>0 g\<^bold>\<rangle>, \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>1\<^bold>\<rangle>\<^bold>] \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) \<^bold>\<cdot> \<^bold>\<a>\<^sup>-\<^sup>1\<^bold>[(\<^bold>\<langle>g\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>tab\<^sub>0 g\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>1\<^bold>\<rangle>, \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>, \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>\<^bold>] \<^bold>\<cdot> \<^bold>\<a>\<^sup>-\<^sup>1\<^bold>[\<^bold>\<langle>g\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>tab\<^sub>0 g\<^bold>\<rangle>, \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>1\<^bold>\<rangle>, \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>\<^bold>] \<^bold>\<cdot> \<^bold>\<a>\<^sup>-\<^sup>1\<^bold>[\<^bold>\<langle>g\<^bold>\<rangle>, \<^bold>\<langle>tab\<^sub>0 g\<^bold>\<rangle>, \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>1\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>\<^bold>]\<rbrace>
[PROOF STEP]
by simp
[PROOF STATE]
proof (state)
this:
\<a>[g, tab\<^sub>0 g \<star> g\<^sub>0h\<^sub>1.p\<^sub>1, f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine] \<cdot> \<a>[g \<star> tab\<^sub>0 g \<star> g\<^sub>0h\<^sub>1.p\<^sub>1, f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine] \<cdot> ((\<a>[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> \<a>\<^sup>-\<^sup>1[(g \<star> tab\<^sub>0 g) \<star> g\<^sub>0h\<^sub>1.p\<^sub>1, f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine] \<cdot> \<a>\<^sup>-\<^sup>1[g \<star> tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1, f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine] \<cdot> \<a>\<^sup>-\<^sup>1[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1 \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine] = \<lbrace>\<^bold>\<a>\<^bold>[\<^bold>\<langle>g\<^bold>\<rangle>, \<^bold>\<langle>tab\<^sub>0 g\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>1\<^bold>\<rangle>, \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>\<^bold>] \<^bold>\<cdot> \<^bold>\<a>\<^bold>[\<^bold>\<langle>g\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>tab\<^sub>0 g\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>1\<^bold>\<rangle>, \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>, \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>\<^bold>] \<^bold>\<cdot> ((\<^bold>\<a>\<^bold>[\<^bold>\<langle>g\<^bold>\<rangle>, \<^bold>\<langle>tab\<^sub>0 g\<^bold>\<rangle>, \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>1\<^bold>\<rangle>\<^bold>] \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) \<^bold>\<cdot> \<^bold>\<a>\<^sup>-\<^sup>1\<^bold>[(\<^bold>\<langle>g\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>tab\<^sub>0 g\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>1\<^bold>\<rangle>, \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>, \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>\<^bold>] \<^bold>\<cdot> \<^bold>\<a>\<^sup>-\<^sup>1\<^bold>[\<^bold>\<langle>g\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>tab\<^sub>0 g\<^bold>\<rangle>, \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>1\<^bold>\<rangle>, \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>\<^bold>] \<^bold>\<cdot> \<^bold>\<a>\<^sup>-\<^sup>1\<^bold>[\<^bold>\<langle>g\<^bold>\<rangle>, \<^bold>\<langle>tab\<^sub>0 g\<^bold>\<rangle>, \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>1\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>\<^bold>]\<rbrace>
goal (1 subgoal):
1. \<a>[g, tab\<^sub>0 g \<star> g\<^sub>0h\<^sub>1.p\<^sub>1, f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine] \<cdot> \<a>[g \<star> tab\<^sub>0 g \<star> g\<^sub>0h\<^sub>1.p\<^sub>1, f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine] \<cdot> ((\<a>[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> \<a>\<^sup>-\<^sup>1[(g \<star> tab\<^sub>0 g) \<star> g\<^sub>0h\<^sub>1.p\<^sub>1, f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine] \<cdot> \<a>\<^sup>-\<^sup>1[g \<star> tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1, f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine] \<cdot> \<a>\<^sup>-\<^sup>1[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1 \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine] = g \<star> \<a>\<^sup>-\<^sup>1[tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1, f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine]
[PROOF STEP]
also
[PROOF STATE]
proof (state)
this:
\<a>[g, tab\<^sub>0 g \<star> g\<^sub>0h\<^sub>1.p\<^sub>1, f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine] \<cdot> \<a>[g \<star> tab\<^sub>0 g \<star> g\<^sub>0h\<^sub>1.p\<^sub>1, f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine] \<cdot> ((\<a>[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> \<a>\<^sup>-\<^sup>1[(g \<star> tab\<^sub>0 g) \<star> g\<^sub>0h\<^sub>1.p\<^sub>1, f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine] \<cdot> \<a>\<^sup>-\<^sup>1[g \<star> tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1, f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine] \<cdot> \<a>\<^sup>-\<^sup>1[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1 \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine] = \<lbrace>\<^bold>\<a>\<^bold>[\<^bold>\<langle>g\<^bold>\<rangle>, \<^bold>\<langle>tab\<^sub>0 g\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>1\<^bold>\<rangle>, \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>\<^bold>] \<^bold>\<cdot> \<^bold>\<a>\<^bold>[\<^bold>\<langle>g\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>tab\<^sub>0 g\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>1\<^bold>\<rangle>, \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>, \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>\<^bold>] \<^bold>\<cdot> ((\<^bold>\<a>\<^bold>[\<^bold>\<langle>g\<^bold>\<rangle>, \<^bold>\<langle>tab\<^sub>0 g\<^bold>\<rangle>, \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>1\<^bold>\<rangle>\<^bold>] \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) \<^bold>\<cdot> \<^bold>\<a>\<^sup>-\<^sup>1\<^bold>[(\<^bold>\<langle>g\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>tab\<^sub>0 g\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>1\<^bold>\<rangle>, \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>, \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>\<^bold>] \<^bold>\<cdot> \<^bold>\<a>\<^sup>-\<^sup>1\<^bold>[\<^bold>\<langle>g\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>tab\<^sub>0 g\<^bold>\<rangle>, \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>1\<^bold>\<rangle>, \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>\<^bold>] \<^bold>\<cdot> \<^bold>\<a>\<^sup>-\<^sup>1\<^bold>[\<^bold>\<langle>g\<^bold>\<rangle>, \<^bold>\<langle>tab\<^sub>0 g\<^bold>\<rangle>, \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>1\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>\<^bold>]\<rbrace>
goal (1 subgoal):
1. \<a>[g, tab\<^sub>0 g \<star> g\<^sub>0h\<^sub>1.p\<^sub>1, f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine] \<cdot> \<a>[g \<star> tab\<^sub>0 g \<star> g\<^sub>0h\<^sub>1.p\<^sub>1, f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine] \<cdot> ((\<a>[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> \<a>\<^sup>-\<^sup>1[(g \<star> tab\<^sub>0 g) \<star> g\<^sub>0h\<^sub>1.p\<^sub>1, f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine] \<cdot> \<a>\<^sup>-\<^sup>1[g \<star> tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1, f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine] \<cdot> \<a>\<^sup>-\<^sup>1[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1 \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine] = g \<star> \<a>\<^sup>-\<^sup>1[tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1, f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine]
[PROOF STEP]
have "... = \<lbrace>\<^bold>\<langle>g\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<a>\<^sup>-\<^sup>1\<^bold>[\<^bold>\<langle>tab\<^sub>0 g\<^bold>\<rangle>, \<^bold>\<langle>Tgh.p\<^sub>1\<^bold>\<rangle>, \<^bold>\<langle>TfTgh.p\<^sub>0\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>\<^bold>]\<rbrace>"
[PROOF STATE]
proof (prove)
goal (1 subgoal):
1. \<lbrace>\<^bold>\<a>\<^bold>[\<^bold>\<langle>g\<^bold>\<rangle>, \<^bold>\<langle>tab\<^sub>0 g\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>1\<^bold>\<rangle>, \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>\<^bold>] \<^bold>\<cdot> \<^bold>\<a>\<^bold>[\<^bold>\<langle>g\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>tab\<^sub>0 g\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>1\<^bold>\<rangle>, \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>, \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>\<^bold>] \<^bold>\<cdot> ((\<^bold>\<a>\<^bold>[\<^bold>\<langle>g\<^bold>\<rangle>, \<^bold>\<langle>tab\<^sub>0 g\<^bold>\<rangle>, \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>1\<^bold>\<rangle>\<^bold>] \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) \<^bold>\<cdot> \<^bold>\<a>\<^sup>-\<^sup>1\<^bold>[(\<^bold>\<langle>g\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>tab\<^sub>0 g\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>1\<^bold>\<rangle>, \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>, \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>\<^bold>] \<^bold>\<cdot> \<^bold>\<a>\<^sup>-\<^sup>1\<^bold>[\<^bold>\<langle>g\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>tab\<^sub>0 g\<^bold>\<rangle>, \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>1\<^bold>\<rangle>, \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>\<^bold>] \<^bold>\<cdot> \<^bold>\<a>\<^sup>-\<^sup>1\<^bold>[\<^bold>\<langle>g\<^bold>\<rangle>, \<^bold>\<langle>tab\<^sub>0 g\<^bold>\<rangle>, \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>1\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>\<^bold>]\<rbrace> = \<lbrace>\<^bold>\<langle>g\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<a>\<^sup>-\<^sup>1\<^bold>[\<^bold>\<langle>tab\<^sub>0 g\<^bold>\<rangle>, \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>1\<^bold>\<rangle>, \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>\<^bold>]\<rbrace>
[PROOF STEP]
apply (intro E.eval_eqI)
[PROOF STATE]
proof (prove)
goal (2 subgoals):
1. E.VPar (\<^bold>\<a>\<^bold>[\<^bold>\<langle>g\<^bold>\<rangle>, \<^bold>\<langle>tab\<^sub>0 g\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>1\<^bold>\<rangle>, \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>\<^bold>] \<^bold>\<cdot> \<^bold>\<a>\<^bold>[\<^bold>\<langle>g\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>tab\<^sub>0 g\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>1\<^bold>\<rangle>, \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>, \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>\<^bold>] \<^bold>\<cdot> ((\<^bold>\<a>\<^bold>[\<^bold>\<langle>g\<^bold>\<rangle>, \<^bold>\<langle>tab\<^sub>0 g\<^bold>\<rangle>, \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>1\<^bold>\<rangle>\<^bold>] \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) \<^bold>\<cdot> \<^bold>\<a>\<^sup>-\<^sup>1\<^bold>[(\<^bold>\<langle>g\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>tab\<^sub>0 g\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>1\<^bold>\<rangle>, \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>, \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>\<^bold>] \<^bold>\<cdot> \<^bold>\<a>\<^sup>-\<^sup>1\<^bold>[\<^bold>\<langle>g\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>tab\<^sub>0 g\<^bold>\<rangle>, \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>1\<^bold>\<rangle>, \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>\<^bold>] \<^bold>\<cdot> \<^bold>\<a>\<^sup>-\<^sup>1\<^bold>[\<^bold>\<langle>g\<^bold>\<rangle>, \<^bold>\<langle>tab\<^sub>0 g\<^bold>\<rangle>, \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>1\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>\<^bold>]) (\<^bold>\<langle>g\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<a>\<^sup>-\<^sup>1\<^bold>[\<^bold>\<langle>tab\<^sub>0 g\<^bold>\<rangle>, \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>1\<^bold>\<rangle>, \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>\<^bold>])
2. \<^bold>\<lfloor>\<^bold>\<a>\<^bold>[\<^bold>\<langle>g\<^bold>\<rangle>, \<^bold>\<langle>tab\<^sub>0 g\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>1\<^bold>\<rangle>, \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>\<^bold>] \<^bold>\<cdot> \<^bold>\<a>\<^bold>[\<^bold>\<langle>g\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>tab\<^sub>0 g\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>1\<^bold>\<rangle>, \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>, \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>\<^bold>] \<^bold>\<cdot> ((\<^bold>\<a>\<^bold>[\<^bold>\<langle>g\<^bold>\<rangle>, \<^bold>\<langle>tab\<^sub>0 g\<^bold>\<rangle>, \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>1\<^bold>\<rangle>\<^bold>] \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) \<^bold>\<cdot> \<^bold>\<a>\<^sup>-\<^sup>1\<^bold>[(\<^bold>\<langle>g\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>tab\<^sub>0 g\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>1\<^bold>\<rangle>, \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>, \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>\<^bold>] \<^bold>\<cdot> \<^bold>\<a>\<^sup>-\<^sup>1\<^bold>[\<^bold>\<langle>g\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>tab\<^sub>0 g\<^bold>\<rangle>, \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>1\<^bold>\<rangle>, \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>\<^bold>] \<^bold>\<cdot> \<^bold>\<a>\<^sup>-\<^sup>1\<^bold>[\<^bold>\<langle>g\<^bold>\<rangle>, \<^bold>\<langle>tab\<^sub>0 g\<^bold>\<rangle>, \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>1\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>\<^bold>]\<^bold>\<rfloor> = \<^bold>\<lfloor>\<^bold>\<langle>g\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<a>\<^sup>-\<^sup>1\<^bold>[\<^bold>\<langle>tab\<^sub>0 g\<^bold>\<rangle>, \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>1\<^bold>\<rangle>, \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>\<^bold>]\<^bold>\<rfloor>
[PROOF STEP]
by simp_all
[PROOF STATE]
proof (state)
this:
\<lbrace>\<^bold>\<a>\<^bold>[\<^bold>\<langle>g\<^bold>\<rangle>, \<^bold>\<langle>tab\<^sub>0 g\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>1\<^bold>\<rangle>, \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>\<^bold>] \<^bold>\<cdot> \<^bold>\<a>\<^bold>[\<^bold>\<langle>g\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>tab\<^sub>0 g\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>1\<^bold>\<rangle>, \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>, \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>\<^bold>] \<^bold>\<cdot> ((\<^bold>\<a>\<^bold>[\<^bold>\<langle>g\<^bold>\<rangle>, \<^bold>\<langle>tab\<^sub>0 g\<^bold>\<rangle>, \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>1\<^bold>\<rangle>\<^bold>] \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) \<^bold>\<cdot> \<^bold>\<a>\<^sup>-\<^sup>1\<^bold>[(\<^bold>\<langle>g\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>tab\<^sub>0 g\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>1\<^bold>\<rangle>, \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>, \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>\<^bold>] \<^bold>\<cdot> \<^bold>\<a>\<^sup>-\<^sup>1\<^bold>[\<^bold>\<langle>g\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>tab\<^sub>0 g\<^bold>\<rangle>, \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>1\<^bold>\<rangle>, \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>\<^bold>] \<^bold>\<cdot> \<^bold>\<a>\<^sup>-\<^sup>1\<^bold>[\<^bold>\<langle>g\<^bold>\<rangle>, \<^bold>\<langle>tab\<^sub>0 g\<^bold>\<rangle>, \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>1\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>\<^bold>]\<rbrace> = \<lbrace>\<^bold>\<langle>g\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<a>\<^sup>-\<^sup>1\<^bold>[\<^bold>\<langle>tab\<^sub>0 g\<^bold>\<rangle>, \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>1\<^bold>\<rangle>, \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>\<^bold>]\<rbrace>
goal (1 subgoal):
1. \<a>[g, tab\<^sub>0 g \<star> g\<^sub>0h\<^sub>1.p\<^sub>1, f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine] \<cdot> \<a>[g \<star> tab\<^sub>0 g \<star> g\<^sub>0h\<^sub>1.p\<^sub>1, f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine] \<cdot> ((\<a>[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> \<a>\<^sup>-\<^sup>1[(g \<star> tab\<^sub>0 g) \<star> g\<^sub>0h\<^sub>1.p\<^sub>1, f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine] \<cdot> \<a>\<^sup>-\<^sup>1[g \<star> tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1, f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine] \<cdot> \<a>\<^sup>-\<^sup>1[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1 \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine] = g \<star> \<a>\<^sup>-\<^sup>1[tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1, f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine]
[PROOF STEP]
also
[PROOF STATE]
proof (state)
this:
\<lbrace>\<^bold>\<a>\<^bold>[\<^bold>\<langle>g\<^bold>\<rangle>, \<^bold>\<langle>tab\<^sub>0 g\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>1\<^bold>\<rangle>, \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>\<^bold>] \<^bold>\<cdot> \<^bold>\<a>\<^bold>[\<^bold>\<langle>g\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>tab\<^sub>0 g\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>1\<^bold>\<rangle>, \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>, \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>\<^bold>] \<^bold>\<cdot> ((\<^bold>\<a>\<^bold>[\<^bold>\<langle>g\<^bold>\<rangle>, \<^bold>\<langle>tab\<^sub>0 g\<^bold>\<rangle>, \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>1\<^bold>\<rangle>\<^bold>] \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) \<^bold>\<cdot> \<^bold>\<a>\<^sup>-\<^sup>1\<^bold>[(\<^bold>\<langle>g\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>tab\<^sub>0 g\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>1\<^bold>\<rangle>, \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>, \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>\<^bold>] \<^bold>\<cdot> \<^bold>\<a>\<^sup>-\<^sup>1\<^bold>[\<^bold>\<langle>g\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>tab\<^sub>0 g\<^bold>\<rangle>, \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>1\<^bold>\<rangle>, \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>\<^bold>] \<^bold>\<cdot> \<^bold>\<a>\<^sup>-\<^sup>1\<^bold>[\<^bold>\<langle>g\<^bold>\<rangle>, \<^bold>\<langle>tab\<^sub>0 g\<^bold>\<rangle>, \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>1\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>\<^bold>]\<rbrace> = \<lbrace>\<^bold>\<langle>g\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<a>\<^sup>-\<^sup>1\<^bold>[\<^bold>\<langle>tab\<^sub>0 g\<^bold>\<rangle>, \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>1\<^bold>\<rangle>, \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>\<^bold>]\<rbrace>
goal (1 subgoal):
1. \<a>[g, tab\<^sub>0 g \<star> g\<^sub>0h\<^sub>1.p\<^sub>1, f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine] \<cdot> \<a>[g \<star> tab\<^sub>0 g \<star> g\<^sub>0h\<^sub>1.p\<^sub>1, f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine] \<cdot> ((\<a>[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> \<a>\<^sup>-\<^sup>1[(g \<star> tab\<^sub>0 g) \<star> g\<^sub>0h\<^sub>1.p\<^sub>1, f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine] \<cdot> \<a>\<^sup>-\<^sup>1[g \<star> tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1, f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine] \<cdot> \<a>\<^sup>-\<^sup>1[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1 \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine] = g \<star> \<a>\<^sup>-\<^sup>1[tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1, f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine]
[PROOF STEP]
have "... = g \<star> \<a>\<^sup>-\<^sup>1[tab\<^sub>0 g, Tgh.p\<^sub>1, TfTgh.p\<^sub>0 \<star> TTfgh_TfTgh.chine]"
[PROOF STATE]
proof (prove)
goal (1 subgoal):
1. \<lbrace>\<^bold>\<langle>g\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<a>\<^sup>-\<^sup>1\<^bold>[\<^bold>\<langle>tab\<^sub>0 g\<^bold>\<rangle>, \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>1\<^bold>\<rangle>, \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>\<^bold>]\<rbrace> = g \<star> \<a>\<^sup>-\<^sup>1[tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1, f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine]
[PROOF STEP]
using gh g\<^sub>0h\<^sub>1.p\<^sub>1_simps f\<^sub>0gh\<^sub>1.p\<^sub>0_simps \<a>'_def \<alpha>_def
[PROOF STATE]
proof (prove)
using this:
src g = trg h
trg g\<^sub>0h\<^sub>1.p\<^sub>1 = src (tab\<^sub>0 g)
trg f\<^sub>0gh\<^sub>1.p\<^sub>0 = src (tab\<^sub>1 g \<star> g\<^sub>0h\<^sub>1.p\<^sub>1)
\<a>\<^sup>-\<^sup>1[?\<mu>, ?\<nu>, ?\<tau>] \<equiv> \<alpha>' (?\<mu>, ?\<nu>, ?\<tau>)
\<alpha> ?\<mu>\<nu>\<tau> \<equiv> \<a>[fst ?\<mu>\<nu>\<tau>, fst (snd ?\<mu>\<nu>\<tau>), snd (snd ?\<mu>\<nu>\<tau>)]
goal (1 subgoal):
1. \<lbrace>\<^bold>\<langle>g\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<a>\<^sup>-\<^sup>1\<^bold>[\<^bold>\<langle>tab\<^sub>0 g\<^bold>\<rangle>, \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>1\<^bold>\<rangle>, \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>\<^bold>]\<rbrace> = g \<star> \<a>\<^sup>-\<^sup>1[tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1, f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine]
[PROOF STEP]
by simp
[PROOF STATE]
proof (state)
this:
\<lbrace>\<^bold>\<langle>g\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<a>\<^sup>-\<^sup>1\<^bold>[\<^bold>\<langle>tab\<^sub>0 g\<^bold>\<rangle>, \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>1\<^bold>\<rangle>, \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>\<^bold>]\<rbrace> = g \<star> \<a>\<^sup>-\<^sup>1[tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1, f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine]
goal (1 subgoal):
1. \<a>[g, tab\<^sub>0 g \<star> g\<^sub>0h\<^sub>1.p\<^sub>1, f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine] \<cdot> \<a>[g \<star> tab\<^sub>0 g \<star> g\<^sub>0h\<^sub>1.p\<^sub>1, f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine] \<cdot> ((\<a>[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> \<a>\<^sup>-\<^sup>1[(g \<star> tab\<^sub>0 g) \<star> g\<^sub>0h\<^sub>1.p\<^sub>1, f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine] \<cdot> \<a>\<^sup>-\<^sup>1[g \<star> tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1, f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine] \<cdot> \<a>\<^sup>-\<^sup>1[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1 \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine] = g \<star> \<a>\<^sup>-\<^sup>1[tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1, f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine]
[PROOF STEP]
finally
[PROOF STATE]
proof (chain)
picking this:
\<a>[g, tab\<^sub>0 g \<star> g\<^sub>0h\<^sub>1.p\<^sub>1, f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine] \<cdot> \<a>[g \<star> tab\<^sub>0 g \<star> g\<^sub>0h\<^sub>1.p\<^sub>1, f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine] \<cdot> ((\<a>[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> \<a>\<^sup>-\<^sup>1[(g \<star> tab\<^sub>0 g) \<star> g\<^sub>0h\<^sub>1.p\<^sub>1, f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine] \<cdot> \<a>\<^sup>-\<^sup>1[g \<star> tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1, f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine] \<cdot> \<a>\<^sup>-\<^sup>1[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1 \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine] = g \<star> \<a>\<^sup>-\<^sup>1[tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1, f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine]
[PROOF STEP]
show ?thesis
[PROOF STATE]
proof (prove)
using this:
\<a>[g, tab\<^sub>0 g \<star> g\<^sub>0h\<^sub>1.p\<^sub>1, f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine] \<cdot> \<a>[g \<star> tab\<^sub>0 g \<star> g\<^sub>0h\<^sub>1.p\<^sub>1, f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine] \<cdot> ((\<a>[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> \<a>\<^sup>-\<^sup>1[(g \<star> tab\<^sub>0 g) \<star> g\<^sub>0h\<^sub>1.p\<^sub>1, f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine] \<cdot> \<a>\<^sup>-\<^sup>1[g \<star> tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1, f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine] \<cdot> \<a>\<^sup>-\<^sup>1[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1 \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine] = g \<star> \<a>\<^sup>-\<^sup>1[tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1, f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine]
goal (1 subgoal):
1. \<a>[g, tab\<^sub>0 g \<star> g\<^sub>0h\<^sub>1.p\<^sub>1, f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine] \<cdot> \<a>[g \<star> tab\<^sub>0 g \<star> g\<^sub>0h\<^sub>1.p\<^sub>1, f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine] \<cdot> ((\<a>[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> \<a>\<^sup>-\<^sup>1[(g \<star> tab\<^sub>0 g) \<star> g\<^sub>0h\<^sub>1.p\<^sub>1, f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine] \<cdot> \<a>\<^sup>-\<^sup>1[g \<star> tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1, f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine] \<cdot> \<a>\<^sup>-\<^sup>1[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1 \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine] = g \<star> \<a>\<^sup>-\<^sup>1[tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1, f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine]
[PROOF STEP]
by simp
[PROOF STATE]
proof (state)
this:
\<a>[g, tab\<^sub>0 g \<star> g\<^sub>0h\<^sub>1.p\<^sub>1, f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine] \<cdot> \<a>[g \<star> tab\<^sub>0 g \<star> g\<^sub>0h\<^sub>1.p\<^sub>1, f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine] \<cdot> ((\<a>[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> \<a>\<^sup>-\<^sup>1[(g \<star> tab\<^sub>0 g) \<star> g\<^sub>0h\<^sub>1.p\<^sub>1, f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine] \<cdot> \<a>\<^sup>-\<^sup>1[g \<star> tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1, f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine] \<cdot> \<a>\<^sup>-\<^sup>1[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1 \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine] = g \<star> \<a>\<^sup>-\<^sup>1[tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1, f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine]
goal:
No subgoals!
[PROOF STEP]
qed
[PROOF STATE]
proof (state)
this:
\<a>[g, tab\<^sub>0 g \<star> g\<^sub>0h\<^sub>1.p\<^sub>1, f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine] \<cdot> \<a>[g \<star> tab\<^sub>0 g \<star> g\<^sub>0h\<^sub>1.p\<^sub>1, f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine] \<cdot> ((\<a>[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> \<a>\<^sup>-\<^sup>1[(g \<star> tab\<^sub>0 g) \<star> g\<^sub>0h\<^sub>1.p\<^sub>1, f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine] \<cdot> \<a>\<^sup>-\<^sup>1[g \<star> tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1, f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine] \<cdot> \<a>\<^sup>-\<^sup>1[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1 \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine] = g \<star> \<a>\<^sup>-\<^sup>1[tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1, f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine]
goal (1 subgoal):
1. (g \<star> h \<star> TTfgh_TfTgh.the_\<theta>) \<cdot> (can (\<^bold>\<langle>g\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> ((\<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) (((\<^bold>\<langle>g\<^bold>\<rangle> \<^bold>\<star> (\<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) \<cdot> \<a>\<^sup>-\<^sup>1[g \<star> (h \<star> tab\<^sub>0 h) \<star> g\<^sub>0h\<^sub>1.p\<^sub>0, f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine] \<cdot> \<a>\<^sup>-\<^sup>1[g, (h \<star> tab\<^sub>0 h) \<star> g\<^sub>0h\<^sub>1.p\<^sub>0, f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine]) \<cdot> (g \<star> (h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine) \<cdot> (g \<star> g\<^sub>0h\<^sub>1.\<phi> \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine) \<cdot> (\<a>[g, tab\<^sub>0 g \<star> g\<^sub>0h\<^sub>1.p\<^sub>1, f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine] \<cdot> \<a>[g \<star> tab\<^sub>0 g \<star> g\<^sub>0h\<^sub>1.p\<^sub>1, f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine] \<cdot> ((\<a>[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> \<a>\<^sup>-\<^sup>1[(g \<star> tab\<^sub>0 g) \<star> g\<^sub>0h\<^sub>1.p\<^sub>1, f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine] \<cdot> \<a>\<^sup>-\<^sup>1[g \<star> tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1, f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine] \<cdot> \<a>\<^sup>-\<^sup>1[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1 \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine]) \<cdot> \<a>[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1 \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine] \<cdot> (g.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>1 \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine) \<cdot> \<beta>\<^sub>g = ((g \<star> h \<star> TTfgh_TfTgh.the_\<theta>) \<cdot> (g \<star> can (\<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> ((\<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) (((\<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>)) \<cdot> (g \<star> (h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine) \<cdot> (g \<star> g\<^sub>0h\<^sub>1.\<phi> \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine) \<cdot> (g \<star> \<a>\<^sup>-\<^sup>1[tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1, f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine])) \<cdot> \<a>[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1 \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine] \<cdot> (g.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>1 \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine) \<cdot> \<beta>\<^sub>g
[PROOF STEP]
ultimately
[PROOF STATE]
proof (chain)
picking this:
can (\<^bold>\<langle>g\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> ((\<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) (((\<^bold>\<langle>g\<^bold>\<rangle> \<^bold>\<star> (\<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) \<cdot> \<a>\<^sup>-\<^sup>1[g \<star> (h \<star> tab\<^sub>0 h) \<star> g\<^sub>0h\<^sub>1.p\<^sub>0, f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine] \<cdot> \<a>\<^sup>-\<^sup>1[g, (h \<star> tab\<^sub>0 h) \<star> g\<^sub>0h\<^sub>1.p\<^sub>0, f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine] = g \<star> can (\<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> ((\<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) (((\<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>)
\<a>[g, tab\<^sub>0 g \<star> g\<^sub>0h\<^sub>1.p\<^sub>1, f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine] \<cdot> \<a>[g \<star> tab\<^sub>0 g \<star> g\<^sub>0h\<^sub>1.p\<^sub>1, f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine] \<cdot> ((\<a>[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> \<a>\<^sup>-\<^sup>1[(g \<star> tab\<^sub>0 g) \<star> g\<^sub>0h\<^sub>1.p\<^sub>1, f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine] \<cdot> \<a>\<^sup>-\<^sup>1[g \<star> tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1, f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine] \<cdot> \<a>\<^sup>-\<^sup>1[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1 \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine] = g \<star> \<a>\<^sup>-\<^sup>1[tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1, f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine]
[PROOF STEP]
show ?thesis
[PROOF STATE]
proof (prove)
using this:
can (\<^bold>\<langle>g\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> ((\<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) (((\<^bold>\<langle>g\<^bold>\<rangle> \<^bold>\<star> (\<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) \<cdot> \<a>\<^sup>-\<^sup>1[g \<star> (h \<star> tab\<^sub>0 h) \<star> g\<^sub>0h\<^sub>1.p\<^sub>0, f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine] \<cdot> \<a>\<^sup>-\<^sup>1[g, (h \<star> tab\<^sub>0 h) \<star> g\<^sub>0h\<^sub>1.p\<^sub>0, f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine] = g \<star> can (\<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> ((\<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) (((\<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>)
\<a>[g, tab\<^sub>0 g \<star> g\<^sub>0h\<^sub>1.p\<^sub>1, f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine] \<cdot> \<a>[g \<star> tab\<^sub>0 g \<star> g\<^sub>0h\<^sub>1.p\<^sub>1, f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine] \<cdot> ((\<a>[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> \<a>\<^sup>-\<^sup>1[(g \<star> tab\<^sub>0 g) \<star> g\<^sub>0h\<^sub>1.p\<^sub>1, f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine] \<cdot> \<a>\<^sup>-\<^sup>1[g \<star> tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1, f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine] \<cdot> \<a>\<^sup>-\<^sup>1[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1 \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine] = g \<star> \<a>\<^sup>-\<^sup>1[tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1, f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine]
goal (1 subgoal):
1. (g \<star> h \<star> TTfgh_TfTgh.the_\<theta>) \<cdot> (can (\<^bold>\<langle>g\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> ((\<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) (((\<^bold>\<langle>g\<^bold>\<rangle> \<^bold>\<star> (\<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) \<cdot> \<a>\<^sup>-\<^sup>1[g \<star> (h \<star> tab\<^sub>0 h) \<star> g\<^sub>0h\<^sub>1.p\<^sub>0, f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine] \<cdot> \<a>\<^sup>-\<^sup>1[g, (h \<star> tab\<^sub>0 h) \<star> g\<^sub>0h\<^sub>1.p\<^sub>0, f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine]) \<cdot> (g \<star> (h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine) \<cdot> (g \<star> g\<^sub>0h\<^sub>1.\<phi> \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine) \<cdot> (\<a>[g, tab\<^sub>0 g \<star> g\<^sub>0h\<^sub>1.p\<^sub>1, f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine] \<cdot> \<a>[g \<star> tab\<^sub>0 g \<star> g\<^sub>0h\<^sub>1.p\<^sub>1, f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine] \<cdot> ((\<a>[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> \<a>\<^sup>-\<^sup>1[(g \<star> tab\<^sub>0 g) \<star> g\<^sub>0h\<^sub>1.p\<^sub>1, f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine] \<cdot> \<a>\<^sup>-\<^sup>1[g \<star> tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1, f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine] \<cdot> \<a>\<^sup>-\<^sup>1[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1 \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine]) \<cdot> \<a>[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1 \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine] \<cdot> (g.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>1 \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine) \<cdot> \<beta>\<^sub>g = ((g \<star> h \<star> TTfgh_TfTgh.the_\<theta>) \<cdot> (g \<star> can (\<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> ((\<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) (((\<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>)) \<cdot> (g \<star> (h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine) \<cdot> (g \<star> g\<^sub>0h\<^sub>1.\<phi> \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine) \<cdot> (g \<star> \<a>\<^sup>-\<^sup>1[tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1, f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine])) \<cdot> \<a>[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1 \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine] \<cdot> (g.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>1 \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine) \<cdot> \<beta>\<^sub>g
[PROOF STEP]
using comp_assoc
[PROOF STATE]
proof (prove)
using this:
can (\<^bold>\<langle>g\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> ((\<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) (((\<^bold>\<langle>g\<^bold>\<rangle> \<^bold>\<star> (\<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) \<cdot> \<a>\<^sup>-\<^sup>1[g \<star> (h \<star> tab\<^sub>0 h) \<star> g\<^sub>0h\<^sub>1.p\<^sub>0, f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine] \<cdot> \<a>\<^sup>-\<^sup>1[g, (h \<star> tab\<^sub>0 h) \<star> g\<^sub>0h\<^sub>1.p\<^sub>0, f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine] = g \<star> can (\<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> ((\<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) (((\<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>)
\<a>[g, tab\<^sub>0 g \<star> g\<^sub>0h\<^sub>1.p\<^sub>1, f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine] \<cdot> \<a>[g \<star> tab\<^sub>0 g \<star> g\<^sub>0h\<^sub>1.p\<^sub>1, f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine] \<cdot> ((\<a>[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> \<a>\<^sup>-\<^sup>1[(g \<star> tab\<^sub>0 g) \<star> g\<^sub>0h\<^sub>1.p\<^sub>1, f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine] \<cdot> \<a>\<^sup>-\<^sup>1[g \<star> tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1, f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine] \<cdot> \<a>\<^sup>-\<^sup>1[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1 \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine] = g \<star> \<a>\<^sup>-\<^sup>1[tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1, f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine]
(?h \<cdot> ?g) \<cdot> ?f = ?h \<cdot> ?g \<cdot> ?f
goal (1 subgoal):
1. (g \<star> h \<star> TTfgh_TfTgh.the_\<theta>) \<cdot> (can (\<^bold>\<langle>g\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> ((\<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) (((\<^bold>\<langle>g\<^bold>\<rangle> \<^bold>\<star> (\<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) \<cdot> \<a>\<^sup>-\<^sup>1[g \<star> (h \<star> tab\<^sub>0 h) \<star> g\<^sub>0h\<^sub>1.p\<^sub>0, f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine] \<cdot> \<a>\<^sup>-\<^sup>1[g, (h \<star> tab\<^sub>0 h) \<star> g\<^sub>0h\<^sub>1.p\<^sub>0, f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine]) \<cdot> (g \<star> (h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine) \<cdot> (g \<star> g\<^sub>0h\<^sub>1.\<phi> \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine) \<cdot> (\<a>[g, tab\<^sub>0 g \<star> g\<^sub>0h\<^sub>1.p\<^sub>1, f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine] \<cdot> \<a>[g \<star> tab\<^sub>0 g \<star> g\<^sub>0h\<^sub>1.p\<^sub>1, f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine] \<cdot> ((\<a>[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> \<a>\<^sup>-\<^sup>1[(g \<star> tab\<^sub>0 g) \<star> g\<^sub>0h\<^sub>1.p\<^sub>1, f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine] \<cdot> \<a>\<^sup>-\<^sup>1[g \<star> tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1, f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine] \<cdot> \<a>\<^sup>-\<^sup>1[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1 \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine]) \<cdot> \<a>[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1 \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine] \<cdot> (g.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>1 \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine) \<cdot> \<beta>\<^sub>g = ((g \<star> h \<star> TTfgh_TfTgh.the_\<theta>) \<cdot> (g \<star> can (\<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> ((\<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) (((\<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>)) \<cdot> (g \<star> (h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine) \<cdot> (g \<star> g\<^sub>0h\<^sub>1.\<phi> \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine) \<cdot> (g \<star> \<a>\<^sup>-\<^sup>1[tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1, f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine])) \<cdot> \<a>[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1 \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine] \<cdot> (g.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>1 \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine) \<cdot> \<beta>\<^sub>g
[PROOF STEP]
by presburger
[PROOF STATE]
proof (state)
this:
(g \<star> h \<star> TTfgh_TfTgh.the_\<theta>) \<cdot> (can (\<^bold>\<langle>g\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> ((\<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) (((\<^bold>\<langle>g\<^bold>\<rangle> \<^bold>\<star> (\<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) \<cdot> \<a>\<^sup>-\<^sup>1[g \<star> (h \<star> tab\<^sub>0 h) \<star> g\<^sub>0h\<^sub>1.p\<^sub>0, f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine] \<cdot> \<a>\<^sup>-\<^sup>1[g, (h \<star> tab\<^sub>0 h) \<star> g\<^sub>0h\<^sub>1.p\<^sub>0, f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine]) \<cdot> (g \<star> (h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine) \<cdot> (g \<star> g\<^sub>0h\<^sub>1.\<phi> \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine) \<cdot> (\<a>[g, tab\<^sub>0 g \<star> g\<^sub>0h\<^sub>1.p\<^sub>1, f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine] \<cdot> \<a>[g \<star> tab\<^sub>0 g \<star> g\<^sub>0h\<^sub>1.p\<^sub>1, f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine] \<cdot> ((\<a>[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> \<a>\<^sup>-\<^sup>1[(g \<star> tab\<^sub>0 g) \<star> g\<^sub>0h\<^sub>1.p\<^sub>1, f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine] \<cdot> \<a>\<^sup>-\<^sup>1[g \<star> tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1, f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine] \<cdot> \<a>\<^sup>-\<^sup>1[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1 \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine]) \<cdot> \<a>[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1 \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine] \<cdot> (g.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>1 \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine) \<cdot> \<beta>\<^sub>g = ((g \<star> h \<star> TTfgh_TfTgh.the_\<theta>) \<cdot> (g \<star> can (\<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> ((\<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) (((\<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>)) \<cdot> (g \<star> (h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine) \<cdot> (g \<star> g\<^sub>0h\<^sub>1.\<phi> \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine) \<cdot> (g \<star> \<a>\<^sup>-\<^sup>1[tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1, f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine])) \<cdot> \<a>[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1 \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine] \<cdot> (g.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>1 \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine) \<cdot> \<beta>\<^sub>g
goal:
No subgoals!
[PROOF STEP]
qed
[PROOF STATE]
proof (state)
this:
(g \<star> h \<star> TTfgh_TfTgh.the_\<theta>) \<cdot> (can (\<^bold>\<langle>g\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> ((\<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) (((\<^bold>\<langle>g\<^bold>\<rangle> \<^bold>\<star> (\<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) \<cdot> \<a>\<^sup>-\<^sup>1[g \<star> (h \<star> tab\<^sub>0 h) \<star> g\<^sub>0h\<^sub>1.p\<^sub>0, f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine] \<cdot> \<a>\<^sup>-\<^sup>1[g, (h \<star> tab\<^sub>0 h) \<star> g\<^sub>0h\<^sub>1.p\<^sub>0, f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine]) \<cdot> (g \<star> (h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine) \<cdot> (g \<star> g\<^sub>0h\<^sub>1.\<phi> \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine) \<cdot> (\<a>[g, tab\<^sub>0 g \<star> g\<^sub>0h\<^sub>1.p\<^sub>1, f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine] \<cdot> \<a>[g \<star> tab\<^sub>0 g \<star> g\<^sub>0h\<^sub>1.p\<^sub>1, f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine] \<cdot> ((\<a>[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> \<a>\<^sup>-\<^sup>1[(g \<star> tab\<^sub>0 g) \<star> g\<^sub>0h\<^sub>1.p\<^sub>1, f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine] \<cdot> \<a>\<^sup>-\<^sup>1[g \<star> tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1, f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine] \<cdot> \<a>\<^sup>-\<^sup>1[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1 \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine]) \<cdot> \<a>[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1 \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine] \<cdot> (g.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>1 \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine) \<cdot> \<beta>\<^sub>g = ((g \<star> h \<star> TTfgh_TfTgh.the_\<theta>) \<cdot> (g \<star> can (\<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> ((\<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) (((\<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>)) \<cdot> (g \<star> (h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine) \<cdot> (g \<star> g\<^sub>0h\<^sub>1.\<phi> \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine) \<cdot> (g \<star> \<a>\<^sup>-\<^sup>1[tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1, f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine])) \<cdot> \<a>[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1 \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine] \<cdot> (g.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>1 \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine) \<cdot> \<beta>\<^sub>g
goal (1 subgoal):
1. (g \<star> \<theta>\<^sub>g) \<cdot> \<a>[g, tab\<^sub>0 g, w\<^sub>g] \<cdot> (g.tab \<star> w\<^sub>g) = ((g \<star> \<theta>\<^sub>g') \<cdot> \<a>[g, tab\<^sub>0 g, w\<^sub>g'] \<cdot> (g.tab \<star> w\<^sub>g')) \<cdot> \<beta>\<^sub>g
[PROOF STEP]
also
[PROOF STATE]
proof (state)
this:
(g \<star> h \<star> TTfgh_TfTgh.the_\<theta>) \<cdot> (can (\<^bold>\<langle>g\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> ((\<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) (((\<^bold>\<langle>g\<^bold>\<rangle> \<^bold>\<star> (\<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) \<cdot> \<a>\<^sup>-\<^sup>1[g \<star> (h \<star> tab\<^sub>0 h) \<star> g\<^sub>0h\<^sub>1.p\<^sub>0, f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine] \<cdot> \<a>\<^sup>-\<^sup>1[g, (h \<star> tab\<^sub>0 h) \<star> g\<^sub>0h\<^sub>1.p\<^sub>0, f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine]) \<cdot> (g \<star> (h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine) \<cdot> (g \<star> g\<^sub>0h\<^sub>1.\<phi> \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine) \<cdot> (\<a>[g, tab\<^sub>0 g \<star> g\<^sub>0h\<^sub>1.p\<^sub>1, f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine] \<cdot> \<a>[g \<star> tab\<^sub>0 g \<star> g\<^sub>0h\<^sub>1.p\<^sub>1, f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine] \<cdot> ((\<a>[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1] \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<star> TTfgh_TfTgh.chine) \<cdot> \<a>\<^sup>-\<^sup>1[(g \<star> tab\<^sub>0 g) \<star> g\<^sub>0h\<^sub>1.p\<^sub>1, f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine] \<cdot> \<a>\<^sup>-\<^sup>1[g \<star> tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1, f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine] \<cdot> \<a>\<^sup>-\<^sup>1[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1 \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine]) \<cdot> \<a>[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1 \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine] \<cdot> (g.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>1 \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine) \<cdot> \<beta>\<^sub>g = ((g \<star> h \<star> TTfgh_TfTgh.the_\<theta>) \<cdot> (g \<star> can (\<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> ((\<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) (((\<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>)) \<cdot> (g \<star> (h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine) \<cdot> (g \<star> g\<^sub>0h\<^sub>1.\<phi> \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine) \<cdot> (g \<star> \<a>\<^sup>-\<^sup>1[tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1, f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine])) \<cdot> \<a>[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1 \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine] \<cdot> (g.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>1 \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine) \<cdot> \<beta>\<^sub>g
goal (1 subgoal):
1. (g \<star> \<theta>\<^sub>g) \<cdot> \<a>[g, tab\<^sub>0 g, w\<^sub>g] \<cdot> (g.tab \<star> w\<^sub>g) = ((g \<star> \<theta>\<^sub>g') \<cdot> \<a>[g, tab\<^sub>0 g, w\<^sub>g'] \<cdot> (g.tab \<star> w\<^sub>g')) \<cdot> \<beta>\<^sub>g
[PROOF STEP]
have "... = (g \<star>
(h \<star> TTfgh_TfTgh.the_\<theta>) \<cdot>
(can (\<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> ((\<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>Tgh.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TfTgh.p\<^sub>0\<^bold>\<rangle>)
\<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>)
(((\<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>Tgh.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TfTgh.p\<^sub>0\<^bold>\<rangle>
\<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>)) \<cdot>
((h.tab \<star> Tgh.p\<^sub>0) \<star> TfTgh.p\<^sub>0 \<star> TTfgh_TfTgh.chine) \<cdot>
(g\<^sub>0h\<^sub>1.\<phi> \<star> TfTgh.p\<^sub>0 \<star> TTfgh_TfTgh.chine) \<cdot>
\<a>\<^sup>-\<^sup>1[tab\<^sub>0 g, Tgh.p\<^sub>1, TfTgh.p\<^sub>0 \<star> TTfgh_TfTgh.chine]) \<cdot>
\<a>[g, tab\<^sub>0 g, Tgh.p\<^sub>1 \<star> TfTgh.p\<^sub>0 \<star> TTfgh_TfTgh.chine] \<cdot>
(g.tab \<star> Tgh.p\<^sub>1 \<star> TfTgh.p\<^sub>0 \<star> TTfgh_TfTgh.chine) \<cdot>
\<beta>\<^sub>g"
[PROOF STATE]
proof (prove)
goal (1 subgoal):
1. ((g \<star> h \<star> TTfgh_TfTgh.the_\<theta>) \<cdot> (g \<star> can (\<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> ((\<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) (((\<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>)) \<cdot> (g \<star> (h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine) \<cdot> (g \<star> g\<^sub>0h\<^sub>1.\<phi> \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine) \<cdot> (g \<star> \<a>\<^sup>-\<^sup>1[tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1, f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine])) \<cdot> \<a>[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1 \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine] \<cdot> (g.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>1 \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine) \<cdot> \<beta>\<^sub>g = (g \<star> (h \<star> TTfgh_TfTgh.the_\<theta>) \<cdot> can (\<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> ((\<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) (((\<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) \<cdot> ((h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine) \<cdot> (g\<^sub>0h\<^sub>1.\<phi> \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine) \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1, f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine]) \<cdot> \<a>[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1 \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine] \<cdot> (g.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>1 \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine) \<cdot> \<beta>\<^sub>g
[PROOF STEP]
using gh whisker_left
[PROOF STATE]
proof (prove)
using this:
src g = trg h
\<lbrakk>ide ?f; seq ?\<nu> ?\<mu>\<rbrakk> \<Longrightarrow> ?f \<star> ?\<nu> \<cdot> ?\<mu> = (?f \<star> ?\<nu>) \<cdot> (?f \<star> ?\<mu>)
goal (1 subgoal):
1. ((g \<star> h \<star> TTfgh_TfTgh.the_\<theta>) \<cdot> (g \<star> can (\<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> ((\<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) (((\<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>)) \<cdot> (g \<star> (h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine) \<cdot> (g \<star> g\<^sub>0h\<^sub>1.\<phi> \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine) \<cdot> (g \<star> \<a>\<^sup>-\<^sup>1[tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1, f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine])) \<cdot> \<a>[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1 \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine] \<cdot> (g.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>1 \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine) \<cdot> \<beta>\<^sub>g = (g \<star> (h \<star> TTfgh_TfTgh.the_\<theta>) \<cdot> can (\<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> ((\<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) (((\<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) \<cdot> ((h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine) \<cdot> (g\<^sub>0h\<^sub>1.\<phi> \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine) \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1, f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine]) \<cdot> \<a>[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1 \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine] \<cdot> (g.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>1 \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine) \<cdot> \<beta>\<^sub>g
[PROOF STEP]
by auto
[PROOF STATE]
proof (state)
this:
((g \<star> h \<star> TTfgh_TfTgh.the_\<theta>) \<cdot> (g \<star> can (\<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> ((\<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) (((\<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>)) \<cdot> (g \<star> (h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine) \<cdot> (g \<star> g\<^sub>0h\<^sub>1.\<phi> \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine) \<cdot> (g \<star> \<a>\<^sup>-\<^sup>1[tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1, f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine])) \<cdot> \<a>[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1 \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine] \<cdot> (g.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>1 \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine) \<cdot> \<beta>\<^sub>g = (g \<star> (h \<star> TTfgh_TfTgh.the_\<theta>) \<cdot> can (\<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> ((\<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) (((\<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) \<cdot> ((h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine) \<cdot> (g\<^sub>0h\<^sub>1.\<phi> \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine) \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1, f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine]) \<cdot> \<a>[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1 \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine] \<cdot> (g.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>1 \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine) \<cdot> \<beta>\<^sub>g
goal (1 subgoal):
1. (g \<star> \<theta>\<^sub>g) \<cdot> \<a>[g, tab\<^sub>0 g, w\<^sub>g] \<cdot> (g.tab \<star> w\<^sub>g) = ((g \<star> \<theta>\<^sub>g') \<cdot> \<a>[g, tab\<^sub>0 g, w\<^sub>g'] \<cdot> (g.tab \<star> w\<^sub>g')) \<cdot> \<beta>\<^sub>g
[PROOF STEP]
also
[PROOF STATE]
proof (state)
this:
((g \<star> h \<star> TTfgh_TfTgh.the_\<theta>) \<cdot> (g \<star> can (\<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> ((\<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) (((\<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>)) \<cdot> (g \<star> (h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine) \<cdot> (g \<star> g\<^sub>0h\<^sub>1.\<phi> \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine) \<cdot> (g \<star> \<a>\<^sup>-\<^sup>1[tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1, f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine])) \<cdot> \<a>[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1 \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine] \<cdot> (g.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>1 \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine) \<cdot> \<beta>\<^sub>g = (g \<star> (h \<star> TTfgh_TfTgh.the_\<theta>) \<cdot> can (\<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> ((\<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) (((\<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) \<cdot> ((h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine) \<cdot> (g\<^sub>0h\<^sub>1.\<phi> \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine) \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1, f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine]) \<cdot> \<a>[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1 \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine] \<cdot> (g.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>1 \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine) \<cdot> \<beta>\<^sub>g
goal (1 subgoal):
1. (g \<star> \<theta>\<^sub>g) \<cdot> \<a>[g, tab\<^sub>0 g, w\<^sub>g] \<cdot> (g.tab \<star> w\<^sub>g) = ((g \<star> \<theta>\<^sub>g') \<cdot> \<a>[g, tab\<^sub>0 g, w\<^sub>g'] \<cdot> (g.tab \<star> w\<^sub>g')) \<cdot> \<beta>\<^sub>g
[PROOF STEP]
have "... = g.composite_cell w\<^sub>g' \<theta>\<^sub>g' \<cdot> \<beta>\<^sub>g"
[PROOF STATE]
proof (prove)
goal (1 subgoal):
1. (g \<star> (h \<star> TTfgh_TfTgh.the_\<theta>) \<cdot> can (\<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> ((\<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) (((\<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) \<cdot> ((h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine) \<cdot> (g\<^sub>0h\<^sub>1.\<phi> \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine) \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1, f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine]) \<cdot> \<a>[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1 \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine] \<cdot> (g.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>1 \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine) \<cdot> \<beta>\<^sub>g = ((g \<star> \<theta>\<^sub>g') \<cdot> \<a>[g, tab\<^sub>0 g, w\<^sub>g'] \<cdot> (g.tab \<star> w\<^sub>g')) \<cdot> \<beta>\<^sub>g
[PROOF STEP]
unfolding w\<^sub>g'_def \<theta>\<^sub>g'_def
[PROOF STATE]
proof (prove)
goal (1 subgoal):
1. (g \<star> (h \<star> TTfgh_TfTgh.the_\<theta>) \<cdot> can (\<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> ((\<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) (((\<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) \<cdot> ((h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine) \<cdot> (g\<^sub>0h\<^sub>1.\<phi> \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine) \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1, f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine]) \<cdot> \<a>[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1 \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine] \<cdot> (g.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>1 \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine) \<cdot> \<beta>\<^sub>g = ((g \<star> (h \<star> TTfgh_TfTgh.the_\<theta>) \<cdot> can (\<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> ((\<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) (((\<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) \<cdot> ((h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine) \<cdot> (g\<^sub>0h\<^sub>1.\<phi> \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine) \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1, f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine]) \<cdot> \<a>[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1 \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine] \<cdot> (g.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>1 \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine)) \<cdot> \<beta>\<^sub>g
[PROOF STEP]
using comp_assoc
[PROOF STATE]
proof (prove)
using this:
(?h \<cdot> ?g) \<cdot> ?f = ?h \<cdot> ?g \<cdot> ?f
goal (1 subgoal):
1. (g \<star> (h \<star> TTfgh_TfTgh.the_\<theta>) \<cdot> can (\<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> ((\<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) (((\<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) \<cdot> ((h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine) \<cdot> (g\<^sub>0h\<^sub>1.\<phi> \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine) \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1, f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine]) \<cdot> \<a>[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1 \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine] \<cdot> (g.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>1 \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine) \<cdot> \<beta>\<^sub>g = ((g \<star> (h \<star> TTfgh_TfTgh.the_\<theta>) \<cdot> can (\<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> ((\<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) (((\<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) \<cdot> ((h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine) \<cdot> (g\<^sub>0h\<^sub>1.\<phi> \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine) \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1, f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine]) \<cdot> \<a>[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1 \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine] \<cdot> (g.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>1 \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine)) \<cdot> \<beta>\<^sub>g
[PROOF STEP]
by presburger
[PROOF STATE]
proof (state)
this:
(g \<star> (h \<star> TTfgh_TfTgh.the_\<theta>) \<cdot> can (\<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> ((\<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) (((\<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) \<cdot> ((h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine) \<cdot> (g\<^sub>0h\<^sub>1.\<phi> \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine) \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1, f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine]) \<cdot> \<a>[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1 \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine] \<cdot> (g.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>1 \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine) \<cdot> \<beta>\<^sub>g = ((g \<star> \<theta>\<^sub>g') \<cdot> \<a>[g, tab\<^sub>0 g, w\<^sub>g'] \<cdot> (g.tab \<star> w\<^sub>g')) \<cdot> \<beta>\<^sub>g
goal (1 subgoal):
1. (g \<star> \<theta>\<^sub>g) \<cdot> \<a>[g, tab\<^sub>0 g, w\<^sub>g] \<cdot> (g.tab \<star> w\<^sub>g) = ((g \<star> \<theta>\<^sub>g') \<cdot> \<a>[g, tab\<^sub>0 g, w\<^sub>g'] \<cdot> (g.tab \<star> w\<^sub>g')) \<cdot> \<beta>\<^sub>g
[PROOF STEP]
finally
[PROOF STATE]
proof (chain)
picking this:
(g \<star> \<theta>\<^sub>g) \<cdot> \<a>[g, tab\<^sub>0 g, w\<^sub>g] \<cdot> (g.tab \<star> w\<^sub>g) = ((g \<star> \<theta>\<^sub>g') \<cdot> \<a>[g, tab\<^sub>0 g, w\<^sub>g'] \<cdot> (g.tab \<star> w\<^sub>g')) \<cdot> \<beta>\<^sub>g
[PROOF STEP]
show ?thesis
[PROOF STATE]
proof (prove)
using this:
(g \<star> \<theta>\<^sub>g) \<cdot> \<a>[g, tab\<^sub>0 g, w\<^sub>g] \<cdot> (g.tab \<star> w\<^sub>g) = ((g \<star> \<theta>\<^sub>g') \<cdot> \<a>[g, tab\<^sub>0 g, w\<^sub>g'] \<cdot> (g.tab \<star> w\<^sub>g')) \<cdot> \<beta>\<^sub>g
goal (1 subgoal):
1. (g \<star> \<theta>\<^sub>g) \<cdot> \<a>[g, tab\<^sub>0 g, w\<^sub>g] \<cdot> (g.tab \<star> w\<^sub>g) = ((g \<star> \<theta>\<^sub>g') \<cdot> \<a>[g, tab\<^sub>0 g, w\<^sub>g'] \<cdot> (g.tab \<star> w\<^sub>g')) \<cdot> \<beta>\<^sub>g
[PROOF STEP]
by blast
[PROOF STATE]
proof (state)
this:
(g \<star> \<theta>\<^sub>g) \<cdot> \<a>[g, tab\<^sub>0 g, w\<^sub>g] \<cdot> (g.tab \<star> w\<^sub>g) = ((g \<star> \<theta>\<^sub>g') \<cdot> \<a>[g, tab\<^sub>0 g, w\<^sub>g'] \<cdot> (g.tab \<star> w\<^sub>g')) \<cdot> \<beta>\<^sub>g
goal:
No subgoals!
[PROOF STEP]
qed
[PROOF STATE]
proof (state)
this:
(g \<star> \<theta>\<^sub>g) \<cdot> \<a>[g, tab\<^sub>0 g, w\<^sub>g] \<cdot> (g.tab \<star> w\<^sub>g) = ((g \<star> \<theta>\<^sub>g') \<cdot> \<a>[g, tab\<^sub>0 g, w\<^sub>g'] \<cdot> (g.tab \<star> w\<^sub>g')) \<cdot> \<beta>\<^sub>g
goal (2 subgoals):
1. \<lbrakk>\<lbrakk>g\<^sub>0h\<^sub>1.p\<^sub>1 \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine\<rbrakk>\<rbrakk> = \<lbrakk>\<lbrakk>f\<^sub>0g\<^sub>1.p\<^sub>0 \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1\<rbrakk>\<rbrakk>
2. \<lbrakk>\<lbrakk>g\<^sub>0h\<^sub>1.p\<^sub>0 \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine\<rbrakk>\<rbrakk> = \<lbrakk>\<lbrakk>fg\<^sub>0h\<^sub>1.p\<^sub>0\<rbrakk>\<rbrakk>
[PROOF STEP]
have 6: "\<exists>!\<gamma>. \<guillemotleft>\<gamma> : w\<^sub>g \<Rightarrow> w\<^sub>g'\<guillemotright> \<and> \<beta>\<^sub>g = tab\<^sub>1 g \<star> \<gamma> \<and> \<theta>\<^sub>g = \<theta>\<^sub>g' \<cdot> (tab\<^sub>0 g \<star> \<gamma>)"
[PROOF STATE]
proof (prove)
goal (1 subgoal):
1. \<exists>!\<gamma>. \<guillemotleft>\<gamma> : w\<^sub>g \<rightarrow> w\<^sub>g'\<guillemotright> \<and> \<beta>\<^sub>g = tab\<^sub>1 g \<star> \<gamma> \<and> \<theta>\<^sub>g = \<theta>\<^sub>g' \<cdot> (tab\<^sub>0 g \<star> \<gamma>)
[PROOF STEP]
using w\<^sub>g w\<^sub>g' \<theta>\<^sub>g \<theta>\<^sub>g' \<beta>\<^sub>g eq\<^sub>g g.T2 [of w\<^sub>g w\<^sub>g' \<theta>\<^sub>g u\<^sub>g \<theta>\<^sub>g' \<beta>\<^sub>g]
[PROOF STATE]
proof (prove)
using this:
ide w\<^sub>g
ide w\<^sub>g'
\<guillemotleft>\<theta>\<^sub>g : tab\<^sub>0 g \<star> w\<^sub>g \<rightarrow> u\<^sub>g\<guillemotright>
\<guillemotleft>\<theta>\<^sub>g' : tab\<^sub>0 g \<star> w\<^sub>g' \<rightarrow> u\<^sub>g\<guillemotright>
\<guillemotleft>\<beta>\<^sub>g : tab\<^sub>1 g \<star> w\<^sub>g \<rightarrow> tab\<^sub>1 g \<star> w\<^sub>g'\<guillemotright>
(g \<star> \<theta>\<^sub>g) \<cdot> \<a>[g, tab\<^sub>0 g, w\<^sub>g] \<cdot> (g.tab \<star> w\<^sub>g) = ((g \<star> \<theta>\<^sub>g') \<cdot> \<a>[g, tab\<^sub>0 g, w\<^sub>g'] \<cdot> (g.tab \<star> w\<^sub>g')) \<cdot> \<beta>\<^sub>g
\<lbrakk>ide w\<^sub>g; ide w\<^sub>g'; \<guillemotleft>\<theta>\<^sub>g : tab\<^sub>0 g \<star> w\<^sub>g \<rightarrow> u\<^sub>g\<guillemotright>; \<guillemotleft>\<theta>\<^sub>g' : tab\<^sub>0 g \<star> w\<^sub>g' \<rightarrow> u\<^sub>g\<guillemotright>; \<guillemotleft>\<beta>\<^sub>g : tab\<^sub>1 g \<star> w\<^sub>g \<rightarrow> tab\<^sub>1 g \<star> w\<^sub>g'\<guillemotright>; (g \<star> \<theta>\<^sub>g) \<cdot> \<a>[g, tab\<^sub>0 g, w\<^sub>g] \<cdot> (g.tab \<star> w\<^sub>g) = ((g \<star> \<theta>\<^sub>g') \<cdot> \<a>[g, tab\<^sub>0 g, w\<^sub>g'] \<cdot> (g.tab \<star> w\<^sub>g')) \<cdot> \<beta>\<^sub>g\<rbrakk> \<Longrightarrow> \<exists>!\<gamma>. \<guillemotleft>\<gamma> : w\<^sub>g \<rightarrow> w\<^sub>g'\<guillemotright> \<and> \<beta>\<^sub>g = tab\<^sub>1 g \<star> \<gamma> \<and> \<theta>\<^sub>g = \<theta>\<^sub>g' \<cdot> (tab\<^sub>0 g \<star> \<gamma>)
goal (1 subgoal):
1. \<exists>!\<gamma>. \<guillemotleft>\<gamma> : w\<^sub>g \<rightarrow> w\<^sub>g'\<guillemotright> \<and> \<beta>\<^sub>g = tab\<^sub>1 g \<star> \<gamma> \<and> \<theta>\<^sub>g = \<theta>\<^sub>g' \<cdot> (tab\<^sub>0 g \<star> \<gamma>)
[PROOF STEP]
by blast
[PROOF STATE]
proof (state)
this:
\<exists>!\<gamma>. \<guillemotleft>\<gamma> : w\<^sub>g \<rightarrow> w\<^sub>g'\<guillemotright> \<and> \<beta>\<^sub>g = tab\<^sub>1 g \<star> \<gamma> \<and> \<theta>\<^sub>g = \<theta>\<^sub>g' \<cdot> (tab\<^sub>0 g \<star> \<gamma>)
goal (2 subgoals):
1. \<lbrakk>\<lbrakk>g\<^sub>0h\<^sub>1.p\<^sub>1 \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine\<rbrakk>\<rbrakk> = \<lbrakk>\<lbrakk>f\<^sub>0g\<^sub>1.p\<^sub>0 \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1\<rbrakk>\<rbrakk>
2. \<lbrakk>\<lbrakk>g\<^sub>0h\<^sub>1.p\<^sub>0 \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine\<rbrakk>\<rbrakk> = \<lbrakk>\<lbrakk>fg\<^sub>0h\<^sub>1.p\<^sub>0\<rbrakk>\<rbrakk>
[PROOF STEP]
obtain \<gamma>\<^sub>g where \<gamma>\<^sub>g: "\<guillemotleft>\<gamma>\<^sub>g : w\<^sub>g \<Rightarrow> w\<^sub>g'\<guillemotright> \<and> \<beta>\<^sub>g = tab\<^sub>1 g \<star> \<gamma>\<^sub>g \<and> \<theta>\<^sub>g = \<theta>\<^sub>g' \<cdot> (tab\<^sub>0 g \<star> \<gamma>\<^sub>g)"
[PROOF STATE]
proof (prove)
goal (1 subgoal):
1. (\<And>\<gamma>\<^sub>g. \<guillemotleft>\<gamma>\<^sub>g : w\<^sub>g \<rightarrow> w\<^sub>g'\<guillemotright> \<and> \<beta>\<^sub>g = tab\<^sub>1 g \<star> \<gamma>\<^sub>g \<and> \<theta>\<^sub>g = \<theta>\<^sub>g' \<cdot> (tab\<^sub>0 g \<star> \<gamma>\<^sub>g) \<Longrightarrow> thesis) \<Longrightarrow> thesis
[PROOF STEP]
using 6
[PROOF STATE]
proof (prove)
using this:
\<exists>!\<gamma>. \<guillemotleft>\<gamma> : w\<^sub>g \<rightarrow> w\<^sub>g'\<guillemotright> \<and> \<beta>\<^sub>g = tab\<^sub>1 g \<star> \<gamma> \<and> \<theta>\<^sub>g = \<theta>\<^sub>g' \<cdot> (tab\<^sub>0 g \<star> \<gamma>)
goal (1 subgoal):
1. (\<And>\<gamma>\<^sub>g. \<guillemotleft>\<gamma>\<^sub>g : w\<^sub>g \<rightarrow> w\<^sub>g'\<guillemotright> \<and> \<beta>\<^sub>g = tab\<^sub>1 g \<star> \<gamma>\<^sub>g \<and> \<theta>\<^sub>g = \<theta>\<^sub>g' \<cdot> (tab\<^sub>0 g \<star> \<gamma>\<^sub>g) \<Longrightarrow> thesis) \<Longrightarrow> thesis
[PROOF STEP]
by auto
[PROOF STATE]
proof (state)
this:
\<guillemotleft>\<gamma>\<^sub>g : w\<^sub>g \<rightarrow> w\<^sub>g'\<guillemotright> \<and> \<beta>\<^sub>g = tab\<^sub>1 g \<star> \<gamma>\<^sub>g \<and> \<theta>\<^sub>g = \<theta>\<^sub>g' \<cdot> (tab\<^sub>0 g \<star> \<gamma>\<^sub>g)
goal (2 subgoals):
1. \<lbrakk>\<lbrakk>g\<^sub>0h\<^sub>1.p\<^sub>1 \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine\<rbrakk>\<rbrakk> = \<lbrakk>\<lbrakk>f\<^sub>0g\<^sub>1.p\<^sub>0 \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1\<rbrakk>\<rbrakk>
2. \<lbrakk>\<lbrakk>g\<^sub>0h\<^sub>1.p\<^sub>0 \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine\<rbrakk>\<rbrakk> = \<lbrakk>\<lbrakk>fg\<^sub>0h\<^sub>1.p\<^sub>0\<rbrakk>\<rbrakk>
[PROOF STEP]
show "\<lbrakk>\<lbrakk>Tgh.p\<^sub>1 \<star> TfTgh.p\<^sub>0 \<star> TTfgh_TfTgh.chine\<rbrakk>\<rbrakk> = \<lbrakk>\<lbrakk>Tfg.p\<^sub>0 \<star> TTfgh.p\<^sub>1\<rbrakk>\<rbrakk>"
[PROOF STATE]
proof (prove)
goal (1 subgoal):
1. \<lbrakk>\<lbrakk>g\<^sub>0h\<^sub>1.p\<^sub>1 \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine\<rbrakk>\<rbrakk> = \<lbrakk>\<lbrakk>f\<^sub>0g\<^sub>1.p\<^sub>0 \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1\<rbrakk>\<rbrakk>
[PROOF STEP]
proof -
[PROOF STATE]
proof (state)
goal (1 subgoal):
1. \<lbrakk>\<lbrakk>g\<^sub>0h\<^sub>1.p\<^sub>1 \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine\<rbrakk>\<rbrakk> = \<lbrakk>\<lbrakk>f\<^sub>0g\<^sub>1.p\<^sub>0 \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1\<rbrakk>\<rbrakk>
[PROOF STEP]
have "iso \<gamma>\<^sub>g"
[PROOF STATE]
proof (prove)
goal (1 subgoal):
1. local.iso \<gamma>\<^sub>g
[PROOF STEP]
using \<gamma>\<^sub>g BS3 w\<^sub>g_is_map w\<^sub>g'_is_map
[PROOF STATE]
proof (prove)
using this:
\<guillemotleft>\<gamma>\<^sub>g : w\<^sub>g \<rightarrow> w\<^sub>g'\<guillemotright> \<and> \<beta>\<^sub>g = tab\<^sub>1 g \<star> \<gamma>\<^sub>g \<and> \<theta>\<^sub>g = \<theta>\<^sub>g' \<cdot> (tab\<^sub>0 g \<star> \<gamma>\<^sub>g)
\<lbrakk>is_left_adjoint ?f; is_left_adjoint ?f'; \<guillemotleft>?\<mu> : ?f \<rightarrow> ?f'\<guillemotright>; \<guillemotleft>?\<mu>' : ?f \<rightarrow> ?f'\<guillemotright>\<rbrakk> \<Longrightarrow> local.iso ?\<mu> \<and> local.iso ?\<mu>' \<and> ?\<mu> = ?\<mu>'
is_left_adjoint w\<^sub>g
is_left_adjoint w\<^sub>g'
goal (1 subgoal):
1. local.iso \<gamma>\<^sub>g
[PROOF STEP]
by blast
[PROOF STATE]
proof (state)
this:
local.iso \<gamma>\<^sub>g
goal (1 subgoal):
1. \<lbrakk>\<lbrakk>g\<^sub>0h\<^sub>1.p\<^sub>1 \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine\<rbrakk>\<rbrakk> = \<lbrakk>\<lbrakk>f\<^sub>0g\<^sub>1.p\<^sub>0 \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1\<rbrakk>\<rbrakk>
[PROOF STEP]
hence "isomorphic w\<^sub>g w\<^sub>g'"
[PROOF STATE]
proof (prove)
using this:
local.iso \<gamma>\<^sub>g
goal (1 subgoal):
1. w\<^sub>g \<cong> w\<^sub>g'
[PROOF STEP]
using \<gamma>\<^sub>g isomorphic_def isomorphic_symmetric
[PROOF STATE]
proof (prove)
using this:
local.iso \<gamma>\<^sub>g
\<guillemotleft>\<gamma>\<^sub>g : w\<^sub>g \<rightarrow> w\<^sub>g'\<guillemotright> \<and> \<beta>\<^sub>g = tab\<^sub>1 g \<star> \<gamma>\<^sub>g \<and> \<theta>\<^sub>g = \<theta>\<^sub>g' \<cdot> (tab\<^sub>0 g \<star> \<gamma>\<^sub>g)
(?a \<cong> ?a') = (\<exists>f. \<guillemotleft>f : ?a \<rightarrow> ?a'\<guillemotright> \<and> local.iso f)
?f \<cong> ?g \<Longrightarrow> ?g \<cong> ?f
goal (1 subgoal):
1. w\<^sub>g \<cong> w\<^sub>g'
[PROOF STEP]
by auto
[PROOF STATE]
proof (state)
this:
w\<^sub>g \<cong> w\<^sub>g'
goal (1 subgoal):
1. \<lbrakk>\<lbrakk>g\<^sub>0h\<^sub>1.p\<^sub>1 \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine\<rbrakk>\<rbrakk> = \<lbrakk>\<lbrakk>f\<^sub>0g\<^sub>1.p\<^sub>0 \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1\<rbrakk>\<rbrakk>
[PROOF STEP]
thus ?thesis
[PROOF STATE]
proof (prove)
using this:
w\<^sub>g \<cong> w\<^sub>g'
goal (1 subgoal):
1. \<lbrakk>\<lbrakk>g\<^sub>0h\<^sub>1.p\<^sub>1 \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine\<rbrakk>\<rbrakk> = \<lbrakk>\<lbrakk>f\<^sub>0g\<^sub>1.p\<^sub>0 \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1\<rbrakk>\<rbrakk>
[PROOF STEP]
using w\<^sub>g w\<^sub>g' w\<^sub>g_def w\<^sub>g'_def Maps.CLS_eqI
[PROOF STATE]
proof (prove)
using this:
w\<^sub>g \<cong> w\<^sub>g'
ide w\<^sub>g
ide w\<^sub>g'
w\<^sub>g = f\<^sub>0g\<^sub>1.p\<^sub>0 \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1
w\<^sub>g' = g\<^sub>0h\<^sub>1.p\<^sub>1 \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine
ide ?f \<Longrightarrow> (\<lbrakk>\<lbrakk>?f\<rbrakk>\<rbrakk> = \<lbrakk>\<lbrakk>?g\<rbrakk>\<rbrakk>) = (?f \<cong> ?g)
goal (1 subgoal):
1. \<lbrakk>\<lbrakk>g\<^sub>0h\<^sub>1.p\<^sub>1 \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine\<rbrakk>\<rbrakk> = \<lbrakk>\<lbrakk>f\<^sub>0g\<^sub>1.p\<^sub>0 \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1\<rbrakk>\<rbrakk>
[PROOF STEP]
by auto
[PROOF STATE]
proof (state)
this:
\<lbrakk>\<lbrakk>g\<^sub>0h\<^sub>1.p\<^sub>1 \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine\<rbrakk>\<rbrakk> = \<lbrakk>\<lbrakk>f\<^sub>0g\<^sub>1.p\<^sub>0 \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1\<rbrakk>\<rbrakk>
goal:
No subgoals!
[PROOF STEP]
qed
[PROOF STATE]
proof (state)
this:
\<lbrakk>\<lbrakk>g\<^sub>0h\<^sub>1.p\<^sub>1 \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine\<rbrakk>\<rbrakk> = \<lbrakk>\<lbrakk>f\<^sub>0g\<^sub>1.p\<^sub>0 \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1\<rbrakk>\<rbrakk>
goal (1 subgoal):
1. \<lbrakk>\<lbrakk>g\<^sub>0h\<^sub>1.p\<^sub>0 \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine\<rbrakk>\<rbrakk> = \<lbrakk>\<lbrakk>fg\<^sub>0h\<^sub>1.p\<^sub>0\<rbrakk>\<rbrakk>
[PROOF STEP]
define u\<^sub>h where "u\<^sub>h = tab\<^sub>0 h \<star> TTfgh.p\<^sub>0"
[PROOF STATE]
proof (state)
this:
u\<^sub>h = tab\<^sub>0 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0
goal (1 subgoal):
1. \<lbrakk>\<lbrakk>g\<^sub>0h\<^sub>1.p\<^sub>0 \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine\<rbrakk>\<rbrakk> = \<lbrakk>\<lbrakk>fg\<^sub>0h\<^sub>1.p\<^sub>0\<rbrakk>\<rbrakk>
[PROOF STEP]
define w\<^sub>h where "w\<^sub>h = TTfgh.p\<^sub>0"
[PROOF STATE]
proof (state)
this:
w\<^sub>h = fg\<^sub>0h\<^sub>1.p\<^sub>0
goal (1 subgoal):
1. \<lbrakk>\<lbrakk>g\<^sub>0h\<^sub>1.p\<^sub>0 \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine\<rbrakk>\<rbrakk> = \<lbrakk>\<lbrakk>fg\<^sub>0h\<^sub>1.p\<^sub>0\<rbrakk>\<rbrakk>
[PROOF STEP]
define w\<^sub>h' where "w\<^sub>h' = Tgh.p\<^sub>0 \<star> TfTgh.p\<^sub>0 \<star> TTfgh_TfTgh.chine"
[PROOF STATE]
proof (state)
this:
w\<^sub>h' = g\<^sub>0h\<^sub>1.p\<^sub>0 \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine
goal (1 subgoal):
1. \<lbrakk>\<lbrakk>g\<^sub>0h\<^sub>1.p\<^sub>0 \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine\<rbrakk>\<rbrakk> = \<lbrakk>\<lbrakk>fg\<^sub>0h\<^sub>1.p\<^sub>0\<rbrakk>\<rbrakk>
[PROOF STEP]
define \<theta>\<^sub>h
where "\<theta>\<^sub>h = tab\<^sub>0 h \<star> TTfgh.p\<^sub>0"
[PROOF STATE]
proof (state)
this:
\<theta>\<^sub>h = tab\<^sub>0 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0
goal (1 subgoal):
1. \<lbrakk>\<lbrakk>g\<^sub>0h\<^sub>1.p\<^sub>0 \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine\<rbrakk>\<rbrakk> = \<lbrakk>\<lbrakk>fg\<^sub>0h\<^sub>1.p\<^sub>0\<rbrakk>\<rbrakk>
[PROOF STEP]
define \<theta>\<^sub>h'
where "\<theta>\<^sub>h' = TTfgh_TfTgh.the_\<theta> \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>0 h \<star> Tgh.p\<^sub>0, TfTgh.p\<^sub>0, TTfgh_TfTgh.chine] \<cdot>
\<a>\<^sup>-\<^sup>1[tab\<^sub>0 h, Tgh.p\<^sub>0, TfTgh.p\<^sub>0 \<star> TTfgh_TfTgh.chine]"
[PROOF STATE]
proof (state)
this:
\<theta>\<^sub>h' = TTfgh_TfTgh.the_\<theta> \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0, f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine] \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>0 h, g\<^sub>0h\<^sub>1.p\<^sub>0, f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine]
goal (1 subgoal):
1. \<lbrakk>\<lbrakk>g\<^sub>0h\<^sub>1.p\<^sub>0 \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine\<rbrakk>\<rbrakk> = \<lbrakk>\<lbrakk>fg\<^sub>0h\<^sub>1.p\<^sub>0\<rbrakk>\<rbrakk>
[PROOF STEP]
define \<beta>\<^sub>h
where "\<beta>\<^sub>h = \<a>[tab\<^sub>1 h, Tgh.p\<^sub>0, TfTgh.p\<^sub>0 \<star> TTfgh_TfTgh.chine] \<cdot>
(g\<^sub>0h\<^sub>1.\<phi> \<star> TfTgh.p\<^sub>0 \<star> TTfgh_TfTgh.chine) \<cdot>
\<a>\<^sup>-\<^sup>1[tab\<^sub>0 g, Tgh.p\<^sub>1, TfTgh.p\<^sub>0 \<star> TTfgh_TfTgh.chine] \<cdot> (tab\<^sub>0 g \<star> \<gamma>\<^sub>g) \<cdot>
\<a>[tab\<^sub>0 g, Tfg.p\<^sub>0, TTfgh.p\<^sub>1] \<cdot> inv fg\<^sub>0h\<^sub>1.\<phi>"
[PROOF STATE]
proof (state)
this:
\<beta>\<^sub>h = \<a>[tab\<^sub>1 h, g\<^sub>0h\<^sub>1.p\<^sub>0, f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine] \<cdot> (g\<^sub>0h\<^sub>1.\<phi> \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine) \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1, f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine] \<cdot> (tab\<^sub>0 g \<star> \<gamma>\<^sub>g) \<cdot> \<a>[tab\<^sub>0 g, f\<^sub>0g\<^sub>1.p\<^sub>0, fg\<^sub>0h\<^sub>1.p\<^sub>1] \<cdot> local.inv fg\<^sub>0h\<^sub>1.\<phi>
goal (1 subgoal):
1. \<lbrakk>\<lbrakk>g\<^sub>0h\<^sub>1.p\<^sub>0 \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine\<rbrakk>\<rbrakk> = \<lbrakk>\<lbrakk>fg\<^sub>0h\<^sub>1.p\<^sub>0\<rbrakk>\<rbrakk>
[PROOF STEP]
have u\<^sub>h: "ide u\<^sub>h"
[PROOF STATE]
proof (prove)
goal (1 subgoal):
1. ide u\<^sub>h
[PROOF STEP]
unfolding u\<^sub>h_def
[PROOF STATE]
proof (prove)
goal (1 subgoal):
1. ide (tab\<^sub>0 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0)
[PROOF STEP]
by simp
[PROOF STATE]
proof (state)
this:
ide u\<^sub>h
goal (1 subgoal):
1. \<lbrakk>\<lbrakk>g\<^sub>0h\<^sub>1.p\<^sub>0 \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine\<rbrakk>\<rbrakk> = \<lbrakk>\<lbrakk>fg\<^sub>0h\<^sub>1.p\<^sub>0\<rbrakk>\<rbrakk>
[PROOF STEP]
have w\<^sub>h: "ide w\<^sub>h"
[PROOF STATE]
proof (prove)
goal (1 subgoal):
1. ide w\<^sub>h
[PROOF STEP]
unfolding w\<^sub>h_def
[PROOF STATE]
proof (prove)
goal (1 subgoal):
1. ide fg\<^sub>0h\<^sub>1.p\<^sub>0
[PROOF STEP]
by simp
[PROOF STATE]
proof (state)
this:
ide w\<^sub>h
goal (1 subgoal):
1. \<lbrakk>\<lbrakk>g\<^sub>0h\<^sub>1.p\<^sub>0 \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine\<rbrakk>\<rbrakk> = \<lbrakk>\<lbrakk>fg\<^sub>0h\<^sub>1.p\<^sub>0\<rbrakk>\<rbrakk>
[PROOF STEP]
have w\<^sub>h_is_map: "is_left_adjoint w\<^sub>h"
[PROOF STATE]
proof (prove)
goal (1 subgoal):
1. is_left_adjoint w\<^sub>h
[PROOF STEP]
unfolding w\<^sub>h_def
[PROOF STATE]
proof (prove)
goal (1 subgoal):
1. is_left_adjoint fg\<^sub>0h\<^sub>1.p\<^sub>0
[PROOF STEP]
by simp
[PROOF STATE]
proof (state)
this:
is_left_adjoint w\<^sub>h
goal (1 subgoal):
1. \<lbrakk>\<lbrakk>g\<^sub>0h\<^sub>1.p\<^sub>0 \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine\<rbrakk>\<rbrakk> = \<lbrakk>\<lbrakk>fg\<^sub>0h\<^sub>1.p\<^sub>0\<rbrakk>\<rbrakk>
[PROOF STEP]
have w\<^sub>h': "ide w\<^sub>h'"
[PROOF STATE]
proof (prove)
goal (1 subgoal):
1. ide w\<^sub>h'
[PROOF STEP]
unfolding w\<^sub>h'_def
[PROOF STATE]
proof (prove)
goal (1 subgoal):
1. ide (g\<^sub>0h\<^sub>1.p\<^sub>0 \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine)
[PROOF STEP]
by simp
[PROOF STATE]
proof (state)
this:
ide w\<^sub>h'
goal (1 subgoal):
1. \<lbrakk>\<lbrakk>g\<^sub>0h\<^sub>1.p\<^sub>0 \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine\<rbrakk>\<rbrakk> = \<lbrakk>\<lbrakk>fg\<^sub>0h\<^sub>1.p\<^sub>0\<rbrakk>\<rbrakk>
[PROOF STEP]
have w\<^sub>h'_is_map: "is_left_adjoint w\<^sub>h'"
[PROOF STATE]
proof (prove)
goal (1 subgoal):
1. is_left_adjoint w\<^sub>h'
[PROOF STEP]
unfolding w\<^sub>h'_def
[PROOF STATE]
proof (prove)
goal (1 subgoal):
1. is_left_adjoint (g\<^sub>0h\<^sub>1.p\<^sub>0 \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine)
[PROOF STEP]
using g\<^sub>0h\<^sub>1.p\<^sub>0_simps f\<^sub>0gh\<^sub>1.p\<^sub>0_simps TTfgh_TfTgh.is_map left_adjoints_compose
[PROOF STATE]
proof (prove)
using this:
trg g\<^sub>0h\<^sub>1.p\<^sub>0 = src (tab\<^sub>1 h)
trg f\<^sub>0gh\<^sub>1.p\<^sub>0 = src (tab\<^sub>1 g \<star> g\<^sub>0h\<^sub>1.p\<^sub>1)
is_left_adjoint TTfgh_TfTgh.chine
\<lbrakk>is_left_adjoint ?f; is_left_adjoint ?f'; src ?f = trg ?f'\<rbrakk> \<Longrightarrow> is_left_adjoint (?f \<star> ?f')
goal (1 subgoal):
1. is_left_adjoint (g\<^sub>0h\<^sub>1.p\<^sub>0 \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine)
[PROOF STEP]
by simp
[PROOF STATE]
proof (state)
this:
is_left_adjoint w\<^sub>h'
goal (1 subgoal):
1. \<lbrakk>\<lbrakk>g\<^sub>0h\<^sub>1.p\<^sub>0 \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine\<rbrakk>\<rbrakk> = \<lbrakk>\<lbrakk>fg\<^sub>0h\<^sub>1.p\<^sub>0\<rbrakk>\<rbrakk>
[PROOF STEP]
have \<theta>\<^sub>h: "\<guillemotleft>\<theta>\<^sub>h : tab\<^sub>0 h \<star> w\<^sub>h \<Rightarrow> u\<^sub>h\<guillemotright>"
[PROOF STATE]
proof (prove)
goal (1 subgoal):
1. \<guillemotleft>\<theta>\<^sub>h : tab\<^sub>0 h \<star> w\<^sub>h \<rightarrow> u\<^sub>h\<guillemotright>
[PROOF STEP]
unfolding \<theta>\<^sub>h_def w\<^sub>h_def u\<^sub>h_def
[PROOF STATE]
proof (prove)
goal (1 subgoal):
1. \<guillemotleft>tab\<^sub>0 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0 : tab\<^sub>0 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0 \<rightarrow> tab\<^sub>0 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0\<guillemotright>
[PROOF STEP]
by auto
[PROOF STATE]
proof (state)
this:
\<guillemotleft>\<theta>\<^sub>h : tab\<^sub>0 h \<star> w\<^sub>h \<rightarrow> u\<^sub>h\<guillemotright>
goal (1 subgoal):
1. \<lbrakk>\<lbrakk>g\<^sub>0h\<^sub>1.p\<^sub>0 \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine\<rbrakk>\<rbrakk> = \<lbrakk>\<lbrakk>fg\<^sub>0h\<^sub>1.p\<^sub>0\<rbrakk>\<rbrakk>
[PROOF STEP]
have \<theta>\<^sub>h': "\<guillemotleft>\<theta>\<^sub>h' : tab\<^sub>0 h \<star> w\<^sub>h' \<Rightarrow> u\<^sub>h\<guillemotright>"
[PROOF STATE]
proof (prove)
goal (1 subgoal):
1. \<guillemotleft>\<theta>\<^sub>h' : tab\<^sub>0 h \<star> w\<^sub>h' \<rightarrow> u\<^sub>h\<guillemotright>
[PROOF STEP]
unfolding \<theta>\<^sub>h'_def w\<^sub>h'_def u\<^sub>h_def
[PROOF STATE]
proof (prove)
goal (1 subgoal):
1. \<guillemotleft>TTfgh_TfTgh.the_\<theta> \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0, f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine] \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>0 h, g\<^sub>0h\<^sub>1.p\<^sub>0, f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine] : tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0 \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine \<rightarrow> tab\<^sub>0 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0\<guillemotright>
[PROOF STEP]
using g\<^sub>0h\<^sub>1.p\<^sub>0_simps f\<^sub>0gh\<^sub>1.p\<^sub>0_simps
[PROOF STATE]
proof (prove)
using this:
trg g\<^sub>0h\<^sub>1.p\<^sub>0 = src (tab\<^sub>1 h)
trg f\<^sub>0gh\<^sub>1.p\<^sub>0 = src (tab\<^sub>1 g \<star> g\<^sub>0h\<^sub>1.p\<^sub>1)
goal (1 subgoal):
1. \<guillemotleft>TTfgh_TfTgh.the_\<theta> \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0, f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine] \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>0 h, g\<^sub>0h\<^sub>1.p\<^sub>0, f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine] : tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0 \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine \<rightarrow> tab\<^sub>0 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0\<guillemotright>
[PROOF STEP]
by (intro comp_in_homI) auto
[PROOF STATE]
proof (state)
this:
\<guillemotleft>\<theta>\<^sub>h' : tab\<^sub>0 h \<star> w\<^sub>h' \<rightarrow> u\<^sub>h\<guillemotright>
goal (1 subgoal):
1. \<lbrakk>\<lbrakk>g\<^sub>0h\<^sub>1.p\<^sub>0 \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine\<rbrakk>\<rbrakk> = \<lbrakk>\<lbrakk>fg\<^sub>0h\<^sub>1.p\<^sub>0\<rbrakk>\<rbrakk>
[PROOF STEP]
have \<beta>\<^sub>h: "\<guillemotleft>\<beta>\<^sub>h : tab\<^sub>1 h \<star> w\<^sub>h \<Rightarrow> tab\<^sub>1 h \<star> w\<^sub>h'\<guillemotright>"
[PROOF STATE]
proof (prove)
goal (1 subgoal):
1. \<guillemotleft>\<beta>\<^sub>h : tab\<^sub>1 h \<star> w\<^sub>h \<rightarrow> tab\<^sub>1 h \<star> w\<^sub>h'\<guillemotright>
[PROOF STEP]
proof (unfold \<beta>\<^sub>h_def w\<^sub>h_def w\<^sub>h'_def, intro comp_in_homI)
[PROOF STATE]
proof (state)
goal (6 subgoals):
1. \<guillemotleft>local.inv fg\<^sub>0h\<^sub>1.\<phi> : tab\<^sub>1 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0 \<rightarrow> ?b4\<guillemotright>
2. \<guillemotleft>\<a>[tab\<^sub>0 g, f\<^sub>0g\<^sub>1.p\<^sub>0, fg\<^sub>0h\<^sub>1.p\<^sub>1] : ?b4 \<rightarrow> ?b3\<guillemotright>
3. \<guillemotleft>tab\<^sub>0 g \<star> \<gamma>\<^sub>g : ?b3 \<rightarrow> ?b2\<guillemotright>
4. \<guillemotleft>\<a>\<^sup>-\<^sup>1[tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1, f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine] : ?b2 \<rightarrow> ?b1\<guillemotright>
5. \<guillemotleft>g\<^sub>0h\<^sub>1.\<phi> \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine : ?b1 \<rightarrow> ?b\<guillemotright>
6. \<guillemotleft>\<a>[tab\<^sub>1 h, g\<^sub>0h\<^sub>1.p\<^sub>0, f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine] : ?b \<rightarrow> tab\<^sub>1 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0 \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine\<guillemotright>
[PROOF STEP]
show "\<guillemotleft>inv fg\<^sub>0h\<^sub>1.\<phi> : tab\<^sub>1 h \<star> TTfgh.p\<^sub>0 \<Rightarrow> (tab\<^sub>0 g \<star> Tfg.p\<^sub>0) \<star> TTfgh.p\<^sub>1\<guillemotright>"
[PROOF STATE]
proof (prove)
goal (1 subgoal):
1. \<guillemotleft>local.inv fg\<^sub>0h\<^sub>1.\<phi> : tab\<^sub>1 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0 \<rightarrow> (tab\<^sub>0 g \<star> f\<^sub>0g\<^sub>1.p\<^sub>0) \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1\<guillemotright>
[PROOF STEP]
using fg\<^sub>0h\<^sub>1.\<phi>_uniqueness
[PROOF STATE]
proof (prove)
using this:
\<guillemotleft>?\<mu> : (tab\<^sub>0 g \<star> f\<^sub>0g\<^sub>1.p\<^sub>0) \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1 \<rightarrow> tab\<^sub>1 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0\<guillemotright> \<Longrightarrow> ?\<mu> = fg\<^sub>0h\<^sub>1.\<phi>
local.iso fg\<^sub>0h\<^sub>1.\<phi>
goal (1 subgoal):
1. \<guillemotleft>local.inv fg\<^sub>0h\<^sub>1.\<phi> : tab\<^sub>1 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0 \<rightarrow> (tab\<^sub>0 g \<star> f\<^sub>0g\<^sub>1.p\<^sub>0) \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1\<guillemotright>
[PROOF STEP]
by blast
[PROOF STATE]
proof (state)
this:
\<guillemotleft>local.inv fg\<^sub>0h\<^sub>1.\<phi> : tab\<^sub>1 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0 \<rightarrow> (tab\<^sub>0 g \<star> f\<^sub>0g\<^sub>1.p\<^sub>0) \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1\<guillemotright>
goal (5 subgoals):
1. \<guillemotleft>\<a>[tab\<^sub>0 g, f\<^sub>0g\<^sub>1.p\<^sub>0, fg\<^sub>0h\<^sub>1.p\<^sub>1] : (tab\<^sub>0 g \<star> f\<^sub>0g\<^sub>1.p\<^sub>0) \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1 \<rightarrow> ?b3\<guillemotright>
2. \<guillemotleft>tab\<^sub>0 g \<star> \<gamma>\<^sub>g : ?b3 \<rightarrow> ?b2\<guillemotright>
3. \<guillemotleft>\<a>\<^sup>-\<^sup>1[tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1, f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine] : ?b2 \<rightarrow> ?b1\<guillemotright>
4. \<guillemotleft>g\<^sub>0h\<^sub>1.\<phi> \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine : ?b1 \<rightarrow> ?b\<guillemotright>
5. \<guillemotleft>\<a>[tab\<^sub>1 h, g\<^sub>0h\<^sub>1.p\<^sub>0, f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine] : ?b \<rightarrow> tab\<^sub>1 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0 \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine\<guillemotright>
[PROOF STEP]
show "\<guillemotleft>\<a>[tab\<^sub>0 g, Tfg.p\<^sub>0, TTfgh.p\<^sub>1] :
(tab\<^sub>0 g \<star> Tfg.p\<^sub>0) \<star> TTfgh.p\<^sub>1 \<Rightarrow> tab\<^sub>0 g \<star> Tfg.p\<^sub>0 \<star> TTfgh.p\<^sub>1\<guillemotright>"
[PROOF STATE]
proof (prove)
goal (1 subgoal):
1. \<guillemotleft>\<a>[tab\<^sub>0 g, f\<^sub>0g\<^sub>1.p\<^sub>0, fg\<^sub>0h\<^sub>1.p\<^sub>1] : (tab\<^sub>0 g \<star> f\<^sub>0g\<^sub>1.p\<^sub>0) \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1 \<rightarrow> tab\<^sub>0 g \<star> f\<^sub>0g\<^sub>1.p\<^sub>0 \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1\<guillemotright>
[PROOF STEP]
using fg\<^sub>0h\<^sub>1.p\<^sub>1_simps
[PROOF STATE]
proof (prove)
using this:
trg fg\<^sub>0h\<^sub>1.p\<^sub>1 = src (tab\<^sub>0 g \<star> f\<^sub>0g\<^sub>1.p\<^sub>0)
goal (1 subgoal):
1. \<guillemotleft>\<a>[tab\<^sub>0 g, f\<^sub>0g\<^sub>1.p\<^sub>0, fg\<^sub>0h\<^sub>1.p\<^sub>1] : (tab\<^sub>0 g \<star> f\<^sub>0g\<^sub>1.p\<^sub>0) \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1 \<rightarrow> tab\<^sub>0 g \<star> f\<^sub>0g\<^sub>1.p\<^sub>0 \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1\<guillemotright>
[PROOF STEP]
by auto
[PROOF STATE]
proof (state)
this:
\<guillemotleft>\<a>[tab\<^sub>0 g, f\<^sub>0g\<^sub>1.p\<^sub>0, fg\<^sub>0h\<^sub>1.p\<^sub>1] : (tab\<^sub>0 g \<star> f\<^sub>0g\<^sub>1.p\<^sub>0) \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1 \<rightarrow> tab\<^sub>0 g \<star> f\<^sub>0g\<^sub>1.p\<^sub>0 \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1\<guillemotright>
goal (4 subgoals):
1. \<guillemotleft>tab\<^sub>0 g \<star> \<gamma>\<^sub>g : tab\<^sub>0 g \<star> f\<^sub>0g\<^sub>1.p\<^sub>0 \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1 \<rightarrow> ?b2\<guillemotright>
2. \<guillemotleft>\<a>\<^sup>-\<^sup>1[tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1, f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine] : ?b2 \<rightarrow> ?b1\<guillemotright>
3. \<guillemotleft>g\<^sub>0h\<^sub>1.\<phi> \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine : ?b1 \<rightarrow> ?b\<guillemotright>
4. \<guillemotleft>\<a>[tab\<^sub>1 h, g\<^sub>0h\<^sub>1.p\<^sub>0, f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine] : ?b \<rightarrow> tab\<^sub>1 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0 \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine\<guillemotright>
[PROOF STEP]
show "\<guillemotleft>tab\<^sub>0 g \<star> \<gamma>\<^sub>g :
tab\<^sub>0 g \<star> Tfg.p\<^sub>0 \<star> TTfgh.p\<^sub>1 \<Rightarrow> tab\<^sub>0 g \<star> Tgh.p\<^sub>1 \<star> TfTgh.p\<^sub>0 \<star> TTfgh_TfTgh.chine\<guillemotright>"
[PROOF STATE]
proof (prove)
goal (1 subgoal):
1. \<guillemotleft>tab\<^sub>0 g \<star> \<gamma>\<^sub>g : tab\<^sub>0 g \<star> f\<^sub>0g\<^sub>1.p\<^sub>0 \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1 \<rightarrow> tab\<^sub>0 g \<star> g\<^sub>0h\<^sub>1.p\<^sub>1 \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine\<guillemotright>
[PROOF STEP]
using \<gamma>\<^sub>g w\<^sub>g_def w\<^sub>g'_def fg\<^sub>0h\<^sub>1.p\<^sub>1_simps
[PROOF STATE]
proof (prove)
using this:
\<guillemotleft>\<gamma>\<^sub>g : w\<^sub>g \<rightarrow> w\<^sub>g'\<guillemotright> \<and> \<beta>\<^sub>g = tab\<^sub>1 g \<star> \<gamma>\<^sub>g \<and> \<theta>\<^sub>g = \<theta>\<^sub>g' \<cdot> (tab\<^sub>0 g \<star> \<gamma>\<^sub>g)
w\<^sub>g = f\<^sub>0g\<^sub>1.p\<^sub>0 \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1
w\<^sub>g' = g\<^sub>0h\<^sub>1.p\<^sub>1 \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine
trg fg\<^sub>0h\<^sub>1.p\<^sub>1 = src (tab\<^sub>0 g \<star> f\<^sub>0g\<^sub>1.p\<^sub>0)
goal (1 subgoal):
1. \<guillemotleft>tab\<^sub>0 g \<star> \<gamma>\<^sub>g : tab\<^sub>0 g \<star> f\<^sub>0g\<^sub>1.p\<^sub>0 \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1 \<rightarrow> tab\<^sub>0 g \<star> g\<^sub>0h\<^sub>1.p\<^sub>1 \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine\<guillemotright>
[PROOF STEP]
by auto
[PROOF STATE]
proof (state)
this:
\<guillemotleft>tab\<^sub>0 g \<star> \<gamma>\<^sub>g : tab\<^sub>0 g \<star> f\<^sub>0g\<^sub>1.p\<^sub>0 \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1 \<rightarrow> tab\<^sub>0 g \<star> g\<^sub>0h\<^sub>1.p\<^sub>1 \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine\<guillemotright>
goal (3 subgoals):
1. \<guillemotleft>\<a>\<^sup>-\<^sup>1[tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1, f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine] : tab\<^sub>0 g \<star> g\<^sub>0h\<^sub>1.p\<^sub>1 \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine \<rightarrow> ?b1\<guillemotright>
2. \<guillemotleft>g\<^sub>0h\<^sub>1.\<phi> \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine : ?b1 \<rightarrow> ?b\<guillemotright>
3. \<guillemotleft>\<a>[tab\<^sub>1 h, g\<^sub>0h\<^sub>1.p\<^sub>0, f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine] : ?b \<rightarrow> tab\<^sub>1 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0 \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine\<guillemotright>
[PROOF STEP]
show "\<guillemotleft>\<a>\<^sup>-\<^sup>1[tab\<^sub>0 g, Tgh.p\<^sub>1, TfTgh.p\<^sub>0 \<star> TTfgh_TfTgh.chine] :
tab\<^sub>0 g \<star> Tgh.p\<^sub>1 \<star> TfTgh.p\<^sub>0 \<star> TTfgh_TfTgh.chine
\<Rightarrow> (tab\<^sub>0 g \<star> Tgh.p\<^sub>1) \<star> TfTgh.p\<^sub>0 \<star> TTfgh_TfTgh.chine\<guillemotright>"
[PROOF STATE]
proof (prove)
goal (1 subgoal):
1. \<guillemotleft>\<a>\<^sup>-\<^sup>1[tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1, f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine] : tab\<^sub>0 g \<star> g\<^sub>0h\<^sub>1.p\<^sub>1 \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine \<rightarrow> (tab\<^sub>0 g \<star> g\<^sub>0h\<^sub>1.p\<^sub>1) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine\<guillemotright>
[PROOF STEP]
using fg\<^sub>0h\<^sub>1.p\<^sub>1_simps
[PROOF STATE]
proof (prove)
using this:
trg fg\<^sub>0h\<^sub>1.p\<^sub>1 = src (tab\<^sub>0 g \<star> f\<^sub>0g\<^sub>1.p\<^sub>0)
goal (1 subgoal):
1. \<guillemotleft>\<a>\<^sup>-\<^sup>1[tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1, f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine] : tab\<^sub>0 g \<star> g\<^sub>0h\<^sub>1.p\<^sub>1 \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine \<rightarrow> (tab\<^sub>0 g \<star> g\<^sub>0h\<^sub>1.p\<^sub>1) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine\<guillemotright>
[PROOF STEP]
by auto
[PROOF STATE]
proof (state)
this:
\<guillemotleft>\<a>\<^sup>-\<^sup>1[tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1, f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine] : tab\<^sub>0 g \<star> g\<^sub>0h\<^sub>1.p\<^sub>1 \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine \<rightarrow> (tab\<^sub>0 g \<star> g\<^sub>0h\<^sub>1.p\<^sub>1) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine\<guillemotright>
goal (2 subgoals):
1. \<guillemotleft>g\<^sub>0h\<^sub>1.\<phi> \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine : (tab\<^sub>0 g \<star> g\<^sub>0h\<^sub>1.p\<^sub>1) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine \<rightarrow> ?b\<guillemotright>
2. \<guillemotleft>\<a>[tab\<^sub>1 h, g\<^sub>0h\<^sub>1.p\<^sub>0, f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine] : ?b \<rightarrow> tab\<^sub>1 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0 \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine\<guillemotright>
[PROOF STEP]
show "\<guillemotleft>g\<^sub>0h\<^sub>1.\<phi> \<star> TfTgh.p\<^sub>0 \<star> TTfgh_TfTgh.chine :
(tab\<^sub>0 g \<star> Tgh.p\<^sub>1) \<star> TfTgh.p\<^sub>0 \<star> TTfgh_TfTgh.chine
\<Rightarrow> (tab\<^sub>1 h \<star> Tgh.p\<^sub>0) \<star> TfTgh.p\<^sub>0 \<star> TTfgh_TfTgh.chine\<guillemotright>"
[PROOF STATE]
proof (prove)
goal (1 subgoal):
1. \<guillemotleft>g\<^sub>0h\<^sub>1.\<phi> \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine : (tab\<^sub>0 g \<star> g\<^sub>0h\<^sub>1.p\<^sub>1) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine \<rightarrow> (tab\<^sub>1 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine\<guillemotright>
[PROOF STEP]
using fg\<^sub>0h\<^sub>1.p\<^sub>1_simps
[PROOF STATE]
proof (prove)
using this:
trg fg\<^sub>0h\<^sub>1.p\<^sub>1 = src (tab\<^sub>0 g \<star> f\<^sub>0g\<^sub>1.p\<^sub>0)
goal (1 subgoal):
1. \<guillemotleft>g\<^sub>0h\<^sub>1.\<phi> \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine : (tab\<^sub>0 g \<star> g\<^sub>0h\<^sub>1.p\<^sub>1) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine \<rightarrow> (tab\<^sub>1 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine\<guillemotright>
[PROOF STEP]
by force
[PROOF STATE]
proof (state)
this:
\<guillemotleft>g\<^sub>0h\<^sub>1.\<phi> \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine : (tab\<^sub>0 g \<star> g\<^sub>0h\<^sub>1.p\<^sub>1) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine \<rightarrow> (tab\<^sub>1 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine\<guillemotright>
goal (1 subgoal):
1. \<guillemotleft>\<a>[tab\<^sub>1 h, g\<^sub>0h\<^sub>1.p\<^sub>0, f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine] : (tab\<^sub>1 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine \<rightarrow> tab\<^sub>1 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0 \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine\<guillemotright>
[PROOF STEP]
show "\<guillemotleft>\<a>[tab\<^sub>1 h, Tgh.p\<^sub>0, TfTgh.p\<^sub>0 \<star> TTfgh_TfTgh.chine] :
(tab\<^sub>1 h \<star> Tgh.p\<^sub>0) \<star> TfTgh.p\<^sub>0 \<star> TTfgh_TfTgh.chine
\<Rightarrow> tab\<^sub>1 h \<star> Tgh.p\<^sub>0 \<star> TfTgh.p\<^sub>0 \<star> TTfgh_TfTgh.chine\<guillemotright>"
[PROOF STATE]
proof (prove)
goal (1 subgoal):
1. \<guillemotleft>\<a>[tab\<^sub>1 h, g\<^sub>0h\<^sub>1.p\<^sub>0, f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine] : (tab\<^sub>1 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine \<rightarrow> tab\<^sub>1 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0 \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine\<guillemotright>
[PROOF STEP]
using fg\<^sub>0h\<^sub>1.p\<^sub>1_simps
[PROOF STATE]
proof (prove)
using this:
trg fg\<^sub>0h\<^sub>1.p\<^sub>1 = src (tab\<^sub>0 g \<star> f\<^sub>0g\<^sub>1.p\<^sub>0)
goal (1 subgoal):
1. \<guillemotleft>\<a>[tab\<^sub>1 h, g\<^sub>0h\<^sub>1.p\<^sub>0, f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine] : (tab\<^sub>1 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine \<rightarrow> tab\<^sub>1 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0 \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine\<guillemotright>
[PROOF STEP]
by auto
[PROOF STATE]
proof (state)
this:
\<guillemotleft>\<a>[tab\<^sub>1 h, g\<^sub>0h\<^sub>1.p\<^sub>0, f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine] : (tab\<^sub>1 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine \<rightarrow> tab\<^sub>1 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0 \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine\<guillemotright>
goal:
No subgoals!
[PROOF STEP]
qed
[PROOF STATE]
proof (state)
this:
\<guillemotleft>\<beta>\<^sub>h : tab\<^sub>1 h \<star> w\<^sub>h \<rightarrow> tab\<^sub>1 h \<star> w\<^sub>h'\<guillemotright>
goal (1 subgoal):
1. \<lbrakk>\<lbrakk>g\<^sub>0h\<^sub>1.p\<^sub>0 \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine\<rbrakk>\<rbrakk> = \<lbrakk>\<lbrakk>fg\<^sub>0h\<^sub>1.p\<^sub>0\<rbrakk>\<rbrakk>
[PROOF STEP]
have eq\<^sub>h: "h.composite_cell w\<^sub>h \<theta>\<^sub>h = h.composite_cell w\<^sub>h' \<theta>\<^sub>h' \<cdot> \<beta>\<^sub>h"
[PROOF STATE]
proof (prove)
goal (1 subgoal):
1. (h \<star> \<theta>\<^sub>h) \<cdot> \<a>[h, tab\<^sub>0 h, w\<^sub>h] \<cdot> (h.tab \<star> w\<^sub>h) = ((h \<star> \<theta>\<^sub>h') \<cdot> \<a>[h, tab\<^sub>0 h, w\<^sub>h'] \<cdot> (h.tab \<star> w\<^sub>h')) \<cdot> \<beta>\<^sub>h
[PROOF STEP]
proof -
[PROOF STATE]
proof (state)
goal (1 subgoal):
1. (h \<star> \<theta>\<^sub>h) \<cdot> \<a>[h, tab\<^sub>0 h, w\<^sub>h] \<cdot> (h.tab \<star> w\<^sub>h) = ((h \<star> \<theta>\<^sub>h') \<cdot> \<a>[h, tab\<^sub>0 h, w\<^sub>h'] \<cdot> (h.tab \<star> w\<^sub>h')) \<cdot> \<beta>\<^sub>h
[PROOF STEP]
have "h.composite_cell w\<^sub>h \<theta>\<^sub>h =
(h \<star> tab\<^sub>0 h \<star> TTfgh.p\<^sub>0) \<cdot> \<a>[h, tab\<^sub>0 h, TTfgh.p\<^sub>0] \<cdot> (h.tab \<star> TTfgh.p\<^sub>0)"
[PROOF STATE]
proof (prove)
goal (1 subgoal):
1. (h \<star> \<theta>\<^sub>h) \<cdot> \<a>[h, tab\<^sub>0 h, w\<^sub>h] \<cdot> (h.tab \<star> w\<^sub>h) = (h \<star> tab\<^sub>0 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0) \<cdot> \<a>[h, tab\<^sub>0 h, fg\<^sub>0h\<^sub>1.p\<^sub>0] \<cdot> (h.tab \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0)
[PROOF STEP]
unfolding w\<^sub>h_def \<theta>\<^sub>h_def
[PROOF STATE]
proof (prove)
goal (1 subgoal):
1. (h \<star> tab\<^sub>0 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0) \<cdot> \<a>[h, tab\<^sub>0 h, fg\<^sub>0h\<^sub>1.p\<^sub>0] \<cdot> (h.tab \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0) = (h \<star> tab\<^sub>0 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0) \<cdot> \<a>[h, tab\<^sub>0 h, fg\<^sub>0h\<^sub>1.p\<^sub>0] \<cdot> (h.tab \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0)
[PROOF STEP]
by simp
[PROOF STATE]
proof (state)
this:
(h \<star> \<theta>\<^sub>h) \<cdot> \<a>[h, tab\<^sub>0 h, w\<^sub>h] \<cdot> (h.tab \<star> w\<^sub>h) = (h \<star> tab\<^sub>0 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0) \<cdot> \<a>[h, tab\<^sub>0 h, fg\<^sub>0h\<^sub>1.p\<^sub>0] \<cdot> (h.tab \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0)
goal (1 subgoal):
1. (h \<star> \<theta>\<^sub>h) \<cdot> \<a>[h, tab\<^sub>0 h, w\<^sub>h] \<cdot> (h.tab \<star> w\<^sub>h) = ((h \<star> \<theta>\<^sub>h') \<cdot> \<a>[h, tab\<^sub>0 h, w\<^sub>h'] \<cdot> (h.tab \<star> w\<^sub>h')) \<cdot> \<beta>\<^sub>h
[PROOF STEP]
also
[PROOF STATE]
proof (state)
this:
(h \<star> \<theta>\<^sub>h) \<cdot> \<a>[h, tab\<^sub>0 h, w\<^sub>h] \<cdot> (h.tab \<star> w\<^sub>h) = (h \<star> tab\<^sub>0 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0) \<cdot> \<a>[h, tab\<^sub>0 h, fg\<^sub>0h\<^sub>1.p\<^sub>0] \<cdot> (h.tab \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0)
goal (1 subgoal):
1. (h \<star> \<theta>\<^sub>h) \<cdot> \<a>[h, tab\<^sub>0 h, w\<^sub>h] \<cdot> (h.tab \<star> w\<^sub>h) = ((h \<star> \<theta>\<^sub>h') \<cdot> \<a>[h, tab\<^sub>0 h, w\<^sub>h'] \<cdot> (h.tab \<star> w\<^sub>h')) \<cdot> \<beta>\<^sub>h
[PROOF STEP]
have "... = \<a>[h, tab\<^sub>0 h, TTfgh.p\<^sub>0] \<cdot> (h.tab \<star> TTfgh.p\<^sub>0)"
[PROOF STATE]
proof (prove)
goal (1 subgoal):
1. (h \<star> tab\<^sub>0 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0) \<cdot> \<a>[h, tab\<^sub>0 h, fg\<^sub>0h\<^sub>1.p\<^sub>0] \<cdot> (h.tab \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0) = \<a>[h, tab\<^sub>0 h, fg\<^sub>0h\<^sub>1.p\<^sub>0] \<cdot> (h.tab \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0)
[PROOF STEP]
proof -
[PROOF STATE]
proof (state)
goal (1 subgoal):
1. (h \<star> tab\<^sub>0 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0) \<cdot> \<a>[h, tab\<^sub>0 h, fg\<^sub>0h\<^sub>1.p\<^sub>0] \<cdot> (h.tab \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0) = \<a>[h, tab\<^sub>0 h, fg\<^sub>0h\<^sub>1.p\<^sub>0] \<cdot> (h.tab \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0)
[PROOF STEP]
have "(h \<star> tab\<^sub>0 h \<star> TTfgh.p\<^sub>0) \<cdot> \<a>[h, tab\<^sub>0 h, TTfgh.p\<^sub>0] = \<a>[h, tab\<^sub>0 h, TTfgh.p\<^sub>0]"
[PROOF STATE]
proof (prove)
goal (1 subgoal):
1. (h \<star> tab\<^sub>0 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0) \<cdot> \<a>[h, tab\<^sub>0 h, fg\<^sub>0h\<^sub>1.p\<^sub>0] = \<a>[h, tab\<^sub>0 h, fg\<^sub>0h\<^sub>1.p\<^sub>0]
[PROOF STEP]
using comp_cod_arr
[PROOF STATE]
proof (prove)
using this:
\<lbrakk>arr ?f; cod ?f = ?b\<rbrakk> \<Longrightarrow> ?b \<cdot> ?f = ?f
goal (1 subgoal):
1. (h \<star> tab\<^sub>0 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0) \<cdot> \<a>[h, tab\<^sub>0 h, fg\<^sub>0h\<^sub>1.p\<^sub>0] = \<a>[h, tab\<^sub>0 h, fg\<^sub>0h\<^sub>1.p\<^sub>0]
[PROOF STEP]
by simp
[PROOF STATE]
proof (state)
this:
(h \<star> tab\<^sub>0 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0) \<cdot> \<a>[h, tab\<^sub>0 h, fg\<^sub>0h\<^sub>1.p\<^sub>0] = \<a>[h, tab\<^sub>0 h, fg\<^sub>0h\<^sub>1.p\<^sub>0]
goal (1 subgoal):
1. (h \<star> tab\<^sub>0 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0) \<cdot> \<a>[h, tab\<^sub>0 h, fg\<^sub>0h\<^sub>1.p\<^sub>0] \<cdot> (h.tab \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0) = \<a>[h, tab\<^sub>0 h, fg\<^sub>0h\<^sub>1.p\<^sub>0] \<cdot> (h.tab \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0)
[PROOF STEP]
thus ?thesis
[PROOF STATE]
proof (prove)
using this:
(h \<star> tab\<^sub>0 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0) \<cdot> \<a>[h, tab\<^sub>0 h, fg\<^sub>0h\<^sub>1.p\<^sub>0] = \<a>[h, tab\<^sub>0 h, fg\<^sub>0h\<^sub>1.p\<^sub>0]
goal (1 subgoal):
1. (h \<star> tab\<^sub>0 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0) \<cdot> \<a>[h, tab\<^sub>0 h, fg\<^sub>0h\<^sub>1.p\<^sub>0] \<cdot> (h.tab \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0) = \<a>[h, tab\<^sub>0 h, fg\<^sub>0h\<^sub>1.p\<^sub>0] \<cdot> (h.tab \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0)
[PROOF STEP]
using comp_assoc
[PROOF STATE]
proof (prove)
using this:
(h \<star> tab\<^sub>0 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0) \<cdot> \<a>[h, tab\<^sub>0 h, fg\<^sub>0h\<^sub>1.p\<^sub>0] = \<a>[h, tab\<^sub>0 h, fg\<^sub>0h\<^sub>1.p\<^sub>0]
(?h \<cdot> ?g) \<cdot> ?f = ?h \<cdot> ?g \<cdot> ?f
goal (1 subgoal):
1. (h \<star> tab\<^sub>0 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0) \<cdot> \<a>[h, tab\<^sub>0 h, fg\<^sub>0h\<^sub>1.p\<^sub>0] \<cdot> (h.tab \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0) = \<a>[h, tab\<^sub>0 h, fg\<^sub>0h\<^sub>1.p\<^sub>0] \<cdot> (h.tab \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0)
[PROOF STEP]
by metis
[PROOF STATE]
proof (state)
this:
(h \<star> tab\<^sub>0 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0) \<cdot> \<a>[h, tab\<^sub>0 h, fg\<^sub>0h\<^sub>1.p\<^sub>0] \<cdot> (h.tab \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0) = \<a>[h, tab\<^sub>0 h, fg\<^sub>0h\<^sub>1.p\<^sub>0] \<cdot> (h.tab \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0)
goal:
No subgoals!
[PROOF STEP]
qed
[PROOF STATE]
proof (state)
this:
(h \<star> tab\<^sub>0 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0) \<cdot> \<a>[h, tab\<^sub>0 h, fg\<^sub>0h\<^sub>1.p\<^sub>0] \<cdot> (h.tab \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0) = \<a>[h, tab\<^sub>0 h, fg\<^sub>0h\<^sub>1.p\<^sub>0] \<cdot> (h.tab \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0)
goal (1 subgoal):
1. (h \<star> \<theta>\<^sub>h) \<cdot> \<a>[h, tab\<^sub>0 h, w\<^sub>h] \<cdot> (h.tab \<star> w\<^sub>h) = ((h \<star> \<theta>\<^sub>h') \<cdot> \<a>[h, tab\<^sub>0 h, w\<^sub>h'] \<cdot> (h.tab \<star> w\<^sub>h')) \<cdot> \<beta>\<^sub>h
[PROOF STEP]
also
[PROOF STATE]
proof (state)
this:
(h \<star> tab\<^sub>0 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0) \<cdot> \<a>[h, tab\<^sub>0 h, fg\<^sub>0h\<^sub>1.p\<^sub>0] \<cdot> (h.tab \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0) = \<a>[h, tab\<^sub>0 h, fg\<^sub>0h\<^sub>1.p\<^sub>0] \<cdot> (h.tab \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0)
goal (1 subgoal):
1. (h \<star> \<theta>\<^sub>h) \<cdot> \<a>[h, tab\<^sub>0 h, w\<^sub>h] \<cdot> (h.tab \<star> w\<^sub>h) = ((h \<star> \<theta>\<^sub>h') \<cdot> \<a>[h, tab\<^sub>0 h, w\<^sub>h'] \<cdot> (h.tab \<star> w\<^sub>h')) \<cdot> \<beta>\<^sub>h
[PROOF STEP]
have "... = (\<a>[h, tab\<^sub>0 h, TTfgh.p\<^sub>0] \<cdot> (h.tab \<star> TTfgh.p\<^sub>0) \<cdot>
fg\<^sub>0h\<^sub>1.\<phi> \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>0 g, Tfg.p\<^sub>0, TTfgh.p\<^sub>1]) \<cdot>
\<a>[tab\<^sub>0 g, Tfg.p\<^sub>0, TTfgh.p\<^sub>1] \<cdot> inv fg\<^sub>0h\<^sub>1.\<phi>"
[PROOF STATE]
proof (prove)
goal (1 subgoal):
1. \<a>[h, tab\<^sub>0 h, fg\<^sub>0h\<^sub>1.p\<^sub>0] \<cdot> (h.tab \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0) = (\<a>[h, tab\<^sub>0 h, fg\<^sub>0h\<^sub>1.p\<^sub>0] \<cdot> (h.tab \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0) \<cdot> fg\<^sub>0h\<^sub>1.\<phi> \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>0 g, f\<^sub>0g\<^sub>1.p\<^sub>0, fg\<^sub>0h\<^sub>1.p\<^sub>1]) \<cdot> \<a>[tab\<^sub>0 g, f\<^sub>0g\<^sub>1.p\<^sub>0, fg\<^sub>0h\<^sub>1.p\<^sub>1] \<cdot> local.inv fg\<^sub>0h\<^sub>1.\<phi>
[PROOF STEP]
proof -
[PROOF STATE]
proof (state)
goal (1 subgoal):
1. \<a>[h, tab\<^sub>0 h, fg\<^sub>0h\<^sub>1.p\<^sub>0] \<cdot> (h.tab \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0) = (\<a>[h, tab\<^sub>0 h, fg\<^sub>0h\<^sub>1.p\<^sub>0] \<cdot> (h.tab \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0) \<cdot> fg\<^sub>0h\<^sub>1.\<phi> \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>0 g, f\<^sub>0g\<^sub>1.p\<^sub>0, fg\<^sub>0h\<^sub>1.p\<^sub>1]) \<cdot> \<a>[tab\<^sub>0 g, f\<^sub>0g\<^sub>1.p\<^sub>0, fg\<^sub>0h\<^sub>1.p\<^sub>1] \<cdot> local.inv fg\<^sub>0h\<^sub>1.\<phi>
[PROOF STEP]
have "(h.tab \<star> TTfgh.p\<^sub>0) \<cdot> fg\<^sub>0h\<^sub>1.\<phi> \<cdot> (\<a>\<^sup>-\<^sup>1[tab\<^sub>0 g, Tfg.p\<^sub>0, TTfgh.p\<^sub>1] \<cdot>
\<a>[tab\<^sub>0 g, Tfg.p\<^sub>0, TTfgh.p\<^sub>1]) \<cdot> inv fg\<^sub>0h\<^sub>1.\<phi> =
(h.tab \<star> TTfgh.p\<^sub>0) \<cdot> fg\<^sub>0h\<^sub>1.\<phi> \<cdot> ((tab\<^sub>0 g \<star> Tfg.p\<^sub>0) \<star> TTfgh.p\<^sub>1) \<cdot> inv fg\<^sub>0h\<^sub>1.\<phi>"
[PROOF STATE]
proof (prove)
goal (1 subgoal):
1. (h.tab \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0) \<cdot> fg\<^sub>0h\<^sub>1.\<phi> \<cdot> (\<a>\<^sup>-\<^sup>1[tab\<^sub>0 g, f\<^sub>0g\<^sub>1.p\<^sub>0, fg\<^sub>0h\<^sub>1.p\<^sub>1] \<cdot> \<a>[tab\<^sub>0 g, f\<^sub>0g\<^sub>1.p\<^sub>0, fg\<^sub>0h\<^sub>1.p\<^sub>1]) \<cdot> local.inv fg\<^sub>0h\<^sub>1.\<phi> = (h.tab \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0) \<cdot> fg\<^sub>0h\<^sub>1.\<phi> \<cdot> ((tab\<^sub>0 g \<star> f\<^sub>0g\<^sub>1.p\<^sub>0) \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> local.inv fg\<^sub>0h\<^sub>1.\<phi>
[PROOF STEP]
using fg\<^sub>0h\<^sub>1.p\<^sub>1_simps comp_assoc_assoc'
[PROOF STATE]
proof (prove)
using this:
trg fg\<^sub>0h\<^sub>1.p\<^sub>1 = src (tab\<^sub>0 g \<star> f\<^sub>0g\<^sub>1.p\<^sub>0)
\<lbrakk>ide ?f; ide ?g; ide ?h; src ?f = trg ?g; src ?g = trg ?h\<rbrakk> \<Longrightarrow> \<a>[?f, ?g, ?h] \<cdot> \<a>\<^sup>-\<^sup>1[?f, ?g, ?h] = ?f \<star> ?g \<star> ?h
\<lbrakk>ide ?f; ide ?g; ide ?h; src ?f = trg ?g; src ?g = trg ?h\<rbrakk> \<Longrightarrow> \<a>\<^sup>-\<^sup>1[?f, ?g, ?h] \<cdot> \<a>[?f, ?g, ?h] = (?f \<star> ?g) \<star> ?h
goal (1 subgoal):
1. (h.tab \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0) \<cdot> fg\<^sub>0h\<^sub>1.\<phi> \<cdot> (\<a>\<^sup>-\<^sup>1[tab\<^sub>0 g, f\<^sub>0g\<^sub>1.p\<^sub>0, fg\<^sub>0h\<^sub>1.p\<^sub>1] \<cdot> \<a>[tab\<^sub>0 g, f\<^sub>0g\<^sub>1.p\<^sub>0, fg\<^sub>0h\<^sub>1.p\<^sub>1]) \<cdot> local.inv fg\<^sub>0h\<^sub>1.\<phi> = (h.tab \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0) \<cdot> fg\<^sub>0h\<^sub>1.\<phi> \<cdot> ((tab\<^sub>0 g \<star> f\<^sub>0g\<^sub>1.p\<^sub>0) \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> local.inv fg\<^sub>0h\<^sub>1.\<phi>
[PROOF STEP]
by simp
[PROOF STATE]
proof (state)
this:
(h.tab \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0) \<cdot> fg\<^sub>0h\<^sub>1.\<phi> \<cdot> (\<a>\<^sup>-\<^sup>1[tab\<^sub>0 g, f\<^sub>0g\<^sub>1.p\<^sub>0, fg\<^sub>0h\<^sub>1.p\<^sub>1] \<cdot> \<a>[tab\<^sub>0 g, f\<^sub>0g\<^sub>1.p\<^sub>0, fg\<^sub>0h\<^sub>1.p\<^sub>1]) \<cdot> local.inv fg\<^sub>0h\<^sub>1.\<phi> = (h.tab \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0) \<cdot> fg\<^sub>0h\<^sub>1.\<phi> \<cdot> ((tab\<^sub>0 g \<star> f\<^sub>0g\<^sub>1.p\<^sub>0) \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> local.inv fg\<^sub>0h\<^sub>1.\<phi>
goal (1 subgoal):
1. \<a>[h, tab\<^sub>0 h, fg\<^sub>0h\<^sub>1.p\<^sub>0] \<cdot> (h.tab \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0) = (\<a>[h, tab\<^sub>0 h, fg\<^sub>0h\<^sub>1.p\<^sub>0] \<cdot> (h.tab \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0) \<cdot> fg\<^sub>0h\<^sub>1.\<phi> \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>0 g, f\<^sub>0g\<^sub>1.p\<^sub>0, fg\<^sub>0h\<^sub>1.p\<^sub>1]) \<cdot> \<a>[tab\<^sub>0 g, f\<^sub>0g\<^sub>1.p\<^sub>0, fg\<^sub>0h\<^sub>1.p\<^sub>1] \<cdot> local.inv fg\<^sub>0h\<^sub>1.\<phi>
[PROOF STEP]
also
[PROOF STATE]
proof (state)
this:
(h.tab \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0) \<cdot> fg\<^sub>0h\<^sub>1.\<phi> \<cdot> (\<a>\<^sup>-\<^sup>1[tab\<^sub>0 g, f\<^sub>0g\<^sub>1.p\<^sub>0, fg\<^sub>0h\<^sub>1.p\<^sub>1] \<cdot> \<a>[tab\<^sub>0 g, f\<^sub>0g\<^sub>1.p\<^sub>0, fg\<^sub>0h\<^sub>1.p\<^sub>1]) \<cdot> local.inv fg\<^sub>0h\<^sub>1.\<phi> = (h.tab \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0) \<cdot> fg\<^sub>0h\<^sub>1.\<phi> \<cdot> ((tab\<^sub>0 g \<star> f\<^sub>0g\<^sub>1.p\<^sub>0) \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> local.inv fg\<^sub>0h\<^sub>1.\<phi>
goal (1 subgoal):
1. \<a>[h, tab\<^sub>0 h, fg\<^sub>0h\<^sub>1.p\<^sub>0] \<cdot> (h.tab \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0) = (\<a>[h, tab\<^sub>0 h, fg\<^sub>0h\<^sub>1.p\<^sub>0] \<cdot> (h.tab \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0) \<cdot> fg\<^sub>0h\<^sub>1.\<phi> \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>0 g, f\<^sub>0g\<^sub>1.p\<^sub>0, fg\<^sub>0h\<^sub>1.p\<^sub>1]) \<cdot> \<a>[tab\<^sub>0 g, f\<^sub>0g\<^sub>1.p\<^sub>0, fg\<^sub>0h\<^sub>1.p\<^sub>1] \<cdot> local.inv fg\<^sub>0h\<^sub>1.\<phi>
[PROOF STEP]
have "... = (h.tab \<star> TTfgh.p\<^sub>0) \<cdot> fg\<^sub>0h\<^sub>1.\<phi> \<cdot> inv fg\<^sub>0h\<^sub>1.\<phi>"
[PROOF STATE]
proof (prove)
goal (1 subgoal):
1. (h.tab \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0) \<cdot> fg\<^sub>0h\<^sub>1.\<phi> \<cdot> ((tab\<^sub>0 g \<star> f\<^sub>0g\<^sub>1.p\<^sub>0) \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> local.inv fg\<^sub>0h\<^sub>1.\<phi> = (h.tab \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0) \<cdot> fg\<^sub>0h\<^sub>1.\<phi> \<cdot> local.inv fg\<^sub>0h\<^sub>1.\<phi>
[PROOF STEP]
using fg\<^sub>0h\<^sub>1.p\<^sub>1_simps fg\<^sub>0h\<^sub>1.\<phi>_uniqueness comp_cod_arr
[PROOF STATE]
proof (prove)
using this:
trg fg\<^sub>0h\<^sub>1.p\<^sub>1 = src (tab\<^sub>0 g \<star> f\<^sub>0g\<^sub>1.p\<^sub>0)
\<guillemotleft>?\<mu> : (tab\<^sub>0 g \<star> f\<^sub>0g\<^sub>1.p\<^sub>0) \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1 \<rightarrow> tab\<^sub>1 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0\<guillemotright> \<Longrightarrow> ?\<mu> = fg\<^sub>0h\<^sub>1.\<phi>
local.iso fg\<^sub>0h\<^sub>1.\<phi>
\<lbrakk>arr ?f; cod ?f = ?b\<rbrakk> \<Longrightarrow> ?b \<cdot> ?f = ?f
goal (1 subgoal):
1. (h.tab \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0) \<cdot> fg\<^sub>0h\<^sub>1.\<phi> \<cdot> ((tab\<^sub>0 g \<star> f\<^sub>0g\<^sub>1.p\<^sub>0) \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> local.inv fg\<^sub>0h\<^sub>1.\<phi> = (h.tab \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0) \<cdot> fg\<^sub>0h\<^sub>1.\<phi> \<cdot> local.inv fg\<^sub>0h\<^sub>1.\<phi>
[PROOF STEP]
by simp
[PROOF STATE]
proof (state)
this:
(h.tab \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0) \<cdot> fg\<^sub>0h\<^sub>1.\<phi> \<cdot> ((tab\<^sub>0 g \<star> f\<^sub>0g\<^sub>1.p\<^sub>0) \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> local.inv fg\<^sub>0h\<^sub>1.\<phi> = (h.tab \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0) \<cdot> fg\<^sub>0h\<^sub>1.\<phi> \<cdot> local.inv fg\<^sub>0h\<^sub>1.\<phi>
goal (1 subgoal):
1. \<a>[h, tab\<^sub>0 h, fg\<^sub>0h\<^sub>1.p\<^sub>0] \<cdot> (h.tab \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0) = (\<a>[h, tab\<^sub>0 h, fg\<^sub>0h\<^sub>1.p\<^sub>0] \<cdot> (h.tab \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0) \<cdot> fg\<^sub>0h\<^sub>1.\<phi> \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>0 g, f\<^sub>0g\<^sub>1.p\<^sub>0, fg\<^sub>0h\<^sub>1.p\<^sub>1]) \<cdot> \<a>[tab\<^sub>0 g, f\<^sub>0g\<^sub>1.p\<^sub>0, fg\<^sub>0h\<^sub>1.p\<^sub>1] \<cdot> local.inv fg\<^sub>0h\<^sub>1.\<phi>
[PROOF STEP]
also
[PROOF STATE]
proof (state)
this:
(h.tab \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0) \<cdot> fg\<^sub>0h\<^sub>1.\<phi> \<cdot> ((tab\<^sub>0 g \<star> f\<^sub>0g\<^sub>1.p\<^sub>0) \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> local.inv fg\<^sub>0h\<^sub>1.\<phi> = (h.tab \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0) \<cdot> fg\<^sub>0h\<^sub>1.\<phi> \<cdot> local.inv fg\<^sub>0h\<^sub>1.\<phi>
goal (1 subgoal):
1. \<a>[h, tab\<^sub>0 h, fg\<^sub>0h\<^sub>1.p\<^sub>0] \<cdot> (h.tab \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0) = (\<a>[h, tab\<^sub>0 h, fg\<^sub>0h\<^sub>1.p\<^sub>0] \<cdot> (h.tab \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0) \<cdot> fg\<^sub>0h\<^sub>1.\<phi> \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>0 g, f\<^sub>0g\<^sub>1.p\<^sub>0, fg\<^sub>0h\<^sub>1.p\<^sub>1]) \<cdot> \<a>[tab\<^sub>0 g, f\<^sub>0g\<^sub>1.p\<^sub>0, fg\<^sub>0h\<^sub>1.p\<^sub>1] \<cdot> local.inv fg\<^sub>0h\<^sub>1.\<phi>
[PROOF STEP]
have "... = (h.tab \<star> TTfgh.p\<^sub>0) \<cdot> (tab\<^sub>1 h \<star> TTfgh.p\<^sub>0)"
[PROOF STATE]
proof (prove)
goal (1 subgoal):
1. (h.tab \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0) \<cdot> fg\<^sub>0h\<^sub>1.\<phi> \<cdot> local.inv fg\<^sub>0h\<^sub>1.\<phi> = (h.tab \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0) \<cdot> (tab\<^sub>1 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0)
[PROOF STEP]
using comp_arr_inv' fg\<^sub>0h\<^sub>1.\<phi>_uniqueness
[PROOF STATE]
proof (prove)
using this:
local.iso ?f \<Longrightarrow> ?f \<cdot> local.inv ?f = cod ?f
\<guillemotleft>?\<mu> : (tab\<^sub>0 g \<star> f\<^sub>0g\<^sub>1.p\<^sub>0) \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1 \<rightarrow> tab\<^sub>1 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0\<guillemotright> \<Longrightarrow> ?\<mu> = fg\<^sub>0h\<^sub>1.\<phi>
local.iso fg\<^sub>0h\<^sub>1.\<phi>
goal (1 subgoal):
1. (h.tab \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0) \<cdot> fg\<^sub>0h\<^sub>1.\<phi> \<cdot> local.inv fg\<^sub>0h\<^sub>1.\<phi> = (h.tab \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0) \<cdot> (tab\<^sub>1 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0)
[PROOF STEP]
by simp
[PROOF STATE]
proof (state)
this:
(h.tab \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0) \<cdot> fg\<^sub>0h\<^sub>1.\<phi> \<cdot> local.inv fg\<^sub>0h\<^sub>1.\<phi> = (h.tab \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0) \<cdot> (tab\<^sub>1 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0)
goal (1 subgoal):
1. \<a>[h, tab\<^sub>0 h, fg\<^sub>0h\<^sub>1.p\<^sub>0] \<cdot> (h.tab \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0) = (\<a>[h, tab\<^sub>0 h, fg\<^sub>0h\<^sub>1.p\<^sub>0] \<cdot> (h.tab \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0) \<cdot> fg\<^sub>0h\<^sub>1.\<phi> \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>0 g, f\<^sub>0g\<^sub>1.p\<^sub>0, fg\<^sub>0h\<^sub>1.p\<^sub>1]) \<cdot> \<a>[tab\<^sub>0 g, f\<^sub>0g\<^sub>1.p\<^sub>0, fg\<^sub>0h\<^sub>1.p\<^sub>1] \<cdot> local.inv fg\<^sub>0h\<^sub>1.\<phi>
[PROOF STEP]
also
[PROOF STATE]
proof (state)
this:
(h.tab \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0) \<cdot> fg\<^sub>0h\<^sub>1.\<phi> \<cdot> local.inv fg\<^sub>0h\<^sub>1.\<phi> = (h.tab \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0) \<cdot> (tab\<^sub>1 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0)
goal (1 subgoal):
1. \<a>[h, tab\<^sub>0 h, fg\<^sub>0h\<^sub>1.p\<^sub>0] \<cdot> (h.tab \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0) = (\<a>[h, tab\<^sub>0 h, fg\<^sub>0h\<^sub>1.p\<^sub>0] \<cdot> (h.tab \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0) \<cdot> fg\<^sub>0h\<^sub>1.\<phi> \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>0 g, f\<^sub>0g\<^sub>1.p\<^sub>0, fg\<^sub>0h\<^sub>1.p\<^sub>1]) \<cdot> \<a>[tab\<^sub>0 g, f\<^sub>0g\<^sub>1.p\<^sub>0, fg\<^sub>0h\<^sub>1.p\<^sub>1] \<cdot> local.inv fg\<^sub>0h\<^sub>1.\<phi>
[PROOF STEP]
have "... = h.tab \<star> TTfgh.p\<^sub>0"
[PROOF STATE]
proof (prove)
goal (1 subgoal):
1. (h.tab \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0) \<cdot> (tab\<^sub>1 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0) = h.tab \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0
[PROOF STEP]
using comp_arr_dom fg\<^sub>0h\<^sub>1.p\<^sub>0_simps
[PROOF STATE]
proof (prove)
using this:
\<lbrakk>arr ?f; local.dom ?f = ?a\<rbrakk> \<Longrightarrow> ?f \<cdot> ?a = ?f
trg fg\<^sub>0h\<^sub>1.p\<^sub>0 = src (tab\<^sub>1 h)
goal (1 subgoal):
1. (h.tab \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0) \<cdot> (tab\<^sub>1 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0) = h.tab \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0
[PROOF STEP]
by simp
[PROOF STATE]
proof (state)
this:
(h.tab \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0) \<cdot> (tab\<^sub>1 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0) = h.tab \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0
goal (1 subgoal):
1. \<a>[h, tab\<^sub>0 h, fg\<^sub>0h\<^sub>1.p\<^sub>0] \<cdot> (h.tab \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0) = (\<a>[h, tab\<^sub>0 h, fg\<^sub>0h\<^sub>1.p\<^sub>0] \<cdot> (h.tab \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0) \<cdot> fg\<^sub>0h\<^sub>1.\<phi> \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>0 g, f\<^sub>0g\<^sub>1.p\<^sub>0, fg\<^sub>0h\<^sub>1.p\<^sub>1]) \<cdot> \<a>[tab\<^sub>0 g, f\<^sub>0g\<^sub>1.p\<^sub>0, fg\<^sub>0h\<^sub>1.p\<^sub>1] \<cdot> local.inv fg\<^sub>0h\<^sub>1.\<phi>
[PROOF STEP]
finally
[PROOF STATE]
proof (chain)
picking this:
(h.tab \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0) \<cdot> fg\<^sub>0h\<^sub>1.\<phi> \<cdot> (\<a>\<^sup>-\<^sup>1[tab\<^sub>0 g, f\<^sub>0g\<^sub>1.p\<^sub>0, fg\<^sub>0h\<^sub>1.p\<^sub>1] \<cdot> \<a>[tab\<^sub>0 g, f\<^sub>0g\<^sub>1.p\<^sub>0, fg\<^sub>0h\<^sub>1.p\<^sub>1]) \<cdot> local.inv fg\<^sub>0h\<^sub>1.\<phi> = h.tab \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0
[PROOF STEP]
have "(h.tab \<star> TTfgh.p\<^sub>0) \<cdot> fg\<^sub>0h\<^sub>1.\<phi> \<cdot> (\<a>\<^sup>-\<^sup>1[tab\<^sub>0 g, Tfg.p\<^sub>0, TTfgh.p\<^sub>1] \<cdot>
\<a>[tab\<^sub>0 g, Tfg.p\<^sub>0, TTfgh.p\<^sub>1]) \<cdot> inv fg\<^sub>0h\<^sub>1.\<phi> =
h.tab \<star> TTfgh.p\<^sub>0"
[PROOF STATE]
proof (prove)
using this:
(h.tab \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0) \<cdot> fg\<^sub>0h\<^sub>1.\<phi> \<cdot> (\<a>\<^sup>-\<^sup>1[tab\<^sub>0 g, f\<^sub>0g\<^sub>1.p\<^sub>0, fg\<^sub>0h\<^sub>1.p\<^sub>1] \<cdot> \<a>[tab\<^sub>0 g, f\<^sub>0g\<^sub>1.p\<^sub>0, fg\<^sub>0h\<^sub>1.p\<^sub>1]) \<cdot> local.inv fg\<^sub>0h\<^sub>1.\<phi> = h.tab \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0
goal (1 subgoal):
1. (h.tab \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0) \<cdot> fg\<^sub>0h\<^sub>1.\<phi> \<cdot> (\<a>\<^sup>-\<^sup>1[tab\<^sub>0 g, f\<^sub>0g\<^sub>1.p\<^sub>0, fg\<^sub>0h\<^sub>1.p\<^sub>1] \<cdot> \<a>[tab\<^sub>0 g, f\<^sub>0g\<^sub>1.p\<^sub>0, fg\<^sub>0h\<^sub>1.p\<^sub>1]) \<cdot> local.inv fg\<^sub>0h\<^sub>1.\<phi> = h.tab \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0
[PROOF STEP]
by simp
[PROOF STATE]
proof (state)
this:
(h.tab \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0) \<cdot> fg\<^sub>0h\<^sub>1.\<phi> \<cdot> (\<a>\<^sup>-\<^sup>1[tab\<^sub>0 g, f\<^sub>0g\<^sub>1.p\<^sub>0, fg\<^sub>0h\<^sub>1.p\<^sub>1] \<cdot> \<a>[tab\<^sub>0 g, f\<^sub>0g\<^sub>1.p\<^sub>0, fg\<^sub>0h\<^sub>1.p\<^sub>1]) \<cdot> local.inv fg\<^sub>0h\<^sub>1.\<phi> = h.tab \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0
goal (1 subgoal):
1. \<a>[h, tab\<^sub>0 h, fg\<^sub>0h\<^sub>1.p\<^sub>0] \<cdot> (h.tab \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0) = (\<a>[h, tab\<^sub>0 h, fg\<^sub>0h\<^sub>1.p\<^sub>0] \<cdot> (h.tab \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0) \<cdot> fg\<^sub>0h\<^sub>1.\<phi> \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>0 g, f\<^sub>0g\<^sub>1.p\<^sub>0, fg\<^sub>0h\<^sub>1.p\<^sub>1]) \<cdot> \<a>[tab\<^sub>0 g, f\<^sub>0g\<^sub>1.p\<^sub>0, fg\<^sub>0h\<^sub>1.p\<^sub>1] \<cdot> local.inv fg\<^sub>0h\<^sub>1.\<phi>
[PROOF STEP]
thus ?thesis
[PROOF STATE]
proof (prove)
using this:
(h.tab \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0) \<cdot> fg\<^sub>0h\<^sub>1.\<phi> \<cdot> (\<a>\<^sup>-\<^sup>1[tab\<^sub>0 g, f\<^sub>0g\<^sub>1.p\<^sub>0, fg\<^sub>0h\<^sub>1.p\<^sub>1] \<cdot> \<a>[tab\<^sub>0 g, f\<^sub>0g\<^sub>1.p\<^sub>0, fg\<^sub>0h\<^sub>1.p\<^sub>1]) \<cdot> local.inv fg\<^sub>0h\<^sub>1.\<phi> = h.tab \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0
goal (1 subgoal):
1. \<a>[h, tab\<^sub>0 h, fg\<^sub>0h\<^sub>1.p\<^sub>0] \<cdot> (h.tab \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0) = (\<a>[h, tab\<^sub>0 h, fg\<^sub>0h\<^sub>1.p\<^sub>0] \<cdot> (h.tab \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0) \<cdot> fg\<^sub>0h\<^sub>1.\<phi> \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>0 g, f\<^sub>0g\<^sub>1.p\<^sub>0, fg\<^sub>0h\<^sub>1.p\<^sub>1]) \<cdot> \<a>[tab\<^sub>0 g, f\<^sub>0g\<^sub>1.p\<^sub>0, fg\<^sub>0h\<^sub>1.p\<^sub>1] \<cdot> local.inv fg\<^sub>0h\<^sub>1.\<phi>
[PROOF STEP]
using comp_assoc
[PROOF STATE]
proof (prove)
using this:
(h.tab \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0) \<cdot> fg\<^sub>0h\<^sub>1.\<phi> \<cdot> (\<a>\<^sup>-\<^sup>1[tab\<^sub>0 g, f\<^sub>0g\<^sub>1.p\<^sub>0, fg\<^sub>0h\<^sub>1.p\<^sub>1] \<cdot> \<a>[tab\<^sub>0 g, f\<^sub>0g\<^sub>1.p\<^sub>0, fg\<^sub>0h\<^sub>1.p\<^sub>1]) \<cdot> local.inv fg\<^sub>0h\<^sub>1.\<phi> = h.tab \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0
(?h \<cdot> ?g) \<cdot> ?f = ?h \<cdot> ?g \<cdot> ?f
goal (1 subgoal):
1. \<a>[h, tab\<^sub>0 h, fg\<^sub>0h\<^sub>1.p\<^sub>0] \<cdot> (h.tab \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0) = (\<a>[h, tab\<^sub>0 h, fg\<^sub>0h\<^sub>1.p\<^sub>0] \<cdot> (h.tab \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0) \<cdot> fg\<^sub>0h\<^sub>1.\<phi> \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>0 g, f\<^sub>0g\<^sub>1.p\<^sub>0, fg\<^sub>0h\<^sub>1.p\<^sub>1]) \<cdot> \<a>[tab\<^sub>0 g, f\<^sub>0g\<^sub>1.p\<^sub>0, fg\<^sub>0h\<^sub>1.p\<^sub>1] \<cdot> local.inv fg\<^sub>0h\<^sub>1.\<phi>
[PROOF STEP]
by simp
[PROOF STATE]
proof (state)
this:
\<a>[h, tab\<^sub>0 h, fg\<^sub>0h\<^sub>1.p\<^sub>0] \<cdot> (h.tab \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0) = (\<a>[h, tab\<^sub>0 h, fg\<^sub>0h\<^sub>1.p\<^sub>0] \<cdot> (h.tab \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0) \<cdot> fg\<^sub>0h\<^sub>1.\<phi> \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>0 g, f\<^sub>0g\<^sub>1.p\<^sub>0, fg\<^sub>0h\<^sub>1.p\<^sub>1]) \<cdot> \<a>[tab\<^sub>0 g, f\<^sub>0g\<^sub>1.p\<^sub>0, fg\<^sub>0h\<^sub>1.p\<^sub>1] \<cdot> local.inv fg\<^sub>0h\<^sub>1.\<phi>
goal:
No subgoals!
[PROOF STEP]
qed
[PROOF STATE]
proof (state)
this:
\<a>[h, tab\<^sub>0 h, fg\<^sub>0h\<^sub>1.p\<^sub>0] \<cdot> (h.tab \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0) = (\<a>[h, tab\<^sub>0 h, fg\<^sub>0h\<^sub>1.p\<^sub>0] \<cdot> (h.tab \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0) \<cdot> fg\<^sub>0h\<^sub>1.\<phi> \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>0 g, f\<^sub>0g\<^sub>1.p\<^sub>0, fg\<^sub>0h\<^sub>1.p\<^sub>1]) \<cdot> \<a>[tab\<^sub>0 g, f\<^sub>0g\<^sub>1.p\<^sub>0, fg\<^sub>0h\<^sub>1.p\<^sub>1] \<cdot> local.inv fg\<^sub>0h\<^sub>1.\<phi>
goal (1 subgoal):
1. (h \<star> \<theta>\<^sub>h) \<cdot> \<a>[h, tab\<^sub>0 h, w\<^sub>h] \<cdot> (h.tab \<star> w\<^sub>h) = ((h \<star> \<theta>\<^sub>h') \<cdot> \<a>[h, tab\<^sub>0 h, w\<^sub>h'] \<cdot> (h.tab \<star> w\<^sub>h')) \<cdot> \<beta>\<^sub>h
[PROOF STEP]
also
[PROOF STATE]
proof (state)
this:
\<a>[h, tab\<^sub>0 h, fg\<^sub>0h\<^sub>1.p\<^sub>0] \<cdot> (h.tab \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0) = (\<a>[h, tab\<^sub>0 h, fg\<^sub>0h\<^sub>1.p\<^sub>0] \<cdot> (h.tab \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0) \<cdot> fg\<^sub>0h\<^sub>1.\<phi> \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>0 g, f\<^sub>0g\<^sub>1.p\<^sub>0, fg\<^sub>0h\<^sub>1.p\<^sub>1]) \<cdot> \<a>[tab\<^sub>0 g, f\<^sub>0g\<^sub>1.p\<^sub>0, fg\<^sub>0h\<^sub>1.p\<^sub>1] \<cdot> local.inv fg\<^sub>0h\<^sub>1.\<phi>
goal (1 subgoal):
1. (h \<star> \<theta>\<^sub>h) \<cdot> \<a>[h, tab\<^sub>0 h, w\<^sub>h] \<cdot> (h.tab \<star> w\<^sub>h) = ((h \<star> \<theta>\<^sub>h') \<cdot> \<a>[h, tab\<^sub>0 h, w\<^sub>h'] \<cdot> (h.tab \<star> w\<^sub>h')) \<cdot> \<beta>\<^sub>h
[PROOF STEP]
have "... = \<theta>\<^sub>g \<cdot> \<a>[tab\<^sub>0 g, Tfg.p\<^sub>0, TTfgh.p\<^sub>1] \<cdot> inv fg\<^sub>0h\<^sub>1.\<phi>"
[PROOF STATE]
proof (prove)
goal (1 subgoal):
1. (\<a>[h, tab\<^sub>0 h, fg\<^sub>0h\<^sub>1.p\<^sub>0] \<cdot> (h.tab \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0) \<cdot> fg\<^sub>0h\<^sub>1.\<phi> \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>0 g, f\<^sub>0g\<^sub>1.p\<^sub>0, fg\<^sub>0h\<^sub>1.p\<^sub>1]) \<cdot> \<a>[tab\<^sub>0 g, f\<^sub>0g\<^sub>1.p\<^sub>0, fg\<^sub>0h\<^sub>1.p\<^sub>1] \<cdot> local.inv fg\<^sub>0h\<^sub>1.\<phi> = \<theta>\<^sub>g \<cdot> \<a>[tab\<^sub>0 g, f\<^sub>0g\<^sub>1.p\<^sub>0, fg\<^sub>0h\<^sub>1.p\<^sub>1] \<cdot> local.inv fg\<^sub>0h\<^sub>1.\<phi>
[PROOF STEP]
unfolding \<theta>\<^sub>g_def
[PROOF STATE]
proof (prove)
goal (1 subgoal):
1. (\<a>[h, tab\<^sub>0 h, fg\<^sub>0h\<^sub>1.p\<^sub>0] \<cdot> (h.tab \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0) \<cdot> fg\<^sub>0h\<^sub>1.\<phi> \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>0 g, f\<^sub>0g\<^sub>1.p\<^sub>0, fg\<^sub>0h\<^sub>1.p\<^sub>1]) \<cdot> \<a>[tab\<^sub>0 g, f\<^sub>0g\<^sub>1.p\<^sub>0, fg\<^sub>0h\<^sub>1.p\<^sub>1] \<cdot> local.inv fg\<^sub>0h\<^sub>1.\<phi> = (\<a>[h, tab\<^sub>0 h, fg\<^sub>0h\<^sub>1.p\<^sub>0] \<cdot> (h.tab \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0) \<cdot> fg\<^sub>0h\<^sub>1.\<phi> \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>0 g, f\<^sub>0g\<^sub>1.p\<^sub>0, fg\<^sub>0h\<^sub>1.p\<^sub>1]) \<cdot> \<a>[tab\<^sub>0 g, f\<^sub>0g\<^sub>1.p\<^sub>0, fg\<^sub>0h\<^sub>1.p\<^sub>1] \<cdot> local.inv fg\<^sub>0h\<^sub>1.\<phi>
[PROOF STEP]
by simp
[PROOF STATE]
proof (state)
this:
(\<a>[h, tab\<^sub>0 h, fg\<^sub>0h\<^sub>1.p\<^sub>0] \<cdot> (h.tab \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0) \<cdot> fg\<^sub>0h\<^sub>1.\<phi> \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>0 g, f\<^sub>0g\<^sub>1.p\<^sub>0, fg\<^sub>0h\<^sub>1.p\<^sub>1]) \<cdot> \<a>[tab\<^sub>0 g, f\<^sub>0g\<^sub>1.p\<^sub>0, fg\<^sub>0h\<^sub>1.p\<^sub>1] \<cdot> local.inv fg\<^sub>0h\<^sub>1.\<phi> = \<theta>\<^sub>g \<cdot> \<a>[tab\<^sub>0 g, f\<^sub>0g\<^sub>1.p\<^sub>0, fg\<^sub>0h\<^sub>1.p\<^sub>1] \<cdot> local.inv fg\<^sub>0h\<^sub>1.\<phi>
goal (1 subgoal):
1. (h \<star> \<theta>\<^sub>h) \<cdot> \<a>[h, tab\<^sub>0 h, w\<^sub>h] \<cdot> (h.tab \<star> w\<^sub>h) = ((h \<star> \<theta>\<^sub>h') \<cdot> \<a>[h, tab\<^sub>0 h, w\<^sub>h'] \<cdot> (h.tab \<star> w\<^sub>h')) \<cdot> \<beta>\<^sub>h
[PROOF STEP]
also
[PROOF STATE]
proof (state)
this:
(\<a>[h, tab\<^sub>0 h, fg\<^sub>0h\<^sub>1.p\<^sub>0] \<cdot> (h.tab \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0) \<cdot> fg\<^sub>0h\<^sub>1.\<phi> \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>0 g, f\<^sub>0g\<^sub>1.p\<^sub>0, fg\<^sub>0h\<^sub>1.p\<^sub>1]) \<cdot> \<a>[tab\<^sub>0 g, f\<^sub>0g\<^sub>1.p\<^sub>0, fg\<^sub>0h\<^sub>1.p\<^sub>1] \<cdot> local.inv fg\<^sub>0h\<^sub>1.\<phi> = \<theta>\<^sub>g \<cdot> \<a>[tab\<^sub>0 g, f\<^sub>0g\<^sub>1.p\<^sub>0, fg\<^sub>0h\<^sub>1.p\<^sub>1] \<cdot> local.inv fg\<^sub>0h\<^sub>1.\<phi>
goal (1 subgoal):
1. (h \<star> \<theta>\<^sub>h) \<cdot> \<a>[h, tab\<^sub>0 h, w\<^sub>h] \<cdot> (h.tab \<star> w\<^sub>h) = ((h \<star> \<theta>\<^sub>h') \<cdot> \<a>[h, tab\<^sub>0 h, w\<^sub>h'] \<cdot> (h.tab \<star> w\<^sub>h')) \<cdot> \<beta>\<^sub>h
[PROOF STEP]
have "... = (\<theta>\<^sub>g' \<cdot> (tab\<^sub>0 g \<star> \<gamma>\<^sub>g)) \<cdot> \<a>[tab\<^sub>0 g, Tfg.p\<^sub>0, TTfgh.p\<^sub>1] \<cdot> inv fg\<^sub>0h\<^sub>1.\<phi>"
[PROOF STATE]
proof (prove)
goal (1 subgoal):
1. \<theta>\<^sub>g \<cdot> \<a>[tab\<^sub>0 g, f\<^sub>0g\<^sub>1.p\<^sub>0, fg\<^sub>0h\<^sub>1.p\<^sub>1] \<cdot> local.inv fg\<^sub>0h\<^sub>1.\<phi> = (\<theta>\<^sub>g' \<cdot> (tab\<^sub>0 g \<star> \<gamma>\<^sub>g)) \<cdot> \<a>[tab\<^sub>0 g, f\<^sub>0g\<^sub>1.p\<^sub>0, fg\<^sub>0h\<^sub>1.p\<^sub>1] \<cdot> local.inv fg\<^sub>0h\<^sub>1.\<phi>
[PROOF STEP]
using \<gamma>\<^sub>g
[PROOF STATE]
proof (prove)
using this:
\<guillemotleft>\<gamma>\<^sub>g : w\<^sub>g \<rightarrow> w\<^sub>g'\<guillemotright> \<and> \<beta>\<^sub>g = tab\<^sub>1 g \<star> \<gamma>\<^sub>g \<and> \<theta>\<^sub>g = \<theta>\<^sub>g' \<cdot> (tab\<^sub>0 g \<star> \<gamma>\<^sub>g)
goal (1 subgoal):
1. \<theta>\<^sub>g \<cdot> \<a>[tab\<^sub>0 g, f\<^sub>0g\<^sub>1.p\<^sub>0, fg\<^sub>0h\<^sub>1.p\<^sub>1] \<cdot> local.inv fg\<^sub>0h\<^sub>1.\<phi> = (\<theta>\<^sub>g' \<cdot> (tab\<^sub>0 g \<star> \<gamma>\<^sub>g)) \<cdot> \<a>[tab\<^sub>0 g, f\<^sub>0g\<^sub>1.p\<^sub>0, fg\<^sub>0h\<^sub>1.p\<^sub>1] \<cdot> local.inv fg\<^sub>0h\<^sub>1.\<phi>
[PROOF STEP]
by simp
[PROOF STATE]
proof (state)
this:
\<theta>\<^sub>g \<cdot> \<a>[tab\<^sub>0 g, f\<^sub>0g\<^sub>1.p\<^sub>0, fg\<^sub>0h\<^sub>1.p\<^sub>1] \<cdot> local.inv fg\<^sub>0h\<^sub>1.\<phi> = (\<theta>\<^sub>g' \<cdot> (tab\<^sub>0 g \<star> \<gamma>\<^sub>g)) \<cdot> \<a>[tab\<^sub>0 g, f\<^sub>0g\<^sub>1.p\<^sub>0, fg\<^sub>0h\<^sub>1.p\<^sub>1] \<cdot> local.inv fg\<^sub>0h\<^sub>1.\<phi>
goal (1 subgoal):
1. (h \<star> \<theta>\<^sub>h) \<cdot> \<a>[h, tab\<^sub>0 h, w\<^sub>h] \<cdot> (h.tab \<star> w\<^sub>h) = ((h \<star> \<theta>\<^sub>h') \<cdot> \<a>[h, tab\<^sub>0 h, w\<^sub>h'] \<cdot> (h.tab \<star> w\<^sub>h')) \<cdot> \<beta>\<^sub>h
[PROOF STEP]
also
[PROOF STATE]
proof (state)
this:
\<theta>\<^sub>g \<cdot> \<a>[tab\<^sub>0 g, f\<^sub>0g\<^sub>1.p\<^sub>0, fg\<^sub>0h\<^sub>1.p\<^sub>1] \<cdot> local.inv fg\<^sub>0h\<^sub>1.\<phi> = (\<theta>\<^sub>g' \<cdot> (tab\<^sub>0 g \<star> \<gamma>\<^sub>g)) \<cdot> \<a>[tab\<^sub>0 g, f\<^sub>0g\<^sub>1.p\<^sub>0, fg\<^sub>0h\<^sub>1.p\<^sub>1] \<cdot> local.inv fg\<^sub>0h\<^sub>1.\<phi>
goal (1 subgoal):
1. (h \<star> \<theta>\<^sub>h) \<cdot> \<a>[h, tab\<^sub>0 h, w\<^sub>h] \<cdot> (h.tab \<star> w\<^sub>h) = ((h \<star> \<theta>\<^sub>h') \<cdot> \<a>[h, tab\<^sub>0 h, w\<^sub>h'] \<cdot> (h.tab \<star> w\<^sub>h')) \<cdot> \<beta>\<^sub>h
[PROOF STEP]
have "... = (h \<star> TTfgh_TfTgh.the_\<theta>) \<cdot>
can (\<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> ((\<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>Tgh.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TfTgh.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>)
(((\<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>Tgh.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TfTgh.p\<^sub>0\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) \<cdot>
((h.tab \<star> Tgh.p\<^sub>0) \<star> TfTgh.p\<^sub>0 \<star> TTfgh_TfTgh.chine) \<cdot>
(g\<^sub>0h\<^sub>1.\<phi> \<star> TfTgh.p\<^sub>0 \<star> TTfgh_TfTgh.chine) \<cdot>
\<a>\<^sup>-\<^sup>1[tab\<^sub>0 g, Tgh.p\<^sub>1, TfTgh.p\<^sub>0 \<star> TTfgh_TfTgh.chine] \<cdot>
(tab\<^sub>0 g \<star> \<gamma>\<^sub>g) \<cdot>
\<a>[tab\<^sub>0 g, Tfg.p\<^sub>0, TTfgh.p\<^sub>1] \<cdot>
inv fg\<^sub>0h\<^sub>1.\<phi>"
[PROOF STATE]
proof (prove)
goal (1 subgoal):
1. (\<theta>\<^sub>g' \<cdot> (tab\<^sub>0 g \<star> \<gamma>\<^sub>g)) \<cdot> \<a>[tab\<^sub>0 g, f\<^sub>0g\<^sub>1.p\<^sub>0, fg\<^sub>0h\<^sub>1.p\<^sub>1] \<cdot> local.inv fg\<^sub>0h\<^sub>1.\<phi> = (h \<star> TTfgh_TfTgh.the_\<theta>) \<cdot> can (\<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> ((\<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) (((\<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) \<cdot> ((h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine) \<cdot> (g\<^sub>0h\<^sub>1.\<phi> \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine) \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1, f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine] \<cdot> (tab\<^sub>0 g \<star> \<gamma>\<^sub>g) \<cdot> \<a>[tab\<^sub>0 g, f\<^sub>0g\<^sub>1.p\<^sub>0, fg\<^sub>0h\<^sub>1.p\<^sub>1] \<cdot> local.inv fg\<^sub>0h\<^sub>1.\<phi>
[PROOF STEP]
unfolding \<theta>\<^sub>g'_def
[PROOF STATE]
proof (prove)
goal (1 subgoal):
1. (((h \<star> TTfgh_TfTgh.the_\<theta>) \<cdot> can (\<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> ((\<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) (((\<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) \<cdot> ((h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine) \<cdot> (g\<^sub>0h\<^sub>1.\<phi> \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine) \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1, f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine]) \<cdot> (tab\<^sub>0 g \<star> \<gamma>\<^sub>g)) \<cdot> \<a>[tab\<^sub>0 g, f\<^sub>0g\<^sub>1.p\<^sub>0, fg\<^sub>0h\<^sub>1.p\<^sub>1] \<cdot> local.inv fg\<^sub>0h\<^sub>1.\<phi> = (h \<star> TTfgh_TfTgh.the_\<theta>) \<cdot> can (\<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> ((\<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) (((\<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) \<cdot> ((h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine) \<cdot> (g\<^sub>0h\<^sub>1.\<phi> \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine) \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1, f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine] \<cdot> (tab\<^sub>0 g \<star> \<gamma>\<^sub>g) \<cdot> \<a>[tab\<^sub>0 g, f\<^sub>0g\<^sub>1.p\<^sub>0, fg\<^sub>0h\<^sub>1.p\<^sub>1] \<cdot> local.inv fg\<^sub>0h\<^sub>1.\<phi>
[PROOF STEP]
using comp_assoc
[PROOF STATE]
proof (prove)
using this:
(?h \<cdot> ?g) \<cdot> ?f = ?h \<cdot> ?g \<cdot> ?f
goal (1 subgoal):
1. (((h \<star> TTfgh_TfTgh.the_\<theta>) \<cdot> can (\<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> ((\<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) (((\<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) \<cdot> ((h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine) \<cdot> (g\<^sub>0h\<^sub>1.\<phi> \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine) \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1, f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine]) \<cdot> (tab\<^sub>0 g \<star> \<gamma>\<^sub>g)) \<cdot> \<a>[tab\<^sub>0 g, f\<^sub>0g\<^sub>1.p\<^sub>0, fg\<^sub>0h\<^sub>1.p\<^sub>1] \<cdot> local.inv fg\<^sub>0h\<^sub>1.\<phi> = (h \<star> TTfgh_TfTgh.the_\<theta>) \<cdot> can (\<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> ((\<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) (((\<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) \<cdot> ((h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine) \<cdot> (g\<^sub>0h\<^sub>1.\<phi> \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine) \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1, f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine] \<cdot> (tab\<^sub>0 g \<star> \<gamma>\<^sub>g) \<cdot> \<a>[tab\<^sub>0 g, f\<^sub>0g\<^sub>1.p\<^sub>0, fg\<^sub>0h\<^sub>1.p\<^sub>1] \<cdot> local.inv fg\<^sub>0h\<^sub>1.\<phi>
[PROOF STEP]
by presburger
[PROOF STATE]
proof (state)
this:
(\<theta>\<^sub>g' \<cdot> (tab\<^sub>0 g \<star> \<gamma>\<^sub>g)) \<cdot> \<a>[tab\<^sub>0 g, f\<^sub>0g\<^sub>1.p\<^sub>0, fg\<^sub>0h\<^sub>1.p\<^sub>1] \<cdot> local.inv fg\<^sub>0h\<^sub>1.\<phi> = (h \<star> TTfgh_TfTgh.the_\<theta>) \<cdot> can (\<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> ((\<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) (((\<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) \<cdot> ((h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine) \<cdot> (g\<^sub>0h\<^sub>1.\<phi> \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine) \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1, f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine] \<cdot> (tab\<^sub>0 g \<star> \<gamma>\<^sub>g) \<cdot> \<a>[tab\<^sub>0 g, f\<^sub>0g\<^sub>1.p\<^sub>0, fg\<^sub>0h\<^sub>1.p\<^sub>1] \<cdot> local.inv fg\<^sub>0h\<^sub>1.\<phi>
goal (1 subgoal):
1. (h \<star> \<theta>\<^sub>h) \<cdot> \<a>[h, tab\<^sub>0 h, w\<^sub>h] \<cdot> (h.tab \<star> w\<^sub>h) = ((h \<star> \<theta>\<^sub>h') \<cdot> \<a>[h, tab\<^sub>0 h, w\<^sub>h'] \<cdot> (h.tab \<star> w\<^sub>h')) \<cdot> \<beta>\<^sub>h
[PROOF STEP]
also
[PROOF STATE]
proof (state)
this:
(\<theta>\<^sub>g' \<cdot> (tab\<^sub>0 g \<star> \<gamma>\<^sub>g)) \<cdot> \<a>[tab\<^sub>0 g, f\<^sub>0g\<^sub>1.p\<^sub>0, fg\<^sub>0h\<^sub>1.p\<^sub>1] \<cdot> local.inv fg\<^sub>0h\<^sub>1.\<phi> = (h \<star> TTfgh_TfTgh.the_\<theta>) \<cdot> can (\<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> ((\<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) (((\<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) \<cdot> ((h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine) \<cdot> (g\<^sub>0h\<^sub>1.\<phi> \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine) \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1, f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine] \<cdot> (tab\<^sub>0 g \<star> \<gamma>\<^sub>g) \<cdot> \<a>[tab\<^sub>0 g, f\<^sub>0g\<^sub>1.p\<^sub>0, fg\<^sub>0h\<^sub>1.p\<^sub>1] \<cdot> local.inv fg\<^sub>0h\<^sub>1.\<phi>
goal (1 subgoal):
1. (h \<star> \<theta>\<^sub>h) \<cdot> \<a>[h, tab\<^sub>0 h, w\<^sub>h] \<cdot> (h.tab \<star> w\<^sub>h) = ((h \<star> \<theta>\<^sub>h') \<cdot> \<a>[h, tab\<^sub>0 h, w\<^sub>h'] \<cdot> (h.tab \<star> w\<^sub>h')) \<cdot> \<beta>\<^sub>h
[PROOF STEP]
have "... = (h \<star> TTfgh_TfTgh.the_\<theta>) \<cdot>
can (\<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> ((\<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>Tgh.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TfTgh.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>)
(((\<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>Tgh.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TfTgh.p\<^sub>0\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) \<cdot>
((\<a>\<^sup>-\<^sup>1[h \<star> tab\<^sub>0 h, Tgh.p\<^sub>0, TfTgh.p\<^sub>0 \<star> TTfgh_TfTgh.chine] \<cdot>
\<a>[h \<star> tab\<^sub>0 h, Tgh.p\<^sub>0, TfTgh.p\<^sub>0 \<star> TTfgh_TfTgh.chine]) \<cdot>
((h.tab \<star> Tgh.p\<^sub>0) \<star> TfTgh.p\<^sub>0 \<star> TTfgh_TfTgh.chine)) \<cdot>
(g\<^sub>0h\<^sub>1.\<phi> \<star> TfTgh.p\<^sub>0 \<star> TTfgh_TfTgh.chine) \<cdot>
\<a>\<^sup>-\<^sup>1[tab\<^sub>0 g, Tgh.p\<^sub>1, TfTgh.p\<^sub>0 \<star> TTfgh_TfTgh.chine] \<cdot>
(tab\<^sub>0 g \<star> \<gamma>\<^sub>g) \<cdot>
\<a>[tab\<^sub>0 g, Tfg.p\<^sub>0, TTfgh.p\<^sub>1] \<cdot>
inv fg\<^sub>0h\<^sub>1.\<phi>"
[PROOF STATE]
proof (prove)
goal (1 subgoal):
1. (h \<star> TTfgh_TfTgh.the_\<theta>) \<cdot> can (\<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> ((\<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) (((\<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) \<cdot> ((h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine) \<cdot> (g\<^sub>0h\<^sub>1.\<phi> \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine) \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1, f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine] \<cdot> (tab\<^sub>0 g \<star> \<gamma>\<^sub>g) \<cdot> \<a>[tab\<^sub>0 g, f\<^sub>0g\<^sub>1.p\<^sub>0, fg\<^sub>0h\<^sub>1.p\<^sub>1] \<cdot> local.inv fg\<^sub>0h\<^sub>1.\<phi> = (h \<star> TTfgh_TfTgh.the_\<theta>) \<cdot> can (\<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> ((\<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) (((\<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) \<cdot> ((\<a>\<^sup>-\<^sup>1[h \<star> tab\<^sub>0 h, g\<^sub>0h\<^sub>1.p\<^sub>0, f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine] \<cdot> \<a>[h \<star> tab\<^sub>0 h, g\<^sub>0h\<^sub>1.p\<^sub>0, f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine]) \<cdot> ((h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine)) \<cdot> (g\<^sub>0h\<^sub>1.\<phi> \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine) \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1, f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine] \<cdot> (tab\<^sub>0 g \<star> \<gamma>\<^sub>g) \<cdot> \<a>[tab\<^sub>0 g, f\<^sub>0g\<^sub>1.p\<^sub>0, fg\<^sub>0h\<^sub>1.p\<^sub>1] \<cdot> local.inv fg\<^sub>0h\<^sub>1.\<phi>
[PROOF STEP]
proof -
[PROOF STATE]
proof (state)
goal (1 subgoal):
1. (h \<star> TTfgh_TfTgh.the_\<theta>) \<cdot> can (\<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> ((\<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) (((\<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) \<cdot> ((h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine) \<cdot> (g\<^sub>0h\<^sub>1.\<phi> \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine) \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1, f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine] \<cdot> (tab\<^sub>0 g \<star> \<gamma>\<^sub>g) \<cdot> \<a>[tab\<^sub>0 g, f\<^sub>0g\<^sub>1.p\<^sub>0, fg\<^sub>0h\<^sub>1.p\<^sub>1] \<cdot> local.inv fg\<^sub>0h\<^sub>1.\<phi> = (h \<star> TTfgh_TfTgh.the_\<theta>) \<cdot> can (\<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> ((\<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) (((\<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) \<cdot> ((\<a>\<^sup>-\<^sup>1[h \<star> tab\<^sub>0 h, g\<^sub>0h\<^sub>1.p\<^sub>0, f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine] \<cdot> \<a>[h \<star> tab\<^sub>0 h, g\<^sub>0h\<^sub>1.p\<^sub>0, f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine]) \<cdot> ((h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine)) \<cdot> (g\<^sub>0h\<^sub>1.\<phi> \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine) \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1, f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine] \<cdot> (tab\<^sub>0 g \<star> \<gamma>\<^sub>g) \<cdot> \<a>[tab\<^sub>0 g, f\<^sub>0g\<^sub>1.p\<^sub>0, fg\<^sub>0h\<^sub>1.p\<^sub>1] \<cdot> local.inv fg\<^sub>0h\<^sub>1.\<phi>
[PROOF STEP]
have "(\<a>\<^sup>-\<^sup>1[h \<star> tab\<^sub>0 h, Tgh.p\<^sub>0, TfTgh.p\<^sub>0 \<star> TTfgh_TfTgh.chine] \<cdot>
\<a>[h \<star> tab\<^sub>0 h, Tgh.p\<^sub>0, TfTgh.p\<^sub>0 \<star> TTfgh_TfTgh.chine]) \<cdot>
((h.tab \<star> Tgh.p\<^sub>0) \<star> TfTgh.p\<^sub>0 \<star> TTfgh_TfTgh.chine) =
(h.tab \<star> Tgh.p\<^sub>0) \<star> TfTgh.p\<^sub>0 \<star> TTfgh_TfTgh.chine"
[PROOF STATE]
proof (prove)
goal (1 subgoal):
1. (\<a>\<^sup>-\<^sup>1[h \<star> tab\<^sub>0 h, g\<^sub>0h\<^sub>1.p\<^sub>0, f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine] \<cdot> \<a>[h \<star> tab\<^sub>0 h, g\<^sub>0h\<^sub>1.p\<^sub>0, f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine]) \<cdot> ((h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine) = (h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine
[PROOF STEP]
using comp_cod_arr comp_assoc_assoc'
[PROOF STATE]
proof (prove)
using this:
\<lbrakk>arr ?f; cod ?f = ?b\<rbrakk> \<Longrightarrow> ?b \<cdot> ?f = ?f
\<lbrakk>ide ?f; ide ?g; ide ?h; src ?f = trg ?g; src ?g = trg ?h\<rbrakk> \<Longrightarrow> \<a>[?f, ?g, ?h] \<cdot> \<a>\<^sup>-\<^sup>1[?f, ?g, ?h] = ?f \<star> ?g \<star> ?h
\<lbrakk>ide ?f; ide ?g; ide ?h; src ?f = trg ?g; src ?g = trg ?h\<rbrakk> \<Longrightarrow> \<a>\<^sup>-\<^sup>1[?f, ?g, ?h] \<cdot> \<a>[?f, ?g, ?h] = (?f \<star> ?g) \<star> ?h
goal (1 subgoal):
1. (\<a>\<^sup>-\<^sup>1[h \<star> tab\<^sub>0 h, g\<^sub>0h\<^sub>1.p\<^sub>0, f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine] \<cdot> \<a>[h \<star> tab\<^sub>0 h, g\<^sub>0h\<^sub>1.p\<^sub>0, f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine]) \<cdot> ((h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine) = (h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine
[PROOF STEP]
by simp
[PROOF STATE]
proof (state)
this:
(\<a>\<^sup>-\<^sup>1[h \<star> tab\<^sub>0 h, g\<^sub>0h\<^sub>1.p\<^sub>0, f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine] \<cdot> \<a>[h \<star> tab\<^sub>0 h, g\<^sub>0h\<^sub>1.p\<^sub>0, f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine]) \<cdot> ((h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine) = (h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine
goal (1 subgoal):
1. (h \<star> TTfgh_TfTgh.the_\<theta>) \<cdot> can (\<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> ((\<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) (((\<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) \<cdot> ((h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine) \<cdot> (g\<^sub>0h\<^sub>1.\<phi> \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine) \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1, f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine] \<cdot> (tab\<^sub>0 g \<star> \<gamma>\<^sub>g) \<cdot> \<a>[tab\<^sub>0 g, f\<^sub>0g\<^sub>1.p\<^sub>0, fg\<^sub>0h\<^sub>1.p\<^sub>1] \<cdot> local.inv fg\<^sub>0h\<^sub>1.\<phi> = (h \<star> TTfgh_TfTgh.the_\<theta>) \<cdot> can (\<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> ((\<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) (((\<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) \<cdot> ((\<a>\<^sup>-\<^sup>1[h \<star> tab\<^sub>0 h, g\<^sub>0h\<^sub>1.p\<^sub>0, f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine] \<cdot> \<a>[h \<star> tab\<^sub>0 h, g\<^sub>0h\<^sub>1.p\<^sub>0, f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine]) \<cdot> ((h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine)) \<cdot> (g\<^sub>0h\<^sub>1.\<phi> \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine) \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1, f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine] \<cdot> (tab\<^sub>0 g \<star> \<gamma>\<^sub>g) \<cdot> \<a>[tab\<^sub>0 g, f\<^sub>0g\<^sub>1.p\<^sub>0, fg\<^sub>0h\<^sub>1.p\<^sub>1] \<cdot> local.inv fg\<^sub>0h\<^sub>1.\<phi>
[PROOF STEP]
thus ?thesis
[PROOF STATE]
proof (prove)
using this:
(\<a>\<^sup>-\<^sup>1[h \<star> tab\<^sub>0 h, g\<^sub>0h\<^sub>1.p\<^sub>0, f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine] \<cdot> \<a>[h \<star> tab\<^sub>0 h, g\<^sub>0h\<^sub>1.p\<^sub>0, f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine]) \<cdot> ((h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine) = (h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine
goal (1 subgoal):
1. (h \<star> TTfgh_TfTgh.the_\<theta>) \<cdot> can (\<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> ((\<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) (((\<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) \<cdot> ((h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine) \<cdot> (g\<^sub>0h\<^sub>1.\<phi> \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine) \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1, f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine] \<cdot> (tab\<^sub>0 g \<star> \<gamma>\<^sub>g) \<cdot> \<a>[tab\<^sub>0 g, f\<^sub>0g\<^sub>1.p\<^sub>0, fg\<^sub>0h\<^sub>1.p\<^sub>1] \<cdot> local.inv fg\<^sub>0h\<^sub>1.\<phi> = (h \<star> TTfgh_TfTgh.the_\<theta>) \<cdot> can (\<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> ((\<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) (((\<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) \<cdot> ((\<a>\<^sup>-\<^sup>1[h \<star> tab\<^sub>0 h, g\<^sub>0h\<^sub>1.p\<^sub>0, f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine] \<cdot> \<a>[h \<star> tab\<^sub>0 h, g\<^sub>0h\<^sub>1.p\<^sub>0, f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine]) \<cdot> ((h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine)) \<cdot> (g\<^sub>0h\<^sub>1.\<phi> \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine) \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1, f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine] \<cdot> (tab\<^sub>0 g \<star> \<gamma>\<^sub>g) \<cdot> \<a>[tab\<^sub>0 g, f\<^sub>0g\<^sub>1.p\<^sub>0, fg\<^sub>0h\<^sub>1.p\<^sub>1] \<cdot> local.inv fg\<^sub>0h\<^sub>1.\<phi>
[PROOF STEP]
using comp_assoc
[PROOF STATE]
proof (prove)
using this:
(\<a>\<^sup>-\<^sup>1[h \<star> tab\<^sub>0 h, g\<^sub>0h\<^sub>1.p\<^sub>0, f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine] \<cdot> \<a>[h \<star> tab\<^sub>0 h, g\<^sub>0h\<^sub>1.p\<^sub>0, f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine]) \<cdot> ((h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine) = (h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine
(?h \<cdot> ?g) \<cdot> ?f = ?h \<cdot> ?g \<cdot> ?f
goal (1 subgoal):
1. (h \<star> TTfgh_TfTgh.the_\<theta>) \<cdot> can (\<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> ((\<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) (((\<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) \<cdot> ((h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine) \<cdot> (g\<^sub>0h\<^sub>1.\<phi> \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine) \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1, f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine] \<cdot> (tab\<^sub>0 g \<star> \<gamma>\<^sub>g) \<cdot> \<a>[tab\<^sub>0 g, f\<^sub>0g\<^sub>1.p\<^sub>0, fg\<^sub>0h\<^sub>1.p\<^sub>1] \<cdot> local.inv fg\<^sub>0h\<^sub>1.\<phi> = (h \<star> TTfgh_TfTgh.the_\<theta>) \<cdot> can (\<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> ((\<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) (((\<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) \<cdot> ((\<a>\<^sup>-\<^sup>1[h \<star> tab\<^sub>0 h, g\<^sub>0h\<^sub>1.p\<^sub>0, f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine] \<cdot> \<a>[h \<star> tab\<^sub>0 h, g\<^sub>0h\<^sub>1.p\<^sub>0, f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine]) \<cdot> ((h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine)) \<cdot> (g\<^sub>0h\<^sub>1.\<phi> \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine) \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1, f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine] \<cdot> (tab\<^sub>0 g \<star> \<gamma>\<^sub>g) \<cdot> \<a>[tab\<^sub>0 g, f\<^sub>0g\<^sub>1.p\<^sub>0, fg\<^sub>0h\<^sub>1.p\<^sub>1] \<cdot> local.inv fg\<^sub>0h\<^sub>1.\<phi>
[PROOF STEP]
by simp
[PROOF STATE]
proof (state)
this:
(h \<star> TTfgh_TfTgh.the_\<theta>) \<cdot> can (\<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> ((\<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) (((\<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) \<cdot> ((h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine) \<cdot> (g\<^sub>0h\<^sub>1.\<phi> \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine) \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1, f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine] \<cdot> (tab\<^sub>0 g \<star> \<gamma>\<^sub>g) \<cdot> \<a>[tab\<^sub>0 g, f\<^sub>0g\<^sub>1.p\<^sub>0, fg\<^sub>0h\<^sub>1.p\<^sub>1] \<cdot> local.inv fg\<^sub>0h\<^sub>1.\<phi> = (h \<star> TTfgh_TfTgh.the_\<theta>) \<cdot> can (\<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> ((\<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) (((\<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) \<cdot> ((\<a>\<^sup>-\<^sup>1[h \<star> tab\<^sub>0 h, g\<^sub>0h\<^sub>1.p\<^sub>0, f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine] \<cdot> \<a>[h \<star> tab\<^sub>0 h, g\<^sub>0h\<^sub>1.p\<^sub>0, f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine]) \<cdot> ((h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine)) \<cdot> (g\<^sub>0h\<^sub>1.\<phi> \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine) \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1, f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine] \<cdot> (tab\<^sub>0 g \<star> \<gamma>\<^sub>g) \<cdot> \<a>[tab\<^sub>0 g, f\<^sub>0g\<^sub>1.p\<^sub>0, fg\<^sub>0h\<^sub>1.p\<^sub>1] \<cdot> local.inv fg\<^sub>0h\<^sub>1.\<phi>
goal:
No subgoals!
[PROOF STEP]
qed
[PROOF STATE]
proof (state)
this:
(h \<star> TTfgh_TfTgh.the_\<theta>) \<cdot> can (\<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> ((\<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) (((\<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) \<cdot> ((h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine) \<cdot> (g\<^sub>0h\<^sub>1.\<phi> \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine) \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1, f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine] \<cdot> (tab\<^sub>0 g \<star> \<gamma>\<^sub>g) \<cdot> \<a>[tab\<^sub>0 g, f\<^sub>0g\<^sub>1.p\<^sub>0, fg\<^sub>0h\<^sub>1.p\<^sub>1] \<cdot> local.inv fg\<^sub>0h\<^sub>1.\<phi> = (h \<star> TTfgh_TfTgh.the_\<theta>) \<cdot> can (\<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> ((\<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) (((\<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) \<cdot> ((\<a>\<^sup>-\<^sup>1[h \<star> tab\<^sub>0 h, g\<^sub>0h\<^sub>1.p\<^sub>0, f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine] \<cdot> \<a>[h \<star> tab\<^sub>0 h, g\<^sub>0h\<^sub>1.p\<^sub>0, f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine]) \<cdot> ((h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine)) \<cdot> (g\<^sub>0h\<^sub>1.\<phi> \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine) \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1, f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine] \<cdot> (tab\<^sub>0 g \<star> \<gamma>\<^sub>g) \<cdot> \<a>[tab\<^sub>0 g, f\<^sub>0g\<^sub>1.p\<^sub>0, fg\<^sub>0h\<^sub>1.p\<^sub>1] \<cdot> local.inv fg\<^sub>0h\<^sub>1.\<phi>
goal (1 subgoal):
1. (h \<star> \<theta>\<^sub>h) \<cdot> \<a>[h, tab\<^sub>0 h, w\<^sub>h] \<cdot> (h.tab \<star> w\<^sub>h) = ((h \<star> \<theta>\<^sub>h') \<cdot> \<a>[h, tab\<^sub>0 h, w\<^sub>h'] \<cdot> (h.tab \<star> w\<^sub>h')) \<cdot> \<beta>\<^sub>h
[PROOF STEP]
also
[PROOF STATE]
proof (state)
this:
(h \<star> TTfgh_TfTgh.the_\<theta>) \<cdot> can (\<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> ((\<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) (((\<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) \<cdot> ((h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine) \<cdot> (g\<^sub>0h\<^sub>1.\<phi> \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine) \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1, f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine] \<cdot> (tab\<^sub>0 g \<star> \<gamma>\<^sub>g) \<cdot> \<a>[tab\<^sub>0 g, f\<^sub>0g\<^sub>1.p\<^sub>0, fg\<^sub>0h\<^sub>1.p\<^sub>1] \<cdot> local.inv fg\<^sub>0h\<^sub>1.\<phi> = (h \<star> TTfgh_TfTgh.the_\<theta>) \<cdot> can (\<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> ((\<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) (((\<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) \<cdot> ((\<a>\<^sup>-\<^sup>1[h \<star> tab\<^sub>0 h, g\<^sub>0h\<^sub>1.p\<^sub>0, f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine] \<cdot> \<a>[h \<star> tab\<^sub>0 h, g\<^sub>0h\<^sub>1.p\<^sub>0, f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine]) \<cdot> ((h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine)) \<cdot> (g\<^sub>0h\<^sub>1.\<phi> \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine) \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1, f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine] \<cdot> (tab\<^sub>0 g \<star> \<gamma>\<^sub>g) \<cdot> \<a>[tab\<^sub>0 g, f\<^sub>0g\<^sub>1.p\<^sub>0, fg\<^sub>0h\<^sub>1.p\<^sub>1] \<cdot> local.inv fg\<^sub>0h\<^sub>1.\<phi>
goal (1 subgoal):
1. (h \<star> \<theta>\<^sub>h) \<cdot> \<a>[h, tab\<^sub>0 h, w\<^sub>h] \<cdot> (h.tab \<star> w\<^sub>h) = ((h \<star> \<theta>\<^sub>h') \<cdot> \<a>[h, tab\<^sub>0 h, w\<^sub>h'] \<cdot> (h.tab \<star> w\<^sub>h')) \<cdot> \<beta>\<^sub>h
[PROOF STEP]
have "... = (h \<star> TTfgh_TfTgh.the_\<theta>) \<cdot>
can (\<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> ((\<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>Tgh.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TfTgh.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>)
(((\<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>Tgh.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TfTgh.p\<^sub>0\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) \<cdot>
\<a>\<^sup>-\<^sup>1[h \<star> tab\<^sub>0 h, Tgh.p\<^sub>0, TfTgh.p\<^sub>0 \<star> TTfgh_TfTgh.chine] \<cdot>
(\<a>[h \<star> tab\<^sub>0 h, Tgh.p\<^sub>0, TfTgh.p\<^sub>0 \<star> TTfgh_TfTgh.chine] \<cdot>
((h.tab \<star> Tgh.p\<^sub>0) \<star> TfTgh.p\<^sub>0 \<star> TTfgh_TfTgh.chine)) \<cdot>
(g\<^sub>0h\<^sub>1.\<phi> \<star> TfTgh.p\<^sub>0 \<star> TTfgh_TfTgh.chine) \<cdot>
\<a>\<^sup>-\<^sup>1[tab\<^sub>0 g, Tgh.p\<^sub>1, TfTgh.p\<^sub>0 \<star> TTfgh_TfTgh.chine] \<cdot>
(tab\<^sub>0 g \<star> \<gamma>\<^sub>g) \<cdot>
\<a>[tab\<^sub>0 g, Tfg.p\<^sub>0, TTfgh.p\<^sub>1] \<cdot>
inv fg\<^sub>0h\<^sub>1.\<phi>"
[PROOF STATE]
proof (prove)
goal (1 subgoal):
1. (h \<star> TTfgh_TfTgh.the_\<theta>) \<cdot> can (\<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> ((\<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) (((\<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) \<cdot> ((\<a>\<^sup>-\<^sup>1[h \<star> tab\<^sub>0 h, g\<^sub>0h\<^sub>1.p\<^sub>0, f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine] \<cdot> \<a>[h \<star> tab\<^sub>0 h, g\<^sub>0h\<^sub>1.p\<^sub>0, f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine]) \<cdot> ((h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine)) \<cdot> (g\<^sub>0h\<^sub>1.\<phi> \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine) \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1, f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine] \<cdot> (tab\<^sub>0 g \<star> \<gamma>\<^sub>g) \<cdot> \<a>[tab\<^sub>0 g, f\<^sub>0g\<^sub>1.p\<^sub>0, fg\<^sub>0h\<^sub>1.p\<^sub>1] \<cdot> local.inv fg\<^sub>0h\<^sub>1.\<phi> = (h \<star> TTfgh_TfTgh.the_\<theta>) \<cdot> can (\<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> ((\<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) (((\<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) \<cdot> \<a>\<^sup>-\<^sup>1[h \<star> tab\<^sub>0 h, g\<^sub>0h\<^sub>1.p\<^sub>0, f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine] \<cdot> (\<a>[h \<star> tab\<^sub>0 h, g\<^sub>0h\<^sub>1.p\<^sub>0, f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine] \<cdot> ((h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine)) \<cdot> (g\<^sub>0h\<^sub>1.\<phi> \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine) \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1, f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine] \<cdot> (tab\<^sub>0 g \<star> \<gamma>\<^sub>g) \<cdot> \<a>[tab\<^sub>0 g, f\<^sub>0g\<^sub>1.p\<^sub>0, fg\<^sub>0h\<^sub>1.p\<^sub>1] \<cdot> local.inv fg\<^sub>0h\<^sub>1.\<phi>
[PROOF STEP]
using comp_assoc
[PROOF STATE]
proof (prove)
using this:
(?h \<cdot> ?g) \<cdot> ?f = ?h \<cdot> ?g \<cdot> ?f
goal (1 subgoal):
1. (h \<star> TTfgh_TfTgh.the_\<theta>) \<cdot> can (\<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> ((\<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) (((\<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) \<cdot> ((\<a>\<^sup>-\<^sup>1[h \<star> tab\<^sub>0 h, g\<^sub>0h\<^sub>1.p\<^sub>0, f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine] \<cdot> \<a>[h \<star> tab\<^sub>0 h, g\<^sub>0h\<^sub>1.p\<^sub>0, f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine]) \<cdot> ((h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine)) \<cdot> (g\<^sub>0h\<^sub>1.\<phi> \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine) \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1, f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine] \<cdot> (tab\<^sub>0 g \<star> \<gamma>\<^sub>g) \<cdot> \<a>[tab\<^sub>0 g, f\<^sub>0g\<^sub>1.p\<^sub>0, fg\<^sub>0h\<^sub>1.p\<^sub>1] \<cdot> local.inv fg\<^sub>0h\<^sub>1.\<phi> = (h \<star> TTfgh_TfTgh.the_\<theta>) \<cdot> can (\<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> ((\<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) (((\<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) \<cdot> \<a>\<^sup>-\<^sup>1[h \<star> tab\<^sub>0 h, g\<^sub>0h\<^sub>1.p\<^sub>0, f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine] \<cdot> (\<a>[h \<star> tab\<^sub>0 h, g\<^sub>0h\<^sub>1.p\<^sub>0, f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine] \<cdot> ((h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine)) \<cdot> (g\<^sub>0h\<^sub>1.\<phi> \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine) \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1, f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine] \<cdot> (tab\<^sub>0 g \<star> \<gamma>\<^sub>g) \<cdot> \<a>[tab\<^sub>0 g, f\<^sub>0g\<^sub>1.p\<^sub>0, fg\<^sub>0h\<^sub>1.p\<^sub>1] \<cdot> local.inv fg\<^sub>0h\<^sub>1.\<phi>
[PROOF STEP]
by presburger
[PROOF STATE]
proof (state)
this:
(h \<star> TTfgh_TfTgh.the_\<theta>) \<cdot> can (\<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> ((\<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) (((\<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) \<cdot> ((\<a>\<^sup>-\<^sup>1[h \<star> tab\<^sub>0 h, g\<^sub>0h\<^sub>1.p\<^sub>0, f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine] \<cdot> \<a>[h \<star> tab\<^sub>0 h, g\<^sub>0h\<^sub>1.p\<^sub>0, f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine]) \<cdot> ((h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine)) \<cdot> (g\<^sub>0h\<^sub>1.\<phi> \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine) \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1, f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine] \<cdot> (tab\<^sub>0 g \<star> \<gamma>\<^sub>g) \<cdot> \<a>[tab\<^sub>0 g, f\<^sub>0g\<^sub>1.p\<^sub>0, fg\<^sub>0h\<^sub>1.p\<^sub>1] \<cdot> local.inv fg\<^sub>0h\<^sub>1.\<phi> = (h \<star> TTfgh_TfTgh.the_\<theta>) \<cdot> can (\<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> ((\<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) (((\<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) \<cdot> \<a>\<^sup>-\<^sup>1[h \<star> tab\<^sub>0 h, g\<^sub>0h\<^sub>1.p\<^sub>0, f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine] \<cdot> (\<a>[h \<star> tab\<^sub>0 h, g\<^sub>0h\<^sub>1.p\<^sub>0, f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine] \<cdot> ((h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine)) \<cdot> (g\<^sub>0h\<^sub>1.\<phi> \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine) \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1, f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine] \<cdot> (tab\<^sub>0 g \<star> \<gamma>\<^sub>g) \<cdot> \<a>[tab\<^sub>0 g, f\<^sub>0g\<^sub>1.p\<^sub>0, fg\<^sub>0h\<^sub>1.p\<^sub>1] \<cdot> local.inv fg\<^sub>0h\<^sub>1.\<phi>
goal (1 subgoal):
1. (h \<star> \<theta>\<^sub>h) \<cdot> \<a>[h, tab\<^sub>0 h, w\<^sub>h] \<cdot> (h.tab \<star> w\<^sub>h) = ((h \<star> \<theta>\<^sub>h') \<cdot> \<a>[h, tab\<^sub>0 h, w\<^sub>h'] \<cdot> (h.tab \<star> w\<^sub>h')) \<cdot> \<beta>\<^sub>h
[PROOF STEP]
also
[PROOF STATE]
proof (state)
this:
(h \<star> TTfgh_TfTgh.the_\<theta>) \<cdot> can (\<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> ((\<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) (((\<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) \<cdot> ((\<a>\<^sup>-\<^sup>1[h \<star> tab\<^sub>0 h, g\<^sub>0h\<^sub>1.p\<^sub>0, f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine] \<cdot> \<a>[h \<star> tab\<^sub>0 h, g\<^sub>0h\<^sub>1.p\<^sub>0, f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine]) \<cdot> ((h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine)) \<cdot> (g\<^sub>0h\<^sub>1.\<phi> \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine) \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1, f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine] \<cdot> (tab\<^sub>0 g \<star> \<gamma>\<^sub>g) \<cdot> \<a>[tab\<^sub>0 g, f\<^sub>0g\<^sub>1.p\<^sub>0, fg\<^sub>0h\<^sub>1.p\<^sub>1] \<cdot> local.inv fg\<^sub>0h\<^sub>1.\<phi> = (h \<star> TTfgh_TfTgh.the_\<theta>) \<cdot> can (\<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> ((\<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) (((\<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) \<cdot> \<a>\<^sup>-\<^sup>1[h \<star> tab\<^sub>0 h, g\<^sub>0h\<^sub>1.p\<^sub>0, f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine] \<cdot> (\<a>[h \<star> tab\<^sub>0 h, g\<^sub>0h\<^sub>1.p\<^sub>0, f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine] \<cdot> ((h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine)) \<cdot> (g\<^sub>0h\<^sub>1.\<phi> \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine) \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1, f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine] \<cdot> (tab\<^sub>0 g \<star> \<gamma>\<^sub>g) \<cdot> \<a>[tab\<^sub>0 g, f\<^sub>0g\<^sub>1.p\<^sub>0, fg\<^sub>0h\<^sub>1.p\<^sub>1] \<cdot> local.inv fg\<^sub>0h\<^sub>1.\<phi>
goal (1 subgoal):
1. (h \<star> \<theta>\<^sub>h) \<cdot> \<a>[h, tab\<^sub>0 h, w\<^sub>h] \<cdot> (h.tab \<star> w\<^sub>h) = ((h \<star> \<theta>\<^sub>h') \<cdot> \<a>[h, tab\<^sub>0 h, w\<^sub>h'] \<cdot> (h.tab \<star> w\<^sub>h')) \<cdot> \<beta>\<^sub>h
[PROOF STEP]
have "... = (h \<star> TTfgh_TfTgh.the_\<theta>) \<cdot>
can (\<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> ((\<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>Tgh.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TfTgh.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>)
(((\<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>Tgh.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TfTgh.p\<^sub>0\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) \<cdot>
\<a>\<^sup>-\<^sup>1[h \<star> tab\<^sub>0 h, Tgh.p\<^sub>0, TfTgh.p\<^sub>0 \<star> TTfgh_TfTgh.chine] \<cdot>
(h.tab \<star> Tgh.p\<^sub>0 \<star> TfTgh.p\<^sub>0 \<star> TTfgh_TfTgh.chine) \<cdot>
\<a>[tab\<^sub>1 h, Tgh.p\<^sub>0, TfTgh.p\<^sub>0 \<star> TTfgh_TfTgh.chine] \<cdot>
(g\<^sub>0h\<^sub>1.\<phi> \<star> TfTgh.p\<^sub>0 \<star> TTfgh_TfTgh.chine) \<cdot>
\<a>\<^sup>-\<^sup>1[tab\<^sub>0 g, Tgh.p\<^sub>1, TfTgh.p\<^sub>0 \<star> TTfgh_TfTgh.chine] \<cdot>
(tab\<^sub>0 g \<star> \<gamma>\<^sub>g) \<cdot>
\<a>[tab\<^sub>0 g, Tfg.p\<^sub>0, TTfgh.p\<^sub>1] \<cdot>
inv fg\<^sub>0h\<^sub>1.\<phi>"
[PROOF STATE]
proof (prove)
goal (1 subgoal):
1. (h \<star> TTfgh_TfTgh.the_\<theta>) \<cdot> can (\<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> ((\<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) (((\<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) \<cdot> \<a>\<^sup>-\<^sup>1[h \<star> tab\<^sub>0 h, g\<^sub>0h\<^sub>1.p\<^sub>0, f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine] \<cdot> (\<a>[h \<star> tab\<^sub>0 h, g\<^sub>0h\<^sub>1.p\<^sub>0, f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine] \<cdot> ((h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine)) \<cdot> (g\<^sub>0h\<^sub>1.\<phi> \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine) \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1, f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine] \<cdot> (tab\<^sub>0 g \<star> \<gamma>\<^sub>g) \<cdot> \<a>[tab\<^sub>0 g, f\<^sub>0g\<^sub>1.p\<^sub>0, fg\<^sub>0h\<^sub>1.p\<^sub>1] \<cdot> local.inv fg\<^sub>0h\<^sub>1.\<phi> = (h \<star> TTfgh_TfTgh.the_\<theta>) \<cdot> can (\<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> ((\<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) (((\<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) \<cdot> \<a>\<^sup>-\<^sup>1[h \<star> tab\<^sub>0 h, g\<^sub>0h\<^sub>1.p\<^sub>0, f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine] \<cdot> (h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0 \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine) \<cdot> \<a>[tab\<^sub>1 h, g\<^sub>0h\<^sub>1.p\<^sub>0, f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine] \<cdot> (g\<^sub>0h\<^sub>1.\<phi> \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine) \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1, f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine] \<cdot> (tab\<^sub>0 g \<star> \<gamma>\<^sub>g) \<cdot> \<a>[tab\<^sub>0 g, f\<^sub>0g\<^sub>1.p\<^sub>0, fg\<^sub>0h\<^sub>1.p\<^sub>1] \<cdot> local.inv fg\<^sub>0h\<^sub>1.\<phi>
[PROOF STEP]
using assoc_naturality [of h.tab Tgh.p\<^sub>0 "TfTgh.p\<^sub>0 \<star> TTfgh_TfTgh.chine"] comp_assoc
[PROOF STATE]
proof (prove)
using this:
\<lbrakk>arr h.tab; arr g\<^sub>0h\<^sub>1.p\<^sub>0; hseq f\<^sub>0gh\<^sub>1.p\<^sub>0 TTfgh_TfTgh.chine; src h.tab = trg g\<^sub>0h\<^sub>1.p\<^sub>0; src g\<^sub>0h\<^sub>1.p\<^sub>0 = trg (f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine)\<rbrakk> \<Longrightarrow> \<a>[cod h.tab, cod g\<^sub>0h\<^sub>1.p\<^sub>0, cod (f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine)] \<cdot> ((h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine) = (h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0 \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine) \<cdot> \<a>[local.dom h.tab, local.dom g\<^sub>0h\<^sub>1.p\<^sub>0, local.dom (f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine)]
(?h \<cdot> ?g) \<cdot> ?f = ?h \<cdot> ?g \<cdot> ?f
goal (1 subgoal):
1. (h \<star> TTfgh_TfTgh.the_\<theta>) \<cdot> can (\<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> ((\<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) (((\<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) \<cdot> \<a>\<^sup>-\<^sup>1[h \<star> tab\<^sub>0 h, g\<^sub>0h\<^sub>1.p\<^sub>0, f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine] \<cdot> (\<a>[h \<star> tab\<^sub>0 h, g\<^sub>0h\<^sub>1.p\<^sub>0, f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine] \<cdot> ((h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine)) \<cdot> (g\<^sub>0h\<^sub>1.\<phi> \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine) \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1, f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine] \<cdot> (tab\<^sub>0 g \<star> \<gamma>\<^sub>g) \<cdot> \<a>[tab\<^sub>0 g, f\<^sub>0g\<^sub>1.p\<^sub>0, fg\<^sub>0h\<^sub>1.p\<^sub>1] \<cdot> local.inv fg\<^sub>0h\<^sub>1.\<phi> = (h \<star> TTfgh_TfTgh.the_\<theta>) \<cdot> can (\<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> ((\<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) (((\<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) \<cdot> \<a>\<^sup>-\<^sup>1[h \<star> tab\<^sub>0 h, g\<^sub>0h\<^sub>1.p\<^sub>0, f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine] \<cdot> (h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0 \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine) \<cdot> \<a>[tab\<^sub>1 h, g\<^sub>0h\<^sub>1.p\<^sub>0, f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine] \<cdot> (g\<^sub>0h\<^sub>1.\<phi> \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine) \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1, f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine] \<cdot> (tab\<^sub>0 g \<star> \<gamma>\<^sub>g) \<cdot> \<a>[tab\<^sub>0 g, f\<^sub>0g\<^sub>1.p\<^sub>0, fg\<^sub>0h\<^sub>1.p\<^sub>1] \<cdot> local.inv fg\<^sub>0h\<^sub>1.\<phi>
[PROOF STEP]
by simp
[PROOF STATE]
proof (state)
this:
(h \<star> TTfgh_TfTgh.the_\<theta>) \<cdot> can (\<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> ((\<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) (((\<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) \<cdot> \<a>\<^sup>-\<^sup>1[h \<star> tab\<^sub>0 h, g\<^sub>0h\<^sub>1.p\<^sub>0, f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine] \<cdot> (\<a>[h \<star> tab\<^sub>0 h, g\<^sub>0h\<^sub>1.p\<^sub>0, f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine] \<cdot> ((h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine)) \<cdot> (g\<^sub>0h\<^sub>1.\<phi> \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine) \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1, f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine] \<cdot> (tab\<^sub>0 g \<star> \<gamma>\<^sub>g) \<cdot> \<a>[tab\<^sub>0 g, f\<^sub>0g\<^sub>1.p\<^sub>0, fg\<^sub>0h\<^sub>1.p\<^sub>1] \<cdot> local.inv fg\<^sub>0h\<^sub>1.\<phi> = (h \<star> TTfgh_TfTgh.the_\<theta>) \<cdot> can (\<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> ((\<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) (((\<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) \<cdot> \<a>\<^sup>-\<^sup>1[h \<star> tab\<^sub>0 h, g\<^sub>0h\<^sub>1.p\<^sub>0, f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine] \<cdot> (h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0 \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine) \<cdot> \<a>[tab\<^sub>1 h, g\<^sub>0h\<^sub>1.p\<^sub>0, f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine] \<cdot> (g\<^sub>0h\<^sub>1.\<phi> \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine) \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1, f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine] \<cdot> (tab\<^sub>0 g \<star> \<gamma>\<^sub>g) \<cdot> \<a>[tab\<^sub>0 g, f\<^sub>0g\<^sub>1.p\<^sub>0, fg\<^sub>0h\<^sub>1.p\<^sub>1] \<cdot> local.inv fg\<^sub>0h\<^sub>1.\<phi>
goal (1 subgoal):
1. (h \<star> \<theta>\<^sub>h) \<cdot> \<a>[h, tab\<^sub>0 h, w\<^sub>h] \<cdot> (h.tab \<star> w\<^sub>h) = ((h \<star> \<theta>\<^sub>h') \<cdot> \<a>[h, tab\<^sub>0 h, w\<^sub>h'] \<cdot> (h.tab \<star> w\<^sub>h')) \<cdot> \<beta>\<^sub>h
[PROOF STEP]
also
[PROOF STATE]
proof (state)
this:
(h \<star> TTfgh_TfTgh.the_\<theta>) \<cdot> can (\<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> ((\<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) (((\<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) \<cdot> \<a>\<^sup>-\<^sup>1[h \<star> tab\<^sub>0 h, g\<^sub>0h\<^sub>1.p\<^sub>0, f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine] \<cdot> (\<a>[h \<star> tab\<^sub>0 h, g\<^sub>0h\<^sub>1.p\<^sub>0, f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine] \<cdot> ((h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine)) \<cdot> (g\<^sub>0h\<^sub>1.\<phi> \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine) \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1, f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine] \<cdot> (tab\<^sub>0 g \<star> \<gamma>\<^sub>g) \<cdot> \<a>[tab\<^sub>0 g, f\<^sub>0g\<^sub>1.p\<^sub>0, fg\<^sub>0h\<^sub>1.p\<^sub>1] \<cdot> local.inv fg\<^sub>0h\<^sub>1.\<phi> = (h \<star> TTfgh_TfTgh.the_\<theta>) \<cdot> can (\<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> ((\<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) (((\<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) \<cdot> \<a>\<^sup>-\<^sup>1[h \<star> tab\<^sub>0 h, g\<^sub>0h\<^sub>1.p\<^sub>0, f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine] \<cdot> (h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0 \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine) \<cdot> \<a>[tab\<^sub>1 h, g\<^sub>0h\<^sub>1.p\<^sub>0, f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine] \<cdot> (g\<^sub>0h\<^sub>1.\<phi> \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine) \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1, f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine] \<cdot> (tab\<^sub>0 g \<star> \<gamma>\<^sub>g) \<cdot> \<a>[tab\<^sub>0 g, f\<^sub>0g\<^sub>1.p\<^sub>0, fg\<^sub>0h\<^sub>1.p\<^sub>1] \<cdot> local.inv fg\<^sub>0h\<^sub>1.\<phi>
goal (1 subgoal):
1. (h \<star> \<theta>\<^sub>h) \<cdot> \<a>[h, tab\<^sub>0 h, w\<^sub>h] \<cdot> (h.tab \<star> w\<^sub>h) = ((h \<star> \<theta>\<^sub>h') \<cdot> \<a>[h, tab\<^sub>0 h, w\<^sub>h'] \<cdot> (h.tab \<star> w\<^sub>h')) \<cdot> \<beta>\<^sub>h
[PROOF STEP]
have "... = (h \<star> TTfgh_TfTgh.the_\<theta>) \<cdot>
can (\<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> ((\<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>Tgh.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TfTgh.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>)
(((\<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>Tgh.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TfTgh.p\<^sub>0\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) \<cdot>
\<a>\<^sup>-\<^sup>1[h \<star> tab\<^sub>0 h, Tgh.p\<^sub>0, TfTgh.p\<^sub>0 \<star> TTfgh_TfTgh.chine] \<cdot>
((\<a>\<^sup>-\<^sup>1[h, tab\<^sub>0 h, Tgh.p\<^sub>0 \<star> TfTgh.p\<^sub>0 \<star> TTfgh_TfTgh.chine] \<cdot>
\<a>[h, tab\<^sub>0 h, Tgh.p\<^sub>0 \<star> TfTgh.p\<^sub>0 \<star> TTfgh_TfTgh.chine]) \<cdot>
(h.tab \<star> Tgh.p\<^sub>0 \<star> TfTgh.p\<^sub>0 \<star> TTfgh_TfTgh.chine)) \<cdot>
\<a>[tab\<^sub>1 h, Tgh.p\<^sub>0, TfTgh.p\<^sub>0 \<star> TTfgh_TfTgh.chine] \<cdot>
(g\<^sub>0h\<^sub>1.\<phi> \<star> TfTgh.p\<^sub>0 \<star> TTfgh_TfTgh.chine) \<cdot>
\<a>\<^sup>-\<^sup>1[tab\<^sub>0 g, Tgh.p\<^sub>1, TfTgh.p\<^sub>0 \<star> TTfgh_TfTgh.chine] \<cdot>
(tab\<^sub>0 g \<star> \<gamma>\<^sub>g) \<cdot>
\<a>[tab\<^sub>0 g, Tfg.p\<^sub>0, TTfgh.p\<^sub>1] \<cdot>
inv fg\<^sub>0h\<^sub>1.\<phi>"
[PROOF STATE]
proof (prove)
goal (1 subgoal):
1. (h \<star> TTfgh_TfTgh.the_\<theta>) \<cdot> can (\<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> ((\<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) (((\<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) \<cdot> \<a>\<^sup>-\<^sup>1[h \<star> tab\<^sub>0 h, g\<^sub>0h\<^sub>1.p\<^sub>0, f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine] \<cdot> (h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0 \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine) \<cdot> \<a>[tab\<^sub>1 h, g\<^sub>0h\<^sub>1.p\<^sub>0, f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine] \<cdot> (g\<^sub>0h\<^sub>1.\<phi> \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine) \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1, f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine] \<cdot> (tab\<^sub>0 g \<star> \<gamma>\<^sub>g) \<cdot> \<a>[tab\<^sub>0 g, f\<^sub>0g\<^sub>1.p\<^sub>0, fg\<^sub>0h\<^sub>1.p\<^sub>1] \<cdot> local.inv fg\<^sub>0h\<^sub>1.\<phi> = (h \<star> TTfgh_TfTgh.the_\<theta>) \<cdot> can (\<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> ((\<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) (((\<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) \<cdot> \<a>\<^sup>-\<^sup>1[h \<star> tab\<^sub>0 h, g\<^sub>0h\<^sub>1.p\<^sub>0, f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine] \<cdot> ((\<a>\<^sup>-\<^sup>1[h, tab\<^sub>0 h, g\<^sub>0h\<^sub>1.p\<^sub>0 \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine] \<cdot> \<a>[h, tab\<^sub>0 h, g\<^sub>0h\<^sub>1.p\<^sub>0 \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine]) \<cdot> (h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0 \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine)) \<cdot> \<a>[tab\<^sub>1 h, g\<^sub>0h\<^sub>1.p\<^sub>0, f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine] \<cdot> (g\<^sub>0h\<^sub>1.\<phi> \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine) \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1, f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine] \<cdot> (tab\<^sub>0 g \<star> \<gamma>\<^sub>g) \<cdot> \<a>[tab\<^sub>0 g, f\<^sub>0g\<^sub>1.p\<^sub>0, fg\<^sub>0h\<^sub>1.p\<^sub>1] \<cdot> local.inv fg\<^sub>0h\<^sub>1.\<phi>
[PROOF STEP]
proof -
[PROOF STATE]
proof (state)
goal (1 subgoal):
1. (h \<star> TTfgh_TfTgh.the_\<theta>) \<cdot> can (\<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> ((\<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) (((\<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) \<cdot> \<a>\<^sup>-\<^sup>1[h \<star> tab\<^sub>0 h, g\<^sub>0h\<^sub>1.p\<^sub>0, f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine] \<cdot> (h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0 \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine) \<cdot> \<a>[tab\<^sub>1 h, g\<^sub>0h\<^sub>1.p\<^sub>0, f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine] \<cdot> (g\<^sub>0h\<^sub>1.\<phi> \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine) \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1, f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine] \<cdot> (tab\<^sub>0 g \<star> \<gamma>\<^sub>g) \<cdot> \<a>[tab\<^sub>0 g, f\<^sub>0g\<^sub>1.p\<^sub>0, fg\<^sub>0h\<^sub>1.p\<^sub>1] \<cdot> local.inv fg\<^sub>0h\<^sub>1.\<phi> = (h \<star> TTfgh_TfTgh.the_\<theta>) \<cdot> can (\<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> ((\<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) (((\<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) \<cdot> \<a>\<^sup>-\<^sup>1[h \<star> tab\<^sub>0 h, g\<^sub>0h\<^sub>1.p\<^sub>0, f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine] \<cdot> ((\<a>\<^sup>-\<^sup>1[h, tab\<^sub>0 h, g\<^sub>0h\<^sub>1.p\<^sub>0 \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine] \<cdot> \<a>[h, tab\<^sub>0 h, g\<^sub>0h\<^sub>1.p\<^sub>0 \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine]) \<cdot> (h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0 \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine)) \<cdot> \<a>[tab\<^sub>1 h, g\<^sub>0h\<^sub>1.p\<^sub>0, f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine] \<cdot> (g\<^sub>0h\<^sub>1.\<phi> \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine) \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1, f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine] \<cdot> (tab\<^sub>0 g \<star> \<gamma>\<^sub>g) \<cdot> \<a>[tab\<^sub>0 g, f\<^sub>0g\<^sub>1.p\<^sub>0, fg\<^sub>0h\<^sub>1.p\<^sub>1] \<cdot> local.inv fg\<^sub>0h\<^sub>1.\<phi>
[PROOF STEP]
have "(\<a>\<^sup>-\<^sup>1[h, tab\<^sub>0 h, Tgh.p\<^sub>0 \<star> TfTgh.p\<^sub>0 \<star> TTfgh_TfTgh.chine] \<cdot>
\<a>[h, tab\<^sub>0 h, Tgh.p\<^sub>0 \<star> TfTgh.p\<^sub>0 \<star> TTfgh_TfTgh.chine]) \<cdot>
(h.tab \<star> Tgh.p\<^sub>0 \<star> TfTgh.p\<^sub>0 \<star> TTfgh_TfTgh.chine) =
h.tab \<star> Tgh.p\<^sub>0 \<star> TfTgh.p\<^sub>0 \<star> TTfgh_TfTgh.chine"
[PROOF STATE]
proof (prove)
goal (1 subgoal):
1. (\<a>\<^sup>-\<^sup>1[h, tab\<^sub>0 h, g\<^sub>0h\<^sub>1.p\<^sub>0 \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine] \<cdot> \<a>[h, tab\<^sub>0 h, g\<^sub>0h\<^sub>1.p\<^sub>0 \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine]) \<cdot> (h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0 \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine) = h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0 \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine
[PROOF STEP]
using comp_cod_arr comp_assoc_assoc'
[PROOF STATE]
proof (prove)
using this:
\<lbrakk>arr ?f; cod ?f = ?b\<rbrakk> \<Longrightarrow> ?b \<cdot> ?f = ?f
\<lbrakk>ide ?f; ide ?g; ide ?h; src ?f = trg ?g; src ?g = trg ?h\<rbrakk> \<Longrightarrow> \<a>[?f, ?g, ?h] \<cdot> \<a>\<^sup>-\<^sup>1[?f, ?g, ?h] = ?f \<star> ?g \<star> ?h
\<lbrakk>ide ?f; ide ?g; ide ?h; src ?f = trg ?g; src ?g = trg ?h\<rbrakk> \<Longrightarrow> \<a>\<^sup>-\<^sup>1[?f, ?g, ?h] \<cdot> \<a>[?f, ?g, ?h] = (?f \<star> ?g) \<star> ?h
goal (1 subgoal):
1. (\<a>\<^sup>-\<^sup>1[h, tab\<^sub>0 h, g\<^sub>0h\<^sub>1.p\<^sub>0 \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine] \<cdot> \<a>[h, tab\<^sub>0 h, g\<^sub>0h\<^sub>1.p\<^sub>0 \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine]) \<cdot> (h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0 \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine) = h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0 \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine
[PROOF STEP]
by simp
[PROOF STATE]
proof (state)
this:
(\<a>\<^sup>-\<^sup>1[h, tab\<^sub>0 h, g\<^sub>0h\<^sub>1.p\<^sub>0 \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine] \<cdot> \<a>[h, tab\<^sub>0 h, g\<^sub>0h\<^sub>1.p\<^sub>0 \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine]) \<cdot> (h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0 \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine) = h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0 \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine
goal (1 subgoal):
1. (h \<star> TTfgh_TfTgh.the_\<theta>) \<cdot> can (\<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> ((\<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) (((\<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) \<cdot> \<a>\<^sup>-\<^sup>1[h \<star> tab\<^sub>0 h, g\<^sub>0h\<^sub>1.p\<^sub>0, f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine] \<cdot> (h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0 \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine) \<cdot> \<a>[tab\<^sub>1 h, g\<^sub>0h\<^sub>1.p\<^sub>0, f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine] \<cdot> (g\<^sub>0h\<^sub>1.\<phi> \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine) \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1, f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine] \<cdot> (tab\<^sub>0 g \<star> \<gamma>\<^sub>g) \<cdot> \<a>[tab\<^sub>0 g, f\<^sub>0g\<^sub>1.p\<^sub>0, fg\<^sub>0h\<^sub>1.p\<^sub>1] \<cdot> local.inv fg\<^sub>0h\<^sub>1.\<phi> = (h \<star> TTfgh_TfTgh.the_\<theta>) \<cdot> can (\<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> ((\<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) (((\<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) \<cdot> \<a>\<^sup>-\<^sup>1[h \<star> tab\<^sub>0 h, g\<^sub>0h\<^sub>1.p\<^sub>0, f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine] \<cdot> ((\<a>\<^sup>-\<^sup>1[h, tab\<^sub>0 h, g\<^sub>0h\<^sub>1.p\<^sub>0 \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine] \<cdot> \<a>[h, tab\<^sub>0 h, g\<^sub>0h\<^sub>1.p\<^sub>0 \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine]) \<cdot> (h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0 \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine)) \<cdot> \<a>[tab\<^sub>1 h, g\<^sub>0h\<^sub>1.p\<^sub>0, f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine] \<cdot> (g\<^sub>0h\<^sub>1.\<phi> \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine) \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1, f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine] \<cdot> (tab\<^sub>0 g \<star> \<gamma>\<^sub>g) \<cdot> \<a>[tab\<^sub>0 g, f\<^sub>0g\<^sub>1.p\<^sub>0, fg\<^sub>0h\<^sub>1.p\<^sub>1] \<cdot> local.inv fg\<^sub>0h\<^sub>1.\<phi>
[PROOF STEP]
thus ?thesis
[PROOF STATE]
proof (prove)
using this:
(\<a>\<^sup>-\<^sup>1[h, tab\<^sub>0 h, g\<^sub>0h\<^sub>1.p\<^sub>0 \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine] \<cdot> \<a>[h, tab\<^sub>0 h, g\<^sub>0h\<^sub>1.p\<^sub>0 \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine]) \<cdot> (h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0 \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine) = h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0 \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine
goal (1 subgoal):
1. (h \<star> TTfgh_TfTgh.the_\<theta>) \<cdot> can (\<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> ((\<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) (((\<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) \<cdot> \<a>\<^sup>-\<^sup>1[h \<star> tab\<^sub>0 h, g\<^sub>0h\<^sub>1.p\<^sub>0, f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine] \<cdot> (h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0 \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine) \<cdot> \<a>[tab\<^sub>1 h, g\<^sub>0h\<^sub>1.p\<^sub>0, f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine] \<cdot> (g\<^sub>0h\<^sub>1.\<phi> \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine) \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1, f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine] \<cdot> (tab\<^sub>0 g \<star> \<gamma>\<^sub>g) \<cdot> \<a>[tab\<^sub>0 g, f\<^sub>0g\<^sub>1.p\<^sub>0, fg\<^sub>0h\<^sub>1.p\<^sub>1] \<cdot> local.inv fg\<^sub>0h\<^sub>1.\<phi> = (h \<star> TTfgh_TfTgh.the_\<theta>) \<cdot> can (\<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> ((\<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) (((\<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) \<cdot> \<a>\<^sup>-\<^sup>1[h \<star> tab\<^sub>0 h, g\<^sub>0h\<^sub>1.p\<^sub>0, f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine] \<cdot> ((\<a>\<^sup>-\<^sup>1[h, tab\<^sub>0 h, g\<^sub>0h\<^sub>1.p\<^sub>0 \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine] \<cdot> \<a>[h, tab\<^sub>0 h, g\<^sub>0h\<^sub>1.p\<^sub>0 \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine]) \<cdot> (h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0 \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine)) \<cdot> \<a>[tab\<^sub>1 h, g\<^sub>0h\<^sub>1.p\<^sub>0, f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine] \<cdot> (g\<^sub>0h\<^sub>1.\<phi> \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine) \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1, f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine] \<cdot> (tab\<^sub>0 g \<star> \<gamma>\<^sub>g) \<cdot> \<a>[tab\<^sub>0 g, f\<^sub>0g\<^sub>1.p\<^sub>0, fg\<^sub>0h\<^sub>1.p\<^sub>1] \<cdot> local.inv fg\<^sub>0h\<^sub>1.\<phi>
[PROOF STEP]
using comp_assoc
[PROOF STATE]
proof (prove)
using this:
(\<a>\<^sup>-\<^sup>1[h, tab\<^sub>0 h, g\<^sub>0h\<^sub>1.p\<^sub>0 \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine] \<cdot> \<a>[h, tab\<^sub>0 h, g\<^sub>0h\<^sub>1.p\<^sub>0 \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine]) \<cdot> (h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0 \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine) = h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0 \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine
(?h \<cdot> ?g) \<cdot> ?f = ?h \<cdot> ?g \<cdot> ?f
goal (1 subgoal):
1. (h \<star> TTfgh_TfTgh.the_\<theta>) \<cdot> can (\<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> ((\<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) (((\<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) \<cdot> \<a>\<^sup>-\<^sup>1[h \<star> tab\<^sub>0 h, g\<^sub>0h\<^sub>1.p\<^sub>0, f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine] \<cdot> (h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0 \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine) \<cdot> \<a>[tab\<^sub>1 h, g\<^sub>0h\<^sub>1.p\<^sub>0, f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine] \<cdot> (g\<^sub>0h\<^sub>1.\<phi> \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine) \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1, f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine] \<cdot> (tab\<^sub>0 g \<star> \<gamma>\<^sub>g) \<cdot> \<a>[tab\<^sub>0 g, f\<^sub>0g\<^sub>1.p\<^sub>0, fg\<^sub>0h\<^sub>1.p\<^sub>1] \<cdot> local.inv fg\<^sub>0h\<^sub>1.\<phi> = (h \<star> TTfgh_TfTgh.the_\<theta>) \<cdot> can (\<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> ((\<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) (((\<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) \<cdot> \<a>\<^sup>-\<^sup>1[h \<star> tab\<^sub>0 h, g\<^sub>0h\<^sub>1.p\<^sub>0, f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine] \<cdot> ((\<a>\<^sup>-\<^sup>1[h, tab\<^sub>0 h, g\<^sub>0h\<^sub>1.p\<^sub>0 \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine] \<cdot> \<a>[h, tab\<^sub>0 h, g\<^sub>0h\<^sub>1.p\<^sub>0 \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine]) \<cdot> (h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0 \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine)) \<cdot> \<a>[tab\<^sub>1 h, g\<^sub>0h\<^sub>1.p\<^sub>0, f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine] \<cdot> (g\<^sub>0h\<^sub>1.\<phi> \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine) \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1, f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine] \<cdot> (tab\<^sub>0 g \<star> \<gamma>\<^sub>g) \<cdot> \<a>[tab\<^sub>0 g, f\<^sub>0g\<^sub>1.p\<^sub>0, fg\<^sub>0h\<^sub>1.p\<^sub>1] \<cdot> local.inv fg\<^sub>0h\<^sub>1.\<phi>
[PROOF STEP]
by simp
[PROOF STATE]
proof (state)
this:
(h \<star> TTfgh_TfTgh.the_\<theta>) \<cdot> can (\<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> ((\<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) (((\<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) \<cdot> \<a>\<^sup>-\<^sup>1[h \<star> tab\<^sub>0 h, g\<^sub>0h\<^sub>1.p\<^sub>0, f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine] \<cdot> (h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0 \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine) \<cdot> \<a>[tab\<^sub>1 h, g\<^sub>0h\<^sub>1.p\<^sub>0, f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine] \<cdot> (g\<^sub>0h\<^sub>1.\<phi> \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine) \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1, f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine] \<cdot> (tab\<^sub>0 g \<star> \<gamma>\<^sub>g) \<cdot> \<a>[tab\<^sub>0 g, f\<^sub>0g\<^sub>1.p\<^sub>0, fg\<^sub>0h\<^sub>1.p\<^sub>1] \<cdot> local.inv fg\<^sub>0h\<^sub>1.\<phi> = (h \<star> TTfgh_TfTgh.the_\<theta>) \<cdot> can (\<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> ((\<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) (((\<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) \<cdot> \<a>\<^sup>-\<^sup>1[h \<star> tab\<^sub>0 h, g\<^sub>0h\<^sub>1.p\<^sub>0, f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine] \<cdot> ((\<a>\<^sup>-\<^sup>1[h, tab\<^sub>0 h, g\<^sub>0h\<^sub>1.p\<^sub>0 \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine] \<cdot> \<a>[h, tab\<^sub>0 h, g\<^sub>0h\<^sub>1.p\<^sub>0 \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine]) \<cdot> (h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0 \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine)) \<cdot> \<a>[tab\<^sub>1 h, g\<^sub>0h\<^sub>1.p\<^sub>0, f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine] \<cdot> (g\<^sub>0h\<^sub>1.\<phi> \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine) \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1, f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine] \<cdot> (tab\<^sub>0 g \<star> \<gamma>\<^sub>g) \<cdot> \<a>[tab\<^sub>0 g, f\<^sub>0g\<^sub>1.p\<^sub>0, fg\<^sub>0h\<^sub>1.p\<^sub>1] \<cdot> local.inv fg\<^sub>0h\<^sub>1.\<phi>
goal:
No subgoals!
[PROOF STEP]
qed
[PROOF STATE]
proof (state)
this:
(h \<star> TTfgh_TfTgh.the_\<theta>) \<cdot> can (\<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> ((\<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) (((\<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) \<cdot> \<a>\<^sup>-\<^sup>1[h \<star> tab\<^sub>0 h, g\<^sub>0h\<^sub>1.p\<^sub>0, f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine] \<cdot> (h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0 \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine) \<cdot> \<a>[tab\<^sub>1 h, g\<^sub>0h\<^sub>1.p\<^sub>0, f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine] \<cdot> (g\<^sub>0h\<^sub>1.\<phi> \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine) \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1, f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine] \<cdot> (tab\<^sub>0 g \<star> \<gamma>\<^sub>g) \<cdot> \<a>[tab\<^sub>0 g, f\<^sub>0g\<^sub>1.p\<^sub>0, fg\<^sub>0h\<^sub>1.p\<^sub>1] \<cdot> local.inv fg\<^sub>0h\<^sub>1.\<phi> = (h \<star> TTfgh_TfTgh.the_\<theta>) \<cdot> can (\<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> ((\<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) (((\<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) \<cdot> \<a>\<^sup>-\<^sup>1[h \<star> tab\<^sub>0 h, g\<^sub>0h\<^sub>1.p\<^sub>0, f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine] \<cdot> ((\<a>\<^sup>-\<^sup>1[h, tab\<^sub>0 h, g\<^sub>0h\<^sub>1.p\<^sub>0 \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine] \<cdot> \<a>[h, tab\<^sub>0 h, g\<^sub>0h\<^sub>1.p\<^sub>0 \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine]) \<cdot> (h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0 \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine)) \<cdot> \<a>[tab\<^sub>1 h, g\<^sub>0h\<^sub>1.p\<^sub>0, f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine] \<cdot> (g\<^sub>0h\<^sub>1.\<phi> \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine) \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1, f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine] \<cdot> (tab\<^sub>0 g \<star> \<gamma>\<^sub>g) \<cdot> \<a>[tab\<^sub>0 g, f\<^sub>0g\<^sub>1.p\<^sub>0, fg\<^sub>0h\<^sub>1.p\<^sub>1] \<cdot> local.inv fg\<^sub>0h\<^sub>1.\<phi>
goal (1 subgoal):
1. (h \<star> \<theta>\<^sub>h) \<cdot> \<a>[h, tab\<^sub>0 h, w\<^sub>h] \<cdot> (h.tab \<star> w\<^sub>h) = ((h \<star> \<theta>\<^sub>h') \<cdot> \<a>[h, tab\<^sub>0 h, w\<^sub>h'] \<cdot> (h.tab \<star> w\<^sub>h')) \<cdot> \<beta>\<^sub>h
[PROOF STEP]
also
[PROOF STATE]
proof (state)
this:
(h \<star> TTfgh_TfTgh.the_\<theta>) \<cdot> can (\<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> ((\<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) (((\<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) \<cdot> \<a>\<^sup>-\<^sup>1[h \<star> tab\<^sub>0 h, g\<^sub>0h\<^sub>1.p\<^sub>0, f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine] \<cdot> (h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0 \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine) \<cdot> \<a>[tab\<^sub>1 h, g\<^sub>0h\<^sub>1.p\<^sub>0, f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine] \<cdot> (g\<^sub>0h\<^sub>1.\<phi> \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine) \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1, f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine] \<cdot> (tab\<^sub>0 g \<star> \<gamma>\<^sub>g) \<cdot> \<a>[tab\<^sub>0 g, f\<^sub>0g\<^sub>1.p\<^sub>0, fg\<^sub>0h\<^sub>1.p\<^sub>1] \<cdot> local.inv fg\<^sub>0h\<^sub>1.\<phi> = (h \<star> TTfgh_TfTgh.the_\<theta>) \<cdot> can (\<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> ((\<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) (((\<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) \<cdot> \<a>\<^sup>-\<^sup>1[h \<star> tab\<^sub>0 h, g\<^sub>0h\<^sub>1.p\<^sub>0, f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine] \<cdot> ((\<a>\<^sup>-\<^sup>1[h, tab\<^sub>0 h, g\<^sub>0h\<^sub>1.p\<^sub>0 \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine] \<cdot> \<a>[h, tab\<^sub>0 h, g\<^sub>0h\<^sub>1.p\<^sub>0 \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine]) \<cdot> (h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0 \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine)) \<cdot> \<a>[tab\<^sub>1 h, g\<^sub>0h\<^sub>1.p\<^sub>0, f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine] \<cdot> (g\<^sub>0h\<^sub>1.\<phi> \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine) \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1, f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine] \<cdot> (tab\<^sub>0 g \<star> \<gamma>\<^sub>g) \<cdot> \<a>[tab\<^sub>0 g, f\<^sub>0g\<^sub>1.p\<^sub>0, fg\<^sub>0h\<^sub>1.p\<^sub>1] \<cdot> local.inv fg\<^sub>0h\<^sub>1.\<phi>
goal (1 subgoal):
1. (h \<star> \<theta>\<^sub>h) \<cdot> \<a>[h, tab\<^sub>0 h, w\<^sub>h] \<cdot> (h.tab \<star> w\<^sub>h) = ((h \<star> \<theta>\<^sub>h') \<cdot> \<a>[h, tab\<^sub>0 h, w\<^sub>h'] \<cdot> (h.tab \<star> w\<^sub>h')) \<cdot> \<beta>\<^sub>h
[PROOF STEP]
have "... = (h \<star> TTfgh_TfTgh.the_\<theta>) \<cdot>
(can (\<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> ((\<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>Tgh.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TfTgh.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>)
(((\<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>Tgh.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TfTgh.p\<^sub>0\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) \<cdot>
(\<a>\<^sup>-\<^sup>1[h \<star> tab\<^sub>0 h, Tgh.p\<^sub>0, TfTgh.p\<^sub>0 \<star> TTfgh_TfTgh.chine] \<cdot>
\<a>\<^sup>-\<^sup>1[h, tab\<^sub>0 h, Tgh.p\<^sub>0 \<star> TfTgh.p\<^sub>0 \<star> TTfgh_TfTgh.chine])) \<cdot>
\<a>[h, tab\<^sub>0 h, Tgh.p\<^sub>0 \<star> TfTgh.p\<^sub>0 \<star> TTfgh_TfTgh.chine] \<cdot>
(h.tab \<star> Tgh.p\<^sub>0 \<star> TfTgh.p\<^sub>0 \<star> TTfgh_TfTgh.chine) \<cdot>
\<a>[tab\<^sub>1 h, Tgh.p\<^sub>0, TfTgh.p\<^sub>0 \<star> TTfgh_TfTgh.chine] \<cdot>
(g\<^sub>0h\<^sub>1.\<phi> \<star> TfTgh.p\<^sub>0 \<star> TTfgh_TfTgh.chine) \<cdot>
\<a>\<^sup>-\<^sup>1[tab\<^sub>0 g, Tgh.p\<^sub>1, TfTgh.p\<^sub>0 \<star> TTfgh_TfTgh.chine] \<cdot>
(tab\<^sub>0 g \<star> \<gamma>\<^sub>g) \<cdot>
\<a>[tab\<^sub>0 g, Tfg.p\<^sub>0, TTfgh.p\<^sub>1] \<cdot>
inv fg\<^sub>0h\<^sub>1.\<phi>"
[PROOF STATE]
proof (prove)
goal (1 subgoal):
1. (h \<star> TTfgh_TfTgh.the_\<theta>) \<cdot> can (\<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> ((\<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) (((\<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) \<cdot> \<a>\<^sup>-\<^sup>1[h \<star> tab\<^sub>0 h, g\<^sub>0h\<^sub>1.p\<^sub>0, f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine] \<cdot> ((\<a>\<^sup>-\<^sup>1[h, tab\<^sub>0 h, g\<^sub>0h\<^sub>1.p\<^sub>0 \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine] \<cdot> \<a>[h, tab\<^sub>0 h, g\<^sub>0h\<^sub>1.p\<^sub>0 \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine]) \<cdot> (h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0 \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine)) \<cdot> \<a>[tab\<^sub>1 h, g\<^sub>0h\<^sub>1.p\<^sub>0, f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine] \<cdot> (g\<^sub>0h\<^sub>1.\<phi> \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine) \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1, f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine] \<cdot> (tab\<^sub>0 g \<star> \<gamma>\<^sub>g) \<cdot> \<a>[tab\<^sub>0 g, f\<^sub>0g\<^sub>1.p\<^sub>0, fg\<^sub>0h\<^sub>1.p\<^sub>1] \<cdot> local.inv fg\<^sub>0h\<^sub>1.\<phi> = (h \<star> TTfgh_TfTgh.the_\<theta>) \<cdot> (can (\<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> ((\<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) (((\<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) \<cdot> \<a>\<^sup>-\<^sup>1[h \<star> tab\<^sub>0 h, g\<^sub>0h\<^sub>1.p\<^sub>0, f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine] \<cdot> \<a>\<^sup>-\<^sup>1[h, tab\<^sub>0 h, g\<^sub>0h\<^sub>1.p\<^sub>0 \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine]) \<cdot> \<a>[h, tab\<^sub>0 h, g\<^sub>0h\<^sub>1.p\<^sub>0 \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine] \<cdot> (h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0 \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine) \<cdot> \<a>[tab\<^sub>1 h, g\<^sub>0h\<^sub>1.p\<^sub>0, f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine] \<cdot> (g\<^sub>0h\<^sub>1.\<phi> \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine) \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1, f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine] \<cdot> (tab\<^sub>0 g \<star> \<gamma>\<^sub>g) \<cdot> \<a>[tab\<^sub>0 g, f\<^sub>0g\<^sub>1.p\<^sub>0, fg\<^sub>0h\<^sub>1.p\<^sub>1] \<cdot> local.inv fg\<^sub>0h\<^sub>1.\<phi>
[PROOF STEP]
using comp_assoc
[PROOF STATE]
proof (prove)
using this:
(?h \<cdot> ?g) \<cdot> ?f = ?h \<cdot> ?g \<cdot> ?f
goal (1 subgoal):
1. (h \<star> TTfgh_TfTgh.the_\<theta>) \<cdot> can (\<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> ((\<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) (((\<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) \<cdot> \<a>\<^sup>-\<^sup>1[h \<star> tab\<^sub>0 h, g\<^sub>0h\<^sub>1.p\<^sub>0, f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine] \<cdot> ((\<a>\<^sup>-\<^sup>1[h, tab\<^sub>0 h, g\<^sub>0h\<^sub>1.p\<^sub>0 \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine] \<cdot> \<a>[h, tab\<^sub>0 h, g\<^sub>0h\<^sub>1.p\<^sub>0 \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine]) \<cdot> (h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0 \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine)) \<cdot> \<a>[tab\<^sub>1 h, g\<^sub>0h\<^sub>1.p\<^sub>0, f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine] \<cdot> (g\<^sub>0h\<^sub>1.\<phi> \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine) \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1, f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine] \<cdot> (tab\<^sub>0 g \<star> \<gamma>\<^sub>g) \<cdot> \<a>[tab\<^sub>0 g, f\<^sub>0g\<^sub>1.p\<^sub>0, fg\<^sub>0h\<^sub>1.p\<^sub>1] \<cdot> local.inv fg\<^sub>0h\<^sub>1.\<phi> = (h \<star> TTfgh_TfTgh.the_\<theta>) \<cdot> (can (\<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> ((\<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) (((\<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) \<cdot> \<a>\<^sup>-\<^sup>1[h \<star> tab\<^sub>0 h, g\<^sub>0h\<^sub>1.p\<^sub>0, f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine] \<cdot> \<a>\<^sup>-\<^sup>1[h, tab\<^sub>0 h, g\<^sub>0h\<^sub>1.p\<^sub>0 \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine]) \<cdot> \<a>[h, tab\<^sub>0 h, g\<^sub>0h\<^sub>1.p\<^sub>0 \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine] \<cdot> (h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0 \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine) \<cdot> \<a>[tab\<^sub>1 h, g\<^sub>0h\<^sub>1.p\<^sub>0, f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine] \<cdot> (g\<^sub>0h\<^sub>1.\<phi> \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine) \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1, f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine] \<cdot> (tab\<^sub>0 g \<star> \<gamma>\<^sub>g) \<cdot> \<a>[tab\<^sub>0 g, f\<^sub>0g\<^sub>1.p\<^sub>0, fg\<^sub>0h\<^sub>1.p\<^sub>1] \<cdot> local.inv fg\<^sub>0h\<^sub>1.\<phi>
[PROOF STEP]
by presburger
[PROOF STATE]
proof (state)
this:
(h \<star> TTfgh_TfTgh.the_\<theta>) \<cdot> can (\<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> ((\<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) (((\<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) \<cdot> \<a>\<^sup>-\<^sup>1[h \<star> tab\<^sub>0 h, g\<^sub>0h\<^sub>1.p\<^sub>0, f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine] \<cdot> ((\<a>\<^sup>-\<^sup>1[h, tab\<^sub>0 h, g\<^sub>0h\<^sub>1.p\<^sub>0 \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine] \<cdot> \<a>[h, tab\<^sub>0 h, g\<^sub>0h\<^sub>1.p\<^sub>0 \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine]) \<cdot> (h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0 \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine)) \<cdot> \<a>[tab\<^sub>1 h, g\<^sub>0h\<^sub>1.p\<^sub>0, f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine] \<cdot> (g\<^sub>0h\<^sub>1.\<phi> \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine) \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1, f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine] \<cdot> (tab\<^sub>0 g \<star> \<gamma>\<^sub>g) \<cdot> \<a>[tab\<^sub>0 g, f\<^sub>0g\<^sub>1.p\<^sub>0, fg\<^sub>0h\<^sub>1.p\<^sub>1] \<cdot> local.inv fg\<^sub>0h\<^sub>1.\<phi> = (h \<star> TTfgh_TfTgh.the_\<theta>) \<cdot> (can (\<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> ((\<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) (((\<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) \<cdot> \<a>\<^sup>-\<^sup>1[h \<star> tab\<^sub>0 h, g\<^sub>0h\<^sub>1.p\<^sub>0, f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine] \<cdot> \<a>\<^sup>-\<^sup>1[h, tab\<^sub>0 h, g\<^sub>0h\<^sub>1.p\<^sub>0 \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine]) \<cdot> \<a>[h, tab\<^sub>0 h, g\<^sub>0h\<^sub>1.p\<^sub>0 \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine] \<cdot> (h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0 \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine) \<cdot> \<a>[tab\<^sub>1 h, g\<^sub>0h\<^sub>1.p\<^sub>0, f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine] \<cdot> (g\<^sub>0h\<^sub>1.\<phi> \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine) \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1, f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine] \<cdot> (tab\<^sub>0 g \<star> \<gamma>\<^sub>g) \<cdot> \<a>[tab\<^sub>0 g, f\<^sub>0g\<^sub>1.p\<^sub>0, fg\<^sub>0h\<^sub>1.p\<^sub>1] \<cdot> local.inv fg\<^sub>0h\<^sub>1.\<phi>
goal (1 subgoal):
1. (h \<star> \<theta>\<^sub>h) \<cdot> \<a>[h, tab\<^sub>0 h, w\<^sub>h] \<cdot> (h.tab \<star> w\<^sub>h) = ((h \<star> \<theta>\<^sub>h') \<cdot> \<a>[h, tab\<^sub>0 h, w\<^sub>h'] \<cdot> (h.tab \<star> w\<^sub>h')) \<cdot> \<beta>\<^sub>h
[PROOF STEP]
also
[PROOF STATE]
proof (state)
this:
(h \<star> TTfgh_TfTgh.the_\<theta>) \<cdot> can (\<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> ((\<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) (((\<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) \<cdot> \<a>\<^sup>-\<^sup>1[h \<star> tab\<^sub>0 h, g\<^sub>0h\<^sub>1.p\<^sub>0, f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine] \<cdot> ((\<a>\<^sup>-\<^sup>1[h, tab\<^sub>0 h, g\<^sub>0h\<^sub>1.p\<^sub>0 \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine] \<cdot> \<a>[h, tab\<^sub>0 h, g\<^sub>0h\<^sub>1.p\<^sub>0 \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine]) \<cdot> (h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0 \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine)) \<cdot> \<a>[tab\<^sub>1 h, g\<^sub>0h\<^sub>1.p\<^sub>0, f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine] \<cdot> (g\<^sub>0h\<^sub>1.\<phi> \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine) \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1, f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine] \<cdot> (tab\<^sub>0 g \<star> \<gamma>\<^sub>g) \<cdot> \<a>[tab\<^sub>0 g, f\<^sub>0g\<^sub>1.p\<^sub>0, fg\<^sub>0h\<^sub>1.p\<^sub>1] \<cdot> local.inv fg\<^sub>0h\<^sub>1.\<phi> = (h \<star> TTfgh_TfTgh.the_\<theta>) \<cdot> (can (\<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> ((\<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) (((\<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) \<cdot> \<a>\<^sup>-\<^sup>1[h \<star> tab\<^sub>0 h, g\<^sub>0h\<^sub>1.p\<^sub>0, f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine] \<cdot> \<a>\<^sup>-\<^sup>1[h, tab\<^sub>0 h, g\<^sub>0h\<^sub>1.p\<^sub>0 \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine]) \<cdot> \<a>[h, tab\<^sub>0 h, g\<^sub>0h\<^sub>1.p\<^sub>0 \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine] \<cdot> (h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0 \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine) \<cdot> \<a>[tab\<^sub>1 h, g\<^sub>0h\<^sub>1.p\<^sub>0, f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine] \<cdot> (g\<^sub>0h\<^sub>1.\<phi> \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine) \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1, f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine] \<cdot> (tab\<^sub>0 g \<star> \<gamma>\<^sub>g) \<cdot> \<a>[tab\<^sub>0 g, f\<^sub>0g\<^sub>1.p\<^sub>0, fg\<^sub>0h\<^sub>1.p\<^sub>1] \<cdot> local.inv fg\<^sub>0h\<^sub>1.\<phi>
goal (1 subgoal):
1. (h \<star> \<theta>\<^sub>h) \<cdot> \<a>[h, tab\<^sub>0 h, w\<^sub>h] \<cdot> (h.tab \<star> w\<^sub>h) = ((h \<star> \<theta>\<^sub>h') \<cdot> \<a>[h, tab\<^sub>0 h, w\<^sub>h'] \<cdot> (h.tab \<star> w\<^sub>h')) \<cdot> \<beta>\<^sub>h
[PROOF STEP]
have "... = ((h \<star> TTfgh_TfTgh.the_\<theta>) \<cdot>
(h \<star> can (((\<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>Tgh.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TfTgh.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>)
(\<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>Tgh.p\<^sub>0\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>TfTgh.p\<^sub>0\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>))) \<cdot>
\<a>[h, tab\<^sub>0 h, Tgh.p\<^sub>0 \<star> TfTgh.p\<^sub>0 \<star> TTfgh_TfTgh.chine] \<cdot>
(h.tab \<star> Tgh.p\<^sub>0 \<star> TfTgh.p\<^sub>0 \<star> TTfgh_TfTgh.chine) \<cdot>
\<a>[tab\<^sub>1 h, Tgh.p\<^sub>0, TfTgh.p\<^sub>0 \<star> TTfgh_TfTgh.chine] \<cdot>
(g\<^sub>0h\<^sub>1.\<phi> \<star> TfTgh.p\<^sub>0 \<star> TTfgh_TfTgh.chine) \<cdot>
\<a>\<^sup>-\<^sup>1[tab\<^sub>0 g, Tgh.p\<^sub>1, TfTgh.p\<^sub>0 \<star> TTfgh_TfTgh.chine] \<cdot>
(tab\<^sub>0 g \<star> \<gamma>\<^sub>g) \<cdot>
\<a>[tab\<^sub>0 g, Tfg.p\<^sub>0, TTfgh.p\<^sub>1] \<cdot>
inv fg\<^sub>0h\<^sub>1.\<phi>"
[PROOF STATE]
proof (prove)
goal (1 subgoal):
1. (h \<star> TTfgh_TfTgh.the_\<theta>) \<cdot> (can (\<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> ((\<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) (((\<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) \<cdot> \<a>\<^sup>-\<^sup>1[h \<star> tab\<^sub>0 h, g\<^sub>0h\<^sub>1.p\<^sub>0, f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine] \<cdot> \<a>\<^sup>-\<^sup>1[h, tab\<^sub>0 h, g\<^sub>0h\<^sub>1.p\<^sub>0 \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine]) \<cdot> \<a>[h, tab\<^sub>0 h, g\<^sub>0h\<^sub>1.p\<^sub>0 \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine] \<cdot> (h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0 \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine) \<cdot> \<a>[tab\<^sub>1 h, g\<^sub>0h\<^sub>1.p\<^sub>0, f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine] \<cdot> (g\<^sub>0h\<^sub>1.\<phi> \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine) \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1, f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine] \<cdot> (tab\<^sub>0 g \<star> \<gamma>\<^sub>g) \<cdot> \<a>[tab\<^sub>0 g, f\<^sub>0g\<^sub>1.p\<^sub>0, fg\<^sub>0h\<^sub>1.p\<^sub>1] \<cdot> local.inv fg\<^sub>0h\<^sub>1.\<phi> = ((h \<star> TTfgh_TfTgh.the_\<theta>) \<cdot> (h \<star> can (((\<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) (\<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>))) \<cdot> \<a>[h, tab\<^sub>0 h, g\<^sub>0h\<^sub>1.p\<^sub>0 \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine] \<cdot> (h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0 \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine) \<cdot> \<a>[tab\<^sub>1 h, g\<^sub>0h\<^sub>1.p\<^sub>0, f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine] \<cdot> (g\<^sub>0h\<^sub>1.\<phi> \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine) \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1, f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine] \<cdot> (tab\<^sub>0 g \<star> \<gamma>\<^sub>g) \<cdot> \<a>[tab\<^sub>0 g, f\<^sub>0g\<^sub>1.p\<^sub>0, fg\<^sub>0h\<^sub>1.p\<^sub>1] \<cdot> local.inv fg\<^sub>0h\<^sub>1.\<phi>
[PROOF STEP]
proof -
[PROOF STATE]
proof (state)
goal (1 subgoal):
1. (h \<star> TTfgh_TfTgh.the_\<theta>) \<cdot> (can (\<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> ((\<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) (((\<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) \<cdot> \<a>\<^sup>-\<^sup>1[h \<star> tab\<^sub>0 h, g\<^sub>0h\<^sub>1.p\<^sub>0, f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine] \<cdot> \<a>\<^sup>-\<^sup>1[h, tab\<^sub>0 h, g\<^sub>0h\<^sub>1.p\<^sub>0 \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine]) \<cdot> \<a>[h, tab\<^sub>0 h, g\<^sub>0h\<^sub>1.p\<^sub>0 \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine] \<cdot> (h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0 \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine) \<cdot> \<a>[tab\<^sub>1 h, g\<^sub>0h\<^sub>1.p\<^sub>0, f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine] \<cdot> (g\<^sub>0h\<^sub>1.\<phi> \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine) \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1, f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine] \<cdot> (tab\<^sub>0 g \<star> \<gamma>\<^sub>g) \<cdot> \<a>[tab\<^sub>0 g, f\<^sub>0g\<^sub>1.p\<^sub>0, fg\<^sub>0h\<^sub>1.p\<^sub>1] \<cdot> local.inv fg\<^sub>0h\<^sub>1.\<phi> = ((h \<star> TTfgh_TfTgh.the_\<theta>) \<cdot> (h \<star> can (((\<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) (\<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>))) \<cdot> \<a>[h, tab\<^sub>0 h, g\<^sub>0h\<^sub>1.p\<^sub>0 \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine] \<cdot> (h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0 \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine) \<cdot> \<a>[tab\<^sub>1 h, g\<^sub>0h\<^sub>1.p\<^sub>0, f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine] \<cdot> (g\<^sub>0h\<^sub>1.\<phi> \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine) \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1, f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine] \<cdot> (tab\<^sub>0 g \<star> \<gamma>\<^sub>g) \<cdot> \<a>[tab\<^sub>0 g, f\<^sub>0g\<^sub>1.p\<^sub>0, fg\<^sub>0h\<^sub>1.p\<^sub>1] \<cdot> local.inv fg\<^sub>0h\<^sub>1.\<phi>
[PROOF STEP]
have "can (\<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> ((\<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>Tgh.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TfTgh.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>)
(((\<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>Tgh.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TfTgh.p\<^sub>0\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) \<cdot>
(\<a>\<^sup>-\<^sup>1[h \<star> tab\<^sub>0 h, Tgh.p\<^sub>0, TfTgh.p\<^sub>0 \<star> TTfgh_TfTgh.chine] \<cdot>
\<a>\<^sup>-\<^sup>1[h, tab\<^sub>0 h, Tgh.p\<^sub>0 \<star> TfTgh.p\<^sub>0 \<star> TTfgh_TfTgh.chine]) =
can (\<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> ((\<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>Tgh.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TfTgh.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>)
(((\<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>Tgh.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TfTgh.p\<^sub>0\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) \<cdot>
\<lbrace>\<^bold>\<a>\<^sup>-\<^sup>1\<^bold>[\<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle>, \<^bold>\<langle>Tgh.p\<^sub>0\<^bold>\<rangle>, \<^bold>\<langle>TfTgh.p\<^sub>0\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>\<^bold>] \<^bold>\<cdot>
\<^bold>\<a>\<^sup>-\<^sup>1\<^bold>[\<^bold>\<langle>h\<^bold>\<rangle>, \<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle>, \<^bold>\<langle>Tgh.p\<^sub>0\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>TfTgh.p\<^sub>0\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>\<^bold>]\<rbrace>"
[PROOF STATE]
proof (prove)
goal (1 subgoal):
1. can (\<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> ((\<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) (((\<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) \<cdot> \<a>\<^sup>-\<^sup>1[h \<star> tab\<^sub>0 h, g\<^sub>0h\<^sub>1.p\<^sub>0, f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine] \<cdot> \<a>\<^sup>-\<^sup>1[h, tab\<^sub>0 h, g\<^sub>0h\<^sub>1.p\<^sub>0 \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine] = can (\<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> ((\<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) (((\<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) \<cdot> \<lbrace>\<^bold>\<a>\<^sup>-\<^sup>1\<^bold>[\<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle>, \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>, \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>\<^bold>] \<^bold>\<cdot> \<^bold>\<a>\<^sup>-\<^sup>1\<^bold>[\<^bold>\<langle>h\<^bold>\<rangle>, \<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle>, \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>\<^bold>]\<rbrace>
[PROOF STEP]
using \<a>'_def \<alpha>_def
[PROOF STATE]
proof (prove)
using this:
\<a>\<^sup>-\<^sup>1[?\<mu>, ?\<nu>, ?\<tau>] \<equiv> \<alpha>' (?\<mu>, ?\<nu>, ?\<tau>)
\<alpha> ?\<mu>\<nu>\<tau> \<equiv> \<a>[fst ?\<mu>\<nu>\<tau>, fst (snd ?\<mu>\<nu>\<tau>), snd (snd ?\<mu>\<nu>\<tau>)]
goal (1 subgoal):
1. can (\<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> ((\<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) (((\<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) \<cdot> \<a>\<^sup>-\<^sup>1[h \<star> tab\<^sub>0 h, g\<^sub>0h\<^sub>1.p\<^sub>0, f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine] \<cdot> \<a>\<^sup>-\<^sup>1[h, tab\<^sub>0 h, g\<^sub>0h\<^sub>1.p\<^sub>0 \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine] = can (\<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> ((\<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) (((\<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) \<cdot> \<lbrace>\<^bold>\<a>\<^sup>-\<^sup>1\<^bold>[\<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle>, \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>, \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>\<^bold>] \<^bold>\<cdot> \<^bold>\<a>\<^sup>-\<^sup>1\<^bold>[\<^bold>\<langle>h\<^bold>\<rangle>, \<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle>, \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>\<^bold>]\<rbrace>
[PROOF STEP]
by simp
[PROOF STATE]
proof (state)
this:
can (\<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> ((\<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) (((\<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) \<cdot> \<a>\<^sup>-\<^sup>1[h \<star> tab\<^sub>0 h, g\<^sub>0h\<^sub>1.p\<^sub>0, f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine] \<cdot> \<a>\<^sup>-\<^sup>1[h, tab\<^sub>0 h, g\<^sub>0h\<^sub>1.p\<^sub>0 \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine] = can (\<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> ((\<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) (((\<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) \<cdot> \<lbrace>\<^bold>\<a>\<^sup>-\<^sup>1\<^bold>[\<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle>, \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>, \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>\<^bold>] \<^bold>\<cdot> \<^bold>\<a>\<^sup>-\<^sup>1\<^bold>[\<^bold>\<langle>h\<^bold>\<rangle>, \<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle>, \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>\<^bold>]\<rbrace>
goal (1 subgoal):
1. (h \<star> TTfgh_TfTgh.the_\<theta>) \<cdot> (can (\<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> ((\<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) (((\<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) \<cdot> \<a>\<^sup>-\<^sup>1[h \<star> tab\<^sub>0 h, g\<^sub>0h\<^sub>1.p\<^sub>0, f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine] \<cdot> \<a>\<^sup>-\<^sup>1[h, tab\<^sub>0 h, g\<^sub>0h\<^sub>1.p\<^sub>0 \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine]) \<cdot> \<a>[h, tab\<^sub>0 h, g\<^sub>0h\<^sub>1.p\<^sub>0 \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine] \<cdot> (h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0 \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine) \<cdot> \<a>[tab\<^sub>1 h, g\<^sub>0h\<^sub>1.p\<^sub>0, f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine] \<cdot> (g\<^sub>0h\<^sub>1.\<phi> \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine) \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1, f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine] \<cdot> (tab\<^sub>0 g \<star> \<gamma>\<^sub>g) \<cdot> \<a>[tab\<^sub>0 g, f\<^sub>0g\<^sub>1.p\<^sub>0, fg\<^sub>0h\<^sub>1.p\<^sub>1] \<cdot> local.inv fg\<^sub>0h\<^sub>1.\<phi> = ((h \<star> TTfgh_TfTgh.the_\<theta>) \<cdot> (h \<star> can (((\<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) (\<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>))) \<cdot> \<a>[h, tab\<^sub>0 h, g\<^sub>0h\<^sub>1.p\<^sub>0 \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine] \<cdot> (h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0 \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine) \<cdot> \<a>[tab\<^sub>1 h, g\<^sub>0h\<^sub>1.p\<^sub>0, f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine] \<cdot> (g\<^sub>0h\<^sub>1.\<phi> \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine) \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1, f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine] \<cdot> (tab\<^sub>0 g \<star> \<gamma>\<^sub>g) \<cdot> \<a>[tab\<^sub>0 g, f\<^sub>0g\<^sub>1.p\<^sub>0, fg\<^sub>0h\<^sub>1.p\<^sub>1] \<cdot> local.inv fg\<^sub>0h\<^sub>1.\<phi>
[PROOF STEP]
also
[PROOF STATE]
proof (state)
this:
can (\<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> ((\<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) (((\<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) \<cdot> \<a>\<^sup>-\<^sup>1[h \<star> tab\<^sub>0 h, g\<^sub>0h\<^sub>1.p\<^sub>0, f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine] \<cdot> \<a>\<^sup>-\<^sup>1[h, tab\<^sub>0 h, g\<^sub>0h\<^sub>1.p\<^sub>0 \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine] = can (\<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> ((\<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) (((\<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) \<cdot> \<lbrace>\<^bold>\<a>\<^sup>-\<^sup>1\<^bold>[\<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle>, \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>, \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>\<^bold>] \<^bold>\<cdot> \<^bold>\<a>\<^sup>-\<^sup>1\<^bold>[\<^bold>\<langle>h\<^bold>\<rangle>, \<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle>, \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>\<^bold>]\<rbrace>
goal (1 subgoal):
1. (h \<star> TTfgh_TfTgh.the_\<theta>) \<cdot> (can (\<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> ((\<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) (((\<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) \<cdot> \<a>\<^sup>-\<^sup>1[h \<star> tab\<^sub>0 h, g\<^sub>0h\<^sub>1.p\<^sub>0, f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine] \<cdot> \<a>\<^sup>-\<^sup>1[h, tab\<^sub>0 h, g\<^sub>0h\<^sub>1.p\<^sub>0 \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine]) \<cdot> \<a>[h, tab\<^sub>0 h, g\<^sub>0h\<^sub>1.p\<^sub>0 \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine] \<cdot> (h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0 \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine) \<cdot> \<a>[tab\<^sub>1 h, g\<^sub>0h\<^sub>1.p\<^sub>0, f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine] \<cdot> (g\<^sub>0h\<^sub>1.\<phi> \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine) \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1, f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine] \<cdot> (tab\<^sub>0 g \<star> \<gamma>\<^sub>g) \<cdot> \<a>[tab\<^sub>0 g, f\<^sub>0g\<^sub>1.p\<^sub>0, fg\<^sub>0h\<^sub>1.p\<^sub>1] \<cdot> local.inv fg\<^sub>0h\<^sub>1.\<phi> = ((h \<star> TTfgh_TfTgh.the_\<theta>) \<cdot> (h \<star> can (((\<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) (\<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>))) \<cdot> \<a>[h, tab\<^sub>0 h, g\<^sub>0h\<^sub>1.p\<^sub>0 \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine] \<cdot> (h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0 \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine) \<cdot> \<a>[tab\<^sub>1 h, g\<^sub>0h\<^sub>1.p\<^sub>0, f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine] \<cdot> (g\<^sub>0h\<^sub>1.\<phi> \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine) \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1, f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine] \<cdot> (tab\<^sub>0 g \<star> \<gamma>\<^sub>g) \<cdot> \<a>[tab\<^sub>0 g, f\<^sub>0g\<^sub>1.p\<^sub>0, fg\<^sub>0h\<^sub>1.p\<^sub>1] \<cdot> local.inv fg\<^sub>0h\<^sub>1.\<phi>
[PROOF STEP]
have "... = can (\<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> ((\<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>Tgh.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TfTgh.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>)
(((\<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>Tgh.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TfTgh.p\<^sub>0\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) \<cdot>
can (((\<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>Tgh.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TfTgh.p\<^sub>0\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>)
(\<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>Tgh.p\<^sub>0\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>TfTgh.p\<^sub>0\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>)"
[PROOF STATE]
proof (prove)
goal (1 subgoal):
1. can (\<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> ((\<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) (((\<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) \<cdot> \<lbrace>\<^bold>\<a>\<^sup>-\<^sup>1\<^bold>[\<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle>, \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>, \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>\<^bold>] \<^bold>\<cdot> \<^bold>\<a>\<^sup>-\<^sup>1\<^bold>[\<^bold>\<langle>h\<^bold>\<rangle>, \<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle>, \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>\<^bold>]\<rbrace> = can (\<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> ((\<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) (((\<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) \<cdot> can (((\<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) (\<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>)
[PROOF STEP]
proof -
[PROOF STATE]
proof (state)
goal (1 subgoal):
1. can (\<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> ((\<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) (((\<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) \<cdot> \<lbrace>\<^bold>\<a>\<^sup>-\<^sup>1\<^bold>[\<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle>, \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>, \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>\<^bold>] \<^bold>\<cdot> \<^bold>\<a>\<^sup>-\<^sup>1\<^bold>[\<^bold>\<langle>h\<^bold>\<rangle>, \<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle>, \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>\<^bold>]\<rbrace> = can (\<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> ((\<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) (((\<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) \<cdot> can (((\<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) (\<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>)
[PROOF STEP]
have "\<lbrace>\<^bold>\<a>\<^sup>-\<^sup>1\<^bold>[\<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle>, \<^bold>\<langle>Tgh.p\<^sub>0\<^bold>\<rangle>, \<^bold>\<langle>TfTgh.p\<^sub>0\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>\<^bold>] \<^bold>\<cdot>
\<^bold>\<a>\<^sup>-\<^sup>1\<^bold>[\<^bold>\<langle>h\<^bold>\<rangle>, \<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle>, \<^bold>\<langle>Tgh.p\<^sub>0\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>TfTgh.p\<^sub>0\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>\<^bold>]\<rbrace> =
can (((\<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>Tgh.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TfTgh.p\<^sub>0\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>)
(\<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>Tgh.p\<^sub>0\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>TfTgh.p\<^sub>0\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>)"
[PROOF STATE]
proof (prove)
goal (1 subgoal):
1. \<lbrace>\<^bold>\<a>\<^sup>-\<^sup>1\<^bold>[\<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle>, \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>, \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>\<^bold>] \<^bold>\<cdot> \<^bold>\<a>\<^sup>-\<^sup>1\<^bold>[\<^bold>\<langle>h\<^bold>\<rangle>, \<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle>, \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>\<^bold>]\<rbrace> = can (((\<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) (\<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>)
[PROOF STEP]
unfolding can_def
[PROOF STATE]
proof (prove)
goal (1 subgoal):
1. \<lbrace>\<^bold>\<a>\<^sup>-\<^sup>1\<^bold>[\<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle>, \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>, \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>\<^bold>] \<^bold>\<cdot> \<^bold>\<a>\<^sup>-\<^sup>1\<^bold>[\<^bold>\<langle>h\<^bold>\<rangle>, \<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle>, \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>\<^bold>]\<rbrace> = \<lbrace>E.Inv ((((\<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>)\<^bold>\<down>) \<^bold>\<cdot> (\<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>)\<^bold>\<down>\<rbrace>
[PROOF STEP]
apply (intro E.eval_eqI)
[PROOF STATE]
proof (prove)
goal (2 subgoals):
1. E.VPar (\<^bold>\<a>\<^sup>-\<^sup>1\<^bold>[\<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle>, \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>, \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>\<^bold>] \<^bold>\<cdot> \<^bold>\<a>\<^sup>-\<^sup>1\<^bold>[\<^bold>\<langle>h\<^bold>\<rangle>, \<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle>, \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>\<^bold>]) (E.Inv ((((\<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>)\<^bold>\<down>) \<^bold>\<cdot> (\<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>)\<^bold>\<down>)
2. \<^bold>\<lfloor>\<^bold>\<a>\<^sup>-\<^sup>1\<^bold>[\<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle>, \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>, \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>\<^bold>] \<^bold>\<cdot> \<^bold>\<a>\<^sup>-\<^sup>1\<^bold>[\<^bold>\<langle>h\<^bold>\<rangle>, \<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle>, \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>\<^bold>]\<^bold>\<rfloor> = \<^bold>\<lfloor>E.Inv ((((\<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>)\<^bold>\<down>) \<^bold>\<cdot> (\<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>)\<^bold>\<down>\<^bold>\<rfloor>
[PROOF STEP]
by simp_all
[PROOF STATE]
proof (state)
this:
\<lbrace>\<^bold>\<a>\<^sup>-\<^sup>1\<^bold>[\<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle>, \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>, \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>\<^bold>] \<^bold>\<cdot> \<^bold>\<a>\<^sup>-\<^sup>1\<^bold>[\<^bold>\<langle>h\<^bold>\<rangle>, \<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle>, \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>\<^bold>]\<rbrace> = can (((\<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) (\<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>)
goal (1 subgoal):
1. can (\<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> ((\<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) (((\<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) \<cdot> \<lbrace>\<^bold>\<a>\<^sup>-\<^sup>1\<^bold>[\<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle>, \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>, \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>\<^bold>] \<^bold>\<cdot> \<^bold>\<a>\<^sup>-\<^sup>1\<^bold>[\<^bold>\<langle>h\<^bold>\<rangle>, \<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle>, \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>\<^bold>]\<rbrace> = can (\<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> ((\<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) (((\<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) \<cdot> can (((\<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) (\<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>)
[PROOF STEP]
thus ?thesis
[PROOF STATE]
proof (prove)
using this:
\<lbrace>\<^bold>\<a>\<^sup>-\<^sup>1\<^bold>[\<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle>, \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>, \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>\<^bold>] \<^bold>\<cdot> \<^bold>\<a>\<^sup>-\<^sup>1\<^bold>[\<^bold>\<langle>h\<^bold>\<rangle>, \<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle>, \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>\<^bold>]\<rbrace> = can (((\<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) (\<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>)
goal (1 subgoal):
1. can (\<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> ((\<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) (((\<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) \<cdot> \<lbrace>\<^bold>\<a>\<^sup>-\<^sup>1\<^bold>[\<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle>, \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>, \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>\<^bold>] \<^bold>\<cdot> \<^bold>\<a>\<^sup>-\<^sup>1\<^bold>[\<^bold>\<langle>h\<^bold>\<rangle>, \<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle>, \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>\<^bold>]\<rbrace> = can (\<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> ((\<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) (((\<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) \<cdot> can (((\<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) (\<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>)
[PROOF STEP]
by simp
[PROOF STATE]
proof (state)
this:
can (\<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> ((\<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) (((\<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) \<cdot> \<lbrace>\<^bold>\<a>\<^sup>-\<^sup>1\<^bold>[\<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle>, \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>, \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>\<^bold>] \<^bold>\<cdot> \<^bold>\<a>\<^sup>-\<^sup>1\<^bold>[\<^bold>\<langle>h\<^bold>\<rangle>, \<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle>, \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>\<^bold>]\<rbrace> = can (\<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> ((\<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) (((\<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) \<cdot> can (((\<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) (\<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>)
goal:
No subgoals!
[PROOF STEP]
qed
[PROOF STATE]
proof (state)
this:
can (\<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> ((\<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) (((\<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) \<cdot> \<lbrace>\<^bold>\<a>\<^sup>-\<^sup>1\<^bold>[\<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle>, \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>, \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>\<^bold>] \<^bold>\<cdot> \<^bold>\<a>\<^sup>-\<^sup>1\<^bold>[\<^bold>\<langle>h\<^bold>\<rangle>, \<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle>, \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>\<^bold>]\<rbrace> = can (\<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> ((\<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) (((\<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) \<cdot> can (((\<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) (\<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>)
goal (1 subgoal):
1. (h \<star> TTfgh_TfTgh.the_\<theta>) \<cdot> (can (\<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> ((\<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) (((\<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) \<cdot> \<a>\<^sup>-\<^sup>1[h \<star> tab\<^sub>0 h, g\<^sub>0h\<^sub>1.p\<^sub>0, f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine] \<cdot> \<a>\<^sup>-\<^sup>1[h, tab\<^sub>0 h, g\<^sub>0h\<^sub>1.p\<^sub>0 \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine]) \<cdot> \<a>[h, tab\<^sub>0 h, g\<^sub>0h\<^sub>1.p\<^sub>0 \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine] \<cdot> (h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0 \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine) \<cdot> \<a>[tab\<^sub>1 h, g\<^sub>0h\<^sub>1.p\<^sub>0, f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine] \<cdot> (g\<^sub>0h\<^sub>1.\<phi> \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine) \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1, f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine] \<cdot> (tab\<^sub>0 g \<star> \<gamma>\<^sub>g) \<cdot> \<a>[tab\<^sub>0 g, f\<^sub>0g\<^sub>1.p\<^sub>0, fg\<^sub>0h\<^sub>1.p\<^sub>1] \<cdot> local.inv fg\<^sub>0h\<^sub>1.\<phi> = ((h \<star> TTfgh_TfTgh.the_\<theta>) \<cdot> (h \<star> can (((\<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) (\<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>))) \<cdot> \<a>[h, tab\<^sub>0 h, g\<^sub>0h\<^sub>1.p\<^sub>0 \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine] \<cdot> (h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0 \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine) \<cdot> \<a>[tab\<^sub>1 h, g\<^sub>0h\<^sub>1.p\<^sub>0, f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine] \<cdot> (g\<^sub>0h\<^sub>1.\<phi> \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine) \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1, f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine] \<cdot> (tab\<^sub>0 g \<star> \<gamma>\<^sub>g) \<cdot> \<a>[tab\<^sub>0 g, f\<^sub>0g\<^sub>1.p\<^sub>0, fg\<^sub>0h\<^sub>1.p\<^sub>1] \<cdot> local.inv fg\<^sub>0h\<^sub>1.\<phi>
[PROOF STEP]
also
[PROOF STATE]
proof (state)
this:
can (\<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> ((\<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) (((\<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) \<cdot> \<lbrace>\<^bold>\<a>\<^sup>-\<^sup>1\<^bold>[\<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle>, \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>, \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>\<^bold>] \<^bold>\<cdot> \<^bold>\<a>\<^sup>-\<^sup>1\<^bold>[\<^bold>\<langle>h\<^bold>\<rangle>, \<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle>, \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>\<^bold>]\<rbrace> = can (\<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> ((\<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) (((\<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) \<cdot> can (((\<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) (\<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>)
goal (1 subgoal):
1. (h \<star> TTfgh_TfTgh.the_\<theta>) \<cdot> (can (\<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> ((\<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) (((\<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) \<cdot> \<a>\<^sup>-\<^sup>1[h \<star> tab\<^sub>0 h, g\<^sub>0h\<^sub>1.p\<^sub>0, f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine] \<cdot> \<a>\<^sup>-\<^sup>1[h, tab\<^sub>0 h, g\<^sub>0h\<^sub>1.p\<^sub>0 \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine]) \<cdot> \<a>[h, tab\<^sub>0 h, g\<^sub>0h\<^sub>1.p\<^sub>0 \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine] \<cdot> (h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0 \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine) \<cdot> \<a>[tab\<^sub>1 h, g\<^sub>0h\<^sub>1.p\<^sub>0, f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine] \<cdot> (g\<^sub>0h\<^sub>1.\<phi> \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine) \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1, f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine] \<cdot> (tab\<^sub>0 g \<star> \<gamma>\<^sub>g) \<cdot> \<a>[tab\<^sub>0 g, f\<^sub>0g\<^sub>1.p\<^sub>0, fg\<^sub>0h\<^sub>1.p\<^sub>1] \<cdot> local.inv fg\<^sub>0h\<^sub>1.\<phi> = ((h \<star> TTfgh_TfTgh.the_\<theta>) \<cdot> (h \<star> can (((\<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) (\<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>))) \<cdot> \<a>[h, tab\<^sub>0 h, g\<^sub>0h\<^sub>1.p\<^sub>0 \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine] \<cdot> (h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0 \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine) \<cdot> \<a>[tab\<^sub>1 h, g\<^sub>0h\<^sub>1.p\<^sub>0, f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine] \<cdot> (g\<^sub>0h\<^sub>1.\<phi> \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine) \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1, f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine] \<cdot> (tab\<^sub>0 g \<star> \<gamma>\<^sub>g) \<cdot> \<a>[tab\<^sub>0 g, f\<^sub>0g\<^sub>1.p\<^sub>0, fg\<^sub>0h\<^sub>1.p\<^sub>1] \<cdot> local.inv fg\<^sub>0h\<^sub>1.\<phi>
[PROOF STEP]
have "... = can (\<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> ((\<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>Tgh.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TfTgh.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>)
(\<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>Tgh.p\<^sub>0\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>TfTgh.p\<^sub>0\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>)"
[PROOF STATE]
proof (prove)
goal (1 subgoal):
1. can (\<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> ((\<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) (((\<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) \<cdot> can (((\<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) (\<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) = can (\<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> ((\<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) (\<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>)
[PROOF STEP]
by simp
[PROOF STATE]
proof (state)
this:
can (\<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> ((\<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) (((\<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) \<cdot> can (((\<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) (\<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) = can (\<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> ((\<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) (\<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>)
goal (1 subgoal):
1. (h \<star> TTfgh_TfTgh.the_\<theta>) \<cdot> (can (\<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> ((\<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) (((\<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) \<cdot> \<a>\<^sup>-\<^sup>1[h \<star> tab\<^sub>0 h, g\<^sub>0h\<^sub>1.p\<^sub>0, f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine] \<cdot> \<a>\<^sup>-\<^sup>1[h, tab\<^sub>0 h, g\<^sub>0h\<^sub>1.p\<^sub>0 \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine]) \<cdot> \<a>[h, tab\<^sub>0 h, g\<^sub>0h\<^sub>1.p\<^sub>0 \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine] \<cdot> (h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0 \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine) \<cdot> \<a>[tab\<^sub>1 h, g\<^sub>0h\<^sub>1.p\<^sub>0, f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine] \<cdot> (g\<^sub>0h\<^sub>1.\<phi> \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine) \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1, f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine] \<cdot> (tab\<^sub>0 g \<star> \<gamma>\<^sub>g) \<cdot> \<a>[tab\<^sub>0 g, f\<^sub>0g\<^sub>1.p\<^sub>0, fg\<^sub>0h\<^sub>1.p\<^sub>1] \<cdot> local.inv fg\<^sub>0h\<^sub>1.\<phi> = ((h \<star> TTfgh_TfTgh.the_\<theta>) \<cdot> (h \<star> can (((\<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) (\<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>))) \<cdot> \<a>[h, tab\<^sub>0 h, g\<^sub>0h\<^sub>1.p\<^sub>0 \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine] \<cdot> (h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0 \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine) \<cdot> \<a>[tab\<^sub>1 h, g\<^sub>0h\<^sub>1.p\<^sub>0, f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine] \<cdot> (g\<^sub>0h\<^sub>1.\<phi> \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine) \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1, f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine] \<cdot> (tab\<^sub>0 g \<star> \<gamma>\<^sub>g) \<cdot> \<a>[tab\<^sub>0 g, f\<^sub>0g\<^sub>1.p\<^sub>0, fg\<^sub>0h\<^sub>1.p\<^sub>1] \<cdot> local.inv fg\<^sub>0h\<^sub>1.\<phi>
[PROOF STEP]
also
[PROOF STATE]
proof (state)
this:
can (\<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> ((\<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) (((\<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) \<cdot> can (((\<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) (\<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) = can (\<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> ((\<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) (\<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>)
goal (1 subgoal):
1. (h \<star> TTfgh_TfTgh.the_\<theta>) \<cdot> (can (\<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> ((\<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) (((\<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) \<cdot> \<a>\<^sup>-\<^sup>1[h \<star> tab\<^sub>0 h, g\<^sub>0h\<^sub>1.p\<^sub>0, f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine] \<cdot> \<a>\<^sup>-\<^sup>1[h, tab\<^sub>0 h, g\<^sub>0h\<^sub>1.p\<^sub>0 \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine]) \<cdot> \<a>[h, tab\<^sub>0 h, g\<^sub>0h\<^sub>1.p\<^sub>0 \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine] \<cdot> (h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0 \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine) \<cdot> \<a>[tab\<^sub>1 h, g\<^sub>0h\<^sub>1.p\<^sub>0, f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine] \<cdot> (g\<^sub>0h\<^sub>1.\<phi> \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine) \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1, f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine] \<cdot> (tab\<^sub>0 g \<star> \<gamma>\<^sub>g) \<cdot> \<a>[tab\<^sub>0 g, f\<^sub>0g\<^sub>1.p\<^sub>0, fg\<^sub>0h\<^sub>1.p\<^sub>1] \<cdot> local.inv fg\<^sub>0h\<^sub>1.\<phi> = ((h \<star> TTfgh_TfTgh.the_\<theta>) \<cdot> (h \<star> can (((\<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) (\<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>))) \<cdot> \<a>[h, tab\<^sub>0 h, g\<^sub>0h\<^sub>1.p\<^sub>0 \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine] \<cdot> (h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0 \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine) \<cdot> \<a>[tab\<^sub>1 h, g\<^sub>0h\<^sub>1.p\<^sub>0, f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine] \<cdot> (g\<^sub>0h\<^sub>1.\<phi> \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine) \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1, f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine] \<cdot> (tab\<^sub>0 g \<star> \<gamma>\<^sub>g) \<cdot> \<a>[tab\<^sub>0 g, f\<^sub>0g\<^sub>1.p\<^sub>0, fg\<^sub>0h\<^sub>1.p\<^sub>1] \<cdot> local.inv fg\<^sub>0h\<^sub>1.\<phi>
[PROOF STEP]
have "... = h \<star> can (((\<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>Tgh.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TfTgh.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>)
(\<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>Tgh.p\<^sub>0\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>TfTgh.p\<^sub>0\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>)"
[PROOF STATE]
proof (prove)
goal (1 subgoal):
1. can (\<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> ((\<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) (\<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) = h \<star> can (((\<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) (\<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>)
[PROOF STEP]
using whisker_can_left_0
[PROOF STATE]
proof (prove)
using this:
\<lbrakk>E.Ide ?t; E.Ide ?u; \<^bold>\<lfloor>?t\<^bold>\<rfloor> = \<^bold>\<lfloor>?u\<^bold>\<rfloor>; ide ?g; E.Trg ?t = \<^bold>\<langle>src ?g\<^bold>\<rangle>\<^sub>0\<rbrakk> \<Longrightarrow> ?g \<star> can ?u ?t = can (\<^bold>\<langle>?g\<^bold>\<rangle> \<^bold>\<star> ?u) (\<^bold>\<langle>?g\<^bold>\<rangle> \<^bold>\<star> ?t)
goal (1 subgoal):
1. can (\<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> ((\<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) (\<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) = h \<star> can (((\<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) (\<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>)
[PROOF STEP]
by simp
[PROOF STATE]
proof (state)
this:
can (\<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> ((\<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) (\<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) = h \<star> can (((\<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) (\<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>)
goal (1 subgoal):
1. (h \<star> TTfgh_TfTgh.the_\<theta>) \<cdot> (can (\<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> ((\<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) (((\<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) \<cdot> \<a>\<^sup>-\<^sup>1[h \<star> tab\<^sub>0 h, g\<^sub>0h\<^sub>1.p\<^sub>0, f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine] \<cdot> \<a>\<^sup>-\<^sup>1[h, tab\<^sub>0 h, g\<^sub>0h\<^sub>1.p\<^sub>0 \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine]) \<cdot> \<a>[h, tab\<^sub>0 h, g\<^sub>0h\<^sub>1.p\<^sub>0 \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine] \<cdot> (h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0 \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine) \<cdot> \<a>[tab\<^sub>1 h, g\<^sub>0h\<^sub>1.p\<^sub>0, f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine] \<cdot> (g\<^sub>0h\<^sub>1.\<phi> \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine) \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1, f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine] \<cdot> (tab\<^sub>0 g \<star> \<gamma>\<^sub>g) \<cdot> \<a>[tab\<^sub>0 g, f\<^sub>0g\<^sub>1.p\<^sub>0, fg\<^sub>0h\<^sub>1.p\<^sub>1] \<cdot> local.inv fg\<^sub>0h\<^sub>1.\<phi> = ((h \<star> TTfgh_TfTgh.the_\<theta>) \<cdot> (h \<star> can (((\<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) (\<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>))) \<cdot> \<a>[h, tab\<^sub>0 h, g\<^sub>0h\<^sub>1.p\<^sub>0 \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine] \<cdot> (h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0 \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine) \<cdot> \<a>[tab\<^sub>1 h, g\<^sub>0h\<^sub>1.p\<^sub>0, f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine] \<cdot> (g\<^sub>0h\<^sub>1.\<phi> \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine) \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1, f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine] \<cdot> (tab\<^sub>0 g \<star> \<gamma>\<^sub>g) \<cdot> \<a>[tab\<^sub>0 g, f\<^sub>0g\<^sub>1.p\<^sub>0, fg\<^sub>0h\<^sub>1.p\<^sub>1] \<cdot> local.inv fg\<^sub>0h\<^sub>1.\<phi>
[PROOF STEP]
finally
[PROOF STATE]
proof (chain)
picking this:
can (\<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> ((\<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) (((\<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) \<cdot> \<a>\<^sup>-\<^sup>1[h \<star> tab\<^sub>0 h, g\<^sub>0h\<^sub>1.p\<^sub>0, f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine] \<cdot> \<a>\<^sup>-\<^sup>1[h, tab\<^sub>0 h, g\<^sub>0h\<^sub>1.p\<^sub>0 \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine] = h \<star> can (((\<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) (\<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>)
[PROOF STEP]
have "can (\<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> ((\<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>Tgh.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TfTgh.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>)
(((\<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>Tgh.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TfTgh.p\<^sub>0\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) \<cdot>
(\<a>\<^sup>-\<^sup>1[h \<star> tab\<^sub>0 h, Tgh.p\<^sub>0, TfTgh.p\<^sub>0 \<star> TTfgh_TfTgh.chine] \<cdot>
\<a>\<^sup>-\<^sup>1[h, tab\<^sub>0 h, Tgh.p\<^sub>0 \<star> TfTgh.p\<^sub>0 \<star> TTfgh_TfTgh.chine]) =
h \<star> can (((\<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>Tgh.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TfTgh.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>)
(\<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>Tgh.p\<^sub>0\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>TfTgh.p\<^sub>0\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>)"
[PROOF STATE]
proof (prove)
using this:
can (\<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> ((\<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) (((\<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) \<cdot> \<a>\<^sup>-\<^sup>1[h \<star> tab\<^sub>0 h, g\<^sub>0h\<^sub>1.p\<^sub>0, f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine] \<cdot> \<a>\<^sup>-\<^sup>1[h, tab\<^sub>0 h, g\<^sub>0h\<^sub>1.p\<^sub>0 \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine] = h \<star> can (((\<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) (\<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>)
goal (1 subgoal):
1. can (\<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> ((\<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) (((\<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) \<cdot> \<a>\<^sup>-\<^sup>1[h \<star> tab\<^sub>0 h, g\<^sub>0h\<^sub>1.p\<^sub>0, f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine] \<cdot> \<a>\<^sup>-\<^sup>1[h, tab\<^sub>0 h, g\<^sub>0h\<^sub>1.p\<^sub>0 \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine] = h \<star> can (((\<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) (\<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>)
[PROOF STEP]
by simp
[PROOF STATE]
proof (state)
this:
can (\<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> ((\<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) (((\<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) \<cdot> \<a>\<^sup>-\<^sup>1[h \<star> tab\<^sub>0 h, g\<^sub>0h\<^sub>1.p\<^sub>0, f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine] \<cdot> \<a>\<^sup>-\<^sup>1[h, tab\<^sub>0 h, g\<^sub>0h\<^sub>1.p\<^sub>0 \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine] = h \<star> can (((\<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) (\<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>)
goal (1 subgoal):
1. (h \<star> TTfgh_TfTgh.the_\<theta>) \<cdot> (can (\<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> ((\<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) (((\<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) \<cdot> \<a>\<^sup>-\<^sup>1[h \<star> tab\<^sub>0 h, g\<^sub>0h\<^sub>1.p\<^sub>0, f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine] \<cdot> \<a>\<^sup>-\<^sup>1[h, tab\<^sub>0 h, g\<^sub>0h\<^sub>1.p\<^sub>0 \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine]) \<cdot> \<a>[h, tab\<^sub>0 h, g\<^sub>0h\<^sub>1.p\<^sub>0 \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine] \<cdot> (h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0 \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine) \<cdot> \<a>[tab\<^sub>1 h, g\<^sub>0h\<^sub>1.p\<^sub>0, f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine] \<cdot> (g\<^sub>0h\<^sub>1.\<phi> \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine) \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1, f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine] \<cdot> (tab\<^sub>0 g \<star> \<gamma>\<^sub>g) \<cdot> \<a>[tab\<^sub>0 g, f\<^sub>0g\<^sub>1.p\<^sub>0, fg\<^sub>0h\<^sub>1.p\<^sub>1] \<cdot> local.inv fg\<^sub>0h\<^sub>1.\<phi> = ((h \<star> TTfgh_TfTgh.the_\<theta>) \<cdot> (h \<star> can (((\<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) (\<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>))) \<cdot> \<a>[h, tab\<^sub>0 h, g\<^sub>0h\<^sub>1.p\<^sub>0 \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine] \<cdot> (h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0 \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine) \<cdot> \<a>[tab\<^sub>1 h, g\<^sub>0h\<^sub>1.p\<^sub>0, f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine] \<cdot> (g\<^sub>0h\<^sub>1.\<phi> \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine) \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1, f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine] \<cdot> (tab\<^sub>0 g \<star> \<gamma>\<^sub>g) \<cdot> \<a>[tab\<^sub>0 g, f\<^sub>0g\<^sub>1.p\<^sub>0, fg\<^sub>0h\<^sub>1.p\<^sub>1] \<cdot> local.inv fg\<^sub>0h\<^sub>1.\<phi>
[PROOF STEP]
thus ?thesis
[PROOF STATE]
proof (prove)
using this:
can (\<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> ((\<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) (((\<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) \<cdot> \<a>\<^sup>-\<^sup>1[h \<star> tab\<^sub>0 h, g\<^sub>0h\<^sub>1.p\<^sub>0, f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine] \<cdot> \<a>\<^sup>-\<^sup>1[h, tab\<^sub>0 h, g\<^sub>0h\<^sub>1.p\<^sub>0 \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine] = h \<star> can (((\<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) (\<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>)
goal (1 subgoal):
1. (h \<star> TTfgh_TfTgh.the_\<theta>) \<cdot> (can (\<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> ((\<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) (((\<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) \<cdot> \<a>\<^sup>-\<^sup>1[h \<star> tab\<^sub>0 h, g\<^sub>0h\<^sub>1.p\<^sub>0, f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine] \<cdot> \<a>\<^sup>-\<^sup>1[h, tab\<^sub>0 h, g\<^sub>0h\<^sub>1.p\<^sub>0 \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine]) \<cdot> \<a>[h, tab\<^sub>0 h, g\<^sub>0h\<^sub>1.p\<^sub>0 \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine] \<cdot> (h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0 \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine) \<cdot> \<a>[tab\<^sub>1 h, g\<^sub>0h\<^sub>1.p\<^sub>0, f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine] \<cdot> (g\<^sub>0h\<^sub>1.\<phi> \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine) \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1, f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine] \<cdot> (tab\<^sub>0 g \<star> \<gamma>\<^sub>g) \<cdot> \<a>[tab\<^sub>0 g, f\<^sub>0g\<^sub>1.p\<^sub>0, fg\<^sub>0h\<^sub>1.p\<^sub>1] \<cdot> local.inv fg\<^sub>0h\<^sub>1.\<phi> = ((h \<star> TTfgh_TfTgh.the_\<theta>) \<cdot> (h \<star> can (((\<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) (\<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>))) \<cdot> \<a>[h, tab\<^sub>0 h, g\<^sub>0h\<^sub>1.p\<^sub>0 \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine] \<cdot> (h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0 \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine) \<cdot> \<a>[tab\<^sub>1 h, g\<^sub>0h\<^sub>1.p\<^sub>0, f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine] \<cdot> (g\<^sub>0h\<^sub>1.\<phi> \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine) \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1, f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine] \<cdot> (tab\<^sub>0 g \<star> \<gamma>\<^sub>g) \<cdot> \<a>[tab\<^sub>0 g, f\<^sub>0g\<^sub>1.p\<^sub>0, fg\<^sub>0h\<^sub>1.p\<^sub>1] \<cdot> local.inv fg\<^sub>0h\<^sub>1.\<phi>
[PROOF STEP]
using comp_assoc
[PROOF STATE]
proof (prove)
using this:
can (\<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> ((\<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) (((\<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) \<cdot> \<a>\<^sup>-\<^sup>1[h \<star> tab\<^sub>0 h, g\<^sub>0h\<^sub>1.p\<^sub>0, f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine] \<cdot> \<a>\<^sup>-\<^sup>1[h, tab\<^sub>0 h, g\<^sub>0h\<^sub>1.p\<^sub>0 \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine] = h \<star> can (((\<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) (\<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>)
(?h \<cdot> ?g) \<cdot> ?f = ?h \<cdot> ?g \<cdot> ?f
goal (1 subgoal):
1. (h \<star> TTfgh_TfTgh.the_\<theta>) \<cdot> (can (\<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> ((\<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) (((\<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) \<cdot> \<a>\<^sup>-\<^sup>1[h \<star> tab\<^sub>0 h, g\<^sub>0h\<^sub>1.p\<^sub>0, f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine] \<cdot> \<a>\<^sup>-\<^sup>1[h, tab\<^sub>0 h, g\<^sub>0h\<^sub>1.p\<^sub>0 \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine]) \<cdot> \<a>[h, tab\<^sub>0 h, g\<^sub>0h\<^sub>1.p\<^sub>0 \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine] \<cdot> (h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0 \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine) \<cdot> \<a>[tab\<^sub>1 h, g\<^sub>0h\<^sub>1.p\<^sub>0, f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine] \<cdot> (g\<^sub>0h\<^sub>1.\<phi> \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine) \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1, f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine] \<cdot> (tab\<^sub>0 g \<star> \<gamma>\<^sub>g) \<cdot> \<a>[tab\<^sub>0 g, f\<^sub>0g\<^sub>1.p\<^sub>0, fg\<^sub>0h\<^sub>1.p\<^sub>1] \<cdot> local.inv fg\<^sub>0h\<^sub>1.\<phi> = ((h \<star> TTfgh_TfTgh.the_\<theta>) \<cdot> (h \<star> can (((\<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) (\<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>))) \<cdot> \<a>[h, tab\<^sub>0 h, g\<^sub>0h\<^sub>1.p\<^sub>0 \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine] \<cdot> (h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0 \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine) \<cdot> \<a>[tab\<^sub>1 h, g\<^sub>0h\<^sub>1.p\<^sub>0, f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine] \<cdot> (g\<^sub>0h\<^sub>1.\<phi> \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine) \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1, f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine] \<cdot> (tab\<^sub>0 g \<star> \<gamma>\<^sub>g) \<cdot> \<a>[tab\<^sub>0 g, f\<^sub>0g\<^sub>1.p\<^sub>0, fg\<^sub>0h\<^sub>1.p\<^sub>1] \<cdot> local.inv fg\<^sub>0h\<^sub>1.\<phi>
[PROOF STEP]
by presburger
[PROOF STATE]
proof (state)
this:
(h \<star> TTfgh_TfTgh.the_\<theta>) \<cdot> (can (\<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> ((\<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) (((\<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) \<cdot> \<a>\<^sup>-\<^sup>1[h \<star> tab\<^sub>0 h, g\<^sub>0h\<^sub>1.p\<^sub>0, f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine] \<cdot> \<a>\<^sup>-\<^sup>1[h, tab\<^sub>0 h, g\<^sub>0h\<^sub>1.p\<^sub>0 \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine]) \<cdot> \<a>[h, tab\<^sub>0 h, g\<^sub>0h\<^sub>1.p\<^sub>0 \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine] \<cdot> (h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0 \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine) \<cdot> \<a>[tab\<^sub>1 h, g\<^sub>0h\<^sub>1.p\<^sub>0, f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine] \<cdot> (g\<^sub>0h\<^sub>1.\<phi> \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine) \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1, f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine] \<cdot> (tab\<^sub>0 g \<star> \<gamma>\<^sub>g) \<cdot> \<a>[tab\<^sub>0 g, f\<^sub>0g\<^sub>1.p\<^sub>0, fg\<^sub>0h\<^sub>1.p\<^sub>1] \<cdot> local.inv fg\<^sub>0h\<^sub>1.\<phi> = ((h \<star> TTfgh_TfTgh.the_\<theta>) \<cdot> (h \<star> can (((\<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) (\<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>))) \<cdot> \<a>[h, tab\<^sub>0 h, g\<^sub>0h\<^sub>1.p\<^sub>0 \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine] \<cdot> (h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0 \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine) \<cdot> \<a>[tab\<^sub>1 h, g\<^sub>0h\<^sub>1.p\<^sub>0, f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine] \<cdot> (g\<^sub>0h\<^sub>1.\<phi> \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine) \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1, f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine] \<cdot> (tab\<^sub>0 g \<star> \<gamma>\<^sub>g) \<cdot> \<a>[tab\<^sub>0 g, f\<^sub>0g\<^sub>1.p\<^sub>0, fg\<^sub>0h\<^sub>1.p\<^sub>1] \<cdot> local.inv fg\<^sub>0h\<^sub>1.\<phi>
goal:
No subgoals!
[PROOF STEP]
qed
[PROOF STATE]
proof (state)
this:
(h \<star> TTfgh_TfTgh.the_\<theta>) \<cdot> (can (\<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> ((\<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) (((\<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) \<cdot> \<a>\<^sup>-\<^sup>1[h \<star> tab\<^sub>0 h, g\<^sub>0h\<^sub>1.p\<^sub>0, f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine] \<cdot> \<a>\<^sup>-\<^sup>1[h, tab\<^sub>0 h, g\<^sub>0h\<^sub>1.p\<^sub>0 \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine]) \<cdot> \<a>[h, tab\<^sub>0 h, g\<^sub>0h\<^sub>1.p\<^sub>0 \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine] \<cdot> (h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0 \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine) \<cdot> \<a>[tab\<^sub>1 h, g\<^sub>0h\<^sub>1.p\<^sub>0, f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine] \<cdot> (g\<^sub>0h\<^sub>1.\<phi> \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine) \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1, f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine] \<cdot> (tab\<^sub>0 g \<star> \<gamma>\<^sub>g) \<cdot> \<a>[tab\<^sub>0 g, f\<^sub>0g\<^sub>1.p\<^sub>0, fg\<^sub>0h\<^sub>1.p\<^sub>1] \<cdot> local.inv fg\<^sub>0h\<^sub>1.\<phi> = ((h \<star> TTfgh_TfTgh.the_\<theta>) \<cdot> (h \<star> can (((\<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) (\<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>))) \<cdot> \<a>[h, tab\<^sub>0 h, g\<^sub>0h\<^sub>1.p\<^sub>0 \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine] \<cdot> (h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0 \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine) \<cdot> \<a>[tab\<^sub>1 h, g\<^sub>0h\<^sub>1.p\<^sub>0, f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine] \<cdot> (g\<^sub>0h\<^sub>1.\<phi> \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine) \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1, f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine] \<cdot> (tab\<^sub>0 g \<star> \<gamma>\<^sub>g) \<cdot> \<a>[tab\<^sub>0 g, f\<^sub>0g\<^sub>1.p\<^sub>0, fg\<^sub>0h\<^sub>1.p\<^sub>1] \<cdot> local.inv fg\<^sub>0h\<^sub>1.\<phi>
goal (1 subgoal):
1. (h \<star> \<theta>\<^sub>h) \<cdot> \<a>[h, tab\<^sub>0 h, w\<^sub>h] \<cdot> (h.tab \<star> w\<^sub>h) = ((h \<star> \<theta>\<^sub>h') \<cdot> \<a>[h, tab\<^sub>0 h, w\<^sub>h'] \<cdot> (h.tab \<star> w\<^sub>h')) \<cdot> \<beta>\<^sub>h
[PROOF STEP]
also
[PROOF STATE]
proof (state)
this:
(h \<star> TTfgh_TfTgh.the_\<theta>) \<cdot> (can (\<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> ((\<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) (((\<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) \<cdot> \<a>\<^sup>-\<^sup>1[h \<star> tab\<^sub>0 h, g\<^sub>0h\<^sub>1.p\<^sub>0, f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine] \<cdot> \<a>\<^sup>-\<^sup>1[h, tab\<^sub>0 h, g\<^sub>0h\<^sub>1.p\<^sub>0 \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine]) \<cdot> \<a>[h, tab\<^sub>0 h, g\<^sub>0h\<^sub>1.p\<^sub>0 \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine] \<cdot> (h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0 \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine) \<cdot> \<a>[tab\<^sub>1 h, g\<^sub>0h\<^sub>1.p\<^sub>0, f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine] \<cdot> (g\<^sub>0h\<^sub>1.\<phi> \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine) \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1, f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine] \<cdot> (tab\<^sub>0 g \<star> \<gamma>\<^sub>g) \<cdot> \<a>[tab\<^sub>0 g, f\<^sub>0g\<^sub>1.p\<^sub>0, fg\<^sub>0h\<^sub>1.p\<^sub>1] \<cdot> local.inv fg\<^sub>0h\<^sub>1.\<phi> = ((h \<star> TTfgh_TfTgh.the_\<theta>) \<cdot> (h \<star> can (((\<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) (\<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>))) \<cdot> \<a>[h, tab\<^sub>0 h, g\<^sub>0h\<^sub>1.p\<^sub>0 \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine] \<cdot> (h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0 \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine) \<cdot> \<a>[tab\<^sub>1 h, g\<^sub>0h\<^sub>1.p\<^sub>0, f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine] \<cdot> (g\<^sub>0h\<^sub>1.\<phi> \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine) \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1, f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine] \<cdot> (tab\<^sub>0 g \<star> \<gamma>\<^sub>g) \<cdot> \<a>[tab\<^sub>0 g, f\<^sub>0g\<^sub>1.p\<^sub>0, fg\<^sub>0h\<^sub>1.p\<^sub>1] \<cdot> local.inv fg\<^sub>0h\<^sub>1.\<phi>
goal (1 subgoal):
1. (h \<star> \<theta>\<^sub>h) \<cdot> \<a>[h, tab\<^sub>0 h, w\<^sub>h] \<cdot> (h.tab \<star> w\<^sub>h) = ((h \<star> \<theta>\<^sub>h') \<cdot> \<a>[h, tab\<^sub>0 h, w\<^sub>h'] \<cdot> (h.tab \<star> w\<^sub>h')) \<cdot> \<beta>\<^sub>h
[PROOF STEP]
have "... = (h \<star> TTfgh_TfTgh.the_\<theta> \<cdot>
can (((\<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>Tgh.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TfTgh.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>)
(\<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>Tgh.p\<^sub>0\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>TfTgh.p\<^sub>0\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>)) \<cdot>
\<a>[h, tab\<^sub>0 h, Tgh.p\<^sub>0 \<star> TfTgh.p\<^sub>0 \<star> TTfgh_TfTgh.chine] \<cdot>
(h.tab \<star> Tgh.p\<^sub>0 \<star> TfTgh.p\<^sub>0 \<star> TTfgh_TfTgh.chine) \<cdot>
\<a>[tab\<^sub>1 h, Tgh.p\<^sub>0, TfTgh.p\<^sub>0 \<star> TTfgh_TfTgh.chine] \<cdot>
(g\<^sub>0h\<^sub>1.\<phi> \<star> TfTgh.p\<^sub>0 \<star> TTfgh_TfTgh.chine) \<cdot>
\<a>\<^sup>-\<^sup>1[tab\<^sub>0 g, Tgh.p\<^sub>1, TfTgh.p\<^sub>0 \<star> TTfgh_TfTgh.chine] \<cdot>
(tab\<^sub>0 g \<star> \<gamma>\<^sub>g) \<cdot>
\<a>[tab\<^sub>0 g, Tfg.p\<^sub>0, TTfgh.p\<^sub>1] \<cdot>
inv fg\<^sub>0h\<^sub>1.\<phi>"
[PROOF STATE]
proof (prove)
goal (1 subgoal):
1. ((h \<star> TTfgh_TfTgh.the_\<theta>) \<cdot> (h \<star> can (((\<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) (\<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>))) \<cdot> \<a>[h, tab\<^sub>0 h, g\<^sub>0h\<^sub>1.p\<^sub>0 \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine] \<cdot> (h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0 \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine) \<cdot> \<a>[tab\<^sub>1 h, g\<^sub>0h\<^sub>1.p\<^sub>0, f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine] \<cdot> (g\<^sub>0h\<^sub>1.\<phi> \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine) \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1, f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine] \<cdot> (tab\<^sub>0 g \<star> \<gamma>\<^sub>g) \<cdot> \<a>[tab\<^sub>0 g, f\<^sub>0g\<^sub>1.p\<^sub>0, fg\<^sub>0h\<^sub>1.p\<^sub>1] \<cdot> local.inv fg\<^sub>0h\<^sub>1.\<phi> = (h \<star> TTfgh_TfTgh.the_\<theta> \<cdot> can (((\<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) (\<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>)) \<cdot> \<a>[h, tab\<^sub>0 h, g\<^sub>0h\<^sub>1.p\<^sub>0 \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine] \<cdot> (h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0 \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine) \<cdot> \<a>[tab\<^sub>1 h, g\<^sub>0h\<^sub>1.p\<^sub>0, f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine] \<cdot> (g\<^sub>0h\<^sub>1.\<phi> \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine) \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1, f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine] \<cdot> (tab\<^sub>0 g \<star> \<gamma>\<^sub>g) \<cdot> \<a>[tab\<^sub>0 g, f\<^sub>0g\<^sub>1.p\<^sub>0, fg\<^sub>0h\<^sub>1.p\<^sub>1] \<cdot> local.inv fg\<^sub>0h\<^sub>1.\<phi>
[PROOF STEP]
using whisker_left [of h] comp_assoc
[PROOF STATE]
proof (prove)
using this:
\<lbrakk>ide h; seq ?\<nu> ?\<mu>\<rbrakk> \<Longrightarrow> h \<star> ?\<nu> \<cdot> ?\<mu> = (h \<star> ?\<nu>) \<cdot> (h \<star> ?\<mu>)
(?h \<cdot> ?g) \<cdot> ?f = ?h \<cdot> ?g \<cdot> ?f
goal (1 subgoal):
1. ((h \<star> TTfgh_TfTgh.the_\<theta>) \<cdot> (h \<star> can (((\<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) (\<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>))) \<cdot> \<a>[h, tab\<^sub>0 h, g\<^sub>0h\<^sub>1.p\<^sub>0 \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine] \<cdot> (h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0 \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine) \<cdot> \<a>[tab\<^sub>1 h, g\<^sub>0h\<^sub>1.p\<^sub>0, f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine] \<cdot> (g\<^sub>0h\<^sub>1.\<phi> \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine) \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1, f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine] \<cdot> (tab\<^sub>0 g \<star> \<gamma>\<^sub>g) \<cdot> \<a>[tab\<^sub>0 g, f\<^sub>0g\<^sub>1.p\<^sub>0, fg\<^sub>0h\<^sub>1.p\<^sub>1] \<cdot> local.inv fg\<^sub>0h\<^sub>1.\<phi> = (h \<star> TTfgh_TfTgh.the_\<theta> \<cdot> can (((\<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) (\<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>)) \<cdot> \<a>[h, tab\<^sub>0 h, g\<^sub>0h\<^sub>1.p\<^sub>0 \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine] \<cdot> (h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0 \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine) \<cdot> \<a>[tab\<^sub>1 h, g\<^sub>0h\<^sub>1.p\<^sub>0, f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine] \<cdot> (g\<^sub>0h\<^sub>1.\<phi> \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine) \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1, f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine] \<cdot> (tab\<^sub>0 g \<star> \<gamma>\<^sub>g) \<cdot> \<a>[tab\<^sub>0 g, f\<^sub>0g\<^sub>1.p\<^sub>0, fg\<^sub>0h\<^sub>1.p\<^sub>1] \<cdot> local.inv fg\<^sub>0h\<^sub>1.\<phi>
[PROOF STEP]
by simp
[PROOF STATE]
proof (state)
this:
((h \<star> TTfgh_TfTgh.the_\<theta>) \<cdot> (h \<star> can (((\<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) (\<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>))) \<cdot> \<a>[h, tab\<^sub>0 h, g\<^sub>0h\<^sub>1.p\<^sub>0 \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine] \<cdot> (h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0 \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine) \<cdot> \<a>[tab\<^sub>1 h, g\<^sub>0h\<^sub>1.p\<^sub>0, f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine] \<cdot> (g\<^sub>0h\<^sub>1.\<phi> \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine) \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1, f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine] \<cdot> (tab\<^sub>0 g \<star> \<gamma>\<^sub>g) \<cdot> \<a>[tab\<^sub>0 g, f\<^sub>0g\<^sub>1.p\<^sub>0, fg\<^sub>0h\<^sub>1.p\<^sub>1] \<cdot> local.inv fg\<^sub>0h\<^sub>1.\<phi> = (h \<star> TTfgh_TfTgh.the_\<theta> \<cdot> can (((\<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) (\<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>)) \<cdot> \<a>[h, tab\<^sub>0 h, g\<^sub>0h\<^sub>1.p\<^sub>0 \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine] \<cdot> (h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0 \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine) \<cdot> \<a>[tab\<^sub>1 h, g\<^sub>0h\<^sub>1.p\<^sub>0, f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine] \<cdot> (g\<^sub>0h\<^sub>1.\<phi> \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine) \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1, f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine] \<cdot> (tab\<^sub>0 g \<star> \<gamma>\<^sub>g) \<cdot> \<a>[tab\<^sub>0 g, f\<^sub>0g\<^sub>1.p\<^sub>0, fg\<^sub>0h\<^sub>1.p\<^sub>1] \<cdot> local.inv fg\<^sub>0h\<^sub>1.\<phi>
goal (1 subgoal):
1. (h \<star> \<theta>\<^sub>h) \<cdot> \<a>[h, tab\<^sub>0 h, w\<^sub>h] \<cdot> (h.tab \<star> w\<^sub>h) = ((h \<star> \<theta>\<^sub>h') \<cdot> \<a>[h, tab\<^sub>0 h, w\<^sub>h'] \<cdot> (h.tab \<star> w\<^sub>h')) \<cdot> \<beta>\<^sub>h
[PROOF STEP]
also
[PROOF STATE]
proof (state)
this:
((h \<star> TTfgh_TfTgh.the_\<theta>) \<cdot> (h \<star> can (((\<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) (\<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>))) \<cdot> \<a>[h, tab\<^sub>0 h, g\<^sub>0h\<^sub>1.p\<^sub>0 \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine] \<cdot> (h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0 \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine) \<cdot> \<a>[tab\<^sub>1 h, g\<^sub>0h\<^sub>1.p\<^sub>0, f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine] \<cdot> (g\<^sub>0h\<^sub>1.\<phi> \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine) \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1, f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine] \<cdot> (tab\<^sub>0 g \<star> \<gamma>\<^sub>g) \<cdot> \<a>[tab\<^sub>0 g, f\<^sub>0g\<^sub>1.p\<^sub>0, fg\<^sub>0h\<^sub>1.p\<^sub>1] \<cdot> local.inv fg\<^sub>0h\<^sub>1.\<phi> = (h \<star> TTfgh_TfTgh.the_\<theta> \<cdot> can (((\<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) (\<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>)) \<cdot> \<a>[h, tab\<^sub>0 h, g\<^sub>0h\<^sub>1.p\<^sub>0 \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine] \<cdot> (h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0 \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine) \<cdot> \<a>[tab\<^sub>1 h, g\<^sub>0h\<^sub>1.p\<^sub>0, f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine] \<cdot> (g\<^sub>0h\<^sub>1.\<phi> \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine) \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1, f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine] \<cdot> (tab\<^sub>0 g \<star> \<gamma>\<^sub>g) \<cdot> \<a>[tab\<^sub>0 g, f\<^sub>0g\<^sub>1.p\<^sub>0, fg\<^sub>0h\<^sub>1.p\<^sub>1] \<cdot> local.inv fg\<^sub>0h\<^sub>1.\<phi>
goal (1 subgoal):
1. (h \<star> \<theta>\<^sub>h) \<cdot> \<a>[h, tab\<^sub>0 h, w\<^sub>h] \<cdot> (h.tab \<star> w\<^sub>h) = ((h \<star> \<theta>\<^sub>h') \<cdot> \<a>[h, tab\<^sub>0 h, w\<^sub>h'] \<cdot> (h.tab \<star> w\<^sub>h')) \<cdot> \<beta>\<^sub>h
[PROOF STEP]
have "... = (h \<star> TTfgh_TfTgh.the_\<theta> \<cdot>
\<a>\<^sup>-\<^sup>1[tab\<^sub>0 h \<star> Tgh.p\<^sub>0, TfTgh.p\<^sub>0, TTfgh_TfTgh.chine] \<cdot>
\<a>\<^sup>-\<^sup>1[tab\<^sub>0 h, Tgh.p\<^sub>0, TfTgh.p\<^sub>0 \<star> TTfgh_TfTgh.chine]) \<cdot>
\<a>[h, tab\<^sub>0 h, Tgh.p\<^sub>0 \<star> TfTgh.p\<^sub>0 \<star> TTfgh_TfTgh.chine] \<cdot>
(h.tab \<star> Tgh.p\<^sub>0 \<star> TfTgh.p\<^sub>0 \<star> TTfgh_TfTgh.chine) \<cdot>
\<a>[tab\<^sub>1 h, Tgh.p\<^sub>0, TfTgh.p\<^sub>0 \<star> TTfgh_TfTgh.chine] \<cdot>
(g\<^sub>0h\<^sub>1.\<phi> \<star> TfTgh.p\<^sub>0 \<star> TTfgh_TfTgh.chine) \<cdot>
\<a>\<^sup>-\<^sup>1[tab\<^sub>0 g, Tgh.p\<^sub>1, TfTgh.p\<^sub>0 \<star> TTfgh_TfTgh.chine] \<cdot>
(tab\<^sub>0 g \<star> \<gamma>\<^sub>g) \<cdot>
\<a>[tab\<^sub>0 g, Tfg.p\<^sub>0, TTfgh.p\<^sub>1] \<cdot>
inv fg\<^sub>0h\<^sub>1.\<phi>"
[PROOF STATE]
proof (prove)
goal (1 subgoal):
1. (h \<star> TTfgh_TfTgh.the_\<theta> \<cdot> can (((\<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) (\<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>)) \<cdot> \<a>[h, tab\<^sub>0 h, g\<^sub>0h\<^sub>1.p\<^sub>0 \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine] \<cdot> (h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0 \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine) \<cdot> \<a>[tab\<^sub>1 h, g\<^sub>0h\<^sub>1.p\<^sub>0, f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine] \<cdot> (g\<^sub>0h\<^sub>1.\<phi> \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine) \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1, f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine] \<cdot> (tab\<^sub>0 g \<star> \<gamma>\<^sub>g) \<cdot> \<a>[tab\<^sub>0 g, f\<^sub>0g\<^sub>1.p\<^sub>0, fg\<^sub>0h\<^sub>1.p\<^sub>1] \<cdot> local.inv fg\<^sub>0h\<^sub>1.\<phi> = (h \<star> TTfgh_TfTgh.the_\<theta> \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0, f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine] \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>0 h, g\<^sub>0h\<^sub>1.p\<^sub>0, f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine]) \<cdot> \<a>[h, tab\<^sub>0 h, g\<^sub>0h\<^sub>1.p\<^sub>0 \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine] \<cdot> (h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0 \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine) \<cdot> \<a>[tab\<^sub>1 h, g\<^sub>0h\<^sub>1.p\<^sub>0, f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine] \<cdot> (g\<^sub>0h\<^sub>1.\<phi> \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine) \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1, f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine] \<cdot> (tab\<^sub>0 g \<star> \<gamma>\<^sub>g) \<cdot> \<a>[tab\<^sub>0 g, f\<^sub>0g\<^sub>1.p\<^sub>0, fg\<^sub>0h\<^sub>1.p\<^sub>1] \<cdot> local.inv fg\<^sub>0h\<^sub>1.\<phi>
[PROOF STEP]
proof -
[PROOF STATE]
proof (state)
goal (1 subgoal):
1. (h \<star> TTfgh_TfTgh.the_\<theta> \<cdot> can (((\<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) (\<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>)) \<cdot> \<a>[h, tab\<^sub>0 h, g\<^sub>0h\<^sub>1.p\<^sub>0 \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine] \<cdot> (h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0 \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine) \<cdot> \<a>[tab\<^sub>1 h, g\<^sub>0h\<^sub>1.p\<^sub>0, f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine] \<cdot> (g\<^sub>0h\<^sub>1.\<phi> \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine) \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1, f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine] \<cdot> (tab\<^sub>0 g \<star> \<gamma>\<^sub>g) \<cdot> \<a>[tab\<^sub>0 g, f\<^sub>0g\<^sub>1.p\<^sub>0, fg\<^sub>0h\<^sub>1.p\<^sub>1] \<cdot> local.inv fg\<^sub>0h\<^sub>1.\<phi> = (h \<star> TTfgh_TfTgh.the_\<theta> \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0, f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine] \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>0 h, g\<^sub>0h\<^sub>1.p\<^sub>0, f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine]) \<cdot> \<a>[h, tab\<^sub>0 h, g\<^sub>0h\<^sub>1.p\<^sub>0 \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine] \<cdot> (h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0 \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine) \<cdot> \<a>[tab\<^sub>1 h, g\<^sub>0h\<^sub>1.p\<^sub>0, f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine] \<cdot> (g\<^sub>0h\<^sub>1.\<phi> \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine) \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1, f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine] \<cdot> (tab\<^sub>0 g \<star> \<gamma>\<^sub>g) \<cdot> \<a>[tab\<^sub>0 g, f\<^sub>0g\<^sub>1.p\<^sub>0, fg\<^sub>0h\<^sub>1.p\<^sub>1] \<cdot> local.inv fg\<^sub>0h\<^sub>1.\<phi>
[PROOF STEP]
have "can (((\<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>Tgh.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TfTgh.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>)
(\<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>Tgh.p\<^sub>0\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>TfTgh.p\<^sub>0\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) =
\<lbrace>\<^bold>\<a>\<^sup>-\<^sup>1\<^bold>[\<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>Tgh.p\<^sub>0\<^bold>\<rangle>, \<^bold>\<langle>TfTgh.p\<^sub>0\<^bold>\<rangle>, \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>\<^bold>] \<^bold>\<cdot>
\<^bold>\<a>\<^sup>-\<^sup>1\<^bold>[\<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle>, \<^bold>\<langle>Tgh.p\<^sub>0\<^bold>\<rangle>, \<^bold>\<langle>TfTgh.p\<^sub>0\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>\<^bold>]\<rbrace>"
[PROOF STATE]
proof (prove)
goal (1 subgoal):
1. can (((\<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) (\<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) = \<lbrace>\<^bold>\<a>\<^sup>-\<^sup>1\<^bold>[\<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>, \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>, \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>\<^bold>] \<^bold>\<cdot> \<^bold>\<a>\<^sup>-\<^sup>1\<^bold>[\<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle>, \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>, \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>\<^bold>]\<rbrace>
[PROOF STEP]
unfolding can_def
[PROOF STATE]
proof (prove)
goal (1 subgoal):
1. \<lbrace>E.Inv ((((\<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>)\<^bold>\<down>) \<^bold>\<cdot> (\<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>)\<^bold>\<down>\<rbrace> = \<lbrace>\<^bold>\<a>\<^sup>-\<^sup>1\<^bold>[\<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>, \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>, \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>\<^bold>] \<^bold>\<cdot> \<^bold>\<a>\<^sup>-\<^sup>1\<^bold>[\<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle>, \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>, \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>\<^bold>]\<rbrace>
[PROOF STEP]
apply (intro E.eval_eqI)
[PROOF STATE]
proof (prove)
goal (2 subgoals):
1. E.VPar (E.Inv ((((\<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>)\<^bold>\<down>) \<^bold>\<cdot> (\<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>)\<^bold>\<down>) (\<^bold>\<a>\<^sup>-\<^sup>1\<^bold>[\<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>, \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>, \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>\<^bold>] \<^bold>\<cdot> \<^bold>\<a>\<^sup>-\<^sup>1\<^bold>[\<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle>, \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>, \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>\<^bold>])
2. \<^bold>\<lfloor>E.Inv ((((\<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>)\<^bold>\<down>) \<^bold>\<cdot> (\<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>)\<^bold>\<down>\<^bold>\<rfloor> = \<^bold>\<lfloor>\<^bold>\<a>\<^sup>-\<^sup>1\<^bold>[\<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>, \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>, \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>\<^bold>] \<^bold>\<cdot> \<^bold>\<a>\<^sup>-\<^sup>1\<^bold>[\<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle>, \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>, \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>\<^bold>]\<^bold>\<rfloor>
[PROOF STEP]
by auto
[PROOF STATE]
proof (state)
this:
can (((\<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) (\<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) = \<lbrace>\<^bold>\<a>\<^sup>-\<^sup>1\<^bold>[\<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>, \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>, \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>\<^bold>] \<^bold>\<cdot> \<^bold>\<a>\<^sup>-\<^sup>1\<^bold>[\<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle>, \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>, \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>\<^bold>]\<rbrace>
goal (1 subgoal):
1. (h \<star> TTfgh_TfTgh.the_\<theta> \<cdot> can (((\<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) (\<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>)) \<cdot> \<a>[h, tab\<^sub>0 h, g\<^sub>0h\<^sub>1.p\<^sub>0 \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine] \<cdot> (h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0 \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine) \<cdot> \<a>[tab\<^sub>1 h, g\<^sub>0h\<^sub>1.p\<^sub>0, f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine] \<cdot> (g\<^sub>0h\<^sub>1.\<phi> \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine) \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1, f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine] \<cdot> (tab\<^sub>0 g \<star> \<gamma>\<^sub>g) \<cdot> \<a>[tab\<^sub>0 g, f\<^sub>0g\<^sub>1.p\<^sub>0, fg\<^sub>0h\<^sub>1.p\<^sub>1] \<cdot> local.inv fg\<^sub>0h\<^sub>1.\<phi> = (h \<star> TTfgh_TfTgh.the_\<theta> \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0, f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine] \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>0 h, g\<^sub>0h\<^sub>1.p\<^sub>0, f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine]) \<cdot> \<a>[h, tab\<^sub>0 h, g\<^sub>0h\<^sub>1.p\<^sub>0 \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine] \<cdot> (h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0 \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine) \<cdot> \<a>[tab\<^sub>1 h, g\<^sub>0h\<^sub>1.p\<^sub>0, f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine] \<cdot> (g\<^sub>0h\<^sub>1.\<phi> \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine) \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1, f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine] \<cdot> (tab\<^sub>0 g \<star> \<gamma>\<^sub>g) \<cdot> \<a>[tab\<^sub>0 g, f\<^sub>0g\<^sub>1.p\<^sub>0, fg\<^sub>0h\<^sub>1.p\<^sub>1] \<cdot> local.inv fg\<^sub>0h\<^sub>1.\<phi>
[PROOF STEP]
also
[PROOF STATE]
proof (state)
this:
can (((\<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) (\<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) = \<lbrace>\<^bold>\<a>\<^sup>-\<^sup>1\<^bold>[\<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>, \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>, \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>\<^bold>] \<^bold>\<cdot> \<^bold>\<a>\<^sup>-\<^sup>1\<^bold>[\<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle>, \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>, \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>\<^bold>]\<rbrace>
goal (1 subgoal):
1. (h \<star> TTfgh_TfTgh.the_\<theta> \<cdot> can (((\<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) (\<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>)) \<cdot> \<a>[h, tab\<^sub>0 h, g\<^sub>0h\<^sub>1.p\<^sub>0 \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine] \<cdot> (h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0 \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine) \<cdot> \<a>[tab\<^sub>1 h, g\<^sub>0h\<^sub>1.p\<^sub>0, f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine] \<cdot> (g\<^sub>0h\<^sub>1.\<phi> \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine) \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1, f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine] \<cdot> (tab\<^sub>0 g \<star> \<gamma>\<^sub>g) \<cdot> \<a>[tab\<^sub>0 g, f\<^sub>0g\<^sub>1.p\<^sub>0, fg\<^sub>0h\<^sub>1.p\<^sub>1] \<cdot> local.inv fg\<^sub>0h\<^sub>1.\<phi> = (h \<star> TTfgh_TfTgh.the_\<theta> \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0, f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine] \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>0 h, g\<^sub>0h\<^sub>1.p\<^sub>0, f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine]) \<cdot> \<a>[h, tab\<^sub>0 h, g\<^sub>0h\<^sub>1.p\<^sub>0 \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine] \<cdot> (h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0 \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine) \<cdot> \<a>[tab\<^sub>1 h, g\<^sub>0h\<^sub>1.p\<^sub>0, f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine] \<cdot> (g\<^sub>0h\<^sub>1.\<phi> \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine) \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1, f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine] \<cdot> (tab\<^sub>0 g \<star> \<gamma>\<^sub>g) \<cdot> \<a>[tab\<^sub>0 g, f\<^sub>0g\<^sub>1.p\<^sub>0, fg\<^sub>0h\<^sub>1.p\<^sub>1] \<cdot> local.inv fg\<^sub>0h\<^sub>1.\<phi>
[PROOF STEP]
have "... = \<a>\<^sup>-\<^sup>1[tab\<^sub>0 h \<star> Tgh.p\<^sub>0, TfTgh.p\<^sub>0, TTfgh_TfTgh.chine] \<cdot>
\<a>\<^sup>-\<^sup>1[tab\<^sub>0 h, Tgh.p\<^sub>0, TfTgh.p\<^sub>0 \<star> TTfgh_TfTgh.chine]"
[PROOF STATE]
proof (prove)
goal (1 subgoal):
1. \<lbrace>\<^bold>\<a>\<^sup>-\<^sup>1\<^bold>[\<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>, \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>, \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>\<^bold>] \<^bold>\<cdot> \<^bold>\<a>\<^sup>-\<^sup>1\<^bold>[\<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle>, \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>, \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>\<^bold>]\<rbrace> = \<a>\<^sup>-\<^sup>1[tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0, f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine] \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>0 h, g\<^sub>0h\<^sub>1.p\<^sub>0, f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine]
[PROOF STEP]
using \<a>'_def \<alpha>_def
[PROOF STATE]
proof (prove)
using this:
\<a>\<^sup>-\<^sup>1[?\<mu>, ?\<nu>, ?\<tau>] \<equiv> \<alpha>' (?\<mu>, ?\<nu>, ?\<tau>)
\<alpha> ?\<mu>\<nu>\<tau> \<equiv> \<a>[fst ?\<mu>\<nu>\<tau>, fst (snd ?\<mu>\<nu>\<tau>), snd (snd ?\<mu>\<nu>\<tau>)]
goal (1 subgoal):
1. \<lbrace>\<^bold>\<a>\<^sup>-\<^sup>1\<^bold>[\<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>, \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>, \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>\<^bold>] \<^bold>\<cdot> \<^bold>\<a>\<^sup>-\<^sup>1\<^bold>[\<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle>, \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>, \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>\<^bold>]\<rbrace> = \<a>\<^sup>-\<^sup>1[tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0, f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine] \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>0 h, g\<^sub>0h\<^sub>1.p\<^sub>0, f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine]
[PROOF STEP]
by simp
[PROOF STATE]
proof (state)
this:
\<lbrace>\<^bold>\<a>\<^sup>-\<^sup>1\<^bold>[\<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>, \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>, \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>\<^bold>] \<^bold>\<cdot> \<^bold>\<a>\<^sup>-\<^sup>1\<^bold>[\<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle>, \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>, \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>\<^bold>]\<rbrace> = \<a>\<^sup>-\<^sup>1[tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0, f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine] \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>0 h, g\<^sub>0h\<^sub>1.p\<^sub>0, f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine]
goal (1 subgoal):
1. (h \<star> TTfgh_TfTgh.the_\<theta> \<cdot> can (((\<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) (\<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>)) \<cdot> \<a>[h, tab\<^sub>0 h, g\<^sub>0h\<^sub>1.p\<^sub>0 \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine] \<cdot> (h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0 \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine) \<cdot> \<a>[tab\<^sub>1 h, g\<^sub>0h\<^sub>1.p\<^sub>0, f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine] \<cdot> (g\<^sub>0h\<^sub>1.\<phi> \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine) \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1, f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine] \<cdot> (tab\<^sub>0 g \<star> \<gamma>\<^sub>g) \<cdot> \<a>[tab\<^sub>0 g, f\<^sub>0g\<^sub>1.p\<^sub>0, fg\<^sub>0h\<^sub>1.p\<^sub>1] \<cdot> local.inv fg\<^sub>0h\<^sub>1.\<phi> = (h \<star> TTfgh_TfTgh.the_\<theta> \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0, f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine] \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>0 h, g\<^sub>0h\<^sub>1.p\<^sub>0, f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine]) \<cdot> \<a>[h, tab\<^sub>0 h, g\<^sub>0h\<^sub>1.p\<^sub>0 \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine] \<cdot> (h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0 \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine) \<cdot> \<a>[tab\<^sub>1 h, g\<^sub>0h\<^sub>1.p\<^sub>0, f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine] \<cdot> (g\<^sub>0h\<^sub>1.\<phi> \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine) \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1, f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine] \<cdot> (tab\<^sub>0 g \<star> \<gamma>\<^sub>g) \<cdot> \<a>[tab\<^sub>0 g, f\<^sub>0g\<^sub>1.p\<^sub>0, fg\<^sub>0h\<^sub>1.p\<^sub>1] \<cdot> local.inv fg\<^sub>0h\<^sub>1.\<phi>
[PROOF STEP]
finally
[PROOF STATE]
proof (chain)
picking this:
can (((\<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) (\<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) = \<a>\<^sup>-\<^sup>1[tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0, f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine] \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>0 h, g\<^sub>0h\<^sub>1.p\<^sub>0, f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine]
[PROOF STEP]
have "can (((\<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>Tgh.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TfTgh.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>)
(\<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>Tgh.p\<^sub>0\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>TfTgh.p\<^sub>0\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) =
\<a>\<^sup>-\<^sup>1[tab\<^sub>0 h \<star> Tgh.p\<^sub>0, TfTgh.p\<^sub>0, TTfgh_TfTgh.chine] \<cdot>
\<a>\<^sup>-\<^sup>1[tab\<^sub>0 h, Tgh.p\<^sub>0, TfTgh.p\<^sub>0 \<star> TTfgh_TfTgh.chine]"
[PROOF STATE]
proof (prove)
using this:
can (((\<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) (\<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) = \<a>\<^sup>-\<^sup>1[tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0, f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine] \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>0 h, g\<^sub>0h\<^sub>1.p\<^sub>0, f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine]
goal (1 subgoal):
1. can (((\<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) (\<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) = \<a>\<^sup>-\<^sup>1[tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0, f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine] \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>0 h, g\<^sub>0h\<^sub>1.p\<^sub>0, f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine]
[PROOF STEP]
by simp
[PROOF STATE]
proof (state)
this:
can (((\<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) (\<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) = \<a>\<^sup>-\<^sup>1[tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0, f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine] \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>0 h, g\<^sub>0h\<^sub>1.p\<^sub>0, f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine]
goal (1 subgoal):
1. (h \<star> TTfgh_TfTgh.the_\<theta> \<cdot> can (((\<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) (\<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>)) \<cdot> \<a>[h, tab\<^sub>0 h, g\<^sub>0h\<^sub>1.p\<^sub>0 \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine] \<cdot> (h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0 \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine) \<cdot> \<a>[tab\<^sub>1 h, g\<^sub>0h\<^sub>1.p\<^sub>0, f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine] \<cdot> (g\<^sub>0h\<^sub>1.\<phi> \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine) \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1, f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine] \<cdot> (tab\<^sub>0 g \<star> \<gamma>\<^sub>g) \<cdot> \<a>[tab\<^sub>0 g, f\<^sub>0g\<^sub>1.p\<^sub>0, fg\<^sub>0h\<^sub>1.p\<^sub>1] \<cdot> local.inv fg\<^sub>0h\<^sub>1.\<phi> = (h \<star> TTfgh_TfTgh.the_\<theta> \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0, f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine] \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>0 h, g\<^sub>0h\<^sub>1.p\<^sub>0, f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine]) \<cdot> \<a>[h, tab\<^sub>0 h, g\<^sub>0h\<^sub>1.p\<^sub>0 \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine] \<cdot> (h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0 \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine) \<cdot> \<a>[tab\<^sub>1 h, g\<^sub>0h\<^sub>1.p\<^sub>0, f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine] \<cdot> (g\<^sub>0h\<^sub>1.\<phi> \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine) \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1, f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine] \<cdot> (tab\<^sub>0 g \<star> \<gamma>\<^sub>g) \<cdot> \<a>[tab\<^sub>0 g, f\<^sub>0g\<^sub>1.p\<^sub>0, fg\<^sub>0h\<^sub>1.p\<^sub>1] \<cdot> local.inv fg\<^sub>0h\<^sub>1.\<phi>
[PROOF STEP]
thus ?thesis
[PROOF STATE]
proof (prove)
using this:
can (((\<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) (\<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) = \<a>\<^sup>-\<^sup>1[tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0, f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine] \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>0 h, g\<^sub>0h\<^sub>1.p\<^sub>0, f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine]
goal (1 subgoal):
1. (h \<star> TTfgh_TfTgh.the_\<theta> \<cdot> can (((\<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) (\<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>)) \<cdot> \<a>[h, tab\<^sub>0 h, g\<^sub>0h\<^sub>1.p\<^sub>0 \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine] \<cdot> (h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0 \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine) \<cdot> \<a>[tab\<^sub>1 h, g\<^sub>0h\<^sub>1.p\<^sub>0, f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine] \<cdot> (g\<^sub>0h\<^sub>1.\<phi> \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine) \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1, f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine] \<cdot> (tab\<^sub>0 g \<star> \<gamma>\<^sub>g) \<cdot> \<a>[tab\<^sub>0 g, f\<^sub>0g\<^sub>1.p\<^sub>0, fg\<^sub>0h\<^sub>1.p\<^sub>1] \<cdot> local.inv fg\<^sub>0h\<^sub>1.\<phi> = (h \<star> TTfgh_TfTgh.the_\<theta> \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0, f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine] \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>0 h, g\<^sub>0h\<^sub>1.p\<^sub>0, f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine]) \<cdot> \<a>[h, tab\<^sub>0 h, g\<^sub>0h\<^sub>1.p\<^sub>0 \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine] \<cdot> (h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0 \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine) \<cdot> \<a>[tab\<^sub>1 h, g\<^sub>0h\<^sub>1.p\<^sub>0, f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine] \<cdot> (g\<^sub>0h\<^sub>1.\<phi> \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine) \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1, f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine] \<cdot> (tab\<^sub>0 g \<star> \<gamma>\<^sub>g) \<cdot> \<a>[tab\<^sub>0 g, f\<^sub>0g\<^sub>1.p\<^sub>0, fg\<^sub>0h\<^sub>1.p\<^sub>1] \<cdot> local.inv fg\<^sub>0h\<^sub>1.\<phi>
[PROOF STEP]
by simp
[PROOF STATE]
proof (state)
this:
(h \<star> TTfgh_TfTgh.the_\<theta> \<cdot> can (((\<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) (\<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>)) \<cdot> \<a>[h, tab\<^sub>0 h, g\<^sub>0h\<^sub>1.p\<^sub>0 \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine] \<cdot> (h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0 \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine) \<cdot> \<a>[tab\<^sub>1 h, g\<^sub>0h\<^sub>1.p\<^sub>0, f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine] \<cdot> (g\<^sub>0h\<^sub>1.\<phi> \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine) \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1, f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine] \<cdot> (tab\<^sub>0 g \<star> \<gamma>\<^sub>g) \<cdot> \<a>[tab\<^sub>0 g, f\<^sub>0g\<^sub>1.p\<^sub>0, fg\<^sub>0h\<^sub>1.p\<^sub>1] \<cdot> local.inv fg\<^sub>0h\<^sub>1.\<phi> = (h \<star> TTfgh_TfTgh.the_\<theta> \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0, f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine] \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>0 h, g\<^sub>0h\<^sub>1.p\<^sub>0, f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine]) \<cdot> \<a>[h, tab\<^sub>0 h, g\<^sub>0h\<^sub>1.p\<^sub>0 \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine] \<cdot> (h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0 \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine) \<cdot> \<a>[tab\<^sub>1 h, g\<^sub>0h\<^sub>1.p\<^sub>0, f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine] \<cdot> (g\<^sub>0h\<^sub>1.\<phi> \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine) \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1, f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine] \<cdot> (tab\<^sub>0 g \<star> \<gamma>\<^sub>g) \<cdot> \<a>[tab\<^sub>0 g, f\<^sub>0g\<^sub>1.p\<^sub>0, fg\<^sub>0h\<^sub>1.p\<^sub>1] \<cdot> local.inv fg\<^sub>0h\<^sub>1.\<phi>
goal:
No subgoals!
[PROOF STEP]
qed
[PROOF STATE]
proof (state)
this:
(h \<star> TTfgh_TfTgh.the_\<theta> \<cdot> can (((\<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) (\<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>)) \<cdot> \<a>[h, tab\<^sub>0 h, g\<^sub>0h\<^sub>1.p\<^sub>0 \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine] \<cdot> (h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0 \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine) \<cdot> \<a>[tab\<^sub>1 h, g\<^sub>0h\<^sub>1.p\<^sub>0, f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine] \<cdot> (g\<^sub>0h\<^sub>1.\<phi> \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine) \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1, f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine] \<cdot> (tab\<^sub>0 g \<star> \<gamma>\<^sub>g) \<cdot> \<a>[tab\<^sub>0 g, f\<^sub>0g\<^sub>1.p\<^sub>0, fg\<^sub>0h\<^sub>1.p\<^sub>1] \<cdot> local.inv fg\<^sub>0h\<^sub>1.\<phi> = (h \<star> TTfgh_TfTgh.the_\<theta> \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0, f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine] \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>0 h, g\<^sub>0h\<^sub>1.p\<^sub>0, f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine]) \<cdot> \<a>[h, tab\<^sub>0 h, g\<^sub>0h\<^sub>1.p\<^sub>0 \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine] \<cdot> (h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0 \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine) \<cdot> \<a>[tab\<^sub>1 h, g\<^sub>0h\<^sub>1.p\<^sub>0, f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine] \<cdot> (g\<^sub>0h\<^sub>1.\<phi> \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine) \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1, f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine] \<cdot> (tab\<^sub>0 g \<star> \<gamma>\<^sub>g) \<cdot> \<a>[tab\<^sub>0 g, f\<^sub>0g\<^sub>1.p\<^sub>0, fg\<^sub>0h\<^sub>1.p\<^sub>1] \<cdot> local.inv fg\<^sub>0h\<^sub>1.\<phi>
goal (1 subgoal):
1. (h \<star> \<theta>\<^sub>h) \<cdot> \<a>[h, tab\<^sub>0 h, w\<^sub>h] \<cdot> (h.tab \<star> w\<^sub>h) = ((h \<star> \<theta>\<^sub>h') \<cdot> \<a>[h, tab\<^sub>0 h, w\<^sub>h'] \<cdot> (h.tab \<star> w\<^sub>h')) \<cdot> \<beta>\<^sub>h
[PROOF STEP]
also
[PROOF STATE]
proof (state)
this:
(h \<star> TTfgh_TfTgh.the_\<theta> \<cdot> can (((\<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>) (\<^bold>\<langle>tab\<^sub>0 h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>g\<^sub>0h\<^sub>1.p\<^sub>0\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>f\<^sub>0gh\<^sub>1.p\<^sub>0\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>TTfgh_TfTgh.chine\<^bold>\<rangle>)) \<cdot> \<a>[h, tab\<^sub>0 h, g\<^sub>0h\<^sub>1.p\<^sub>0 \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine] \<cdot> (h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0 \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine) \<cdot> \<a>[tab\<^sub>1 h, g\<^sub>0h\<^sub>1.p\<^sub>0, f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine] \<cdot> (g\<^sub>0h\<^sub>1.\<phi> \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine) \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1, f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine] \<cdot> (tab\<^sub>0 g \<star> \<gamma>\<^sub>g) \<cdot> \<a>[tab\<^sub>0 g, f\<^sub>0g\<^sub>1.p\<^sub>0, fg\<^sub>0h\<^sub>1.p\<^sub>1] \<cdot> local.inv fg\<^sub>0h\<^sub>1.\<phi> = (h \<star> TTfgh_TfTgh.the_\<theta> \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0, f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine] \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>0 h, g\<^sub>0h\<^sub>1.p\<^sub>0, f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine]) \<cdot> \<a>[h, tab\<^sub>0 h, g\<^sub>0h\<^sub>1.p\<^sub>0 \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine] \<cdot> (h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0 \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine) \<cdot> \<a>[tab\<^sub>1 h, g\<^sub>0h\<^sub>1.p\<^sub>0, f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine] \<cdot> (g\<^sub>0h\<^sub>1.\<phi> \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine) \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1, f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine] \<cdot> (tab\<^sub>0 g \<star> \<gamma>\<^sub>g) \<cdot> \<a>[tab\<^sub>0 g, f\<^sub>0g\<^sub>1.p\<^sub>0, fg\<^sub>0h\<^sub>1.p\<^sub>1] \<cdot> local.inv fg\<^sub>0h\<^sub>1.\<phi>
goal (1 subgoal):
1. (h \<star> \<theta>\<^sub>h) \<cdot> \<a>[h, tab\<^sub>0 h, w\<^sub>h] \<cdot> (h.tab \<star> w\<^sub>h) = ((h \<star> \<theta>\<^sub>h') \<cdot> \<a>[h, tab\<^sub>0 h, w\<^sub>h'] \<cdot> (h.tab \<star> w\<^sub>h')) \<cdot> \<beta>\<^sub>h
[PROOF STEP]
have "... = h.composite_cell w\<^sub>h' \<theta>\<^sub>h' \<cdot> \<beta>\<^sub>h"
[PROOF STATE]
proof (prove)
goal (1 subgoal):
1. (h \<star> TTfgh_TfTgh.the_\<theta> \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0, f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine] \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>0 h, g\<^sub>0h\<^sub>1.p\<^sub>0, f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine]) \<cdot> \<a>[h, tab\<^sub>0 h, g\<^sub>0h\<^sub>1.p\<^sub>0 \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine] \<cdot> (h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0 \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine) \<cdot> \<a>[tab\<^sub>1 h, g\<^sub>0h\<^sub>1.p\<^sub>0, f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine] \<cdot> (g\<^sub>0h\<^sub>1.\<phi> \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine) \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1, f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine] \<cdot> (tab\<^sub>0 g \<star> \<gamma>\<^sub>g) \<cdot> \<a>[tab\<^sub>0 g, f\<^sub>0g\<^sub>1.p\<^sub>0, fg\<^sub>0h\<^sub>1.p\<^sub>1] \<cdot> local.inv fg\<^sub>0h\<^sub>1.\<phi> = ((h \<star> \<theta>\<^sub>h') \<cdot> \<a>[h, tab\<^sub>0 h, w\<^sub>h'] \<cdot> (h.tab \<star> w\<^sub>h')) \<cdot> \<beta>\<^sub>h
[PROOF STEP]
unfolding w\<^sub>h'_def \<theta>\<^sub>h'_def \<beta>\<^sub>h_def
[PROOF STATE]
proof (prove)
goal (1 subgoal):
1. (h \<star> TTfgh_TfTgh.the_\<theta> \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0, f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine] \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>0 h, g\<^sub>0h\<^sub>1.p\<^sub>0, f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine]) \<cdot> \<a>[h, tab\<^sub>0 h, g\<^sub>0h\<^sub>1.p\<^sub>0 \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine] \<cdot> (h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0 \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine) \<cdot> \<a>[tab\<^sub>1 h, g\<^sub>0h\<^sub>1.p\<^sub>0, f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine] \<cdot> (g\<^sub>0h\<^sub>1.\<phi> \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine) \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1, f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine] \<cdot> (tab\<^sub>0 g \<star> \<gamma>\<^sub>g) \<cdot> \<a>[tab\<^sub>0 g, f\<^sub>0g\<^sub>1.p\<^sub>0, fg\<^sub>0h\<^sub>1.p\<^sub>1] \<cdot> local.inv fg\<^sub>0h\<^sub>1.\<phi> = ((h \<star> TTfgh_TfTgh.the_\<theta> \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0, f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine] \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>0 h, g\<^sub>0h\<^sub>1.p\<^sub>0, f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine]) \<cdot> \<a>[h, tab\<^sub>0 h, g\<^sub>0h\<^sub>1.p\<^sub>0 \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine] \<cdot> (h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0 \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine)) \<cdot> \<a>[tab\<^sub>1 h, g\<^sub>0h\<^sub>1.p\<^sub>0, f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine] \<cdot> (g\<^sub>0h\<^sub>1.\<phi> \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine) \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1, f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine] \<cdot> (tab\<^sub>0 g \<star> \<gamma>\<^sub>g) \<cdot> \<a>[tab\<^sub>0 g, f\<^sub>0g\<^sub>1.p\<^sub>0, fg\<^sub>0h\<^sub>1.p\<^sub>1] \<cdot> local.inv fg\<^sub>0h\<^sub>1.\<phi>
[PROOF STEP]
using comp_assoc
[PROOF STATE]
proof (prove)
using this:
(?h \<cdot> ?g) \<cdot> ?f = ?h \<cdot> ?g \<cdot> ?f
goal (1 subgoal):
1. (h \<star> TTfgh_TfTgh.the_\<theta> \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0, f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine] \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>0 h, g\<^sub>0h\<^sub>1.p\<^sub>0, f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine]) \<cdot> \<a>[h, tab\<^sub>0 h, g\<^sub>0h\<^sub>1.p\<^sub>0 \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine] \<cdot> (h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0 \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine) \<cdot> \<a>[tab\<^sub>1 h, g\<^sub>0h\<^sub>1.p\<^sub>0, f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine] \<cdot> (g\<^sub>0h\<^sub>1.\<phi> \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine) \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1, f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine] \<cdot> (tab\<^sub>0 g \<star> \<gamma>\<^sub>g) \<cdot> \<a>[tab\<^sub>0 g, f\<^sub>0g\<^sub>1.p\<^sub>0, fg\<^sub>0h\<^sub>1.p\<^sub>1] \<cdot> local.inv fg\<^sub>0h\<^sub>1.\<phi> = ((h \<star> TTfgh_TfTgh.the_\<theta> \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0, f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine] \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>0 h, g\<^sub>0h\<^sub>1.p\<^sub>0, f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine]) \<cdot> \<a>[h, tab\<^sub>0 h, g\<^sub>0h\<^sub>1.p\<^sub>0 \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine] \<cdot> (h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0 \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine)) \<cdot> \<a>[tab\<^sub>1 h, g\<^sub>0h\<^sub>1.p\<^sub>0, f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine] \<cdot> (g\<^sub>0h\<^sub>1.\<phi> \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine) \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1, f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine] \<cdot> (tab\<^sub>0 g \<star> \<gamma>\<^sub>g) \<cdot> \<a>[tab\<^sub>0 g, f\<^sub>0g\<^sub>1.p\<^sub>0, fg\<^sub>0h\<^sub>1.p\<^sub>1] \<cdot> local.inv fg\<^sub>0h\<^sub>1.\<phi>
[PROOF STEP]
by presburger
[PROOF STATE]
proof (state)
this:
(h \<star> TTfgh_TfTgh.the_\<theta> \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0, f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine] \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>0 h, g\<^sub>0h\<^sub>1.p\<^sub>0, f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine]) \<cdot> \<a>[h, tab\<^sub>0 h, g\<^sub>0h\<^sub>1.p\<^sub>0 \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine] \<cdot> (h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0 \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine) \<cdot> \<a>[tab\<^sub>1 h, g\<^sub>0h\<^sub>1.p\<^sub>0, f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine] \<cdot> (g\<^sub>0h\<^sub>1.\<phi> \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine) \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1, f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine] \<cdot> (tab\<^sub>0 g \<star> \<gamma>\<^sub>g) \<cdot> \<a>[tab\<^sub>0 g, f\<^sub>0g\<^sub>1.p\<^sub>0, fg\<^sub>0h\<^sub>1.p\<^sub>1] \<cdot> local.inv fg\<^sub>0h\<^sub>1.\<phi> = ((h \<star> \<theta>\<^sub>h') \<cdot> \<a>[h, tab\<^sub>0 h, w\<^sub>h'] \<cdot> (h.tab \<star> w\<^sub>h')) \<cdot> \<beta>\<^sub>h
goal (1 subgoal):
1. (h \<star> \<theta>\<^sub>h) \<cdot> \<a>[h, tab\<^sub>0 h, w\<^sub>h] \<cdot> (h.tab \<star> w\<^sub>h) = ((h \<star> \<theta>\<^sub>h') \<cdot> \<a>[h, tab\<^sub>0 h, w\<^sub>h'] \<cdot> (h.tab \<star> w\<^sub>h')) \<cdot> \<beta>\<^sub>h
[PROOF STEP]
finally
[PROOF STATE]
proof (chain)
picking this:
(h \<star> \<theta>\<^sub>h) \<cdot> \<a>[h, tab\<^sub>0 h, w\<^sub>h] \<cdot> (h.tab \<star> w\<^sub>h) = ((h \<star> \<theta>\<^sub>h') \<cdot> \<a>[h, tab\<^sub>0 h, w\<^sub>h'] \<cdot> (h.tab \<star> w\<^sub>h')) \<cdot> \<beta>\<^sub>h
[PROOF STEP]
show ?thesis
[PROOF STATE]
proof (prove)
using this:
(h \<star> \<theta>\<^sub>h) \<cdot> \<a>[h, tab\<^sub>0 h, w\<^sub>h] \<cdot> (h.tab \<star> w\<^sub>h) = ((h \<star> \<theta>\<^sub>h') \<cdot> \<a>[h, tab\<^sub>0 h, w\<^sub>h'] \<cdot> (h.tab \<star> w\<^sub>h')) \<cdot> \<beta>\<^sub>h
goal (1 subgoal):
1. (h \<star> \<theta>\<^sub>h) \<cdot> \<a>[h, tab\<^sub>0 h, w\<^sub>h] \<cdot> (h.tab \<star> w\<^sub>h) = ((h \<star> \<theta>\<^sub>h') \<cdot> \<a>[h, tab\<^sub>0 h, w\<^sub>h'] \<cdot> (h.tab \<star> w\<^sub>h')) \<cdot> \<beta>\<^sub>h
[PROOF STEP]
by simp
[PROOF STATE]
proof (state)
this:
(h \<star> \<theta>\<^sub>h) \<cdot> \<a>[h, tab\<^sub>0 h, w\<^sub>h] \<cdot> (h.tab \<star> w\<^sub>h) = ((h \<star> \<theta>\<^sub>h') \<cdot> \<a>[h, tab\<^sub>0 h, w\<^sub>h'] \<cdot> (h.tab \<star> w\<^sub>h')) \<cdot> \<beta>\<^sub>h
goal:
No subgoals!
[PROOF STEP]
qed
[PROOF STATE]
proof (state)
this:
(h \<star> \<theta>\<^sub>h) \<cdot> \<a>[h, tab\<^sub>0 h, w\<^sub>h] \<cdot> (h.tab \<star> w\<^sub>h) = ((h \<star> \<theta>\<^sub>h') \<cdot> \<a>[h, tab\<^sub>0 h, w\<^sub>h'] \<cdot> (h.tab \<star> w\<^sub>h')) \<cdot> \<beta>\<^sub>h
goal (1 subgoal):
1. \<lbrakk>\<lbrakk>g\<^sub>0h\<^sub>1.p\<^sub>0 \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine\<rbrakk>\<rbrakk> = \<lbrakk>\<lbrakk>fg\<^sub>0h\<^sub>1.p\<^sub>0\<rbrakk>\<rbrakk>
[PROOF STEP]
have 7: "\<exists>!\<gamma>. \<guillemotleft>\<gamma> : w\<^sub>h \<Rightarrow> w\<^sub>h'\<guillemotright> \<and> \<beta>\<^sub>h = tab\<^sub>1 h \<star> \<gamma> \<and> \<theta>\<^sub>h = \<theta>\<^sub>h' \<cdot> (tab\<^sub>0 h \<star> \<gamma>)"
[PROOF STATE]
proof (prove)
goal (1 subgoal):
1. \<exists>!\<gamma>. \<guillemotleft>\<gamma> : w\<^sub>h \<rightarrow> w\<^sub>h'\<guillemotright> \<and> \<beta>\<^sub>h = tab\<^sub>1 h \<star> \<gamma> \<and> \<theta>\<^sub>h = \<theta>\<^sub>h' \<cdot> (tab\<^sub>0 h \<star> \<gamma>)
[PROOF STEP]
using w\<^sub>h w\<^sub>h' \<theta>\<^sub>h \<theta>\<^sub>h' \<beta>\<^sub>h eq\<^sub>h h.T2 [of w\<^sub>h w\<^sub>h' \<theta>\<^sub>h u\<^sub>h \<theta>\<^sub>h' \<beta>\<^sub>h]
[PROOF STATE]
proof (prove)
using this:
ide w\<^sub>h
ide w\<^sub>h'
\<guillemotleft>\<theta>\<^sub>h : tab\<^sub>0 h \<star> w\<^sub>h \<rightarrow> u\<^sub>h\<guillemotright>
\<guillemotleft>\<theta>\<^sub>h' : tab\<^sub>0 h \<star> w\<^sub>h' \<rightarrow> u\<^sub>h\<guillemotright>
\<guillemotleft>\<beta>\<^sub>h : tab\<^sub>1 h \<star> w\<^sub>h \<rightarrow> tab\<^sub>1 h \<star> w\<^sub>h'\<guillemotright>
(h \<star> \<theta>\<^sub>h) \<cdot> \<a>[h, tab\<^sub>0 h, w\<^sub>h] \<cdot> (h.tab \<star> w\<^sub>h) = ((h \<star> \<theta>\<^sub>h') \<cdot> \<a>[h, tab\<^sub>0 h, w\<^sub>h'] \<cdot> (h.tab \<star> w\<^sub>h')) \<cdot> \<beta>\<^sub>h
\<lbrakk>ide w\<^sub>h; ide w\<^sub>h'; \<guillemotleft>\<theta>\<^sub>h : tab\<^sub>0 h \<star> w\<^sub>h \<rightarrow> u\<^sub>h\<guillemotright>; \<guillemotleft>\<theta>\<^sub>h' : tab\<^sub>0 h \<star> w\<^sub>h' \<rightarrow> u\<^sub>h\<guillemotright>; \<guillemotleft>\<beta>\<^sub>h : tab\<^sub>1 h \<star> w\<^sub>h \<rightarrow> tab\<^sub>1 h \<star> w\<^sub>h'\<guillemotright>; (h \<star> \<theta>\<^sub>h) \<cdot> \<a>[h, tab\<^sub>0 h, w\<^sub>h] \<cdot> (h.tab \<star> w\<^sub>h) = ((h \<star> \<theta>\<^sub>h') \<cdot> \<a>[h, tab\<^sub>0 h, w\<^sub>h'] \<cdot> (h.tab \<star> w\<^sub>h')) \<cdot> \<beta>\<^sub>h\<rbrakk> \<Longrightarrow> \<exists>!\<gamma>. \<guillemotleft>\<gamma> : w\<^sub>h \<rightarrow> w\<^sub>h'\<guillemotright> \<and> \<beta>\<^sub>h = tab\<^sub>1 h \<star> \<gamma> \<and> \<theta>\<^sub>h = \<theta>\<^sub>h' \<cdot> (tab\<^sub>0 h \<star> \<gamma>)
goal (1 subgoal):
1. \<exists>!\<gamma>. \<guillemotleft>\<gamma> : w\<^sub>h \<rightarrow> w\<^sub>h'\<guillemotright> \<and> \<beta>\<^sub>h = tab\<^sub>1 h \<star> \<gamma> \<and> \<theta>\<^sub>h = \<theta>\<^sub>h' \<cdot> (tab\<^sub>0 h \<star> \<gamma>)
[PROOF STEP]
by blast
[PROOF STATE]
proof (state)
this:
\<exists>!\<gamma>. \<guillemotleft>\<gamma> : w\<^sub>h \<rightarrow> w\<^sub>h'\<guillemotright> \<and> \<beta>\<^sub>h = tab\<^sub>1 h \<star> \<gamma> \<and> \<theta>\<^sub>h = \<theta>\<^sub>h' \<cdot> (tab\<^sub>0 h \<star> \<gamma>)
goal (1 subgoal):
1. \<lbrakk>\<lbrakk>g\<^sub>0h\<^sub>1.p\<^sub>0 \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine\<rbrakk>\<rbrakk> = \<lbrakk>\<lbrakk>fg\<^sub>0h\<^sub>1.p\<^sub>0\<rbrakk>\<rbrakk>
[PROOF STEP]
obtain \<gamma>\<^sub>h where \<gamma>\<^sub>h: "\<guillemotleft>\<gamma>\<^sub>h : w\<^sub>h \<Rightarrow> w\<^sub>h'\<guillemotright> \<and> \<beta>\<^sub>h = tab\<^sub>1 h \<star> \<gamma>\<^sub>h \<and> \<theta>\<^sub>h = \<theta>\<^sub>h' \<cdot> (tab\<^sub>0 h \<star> \<gamma>\<^sub>h)"
[PROOF STATE]
proof (prove)
goal (1 subgoal):
1. (\<And>\<gamma>\<^sub>h. \<guillemotleft>\<gamma>\<^sub>h : w\<^sub>h \<rightarrow> w\<^sub>h'\<guillemotright> \<and> \<beta>\<^sub>h = tab\<^sub>1 h \<star> \<gamma>\<^sub>h \<and> \<theta>\<^sub>h = \<theta>\<^sub>h' \<cdot> (tab\<^sub>0 h \<star> \<gamma>\<^sub>h) \<Longrightarrow> thesis) \<Longrightarrow> thesis
[PROOF STEP]
using 7
[PROOF STATE]
proof (prove)
using this:
\<exists>!\<gamma>. \<guillemotleft>\<gamma> : w\<^sub>h \<rightarrow> w\<^sub>h'\<guillemotright> \<and> \<beta>\<^sub>h = tab\<^sub>1 h \<star> \<gamma> \<and> \<theta>\<^sub>h = \<theta>\<^sub>h' \<cdot> (tab\<^sub>0 h \<star> \<gamma>)
goal (1 subgoal):
1. (\<And>\<gamma>\<^sub>h. \<guillemotleft>\<gamma>\<^sub>h : w\<^sub>h \<rightarrow> w\<^sub>h'\<guillemotright> \<and> \<beta>\<^sub>h = tab\<^sub>1 h \<star> \<gamma>\<^sub>h \<and> \<theta>\<^sub>h = \<theta>\<^sub>h' \<cdot> (tab\<^sub>0 h \<star> \<gamma>\<^sub>h) \<Longrightarrow> thesis) \<Longrightarrow> thesis
[PROOF STEP]
by auto
[PROOF STATE]
proof (state)
this:
\<guillemotleft>\<gamma>\<^sub>h : w\<^sub>h \<rightarrow> w\<^sub>h'\<guillemotright> \<and> \<beta>\<^sub>h = tab\<^sub>1 h \<star> \<gamma>\<^sub>h \<and> \<theta>\<^sub>h = \<theta>\<^sub>h' \<cdot> (tab\<^sub>0 h \<star> \<gamma>\<^sub>h)
goal (1 subgoal):
1. \<lbrakk>\<lbrakk>g\<^sub>0h\<^sub>1.p\<^sub>0 \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine\<rbrakk>\<rbrakk> = \<lbrakk>\<lbrakk>fg\<^sub>0h\<^sub>1.p\<^sub>0\<rbrakk>\<rbrakk>
[PROOF STEP]
show "\<lbrakk>\<lbrakk>Tgh.p\<^sub>0 \<star> TfTgh.p\<^sub>0 \<star> TTfgh_TfTgh.chine\<rbrakk>\<rbrakk> = \<lbrakk>\<lbrakk>TTfgh.p\<^sub>0\<rbrakk>\<rbrakk>"
[PROOF STATE]
proof (prove)
goal (1 subgoal):
1. \<lbrakk>\<lbrakk>g\<^sub>0h\<^sub>1.p\<^sub>0 \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine\<rbrakk>\<rbrakk> = \<lbrakk>\<lbrakk>fg\<^sub>0h\<^sub>1.p\<^sub>0\<rbrakk>\<rbrakk>
[PROOF STEP]
proof -
[PROOF STATE]
proof (state)
goal (1 subgoal):
1. \<lbrakk>\<lbrakk>g\<^sub>0h\<^sub>1.p\<^sub>0 \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine\<rbrakk>\<rbrakk> = \<lbrakk>\<lbrakk>fg\<^sub>0h\<^sub>1.p\<^sub>0\<rbrakk>\<rbrakk>
[PROOF STEP]
have "iso \<gamma>\<^sub>h"
[PROOF STATE]
proof (prove)
goal (1 subgoal):
1. local.iso \<gamma>\<^sub>h
[PROOF STEP]
using \<gamma>\<^sub>h BS3 w\<^sub>h_is_map w\<^sub>h'_is_map
[PROOF STATE]
proof (prove)
using this:
\<guillemotleft>\<gamma>\<^sub>h : w\<^sub>h \<rightarrow> w\<^sub>h'\<guillemotright> \<and> \<beta>\<^sub>h = tab\<^sub>1 h \<star> \<gamma>\<^sub>h \<and> \<theta>\<^sub>h = \<theta>\<^sub>h' \<cdot> (tab\<^sub>0 h \<star> \<gamma>\<^sub>h)
\<lbrakk>is_left_adjoint ?f; is_left_adjoint ?f'; \<guillemotleft>?\<mu> : ?f \<rightarrow> ?f'\<guillemotright>; \<guillemotleft>?\<mu>' : ?f \<rightarrow> ?f'\<guillemotright>\<rbrakk> \<Longrightarrow> local.iso ?\<mu> \<and> local.iso ?\<mu>' \<and> ?\<mu> = ?\<mu>'
is_left_adjoint w\<^sub>h
is_left_adjoint w\<^sub>h'
goal (1 subgoal):
1. local.iso \<gamma>\<^sub>h
[PROOF STEP]
by blast
[PROOF STATE]
proof (state)
this:
local.iso \<gamma>\<^sub>h
goal (1 subgoal):
1. \<lbrakk>\<lbrakk>g\<^sub>0h\<^sub>1.p\<^sub>0 \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine\<rbrakk>\<rbrakk> = \<lbrakk>\<lbrakk>fg\<^sub>0h\<^sub>1.p\<^sub>0\<rbrakk>\<rbrakk>
[PROOF STEP]
hence "isomorphic w\<^sub>h w\<^sub>h'"
[PROOF STATE]
proof (prove)
using this:
local.iso \<gamma>\<^sub>h
goal (1 subgoal):
1. w\<^sub>h \<cong> w\<^sub>h'
[PROOF STEP]
using \<gamma>\<^sub>h isomorphic_def isomorphic_symmetric
[PROOF STATE]
proof (prove)
using this:
local.iso \<gamma>\<^sub>h
\<guillemotleft>\<gamma>\<^sub>h : w\<^sub>h \<rightarrow> w\<^sub>h'\<guillemotright> \<and> \<beta>\<^sub>h = tab\<^sub>1 h \<star> \<gamma>\<^sub>h \<and> \<theta>\<^sub>h = \<theta>\<^sub>h' \<cdot> (tab\<^sub>0 h \<star> \<gamma>\<^sub>h)
(?a \<cong> ?a') = (\<exists>f. \<guillemotleft>f : ?a \<rightarrow> ?a'\<guillemotright> \<and> local.iso f)
?f \<cong> ?g \<Longrightarrow> ?g \<cong> ?f
goal (1 subgoal):
1. w\<^sub>h \<cong> w\<^sub>h'
[PROOF STEP]
by auto
[PROOF STATE]
proof (state)
this:
w\<^sub>h \<cong> w\<^sub>h'
goal (1 subgoal):
1. \<lbrakk>\<lbrakk>g\<^sub>0h\<^sub>1.p\<^sub>0 \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine\<rbrakk>\<rbrakk> = \<lbrakk>\<lbrakk>fg\<^sub>0h\<^sub>1.p\<^sub>0\<rbrakk>\<rbrakk>
[PROOF STEP]
thus ?thesis
[PROOF STATE]
proof (prove)
using this:
w\<^sub>h \<cong> w\<^sub>h'
goal (1 subgoal):
1. \<lbrakk>\<lbrakk>g\<^sub>0h\<^sub>1.p\<^sub>0 \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine\<rbrakk>\<rbrakk> = \<lbrakk>\<lbrakk>fg\<^sub>0h\<^sub>1.p\<^sub>0\<rbrakk>\<rbrakk>
[PROOF STEP]
using w\<^sub>h w\<^sub>h' w\<^sub>h_def w\<^sub>h'_def Maps.CLS_eqI [of w\<^sub>h w\<^sub>h']
[PROOF STATE]
proof (prove)
using this:
w\<^sub>h \<cong> w\<^sub>h'
ide w\<^sub>h
ide w\<^sub>h'
w\<^sub>h = fg\<^sub>0h\<^sub>1.p\<^sub>0
w\<^sub>h' = g\<^sub>0h\<^sub>1.p\<^sub>0 \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine
ide w\<^sub>h \<Longrightarrow> (\<lbrakk>\<lbrakk>w\<^sub>h\<rbrakk>\<rbrakk> = \<lbrakk>\<lbrakk>w\<^sub>h'\<rbrakk>\<rbrakk>) = (w\<^sub>h \<cong> w\<^sub>h')
goal (1 subgoal):
1. \<lbrakk>\<lbrakk>g\<^sub>0h\<^sub>1.p\<^sub>0 \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine\<rbrakk>\<rbrakk> = \<lbrakk>\<lbrakk>fg\<^sub>0h\<^sub>1.p\<^sub>0\<rbrakk>\<rbrakk>
[PROOF STEP]
by simp
[PROOF STATE]
proof (state)
this:
\<lbrakk>\<lbrakk>g\<^sub>0h\<^sub>1.p\<^sub>0 \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine\<rbrakk>\<rbrakk> = \<lbrakk>\<lbrakk>fg\<^sub>0h\<^sub>1.p\<^sub>0\<rbrakk>\<rbrakk>
goal:
No subgoals!
[PROOF STEP]
qed
[PROOF STATE]
proof (state)
this:
\<lbrakk>\<lbrakk>g\<^sub>0h\<^sub>1.p\<^sub>0 \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine\<rbrakk>\<rbrakk> = \<lbrakk>\<lbrakk>fg\<^sub>0h\<^sub>1.p\<^sub>0\<rbrakk>\<rbrakk>
goal:
No subgoals!
[PROOF STEP]
qed |
/-
Copyright © 2020 Nicolò Cavalleri. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Nicolò Cavalleri, Sebastien Gouezel, Heather Macbeth, Patrick Massot, Floris van Doorn
! This file was ported from Lean 3 source module topology.vector_bundle.basic
! leanprover-community/mathlib commit 0187644979f2d3e10a06e916a869c994facd9a87
! Please do not edit these lines, except to modify the commit id
! if you have ported upstream changes.
-/
import Mathbin.Analysis.NormedSpace.BoundedLinearMaps
import Mathbin.Topology.FiberBundle.Basic
/-!
# Vector bundles
In this file we define (topological) vector bundles.
Let `B` be the base space, let `F` be a normed space over a normed field `R`, and let
`E : B → Type*` be a `fiber_bundle` with fiber `F`, in which, for each `x`, the fiber `E x` is a
topological vector space over `R`.
To have a vector bundle structure on `bundle.total_space E`, one should additionally have the
following properties:
* The bundle trivializations in the trivialization atlas should be continuous linear equivs in the
fibers;
* For any two trivializations `e`, `e'` in the atlas the transition function considered as a map
from `B` into `F →L[R] F` is continuous on `e.base_set ∩ e'.base_set` with respect to the operator
norm topology on `F →L[R] F`.
If these conditions are satisfied, we register the typeclass `vector_bundle R F E`.
We define constructions on vector bundles like pullbacks and direct sums in other files.
## Implementation notes
The implementation choices in the vector bundle definition are discussed in the "Implementation
notes" section of `topology.fiber_bundle.basic`.
## Tags
Vector bundle
-/
noncomputable section
open Bundle Set
open Classical Bundle
variable (R 𝕜 : Type _) {B : Type _} (F : Type _) (E : B → Type _)
section TopologicalVectorSpace
variable {B F E} [Semiring R] [TopologicalSpace F] [TopologicalSpace B]
/-- A mixin class for `pretrivialization`, stating that a pretrivialization is fiberwise linear with
respect to given module structures on its fibers and the model fiber. -/
protected class Pretrivialization.IsLinear [AddCommMonoid F] [Module R F] [∀ x, AddCommMonoid (E x)]
[∀ x, Module R (E x)] (e : Pretrivialization F (π E)) : Prop where
linear : ∀ b ∈ e.baseSet, IsLinearMap R fun x : E b => (e (totalSpaceMk b x)).2
#align pretrivialization.is_linear Pretrivialization.IsLinear
namespace Pretrivialization
variable {F E} (e : Pretrivialization F (π E)) {x : TotalSpace E} {b : B} {y : E b}
theorem linear [AddCommMonoid F] [Module R F] [∀ x, AddCommMonoid (E x)] [∀ x, Module R (E x)]
[e.isLinear R] {b : B} (hb : b ∈ e.baseSet) :
IsLinearMap R fun x : E b => (e (totalSpaceMk b x)).2 :=
Pretrivialization.IsLinear.linear b hb
#align pretrivialization.linear Pretrivialization.linear
variable [AddCommMonoid F] [Module R F] [∀ x, AddCommMonoid (E x)] [∀ x, Module R (E x)]
/-- A fiberwise linear inverse to `e`. -/
@[simps]
protected def symmₗ (e : Pretrivialization F (π E)) [e.isLinear R] (b : B) : F →ₗ[R] E b :=
by
refine' IsLinearMap.mk' (e.symm b) _
by_cases hb : b ∈ e.base_set
·
exact
(((e.linear R hb).mk' _).inverse (e.symm b) (e.symm_apply_apply_mk hb) fun v =>
congr_arg Prod.snd <| e.apply_mk_symm hb v).isLinear
· rw [e.coe_symm_of_not_mem hb]
exact (0 : F →ₗ[R] E b).isLinear
#align pretrivialization.symmₗ Pretrivialization.symmₗ
/-- A pretrivialization for a vector bundle defines linear equivalences between the
fibers and the model space. -/
@[simps (config := { fullyApplied := false })]
def linearEquivAt (e : Pretrivialization F (π E)) [e.isLinear R] (b : B) (hb : b ∈ e.baseSet) :
E b ≃ₗ[R] F where
toFun y := (e (totalSpaceMk b y)).2
invFun := e.symm b
left_inv := e.symm_apply_apply_mk hb
right_inv v := by simp_rw [e.apply_mk_symm hb v]
map_add' v w := (e.linear R hb).map_add v w
map_smul' c v := (e.linear R hb).map_smul c v
#align pretrivialization.linear_equiv_at Pretrivialization.linearEquivAt
/-- A fiberwise linear map equal to `e` on `e.base_set`. -/
protected def linearMapAt (e : Pretrivialization F (π E)) [e.isLinear R] (b : B) : E b →ₗ[R] F :=
if hb : b ∈ e.baseSet then e.linearEquivAt R b hb else 0
#align pretrivialization.linear_map_at Pretrivialization.linearMapAt
variable {R}
theorem coe_linearMapAt (e : Pretrivialization F (π E)) [e.isLinear R] (b : B) :
⇑(e.linearMapAt R b) = fun y => if b ∈ e.baseSet then (e (totalSpaceMk b y)).2 else 0 :=
by
rw [Pretrivialization.linearMapAt]
split_ifs <;> rfl
#align pretrivialization.coe_linear_map_at Pretrivialization.coe_linearMapAt
theorem coe_linearMapAt_of_mem (e : Pretrivialization F (π E)) [e.isLinear R] {b : B}
(hb : b ∈ e.baseSet) : ⇑(e.linearMapAt R b) = fun y => (e (totalSpaceMk b y)).2 := by
simp_rw [coe_linear_map_at, if_pos hb]
#align pretrivialization.coe_linear_map_at_of_mem Pretrivialization.coe_linearMapAt_of_mem
theorem linearMapAt_apply (e : Pretrivialization F (π E)) [e.isLinear R] {b : B} (y : E b) :
e.linearMapAt R b y = if b ∈ e.baseSet then (e (totalSpaceMk b y)).2 else 0 := by
rw [coe_linear_map_at]
#align pretrivialization.linear_map_at_apply Pretrivialization.linearMapAt_apply
theorem linearMapAt_def_of_mem (e : Pretrivialization F (π E)) [e.isLinear R] {b : B}
(hb : b ∈ e.baseSet) : e.linearMapAt R b = e.linearEquivAt R b hb :=
dif_pos hb
#align pretrivialization.linear_map_at_def_of_mem Pretrivialization.linearMapAt_def_of_mem
theorem linearMapAt_def_of_not_mem (e : Pretrivialization F (π E)) [e.isLinear R] {b : B}
(hb : b ∉ e.baseSet) : e.linearMapAt R b = 0 :=
dif_neg hb
#align pretrivialization.linear_map_at_def_of_not_mem Pretrivialization.linearMapAt_def_of_not_mem
theorem linearMapAt_eq_zero (e : Pretrivialization F (π E)) [e.isLinear R] {b : B}
(hb : b ∉ e.baseSet) : e.linearMapAt R b = 0 :=
dif_neg hb
#align pretrivialization.linear_map_at_eq_zero Pretrivialization.linearMapAt_eq_zero
theorem symmₗ_linearMapAt (e : Pretrivialization F (π E)) [e.isLinear R] {b : B}
(hb : b ∈ e.baseSet) (y : E b) : e.symmₗ R b (e.linearMapAt R b y) = y :=
by
rw [e.linear_map_at_def_of_mem hb]
exact (e.linear_equiv_at R b hb).left_inv y
#align pretrivialization.symmₗ_linear_map_at Pretrivialization.symmₗ_linearMapAt
theorem linearMapAt_symmₗ (e : Pretrivialization F (π E)) [e.isLinear R] {b : B}
(hb : b ∈ e.baseSet) (y : F) : e.linearMapAt R b (e.symmₗ R b y) = y :=
by
rw [e.linear_map_at_def_of_mem hb]
exact (e.linear_equiv_at R b hb).right_inv y
#align pretrivialization.linear_map_at_symmₗ Pretrivialization.linearMapAt_symmₗ
end Pretrivialization
variable (R) [TopologicalSpace (TotalSpace E)]
/-- A mixin class for `trivialization`, stating that a trivialization is fiberwise linear with
respect to given module structures on its fibers and the model fiber. -/
protected class Trivialization.IsLinear [AddCommMonoid F] [Module R F] [∀ x, AddCommMonoid (E x)]
[∀ x, Module R (E x)] (e : Trivialization F (π E)) : Prop where
linear : ∀ b ∈ e.baseSet, IsLinearMap R fun x : E b => (e (totalSpaceMk b x)).2
#align trivialization.is_linear Trivialization.IsLinear
namespace Trivialization
variable (e : Trivialization F (π E)) {x : TotalSpace E} {b : B} {y : E b}
protected theorem linear [AddCommMonoid F] [Module R F] [∀ x, AddCommMonoid (E x)]
[∀ x, Module R (E x)] [e.isLinear R] {b : B} (hb : b ∈ e.baseSet) :
IsLinearMap R fun y : E b => (e (totalSpaceMk b y)).2 :=
Trivialization.IsLinear.linear b hb
#align trivialization.linear Trivialization.linear
instance toPretrivialization.isLinear [AddCommMonoid F] [Module R F] [∀ x, AddCommMonoid (E x)]
[∀ x, Module R (E x)] [e.isLinear R] : e.toPretrivialization.isLinear R :=
{ (‹_› : e.isLinear R) with }
#align trivialization.to_pretrivialization.is_linear Trivialization.toPretrivialization.isLinear
variable [AddCommMonoid F] [Module R F] [∀ x, AddCommMonoid (E x)] [∀ x, Module R (E x)]
/-- A trivialization for a vector bundle defines linear equivalences between the
fibers and the model space. -/
def linearEquivAt (e : Trivialization F (π E)) [e.isLinear R] (b : B) (hb : b ∈ e.baseSet) :
E b ≃ₗ[R] F :=
e.toPretrivialization.linearEquivAt R b hb
#align trivialization.linear_equiv_at Trivialization.linearEquivAt
variable {R}
@[simp]
theorem linearEquivAt_apply (e : Trivialization F (π E)) [e.isLinear R] (b : B) (hb : b ∈ e.baseSet)
(v : E b) : e.linearEquivAt R b hb v = (e (totalSpaceMk b v)).2 :=
rfl
#align trivialization.linear_equiv_at_apply Trivialization.linearEquivAt_apply
@[simp]
theorem linearEquivAt_symm_apply (e : Trivialization F (π E)) [e.isLinear R] (b : B)
(hb : b ∈ e.baseSet) (v : F) : (e.linearEquivAt R b hb).symm v = e.symm b v :=
rfl
#align trivialization.linear_equiv_at_symm_apply Trivialization.linearEquivAt_symm_apply
variable (R)
/-- A fiberwise linear inverse to `e`. -/
protected def symmₗ (e : Trivialization F (π E)) [e.isLinear R] (b : B) : F →ₗ[R] E b :=
e.toPretrivialization.symmₗ R b
#align trivialization.symmₗ Trivialization.symmₗ
variable {R}
theorem coe_symmₗ (e : Trivialization F (π E)) [e.isLinear R] (b : B) : ⇑(e.symmₗ R b) = e.symm b :=
rfl
#align trivialization.coe_symmₗ Trivialization.coe_symmₗ
variable (R)
/-- A fiberwise linear map equal to `e` on `e.base_set`. -/
protected def linearMapAt (e : Trivialization F (π E)) [e.isLinear R] (b : B) : E b →ₗ[R] F :=
e.toPretrivialization.linearMapAt R b
#align trivialization.linear_map_at Trivialization.linearMapAt
variable {R}
theorem coe_linearMapAt (e : Trivialization F (π E)) [e.isLinear R] (b : B) :
⇑(e.linearMapAt R b) = fun y => if b ∈ e.baseSet then (e (totalSpaceMk b y)).2 else 0 :=
e.toPretrivialization.coe_linearMapAt b
#align trivialization.coe_linear_map_at Trivialization.coe_linearMapAt
theorem coe_linearMapAt_of_mem (e : Trivialization F (π E)) [e.isLinear R] {b : B}
(hb : b ∈ e.baseSet) : ⇑(e.linearMapAt R b) = fun y => (e (totalSpaceMk b y)).2 := by
simp_rw [coe_linear_map_at, if_pos hb]
#align trivialization.coe_linear_map_at_of_mem Trivialization.coe_linearMapAt_of_mem
theorem linearMapAt_apply (e : Trivialization F (π E)) [e.isLinear R] {b : B} (y : E b) :
e.linearMapAt R b y = if b ∈ e.baseSet then (e (totalSpaceMk b y)).2 else 0 := by
rw [coe_linear_map_at]
#align trivialization.linear_map_at_apply Trivialization.linearMapAt_apply
theorem linearMapAt_def_of_mem (e : Trivialization F (π E)) [e.isLinear R] {b : B}
(hb : b ∈ e.baseSet) : e.linearMapAt R b = e.linearEquivAt R b hb :=
dif_pos hb
#align trivialization.linear_map_at_def_of_mem Trivialization.linearMapAt_def_of_mem
theorem linearMapAt_def_of_not_mem (e : Trivialization F (π E)) [e.isLinear R] {b : B}
(hb : b ∉ e.baseSet) : e.linearMapAt R b = 0 :=
dif_neg hb
#align trivialization.linear_map_at_def_of_not_mem Trivialization.linearMapAt_def_of_not_mem
theorem symmₗ_linearMapAt (e : Trivialization F (π E)) [e.isLinear R] {b : B} (hb : b ∈ e.baseSet)
(y : E b) : e.symmₗ R b (e.linearMapAt R b y) = y :=
e.toPretrivialization.symmₗ_linearMapAt hb y
#align trivialization.symmₗ_linear_map_at Trivialization.symmₗ_linearMapAt
theorem linearMapAt_symmₗ (e : Trivialization F (π E)) [e.isLinear R] {b : B} (hb : b ∈ e.baseSet)
(y : F) : e.linearMapAt R b (e.symmₗ R b y) = y :=
e.toPretrivialization.linearMapAt_symmₗ hb y
#align trivialization.linear_map_at_symmₗ Trivialization.linearMapAt_symmₗ
variable (R)
/-- A coordinate change function between two trivializations, as a continuous linear equivalence.
Defined to be the identity when `b` does not lie in the base set of both trivializations. -/
def coordChangeL (e e' : Trivialization F (π E)) [e.isLinear R] [e'.isLinear R] (b : B) :
F ≃L[R] F :=
{
if hb : b ∈ e.baseSet ∩ e'.baseSet then
(e.linearEquivAt R b (hb.1 : _)).symm.trans (e'.linearEquivAt R b hb.2)
else
LinearEquiv.refl R
F with
continuous_toFun := by
by_cases hb : b ∈ e.base_set ∩ e'.base_set
· simp_rw [dif_pos hb]
refine' (e'.continuous_on.comp_continuous _ _).snd
exact
e.continuous_on_symm.comp_continuous (Continuous.Prod.mk b) fun y =>
mk_mem_prod hb.1 (mem_univ y)
exact fun y => e'.mem_source.mpr hb.2
· rw [dif_neg hb]
exact continuous_id
continuous_invFun := by
by_cases hb : b ∈ e.base_set ∩ e'.base_set
· simp_rw [dif_pos hb]
refine' (e.continuous_on.comp_continuous _ _).snd
exact
e'.continuous_on_symm.comp_continuous (Continuous.Prod.mk b) fun y =>
mk_mem_prod hb.2 (mem_univ y)
exact fun y => e.mem_source.mpr hb.1
· rw [dif_neg hb]
exact continuous_id }
#align trivialization.coord_changeL Trivialization.coordChangeL
variable {R}
theorem coe_coordChangeL (e e' : Trivialization F (π E)) [e.isLinear R] [e'.isLinear R] {b : B}
(hb : b ∈ e.baseSet ∩ e'.baseSet) :
⇑(coordChangeL R e e' b) = (e.linearEquivAt R b hb.1).symm.trans (e'.linearEquivAt R b hb.2) :=
congr_arg LinearEquiv.toFun (dif_pos hb)
#align trivialization.coe_coord_changeL Trivialization.coe_coordChangeL
theorem coe_coord_changeL' (e e' : Trivialization F (π E)) [e.isLinear R] [e'.isLinear R] {b : B}
(hb : b ∈ e.baseSet ∩ e'.baseSet) :
(coordChangeL R e e' b).toLinearEquiv =
(e.linearEquivAt R b hb.1).symm.trans (e'.linearEquivAt R b hb.2) :=
LinearEquiv.coe_injective (coe_coordChangeL _ _ _)
#align trivialization.coe_coord_changeL' Trivialization.coe_coord_changeL'
theorem symm_coordChangeL (e e' : Trivialization F (π E)) [e.isLinear R] [e'.isLinear R] {b : B}
(hb : b ∈ e'.baseSet ∩ e.baseSet) : (e.coordChangeL R e' b).symm = e'.coordChangeL R e b :=
by
apply ContinuousLinearEquiv.toLinearEquiv_injective
rw [coe_coord_changeL' e' e hb, (coord_changeL R e e' b).symm_toLinearEquiv,
coe_coord_changeL' e e' hb.symm, LinearEquiv.trans_symm, LinearEquiv.symm_symm]
#align trivialization.symm_coord_changeL Trivialization.symm_coordChangeL
theorem coordChangeL_apply (e e' : Trivialization F (π E)) [e.isLinear R] [e'.isLinear R] {b : B}
(hb : b ∈ e.baseSet ∩ e'.baseSet) (y : F) :
coordChangeL R e e' b y = (e' (totalSpaceMk b (e.symm b y))).2 :=
congr_arg (fun f => LinearEquiv.toFun f y) (dif_pos hb)
#align trivialization.coord_changeL_apply Trivialization.coordChangeL_apply
theorem mk_coordChangeL (e e' : Trivialization F (π E)) [e.isLinear R] [e'.isLinear R] {b : B}
(hb : b ∈ e.baseSet ∩ e'.baseSet) (y : F) :
(b, coordChangeL R e e' b y) = e' (totalSpaceMk b (e.symm b y)) :=
by
ext
· rw [e.mk_symm hb.1 y, e'.coe_fst', e.proj_symm_apply' hb.1]
rw [e.proj_symm_apply' hb.1]
exact hb.2
· exact e.coord_changeL_apply e' hb y
#align trivialization.mk_coord_changeL Trivialization.mk_coordChangeL
theorem apply_symm_apply_eq_coordChangeL (e e' : Trivialization F (π E)) [e.isLinear R]
[e'.isLinear R] {b : B} (hb : b ∈ e.baseSet ∩ e'.baseSet) (v : F) :
e' (e.toLocalHomeomorph.symm (b, v)) = (b, e.coordChangeL R e' b v) := by
rw [e.mk_coord_changeL e' hb, e.mk_symm hb.1]
#align trivialization.apply_symm_apply_eq_coord_changeL Trivialization.apply_symm_apply_eq_coordChangeL
/-- A version of `coord_change_apply` that fully unfolds `coord_change`. The right-hand side is
ugly, but has good definitional properties for specifically defined trivializations. -/
theorem coordChangeL_apply' (e e' : Trivialization F (π E)) [e.isLinear R] [e'.isLinear R] {b : B}
(hb : b ∈ e.baseSet ∩ e'.baseSet) (y : F) :
coordChangeL R e e' b y = (e' (e.toLocalHomeomorph.symm (b, y))).2 := by
rw [e.coord_changeL_apply e' hb, e.mk_symm hb.1]
#align trivialization.coord_changeL_apply' Trivialization.coordChangeL_apply'
theorem coordChangeL_symm_apply (e e' : Trivialization F (π E)) [e.isLinear R] [e'.isLinear R]
{b : B} (hb : b ∈ e.baseSet ∩ e'.baseSet) :
⇑(coordChangeL R e e' b).symm =
(e'.linearEquivAt R b hb.2).symm.trans (e.linearEquivAt R b hb.1) :=
congr_arg LinearEquiv.invFun (dif_pos hb)
#align trivialization.coord_changeL_symm_apply Trivialization.coordChangeL_symm_apply
end Trivialization
end TopologicalVectorSpace
section
namespace Bundle
/-- The zero section of a vector bundle -/
def zeroSection [∀ x, Zero (E x)] : B → TotalSpace E := fun x => totalSpaceMk x 0
#align bundle.zero_section Bundle.zeroSection
@[simp, mfld_simps]
theorem zeroSection_proj [∀ x, Zero (E x)] (x : B) : (zeroSection E x).proj = x :=
rfl
#align bundle.zero_section_proj Bundle.zeroSection_proj
@[simp, mfld_simps]
theorem zeroSection_snd [∀ x, Zero (E x)] (x : B) : (zeroSection E x).2 = 0 :=
rfl
#align bundle.zero_section_snd Bundle.zeroSection_snd
end Bundle
open Bundle
variable [NontriviallyNormedField R] [∀ x, AddCommMonoid (E x)] [∀ x, Module R (E x)]
[NormedAddCommGroup F] [NormedSpace R F] [TopologicalSpace B] [TopologicalSpace (TotalSpace E)]
[∀ x, TopologicalSpace (E x)] [FiberBundle F E]
/- ./././Mathport/Syntax/Translate/Command.lean:388:30: infer kinds are unsupported in Lean 4: #[`continuousOn_coord_change'] [] -/
/-- The space `total_space E` (for `E : B → Type*` such that each `E x` is a topological vector
space) has a topological vector space structure with fiber `F` (denoted with
`vector_bundle R F E`) if around every point there is a fiber bundle trivialization
which is linear in the fibers. -/
class VectorBundle : Prop where
trivialization_linear' : ∀ (e : Trivialization F (π E)) [MemTrivializationAtlas e], e.isLinear R
continuousOn_coord_change' :
∀ (e e' : Trivialization F (π E)) [MemTrivializationAtlas e] [MemTrivializationAtlas e'],
ContinuousOn (fun b => Trivialization.coordChangeL R e e' b : B → F →L[R] F)
(e.baseSet ∩ e'.baseSet)
#align vector_bundle VectorBundle
variable {F E}
instance (priority := 100) trivialization_linear [VectorBundle R F E] (e : Trivialization F (π E))
[MemTrivializationAtlas e] : e.isLinear R :=
VectorBundle.trivialization_linear' e
#align trivialization_linear trivialization_linear
theorem continuousOn_coord_change [VectorBundle R F E] (e e' : Trivialization F (π E))
[he : MemTrivializationAtlas e] [he' : MemTrivializationAtlas e'] :
ContinuousOn (fun b => Trivialization.coordChangeL R e e' b : B → F →L[R] F)
(e.baseSet ∩ e'.baseSet) :=
VectorBundle.continuousOn_coord_change' R e e'
#align continuous_on_coord_change continuousOn_coord_change
namespace Trivialization
/-- Forward map of `continuous_linear_equiv_at` (only propositionally equal),
defined everywhere (`0` outside domain). -/
@[simps (config := { fullyApplied := false }) apply]
def continuousLinearMapAt (e : Trivialization F (π E)) [e.isLinear R] (b : B) : E b →L[R] F :=
{-- given explicitly to help `simps`
e.linearMapAt
R b with
toFun := e.linearMapAt R b
cont := by
dsimp
rw [e.coe_linear_map_at b]
refine' continuous_if_const _ (fun hb => _) fun _ => continuous_zero
exact
continuous_snd.comp
(e.continuous_on.comp_continuous (FiberBundle.totalSpaceMk_inducing F E b).Continuous
fun x => e.mem_source.mpr hb) }
#align trivialization.continuous_linear_map_at Trivialization.continuousLinearMapAt
/-- Backwards map of `continuous_linear_equiv_at`, defined everywhere. -/
@[simps (config := { fullyApplied := false }) apply]
def symmL (e : Trivialization F (π E)) [e.isLinear R] (b : B) : F →L[R] E b :=
{-- given explicitly to help `simps`
e.symmₗ
R b with
toFun := e.symm b
cont := by
by_cases hb : b ∈ e.base_set
· rw [(FiberBundle.totalSpaceMk_inducing F E b).continuous_iff]
exact
e.continuous_on_symm.comp_continuous (continuous_const.prod_mk continuous_id) fun x =>
mk_mem_prod hb (mem_univ x)
· refine' continuous_zero.congr fun x => (e.symm_apply_of_not_mem hb x).symm }
#align trivialization.symmL Trivialization.symmL
variable {R}
theorem symmL_continuousLinearMapAt (e : Trivialization F (π E)) [e.isLinear R] {b : B}
(hb : b ∈ e.baseSet) (y : E b) : e.symmL R b (e.continuousLinearMapAt R b y) = y :=
e.symmₗ_linearMapAt hb y
#align trivialization.symmL_continuous_linear_map_at Trivialization.symmL_continuousLinearMapAt
theorem continuousLinearMapAt_symmL (e : Trivialization F (π E)) [e.isLinear R] {b : B}
(hb : b ∈ e.baseSet) (y : F) : e.continuousLinearMapAt R b (e.symmL R b y) = y :=
e.linearMapAt_symmₗ hb y
#align trivialization.continuous_linear_map_at_symmL Trivialization.continuousLinearMapAt_symmL
variable (R)
/-- In a vector bundle, a trivialization in the fiber (which is a priori only linear)
is in fact a continuous linear equiv between the fibers and the model fiber. -/
@[simps (config := { fullyApplied := false }) apply symm_apply]
def continuousLinearEquivAt (e : Trivialization F (π E)) [e.isLinear R] (b : B)
(hb : b ∈ e.baseSet) : E b ≃L[R] F :=
{-- given explicitly to help `simps`
-- given explicitly to help `simps`
e.toPretrivialization.linearEquivAt
R b hb with
toFun := fun y => (e (totalSpaceMk b y)).2
invFun := e.symm b
continuous_toFun :=
continuous_snd.comp
(e.ContinuousOn.comp_continuous (FiberBundle.totalSpaceMk_inducing F E b).Continuous
fun x => e.mem_source.mpr hb)
continuous_invFun := (e.symmL R b).Continuous }
#align trivialization.continuous_linear_equiv_at Trivialization.continuousLinearEquivAt
variable {R}
theorem coe_continuousLinearEquivAt_eq (e : Trivialization F (π E)) [e.isLinear R] {b : B}
(hb : b ∈ e.baseSet) :
(e.continuousLinearEquivAt R b hb : E b → F) = e.continuousLinearMapAt R b :=
(e.coe_linearMapAt_of_mem hb).symm
#align trivialization.coe_continuous_linear_equiv_at_eq Trivialization.coe_continuousLinearEquivAt_eq
theorem symm_continuousLinearEquivAt_eq (e : Trivialization F (π E)) [e.isLinear R] {b : B}
(hb : b ∈ e.baseSet) : ((e.continuousLinearEquivAt R b hb).symm : F → E b) = e.symmL R b :=
rfl
#align trivialization.symm_continuous_linear_equiv_at_eq Trivialization.symm_continuousLinearEquivAt_eq
@[simp]
theorem continuousLinearEquivAt_apply' (e : Trivialization F (π E)) [e.isLinear R]
(x : TotalSpace E) (hx : x ∈ e.source) :
e.continuousLinearEquivAt R x.proj (e.mem_source.1 hx) x.2 = (e x).2 :=
by
cases x
rfl
#align trivialization.continuous_linear_equiv_at_apply' Trivialization.continuousLinearEquivAt_apply'
variable (R)
theorem apply_eq_prod_continuousLinearEquivAt (e : Trivialization F (π E)) [e.isLinear R] (b : B)
(hb : b ∈ e.baseSet) (z : E b) : e ⟨b, z⟩ = (b, e.continuousLinearEquivAt R b hb z) :=
by
ext
· refine' e.coe_fst _
rw [e.source_eq]
exact hb
· simp only [coe_coe, continuous_linear_equiv_at_apply]
#align trivialization.apply_eq_prod_continuous_linear_equiv_at Trivialization.apply_eq_prod_continuousLinearEquivAt
protected theorem zeroSection (e : Trivialization F (π E)) [e.isLinear R] {x : B}
(hx : x ∈ e.baseSet) : e (zeroSection E x) = (x, 0) := by
simp_rw [zero_section, total_space_mk, e.apply_eq_prod_continuous_linear_equiv_at R x hx 0,
map_zero]
#align trivialization.zero_section Trivialization.zeroSection
variable {R}
theorem symm_apply_eq_mk_continuousLinearEquivAt_symm (e : Trivialization F (π E)) [e.isLinear R]
(b : B) (hb : b ∈ e.baseSet) (z : F) :
e.toLocalHomeomorph.symm ⟨b, z⟩ = totalSpaceMk b ((e.continuousLinearEquivAt R b hb).symm z) :=
by
have h : (b, z) ∈ e.target := by
rw [e.target_eq]
exact ⟨hb, mem_univ _⟩
apply e.inj_on (e.map_target h)
· simp only [e.source_eq, hb, mem_preimage]
simp_rw [e.right_inv h, coe_coe, e.apply_eq_prod_continuous_linear_equiv_at R b hb,
ContinuousLinearEquiv.apply_symm_apply]
#align trivialization.symm_apply_eq_mk_continuous_linear_equiv_at_symm Trivialization.symm_apply_eq_mk_continuousLinearEquivAt_symm
theorem comp_continuousLinearEquivAt_eq_coord_change (e e' : Trivialization F (π E)) [e.isLinear R]
[e'.isLinear R] {b : B} (hb : b ∈ e.baseSet ∩ e'.baseSet) :
(e.continuousLinearEquivAt R b hb.1).symm.trans (e'.continuousLinearEquivAt R b hb.2) =
coordChangeL R e e' b :=
by
ext v
rw [coord_changeL_apply e e' hb]
rfl
#align trivialization.comp_continuous_linear_equiv_at_eq_coord_change Trivialization.comp_continuousLinearEquivAt_eq_coord_change
end Trivialization
include R F
/-! ### Constructing vector bundles -/
variable (R B F)
/-- Analogous construction of `fiber_bundle_core` for vector bundles. This
construction gives a way to construct vector bundles from a structure registering how
trivialization changes act on fibers. -/
structure VectorBundleCore (ι : Type _) where
baseSet : ι → Set B
isOpen_baseSet : ∀ i, IsOpen (base_set i)
indexAt : B → ι
mem_baseSet_at : ∀ x, x ∈ base_set (index_at x)
coordChange : ι → ι → B → F →L[R] F
coordChange_self : ∀ i, ∀ x ∈ base_set i, ∀ v, coord_change i i x v = v
continuousOn_coordChange : ∀ i j, ContinuousOn (coord_change i j) (base_set i ∩ base_set j)
coordChange_comp :
∀ i j k,
∀ x ∈ base_set i ∩ base_set j ∩ base_set k,
∀ v, (coord_change j k x) (coord_change i j x v) = coord_change i k x v
#align vector_bundle_core VectorBundleCore
/-- The trivial vector bundle core, in which all the changes of coordinates are the
identity. -/
def trivialVectorBundleCore (ι : Type _) [Inhabited ι] : VectorBundleCore R B F ι
where
baseSet ι := univ
isOpen_baseSet i := isOpen_univ
indexAt := default
mem_baseSet_at x := mem_univ x
coordChange i j x := ContinuousLinearMap.id R F
coordChange_self i x hx v := rfl
coordChange_comp i j k x hx v := rfl
continuousOn_coordChange i j := continuousOn_const
#align trivial_vector_bundle_core trivialVectorBundleCore
instance (ι : Type _) [Inhabited ι] : Inhabited (VectorBundleCore R B F ι) :=
⟨trivialVectorBundleCore R B F ι⟩
namespace VectorBundleCore
variable {R B F} {ι : Type _} (Z : VectorBundleCore R B F ι)
/-- Natural identification to a `fiber_bundle_core`. -/
@[simps (config := mfld_cfg)]
def toFiberBundleCore : FiberBundleCore ι B F :=
{ Z with
coordChange := fun i j b => Z.coordChange i j b
continuousOn_coordChange := fun i j =>
isBoundedBilinearMapApply.Continuous.comp_continuousOn
((Z.continuousOn_coordChange i j).Prod_map continuousOn_id) }
#align vector_bundle_core.to_fiber_bundle_core VectorBundleCore.toFiberBundleCore
instance toFiberBundleCoreCoe : Coe (VectorBundleCore R B F ι) (FiberBundleCore ι B F) :=
⟨toFiberBundleCore⟩
#align vector_bundle_core.to_fiber_bundle_core_coe VectorBundleCore.toFiberBundleCoreCoe
include Z
theorem coordChange_linear_comp (i j k : ι) :
∀ x ∈ Z.baseSet i ∩ Z.baseSet j ∩ Z.baseSet k,
(Z.coordChange j k x).comp (Z.coordChange i j x) = Z.coordChange i k x :=
fun x hx => by
ext v
exact Z.coord_change_comp i j k x hx v
#align vector_bundle_core.coord_change_linear_comp VectorBundleCore.coordChange_linear_comp
/-- The index set of a vector bundle core, as a convenience function for dot notation -/
@[nolint unused_arguments has_nonempty_instance]
def Index :=
ι
#align vector_bundle_core.index VectorBundleCore.Index
/-- The base space of a vector bundle core, as a convenience function for dot notation-/
@[nolint unused_arguments, reducible]
def Base :=
B
#align vector_bundle_core.base VectorBundleCore.Base
/-- The fiber of a vector bundle core, as a convenience function for dot notation and
typeclass inference -/
@[nolint unused_arguments has_nonempty_instance]
def Fiber : B → Type _ :=
Z.toFiberBundleCore.Fiber
#align vector_bundle_core.fiber VectorBundleCore.Fiber
instance topologicalSpaceFiber (x : B) : TopologicalSpace (Z.Fiber x) := by
delta_instance vector_bundle_core.fiber
#align vector_bundle_core.topological_space_fiber VectorBundleCore.topologicalSpaceFiber
instance addCommMonoidFiber : ∀ x : B, AddCommMonoid (Z.Fiber x) := by
dsimp [VectorBundleCore.Fiber] <;> delta_instance fiber_bundle_core.fiber
#align vector_bundle_core.add_comm_monoid_fiber VectorBundleCore.addCommMonoidFiber
instance moduleFiber : ∀ x : B, Module R (Z.Fiber x) := by
dsimp [VectorBundleCore.Fiber] <;> delta_instance fiber_bundle_core.fiber
#align vector_bundle_core.module_fiber VectorBundleCore.moduleFiber
instance addCommGroupFiber [AddCommGroup F] : ∀ x : B, AddCommGroup (Z.Fiber x) := by
dsimp [VectorBundleCore.Fiber] <;> delta_instance fiber_bundle_core.fiber
#align vector_bundle_core.add_comm_group_fiber VectorBundleCore.addCommGroupFiber
/-- The projection from the total space of a fiber bundle core, on its base. -/
@[reducible, simp, mfld_simps]
def proj : TotalSpace Z.Fiber → B :=
TotalSpace.proj
#align vector_bundle_core.proj VectorBundleCore.proj
/-- The total space of the vector bundle, as a convenience function for dot notation.
It is by definition equal to `bundle.total_space Z.fiber`, a.k.a. `Σ x, Z.fiber x` but with a
different name for typeclass inference. -/
@[nolint unused_arguments, reducible]
def TotalSpace :=
Bundle.TotalSpace Z.Fiber
#align vector_bundle_core.total_space VectorBundleCore.TotalSpace
/-- Local homeomorphism version of the trivialization change. -/
def trivChange (i j : ι) : LocalHomeomorph (B × F) (B × F) :=
FiberBundleCore.trivChange (↑Z) i j
#align vector_bundle_core.triv_change VectorBundleCore.trivChange
@[simp, mfld_simps]
theorem mem_trivChange_source (i j : ι) (p : B × F) :
p ∈ (Z.trivChange i j).source ↔ p.1 ∈ Z.baseSet i ∩ Z.baseSet j :=
FiberBundleCore.mem_trivChange_source (↑Z) i j p
#align vector_bundle_core.mem_triv_change_source VectorBundleCore.mem_trivChange_source
/-- Topological structure on the total space of a vector bundle created from core, designed so
that all the local trivialization are continuous. -/
instance toTopologicalSpace : TopologicalSpace Z.TotalSpace :=
Z.toFiberBundleCore.toTopologicalSpace
#align vector_bundle_core.to_topological_space VectorBundleCore.toTopologicalSpace
variable (b : B) (a : F)
@[simp, mfld_simps]
theorem coe_coordChange (i j : ι) : Z.toFiberBundleCore.coordChange i j b = Z.coordChange i j b :=
rfl
#align vector_bundle_core.coe_coord_change VectorBundleCore.coe_coordChange
/-- One of the standard local trivializations of a vector bundle constructed from core, taken by
considering this in particular as a fiber bundle constructed from core. -/
def localTriv (i : ι) : Trivialization F (π Z.Fiber) := by
dsimp [VectorBundleCore.TotalSpace, VectorBundleCore.Fiber] <;>
exact Z.to_fiber_bundle_core.local_triv i
#align vector_bundle_core.local_triv VectorBundleCore.localTriv
/-- The standard local trivializations of a vector bundle constructed from core are linear. -/
instance localTriv.isLinear (i : ι) : (Z.localTriv i).isLinear R
where linear x hx := by
dsimp [VectorBundleCore.localTriv] <;>
exact
{ map_add := fun v w => by simp only [ContinuousLinearMap.map_add, mfld_simps]
map_smul := fun r v => by simp only [ContinuousLinearMap.map_smul, mfld_simps] }
#align vector_bundle_core.local_triv.is_linear VectorBundleCore.localTriv.isLinear
variable (i j : ι)
@[simp, mfld_simps]
theorem mem_localTriv_source (p : Z.TotalSpace) : p ∈ (Z.localTriv i).source ↔ p.1 ∈ Z.baseSet i :=
by dsimp [VectorBundleCore.Fiber] <;> exact Iff.rfl
#align vector_bundle_core.mem_local_triv_source VectorBundleCore.mem_localTriv_source
@[simp, mfld_simps]
theorem baseSet_at : Z.baseSet i = (Z.localTriv i).baseSet :=
rfl
#align vector_bundle_core.base_set_at VectorBundleCore.baseSet_at
@[simp, mfld_simps]
theorem localTriv_apply (p : Z.TotalSpace) :
(Z.localTriv i) p = ⟨p.1, Z.coordChange (Z.indexAt p.1) i p.1 p.2⟩ :=
rfl
#align vector_bundle_core.local_triv_apply VectorBundleCore.localTriv_apply
@[simp, mfld_simps]
theorem mem_localTriv_target (p : B × F) :
p ∈ (Z.localTriv i).target ↔ p.1 ∈ (Z.localTriv i).baseSet :=
Z.toFiberBundleCore.mem_localTriv_target i p
#align vector_bundle_core.mem_local_triv_target VectorBundleCore.mem_localTriv_target
@[simp, mfld_simps]
theorem localTriv_symm_fst (p : B × F) :
(Z.localTriv i).toLocalHomeomorph.symm p = ⟨p.1, Z.coordChange i (Z.indexAt p.1) p.1 p.2⟩ :=
rfl
#align vector_bundle_core.local_triv_symm_fst VectorBundleCore.localTriv_symm_fst
@[simp, mfld_simps]
theorem localTriv_symm_apply {b : B} (hb : b ∈ Z.baseSet i) (v : F) :
(Z.localTriv i).symm b v = Z.coordChange i (Z.indexAt b) b v := by
apply (Z.local_triv i).symm_apply hb v
#align vector_bundle_core.local_triv_symm_apply VectorBundleCore.localTriv_symm_apply
@[simp, mfld_simps]
theorem localTriv_coordChange_eq {b : B} (hb : b ∈ Z.baseSet i ∩ Z.baseSet j) (v : F) :
(Z.localTriv i).coordChangeL R (Z.localTriv j) b v = Z.coordChange i j b v :=
by
rw [Trivialization.coordChangeL_apply', local_triv_symm_fst, local_triv_apply, coord_change_comp]
exacts[⟨⟨hb.1, Z.mem_base_set_at b⟩, hb.2⟩, hb]
#align vector_bundle_core.local_triv_coord_change_eq VectorBundleCore.localTriv_coordChange_eq
/-- Preferred local trivialization of a vector bundle constructed from core, at a given point, as
a bundle trivialization -/
def localTrivAt (b : B) : Trivialization F (π Z.Fiber) :=
Z.localTriv (Z.indexAt b)
#align vector_bundle_core.local_triv_at VectorBundleCore.localTrivAt
@[simp, mfld_simps]
theorem localTrivAt_def : Z.localTriv (Z.indexAt b) = Z.localTrivAt b :=
rfl
#align vector_bundle_core.local_triv_at_def VectorBundleCore.localTrivAt_def
@[simp, mfld_simps]
theorem mem_source_at : (⟨b, a⟩ : Z.TotalSpace) ∈ (Z.localTrivAt b).source :=
by
rw [local_triv_at, mem_local_triv_source]
exact Z.mem_base_set_at b
#align vector_bundle_core.mem_source_at VectorBundleCore.mem_source_at
@[simp, mfld_simps]
theorem localTrivAt_apply (p : Z.TotalSpace) : (Z.localTrivAt p.1) p = ⟨p.1, p.2⟩ :=
FiberBundleCore.localTrivAt_apply Z p
#align vector_bundle_core.local_triv_at_apply VectorBundleCore.localTrivAt_apply
@[simp, mfld_simps]
theorem localTrivAt_apply_mk (b : B) (a : F) : (Z.localTrivAt b) ⟨b, a⟩ = ⟨b, a⟩ :=
Z.localTrivAt_apply _
#align vector_bundle_core.local_triv_at_apply_mk VectorBundleCore.localTrivAt_apply_mk
@[simp, mfld_simps]
theorem mem_localTrivAt_baseSet : b ∈ (Z.localTrivAt b).baseSet :=
FiberBundleCore.mem_localTrivAt_baseSet Z b
#align vector_bundle_core.mem_local_triv_at_base_set VectorBundleCore.mem_localTrivAt_baseSet
instance fiberBundle : FiberBundle F Z.Fiber :=
Z.toFiberBundleCore.FiberBundle
#align vector_bundle_core.fiber_bundle VectorBundleCore.fiberBundle
instance vectorBundle : VectorBundle R F Z.Fiber
where
trivialization_linear' := by
rintro _ ⟨i, rfl⟩
apply local_triv.is_linear
continuousOn_coord_change' := by
rintro _ _ ⟨i, rfl⟩ ⟨i', rfl⟩
refine' (Z.continuous_on_coord_change i i').congr fun b hb => _
ext v
exact Z.local_triv_coord_change_eq i i' hb v
#align vector_bundle_core.vector_bundle VectorBundleCore.vectorBundle
/-- The projection on the base of a vector bundle created from core is continuous -/
@[continuity]
theorem continuous_proj : Continuous Z.proj :=
FiberBundleCore.continuous_proj Z
#align vector_bundle_core.continuous_proj VectorBundleCore.continuous_proj
/-- The projection on the base of a vector bundle created from core is an open map -/
theorem isOpenMap_proj : IsOpenMap Z.proj :=
FiberBundleCore.isOpenMap_proj Z
#align vector_bundle_core.is_open_map_proj VectorBundleCore.isOpenMap_proj
end VectorBundleCore
end
/-! ### Vector prebundle -/
section
variable [NontriviallyNormedField R] [∀ x, AddCommMonoid (E x)] [∀ x, Module R (E x)]
[NormedAddCommGroup F] [NormedSpace R F] [TopologicalSpace B]
open TopologicalSpace
open VectorBundle
/- ./././Mathport/Syntax/Translate/Basic.lean:635:2: warning: expanding binder collection (e e' «expr ∈ » pretrivialization_atlas) -/
/-- This structure permits to define a vector bundle when trivializations are given as local
equivalences but there is not yet a topology on the total space or the fibers.
The total space is hence given a topology in such a way that there is a fiber bundle structure for
which the local equivalences are also local homeomorphisms and hence vector bundle trivializations.
The topology on the fibers is induced from the one on the total space.
The field `exists_coord_change` is stated as an existential statement (instead of 3 separate
fields), since it depends on propositional information (namely `e e' ∈ pretrivialization_atlas`).
This makes it inconvenient to explicitly define a `coord_change` function when constructing a
`vector_prebundle`. -/
@[nolint has_nonempty_instance]
structure VectorPrebundle where
pretrivializationAtlas : Set (Pretrivialization F (π E))
pretrivialization_linear' :
∀ (e : Pretrivialization F (π E)) (he : e ∈ pretrivialization_atlas), e.isLinear R
pretrivializationAt : B → Pretrivialization F (π E)
mem_base_pretrivializationAt : ∀ x : B, x ∈ (pretrivialization_at x).baseSet
pretrivialization_mem_atlas : ∀ x : B, pretrivialization_at x ∈ pretrivialization_atlas
exists_coord_change :
∀ (e) (_ : e ∈ pretrivialization_atlas) (e') (_ : e' ∈ pretrivialization_atlas),
∃ f : B → F →L[R] F,
ContinuousOn f (e.baseSet ∩ e'.baseSet) ∧
∀ (b : B) (hb : b ∈ e.baseSet ∩ e'.baseSet) (v : F),
f b v = (e' (totalSpaceMk b (e.symm b v))).2
#align vector_prebundle VectorPrebundle
namespace VectorPrebundle
variable {R E F}
/-- A randomly chosen coordinate change on a `vector_prebundle`, given by
the field `exists_coord_change`. -/
def coordChange (a : VectorPrebundle R F E) {e e' : Pretrivialization F (π E)}
(he : e ∈ a.pretrivializationAtlas) (he' : e' ∈ a.pretrivializationAtlas) (b : B) : F →L[R] F :=
Classical.choose (a.exists_coord_change e he e' he') b
#align vector_prebundle.coord_change VectorPrebundle.coordChange
theorem continuousOn_coordChange (a : VectorPrebundle R F E) {e e' : Pretrivialization F (π E)}
(he : e ∈ a.pretrivializationAtlas) (he' : e' ∈ a.pretrivializationAtlas) :
ContinuousOn (a.coordChange he he') (e.baseSet ∩ e'.baseSet) :=
(Classical.choose_spec (a.exists_coord_change e he e' he')).1
#align vector_prebundle.continuous_on_coord_change VectorPrebundle.continuousOn_coordChange
theorem coordChange_apply (a : VectorPrebundle R F E) {e e' : Pretrivialization F (π E)}
(he : e ∈ a.pretrivializationAtlas) (he' : e' ∈ a.pretrivializationAtlas) {b : B}
(hb : b ∈ e.baseSet ∩ e'.baseSet) (v : F) :
a.coordChange he he' b v = (e' (totalSpaceMk b (e.symm b v))).2 :=
(Classical.choose_spec (a.exists_coord_change e he e' he')).2 b hb v
#align vector_prebundle.coord_change_apply VectorPrebundle.coordChange_apply
theorem mk_coordChange (a : VectorPrebundle R F E) {e e' : Pretrivialization F (π E)}
(he : e ∈ a.pretrivializationAtlas) (he' : e' ∈ a.pretrivializationAtlas) {b : B}
(hb : b ∈ e.baseSet ∩ e'.baseSet) (v : F) :
(b, a.coordChange he he' b v) = e' (totalSpaceMk b (e.symm b v)) :=
by
ext
· rw [e.mk_symm hb.1 v, e'.coe_fst', e.proj_symm_apply' hb.1]
rw [e.proj_symm_apply' hb.1]
exact hb.2
· exact a.coord_change_apply he he' hb v
#align vector_prebundle.mk_coord_change VectorPrebundle.mk_coordChange
/- ./././Mathport/Syntax/Translate/Expr.lean:177:8: unsupported: ambiguous notation -/
/-- Natural identification of `vector_prebundle` as a `fiber_prebundle`. -/
def toFiberPrebundle (a : VectorPrebundle R F E) : FiberPrebundle F E :=
{ a with
continuous_triv_change := by
intro e he e' he'
have :=
is_bounded_bilinear_map_apply.continuous.comp_continuous_on
((a.continuous_on_coord_change he' he).Prod_map continuousOn_id)
have H :
e'.to_local_equiv.target ∩ e'.to_local_equiv.symm ⁻¹' e.to_local_equiv.source =
(e'.base_set ∩ e.base_set) ×ˢ univ :=
by
rw [e'.target_eq, e.source_eq]
ext ⟨b, f⟩
simp (config := { contextual := true }) only [-total_space.proj, and_congr_right_iff,
e'.proj_symm_apply', iff_self_iff, imp_true_iff, mfld_simps]
rw [H]
refine' (continuous_on_fst.prod this).congr _
rintro ⟨b, f⟩ ⟨hb, -⟩
dsimp only [Function.comp, Prod.map]
rw [a.mk_coord_change _ _ hb, e'.mk_symm hb.1]
rfl }
#align vector_prebundle.to_fiber_prebundle VectorPrebundle.toFiberPrebundle
/-- Topology on the total space that will make the prebundle into a bundle. -/
def totalSpaceTopology (a : VectorPrebundle R F E) : TopologicalSpace (TotalSpace E) :=
a.toFiberPrebundle.totalSpaceTopology
#align vector_prebundle.total_space_topology VectorPrebundle.totalSpaceTopology
/-- Promotion from a `trivialization` in the `pretrivialization_atlas` of a
`vector_prebundle` to a `trivialization`. -/
def trivializationOfMemPretrivializationAtlas (a : VectorPrebundle R F E)
{e : Pretrivialization F (π E)} (he : e ∈ a.pretrivializationAtlas) :
@Trivialization B F _ _ _ a.totalSpaceTopology (π E) :=
a.toFiberPrebundle.trivializationOfMemPretrivializationAtlas he
#align vector_prebundle.trivialization_of_mem_pretrivialization_atlas VectorPrebundle.trivializationOfMemPretrivializationAtlas
theorem linear_of_mem_pretrivializationAtlas (a : VectorPrebundle R F E)
{e : Pretrivialization F (π E)} (he : e ∈ a.pretrivializationAtlas) :
@Trivialization.IsLinear R B F _ _ _ _ a.totalSpaceTopology _ _ _ _
(trivializationOfMemPretrivializationAtlas a he) :=
{ linear := (a.pretrivialization_linear' e he).linear }
#align vector_prebundle.linear_of_mem_pretrivialization_atlas VectorPrebundle.linear_of_mem_pretrivializationAtlas
variable (a : VectorPrebundle R F E)
theorem mem_trivialization_at_source (b : B) (x : E b) :
totalSpaceMk b x ∈ (a.pretrivializationAt b).source :=
a.toFiberPrebundle.mem_pretrivializationAt_source b x
#align vector_prebundle.mem_trivialization_at_source VectorPrebundle.mem_trivialization_at_source
@[simp]
theorem totalSpaceMk_preimage_source (b : B) :
totalSpaceMk b ⁻¹' (a.pretrivializationAt b).source = univ :=
a.toFiberPrebundle.totalSpaceMk_preimage_source b
#align vector_prebundle.total_space_mk_preimage_source VectorPrebundle.totalSpaceMk_preimage_source
/-- Topology on the fibers `E b` induced by the map `E b → E.total_space`. -/
def fiberTopology (b : B) : TopologicalSpace (E b) :=
a.toFiberPrebundle.fiberTopology b
#align vector_prebundle.fiber_topology VectorPrebundle.fiberTopology
@[continuity]
theorem inducing_totalSpaceMk (b : B) :
@Inducing _ _ (a.fiberTopology b) a.totalSpaceTopology (totalSpaceMk b) :=
a.toFiberPrebundle.inducing_totalSpaceMk b
#align vector_prebundle.inducing_total_space_mk VectorPrebundle.inducing_totalSpaceMk
@[continuity]
theorem continuous_totalSpaceMk (b : B) :
@Continuous _ _ (a.fiberTopology b) a.totalSpaceTopology (totalSpaceMk b) :=
a.toFiberPrebundle.continuous_totalSpaceMk b
#align vector_prebundle.continuous_total_space_mk VectorPrebundle.continuous_totalSpaceMk
/-- Make a `fiber_bundle` from a `vector_prebundle`; auxiliary construction for
`vector_prebundle.vector_bundle`. -/
def toFiberBundle : @FiberBundle B F _ _ _ a.totalSpaceTopology a.fiberTopology :=
a.toFiberPrebundle.toFiberBundle
#align vector_prebundle.to_fiber_bundle VectorPrebundle.toFiberBundle
/-- Make a `vector_bundle` from a `vector_prebundle`. Concretely this means
that, given a `vector_prebundle` structure for a sigma-type `E` -- which consists of a
number of "pretrivializations" identifying parts of `E` with product spaces `U × F` -- one
establishes that for the topology constructed on the sigma-type using
`vector_prebundle.total_space_topology`, these "pretrivializations" are actually
"trivializations" (i.e., homeomorphisms with respect to the constructed topology). -/
theorem to_vectorBundle :
@VectorBundle R _ F E _ _ _ _ _ _ a.totalSpaceTopology a.fiberTopology a.toFiberBundle :=
{ trivialization_linear' := by
rintro _ ⟨e, he, rfl⟩
apply linear_of_mem_pretrivialization_atlas
continuousOn_coord_change' :=
by
rintro _ _ ⟨e, he, rfl⟩ ⟨e', he', rfl⟩
refine' (a.continuous_on_coord_change he he').congr _
intro b hb
ext v
rw [a.coord_change_apply he he' hb v, ContinuousLinearEquiv.coe_coe,
Trivialization.coordChangeL_apply]
exacts[rfl, hb] }
#align vector_prebundle.to_vector_bundle VectorPrebundle.to_vectorBundle
end VectorPrebundle
end
|
open import Agda.Primitive
open import Agda.Builtin.List
open import Agda.Builtin.Equality
private
variable
a p : Level
A : Set a
P Q : A → Set p
data Any {a p} {A : Set a} (P : A → Set p) : List A → Set (a ⊔ p) where
here : ∀ {x xs} (px : P x) → Any P (x ∷ xs)
there : ∀ {x xs} (pxs : Any P xs) → Any P (x ∷ xs)
map : (∀ {x} → P x → Q x) → (∀ {xs} → Any P xs → Any Q xs)
map g (here px) = here (g px)
map g (there pxs) = there (map g pxs)
postulate
map-id : ∀ (f : ∀ {x} → P x → P x) → (∀ {x} (p : P x) → f p ≡ p) →
∀ {xs} → (p : Any P xs) → map f p ≡ p
|
theory Set_Example
imports Sepreftime
begin
fun makeset where
"makeset (S,[]) = S" |
"makeset (S,(x#xs)) = makeset (insert x S, xs)"
lemma makeset_set_aux: "makeset (S,xs) = S \<union> set xs"
apply(induct xs arbitrary: S) by auto
corollary Z: "makeset ({},xs) = set xs" using makeset_set_aux by fast
fun t_insert :: "nat \<Rightarrow> nat" where "t_insert n = 3*n"
fun t_makeset :: "nat \<Rightarrow> nat" where "t_makeset n = n * t_insert n"
definition "makeset_SPEC xs = SPECT [set xs \<mapsto> t_makeset (length xs)]"
definition "makeset_impl xs \<equiv> do {
as \<leftarrow> RETURNT xs;
S \<leftarrow> RETURNT {};
s \<leftarrow> RETURNT (S,xs);
r \<leftarrow> whileT (\<lambda>(S,as). as \<noteq> [])
(\<lambda>(S,as). do {
(* ASSERT (as\<noteq>[]);
ASSERT (length as \<le> length xs);*)
x \<leftarrow> RETURNT (hd as);
as' \<leftarrow> RETURNT (tl as);
S' \<leftarrow> REST [insert x S \<mapsto> t_insert (length xs)];
RETURNT (S',as')
} ) s;
RETURNT (fst r)
}"
thm whileT_rule'''
(*
lemma ASSERT_bindT: "ASSERT B \<bind> f = SPECT M \<longleftrightarrow> (B \<and> f () = SPECT M)"
unfolding ASSERT_def iASSERT_def by simp
*)
lemma k: "y\<noteq>[] \<Longrightarrow> makeset (insert (hd y) x, tl y) = makeset (x, y)"
apply (cases y) by auto
lemma "makeset_impl xs \<le> makeset_SPEC xs"
unfolding makeset_SPEC_def makeset_impl_def apply simp
apply(rule Refinement_by_T)
apply (simp only: T_bindT)
apply(simp only: T_RETURNT)
apply(rule T_conseq4)
apply(rule whileT_rule'''[where I="\<lambda>s. if makeset s = makeset ({},xs) then Some (length (snd s) * t_insert (length xs) ) else None" ])
apply simp
subgoal (* progress *)
apply(auto split: prod.splits)
apply(auto simp add: ASSERT_bindT)
sorry
subgoal (* IS *)
apply (auto split: if_splits)
apply (simp only: T_bindT)
apply(simp only: T_RETURNT)
apply(simp only: T_REST)
apply(subst k) apply simp apply simp
apply(auto simp: mm3_def mm2_def)
by (simp add: left_diff_distrib')
apply simp
(* post condition *)
apply (auto split: if_splits)
unfolding Z apply simp apply simp
unfolding mm3_def by simp
end |
theory Mk_Record_Simp
imports Refine_Util Mpat_Antiquot
begin
(*
mk_record_simp: Converts a lemma of the form
"f s = x" to the form "r \<equiv> s \<Longrightarrow> f r = x"
This is used to fold the x.simp - lemmas of a record x with a definition
of the form "r \<equiv> \<lparr> ... \<rparr>".
Usage example:
record foo = ...
definition c :: foo where "c \<equiv> \<lparr> ... \<rparr>"
lemmas c_simps[simp] = foo.simps[mk_record_simp, OF c_def]
*)
lemma mk_record_simp_thm:
fixes f :: "'a \<Rightarrow> 'b"
assumes "f s = x"
assumes "r \<equiv> s"
shows "f r = x"
using assms by simp
ML {*
fun mk_record_simp context thm = let
val ctxt = Context.proof_of context
val cert = Thm.cterm_of ctxt
in
case Thm.concl_of thm of
@{mpat "Trueprop (?f _=_)"} =>
let
val cf = cert f
val r = infer_instantiate ctxt [(("f", 0), cf)] @{thm mk_record_simp_thm}
val r = r OF [thm]
in r end
| _ => raise THM("",~1,[thm])
end
*}
attribute_setup mk_record_simp =
{* Scan.succeed (Thm.rule_attribute [] (mk_record_simp)) *}
"Make simplification rule for record definition"
end
|
[GOAL]
α β γ : Type u
F : Type u → Type v
inst✝¹ : Applicative F
inst✝ : LawfulApplicative F
x : F (α → β)
f : γ → α
y : F γ
⊢ (Seq.seq x fun x => f <$> y) = Seq.seq ((fun x => x ∘ f) <$> x) fun x => y
[PROOFSTEP]
simp [← pure_seq]
[GOAL]
α β γ : Type u
F : Type u → Type v
inst✝¹ : Applicative F
inst✝ : LawfulApplicative F
x : F (α → β)
f : γ → α
y : F γ
⊢ (Seq.seq x fun x => Seq.seq (pure f) fun x => y) = Seq.seq (Seq.seq (pure fun x => x ∘ f) fun x_1 => x) fun x => y
[PROOFSTEP]
simp [seq_assoc, ← comp_map, (· ∘ ·)]
[GOAL]
α β γ : Type u
F : Type u → Type v
inst✝¹ : Applicative F
inst✝ : LawfulApplicative F
x : F (α → β)
f : γ → α
y : F γ
⊢ (Seq.seq ((fun x x_1 => x (f x_1)) <$> x) fun x => y) =
Seq.seq (Seq.seq (pure fun x x_1 => x (f x_1)) fun x_1 => x) fun x => y
[PROOFSTEP]
simp [pure_seq]
[GOAL]
α β γ : Type u
F : Type u → Type v
inst✝¹ : Applicative F
inst✝ : LawfulApplicative F
f : β → γ
x : F (α → β)
y : F α
⊢ (f <$> Seq.seq x fun x => y) = Seq.seq ((fun x => f ∘ x) <$> x) fun x => y
[PROOFSTEP]
simp only [← pure_seq]
[GOAL]
α β γ : Type u
F : Type u → Type v
inst✝¹ : Applicative F
inst✝ : LawfulApplicative F
f : β → γ
x : F (α → β)
y : F α
⊢ (Seq.seq (pure f) fun x_1 => Seq.seq x fun x => y) = Seq.seq (Seq.seq (pure fun x => f ∘ x) fun x_1 => x) fun x => y
[PROOFSTEP]
simp [seq_assoc]
[GOAL]
α β γ : Type u
m : Type u → Type v
inst✝¹ : Monad m
inst✝ : LawfulMonad m
x : m α
g : α → m β
f : β → γ
⊢ f <$> (x >>= g) = do
let a ← x
f <$> g a
[PROOFSTEP]
rw [← bind_pure_comp, bind_assoc]
[GOAL]
α β γ : Type u
m : Type u → Type v
inst✝¹ : Monad m
inst✝ : LawfulMonad m
x : m α
g : α → m β
f : β → γ
⊢ (do
let x ← x
let a ← g x
pure (f a)) =
do
let a ← x
f <$> g a
[PROOFSTEP]
simp [bind_pure_comp]
[GOAL]
α β γ : Type u
m : Type u → Type v
inst✝¹ : Monad m
inst✝ : LawfulMonad m
x : m α
g : β → m γ
f : α → β
⊢ f <$> x >>= g = x >>= g ∘ f
[PROOFSTEP]
rw [← bind_pure_comp, bind_assoc]
[GOAL]
α β γ : Type u
m : Type u → Type v
inst✝¹ : Monad m
inst✝ : LawfulMonad m
x : m α
g : β → m γ
f : α → β
⊢ (x >>= fun x => pure (f x) >>= g) = x >>= g ∘ f
[PROOFSTEP]
simp [pure_bind, (· ∘ ·)]
[GOAL]
α✝ β✝ γ : Type u
m : Type u → Type v
inst✝¹ : Monad m
inst✝ : LawfulMonad m
α : Type u_1
β : Type u
f : α → m β
⊢ f >=> pure = f
[PROOFSTEP]
simp only [(· >=> ·), functor_norm]
[GOAL]
α✝ β✝ γ : Type u
m : Type u → Type v
inst✝¹ : Monad m
inst✝ : LawfulMonad m
α β : Type u
f : α → m β
⊢ pure >=> f = f
[PROOFSTEP]
simp only [(· >=> ·), functor_norm]
[GOAL]
α✝ β✝ γ✝ : Type u
m : Type u → Type v
inst✝¹ : Monad m
inst✝ : LawfulMonad m
α : Type u_1
β γ φ : Type u
f : α → m β
g : β → m γ
h : γ → m φ
⊢ (f >=> g) >=> h = f >=> g >=> h
[PROOFSTEP]
simp only [(· >=> ·), functor_norm]
[GOAL]
α✝ β✝ γ : Type u
m : Type u → Type u
inst✝¹ : Monad m
inst✝ : LawfulMonad m
α β : Type u
f : α → β
a : m (m α)
⊢ joinM (Functor.map f <$> a) = f <$> joinM a
[PROOFSTEP]
simp only [joinM, (· ∘ ·), id.def, ← bind_pure_comp, bind_assoc, map_bind, pure_bind]
[GOAL]
α✝ β γ : Type u
m : Type u → Type u
inst✝¹ : Monad m
inst✝ : LawfulMonad m
α : Type u
a : m (m (m α))
⊢ joinM (joinM <$> a) = joinM (joinM a)
[PROOFSTEP]
simp only [joinM, (· ∘ ·), id.def, map_bind, ← bind_pure_comp, bind_assoc, pure_bind]
[GOAL]
α✝ β γ : Type u
m : Type u → Type u
inst✝¹ : Monad m
inst✝ : LawfulMonad m
α : Type u
a : m α
⊢ joinM (pure <$> a) = a
[PROOFSTEP]
simp only [joinM, (· ∘ ·), id.def, map_bind, ← bind_pure_comp, bind_assoc, pure_bind, bind_pure]
[GOAL]
α β γ : Type u
F : Type → Type v
inst✝ : Alternative F
h : Decidable True
⊢ guard True = pure ()
[PROOFSTEP]
simp [guard, if_pos]
[GOAL]
α β γ : Type u
F : Type → Type v
inst✝ : Alternative F
h : Decidable False
⊢ guard False = failure
[PROOFSTEP]
simp [guard, if_neg not_false]
[GOAL]
α β γ : Type u
e : Type v
⊢ LawfulFunctor (Sum e)
[PROOFSTEP]
refine' { .. }
[GOAL]
case refine'_1
α β γ : Type u
e : Type v
⊢ ∀ {α β : Type u}, Functor.mapConst = Functor.map ∘ Function.const β
[PROOFSTEP]
intros
[GOAL]
case refine'_2
α β γ : Type u
e : Type v
⊢ ∀ {α : Type u} (x : e ⊕ α), id <$> x = x
[PROOFSTEP]
intros
[GOAL]
case refine'_3
α β γ : Type u
e : Type v
⊢ ∀ {α β γ : Type u} (g : α → β) (h : β → γ) (x : e ⊕ α), (h ∘ g) <$> x = h <$> g <$> x
[PROOFSTEP]
intros
[GOAL]
case refine'_1
α β γ : Type u
e : Type v
α✝ β✝ : Type u
⊢ Functor.mapConst = Functor.map ∘ Function.const β✝
[PROOFSTEP]
try casesm Sum _ _
[GOAL]
case refine'_1
α β γ : Type u
e : Type v
α✝ β✝ : Type u
⊢ Functor.mapConst = Functor.map ∘ Function.const β✝
[PROOFSTEP]
casesm Sum _ _
[GOAL]
case refine'_2
α β γ : Type u
e : Type v
α✝ : Type u
x✝ : e ⊕ α✝
⊢ id <$> x✝ = x✝
[PROOFSTEP]
try casesm Sum _ _
[GOAL]
case refine'_2
α β γ : Type u
e : Type v
α✝ : Type u
x✝ : e ⊕ α✝
⊢ id <$> x✝ = x✝
[PROOFSTEP]
casesm Sum _ _
[GOAL]
case refine'_3
α β γ : Type u
e : Type v
α✝ β✝ γ✝ : Type u
g✝ : α✝ → β✝
h✝ : β✝ → γ✝
x✝ : e ⊕ α✝
⊢ (h✝ ∘ g✝) <$> x✝ = h✝ <$> g✝ <$> x✝
[PROOFSTEP]
try casesm Sum _ _
[GOAL]
case refine'_3
α β γ : Type u
e : Type v
α✝ β✝ γ✝ : Type u
g✝ : α✝ → β✝
h✝ : β✝ → γ✝
x✝ : e ⊕ α✝
⊢ (h✝ ∘ g✝) <$> x✝ = h✝ <$> g✝ <$> x✝
[PROOFSTEP]
casesm Sum _ _
[GOAL]
case refine'_1
α β γ : Type u
e : Type v
α✝ β✝ : Type u
⊢ Functor.mapConst = Functor.map ∘ Function.const β✝
[PROOFSTEP]
rfl
[GOAL]
case refine'_2.inl
α β γ : Type u
e : Type v
α✝ : Type u
val✝ : e
⊢ id <$> inl val✝ = inl val✝
[PROOFSTEP]
rfl
[GOAL]
case refine'_2.inr
α β γ : Type u
e : Type v
α✝ : Type u
val✝ : α✝
⊢ id <$> inr val✝ = inr val✝
[PROOFSTEP]
rfl
[GOAL]
case refine'_3.inl
α β γ : Type u
e : Type v
α✝ β✝ γ✝ : Type u
g✝ : α✝ → β✝
h✝ : β✝ → γ✝
val✝ : e
⊢ (h✝ ∘ g✝) <$> inl val✝ = h✝ <$> g✝ <$> inl val✝
[PROOFSTEP]
rfl
[GOAL]
case refine'_3.inr
α β γ : Type u
e : Type v
α✝ β✝ γ✝ : Type u
g✝ : α✝ → β✝
h✝ : β✝ → γ✝
val✝ : α✝
⊢ (h✝ ∘ g✝) <$> inr val✝ = h✝ <$> g✝ <$> inr val✝
[PROOFSTEP]
rfl
[GOAL]
α β γ : Type u
e : Type v
⊢ ∀ {α β : Type u} (x : e ⊕ α) (y : e ⊕ β), (SeqLeft.seqLeft x fun x => y) = Seq.seq (Function.const β <$> x) fun x => y
[PROOFSTEP]
intros
[GOAL]
α β γ : Type u
e : Type v
α✝ β✝ : Type u
x✝ : e ⊕ α✝
y✝ : e ⊕ β✝
⊢ (SeqLeft.seqLeft x✝ fun x => y✝) = Seq.seq (Function.const β✝ <$> x✝) fun x => y✝
[PROOFSTEP]
casesm Sum _ _
[GOAL]
case inl
α β γ : Type u
e : Type v
α✝ β✝ : Type u
y✝ : e ⊕ β✝
val✝ : e
⊢ (SeqLeft.seqLeft (inl val✝) fun x => y✝) = Seq.seq (Function.const β✝ <$> inl val✝) fun x => y✝
[PROOFSTEP]
rfl
[GOAL]
case inr
α β γ : Type u
e : Type v
α✝ β✝ : Type u
y✝ : e ⊕ β✝
val✝ : α✝
⊢ (SeqLeft.seqLeft (inr val✝) fun x => y✝) = Seq.seq (Function.const β✝ <$> inr val✝) fun x => y✝
[PROOFSTEP]
rfl
[GOAL]
α β γ : Type u
e : Type v
⊢ ∀ {α β : Type u} (x : e ⊕ α) (y : e ⊕ β),
(SeqRight.seqRight x fun x => y) = Seq.seq (Function.const α id <$> x) fun x => y
[PROOFSTEP]
intros
[GOAL]
α β γ : Type u
e : Type v
α✝ β✝ : Type u
x✝ : e ⊕ α✝
y✝ : e ⊕ β✝
⊢ (SeqRight.seqRight x✝ fun x => y✝) = Seq.seq (Function.const α✝ id <$> x✝) fun x => y✝
[PROOFSTEP]
casesm Sum _ _
[GOAL]
case inl
α β γ : Type u
e : Type v
α✝ β✝ : Type u
y✝ : e ⊕ β✝
val✝ : e
⊢ (SeqRight.seqRight (inl val✝) fun x => y✝) = Seq.seq (Function.const α✝ id <$> inl val✝) fun x => y✝
[PROOFSTEP]
casesm Sum _ _
[GOAL]
case inr
α β γ : Type u
e : Type v
α✝ β✝ : Type u
y✝ : e ⊕ β✝
val✝ : α✝
⊢ (SeqRight.seqRight (inr val✝) fun x => y✝) = Seq.seq (Function.const α✝ id <$> inr val✝) fun x => y✝
[PROOFSTEP]
casesm Sum _ _
[GOAL]
case inl.inl
α β γ : Type u
e : Type v
α✝ β✝ : Type u
val✝¹ val✝ : e
⊢ (SeqRight.seqRight (inl val✝¹) fun x => inl val✝) = Seq.seq (Function.const α✝ id <$> inl val✝¹) fun x => inl val✝
[PROOFSTEP]
rfl
[GOAL]
case inl.inr
α β γ : Type u
e : Type v
α✝ β✝ : Type u
val✝¹ : e
val✝ : β✝
⊢ (SeqRight.seqRight (inl val✝¹) fun x => inr val✝) = Seq.seq (Function.const α✝ id <$> inl val✝¹) fun x => inr val✝
[PROOFSTEP]
rfl
[GOAL]
case inr.inl
α β γ : Type u
e : Type v
α✝ β✝ : Type u
val✝¹ : α✝
val✝ : e
⊢ (SeqRight.seqRight (inr val✝¹) fun x => inl val✝) = Seq.seq (Function.const α✝ id <$> inr val✝¹) fun x => inl val✝
[PROOFSTEP]
rfl
[GOAL]
case inr.inr
α β γ : Type u
e : Type v
α✝ β✝ : Type u
val✝¹ : α✝
val✝ : β✝
⊢ (SeqRight.seqRight (inr val✝¹) fun x => inr val✝) = Seq.seq (Function.const α✝ id <$> inr val✝¹) fun x => inr val✝
[PROOFSTEP]
rfl
[GOAL]
α β γ : Type u
e : Type v
⊢ ∀ {α β : Type u} (g : α → β) (x : e ⊕ α), (Seq.seq (pure g) fun x_1 => x) = g <$> x
[PROOFSTEP]
intros
[GOAL]
α β γ : Type u
e : Type v
α✝ β✝ : Type u
g✝ : α✝ → β✝
x✝ : e ⊕ α✝
⊢ (Seq.seq (pure g✝) fun x => x✝) = g✝ <$> x✝
[PROOFSTEP]
rfl
[GOAL]
α β γ : Type u
e : Type v
⊢ ∀ {α β : Type u} (f : α → β) (x : e ⊕ α),
(do
let a ← x
pure (f a)) =
f <$> x
[PROOFSTEP]
intros
[GOAL]
α β γ : Type u
e : Type v
α✝ β✝ : Type u
f✝ : α✝ → β✝
x✝ : e ⊕ α✝
⊢ (do
let a ← x✝
pure (f✝ a)) =
f✝ <$> x✝
[PROOFSTEP]
casesm Sum _ _
[GOAL]
case inl
α β γ : Type u
e : Type v
α✝ β✝ : Type u
f✝ : α✝ → β✝
val✝ : e
⊢ (do
let a ← inl val✝
pure (f✝ a)) =
f✝ <$> inl val✝
[PROOFSTEP]
rfl
[GOAL]
case inr
α β γ : Type u
e : Type v
α✝ β✝ : Type u
f✝ : α✝ → β✝
val✝ : α✝
⊢ (do
let a ← inr val✝
pure (f✝ a)) =
f✝ <$> inr val✝
[PROOFSTEP]
rfl
[GOAL]
α β γ : Type u
e : Type v
⊢ ∀ {α β : Type u} (f : e ⊕ (α → β)) (x : e ⊕ α),
(do
let x_1 ← f
x_1 <$> x) =
Seq.seq f fun x_1 => x
[PROOFSTEP]
intros
[GOAL]
α β γ : Type u
e : Type v
α✝ β✝ : Type u
f✝ : e ⊕ (α✝ → β✝)
x✝ : e ⊕ α✝
⊢ (do
let x ← f✝
x <$> x✝) =
Seq.seq f✝ fun x => x✝
[PROOFSTEP]
casesm Sum _ _
[GOAL]
case inl
α β γ : Type u
e : Type v
α✝ β✝ : Type u
x✝ : e ⊕ α✝
val✝ : e
⊢ (do
let x ← inl val✝
x <$> x✝) =
Seq.seq (inl val✝) fun x => x✝
[PROOFSTEP]
rfl
[GOAL]
case inr
α β γ : Type u
e : Type v
α✝ β✝ : Type u
x✝ : e ⊕ α✝
val✝ : α✝ → β✝
⊢ (do
let x ← inr val✝
x <$> x✝) =
Seq.seq (inr val✝) fun x => x✝
[PROOFSTEP]
rfl
[GOAL]
α β γ : Type u
e : Type v
⊢ ∀ {α β : Type u} (x : α) (f : α → e ⊕ β), pure x >>= f = f x
[PROOFSTEP]
intros
[GOAL]
α β γ : Type u
e : Type v
α✝ β✝ : Type u
x✝ : α✝
f✝ : α✝ → e ⊕ β✝
⊢ pure x✝ >>= f✝ = f✝ x✝
[PROOFSTEP]
rfl
[GOAL]
α β γ : Type u
e : Type v
⊢ ∀ {α β γ : Type u} (x : e ⊕ α) (f : α → e ⊕ β) (g : β → e ⊕ γ), x >>= f >>= g = x >>= fun x => f x >>= g
[PROOFSTEP]
intros
[GOAL]
α β γ : Type u
e : Type v
α✝ β✝ γ✝ : Type u
x✝ : e ⊕ α✝
f✝ : α✝ → e ⊕ β✝
g✝ : β✝ → e ⊕ γ✝
⊢ x✝ >>= f✝ >>= g✝ = x✝ >>= fun x => f✝ x >>= g✝
[PROOFSTEP]
casesm Sum _ _
[GOAL]
case inl
α β γ : Type u
e : Type v
α✝ β✝ γ✝ : Type u
f✝ : α✝ → e ⊕ β✝
g✝ : β✝ → e ⊕ γ✝
val✝ : e
⊢ inl val✝ >>= f✝ >>= g✝ = inl val✝ >>= fun x => f✝ x >>= g✝
[PROOFSTEP]
rfl
[GOAL]
case inr
α β γ : Type u
e : Type v
α✝ β✝ γ✝ : Type u
f✝ : α✝ → e ⊕ β✝
g✝ : β✝ → e ⊕ γ✝
val✝ : α✝
⊢ inr val✝ >>= f✝ >>= g✝ = inr val✝ >>= fun x => f✝ x >>= g✝
[PROOFSTEP]
rfl
[GOAL]
α✝ β✝ γ✝ : Type u
m✝ : ?m.20624
m : Type u → Type v
h : Applicative m
inst✝ : CommApplicative m
α β γ : Type u
a : m α
b : m β
f : α → β → γ
⊢ (Seq.seq (f <$> a) fun x => b) = (fun p => f p.fst p.snd) <$> Seq.seq (Prod.mk <$> a) fun x => b
[PROOFSTEP]
simp [seq_map_assoc, map_seq, seq_assoc, seq_pure, map_map]
[GOAL]
α✝ β✝ γ✝ : Type u
m✝ : ?m.20624
m : Type u → Type v
h : Applicative m
inst✝ : CommApplicative m
α β γ : Type u
a : m α
b : m β
f : α → β → γ
⊢ (Seq.seq (f <$> a) fun x => b) = Seq.seq (((fun x => (fun p => f p.fst p.snd) ∘ x) ∘ Prod.mk) <$> a) fun x => b
[PROOFSTEP]
rfl
[GOAL]
α✝ β✝ γ✝ : Type u
m✝ : ?m.20624
m : Type u → Type v
h : Applicative m
inst✝ : CommApplicative m
α β γ : Type u
a : m α
b : m β
f : α → β → γ
⊢ ((fun p => f p.fst p.snd) <$> Seq.seq (Prod.mk <$> a) fun x => b) = Seq.seq ((fun b a => f a b) <$> b) fun x => a
[PROOFSTEP]
rw [@CommApplicative.commutative_prod m h]
[GOAL]
α✝ β✝ γ✝ : Type u
m✝ : ?m.20624
m : Type u → Type v
h : Applicative m
inst✝ : CommApplicative m
α β γ : Type u
a : m α
b : m β
f : α → β → γ
⊢ ((fun p => f p.fst p.snd) <$> Seq.seq ((fun b a => (a, b)) <$> b) fun x => a) =
Seq.seq ((fun b a => f a b) <$> b) fun x => a
[PROOFSTEP]
simp [seq_map_assoc, map_seq, seq_assoc, seq_pure, map_map, (· ∘ ·)]
|
function y=stdtpdf(x,mu,sigma2,nu)
% Probability Density Function (PDF) for the Standardized T distribution
%
% USAGE:
% Y = stdtpdf(X,MU,SIGMA2,NU)
%
% INPUTS:
% X - Standardized T random variables
% MU - Mean of X, either scalar or size(x)
% SIGMA2 - Variance of X, either scalar or size(x)
% NU - Degree of freedom parameters, either scalar or size(x)
%
% OUTPUTS:
% Y - Probability density evaluated at X
%
% COMMENTS:
% NU>2
%
% REFERENCES:
% [1] Cassella and Berger (1990) 'Statistical Inference'
%
% See also STDTCDF, STDTINV, STDTRND, STDTLOGLIK, TPDF
% Copyright:
% Kevin Sheppard
% [email protected]
% Revision: 6 Date: 8/21/2014
[T,K]=size(x);
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% Input Checking
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
if K~=1
error('X must be a column vector');
end
if nargin==4
if length(mu)~=1 && ~all(size(mu)==[T K])
error('mu must be either a scalar or the same size as X');
end
if any(sigma2<=0)
error('sigma2 must contain only positive elements')
end
if length(sigma2)==1
sigma2=sigma2*ones(T,K);
elseif size(sigma2,1)~=T || size(sigma2,2)~=1
error('sigma2 must be a scalar or a vector with the same dimensions as X');
end
if length(nu)>1 || nu<=2
error('nu must be a scalar greater than 2');
end
x=x-mu;
else
error('Only 4 inputs supported');
end
constant = exp(gammaln( 0.5 * (nu + 1)) - gammaln(0.5 * nu));
y = constant ./ sqrt(pi * (nu - 2) * sigma2) .* (1 + (x-mu) .^ 2.0 / (sigma2 * (nu - 2))) .^ (-(nu + 1) / 2);
|
/-
Copyright © 2018 François G. Dorais. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
-/
import .basic .subst
universe u
variables {σ : Type u} {sig : σ → ℕ} {I : Type*} (ax : I → eqn sig)
include sig ax
definition models (e : eqn sig) : Prop :=
∀ {α : Type (u+1)} (a : alg sig α), (∀ i, a ⊧ ax i) → (a ⊧ e)
infix ` ⊨ `:20 := models
lemma model.axiom (i : I) : ax ⊨ ax i := λ _ _ ha, ha i
namespace model
lemma reflexivity (t : term sig) : ax ⊨ t ≡ t :=
λ _ _ _ _, rfl
lemma symmetry {{t u : term sig}} : (ax ⊨ t ≡ u) → (ax ⊨ u ≡ t) :=
λ h _ a ha val, eq.symm (h a ha val)
lemma transitivity {{t u v : term sig}} : (ax ⊨ t ≡ u) → (ax ⊨ u ≡ v) → (ax ⊨ t ≡ v) :=
λ htu huv α a ha val, eq.trans (htu a ha val) (huv a ha val)
lemma substitiution (sub : ℕ → term sig) {{t u : term sig}} :
(ax ⊨ t ≡ u) → (ax ⊨ subst sub t ≡ subst sub u) :=
λ h _ a ha val,
calc
eval a val (subst sub t)
= eval a (λ n, eval a val (sub n)) t : by rw subst_eval
... = eval a (λ n, eval a val (sub n)) u : by rw h a ha
... = eval a val (subst sub u) : by rw subst_eval
lemma replacement {{sub₁ sub₂ : ℕ → term sig}} :
(∀ n, ax ⊨ (sub₁ n) ≡ (sub₂ n)) → (∀ t, ax ⊨ subst sub₁ t ≡ subst sub₂ t)
| h (term.var n) :=
λ _ a ha val, h n a ha val
| h (term.app s ts) :=
λ _ a ha val,
have tup.map (eval a val) (tup.map (subst sub₁) ts) = tup.map (eval a val) (tup.map (subst sub₂) ts),
from tup.ext (λ i, replacement h (ts i) a ha val),
calc
eval a val (subst sub₁ (term.app s ts))
= eval a val (term.app s (tup.map (subst sub₁) ts)) : by rw subst_app
... = a.app s (tup.map (eval a val) (tup.map (subst sub₁) ts)) : by rw eval_app
... = a.app s (tup.map (eval a val) (tup.map (subst sub₂) ts)) : by rw this
... = eval a val (term.app s (tup.map (subst sub₂) ts)) : by rw eval_app
... = eval a val (subst sub₂ (term.app s ts)) : by rw subst_app
end model |
[STATEMENT]
lemma less_nat_qbs_morphism:
"(<) \<in> \<nat>\<^sub>Q \<rightarrow>\<^sub>Q exp_qbs \<nat>\<^sub>Q \<bool>\<^sub>Q"
[PROOF STATE]
proof (prove)
goal (1 subgoal):
1. (<) \<in> \<nat>\<^sub>Q \<rightarrow>\<^sub>Q \<nat>\<^sub>Q \<Rightarrow>\<^sub>Q \<bool>\<^sub>Q
[PROOF STEP]
by(auto intro!: nat_qbs_morphism) |
theory State_Networks_Impl
imports TA_Impl.Normalized_Zone_Semantics_Impl State_Networks
begin
(* XXX Move *)
lemma finite_lists_length_eq:
"finite {s. length s = r \<and> set s \<subseteq> S}" if "finite S"
by (rule finite_lists_length_le[OF that, THEN finite_subset[rotated], where n1 = r]) auto
(* XXX Move *)
lemma finite_lists_boundedI:
assumes "\<forall> i < r. finite (S i)"
shows "finite {s. length s = r \<and> (\<forall>i<r. s ! i \<in> S i)}" (is "finite ?R")
proof -
let ?S = "\<Union> {S i | i. i < r}"
have "?R \<subseteq> {s. length s = r \<and> set s \<subseteq> ?S}"
by (auto dest!: aux)
moreover have "finite \<dots>" by (rule finite_lists_length_eq) (use assms in auto)
ultimately show ?thesis by (rule finite_subset)
qed
abbreviation "repeat x n \<equiv> map (\<lambda> _. x) [0..<n]"
subsection \<open>Pre-compiled networks with states and clocks as natural numbers\<close>
locale State_Network_Reachability_Problem_precompiled_defs =
fixes p :: nat \<comment> \<open>Number of processes\<close>
and m :: nat \<comment> \<open>Number of clocks\<close>
and k :: "nat list" \<comment> \<open>Clock ceiling. Maximal constant appearing in automaton for each state\<close>
and inv :: "(nat, int) cconstraint list list" \<comment> \<open>Clock invariants on states per process\<close>
and pred :: "('st \<Rightarrow> bool) list list" \<comment> \<open>Clock invariants on states per process\<close>
and trans ::
"((nat, int) cconstraint * ('st \<Rightarrow> bool) * nat act * nat list * ('st \<Rightarrow> 'st) * nat) list list list"
\<comment> \<open>Transitions between states per process\<close>
and final :: "nat list list" \<comment> \<open>Final states per process. Initial location is 0\<close>
begin
definition "clkp_set' \<equiv> \<Union>
(collect_clock_pairs ` set (concat inv)
\<union> (\<lambda> (g, _). collect_clock_pairs g) ` set (concat (concat trans)))"
definition clk_set'_def: "clk_set' =
(fst ` clkp_set' \<union> \<Union> ((\<lambda> (_, _, _, r, _). set r) ` set (concat (concat trans))))"
text \<open>Definition of the corresponding network\<close>
definition "make_trans \<equiv> \<lambda> (g, c, a, r, m, l'). (g, (a, (c, m)), r, l')"
definition "I i l \<equiv> if l < length (inv ! i) then inv ! i ! l else []"
definition "T i \<equiv>
{(l, make_trans (trans ! i ! l ! j)) | l j. l < length (trans ! i) \<and> j < length (trans ! i ! l)}"
definition "P \<equiv> map (\<lambda> P l. P ! l) pred"
definition N :: "(nat, nat, int, nat, 'st) snta" where
"N \<equiv> (map (\<lambda> i. (T i, I i)) [0..<p], P)"
definition "init \<equiv> repeat (0::nat) p"
definition "F s \<equiv> \<exists> i < length s. s ! i \<in> set (final ! i)"
definition "k_fun \<equiv> \<lambda> i. if i \<le> m then k ! i else 0"
sublocale product: Prod_TA_Defs N .
abbreviation "A \<equiv> product.prod_ta"
term state_set
end
lemma snd_comp[simp]:
"snd o (\<lambda> i. (f i, g i)) = g"
by auto
locale State_Network_Reachability_Problem_precompiled_raw =
State_Network_Reachability_Problem_precompiled_defs +
assumes process_length: "length inv = p" "length trans = p" "length pred = p"
and lengths:
"\<forall> i < p. length (pred ! i) = length (trans ! i) \<and> length (inv ! i) = length (trans ! i)"
and state_set: "\<forall> T \<in> set trans. \<forall> xs \<in> set T. \<forall> (_, _, _, _, _, l) \<in> set xs. l < length T"
and k_length: "length k = m + 1" \<comment> \<open>Zero entry is just a dummy for the zero clock\<close>
(* XXX Make this an abbreviation? *)
assumes k_ceiling:
(* "\<forall> c \<in> {1..m}. k ! c = Max ({d. (c, d) \<in> clkp_set'} \<union> {0})" *)
"\<forall> (c, d) \<in> clkp_set'. k ! c \<ge> d"
"k ! 0 = 0"
assumes consts_nats: "snd ` clkp_set' \<subseteq> \<nat>"
assumes clock_set: "clk_set' = {1..m}"
and p_gt_0: "p > 0"
and m_gt_0: "m > 0"
(* XXX Can get rid of these two? *)
and processes_have_trans: "\<forall> i < p. trans ! i \<noteq> []" \<comment> \<open>Necessary for refinement\<close>
and start_has_trans: "\<forall> q < p. trans ! q ! 0 \<noteq> []" \<comment> \<open>Necessary for refinement\<close>
locale State_Network_Reachability_Problem_precompiled =
State_Network_Reachability_Problem_precompiled_raw +
assumes discrete_state_finite: "\<forall> i < p. \<forall> l < length (trans ! i). finite {s. (pred ! i ! l) s}"
begin
lemma consts_nats':
"\<forall> I \<in> set inv. \<forall> cc \<in> set I. \<forall> (c, d) \<in> collect_clock_pairs cc. d \<in> \<nat>"
"\<forall> T \<in> set trans. \<forall> xs \<in> set T. \<forall> (g, _) \<in> set xs. \<forall> (c, d) \<in> collect_clock_pairs g. d \<in> \<nat>"
using consts_nats unfolding clkp_set'_def by fastforce+
lemma clkp_set_simp_1:
"\<Union> (collect_clock_pairs ` set (concat inv)) = collect_clki (inv_of A)"
apply (simp add:
product.prod_ta_def inv_of_def product.collect_clki_prod_invariant
product.collect_clki_product_invariant
)
unfolding inv_of_def collect_clki_alt_def I_def[abs_def] N_def I_def
using process_length(1)
apply (simp add: image_Union inv_of_def)
apply safe
apply (fastforce dest!: aux)
by (fastforce dest!: nth_mem)
(* XXX Unused *)
lemma processes_have_trans_alt:
"\<forall> i < p. length (trans ! i) > 0"
using processes_have_trans by auto
lemma init_states:
"init \<in> Product_TA_Defs.states (fst N)"
unfolding Product_TA_Defs.states_def
unfolding N_def trans_of_def T_def init_def using processes_have_trans p_gt_0 start_has_trans
by force
lemma states_not_empty:
"Product_TA_Defs.states (fst N) \<noteq> {}"
using init_states by blast
lemma length_prod_T [simp]: "length product.T = p"
unfolding N_def by auto
lemma length_N [simp]: "length (fst N) = p"
unfolding N_def by auto
lemma length_P [simp]: "length P = p"
unfolding N_def P_def using process_length(3) by auto
(*
lemma trans_length_simp:
assumes "xs \<in> set trans"
shows "n = length xs"
using assms trans_length by auto
*)
lemma [simp]:
"product.T' = product.product_trans"
unfolding product.product_ta_def trans_of_def by simp
lemma clk_set_simp_2:
"\<Union> ((\<lambda> (g, _, _, r, _). set r) ` set (concat (concat trans))) \<supseteq> collect_clkvt (trans_of A)"
apply (simp add: product.product_ta_def trans_of_def)
apply (rule subset_trans)
apply (rule product.collect_clkvt_prod_trans_subs)
apply simp
apply (rule subset_trans)
apply (rule product.collect_clkvt_product_trans_subs)
unfolding collect_clkvt_alt_def trans_of_def N_def T_def make_trans_def
using process_length(2)
by (fastforce dest!: nth_mem elim: bexI[rotated]) (* XXX Magic *)
lemma clkp_set_simp_3:
"\<Union> ((\<lambda> (g, _). collect_clock_pairs g) ` set (concat (concat trans))) \<supseteq> collect_clkt (trans_of A)"
apply (simp add: product.product_ta_def trans_of_def)
apply (rule subset_trans)
apply (rule product.collect_clkt_prod_trans_subs)
apply simp
apply (rule subset_trans)
apply (rule product.collect_clkt_product_trans_subs)
unfolding collect_clkt_alt_def trans_of_def N_def T_def make_trans_def
using process_length(2)
by (fastforce dest!: nth_mem elim: bexI[rotated]) (* XXX Magic *)
lemma clkp_set'_subs:
"clkp_set A \<subseteq> clkp_set'"
using clkp_set_simp_1 clkp_set_simp_3 by (fastforce simp add: clkp_set'_def clkp_set_def)
lemma clk_set'_subs:
"clk_set A \<subseteq> clk_set'"
using clkp_set'_subs clk_set_simp_2 by (auto simp: clk_set'_def)
(* XXX Interesting for finiteness *)
(* XXX Move *)
lemma Collect_fold_pair:
"{f a b | a b. P a b} = (\<lambda> (a, b). f a b) ` {(a, b). P a b}" for P
by auto
lemma [simp]:
"product.p = p"
unfolding product.p_def by simp
(* XXX Interesting case of proving finiteness *)
lemma finite_T[intro, simp]:
"finite (trans_of A)"
unfolding product.prod_ta_def trans_of_def
proof (simp, rule product.finite_prod_trans, goal_cases)
case 1
have *: "l < length (trans ! q)" if "l \<in> state_set (trans_of (product.N ! q))" "q < p" for l q
using that state_set
unfolding trans_of_def apply simp
apply (erule disjE)
unfolding N_def
apply simp
unfolding T_def
apply force
unfolding make_trans_def
apply clarsimp
using process_length(2)
apply (fastforce dest!: nth_mem split: prod.split_asm)
done
with process_length(3) discrete_state_finite show ?case by simp (auto simp: N_def P_def)
next
case 2
show ?case
proof
fix A assume A: "A \<in> set product.N"
have
"{(l, j). l < length (trans ! i) \<and> j < length (trans ! i ! l)}
= \<Union> ((\<lambda> l. {(l, j) | j. j < length (trans ! i ! l)}) ` {l. l < length (trans ! i)})" for i
by auto
then show "finite (trans_of A)" using A unfolding N_def T_def trans_of_def
by (fastforce simp: Collect_fold_pair)
qed
next
case 3
then show ?case unfolding product.p_def unfolding N_def using p_gt_0 by simp
qed
(* XXX *)
lemma
"clk_set' \<noteq> {}"
using clock_set m_gt_0 by auto
lemma clk_set:
"clk_set A \<subseteq> {1..m}"
using clock_set m_gt_0 clk_set'_subs by auto
lemma
"\<forall>(_, d)\<in>clkp_set A. d \<in> \<int>"
unfolding Ints_def by auto
lemma clkp_set_consts_nat:
"\<forall>(_, d)\<in>clkp_set A. d \<in> \<nat>"
using clkp_set'_subs consts_nats' unfolding clkp_set'_def by force
lemma finite_range_I':
"finite (range product.I')"
apply (rule product.finite_invariant_of_product)
unfolding N_def inv_of_def I_def by (auto intro: finite_subset[where B = "{[]}"])
lemma finite_range_inv_of_A[intro, simp]:
"finite (range (inv_of A))"
proof -
have "range (inv_of A) \<subseteq> range (product.I')" by (auto simp: product.inv_of_simp)
then show ?thesis by (rule finite_subset) (rule finite_range_I')
qed
lemma finite_clkp_set_A[intro, simp]:
"finite (clkp_set A)"
unfolding clkp_set_def collect_clki_alt_def collect_clkt_alt_def by fast
lemma [intro, simp]:
"k_fun 0 = 0"
unfolding k_fun_def using k_ceiling by simp
lemma [intro, simp]:
"k_fun i = 0" if "i > m"
unfolding k_fun_def using that by simp
lemma clkp_set'_bounds:
"a \<in> {Suc 0..m}" if "(a, b) \<in> clkp_set'"
using that clock_set unfolding clk_set'_def by auto
lemma [intro]:
"b \<le> int (k_fun a)" if "(a, b) \<in> clkp_set A"
using that k_ceiling clkp_set'_subs k_length clkp_set'_bounds unfolding k_fun_def by force
end
locale State_Network_Reachability_Problem_precompiled_start_state =
State_Network_Reachability_Problem_precompiled _ _ _ _ pred
for pred :: "('st \<Rightarrow> bool) list list" +
fixes s\<^sub>0 :: "'st"
assumes start_pred: "(\<forall> i < p. (pred ! i ! 0) s\<^sub>0)"
begin
sublocale Reachability_Problem A "(init, s\<^sub>0)" "PR_CONST (\<lambda> (l, s). F l)" m k_fun
using clkp_set_consts_nat clk_set m_gt_0 by - (standard; blast)
lemma [simp]:
"fst ` (\<lambda>(l, g, a, r, l'). (l, map conv_ac g, a, r, l')) ` S = fst ` S"
by force
lemma [simp]:
"(snd \<circ> snd \<circ> snd \<circ> snd) ` (\<lambda>(l, g, a, r, l'). (l, map conv_ac g, a, r, l')) ` S
= (snd \<circ> snd \<circ> snd \<circ> snd) ` S"
by force
lemma map_trans_of:
"map trans_of (map conv_A (fst N)) = map ((`) conv_t) (map trans_of (fst N))"
by (simp add: trans_of_def split: prod.split)
lemma [simp]:
"Product_TA_Defs.states (map conv_A (fst N)) = Product_TA_Defs.states (fst N)"
unfolding Product_TA_Defs.states_def map_trans_of by simp
lemma [simp]:
"product.P = P"
unfolding N_def by simp
lemma start_pred':
"\<forall> i < p. (pred ! i ! (init ! i)) s\<^sub>0"
using start_pred unfolding init_def by auto
lemma start_pred'':
"\<forall> i < p. ((P ! i) (init ! i)) s\<^sub>0"
using start_pred' process_length(3) unfolding P_def by auto
sublocale product': Prod_TA "(map conv_A (fst N), snd N)" init s\<^sub>0
by (standard; simp add: init_states start_pred'')
end (* End of locale *)
datatype ('c, 't) constr =
lt 'c 't |
le 'c 't |
eq 'c 't |
gt 'c 't |
ge 'c 't
type_synonym int_var_constr = "(nat, int) cconstraint"
definition check :: "int_var_constr \<Rightarrow> (nat \<Rightarrow> int) \<Rightarrow> bool" where
"check c x \<equiv> list_all (clock_val_a x) c"
datatype ('c, 't) upd =
upd 'c 't |
inc 'c |
dec 'c
type_synonym int_var_upd = "(nat, int) upd"
fun modify :: "(nat, int) upd \<Rightarrow> int list \<Rightarrow> int list" where
"modify (upd i x) s = s[i := x]"
| "modify (inc i) s = s[i := s ! i + 1]"
| "modify (dec i) s = s[i := s ! i - 1]"
locale State_Network_Reachability_Problem_precompiled_int_vars_defs =
fixes p :: nat \<comment> \<open>Number of processes\<close>
and m :: nat \<comment> \<open>Number of clocks\<close>
and k :: "nat list" \<comment> \<open>Clock ceiling. Maximal constant appearing in automaton for each state\<close>
and inv :: "(nat, int) cconstraint list list" \<comment> \<open>Clock invariants on states per process\<close>
and pred :: "int_var_constr list list" \<comment> \<open>Clock invariants on states per process\<close>
and trans ::
"((nat, int) cconstraint * int_var_constr * nat act * nat list * int_var_upd * nat) list list list"
\<comment> \<open>Transitions between states per process\<close>
and final :: "nat list list" \<comment> \<open>Final states per process. Initial location is 0\<close>
fixes r :: nat \<comment> \<open>Number of integer variables\<close>
and bounds :: "(int \<times> int) list" \<comment> \<open>Lower and upper bounds for the variables\<close>
begin
definition
"checkb c s \<equiv>
check c ((!) s) \<and> length s = r \<and> (\<forall> i < r. fst (bounds ! i) < s ! i \<and> s ! i < snd (bounds ! i))"
definition pred' where "pred' = map (map checkb) pred"
definition trans' where "trans' =
map (map (map (\<lambda> (g, c, a, r, m, l). (g, \<lambda> s. check c ((!) s), a, r, modify m, l)))) trans"
definition "s\<^sub>0 \<equiv> repeat 0 r"
end
locale State_Network_Reachability_Problem_precompiled_int_vars =
State_Network_Reachability_Problem_precompiled_int_vars_defs p m k inv pred trans final r bounds +
State_Network_Reachability_Problem_precompiled_raw p m k inv pred' trans' final
for p m k inv pred trans final r bounds +
fixes na :: nat \<comment> \<open>Number of action labels\<close>
assumes init_pred: "\<forall>i<p. (pred' ! i ! 0) s\<^sub>0"
and actions_bounded:
"\<forall>T\<in>set trans'. \<forall>xs\<in>set T. \<forall>(_, _, a, _)\<in>set xs. pred_act (\<lambda>a. a < na) a"
begin
lemma trans'_length:
"length trans' = length trans"
unfolding trans'_def by simp
lemma trans'_lengths:
"length (trans' ! i) = length (trans ! i)" if "i < p"
unfolding trans'_def using process_length(2)[unfolded trans'_length] that by simp
lemma pred'_length:
"length pred' = length pred"
unfolding pred'_def by simp
lemma pred'_lengths:
"length (pred' ! i) = length (pred ! i)" if "i < p"
unfolding pred'_def using process_length(3)[unfolded pred'_length] that by simp
lemma trans'_length_pred:
"length (trans' ! i) = length (pred ! i)" if "i < p"
using pred'_lengths lengths that by simp
lemma
"finite {s. length s = r \<and> (\<forall>i<r. fst (bounds ! i) < s ! i \<and> s ! i < snd (bounds ! i))}"
using finite_lists_boundedI by force
sublocale State_Network_Reachability_Problem_precompiled p m k inv pred' trans' final
apply standard
apply safe
apply (simp only: trans'_length_pred)
unfolding pred'_def trans'_def checkb_def
using process_length(3)[unfolded pred'_length] finite_lists_boundedI by force (* XXX Slow *)
end
end (* End of theory *)
|
#pragma once
#include "kl/type_traits.hpp"
#include "kl/reflect_struct.hpp"
#include <boost/type_index.hpp>
#include <type_traits>
#include <string>
namespace kl {
namespace detail {
KL_VALID_EXPR_HELPER(has_reflect_struct,
reflect_struct(0, std::declval<T&>(), record<T>))
} // namespace detail
template <typename T>
using is_reflectable = detail::has_reflect_struct<T>;
template <typename T>
inline constexpr bool is_reflectable_v = is_reflectable<T>::value;
struct ctti
{
template <typename T>
static constexpr bool is_reflectable = kl::is_reflectable_v<T>;
template <typename Reflected>
static std::string name()
{
return boost::typeindex::type_id<Reflected>().pretty_name();
}
template <typename Reflected, typename Visitor>
static constexpr void reflect(Reflected&& r, Visitor&& v)
{
using R = remove_cvref_t<Reflected>;
static_assert(
detail::has_reflect_struct_v<R>,
"Can't reflect this type. Define reflect_struct function");
reflect_struct(std::forward<Visitor>(v), std::forward<Reflected>(r),
record<R>);
}
template <typename Reflected>
static constexpr std::size_t num_fields() noexcept
{
using R = remove_cvref_t<Reflected>;
static_assert(
detail::has_reflect_struct_v<R>,
"Can't reflect this type. Define reflect_struct function");
return reflect_num_fields(record<R>);
}
};
} // namespace kl
|
# Write JavaSeis with Julia package TeaSeis.jl
# Run from Julia REPL
# julia> include("C:/Users/xinfa/Documents/code/pieseis/examples/write_file.jl")
# Run from command line (Linux terminal or Windows command prompt)
# C:\Users\xinfa\AppData\Local\Julia-1.0.1\bin\julia.exe C:\Users\xinfa\Documents\code\pieseis\examples\write_file.jl
println("start")
push!(LOAD_PATH, "C:/Users/xinfa/Documents/code/TeaSeis.jl/src")
using TeaSeis
println("done load")
filename = "C:/Users/xinfa/Documents/181116_julia.js"
io = jsopen(filename, "w", axis_lengths=[101, 101, 101], dataformat=Int16)
println("done open")
trcs, hdrs = allocframe(io)
map(i->set!(prop(io, stockprop[:TRC_TYPE]), hdrs, i, tracetype[:live]), 1:size(io,2))
map(i->set!(prop(io, stockprop[:TRACE] ), hdrs, i, i ), 1:size(io,2))
map(i->set!(prop(io, stockprop[:FRAME] ), hdrs, i, 1 ), 1:size(io,2))
println("done write headers")
using Random
rand!(trcs)
writeframe(io, trcs, hdrs)
println("done write traces")
close(io)
|
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.