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Load LFindLoad.
From lfind Require Import LFind.
From QuickChick Require Import QuickChick.
From adtind Require Import goal33.
Derive Show for natural.
Derive Arbitrary for natural.
Instance Dec_Eq_natural : Dec_Eq natural.
Proof. dec_eq. Qed.
Lemma conj8synthconj3 : forall (lv0 : natural) (lv1 : natural), (@eq natural (plus (mult lv0 lv1) lv0) (plus lv0 (mult lv0 lv1))).
Admitted.
QuickChick conj8synthconj3.
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#' Vaginal microbiome OTU table metadata
#'
#' Associated metadata to \code{\link{otu_table}}.
#'
#' @name metadata
#'
#' @docType data
#'
#' @usage data(metadata)
#'
#' @references Macklaim et al (2014). Microb Ecol Health Dis.
#' doi: http://dx.doi.org/10.3402/mehd.v26.27799
#'
#' @format A data frame with 297 rows and 35 columns, where rows are samples
#' and columns are collected metadata.
#'
#' @seealso \code{\link{otu_table}}
#'
NULL
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lemma LIMSEQ_linear: "X \<longlonglongrightarrow> x \<Longrightarrow> l > 0 \<Longrightarrow> (\<lambda> n. X (n * l)) \<longlonglongrightarrow> x"
|
module BFF where
open import Data.Nat using (ℕ)
open import Data.Fin using (Fin)
open import Level using () renaming (zero to ℓ₀)
import Category.Monad
import Category.Functor
open import Data.Maybe using (Maybe ; just ; nothing ; maybe′)
open Category.Monad.RawMonad {Level.zero} Data.Maybe.monad using (_>>=_)
open Category.Functor.RawFunctor {Level.zero} Data.Maybe.functor using (_<$>_)
open import Data.List using (List ; [] ; _∷_ ; map ; length)
open import Data.Vec using (Vec ; toList ; fromList ; allFin) renaming (lookup to lookupV ; map to mapV ; [] to []V ; _∷_ to _∷V_)
open import Function using (_∘_ ; flip)
open import Relation.Binary using (Setoid ; DecSetoid ; module DecSetoid)
open import FinMap
open import Generic using (sequenceV ; ≡-to-Π)
open import Structures using (Shaped ; module Shaped)
open import Instances using (VecShaped)
import CheckInsert
open import GetTypes using (VecVec-to-PartialVecVec ; PartialVecVec-to-PartialShapeShape)
module PartialShapeBFF (A : DecSetoid ℓ₀ ℓ₀) where
open GetTypes.PartialShapeShape public using (Get ; module Get)
open module A = DecSetoid A using (Carrier) renaming (_≟_ to deq)
open CheckInsert A
assoc : {n m : ℕ} → Vec (Fin n) m → Vec Carrier m → Maybe (FinMapMaybe n Carrier)
assoc []V []V = just empty
assoc (i ∷V is) (b ∷V bs) = (assoc is bs) >>= (checkInsert i b)
enumerate : {S : Set} {C : Set → S → Set} → (ShapeT : Shaped S C) → (s : S) → C (Fin (Shaped.arity ShapeT s)) s
enumerate ShapeT s = fill s (allFin (arity s))
where open Shaped ShapeT
denumerate : {S : Set} {C : Set → S → Set} → (ShapeT : Shaped S C) → {α : Set} {s : S} → (c : C α s) → Fin (Shaped.arity ShapeT s) → α
denumerate ShapeT c = flip lookupV (Shaped.content ShapeT c)
bff : (G : Get) → {i : Get.I G} → (j : Get.I G) → Get.SourceContainer G Carrier (Get.gl₁ G i) → Get.ViewContainer G Carrier (Get.gl₂ G j) → Maybe (Get.SourceContainer G (Maybe Carrier) (Get.gl₁ G j))
bff G {i} j s v = let s′ = enumerate SourceShapeT (gl₁ i)
t′ = get s′
g = fromFunc (denumerate SourceShapeT s)
g′ = delete-many (Shaped.content ViewShapeT t′) g
t = enumerate SourceShapeT (gl₁ j)
h = assoc (Shaped.content ViewShapeT (get t)) (Shaped.content ViewShapeT v)
h′ = (flip union (reshape g′ (Shaped.arity SourceShapeT (gl₁ j)))) <$> h
in ((λ f → fmapS f t) ∘ flip lookupM) <$> h′
where open Get G
sbff : (G : Get) → {i : Get.I G} → (j : Get.I G) → Get.SourceContainer G Carrier (Get.gl₁ G i) → Get.ViewContainer G Carrier (Get.gl₂ G j) → Maybe (Get.SourceContainer G Carrier (Get.gl₁ G j))
sbff G j s v = bff G j s v >>= Shaped.sequence (Get.SourceShapeT G)
module PartialVecBFF (A : DecSetoid ℓ₀ ℓ₀) where
open GetTypes.PartialVecVec public using (Get)
open module A = DecSetoid A using (Carrier) renaming (_≟_ to deq)
open CheckInsert A
open PartialShapeBFF A public using (assoc)
enumerate : {n : ℕ} → Vec Carrier n → Vec (Fin n) n
enumerate {n} _ = PartialShapeBFF.enumerate A VecShaped n
enumeratel : (n : ℕ) → Vec (Fin n) n
enumeratel = PartialShapeBFF.enumerate A VecShaped
denumerate : {n : ℕ} → Vec Carrier n → Fin n → Carrier
denumerate = PartialShapeBFF.denumerate A VecShaped
bff : (G : Get) → {i : Get.I G} → (j : Get.I G) → Vec Carrier (Get.gl₁ G i) → Vec Carrier (Get.gl₂ G j) → Maybe (Vec (Maybe Carrier) (Get.gl₁ G j))
bff G j s v = PartialShapeBFF.bff A (PartialVecVec-to-PartialShapeShape G) j s v
sbff : (G : Get) → {i : Get.I G} → (j : Get.I G) → Vec Carrier (Get.gl₁ G i) → Vec Carrier (Get.gl₂ G j) → Maybe (Vec Carrier (Get.gl₁ G j))
sbff G j s v = PartialShapeBFF.sbff A (PartialVecVec-to-PartialShapeShape G) j s v
module VecBFF (A : DecSetoid ℓ₀ ℓ₀) where
open GetTypes.VecVec public using (Get)
open module A = DecSetoid A using (Carrier) renaming (_≟_ to deq)
open CheckInsert A
open PartialVecBFF A public using (assoc ; enumerate ; denumerate)
bff : (G : Get) → {n : ℕ} → (m : ℕ) → Vec Carrier n → Vec Carrier (Get.getlen G m) → Maybe (Vec (Maybe Carrier) m)
bff G = PartialVecBFF.bff A (VecVec-to-PartialVecVec G)
sbff : (G : Get) → {n : ℕ} → (m : ℕ) → Vec Carrier n → Vec Carrier (Get.getlen G m) → Maybe (Vec Carrier m)
sbff G = PartialVecBFF.sbff A (VecVec-to-PartialVecVec G)
|
Set Implicit Arguments.
Require Import Variable_Sets.
Require Import deBruijn_Isomorphism.
(** This file provides a connection between the generic meta library
and some simple typed languages such as STLC or ML
based on some isomorphisms.
A concrete type/term class [TT] is isomorphic to the generic
representation [RR] when there is an isomorphism
between [TT] and [Interpret RR].
In the following, two modules [MT] and [MY] are used.
- MT stands for module for terms of e.g. STLC.
- MY stands for module for types of e.g. STLC
*)
Module dBTemplate (iso1 iso2 : Iso_full).
Hint Rewrite iso1.To_From iso1.From_To iso2.To_From iso2.From_To : isorew.
Hint Resolve iso1.To_From iso1.From_To iso2.To_From iso2.From_To.
(**************************************************************)
(** * Shifting *)
(**************************************************************)
(** Shifting for a iso1 variable in a iso2 *)
Definition Tshift (X : atom) (T : iso2.TT) : iso2.TT :=
iso2.To (shift X iso1.RR (iso2.From T)).
(**************************************************************)
(** * Substitution *)
(**************************************************************)
(** Substitution for a iso1 variable in a iso2 *)
Definition Tsubst (T:iso2.TT) (m:atom) (U:iso1.TT) : iso2.TT :=
iso2.To (subst (iso2.From T) m (iso1.From U)).
(**************************************************************)
(** * Term size *)
(**************************************************************)
Definition Ysize (T:iso1.TT) : nat := size (iso1.From T).
(**************************************************************)
(** * A tactic unfolding everything *)
(**************************************************************)
Ltac gunfold :=
unfold Tshift, Tsubst in *;
unfold Ysize in *;
intros;
repeat rewrite iso2.To_From in *;
repeat rewrite iso2.From_To in *;
repeat rewrite iso1.From_To in *;
repeat rewrite iso1.From_To in *;
simpl in *.
(**************************************************************)
(** * Homomorphisms *)
(**************************************************************)
(** [From] is a homomorphism w.r.t. substitutions. *)
Lemma From_Tshift : forall (T:iso2.TT) (a:atom),
iso2.From (Tshift a T) = shift a iso1.RR (iso2.From T).
Proof.
unfold Tshift; intros; autorewrite with isorew; auto.
Qed.
(** [From] is a homomorphism w.r.t. substitutions. *)
Lemma From_Tsubst : forall (T:iso2.TT) (a:atom) (U:iso1.TT),
iso2.From (Tsubst T a U) = subst (iso2.From T) a (iso1.From U).
Proof.
unfold Tsubst; intros; autorewrite with isorew; auto.
Qed.
(**************************************************************)
(** Tshift and Tsubst are identity function when no [Repr] occurs. *)
(**************************************************************)
Lemma Tshift_id: forall n T,
noRepr iso2.RR ->
iso1.RR <> iso2.RR ->
Tshift n T = T.
Proof.
gunfold; rewrite <- noRepr_shift_hetero; autorewrite with isorew; auto.
Qed.
Lemma Tsubst_id : forall n T U,
noRepr iso2.RR ->
iso1.RR <> iso2.RR ->
Tsubst T n U = T.
Proof.
gunfold; rewrite <- noRepr_subst_hetero; autorewrite with isorew; auto.
Qed.
(**************************************************************)
(** * Environments *)
(**************************************************************)
(** TEnv is (TT * TT) list.
- [inl] is used for type variable binding.
- [inr] is used for term variable binding. *)
Notation TEnv := (Env iso1.TT).
Fixpoint From_TEnv (e:TEnv) : (ENV iso1.RR) :=
match e with
| nil => nil
| inl T :: e' => inl (iso1.From T) :: (From_TEnv e')
| inr T :: e' => inr (iso1.From T) :: (From_TEnv e')
end.
Fixpoint To_TEnv (e:ENV iso1.RR) : TEnv :=
match e with
| nil => nil
| inl T :: e' => inl (iso1.To T) :: (To_TEnv e')
| inr T :: e' => inr (iso1.To T) :: (To_TEnv e')
end.
Lemma From_To_TEnv : forall e : ENV iso1.RR, From_TEnv (To_TEnv e) = e.
Proof.
induction e;
[ simpl; auto
| destruct a;
simpl; rewrite IHe; rewrite iso1.From_To; auto
].
Qed.
Lemma To_From_TEnv : forall e : TEnv, To_TEnv (From_TEnv e) = e.
Proof.
induction e; simpl; auto.
destruct a; simpl; rewrite IHe, iso1.To_From; auto.
Qed.
Hint Resolve From_To_TEnv.
Hint Rewrite From_To_TEnv : isorew.
Definition Tlth (e : TEnv) : atom := lth (From_TEnv e).
Notation "[[[ e ]]]" := (Tlth e).
(** Tremove_right removes the x th element in environment. *)
Fixpoint Tremove_right (e : TEnv) (x : nat) {struct e} : TEnv :=
match e with
| nil => nil
| (inl T)::e' => (inl T)::(Tremove_right e' x)
| (inr T)::e' =>
match x with
| O => e'
| S x => (inr T::(Tremove_right e' x))
end
end.
(**************************************************************)
(** To and From function on (option TT) *)
(**************************************************************)
Fixpoint opt_To (T : option (Interpret iso1.RR)) : option iso1.TT :=
match T with
| None => None
| Some T' => Some (iso1.To T')
end.
Fixpoint opt_From (T : option (iso1.TT)) : option (Interpret iso1.RR) :=
match T with
| None => None
| Some T' => Some (iso1.From T')
end.
Lemma opt_To_preserving_none (T : option (Interpret iso1.RR)) :
T = None -> opt_To T = None.
Proof.
intros; rewrite H; auto.
Qed.
Lemma opt_To_preserving_none_rev (T : option (Interpret iso1.RR)) :
opt_To T = None -> T = None.
Proof.
induction T; simpl; intros; [discriminate | auto].
Qed.
Lemma opt_From_preserving_some (T : option iso1.TT) (t : iso1.TT):
T = Some t -> opt_From T = Some (iso1.From t).
Proof.
intros; rewrite H; auto.
Qed.
Lemma opt_To_preserving_some (T : option (Interpret iso1.RR)) (t : Interpret iso1.RR):
T = Some t -> opt_To T = Some (iso1.To t).
Proof.
intros; rewrite H; auto.
Qed.
Lemma opt_To_preserving_some_rev (T : option (Interpret iso1.RR)) (t : Interpret iso1.RR):
opt_To T = Some (iso1.To t) -> T = Some t.
Proof.
induction T; simpl; intros; [idtac | discriminate].
rewrite <- (iso1.From_To a);rewrite <- (iso1.From_To t0).
rewrite <- opt_To_preserving_some with (T:= Some a) in H;auto.
rewrite <- opt_To_preserving_some with (T:= Some t0) in H;auto.
inversion H; rewrite H1;reflexivity.
Qed.
Lemma opt_To_preserving_eq (T U : option (Interpret iso1.RR)) :
T = U -> opt_To T = opt_To U.
Proof.
intros; rewrite H; auto.
Qed.
Lemma opt_To_preserving_eq_rev (T U : option (Interpret iso1.RR)) :
opt_To T = opt_To U -> T = U.
Proof.
induction T; induction U; intros;
[ inversion H;
rewrite <- (iso1.From_To a); rewrite <- (iso1.From_To a0);
rewrite H1; reflexivity
| inversion H
| inversion H
| auto
].
Qed.
Hint Resolve
opt_To_preserving_none opt_To_preserving_none_rev
opt_To_preserving_some opt_To_preserving_some_rev
opt_To_preserving_eq opt_To_preserving_eq_rev
opt_From_preserving_some
: opt.
(**************************************************************)
(** Generic versions of [Tget_left] and [Tget_right] *)
(**************************************************************)
Definition gTget_left (e : TEnv) (X : atom) : option (iso1.TT) :=
opt_To (get_left (From_TEnv e) X).
Definition gTget_right (e : TEnv) (X : atom) : option (iso1.TT) :=
opt_To (get_right (From_TEnv e) X).
(**************************************************************)
(** * Well-formedness in an environment *)
(**************************************************************)
(** Well-formed types in an environment *)
Definition Twf_typ (e : TEnv) (T : iso1.TT) : Prop :=
HO_wf [[From_TEnv e]] (iso1.From T).
Fixpoint Twf_env (e : TEnv) : Prop :=
match e with
nil => True
| (inr T)::e => Twf_typ e T /\ Twf_env e
| (inl T)::e => Twf_typ e T /\ Twf_env e
end.
Definition gTwf_env (e : TEnv) : Prop :=
wf_env (From_TEnv e).
Lemma Twf_env_gTwf_env : forall (e: TEnv),
Twf_env e -> gTwf_env e.
Proof.
unfold gTwf_env;induction e;auto.
simpl;destruct a;intros;destruct H;simpl;
(split;
[ unfold Twf_typ in H;auto
| auto
]).
Qed.
Lemma gTwf_env_Twf_env : forall (e: TEnv),
gTwf_env e -> Twf_env e.
Proof.
unfold gTwf_env;induction e;auto.
simpl;destruct a;intros;destruct H;simpl;
(split;
[ unfold Twf_typ in H;auto
| auto
]).
Qed.
Lemma Twf_env_weaken : forall (T : iso1.TT) (n m : TEnv),
[[[n]]] <= [[[m]]] ->
Twf_typ n T ->
Twf_typ m T.
Proof.
unfold Twf_typ.
eauto using HO_wf_weaken.
Qed.
(** Generic version of [Tremove_right] *)
Definition gTremove_right (e : TEnv) (x : nat) : TEnv :=
To_TEnv (remove_right (From_TEnv e) x).
Lemma Tremove_right_gTremove_right: forall (e:TEnv)(x:nat),
Tremove_right e x = gTremove_right e x.
Proof.
unfold gTremove_right.
induction e; simpl; intros; auto.
destruct a;
[ simpl; rewrite IHe, iso1.To_From; auto
| destruct x;
[ simpl; rewrite To_From_TEnv; auto
| simpl; rewrite iso1.To_From; rewrite IHe; auto
]
].
Qed.
Lemma Twf_typ_remove_right : forall (e : TEnv) (x : nat) (T : iso1.TT),
Twf_typ e T -> Twf_typ (Tremove_right e x) T.
Proof.
intros.
rewrite Tremove_right_gTremove_right.
unfold Twf_typ, gTremove_right; intros.
autorewrite with isorew.
auto using HO_wf_remove_right.
Qed.
Lemma Twf_typ_insert_right : forall (e : TEnv) (n : nat) (T : iso1.TT),
Twf_typ (Tremove_right e n) T -> Twf_typ e T.
Proof.
intros.
rewrite Tremove_right_gTremove_right in H.
unfold Twf_typ, gTremove_right in *; intros.
autorewrite with isorew in *.
eauto using HO_wf_insert_right.
Qed.
Lemma Twf_env_remove_right : forall (e : TEnv) (x : nat),
Twf_env e ->
Twf_env (Tremove_right e x).
Proof.
intros.
rewrite Tremove_right_gTremove_right.
apply gTwf_env_Twf_env.
apply Twf_env_gTwf_env in H.
unfold gTwf_env, gTremove_right in *; intros.
autorewrite with isorew.
eauto using wf_env_remove_right.
Qed.
(** Generic version of [Tinsert_left] *)
Definition gTinsert_left n (e e':TEnv) :=
insert_left n (From_TEnv e) (From_TEnv e').
(** Isomorphisms between generic well-formedness and specific well-formedness *)
Definition Tinsert (n: nat) (e: TEnv) (T: iso1.TT) (H:Twf_typ nil T) : TEnv :=
To_TEnv (insert n (From_TEnv e) (iso1.From T) H).
Lemma Tinsert_S : forall (e:TEnv) (n:nat) U H,
S [[[e]]] = [[[Tinsert n e U H]]].
Proof.
unfold Tinsert, Tlth; intros.
autorewrite with isorew.
apply insert_S.
Qed.
Hint Resolve Tinsert_S.
Lemma Twf_typ_weakening_right : forall (e : TEnv) (T U : iso1.TT),
Twf_typ e U -> Twf_typ ((inr T)::e) U.
Proof.
unfold Twf_typ.
auto using HO_wf_weakening_right.
Qed.
Lemma Twf_typ_strengthening_right : forall (e : TEnv) (T U : iso1.TT),
Twf_typ ((inr T)::e) U -> Twf_typ e U.
Proof.
unfold Twf_typ.
eauto using HO_wf_strengthening_right.
Qed.
Lemma Twf_typ_eleft : forall (T U V : iso1.TT) (e : TEnv),
Twf_typ ((inl U)::e) T -> Twf_typ ((inl V)::e) T.
Proof.
unfold Twf_typ.
eauto using HO_wf_left.
Qed.
End dBTemplate.
|
<a href="https://colab.research.google.com/github/starhou/Algorithm/blob/master/Classic_MLmodel.ipynb" target="_parent"></a>
# 损失函数
### **KL散度**
$D_{\mathrm{KL}}(P \| Q)=-\sum_{i} P(i) \ln \frac{Q(i)}{P(i)}=\sum_{i} P(i) \ln \frac{P(i)}{Q(i)}=\sum_{i}P(i)(\log{P(i)}-\log{Q(i)})$
其中$Q(i)>0,P(i)>0$, 其中P为原始分布，Q为近似分布，KL散度是两个分布对数差值的期望, 并非距离度量，没有对称性
### **交叉熵** = 熵+KL散度
衡量的是同个变量的不同分布
样本集的两个概率分布 p(x) 和 q(x)，其中 p(x) 为真实分布， q(x)非真实分布
$\mathbf{H}(p, q)=\mathbf{E}_{p}[-\log q]=\mathbf{H}(p)+D_{\mathrm{KL}}(p \| q)=\sum_{x} p(x) \log \frac{1}{q(x)}$
### **条件熵：**
已知随机变量 X 的条件下随机变量 Y 的不确定性
$\begin{aligned} H(Y | X) &=\sum_{x} p(x) H(Y | X=x) \\ &=-\sum_{x} p(x) \sum_{y} p(y | x) \log p(y | x) \\ &=-\sum_{x} \sum_{y} p(x, y) \log p(y | x) \\ &=-\sum_{x, y} p(x, y) \log p(y | x) \end{aligned}$
### **联合熵**：
衡量的是不同的变量
$H(X, Y)=-\sum_{x, y} p(x, y) \log p(x, y)=-\sum_{i=1}^{n} \sum_{j=1}^{m} p\left(x_{i}, y_{i}\right) \log p\left(x_{i}, y_{i}\right)$
# 支撑向量机（SVM）
一般优化表示，
\begin{equation}
\begin{aligned}
&\min \quad f_{0}(x)\\
&\text { s.t. } \quad f_{i}(x) \leq 0, \quad i=1, \cdots, m\\
&\quad \quad \quad h_{i}(x)=0, i=1, \cdots, p
\end{aligned}
\end{equation}
1. 不管原命题形式如何，其对偶问题都是凸函数
2. 不管什么问题，都可以转化为拉格朗日对偶问题求解
3. 对偶问题是原问题最优解的下确界
4. slater条件：存在一点 $x \in$relint $D$ ($D$ 的相对内点集）满足$f_{i}(x)<0, i=1, \ldots, m, A x=b$, 这样的点称为严格可行的点
5. slater定理：满足slater条件且原问题是凸优化问题时，强对偶性成立
6. slater条件确保了鞍点的存在，KKT条件是鞍点是最优解的充分条件。当原问题是凸优化问题时，KKT条件是充要条件。
7. KKT条件，其中$h_{i}(x)$是等式约束，$g_{i}(x)$是不等式约束
$$
\begin{array}{c}
\nabla f(x)+\sum_{i=1}^{n} \lambda_{i} \nabla h_{i}(x)+\sum_{i=1}^{n} \mu_{i} \nabla g_{i}(x)=0 \\
\mu_{i} g(x)_{i}=0 \\
\mu_{i} \geq 0 \\
h_{i}(x)=0 \\
g_{i}(x) \leq 0 \\
i=1,2, \ldots, n
\end{array}
$$
8. 凸优化就是这个优化问题的优化函数是凸函数，并且可行域是凸集
#GDBT
```python
```
|
% Isoparametric Formulation Implementation
% clear memory
clear all
close all
clc
% E: modulus of elasticity
% A: area of cross section
% L: length of bar
E=8; L=4;
u_exact=@(x) (56-8*(x-2)-24*heaviside(x-5))/2/x;
hold on;
ezplot(u_exact,[2 6])
title('Exact Solution v.s. FEM','interpreter','latex');
xlabel('x','interpreter','latex');
ylabel('Axial stress, $\it{\sigma}_{x}$','interpreter','latex','FontSize',12);
for i=1:4 %For NEL = 1, 2, 4, 8
fprintf( '\nNumber of elements:%d\n\n',2^(i-1) );
% numberElements: number of elements
numberElements=2^(i-1);
% numberNodes: number of nodes
numberNodes=2*numberElements+1;
A=zeros(1,numberElements);
% generation of coordinates and connectivities
NNOD=3;
nodeCoordinates=linspace(2,L+2,numberNodes);
%Generate element length vector
for i=1:numberElements
Le(i)=L/numberElements;
elementNodes(i,:)=[(i-1)*2+1 (i-1)*2+2 (i-1)*2+3];
A(i)=(nodeCoordinates(2*i-1)+Le(i)/2)*2;
end
% for structure:
% displacements: displacement vector
% force : force vector
% stiffness: stiffness matrix
force=zeros(numberNodes,1);
stiffness=zeros(numberNodes,numberNodes);
% computation of the system stiffness matrix and force vector
for e=1:numberElements;
% elementDof: element degrees of freedom (Dof)
elementDof=elementNodes(e,:) ;
detJacobian=Le(e)/2;
invJacobian=1/detJacobian;
ngp = 3;
[w,xi]=gauss1d(ngp);
xc=0.5*(nodeCoordinates(elementDof(1))+nodeCoordinates(elementDof(end)));
for ip=1:ngp;
[shape,naturalDerivatives]=shapeFunctionL3(xi(ip));
B=naturalDerivatives*invJacobian;
stiffness(elementDof,elementDof)=...
stiffness(elementDof,elementDof)+ B'*B*w(ip)*detJacobian*E*A(e);
force(elementDof)=force(elementDof)+...
8*shape'*detJacobian*w(ip);
end
if(nodeCoordinates(elementDof(end))==5)
x=(5-xc)/detJacobian;
[s,n]=shapeFunctionL3(x);
force(elementDof)= force(elementDof)+...
24*s';
end
if(nodeCoordinates(elementDof(1))<5&&...
nodeCoordinates(elementDof(3))>5)
x=(5-xc)/detJacobian;
[s,n]=shapeFunctionL3(x);
force(elementDof)= force(elementDof)+...
24*s';
end
end
% boundary conditions and solution
% prescribed dofs
prescribedDof=[1];
% solution
GDof=numberNodes;
displacements=solution(GDof,prescribedDof,stiffness,force);
% output displacements/reactions
outputDisplacementsReactionsPretty(displacements,stiffness, ...
numberNodes,prescribedDof,force)
fprintf('Axial stress\n')
fprintf('element\t\taxial stress\n')
%Stress and strain recovery
ngp = 3;
elementNodeCoor=zeros(numberElements*NNOD,1);
elementNodeStr=zeros(numberElements*NNOD,1);
for e=1:numberElements;
% elementDof: element degrees of freedom (Dof)
elementDof=elementNodes(e,:);
detJacobian=Le(e)/2;
invJacobian=1/detJacobian;
xi=[-1 0 1];
for ip=1:NNOD;
[shape,naturalDerivatives]=shapeFunctionL3(xi(ip));
B=naturalDerivatives*invJacobian;
elementNodeCoor(ip+(e-1)*NNOD,1)=nodeCoordinates(elementDof(ip));
elementNodeStr(ip+(e-1)*NNOD,1)=E*B*displacements(elementDof,1);
fprintf('%2.0f\t%2.0fth node\t%10.4e\n', e, ip,elementNodeStr(ip+(e-1)*NNOD,1))
end
end
%post process
switch numberElements
case 1
str='r*--';
case 2
str='g*--';
case 4
str='k*--';
case 8
str='m*--';
end
plot(elementNodeCoor,elementNodeStr,str)
hold on;
end
legend('Exact solution','NEL=1','NEL=2','NEL=4','NEL=8','interpreter','latex');
|
Require Import Terms.
Require Import LNaVSyntax.
Require Import LNaVBigStep.
(** * Equivalences *)
(** Low-equivalence judgments. *)
Inductive eq_atom : Atom -> Atom -> Lab -> Prop :=
| eq_a : forall b1 l1 b2 l2 l,
l1 = l2 ->
((l1 <: l \/ l2 <: l) -> eq_box b1 b2 l) ->
eq_atom (b1,l1) (b2,l2) l
with eq_box : Box -> Box -> Lab -> Prop :=
| eq_v : forall v1 v2 l, eq_val v1 v2 l -> eq_box (V v1) (V v2) l
| eq_d : forall e l, eq_box (D e) (D e) l
with eq_val : Val -> Val -> Lab -> Prop :=
| eq_vconst : forall c' l, eq_val (VConst c') (VConst c') l
| eq_vinx : forall d a a' l,
eq_atom a a' l ->
eq_val (VInx d a) (VInx d a') l
| eq_vclos : forall r1 r2 x t l,
eq_env r1 r2 l ->
eq_val (VClos r1 x t) (VClos r2 x t) l
with eq_env : Env -> Env -> Lab -> Prop :=
| eq_e_nil : forall l, eq_env nil nil l
| eq_e_cons : forall x a1 a2 r1 r2 l,
eq_atom a1 a2 l -> eq_env r1 r2 l -> eq_env ((x,a1)::r1) ((x,a2)::r2) l.
Hint Constructors eq_atom eq_box eq_val eq_env.
Scheme eq_atom_ind' := Minimality for eq_atom Sort Prop
with eq_box_ind' := Minimality for eq_box Sort Prop
with eq_val_ind' := Minimality for eq_val Sort Prop
with eq_env_ind' := Minimality for eq_env Sort Prop.
Combined Scheme eq_mutind from eq_val_ind', eq_box_ind', eq_atom_ind', eq_env_ind'.
(** Low-equivalence judgments, that take the pc label into account. *)
Definition eq_atom' a1 pc1 a2 pc2 l :=
(pc1 <: l \/ pc2 <: l) ->
(pc1 = pc2 /\ eq_atom a1 a2 l).
Definition eq_env' r1 pc1 r2 pc2 l :=
(pc1 <: l \/ pc2 <: l) ->
(pc1 = pc2 /\ eq_env r1 r2 l).
(** * Preliminary lemmas. *)
Lemma maps_eq_env : forall x a1 a2 r1 r2 l,
eq_env r1 r2 l ->
maps r1 x a1 ->
maps r2 x a2 ->
eq_atom a1 a2 l.
Proof.
intros x a1 a2 r1 r2 l Heq_env Hmaps1 Hmaps2.
induction Heq_env. invs Hmaps1.
invs Hmaps1; invs Hmaps2. destruct (beq_nat x x0).
congruence. auto.
Qed.
Lemma maps_eq_env' : forall x a1 a2 r1 r2 pc1 pc2 l,
eq_env' r1 pc1 r2 pc2 l ->
maps r1 x a1 ->
maps r2 x a2 ->
eq_atom' a1 pc1 a2 pc2 l.
Proof.
intros x a1 a2 r1 r2 pc1 pc2 l Heq_env' Hmaps1 Hmaps2 Hflows.
apply Heq_env' in Hflows. intuition eauto using maps_eq_env.
Qed.
Lemma eq_env'_cons_inv_env : forall x a1 a2 r1 r2 pc1 pc2 l,
eq_env' ((x,a1) :: r1) pc1 ((x,a2) :: r2) pc2 l ->
eq_env' r1 pc1 r2 pc2 l.
Proof.
intros x a1 a2 r1 r2 pc1 pc2 l H Hflows. apply H in Hflows as [? Henv].
invsc Henv. auto.
Qed.
Lemma eq_env'_cons_inv_atom : forall x a1 a2 r1 r2 pc1 pc2 l,
eq_env' ((x,a1) :: r1) pc1 ((x,a2) :: r2) pc2 l ->
eq_atom' a1 pc1 a2 pc2 l.
Proof.
intros x a1 a2 r1 r2 pc1 pc2 l H Hflows. apply H in Hflows as [? Henv].
invsc Henv. auto.
Qed.
Lemma cons_eq_env' : forall x a1 a2 r1 r2 pc1 pc2 l,
eq_atom' a1 pc1 a2 pc2 l ->
eq_env' r1 pc1 r2 pc2 l ->
eq_env' ((x,a1) :: r1) pc1 ((x,a2) :: r2) pc2 l.
Proof.
intros x a1 a2 r1 r2 pc1 pc2 l Hatom Henv Hflows.
specialize (Hatom Hflows). specialize (Henv Hflows).
intuition auto.
Qed.
(** Reflexivity of low-equivalence. Later, we prove this is actually an
equivalence relation. *)
Lemma eq_refl :
(forall v l, eq_val v v l) /\
(forall b l, eq_box b b l) /\
(forall a l, eq_atom a a l) /\
(forall r l, eq_env r r l).
Proof. apply val_box_atom_env_mutind; eauto. Qed.
Lemma eq_val_refl : forall v l, eq_val v v l.
Proof. pose proof eq_refl. intuition. Qed.
Lemma eq_box_refl : forall b l, eq_box b b l.
Proof. pose proof eq_refl. intuition. Qed.
Lemma eq_atom_refl : forall a l, eq_atom a a l.
Proof. pose proof eq_refl. intuition. Qed.
Lemma eq_env_refl : forall r l, eq_env r r l.
Proof. pose proof eq_refl. intuition. Qed.
Lemma intro_and2 : forall (P P1 P2: Prop),
(P -> (P1 /\ P2)) ->
(P -> P1) /\ (P -> P2).
Proof. tauto. Qed.
Lemma eq_val_eq_tag : forall v1 v2 l,
eq_val v1 v2 l ->
tag_of v1 = tag_of v2.
Proof.
intros v1 v2 l Heq.
inversion Heq; eauto.
Qed.
Lemma or_introrefl : forall (P : Prop),
P -> P \/ P.
Proof. left. assumption. Qed.
Lemma or_elimrefl : forall (P : Prop),
P \/ P -> P.
Proof. intros P H. destruct H; assumption. Qed.
(** Monotonicity of the pc label. This property is essential for the
non-interference proof to go through. *)
Lemma pc_eval_monotonic : forall r t pc a pc',
r |- t, pc ==> a, pc' -> pc <: pc'.
Proof.
intros r t pc a pc' Heval.
(eval_cases (induction Heval) Case);
eauto 4 using flows_refl, flows_trans, join_1_rev.
Qed.
(* Binary operations respect equality *)
Lemma eq_box_eq_bop : forall b b11 b12 b21 b22 l,
eq_box b11 b21 l ->
eq_box b12 b22 l ->
bop_box b b11 b12 = bop_box b b21 b22.
Proof.
intros b b11 b12 b21 b22 l Heq1 Heq2.
invsc Heq1; eauto; invsc Heq2; eauto;
destruct b; invsc H; try invsc H0; eauto.
Qed.
(** * Non-interference *)
(** Strengthened version of non-interference. *)
Lemma non_interference_strong :
forall r1 t pc1 a1 pc1',
r1 |- t, pc1 ==> a1, pc1' ->
forall r2 pc2 a2 pc2' l,
r2 |- t, pc2 ==> a2, pc2' ->
eq_env' r1 pc1 r2 pc2 l ->
(eq_env' r1 pc1' r2 pc2' l /\
eq_atom' a1 pc1' a2 pc2' l).
Proof.
intros r1 t pc1 a1 pc1' H.
(eval_cases (induction H) Case);
intros r2 pc2 a2 pc2'; try (rename l into l1); intro l;
intros Heval2 Heq_env; invsc Heval2.
Case "eval_var". eauto using maps_eq_env'.
Case "eval_const". intuition.
intros Hpc. specialize (Heq_env Hpc). invsc Heq_env; eauto.
Case "eval_let".
apply intro_and2. intro Hpc.
pose proof (pc_eval_monotonic _ _ _ _ _ H).
pose proof (pc_eval_monotonic _ _ _ _ _ H0).
pose proof (pc_eval_monotonic _ _ _ _ _ H8).
pose proof (pc_eval_monotonic _ _ _ _ _ H9).
assert (pc <: l \/ pc2 <: l) by (destruct Hpc; eauto using flows_trans).
assert (pc' <: l \/ pc2' <: l) by (destruct Hpc; eauto using flows_trans).
specialize (Heq_env H5). invsc Heq_env.
apply IHeval1 with (l:=l) in H8; try (intro; eauto). destruct H8.
apply IHeval2 with (l:=l) in H9; eauto using cons_eq_env'. destruct H9.
specialize (H11 Hpc). invsc H11; eauto.
Case "eval_abs". intuition.
intros Hpc. specialize (Heq_env Hpc). invsc Heq_env; eauto.
apply intro_and2. intro Hpc.
pose proof (pc_eval_monotonic _ _ _ _ _ H1).
pose proof (pc_eval_monotonic _ _ _ _ _ H10).
pose proof (maps_eq_env' _ _ _ _ _ _ _ _ Heq_env H H4).
pose proof (maps_eq_env' _ _ _ _ _ _ _ _ Heq_env H0 H6).
assert (pc <: l \/ pc2 <: l) by
(destruct Hpc; eauto using join_1_rev, flows_trans).
specialize (H5 H8). invsc H5. specialize (H7 H8). invsc H7.
invsc H11. assert (l0 <: l \/ l0 <: l) by
(destruct Hpc; eauto using join_2_rev, flows_trans).
specialize (H17 H7). invsc H17. invsc H13.
specialize (Heq_env H8). invsc Heq_env.
apply IHeval with (l:=l) in H10. invsc H10.
specialize (H14 Hpc). invsc H14; eauto.
apply cons_eq_env'; intro; eauto.
SCase "type error". (* spurious *)
apply intro_and2. intro Hpc.
pose proof (pc_eval_monotonic _ _ _ _ _ H1).
assert (pc <: l \/ pc2 <: l) by
(destruct Hpc; eauto using join_1_rev, flows_trans).
pose proof (maps_eq_env' _ _ _ _ _ _ _ _ Heq_env H H5).
specialize (H4 H3). invsc H4. invsc H7.
assert (l0 <: l \/ l0 <: l) by
(destruct Hpc; eauto using join_2_rev, flows_trans).
specialize (H13 H4). invsc H13. invsc H7.
simpl in H9. exfalso; auto.
Case "eval_app_no_abs".
SCase "no type error". (* spurious *)
apply intro_and2. intro Hpc.
pose proof (pc_eval_monotonic _ _ _ _ _ H9).
assert (pc <: l \/ pc2 <: l) by
(destruct Hpc; eauto using join_1_rev, flows_trans).
pose proof (maps_eq_env' _ _ _ _ _ _ _ _ Heq_env H H3).
specialize (H4 H2). invsc H4. invsc H7.
assert (l0 <: l \/ l0 <: l) by
(destruct Hpc; eauto using join_2_rev, flows_trans).
specialize (H13 H4). invsc H13. invsc H8.
simpl in H0. exfalso; auto.
SCase "eval_app_no_abs".
apply intro_and2. intro Hpc.
assert (pc <: l \/ pc2 <: l) by
(destruct Hpc; eauto using join_1_rev, flows_trans).
specialize (Heq_env H1). invsc Heq_env.
pose proof (maps_eq_env _ _ _ _ _ _ H3 H H4).
invsc H2.
assert (l0 <: l \/ l0 <: l) by
(destruct Hpc; eauto using join_2_rev, flows_trans).
specialize (H12 H2). invsc H12; eauto.
Case "eval_inx". intuition.
intros Hpc. specialize (Heq_env Hpc). invsc Heq_env.
pose proof (maps_eq_env _ _ _ _ _ _ H1 H H6). eauto using maps_eq_env.
Case "eval_match".
pose proof (pc_eval_monotonic _ _ _ _ _ H0).
pose proof (pc_eval_monotonic _ _ _ _ _ H10).
apply intro_and2. intro Hpc.
assert (pc <: l \/ pc2 <: l) by
(destruct Hpc; eauto using flows_trans, join_1_rev).
specialize (Heq_env H3). invsc Heq_env.
pose proof (maps_eq_env _ _ _ _ _ _ H5 H H9).
invsc H4. assert (l0 <: l \/ l0 <: l) by
(destruct Hpc; eauto using flows_trans, join_2_rev).
specialize (H14 H4). invsc H14. invsc H8.
apply IHeval with (l:=l) in H10. invsc H10. specialize (H7 Hpc).
invsc H7. eauto. eapply cons_eq_env'; intro; eauto.
SCase "type error". (* spurious *)
pose proof (pc_eval_monotonic _ _ _ _ _ H0).
apply intro_and2. intro Hpc.
assert (pc <: l \/ pc2 <: l) by
(destruct Hpc; eauto using flows_trans, join_1_rev).
specialize (Heq_env H2). invsc Heq_env.
pose proof (maps_eq_env _ _ _ _ _ _ H4 H H9).
invsc H3. assert (l0 <: l \/ l0 <: l) by
(destruct Hpc; eauto using flows_trans, join_2_rev).
specialize (H13 H3). invsc H13. invsc H6.
simpl in H10. exfalso; auto.
Case "eval_match_no_sum". SCase "no type error". (* spurious *)
pose proof (pc_eval_monotonic _ _ _ _ _ H10).
apply intro_and2. intro Hpc.
assert (pc <: l \/ pc2 <: l) by
(destruct Hpc; eauto using flows_trans, join_1_rev).
specialize (Heq_env H2). invsc Heq_env.
pose proof (maps_eq_env _ _ _ _ _ _ H4 H H9).
invsc H3. assert (l0 <: l \/ l0 <: l) by
(destruct Hpc; eauto using flows_trans, join_2_rev).
specialize (H13 H3). invsc H13. invsc H7.
simpl in H0. exfalso; auto.
SCase "type error".
apply intro_and2. intro Hpc.
assert (pc <: l \/ pc2 <: l) by
(destruct Hpc; eauto using flows_trans, join_1_rev).
specialize (Heq_env H1). invsc Heq_env.
pose proof (maps_eq_env _ _ _ _ _ _ H3 H H9).
invsc H2. assert (l0 <: l \/ l0 <: l) by
(destruct Hpc; eauto using flows_trans, join_2_rev).
specialize (H12 H2). invsc H12; eauto.
Case "eval_tag".
apply intro_and2. intro Hpc. specialize (Heq_env Hpc).
invsc Heq_env. pose proof (maps_eq_env _ _ _ _ _ _ H1 H H2).
invsc H0. split. auto. split. reflexivity.
constructor. reflexivity. intro. specialize (H9 H0).
unfold tag_box; invsc H9; eauto. invsc H3; eauto.
Case "eval_bop".
apply intro_and2. intro Hpc. specialize (Heq_env Hpc). invsc Heq_env.
pose proof (maps_eq_env _ _ _ _ _ _ H2 H H8).
pose proof (maps_eq_env _ _ _ _ _ _ H2 H0 H9).
split. intuition. split. reflexivity. invsc H1. invsc H3.
constructor. reflexivity. intro.
assert (l'0 <: l \/ l'0 <: l) by
(destruct H1; eauto using flows_trans, join_1_rev).
assert (l''0 <: l \/ l''0 <: l) by
(destruct H1; eauto using flows_trans, join_2_rev).
specialize (H12 H3). specialize (H11 H4).
pose proof (eq_box_eq_bop bo _ _ _ _ l H12 H11).
rewrite H5. apply eq_box_refl.
Case "eval_bracket".
pose proof (pc_eval_monotonic _ _ _ _ _ H0).
pose proof (pc_eval_monotonic _ _ _ _ _ H8).
apply intro_and2. intro Hpc.
assert (pc <: l \/ pc2 <: l) by (destruct Hpc; eauto using join_1_rev).
specialize (Heq_env H3). invsc Heq_env.
pose proof (maps_eq_env _ _ _ _ _ _ H6 H H4).
invsc H5.
assert (l'0 <: l \/ l'0 <: l) by (destruct Hpc; eauto using join_2_rev).
specialize (H14 H5). invsc H14. invsc H10. split. intuition. split. reflexivity.
apply IHeval with (l:=l) in H8; try (intro; eauto). invsc H8.
remember (flows_dec (l'' \_/ pc') (l0 \_/ (pc2 \_/ l'0))) as f1.
remember (flows_dec (l''0 \_/ pc'0) (l0 \_/ (pc2 \_/ l'0))) as f2.
destruct f1; destruct f2; constructor; auto; intro.
SCase "flow, flow".
assert (pc' <: l \/ pc'0 <: l) by
(destruct H8; destruct Hpc; eauto using join_2_rev, flows_trans, join_minimal).
specialize (H9 H10). invsc H9. invsc H12.
apply H17.
destruct H8; destruct Hpc; eauto using join_1_rev, flows_trans, join_minimal.
SCase "flow, no flow". (* spurious *)
assert (pc' <: l \/ pc'0 <: l) by
(destruct H8; destruct Hpc; eauto using join_2_rev, flows_trans, join_minimal).
specialize (H9 H10). invsc H9. invsc H12.
contradiction.
SCase "no flow, flow". (* spurious *)
assert (pc' <: l \/ pc'0 <: l) by
(destruct H8; destruct Hpc; eauto using join_2_rev, flows_trans, join_minimal).
specialize (H9 H10). invsc H9. invsc H12.
contradiction.
Case "eval_bracket". SCase "type error". (* spurious *)
apply intro_and2. intro Hpc.
pose proof (pc_eval_monotonic _ _ _ _ _ H0).
assert (pc <: l \/ pc2 <: l) by
(destruct Hpc; eauto using join_1_rev, flows_trans).
specialize (Heq_env H2). invsc Heq_env.
pose proof (maps_eq_env _ _ _ _ _ _ H5 H H4).
invsc H3.
assert (l'0 <: l \/ l'0 <: l) by
(destruct Hpc; eauto using join_2_rev, flows_trans).
specialize (H13 H3). invsc H13. invsc H7.
simpl in H8. exfalso; auto.
Case "eval_bracket_no_lab". SCase "no type error". (* spurious *)
apply intro_and2. intro Hpc.
pose proof (pc_eval_monotonic _ _ _ _ _ H8).
assert (pc <: l \/ pc2 <: l) by
(destruct Hpc; eauto using join_1_rev, flows_trans).
specialize (Heq_env H2). invsc Heq_env.
pose proof (maps_eq_env _ _ _ _ _ _ H5 H H4).
invsc H3.
assert (l'0 <: l \/ l'0 <: l) by
(destruct Hpc; eauto using join_2_rev, flows_trans).
specialize (H13 H3). invsc H13. invsc H9.
simpl in H0. exfalso; auto.
Case "eval_bracket_no_lab".
apply intro_and2. intro Hpc.
assert (pc <: l \/ pc2 <: l) by (destruct Hpc; eauto using join_1_rev).
specialize (Heq_env H1). invsc Heq_env.
pose proof (maps_eq_env _ _ _ _ _ _ H3 H H4).
invsc H2.
assert (l'0 <: l \/ l'0 <: l) by (destruct Hpc; eauto using join_2_rev).
specialize (H12 H2). invsc H12; eauto.
Case "eval_label_of".
apply intro_and2. intro Hpc.
specialize (Heq_env Hpc). invsc Heq_env.
pose proof (maps_eq_env _ _ _ _ _ _ H1 H H2).
invsc H0. split; eauto.
Case "eval_get_pc".
apply intro_and2. intro Hpc.
specialize (Heq_env Hpc). invsc Heq_env.
split; eauto.
Case "eval_mk_nav".
apply intro_and2. intro Hpc.
specialize (Heq_env Hpc). invsc Heq_env.
pose proof (maps_eq_env _ _ _ _ _ _ H1 H H2).
invsc H0. split. intuition. split. reflexivity. constructor. reflexivity.
intro. specialize (H9 H0). invsc H9; unfold mk_nav_box; try invsc H3; eauto;
destruct c'; auto.
Case "eval_to_sum".
apply intro_and2. intro Hpc. specialize (Heq_env Hpc). invsc Heq_env.
pose proof (maps_eq_env _ _ _ _ _ _ H1 H H2).
invsc H0. split. intuition. split. reflexivity. constructor. reflexivity.
intro. specialize (H9 H0). unfold to_sum_box. invsc H9; constructor; constructor; eauto.
Qed.
(** Finally, non-interference. *)
Theorem non_interference: forall r1 t pc1 a1 pc1' r2 pc2 a2 pc2' l,
eq_env' r1 pc1 r2 pc2 l ->
r1 |- t, pc1 ==> a1, pc1' ->
r2 |- t, pc2 ==> a2, pc2' ->
eq_atom' a1 pc1' a2 pc2' l.
Proof.
intros r1 t pc1 a1 pc1' r2 pc2 a2 pc2' l
Hequiv_env Heval1 Heval2.
assert (eq_env' r1 pc1' r2 pc2' l /\ eq_atom' a1 pc1' a2 pc2' l) by
eauto using non_interference_strong.
intuition.
Qed.
|
import tactic.basic
import .ch07_indprop
open nat (
le less_than_or_equal.refl less_than_or_equal.step lt
)
open indprop (next_nat total_relation total_relation.intro)
open indprop.next_nat
variables {α : Type}
variables {n m o p: ℕ}
namespace rel
/-
Definition relation (X: Type) := X → X → Prop.
-/
def relation (α : Type) := α → α → Prop
/-
Print le.
(* ====> Inductive le (n : nat) : nat -> Prop :=
le_n : n <= n
| le_S : forall m : nat, n <= m -> n <= S m *)
Check le : nat → nat → Prop.
Check le : relation nat.
-/
#print le
#check (le : ℕ → ℕ → Prop)
#check (le : relation ℕ)
/-
Definition partial_function {X: Type} (R: relation X) :=
∀x y1 y2 : X, R x y1 → R x y2 → y1 = y2.
-/
def partial_function (R : relation α) :=
∀{x y₁ y₂} (h₁ : R x y₁) (h₂ : R x y₂), y₁ = y₂
/-
Print next_nat.
(* ====> Inductive next_nat (n : nat) : nat -> Prop :=
nn : next_nat n (S n) *)
Check next_nat : relation nat.
Theorem next_nat_partial_function :
partial_function next_nat.
Proof.
unfold partial_function.
intros x y1 y2 H1 H2.
inversion H1. inversion H2.
reflexivity. Qed.
-/
#print next_nat
#check (next_nat : relation ℕ)
theorem next_nat_partial_function : partial_function next_nat :=
begin
unfold partial_function,
intros,
cases h₁,
cases h₂,
refl,
end
/-
Theorem le_not_a_partial_function :
¬(partial_function le).
Proof.
unfold not. unfold partial_function. intros Hc.
assert (0 = 1) as Nonsense. {
apply Hc with (x := 0).
- apply le_n.
- apply le_S. apply le_n. }
discriminate Nonsense. Qed.
-/
theorem le_not_a_partial_function : ¬partial_function le :=
begin
unfold partial_function,
by_contradiction c,
have nonsense : 0 = 1,
apply @c 0 0 1,
apply less_than_or_equal.refl,
apply less_than_or_equal.step,
apply less_than_or_equal.refl,
cases nonsense,
end
theorem total_relation_not_partial : ¬partial_function total_relation :=
begin
unfold partial_function,
by_contradiction c,
cases c (total_relation.intro 0 0) (total_relation.intro 0 1),
end
theorem empty_relation_partial : partial_function (@empty_relation α) :=
begin
unfold partial_function,
intros,
cases h₁,
end
/-
Definition reflexive {X: Type} (R: relation X) :=
∀a : X, R a a.
Theorem le_reflexive :
reflexive le.
Proof.
unfold reflexive. intros n. apply le_n. Qed.
-/
def reflexive (R : relation α) := ∀a, R a a
theorem le_refl : reflexive le :=
begin
unfold reflexive,
intro n,
apply less_than_or_equal.refl,
end
/-
Definition transitive {X: Type} (R: relation X) :=
∀a b c : X, (R a b) → (R b c) → (R a c).
Theorem le_trans :
transitive le.
Proof.
intros n m o Hnm Hmo.
induction Hmo.
- (* le_n *) apply Hnm.
- (* le_S *) apply le_S. apply IHHmo. Qed.
Theorem lt_trans:
transitive lt.
Proof.
unfold lt. unfold transitive.
intros n m o Hnm Hmo.
apply le_S in Hnm.
apply le_trans with (a := (S n)) (b := (S m)) (c := o).
apply Hnm.
apply Hmo. Qed.
-/
#check transitive
def transitive (R : relation α) :=
∀{a b c : α} (hab : R a b) (hbc : R b c), R a c
theorem le_trans: transitive le :=
begin
unfold transitive,
intros,
induction hbc with b' h ih,
exact hab,
apply less_than_or_equal.step,
exact ih,
end
theorem lt_trans : transitive lt :=
begin
unfold transitive,
intros,
exact le_trans (less_than_or_equal.step hab) hbc,
end
/-
Theorem lt_trans' :
transitive lt.
Proof.
(* Prove this by induction on evidence that m is less than o. *)
unfold lt. unfold transitive.
intros n m o Hnm Hmo.
induction Hmo as [| m' Hm'o].
(* FILL IN HERE *) Admitted.
-/
theorem lt_trans' : transitive lt :=
begin
unfold transitive,
intros,
induction hbc with b' h ih,
exact less_than_or_equal.step hab,
exact less_than_or_equal.step ih,
end
/-
Theorem lt_trans'' :
transitive lt.
Proof.
unfold lt. unfold transitive.
intros n m o Hnm Hmo.
induction o as [| o'].
(* FILL IN HERE *) Admitted.
-/
theorem lt_trans'' : transitive lt :=
begin
unfold transitive,
intros,
induction c with c ih,
cases hbc,
cases hbc with _ h,
exact less_than_or_equal.step hab,
exact less_than_or_equal.step (ih h),
end
/-
Theorem le_Sn_le : ∀n m, S n ≤ m → n ≤ m.
Proof.
intros n m H. apply le_trans with (S n).
- apply le_S. apply le_n.
- apply H.
Qed.
-/
theorem nat.le_of_succ_le (h : n + 1 ≤ m) : n ≤ m :=
begin
apply le_trans,
apply less_than_or_equal.step,
apply less_than_or_equal.refl,
exact h,
end
/-
Theorem le_S_n : ∀n m,
(S n ≤ S m) → (n ≤ m).
Proof.
(* FILL IN HERE *) Admitted.
-/
theorem nat.le_of_succ_le_succ (h : n + 1 ≤ m + 1) : n ≤ m :=
begin
cases h with _ h,
exact less_than_or_equal.refl,
exact nat.le_of_succ_le h,
end
/-
Theorem le_Sn_n : ∀n,
¬(S n ≤ n).
Proof.
(* FILL IN HERE *) Admitted.
-/
theorem nat.not_succ_le_self (n) : ¬(n + 1 ≤ n) :=
begin
by_contra c,
induction n with n ih,
cases c,
exact ih (nat.le_of_succ_le_succ c),
end
/-
Definition symmetric {X: Type} (R: relation X) :=
∀a b : X, (R a b) → (R b a).
-/
def symmetric (R : relation α) := ∀{a b : α} (h : R a b), R b a
/-
Theorem le_not_symmetric :
¬(symmetric le).
Proof.
(* FILL IN HERE *) Admitted.
-/
theorem le_not_symmetric : ¬symmetric le :=
begin
unfold symmetric,
by_contra c,
cases c (less_than_or_equal.step (@less_than_or_equal.refl 0)),
end
/-
Definition antisymmetric {X: Type} (R: relation X) :=
∀a b : X, (R a b) → (R b a) → a = b.
-/
def anti_symmetric (R : relation α) := ∀{a b} (hab : R a b) (hba : R b a), a = b
/-
Theorem le_antisymmetric :
antisymmetric le.
Proof.
(* FILL IN HERE *) Admitted.
-/
theorem le_antisymm : anti_symmetric le :=
begin
unfold anti_symmetric,
intros,
induction a with a ih generalizing b,
cases hba,
refl,
cases b,
cases hab,
apply congr_arg,
apply ih,
exact nat.le_of_succ_le_succ hab,
exact nat.le_of_succ_le_succ hba,
end
/-
Theorem le_step : ∀n m p,
n < m →
m ≤ S p →
n ≤ p.
Proof.
(* FILL IN HERE *) Admitted.
-/
theorem le_step (hn : n < m) (hm : m ≤ p + 1) : n ≤ p :=
nat.le_of_succ_le_succ $ le_trans hn hm
/-
Definition equivalence {X:Type} (R: relation X) :=
(reflexive R) ∧ (symmetric R) ∧ (transitive R).
-/
def equivalence (R : relation α) := reflexive R ∧ symmetric R ∧ transitive R
/-
Definition order {X:Type} (R: relation X) :=
(reflexive R) ∧ (antisymmetric R) ∧ (transitive R).
-/
def partial_order (R : relation α) :=
reflexive R ∧ anti_symmetric R ∧ transitive R
/-
Definition preorder {X:Type} (R: relation X) :=
(reflexive R) ∧ (transitive R).
Theorem le_order :
order le.
Proof.
unfold order. split.
- (* refl *) apply le_reflexive.
- split.
+ (* antisym *) apply le_antisymmetric.
+ (* transitive. *) apply le_trans. Qed.
-/
def preorder (R : relation α) := reflexive R ∧ transitive R
theorem le_order : partial_order le :=
begin
unfold partial_order,
split,
apply le_refl,
split,
apply le_antisymm,
apply le_trans,
end
/-
Inductive clos_refl_trans {A: Type} (R: relation A) : relation A :=
| rt_step x y (H : R x y) : clos_refl_trans R x y
| rt_refl x : clos_refl_trans R x x
| rt_trans x y z
(Hxy : clos_refl_trans R x y)
(Hyz : clos_refl_trans R y z) :
clos_refl_trans R x z.
-/
inductive clos_refl_trans (R : relation α) : relation α
| rt_step {x y} (h : R x y) : clos_refl_trans x y
| rt_refl (x) : clos_refl_trans x x
| rt_trans {x y z}
(hxy : clos_refl_trans x y)
(hyz : clos_refl_trans y z) : clos_refl_trans x z
open clos_refl_trans
/-
Theorem next_nat_closure_is_le : ∀n m,
(n ≤ m) ↔ ((clos_refl_trans next_nat) n m).
Proof.
intros n m. split.
- (* -> *)
intro H. induction H.
+ (* le_n *) apply rt_refl.
+ (* le_S *)
apply rt_trans with m. apply IHle. apply rt_step.
apply nn.
- (* <- *)
intro H. induction H.
+ (* rt_step *) inversion H. apply le_S. apply le_n.
+ (* rt_refl *) apply le_n.
+ (* rt_trans *)
apply le_trans with y.
apply IHclos_refl_trans1.
apply IHclos_refl_trans2. Qed.
-/
theorem next_nat_closure_is_le : n ≤ m ↔ (clos_refl_trans next_nat) n m :=
begin
split,
intro h,
induction h with m h ih,
apply rt_refl,
apply rt_trans,
apply ih,
apply rt_step,
apply nn,
intro h,
induction h,
case rt_step : x y h {
cases h,
apply less_than_or_equal.step,
exact less_than_or_equal.refl,
},
case rt_refl : x { exact less_than_or_equal.refl, },
case rt_trans : x y z hxy hyz ihx ihy { exact le_trans ihx ihy, },
end
/-
Inductive clos_refl_trans_1n {A : Type}
(R : relation A) (x : A)
: A → Prop :=
| rt1n_refl : clos_refl_trans_1n R x x
| rt1n_trans (y z : A)
(Hxy : R x y) (Hrest : clos_refl_trans_1n R y z) :
clos_refl_trans_1n R x z.
-/
inductive clos_refl_trans_1n (R : relation α) : α → α → Prop
| rt1n_refl (x) : clos_refl_trans_1n x x
| rt1n_trans {x y z} (hxy : R x y) (hyz : clos_refl_trans_1n y z)
: clos_refl_trans_1n x z
open clos_refl_trans_1n
/-
Lemma rsc_R : ∀(X:Type) (R:relation X) (x y : X),
R x y → clos_refl_trans_1n R x y.
Proof.
intros X R x y H.
apply rt1n_trans with y. apply H. apply rt1n_refl. Qed.
-/
lemma rsc_R {R : relation α} {x y} (h : R x y) : clos_refl_trans_1n R x y
:= rt1n_trans h (rt1n_refl y)
/-
Lemma rsc_trans :
∀(X:Type) (R: relation X) (x y z : X),
clos_refl_trans_1n R x y →
clos_refl_trans_1n R y z →
clos_refl_trans_1n R x z.
Proof.
(* FILL IN HERE *) Admitted.
-/
lemma rsc_trans {R : relation α} {x y z}
(hxy : clos_refl_trans_1n R x y) (hyz : clos_refl_trans_1n R y z)
: clos_refl_trans_1n R x z :=
begin
induction hxy,
case rt1n_refl { exact hyz, },
case rt1n_trans : x' y' z' hxy' hyz' ih { exact rt1n_trans hxy' (ih hyz), },
end
/-
Theorem rtc_rsc_coincide :
∀(X:Type) (R: relation X) (x y : X),
clos_refl_trans R x y ↔ clos_refl_trans_1n R x y.
Proof.
(* FILL IN HERE *) Admitted.
-/
theorem rtc_rsc_coincide {R : relation α} {x y}
: clos_refl_trans R x y ↔ clos_refl_trans_1n R x y :=
begin
split,
intro h,
induction h,
case rt_step : x' y' h' { exact rsc_R h', },
case rt_refl { exact rt1n_refl h, },
case rt_trans : x' y' z' hxy hyz ihy ihz { exact rsc_trans ihy ihz, },
intro h,
induction h,
case rt1n_refl { exact rt_refl h },
case rt1n_trans : x' y' z' hxy hyz ih { exact rt_trans (rt_step hxy) ih, },
end
end rel
|
-- --------------------------------------------------------------- [ Model.idr ]
-- Module : Model.idr
-- Copyright : (c) Jan de Muijnck-Hughes
-- License : see LICENSE
-- --------------------------------------------------------------------- [ EOH ]
||| Example of using PML to model a paper.
module GRL.Lang.Test.PML
import GRL.Lang.PML
-- ------------------------------------------------------------------- [ Paper ]
paper : PAPER
paper = MkPaper "My First Paper"
abst : ABSTRACT
abst = MkAbs
bib : BIB
bib = MkBib
intr : SECT
intr = MkSect "Introduction"
meth : SECT
meth = MkSect "Methodology"
res : SECT
res = MkSect "Results"
disc : SECT
disc = MkSect "Discussion"
-- ------------------------------------------------------------------- [ Tasks ]
wabs : WRITING
wabs = MkAuth "Abstract" SATISFIED
rabs : REVIEW
rabs = MkRev "Abstract" WEAKSATIS
wbib : WRITING
wbib = MkAuth "Bib" WEAKSATIS
rbib : REVIEW
rbib = MkRev "Bib" WEAKSATIS
wIntro : WRITING
wIntro = MkAuth "Intro" DENIED
rIntro : REVIEW
rIntro = MkRev "Intro" DENIED
wMeth : WRITING
wMeth = MkAuth "Meth" DENIED
rMeth : REVIEW
rMeth = MkRev "Meth" DENIED
wRes : WRITING
wRes = MkAuth "Res" DENIED
rRes : REVIEW
rRes = MkRev "Res" DENIED
wDis : WRITING
wDis = MkAuth "Dis" DENIED
rDis : REVIEW
rDis = MkRev "Dis" DENIED
-- ------------------------------------------------------------- [ Build Model ]
paperPlan : GModel
paperPlan = emptyModel
\= paper
\= abst \= wabs \= rabs \= bib \= wbib \= rbib
\= intr \= wIntro \= rIntro \= meth \= wMeth \= rMeth
\= res \= wRes \= rRes \= disc \= wDis \= rDis
\= (paper &= abst) \= (wabs ==> abst) \= (rabs ==> abst)
\= (paper &= bib) \= (wbib ==> bib) \= (rbib ==> bib)
\= (paper &= intr) \= (wIntro ==> intr) \= (rIntro ==> intr)
\= (paper &= meth) \= (wMeth ==> meth) \= (rMeth ==> meth)
\= (paper &= res) \= (wRes ==> res) \= (rRes ==> res)
\= (paper &= disc) \= (wDis ==> disc) \= (rDis ==> disc)
-- -------------------------------------------------------------------- [ Test ]
export
runTest : IO ()
runTest = do
putStrLn $ prettyModel paperPlan
-- --------------------------------------------------------------------- [ EOF ]
|
{-# OPTIONS --cubical --no-import-sorts --safe #-}
module Cubical.DStructures.Structures.Higher where
open import Cubical.Foundations.Prelude
open import Cubical.Foundations.Equiv
open import Cubical.Foundations.HLevels
open import Cubical.Foundations.Isomorphism
open import Cubical.Foundations.Function
open import Cubical.Foundations.Pointed
open import Cubical.Foundations.Univalence
open import Cubical.Functions.FunExtEquiv
open import Cubical.Homotopy.Base
open import Cubical.Homotopy.Connected
open import Cubical.Data.Sigma
open import Cubical.Data.Nat
open import Cubical.Relation.Binary
open import Cubical.Algebra.Group
open import Cubical.Algebra.Group.Higher
open import Cubical.Algebra.Group.EilenbergMacLane1
open import Cubical.HITs.EilenbergMacLane1
open import Cubical.DStructures.Base
open import Cubical.DStructures.Meta.Properties
open import Cubical.DStructures.Meta.Isomorphism
open import Cubical.DStructures.Structures.Universe
open import Cubical.DStructures.Structures.Type
open import Cubical.DStructures.Structures.Group
open import Cubical.DStructures.Structures.Constant
private
variable
ℓ ℓ' : Level
𝒮ᴰ-connected : {ℓ : Level} (k : ℕ) → URGStrᴰ (𝒮-universe {ℓ}) (isConnected k) ℓ-zero
𝒮ᴰ-connected k =
Subtype→Sub-𝒮ᴰ (λ A → isConnected k A , isPropIsContr)
𝒮-universe
𝒮ᴰ-truncated : {ℓ : Level} (n : ℕ) → URGStrᴰ (𝒮-universe {ℓ}) (isOfHLevel n) ℓ-zero
𝒮ᴰ-truncated n =
Subtype→Sub-𝒮ᴰ (λ A → isOfHLevel n A , isPropIsOfHLevel n)
𝒮-universe
𝒮ᴰ-BGroup : (n k : ℕ)
→ URGStrᴰ (𝒮-universe {ℓ})
(λ A → A × (isConnected (k + 1) A) × (isOfHLevel (n + k + 2) A))
ℓ
𝒮ᴰ-BGroup n k =
combine-𝒮ᴰ 𝒮ᴰ-pointed
(combine-𝒮ᴰ (𝒮ᴰ-connected (k + 1))
(𝒮ᴰ-truncated (n + k + 2)))
𝒮-BGroup : (n k : ℕ) → URGStr (Σ[ A ∈ Type ℓ ] A × (isConnected (k + 1) A) × (isOfHLevel (n + k + 2) A)) ℓ
𝒮-BGroup n k = ∫⟨ 𝒮-universe ⟩ 𝒮ᴰ-BGroup n k
𝒮-1BGroup : URGStr 1BGroupΣ ℓ
𝒮-1BGroup = 𝒮-BGroup 0 1
𝒮-Iso-BGroup-Group : {ℓ : Level} → 𝒮-PIso (𝒮-group ℓ) 𝒮-1BGroup
RelIso.fun 𝒮-Iso-BGroup-Group G = EM₁ G , embase , EM₁Connected G , EM₁Groupoid G
RelIso.inv 𝒮-Iso-BGroup-Group = π₁-1BGroupΣ
RelIso.leftInv 𝒮-Iso-BGroup-Group = π₁EM₁≃
RelIso.rightInv 𝒮-Iso-BGroup-Group BG = basetype-≅ , basepoint-≅ , tt , tt
where
-- notation
type = fst BG
* = fst (snd BG)
conn = fst (snd (snd BG))
trunc = snd (snd (snd BG))
BG' = (bgroup (type , *) conn trunc)
π₁BG : Group
π₁BG = π₁-1BGroupΣ BG
EM₁π₁BG : 1BGroupΣ
EM₁π₁BG = EM₁ π₁BG , embase , EM₁Connected π₁BG , EM₁Groupoid π₁BG
-- equivalences
basetype-≅ : EM₁ π₁BG ≃ type
fst basetype-≅ = EM₁-functor-lInv-function π₁BG BG' (GroupEquiv.hom (π₁EM₁≃ π₁BG))
snd basetype-≅ = EM₁-functor-lInv-onIso-isEquiv π₁BG BG' (π₁EM₁≃ π₁BG)
basepoint-≅ : * ≡ *
basepoint-≅ = refl
𝒮ᴰ-BGroupHom : (n k : ℕ) → URGStrᴰ (𝒮-BGroup {ℓ} n k ×𝒮 𝒮-BGroup {ℓ'} n k)
(λ (BG , BH) → BGroupHomΣ BG BH)
(ℓ-max ℓ ℓ')
𝒮ᴰ-BGroupHom n k =
make-𝒮ᴰ
(λ {(BG , BH)} {(BG' , BH')} f (((eᴳ , _) , eᴳ-pt , _), ((eᴴ , _) , eᴴ-pt , _)) f'
→ ((eᴴ , eᴴ-pt) ∘∙ f) ∙∼ (f' ∘∙ (eᴳ , eᴳ-pt)))
(λ {(BG , BH)} f → q {(BG , BH)} f)
contrSingl
where
module _ {(BG , BH) : BGroupΣ n k × BGroupΣ n k} (f : BGroupHomΣ BG BH) where
q : (id∙ (baseΣ BH) ∘∙ f) ∙∼ (f ∘∙ id∙ (baseΣ BG))
q = funExt∙⁻ (id∙ (baseΣ BH) ∘∙ f
≡⟨ ∘∙-idʳ f ⟩
f
≡⟨ sym (∘∙-idˡ f) ⟩
(f ∘∙ id∙ (baseΣ BG)) ∎)
module _ ((BG , BH) : BGroupΣ n k × BGroupΣ n k) (f : BGroupHomΣ BG BH) where
contrSingl : isContr (Σ[ f' ∈ BGroupHomΣ BG BH ] ((id∙ (baseΣ BH) ∘∙ f) ∙∼ (f' ∘∙ id∙ (baseΣ BG))))
contrSingl =
isContrRespectEquiv (Σ-cong-equiv-snd (λ f' → f ≡ f'
≃⟨ invEquiv (funExt∙≃ f f') ⟩
f ∙∼ f'
≃⟨ pathToEquiv (cong (_∙∼ f')
(sym (∘∙-idʳ f))
∙ cong ((id∙ (baseΣ BH) ∘∙ f) ∙∼_)
(sym (∘∙-idˡ f'))) ⟩
(id∙ (baseΣ BH) ∘∙ f) ∙∼ (f' ∘∙ id∙ (baseΣ BG)) ■))
(isContrSingl f)
|
lemma subset_snd_imageI: "A \<times> B \<subseteq> S \<Longrightarrow> x \<in> A \<Longrightarrow> B \<subseteq> snd ` S"
|
(* Title: A state based hotel key card system
Author: Tobias Nipkow, TU Muenchen
*)
(*<*)
theory State
imports Basis
begin
declare if_split_asm[split]
(*>*)
section\<open>A state based model\<close>
text\<open>The model is based on three opaque types @{typ guest},
@{typ key} and @{typ room}. Type @{typ card} is just an abbreviation
for @{typ"key \<times> key"}.
The state of the system is modelled as a record which combines the
information about the front desk, the rooms and the guests.
\<close>
record state =
owns :: "room \<Rightarrow> guest option"
currk :: "room \<Rightarrow> key"
issued :: "key set"
cards :: "guest \<Rightarrow> card set"
roomk :: "room \<Rightarrow> key"
isin :: "room \<Rightarrow> guest set"
safe :: "room \<Rightarrow> bool"
text\<open>\noindent Reception records who @{const owns} a room (if anybody, hence
@{typ"guest option"}), the current key @{const currk} that has been
issued for a room, and which keys have been @{const issued} so
far. Each guest has a set of @{const cards}. Each room has a key
@{const roomk} recorded in the lock and a set @{const isin} of
occupants. The auxiliary variable @{const safe} is explained further
below; we ignore it for now.
In specification languages like Z, VDM and B we would now define a
number of operations on this state space. Since they are the only
permissible operations on the state, this defines a set of
\emph{reachable} states. In a purely logical environment like
Isabelle/HOL this set can be defined directly by an inductive
definition. Each clause of the definition corresponds to a
transition/operation/event. This is the standard approach to modelling
state machines in theorem provers.
The set of reachable states of the system (called \<open>reach\<close>) is
defined by four transitions: initialization, checking in, entering a room,
and leaving a room:\<close>
(*<*)
inductive_set reach :: "state set"
where
(*>*)
init:
"inj initk \<Longrightarrow>
\<lparr> owns = (\<lambda>r. None), currk = initk, issued = range initk,
cards = (\<lambda>g. {}), roomk = initk, isin = (\<lambda>r. {}),
safe = (\<lambda>r. True) \<rparr> \<in> reach"
| check_in:
"\<lbrakk> s \<in> reach; k \<notin> issued s \<rbrakk> \<Longrightarrow>
s\<lparr> currk := (currk s)(r := k), issued := issued s \<union> {k},
cards := (cards s)(g := cards s g \<union> {(currk s r, k)}),
owns := (owns s)(r := Some g),
safe := (safe s)(r := False) \<rparr> \<in> reach"
| enter_room:
"\<lbrakk> s \<in> reach; (k,k') \<in> cards s g; roomk s r \<in> {k,k'} \<rbrakk> \<Longrightarrow>
s\<lparr> isin := (isin s)(r := isin s r \<union> {g}),
roomk := (roomk s)(r := k'),
safe := (safe s)(r := owns s r = \<lfloor>g\<rfloor> \<and> isin s r = {} \<and> k' = currk s r
\<or> safe s r)
\<rparr> \<in> reach"
| exit_room:
"\<lbrakk> s \<in> reach; g \<in> isin s r \<rbrakk> \<Longrightarrow>
s\<lparr> isin := (isin s)(r := isin s r - {g}) \<rparr> \<in> reach"
text\<open>\bigskip There is no check-out event because it is implicit in the next
check-in for that room: this covers the cases where a guest leaves without checking out (in which case the room should not be blocked forever) or where
the hotel decides to rent out a room prematurely, probably by accident.
Neither do guests have to return their cards at any point because they may
loose cards or may pretended to have lost them.
We will now explain the events.
\begin{description}
\item[\<open>init\<close>]
Initialization requires that every room has a different key, i.e.\
that @{const currk} is injective. Nobody
owns a room, the keys of all rooms are recorded as issued, nobody has
a card, and all rooms are empty.
\item[@{thm[source] enter_room}]
A guest may enter if either of the two keys on his card equal the room key.
Then \<open>g\<close> is added to the occupants of \<open>r\<close> and
the room key is set to the second key on the card.
Normally this has no effect because the second key is already the room key.
But when entering for the first time, the first key on the card equals
the room key and then the lock is actually recoded.
\item[\<open>exit_room\<close>]
removes an occupant from the occupants of a room.
\item[\<open>check_in\<close>] for room \<open>r\<close> and guest \<open>g\<close>
issues the card @{term"(currk s r, k)"}
to \<open>g\<close>, where \<open>k\<close> is new, makes \<open>g\<close> the owner of the room,
and sets @{term"currk s r"} to the new key \<open>k\<close>.
\end{description}
The reader can easily check that our specification allows the intended
distributed implementation: entering only reads and writes the key in
that lock, and check-in only reads and writes the information at
reception.
In contrast to Jackson we require that initially distinct rooms have
distinct keys. This protects the hotel from its guests: otherwise a
guest may be able to enter rooms he does not own, potentially stealing
objects from those rooms. Of course he can also steal objects from his
own room, but in that case it is easier to hold him responsible. In
general, the hotel may just want to minimize the opportunity for
theft.
The main difference to Jackson's model is that his can talk about
transitions between states rather than merely about reachable
states. This means that he can specify that unauthorized entry into a
room should not occur. Because our specification does not formalize
the transition relation itself, we need to include the \<open>isin\<close>
component in order to
express the same requirement. In the end, we would like to establish
that the system is \emph{safe}: only the owner of a room can be in a
room:
\begin{center}
@{prop"s \<in> reach \<Longrightarrow> g \<in> isin s r \<Longrightarrow> owns s r = Some g"}
\end{center}
Unfortunately, this is just not true. It does not take a PhD in
computer science to come up with the following scenario: because
guests can retain their cards, there is nothing to stop a guest from
reentering his old room after he has checked out (in our model: after
the next guest has checked in), but before the next guest has entered
his room. Hence the best we can do is to prove a conditional safety
property: under certain conditions, the above safety property
holds. The question is: which conditions? It is clear that the room
must be empty when its owner enters it, or all bets
are off. But is that sufficient? Unfortunately not. Jackson's Alloy tool
took 2 seconds~\cite[p.~303]{Jackson06}
to find the following ``guest-in-the-middle'' attack:
\begin{enumerate}
\item Guest 1 checks in and obtains a card $(k_1,k_2)$ for room 1 (whose key
in the lock is $k_1$). Guest 1 does not enter room 1.
\item Guest 2 checks in, obtains a card $(k_2,k_3)$ for room 1, but
does not enter room 1 either.
\item Guest 1 checks in again, obtains a card $(k_3,k_4)$, goes to
room 1, opens it with his old card $(k_1,k_2)$, finds the room empty,
and feels safe \ldots
\end{enumerate}
After Guest~1 has left his room, Guest~2 enters and makes off with the
luggage.
Jackson now assumes that guests return their cards upon
check-out, which can be modelled as follows: upon check-in, the new card
is not added to the guest's set of cards but it replaces his previous
set of cards, i.e.\ guests return old cards the next time they check
in. Under this assumption, Alloy finds no more counterexamples to
safety --- at least not up to 6 cards and keys and 3 guests and
rooms. This is not a proof but a strong indication that the given
assumptions suffice for safety. We prove that this is indeed the case.
It should be noted that the system also suffers from a liveness
problem: if a guest never enters the room he checked in to, that room
is forever blocked. In practice this is dealt with by a master key. We
ignore liveness.
\subsection{Formalizing safety}
\label{sec:formalizing-safety}
It should be clear that one cannot force guests to always return their
cards (or, equivalently, never to use an old card). We can only prove that if
they do, their room is safe. However, we do not follow Jackson's
approach of globally assuming everybody returns their old cards upon
check-in. Instead we would like to take a local approach where it is up
to each guest whether he follows this safety policy. We allow
guests to keep their cards but make safety dependent on how they use
them. This generality requires a finer grained model: we need to
record if a guest has entered his room in a safe manner,
i.e.\ if it was empty and if he used the latest key for the room, the
one stored at reception.
The auxiliary variable @{const safe} records for each room if this
was the case at some point between his last check-in and now.
The main theorem will be that if a room is safe in this
manner, then only the owner can be in the room.
Now we explain how @{const safe} is modified with each event:
\begin{description}
\item[\<open>init\<close>] sets @{const safe} to @{const True} for every room.
\item[\<open>check_in\<close>] for room \<open>r\<close> resets @{prop"safe s r"}
because it is not safe for the new owner yet.
\item[@{thm[source] enter_room}] for room \<open>r\<close> sets @{prop"safe s r"} if
the owner entered an empty room using the latest card issued for that room
by reception, or if the room was already safe.
\item[\<open>exit_room\<close>] does not modify @{const safe}.
\end{description}
The reader should convince his or herself that @{const safe}
corresponds to the informal safety policy set out above. Note that a
guest may find his room non-empty the first time he enters, and
@{const safe} will not be set, but he may come back later, find the
room empty, and then @{const safe} will be set. Furthermore, it is
important that @{thm[source] enter_room} cannot reset @{const safe}
due to the disjunct \<open>\<or> safe s r\<close>. Hence \<open>check_in\<close> is
the only event that can reset @{const safe}. That is, a room stays
safe until the next \<open>check_in\<close>. Additionally @{const safe} is
initially @{const True}, which is fine because initially injectivity
of \<open>initk\<close> prohibits illegal entries by non-owners.
Note that because none of the other state components depend on @{const
safe}, it is truly auxiliary: it can be deleted from the system and
the same set of reachable states is obtained, modulo the absence of
@{const safe}.
We have formalized a very general safety policy of always using the
latest card. A special case of this policy is the one called
\emph{NoIntervening} by Jackson~\cite[p.~200]{Jackson06}: every \<open>check_in\<close> must immediately be followed by the corresponding @{thm[source]
enter_room}.
\<close>
(*<*)
lemma currk_issued[simp]: "s : reach \<Longrightarrow> currk s r : issued s"
by (induct set: reach) auto
lemma key1_issued[simp]: "s : reach \<Longrightarrow> (k,k') : cards s g \<Longrightarrow> k : issued s"
by (induct set: reach) auto
lemma key2_issued[simp]: "s : reach \<Longrightarrow> (k,k') : cards s g \<Longrightarrow> k' : issued s"
by (induct set: reach) auto
lemma roomk_issued[simp]: "s : reach \<Longrightarrow> roomk s k : issued s"
by (induct set: reach) auto
lemma currk_inj[simp]:
"s : reach \<Longrightarrow> \<forall>r r'. (currk s r = currk s r') = (r = r')"
by (induct set: reach) (auto simp:inj_on_def)
lemma key1_not_currk[simp]:
"s : reach \<Longrightarrow> (currk s r,k') \<notin> cards s g"
by (induct set: reach) auto
lemma guest_key2_disj[simp]:
"\<lbrakk> s : reach; (k\<^sub>1,k) \<in> cards s g\<^sub>1; (k\<^sub>2,k) \<in> cards s g\<^sub>2 \<rbrakk> \<Longrightarrow> g\<^sub>1=g\<^sub>2"
by (induct set: reach) auto
lemma safe_roomk_currk[simp]:
"s : reach \<Longrightarrow> safe s r \<Longrightarrow> roomk s r = currk s r"
by (induct set: reach) auto
lemma safe_only_owner_enter_normal_aux[simp]:
"\<lbrakk> s : reach; safe s r; (k',roomk s r) \<in> cards s g \<rbrakk> \<Longrightarrow> owns s r = Some g"
by (induct set: reach) (auto)
lemma safe_only_owner_enter_normal:
assumes "s : reach"
shows "\<lbrakk> safe s r; (k',roomk s r) \<in> cards s g \<rbrakk> \<Longrightarrow> owns s r = Some g"
using assms
proof induct
case (enter_room s k k1 g1 r1)
let ?s' = "s\<lparr>isin := (isin s)(r1 := isin s r1 \<union> {g1}),
roomk := (roomk s)(r1 := k1),
safe := (safe s)
(r1 :=
owns s r1 = Some g1 \<and> isin s r1 = {} \<and> k1 = currk s r1 \<or>
safe s r1)\<rparr>"
note s = \<open>s \<in> reach\<close>
and IH = \<open>\<lbrakk> safe s r; (k', roomk s r) \<in> cards s g \<rbrakk> \<Longrightarrow> owns s r = Some g\<close>
and card_g1 = \<open>(k,k1) \<in> cards s g1\<close> and safe = \<open>safe ?s' r\<close>
and card_g = \<open>(k',roomk ?s' r) \<in> cards ?s' g\<close>
have "roomk s r1 = k \<or> roomk s r1 = k1" using \<open>roomk s r1 \<in> {k,k1}\<close> by simp
thus ?case
proof
assume [symmetric,simp]: "roomk s r1 = k"
show ?thesis
proof (cases "r1 = r")
assume "r1 \<noteq> r" with IH safe card_g show ?thesis by simp
next
assume [simp]: "r1 = r"
hence safe': "owns s r = Some g1 \<or> safe s r"
using safe by auto
thus ?thesis
proof
assume "safe s r"
with s card_g1 have False by simp
thus ?thesis ..
next
assume [simp]: "owns s r = Some g1"
thus "owns ?s' r = Some g"
using s card_g card_g1 by simp
qed
qed
next
assume "roomk s r1 = k1"
with enter_room show ?case by auto
qed
qed auto
theorem "s : reach \<Longrightarrow> safe s r \<Longrightarrow> g : isin s r \<Longrightarrow> owns s r = Some g"
by (induct set: reach) auto
theorem safe: assumes "s : reach"
shows "safe s r \<Longrightarrow> g : isin s r \<Longrightarrow> owns s r = Some g"
using assms
proof induct
case (enter_room s k1 k2 g1 r1)
let ?s' = "s\<lparr>isin := (isin s)(r1 := isin s r1 \<union> {g1}),
roomk := (roomk s)(r1 := k2),
safe := (safe s)
(r1 :=
owns s r1 = Some g1 \<and> isin s r1 = {} \<and> k2 = currk s r1 \<or>
safe s r1)\<rparr>"
note s = \<open>s \<in> reach\<close>
and IH = \<open>\<lbrakk> safe s r; g \<in> isin s r \<rbrakk> \<Longrightarrow> owns s r = Some g\<close>
and card_g1 = \<open>(k1,k2) \<in> cards s g1\<close> and safe = \<open>safe ?s' r\<close>
and isin = \<open>g \<in> isin ?s' r\<close>
show ?case
proof (cases "r1 = r")
assume "r1 \<noteq> r" with IH isin safe show ?thesis by simp
next
assume [simp]: "r1 = r"
have "g \<in> isin s r \<or> g = g1" using isin by auto
thus ?thesis
proof
assume g: "g \<in> isin s r"
then have "safe s r" using safe by auto
with g show ?thesis using IH by simp
next
assume [simp]: "g = g1"
have "k2 = roomk s r1 \<or> k1 = roomk s r1"
using \<open>roomk s r1 \<in> {k1,k2}\<close> by auto
thus ?thesis
proof
assume "k2 = roomk s r1"
with card_g1 s safe show ?thesis
by auto
next
assume [simp]: "k1 = roomk s r1"
have "owns s r = Some g1 \<or> safe s r" using safe by auto
thus ?thesis
proof
assume "owns s r = Some g1" thus ?thesis by simp
next
assume "safe s r"
hence False using s card_g1 by auto
thus ?thesis ..
qed
qed
qed
qed
qed auto
(*>*)
text\<open>
\subsection{Verifying safety}
\label{sec:verisafe}
All of our lemmas are invariants of @{const reach}.
The complete list, culminating in the main theorem, is this:
\begin{lemma}\label{state-lemmas}
\begin{enumerate}
\item @{thm currk_issued}
\item @{thm key1_issued}
\item @{thm key2_issued}
\item @{thm roomk_issued}
\item \label{currk_inj} @{thm currk_inj}
\item \label{key1_not_currk} @{thm key1_not_currk}
\item @{thm guest_key2_disj}
\item \label{safe_roomk_currk} @{thm[display] safe_roomk_currk}
\item \label{safe_only_owner_enter_normal} @{thm safe_only_owner_enter_normal}
\end{enumerate}
\end{lemma}
\begin{theorem}\label{safe-state}
@{thm[mode=IfThen] safe}
\end{theorem}
The lemmas and the theorem are proved in this order, each one is marked as a
simplification rule, and each proof is a one-liner: induction on
@{prop"s \<in> reach"} followed by \<open>auto\<close>.
Although, or maybe even because these proofs work so smoothly one may
like to understand why. Hence we examine the proof of
Theorem~\ref{safe-state} in more detail. The only interesting case is
@{thm[source] enter_room}. We assume that guest \<open>g\<^sub>1\<close> enters room
\<open>r\<^sub>1\<close> with card @{term"(k\<^sub>1,k\<^sub>2)"} and call the new state \<open>t\<close>.
We assume @{prop"safe t r"} and @{prop"g \<in> isin t r"} and prove
@{prop"owns t r = \<lfloor>g\<rfloor>"} by case distinction.
If @{prop"r\<^sub>1 \<noteq> r"}, the claim follows directly from the induction hypothesis
using \mbox{@{prop"safe s r"}} and @{prop"g \<in> isin t r"}
because @{prop"owns t r = owns s r"} and @{prop"safe t r = safe s r"}.
If @{prop"r\<^sub>1 = r"} then @{prop"g \<in> isin t r"} is equivalent with
@{prop"g \<in> isin s r \<or> g = g\<^sub>1"}. If @{prop"g \<in> isin s r"} then
\mbox{@{prop"safe s r"}} follows from @{prop"safe t r"} by
definition of @{thm[source]enter_room} because @{prop"g \<in> isin s r"}
implies @{prop"isin s r \<noteq> {}"}. Hence the induction hypothesis implies the
claim. If @{prop"g = g\<^sub>1"} we make another case distinction.
If @{prop"k\<^sub>2 = roomk s r"}, the claim follows immediately from
Lemma~\ref{state-lemmas}.\ref{safe_only_owner_enter_normal} above:
only the owner of a room can possess a card where the second
key is the room key.
If @{prop"k\<^sub>1 = roomk s r"} then, by definition of @{thm[source]enter_room},
@{prop"safe t r"} implies @{prop"owns s r = \<lfloor>g\<rfloor> \<or> safe s r"}.
In the first case the claim is immediate. If @{prop"safe s r"}
then @{prop"roomk s r = currk s r"}
(by Lemma~\ref{state-lemmas}.\ref{safe_roomk_currk})
and thus @{prop"(currk s r, k\<^sub>2) \<in> cards s g"} by assumption
@{prop"(k\<^sub>1,k\<^sub>2) \<in> cards s g\<^sub>1"}, thus contradicting
Lemma~\ref{state-lemmas}.\ref{key1_not_currk}.
This detailed proof shows that a number of case distinctions are
required. Luckily, they all suggest themselves to Isabelle via the
definition of function update (\<open>:=\<close>) or via disjunctions that
arise automatically.
\<close>
(*<*)
end
(*>*)
|
[STATEMENT]
lemma parCasesOutputFrame[consumes 11, case_names cPar1 cPar2]:
fixes \<Psi> :: 'b
and P :: "('a, 'b, 'c) psi"
and Q :: "('a, 'b, 'c) psi"
and M :: 'a
and xvec :: "name list"
and N :: 'a
and T :: "('a, 'b, 'c) psi"
and C :: "'d::fs_name"
assumes Trans: "\<Psi> \<rhd> P \<parallel> Q \<longmapsto>M\<lparr>\<nu>*xvec\<rparr>\<langle>N\<rangle> \<prec> T"
and "xvec \<sharp>* \<Psi>"
and "xvec \<sharp>* P"
and "xvec \<sharp>* Q"
and "xvec \<sharp>* M"
and "extractFrame(P \<parallel> Q) = \<langle>A\<^sub>P\<^sub>Q, \<Psi>\<^sub>P\<^sub>Q\<rangle>"
and "distinct A\<^sub>P\<^sub>Q"
and "A\<^sub>P\<^sub>Q \<sharp>* \<Psi>"
and "A\<^sub>P\<^sub>Q \<sharp>* P"
and "A\<^sub>P\<^sub>Q \<sharp>* Q"
and "A\<^sub>P\<^sub>Q \<sharp>* M"
and rPar1: "\<And>P' A\<^sub>P \<Psi>\<^sub>P A\<^sub>Q \<Psi>\<^sub>Q. \<lbrakk>\<Psi> \<otimes> \<Psi>\<^sub>Q \<rhd> P \<longmapsto>M\<lparr>\<nu>*xvec\<rparr>\<langle>N\<rangle> \<prec> P'; extractFrame P = \<langle>A\<^sub>P, \<Psi>\<^sub>P\<rangle>; extractFrame Q = \<langle>A\<^sub>Q, \<Psi>\<^sub>Q\<rangle>;
distinct A\<^sub>P; distinct A\<^sub>Q; A\<^sub>P \<sharp>* \<Psi>; A\<^sub>P \<sharp>* P; A\<^sub>P \<sharp>* Q; A\<^sub>P \<sharp>* M; A\<^sub>Q \<sharp>* \<Psi>; A\<^sub>Q \<sharp>* P; A\<^sub>Q \<sharp>* Q; A\<^sub>Q \<sharp>* M;
A\<^sub>P \<sharp>* \<Psi>\<^sub>Q; A\<^sub>Q \<sharp>* \<Psi>\<^sub>P; A\<^sub>P \<sharp>* A\<^sub>Q; A\<^sub>P\<^sub>Q = A\<^sub>P@A\<^sub>Q; \<Psi>\<^sub>P\<^sub>Q = \<Psi>\<^sub>P \<otimes> \<Psi>\<^sub>Q\<rbrakk> \<Longrightarrow> Prop (P' \<parallel> Q)"
and rPar2: "\<And>Q' A\<^sub>P \<Psi>\<^sub>P A\<^sub>Q \<Psi>\<^sub>Q. \<lbrakk>\<Psi> \<otimes> \<Psi>\<^sub>P \<rhd> Q \<longmapsto>M\<lparr>\<nu>*xvec\<rparr>\<langle>N\<rangle> \<prec> Q'; extractFrame P = \<langle>A\<^sub>P, \<Psi>\<^sub>P\<rangle>; extractFrame Q = \<langle>A\<^sub>Q, \<Psi>\<^sub>Q\<rangle>;
distinct A\<^sub>P; distinct A\<^sub>Q; A\<^sub>P \<sharp>* \<Psi>; A\<^sub>P \<sharp>* P; A\<^sub>P \<sharp>* Q; A\<^sub>P \<sharp>* M; A\<^sub>Q \<sharp>* \<Psi>; A\<^sub>Q \<sharp>* P; A\<^sub>Q \<sharp>* Q; A\<^sub>Q \<sharp>* M;
A\<^sub>P \<sharp>* \<Psi>\<^sub>Q; A\<^sub>Q \<sharp>* \<Psi>\<^sub>P; A\<^sub>P \<sharp>* A\<^sub>Q; A\<^sub>P\<^sub>Q = A\<^sub>P@A\<^sub>Q; \<Psi>\<^sub>P\<^sub>Q = \<Psi>\<^sub>P \<otimes> \<Psi>\<^sub>Q\<rbrakk> \<Longrightarrow> Prop (P \<parallel> Q')"
shows "Prop T"
[PROOF STATE]
proof (prove)
goal (1 subgoal):
1. Prop T
[PROOF STEP]
using Trans \<open>xvec \<sharp>* \<Psi>\<close> \<open>xvec \<sharp>* P\<close> \<open>xvec \<sharp>* Q\<close> \<open>xvec \<sharp>* M\<close>
[PROOF STATE]
proof (prove)
using this:
\<Psi> \<rhd> P \<parallel> Q \<longmapsto> M\<lparr>\<nu>*xvec\<rparr>\<langle>N\<rangle> \<prec> T
xvec \<sharp>* \<Psi>
xvec \<sharp>* P
xvec \<sharp>* Q
xvec \<sharp>* M
goal (1 subgoal):
1. Prop T
[PROOF STEP]
proof(induct rule: parOutputCases[of _ _ _ _ _ _ _ "(A\<^sub>P\<^sub>Q, \<Psi>\<^sub>P\<^sub>Q)"])
[PROOF STATE]
proof (state)
goal (2 subgoals):
1. \<And>P' A\<^sub>Q \<Psi>\<^sub>Q. \<lbrakk>\<Psi> \<otimes> \<Psi>\<^sub>Q \<rhd> P \<longmapsto> M\<lparr>\<nu>*xvec\<rparr>\<langle>N\<rangle> \<prec> P'; extractFrame Q = \<langle>A\<^sub>Q, \<Psi>\<^sub>Q\<rangle>; distinct A\<^sub>Q; A\<^sub>Q \<sharp>* \<Psi>; A\<^sub>Q \<sharp>* P; A\<^sub>Q \<sharp>* Q; A\<^sub>Q \<sharp>* M; A\<^sub>Q \<sharp>* xvec; A\<^sub>Q \<sharp>* N; A\<^sub>Q \<sharp>* (A\<^sub>P\<^sub>Q, \<Psi>\<^sub>P\<^sub>Q); A\<^sub>Q \<sharp>* xvec; distinct xvec\<rbrakk> \<Longrightarrow> Prop (P' \<parallel> Q)
2. \<And>Q' A\<^sub>P \<Psi>\<^sub>P. \<lbrakk>\<Psi> \<otimes> \<Psi>\<^sub>P \<rhd> Q \<longmapsto> M\<lparr>\<nu>*xvec\<rparr>\<langle>N\<rangle> \<prec> Q'; extractFrame P = \<langle>A\<^sub>P, \<Psi>\<^sub>P\<rangle>; distinct A\<^sub>P; A\<^sub>P \<sharp>* \<Psi>; A\<^sub>P \<sharp>* P; A\<^sub>P \<sharp>* Q; A\<^sub>P \<sharp>* M; A\<^sub>P \<sharp>* xvec; A\<^sub>P \<sharp>* N; A\<^sub>P \<sharp>* (A\<^sub>P\<^sub>Q, \<Psi>\<^sub>P\<^sub>Q); A\<^sub>P \<sharp>* xvec; distinct xvec\<rbrakk> \<Longrightarrow> Prop (P \<parallel> Q')
[PROOF STEP]
case(cPar1 P' A\<^sub>Q \<Psi>\<^sub>Q)
[PROOF STATE]
proof (state)
this:
\<Psi> \<otimes> \<Psi>\<^sub>Q \<rhd> P \<longmapsto> M\<lparr>\<nu>*xvec\<rparr>\<langle>N\<rangle> \<prec> P'
extractFrame Q = \<langle>A\<^sub>Q, \<Psi>\<^sub>Q\<rangle>
distinct A\<^sub>Q
A\<^sub>Q \<sharp>* \<Psi>
A\<^sub>Q \<sharp>* P
A\<^sub>Q \<sharp>* Q
A\<^sub>Q \<sharp>* M
A\<^sub>Q \<sharp>* xvec
A\<^sub>Q \<sharp>* N
A\<^sub>Q \<sharp>* (A\<^sub>P\<^sub>Q, \<Psi>\<^sub>P\<^sub>Q)
A\<^sub>Q \<sharp>* xvec
distinct xvec
goal (2 subgoals):
1. \<And>P' A\<^sub>Q \<Psi>\<^sub>Q. \<lbrakk>\<Psi> \<otimes> \<Psi>\<^sub>Q \<rhd> P \<longmapsto> M\<lparr>\<nu>*xvec\<rparr>\<langle>N\<rangle> \<prec> P'; extractFrame Q = \<langle>A\<^sub>Q, \<Psi>\<^sub>Q\<rangle>; distinct A\<^sub>Q; A\<^sub>Q \<sharp>* \<Psi>; A\<^sub>Q \<sharp>* P; A\<^sub>Q \<sharp>* Q; A\<^sub>Q \<sharp>* M; A\<^sub>Q \<sharp>* xvec; A\<^sub>Q \<sharp>* N; A\<^sub>Q \<sharp>* (A\<^sub>P\<^sub>Q, \<Psi>\<^sub>P\<^sub>Q); A\<^sub>Q \<sharp>* xvec; distinct xvec\<rbrakk> \<Longrightarrow> Prop (P' \<parallel> Q)
2. \<And>Q' A\<^sub>P \<Psi>\<^sub>P. \<lbrakk>\<Psi> \<otimes> \<Psi>\<^sub>P \<rhd> Q \<longmapsto> M\<lparr>\<nu>*xvec\<rparr>\<langle>N\<rangle> \<prec> Q'; extractFrame P = \<langle>A\<^sub>P, \<Psi>\<^sub>P\<rangle>; distinct A\<^sub>P; A\<^sub>P \<sharp>* \<Psi>; A\<^sub>P \<sharp>* P; A\<^sub>P \<sharp>* Q; A\<^sub>P \<sharp>* M; A\<^sub>P \<sharp>* xvec; A\<^sub>P \<sharp>* N; A\<^sub>P \<sharp>* (A\<^sub>P\<^sub>Q, \<Psi>\<^sub>P\<^sub>Q); A\<^sub>P \<sharp>* xvec; distinct xvec\<rbrakk> \<Longrightarrow> Prop (P \<parallel> Q')
[PROOF STEP]
from \<open>A\<^sub>Q \<sharp>* (A\<^sub>P\<^sub>Q, \<Psi>\<^sub>P\<^sub>Q)\<close>
[PROOF STATE]
proof (chain)
picking this:
A\<^sub>Q \<sharp>* (A\<^sub>P\<^sub>Q, \<Psi>\<^sub>P\<^sub>Q)
[PROOF STEP]
have "A\<^sub>Q \<sharp>* A\<^sub>P\<^sub>Q" and "A\<^sub>Q \<sharp>* \<Psi>\<^sub>P\<^sub>Q"
[PROOF STATE]
proof (prove)
using this:
A\<^sub>Q \<sharp>* (A\<^sub>P\<^sub>Q, \<Psi>\<^sub>P\<^sub>Q)
goal (1 subgoal):
1. A\<^sub>Q \<sharp>* A\<^sub>P\<^sub>Q &&& A\<^sub>Q \<sharp>* \<Psi>\<^sub>P\<^sub>Q
[PROOF STEP]
by simp+
[PROOF STATE]
proof (state)
this:
A\<^sub>Q \<sharp>* A\<^sub>P\<^sub>Q
A\<^sub>Q \<sharp>* \<Psi>\<^sub>P\<^sub>Q
goal (2 subgoals):
1. \<And>P' A\<^sub>Q \<Psi>\<^sub>Q. \<lbrakk>\<Psi> \<otimes> \<Psi>\<^sub>Q \<rhd> P \<longmapsto> M\<lparr>\<nu>*xvec\<rparr>\<langle>N\<rangle> \<prec> P'; extractFrame Q = \<langle>A\<^sub>Q, \<Psi>\<^sub>Q\<rangle>; distinct A\<^sub>Q; A\<^sub>Q \<sharp>* \<Psi>; A\<^sub>Q \<sharp>* P; A\<^sub>Q \<sharp>* Q; A\<^sub>Q \<sharp>* M; A\<^sub>Q \<sharp>* xvec; A\<^sub>Q \<sharp>* N; A\<^sub>Q \<sharp>* (A\<^sub>P\<^sub>Q, \<Psi>\<^sub>P\<^sub>Q); A\<^sub>Q \<sharp>* xvec; distinct xvec\<rbrakk> \<Longrightarrow> Prop (P' \<parallel> Q)
2. \<And>Q' A\<^sub>P \<Psi>\<^sub>P. \<lbrakk>\<Psi> \<otimes> \<Psi>\<^sub>P \<rhd> Q \<longmapsto> M\<lparr>\<nu>*xvec\<rparr>\<langle>N\<rangle> \<prec> Q'; extractFrame P = \<langle>A\<^sub>P, \<Psi>\<^sub>P\<rangle>; distinct A\<^sub>P; A\<^sub>P \<sharp>* \<Psi>; A\<^sub>P \<sharp>* P; A\<^sub>P \<sharp>* Q; A\<^sub>P \<sharp>* M; A\<^sub>P \<sharp>* xvec; A\<^sub>P \<sharp>* N; A\<^sub>P \<sharp>* (A\<^sub>P\<^sub>Q, \<Psi>\<^sub>P\<^sub>Q); A\<^sub>P \<sharp>* xvec; distinct xvec\<rbrakk> \<Longrightarrow> Prop (P \<parallel> Q')
[PROOF STEP]
obtain A\<^sub>P \<Psi>\<^sub>P where FrP: "extractFrame P = \<langle>A\<^sub>P, \<Psi>\<^sub>P\<rangle>" and "distinct A\<^sub>P"
"A\<^sub>P \<sharp>* (P, Q, \<Psi>, M, A\<^sub>Q, A\<^sub>P\<^sub>Q, \<Psi>\<^sub>Q)"
[PROOF STATE]
proof (prove)
goal (1 subgoal):
1. (\<And>A\<^sub>P \<Psi>\<^sub>P. \<lbrakk>extractFrame P = \<langle>A\<^sub>P, \<Psi>\<^sub>P\<rangle>; distinct A\<^sub>P; A\<^sub>P \<sharp>* (P, Q, \<Psi>, M, A\<^sub>Q, A\<^sub>P\<^sub>Q, \<Psi>\<^sub>Q)\<rbrakk> \<Longrightarrow> thesis) \<Longrightarrow> thesis
[PROOF STEP]
by(rule freshFrame)
[PROOF STATE]
proof (state)
this:
extractFrame P = \<langle>A\<^sub>P, \<Psi>\<^sub>P\<rangle>
distinct A\<^sub>P
A\<^sub>P \<sharp>* (P, Q, \<Psi>, M, A\<^sub>Q, A\<^sub>P\<^sub>Q, \<Psi>\<^sub>Q)
goal (2 subgoals):
1. \<And>P' A\<^sub>Q \<Psi>\<^sub>Q. \<lbrakk>\<Psi> \<otimes> \<Psi>\<^sub>Q \<rhd> P \<longmapsto> M\<lparr>\<nu>*xvec\<rparr>\<langle>N\<rangle> \<prec> P'; extractFrame Q = \<langle>A\<^sub>Q, \<Psi>\<^sub>Q\<rangle>; distinct A\<^sub>Q; A\<^sub>Q \<sharp>* \<Psi>; A\<^sub>Q \<sharp>* P; A\<^sub>Q \<sharp>* Q; A\<^sub>Q \<sharp>* M; A\<^sub>Q \<sharp>* xvec; A\<^sub>Q \<sharp>* N; A\<^sub>Q \<sharp>* (A\<^sub>P\<^sub>Q, \<Psi>\<^sub>P\<^sub>Q); A\<^sub>Q \<sharp>* xvec; distinct xvec\<rbrakk> \<Longrightarrow> Prop (P' \<parallel> Q)
2. \<And>Q' A\<^sub>P \<Psi>\<^sub>P. \<lbrakk>\<Psi> \<otimes> \<Psi>\<^sub>P \<rhd> Q \<longmapsto> M\<lparr>\<nu>*xvec\<rparr>\<langle>N\<rangle> \<prec> Q'; extractFrame P = \<langle>A\<^sub>P, \<Psi>\<^sub>P\<rangle>; distinct A\<^sub>P; A\<^sub>P \<sharp>* \<Psi>; A\<^sub>P \<sharp>* P; A\<^sub>P \<sharp>* Q; A\<^sub>P \<sharp>* M; A\<^sub>P \<sharp>* xvec; A\<^sub>P \<sharp>* N; A\<^sub>P \<sharp>* (A\<^sub>P\<^sub>Q, \<Psi>\<^sub>P\<^sub>Q); A\<^sub>P \<sharp>* xvec; distinct xvec\<rbrakk> \<Longrightarrow> Prop (P \<parallel> Q')
[PROOF STEP]
hence "A\<^sub>P \<sharp>* P" and "A\<^sub>P \<sharp>* Q" and "A\<^sub>P \<sharp>* \<Psi>" and "A\<^sub>P \<sharp>* M" and "A\<^sub>P \<sharp>* A\<^sub>Q" and "A\<^sub>P \<sharp>* A\<^sub>P\<^sub>Q" and "A\<^sub>P \<sharp>* \<Psi>\<^sub>Q"
[PROOF STATE]
proof (prove)
using this:
extractFrame P = \<langle>A\<^sub>P, \<Psi>\<^sub>P\<rangle>
distinct A\<^sub>P
A\<^sub>P \<sharp>* (P, Q, \<Psi>, M, A\<^sub>Q, A\<^sub>P\<^sub>Q, \<Psi>\<^sub>Q)
goal (1 subgoal):
1. (A\<^sub>P \<sharp>* P &&& A\<^sub>P \<sharp>* Q &&& A\<^sub>P \<sharp>* \<Psi>) &&& (A\<^sub>P \<sharp>* M &&& A\<^sub>P \<sharp>* A\<^sub>Q) &&& A\<^sub>P \<sharp>* A\<^sub>P\<^sub>Q &&& A\<^sub>P \<sharp>* \<Psi>\<^sub>Q
[PROOF STEP]
by simp+
[PROOF STATE]
proof (state)
this:
A\<^sub>P \<sharp>* P
A\<^sub>P \<sharp>* Q
A\<^sub>P \<sharp>* \<Psi>
A\<^sub>P \<sharp>* M
A\<^sub>P \<sharp>* A\<^sub>Q
A\<^sub>P \<sharp>* A\<^sub>P\<^sub>Q
A\<^sub>P \<sharp>* \<Psi>\<^sub>Q
goal (2 subgoals):
1. \<And>P' A\<^sub>Q \<Psi>\<^sub>Q. \<lbrakk>\<Psi> \<otimes> \<Psi>\<^sub>Q \<rhd> P \<longmapsto> M\<lparr>\<nu>*xvec\<rparr>\<langle>N\<rangle> \<prec> P'; extractFrame Q = \<langle>A\<^sub>Q, \<Psi>\<^sub>Q\<rangle>; distinct A\<^sub>Q; A\<^sub>Q \<sharp>* \<Psi>; A\<^sub>Q \<sharp>* P; A\<^sub>Q \<sharp>* Q; A\<^sub>Q \<sharp>* M; A\<^sub>Q \<sharp>* xvec; A\<^sub>Q \<sharp>* N; A\<^sub>Q \<sharp>* (A\<^sub>P\<^sub>Q, \<Psi>\<^sub>P\<^sub>Q); A\<^sub>Q \<sharp>* xvec; distinct xvec\<rbrakk> \<Longrightarrow> Prop (P' \<parallel> Q)
2. \<And>Q' A\<^sub>P \<Psi>\<^sub>P. \<lbrakk>\<Psi> \<otimes> \<Psi>\<^sub>P \<rhd> Q \<longmapsto> M\<lparr>\<nu>*xvec\<rparr>\<langle>N\<rangle> \<prec> Q'; extractFrame P = \<langle>A\<^sub>P, \<Psi>\<^sub>P\<rangle>; distinct A\<^sub>P; A\<^sub>P \<sharp>* \<Psi>; A\<^sub>P \<sharp>* P; A\<^sub>P \<sharp>* Q; A\<^sub>P \<sharp>* M; A\<^sub>P \<sharp>* xvec; A\<^sub>P \<sharp>* N; A\<^sub>P \<sharp>* (A\<^sub>P\<^sub>Q, \<Psi>\<^sub>P\<^sub>Q); A\<^sub>P \<sharp>* xvec; distinct xvec\<rbrakk> \<Longrightarrow> Prop (P \<parallel> Q')
[PROOF STEP]
have FrQ: "extractFrame Q = \<langle>A\<^sub>Q, \<Psi>\<^sub>Q\<rangle>"
[PROOF STATE]
proof (prove)
goal (1 subgoal):
1. extractFrame Q = \<langle>A\<^sub>Q, \<Psi>\<^sub>Q\<rangle>
[PROOF STEP]
by fact
[PROOF STATE]
proof (state)
this:
extractFrame Q = \<langle>A\<^sub>Q, \<Psi>\<^sub>Q\<rangle>
goal (2 subgoals):
1. \<And>P' A\<^sub>Q \<Psi>\<^sub>Q. \<lbrakk>\<Psi> \<otimes> \<Psi>\<^sub>Q \<rhd> P \<longmapsto> M\<lparr>\<nu>*xvec\<rparr>\<langle>N\<rangle> \<prec> P'; extractFrame Q = \<langle>A\<^sub>Q, \<Psi>\<^sub>Q\<rangle>; distinct A\<^sub>Q; A\<^sub>Q \<sharp>* \<Psi>; A\<^sub>Q \<sharp>* P; A\<^sub>Q \<sharp>* Q; A\<^sub>Q \<sharp>* M; A\<^sub>Q \<sharp>* xvec; A\<^sub>Q \<sharp>* N; A\<^sub>Q \<sharp>* (A\<^sub>P\<^sub>Q, \<Psi>\<^sub>P\<^sub>Q); A\<^sub>Q \<sharp>* xvec; distinct xvec\<rbrakk> \<Longrightarrow> Prop (P' \<parallel> Q)
2. \<And>Q' A\<^sub>P \<Psi>\<^sub>P. \<lbrakk>\<Psi> \<otimes> \<Psi>\<^sub>P \<rhd> Q \<longmapsto> M\<lparr>\<nu>*xvec\<rparr>\<langle>N\<rangle> \<prec> Q'; extractFrame P = \<langle>A\<^sub>P, \<Psi>\<^sub>P\<rangle>; distinct A\<^sub>P; A\<^sub>P \<sharp>* \<Psi>; A\<^sub>P \<sharp>* P; A\<^sub>P \<sharp>* Q; A\<^sub>P \<sharp>* M; A\<^sub>P \<sharp>* xvec; A\<^sub>P \<sharp>* N; A\<^sub>P \<sharp>* (A\<^sub>P\<^sub>Q, \<Psi>\<^sub>P\<^sub>Q); A\<^sub>P \<sharp>* xvec; distinct xvec\<rbrakk> \<Longrightarrow> Prop (P \<parallel> Q')
[PROOF STEP]
from \<open>A\<^sub>Q \<sharp>* P\<close> \<open>A\<^sub>P \<sharp>* A\<^sub>Q\<close> FrP
[PROOF STATE]
proof (chain)
picking this:
A\<^sub>Q \<sharp>* P
A\<^sub>P \<sharp>* A\<^sub>Q
extractFrame P = \<langle>A\<^sub>P, \<Psi>\<^sub>P\<rangle>
[PROOF STEP]
have "A\<^sub>Q \<sharp>* \<Psi>\<^sub>P"
[PROOF STATE]
proof (prove)
using this:
A\<^sub>Q \<sharp>* P
A\<^sub>P \<sharp>* A\<^sub>Q
extractFrame P = \<langle>A\<^sub>P, \<Psi>\<^sub>P\<rangle>
goal (1 subgoal):
1. A\<^sub>Q \<sharp>* \<Psi>\<^sub>P
[PROOF STEP]
by(force dest: extractFrameFreshChain)
[PROOF STATE]
proof (state)
this:
A\<^sub>Q \<sharp>* \<Psi>\<^sub>P
goal (2 subgoals):
1. \<And>P' A\<^sub>Q \<Psi>\<^sub>Q. \<lbrakk>\<Psi> \<otimes> \<Psi>\<^sub>Q \<rhd> P \<longmapsto> M\<lparr>\<nu>*xvec\<rparr>\<langle>N\<rangle> \<prec> P'; extractFrame Q = \<langle>A\<^sub>Q, \<Psi>\<^sub>Q\<rangle>; distinct A\<^sub>Q; A\<^sub>Q \<sharp>* \<Psi>; A\<^sub>Q \<sharp>* P; A\<^sub>Q \<sharp>* Q; A\<^sub>Q \<sharp>* M; A\<^sub>Q \<sharp>* xvec; A\<^sub>Q \<sharp>* N; A\<^sub>Q \<sharp>* (A\<^sub>P\<^sub>Q, \<Psi>\<^sub>P\<^sub>Q); A\<^sub>Q \<sharp>* xvec; distinct xvec\<rbrakk> \<Longrightarrow> Prop (P' \<parallel> Q)
2. \<And>Q' A\<^sub>P \<Psi>\<^sub>P. \<lbrakk>\<Psi> \<otimes> \<Psi>\<^sub>P \<rhd> Q \<longmapsto> M\<lparr>\<nu>*xvec\<rparr>\<langle>N\<rangle> \<prec> Q'; extractFrame P = \<langle>A\<^sub>P, \<Psi>\<^sub>P\<rangle>; distinct A\<^sub>P; A\<^sub>P \<sharp>* \<Psi>; A\<^sub>P \<sharp>* P; A\<^sub>P \<sharp>* Q; A\<^sub>P \<sharp>* M; A\<^sub>P \<sharp>* xvec; A\<^sub>P \<sharp>* N; A\<^sub>P \<sharp>* (A\<^sub>P\<^sub>Q, \<Psi>\<^sub>P\<^sub>Q); A\<^sub>P \<sharp>* xvec; distinct xvec\<rbrakk> \<Longrightarrow> Prop (P \<parallel> Q')
[PROOF STEP]
from \<open>extractFrame(P \<parallel> Q) = \<langle>A\<^sub>P\<^sub>Q, \<Psi>\<^sub>P\<^sub>Q\<rangle>\<close> FrP FrQ \<open>A\<^sub>P \<sharp>* A\<^sub>Q\<close> \<open>A\<^sub>P \<sharp>* \<Psi>\<^sub>Q\<close> \<open>A\<^sub>Q \<sharp>* \<Psi>\<^sub>P\<close>
[PROOF STATE]
proof (chain)
picking this:
extractFrame (P \<parallel> Q) = \<langle>A\<^sub>P\<^sub>Q, \<Psi>\<^sub>P\<^sub>Q\<rangle>
extractFrame P = \<langle>A\<^sub>P, \<Psi>\<^sub>P\<rangle>
extractFrame Q = \<langle>A\<^sub>Q, \<Psi>\<^sub>Q\<rangle>
A\<^sub>P \<sharp>* A\<^sub>Q
A\<^sub>P \<sharp>* \<Psi>\<^sub>Q
A\<^sub>Q \<sharp>* \<Psi>\<^sub>P
[PROOF STEP]
have "\<langle>(A\<^sub>P@A\<^sub>Q), \<Psi>\<^sub>P \<otimes> \<Psi>\<^sub>Q\<rangle> = \<langle>A\<^sub>P\<^sub>Q, \<Psi>\<^sub>P\<^sub>Q\<rangle>"
[PROOF STATE]
proof (prove)
using this:
extractFrame (P \<parallel> Q) = \<langle>A\<^sub>P\<^sub>Q, \<Psi>\<^sub>P\<^sub>Q\<rangle>
extractFrame P = \<langle>A\<^sub>P, \<Psi>\<^sub>P\<rangle>
extractFrame Q = \<langle>A\<^sub>Q, \<Psi>\<^sub>Q\<rangle>
A\<^sub>P \<sharp>* A\<^sub>Q
A\<^sub>P \<sharp>* \<Psi>\<^sub>Q
A\<^sub>Q \<sharp>* \<Psi>\<^sub>P
goal (1 subgoal):
1. \<langle>(A\<^sub>P @ A\<^sub>Q), \<Psi>\<^sub>P \<otimes> \<Psi>\<^sub>Q\<rangle> = \<langle>A\<^sub>P\<^sub>Q, \<Psi>\<^sub>P\<^sub>Q\<rangle>
[PROOF STEP]
by simp
[PROOF STATE]
proof (state)
this:
\<langle>(A\<^sub>P @ A\<^sub>Q), \<Psi>\<^sub>P \<otimes> \<Psi>\<^sub>Q\<rangle> = \<langle>A\<^sub>P\<^sub>Q, \<Psi>\<^sub>P\<^sub>Q\<rangle>
goal (2 subgoals):
1. \<And>P' A\<^sub>Q \<Psi>\<^sub>Q. \<lbrakk>\<Psi> \<otimes> \<Psi>\<^sub>Q \<rhd> P \<longmapsto> M\<lparr>\<nu>*xvec\<rparr>\<langle>N\<rangle> \<prec> P'; extractFrame Q = \<langle>A\<^sub>Q, \<Psi>\<^sub>Q\<rangle>; distinct A\<^sub>Q; A\<^sub>Q \<sharp>* \<Psi>; A\<^sub>Q \<sharp>* P; A\<^sub>Q \<sharp>* Q; A\<^sub>Q \<sharp>* M; A\<^sub>Q \<sharp>* xvec; A\<^sub>Q \<sharp>* N; A\<^sub>Q \<sharp>* (A\<^sub>P\<^sub>Q, \<Psi>\<^sub>P\<^sub>Q); A\<^sub>Q \<sharp>* xvec; distinct xvec\<rbrakk> \<Longrightarrow> Prop (P' \<parallel> Q)
2. \<And>Q' A\<^sub>P \<Psi>\<^sub>P. \<lbrakk>\<Psi> \<otimes> \<Psi>\<^sub>P \<rhd> Q \<longmapsto> M\<lparr>\<nu>*xvec\<rparr>\<langle>N\<rangle> \<prec> Q'; extractFrame P = \<langle>A\<^sub>P, \<Psi>\<^sub>P\<rangle>; distinct A\<^sub>P; A\<^sub>P \<sharp>* \<Psi>; A\<^sub>P \<sharp>* P; A\<^sub>P \<sharp>* Q; A\<^sub>P \<sharp>* M; A\<^sub>P \<sharp>* xvec; A\<^sub>P \<sharp>* N; A\<^sub>P \<sharp>* (A\<^sub>P\<^sub>Q, \<Psi>\<^sub>P\<^sub>Q); A\<^sub>P \<sharp>* xvec; distinct xvec\<rbrakk> \<Longrightarrow> Prop (P \<parallel> Q')
[PROOF STEP]
moreover
[PROOF STATE]
proof (state)
this:
\<langle>(A\<^sub>P @ A\<^sub>Q), \<Psi>\<^sub>P \<otimes> \<Psi>\<^sub>Q\<rangle> = \<langle>A\<^sub>P\<^sub>Q, \<Psi>\<^sub>P\<^sub>Q\<rangle>
goal (2 subgoals):
1. \<And>P' A\<^sub>Q \<Psi>\<^sub>Q. \<lbrakk>\<Psi> \<otimes> \<Psi>\<^sub>Q \<rhd> P \<longmapsto> M\<lparr>\<nu>*xvec\<rparr>\<langle>N\<rangle> \<prec> P'; extractFrame Q = \<langle>A\<^sub>Q, \<Psi>\<^sub>Q\<rangle>; distinct A\<^sub>Q; A\<^sub>Q \<sharp>* \<Psi>; A\<^sub>Q \<sharp>* P; A\<^sub>Q \<sharp>* Q; A\<^sub>Q \<sharp>* M; A\<^sub>Q \<sharp>* xvec; A\<^sub>Q \<sharp>* N; A\<^sub>Q \<sharp>* (A\<^sub>P\<^sub>Q, \<Psi>\<^sub>P\<^sub>Q); A\<^sub>Q \<sharp>* xvec; distinct xvec\<rbrakk> \<Longrightarrow> Prop (P' \<parallel> Q)
2. \<And>Q' A\<^sub>P \<Psi>\<^sub>P. \<lbrakk>\<Psi> \<otimes> \<Psi>\<^sub>P \<rhd> Q \<longmapsto> M\<lparr>\<nu>*xvec\<rparr>\<langle>N\<rangle> \<prec> Q'; extractFrame P = \<langle>A\<^sub>P, \<Psi>\<^sub>P\<rangle>; distinct A\<^sub>P; A\<^sub>P \<sharp>* \<Psi>; A\<^sub>P \<sharp>* P; A\<^sub>P \<sharp>* Q; A\<^sub>P \<sharp>* M; A\<^sub>P \<sharp>* xvec; A\<^sub>P \<sharp>* N; A\<^sub>P \<sharp>* (A\<^sub>P\<^sub>Q, \<Psi>\<^sub>P\<^sub>Q); A\<^sub>P \<sharp>* xvec; distinct xvec\<rbrakk> \<Longrightarrow> Prop (P \<parallel> Q')
[PROOF STEP]
from \<open>distinct A\<^sub>P\<close> \<open>distinct A\<^sub>Q\<close> \<open>A\<^sub>P \<sharp>* A\<^sub>Q\<close>
[PROOF STATE]
proof (chain)
picking this:
distinct A\<^sub>P
distinct A\<^sub>Q
A\<^sub>P \<sharp>* A\<^sub>Q
[PROOF STEP]
have "distinct(A\<^sub>P@A\<^sub>Q)"
[PROOF STATE]
proof (prove)
using this:
distinct A\<^sub>P
distinct A\<^sub>Q
A\<^sub>P \<sharp>* A\<^sub>Q
goal (1 subgoal):
1. distinct (A\<^sub>P @ A\<^sub>Q)
[PROOF STEP]
by(auto simp add: fresh_star_def fresh_def name_list_supp)
[PROOF STATE]
proof (state)
this:
distinct (A\<^sub>P @ A\<^sub>Q)
goal (2 subgoals):
1. \<And>P' A\<^sub>Q \<Psi>\<^sub>Q. \<lbrakk>\<Psi> \<otimes> \<Psi>\<^sub>Q \<rhd> P \<longmapsto> M\<lparr>\<nu>*xvec\<rparr>\<langle>N\<rangle> \<prec> P'; extractFrame Q = \<langle>A\<^sub>Q, \<Psi>\<^sub>Q\<rangle>; distinct A\<^sub>Q; A\<^sub>Q \<sharp>* \<Psi>; A\<^sub>Q \<sharp>* P; A\<^sub>Q \<sharp>* Q; A\<^sub>Q \<sharp>* M; A\<^sub>Q \<sharp>* xvec; A\<^sub>Q \<sharp>* N; A\<^sub>Q \<sharp>* (A\<^sub>P\<^sub>Q, \<Psi>\<^sub>P\<^sub>Q); A\<^sub>Q \<sharp>* xvec; distinct xvec\<rbrakk> \<Longrightarrow> Prop (P' \<parallel> Q)
2. \<And>Q' A\<^sub>P \<Psi>\<^sub>P. \<lbrakk>\<Psi> \<otimes> \<Psi>\<^sub>P \<rhd> Q \<longmapsto> M\<lparr>\<nu>*xvec\<rparr>\<langle>N\<rangle> \<prec> Q'; extractFrame P = \<langle>A\<^sub>P, \<Psi>\<^sub>P\<rangle>; distinct A\<^sub>P; A\<^sub>P \<sharp>* \<Psi>; A\<^sub>P \<sharp>* P; A\<^sub>P \<sharp>* Q; A\<^sub>P \<sharp>* M; A\<^sub>P \<sharp>* xvec; A\<^sub>P \<sharp>* N; A\<^sub>P \<sharp>* (A\<^sub>P\<^sub>Q, \<Psi>\<^sub>P\<^sub>Q); A\<^sub>P \<sharp>* xvec; distinct xvec\<rbrakk> \<Longrightarrow> Prop (P \<parallel> Q')
[PROOF STEP]
ultimately
[PROOF STATE]
proof (chain)
picking this:
\<langle>(A\<^sub>P @ A\<^sub>Q), \<Psi>\<^sub>P \<otimes> \<Psi>\<^sub>Q\<rangle> = \<langle>A\<^sub>P\<^sub>Q, \<Psi>\<^sub>P\<^sub>Q\<rangle>
distinct (A\<^sub>P @ A\<^sub>Q)
[PROOF STEP]
obtain p where S: "set p \<subseteq> set(A\<^sub>P@A\<^sub>Q) \<times> set((p \<bullet> A\<^sub>P)@(p \<bullet> A\<^sub>Q))" and "distinctPerm p"
and \<Psi>eq: "\<Psi>\<^sub>P\<^sub>Q = (p \<bullet> \<Psi>\<^sub>P) \<otimes> (p \<bullet> \<Psi>\<^sub>Q)" and Aeq: "A\<^sub>P\<^sub>Q = (p \<bullet> A\<^sub>P)@(p \<bullet> A\<^sub>Q)"
[PROOF STATE]
proof (prove)
using this:
\<langle>(A\<^sub>P @ A\<^sub>Q), \<Psi>\<^sub>P \<otimes> \<Psi>\<^sub>Q\<rangle> = \<langle>A\<^sub>P\<^sub>Q, \<Psi>\<^sub>P\<^sub>Q\<rangle>
distinct (A\<^sub>P @ A\<^sub>Q)
goal (1 subgoal):
1. (\<And>p. \<lbrakk>set p \<subseteq> set (A\<^sub>P @ A\<^sub>Q) \<times> set (p \<bullet> A\<^sub>P @ p \<bullet> A\<^sub>Q); distinctPerm p; \<Psi>\<^sub>P\<^sub>Q = (p \<bullet> \<Psi>\<^sub>P) \<otimes> (p \<bullet> \<Psi>\<^sub>Q); A\<^sub>P\<^sub>Q = p \<bullet> A\<^sub>P @ p \<bullet> A\<^sub>Q\<rbrakk> \<Longrightarrow> thesis) \<Longrightarrow> thesis
[PROOF STEP]
using \<open>A\<^sub>P \<sharp>* A\<^sub>P\<^sub>Q\<close> \<open>A\<^sub>Q \<sharp>* A\<^sub>P\<^sub>Q\<close> \<open>distinct A\<^sub>P\<^sub>Q\<close>
[PROOF STATE]
proof (prove)
using this:
\<langle>(A\<^sub>P @ A\<^sub>Q), \<Psi>\<^sub>P \<otimes> \<Psi>\<^sub>Q\<rangle> = \<langle>A\<^sub>P\<^sub>Q, \<Psi>\<^sub>P\<^sub>Q\<rangle>
distinct (A\<^sub>P @ A\<^sub>Q)
A\<^sub>P \<sharp>* A\<^sub>P\<^sub>Q
A\<^sub>Q \<sharp>* A\<^sub>P\<^sub>Q
distinct A\<^sub>P\<^sub>Q
goal (1 subgoal):
1. (\<And>p. \<lbrakk>set p \<subseteq> set (A\<^sub>P @ A\<^sub>Q) \<times> set (p \<bullet> A\<^sub>P @ p \<bullet> A\<^sub>Q); distinctPerm p; \<Psi>\<^sub>P\<^sub>Q = (p \<bullet> \<Psi>\<^sub>P) \<otimes> (p \<bullet> \<Psi>\<^sub>Q); A\<^sub>P\<^sub>Q = p \<bullet> A\<^sub>P @ p \<bullet> A\<^sub>Q\<rbrakk> \<Longrightarrow> thesis) \<Longrightarrow> thesis
[PROOF STEP]
by(rule_tac frameChainEq') (assumption | simp add: eqvts)+
[PROOF STATE]
proof (state)
this:
set p \<subseteq> set (A\<^sub>P @ A\<^sub>Q) \<times> set (p \<bullet> A\<^sub>P @ p \<bullet> A\<^sub>Q)
distinctPerm p
\<Psi>\<^sub>P\<^sub>Q = (p \<bullet> \<Psi>\<^sub>P) \<otimes> (p \<bullet> \<Psi>\<^sub>Q)
A\<^sub>P\<^sub>Q = p \<bullet> A\<^sub>P @ p \<bullet> A\<^sub>Q
goal (2 subgoals):
1. \<And>P' A\<^sub>Q \<Psi>\<^sub>Q. \<lbrakk>\<Psi> \<otimes> \<Psi>\<^sub>Q \<rhd> P \<longmapsto> M\<lparr>\<nu>*xvec\<rparr>\<langle>N\<rangle> \<prec> P'; extractFrame Q = \<langle>A\<^sub>Q, \<Psi>\<^sub>Q\<rangle>; distinct A\<^sub>Q; A\<^sub>Q \<sharp>* \<Psi>; A\<^sub>Q \<sharp>* P; A\<^sub>Q \<sharp>* Q; A\<^sub>Q \<sharp>* M; A\<^sub>Q \<sharp>* xvec; A\<^sub>Q \<sharp>* N; A\<^sub>Q \<sharp>* (A\<^sub>P\<^sub>Q, \<Psi>\<^sub>P\<^sub>Q); A\<^sub>Q \<sharp>* xvec; distinct xvec\<rbrakk> \<Longrightarrow> Prop (P' \<parallel> Q)
2. \<And>Q' A\<^sub>P \<Psi>\<^sub>P. \<lbrakk>\<Psi> \<otimes> \<Psi>\<^sub>P \<rhd> Q \<longmapsto> M\<lparr>\<nu>*xvec\<rparr>\<langle>N\<rangle> \<prec> Q'; extractFrame P = \<langle>A\<^sub>P, \<Psi>\<^sub>P\<rangle>; distinct A\<^sub>P; A\<^sub>P \<sharp>* \<Psi>; A\<^sub>P \<sharp>* P; A\<^sub>P \<sharp>* Q; A\<^sub>P \<sharp>* M; A\<^sub>P \<sharp>* xvec; A\<^sub>P \<sharp>* N; A\<^sub>P \<sharp>* (A\<^sub>P\<^sub>Q, \<Psi>\<^sub>P\<^sub>Q); A\<^sub>P \<sharp>* xvec; distinct xvec\<rbrakk> \<Longrightarrow> Prop (P \<parallel> Q')
[PROOF STEP]
from \<open>\<Psi> \<otimes> \<Psi>\<^sub>Q \<rhd> P \<longmapsto>M\<lparr>\<nu>*xvec\<rparr>\<langle>N\<rangle> \<prec> P'\<close> S \<open>A\<^sub>P\<^sub>Q \<sharp>* P\<close> \<open>A\<^sub>P \<sharp>* P\<close> \<open>A\<^sub>Q \<sharp>* P\<close> \<open>A\<^sub>P\<^sub>Q \<sharp>* M\<close> \<open>A\<^sub>P \<sharp>* M\<close> \<open>A\<^sub>Q \<sharp>* M\<close> Aeq
[PROOF STATE]
proof (chain)
picking this:
\<Psi> \<otimes> \<Psi>\<^sub>Q \<rhd> P \<longmapsto> M\<lparr>\<nu>*xvec\<rparr>\<langle>N\<rangle> \<prec> P'
set p \<subseteq> set (A\<^sub>P @ A\<^sub>Q) \<times> set (p \<bullet> A\<^sub>P @ p \<bullet> A\<^sub>Q)
A\<^sub>P\<^sub>Q \<sharp>* P
A\<^sub>P \<sharp>* P
A\<^sub>Q \<sharp>* P
A\<^sub>P\<^sub>Q \<sharp>* M
A\<^sub>P \<sharp>* M
A\<^sub>Q \<sharp>* M
A\<^sub>P\<^sub>Q = p \<bullet> A\<^sub>P @ p \<bullet> A\<^sub>Q
[PROOF STEP]
have "(p \<bullet> (\<Psi> \<otimes> \<Psi>\<^sub>Q)) \<rhd> P \<longmapsto>M\<lparr>\<nu>*xvec\<rparr>\<langle>N\<rangle> \<prec> P'"
[PROOF STATE]
proof (prove)
using this:
\<Psi> \<otimes> \<Psi>\<^sub>Q \<rhd> P \<longmapsto> M\<lparr>\<nu>*xvec\<rparr>\<langle>N\<rangle> \<prec> P'
set p \<subseteq> set (A\<^sub>P @ A\<^sub>Q) \<times> set (p \<bullet> A\<^sub>P @ p \<bullet> A\<^sub>Q)
A\<^sub>P\<^sub>Q \<sharp>* P
A\<^sub>P \<sharp>* P
A\<^sub>Q \<sharp>* P
A\<^sub>P\<^sub>Q \<sharp>* M
A\<^sub>P \<sharp>* M
A\<^sub>Q \<sharp>* M
A\<^sub>P\<^sub>Q = p \<bullet> A\<^sub>P @ p \<bullet> A\<^sub>Q
goal (1 subgoal):
1. p \<bullet> \<Psi> \<otimes> \<Psi>\<^sub>Q \<rhd> P \<longmapsto> M\<lparr>\<nu>*xvec\<rparr>\<langle>N\<rangle> \<prec> P'
[PROOF STEP]
by(rule_tac outputPermFrame) (assumption | simp)+
[PROOF STATE]
proof (state)
this:
p \<bullet> \<Psi> \<otimes> \<Psi>\<^sub>Q \<rhd> P \<longmapsto> M\<lparr>\<nu>*xvec\<rparr>\<langle>N\<rangle> \<prec> P'
goal (2 subgoals):
1. \<And>P' A\<^sub>Q \<Psi>\<^sub>Q. \<lbrakk>\<Psi> \<otimes> \<Psi>\<^sub>Q \<rhd> P \<longmapsto> M\<lparr>\<nu>*xvec\<rparr>\<langle>N\<rangle> \<prec> P'; extractFrame Q = \<langle>A\<^sub>Q, \<Psi>\<^sub>Q\<rangle>; distinct A\<^sub>Q; A\<^sub>Q \<sharp>* \<Psi>; A\<^sub>Q \<sharp>* P; A\<^sub>Q \<sharp>* Q; A\<^sub>Q \<sharp>* M; A\<^sub>Q \<sharp>* xvec; A\<^sub>Q \<sharp>* N; A\<^sub>Q \<sharp>* (A\<^sub>P\<^sub>Q, \<Psi>\<^sub>P\<^sub>Q); A\<^sub>Q \<sharp>* xvec; distinct xvec\<rbrakk> \<Longrightarrow> Prop (P' \<parallel> Q)
2. \<And>Q' A\<^sub>P \<Psi>\<^sub>P. \<lbrakk>\<Psi> \<otimes> \<Psi>\<^sub>P \<rhd> Q \<longmapsto> M\<lparr>\<nu>*xvec\<rparr>\<langle>N\<rangle> \<prec> Q'; extractFrame P = \<langle>A\<^sub>P, \<Psi>\<^sub>P\<rangle>; distinct A\<^sub>P; A\<^sub>P \<sharp>* \<Psi>; A\<^sub>P \<sharp>* P; A\<^sub>P \<sharp>* Q; A\<^sub>P \<sharp>* M; A\<^sub>P \<sharp>* xvec; A\<^sub>P \<sharp>* N; A\<^sub>P \<sharp>* (A\<^sub>P\<^sub>Q, \<Psi>\<^sub>P\<^sub>Q); A\<^sub>P \<sharp>* xvec; distinct xvec\<rbrakk> \<Longrightarrow> Prop (P \<parallel> Q')
[PROOF STEP]
with S \<open>A\<^sub>P\<^sub>Q \<sharp>* \<Psi>\<close> \<open>A\<^sub>P \<sharp>* \<Psi>\<close> \<open>A\<^sub>Q \<sharp>* \<Psi>\<close> Aeq
[PROOF STATE]
proof (chain)
picking this:
set p \<subseteq> set (A\<^sub>P @ A\<^sub>Q) \<times> set (p \<bullet> A\<^sub>P @ p \<bullet> A\<^sub>Q)
A\<^sub>P\<^sub>Q \<sharp>* \<Psi>
A\<^sub>P \<sharp>* \<Psi>
A\<^sub>Q \<sharp>* \<Psi>
A\<^sub>P\<^sub>Q = p \<bullet> A\<^sub>P @ p \<bullet> A\<^sub>Q
p \<bullet> \<Psi> \<otimes> \<Psi>\<^sub>Q \<rhd> P \<longmapsto> M\<lparr>\<nu>*xvec\<rparr>\<langle>N\<rangle> \<prec> P'
[PROOF STEP]
have "\<Psi> \<otimes> (p \<bullet> \<Psi>\<^sub>Q) \<rhd> P \<longmapsto>M\<lparr>\<nu>*xvec\<rparr>\<langle>N\<rangle> \<prec> P'"
[PROOF STATE]
proof (prove)
using this:
set p \<subseteq> set (A\<^sub>P @ A\<^sub>Q) \<times> set (p \<bullet> A\<^sub>P @ p \<bullet> A\<^sub>Q)
A\<^sub>P\<^sub>Q \<sharp>* \<Psi>
A\<^sub>P \<sharp>* \<Psi>
A\<^sub>Q \<sharp>* \<Psi>
A\<^sub>P\<^sub>Q = p \<bullet> A\<^sub>P @ p \<bullet> A\<^sub>Q
p \<bullet> \<Psi> \<otimes> \<Psi>\<^sub>Q \<rhd> P \<longmapsto> M\<lparr>\<nu>*xvec\<rparr>\<langle>N\<rangle> \<prec> P'
goal (1 subgoal):
1. \<Psi> \<otimes> (p \<bullet> \<Psi>\<^sub>Q) \<rhd> P \<longmapsto> M\<lparr>\<nu>*xvec\<rparr>\<langle>N\<rangle> \<prec> P'
[PROOF STEP]
by(simp add: eqvts)
[PROOF STATE]
proof (state)
this:
\<Psi> \<otimes> (p \<bullet> \<Psi>\<^sub>Q) \<rhd> P \<longmapsto> M\<lparr>\<nu>*xvec\<rparr>\<langle>N\<rangle> \<prec> P'
goal (2 subgoals):
1. \<And>P' A\<^sub>Q \<Psi>\<^sub>Q. \<lbrakk>\<Psi> \<otimes> \<Psi>\<^sub>Q \<rhd> P \<longmapsto> M\<lparr>\<nu>*xvec\<rparr>\<langle>N\<rangle> \<prec> P'; extractFrame Q = \<langle>A\<^sub>Q, \<Psi>\<^sub>Q\<rangle>; distinct A\<^sub>Q; A\<^sub>Q \<sharp>* \<Psi>; A\<^sub>Q \<sharp>* P; A\<^sub>Q \<sharp>* Q; A\<^sub>Q \<sharp>* M; A\<^sub>Q \<sharp>* xvec; A\<^sub>Q \<sharp>* N; A\<^sub>Q \<sharp>* (A\<^sub>P\<^sub>Q, \<Psi>\<^sub>P\<^sub>Q); A\<^sub>Q \<sharp>* xvec; distinct xvec\<rbrakk> \<Longrightarrow> Prop (P' \<parallel> Q)
2. \<And>Q' A\<^sub>P \<Psi>\<^sub>P. \<lbrakk>\<Psi> \<otimes> \<Psi>\<^sub>P \<rhd> Q \<longmapsto> M\<lparr>\<nu>*xvec\<rparr>\<langle>N\<rangle> \<prec> Q'; extractFrame P = \<langle>A\<^sub>P, \<Psi>\<^sub>P\<rangle>; distinct A\<^sub>P; A\<^sub>P \<sharp>* \<Psi>; A\<^sub>P \<sharp>* P; A\<^sub>P \<sharp>* Q; A\<^sub>P \<sharp>* M; A\<^sub>P \<sharp>* xvec; A\<^sub>P \<sharp>* N; A\<^sub>P \<sharp>* (A\<^sub>P\<^sub>Q, \<Psi>\<^sub>P\<^sub>Q); A\<^sub>P \<sharp>* xvec; distinct xvec\<rbrakk> \<Longrightarrow> Prop (P \<parallel> Q')
[PROOF STEP]
moreover
[PROOF STATE]
proof (state)
this:
\<Psi> \<otimes> (p \<bullet> \<Psi>\<^sub>Q) \<rhd> P \<longmapsto> M\<lparr>\<nu>*xvec\<rparr>\<langle>N\<rangle> \<prec> P'
goal (2 subgoals):
1. \<And>P' A\<^sub>Q \<Psi>\<^sub>Q. \<lbrakk>\<Psi> \<otimes> \<Psi>\<^sub>Q \<rhd> P \<longmapsto> M\<lparr>\<nu>*xvec\<rparr>\<langle>N\<rangle> \<prec> P'; extractFrame Q = \<langle>A\<^sub>Q, \<Psi>\<^sub>Q\<rangle>; distinct A\<^sub>Q; A\<^sub>Q \<sharp>* \<Psi>; A\<^sub>Q \<sharp>* P; A\<^sub>Q \<sharp>* Q; A\<^sub>Q \<sharp>* M; A\<^sub>Q \<sharp>* xvec; A\<^sub>Q \<sharp>* N; A\<^sub>Q \<sharp>* (A\<^sub>P\<^sub>Q, \<Psi>\<^sub>P\<^sub>Q); A\<^sub>Q \<sharp>* xvec; distinct xvec\<rbrakk> \<Longrightarrow> Prop (P' \<parallel> Q)
2. \<And>Q' A\<^sub>P \<Psi>\<^sub>P. \<lbrakk>\<Psi> \<otimes> \<Psi>\<^sub>P \<rhd> Q \<longmapsto> M\<lparr>\<nu>*xvec\<rparr>\<langle>N\<rangle> \<prec> Q'; extractFrame P = \<langle>A\<^sub>P, \<Psi>\<^sub>P\<rangle>; distinct A\<^sub>P; A\<^sub>P \<sharp>* \<Psi>; A\<^sub>P \<sharp>* P; A\<^sub>P \<sharp>* Q; A\<^sub>P \<sharp>* M; A\<^sub>P \<sharp>* xvec; A\<^sub>P \<sharp>* N; A\<^sub>P \<sharp>* (A\<^sub>P\<^sub>Q, \<Psi>\<^sub>P\<^sub>Q); A\<^sub>P \<sharp>* xvec; distinct xvec\<rbrakk> \<Longrightarrow> Prop (P \<parallel> Q')
[PROOF STEP]
from FrP
[PROOF STATE]
proof (chain)
picking this:
extractFrame P = \<langle>A\<^sub>P, \<Psi>\<^sub>P\<rangle>
[PROOF STEP]
have "(p \<bullet> extractFrame P) = p \<bullet> \<langle>A\<^sub>P, \<Psi>\<^sub>P\<rangle>"
[PROOF STATE]
proof (prove)
using this:
extractFrame P = \<langle>A\<^sub>P, \<Psi>\<^sub>P\<rangle>
goal (1 subgoal):
1. p \<bullet> extractFrame P = p \<bullet> \<langle>A\<^sub>P, \<Psi>\<^sub>P\<rangle>
[PROOF STEP]
by simp
[PROOF STATE]
proof (state)
this:
p \<bullet> extractFrame P = p \<bullet> \<langle>A\<^sub>P, \<Psi>\<^sub>P\<rangle>
goal (2 subgoals):
1. \<And>P' A\<^sub>Q \<Psi>\<^sub>Q. \<lbrakk>\<Psi> \<otimes> \<Psi>\<^sub>Q \<rhd> P \<longmapsto> M\<lparr>\<nu>*xvec\<rparr>\<langle>N\<rangle> \<prec> P'; extractFrame Q = \<langle>A\<^sub>Q, \<Psi>\<^sub>Q\<rangle>; distinct A\<^sub>Q; A\<^sub>Q \<sharp>* \<Psi>; A\<^sub>Q \<sharp>* P; A\<^sub>Q \<sharp>* Q; A\<^sub>Q \<sharp>* M; A\<^sub>Q \<sharp>* xvec; A\<^sub>Q \<sharp>* N; A\<^sub>Q \<sharp>* (A\<^sub>P\<^sub>Q, \<Psi>\<^sub>P\<^sub>Q); A\<^sub>Q \<sharp>* xvec; distinct xvec\<rbrakk> \<Longrightarrow> Prop (P' \<parallel> Q)
2. \<And>Q' A\<^sub>P \<Psi>\<^sub>P. \<lbrakk>\<Psi> \<otimes> \<Psi>\<^sub>P \<rhd> Q \<longmapsto> M\<lparr>\<nu>*xvec\<rparr>\<langle>N\<rangle> \<prec> Q'; extractFrame P = \<langle>A\<^sub>P, \<Psi>\<^sub>P\<rangle>; distinct A\<^sub>P; A\<^sub>P \<sharp>* \<Psi>; A\<^sub>P \<sharp>* P; A\<^sub>P \<sharp>* Q; A\<^sub>P \<sharp>* M; A\<^sub>P \<sharp>* xvec; A\<^sub>P \<sharp>* N; A\<^sub>P \<sharp>* (A\<^sub>P\<^sub>Q, \<Psi>\<^sub>P\<^sub>Q); A\<^sub>P \<sharp>* xvec; distinct xvec\<rbrakk> \<Longrightarrow> Prop (P \<parallel> Q')
[PROOF STEP]
with S \<open>A\<^sub>P\<^sub>Q \<sharp>* P\<close> \<open>A\<^sub>P \<sharp>* P\<close> \<open>A\<^sub>Q \<sharp>* P\<close> Aeq
[PROOF STATE]
proof (chain)
picking this:
set p \<subseteq> set (A\<^sub>P @ A\<^sub>Q) \<times> set (p \<bullet> A\<^sub>P @ p \<bullet> A\<^sub>Q)
A\<^sub>P\<^sub>Q \<sharp>* P
A\<^sub>P \<sharp>* P
A\<^sub>Q \<sharp>* P
A\<^sub>P\<^sub>Q = p \<bullet> A\<^sub>P @ p \<bullet> A\<^sub>Q
p \<bullet> extractFrame P = p \<bullet> \<langle>A\<^sub>P, \<Psi>\<^sub>P\<rangle>
[PROOF STEP]
have "extractFrame P = \<langle>(p \<bullet> A\<^sub>P), p \<bullet> \<Psi>\<^sub>P\<rangle>"
[PROOF STATE]
proof (prove)
using this:
set p \<subseteq> set (A\<^sub>P @ A\<^sub>Q) \<times> set (p \<bullet> A\<^sub>P @ p \<bullet> A\<^sub>Q)
A\<^sub>P\<^sub>Q \<sharp>* P
A\<^sub>P \<sharp>* P
A\<^sub>Q \<sharp>* P
A\<^sub>P\<^sub>Q = p \<bullet> A\<^sub>P @ p \<bullet> A\<^sub>Q
p \<bullet> extractFrame P = p \<bullet> \<langle>A\<^sub>P, \<Psi>\<^sub>P\<rangle>
goal (1 subgoal):
1. extractFrame P = \<langle>p \<bullet> A\<^sub>P, p \<bullet> \<Psi>\<^sub>P\<rangle>
[PROOF STEP]
by(simp add: eqvts)
[PROOF STATE]
proof (state)
this:
extractFrame P = \<langle>p \<bullet> A\<^sub>P, p \<bullet> \<Psi>\<^sub>P\<rangle>
goal (2 subgoals):
1. \<And>P' A\<^sub>Q \<Psi>\<^sub>Q. \<lbrakk>\<Psi> \<otimes> \<Psi>\<^sub>Q \<rhd> P \<longmapsto> M\<lparr>\<nu>*xvec\<rparr>\<langle>N\<rangle> \<prec> P'; extractFrame Q = \<langle>A\<^sub>Q, \<Psi>\<^sub>Q\<rangle>; distinct A\<^sub>Q; A\<^sub>Q \<sharp>* \<Psi>; A\<^sub>Q \<sharp>* P; A\<^sub>Q \<sharp>* Q; A\<^sub>Q \<sharp>* M; A\<^sub>Q \<sharp>* xvec; A\<^sub>Q \<sharp>* N; A\<^sub>Q \<sharp>* (A\<^sub>P\<^sub>Q, \<Psi>\<^sub>P\<^sub>Q); A\<^sub>Q \<sharp>* xvec; distinct xvec\<rbrakk> \<Longrightarrow> Prop (P' \<parallel> Q)
2. \<And>Q' A\<^sub>P \<Psi>\<^sub>P. \<lbrakk>\<Psi> \<otimes> \<Psi>\<^sub>P \<rhd> Q \<longmapsto> M\<lparr>\<nu>*xvec\<rparr>\<langle>N\<rangle> \<prec> Q'; extractFrame P = \<langle>A\<^sub>P, \<Psi>\<^sub>P\<rangle>; distinct A\<^sub>P; A\<^sub>P \<sharp>* \<Psi>; A\<^sub>P \<sharp>* P; A\<^sub>P \<sharp>* Q; A\<^sub>P \<sharp>* M; A\<^sub>P \<sharp>* xvec; A\<^sub>P \<sharp>* N; A\<^sub>P \<sharp>* (A\<^sub>P\<^sub>Q, \<Psi>\<^sub>P\<^sub>Q); A\<^sub>P \<sharp>* xvec; distinct xvec\<rbrakk> \<Longrightarrow> Prop (P \<parallel> Q')
[PROOF STEP]
moreover
[PROOF STATE]
proof (state)
this:
extractFrame P = \<langle>p \<bullet> A\<^sub>P, p \<bullet> \<Psi>\<^sub>P\<rangle>
goal (2 subgoals):
1. \<And>P' A\<^sub>Q \<Psi>\<^sub>Q. \<lbrakk>\<Psi> \<otimes> \<Psi>\<^sub>Q \<rhd> P \<longmapsto> M\<lparr>\<nu>*xvec\<rparr>\<langle>N\<rangle> \<prec> P'; extractFrame Q = \<langle>A\<^sub>Q, \<Psi>\<^sub>Q\<rangle>; distinct A\<^sub>Q; A\<^sub>Q \<sharp>* \<Psi>; A\<^sub>Q \<sharp>* P; A\<^sub>Q \<sharp>* Q; A\<^sub>Q \<sharp>* M; A\<^sub>Q \<sharp>* xvec; A\<^sub>Q \<sharp>* N; A\<^sub>Q \<sharp>* (A\<^sub>P\<^sub>Q, \<Psi>\<^sub>P\<^sub>Q); A\<^sub>Q \<sharp>* xvec; distinct xvec\<rbrakk> \<Longrightarrow> Prop (P' \<parallel> Q)
2. \<And>Q' A\<^sub>P \<Psi>\<^sub>P. \<lbrakk>\<Psi> \<otimes> \<Psi>\<^sub>P \<rhd> Q \<longmapsto> M\<lparr>\<nu>*xvec\<rparr>\<langle>N\<rangle> \<prec> Q'; extractFrame P = \<langle>A\<^sub>P, \<Psi>\<^sub>P\<rangle>; distinct A\<^sub>P; A\<^sub>P \<sharp>* \<Psi>; A\<^sub>P \<sharp>* P; A\<^sub>P \<sharp>* Q; A\<^sub>P \<sharp>* M; A\<^sub>P \<sharp>* xvec; A\<^sub>P \<sharp>* N; A\<^sub>P \<sharp>* (A\<^sub>P\<^sub>Q, \<Psi>\<^sub>P\<^sub>Q); A\<^sub>P \<sharp>* xvec; distinct xvec\<rbrakk> \<Longrightarrow> Prop (P \<parallel> Q')
[PROOF STEP]
from FrQ
[PROOF STATE]
proof (chain)
picking this:
extractFrame Q = \<langle>A\<^sub>Q, \<Psi>\<^sub>Q\<rangle>
[PROOF STEP]
have "(p \<bullet> extractFrame Q) = p \<bullet> \<langle>A\<^sub>Q, \<Psi>\<^sub>Q\<rangle>"
[PROOF STATE]
proof (prove)
using this:
extractFrame Q = \<langle>A\<^sub>Q, \<Psi>\<^sub>Q\<rangle>
goal (1 subgoal):
1. p \<bullet> extractFrame Q = p \<bullet> \<langle>A\<^sub>Q, \<Psi>\<^sub>Q\<rangle>
[PROOF STEP]
by simp
[PROOF STATE]
proof (state)
this:
p \<bullet> extractFrame Q = p \<bullet> \<langle>A\<^sub>Q, \<Psi>\<^sub>Q\<rangle>
goal (2 subgoals):
1. \<And>P' A\<^sub>Q \<Psi>\<^sub>Q. \<lbrakk>\<Psi> \<otimes> \<Psi>\<^sub>Q \<rhd> P \<longmapsto> M\<lparr>\<nu>*xvec\<rparr>\<langle>N\<rangle> \<prec> P'; extractFrame Q = \<langle>A\<^sub>Q, \<Psi>\<^sub>Q\<rangle>; distinct A\<^sub>Q; A\<^sub>Q \<sharp>* \<Psi>; A\<^sub>Q \<sharp>* P; A\<^sub>Q \<sharp>* Q; A\<^sub>Q \<sharp>* M; A\<^sub>Q \<sharp>* xvec; A\<^sub>Q \<sharp>* N; A\<^sub>Q \<sharp>* (A\<^sub>P\<^sub>Q, \<Psi>\<^sub>P\<^sub>Q); A\<^sub>Q \<sharp>* xvec; distinct xvec\<rbrakk> \<Longrightarrow> Prop (P' \<parallel> Q)
2. \<And>Q' A\<^sub>P \<Psi>\<^sub>P. \<lbrakk>\<Psi> \<otimes> \<Psi>\<^sub>P \<rhd> Q \<longmapsto> M\<lparr>\<nu>*xvec\<rparr>\<langle>N\<rangle> \<prec> Q'; extractFrame P = \<langle>A\<^sub>P, \<Psi>\<^sub>P\<rangle>; distinct A\<^sub>P; A\<^sub>P \<sharp>* \<Psi>; A\<^sub>P \<sharp>* P; A\<^sub>P \<sharp>* Q; A\<^sub>P \<sharp>* M; A\<^sub>P \<sharp>* xvec; A\<^sub>P \<sharp>* N; A\<^sub>P \<sharp>* (A\<^sub>P\<^sub>Q, \<Psi>\<^sub>P\<^sub>Q); A\<^sub>P \<sharp>* xvec; distinct xvec\<rbrakk> \<Longrightarrow> Prop (P \<parallel> Q')
[PROOF STEP]
with S \<open>A\<^sub>P\<^sub>Q \<sharp>* Q\<close> \<open>A\<^sub>P \<sharp>* Q\<close> \<open>A\<^sub>Q \<sharp>* Q\<close> Aeq
[PROOF STATE]
proof (chain)
picking this:
set p \<subseteq> set (A\<^sub>P @ A\<^sub>Q) \<times> set (p \<bullet> A\<^sub>P @ p \<bullet> A\<^sub>Q)
A\<^sub>P\<^sub>Q \<sharp>* Q
A\<^sub>P \<sharp>* Q
A\<^sub>Q \<sharp>* Q
A\<^sub>P\<^sub>Q = p \<bullet> A\<^sub>P @ p \<bullet> A\<^sub>Q
p \<bullet> extractFrame Q = p \<bullet> \<langle>A\<^sub>Q, \<Psi>\<^sub>Q\<rangle>
[PROOF STEP]
have "extractFrame Q = \<langle>(p \<bullet> A\<^sub>Q), p \<bullet> \<Psi>\<^sub>Q\<rangle>"
[PROOF STATE]
proof (prove)
using this:
set p \<subseteq> set (A\<^sub>P @ A\<^sub>Q) \<times> set (p \<bullet> A\<^sub>P @ p \<bullet> A\<^sub>Q)
A\<^sub>P\<^sub>Q \<sharp>* Q
A\<^sub>P \<sharp>* Q
A\<^sub>Q \<sharp>* Q
A\<^sub>P\<^sub>Q = p \<bullet> A\<^sub>P @ p \<bullet> A\<^sub>Q
p \<bullet> extractFrame Q = p \<bullet> \<langle>A\<^sub>Q, \<Psi>\<^sub>Q\<rangle>
goal (1 subgoal):
1. extractFrame Q = \<langle>p \<bullet> A\<^sub>Q, p \<bullet> \<Psi>\<^sub>Q\<rangle>
[PROOF STEP]
by(simp add: eqvts)
[PROOF STATE]
proof (state)
this:
extractFrame Q = \<langle>p \<bullet> A\<^sub>Q, p \<bullet> \<Psi>\<^sub>Q\<rangle>
goal (2 subgoals):
1. \<And>P' A\<^sub>Q \<Psi>\<^sub>Q. \<lbrakk>\<Psi> \<otimes> \<Psi>\<^sub>Q \<rhd> P \<longmapsto> M\<lparr>\<nu>*xvec\<rparr>\<langle>N\<rangle> \<prec> P'; extractFrame Q = \<langle>A\<^sub>Q, \<Psi>\<^sub>Q\<rangle>; distinct A\<^sub>Q; A\<^sub>Q \<sharp>* \<Psi>; A\<^sub>Q \<sharp>* P; A\<^sub>Q \<sharp>* Q; A\<^sub>Q \<sharp>* M; A\<^sub>Q \<sharp>* xvec; A\<^sub>Q \<sharp>* N; A\<^sub>Q \<sharp>* (A\<^sub>P\<^sub>Q, \<Psi>\<^sub>P\<^sub>Q); A\<^sub>Q \<sharp>* xvec; distinct xvec\<rbrakk> \<Longrightarrow> Prop (P' \<parallel> Q)
2. \<And>Q' A\<^sub>P \<Psi>\<^sub>P. \<lbrakk>\<Psi> \<otimes> \<Psi>\<^sub>P \<rhd> Q \<longmapsto> M\<lparr>\<nu>*xvec\<rparr>\<langle>N\<rangle> \<prec> Q'; extractFrame P = \<langle>A\<^sub>P, \<Psi>\<^sub>P\<rangle>; distinct A\<^sub>P; A\<^sub>P \<sharp>* \<Psi>; A\<^sub>P \<sharp>* P; A\<^sub>P \<sharp>* Q; A\<^sub>P \<sharp>* M; A\<^sub>P \<sharp>* xvec; A\<^sub>P \<sharp>* N; A\<^sub>P \<sharp>* (A\<^sub>P\<^sub>Q, \<Psi>\<^sub>P\<^sub>Q); A\<^sub>P \<sharp>* xvec; distinct xvec\<rbrakk> \<Longrightarrow> Prop (P \<parallel> Q')
[PROOF STEP]
moreover
[PROOF STATE]
proof (state)
this:
extractFrame Q = \<langle>p \<bullet> A\<^sub>Q, p \<bullet> \<Psi>\<^sub>Q\<rangle>
goal (2 subgoals):
1. \<And>P' A\<^sub>Q \<Psi>\<^sub>Q. \<lbrakk>\<Psi> \<otimes> \<Psi>\<^sub>Q \<rhd> P \<longmapsto> M\<lparr>\<nu>*xvec\<rparr>\<langle>N\<rangle> \<prec> P'; extractFrame Q = \<langle>A\<^sub>Q, \<Psi>\<^sub>Q\<rangle>; distinct A\<^sub>Q; A\<^sub>Q \<sharp>* \<Psi>; A\<^sub>Q \<sharp>* P; A\<^sub>Q \<sharp>* Q; A\<^sub>Q \<sharp>* M; A\<^sub>Q \<sharp>* xvec; A\<^sub>Q \<sharp>* N; A\<^sub>Q \<sharp>* (A\<^sub>P\<^sub>Q, \<Psi>\<^sub>P\<^sub>Q); A\<^sub>Q \<sharp>* xvec; distinct xvec\<rbrakk> \<Longrightarrow> Prop (P' \<parallel> Q)
2. \<And>Q' A\<^sub>P \<Psi>\<^sub>P. \<lbrakk>\<Psi> \<otimes> \<Psi>\<^sub>P \<rhd> Q \<longmapsto> M\<lparr>\<nu>*xvec\<rparr>\<langle>N\<rangle> \<prec> Q'; extractFrame P = \<langle>A\<^sub>P, \<Psi>\<^sub>P\<rangle>; distinct A\<^sub>P; A\<^sub>P \<sharp>* \<Psi>; A\<^sub>P \<sharp>* P; A\<^sub>P \<sharp>* Q; A\<^sub>P \<sharp>* M; A\<^sub>P \<sharp>* xvec; A\<^sub>P \<sharp>* N; A\<^sub>P \<sharp>* (A\<^sub>P\<^sub>Q, \<Psi>\<^sub>P\<^sub>Q); A\<^sub>P \<sharp>* xvec; distinct xvec\<rbrakk> \<Longrightarrow> Prop (P \<parallel> Q')
[PROOF STEP]
from \<open>distinct A\<^sub>P\<close> \<open>distinct A\<^sub>Q\<close>
[PROOF STATE]
proof (chain)
picking this:
distinct A\<^sub>P
distinct A\<^sub>Q
[PROOF STEP]
have "distinct(p \<bullet> A\<^sub>P)" and "distinct(p \<bullet> A\<^sub>Q)"
[PROOF STATE]
proof (prove)
using this:
distinct A\<^sub>P
distinct A\<^sub>Q
goal (1 subgoal):
1. distinct (p \<bullet> A\<^sub>P) &&& distinct (p \<bullet> A\<^sub>Q)
[PROOF STEP]
by simp+
[PROOF STATE]
proof (state)
this:
distinct (p \<bullet> A\<^sub>P)
distinct (p \<bullet> A\<^sub>Q)
goal (2 subgoals):
1. \<And>P' A\<^sub>Q \<Psi>\<^sub>Q. \<lbrakk>\<Psi> \<otimes> \<Psi>\<^sub>Q \<rhd> P \<longmapsto> M\<lparr>\<nu>*xvec\<rparr>\<langle>N\<rangle> \<prec> P'; extractFrame Q = \<langle>A\<^sub>Q, \<Psi>\<^sub>Q\<rangle>; distinct A\<^sub>Q; A\<^sub>Q \<sharp>* \<Psi>; A\<^sub>Q \<sharp>* P; A\<^sub>Q \<sharp>* Q; A\<^sub>Q \<sharp>* M; A\<^sub>Q \<sharp>* xvec; A\<^sub>Q \<sharp>* N; A\<^sub>Q \<sharp>* (A\<^sub>P\<^sub>Q, \<Psi>\<^sub>P\<^sub>Q); A\<^sub>Q \<sharp>* xvec; distinct xvec\<rbrakk> \<Longrightarrow> Prop (P' \<parallel> Q)
2. \<And>Q' A\<^sub>P \<Psi>\<^sub>P. \<lbrakk>\<Psi> \<otimes> \<Psi>\<^sub>P \<rhd> Q \<longmapsto> M\<lparr>\<nu>*xvec\<rparr>\<langle>N\<rangle> \<prec> Q'; extractFrame P = \<langle>A\<^sub>P, \<Psi>\<^sub>P\<rangle>; distinct A\<^sub>P; A\<^sub>P \<sharp>* \<Psi>; A\<^sub>P \<sharp>* P; A\<^sub>P \<sharp>* Q; A\<^sub>P \<sharp>* M; A\<^sub>P \<sharp>* xvec; A\<^sub>P \<sharp>* N; A\<^sub>P \<sharp>* (A\<^sub>P\<^sub>Q, \<Psi>\<^sub>P\<^sub>Q); A\<^sub>P \<sharp>* xvec; distinct xvec\<rbrakk> \<Longrightarrow> Prop (P \<parallel> Q')
[PROOF STEP]
moreover
[PROOF STATE]
proof (state)
this:
distinct (p \<bullet> A\<^sub>P)
distinct (p \<bullet> A\<^sub>Q)
goal (2 subgoals):
1. \<And>P' A\<^sub>Q \<Psi>\<^sub>Q. \<lbrakk>\<Psi> \<otimes> \<Psi>\<^sub>Q \<rhd> P \<longmapsto> M\<lparr>\<nu>*xvec\<rparr>\<langle>N\<rangle> \<prec> P'; extractFrame Q = \<langle>A\<^sub>Q, \<Psi>\<^sub>Q\<rangle>; distinct A\<^sub>Q; A\<^sub>Q \<sharp>* \<Psi>; A\<^sub>Q \<sharp>* P; A\<^sub>Q \<sharp>* Q; A\<^sub>Q \<sharp>* M; A\<^sub>Q \<sharp>* xvec; A\<^sub>Q \<sharp>* N; A\<^sub>Q \<sharp>* (A\<^sub>P\<^sub>Q, \<Psi>\<^sub>P\<^sub>Q); A\<^sub>Q \<sharp>* xvec; distinct xvec\<rbrakk> \<Longrightarrow> Prop (P' \<parallel> Q)
2. \<And>Q' A\<^sub>P \<Psi>\<^sub>P. \<lbrakk>\<Psi> \<otimes> \<Psi>\<^sub>P \<rhd> Q \<longmapsto> M\<lparr>\<nu>*xvec\<rparr>\<langle>N\<rangle> \<prec> Q'; extractFrame P = \<langle>A\<^sub>P, \<Psi>\<^sub>P\<rangle>; distinct A\<^sub>P; A\<^sub>P \<sharp>* \<Psi>; A\<^sub>P \<sharp>* P; A\<^sub>P \<sharp>* Q; A\<^sub>P \<sharp>* M; A\<^sub>P \<sharp>* xvec; A\<^sub>P \<sharp>* N; A\<^sub>P \<sharp>* (A\<^sub>P\<^sub>Q, \<Psi>\<^sub>P\<^sub>Q); A\<^sub>P \<sharp>* xvec; distinct xvec\<rbrakk> \<Longrightarrow> Prop (P \<parallel> Q')
[PROOF STEP]
from \<open>A\<^sub>P \<sharp>* A\<^sub>Q\<close>
[PROOF STATE]
proof (chain)
picking this:
A\<^sub>P \<sharp>* A\<^sub>Q
[PROOF STEP]
have "(p \<bullet> A\<^sub>P) \<sharp>* (p \<bullet> A\<^sub>Q)"
[PROOF STATE]
proof (prove)
using this:
A\<^sub>P \<sharp>* A\<^sub>Q
goal (1 subgoal):
1. (p \<bullet> A\<^sub>P) \<sharp>* (p \<bullet> A\<^sub>Q)
[PROOF STEP]
by(simp add: pt_fresh_star_bij[OF pt_name_inst, OF at_name_inst])
[PROOF STATE]
proof (state)
this:
(p \<bullet> A\<^sub>P) \<sharp>* (p \<bullet> A\<^sub>Q)
goal (2 subgoals):
1. \<And>P' A\<^sub>Q \<Psi>\<^sub>Q. \<lbrakk>\<Psi> \<otimes> \<Psi>\<^sub>Q \<rhd> P \<longmapsto> M\<lparr>\<nu>*xvec\<rparr>\<langle>N\<rangle> \<prec> P'; extractFrame Q = \<langle>A\<^sub>Q, \<Psi>\<^sub>Q\<rangle>; distinct A\<^sub>Q; A\<^sub>Q \<sharp>* \<Psi>; A\<^sub>Q \<sharp>* P; A\<^sub>Q \<sharp>* Q; A\<^sub>Q \<sharp>* M; A\<^sub>Q \<sharp>* xvec; A\<^sub>Q \<sharp>* N; A\<^sub>Q \<sharp>* (A\<^sub>P\<^sub>Q, \<Psi>\<^sub>P\<^sub>Q); A\<^sub>Q \<sharp>* xvec; distinct xvec\<rbrakk> \<Longrightarrow> Prop (P' \<parallel> Q)
2. \<And>Q' A\<^sub>P \<Psi>\<^sub>P. \<lbrakk>\<Psi> \<otimes> \<Psi>\<^sub>P \<rhd> Q \<longmapsto> M\<lparr>\<nu>*xvec\<rparr>\<langle>N\<rangle> \<prec> Q'; extractFrame P = \<langle>A\<^sub>P, \<Psi>\<^sub>P\<rangle>; distinct A\<^sub>P; A\<^sub>P \<sharp>* \<Psi>; A\<^sub>P \<sharp>* P; A\<^sub>P \<sharp>* Q; A\<^sub>P \<sharp>* M; A\<^sub>P \<sharp>* xvec; A\<^sub>P \<sharp>* N; A\<^sub>P \<sharp>* (A\<^sub>P\<^sub>Q, \<Psi>\<^sub>P\<^sub>Q); A\<^sub>P \<sharp>* xvec; distinct xvec\<rbrakk> \<Longrightarrow> Prop (P \<parallel> Q')
[PROOF STEP]
moreover
[PROOF STATE]
proof (state)
this:
(p \<bullet> A\<^sub>P) \<sharp>* (p \<bullet> A\<^sub>Q)
goal (2 subgoals):
1. \<And>P' A\<^sub>Q \<Psi>\<^sub>Q. \<lbrakk>\<Psi> \<otimes> \<Psi>\<^sub>Q \<rhd> P \<longmapsto> M\<lparr>\<nu>*xvec\<rparr>\<langle>N\<rangle> \<prec> P'; extractFrame Q = \<langle>A\<^sub>Q, \<Psi>\<^sub>Q\<rangle>; distinct A\<^sub>Q; A\<^sub>Q \<sharp>* \<Psi>; A\<^sub>Q \<sharp>* P; A\<^sub>Q \<sharp>* Q; A\<^sub>Q \<sharp>* M; A\<^sub>Q \<sharp>* xvec; A\<^sub>Q \<sharp>* N; A\<^sub>Q \<sharp>* (A\<^sub>P\<^sub>Q, \<Psi>\<^sub>P\<^sub>Q); A\<^sub>Q \<sharp>* xvec; distinct xvec\<rbrakk> \<Longrightarrow> Prop (P' \<parallel> Q)
2. \<And>Q' A\<^sub>P \<Psi>\<^sub>P. \<lbrakk>\<Psi> \<otimes> \<Psi>\<^sub>P \<rhd> Q \<longmapsto> M\<lparr>\<nu>*xvec\<rparr>\<langle>N\<rangle> \<prec> Q'; extractFrame P = \<langle>A\<^sub>P, \<Psi>\<^sub>P\<rangle>; distinct A\<^sub>P; A\<^sub>P \<sharp>* \<Psi>; A\<^sub>P \<sharp>* P; A\<^sub>P \<sharp>* Q; A\<^sub>P \<sharp>* M; A\<^sub>P \<sharp>* xvec; A\<^sub>P \<sharp>* N; A\<^sub>P \<sharp>* (A\<^sub>P\<^sub>Q, \<Psi>\<^sub>P\<^sub>Q); A\<^sub>P \<sharp>* xvec; distinct xvec\<rbrakk> \<Longrightarrow> Prop (P \<parallel> Q')
[PROOF STEP]
from \<open>A\<^sub>P \<sharp>* \<Psi>\<^sub>Q\<close>
[PROOF STATE]
proof (chain)
picking this:
A\<^sub>P \<sharp>* \<Psi>\<^sub>Q
[PROOF STEP]
have "(p \<bullet> A\<^sub>P) \<sharp>* (p \<bullet> \<Psi>\<^sub>Q)"
[PROOF STATE]
proof (prove)
using this:
A\<^sub>P \<sharp>* \<Psi>\<^sub>Q
goal (1 subgoal):
1. (p \<bullet> A\<^sub>P) \<sharp>* (p \<bullet> \<Psi>\<^sub>Q)
[PROOF STEP]
by(simp add: pt_fresh_star_bij[OF pt_name_inst, OF at_name_inst])
[PROOF STATE]
proof (state)
this:
(p \<bullet> A\<^sub>P) \<sharp>* (p \<bullet> \<Psi>\<^sub>Q)
goal (2 subgoals):
1. \<And>P' A\<^sub>Q \<Psi>\<^sub>Q. \<lbrakk>\<Psi> \<otimes> \<Psi>\<^sub>Q \<rhd> P \<longmapsto> M\<lparr>\<nu>*xvec\<rparr>\<langle>N\<rangle> \<prec> P'; extractFrame Q = \<langle>A\<^sub>Q, \<Psi>\<^sub>Q\<rangle>; distinct A\<^sub>Q; A\<^sub>Q \<sharp>* \<Psi>; A\<^sub>Q \<sharp>* P; A\<^sub>Q \<sharp>* Q; A\<^sub>Q \<sharp>* M; A\<^sub>Q \<sharp>* xvec; A\<^sub>Q \<sharp>* N; A\<^sub>Q \<sharp>* (A\<^sub>P\<^sub>Q, \<Psi>\<^sub>P\<^sub>Q); A\<^sub>Q \<sharp>* xvec; distinct xvec\<rbrakk> \<Longrightarrow> Prop (P' \<parallel> Q)
2. \<And>Q' A\<^sub>P \<Psi>\<^sub>P. \<lbrakk>\<Psi> \<otimes> \<Psi>\<^sub>P \<rhd> Q \<longmapsto> M\<lparr>\<nu>*xvec\<rparr>\<langle>N\<rangle> \<prec> Q'; extractFrame P = \<langle>A\<^sub>P, \<Psi>\<^sub>P\<rangle>; distinct A\<^sub>P; A\<^sub>P \<sharp>* \<Psi>; A\<^sub>P \<sharp>* P; A\<^sub>P \<sharp>* Q; A\<^sub>P \<sharp>* M; A\<^sub>P \<sharp>* xvec; A\<^sub>P \<sharp>* N; A\<^sub>P \<sharp>* (A\<^sub>P\<^sub>Q, \<Psi>\<^sub>P\<^sub>Q); A\<^sub>P \<sharp>* xvec; distinct xvec\<rbrakk> \<Longrightarrow> Prop (P \<parallel> Q')
[PROOF STEP]
moreover
[PROOF STATE]
proof (state)
this:
(p \<bullet> A\<^sub>P) \<sharp>* (p \<bullet> \<Psi>\<^sub>Q)
goal (2 subgoals):
1. \<And>P' A\<^sub>Q \<Psi>\<^sub>Q. \<lbrakk>\<Psi> \<otimes> \<Psi>\<^sub>Q \<rhd> P \<longmapsto> M\<lparr>\<nu>*xvec\<rparr>\<langle>N\<rangle> \<prec> P'; extractFrame Q = \<langle>A\<^sub>Q, \<Psi>\<^sub>Q\<rangle>; distinct A\<^sub>Q; A\<^sub>Q \<sharp>* \<Psi>; A\<^sub>Q \<sharp>* P; A\<^sub>Q \<sharp>* Q; A\<^sub>Q \<sharp>* M; A\<^sub>Q \<sharp>* xvec; A\<^sub>Q \<sharp>* N; A\<^sub>Q \<sharp>* (A\<^sub>P\<^sub>Q, \<Psi>\<^sub>P\<^sub>Q); A\<^sub>Q \<sharp>* xvec; distinct xvec\<rbrakk> \<Longrightarrow> Prop (P' \<parallel> Q)
2. \<And>Q' A\<^sub>P \<Psi>\<^sub>P. \<lbrakk>\<Psi> \<otimes> \<Psi>\<^sub>P \<rhd> Q \<longmapsto> M\<lparr>\<nu>*xvec\<rparr>\<langle>N\<rangle> \<prec> Q'; extractFrame P = \<langle>A\<^sub>P, \<Psi>\<^sub>P\<rangle>; distinct A\<^sub>P; A\<^sub>P \<sharp>* \<Psi>; A\<^sub>P \<sharp>* P; A\<^sub>P \<sharp>* Q; A\<^sub>P \<sharp>* M; A\<^sub>P \<sharp>* xvec; A\<^sub>P \<sharp>* N; A\<^sub>P \<sharp>* (A\<^sub>P\<^sub>Q, \<Psi>\<^sub>P\<^sub>Q); A\<^sub>P \<sharp>* xvec; distinct xvec\<rbrakk> \<Longrightarrow> Prop (P \<parallel> Q')
[PROOF STEP]
from \<open>A\<^sub>Q \<sharp>* \<Psi>\<^sub>P\<close>
[PROOF STATE]
proof (chain)
picking this:
A\<^sub>Q \<sharp>* \<Psi>\<^sub>P
[PROOF STEP]
have "(p \<bullet> A\<^sub>Q) \<sharp>* (p \<bullet> \<Psi>\<^sub>P)"
[PROOF STATE]
proof (prove)
using this:
A\<^sub>Q \<sharp>* \<Psi>\<^sub>P
goal (1 subgoal):
1. (p \<bullet> A\<^sub>Q) \<sharp>* (p \<bullet> \<Psi>\<^sub>P)
[PROOF STEP]
by(simp add: pt_fresh_star_bij[OF pt_name_inst, OF at_name_inst])
[PROOF STATE]
proof (state)
this:
(p \<bullet> A\<^sub>Q) \<sharp>* (p \<bullet> \<Psi>\<^sub>P)
goal (2 subgoals):
1. \<And>P' A\<^sub>Q \<Psi>\<^sub>Q. \<lbrakk>\<Psi> \<otimes> \<Psi>\<^sub>Q \<rhd> P \<longmapsto> M\<lparr>\<nu>*xvec\<rparr>\<langle>N\<rangle> \<prec> P'; extractFrame Q = \<langle>A\<^sub>Q, \<Psi>\<^sub>Q\<rangle>; distinct A\<^sub>Q; A\<^sub>Q \<sharp>* \<Psi>; A\<^sub>Q \<sharp>* P; A\<^sub>Q \<sharp>* Q; A\<^sub>Q \<sharp>* M; A\<^sub>Q \<sharp>* xvec; A\<^sub>Q \<sharp>* N; A\<^sub>Q \<sharp>* (A\<^sub>P\<^sub>Q, \<Psi>\<^sub>P\<^sub>Q); A\<^sub>Q \<sharp>* xvec; distinct xvec\<rbrakk> \<Longrightarrow> Prop (P' \<parallel> Q)
2. \<And>Q' A\<^sub>P \<Psi>\<^sub>P. \<lbrakk>\<Psi> \<otimes> \<Psi>\<^sub>P \<rhd> Q \<longmapsto> M\<lparr>\<nu>*xvec\<rparr>\<langle>N\<rangle> \<prec> Q'; extractFrame P = \<langle>A\<^sub>P, \<Psi>\<^sub>P\<rangle>; distinct A\<^sub>P; A\<^sub>P \<sharp>* \<Psi>; A\<^sub>P \<sharp>* P; A\<^sub>P \<sharp>* Q; A\<^sub>P \<sharp>* M; A\<^sub>P \<sharp>* xvec; A\<^sub>P \<sharp>* N; A\<^sub>P \<sharp>* (A\<^sub>P\<^sub>Q, \<Psi>\<^sub>P\<^sub>Q); A\<^sub>P \<sharp>* xvec; distinct xvec\<rbrakk> \<Longrightarrow> Prop (P \<parallel> Q')
[PROOF STEP]
ultimately
[PROOF STATE]
proof (chain)
picking this:
\<Psi> \<otimes> (p \<bullet> \<Psi>\<^sub>Q) \<rhd> P \<longmapsto> M\<lparr>\<nu>*xvec\<rparr>\<langle>N\<rangle> \<prec> P'
extractFrame P = \<langle>p \<bullet> A\<^sub>P, p \<bullet> \<Psi>\<^sub>P\<rangle>
extractFrame Q = \<langle>p \<bullet> A\<^sub>Q, p \<bullet> \<Psi>\<^sub>Q\<rangle>
distinct (p \<bullet> A\<^sub>P)
distinct (p \<bullet> A\<^sub>Q)
(p \<bullet> A\<^sub>P) \<sharp>* (p \<bullet> A\<^sub>Q)
(p \<bullet> A\<^sub>P) \<sharp>* (p \<bullet> \<Psi>\<^sub>Q)
(p \<bullet> A\<^sub>Q) \<sharp>* (p \<bullet> \<Psi>\<^sub>P)
[PROOF STEP]
show ?case
[PROOF STATE]
proof (prove)
using this:
\<Psi> \<otimes> (p \<bullet> \<Psi>\<^sub>Q) \<rhd> P \<longmapsto> M\<lparr>\<nu>*xvec\<rparr>\<langle>N\<rangle> \<prec> P'
extractFrame P = \<langle>p \<bullet> A\<^sub>P, p \<bullet> \<Psi>\<^sub>P\<rangle>
extractFrame Q = \<langle>p \<bullet> A\<^sub>Q, p \<bullet> \<Psi>\<^sub>Q\<rangle>
distinct (p \<bullet> A\<^sub>P)
distinct (p \<bullet> A\<^sub>Q)
(p \<bullet> A\<^sub>P) \<sharp>* (p \<bullet> A\<^sub>Q)
(p \<bullet> A\<^sub>P) \<sharp>* (p \<bullet> \<Psi>\<^sub>Q)
(p \<bullet> A\<^sub>Q) \<sharp>* (p \<bullet> \<Psi>\<^sub>P)
goal (1 subgoal):
1. Prop (P' \<parallel> Q)
[PROOF STEP]
using \<open>A\<^sub>P\<^sub>Q \<sharp>* \<Psi>\<close> \<open>A\<^sub>P\<^sub>Q \<sharp>* P\<close> \<open>A\<^sub>P\<^sub>Q \<sharp>* Q\<close> \<open>A\<^sub>P\<^sub>Q \<sharp>* M\<close> Aeq \<Psi>eq
[PROOF STATE]
proof (prove)
using this:
\<Psi> \<otimes> (p \<bullet> \<Psi>\<^sub>Q) \<rhd> P \<longmapsto> M\<lparr>\<nu>*xvec\<rparr>\<langle>N\<rangle> \<prec> P'
extractFrame P = \<langle>p \<bullet> A\<^sub>P, p \<bullet> \<Psi>\<^sub>P\<rangle>
extractFrame Q = \<langle>p \<bullet> A\<^sub>Q, p \<bullet> \<Psi>\<^sub>Q\<rangle>
distinct (p \<bullet> A\<^sub>P)
distinct (p \<bullet> A\<^sub>Q)
(p \<bullet> A\<^sub>P) \<sharp>* (p \<bullet> A\<^sub>Q)
(p \<bullet> A\<^sub>P) \<sharp>* (p \<bullet> \<Psi>\<^sub>Q)
(p \<bullet> A\<^sub>Q) \<sharp>* (p \<bullet> \<Psi>\<^sub>P)
A\<^sub>P\<^sub>Q \<sharp>* \<Psi>
A\<^sub>P\<^sub>Q \<sharp>* P
A\<^sub>P\<^sub>Q \<sharp>* Q
A\<^sub>P\<^sub>Q \<sharp>* M
A\<^sub>P\<^sub>Q = p \<bullet> A\<^sub>P @ p \<bullet> A\<^sub>Q
\<Psi>\<^sub>P\<^sub>Q = (p \<bullet> \<Psi>\<^sub>P) \<otimes> (p \<bullet> \<Psi>\<^sub>Q)
goal (1 subgoal):
1. Prop (P' \<parallel> Q)
[PROOF STEP]
by(rule_tac rPar1) (assumption | simp)+
[PROOF STATE]
proof (state)
this:
Prop (P' \<parallel> Q)
goal (1 subgoal):
1. \<And>Q' A\<^sub>P \<Psi>\<^sub>P. \<lbrakk>\<Psi> \<otimes> \<Psi>\<^sub>P \<rhd> Q \<longmapsto> M\<lparr>\<nu>*xvec\<rparr>\<langle>N\<rangle> \<prec> Q'; extractFrame P = \<langle>A\<^sub>P, \<Psi>\<^sub>P\<rangle>; distinct A\<^sub>P; A\<^sub>P \<sharp>* \<Psi>; A\<^sub>P \<sharp>* P; A\<^sub>P \<sharp>* Q; A\<^sub>P \<sharp>* M; A\<^sub>P \<sharp>* xvec; A\<^sub>P \<sharp>* N; A\<^sub>P \<sharp>* (A\<^sub>P\<^sub>Q, \<Psi>\<^sub>P\<^sub>Q); A\<^sub>P \<sharp>* xvec; distinct xvec\<rbrakk> \<Longrightarrow> Prop (P \<parallel> Q')
[PROOF STEP]
next
[PROOF STATE]
proof (state)
goal (1 subgoal):
1. \<And>Q' A\<^sub>P \<Psi>\<^sub>P. \<lbrakk>\<Psi> \<otimes> \<Psi>\<^sub>P \<rhd> Q \<longmapsto> M\<lparr>\<nu>*xvec\<rparr>\<langle>N\<rangle> \<prec> Q'; extractFrame P = \<langle>A\<^sub>P, \<Psi>\<^sub>P\<rangle>; distinct A\<^sub>P; A\<^sub>P \<sharp>* \<Psi>; A\<^sub>P \<sharp>* P; A\<^sub>P \<sharp>* Q; A\<^sub>P \<sharp>* M; A\<^sub>P \<sharp>* xvec; A\<^sub>P \<sharp>* N; A\<^sub>P \<sharp>* (A\<^sub>P\<^sub>Q, \<Psi>\<^sub>P\<^sub>Q); A\<^sub>P \<sharp>* xvec; distinct xvec\<rbrakk> \<Longrightarrow> Prop (P \<parallel> Q')
[PROOF STEP]
case(cPar2 Q' A\<^sub>P \<Psi>\<^sub>P)
[PROOF STATE]
proof (state)
this:
\<Psi> \<otimes> \<Psi>\<^sub>P \<rhd> Q \<longmapsto> M\<lparr>\<nu>*xvec\<rparr>\<langle>N\<rangle> \<prec> Q'
extractFrame P = \<langle>A\<^sub>P, \<Psi>\<^sub>P\<rangle>
distinct A\<^sub>P
A\<^sub>P \<sharp>* \<Psi>
A\<^sub>P \<sharp>* P
A\<^sub>P \<sharp>* Q
A\<^sub>P \<sharp>* M
A\<^sub>P \<sharp>* xvec
A\<^sub>P \<sharp>* N
A\<^sub>P \<sharp>* (A\<^sub>P\<^sub>Q, \<Psi>\<^sub>P\<^sub>Q)
A\<^sub>P \<sharp>* xvec
distinct xvec
goal (1 subgoal):
1. \<And>Q' A\<^sub>P \<Psi>\<^sub>P. \<lbrakk>\<Psi> \<otimes> \<Psi>\<^sub>P \<rhd> Q \<longmapsto> M\<lparr>\<nu>*xvec\<rparr>\<langle>N\<rangle> \<prec> Q'; extractFrame P = \<langle>A\<^sub>P, \<Psi>\<^sub>P\<rangle>; distinct A\<^sub>P; A\<^sub>P \<sharp>* \<Psi>; A\<^sub>P \<sharp>* P; A\<^sub>P \<sharp>* Q; A\<^sub>P \<sharp>* M; A\<^sub>P \<sharp>* xvec; A\<^sub>P \<sharp>* N; A\<^sub>P \<sharp>* (A\<^sub>P\<^sub>Q, \<Psi>\<^sub>P\<^sub>Q); A\<^sub>P \<sharp>* xvec; distinct xvec\<rbrakk> \<Longrightarrow> Prop (P \<parallel> Q')
[PROOF STEP]
from \<open>A\<^sub>P \<sharp>* (A\<^sub>P\<^sub>Q, \<Psi>\<^sub>P\<^sub>Q)\<close>
[PROOF STATE]
proof (chain)
picking this:
A\<^sub>P \<sharp>* (A\<^sub>P\<^sub>Q, \<Psi>\<^sub>P\<^sub>Q)
[PROOF STEP]
have "A\<^sub>P \<sharp>* A\<^sub>P\<^sub>Q" and "A\<^sub>P \<sharp>* \<Psi>\<^sub>P\<^sub>Q"
[PROOF STATE]
proof (prove)
using this:
A\<^sub>P \<sharp>* (A\<^sub>P\<^sub>Q, \<Psi>\<^sub>P\<^sub>Q)
goal (1 subgoal):
1. A\<^sub>P \<sharp>* A\<^sub>P\<^sub>Q &&& A\<^sub>P \<sharp>* \<Psi>\<^sub>P\<^sub>Q
[PROOF STEP]
by simp+
[PROOF STATE]
proof (state)
this:
A\<^sub>P \<sharp>* A\<^sub>P\<^sub>Q
A\<^sub>P \<sharp>* \<Psi>\<^sub>P\<^sub>Q
goal (1 subgoal):
1. \<And>Q' A\<^sub>P \<Psi>\<^sub>P. \<lbrakk>\<Psi> \<otimes> \<Psi>\<^sub>P \<rhd> Q \<longmapsto> M\<lparr>\<nu>*xvec\<rparr>\<langle>N\<rangle> \<prec> Q'; extractFrame P = \<langle>A\<^sub>P, \<Psi>\<^sub>P\<rangle>; distinct A\<^sub>P; A\<^sub>P \<sharp>* \<Psi>; A\<^sub>P \<sharp>* P; A\<^sub>P \<sharp>* Q; A\<^sub>P \<sharp>* M; A\<^sub>P \<sharp>* xvec; A\<^sub>P \<sharp>* N; A\<^sub>P \<sharp>* (A\<^sub>P\<^sub>Q, \<Psi>\<^sub>P\<^sub>Q); A\<^sub>P \<sharp>* xvec; distinct xvec\<rbrakk> \<Longrightarrow> Prop (P \<parallel> Q')
[PROOF STEP]
obtain A\<^sub>Q \<Psi>\<^sub>Q where FrQ: "extractFrame Q = \<langle>A\<^sub>Q, \<Psi>\<^sub>Q\<rangle>" and "distinct A\<^sub>Q"
"A\<^sub>Q \<sharp>* (P, Q, \<Psi>, M, A\<^sub>P, A\<^sub>P\<^sub>Q, \<Psi>\<^sub>P)"
[PROOF STATE]
proof (prove)
goal (1 subgoal):
1. (\<And>A\<^sub>Q \<Psi>\<^sub>Q. \<lbrakk>extractFrame Q = \<langle>A\<^sub>Q, \<Psi>\<^sub>Q\<rangle>; distinct A\<^sub>Q; A\<^sub>Q \<sharp>* (P, Q, \<Psi>, M, A\<^sub>P, A\<^sub>P\<^sub>Q, \<Psi>\<^sub>P)\<rbrakk> \<Longrightarrow> thesis) \<Longrightarrow> thesis
[PROOF STEP]
by(rule freshFrame)
[PROOF STATE]
proof (state)
this:
extractFrame Q = \<langle>A\<^sub>Q, \<Psi>\<^sub>Q\<rangle>
distinct A\<^sub>Q
A\<^sub>Q \<sharp>* (P, Q, \<Psi>, M, A\<^sub>P, A\<^sub>P\<^sub>Q, \<Psi>\<^sub>P)
goal (1 subgoal):
1. \<And>Q' A\<^sub>P \<Psi>\<^sub>P. \<lbrakk>\<Psi> \<otimes> \<Psi>\<^sub>P \<rhd> Q \<longmapsto> M\<lparr>\<nu>*xvec\<rparr>\<langle>N\<rangle> \<prec> Q'; extractFrame P = \<langle>A\<^sub>P, \<Psi>\<^sub>P\<rangle>; distinct A\<^sub>P; A\<^sub>P \<sharp>* \<Psi>; A\<^sub>P \<sharp>* P; A\<^sub>P \<sharp>* Q; A\<^sub>P \<sharp>* M; A\<^sub>P \<sharp>* xvec; A\<^sub>P \<sharp>* N; A\<^sub>P \<sharp>* (A\<^sub>P\<^sub>Q, \<Psi>\<^sub>P\<^sub>Q); A\<^sub>P \<sharp>* xvec; distinct xvec\<rbrakk> \<Longrightarrow> Prop (P \<parallel> Q')
[PROOF STEP]
hence "A\<^sub>Q \<sharp>* P" and "A\<^sub>Q \<sharp>* Q" and "A\<^sub>Q \<sharp>* \<Psi>" and "A\<^sub>Q \<sharp>* M" and "A\<^sub>Q \<sharp>* A\<^sub>P" and "A\<^sub>Q \<sharp>* A\<^sub>P\<^sub>Q" and "A\<^sub>Q \<sharp>* \<Psi>\<^sub>P"
[PROOF STATE]
proof (prove)
using this:
extractFrame Q = \<langle>A\<^sub>Q, \<Psi>\<^sub>Q\<rangle>
distinct A\<^sub>Q
A\<^sub>Q \<sharp>* (P, Q, \<Psi>, M, A\<^sub>P, A\<^sub>P\<^sub>Q, \<Psi>\<^sub>P)
goal (1 subgoal):
1. (A\<^sub>Q \<sharp>* P &&& A\<^sub>Q \<sharp>* Q &&& A\<^sub>Q \<sharp>* \<Psi>) &&& (A\<^sub>Q \<sharp>* M &&& A\<^sub>Q \<sharp>* A\<^sub>P) &&& A\<^sub>Q \<sharp>* A\<^sub>P\<^sub>Q &&& A\<^sub>Q \<sharp>* \<Psi>\<^sub>P
[PROOF STEP]
by simp+
[PROOF STATE]
proof (state)
this:
A\<^sub>Q \<sharp>* P
A\<^sub>Q \<sharp>* Q
A\<^sub>Q \<sharp>* \<Psi>
A\<^sub>Q \<sharp>* M
A\<^sub>Q \<sharp>* A\<^sub>P
A\<^sub>Q \<sharp>* A\<^sub>P\<^sub>Q
A\<^sub>Q \<sharp>* \<Psi>\<^sub>P
goal (1 subgoal):
1. \<And>Q' A\<^sub>P \<Psi>\<^sub>P. \<lbrakk>\<Psi> \<otimes> \<Psi>\<^sub>P \<rhd> Q \<longmapsto> M\<lparr>\<nu>*xvec\<rparr>\<langle>N\<rangle> \<prec> Q'; extractFrame P = \<langle>A\<^sub>P, \<Psi>\<^sub>P\<rangle>; distinct A\<^sub>P; A\<^sub>P \<sharp>* \<Psi>; A\<^sub>P \<sharp>* P; A\<^sub>P \<sharp>* Q; A\<^sub>P \<sharp>* M; A\<^sub>P \<sharp>* xvec; A\<^sub>P \<sharp>* N; A\<^sub>P \<sharp>* (A\<^sub>P\<^sub>Q, \<Psi>\<^sub>P\<^sub>Q); A\<^sub>P \<sharp>* xvec; distinct xvec\<rbrakk> \<Longrightarrow> Prop (P \<parallel> Q')
[PROOF STEP]
have FrP: "extractFrame P = \<langle>A\<^sub>P, \<Psi>\<^sub>P\<rangle>"
[PROOF STATE]
proof (prove)
goal (1 subgoal):
1. extractFrame P = \<langle>A\<^sub>P, \<Psi>\<^sub>P\<rangle>
[PROOF STEP]
by fact
[PROOF STATE]
proof (state)
this:
extractFrame P = \<langle>A\<^sub>P, \<Psi>\<^sub>P\<rangle>
goal (1 subgoal):
1. \<And>Q' A\<^sub>P \<Psi>\<^sub>P. \<lbrakk>\<Psi> \<otimes> \<Psi>\<^sub>P \<rhd> Q \<longmapsto> M\<lparr>\<nu>*xvec\<rparr>\<langle>N\<rangle> \<prec> Q'; extractFrame P = \<langle>A\<^sub>P, \<Psi>\<^sub>P\<rangle>; distinct A\<^sub>P; A\<^sub>P \<sharp>* \<Psi>; A\<^sub>P \<sharp>* P; A\<^sub>P \<sharp>* Q; A\<^sub>P \<sharp>* M; A\<^sub>P \<sharp>* xvec; A\<^sub>P \<sharp>* N; A\<^sub>P \<sharp>* (A\<^sub>P\<^sub>Q, \<Psi>\<^sub>P\<^sub>Q); A\<^sub>P \<sharp>* xvec; distinct xvec\<rbrakk> \<Longrightarrow> Prop (P \<parallel> Q')
[PROOF STEP]
from \<open>A\<^sub>P \<sharp>* Q\<close> \<open>A\<^sub>Q \<sharp>* A\<^sub>P\<close> FrQ
[PROOF STATE]
proof (chain)
picking this:
A\<^sub>P \<sharp>* Q
A\<^sub>Q \<sharp>* A\<^sub>P
extractFrame Q = \<langle>A\<^sub>Q, \<Psi>\<^sub>Q\<rangle>
[PROOF STEP]
have "A\<^sub>P \<sharp>* \<Psi>\<^sub>Q"
[PROOF STATE]
proof (prove)
using this:
A\<^sub>P \<sharp>* Q
A\<^sub>Q \<sharp>* A\<^sub>P
extractFrame Q = \<langle>A\<^sub>Q, \<Psi>\<^sub>Q\<rangle>
goal (1 subgoal):
1. A\<^sub>P \<sharp>* \<Psi>\<^sub>Q
[PROOF STEP]
by(force dest: extractFrameFreshChain)
[PROOF STATE]
proof (state)
this:
A\<^sub>P \<sharp>* \<Psi>\<^sub>Q
goal (1 subgoal):
1. \<And>Q' A\<^sub>P \<Psi>\<^sub>P. \<lbrakk>\<Psi> \<otimes> \<Psi>\<^sub>P \<rhd> Q \<longmapsto> M\<lparr>\<nu>*xvec\<rparr>\<langle>N\<rangle> \<prec> Q'; extractFrame P = \<langle>A\<^sub>P, \<Psi>\<^sub>P\<rangle>; distinct A\<^sub>P; A\<^sub>P \<sharp>* \<Psi>; A\<^sub>P \<sharp>* P; A\<^sub>P \<sharp>* Q; A\<^sub>P \<sharp>* M; A\<^sub>P \<sharp>* xvec; A\<^sub>P \<sharp>* N; A\<^sub>P \<sharp>* (A\<^sub>P\<^sub>Q, \<Psi>\<^sub>P\<^sub>Q); A\<^sub>P \<sharp>* xvec; distinct xvec\<rbrakk> \<Longrightarrow> Prop (P \<parallel> Q')
[PROOF STEP]
from \<open>extractFrame(P \<parallel> Q) = \<langle>A\<^sub>P\<^sub>Q, \<Psi>\<^sub>P\<^sub>Q\<rangle>\<close> FrP FrQ \<open>A\<^sub>Q \<sharp>* A\<^sub>P\<close> \<open>A\<^sub>P \<sharp>* \<Psi>\<^sub>Q\<close> \<open>A\<^sub>Q \<sharp>* \<Psi>\<^sub>P\<close>
[PROOF STATE]
proof (chain)
picking this:
extractFrame (P \<parallel> Q) = \<langle>A\<^sub>P\<^sub>Q, \<Psi>\<^sub>P\<^sub>Q\<rangle>
extractFrame P = \<langle>A\<^sub>P, \<Psi>\<^sub>P\<rangle>
extractFrame Q = \<langle>A\<^sub>Q, \<Psi>\<^sub>Q\<rangle>
A\<^sub>Q \<sharp>* A\<^sub>P
A\<^sub>P \<sharp>* \<Psi>\<^sub>Q
A\<^sub>Q \<sharp>* \<Psi>\<^sub>P
[PROOF STEP]
have "\<langle>(A\<^sub>P@A\<^sub>Q), \<Psi>\<^sub>P \<otimes> \<Psi>\<^sub>Q\<rangle> = \<langle>A\<^sub>P\<^sub>Q, \<Psi>\<^sub>P\<^sub>Q\<rangle>"
[PROOF STATE]
proof (prove)
using this:
extractFrame (P \<parallel> Q) = \<langle>A\<^sub>P\<^sub>Q, \<Psi>\<^sub>P\<^sub>Q\<rangle>
extractFrame P = \<langle>A\<^sub>P, \<Psi>\<^sub>P\<rangle>
extractFrame Q = \<langle>A\<^sub>Q, \<Psi>\<^sub>Q\<rangle>
A\<^sub>Q \<sharp>* A\<^sub>P
A\<^sub>P \<sharp>* \<Psi>\<^sub>Q
A\<^sub>Q \<sharp>* \<Psi>\<^sub>P
goal (1 subgoal):
1. \<langle>(A\<^sub>P @ A\<^sub>Q), \<Psi>\<^sub>P \<otimes> \<Psi>\<^sub>Q\<rangle> = \<langle>A\<^sub>P\<^sub>Q, \<Psi>\<^sub>P\<^sub>Q\<rangle>
[PROOF STEP]
by simp
[PROOF STATE]
proof (state)
this:
\<langle>(A\<^sub>P @ A\<^sub>Q), \<Psi>\<^sub>P \<otimes> \<Psi>\<^sub>Q\<rangle> = \<langle>A\<^sub>P\<^sub>Q, \<Psi>\<^sub>P\<^sub>Q\<rangle>
goal (1 subgoal):
1. \<And>Q' A\<^sub>P \<Psi>\<^sub>P. \<lbrakk>\<Psi> \<otimes> \<Psi>\<^sub>P \<rhd> Q \<longmapsto> M\<lparr>\<nu>*xvec\<rparr>\<langle>N\<rangle> \<prec> Q'; extractFrame P = \<langle>A\<^sub>P, \<Psi>\<^sub>P\<rangle>; distinct A\<^sub>P; A\<^sub>P \<sharp>* \<Psi>; A\<^sub>P \<sharp>* P; A\<^sub>P \<sharp>* Q; A\<^sub>P \<sharp>* M; A\<^sub>P \<sharp>* xvec; A\<^sub>P \<sharp>* N; A\<^sub>P \<sharp>* (A\<^sub>P\<^sub>Q, \<Psi>\<^sub>P\<^sub>Q); A\<^sub>P \<sharp>* xvec; distinct xvec\<rbrakk> \<Longrightarrow> Prop (P \<parallel> Q')
[PROOF STEP]
moreover
[PROOF STATE]
proof (state)
this:
\<langle>(A\<^sub>P @ A\<^sub>Q), \<Psi>\<^sub>P \<otimes> \<Psi>\<^sub>Q\<rangle> = \<langle>A\<^sub>P\<^sub>Q, \<Psi>\<^sub>P\<^sub>Q\<rangle>
goal (1 subgoal):
1. \<And>Q' A\<^sub>P \<Psi>\<^sub>P. \<lbrakk>\<Psi> \<otimes> \<Psi>\<^sub>P \<rhd> Q \<longmapsto> M\<lparr>\<nu>*xvec\<rparr>\<langle>N\<rangle> \<prec> Q'; extractFrame P = \<langle>A\<^sub>P, \<Psi>\<^sub>P\<rangle>; distinct A\<^sub>P; A\<^sub>P \<sharp>* \<Psi>; A\<^sub>P \<sharp>* P; A\<^sub>P \<sharp>* Q; A\<^sub>P \<sharp>* M; A\<^sub>P \<sharp>* xvec; A\<^sub>P \<sharp>* N; A\<^sub>P \<sharp>* (A\<^sub>P\<^sub>Q, \<Psi>\<^sub>P\<^sub>Q); A\<^sub>P \<sharp>* xvec; distinct xvec\<rbrakk> \<Longrightarrow> Prop (P \<parallel> Q')
[PROOF STEP]
from \<open>distinct A\<^sub>P\<close> \<open>distinct A\<^sub>Q\<close> \<open>A\<^sub>Q \<sharp>* A\<^sub>P\<close>
[PROOF STATE]
proof (chain)
picking this:
distinct A\<^sub>P
distinct A\<^sub>Q
A\<^sub>Q \<sharp>* A\<^sub>P
[PROOF STEP]
have "distinct(A\<^sub>P@A\<^sub>Q)"
[PROOF STATE]
proof (prove)
using this:
distinct A\<^sub>P
distinct A\<^sub>Q
A\<^sub>Q \<sharp>* A\<^sub>P
goal (1 subgoal):
1. distinct (A\<^sub>P @ A\<^sub>Q)
[PROOF STEP]
by(auto simp add: fresh_star_def fresh_def name_list_supp)
[PROOF STATE]
proof (state)
this:
distinct (A\<^sub>P @ A\<^sub>Q)
goal (1 subgoal):
1. \<And>Q' A\<^sub>P \<Psi>\<^sub>P. \<lbrakk>\<Psi> \<otimes> \<Psi>\<^sub>P \<rhd> Q \<longmapsto> M\<lparr>\<nu>*xvec\<rparr>\<langle>N\<rangle> \<prec> Q'; extractFrame P = \<langle>A\<^sub>P, \<Psi>\<^sub>P\<rangle>; distinct A\<^sub>P; A\<^sub>P \<sharp>* \<Psi>; A\<^sub>P \<sharp>* P; A\<^sub>P \<sharp>* Q; A\<^sub>P \<sharp>* M; A\<^sub>P \<sharp>* xvec; A\<^sub>P \<sharp>* N; A\<^sub>P \<sharp>* (A\<^sub>P\<^sub>Q, \<Psi>\<^sub>P\<^sub>Q); A\<^sub>P \<sharp>* xvec; distinct xvec\<rbrakk> \<Longrightarrow> Prop (P \<parallel> Q')
[PROOF STEP]
ultimately
[PROOF STATE]
proof (chain)
picking this:
\<langle>(A\<^sub>P @ A\<^sub>Q), \<Psi>\<^sub>P \<otimes> \<Psi>\<^sub>Q\<rangle> = \<langle>A\<^sub>P\<^sub>Q, \<Psi>\<^sub>P\<^sub>Q\<rangle>
distinct (A\<^sub>P @ A\<^sub>Q)
[PROOF STEP]
obtain p where S: "set p \<subseteq> set(A\<^sub>P@A\<^sub>Q) \<times> set((p \<bullet> A\<^sub>P)@(p \<bullet> A\<^sub>Q))" and "distinctPerm p"
and \<Psi>eq: "\<Psi>\<^sub>P\<^sub>Q = (p \<bullet> \<Psi>\<^sub>P) \<otimes> (p \<bullet> \<Psi>\<^sub>Q)" and Aeq: "A\<^sub>P\<^sub>Q = (p \<bullet> A\<^sub>P)@(p \<bullet> A\<^sub>Q)"
[PROOF STATE]
proof (prove)
using this:
\<langle>(A\<^sub>P @ A\<^sub>Q), \<Psi>\<^sub>P \<otimes> \<Psi>\<^sub>Q\<rangle> = \<langle>A\<^sub>P\<^sub>Q, \<Psi>\<^sub>P\<^sub>Q\<rangle>
distinct (A\<^sub>P @ A\<^sub>Q)
goal (1 subgoal):
1. (\<And>p. \<lbrakk>set p \<subseteq> set (A\<^sub>P @ A\<^sub>Q) \<times> set (p \<bullet> A\<^sub>P @ p \<bullet> A\<^sub>Q); distinctPerm p; \<Psi>\<^sub>P\<^sub>Q = (p \<bullet> \<Psi>\<^sub>P) \<otimes> (p \<bullet> \<Psi>\<^sub>Q); A\<^sub>P\<^sub>Q = p \<bullet> A\<^sub>P @ p \<bullet> A\<^sub>Q\<rbrakk> \<Longrightarrow> thesis) \<Longrightarrow> thesis
[PROOF STEP]
using \<open>A\<^sub>P \<sharp>* A\<^sub>P\<^sub>Q\<close> \<open>A\<^sub>Q \<sharp>* A\<^sub>P\<^sub>Q\<close> \<open>distinct A\<^sub>P\<^sub>Q\<close>
[PROOF STATE]
proof (prove)
using this:
\<langle>(A\<^sub>P @ A\<^sub>Q), \<Psi>\<^sub>P \<otimes> \<Psi>\<^sub>Q\<rangle> = \<langle>A\<^sub>P\<^sub>Q, \<Psi>\<^sub>P\<^sub>Q\<rangle>
distinct (A\<^sub>P @ A\<^sub>Q)
A\<^sub>P \<sharp>* A\<^sub>P\<^sub>Q
A\<^sub>Q \<sharp>* A\<^sub>P\<^sub>Q
distinct A\<^sub>P\<^sub>Q
goal (1 subgoal):
1. (\<And>p. \<lbrakk>set p \<subseteq> set (A\<^sub>P @ A\<^sub>Q) \<times> set (p \<bullet> A\<^sub>P @ p \<bullet> A\<^sub>Q); distinctPerm p; \<Psi>\<^sub>P\<^sub>Q = (p \<bullet> \<Psi>\<^sub>P) \<otimes> (p \<bullet> \<Psi>\<^sub>Q); A\<^sub>P\<^sub>Q = p \<bullet> A\<^sub>P @ p \<bullet> A\<^sub>Q\<rbrakk> \<Longrightarrow> thesis) \<Longrightarrow> thesis
[PROOF STEP]
by(rule_tac frameChainEq') (assumption | simp add: eqvts)+
[PROOF STATE]
proof (state)
this:
set p \<subseteq> set (A\<^sub>P @ A\<^sub>Q) \<times> set (p \<bullet> A\<^sub>P @ p \<bullet> A\<^sub>Q)
distinctPerm p
\<Psi>\<^sub>P\<^sub>Q = (p \<bullet> \<Psi>\<^sub>P) \<otimes> (p \<bullet> \<Psi>\<^sub>Q)
A\<^sub>P\<^sub>Q = p \<bullet> A\<^sub>P @ p \<bullet> A\<^sub>Q
goal (1 subgoal):
1. \<And>Q' A\<^sub>P \<Psi>\<^sub>P. \<lbrakk>\<Psi> \<otimes> \<Psi>\<^sub>P \<rhd> Q \<longmapsto> M\<lparr>\<nu>*xvec\<rparr>\<langle>N\<rangle> \<prec> Q'; extractFrame P = \<langle>A\<^sub>P, \<Psi>\<^sub>P\<rangle>; distinct A\<^sub>P; A\<^sub>P \<sharp>* \<Psi>; A\<^sub>P \<sharp>* P; A\<^sub>P \<sharp>* Q; A\<^sub>P \<sharp>* M; A\<^sub>P \<sharp>* xvec; A\<^sub>P \<sharp>* N; A\<^sub>P \<sharp>* (A\<^sub>P\<^sub>Q, \<Psi>\<^sub>P\<^sub>Q); A\<^sub>P \<sharp>* xvec; distinct xvec\<rbrakk> \<Longrightarrow> Prop (P \<parallel> Q')
[PROOF STEP]
from \<open>\<Psi> \<otimes> \<Psi>\<^sub>P \<rhd> Q \<longmapsto>M\<lparr>\<nu>*xvec\<rparr>\<langle>N\<rangle> \<prec> Q'\<close> S \<open>A\<^sub>P\<^sub>Q \<sharp>* Q\<close> \<open>A\<^sub>P \<sharp>* Q\<close> \<open>A\<^sub>Q \<sharp>* Q\<close> \<open>A\<^sub>P\<^sub>Q \<sharp>* M\<close> \<open>A\<^sub>P \<sharp>* M\<close> \<open>A\<^sub>Q \<sharp>* M\<close> Aeq
[PROOF STATE]
proof (chain)
picking this:
\<Psi> \<otimes> \<Psi>\<^sub>P \<rhd> Q \<longmapsto> M\<lparr>\<nu>*xvec\<rparr>\<langle>N\<rangle> \<prec> Q'
set p \<subseteq> set (A\<^sub>P @ A\<^sub>Q) \<times> set (p \<bullet> A\<^sub>P @ p \<bullet> A\<^sub>Q)
A\<^sub>P\<^sub>Q \<sharp>* Q
A\<^sub>P \<sharp>* Q
A\<^sub>Q \<sharp>* Q
A\<^sub>P\<^sub>Q \<sharp>* M
A\<^sub>P \<sharp>* M
A\<^sub>Q \<sharp>* M
A\<^sub>P\<^sub>Q = p \<bullet> A\<^sub>P @ p \<bullet> A\<^sub>Q
[PROOF STEP]
have "(p \<bullet> (\<Psi> \<otimes> \<Psi>\<^sub>P)) \<rhd> Q \<longmapsto>M\<lparr>\<nu>*xvec\<rparr>\<langle>N\<rangle> \<prec> Q'"
[PROOF STATE]
proof (prove)
using this:
\<Psi> \<otimes> \<Psi>\<^sub>P \<rhd> Q \<longmapsto> M\<lparr>\<nu>*xvec\<rparr>\<langle>N\<rangle> \<prec> Q'
set p \<subseteq> set (A\<^sub>P @ A\<^sub>Q) \<times> set (p \<bullet> A\<^sub>P @ p \<bullet> A\<^sub>Q)
A\<^sub>P\<^sub>Q \<sharp>* Q
A\<^sub>P \<sharp>* Q
A\<^sub>Q \<sharp>* Q
A\<^sub>P\<^sub>Q \<sharp>* M
A\<^sub>P \<sharp>* M
A\<^sub>Q \<sharp>* M
A\<^sub>P\<^sub>Q = p \<bullet> A\<^sub>P @ p \<bullet> A\<^sub>Q
goal (1 subgoal):
1. p \<bullet> \<Psi> \<otimes> \<Psi>\<^sub>P \<rhd> Q \<longmapsto> M\<lparr>\<nu>*xvec\<rparr>\<langle>N\<rangle> \<prec> Q'
[PROOF STEP]
by(rule_tac outputPermFrame) (assumption | simp)+
[PROOF STATE]
proof (state)
this:
p \<bullet> \<Psi> \<otimes> \<Psi>\<^sub>P \<rhd> Q \<longmapsto> M\<lparr>\<nu>*xvec\<rparr>\<langle>N\<rangle> \<prec> Q'
goal (1 subgoal):
1. \<And>Q' A\<^sub>P \<Psi>\<^sub>P. \<lbrakk>\<Psi> \<otimes> \<Psi>\<^sub>P \<rhd> Q \<longmapsto> M\<lparr>\<nu>*xvec\<rparr>\<langle>N\<rangle> \<prec> Q'; extractFrame P = \<langle>A\<^sub>P, \<Psi>\<^sub>P\<rangle>; distinct A\<^sub>P; A\<^sub>P \<sharp>* \<Psi>; A\<^sub>P \<sharp>* P; A\<^sub>P \<sharp>* Q; A\<^sub>P \<sharp>* M; A\<^sub>P \<sharp>* xvec; A\<^sub>P \<sharp>* N; A\<^sub>P \<sharp>* (A\<^sub>P\<^sub>Q, \<Psi>\<^sub>P\<^sub>Q); A\<^sub>P \<sharp>* xvec; distinct xvec\<rbrakk> \<Longrightarrow> Prop (P \<parallel> Q')
[PROOF STEP]
with S \<open>A\<^sub>P\<^sub>Q \<sharp>* \<Psi>\<close> \<open>A\<^sub>P \<sharp>* \<Psi>\<close> \<open>A\<^sub>Q \<sharp>* \<Psi>\<close> Aeq
[PROOF STATE]
proof (chain)
picking this:
set p \<subseteq> set (A\<^sub>P @ A\<^sub>Q) \<times> set (p \<bullet> A\<^sub>P @ p \<bullet> A\<^sub>Q)
A\<^sub>P\<^sub>Q \<sharp>* \<Psi>
A\<^sub>P \<sharp>* \<Psi>
A\<^sub>Q \<sharp>* \<Psi>
A\<^sub>P\<^sub>Q = p \<bullet> A\<^sub>P @ p \<bullet> A\<^sub>Q
p \<bullet> \<Psi> \<otimes> \<Psi>\<^sub>P \<rhd> Q \<longmapsto> M\<lparr>\<nu>*xvec\<rparr>\<langle>N\<rangle> \<prec> Q'
[PROOF STEP]
have "\<Psi> \<otimes> (p \<bullet> \<Psi>\<^sub>P) \<rhd> Q \<longmapsto>M\<lparr>\<nu>*xvec\<rparr>\<langle>N\<rangle> \<prec> Q'"
[PROOF STATE]
proof (prove)
using this:
set p \<subseteq> set (A\<^sub>P @ A\<^sub>Q) \<times> set (p \<bullet> A\<^sub>P @ p \<bullet> A\<^sub>Q)
A\<^sub>P\<^sub>Q \<sharp>* \<Psi>
A\<^sub>P \<sharp>* \<Psi>
A\<^sub>Q \<sharp>* \<Psi>
A\<^sub>P\<^sub>Q = p \<bullet> A\<^sub>P @ p \<bullet> A\<^sub>Q
p \<bullet> \<Psi> \<otimes> \<Psi>\<^sub>P \<rhd> Q \<longmapsto> M\<lparr>\<nu>*xvec\<rparr>\<langle>N\<rangle> \<prec> Q'
goal (1 subgoal):
1. \<Psi> \<otimes> (p \<bullet> \<Psi>\<^sub>P) \<rhd> Q \<longmapsto> M\<lparr>\<nu>*xvec\<rparr>\<langle>N\<rangle> \<prec> Q'
[PROOF STEP]
by(simp add: eqvts)
[PROOF STATE]
proof (state)
this:
\<Psi> \<otimes> (p \<bullet> \<Psi>\<^sub>P) \<rhd> Q \<longmapsto> M\<lparr>\<nu>*xvec\<rparr>\<langle>N\<rangle> \<prec> Q'
goal (1 subgoal):
1. \<And>Q' A\<^sub>P \<Psi>\<^sub>P. \<lbrakk>\<Psi> \<otimes> \<Psi>\<^sub>P \<rhd> Q \<longmapsto> M\<lparr>\<nu>*xvec\<rparr>\<langle>N\<rangle> \<prec> Q'; extractFrame P = \<langle>A\<^sub>P, \<Psi>\<^sub>P\<rangle>; distinct A\<^sub>P; A\<^sub>P \<sharp>* \<Psi>; A\<^sub>P \<sharp>* P; A\<^sub>P \<sharp>* Q; A\<^sub>P \<sharp>* M; A\<^sub>P \<sharp>* xvec; A\<^sub>P \<sharp>* N; A\<^sub>P \<sharp>* (A\<^sub>P\<^sub>Q, \<Psi>\<^sub>P\<^sub>Q); A\<^sub>P \<sharp>* xvec; distinct xvec\<rbrakk> \<Longrightarrow> Prop (P \<parallel> Q')
[PROOF STEP]
moreover
[PROOF STATE]
proof (state)
this:
\<Psi> \<otimes> (p \<bullet> \<Psi>\<^sub>P) \<rhd> Q \<longmapsto> M\<lparr>\<nu>*xvec\<rparr>\<langle>N\<rangle> \<prec> Q'
goal (1 subgoal):
1. \<And>Q' A\<^sub>P \<Psi>\<^sub>P. \<lbrakk>\<Psi> \<otimes> \<Psi>\<^sub>P \<rhd> Q \<longmapsto> M\<lparr>\<nu>*xvec\<rparr>\<langle>N\<rangle> \<prec> Q'; extractFrame P = \<langle>A\<^sub>P, \<Psi>\<^sub>P\<rangle>; distinct A\<^sub>P; A\<^sub>P \<sharp>* \<Psi>; A\<^sub>P \<sharp>* P; A\<^sub>P \<sharp>* Q; A\<^sub>P \<sharp>* M; A\<^sub>P \<sharp>* xvec; A\<^sub>P \<sharp>* N; A\<^sub>P \<sharp>* (A\<^sub>P\<^sub>Q, \<Psi>\<^sub>P\<^sub>Q); A\<^sub>P \<sharp>* xvec; distinct xvec\<rbrakk> \<Longrightarrow> Prop (P \<parallel> Q')
[PROOF STEP]
from FrP
[PROOF STATE]
proof (chain)
picking this:
extractFrame P = \<langle>A\<^sub>P, \<Psi>\<^sub>P\<rangle>
[PROOF STEP]
have "(p \<bullet> extractFrame P) = p \<bullet> \<langle>A\<^sub>P, \<Psi>\<^sub>P\<rangle>"
[PROOF STATE]
proof (prove)
using this:
extractFrame P = \<langle>A\<^sub>P, \<Psi>\<^sub>P\<rangle>
goal (1 subgoal):
1. p \<bullet> extractFrame P = p \<bullet> \<langle>A\<^sub>P, \<Psi>\<^sub>P\<rangle>
[PROOF STEP]
by simp
[PROOF STATE]
proof (state)
this:
p \<bullet> extractFrame P = p \<bullet> \<langle>A\<^sub>P, \<Psi>\<^sub>P\<rangle>
goal (1 subgoal):
1. \<And>Q' A\<^sub>P \<Psi>\<^sub>P. \<lbrakk>\<Psi> \<otimes> \<Psi>\<^sub>P \<rhd> Q \<longmapsto> M\<lparr>\<nu>*xvec\<rparr>\<langle>N\<rangle> \<prec> Q'; extractFrame P = \<langle>A\<^sub>P, \<Psi>\<^sub>P\<rangle>; distinct A\<^sub>P; A\<^sub>P \<sharp>* \<Psi>; A\<^sub>P \<sharp>* P; A\<^sub>P \<sharp>* Q; A\<^sub>P \<sharp>* M; A\<^sub>P \<sharp>* xvec; A\<^sub>P \<sharp>* N; A\<^sub>P \<sharp>* (A\<^sub>P\<^sub>Q, \<Psi>\<^sub>P\<^sub>Q); A\<^sub>P \<sharp>* xvec; distinct xvec\<rbrakk> \<Longrightarrow> Prop (P \<parallel> Q')
[PROOF STEP]
with S \<open>A\<^sub>P\<^sub>Q \<sharp>* P\<close> \<open>A\<^sub>P \<sharp>* P\<close> \<open>A\<^sub>Q \<sharp>* P\<close> Aeq
[PROOF STATE]
proof (chain)
picking this:
set p \<subseteq> set (A\<^sub>P @ A\<^sub>Q) \<times> set (p \<bullet> A\<^sub>P @ p \<bullet> A\<^sub>Q)
A\<^sub>P\<^sub>Q \<sharp>* P
A\<^sub>P \<sharp>* P
A\<^sub>Q \<sharp>* P
A\<^sub>P\<^sub>Q = p \<bullet> A\<^sub>P @ p \<bullet> A\<^sub>Q
p \<bullet> extractFrame P = p \<bullet> \<langle>A\<^sub>P, \<Psi>\<^sub>P\<rangle>
[PROOF STEP]
have "extractFrame P = \<langle>(p \<bullet> A\<^sub>P), p \<bullet> \<Psi>\<^sub>P\<rangle>"
[PROOF STATE]
proof (prove)
using this:
set p \<subseteq> set (A\<^sub>P @ A\<^sub>Q) \<times> set (p \<bullet> A\<^sub>P @ p \<bullet> A\<^sub>Q)
A\<^sub>P\<^sub>Q \<sharp>* P
A\<^sub>P \<sharp>* P
A\<^sub>Q \<sharp>* P
A\<^sub>P\<^sub>Q = p \<bullet> A\<^sub>P @ p \<bullet> A\<^sub>Q
p \<bullet> extractFrame P = p \<bullet> \<langle>A\<^sub>P, \<Psi>\<^sub>P\<rangle>
goal (1 subgoal):
1. extractFrame P = \<langle>p \<bullet> A\<^sub>P, p \<bullet> \<Psi>\<^sub>P\<rangle>
[PROOF STEP]
by(simp add: eqvts)
[PROOF STATE]
proof (state)
this:
extractFrame P = \<langle>p \<bullet> A\<^sub>P, p \<bullet> \<Psi>\<^sub>P\<rangle>
goal (1 subgoal):
1. \<And>Q' A\<^sub>P \<Psi>\<^sub>P. \<lbrakk>\<Psi> \<otimes> \<Psi>\<^sub>P \<rhd> Q \<longmapsto> M\<lparr>\<nu>*xvec\<rparr>\<langle>N\<rangle> \<prec> Q'; extractFrame P = \<langle>A\<^sub>P, \<Psi>\<^sub>P\<rangle>; distinct A\<^sub>P; A\<^sub>P \<sharp>* \<Psi>; A\<^sub>P \<sharp>* P; A\<^sub>P \<sharp>* Q; A\<^sub>P \<sharp>* M; A\<^sub>P \<sharp>* xvec; A\<^sub>P \<sharp>* N; A\<^sub>P \<sharp>* (A\<^sub>P\<^sub>Q, \<Psi>\<^sub>P\<^sub>Q); A\<^sub>P \<sharp>* xvec; distinct xvec\<rbrakk> \<Longrightarrow> Prop (P \<parallel> Q')
[PROOF STEP]
moreover
[PROOF STATE]
proof (state)
this:
extractFrame P = \<langle>p \<bullet> A\<^sub>P, p \<bullet> \<Psi>\<^sub>P\<rangle>
goal (1 subgoal):
1. \<And>Q' A\<^sub>P \<Psi>\<^sub>P. \<lbrakk>\<Psi> \<otimes> \<Psi>\<^sub>P \<rhd> Q \<longmapsto> M\<lparr>\<nu>*xvec\<rparr>\<langle>N\<rangle> \<prec> Q'; extractFrame P = \<langle>A\<^sub>P, \<Psi>\<^sub>P\<rangle>; distinct A\<^sub>P; A\<^sub>P \<sharp>* \<Psi>; A\<^sub>P \<sharp>* P; A\<^sub>P \<sharp>* Q; A\<^sub>P \<sharp>* M; A\<^sub>P \<sharp>* xvec; A\<^sub>P \<sharp>* N; A\<^sub>P \<sharp>* (A\<^sub>P\<^sub>Q, \<Psi>\<^sub>P\<^sub>Q); A\<^sub>P \<sharp>* xvec; distinct xvec\<rbrakk> \<Longrightarrow> Prop (P \<parallel> Q')
[PROOF STEP]
from FrQ
[PROOF STATE]
proof (chain)
picking this:
extractFrame Q = \<langle>A\<^sub>Q, \<Psi>\<^sub>Q\<rangle>
[PROOF STEP]
have "(p \<bullet> extractFrame Q) = p \<bullet> \<langle>A\<^sub>Q, \<Psi>\<^sub>Q\<rangle>"
[PROOF STATE]
proof (prove)
using this:
extractFrame Q = \<langle>A\<^sub>Q, \<Psi>\<^sub>Q\<rangle>
goal (1 subgoal):
1. p \<bullet> extractFrame Q = p \<bullet> \<langle>A\<^sub>Q, \<Psi>\<^sub>Q\<rangle>
[PROOF STEP]
by simp
[PROOF STATE]
proof (state)
this:
p \<bullet> extractFrame Q = p \<bullet> \<langle>A\<^sub>Q, \<Psi>\<^sub>Q\<rangle>
goal (1 subgoal):
1. \<And>Q' A\<^sub>P \<Psi>\<^sub>P. \<lbrakk>\<Psi> \<otimes> \<Psi>\<^sub>P \<rhd> Q \<longmapsto> M\<lparr>\<nu>*xvec\<rparr>\<langle>N\<rangle> \<prec> Q'; extractFrame P = \<langle>A\<^sub>P, \<Psi>\<^sub>P\<rangle>; distinct A\<^sub>P; A\<^sub>P \<sharp>* \<Psi>; A\<^sub>P \<sharp>* P; A\<^sub>P \<sharp>* Q; A\<^sub>P \<sharp>* M; A\<^sub>P \<sharp>* xvec; A\<^sub>P \<sharp>* N; A\<^sub>P \<sharp>* (A\<^sub>P\<^sub>Q, \<Psi>\<^sub>P\<^sub>Q); A\<^sub>P \<sharp>* xvec; distinct xvec\<rbrakk> \<Longrightarrow> Prop (P \<parallel> Q')
[PROOF STEP]
with S \<open>A\<^sub>P\<^sub>Q \<sharp>* Q\<close> \<open>A\<^sub>P \<sharp>* Q\<close> \<open>A\<^sub>Q \<sharp>* Q\<close> Aeq
[PROOF STATE]
proof (chain)
picking this:
set p \<subseteq> set (A\<^sub>P @ A\<^sub>Q) \<times> set (p \<bullet> A\<^sub>P @ p \<bullet> A\<^sub>Q)
A\<^sub>P\<^sub>Q \<sharp>* Q
A\<^sub>P \<sharp>* Q
A\<^sub>Q \<sharp>* Q
A\<^sub>P\<^sub>Q = p \<bullet> A\<^sub>P @ p \<bullet> A\<^sub>Q
p \<bullet> extractFrame Q = p \<bullet> \<langle>A\<^sub>Q, \<Psi>\<^sub>Q\<rangle>
[PROOF STEP]
have "extractFrame Q = \<langle>(p \<bullet> A\<^sub>Q), p \<bullet> \<Psi>\<^sub>Q\<rangle>"
[PROOF STATE]
proof (prove)
using this:
set p \<subseteq> set (A\<^sub>P @ A\<^sub>Q) \<times> set (p \<bullet> A\<^sub>P @ p \<bullet> A\<^sub>Q)
A\<^sub>P\<^sub>Q \<sharp>* Q
A\<^sub>P \<sharp>* Q
A\<^sub>Q \<sharp>* Q
A\<^sub>P\<^sub>Q = p \<bullet> A\<^sub>P @ p \<bullet> A\<^sub>Q
p \<bullet> extractFrame Q = p \<bullet> \<langle>A\<^sub>Q, \<Psi>\<^sub>Q\<rangle>
goal (1 subgoal):
1. extractFrame Q = \<langle>p \<bullet> A\<^sub>Q, p \<bullet> \<Psi>\<^sub>Q\<rangle>
[PROOF STEP]
by(simp add: eqvts)
[PROOF STATE]
proof (state)
this:
extractFrame Q = \<langle>p \<bullet> A\<^sub>Q, p \<bullet> \<Psi>\<^sub>Q\<rangle>
goal (1 subgoal):
1. \<And>Q' A\<^sub>P \<Psi>\<^sub>P. \<lbrakk>\<Psi> \<otimes> \<Psi>\<^sub>P \<rhd> Q \<longmapsto> M\<lparr>\<nu>*xvec\<rparr>\<langle>N\<rangle> \<prec> Q'; extractFrame P = \<langle>A\<^sub>P, \<Psi>\<^sub>P\<rangle>; distinct A\<^sub>P; A\<^sub>P \<sharp>* \<Psi>; A\<^sub>P \<sharp>* P; A\<^sub>P \<sharp>* Q; A\<^sub>P \<sharp>* M; A\<^sub>P \<sharp>* xvec; A\<^sub>P \<sharp>* N; A\<^sub>P \<sharp>* (A\<^sub>P\<^sub>Q, \<Psi>\<^sub>P\<^sub>Q); A\<^sub>P \<sharp>* xvec; distinct xvec\<rbrakk> \<Longrightarrow> Prop (P \<parallel> Q')
[PROOF STEP]
moreover
[PROOF STATE]
proof (state)
this:
extractFrame Q = \<langle>p \<bullet> A\<^sub>Q, p \<bullet> \<Psi>\<^sub>Q\<rangle>
goal (1 subgoal):
1. \<And>Q' A\<^sub>P \<Psi>\<^sub>P. \<lbrakk>\<Psi> \<otimes> \<Psi>\<^sub>P \<rhd> Q \<longmapsto> M\<lparr>\<nu>*xvec\<rparr>\<langle>N\<rangle> \<prec> Q'; extractFrame P = \<langle>A\<^sub>P, \<Psi>\<^sub>P\<rangle>; distinct A\<^sub>P; A\<^sub>P \<sharp>* \<Psi>; A\<^sub>P \<sharp>* P; A\<^sub>P \<sharp>* Q; A\<^sub>P \<sharp>* M; A\<^sub>P \<sharp>* xvec; A\<^sub>P \<sharp>* N; A\<^sub>P \<sharp>* (A\<^sub>P\<^sub>Q, \<Psi>\<^sub>P\<^sub>Q); A\<^sub>P \<sharp>* xvec; distinct xvec\<rbrakk> \<Longrightarrow> Prop (P \<parallel> Q')
[PROOF STEP]
from \<open>distinct A\<^sub>P\<close> \<open>distinct A\<^sub>Q\<close>
[PROOF STATE]
proof (chain)
picking this:
distinct A\<^sub>P
distinct A\<^sub>Q
[PROOF STEP]
have "distinct(p \<bullet> A\<^sub>P)" and "distinct(p \<bullet> A\<^sub>Q)"
[PROOF STATE]
proof (prove)
using this:
distinct A\<^sub>P
distinct A\<^sub>Q
goal (1 subgoal):
1. distinct (p \<bullet> A\<^sub>P) &&& distinct (p \<bullet> A\<^sub>Q)
[PROOF STEP]
by simp+
[PROOF STATE]
proof (state)
this:
distinct (p \<bullet> A\<^sub>P)
distinct (p \<bullet> A\<^sub>Q)
goal (1 subgoal):
1. \<And>Q' A\<^sub>P \<Psi>\<^sub>P. \<lbrakk>\<Psi> \<otimes> \<Psi>\<^sub>P \<rhd> Q \<longmapsto> M\<lparr>\<nu>*xvec\<rparr>\<langle>N\<rangle> \<prec> Q'; extractFrame P = \<langle>A\<^sub>P, \<Psi>\<^sub>P\<rangle>; distinct A\<^sub>P; A\<^sub>P \<sharp>* \<Psi>; A\<^sub>P \<sharp>* P; A\<^sub>P \<sharp>* Q; A\<^sub>P \<sharp>* M; A\<^sub>P \<sharp>* xvec; A\<^sub>P \<sharp>* N; A\<^sub>P \<sharp>* (A\<^sub>P\<^sub>Q, \<Psi>\<^sub>P\<^sub>Q); A\<^sub>P \<sharp>* xvec; distinct xvec\<rbrakk> \<Longrightarrow> Prop (P \<parallel> Q')
[PROOF STEP]
moreover
[PROOF STATE]
proof (state)
this:
distinct (p \<bullet> A\<^sub>P)
distinct (p \<bullet> A\<^sub>Q)
goal (1 subgoal):
1. \<And>Q' A\<^sub>P \<Psi>\<^sub>P. \<lbrakk>\<Psi> \<otimes> \<Psi>\<^sub>P \<rhd> Q \<longmapsto> M\<lparr>\<nu>*xvec\<rparr>\<langle>N\<rangle> \<prec> Q'; extractFrame P = \<langle>A\<^sub>P, \<Psi>\<^sub>P\<rangle>; distinct A\<^sub>P; A\<^sub>P \<sharp>* \<Psi>; A\<^sub>P \<sharp>* P; A\<^sub>P \<sharp>* Q; A\<^sub>P \<sharp>* M; A\<^sub>P \<sharp>* xvec; A\<^sub>P \<sharp>* N; A\<^sub>P \<sharp>* (A\<^sub>P\<^sub>Q, \<Psi>\<^sub>P\<^sub>Q); A\<^sub>P \<sharp>* xvec; distinct xvec\<rbrakk> \<Longrightarrow> Prop (P \<parallel> Q')
[PROOF STEP]
from \<open>A\<^sub>Q \<sharp>* A\<^sub>P\<close>
[PROOF STATE]
proof (chain)
picking this:
A\<^sub>Q \<sharp>* A\<^sub>P
[PROOF STEP]
have "(p \<bullet> A\<^sub>P) \<sharp>* (p \<bullet> A\<^sub>Q)"
[PROOF STATE]
proof (prove)
using this:
A\<^sub>Q \<sharp>* A\<^sub>P
goal (1 subgoal):
1. (p \<bullet> A\<^sub>P) \<sharp>* (p \<bullet> A\<^sub>Q)
[PROOF STEP]
by(simp add: pt_fresh_star_bij[OF pt_name_inst, OF at_name_inst])
[PROOF STATE]
proof (state)
this:
(p \<bullet> A\<^sub>P) \<sharp>* (p \<bullet> A\<^sub>Q)
goal (1 subgoal):
1. \<And>Q' A\<^sub>P \<Psi>\<^sub>P. \<lbrakk>\<Psi> \<otimes> \<Psi>\<^sub>P \<rhd> Q \<longmapsto> M\<lparr>\<nu>*xvec\<rparr>\<langle>N\<rangle> \<prec> Q'; extractFrame P = \<langle>A\<^sub>P, \<Psi>\<^sub>P\<rangle>; distinct A\<^sub>P; A\<^sub>P \<sharp>* \<Psi>; A\<^sub>P \<sharp>* P; A\<^sub>P \<sharp>* Q; A\<^sub>P \<sharp>* M; A\<^sub>P \<sharp>* xvec; A\<^sub>P \<sharp>* N; A\<^sub>P \<sharp>* (A\<^sub>P\<^sub>Q, \<Psi>\<^sub>P\<^sub>Q); A\<^sub>P \<sharp>* xvec; distinct xvec\<rbrakk> \<Longrightarrow> Prop (P \<parallel> Q')
[PROOF STEP]
moreover
[PROOF STATE]
proof (state)
this:
(p \<bullet> A\<^sub>P) \<sharp>* (p \<bullet> A\<^sub>Q)
goal (1 subgoal):
1. \<And>Q' A\<^sub>P \<Psi>\<^sub>P. \<lbrakk>\<Psi> \<otimes> \<Psi>\<^sub>P \<rhd> Q \<longmapsto> M\<lparr>\<nu>*xvec\<rparr>\<langle>N\<rangle> \<prec> Q'; extractFrame P = \<langle>A\<^sub>P, \<Psi>\<^sub>P\<rangle>; distinct A\<^sub>P; A\<^sub>P \<sharp>* \<Psi>; A\<^sub>P \<sharp>* P; A\<^sub>P \<sharp>* Q; A\<^sub>P \<sharp>* M; A\<^sub>P \<sharp>* xvec; A\<^sub>P \<sharp>* N; A\<^sub>P \<sharp>* (A\<^sub>P\<^sub>Q, \<Psi>\<^sub>P\<^sub>Q); A\<^sub>P \<sharp>* xvec; distinct xvec\<rbrakk> \<Longrightarrow> Prop (P \<parallel> Q')
[PROOF STEP]
from \<open>A\<^sub>P \<sharp>* \<Psi>\<^sub>Q\<close>
[PROOF STATE]
proof (chain)
picking this:
A\<^sub>P \<sharp>* \<Psi>\<^sub>Q
[PROOF STEP]
have "(p \<bullet> A\<^sub>P) \<sharp>* (p \<bullet> \<Psi>\<^sub>Q)"
[PROOF STATE]
proof (prove)
using this:
A\<^sub>P \<sharp>* \<Psi>\<^sub>Q
goal (1 subgoal):
1. (p \<bullet> A\<^sub>P) \<sharp>* (p \<bullet> \<Psi>\<^sub>Q)
[PROOF STEP]
by(simp add: pt_fresh_star_bij[OF pt_name_inst, OF at_name_inst])
[PROOF STATE]
proof (state)
this:
(p \<bullet> A\<^sub>P) \<sharp>* (p \<bullet> \<Psi>\<^sub>Q)
goal (1 subgoal):
1. \<And>Q' A\<^sub>P \<Psi>\<^sub>P. \<lbrakk>\<Psi> \<otimes> \<Psi>\<^sub>P \<rhd> Q \<longmapsto> M\<lparr>\<nu>*xvec\<rparr>\<langle>N\<rangle> \<prec> Q'; extractFrame P = \<langle>A\<^sub>P, \<Psi>\<^sub>P\<rangle>; distinct A\<^sub>P; A\<^sub>P \<sharp>* \<Psi>; A\<^sub>P \<sharp>* P; A\<^sub>P \<sharp>* Q; A\<^sub>P \<sharp>* M; A\<^sub>P \<sharp>* xvec; A\<^sub>P \<sharp>* N; A\<^sub>P \<sharp>* (A\<^sub>P\<^sub>Q, \<Psi>\<^sub>P\<^sub>Q); A\<^sub>P \<sharp>* xvec; distinct xvec\<rbrakk> \<Longrightarrow> Prop (P \<parallel> Q')
[PROOF STEP]
moreover
[PROOF STATE]
proof (state)
this:
(p \<bullet> A\<^sub>P) \<sharp>* (p \<bullet> \<Psi>\<^sub>Q)
goal (1 subgoal):
1. \<And>Q' A\<^sub>P \<Psi>\<^sub>P. \<lbrakk>\<Psi> \<otimes> \<Psi>\<^sub>P \<rhd> Q \<longmapsto> M\<lparr>\<nu>*xvec\<rparr>\<langle>N\<rangle> \<prec> Q'; extractFrame P = \<langle>A\<^sub>P, \<Psi>\<^sub>P\<rangle>; distinct A\<^sub>P; A\<^sub>P \<sharp>* \<Psi>; A\<^sub>P \<sharp>* P; A\<^sub>P \<sharp>* Q; A\<^sub>P \<sharp>* M; A\<^sub>P \<sharp>* xvec; A\<^sub>P \<sharp>* N; A\<^sub>P \<sharp>* (A\<^sub>P\<^sub>Q, \<Psi>\<^sub>P\<^sub>Q); A\<^sub>P \<sharp>* xvec; distinct xvec\<rbrakk> \<Longrightarrow> Prop (P \<parallel> Q')
[PROOF STEP]
from \<open>A\<^sub>Q \<sharp>* \<Psi>\<^sub>P\<close>
[PROOF STATE]
proof (chain)
picking this:
A\<^sub>Q \<sharp>* \<Psi>\<^sub>P
[PROOF STEP]
have "(p \<bullet> A\<^sub>Q) \<sharp>* (p \<bullet> \<Psi>\<^sub>P)"
[PROOF STATE]
proof (prove)
using this:
A\<^sub>Q \<sharp>* \<Psi>\<^sub>P
goal (1 subgoal):
1. (p \<bullet> A\<^sub>Q) \<sharp>* (p \<bullet> \<Psi>\<^sub>P)
[PROOF STEP]
by(simp add: pt_fresh_star_bij[OF pt_name_inst, OF at_name_inst])
[PROOF STATE]
proof (state)
this:
(p \<bullet> A\<^sub>Q) \<sharp>* (p \<bullet> \<Psi>\<^sub>P)
goal (1 subgoal):
1. \<And>Q' A\<^sub>P \<Psi>\<^sub>P. \<lbrakk>\<Psi> \<otimes> \<Psi>\<^sub>P \<rhd> Q \<longmapsto> M\<lparr>\<nu>*xvec\<rparr>\<langle>N\<rangle> \<prec> Q'; extractFrame P = \<langle>A\<^sub>P, \<Psi>\<^sub>P\<rangle>; distinct A\<^sub>P; A\<^sub>P \<sharp>* \<Psi>; A\<^sub>P \<sharp>* P; A\<^sub>P \<sharp>* Q; A\<^sub>P \<sharp>* M; A\<^sub>P \<sharp>* xvec; A\<^sub>P \<sharp>* N; A\<^sub>P \<sharp>* (A\<^sub>P\<^sub>Q, \<Psi>\<^sub>P\<^sub>Q); A\<^sub>P \<sharp>* xvec; distinct xvec\<rbrakk> \<Longrightarrow> Prop (P \<parallel> Q')
[PROOF STEP]
ultimately
[PROOF STATE]
proof (chain)
picking this:
\<Psi> \<otimes> (p \<bullet> \<Psi>\<^sub>P) \<rhd> Q \<longmapsto> M\<lparr>\<nu>*xvec\<rparr>\<langle>N\<rangle> \<prec> Q'
extractFrame P = \<langle>p \<bullet> A\<^sub>P, p \<bullet> \<Psi>\<^sub>P\<rangle>
extractFrame Q = \<langle>p \<bullet> A\<^sub>Q, p \<bullet> \<Psi>\<^sub>Q\<rangle>
distinct (p \<bullet> A\<^sub>P)
distinct (p \<bullet> A\<^sub>Q)
(p \<bullet> A\<^sub>P) \<sharp>* (p \<bullet> A\<^sub>Q)
(p \<bullet> A\<^sub>P) \<sharp>* (p \<bullet> \<Psi>\<^sub>Q)
(p \<bullet> A\<^sub>Q) \<sharp>* (p \<bullet> \<Psi>\<^sub>P)
[PROOF STEP]
show ?case
[PROOF STATE]
proof (prove)
using this:
\<Psi> \<otimes> (p \<bullet> \<Psi>\<^sub>P) \<rhd> Q \<longmapsto> M\<lparr>\<nu>*xvec\<rparr>\<langle>N\<rangle> \<prec> Q'
extractFrame P = \<langle>p \<bullet> A\<^sub>P, p \<bullet> \<Psi>\<^sub>P\<rangle>
extractFrame Q = \<langle>p \<bullet> A\<^sub>Q, p \<bullet> \<Psi>\<^sub>Q\<rangle>
distinct (p \<bullet> A\<^sub>P)
distinct (p \<bullet> A\<^sub>Q)
(p \<bullet> A\<^sub>P) \<sharp>* (p \<bullet> A\<^sub>Q)
(p \<bullet> A\<^sub>P) \<sharp>* (p \<bullet> \<Psi>\<^sub>Q)
(p \<bullet> A\<^sub>Q) \<sharp>* (p \<bullet> \<Psi>\<^sub>P)
goal (1 subgoal):
1. Prop (P \<parallel> Q')
[PROOF STEP]
using \<open>A\<^sub>P\<^sub>Q \<sharp>* \<Psi>\<close> \<open>A\<^sub>P\<^sub>Q \<sharp>* P\<close> \<open>A\<^sub>P\<^sub>Q \<sharp>* Q\<close> \<open>A\<^sub>P\<^sub>Q \<sharp>* M\<close> Aeq \<Psi>eq
[PROOF STATE]
proof (prove)
using this:
\<Psi> \<otimes> (p \<bullet> \<Psi>\<^sub>P) \<rhd> Q \<longmapsto> M\<lparr>\<nu>*xvec\<rparr>\<langle>N\<rangle> \<prec> Q'
extractFrame P = \<langle>p \<bullet> A\<^sub>P, p \<bullet> \<Psi>\<^sub>P\<rangle>
extractFrame Q = \<langle>p \<bullet> A\<^sub>Q, p \<bullet> \<Psi>\<^sub>Q\<rangle>
distinct (p \<bullet> A\<^sub>P)
distinct (p \<bullet> A\<^sub>Q)
(p \<bullet> A\<^sub>P) \<sharp>* (p \<bullet> A\<^sub>Q)
(p \<bullet> A\<^sub>P) \<sharp>* (p \<bullet> \<Psi>\<^sub>Q)
(p \<bullet> A\<^sub>Q) \<sharp>* (p \<bullet> \<Psi>\<^sub>P)
A\<^sub>P\<^sub>Q \<sharp>* \<Psi>
A\<^sub>P\<^sub>Q \<sharp>* P
A\<^sub>P\<^sub>Q \<sharp>* Q
A\<^sub>P\<^sub>Q \<sharp>* M
A\<^sub>P\<^sub>Q = p \<bullet> A\<^sub>P @ p \<bullet> A\<^sub>Q
\<Psi>\<^sub>P\<^sub>Q = (p \<bullet> \<Psi>\<^sub>P) \<otimes> (p \<bullet> \<Psi>\<^sub>Q)
goal (1 subgoal):
1. Prop (P \<parallel> Q')
[PROOF STEP]
by(rule_tac rPar2) (assumption | simp)+
[PROOF STATE]
proof (state)
this:
Prop (P \<parallel> Q')
goal:
No subgoals!
[PROOF STEP]
qed
|
[STATEMENT]
lemma current_methd:
"\<lbrakk>table_of (methods c) sig = Some new;
ws_prog G; class G C = Some c; C \<noteq> Object;
methd G (super c) sig = Some old\<rbrakk>
\<Longrightarrow> methd G C sig = Some (C,new)"
[PROOF STATE]
proof (prove)
goal (1 subgoal):
1. \<lbrakk>table_of (methods c) sig = Some new; ws_prog G; class G C = Some c; C \<noteq> Object; methd G (super c) sig = Some old\<rbrakk> \<Longrightarrow> methd G C sig = Some (C, new)
[PROOF STEP]
by (auto simp add: methd_rec
intro: filter_tab_SomeI map_add_find_right table_of_map_SomeI)
|
Oregon Duck fans – and Pac-12 fans, for that matter – are sick and tired of hearing about how great the Southeastern Conference (SEC) is. Broadcasters constantly speak about how the SEC is ahead of the Pac-12 in recruiting, quality of play and revenue. Well, that no longer appears to be the case.
The absence of an SEC team in the 2015 NCAA Football National Championship Game is one reason to believe there’s change coming. However, it may surprise you to know that the Pac-12 made more money than the SEC and the B1G in terms of total revenue during the 2013-14 fiscal year (FY14), according to Steve Berkowitz of USA TODAY Sports.
The Pac-12’s revenue nearly doubled from FY13’s $175 million to $334 million in FY14. Berkowitz also reported that Pac-12 Commissioner Larry Scott remained the nation’s most highly-paid conference CEO, earning just over $3.5 million in total compensation for the 2013 calendar year.
Here’s a breakdown of the Pac-12’s total revenues for FY14: Bowl games $41.6 million, NCAA championships $26.3 million, television and media rights $252.7 million, advertising $9 million, investments totaling around $4.2 million – for a grand total of $333,992,599.
For now, the Pac-12 leads all other conferences in revenue. That may change in a few years, due to the SEC’s launch of its own television network with ESPN. However, it’s safe to say that the Pac-12 isn’t content with playing second fiddle to other conferences anymore.
|
classdef testInverseSymmetricFourthOrderTensor < testInverseFourthOrderTensor
methods (Access = protected)
function createRandomFourthOrderTensor(obj)
obj.tensor = Stiffness3DTensor;
obj.tensor.createRandomTensor();
end
end
methods (Static, Access = protected)
function Id = computeIdentityTensor(I,i,j,k,l)
Id = 0.5*(I(i,k)*I(j,l) + I(i,l)*I(j,k));
end
end
end
|
-- | REPL for console version of Minesweeper
module Minesweeper.REPL
import Minesweeper.Game
import Minesweeper.Board
import Minesweeper.Helper
import Control.Monad.State
import Data.Vect
import Effects
import Effect.Random
import Effect.StdIO
import Effect.System
implementation Show Difficulty where
show Easy = "Easy"
show Medium = "Medium"
show Hard = "Hard"
intro : String
intro = """
___ ________ _ _ _____ _____ _ _ _____ ___________ ___________
| \/ |_ _| \ | || ___/ ___|| | | | ___| ___| ___ \ ___| ___ \
| . . | | | | \| || |__ \ `--. | | | | |__ | |__ | |_/ / |__ | |_/ /
| |\/| | | | | . ` || __| `--. \| |/\| | __|| __|| __/| __|| /
| | | |_| |_| |\ || |___/\__/ /\ /\ / |___| |___| | | |___| |\ \
\_| |_/\___/\_| \_/\____/\____/ \/ \/\____/\____/\_| \____/\_| \_|
Written by Ross Meikleham 2015"""
invalid : String -> String
invalid s = "Unknown option \"" ++ s ++ "\". " ++
"Enter h or help to display the list of possible options"
data GameAction =
GQuit
| GHelp
| GShow
| Reveal Pos
| GInvalid String
parseGameAction : String -> GameAction
parseGameAction "q" = GQuit
parseGameAction "quit" = GQuit
parseGameAction "h" = GHelp
parseGameAction "help" = GHelp
parseGameAction "d" = GShow
parseGameAction "display" = GShow
parseGameAction s =
case parseReveal (words s) of
(Just pos) => Reveal pos
(Nothing) => GInvalid (invalid s)
where
parsePos : String -> String -> Maybe Pos
parsePos strRow strCol = do
row <- readNat strRow
col <- readNat strCol
pure $ MkPos col row
parseReveal : List String -> Maybe Pos
parseReveal (command :: row :: col :: []) =
case command of
"reveal" => parsePos row col
"r" => parsePos row col
parseReveal _ = Nothing
gHelp : Nat -> Nat -> String
gHelp (S r) (S c) = ("r [row] [column] or reveal [row] [column] to reveal the given square\n" ++
"\t where row is between 0 and " ++ show r ++ ", and column is between 0 and " ++
show c ++ ".\n" ++
"d or display to display the board\n" ++
"q or quit to exit\n" ++
"h or help to display help\n")
gHelp _ _ = "Error with row/column size\n"
playGame' : Board m n -> IO ()
playGame' {m} {n} board = do
putStr "Enter Option> "
str <- getLine
let option = parseGameAction str
case option of
(Reveal pos) => let (res, newBoard) = runState (revealPos pos) board in
case res of
Playing str => putStrLn (showBoard newBoard) >>= \_ => putStrLn str
>>= \_ => playGame' newBoard
Won => putStrLn (showRevealed newBoard) >>= \_ => putStrLn "You Win!"
Lost => putStrLn (showBoard newBoard) >>= \_ => putStrLn "You Hit a Mine!" >>=
\_ => putStrLn (showRevealed newBoard)
GHelp => putStrLn (gHelp m n) >>= \_ => playGame' board
GQuit => putStrLn "Quitting Game..."
GShow => putStrLn (showBoard board)
GInvalid s => putStrLn s >>= \_ => playGame' board
playGame : Difficulty -> IO()
playGame difficulty = do
let ((MkPos x y), nMines) = getSetupDetails difficulty
mines <- run (generateMines x y nMines)
case mines of
Nothing => putStrLn "More mines than positions available"
Just m => do
let board = createBoard x y m
case board of
Nothing => putStrLn "Error creating board"
Just b => putStrLn (showBoard b) >>= \_ => playGame' b
-- | Difficulty Options/Menu
data DifficultyAction =
DSelected Difficulty
| DHelp
| DQuit
| DInvalid String
difficulty : String
difficulty = "Enter difficulty> "
difficultyHelp : String
difficultyHelp = """
b or beginner to start an easy game
i or intermediate to start a medium difficulty game
e or expert to start a hard game
h or help to display this help option
q or quit to exit to the main menu"""
parseDifficulty : String -> DifficultyAction
parseDifficulty "b" = DSelected Easy
parseDifficulty "beginner" = DSelected Easy
parseDifficulty "i" = DSelected Medium
parseDifficulty "intermediate" = DSelected Medium
parseDifficulty "e" = DSelected Hard
parseDifficulty "expert" = DSelected Hard
parseDifficulty "h" = DHelp
parseDifficulty "help" = DHelp
parseDifficulty "q" = DQuit
parseDifficulty "quit" = DQuit
parseDifficulty s = DInvalid s
difficultyMenu : IO ()
difficultyMenu = do
putStrLn ""
putStr difficulty
optionStr <- getLine
let option = parseDifficulty optionStr
case option of
DSelected d => putStrLn ("Starting " ++ show d ++ " game...") >>= \_ => playGame d
DHelp => putStrLn difficultyHelp >>= \_ => difficultyMenu
DQuit => putStrLn "Returning to main menu..."
DInvalid s => (putStrLn $ invalid s) >>= \_ => difficultyMenu -- | Main Menu options
data MainMenuAction =
Quit
| Play
| Help
| Invalid String
parseOption : String -> MainMenuAction
parseOption "q" = Quit
parseOption "quit" = Quit
parseOption "p" = Play
parseOption "play" = Play
parseOption "h" = Help
parseOption "help" = Help
parseOption i = Invalid i
help : String
help = """
p or play to start a game
q or quit to exit
h or help to display help"""
mainMenu : IO ()
mainMenu = do
putStrLn ""
putStr "Enter option> "
optionStr <- getLine
let option = parseOption optionStr
case option of
Play => difficultyMenu >>= \_ => mainMenu
Help => putStrLn help >>= \_ => mainMenu
Quit => putStrLn "Goodbye :)"
Invalid s => putStrLn (invalid s) >>= \_ => mainMenu
export
repl : IO ()
repl = do
putStrLn intro
mainMenu
|
-- Idris2
import System
import System.Concurrency
-- Test `conditionSignal` works for 1 main and 1 child thread
main : IO ()
main =
do cvMutex <- makeMutex
cv <- makeCondition
t <- fork $ do mutexAcquire cvMutex
conditionWait cv cvMutex
putStrLn "Hello mother"
mutexRelease cvMutex
putStrLn "Hello child"
sleep 1
conditionSignal cv
threadWait t
|
#include "RecordEngine.h"
#include "Serialize/SlamSerialize.pb.h"
#include "Serialize/MessageTypes.h"
#include <opencv2/imgcodecs.hpp>
#include <algorithm>
#include <vector>
#include <sstream>
#include <boost/lexical_cast.hpp>
using namespace LpSlam;
namespace {
template <class TVec3In, class TVec3Out>
TVec3Out * toProtoVec3( TVec3In const& vec3) {
auto outPosition = new TVec3Out();
outPosition->set_x(vec3.value[0]);
outPosition->set_y(vec3.value[1]);
outPosition->set_z(vec3.value[2]);
return outPosition;
}
template <class TPosOrient>
std::pair<LpgfSlamSerialize::Position*, LpgfSlamSerialize::Orientation* >
toProtoPositionOrientation( TPosOrient const& posOrient) {
auto outPosition = new LpgfSlamSerialize::Position();
outPosition->set_x(posOrient.position.value[0]);
outPosition->set_y(posOrient.position.value[1]);
outPosition->set_z(posOrient.position.value[2]);
outPosition->set_x_sigma(posOrient.position.sigma[0]);
outPosition->set_y_sigma(posOrient.position.sigma[1]);
outPosition->set_z_sigma(posOrient.position.sigma[2]);
auto outOrientation = new LpgfSlamSerialize::Orientation();
outOrientation->set_w(posOrient.orientation.value.w());
outOrientation->set_x(posOrient.orientation.value.x());
outOrientation->set_y(posOrient.orientation.value.y());
outOrientation->set_z(posOrient.orientation.value.z());
outOrientation->set_sigma(posOrient.orientation.sigma);
return {outPosition, outOrientation};
}
}
RecordEngine::~RecordEngine() {
stop();
}
RecordEngine::RecordEngine() :
m_recordThread(
[](RecordThreadParams params) -> bool {
const auto lmdStoreState = [] (GlobalState const & state) {
auto [outPosition, outOrientation] = toProtoPositionOrientation(state);
auto outState = new LpgfSlamSerialize::GlobalState();
outState->set_allocated_position(outPosition);
outState->set_allocated_orientation(outOrientation);
LpgfSlamSerialize::Velocity * velocity = nullptr;
if (state.velocityValid) {
velocity = toProtoVec3<Velocity3, LpgfSlamSerialize::Velocity>(
state.velocity);
} else {
velocity = new LpgfSlamSerialize::Velocity();
}
outState->set_allocated_velocity(velocity);
return outState;
};
// check if there is some work on our work queue
try {
RecordQueueEntry recordEntry;
params.m_q.pop(recordEntry);
if (recordEntry.valid == false) {
return false;
}
if (recordEntry.type == EntryType::Camera) {
LpgfSlamSerialize::CameraImage outCamEntry;
outCamEntry.set_timestamp(recordEntry.camera.timestamp.time_since_epoch().count());
std::vector<unsigned char> imgData;
cv::imencode(".jpg", recordEntry.camera.image, imgData);
outCamEntry.set_imagedata(imgData.data(), imgData.size());
outCamEntry.set_cameranumber(recordEntry.camera.cameraNumber);
{
auto [outBasePosition, outBaseOrientation] = toProtoPositionOrientation(recordEntry.camera.base);
auto outBase = new LpgfSlamSerialize::TrackerCoordinateSystem();
outBase->set_allocated_position(outBasePosition);
outBase->set_allocated_orientation(outBaseOrientation);
outCamEntry.set_allocated_imagebase(outBase);
}
// check if there is a second image
if (recordEntry.camera.image_second.has_value()) {
imgData.clear();
cv::imencode(".jpg", recordEntry.camera.image_second.value(), imgData);
outCamEntry.set_imagedata_second(imgData.data(), imgData.size());
outCamEntry.set_cameranumber_second(recordEntry.camera.cameraNumber_second);
{
auto [outBasePosition, outBaseOrientation] = toProtoPositionOrientation(
recordEntry.camera.base_second.value());
auto outBase = new LpgfSlamSerialize::TrackerCoordinateSystem();
outBase->set_allocated_position(outBasePosition);
outBase->set_allocated_orientation(outBaseOrientation);
outCamEntry.set_allocated_imagebase_second(outBase);
}
}
auto outStateOdom = lmdStoreState(recordEntry.state_odom);
outCamEntry.set_allocated_state_odom(outStateOdom);
outCamEntry.set_hasglobalstate_odom(recordEntry.state_odom_valid);
auto outStateMap = lmdStoreState(recordEntry.state_map);
outCamEntry.set_allocated_state_map(outStateMap);
outCamEntry.set_hasglobalstate_map(recordEntry.state_map_valid);
params.m_stream.toStream( Serialization::CameraImage,
outCamEntry, params.m_out.get());
if (params.writeRawFile) {
std::stringstream sFileOut;
std::string rawFileNumber = boost::lexical_cast<std::string>(params.imgCount);
const size_t n_zero = 6;
// fill with leading zeros
rawFileNumber = std::string(n_zero - rawFileNumber.length(), '0') + rawFileNumber;
sFileOut << rawFileNumber << "_left.jpg";
cv::imwrite(sFileOut.str(), recordEntry.camera.image);
if (recordEntry.camera.image_second.has_value()) {
std::stringstream sFileOutRight;
sFileOutRight << rawFileNumber << "_right.jpg";
cv::imwrite(sFileOutRight.str(), recordEntry.camera.image_second.value());
}
}
//cv::imshow("Display Window", recordEntry.camera.image);
params.imgCount++;
} else if (recordEntry.type == EntryType::Sensor) {
if (recordEntry.sensor.getSensorType() == SensorQueueEntry::SensorType::Imu) {
LpgfSlamSerialize::SensorImu outSensorImu;
outSensorImu.set_timestamp(recordEntry.sensor.timestamp.time_since_epoch().count());
auto outAcc = toProtoVec3<Acceleration3, LpgfSlamSerialize::Acceleration>(recordEntry.sensor.getAcceleration());
auto outGyro = toProtoVec3<AngularVelocity3, LpgfSlamSerialize::AngularVelocity>(recordEntry.sensor.getAngluarVelocity());
outSensorImu.set_allocated_acc(outAcc);
outSensorImu.set_allocated_gyro(outGyro);
params.m_stream.toStream( Serialization::SensorImu,
outSensorImu, params.m_out.get());
} else if (recordEntry.sensor.getSensorType() == SensorQueueEntry::SensorType::GlobalState) {
// right now, just using one ProtoBuf message ...
LpgfSlamSerialize::SensorGlobalState outSensorGlobalState;
outSensorGlobalState.set_timestamp(recordEntry.sensor.timestamp.time_since_epoch().count());
outSensorGlobalState.set_reference(recordEntry.sensor.reference);
auto [outPosition, outOrientation] = toProtoPositionOrientation(recordEntry.sensor.getGlobalState());
auto outState = new LpgfSlamSerialize::GlobalState();
outState->set_allocated_position(outPosition);
outState->set_allocated_orientation(outOrientation);
outSensorGlobalState.set_allocated_globalstate(outState);
params.m_stream.toStream( Serialization::SensorGlobalState,
outSensorGlobalState, params.m_out.get());
} else if (recordEntry.sensor.getSensorType() == SensorQueueEntry::SensorType::FeatureList) {
for (auto const& feature: recordEntry.sensor.getFeatureList()) {
LpgfSlamSerialize::SensorFeature outSensorFeature;
//std::cout << feature.m_timestamp.time_since_epoch().count() << std::endl;
outSensorFeature.set_timestamp(feature.m_timestamp.time_since_epoch().count());
outSensorFeature.set_lastobserved(feature.m_lastObserved.time_since_epoch().count());
outSensorFeature.set_observationcount(feature.m_observationCount);
outSensorFeature.set_allocated_position(toProtoVec3<Position3, LpgfSlamSerialize::Position>
(feature.m_position));
outSensorFeature.set_allocated_closestkeyframeposition(toProtoVec3<Position3, LpgfSlamSerialize::Position>
(feature.m_closestKeyframePosition));
outSensorFeature.set_anchorid(feature.m_anchorId);
params.m_stream.toStream( Serialization::SensorFeatureList,
outSensorFeature, params.m_out.get());
}
} else {
spdlog::error("Sensor type not support for recording");
}
} else if (recordEntry.type == EntryType::Result) {
LpgfSlamSerialize::GlobalStateInTime outResult;
auto [res_timestamp, res_gs] = recordEntry.result.globalStateInTime;
outResult.set_timestamp(res_timestamp.system_time.time_since_epoch().count());
auto [outPosition, outOrientation] = toProtoPositionOrientation(res_gs);
auto outState = new LpgfSlamSerialize::GlobalState();
outState->set_allocated_position(outPosition);
outState->set_allocated_orientation(outOrientation);
outResult.set_allocated_globalstate(outState);
params.m_stream.toStream( Serialization::Result,
outResult, params.m_out.get());
} else {
spdlog::error("Recording entry not supported.");
}
// pop all from the sensor values
}
catch (tbb::user_abort &)
{
// all waits on the fusion queue were aborted
return false;
}
// continue thread
return true;
}) {
GOOGLE_PROTOBUF_VERIFY_VERSION;
}
void RecordEngine::setOutputFile(std::string const& outputFile) {
bool startAgain = false;
if (m_out) {
stop();
startAgain = true;
}
m_outputFile = outputFile;
if (startAgain) {
start();
}
}
void RecordEngine::setStoreImages(bool b) {
m_storeImages = b;
}
void RecordEngine::storeSensor( SensorQueueEntry const& sensor) {
if (!m_out) {
// not recording
return;
}
RecordQueueEntry entry;
entry.type = EntryType::Sensor;
entry.sensor = sensor;
m_q.push(entry);
}
void RecordEngine::storeCameraImage( CameraQueueEntry const& camera,
std::optional<GlobalStateInTime> state_odom,
std::optional<GlobalStateInTime> state_map) {
if (!m_out || !m_storeImages) {
// not recording
return;
}
RecordQueueEntry entry;
entry.type = EntryType::Camera;
entry.camera = camera;
if (state_odom.has_value()) {
entry.state_odom = state_odom.value().second;
entry.state_odom_valid = true;
} else {
entry.state_odom_valid = false;
}
if (state_map.has_value()) {
entry.state_map = state_map.value().second;
entry.state_map_valid = true;
} else {
entry.state_map_valid = false;
}
m_q.push(entry);
}
void RecordEngine::storeResult( ResultQueueEntry const& result) {
if (!m_out) {
// not recording
return;
}
RecordQueueEntry entry;
entry.type = EntryType::Result;
entry.result = result;
m_q.push(entry);
}
void RecordEngine::setWriteRawFile(bool b) {
m_writeRawFile = b;
}
void RecordEngine::start() {
if (m_out) {
// already recording ...
return;
}
m_out = std::make_shared<std::ofstream>(m_outputFile, std::ofstream::binary);
m_recordThread.start(RecordThreadParams(m_q, m_stream, m_out, m_imgCount, m_writeRawFile));
std::ofstream (m_outputFile, std::ios::binary);
}
void RecordEngine::stop() {
if (m_recordThread.isRunning()) {
// send the terminate and all previous measurements will be stored
RecordQueueEntry camEntry;
camEntry.valid = false;
m_recordThread.stopAsync();
m_q.push(camEntry);
m_recordThread.stop();
m_q.abort();
m_q.clear();
}
if (m_out) {
if (m_out->is_open()) {
*m_out << std::flush;
m_out->close();
}
m_out.reset();
}
}
|
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|
(*-------------------------------------------*
| CSP-Prover on Isabelle2004 |
| December 2004 |
| July 2005 (modified) |
| September 2005 (modified) |
| |
| CSP-Prover on Isabelle2005 |
| October 2005 (modified) |
| April 2006 (modified) |
| March 2007 (modified) |
| |
| Yoshinao Isobe (AIST JAPAN) |
*-------------------------------------------*)
theory CSP_T_law_SKIP
imports CSP_T_law_basic
begin
(*****************************************************************
1. SKIP |[X]| SKIP
2. SKIP |[X]| P
3. P |[X]| SKIP
4. SKIP -- X
5. SKIP [[r]]
6. SKIP ;; P
7. P ;; SKIP
8. SKIP |. n
*****************************************************************)
(*********************************************************
SKIP |[X]| SKIP
*********************************************************)
lemma cspT_Parallel_term:
"SKIP |[X]| SKIP =T[M1,M2] SKIP"
apply (simp add: cspT_semantics)
apply (rule order_antisym)
(* => *)
apply (rule)
apply (simp add: in_traces)
apply (elim disjE conjE exE)
apply (simp_all)
(* <= *)
apply (rule)
apply (simp add: in_traces)
apply (elim disjE conjE exE)
apply (simp_all)
done
(*********************************************************
SKIP |[X]| P
*********************************************************)
lemma cspT_Parallel_preterm_l:
"SKIP |[X]| (? :Y -> Qf) =T[M,M] ? x:(Y-X) -> (SKIP |[X]| Qf x)"
apply (simp add: cspT_semantics)
apply (rule order_antisym)
(* => *)
apply (rule)
apply (simp add: in_traces)
apply (insert trace_nil_or_Tick_or_Ev)
apply (elim disjE conjE exE)
apply (simp_all)
apply (drule_tac x="t" in spec)
apply (erule disjE, simp)
apply (erule disjE, simp)
apply (elim conjE exE, simp)
apply (simp add: par_tr_head)
apply (rule_tac x="<>" in exI)
apply (rule_tac x="sa" in exI, simp)
apply (drule_tac x="t" in spec)
apply (erule disjE, simp)
apply (erule disjE, simp)
apply (elim conjE exE, simp)
apply (simp add: par_tr_head)
apply (rule_tac x="<Tick>" in exI)
apply (rule_tac x="sa" in exI, simp)
(* <= *)
apply (rule)
apply (simp add: in_traces)
apply (elim disjE conjE exE)
apply (simp_all)
apply (rule_tac x="<>" in exI)
apply (rule_tac x="<Ev a> ^^^ ta" in exI, simp)
apply (simp add: par_tr_head)
apply (rule_tac x="<Tick>" in exI)
apply (rule_tac x="<Ev a> ^^^ ta" in exI, simp)
apply (simp add: par_tr_head)
done
(*********************************************************
P |[X]| SKIP
*********************************************************)
lemma cspT_Parallel_preterm_r:
"(? :Y -> Pf) |[X]| SKIP
=T[M,M] ? x:(Y-X) -> (Pf x |[X]| SKIP)"
apply (rule cspT_trans)
apply (rule cspT_Parallel_commut)
apply (rule cspT_trans)
apply (rule cspT_Parallel_preterm_l)
apply (rule cspT_rm_head, simp)
apply (rule cspT_Parallel_commut)
done
lemmas cspT_Parallel_preterm = cspT_Parallel_preterm_l cspT_Parallel_preterm_r
(*********************************************************
SKIP and Parallel
*********************************************************)
(* p.288 *)
lemma cspT_SKIP_Parallel_Ext_choice_SKIP_l:
"((? :Y -> Pf) [+] SKIP) |[X]| SKIP =T[M,M]
(? x:(Y - X) -> (Pf x |[X]| SKIP)) [+] SKIP"
apply (simp add: cspT_semantics)
apply (rule order_antisym)
(* => *)
apply (rule, simp add: in_traces)
apply (elim conjE exE disjE)
apply (simp_all)
apply (rule disjI2)
apply (rule disjI1)
apply (simp add: par_tr_nil_right)
apply (elim conjE)
apply (simp add: image_iff)
apply (rule_tac x="sa" in exI)
apply (rule_tac x="<>" in exI)
apply (simp add: par_tr_nil_right)
apply (rule disjI2)
apply (rule disjI1)
apply (simp add: par_tr_Tick_right)
apply (elim conjE)
apply (simp add: image_iff)
apply (rule_tac x="sa" in exI)
apply (rule_tac x="<Tick>" in exI)
apply (simp add: par_tr_Tick_right)
(* <= *)
apply (rule, simp add: in_traces)
apply (elim conjE exE disjE)
apply (simp_all)
apply (simp add: par_tr_nil_right)
apply (elim conjE)
apply (rule_tac x="<Ev a> ^^^ sa" in exI)
apply (rule_tac x="<>" in exI)
apply (simp add: par_tr_nil_right)
apply (simp add: image_iff)
apply (simp add: par_tr_Tick_right)
apply (elim conjE)
apply (rule_tac x="<Ev a> ^^^ sa" in exI)
apply (rule_tac x="<Tick>" in exI)
apply (simp add: par_tr_Tick_right)
apply (simp add: image_iff)
done
lemma cspT_SKIP_Parallel_Ext_choice_SKIP_r:
"SKIP |[X]| ((? :Y -> Pf) [+] SKIP) =T[M,M]
(? x:(Y - X) -> (SKIP |[X]| Pf x)) [+] SKIP"
apply (rule cspT_rw_left)
apply (rule cspT_commut)
apply (rule cspT_rw_left)
apply (rule cspT_SKIP_Parallel_Ext_choice_SKIP_l)
apply (rule cspT_rw_left)
apply (rule cspT_decompo)
apply (rule cspT_decompo)
apply (simp)
apply (rule cspT_commut)
apply (rule cspT_reflex)
apply (rule cspT_reflex)
done
lemmas cspT_SKIP_Parallel_Ext_choice_SKIP =
cspT_SKIP_Parallel_Ext_choice_SKIP_l
cspT_SKIP_Parallel_Ext_choice_SKIP_r
(*********************************************************
SKIP -- X
*********************************************************)
lemma cspT_SKIP_Hiding_Id:
"SKIP -- X =T[M1,M2] SKIP"
apply (simp add: cspT_semantics)
apply (rule order_antisym)
(* => *)
apply (rule)
apply (simp add: in_traces)
apply (elim disjE conjE exE)
apply (simp_all)
(* <= *)
apply (rule)
apply (simp add: in_traces)
apply (elim disjE conjE exE)
apply (simp_all)
apply (rule_tac x="<>" in exI)
apply (simp)
apply (rule_tac x="<Tick>" in exI)
apply (simp)
done
(*********************************************************
SKIP and Hiding
*********************************************************)
(* p.288 version
"((? :Y -> Pf) [+] SKIP) -- X =T[M1,M2]
IF (Y Int X = {}) THEN ((? x:Y -> (Pf x -- X)) [+] SKIP)
ELSE (((? x:(Y-X) -> (Pf x -- X)) [+] SKIP)
|~| (! x:(Y Int X) .. (Pf x -- X)))"
*)
lemma cspT_SKIP_Hiding_step:
"((? :Y -> Pf) [+] SKIP) -- X =T[M,M]
((? x:(Y-X) -> (Pf x -- X)) [+] SKIP) |~| (! x:(Y Int X) .. (Pf x -- X))"
apply (simp add: cspT_semantics)
apply (rule order_antisym)
(* => *)
apply (rule, simp add: in_traces)
apply (elim conjE exE disjE)
apply (simp_all)
apply (case_tac "a : X", force)
apply (force)
(* <= *)
apply (rule)
apply (simp add: in_traces)
apply (elim conjE exE bexE disjE)
apply (simp_all)
apply (force)
apply (rule_tac x="<Ev a> ^^^ sa" in exI)
apply (simp)
apply (force)
apply (rule_tac x="<Tick>" in exI)
apply (simp)
apply (force)
apply (rule_tac x="<Ev a> ^^^ s" in exI)
apply (simp)
done
(*********************************************************
SKIP [[r]]
*********************************************************)
lemma cspT_SKIP_Renaming_Id:
"SKIP [[r]] =T[M1,M2] SKIP"
apply (simp add: cspT_semantics)
apply (rule order_antisym)
(* => *)
apply (rule)
apply (simp add: in_traces)
apply (force)
(* <= *)
apply (rule)
apply (simp add: in_traces)
apply (force)
done
(*********************************************************
SKIP ;; P
*********************************************************)
lemma cspT_Seq_compo_unit_l: "SKIP ;; P =T[M,M] P"
apply (simp add: cspT_semantics)
apply (rule order_antisym)
(* => *)
apply (rule, simp add: in_traces)
apply (force)
(* <= *)
apply (rule, simp add: in_traces)
apply (rule disjI2)
apply (rule_tac x="<>" in exI)
apply (rule_tac x="t" in exI)
apply (simp)
done
(*********************************************************
P ;; SKIP
*********************************************************)
lemma cspT_Seq_compo_unit_r: "P ;; SKIP =T[M,M] P"
apply (simp add: cspT_semantics)
apply (rule order_antisym)
(* => *)
apply (rule, simp add: in_traces)
apply (elim conjE exE disjE)
apply (simp_all)
apply (rule memT_prefix_closed, simp)
apply (simp add: rmTick_prefix_rev)
apply (rule memT_prefix_closed, simp, simp)
(* <= *)
apply (rule, simp add: in_traces)
apply (insert trace_last_noTick_or_Tick)
apply (drule_tac x="t" in spec)
apply (erule disjE)
apply (rule disjI1)
apply (rule_tac x="t" in exI, simp)
(* *)
apply (rule disjI2)
apply (elim conjE exE)
apply (rule_tac x="s" in exI)
apply (rule_tac x="<Tick>" in exI)
apply (simp)
done
lemmas cspT_Seq_compo_unit = cspT_Seq_compo_unit_l cspT_Seq_compo_unit_r
(*********************************************************
SKIP and Sequential composition
*********************************************************)
(* p.141 *)
lemma cspT_SKIP_Seq_compo_step:
"((? :X -> Pf) [> SKIP) ;; Q =T[M,M] (? x:X -> (Pf x ;; Q)) [> Q"
apply (simp add: cspT_semantics)
apply (rule order_antisym)
(* => *)
apply (rule, simp add: in_traces)
apply (elim conjE exE disjE)
apply (simp_all)
apply (rule disjI1)
apply (fast)
apply (rule disjI2)
apply (rule disjI1)
apply (insert trace_nil_or_Tick_or_Ev)
apply (drule_tac x="s" in spec)
apply (elim disjE conjE exE)
apply (simp_all)
apply (simp add: appt_assoc)
apply (rule disjI2)
apply (rule_tac x="sb" in exI)
apply (rule_tac x="ta" in exI)
apply (simp)
(* <= *)
apply (rule, simp add: in_traces)
apply (elim conjE exE disjE)
apply (simp_all)
apply (rule disjI1)
apply (rule_tac x="<>" in exI)
apply (simp)
apply (rule disjI1)
apply (rule_tac x="<Ev a> ^^^ sa" in exI)
apply (simp)
apply (rule disjI2)
apply (rule_tac x="<Ev a> ^^^ sa" in exI)
apply (rule_tac x="ta" in exI)
apply (simp add: appt_assoc)
apply (rule disjI1)
apply (rule_tac x="<>" in exI)
apply (simp)
apply (rule disjI2)
apply (rule_tac x="<>" in exI)
apply (rule_tac x="t" in exI)
apply (simp)
done
(*********************************************************
SKIP |. n
*********************************************************)
lemma cspT_SKIP_Depth_rest:
"SKIP |. (Suc n) =T[M1,M2] SKIP"
apply (simp add: cspT_semantics)
apply (rule order_antisym)
(* => *)
apply (rule)
apply (simp add: in_traces)
(* <= *)
apply (rule)
apply (simp add: in_traces)
apply (force)
done
(*********************************************************
cspT_SKIP
*********************************************************)
lemmas cspT_SKIP =
cspT_Parallel_term
cspT_Parallel_preterm
cspT_SKIP_Parallel_Ext_choice_SKIP
cspT_SKIP_Hiding_Id
cspT_SKIP_Hiding_step
cspT_SKIP_Renaming_Id
cspT_Seq_compo_unit
cspT_SKIP_Seq_compo_step
cspT_SKIP_Depth_rest
(*********************************************************
P [+] SKIP
*********************************************************)
(* p.141 *)
lemma cspT_Ext_choice_SKIP_resolve: "P [+] SKIP =T[M,M] P [> SKIP"
apply (simp add: cspT_semantics)
apply (rule order_antisym)
(* => *)
apply (rule, simp add: in_traces)
(* <= *)
apply (rule, simp add: in_traces)
done
lemma cspT_Ext_choice_SKIP_resolve_sym: "P [> SKIP =T[M,M] P [+] SKIP"
apply (rule cspT_sym)
apply (simp add: cspT_Ext_choice_SKIP_resolve)
done
(*********************************************************
SKIP ||| P
*********************************************************)
lemma cspT_Interleave_unit_l:
"SKIP ||| P =T[M,M] P"
apply (simp add: cspT_semantics)
apply (rule order_antisym)
(* => *)
apply (rule)
apply (simp add: in_traces)
apply (elim disjE conjE exE)
apply (simp add: par_tr_nil_left)
apply (simp add: par_tr_Tick_left)
(* <= *)
apply (rule)
apply (simp add: in_traces)
apply (case_tac "noTick t")
apply (rule_tac x="<>" in exI)
apply (rule_tac x="t" in exI)
apply (simp)
apply (simp add: par_tr_nil_left)
apply (simp add: noTick_def)
apply (rule_tac x="<Tick>" in exI)
apply (rule_tac x="t" in exI)
apply (simp)
apply (simp add: par_tr_Tick_left)
apply (simp add: noTick_def)
done
(*********************************************************
P ||| SKIP
*********************************************************)
lemma cspT_Interleave_unit_r:
"P ||| SKIP =T[M,M] P"
apply (rule cspT_rw_left)
apply (rule cspT_commut)
apply (simp add: cspT_Interleave_unit_l)
done
lemmas cspT_Interleave_unit =
cspT_Interleave_unit_l
cspT_Interleave_unit_r
end
|
import datetime
import random
import numpy as np
import torch
from torch.utils import data
class TimeSeries(data.Dataset):
def __init__(self, data_frame, input_time_interval, output_time_interval, output_keyword,
valid_rate=0.2, shuffle_seed=0):
self.data_frame = data_frame
self.data_channels = self.data_frame.head(1).values.shape[1]
self.input_time_interval = input_time_interval
self.output_time_interval = output_time_interval
self.output_keyword = output_keyword
self.get_data_list()
self.dataset_size = len(self.inputs)
self._split(valid_rate, shuffle_seed)
def get_data_list(self):
self.inputs = []
self.targets = []
head = self.data_frame.head(1).index[0]
tail = self.data_frame.tail(1).index[0]
data_head = head - datetime.timedelta(days=1)
while True:
data_head = data_head + datetime.timedelta(days=1)
data_tail = data_head + datetime.timedelta(days=self.input_time_interval - 1)
target_head = data_tail + datetime.timedelta(days=1)
target_tail = target_head + datetime.timedelta(days=self.output_time_interval - 1)
if target_tail > tail:
break
input = self.data_frame[data_head:data_tail]
target = self.data_frame[target_head:target_tail][self.output_keyword]
self.inputs.append(input)
self.targets.append(target)
def _split(self, valid_rate, shuffle_seed):
self.indices = list(range(self.dataset_size))
random.seed(shuffle_seed)
random.shuffle(self.indices)
split = int(np.floor((1 - valid_rate) * self.dataset_size))
self.train_indices, self.valid_indices = self.indices[:split], self.indices[split:]
self.train_dataset = data.Subset(self, self.train_indices)
self.valid_dataset = data.Subset(self, self.valid_indices)
self.train_sampler = data.RandomSampler(self.train_dataset)
self.valid_sampler = data.SequentialSampler(self.valid_dataset)
self.test_sampler = data.SequentialSampler(self)
def get_dataloader(self, batch_size=1, num_workers=0):
train_loader = data.DataLoader(self.train_dataset, batch_size=batch_size,
sampler=self.train_sampler, num_workers=num_workers)
valid_loader = data.DataLoader(self.valid_dataset, batch_size=batch_size,
sampler=self.valid_sampler, num_workers=num_workers)
test_loader = data.DataLoader(self, batch_size=batch_size, sampler=self.test_sampler, num_workers=num_workers)
return train_loader, valid_loader, test_loader
def __getitem__(self, index):
input = self.inputs[index].values.astype(np.float).transpose(1, 0)
target = self.targets[index].values.astype(np.float)
input = torch.from_numpy(input).float()
target = torch.from_numpy(target).float()
return input, target
def __len__(self):
return self.dataset_size
|
-- @@stderr --
dtrace: failed to compile script test/unittest/actions/freopen/err.D_FREOPEN_INVALID.d: [D_FREOPEN_INVALID] line 18: freopen( ) argument #1 cannot be "."
|
#redirect UC Davis Ski and Snowboard Team
|
header "Soundness"
theory Soundness imports Completeness begin
lemma permutation_validS: "fs <~~> gs --> (validS fs = validS gs)"
apply(simp add: validS_def)
apply(simp add: evalS_def)
apply(simp add: perm_set_eq)
done
lemma modelAssigns_vblcase: "phi \<in> modelAssigns M \<Longrightarrow> x \<in> objects M \<Longrightarrow> vblcase x phi \<in> modelAssigns M"
apply (simp add: modelAssigns_def, rule)
apply(erule_tac rangeE)
apply(case_tac xaa rule: vbl_casesE, auto)
done
lemma tmp: "(!x : A. P x | Q) ==> (! x : A. P x) | Q " by blast
lemma soundnessFAll: "!!Gamma.
[| u ~: freeVarsFL (FAll Pos A # Gamma);
validS (instanceF u A # Gamma) |]
==> validS (FAll Pos A # Gamma)"
apply (simp add: validS_def, rule)
apply (drule_tac x=M in spec, rule)
apply(simp add: evalF_instance)
apply (rule tmp, rule)
apply(drule_tac x="% y. if y = u then x else phi y" in bspec)
apply(simp add: modelAssigns_def) apply force
apply(erule disjE)
apply (rule disjI1, simp)
apply(subgoal_tac "evalF M (vblcase x (\<lambda>y. if y = u then x else phi y)) A = evalF M (vblcase x phi) A")
apply force
apply(rule evalF_equiv)
apply(rule equalOn_vblcaseI)
apply(rule,rule)
apply(simp add: freeVarsFL_cons)
apply (rule equalOnI, force)
apply(rule disjI2)
apply(subgoal_tac "evalS M (\<lambda>y. if y = u then x else phi y) Gamma = evalS M phi Gamma")
apply force
apply(rule evalS_equiv)
apply(rule equalOnI)
apply(force simp: freeVarsFL_cons)
done
lemma soundnessFEx: "validS (instanceF x A # Gamma) ==> validS (FAll Neg A # Gamma)"
apply(simp add: validS_def)
apply (simp add: evalF_instance, rule, rule)
apply(drule_tac x=M in spec)
apply (drule_tac x=phi in bspec, assumption)
apply(erule disjE)
apply(rule disjI1)
apply (rule_tac x="phi x" in bexI, assumption)
apply(force dest: modelAssignsD subsetD)
apply (rule disjI2, assumption)
done
lemma soundnessFCut: "[| validS (C # Gamma); validS (FNot C # Delta) |] ==> validS (Gamma @ Delta)"
(* apply(force simp: validS_def evalS_append evalS_cons evalF_FNot)*)
apply (simp add: validS_def, rule, rule)
apply(drule_tac x=M in spec)
apply(drule_tac x=M in spec)
apply(drule_tac x=phi in bspec) apply assumption
apply(drule_tac x=phi in bspec) apply assumption
apply (simp add: evalS_append evalF_FNot, blast)
done
lemma completeness: "fs : deductions (PC) = validS fs"
apply rule
apply(rule soundness) apply assumption
apply(subgoal_tac "fs : deductions CutFreePC")
apply(rule subsetD) prefer 2 apply assumption
apply(rule mono_deductions)
apply(simp add: PC_def CutFreePC_def) apply blast
apply(rule adequacy)
by assumption
end
|
from typing import List, Dict, Union, Tuple
import numpy as np
import pickle
import math
import sympy
from igp2 import AgentState, Lane, VelocityTrajectory, StateTrajectory, Map
from shapely.geometry import Point, LineString, Polygon, MultiPolygon
from shapely.ops import unary_union, split
from grit.core.goal_generator import TypedGoal, GoalGenerator
from grit.core.base import get_occlusions_dir
class FeatureExtractor:
MAX_ONCOMING_VEHICLE_DIST = 100
# Minimum area the occlusion must have to contain a vehicle (assuming a 4m*3m vehicle)
MIN_OCCLUSION_AREA = 12
# Maximum distance the occlusion can be to be considered as significant for creating occlusions.
MAX_OCCLUSION_DISTANCE = 30
FRAME_STEP_SIZE = 25
MISSING = True
NON_MISSING = False
feature_names = {'path_to_goal_length': 'scalar',
'in_correct_lane': 'binary',
'speed': 'scalar',
'acceleration': 'scalar',
'angle_in_lane': 'scalar',
'vehicle_in_front_dist': 'scalar',
'vehicle_in_front_speed': 'scalar',
'oncoming_vehicle_dist': 'scalar',
'oncoming_vehicle_speed': 'scalar',
'road_heading': 'scalar',
'exit_number': 'integer'}
indicator_features = ['exit_number_missing', 'vehicle_in_front_missing', 'oncoming_vehicle_missing']
possibly_missing_features = {'exit_number': 'exit_number_missing',
'oncoming_vehicle_dist': 'oncoming_vehicle_missing',
'oncoming_vehicle_speed': 'oncoming_vehicle_missing',
'vehicle_in_front_dist': 'vehicle_in_front_missing',
'vehicle_in_front_speed': 'vehicle_in_front_missing'}
def __init__(self, scenario_map: Map, *args):
self.scenario_map = scenario_map
# If we want to consider occlusions, we need to provide the scenario map and episode index as parameter,
# in this order.
if len(args) > 1:
self.scenario_name = args[0]
self.episode_idx = args[1]
with open(get_occlusions_dir() + f"{self.scenario_name}_e{self.episode_idx}.p", 'rb') as file:
self.occlusions = pickle.load(file)
def extract(self, agent_id: int, frames: List[Dict[int, AgentState]], goal: TypedGoal, ego_agent_id: int = None,
initial_frame: Dict[int, AgentState] = None) \
-> Dict[str, Union[float, bool]]:
"""Extracts a dict of features describing the observation
Args:
agent_id: identifier for the agent of which we want the features
frames: list of observed frames
goal: goal of the agent
ego_agent_id: id of the ego agent from whose pov the occlusions are taken. Used for indicator features
initial_frame: first frame in which the target agent is visible to the ego. Used for indicator features
Returns: dict of features values
"""
current_frame = frames[-1]
current_state = current_frame[agent_id]
initial_state = frames[0][agent_id]
current_lane = goal.lane_path[0]
lane_path = goal.lane_path
speed = current_state.speed
acceleration = np.linalg.norm(current_state.acceleration)
in_correct_lane = self.in_correct_lane(lane_path)
path_to_goal_length = self.path_to_goal_length(current_state, goal, lane_path)
angle_in_lane = self.angle_in_lane(current_state, current_lane)
road_heading = self.road_heading(lane_path)
exit_number = self.exit_number(initial_state, lane_path)
goal_type = goal.goal_type
vehicle_in_front_id, vehicle_in_front_dist = self.vehicle_in_front(agent_id, lane_path, current_frame)
if vehicle_in_front_id is None:
vehicle_in_front_speed = 20
vehicle_in_front_dist = 100
else:
vehicle_in_front = current_frame[vehicle_in_front_id]
vehicle_in_front_speed = vehicle_in_front.speed
oncoming_vehicle_id, oncoming_vehicle_dist = self.oncoming_vehicle(agent_id, lane_path, current_frame)
if oncoming_vehicle_id is None:
oncoming_vehicle_speed = 20
else:
oncoming_vehicle_speed = current_frame[oncoming_vehicle_id].speed
features = {'path_to_goal_length': path_to_goal_length,
'in_correct_lane': in_correct_lane,
'speed': speed,
'acceleration': acceleration,
'angle_in_lane': angle_in_lane,
'vehicle_in_front_dist': vehicle_in_front_dist,
'vehicle_in_front_speed': vehicle_in_front_speed,
'oncoming_vehicle_dist': oncoming_vehicle_dist,
'oncoming_vehicle_speed': oncoming_vehicle_speed,
'road_heading': road_heading,
'exit_number': exit_number,
'goal_type': goal_type}
# We pass the ego_agent_id only if we want to extract the indicator features.
if ego_agent_id is not None:
occlusion_frame_id = math.ceil(current_state.time / self.FRAME_STEP_SIZE)
frame_occlusions = self.occlusions[occlusion_frame_id]
occlusions = unary_union(self.get_occlusions_ego_polygon(frame_occlusions, ego_agent_id))
vehicle_in_front_occluded = self.is_vehicle_in_front_missing(vehicle_in_front_dist, agent_id, lane_path,
current_frame, occlusions)
oncoming_vehicle_occluded = self.is_oncoming_vehicle_missing(oncoming_vehicle_dist, lane_path, occlusions)
# Get the first state in which both the ego and target vehicles are alive (even if target is occluded).
initial_state = initial_frame[agent_id]
exit_number_occluded = self.is_exit_number_missing(initial_state, goal) \
if self.scenario_name == "round" else False
indicator_features = {'vehicle_in_front_missing': vehicle_in_front_occluded,
'oncoming_vehicle_missing': oncoming_vehicle_occluded,
'exit_number_missing': exit_number_occluded}
features.update(indicator_features)
return features
@staticmethod
def get_vehicles_in_route(ego_agent_id: int, path: List[Lane], frame: Dict[int, AgentState]):
agents = []
for agent_id, agent in frame.items():
agent_point = Point(*agent.position)
if agent_id != ego_agent_id:
for lane in path:
if lane.boundary.contains(agent_point):
agents.append(agent_id)
return agents
@staticmethod
def angle_in_lane(state: AgentState, lane: Lane) -> float:
"""
Get the signed angle between the vehicle heading and the lane heading
Args:
state: current state of the vehicle
lane: : current lane of the vehicle
Returns: angle in radians
"""
lon = lane.distance_at(state.position)
lane_heading = lane.get_heading_at(lon)
angle_diff = np.diff(np.unwrap([lane_heading, state.heading]))[0]
return angle_diff
@staticmethod
def road_heading(lane_path: List[Lane]):
lane = lane_path[-1]
start_heading = lane.get_heading_at(0)
end_heading = lane.get_heading_at(lane.length)
heading_change = np.diff(np.unwrap([start_heading, end_heading]))[0]
return heading_change
@staticmethod
def in_correct_lane(lane_path: List[Lane]):
for idx in range(0, len(lane_path) - 1):
if lane_path[idx].lane_section == lane_path[idx+1].lane_section:
return False
return True
@classmethod
def path_to_goal_length(cls, state: AgentState, goal: TypedGoal, path: List[Lane]) -> float:
end_point = goal.goal.center
return cls.path_to_point_length(state, end_point, path)
@classmethod
def vehicle_in_front(cls, target_agent_id: int, lane_path: List[Lane], frame: Dict[int, AgentState]):
state = frame[target_agent_id]
vehicles_in_route = cls.get_vehicles_in_route(target_agent_id, lane_path, frame)
min_dist = np.inf
vehicle_in_front = None
target_dist_along = cls.dist_along_path(lane_path, state.position)
# find the vehicle in front with closest distance
for agent_id in vehicles_in_route:
agent_point = frame[agent_id].position
agent_dist = cls.dist_along_path(lane_path, agent_point)
dist = agent_dist - target_dist_along
if 1e-4 < dist < min_dist:
vehicle_in_front = agent_id
min_dist = dist
return vehicle_in_front, min_dist
def is_vehicle_in_front_missing(self, dist: float, target_id: int, lane_path: List[Lane],
frame: Dict[int, AgentState], occlusions: MultiPolygon):
"""
Args:
dist: distance of the closest oncoming vehicle, if any.
target_id: id of the vehicle for which we are extracting the features
lane_path: lanes executed by the target vehicle if it had the assigned goal
frame: current state of the world
occlusions: must be unary union of all the occlusions for the ego at that point in time
"""
target_state = frame[target_id]
target_point = Point(*target_state.position)
current_lane = self.scenario_map.best_lane_at(target_state.position, target_state.heading, True)
midline = current_lane.midline
# Remove all the occlusions that are behind the target vehicle as we want possible hidden vehicles in front.
area_before, area_after = self._get_split_at(midline, target_point)
occlusions = self._get_occlusions_past_point(current_lane, lane_path, occlusions, target_point, area_after)
if occlusions is None:
return self.NON_MISSING
occlusions = self._get_significant_occlusions(occlusions)
if occlusions is None:
# The occlusions are not large enough to hide a vehicle.
return self.NON_MISSING
distance_to_occlusion = occlusions.distance(target_point)
if distance_to_occlusion > self.MAX_OCCLUSION_DISTANCE:
# The occlusion is far away, and won't affect the target vehicle decisions.
return self.NON_MISSING
# Otherwise, the feature is missing if there is an occlusion closer than the vehicle in front.
return not dist < distance_to_occlusion + 2.5
@classmethod
def dist_along_path(cls, path: List[Lane], point: np.ndarray):
shapely_point = Point(*point)
midline = cls.get_lane_path_midline(path)
dist = midline.project(shapely_point)
return dist
@staticmethod
def get_current_path_lane_idx(path: List[Lane], point: np.ndarray) -> int:
""" Get the index of the lane closest to a point"""
if type(point) == Point:
shapely_point = point
else:
shapely_point = Point(point[0], point[1])
for idx, lane in enumerate(path):
if lane.boundary.contains(shapely_point):
return idx
closest_lane_dist = np.inf
closest_lane_idx = None
for idx, lane in enumerate(path):
dist = lane.boundary.exterior.distance(shapely_point)
if dist < closest_lane_dist:
closest_lane_dist = dist
closest_lane_idx = idx
return closest_lane_idx
@staticmethod
def path_to_point_length(state: AgentState, point: np.ndarray, path: List[Lane]) -> float:
""" Get the length of a path across multiple lanes
Args:
state: initial state of the vehicle
point: final point to be reached
path: sequence of lanes traversed
Returns: path length
"""
end_lane = path[-1]
end_lane_dist = end_lane.distance_at(point)
start_point = state.position
start_lane = path[0]
start_lane_dist = start_lane.distance_at(start_point)
dist = end_lane_dist - start_lane_dist
if len(path) > 1:
prev_lane = None
for idx in range(len(path) - 1):
lane = path[idx]
lane_change = prev_lane is not None and prev_lane.lane_section == lane.lane_section
if not lane_change:
dist += lane.length
prev_lane = lane
return dist
@staticmethod
def angle_to_goal(state, goal):
goal_heading = np.arctan2(goal[1] - state.y, goal[0] - state.x)
return np.diff(np.unwrap([goal_heading, state.heading]))[0]
@staticmethod
def get_junction_lane(lane_path: List[Lane]) -> Union[Lane, None]:
for lane in lane_path:
if lane.parent_road.junction is not None:
return lane
return None
@staticmethod
def get_lane_path_midline(lane_path: List[Lane]) -> LineString:
midline_points = []
for idx, lane in enumerate(lane_path[:-1]):
# check if next lane is adjacent
if lane_path[idx + 1] not in lane.lane_section.all_lanes:
midline_points.extend(lane.midline.coords[:-1])
midline_points.extend(lane_path[-1].midline.coords)
lane_ls = LineString(midline_points)
return lane_ls
@staticmethod
def _get_split_at(midline, point):
"""
Split the midline at a specific point.
"""
point_on_midline = midline.interpolate(midline.project(point)).buffer(0.0001)
split_lanes = split(midline, point_on_midline)
if len(split_lanes) == 2:
# Handle the case in which the split point is at the start/end of the lane.
line_before, line_after = split_lanes
else:
line_before, _, line_after = split_lanes
return line_before, line_after
def _get_oncoming_vehicles(self, lane_path: List[Lane], ego_agent_id: int, frame: Dict[int, AgentState]) \
-> Dict[int, Tuple[AgentState, float]]:
oncoming_vehicles = {}
ego_junction_lane = self.get_junction_lane(lane_path)
if ego_junction_lane is None:
return oncoming_vehicles
ego_junction_lane_boundary = ego_junction_lane.boundary.buffer(0)
lanes_to_cross = self._get_lanes_to_cross(ego_junction_lane)
agent_lanes = [(i, self.scenario_map.best_lane_at(s.position, s.heading, True)) for i, s in frame.items()]
for lane_to_cross in lanes_to_cross:
lane_sequence = self._get_predecessor_lane_sequence(lane_to_cross)
midline = self.get_lane_path_midline(lane_sequence)
crossing_point = lane_to_cross.boundary.buffer(0).intersection(ego_junction_lane_boundary).centroid
crossing_lon = midline.project(crossing_point)
# find agents in lane to cross
for agent_id, agent_lane in agent_lanes:
agent_state = frame[agent_id]
if agent_id != ego_agent_id and agent_lane in lane_sequence:
agent_lon = midline.project(Point(agent_state.position))
dist = crossing_lon - agent_lon
if 0 < dist < self.MAX_ONCOMING_VEHICLE_DIST:
oncoming_vehicles[agent_id] = (agent_state, dist)
return oncoming_vehicles
def _get_lanes_to_cross(self, ego_lane: Lane) -> List[Lane]:
ego_road = ego_lane.parent_road
ego_incoming_lane = ego_lane.link.predecessor[0]
ego_lane_boundary = ego_lane.boundary.buffer(0)
lanes = []
for connection in ego_road.junction.connections:
for lane_link in connection.lane_links:
lane = lane_link.to_lane
same_predecessor = (ego_incoming_lane.id == lane_link.from_id
and ego_incoming_lane.parent_road.id == connection.incoming_road.id)
if not (same_predecessor or self._has_priority(ego_road, lane.parent_road)):
overlap = ego_lane_boundary.intersection(lane.boundary.buffer(0))
if overlap.area > 1:
lanes.append(lane)
return lanes
def _get_occlusions_past_point(self, current_lane, other_lanes, all_occlusions, point_of_cut, area_to_keep):
"""
Get the occlusions that are both on the 'other_lanes' and on the area_to_keep.
Args:
current_lane: lane on which the vehicle is currently on
other_lanes: lanes for which we want to find the occluded areas
all_occlusions: all the occlusions in the current frame
point_of_cut: point in the current lane at which we want to cut the total occluded areas
area_to_keep: part of the MIDLINE we want the occlusions on
"""
# Find the occlusions that intersect the lanes we want.
possible_occlusions = []
for lane in other_lanes:
o = all_occlusions.intersection(lane.boundary.buffer(0))
if isinstance(o, MultiPolygon):
possible_occlusions.extend(list(o.geoms))
elif isinstance(o, Polygon):
possible_occlusions.append(o)
possible_occlusions = unary_union(possible_occlusions)
if possible_occlusions.area == 0:
return None
# Find the line perpendicular to the current lane that passes through the point_of_cut
ds = current_lane.boundary.boundary.project(point_of_cut)
p = current_lane.boundary.boundary.interpolate(ds)
slope = (p.y - point_of_cut.y) / (p.x - point_of_cut.x)
s_p = sympy.Point(p.x, p.y)
direction1 = Point(p.x - point_of_cut.x, p.y - point_of_cut.y)
direction2 = Point(point_of_cut.x - p.x, point_of_cut.y - p.y)
p1 = self.get_extended_point(30, slope, direction1, s_p)
p2 = self.get_extended_point(30, slope, direction2, s_p)
# Split the occluded areas along the line we just computed.
line = LineString([Point(p1.x, p1.y), Point(p2.x, p2.y)])
intersections = split(possible_occlusions, line)
# Get the occlusions that are on the area_to_keep.
return unary_union([intersection for intersection in intersections.geoms
if intersection.intersection(area_to_keep).length > 1])
@staticmethod
def get_extended_point(length, slope, direction, point):
delta_x = math.sqrt(length ** 2 / (1 + slope ** 2))
delta_y = math.sqrt(length ** 2 - delta_x ** 2)
if direction.x < 0:
delta_x = -delta_x
if direction.y < 0:
delta_y = -delta_y
return point.translate(delta_x, delta_y)
def _get_significant_occlusions(self, occlusions):
"""
Return a Multipolygon or Polygon with the occlusions that are large enough to fit a hidden vehicle.
"""
if isinstance(occlusions, MultiPolygon):
return unary_union([occlusion for occlusion in occlusions.geoms
if occlusion.area > self.MIN_OCCLUSION_AREA])
elif isinstance(occlusions, Polygon):
return occlusions if occlusions.area > self.MIN_OCCLUSION_AREA else None
def _get_min_dist_from_occlusions_oncoming_lanes(self, lanes_to_cross, ego_junction_lane,
ego_junction_lane_boundary, occlusions):
"""
Get the minimum distance from any of the crossing points to the occlusions that could hide an oncoming vehicle.
A crossing point is a point along the target vehicle's path inside a junction.
Args:
lanes_to_cross: list of lanes that the target vehicle will intersect while inside the junction.
ego_junction_lane: lane the target vehicle travels on
ego_junction_lane_boundary: boundary of the ego_junction lane
occlusions: list of all the occlusions in the frame
"""
occluded_oncoming_areas = []
crossing_points = []
for lane_to_cross in lanes_to_cross:
crossing_point = lane_to_cross.boundary.buffer(0).intersection(ego_junction_lane_boundary).centroid
crossing_points.append(crossing_point)
lane_sequence = self._get_predecessor_lane_sequence(lane_to_cross)
midline = self.get_lane_path_midline(lane_sequence)
# Find the occlusions on the lanes that the ego vehicle will cross.
if occlusions:
# Get the part of the midline of the lanes in which there could be oncoming vehicles, that is before
# the crossing point.
# Ignore the occlusions that are "after" (w.r.t traffic direction) the crossing point.
# We only want to check if there is a hidden vehicle that could collide with the ego.
# This can only happen with vehicles that are driving in the lane's direction of traffic
# and have not passed the crossing point that the ego will drive through.
area_before, area_after = self._get_split_at(midline, crossing_point)
# Get the significant occlusions.
lane_occlusions = self._get_occlusions_past_point(ego_junction_lane,
lane_sequence,
occlusions,
crossing_point,
area_before)
if lane_occlusions is None:
continue
occluded_oncoming_areas.append(lane_occlusions)
if occluded_oncoming_areas:
occluded_oncoming_areas = unary_union(occluded_oncoming_areas)
# Only take the occlusions that could fit a hidden vehicle.
occluded_oncoming_areas = self._get_significant_occlusions(occluded_oncoming_areas)
# Get the minimum distance from any of the crossing points and the relevant occlusions.
if occluded_oncoming_areas:
return min([crossing_point.distance(occluded_oncoming_areas) for crossing_point in crossing_points])
# If there are no occlusions large enough to fit a hidden vehicle.
return math.inf
@staticmethod
def get_occlusions_ego_polygon(frame_occlusions, ego_id):
"""
Given the occlusions in a frame, extract the occlusions w.r.t the ego and return them as list of MultiPolygons.
"""
occlusions_vehicle_frame = frame_occlusions[ego_id]
occlusions = []
for road_occlusions in occlusions_vehicle_frame:
for lane_occlusions in occlusions_vehicle_frame[road_occlusions]:
lane_occlusion = occlusions_vehicle_frame[road_occlusions][lane_occlusions]
if lane_occlusion is not None:
occlusions.append(lane_occlusion)
return occlusions
@classmethod
def _get_predecessor_lane_sequence(cls, lane: Lane) -> List[Lane]:
lane_sequence = []
total_length = 0
while lane is not None and total_length < 100:
lane_sequence.insert(0, lane)
total_length += lane.midline.length
lane = lane.link.predecessor[0] if lane.link.predecessor else None
return lane_sequence
@staticmethod
def _has_priority(ego_road, other_road):
for priority in ego_road.junction.priorities:
if (priority.high_id == ego_road.id
and priority.low_id == other_road.id):
return True
return False
def oncoming_vehicle(self, ego_agent_id: int, lane_path: List[Lane], frame: Dict[int, AgentState], max_dist=100):
oncoming_vehicles = self._get_oncoming_vehicles(lane_path, ego_agent_id, frame)
min_dist = max_dist
closest_vehicle_id = None
for agent_id, (agent, dist) in oncoming_vehicles.items():
if dist < min_dist:
min_dist = dist
closest_vehicle_id = agent_id
return closest_vehicle_id, min_dist
def is_oncoming_vehicle_missing(self, min_dist: int, lane_path: List[Lane], occlusions: MultiPolygon):
ego_junction_lane = self.get_junction_lane(lane_path)
if ego_junction_lane is None:
return False
ego_junction_lane_boundary = ego_junction_lane.boundary.buffer(0)
lanes_to_cross = self._get_lanes_to_cross(ego_junction_lane)
min_occlusion_distance = self._get_min_dist_from_occlusions_oncoming_lanes(lanes_to_cross, ego_junction_lane,
ego_junction_lane_boundary,
occlusions)
# If the closest occlusion is too far away (or missing), we say that occlusion is not significant.
if min_occlusion_distance > self.MAX_OCCLUSION_DISTANCE:
return False
# If the closest oncoming vehicle is further away to any of the crossing points that the occlusion,
# then the feature is missing. The 2.5 meters offset is in case the vehicle is partially occluded.
return min_occlusion_distance + 2.5 < min_dist
def exit_number(self, initial_state: AgentState, future_lane_path: List[Lane]):
# get the exit number in a roundabout
if (future_lane_path[-1].parent_road.junction is None
or future_lane_path[-1].parent_road.junction.junction_group is None
or future_lane_path[-1].parent_road.junction.junction_group.type != 'roundabout'):
return 0
position = initial_state.position
heading = initial_state.heading
possible_lanes = self.scenario_map.lanes_within_angle(position, heading, np.pi / 4,
drivable_only=True, max_distance=3)
initial_lane = possible_lanes[GoalGenerator.get_best_lane(possible_lanes, position, heading)]
lane_path = self.path_to_lane(initial_lane, future_lane_path[-1])
# iterate through lane path and count number of junctions
exit_number = 0
entrance_passed = False
if lane_path is not None:
for lane in lane_path:
if self.is_roundabout_entrance(lane):
entrance_passed = True
elif entrance_passed and self.is_roundabout_junction(lane):
exit_number += 1
return exit_number
def is_exit_number_missing(self, initial_state: AgentState, goal: TypedGoal):
"""
The exit number feature is missing if we cannot get the exit number. This happens when:
- the target vehicle is already in the roundabout when it becomes visible to the ego.
- the target vehicle is occluded w.r.t the ego when it enters the roundabout.
Args:
initial_state: state of the target vehicle when it first became visible to the ego
goal: the goal we are trying to get the probability for
"""
return self.exit_number(initial_state, goal.lane_path) == 0
@staticmethod
def is_roundabout_junction(lane: Lane):
junction = lane.parent_road.junction
return (junction is not None and junction.junction_group is not None
and junction.junction_group.type == 'roundabout')
def is_roundabout_entrance(self, lane: Lane) -> bool:
predecessor_in_roundabout = (lane.link.predecessor is not None and len(lane.link.predecessor) == 1
and self.scenario_map.road_in_roundabout(lane.link.predecessor[0].parent_road))
return self.is_roundabout_junction(lane) and not predecessor_in_roundabout
def get_typed_goals(self, trajectory: VelocityTrajectory, goals: List[Tuple[int, int]]):
typed_goals = []
goal_gen = GoalGenerator()
gen_goals = goal_gen.generate(self.scenario_map, trajectory)
for goal in goals:
for gen_goal in gen_goals:
if gen_goal.goal.reached(Point(*goal)):
break
else:
gen_goal = None
typed_goals.append(gen_goal)
return typed_goals
@staticmethod
def goal_type(route: List[Lane]):
return GoalGenerator.get_juction_goal_type(route[-1])
@staticmethod
def path_to_lane(initial_lane: Lane, target_lane: Lane, max_depth=20) -> List[Lane]:
visited_lanes = {initial_lane}
open_set = [[initial_lane]]
while len(open_set) > 0:
lane_sequence = open_set.pop(0)
if len(lane_sequence) > max_depth:
break
lane = lane_sequence[-1]
if lane == target_lane:
return lane_sequence
junction = lane.parent_road.junction
neighbours = lane.traversable_neighbours()
for neighbour in neighbours:
if neighbour not in visited_lanes:
visited_lanes.add(neighbour)
open_set.append(lane_sequence + [neighbour])
return None
class GoalDetector:
""" Detects the goals of agents based on their trajectories"""
def __init__(self, possible_goals, dist_threshold=1.5):
self.dist_threshold = dist_threshold
self.possible_goals = possible_goals
def detect_goals(self, trajectory: StateTrajectory):
goals = []
goal_frame_idxes = []
for point_idx, agent_point in enumerate(trajectory.path):
for goal_idx, goal_point in enumerate(self.possible_goals):
dist = np.linalg.norm(agent_point - goal_point)
if dist <= self.dist_threshold and goal_idx not in goals:
goals.append(goal_idx)
goal_frame_idxes.append(point_idx)
return goals, goal_frame_idxes
def get_agents_goals_ind(self, tracks, static_info, meta_info, map_meta, agent_class='car'):
goal_locations = map_meta.goals
agent_goals = {}
for track_idx in range(len(static_info)):
if static_info[track_idx]['class'] == agent_class:
track = tracks[track_idx]
agent_goals[track_idx] = []
for i in range(static_info[track_idx]['numFrames']):
point = np.array([track['xCenter'][i], track['yCenter'][i]])
for goal_idx, loc in enumerate(goal_locations):
dist = np.linalg.norm(point - loc)
if dist < self.dist_threshold and loc not in agent_goals[track_idx]:
agent_goals[track_idx].append(loc)
return agent_goals
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a : R
⊢ (match { toFinsupp := Finsupp.single n a } with
| { toFinsupp := p } => p.support) ⊆
{n} Tactic: rw [← ofFinsupp_single, support] State Before: R : Type u
a✝ b : R
m n✝ : ℕ
inst✝ : Semiring R
p q : R[X]
n : ℕ
a : R
⊢ (match { toFinsupp := Finsupp.single n a } with
| { toFinsupp := p } => p.support) ⊆
{n} State After: no goals Tactic: exact Finsupp.support_single_subset
|
State Before: α : Type u_1
E : Type ?u.30268
F : Type u_2
𝕜 : Type ?u.30274
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace ℝ F
m✝ : MeasurableSpace α
μ✝ : Measure α
m : MeasurableSpace α
μ : Measure α
c : ℝ≥0∞
s : Set α
⊢ weightedSMul (c • μ) s = ENNReal.toReal c • weightedSMul μ s State After: case h
α : Type u_1
E : Type ?u.30268
F : Type u_2
𝕜 : Type ?u.30274
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace ℝ F
m✝ : MeasurableSpace α
μ✝ : Measure α
m : MeasurableSpace α
μ : Measure α
c : ℝ≥0∞
s : Set α
x : F
⊢ ↑(weightedSMul (c • μ) s) x = ↑(ENNReal.toReal c • weightedSMul μ s) x Tactic: ext1 x State Before: case h
α : Type u_1
E : Type ?u.30268
F : Type u_2
𝕜 : Type ?u.30274
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace ℝ F
m✝ : MeasurableSpace α
μ✝ : Measure α
m : MeasurableSpace α
μ : Measure α
c : ℝ≥0∞
s : Set α
x : F
⊢ ↑(weightedSMul (c • μ) s) x = ↑(ENNReal.toReal c • weightedSMul μ s) x State After: case h
α : Type u_1
E : Type ?u.30268
F : Type u_2
𝕜 : Type ?u.30274
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace ℝ F
m✝ : MeasurableSpace α
μ✝ : Measure α
m : MeasurableSpace α
μ : Measure α
c : ℝ≥0∞
s : Set α
x : F
⊢ ↑(weightedSMul (c • μ) s) x = (ENNReal.toReal c • ↑(weightedSMul μ s)) x Tactic: push_cast State Before: case h
α : Type u_1
E : Type ?u.30268
F : Type u_2
𝕜 : Type ?u.30274
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace ℝ F
m✝ : MeasurableSpace α
μ✝ : Measure α
m : MeasurableSpace α
μ : Measure α
c : ℝ≥0∞
s : Set α
x : F
⊢ ↑(weightedSMul (c • μ) s) x = (ENNReal.toReal c • ↑(weightedSMul μ s)) x State After: case h
α : Type u_1
E : Type ?u.30268
F : Type u_2
𝕜 : Type ?u.30274
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace ℝ F
m✝ : MeasurableSpace α
μ✝ : Measure α
m : MeasurableSpace α
μ : Measure α
c : ℝ≥0∞
s : Set α
x : F
⊢ ENNReal.toReal (↑↑(c • μ) s) • x = ENNReal.toReal c • ENNReal.toReal (↑↑μ s) • x Tactic: simp_rw [Pi.smul_apply, weightedSMul_apply] State Before: case h
α : Type u_1
E : Type ?u.30268
F : Type u_2
𝕜 : Type ?u.30274
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace ℝ F
m✝ : MeasurableSpace α
μ✝ : Measure α
m : MeasurableSpace α
μ : Measure α
c : ℝ≥0∞
s : Set α
x : F
⊢ ENNReal.toReal (↑↑(c • μ) s) • x = ENNReal.toReal c • ENNReal.toReal (↑↑μ s) • x State After: case h
α : Type u_1
E : Type ?u.30268
F : Type u_2
𝕜 : Type ?u.30274
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace ℝ F
m✝ : MeasurableSpace α
μ✝ : Measure α
m : MeasurableSpace α
μ : Measure α
c : ℝ≥0∞
s : Set α
x : F
⊢ ENNReal.toReal ((c • ↑↑μ) s) • x = ENNReal.toReal c • ENNReal.toReal (↑↑μ s) • x Tactic: push_cast State Before: case h
α : Type u_1
E : Type ?u.30268
F : Type u_2
𝕜 : Type ?u.30274
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace ℝ F
m✝ : MeasurableSpace α
μ✝ : Measure α
m : MeasurableSpace α
μ : Measure α
c : ℝ≥0∞
s : Set α
x : F
⊢ ENNReal.toReal ((c • ↑↑μ) s) • x = ENNReal.toReal c • ENNReal.toReal (↑↑μ s) • x State After: no goals Tactic: simp_rw [Pi.smul_apply, smul_eq_mul, toReal_mul, smul_smul]
|
SUBROUTINE FN_TRUNC( FILE_NAME )
!***********************************************************************
!* Truncates a File Name for the Operating System
!*
!* Language: Fortran
!*
!* Author: Stuart G. Mentzer
!*
!* Date: 2003/07/18
!***********************************************************************
! Headers
INCLUDE 'platform.fi'
INCLUDE 'uds_fxn.fi'
! Arguments ______________________________________________________
CHARACTER*(*) FILE_NAME ! File name to truncate
! Variables ______________________________________________________
INTEGER IE, ITV, PFN, L_NAM, L_EXT
CHARACTER TEMP_NAME*256
! Functions ______________________________________________________
CHARACTER LJUST*256
EXTERNAL LJUST
! Find start of path-free file name
FILE_NAME = LJUST( FILE_NAME )
PFN = FN_POSN( FILE_NAME )
IF ( PFN .EQ. 0 ) RETURN
! Truncate name if necessary
IE = FE_POSN( FILE_NAME(PFN:) )
IF ( IE .GT. 0 ) THEN
L_NAM = IE - 1
ELSE
L_NAM = LEN_TRIM( FILE_NAME(PFN:) )
END IF
IF ( L_NAM .GT. LEN_FN_NAM ) THEN ! Truncate name
TEMP_NAME =
& FILE_NAME(:PFN-1+LEN_FN_NAM)//FILE_NAME(PFN+L_NAM:)
FILE_NAME = TEMP_NAME
END IF
! Truncate extension if necessary
IE = FE_POSN( FILE_NAME )
IF ( IE .GT. 0 ) THEN ! Has extension
ITV = FT_POSN( FILE_NAME )
IF ( ITV .EQ. 0 ) ITV = FV_POSN( FILE_NAME )
IF ( ITV .EQ. 0 ) THEN ! No type/version
L_EXT = LEN_TRIM( FILE_NAME(IE+1:) )
ELSE IF ( ITV .GT. IE + 1 ) THEN
L_EXT = LEN_TRIM( FILE_NAME(IE+1:ITV-1) )
ELSE
L_EXT = 0
END IF
IF ( L_EXT .GT. LEN_FN_EXT ) THEN ! Truncate extension
IF ( ITV .EQ. 0 ) THEN ! No type/version
FILE_NAME(IE+LEN_FN_EXT+1:) = ' '
ELSE ! Has type/version
TEMP_NAME =
& FILE_NAME(:IE+LEN_FN_EXT)//FILE_NAME(ITV:)
FILE_NAME = TEMP_NAME
END IF
END IF
END IF
RETURN
END
|
lemma closed_Collect_le: fixes f g :: "'a :: topological_space \<Rightarrow> 'b::linorder_topology" assumes f: "continuous_on UNIV f" and g: "continuous_on UNIV g" shows "closed {x. f x \<le> g x}"
|
Formal statement is: lemma locally_connected_2: assumes "locally connected S" "openin (top_of_set S) t" "x \<in> t" shows "openin (top_of_set S) (connected_component_set t x)" Informal statement is: If $S$ is locally connected and $t$ is an open subset of $S$, then the connected component of $t$ containing $x$ is open in $S$.
|
# Обучение нейрона с помощью функции потерь LogLoss
<h3 style="text-align: center;"><b>Нейрон с сигмоидой</b></h3>
Снова рассмотрим нейрон с сигмоидой, то есть $$f(x) = \sigma(x)=\frac{1}{1+e^{-x}}$$
Ранее мы установили, что **обучение нейрона с сигмоидой с квадратичной функцией потерь**:
$$MSE(w, x) = \frac{1}{2n}\sum_{i=1}^{n} (\hat{y_i} - y_i)^2 = \frac{1}{2n}\sum_{i=1}^{n} (\sigma(w \cdot x_i) - y_i)^2$$
где $w \cdot x_i$ - скалярное произведение, а $\sigma(w \cdot x_i) =\frac{1}{1+e^{-w \cdot x_i}} $ - сигмоида -- **неэффективно**, то есть мы увидели, что даже за большое количество итераций нейрон предсказывает плохо.
Давайте ещё раз взглянем на формулу для градиентного спуска от функции потерь $MSE$ по весам нейрона:
$$ \frac{\partial MSE}{\partial w} = \frac{1}{n} X^T (\sigma(w \cdot X) - y)\sigma(w \cdot X)(1 - \sigma(w \cdot X))$$
А теперь смотрим на график сигмоиды:
**Её значения: числа от 0 до 1.**
Если получше проанализировать формулу, то теперь можно заметить, что, поскольку сигмоида принимает значения между 0 и 1 (а значит (1-$\sigma$) тоже принимает значения от 0 до 1), то мы умножаем $X^T$ на столбец $(\sigma(w \cdot X) - y)$ из чисел от -1 до 1, а потом ещё на столбцы $\sigma(w \cdot X)$ и $(1 - \sigma(w \cdot X))$ из чисел от 0 до 1. Таким образом в лучшем случае $\frac{\partial{Loss}}{\partial{w}}$ будет столбцом из чисел, порядок которых максимум 0.01 (в среднем, понятно, что если сигмоида выдаёт все 0, то будет 0, если все 1, то тоже 0). После этого мы умножаем на шаг градиентного спуска, который обычно порядка 0.001 или 0.1 максимум. То есть мы вычитаем из весов числа порядка ~0.0001. Медленновато спускаемся, не правда ли? Это называют **проблемой затухающих градиентов**.
Чтобы избежать эту проблему в задачах классификации, в которых моделью является нейрон с сигмоидной функцией активации, предсказывающий "вероятности" принадлженостей к классамиспользуют **LogLoss**:
$$J(\hat{y}, y) = -\frac{1}{n} \sum_{i=1}^n y_i \log(\hat{y_i}) + (1 - y_i) \log(1 - \hat{y_i}) = -\frac{1}{n} \sum_{i=1}^n y_i \log(\sigma(w \cdot x_i)) + (1 - y_i) \log(1 - \sigma(w \cdot x_i))$$
где, как и прежде, $y$ - столбец $(n, 1)$ из истинных значений классов, а $\hat{y}$ - столбец $(n, 1)$ из предсказаний нейрона.
```
from matplotlib import pyplot as plt
from matplotlib.colors import ListedColormap
import numpy as np
import pandas as pd
```
```
def loss(y_pred, y):
return -np.mean(y * np.log(y_pred) + (1 - y) * np.log(1 - y_pred))
```
Отметим, что сейчас речь идёт именно о **бинарной классификации (на два класса)**, в многоклассовой классификации используется функция потерь под названием *кросс-энтропия*, которая является обобщением LogLoss'а на случай нескольких классов.
Почему же теперь всё будет лучше? Раньше была проблема умножения маленьких чисел в градиенте. Давайте посмотрим, что теперь:
* Для веса $w_j$:
$$ \frac{\partial Loss}{\partial w_j} =
-\frac{1}{n} \sum_{i=1}^n \left(\frac{y_i}{\sigma(w \cdot x_i)} - \frac{1 - y_i}{1 - \sigma(w \cdot x_i)}\right)(\sigma(w \cdot x_i))_{w_j}' = -\frac{1}{n} \sum_{i=1}^n \left(\frac{y_i}{\sigma(w \cdot x_i)} - \frac{1 - y_i}{1 - \sigma(w \cdot x_i)}\right)\sigma(w \cdot x_i)(1 - \sigma(w \cdot x_i))x_{ij} = $$
$$-\frac{1}{n} \sum_{i=1}^n \left(y_i - \sigma(w \cdot x_i)\right)x_{ij}$$
* Градиент $Loss$'а по вектору весов -- это вектор, $j$-ая компонента которого равна $\frac{\partial Loss}{\partial w_j}$ (помним, что весов всего $m$):
$$\begin{align}
\frac{\partial Loss}{\partial w} &= \begin{bmatrix}
-\frac{1}{n} \sum_{i=1}^n \left(y_i - \sigma(w \cdot x_i)\right)x_{i1} \\
-\frac{1}{n} \sum_{i=1}^n \left(y_i - \sigma(w \cdot x_i)\right)x_{i2} \\
\vdots \\
-\frac{1}{n} \sum_{i=1}^n \left(y_i - \sigma(w \cdot x_i)\right)x_{im}
\end{bmatrix}
\end{align}=\frac{1}{n} X^T \left(\hat{y} - y\right)$$
По аналогии с $w_j$ выведите формулу для свободного члена (bias'а) $b$ (*hint*: можно считать, что при нём есть признак $x_{i0}=1$ на всех $i$):
Получили новое правило для обновления $w$ и $b$.
```
def sigmoid(x):
"""Сигмоидальная функция"""
return 1 / (1 + np.exp(-x))
```
Реализуйте нейрон с функцией потерь LogLoss:
```
class Neuron:
def __init__(self, w=None, b=0):
"""
:param: w -- вектор весов
:param: b -- смещение
"""
# пока что мы не знаем размер матрицы X, а значит не знаем, сколько будет весов
self.w = w
self.b = b
def activate(self, x):
return sigmoid(x)
def forward_pass(self, X):
"""
Эта функция рассчитывает ответ нейрона при предъявлении набора объектов
:param: X -- матрица объектов размера (n, m), каждая строка - отдельный объект
:return: вектор размера (n, 1) из нулей и единиц с ответами перцептрона
"""
# реализуйте forward_pass
n = X.shape[0]
y_pred = np.zeros((n, 1))
y_pred = self.activate(X @ self.w.reshape(X.shape[1], 1) + self.b)
return y_pred.reshape(-1, 1)
def backward_pass(self, X, y, y_pred, learning_rate=0.1):
"""
Обновляет значения весов нейрона в соответствие с этим объектом
:param: X -- матрица объектов размера (n, m)
y -- вектор правильных ответов размера (n, 1)
learning_rate - "скорость обучения" (символ alpha в формулах выше)
В этом методе ничего возвращать не нужно, только правильно поменять веса
с помощью градиентного спуска.
"""
# тут нужно обновить веса по формулам, написанным выше
n = len(y)
y = np.array(y).reshape(-1, 1)
sigma = self.activate(X @ self.w + self.b)
self.w = self.w - learning_rate * (X.T @ (sigma - y)) / n
self.b = self.b - learning_rate * np.mean(sigma - y)
def fit(self, X, y, num_epochs=5000):
"""
Спускаемся в минимум
:param: X -- матрица объектов размера (n, m)
y -- вектор правильных ответов размера (n, 1)
num_epochs -- количество итераций обучения
:return: J_values -- вектор значений функции потерь
"""
self.w = np.zeros((X.shape[1], 1)) # столбец (m, 1)
self.b = 0 # смещение
loss_values = [] # значения функции потерь на различных итерациях обновления весов
for i in range(num_epochs):
# предсказания с текущими весами
y_pred = self.forward_pass(X)
# считаем функцию потерь с текущими весами
loss_values.append(loss(y_pred, y))
# обновляем веса по формуле градиентного спуска
self.backward_pass(X, y, y_pred)
return loss_values
```
<h3 style="text-align: center;"><b>Тестирование</b></h3>
Протестируем нейрон, обученный с новой функцией потерь, на тех же данных, что и в предыдущем ноутбуке:
**Проверка forward_pass()**
```
w = np.array([1., 2.]).reshape(2, 1)
b = 2.
X = np.array([[1., 3.],
[2., 4.],
[-1., -3.2]])
neuron = Neuron(w, b)
y_pred = neuron.forward_pass(X)
print("y_pred = " + str(y_pred))
```
y_pred = [[0.99987661]
[0.99999386]
[0.00449627]]
**Проверка backward_pass()**
```
y = np.array([1, 0, 1]).reshape(3, 1)
```
```
neuron.backward_pass(X, y, y_pred)
print("w = " + str(neuron.w))
print("b = " + str(neuron.b))
```
w = [[0.9001544 ]
[1.76049276]]
b = 1.9998544421863216
Проверьте на наборах данных "яблоки и груши" и "голос".
```
data_apples_pears = pd.read_csv('apples_pears.csv')
```
```
data_apples_pears.head()
```
<div>
<style scoped>
.dataframe tbody tr th:only-of-type {
vertical-align: middle;
}
.dataframe tbody tr th {
vertical-align: top;
}
.dataframe thead th {
text-align: right;
}
</style>
<table border="1" class="dataframe">
<thead>
<tr style="text-align: right;">
<th></th>
<th>yellowness</th>
<th>symmetry</th>
<th>target</th>
</tr>
</thead>
<tbody>
<tr>
<th>0</th>
<td>0.779427</td>
<td>0.257305</td>
<td>1.0</td>
</tr>
<tr>
<th>1</th>
<td>0.777005</td>
<td>0.015915</td>
<td>1.0</td>
</tr>
<tr>
<th>2</th>
<td>0.977092</td>
<td>0.304210</td>
<td>1.0</td>
</tr>
<tr>
<th>3</th>
<td>0.043032</td>
<td>0.140899</td>
<td>0.0</td>
</tr>
<tr>
<th>4</th>
<td>0.760433</td>
<td>0.193123</td>
<td>1.0</td>
</tr>
</tbody>
</table>
</div>
```
plt.figure(figsize=(10, 8))
plt.scatter(data_apples_pears.iloc[:, 0], data_apples_pears.iloc[:, 1], c=data_apples_pears['target'], cmap='rainbow')
plt.title('Яблоки и груши', fontsize=15)
plt.xlabel('симметричность', fontsize=14)
plt.ylabel('желтизна', fontsize=14)
plt.show();
```
```
X = data_apples_pears.iloc[:,:2].values # матрица объекты-признаки
y = data_apples_pears['target'].values.reshape((-1, 1)) # классы (столбец из нулей и единиц)
```
```
%%time
neuron = Neuron(w=np.random.rand(X.shape[1], 1), b=np.random.rand(1))
losses = neuron.fit(X, y, num_epochs=10000)
plt.figure(figsize=(10, 8))
plt.plot(losses)
plt.title('Функция потерь', fontsize=15)
plt.xlabel('номер итерации', fontsize=14)
plt.ylabel('$LogLoss(\hat{y}, y)$', fontsize=14)
plt.show()
```
```
plt.figure(figsize=(10, 8))
plt.scatter(data_apples_pears.iloc[:, 0], data_apples_pears.iloc[:, 1], c=np.array(neuron.forward_pass(X) > 0.7).ravel(), cmap='spring')
plt.title('Яблоки и груши', fontsize=15)
plt.xlabel('симметричность', fontsize=14)
plt.ylabel('желтизна', fontsize=14)
plt.show();
```
```
y_pred = np.array(neuron.forward_pass(X) > 0.7).ravel()
from sklearn.metrics import accuracy_score
print('Точность (доля правильных ответов, из 100%) нашего нейрона: {:.3f} %'.format(
accuracy_score(y, y_pred) * 100))
```
Точность (доля правильных ответов, из 100%) нашего нейрона: 98.800 %
```
data_voice = pd.read_csv("voice.csv")
data_voice['label'] = data_voice['label'].apply(lambda x: 1 if x == 'male' else 0)
```
```
data_voice.head()
```
<div>
<style scoped>
.dataframe tbody tr th:only-of-type {
vertical-align: middle;
}
.dataframe tbody tr th {
vertical-align: top;
}
.dataframe thead th {
text-align: right;
}
</style>
<table border="1" class="dataframe">
<thead>
<tr style="text-align: right;">
<th></th>
<th>meanfreq</th>
<th>sd</th>
<th>median</th>
<th>Q25</th>
<th>Q75</th>
<th>IQR</th>
<th>skew</th>
<th>kurt</th>
<th>sp.ent</th>
<th>sfm</th>
<th>mode</th>
<th>centroid</th>
<th>meanfun</th>
<th>minfun</th>
<th>maxfun</th>
<th>meandom</th>
<th>mindom</th>
<th>maxdom</th>
<th>dfrange</th>
<th>modindx</th>
<th>label</th>
</tr>
</thead>
<tbody>
<tr>
<th>0</th>
<td>0.059781</td>
<td>0.064241</td>
<td>0.032027</td>
<td>0.015071</td>
<td>0.090193</td>
<td>0.075122</td>
<td>12.863462</td>
<td>274.402906</td>
<td>0.893369</td>
<td>0.491918</td>
<td>0.000000</td>
<td>0.059781</td>
<td>0.084279</td>
<td>0.015702</td>
<td>0.275862</td>
<td>0.007812</td>
<td>0.007812</td>
<td>0.007812</td>
<td>0.000000</td>
<td>0.000000</td>
<td>1</td>
</tr>
<tr>
<th>1</th>
<td>0.066009</td>
<td>0.067310</td>
<td>0.040229</td>
<td>0.019414</td>
<td>0.092666</td>
<td>0.073252</td>
<td>22.423285</td>
<td>634.613855</td>
<td>0.892193</td>
<td>0.513724</td>
<td>0.000000</td>
<td>0.066009</td>
<td>0.107937</td>
<td>0.015826</td>
<td>0.250000</td>
<td>0.009014</td>
<td>0.007812</td>
<td>0.054688</td>
<td>0.046875</td>
<td>0.052632</td>
<td>1</td>
</tr>
<tr>
<th>2</th>
<td>0.077316</td>
<td>0.083829</td>
<td>0.036718</td>
<td>0.008701</td>
<td>0.131908</td>
<td>0.123207</td>
<td>30.757155</td>
<td>1024.927705</td>
<td>0.846389</td>
<td>0.478905</td>
<td>0.000000</td>
<td>0.077316</td>
<td>0.098706</td>
<td>0.015656</td>
<td>0.271186</td>
<td>0.007990</td>
<td>0.007812</td>
<td>0.015625</td>
<td>0.007812</td>
<td>0.046512</td>
<td>1</td>
</tr>
<tr>
<th>3</th>
<td>0.151228</td>
<td>0.072111</td>
<td>0.158011</td>
<td>0.096582</td>
<td>0.207955</td>
<td>0.111374</td>
<td>1.232831</td>
<td>4.177296</td>
<td>0.963322</td>
<td>0.727232</td>
<td>0.083878</td>
<td>0.151228</td>
<td>0.088965</td>
<td>0.017798</td>
<td>0.250000</td>
<td>0.201497</td>
<td>0.007812</td>
<td>0.562500</td>
<td>0.554688</td>
<td>0.247119</td>
<td>1</td>
</tr>
<tr>
<th>4</th>
<td>0.135120</td>
<td>0.079146</td>
<td>0.124656</td>
<td>0.078720</td>
<td>0.206045</td>
<td>0.127325</td>
<td>1.101174</td>
<td>4.333713</td>
<td>0.971955</td>
<td>0.783568</td>
<td>0.104261</td>
<td>0.135120</td>
<td>0.106398</td>
<td>0.016931</td>
<td>0.266667</td>
<td>0.712812</td>
<td>0.007812</td>
<td>5.484375</td>
<td>5.476562</td>
<td>0.208274</td>
<td>1</td>
</tr>
</tbody>
</table>
</div>
```
# Чтобы перемешать данные. Изначально там сначала идут все мужчины, потом все женщины
data_voice = data_voice.sample(frac=1)
```
```
X_train = data_voice.iloc[:int(len(data_voice)*0.7), :-1] # матрица объекты-признаки
y_train = data_voice.iloc[:int(len(data_voice)*0.7), -1] # истинные значения пола (мужчина/женщина)
X_test = data_voice.iloc[int(len(data_voice)*0.7):, :-1] # матрица объекты-признаки
y_test = data_voice.iloc[int(len(data_voice)*0.7):, -1] # истинные значения пола (мужчина/женщина)
```
```
from sklearn.preprocessing import StandardScaler
```
```
scaler = StandardScaler()
X_train = scaler.fit_transform(X_train.values)
X_test = scaler.transform(X_test.values)
```
```
plt.figure(figsize=(10, 8))
plt.scatter(X_test[:, 0], X_test[:, 1], c=y_test, cmap='rainbow')
plt.title('Мужские и женские голоса', fontsize=15)
plt.show();
```
```
neuron = Neuron(w=np.random.rand(X.shape[1], 1), b=np.random.rand(1))
losses = neuron.fit(X_train, y_train.values);
```
```
y_pred = neuron.forward_pass(X_test)
y_pred = (y_pred > 0.5).astype(int)
```
```
from sklearn.metrics import accuracy_score
print('Точность (доля правильных ответов, из 100%) нашего нейрона: {:.3f} %'.format(
accuracy_score(y_test, y_pred) * 100))
```
Точность (доля правильных ответов, из 100%) нашего нейрона: 97.476 %
```
plt.figure(figsize=(10, 8))
plt.scatter(X_test[:, 0], X_test[:, 1], c=y_pred, cmap='spring')
plt.title('Мужские и женские голоса', fontsize=15)
plt.show();
```
```
```
|
From Undecidability.L Require Export Datatypes.LNat Datatypes.LBool Tactics.LTactics Computability.Computability Tactics.Lbeta.
Section MuRecursor.
Variable P : term.
Hypothesis P_proc : proc P.
Hint Resolve P_proc : LProc.
Hypothesis dec'_P : forall (n:nat), (exists (b:bool), app P (ext n) == ext b ).
Lemma dec_P : forall n:nat, {b:bool | app P (ext n) == ext b}.
intros. eapply lcomp_comp.
-apply bool_enc_inv_correct.
-apply dec'_P.
Qed.
Section hoas.
Import HOAS_Notations.
Definition mu' := Eval cbn -[enc] in rho (λ mu P n, (P n) (!!K n) (λ Sn, mu P Sn) (!!(ext S) n)).
End hoas.
Import L_Notations.
Lemma mu'_proc : proc mu'.
unfold mu'; Lproc.
Qed.
Hint Resolve mu'_proc : LProc.
Lemma mu'_n_false n: P (ext n) == ext false -> mu' P (ext n) >* mu' P (ext (S n)).
Proof.
intros R. apply equiv_lambda in R;[|Lproc]. recStep mu'. unfold K. Lsimpl.
Qed.
Lemma mu'_0_false n: (forall n', n' < n -> P (ext n') == ext false) -> mu' P (ext 0) >* mu' P (ext n).
Proof.
intros H. induction n.
-reflexivity.
-rewrite IHn.
+apply mu'_n_false. apply H. lia.
+intros. apply H. lia.
Qed.
Lemma mu'_n_true (n:nat): P (ext n) == ext true -> mu' P (ext n) == ext n.
Proof.
intros R. recStep mu'. Lsimpl. rewrite R. unfold K. Lsimpl.
Qed.
(* TODO: mu' sound*)
Lemma mu'_sound v n: proc v -> mu' P (ext (n:nat)) == v ->
(forall n', n' < n -> P (ext n') == ext false) ->
exists n0, n0 >= n /\ P (ext n0) == ext true /\ v == ext n0
/\ forall n', n' < n0 -> P (ext (n':nat)) == ext false.
Proof.
intros pv. intros R. apply equiv_lambda in R;try Lproc. apply star_pow in R. destruct R as [k R]. revert n R. apply complete_induction with (x:=k);clear k;intros k. intros IH n R H.
specialize (dec_P n).
destruct (dec_P n) as [[] eq].
-exists n;intuition. apply pow_star in R. apply star_equiv in R. rewrite <- R. now rewrite mu'_n_true.
-assert (R':=mu'_n_false eq). apply star_pow in R'. destruct R' as [k' R'].
destruct (parametrized_confluence uniform_confluence R R') as [x [l [u [le1 [le2 [R1 [R2 eq']]]]]]]. destruct x.
+inv R1. apply IH in R2 as [n0 [ge1 [Rn0 [eq0 H0]]]].
*exists n0. repeat split;try assumption;lia.
*decide (l=k);[|lia]. subst l. assert (k'=0) by lia. subst k'. inv R'. apply inj_enc in H1. lia.
*intros. decide (n'=n). subst. tauto. apply H. lia.
+destruct R1 as [? [C _]]. destruct pv as [_ [v']]. subst v. inv C.
Qed.
Lemma mu'_complete n0 : P (ext n0) == ext true
-> (forall n', n' < n0 -> P (ext n') == ext false)
-> mu' P (ext 0) == ext n0.
Proof.
intros. rewrite mu'_0_false with (n:=n0);try tauto.
-recStep mu'. Lsimpl. rewrite H. unfold K. Lsimpl.
Qed.
(* the mu combinator:*)
Definition mu :term := lam (mu' #0 (ext 0)).
Lemma mu_proc : proc mu.
unfold mu. Lproc.
Qed.
Hint Resolve mu_proc : LProc.
Lemma mu_sound v : lambda v -> mu P == v -> exists n, v = ext n /\ P (ext n) == ext true /\ (forall n', n' < n -> P (ext n') == ext false).
Proof.
unfold mu. intros lv R. standardizeHypo 100. apply mu'_sound in R.
-destruct R as [n ?]. exists n. intuition. apply unique_normal_forms;try Lproc. assumption.
-split;[|Lproc]. apply equiv_lambda in R;auto. apply closed_star in R;Lproc.
-intros. lia.
Qed.
Lemma mu_complete (n:nat) : P (ext n) == ext true -> exists n0:nat, mu P == ext n0.
Proof.
remember 0 as n0.
assert (forall n':nat, n'< n-(n-n0) -> P (ext n') == ext false) by (intros;lia).
assert ((n-n0)+n0=n) by lia. remember (n-n0) as k. clear Heqk Heqn0 H0 n0. induction k.
-simpl in *. subst. intros. eexists. unfold mu. Lsimpl. apply mu'_complete;eauto. intros. apply H. lia.
-intros. destruct (dec_P (n-S k)) as [y P'].
destruct y.
+eexists. unfold mu. Lsimpl. apply mu'_complete. exact P'. exact H.
+apply IHk. intros. decide (n' = n - (S k)).
*subst. exact P'.
*apply H. lia.
*assumption.
Qed.
Lemma mu_spec : converges (mu P) <-> exists n : nat, P (ext n) == ext true.
Proof.
split.
- intros (? & ? & ?). eapply mu_sound in H as (? & ? & ? & ?); eauto.
- intros []. eapply mu_complete in H as []. exists (ext x0). split. eauto. eapply proc_ext.
Qed.
End MuRecursor.
Hint Resolve mu'_proc : LProc.
Hint Resolve mu_proc : LProc.
|
#! /usr/bin/env python3
import numpy as np
ngenes = 30
nsamps_a = 2
nsamps_b = 3
header = []
header.append("gene")
header.extend([f"sample{1 + int(s) :02}" for s in range(nsamps_a + nsamps_b)])
print("\t".join(header))
for i in range(ngenes):
record = []
record.append(f"gene{1 + i :03}")
record.extend([str(x) for x in np.random.poisson(lam=100, size=nsamps_a)])
record.extend([str(x) for x in np.random.poisson(lam=100, size=nsamps_b)])
print("\t".join(record))
|
function pass = test_sample( )
% Test diskfun sample() command
tol = 100*chebfunpref().cheb2Prefs.chebfun2eps;
% Function to test
f = diskfun(@(x,y) sin(pi*x.*y));
% Ensure the matrix of sampled values is correct.
[m,n] = length(f);
[nn,mm] = size(sample(f));
pass(1) = (m == mm) && (n == nn);
% Sample on fixed grids of various sizes to make sure the right size output
% is given.
m = 120;
n = 121;
[nn,mm] = size(sample(f, m, n));
pass(2) = (m == mm) && (n == nn);
m = 121;
n = 120;
[nn,mm] = size(sample(f, m, n));
pass(3) = (m == mm) && (n == nn);
% Check samples are correct.
% m even and n odd
m = 30;
n = 2*20+1;
cp = chebpts(n);
[t,r] = meshgrid(trigpts(m, [-pi, pi]), cp((n+1)/2:end));
F = f(t,r, 'polar');
G = sample(f, m, (n+1)/2);
pass(4) = norm(F(:) - G(:), inf) < tol;
[U, D, V] = sample(f, m, (n+1)/2);
G = U * D * V.';
pass(5) = norm(F(:) - G(:), inf) < tol;
% m odd and n odd
m = 31;
n = 2*20+1;
cp = chebpts(n);
[t,r] = meshgrid(trigpts(m, [-pi, pi]), cp((n+1)/2:end));
F = f(t,r, 'polar');
G = sample(f, m, (n+1)/2);
pass(6) = norm(F(:) - G(:), inf) < tol;
[U, D, V] = sample(f, m, (n+1)/2);
G = U * D * V.';
pass(7) = norm(F(:) - G(:), inf) < tol;
% m odd and n even
m = 31;
n = 2*20-1;
cp = chebpts(n);
[t,r] = meshgrid(trigpts(m, [-pi, pi]), cp((n+1)/2:end));
F = f(t,r, 'polar');
G = sample(f, m, (n+1)/2);
pass(8) = norm(F(:) - G(:), inf) < tol;
[U, D, V] = sample(f, m, (n+1)/2);
G = U * D * V.';
pass(9) = norm(F(:) - G(:), inf) < tol;
% m even and n even
m = 30;
n = 2*20-1;
cp = chebpts(n);
[t,r] = meshgrid(trigpts(m, [-pi, pi]), cp((n+1)/2:end));
F = f(t,r, 'polar');
G = sample(f, m, (n+1)/2);
pass(10) = norm(F(:) - G(:), inf) < tol;
[U, D, V] = sample(f, m, (n+1)/2);
G = U * D * V.';
pass(11) = norm(F(:) - G(:), inf) < tol;
% Sample should return all ones for the function 1.
f = diskfun(@(x,y) 1 + 0*x);
F = sample(f, 128, 128);
pass(12) = norm(F(:) - 1, inf) < tol;
% Check that errors are caught
try
F = sample(f, 0, 20);
pass(13) = false;
catch ME
pass(13) = strcmp(ME.identifier, 'CHEBFUN:DISKFUN:sample:inputs');
end
try
F = sample(f, 20, 0);
pass(14) = false;
catch ME
pass(14) = strcmp(ME.identifier, 'CHEBFUN:DISKFUN:sample:inputs');
end
end
|
(* (c) Copyright 2006-2018 Microsoft Corporation and Inria. *)
(* Distributed under the terms of CeCILL-B. *)
From mathcomp Require Import ssreflect ssrfun ssrbool ssrnat ssrint.
From fourcolor Require Import part hubcap present.
(******************************************************************************)
(* This file contains the unavoidability proof for cartwheels with a hub *)
(* arity of 6. This proof is a reencoding of the argument that appeared in *)
(* the main text of the Robertson et al. revised proof. *)
(******************************************************************************)
Set Implicit Arguments.
Unset Strict Implicit.
Unset Printing Implicit Defensive.
Lemma exclude6 : excluded_arity 6.
Proof.
Presentation red.
Pcase L0_1: s[1] > 6.
Pcase L1_1: s[3] > 6.
Pcase: s[2] > 5.
Pcase: s[5] > 6.
Pcase: s[4] > 5.
Pcase: s[6] > 5.
Hubcap $[1,2]<=0 $[3,5]<=0 $[4,6]<=0 $.
Hubcap $[1,2]<=(-1) $[3,5]<=(-1) $[4,6]<=2 $.
Pcase: s[6] > 5.
Hubcap $[1,2]<=0 $[3,5]<=(-2) $[4,6]<=2 $.
Hubcap $[1,2]<=(-1) $[3,5]<=(-3) $[4,6]<=4 $.
Pcase L3_1: s[6] > 6.
Pcase: s[4] > 6.
Pcase: s[5] > 5.
Hubcap $[1,2]<=0 $[3,5]<=0 $[4,6]<=0 $.
Hubcap $[1,2]<=0 $[3,5]<=2 $[4,6]<=(-2) $.
Pcase: s[4] > 5.
Pcase: s[5] > 5.
Hubcap $[1,2]<=0 $[3,5]<=0 $[4,6]<=0 $.
Hubcap $[1,2]<=0 $[3,5]<=1 $[4,6]<=(-1) $.
Pcase: s[5] > 5.
Hubcap $[1,2]<=0 $[3,5]<=(-1) $[4,6]<=1 $.
Hubcap $[1,2]<=0 $[3,5]<=0 $[4,6]<=0 $.
Pcase: s[4] > 6.
Similar to *L3_1[3].
Pcase L3_2: s[6] <= 5.
Pcase: s[4] <= 5.
Reducible.
Pcase: s[5] <= 5.
Hubcap $[2,3]<=(-1) $[1,5]<=0 $[4,6]<=1 $.
Hubcap $[2,3]<=0 $[1,5]<=(-1) $[4,6]<=1 $.
Pcase: s[4] <= 5.
Similar to *L3_2[3].
Pcase: s[5] <= 5.
Pcase L4_1: h[6] > 6.
Hubcap $[1,2]<=(-1) $[3,5]<=1 $[4,6]<=0 $.
Pcase: h[5] > 6.
Similar to *L4_1[3].
Hubcap $[2,3]<=(-1) $[1,5]<=1 $[4,6]<=0 $.
Pcase: h[6] > 6.
Hubcap $[1,2]<=0 $[3,5]<=0 $[4,6]<=0 $.
Pcase: h[6] <= 5.
Pcase: f1[6] > 6.
Hubcap $[1,2]<=(-1) $[3,5]<=1 $[4,6]<=0 $.
Pcase: f1[6] <= 5.
Hubcap $[1,2]<=(-2) $[3,5]<=2 $[4,6]<=0 $.
Hubcap $[1,2]<=(-1) $[3,5]<=1 $[4,6]<=0 $.
Pcase: f1[6] > 6.
Hubcap $[1,2]<=0 $[3,5]<=0 $[4,6]<=0 $.
Pcase: f1[6] <= 5.
Pcase: h[1] <= 5.
Hubcap $[1,2]<=(-2) $[3,5]<=2 $[4,6]<=0 $.
Hubcap $[1,2]<=(-1) $[3,5]<=1 $[4,6]<=0 $.
Pcase: h[1] <= 5.
Hubcap $[1,2]<=(-1) $[3,5]<=1 $[4,6]<=0 $.
Hubcap $[1,2]<=0 $[3,5]<=0 $[4,6]<=0 $.
Pcase: s[5] > 6.
Pcase: s[4] > 5.
Pcase: s[6] > 5.
Hubcap $[1,2]<=1 $[3,5]<=(-1) $[4,6]<=0 $.
Hubcap $[1,2]<=0 $[3,5]<=(-2) $[4,6]<=2 $.
Pcase: s[6] > 5.
Hubcap $[1,2]<=1 $[3,5]<=(-3) $[4,6]<=2 $.
Hubcap $[1,2]<=0 $[3,5]<=(-4) $[4,6]<=4 $.
Pcase L3_1: s[6] > 6.
Pcase: s[4] > 6.
Pcase: s[5] > 5.
Hubcap $[1,2]<=1 $[3,5]<=(-1) $[4,6]<=0 $.
Hubcap $[1,2]<=1 $[3,5]<=1 $[4,6]<=(-2) $.
Pcase: s[4] > 5.
Pcase: s[5] > 5.
Hubcap $[1,2]<=1 $[3,5]<=(-1) $[4,6]<=0 $.
Hubcap $[1,2]<=1 $[3,5]<=0 $[4,6]<=(-1) $.
Pcase: s[5] > 5.
Hubcap $[1,2]<=1 $[3,5]<=(-2) $[4,6]<=1 $.
Hubcap $[1,2]<=1 $[3,5]<=(-1) $[4,6]<=0 $.
Pcase: s[4] > 6.
Similar to *L3_1[3].
Pcase L3_2: s[6] <= 5.
Pcase: s[4] <= 5.
Reducible.
Pcase: s[5] <= 5.
Hubcap $[2,3]<=0 $[1,5]<=(-1) $[4,6]<=1 $.
Hubcap $[2,3]<=1 $[1,5]<=(-2) $[4,6]<=1 $.
Pcase: s[4] <= 5.
Similar to *L3_2[3].
Pcase: s[5] <= 5.
Pcase L4_1: h[6] > 6.
Hubcap $[1,2]<=0 $[3,5]<=0 $[4,6]<=0 $.
Pcase: h[5] > 6.
Similar to *L4_1[3].
Hubcap $[2,3]<=0 $[1,5]<=0 $[4,6]<=0 $.
Pcase: h[6] > 6.
Hubcap $[1,2]<=1 $[3,5]<=(-1) $[4,6]<=0 $.
Pcase: h[6] <= 5.
Pcase: f1[6] > 6.
Hubcap $[1,2]<=0 $[3,5]<=0 $[4,6]<=0 $.
Pcase: f1[6] <= 5.
Hubcap $[1,2]<=(-1) $[3,5]<=1 $[4,6]<=0 $.
Hubcap $[1,2]<=0 $[3,5]<=0 $[4,6]<=0 $.
Pcase: f1[6] > 6.
Hubcap $[1,2]<=1 $[3,5]<=(-1) $[4,6]<=0 $.
Pcase: f1[6] <= 5.
Pcase: h[1] <= 5.
Hubcap $[1,2]<=(-1) $[3,5]<=1 $[4,6]<=0 $.
Hubcap $[1,2]<=0 $[3,5]<=0 $[4,6]<=0 $.
Pcase: h[1] <= 5.
Hubcap $[1,2]<=0 $[3,5]<=0 $[4,6]<=0 $.
Hubcap $[1,2]<=1 $[3,5]<=(-1) $[4,6]<=0 $.
Pcase: s[5] > 6.
Similar to L1_1[4].
Pcase: s[4] > 6.
Pcase: s[2] > 6.
Similar to L1_1[1].
Pcase: s[6] > 6.
Similar to L1_1[3].
Pcase L2_1: s[2] <= 5.
Pcase: s[3] <= 5.
Pcase L4_1: s[5] <= 5.
Pcase: s[6] <= 5.
Pcase: h[6] <= 5.
Hubcap $[1,3]<=(-3) $[2,4]<=(-3) $[5,6]<=6 $.
Hubcap $[1,3]<=(-2) $[2,4]<=(-2) $[5,6]<=4 $.
Pcase: h[3] <= 5.
Hubcap $[1,5]<=(-2) $[4,6]<=(-4) $[2,3]<=6 $.
Hubcap $[1,5]<=(-1) $[4,6]<=(-3) $[2,3]<=4 $.
Pcase: s[6] <= 5.
Similar to *L4_1[2].
Pcase: h[3] <= 5.
Hubcap $[1,5]<=(-3) $[4,6]<=(-3) $[2,3]<=6 $.
Hubcap $[1,5]<=(-2) $[4,6]<=(-2) $[2,3]<=4 $.
Pcase: s[5] <= 5.
Pcase: s[6] <= 5.
Pcase: h[6] <= 5.
Hubcap $[1,3]<=(-4) $[2,4]<=(-2) $[5,6]<=6 $.
Hubcap $[1,3]<=(-3) $[2,4]<=(-1) $[5,6]<=4 $.
Pcase: h[6] <= 5.
Pcase: h[5] > 5.
Hubcap $[1,3]<=(-3) $[2,4]<=(-1) $[5,6]<=4 $.
Hubcap $[1,3]<=(-3) $[2,4]<=(-2) $[5,6]<=5 $.
Pcase: h[6] <= 6.
Pcase: h[5] > 5.
Hubcap $[1,3]<=(-2) $[2,4]<=0 $[5,6]<=2 $.
Hubcap $[1,3]<=(-2) $[2,4]<=(-1) $[5,6]<=3 $.
Hubcap $[1,3]<=(-2) $[2,4]<=0 $[5,6]<=2 $.
Pcase: s[6] <= 5.
Pcase: h[6] <= 5.
Pcase: h[1] > 5.
Hubcap $[1,3]<=(-3) $[2,4]<=(-1) $[5,6]<=4 $.
Hubcap $[1,3]<=(-4) $[2,4]<=(-1) $[5,6]<=5 $.
Pcase: h[1] > 5.
Hubcap $[1,3]<=(-2) $[2,4]<=0 $[5,6]<=2 $.
Pcase: h[6] > 6.
Hubcap $[1,3]<=(-2) $[2,4]<=0 $[5,6]<=2 $.
Hubcap $[1,3]<=(-3) $[2,4]<=0 $[5,6]<=3 $.
Pcase: h[3] <= 5.
Pcase: h[2] <= 5.
Hubcap $[1,5]<=(-3) $[4,6]<=(-2) $[2,3]<=5 $.
Hubcap $[1,5]<=(-2) $[4,6]<=(-2) $[2,3]<=4 $.
Pcase: h[3] > 6.
Hubcap $[1,5]<=(-1) $[4,6]<=(-1) $[2,3]<=2 $.
Pcase: h[2] <= 5.
Hubcap $[1,5]<=(-2) $[4,6]<=(-1) $[2,3]<=3 $.
Hubcap $[1,5]<=(-1) $[4,6]<=(-1) $[2,3]<=2 $.
Pcase: s[3] <= 5.
Similar to *L2_1[2].
Pcase: s[5] <= 5.
Similar to L2_1[3].
Pcase: s[6] <= 5.
Similar to *L2_1[5].
Pcase: h[3] > 6.
Hubcap $[1,5]<=0 $[4,6]<=0 $[2,3]<=0 $.
Pcase: h[3] <= 5.
Pcase L3_1: f1[2] <= 5.
Hubcap $[1,5]<=(-2) $[4,6]<=(-1) $[2,3]<=3 $.
Pcase: f1[3] <= 5.
Similar to *L3_1[2].
Hubcap $[1,5]<=(-1) $[4,6]<=(-1) $[2,3]<=2 $.
Pcase L2_2: f1[2] <= 5.
Pcase: h[2] <= 5.
Hubcap $[1,5]<=(-2) $[4,6]<=0 $[2,3]<=2 $.
Hubcap $[1,5]<=(-1) $[4,6]<=0 $[2,3]<=1 $.
Pcase: f1[3] <= 5.
Similar to *L2_2[2].
Pcase L2_3: f1[2] > 6.
Pcase: f1[3] > 6.
Hubcap $[1,5]<=0 $[4,6]<=0 $[2,3]<=0 $.
Pcase: h[4] > 5.
Hubcap $[1,5]<=0 $[4,6]<=0 $[2,3]<=0 $.
Hubcap $[1,5]<=0 $[4,6]<=(-1) $[2,3]<=1 $.
Pcase: f1[3] > 6.
Similar to *L2_3[2].
Pcase L2_4: h[2] <= 5.
Hubcap $[1,5]<=(-1) $[4,6]<=0 $[2,3]<=1 $.
Pcase: h[4] <= 5.
Similar to *L2_4[2].
Hubcap $[1,5]<=0 $[4,6]<=0 $[2,3]<=0 $.
Pcase L1_2: s[2] > 6.
Pcase: s[6] > 6.
Similar to L1_1[5].
Pcase L2_1: s[3] <= 5.
Pcase: s[5] <= 5.
Reducible.
Pcase: s[6] <= 5.
Reducible.
Pcase: s[4] <= 5.
Hubcap $[3,5]<=1 $[2,4]<=0 $[1,6]<=(-1) $.
Hubcap $[3,5]<=1 $[2,4]<=(-1) $[1,6]<=0 $.
Pcase: s[6] <= 5.
Similar to *L2_1[4].
Pcase L2_2: s[4] <= 5.
Pcase: s[5] <= 5.
Hubcap $[1,2]<=(-2) $[3,5]<=1 $[4,6]<=1 $.
Pcase: h[4] > 5.
Hubcap $[1,2]<=(-1) $[3,5]<=0 $[4,6]<=1 $.
Hubcap $[4,2]<=1 $[3,5]<=0 $[1,6]<=(-1) $.
Pcase: s[5] <= 5.
Similar to *L2_2[4].
Pcase L2_3: h[4] > 6.
Pcase: h[5] > 6.
Hubcap $[1,5]<=0 $[2,3]<=0 $[4,6]<=0 $.
Pcase: h[5] <= 5.
Pcase: f1[4] > 5.
Hubcap $[1,5]<=1 $[2,3]<=(-1) $[4,6]<=0 $.
Hubcap $[1,5]<=2 $[2,3]<=(-2) $[4,6]<=0 $.
Pcase: f1[4] <= 5.
Hubcap $[1,5]<=1 $[2,3]<=(-1) $[4,6]<=0 $.
Hubcap $[1,5]<=0 $[2,3]<=0 $[4,6]<=0 $.
Pcase: h[6] > 6.
Similar to *L2_3[4].
Pcase: h[5] > 6.
Pcase: h[4] <= 5.
Pcase: f1[3] <= 5.
Hubcap $[1,5]<=(-1) $[4,6]<=2 $[2,3]<=(-1) $.
Pcase: f1[4] <= 5.
Hubcap $[1,5]<=(-2) $[4,6]<=1 $[2,3]<=1 $.
Hubcap $[1,5]<=(-1) $[4,6]<=1 $[2,3]<=0 $.
Pcase: f1[3] <= 5.
Pcase: h[3] <= 5.
Hubcap $[1,5]<=0 $[4,6]<=2 $[2,3]<=(-2) $.
Hubcap $[1,5]<=0 $[4,6]<=1 $[2,3]<=(-1) $.
Pcase: f1[4] <= 5.
Hubcap $[1,5]<=(-1) $[4,6]<=0 $[2,3]<=1 $.
Pcase: f1[3] > 6.
Hubcap $[1,5]<=0 $[4,6]<=0 $[2,3]<=0 $.
Pcase: h[3] <= 5.
Hubcap $[1,5]<=0 $[4,6]<=1 $[2,3]<=(-1) $.
Hubcap $[1,5]<=0 $[4,6]<=0 $[2,3]<=0 $.
Pcase L2_4: h[6] <= 5.
Pcase: h[4] <= 5.
Reducible.
Pcase: f1[3] <= 5.
Pcase: h[3] <= 5.
Hubcap $[1,5]<=0 $[4,6]<=2 $[2,3]<=(-2) $.
Hubcap $[1,5]<=0 $[4,6]<=1 $[2,3]<=(-1) $.
Pcase: f1[3] > 6.
Hubcap $[1,5]<=0 $[4,6]<=0 $[2,3]<=0 $.
Pcase: h[3] <= 5.
Hubcap $[1,5]<=0 $[4,6]<=1 $[2,3]<=(-1) $.
Hubcap $[1,5]<=0 $[4,6]<=0 $[2,3]<=0 $.
Pcase: h[4] <= 5.
Similar to *L2_4[4].
Pcase L2_5: f1[4] > 6.
Pcase: h[5] > 5.
Pcase: f1[3] > 6.
Hubcap $[1,5]<=0 $[4,6]<=0 $[2,3]<=0 $.
Pcase: f1[3] > 5.
Pcase: h[3] > 5.
Hubcap $[1,5]<=0 $[4,6]<=0 $[2,3]<=0 $.
Hubcap $[1,5]<=0 $[4,6]<=1 $[2,3]<=(-1) $.
Pcase: h[3] > 5.
Hubcap $[1,5]<=0 $[4,6]<=1 $[2,3]<=(-1) $.
Hubcap $[1,5]<=0 $[4,6]<=2 $[2,3]<=(-2) $.
Pcase: f1[3] > 6.
Hubcap $[1,5]<=1 $[4,6]<=0 $[2,3]<=(-1) $.
Pcase: f1[3] > 5.
Pcase: h[3] > 5.
Hubcap $[1,5]<=1 $[4,6]<=0 $[2,3]<=(-1) $.
Hubcap $[1,5]<=1 $[4,6]<=1 $[2,3]<=(-2) $.
Pcase: h[3] > 5.
Hubcap $[1,5]<=1 $[4,6]<=1 $[2,3]<=(-2) $.
Hubcap $[1,5]<=1 $[4,6]<=2 $[2,3]<=(-3) $.
Pcase: f1[5] > 6.
Similar to *L2_5[4].
Pcase: f1[3] > 6.
Hubcap $[1,5]<=0 $[4,6]<=0 $[2,3]<=0 $.
Pcase: f1[3] > 5.
Pcase: h[3] > 5.
Hubcap $[1,5]<=0 $[4,6]<=0 $[2,3]<=0 $.
Hubcap $[1,5]<=0 $[4,6]<=1 $[2,3]<=(-1) $.
Pcase: h[3] > 5.
Hubcap $[1,5]<=0 $[4,6]<=1 $[2,3]<=(-1) $.
Hubcap $[1,5]<=0 $[4,6]<=2 $[2,3]<=(-2) $.
Pcase: s[6] > 6.
Similar to *L1_2[5].
Pcase L1_3: s[2] <= 5.
Pcase: s[4] <= 5.
Reducible.
Pcase: s[5] <= 5.
Reducible.
Pcase: s[6] <= 5.
Reducible.
Pcase: h[6] <= 5.
Reducible.
Pcase: s[3] <= 5.
Hubcap $[1,3]<=0 $[2,4]<=1 $[5,6]<=(-1) $.
Pcase: h[6] > 6.
Hubcap $[1,3]<=(-1) $[2,4]<=1 $[5,6]<=0 $.
Pcase: f1[5] <= 5.
Hubcap $[1,3]<=(-1) $[2,4]<=0 $[5,6]<=1 $.
Pcase: f1[6] > 6.
Hubcap $[1,3]<=(-1) $[2,4]<=1 $[5,6]<=0 $.
Pcase: f1[6] <= 5.
Reducible.
Pcase: h[1] > 5.
Hubcap $[1,3]<=(-1) $[2,4]<=1 $[5,6]<=0 $.
Hubcap $[1,3]<=(-2) $[2,4]<=1 $[5,6]<=1 $.
Pcase: s[6] <= 5.
Similar to *L1_3[5].
Pcase L1_4: s[3] <= 5.
Pcase: s[5] <= 5.
Reducible.
Pcase: s[4] <= 5.
Hubcap $[1,6]<=(-2) $[2,4]<=1 $[3,5]<=1 $.
Pcase: h[3] > 6.
Hubcap $[1,6]<=(-1) $[2,4]<=0 $[3,5]<=1 $.
Hubcap $[1,3]<=1 $[2,4]<=0 $[5,6]<=(-1) $.
Pcase: s[5] <= 5.
Similar to *L1_4[5].
Pcase: s[4] <= 5.
Pcase: h[4] > 6.
Hubcap $[1,2]<=(-1) $[3,5]<=0 $[4,6]<=1 $.
Hubcap $[1,6]<=(-1) $[2,4]<=1 $[3,5]<=0 $.
Pcase L1_5: h[3] <= 5.
Pcase: h[6] <= 5.
Hubcap $[1,3]<=(-1) $[2,4]<=(-1) $[5,6]<=2 $.
Pcase: h[6] > 6.
Pcase: h[5] <= 5.
Reducible.
Pcase: h[5] > 6.
Hubcap $[1,3]<=0 $[2,4]<=0 $[5,6]<=0 $.
Pcase: f1[5] <= 5.
Hubcap $[1,3]<=0 $[2,4]<=1 $[5,6]<=(-1) $.
Hubcap $[1,3]<=0 $[2,4]<=0 $[5,6]<=0 $.
Pcase: f1[6] <= 5.
Reducible.
Pcase: f1[6] > 6.
Hubcap $[1,3]<=0 $[2,4]<=0 $[5,6]<=0 $.
Pcase: h[1] > 5.
Hubcap $[1,3]<=0 $[2,4]<=0 $[5,6]<=0 $.
Hubcap $[1,3]<=(-1) $[2,4]<=0 $[5,6]<=1 $.
Pcase: h[6] <= 5.
Similar to *L1_5[5].
Pcase L1_6: h[4] <= 5.
Pcase: f1[3] <= 5.
Pcase: f1[4] <= 5.
Hubcap $[1,5]<=(-2) $[2,3]<=0 $[4,6]<=2 $.
Hubcap $[1,5]<=(-1) $[2,3]<=(-1) $[4,6]<=2 $.
Pcase: f1[4] <= 5.
Pcase: h[3] > 6.
Hubcap $[1,5]<=(-2) $[2,3]<=1 $[4,6]<=1 $.
Pcase: f1[2] <= 5.
Reducible.
Pcase: f1[2] > 6.
Hubcap $[1,5]<=(-2) $[2,3]<=1 $[4,6]<=1 $.
Pcase: h[2] > 5.
Hubcap $[1,5]<=(-2) $[2,3]<=1 $[4,6]<=1 $.
Hubcap $[1,5]<=(-3) $[2,3]<=2 $[4,6]<=1 $.
Pcase: h[3] > 6.
Hubcap $[1,5]<=(-1) $[2,3]<=0 $[4,6]<=1 $.
Pcase: f1[2] <= 5.
Reducible.
Pcase: f1[2] > 6.
Hubcap $[1,5]<=(-1) $[2,3]<=0 $[4,6]<=1 $.
Pcase: h[2] > 5.
Hubcap $[1,5]<=(-1) $[2,3]<=0 $[4,6]<=1 $.
Hubcap $[1,5]<=(-2) $[2,3]<=1 $[4,6]<=1 $.
Pcase: h[5] <= 5.
Similar to *L1_6[5].
Pcase L1_7: h[4] > 6.
Pcase: h[5] > 6.
Pcase L3_1: h[3] > 6.
Hubcap $[1,5]<=0 $[2,3]<=0 $[4,6]<=0 $.
Pcase: h[6] > 6.
Similar to *L3_1[5].
Pcase: f1[5] <= 5.
Hubcap $[1,3]<=0 $[2,4]<=(-1) $[5,6]<=1 $.
Pcase: f1[6] > 6.
Hubcap $[1,3]<=0 $[2,4]<=0 $[5,6]<=0 $.
Pcase: f1[6] <= 5.
Pcase: h[1] <= 5.
Hubcap $[1,3]<=(-2) $[2,4]<=0 $[5,6]<=2 $.
Hubcap $[1,3]<=(-1) $[2,4]<=0 $[5,6]<=1 $.
Pcase: h[1] <= 5.
Hubcap $[1,3]<=(-1) $[2,4]<=0 $[5,6]<=1 $.
Hubcap $[1,3]<=0 $[2,4]<=0 $[5,6]<=0 $.
Pcase: h[6] > 6.
Pcase: f1[4] > 5.
Pcase: f1[5] > 5.
Hubcap $[1,3]<=0 $[2,4]<=0 $[5,6]<=0 $.
Hubcap $[1,3]<=0 $[2,4]<=1 $[5,6]<=(-1) $.
Pcase: f1[5] > 5.
Hubcap $[1,3]<=(-1) $[2,4]<=0 $[5,6]<=1 $.
Hubcap $[1,3]<=(-1) $[2,4]<=1 $[5,6]<=0 $.
Pcase: f1[4] <= 5.
Pcase: f1[6] <= 5.
Hubcap $[1,3]<=(-2) $[2,4]<=0 $[5,6]<=2 $.
Hubcap $[1,3]<=(-1) $[2,4]<=0 $[5,6]<=1 $.
Pcase: f1[6] > 6.
Hubcap $[1,3]<=0 $[2,4]<=0 $[5,6]<=0 $.
Pcase: f1[6] <= 5.
Pcase: h[1] <= 5.
Hubcap $[1,3]<=(-2) $[2,4]<=0 $[5,6]<=2 $.
Hubcap $[1,3]<=(-1) $[2,4]<=0 $[5,6]<=1 $.
Pcase: h[1] <= 5.
Hubcap $[1,3]<=(-1) $[2,4]<=0 $[5,6]<=1 $.
Hubcap $[1,3]<=0 $[2,4]<=0 $[5,6]<=0 $.
Pcase: h[5] > 6.
Similar to *L1_7[5].
Pcase L1_8: h[6] > 6.
Pcase: f1[5] > 5.
Hubcap $[1,3]<=0 $[2,4]<=0 $[5,6]<=0 $.
Hubcap $[1,3]<=0 $[2,4]<=1 $[5,6]<=(-1) $.
Pcase: h[3] > 6.
Similar to *L1_8[5].
Pcase: f1[6] > 6.
Hubcap $[1,3]<=0 $[2,4]<=0 $[5,6]<=0 $.
Pcase: f1[6] <= 5.
Pcase: h[1] <= 5.
Hubcap $[1,3]<=(-2) $[2,4]<=0 $[5,6]<=2 $.
Hubcap $[1,3]<=(-1) $[2,4]<=0 $[5,6]<=1 $.
Pcase: h[1] <= 5.
Hubcap $[1,3]<=(-1) $[2,4]<=0 $[5,6]<=1 $.
Hubcap $[1,3]<=0 $[2,4]<=0 $[5,6]<=0 $.
Pcase: s[2] > 6.
Similar to L0_1[1].
Pcase: s[3] > 6.
Similar to L0_1[2].
Pcase: s[4] > 6.
Similar to L0_1[3].
Pcase: s[5] > 6.
Similar to L0_1[4].
Pcase: s[6] > 6.
Similar to L0_1[5].
Pcase: s[1] <= 5.
Reducible.
Pcase: s[2] <= 5.
Reducible.
Pcase: s[3] <= 5.
Reducible.
Pcase: s[4] <= 5.
Reducible.
Pcase: s[5] <= 5.
Reducible.
Pcase: s[6] <= 5.
Reducible.
Pcase L0_2: h[1] > 6.
Pcase L1_1: h[3] > 6.
Pcase: h[5] > 6.
Pcase L3_1: h[2] > 6.
Pcase: h[6] > 6.
Hubcap $[1,6]<=0 $[2,4]<=0 $[3,5]<=0 $.
Pcase: h[6] <= 5.
Pcase: f1[5] <= 5.
Hubcap $[1,6]<=1 $[2,4]<=(-2) $[3,5]<=1 $.
Pcase: f1[6] <= 5.
Hubcap $[1,6]<=(-1) $[2,4]<=(-1) $[3,5]<=2 $.
Hubcap $[1,6]<=0 $[2,4]<=(-1) $[3,5]<=1 $.
Pcase: f1[5] <= 5.
Hubcap $[1,6]<=1 $[2,4]<=(-1) $[3,5]<=0 $.
Pcase: f1[6] <= 5.
Hubcap $[1,6]<=(-1) $[2,4]<=0 $[3,5]<=1 $.
Hubcap $[1,6]<=0 $[2,4]<=0 $[3,5]<=0 $.
Pcase: h[6] > 6.
Similar to *L3_1[0].
Pcase L3_2: h[2] <= 5.
Pcase: f1[1] <= 5.
Pcase: f1[5] <= 5.
Hubcap $[1,6]<=0 $[2,4]<=1 $[3,5]<=(-1) $.
Pcase: f1[6] <= 5.
Hubcap $[1,6]<=(-2) $[2,4]<=2 $[3,5]<=0 $.
Hubcap $[1,6]<=(-1) $[2,4]<=2 $[3,5]<=(-1) $.
Pcase: f1[2] <= 5.
Pcase: f1[5] <= 5.
Hubcap $[1,6]<=2 $[2,4]<=0 $[3,5]<=(-2) $.
Pcase: f1[6] <= 5.
Hubcap $[1,6]<=0 $[2,4]<=1 $[3,5]<=(-1) $.
Hubcap $[1,6]<=1 $[2,4]<=1 $[3,5]<=(-2) $.
Pcase: f1[5] <= 5.
Hubcap $[1,6]<=1 $[2,4]<=0 $[3,5]<=(-1) $.
Pcase: f1[6] <= 5.
Hubcap $[1,6]<=(-1) $[2,4]<=1 $[3,5]<=0 $.
Hubcap $[1,6]<=0 $[2,4]<=1 $[3,5]<=(-1) $.
Pcase: h[6] <= 5.
Similar to *L3_2[0].
Pcase L3_3: f1[1] <= 5.
Pcase: f1[5] <= 5.
Hubcap $[1,6]<=0 $[2,4]<=0 $[3,5]<=0 $.
Pcase: f1[6] <= 5.
Hubcap $[1,6]<=(-2) $[2,4]<=1 $[3,5]<=1 $.
Hubcap $[1,6]<=(-1) $[2,4]<=1 $[3,5]<=0 $.
Pcase: f1[6] <= 5.
Similar to *L3_3[0].
Pcase L3_4: f1[2] <= 5.
Pcase: f1[5] <= 5.
Hubcap $[1,6]<=2 $[2,4]<=(-1) $[3,5]<=(-1) $.
Hubcap $[1,6]<=1 $[2,4]<=0 $[3,5]<=(-1) $.
Pcase: f1[5] <= 5.
Similar to *L3_4[0].
Hubcap $[1,6]<=0 $[2,4]<=0 $[3,5]<=0 $.
Pcase L2_1: h[4] > 6.
Pcase: h[6] > 6.
Pcase: h[5] <= 5.
Pcase: f1[5] <= 5.
Hubcap $[1,3]<=(-1) $[2,6]<=(-2) $[4,5]<=3 $.
Pcase: f1[4] <= 5.
Hubcap $[1,3]<=(-2) $[2,6]<=(-1) $[4,5]<=3 $.
Hubcap $[1,3]<=(-1) $[2,6]<=(-1) $[4,5]<=2 $.
Pcase: f1[5] <= 5.
Hubcap $[1,3]<=0 $[2,6]<=(-1) $[4,5]<=1 $.
Pcase: f1[4] <= 5.
Hubcap $[1,3]<=(-1) $[2,6]<=0 $[4,5]<=1 $.
Hubcap $[1,3]<=0 $[2,6]<=0 $[4,5]<=0 $.
Pcase: h[5] <= 5.
Pcase: f1[4] <= 5.
Pcase: f1[6] <= 5.
Hubcap $[1,3]<=(-3) $[2,6]<=(-1) $[4,5]<=4 $.
Hubcap $[1,3]<=(-2) $[2,6]<=(-1) $[4,5]<=3 $.
Pcase: f1[6] <= 5.
Hubcap $[1,3]<=(-2) $[2,6]<=(-1) $[4,5]<=3 $.
Hubcap $[1,3]<=(-1) $[2,6]<=(-1) $[4,5]<=2 $.
Pcase: h[6] <= 5.
Pcase: f1[4] <= 5.
Pcase: f1[6] <= 5.
Hubcap $[1,3]<=(-3) $[2,6]<=1 $[4,5]<=2 $.
Hubcap $[1,3]<=(-2) $[2,6]<=1 $[4,5]<=1 $.
Pcase: f1[6] <= 5.
Hubcap $[1,3]<=(-2) $[2,6]<=1 $[4,5]<=1 $.
Hubcap $[1,3]<=(-1) $[2,6]<=1 $[4,5]<=0 $.
Pcase: f1[4] <= 5.
Pcase: f1[6] <= 5.
Hubcap $[1,3]<=(-2) $[2,6]<=0 $[4,5]<=2 $.
Hubcap $[1,3]<=(-1) $[2,6]<=0 $[4,5]<=1 $.
Pcase: f1[6] <= 5.
Hubcap $[1,3]<=(-1) $[2,6]<=0 $[4,5]<=1 $.
Hubcap $[1,3]<=0 $[2,6]<=0 $[4,5]<=0 $.
Pcase: h[6] > 6.
Similar to *L2_1[4].
Pcase L2_2: h[4] <= 5.
Pcase: f1[3] <= 5.
Pcase: f1[6] <= 5.
Hubcap $[1,3]<=0 $[2,6]<=(-2) $[4,5]<=2 $.
Hubcap $[1,3]<=1 $[2,6]<=(-2) $[4,5]<=1 $.
Pcase: f1[6] <= 5.
Hubcap $[1,3]<=0 $[2,6]<=(-1) $[4,5]<=1 $.
Hubcap $[1,3]<=1 $[2,6]<=(-1) $[4,5]<=0 $.
Pcase: h[6] <= 5.
Similar to *L2_2[4].
Pcase L2_3: f1[3] <= 5.
Pcase: f1[6] <= 5.
Hubcap $[1,3]<=(-1) $[2,6]<=(-1) $[4,5]<=2 $.
Pcase: h[5] <= 5.
Hubcap $[1,3]<=(-1) $[2,6]<=(-2) $[4,5]<=3 $.
Hubcap $[1,3]<=0 $[2,6]<=(-1) $[4,5]<=1 $.
Pcase: f1[6] <= 5.
Similar to *L2_3[4].
Pcase: h[5] <= 5.
Hubcap $[1,3]<=(-1) $[2,6]<=(-1) $[4,5]<=2 $.
Hubcap $[1,3]<=0 $[2,6]<=0 $[4,5]<=0 $.
Pcase: h[5] > 6.
Similar to L1_1[4].
Pcase L1_2: h[2] > 6.
Pcase: h[4] > 6.
Similar to L1_1[1].
Pcase: h[6] > 6.
Similar to L1_1[5].
Pcase L3_1: h[3] <= 5.
Pcase: f1[2] <= 5.
Hubcap $[1,5]<=(-2) $[2,3]<=3 $[4,6]<=(-1) $.
Hubcap $[1,5]<=(-1) $[2,3]<=2 $[4,6]<=(-1) $.
Pcase: h[6] <= 5.
Similar to *L3_1[5].
Pcase L3_2: f1[2] <= 5.
Pcase: f1[6] <= 5.
Hubcap $[1,6]<=(-2) $[2,4]<=0 $[3,5]<=2 $.
Pcase: h[5] <= 5.
Hubcap $[1,6]<=(-2) $[2,4]<=1 $[3,5]<=1 $.
Hubcap $[1,6]<=(-1) $[2,4]<=0 $[3,5]<=1 $.
Pcase: f1[6] <= 5.
Similar to *L3_2[5].
Pcase: h[5] <= 5.
Hubcap $[1,6]<=(-1) $[2,4]<=1 $[3,5]<=0 $.
Hubcap $[1,6]<=0 $[2,4]<=0 $[3,5]<=0 $.
Pcase: h[6] > 6.
Similar to *L1_2[0].
Pcase L1_3: h[2] <= 5.
Pcase: h[5] <= 5.
Hubcap $[1,6]<=(-1) $[2,4]<=2 $[3,5]<=(-1) $.
Pcase: f1[1] <= 5.
Hubcap $[1,6]<=(-1) $[2,4]<=2 $[3,5]<=(-1) $.
Hubcap $[1,6]<=0 $[2,4]<=1 $[3,5]<=(-1) $.
Pcase: h[6] <= 5.
Similar to *L1_3[0].
Pcase L1_4: h[3] <= 5.
Hubcap $[1,6]<=(-1) $[2,4]<=0 $[3,5]<=1 $.
Pcase: h[5] <= 5.
Similar to *L1_4[0].
Pcase L1_5: f1[1] <= 5.
Pcase: f1[6] <= 5.
Hubcap $[1,6]<=(-2) $[2,4]<=1 $[3,5]<=1 $.
Hubcap $[1,6]<=(-1) $[2,4]<=1 $[3,5]<=0 $.
Pcase: f1[6] <= 5.
Similar to *L1_5[0].
Hubcap $[1,6]<=0 $[2,4]<=0 $[3,5]<=0 $.
Pcase: h[2] > 6.
Similar to L0_2[1].
Pcase: h[3] > 6.
Similar to L0_2[2].
Pcase: h[4] > 6.
Similar to L0_2[3].
Pcase: h[5] > 6.
Similar to L0_2[4].
Pcase: h[6] > 6.
Similar to L0_2[5].
Pcase L0_3: h[1] <= 5.
Hubcap $[1,6]<=2 $[2,4]<=(-1) $[3,5]<=(-1) $.
Pcase: h[2] <= 5.
Similar to L0_3[1].
Pcase: h[3] <= 5.
Similar to L0_3[2].
Pcase: h[4] <= 5.
Similar to L0_3[3].
Pcase: h[5] <= 5.
Similar to L0_3[4].
Pcase: h[6] <= 5.
Similar to L0_3[5].
Hubcap $[1,6]<=0 $[2,4]<=0 $[3,5]<=0 $.
Qed.
|
/* ============================================================ *
* halomodel.h *
* Martin Kilbinger 2006-2009 *
* ============================================================ */
#ifndef __HALOMODEL_H
#define __HALOMODEL_H
#include <stdio.h>
#include <stdlib.h>
#include <math.h>
#include <assert.h>
#include <string.h>
#include <fftw3.h>
#include <gsl/gsl_sf_erf.h>
#include "io.h"
#include "errorlist.h"
#include "config.h"
#include "maths.h"
#include "cosmo.h"
#include "nofz.h"
#define hm_base -1900
#define hm_hodtype hm_base + 1
#define hm_Mmin hm_base + 2
#define hm_pofk hm_base + 3
#define hm_nfw hm_base + 4
#define hm_par hm_base + 5
#define hm_overflow hm_base + 6
#define hm_io hm_base + 7
#define hm_zbin hm_base + 8
#define hm_alpha hm_base + 9
#define hm_negative hm_base + 10
#define hm_zmean_2h hm_base + 11
#define hm_halo_bias hm_base + 12
#define hm_undef hm_base + 13
#define hm_gsl_int hm_base + 14
/* Ranges of interpolation tables */
#define k_max_HOD 3336.0
/* Present critical density [M_sol h^2 / Mpc^3] */
#define rho_c0 2.7754e11
/* Mass limits for integration over mass functions */
#define logMmin (3.0*log(10.0))
#define logMmax (16.0*log(10.0))
/* Number of steps for scale-factor-integration (redshift) */
#define Na_hm 20
/* Bit-coded power spectrum types */
typedef enum {pofk_undef=-1, pl=1, pnl=2, p1hdm=4, p2hdm=8, pthdm=16, p1hg=32, p2hg=64, pthg=128,
p1hgcs=256, p1hgss=512, pstellar=1024} pofk_t;
/* Halo mass function type */
typedef enum {ps, st, st2, j01} massfct_t;
#define smassfct_t(i) ( \
i==ps ? "ps" : \
i==st ? "st" : \
i==st2 ? "st2" : \
i==j01 ? "j01" : \
"")
#define Nmassfct_t 4
/* Halo bias type */
typedef enum {halo_bias_sc, halo_bias_tinker05, halo_bias_tinker10} halo_bias_t;
#define shalo_bias_t(i) ( \
i==halo_bias_sc ? "halo_bias_sc" : \
i==halo_bias_tinker05 ? "halo_bias_tinker05" : \
i==halo_bias_tinker10 ? "halo_bias_tinker10" : \
"")
#define Nhalo_bias_t 3
/* HOD (Halo occupation distribution) type */
#define Nhod_t 5
typedef enum {hod_none, hamana04, berwein02, berwein02_hexcl, leauthaud11} hod_t;
#define shod_t(i) ( \
i==hod_none ? "hod_none" : \
i==hamana04 ? "hamana04" : \
i==berwein02 ? "berwein02" : \
i==berwein02_hexcl ? "berwein02_hexcl" :\
i==leauthaud11 ? "leauthaud11" : \
"")
/* ---------------------------------------------------------------- *
* Global variables and functions *
* ---------------------------------------------------------------- */
double FFTLog_TMP;
typedef struct FFTLog_complex
{
double re;
double im;
double amp;
double arg;
} FFTLog_complex;
typedef struct {
int N;
fftw_plan p_forward;
fftw_plan p_backward;
fftw_complex *an;
fftw_complex *ak;
fftw_complex *cm;
fftw_complex *um;
fftw_complex *cmum;
double min;
double max;
double q;
double mu;
double kr;
} FFTLog_config;
typedef struct {
cosmo *cosmo;
redshift_t *redshift;
double zmin, zmax;
/* Dark matter halo profile */
double c0; /* concentration parameter */
double alpha_NFW; /* density slope */
double beta_NFW; /* concentration slope as fct of mass */
massfct_t massfct; /* halo mass function */
halo_bias_t halo_bias; /* Halo bias */
/* Mass function parameters (Sheth&Torman). Do not set manually, they are set *
* in set_massfct() according to enum massfct. */
double nmz_a; /* Called q in CS02 */
double nmz_p; /* a=1, p=1/2 is Press-Schechter mass fct. */
/* HOD (halo occupation distribution) parameters */
hod_t hod; /* HOD type */
double M1, M0, sigma_log_M;
double M_min;
double alpha;
double pi_max;
double eta; /* central galaxy proportion */
/* galaxy-galaxy lensing and wp(rp) */
double log10Mhalo;
double coord_phys;
/* For Leauthaud11 model */
double beta,delta,gamma,Mstar0;
double beta_sat,B_sat,beta_cut,B_cut;
double x; /* any parameter to propagate if needed */
double Mstellar_min, Mstellar_max;
double fcen1, fcen2;
/* Precomputed stuff */
double A; /* Mass function normalisation */
double Mstar; /* M_*(a=1.0) */
interTable2D *Pthdm;
interTable *xir;
interTable *xi_dm;
interTable2D *rhohat;
splineTable* sigRsqr;
double a_xir;
/* FFTLOG flag - OBSOLETE */
int FFTLog;
} cosmo_hm;
typedef struct {
cosmo *cosmo;
cosmo_hm *model;
double a, r, k, ng, ngp, eps, c;
double logMlim, bias_fac, Mh, Mstellar, Mstellar_min, Mstellar_max;
double M, r_vir, *kk;
error **err;
double logrmin, logrmax, rp, xi;
gsl_interp_accel *acc;
gsl_spline *spline;
int i, j, type, asymptotic, logintegrate;
double (*bias_func)(double, void *);
} cosmo_hm_params;
typedef struct gsl_int_params
{
void *params;
funcwithpars func;
error **err;
} gsl_int_params;
typedef struct {
double *z;
double *fac;
double *ypn; /* for spline interpolation */
double zm; /* average weighted redshift*/
int nbins;
} nz_t;
cosmo_hm* init_parameters_hm(double OMEGAM, double OMEGADE, double W0_DE, double W1_DE,
double *W_POLY_DE, int N_POLY_DE,
double H100, double OMEGAB, double OMEGANUMASS,
double NEFFNUMASS, double NORM, double NSPEC,
int Nzbin, const int *Nnz, const nofz_t *nofz, double *par_nz,
double zmin, double zmax,
nonlinear_t NONLINEAR, transfer_t TRANSFER, growth_t GROWTH,
de_param_t DEPARAM, norm_t normmode,
double C0, double ALPHANFW, double BETANFW, massfct_t MASSFCT, halo_bias_t HALO_BIAS,
double M_min, double M1, double M0, double sigma_log_M, double alpha,
double Mstar0, double beta, double delta, double gamma, double B_cut, double B_sat,
double beta_cut, double beta_sat, double Mstellar_min, double Mstellar_max, double eta,
double fcen1, double fcen2,
hod_t HOD, double pi_max, error **err);
cosmo_hm* copy_parameters_hm_only(cosmo_hm* source, error **err);
cosmo_hm *copy_parameters_hm(cosmo_hm *source, error **err);
void read_cosmological_parameters_hm(cosmo_hm **model, FILE *F, error **err);
cosmo_hm *set_cosmological_parameters_to_default_hm(error **err);
void free_parameters_hm(cosmo_hm** model);
void set_massfct(massfct_t massfct, double *nmz_a, double *nmz_p, error **err);
void dump_param_only_hm(cosmo_hm* model, FILE *F);
void dump_param_hm(cosmo_hm* model, FILE *F, error **err);
double sm2_rtbis(double (*func)(double, void *, error **), double x1, double x2,
double xacc, void *param, error **err);
/* From nrcomplex.h,c */
#ifndef _DCOMPLEX_DECLARE_T_
typedef struct DCOMPLEX {double r,i;} dcomplex;
#define _DCOMPLEX_DECLARE_T_
#endif /* _DCOMPLEX_DECLARE_T_ */
dcomplex Complex(double re, double im);
dcomplex Cadd(dcomplex a, dcomplex b);
dcomplex Cmul(dcomplex a, dcomplex b);
dcomplex Cdiv(dcomplex a, dcomplex b);
dcomplex RCmul(double x, dcomplex a);
void sm2_cisi(double x, double *ci, double *si, error **err);
double delta_c(cosmo *model, double a, error **err);
double bis_Mstar(double logM, void *param, error **err);
double bis_Mstar_a(double logM, void *param, error **err);
double Mstar(cosmo_hm *model, error **err);
double Mstar_a(cosmo_hm *model, double a, error **err);
double concentration(cosmo_hm *model, double Mh, double a, error **err);
double Delta_vir(cosmo_hm *model, double a);
double dsigma_R_sqr_dR(cosmo_hm *model, double R, error **err);
double nufnu(cosmo_hm *model, double nu, int asymptotic, error **err);
double nufnu_j01(double x);
double sigma_R_sqr(cosmo_hm *model, double R, error **err);
double sigmasqr_M(cosmo_hm *model, double M, error **err);
double dsigma_m1_dlnM(cosmo_hm *model, double M, error **err);
double dnu_dlnM(cosmo_hm *model, double M, double a, error **err);
double dn_dlnM_lnM(double logM, void *intpar, error **err);
double dn_dlnM_uf(double M, cosmo_hm *model, double a, error **err);
double dn_dlnM(double M, void *intpar, error **err);
double r_vir(cosmo_hm *model, double M, double a, error **err);
double M_vir(cosmo_hm *model, double r_vir, double a, error **err);
double Delta_h(cosmo_hm *model, double a, error **err);
double rho_crit(cosmo_hm *model, double a, error **err);
double rho_crit_halo(cosmo_hm *model, double a, error **err);
double Omega_m_halo(cosmo_hm *model, double a, error **err);
double rho_halo(cosmo_hm *model, double r, double a, double Mh, double c, error **err);
double DeltaSigma_WB2000(cosmo_hm *model, double r, const double a, const double M, double c, double Delta, error **err);
double g_inf(double x, error **err);
double g_sup(double x, error **err);
double int_for_rhohat(double, void *, error **err);
double rhohat_halo(cosmo_hm *model, double k, double M, double a, double c, error **err);
double halo_bias(cosmo_hm *model, double M, double a, int k, error **err);
double bias(cosmo_hm *model, double M, double a, int k, error **err);
double bias_tinker(cosmo_hm *model, double M, double a, error **err);
double bias_tinker10(cosmo_hm *model, double M, double a, error **err);
double int_for_bias_norm(double logM, void *intpar, error **err);
double bias_norm(cosmo_hm *model, double a, error **err);
double int_for_M_ij(double, void *, error **);
double M_ij(cosmo_hm *model, int i, int j, double a, const double *k, error **err);
double P1h_dm(cosmo_hm *model, double a, double k, error **err);
double P2h_dm(cosmo_hm *model, double a, double k, error **err);
double xi_dm_NL_OBSOLETE(cosmo_hm *model, double a, double r, error **err); // non-linear DM xi
double int_for_xi_dm_NL_OBSOLETE(double k, void *intpar, error **err); // non-linear DM xi
#define CHANGE(fct) int change_##fct(cosmo_hm*, cosmo_hm*)
/* ---------------------------------------------------------------- *
* Utils *
* ---------------------------------------------------------------- */
double int_gsl(funcwithpars func,void *params, double a, double b, double eps, error **err);
double integrand_gsl(double x,void *p);
CHANGE(massfct);
CHANGE(massfct_params);
CHANGE(halo_bias);
CHANGE(sigma_R_sqr);
CHANGE(Mstar);
CHANGE(rhohat_halo);
CHANGE(Pth);
#undef CHANGE
#endif
|
\documentclass[11pt,a4paper]{article}
\usepackage[utf8]{inputenc}
\usepackage{amsmath}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{graphicx}
%\usepackage[ddmmyyyy]{datetime}
\usepackage[short,nodayofweek,level,12hr]{datetime}
%\usepackage{cite}
%\usepackage{wrapfig}
%\usepackage[left=2cm,right=2cm,top=2cm,bottom=2cm]{geometry}
\newcommand{\e}{\epsilon}
\newcommand{\dl}{\delta}
\newcommand{\pd}[2]{\frac{\partial #1}{\partial #2}}
\newcommand{\vect}[1]{\underline{#1}}
\newcommand{\uvect}[1]{\hat{#1}}
\newcommand{\1}{\vect{1}}
\newcommand{\grad}{\nabla}
\newcommand{\lc}{l_c}
\title{Surface Tension}
\date{\displaydate{date}}
\newdate{date}{22}{09}{2018}
\author{}
\begin{document}
\maketitle
If a \textit{thin} tube is half-dipped in water, we know that water rises in the tube to a height greater than that of the surrounding fluid. But what do we mean by a thin tube? There must be a critical thickness of the tube beyond which we will expect gravity to dominate and below which surface tension will be important.
Since there is a competition between gravity ($g$) and surface tension ($\Gamma$), we can obtain the critical length ($\lc$) by comparing the pressures exerted by each of these forces-
\begin{align*}
&\rho g \lc \sim \frac{\Gamma}{\lc}\\
\Rightarrow &\lc \sim \bigg(\frac{\Gamma}{\rho g}\bigg)^{1/2}
\end{align*}
This is the length scale over which the effects of surface tension are comparable with those of gravity. At lengths much smaller than this, surface tension will dominate gravity. For water, $\Gamma = 0.07 N/m$ and hence $l_c \approx 2.7$mm. Therefore a \textit{thin} tube, for water, referes to a tube whose diameter is nearly 2.7mm or less.
But why is the surface of a fluid under tension? A fluid consists of a bulk and a surface. The surface, though usually idealized as a sheet with zero thickness, is actually a few molecules thick. For a depth of the order of a few molecules near the water surface, the potential energy of the molecules is substantially higher than that in the bulk. The surface molecules are more energetic, jiggling about more rapidly, often escaping from the liquid to the air. If we wanted to calculate the total potential energy of the liquid by evaluating the potential energy per molecule in the bulk and then multiplying it by the total number of molecules, we will incur an error because potential energy for the surface molecules is much larger. This surface energy is usually incorporated by considering the surface to be an area (even though it is a volume few molecules thick). The excess energy associated with the area is proportional to the area and the coefficient of proportionality is the surface tension.
Increasing the surface area causes more molecules to move from the bulk to the surface which increases their potential energy. We know that force is a derivative of the potential energy ($F=-dU/dx$) and therefore pushing the molecules up the potential energy curve leads to a force. In order to minimize it's potential energy then, a fluid must minimize its surface area which is what we observe in nature.
\section{The Young - Laplace Equation}
The fact that surface of a fluid is under tension leads to a pressure jump between the two fluids. This can be seen by a force balance across the surface. Since the mass of a surface element is zero, the net force must be zero. Therefore, the sum of pressures acting on the two sides plus the surface tension force must add up to zero.
\begin{align*}
&\int(-p\vect n + p \vect n) dA + \oint \Gamma \vect t' ds = 0
\end{align*}
where $\vect n$ is the normal to the surface, $\vect t$ is the tangent to the curve bounding the surface and $\vect t' = \vect n \wedge \vect t$ is a vector perpendicular to the curve. Using Stokes Theorem, this can be written as
\begin{align*}
&\underbrace{\bigg[(\hat p - p)- \Gamma \pd{n_k}{x_k} \bigg]n_i}_{\text{Normal force balance}} + \underbrace{(\dl_{ik} - n_in_k)\pd{\Gamma}{x_k}}_{\text{tangential force balance}} = 0
\end{align*}
If the gradient of surface tension is zero, then second term is vanishes and we get
\begin{align*}
&\hat p - p = \Gamma \pd{n_k}{x_k}\\
&\hat p - p = \Gamma (\grad\cdot \vect{n})
\end{align*}
which is the Young-Laplace Equation. This equation gives us the pressure jump across an interface due to surface tension provided $\Gamma$ is constant everywhere. If $\Gamma$ varies with position, then the gradient of $\Gamma$ will not be zero and there will be flow due the surface tension gradient. Such flows are called Marangoni flows.
We can see a quick implementation of the Young-Laplace Equation by evaluating the pressure jump across a spherical bubble (say air bubble in water). For a sphere, the normal is given by $\vect n = x_i/r$. Hence-
\begin{align*}
\grad\cdot n &= \pd{}{x_i}\frac{x_i}{r}\\
&= \frac{1}{r}\pd{x_i}{x_i} - \frac{x_i x_i}{r^3} \tag{writing $r= (x_ix_i)^{1/2}$ }\\
&= \frac{\dl_{ii}}{r} - \frac{x_i x_i}{r^3}
= \frac{3}{r} - \frac{1}{r}
= \frac{2}{r}
\end{align*}
Therefore, for a spherical bubble, the pressure balance gives us
\begin{align*}
&\hat p - p = \frac{2 \Gamma}{r}
\end{align*}
\section{Shape of a 2D static meniscus}
The Young-Laplce Equation can be used to obtain the shape of a static meniscus. In this section we look at a meniscus close to a plane wall. Just next to the wall, the rise in water level is highest and it tapers off as we move away from the wall. We want to find out the functional form for the interface $z \equiv z(x)$ where $x$ is the distance from the wall and $z$ is the height of the meniscus at a given $x$.
Evaluation of the Young-Laplace Equation requires us to find the divergence of the unit normal. The normal to any surface $F(x,y) = 0$ is given by:
\begin{align*}
& \vect n = \frac{\grad F}{|\grad F|}\\
&\Rightarrow \pd{}{x_i}n_i = \pd{}{x_i}\frac{\pd{F}{x_i}}{|\grad F|}\\
&\Rightarrow \pd{n_i}{x_i} = \frac{1}{|\grad F|}\pd{^2F}{x_i^2} - \frac{1}{|\grad F|^3}\pd{F}{x_i}\pd{F}{x_k}\pd{^2F}{x_ix_k}
\end{align*}
For the 2D static meniscus $z=f(x)$ and hence $F(x,z) = z-f(x) = 0$. Evaluating the necessary derivatives:
\begin{align*}
&\pd{F}{x} = -\pd{f}{x}\\
&\pd{F}{z} = 1\\
&\pd{^2F}{x\partial z} = \pd{^2f}{x^2}\\
\end{align*}
and substituting them in the Young-Laplace equation, we get the following:
\begin{align*}
&\hat p - p = -\Gamma \bigg(\frac{\pd{^2f}{x^2}}{\big(1+(\pd{f}{x})^2\big)^{3/2}} \bigg)\\
\Rightarrow &\hat p - p = -\rho g z = -\Gamma \bigg(\frac{\pd{^2f}{x^2}}{\big(1+(\pd{f}{x})^2\big)^{3/2}} \bigg)
\end{align*}
Here, $\hat p$ is the atmospheric pressure and $p$ is the pressure in the fluid at $z=0$. We can non-dimensionalize the equation by considering the length scale $l_c = \sqrt{\Gamma/\rho g}$. This leaves us with
\begin{align*}
& z = \frac{\pd{^2f}{x^2}}{\big(1+(\pd{f}{x})^2\big)^{3/2}}
\end{align*}
We can greatly simplify the matters if we linearize this equation. Consider the case where the slope of the meniscus is very small everywhere, i.e. $dz/dx \ll 1$. Then the equation becomes
\begin{align*}
& z = \pd{^2f}{x^2}
\end{align*}
whose solution is an exponential
\begin{align*}
&z = z_0 e^{-x}
\end{align*}
or in dimensional terms
\begin{align*}
&z = z_0 e^{-x/l_c}
\end{align*}
which tells us that the height of the meniscus decays exponentially away from the wall with a characterstic length scale of $l_c$. Another quantity of interest here is the maximum height to which water rises along the wall. In order to find it, we must impose the contact angle boundary condition:
\begin{align*}
&\frac{dz}{dx} = \tan(\frac{\pi}{2} + \theta_c) \quad @ \quad z=0\\
\Rightarrow &\frac{-z_0}{l_c} = \tan(\frac{\pi}{2} + \theta_c) \\
\Rightarrow & z_0 =-l_c \tan(\frac{\pi}{2} + \theta_c) \\
\Rightarrow & z_0 =l_c \cot(\theta_c)
\end{align*}
This gives us the height of the meniscus at the wall. But what if $\theta_c = 0$? The height comes out to be infinity. This is obviously wrong but we must remember that linearization was only valid when slope of the meniscus was small. Since we have violated this assumption, the method is expected to give the wrong answer. In order to obtain the height of meniscus at the wall when $\theta_c = 0$, we must solve the full non-linear equation, which leads us to
\begin{align*}
&1-\frac{z^2}{2} = \frac{1}{(1+\frac{dz}{dx}^2)^{1/2}}
\end{align*}
From this, we can obtain the height at the wall to be $\sqrt 2 l_c$ when $\theta_c = 0$.
The equation can be further solved to obtain the complete shape of the meniscus as a transcendental function $x\equiv x(z)$.
\section{Appendix}
\subsection{Force balance interpretation}
Describe
\begin{align*}
&1-\frac{z^2}{2} = \frac{1}{(1+\frac{dz}{dx}^2)^{1/2}}
\end{align*}
as a force balance
\subsection{Solution of the non-linear equation for the 2D meniscus}
\end{document}
|
Formal statement is: lemma at_within_interior: "x \<in> interior S \<Longrightarrow> at x within S = at x" Informal statement is: If $x$ is in the interior of $S$, then the filter at $x$ within $S$ is the same as the filter at $x$.
|
lemma measure_Diff_null_set: "A \<in> sets M \<Longrightarrow> B \<in> null_sets M \<Longrightarrow> measure M (A - B) = measure M A"
|
module LeafClass
use, intrinsic :: iso_fortran_env
use KinematicClass
use FEMDomainClass
use PetiClass
use StemClass
use LightClass
use AirClass
implicit none
type :: Leaf_
type(FEMDomain_) :: FEMDomain
real(real64),allocatable :: LeafSurfaceNode2D(:,:)
real(real64) :: ShapeFactor,Thickness,length,width,center(3)
real(real64) :: MaxThickness,Maxlength,Maxwidth
real(real64) :: center_bottom(3),center_top(3)
real(real64) :: outer_normal_bottom(3),outer_normal_top(3)
real(real64),allocatable :: source(:), ppfd(:),A(:)
integer(int32) :: Division
type(leaf_),pointer :: pleaf
type(Peti_),pointer :: pPeti
real(real64) :: rot_x = 0.0d0
real(real64) :: rot_y = 0.0d0
real(real64) :: rot_z = 0.0d0
real(real64) :: disp_x = 0.0d0
real(real64) :: disp_y = 0.0d0
real(real64) :: disp_z = 0.0d0
real(real64) :: shaperatio = 0.30d0
real(real64) :: minwidth,minlength,MinThickness
integer(int32),allocatable :: I_planeNodeID(:)
integer(int32),allocatable :: I_planeElementID(:)
integer(int32),allocatable :: II_planeNodeID(:)
integer(int32),allocatable :: II_planeElementID(:)
integer(int32) :: A_PointNodeID
integer(int32) :: B_PointNodeID
integer(int32) :: A_PointElementID
integer(int32) :: B_PointElementID
integer(int32) :: xnum = 10
integer(int32) :: ynum = 10
integer(int32) :: znum = 10
! phisiological parameters
real(real64) :: V_cmax = 100.0d0 ! 最大カルボキシル化反応速度, mincro-mol/m-2/s
real(real64) :: V_omax = 100.0d0 ! 最大酸素化反応速度, mincro-mol/m-2/s, lambdaから推定
real(real64) :: O2 = 380.0d0! 酸素濃度, ppm
real(real64) :: CO2=202000.0d0! 二酸化炭素濃度, ppm
real(real64) :: R_d=1.0d0 ! 暗呼吸速度, mincro-mol/m-2/s
real(real64) :: K_c=272.380d0 ! CO2に対するミカエリス定数
real(real64) :: K_o=165820.0d0 ! O2に対するミカエリス定数
real(real64) :: J_=0.0d0 ! 電子伝達速度
real(real64) :: I_=0.0d0 ! 光強度
real(real64) :: phi=0.0d0 ! I-J曲線の初期勾配
real(real64) :: J_max=180.0d0 !最大電子伝達速度,mincro-mol/m-2/s
real(real64) :: theta_r=0.0d0 ! 曲線の凸度
real(real64) :: maxPPFD=1.0d0 ! micro-mol/m^2/s
real(real64) :: Lambda= 37.430d0 ! 暗呼吸速度を無視した時のCO2補償点ppm
real(real64) :: temp=303.0d0 ! temp
real(real64),allocatable :: DryDensity(:)
real(real64),allocatable :: WaterContent(:)
contains
procedure, public :: Init => initLeaf
procedure, public :: rotate => rotateleaf
procedure, public :: move => moveleaf
procedure, public :: curve => curveleaf
procedure, public :: create => createLeaf
procedure,pass :: connectLeafLeaf => connectLeafLeaf
procedure,pass :: connectLeafStem => connectLeafStem
generic :: connect => connectLeafLeaf, connectLeafStem
procedure, public :: photosynthesis => photosynthesisLeaf
procedure, public :: rescale => rescaleleaf
procedure, public :: adjust => adjustLeaf
procedure, public :: resize => resizeleaf
procedure, public :: getCoordinate => getCoordinateleaf
procedure, public :: gmsh => gmshleaf
procedure, public :: msh => mshleaf
procedure, public :: vtk => vtkleaf
procedure, public :: stl => stlleaf
end type
contains
subroutine createLeaf(obj,SurfacePoints,filename,x_num,y_num,x_len,y_len)
class(Leaf_),intent(inout) :: obj
real(real64),optional,intent(in) :: SurfacePoints(:,:),x_len,y_len
character(*),optional,intent(in) :: filename
integer(int32),optional,intent(in) :: x_num,y_num
type(IO_) :: f
type(FEMDomain_) :: domain
type(Math_) :: math
character(:),allocatable :: line
real(real64) :: x, y, r ,theta,x_sum,y_sum,center(2),max_r,coord(2), ret
real(real64),allocatable :: r_data(:),theta_data(:),tx(:),tfx(:)
integer(int32) :: num_ptr, i,id,ids(5),id_n
if(present(filename) )then
call f%open(filename,"r")
! get brief info
num_ptr = 0
x_sum = 0.0d0
y_sum = 0.0d0
do
line = f%readline()
if(f%EOF) exit
num_ptr = num_ptr+1
! read x-y
read(line,*) x, y
x_sum = x_sum + x
y_sum = y_sum + y
enddo
call f%close()
center(1) = x_sum/dble(num_ptr)
center(2) = y_sum/dble(num_ptr)
r_data = zeros(num_ptr)
theta_data = zeros(num_ptr)
! get detail
call f%open(filename,"r")
num_ptr=0
do
line = f%readline()
if(f%EOF) exit
! read x-y
read(line,*) x, y
coord(1) = x - center(1)
coord(2) = y - center(2)
r = sqrt( dot_product(coord,coord) )
theta = angles( coord )
num_ptr = num_ptr + 1
r_data(num_ptr) = r
theta_data(num_ptr) = theta
enddo
max_r = maxval(r_data)
r_data = r_data/max_r
call f%close()
elseif(present(SurfacePoints) )then
num_ptr = size(SurfacePoints,1)
center(1) = x_sum/dble(num_ptr)
center(2) = y_sum/dble(num_ptr)
r_data = zeros(num_ptr)
theta_data = zeros(num_ptr)
num_ptr=0
do i=1,size(SurfacePoints)
! read x-y
x = SurfacePoints(i,1)
y = SurfacePoints(i,2)
coord(1) = x - center(1)
coord(2) = y - center(2)
r = sqrt( dot_product(coord,coord) )
theta = angles( coord )
num_ptr = num_ptr + 1
r_data(num_ptr) = r
theta_data(num_ptr) = theta
enddo
max_r = maxval(r_data)
r_data = r_data/max_r
else
print *, "ERROR :: Leaf%create >> Please import SurfacePoints or Filename"
stop
endif
call obj%femdomain%create("Cylinder3D",x_num=x_num,y_num=y_num)
call obj%femdomain%resize(x=2.0d0)
call obj%femdomain%resize(y=2.0d0)
call obj%femdomain%resize(z=0.010d0)
! ####################################
! test interpolate
!tx = [0.0d0, 1.0d0, 2.0d0, 3.0d0]
!tfx = [0.0d0, 2.0d0, 4.0d0, 8.0d0]
!ret = interpolate(x =tx,Fx=tfx,x_value = -0.50d0)
!print *, ret
!stop
! ####################################
! adjust shape
do i=1,obj%femdomain%nn()
x = obj%femdomain%mesh%nodcoord(i,1)
y = obj%femdomain%mesh%nodcoord(i,2)
r = sqrt(x**2 + y**2)
coord(1:2) = obj%femdomain%mesh%nodcoord(i,1:2)
r = norm(coord)
theta = angles(coord)
! find nearest theta
r = r * interpolate(x=theta_data,Fx=r_data,x_value=theta)
x = r*x
y = r*y
obj%femdomain%mesh%nodcoord(i,1) = x
obj%femdomain%mesh%nodcoord(i,2) = y
enddo
obj%A_PointNodeID = randi(obj%femdomain%nn())
obj%B_PointNodeID = randi(obj%femdomain%nn())
obj%A_PointElementID = randi(obj%femdomain%nn())
obj%B_PointElementID = randi(obj%femdomain%nn())
if(present(x_len) )then
call obj%femdomain%resize(x=x_len)
endif
if(present(y_len) )then
call obj%femdomain%resize(y=y_len)
endif
! ! export data
! call f%open("theta_r_relation.txt","w")
! do i=1,size(r_data)
! call f%write(theta_data(i),r_data(i) )
! enddo
! call f%close()
! call f%plot("theta_r_relation.txt","w l")
! call f%plot(filename,"w l")
end subroutine
! ########################################
subroutine initLeaf(obj,config,regacy,Thickness,length,width,ShapeFactor,&
MaxThickness,Maxlength,Maxwidth,rotx,roty,rotz,location,species,SoyWidthRatio,&
curvature)
class(leaf_),intent(inout) :: obj
real(real64),optional,intent(in) :: Thickness,length,width,ShapeFactor
real(real64),optional,intent(in) :: MaxThickness,Maxlength,Maxwidth
real(real64),optional,intent(in):: rotx,roty,rotz,location(3),SoyWidthRatio,curvature
integer(int32),optional,intent(in) :: species
logical, optional,intent(in) :: regacy
character(*),optional,intent(in) :: config
type(IO_) :: leafconf,f
character(200) :: fn,conf,line
integer(int32),allocatable :: buf(:)
integer(int32) :: id,rmc,n,node_id,node_id2,elemid,blcount,i,j
real(real64) :: loc(3),radius,z,leaf_L
logical :: debug=.false.
! 節を生成するためのスクリプトを開く
if(.not.present(config) .or. index(config,".json")==0 )then
! デフォルトの設定を生成
if(debug) print *, "New leaf-configuration >> leafconfig.json"
call leafconf%open("leafconfig.json")
write(leafconf%fh,*) '{'
write(leafconf%fh,*) ' "type": "leaf",'
write(leafconf%fh,*) ' "minlength": 0.005,'
write(leafconf%fh,*) ' "minwidth": 0.005,'
write(leafconf%fh,*) ' "minthickness": 0.0001,'
write(leafconf%fh,*) ' "maxlength": 0.07,'
write(leafconf%fh,*) ' "maxwidth": 0.045,'
write(leafconf%fh,*) ' "maxthickness": 0.001,'
write(leafconf%fh,*) ' "shaperatio": 0.3,'
write(leafconf%fh,*) ' "drydensity": 0.0,'
write(leafconf%fh,*) ' "watercontent": 0.0,'
write(leafconf%fh,*) ' "xnum": 10,'
write(leafconf%fh,*) ' "ynum": 10,'
write(leafconf%fh,*) ' "znum": 20'
write(leafconf%fh,*) '}'
conf="leafconfig.json"
call leafconf%close()
else
conf = trim(config)
endif
call leafconf%open(trim(conf))
blcount=0
do
read(leafconf%fh,'(a)') line
if(debug) print *, trim(line)
if( adjustl(trim(line))=="{" )then
blcount=1
cycle
endif
if( adjustl(trim(line))=="}" )then
exit
endif
if(blcount==1)then
if(index(line,"type")/=0 .and. index(line,"leaf")==0 )then
print *, "ERROR: This config-file is not for leaf"
return
endif
if(index(line,"maxlength")/=0 )then
! 生育ステージ
rmc=index(line,",")
! カンマがあれば除く
if(rmc /= 0)then
line(rmc:rmc)=" "
endif
id = index(line,":")
read(line(id+1:),*) obj%maxlength
endif
if(index(line,"maxwidth")/=0 )then
! 種子の長さ
rmc=index(line,",")
! カンマがあれば除く
if(rmc /= 0)then
line(rmc:rmc)=" "
endif
id = index(line,":")
read(line(id+1:),*) obj%maxwidth
endif
if(index(line,"maxthickness")/=0 )then
! 種子の長さ
rmc=index(line,",")
! カンマがあれば除く
if(rmc /= 0)then
line(rmc:rmc)=" "
endif
id = index(line,":")
read(line(id+1:),*) obj%maxthickness
endif
if(index(line,"minlength")/=0 )then
! 生育ステージ
rmc=index(line,",")
! カンマがあれば除く
if(rmc /= 0)then
line(rmc:rmc)=" "
endif
id = index(line,":")
read(line(id+1:),*) obj%minlength
endif
if(index(line,"shaperatio")/=0 )then
! 生育ステージ
rmc=index(line,",")
! カンマがあれば除く
if(rmc /= 0)then
line(rmc:rmc)=" "
endif
id = index(line,":")
read(line(id+1:),*) obj%shaperatio
endif
if(index(line,"minwidth")/=0 )then
! 種子の長さ
rmc=index(line,",")
! カンマがあれば除く
if(rmc /= 0)then
line(rmc:rmc)=" "
endif
id = index(line,":")
read(line(id+1:),*) obj%minwidth
endif
if(index(line,"minthickness")/=0 )then
! 種子の長さ
rmc=index(line,",")
! カンマがあれば除く
if(rmc /= 0)then
line(rmc:rmc)=" "
endif
id = index(line,":")
read(line(id+1:),*) obj%minthickness
endif
if(index(line,"xnum")/=0 )then
! 種子の長さ
rmc=index(line,",")
! カンマがあれば除く
if(rmc /= 0)then
line(rmc:rmc)=" "
endif
id = index(line,":")
read(line(id+1:),*) obj%xnum
endif
if(index(line,"ynum")/=0 )then
! 種子の長さ
rmc=index(line,",")
! カンマがあれば除く
if(rmc /= 0)then
line(rmc:rmc)=" "
endif
id = index(line,":")
read(line(id+1:),*) obj%ynum
endif
if(index(line,"znum")/=0 )then
! 種子の長さ
rmc=index(line,",")
! カンマがあれば除く
if(rmc /= 0)then
line(rmc:rmc)=" "
endif
id = index(line,":")
read(line(id+1:),*) obj%znum
endif
cycle
endif
enddo
call leafconf%close()
! グラフ構造とメッシュ構造を生成する。
!
! %%%%%%%%%%%%%%%%%%%%%%%%%%%%% B
! %% % %
! %% % %%
! %% % %%
! %% % %%
! %% % %%
! %% %%
! A %% %%
! <I> %%%%%%%%%%%%%%%%
! メッシュを生成
call obj%FEMdomain%create(meshtype="rectangular3D",x_num=obj%xnum,y_num=obj%ynum,z_num=obj%znum,&
x_len=obj%minwidth/2.0d0,y_len=obj%minwidth/2.0d0,z_len=obj%minlength,shaperatio=obj%shaperatio)
! physical parameters
allocate(obj%A(size(obj%FEMDomain%Mesh%ElemNod,1) ) )
obj%A(:) = 0.0d0
allocate(obj%source(size(obj%FEMDomain%Mesh%ElemNod,1) ) )
obj%source(:) = 0.0d0
allocate(obj%ppfd(size(obj%FEMDomain%Mesh%ElemNod,1) ) )
obj%ppfd(:) = 0.0d0
! initialize physical parameter
obj%DryDensity = zeros( obj%FEMDomain%ne() )
obj%watercontent = zeros(obj%FEMDomain%ne())
obj%DryDensity(:) = freal(leafconf%parse(conf,key1="drydensity"))
obj%watercontent(:) = freal(leafconf%parse(conf,key1="watercontent"))
! <I>面に属する要素番号、節点番号、要素座標、節点座標のリストを生成
obj%I_planeNodeID = obj%FEMdomain%mesh%getNodeList(zmax=0.0d0)
obj%I_planeElementID = obj%FEMdomain%mesh%getElementList(zmax=0.0d0)
! <I>面に属する要素番号、節点番号、要素座標、節点座標のリストを生成
obj%II_planeNodeID = obj%FEMdomain%mesh%getNodeList(zmin=obj%minlength)
obj%II_planeElementID = obj%FEMdomain%mesh%getElementList(zmin=obj%minlength)
buf = obj%FEMDomain%mesh%getNodeList(&
xmin=obj%minwidth/2.0d0 - obj%minwidth/dble(obj%xnum)/2.0d0 ,&
xmax=obj%minwidth/2.0d0 + obj%minwidth/dble(obj%xnum)/2.0d0 ,&
ymin=obj%minwidth/2.0d0 - obj%minwidth/dble(obj%ynum)/2.0d0 ,&
ymax=obj%minwidth/2.0d0 + obj%minwidth/dble(obj%ynum)/2.0d0 ,&
zmax=0.0d0)
obj%A_PointNodeID = buf(1)
buf = obj%FEMDomain%mesh%getNodeList(&
xmin=obj%minwidth/2.0d0 - obj%minwidth/dble(obj%xnum)/2.0d0 ,&
xmax=obj%minwidth/2.0d0 + obj%minwidth/dble(obj%xnum)/2.0d0 ,&
ymin=obj%minwidth/2.0d0 - obj%minwidth/dble(obj%ynum)/2.0d0 ,&
ymax=obj%minwidth/2.0d0 + obj%minwidth/dble(obj%ynum)/2.0d0 ,&
zmin=obj%minlength)
obj%B_PointNodeID = buf(1)
buf = obj%FEMDomain%mesh%getElementList(&
xmin=obj%minwidth/2.0d0 - obj%minwidth/dble(obj%xnum)/2.0d0 ,&
xmax=obj%minwidth/2.0d0 + obj%minwidth/dble(obj%xnum)/2.0d0 ,&
ymin=obj%minwidth/2.0d0 - obj%minwidth/dble(obj%ynum)/2.0d0 ,&
ymax=obj%minwidth/2.0d0 + obj%minwidth/dble(obj%ynum)/2.0d0 ,&
zmax=0.0d0)
obj%A_PointElementID = buf(1)
buf = obj%FEMDomain%mesh%getElementList(&
xmin=obj%minwidth/2.0d0 - obj%minwidth/dble(obj%xnum)/2.0d0 ,&
xmax=obj%minwidth/2.0d0 + obj%minwidth/dble(obj%xnum)/2.0d0 ,&
ymin=obj%minwidth/2.0d0 - obj%minwidth/dble(obj%ynum)/2.0d0 ,&
ymax=obj%minwidth/2.0d0 + obj%minwidth/dble(obj%ynum)/2.0d0 ,&
zmin=obj%minlength)
obj%B_PointElementID = buf(1)
!print *, obj%A_PointNodeID
!print *, obj%B_PointNodeID
!print *, obj%A_PointElementID
!print *, obj%B_PointElementID
!
call obj%FEMdomain%remove()
if(present(species) )then
call obj%FEMdomain%create(meshtype="Leaf3D",x_num=obj%xnum,y_num=obj%ynum,z_num=obj%znum,&
x_len=obj%minwidth/2.0d0,y_len=obj%minthickness/2.0d0,z_len=obj%minlength,species=species,SoyWidthRatio=SoyWidthRatio)
else
call obj%FEMdomain%create(meshtype="Leaf3D",x_num=obj%xnum,y_num=obj%ynum,z_num=obj%znum,&
x_len=obj%minwidth/2.0d0,y_len=obj%minthickness/2.0d0,z_len=obj%minlength,shaperatio=obj%shaperatio)
endif
! デバッグ用
! call f%open("I_phaseNodeID.txt")
! do i=1,size(obj%I_planeNodeID)
! write(f%fh,*) obj%femdomain%mesh%NodCoord( obj%I_planeNodeID(i) ,:)
! enddo
! call f%close()
!
! call f%open("II_phaseNodeID.txt")
! do i=1,size(obj%II_planeNodeID)
! write(f%fh,*) obj%femdomain%mesh%NodCoord( obj%II_planeNodeID(i) ,:)
! enddo
! call f%close()
!
! call f%open("I_phaseElementID.txt")
! do i=1,size(obj%I_planeElementID)
! do j=1,size(obj%femdomain%mesh%elemnod,2)
! write(f%fh,*) obj%femdomain%mesh%NodCoord( &
! obj%femdomain%mesh%elemnod(obj%I_planeElementID(i),j),:)
! enddo
! enddo
! call f%close()
!
! call f%open("II_phaseElementID.txt")
! do i=1,size(obj%II_planeElementID)
! do j=1,size(obj%femdomain%mesh%elemnod,2)
! write(f%fh,*) obj%femdomain%mesh%NodCoord( &
! obj%femdomain%mesh%elemnod(obj%II_planeElementID(i),j),:)
! enddo
! enddo
! call f%close()
! return
! Aについて、要素番号、節点番号、要素座標、節点座標のリストを生成
if( present(regacy))then
if(regacy .eqv. .true.)then
loc(:)=0.0d0
if(present(location) )then
loc(:)=location(:)
endif
obj%ShapeFactor = input(default=0.30d0 ,option= ShapeFactor )
obj%Thickness = input(default=0.10d0,option= Thickness )
obj%length = input(default=0.10d0,option= length )
obj%width = input(default=0.10d0,option= width)
obj%MaxThickness = input(default=0.10d0 ,option= MaxThickness )
obj%Maxlength = input(default=10.0d0 ,option= Maxlength )
obj%Maxwidth = input(default=2.0d0 ,option= Maxwidth)
obj%outer_normal_bottom(:)=0.0d0
obj%outer_normal_bottom(1)=1.0d0
obj%outer_normal_top(:)=0.0d0
obj%outer_normal_top(1)=1.0d0
! rotate
obj%outer_normal_Bottom(:) = Rotation3D(vector=obj%outer_normal_bottom,rotx=rotx,roty=roty,rotz=rotz)
obj%outer_normal_top(:) = Rotation3D(vector=obj%outer_normal_top,rotx=rotx,roty=roty,rotz=rotz)
obj%center_bottom(:)=loc(:)
obj%center_top(:) = obj%center_bottom(:) + obj%length*obj%outer_normal_bottom(:)
endif
endif
end subroutine
! ########################################
subroutine curveleaf(obj,curvature)
! deform by curvature
class(leaf_),intent(inout) :: obj
real(real64),intent(in) :: curvature
real(real64) :: leaf_L,radius,z
integer(int32) :: i
if(curvature < dble(1.0e-5))then
print *, "Caution >> initLeaf >> curvature is too small < 1.0e-5"
print *, "Then, ignored."
return
endif
radius = 1.0d0/curvature
leaf_L = maxval(obj%femdomain%mesh%nodcoord(:,3)) - minval(obj%femdomain%mesh%nodcoord(:,3))
leaf_L = 0.50d0*leaf_L
do i=1, obj%femdomain%nn()
z = obj%femdomain%mesh%nodcoord(i,3)
obj%femdomain%mesh%nodcoord(i,2) = &
obj%femdomain%mesh%nodcoord(i,2) &
- sqrt(radius*radius - leaf_L*leaf_L ) &
+ sqrt(radius*radius - (z - leaf_L)*(z - leaf_L) )
enddo
end subroutine
! ########################################
recursive subroutine rotateleaf(obj,x,y,z,reset)
class(leaf_),intent(inout) :: obj
real(real64),optional,intent(in) :: x,y,z
logical,optional,intent(in) :: reset
real(real64),allocatable :: origin1(:),origin2(:),disp(:)
if(present(reset) )then
if(reset .eqv. .true.)then
call obj%femdomain%rotate(-obj%rot_x,-obj%rot_y,-obj%rot_z)
obj%rot_x = 0.0d0
obj%rot_y = 0.0d0
obj%rot_z = 0.0d0
endif
endif
origin1 = obj%getCoordinate("A")
call obj%femdomain%rotate(x,y,z)
obj%rot_x = obj%rot_x + input(default=0.0d0, option=x)
obj%rot_y = obj%rot_y + input(default=0.0d0, option=y)
obj%rot_z = obj%rot_z + input(default=0.0d0, option=z)
origin2 = obj%getCoordinate("A")
disp = origin1
disp(:) = origin1(:) - origin2(:)
call obj%femdomain%move(x=disp(1),y=disp(2),z=disp(3) )
end subroutine
! ########################################
! ########################################
recursive subroutine moveleaf(obj,x,y,z,reset)
class(leaf_),intent(inout) :: obj
real(real64),optional,intent(in) :: x,y,z
logical,optional,intent(in) :: reset
real(real64),allocatable :: origin1(:),origin2(:),disp(:)
if(present(reset) )then
if(reset .eqv. .true.)then
call obj%femdomain%move(-obj%disp_x,-obj%disp_y,-obj%disp_z)
obj%disp_x = 0.0d0
obj%disp_y = 0.0d0
obj%disp_z = 0.0d0
endif
endif
call obj%femdomain%move(x,y,z)
obj%disp_x = obj%disp_x + input(default=0.0d0, option=x)
obj%disp_y = obj%disp_y + input(default=0.0d0, option=y)
obj%disp_z = obj%disp_z + input(default=0.0d0, option=z)
end subroutine
! ########################################
! ########################################
subroutine connectleafleaf(obj,direct,leaf)
class(leaf_),intent(inout) :: obj
class(leaf_),intent(inout) :: leaf
character(2),intent(in) :: direct
real(real64),allocatable :: x1(:),x2(:),disp(:)
!if(present(Stem) )then
! if(direct=="->" .or. direct=="=>")then
! ! move obj to connect stem (stem is not moved.)
! x1 = leaf%getCoordinate("A")
! x2 = stem%getCoordinate("B")
! disp = x2 - x1
! call leaf%move(x=disp(1),y=disp(2),z=disp(3) )
! endif
!
!
! if(direct=="<-" .or. direct=="<=")then
! ! move obj to connect stem (stem is not moved.)
! x1 = stem%getCoordinate("A")
! x2 = leaf%getCoordinate("B")
! disp = x2 - x1
! call stem%move(x=disp(1),y=disp(2),z=disp(3) )
! endif
! return
!endif
if(direct=="->" .or. direct=="=>")then
! move obj to connect leaf (leaf is not moved.)
x1 = obj%getCoordinate("A")
x2 = leaf%getCoordinate("B")
disp = x2 - x1
call obj%move(x=disp(1),y=disp(2),z=disp(3) )
endif
if(direct=="<-" .or. direct=="<=")then
! move obj to connect leaf (leaf is not moved.)
x1 = leaf%getCoordinate("A")
x2 = obj%getCoordinate("B")
disp = x2 - x1
call leaf%move(x=disp(1),y=disp(2),z=disp(3) )
endif
end subroutine
! ########################################
! ########################################
subroutine connectLeafStem(obj,direct,Stem)
class(leaf_),intent(inout) :: obj
class(Stem_),intent(inout) :: stem
character(2),intent(in) :: direct
real(real64),allocatable :: x1(:),x2(:),disp(:)
if(direct=="->" .or. direct=="=>")then
! move obj to connect stem (stem is not moved.)
x1 = obj%getCoordinate("A")
x2 = stem%getCoordinate("B")
disp = x2 - x1
call obj%move(x=disp(1),y=disp(2),z=disp(3) )
endif
if(direct=="<-" .or. direct=="<=")then
! move obj to connect stem (stem is not moved.)
x1 = stem%getCoordinate("A")
x2 = obj%getCoordinate("B")
disp = x2 - x1
call stem%move(x=disp(1),y=disp(2),z=disp(3) )
endif
end subroutine
! ########################################
! ########################################
function getCoordinateleaf(obj,nodetype) result(ret)
class(leaf_),intent(inout) :: obj
character(*),intent(in) :: nodetype
real(real64),allocatable :: ret(:)
integer(int32) :: dimnum
dimnum = size(obj%femdomain%mesh%nodcoord,2)
allocate(ret(dimnum) )
if( trim(nodetype)=="A" .or. trim(nodetype)=="a")then
ret = obj%femdomain%mesh%nodcoord(obj%A_PointNodeID,:)
endif
if( trim(nodetype)=="B" .or. trim(nodetype)=="B")then
ret = obj%femdomain%mesh%nodcoord(obj%B_PointNodeID,:)
endif
end function
! ########################################
! ########################################
subroutine gmshleaf(obj,name)
class(leaf_),intent(inout) :: obj
character(*),intent(in) ::name
if(obj%femdomain%mesh%empty() )then
return
endif
call obj%femdomain%gmsh(Name=name)
! PPFD を出力
call obj%femdomain%gmsh(Name=name//"_PPFD_",field=obj%PPFD)
! ソース量 を出力
call obj%femdomain%gmsh(Name=name//"_SOURCE_",field=obj%source)
! 光合成速度 を出力
call obj%femdomain%gmsh(Name=name//"_A_",field=obj%A)
end subroutine
! ########################################
! ########################################
subroutine mshleaf(obj,name)
class(leaf_),intent(inout) :: obj
character(*),intent(in) ::name
if(obj%femdomain%mesh%empty() )then
return
endif
call obj%femdomain%msh(Name=name)
! PPFD を出力
!call obj%femdomain%msh(Name=name//"_PPFD_",field=obj%PPFD)
! ソース量 を出力
!call obj%femdomain%msh(Name=name//"_SOURCE_",field=obj%source)
! 光合成速度 を出力
!call obj%femdomain%msh(Name=name//"_A_",field=obj%A)
end subroutine
! ########################################
! ########################################
subroutine vtkleaf(obj,name)
class(leaf_),intent(inout) :: obj
character(*),intent(in) ::name
if(obj%femdomain%mesh%empty() )then
return
endif
call obj%femdomain%vtk(Name=name)
! PPFD を出力
!call obj%femdomain%msh(Name=name//"_PPFD_",field=obj%PPFD)
! ソース量 を出力
!call obj%femdomain%msh(Name=name//"_SOURCE_",field=obj%source)
! 光合成速度 を出力
!call obj%femdomain%msh(Name=name//"_A_",field=obj%A)
end subroutine
! ########################################
! ########################################
subroutine stlleaf(obj,name)
class(leaf_),intent(inout) :: obj
character(*),intent(in) ::name
if(obj%femdomain%mesh%empty() )then
return
endif
call obj%femdomain%stl(Name=name)
! PPFD を出力
!call obj%femdomain%msh(Name=name//"_PPFD_",field=obj%PPFD)
! ソース量 を出力
!call obj%femdomain%msh(Name=name//"_SOURCE_",field=obj%source)
! 光合成速度 を出力
!call obj%femdomain%msh(Name=name//"_A_",field=obj%A)
end subroutine
! ########################################
! ########################################
subroutine resizeleaf(obj,x,y,z)
class(Leaf_),optional,intent(inout) :: obj
real(real64),optional,intent(in) :: x,y,z
real(real64),allocatable :: origin1(:), origin2(:),disp(:)
origin1 = obj%getCoordinate("A")
call obj%femdomain%resize(x_len=x,y_len=y,z_len=z)
origin2 = obj%getCoordinate("A")
disp = origin1 - origin2
call obj%move(x=disp(1),y=disp(2),z=disp(3) )
end subroutine
! ########################################
! ########################################
subroutine rescaleleaf(obj,x,y,z)
class(Leaf_),optional,intent(inout) :: obj
real(real64),optional,intent(in) :: x,y,z
real(real64),allocatable :: origin1(:), origin2(:),disp(:)
origin1 = obj%getCoordinate("A")
call obj%femdomain%resize(x_rate=x,y_rate=y,z_rate=z)
origin2 = obj%getCoordinate("A")
disp = origin1 - origin2
call obj%move(x=disp(1),y=disp(2),z=disp(3) )
end subroutine
! ########################################
! ########################################
!subroutine LayTracingLeaf(obj,maxPPFD,light,)
! class(Leaf_),intent(inout) :: obj
! class(Light_),intent(in) :: light
! real(real64),intent(in) :: maxPPFD
! integer(int32) :: i,j,n,m,node_id
! real(real64) :: lx(3)
! real(real64),allocatable :: Elem_x(:,:)
! ! PPFDを計算する。
! ! Photosynthetic photon flux density (PPFD)
! ! micro-mol/m^2/s
!
! ! 反射、屈折は無視、直線のみ
!
! n=size(obj%FEMDomain%Mesh%ElemNod,2)
! m=size(obj%FEMDomain%Mesh%NodCoord,2)
!
! allocate(Elem_x(n,m) )
! ! 要素ごと
! do i=1, size(obj%FEMDomain%Mesh%ElemNod,1)
! do j=1,size(obj%FEMDomain%Mesh%ElemNod,2)
! node_id = obj%FEMDomain%Mesh%ElemNod(i,j)
! Elem_x(j,:) = obj%FEMDomain%Mesh%NodCoord(node_id,:)
! enddo
! ! 要素座標 >> Elem_x(:,:)
! ! 光源座標 >> lx(:)
!
! enddo
!
!
!
!end subroutine
! ########################################
! ########################################
subroutine photosynthesisLeaf(obj,dt,air)
! https://eprints.lib.hokudai.ac.jp/dspace/bitstream/2115/39102/1/67-013.pdf
class(Leaf_),intent(inout) :: obj
type(Air_),intent(in) :: air
type(IO_) :: f
real(real64),intent(in) :: dt
! Farquhar modelのパラメータ
real(real64) :: A ! CO2吸収速度
real(real64) :: V_c ! カルボキシル化反応速度
real(real64) :: V_o ! 酸素化反応速度
real(real64) :: W_c! RuBPが飽和している場合のCO2吸収速度
real(real64) :: W_j! RuBP供給が律速している場合のCO2吸収速度
real(real64) :: V_cmax ! 最大カルボキシル化反応速度
real(real64) :: V_omax ! 最大酸素化反応速度
real(real64) :: O2 ! 酸素濃度
real(real64) :: CO2 ! 二酸化炭素濃度
real(real64) :: R_d ! なんだっけ
real(real64) :: K_c ! CO2に対するミカエリス定数
real(real64) :: K_o ! O2に対するミカエリス定数
real(real64) :: J_ ! 電子伝達速度
real(real64) :: I_ ! 光強度
real(real64) :: phi ! I-J曲線の初期勾配
real(real64) :: J_max !最大電子伝達速度
real(real64) :: theta_r ! 曲線の凸度
real(real64) :: pfd
real(real64) :: Lambda, volume
integer(int32) :: i, element_id
obj%temp=air%temp
obj%CO2 = air%CO2
obj%O2 = air%O2
! TT-model
do i=1,size(obj%source)
! 要素ごとに電子伝達速度を求める
element_id = i
pfd = obj%ppfd(element_id)
obj%J_ = 0.240d0*pfd/(sqrt(1.0d0 + (0.240d0*0.240d0)*pfd*pfd)/obj%J_max/obj%J_max)
! lambdaからV_omaxを推定
obj%V_omax = obj%Lambda*( 2.0d0 * obj%V_cmax*obj%K_o )/(obj%K_c*O2)
! CO2固定速度の計算
V_c = (obj%V_cmax*obj%CO2)/(obj%CO2 +obj% K_o * (1.0d0+ obj%O2/obj%K_o) )
V_o = (obj%V_omax*obj%O2 )/(obj%O2 + obj%K_o * (1.0d0 + obj%CO2/obj%K_c) )
! RuBPが飽和している場合のCO2吸収速度
W_c = (obj%V_cmax*(obj%CO2 - obj%Lambda))/(obj%CO2 + obj%K_c*(1.0d0 + obj%O2/obj%K_o))
! RuBP供給が律速している場合のCO2吸収速度
W_j = obj%J_ * (obj%CO2 - obj%Lambda)/(4.0d0 * obj%CO2 + 8.0d0 * obj%Lambda ) - obj%R_d
if(W_j >= W_c )then
A = W_c
else
A = W_j
endif
! 要素体積を求める, m^3
obj%A(element_id) = A
volume = obj%femdomain%getVolume(elem=element_id)
!CO2固定量 mincro-mol/m-2/s
! ここ、体積あたりにする必要がある
! 一応、通常の葉の厚さを2mmとして、
! 1 micro-mol/m^2/sを、 1 micro-mol/ 0.002m^3/s= 500micro-mol/m^3/sとして計算
! また、ソース量はC6H12O6の質量gramとして換算する。
! CO2の分子量44.01g/mol
! C6H12O6の分子量180.16g/mol
! 6CO2 + 12H2O => C6H12O6 + 6H2O + 6O2
! よって、生成されるソース量は
! {CO2固定量,mol }× {1/6 してグルコースmol}×グルコース分子量
obj%source(i) =obj%source(i)+ A*dt/500.0d0*volume * 1.0d0/6.0d0 * 180.160d0
enddo
! ! For each elements, estimate photosynthesis by Farquhar model
! do i=1,size(obj%source)
!
! ! 光合成量の計算
! ! Farquhar model
! V_c = (V_cmax*CO2)/(CO2 + K_o * (1.0d0 + O2/K_o) )
! V_o = (V_omax*O2 )/(O2 + K_o * (1.0d0 + CO2/K_c) )
!
! Lambda = (V_omax*K_c*O2)/( 2.0d0 * V_cmax*K_o )
!
! W_c = (V_cmax*(CO2 - Lambda))/(CO2 + K_c*(1.0d0 + O2/K_o) )
!
! J_ = (phi*I_ + J_max - &
! sqrt( (phi*I_ + J_max)**(2.0d0) - 4.0d0*phi*I_*theta_r*J_max)&
! /(2.0d0 * theta_r) )
! W_j = J_ * (CO2 - Lambda)/(4.0d0 * CO2 + 8.0d0 * Lambda ) - R_d
! ! CO2吸収速度
! A = V_c + 0.50d0*V_o - R_d
!
! if(W_j >= W_c )then
! A = W_c
! else
! A = W_j
! endif
!
!
! enddo
!
!
end subroutine
subroutine adjustLeaf(obj,width)
class(Leaf_),intent(inout) :: obj
real(real64),intent(in) :: width(:,:)
end subroutine
end module
|
{-# OPTIONS --cubical --safe --postfix-projections #-}
module Data.Binary.Skew where
open import Prelude
open import Data.Nat
open import Data.List
𝔹 : Type
𝔹 = List ℕ
inc : 𝔹 → 𝔹
inc [] = zero ∷ []
inc (x ∷ []) = zero ∷ x ∷ []
inc (x₁ ∷ zero ∷ xs) = suc x₁ ∷ xs
inc (x₁ ∷ suc x₂ ∷ xs) = zero ∷ x₁ ∷ x₂ ∷ xs
⟦_⇑⟧ : ℕ → 𝔹
⟦ zero ⇑⟧ = []
⟦ suc n ⇑⟧ = inc ⟦ n ⇑⟧
skew : ℕ → ℕ
skew n = suc (n + n)
w : ℕ → ℕ → ℕ
w zero a = a
w (suc n) a = skew (w n a)
⟦_∷_⇓⟧^ : ℕ → (ℕ → ℕ) → ℕ → ℕ
⟦ x ∷ xs ⇓⟧^ a = let a′ = w x a in a′ + xs (skew a′)
⟦_⇓⟧ : 𝔹 → ℕ
⟦ [] ⇓⟧ = zero
⟦ x ∷ xs ⇓⟧ = let a = w x 1 in a + foldr ⟦_∷_⇓⟧^ (const zero) xs a
-- open import Path.Reasoning
-- import Data.Nat.Properties as ℕ
-- inc-suc : ∀ x → ⟦ inc x ⇓⟧ ≡ suc ⟦ x ⇓⟧
-- inc-suc [] = refl
-- inc-suc (x ∷ []) = refl
-- inc-suc (x ∷ zero ∷ xs) = cong suc (ℕ.+-assoc (w x 1) (w x 1) _)
-- inc-suc (x₁ ∷ suc x₂ ∷ xs) = cong suc (cong (w x₁ 1 +_) {!!})
-- 𝔹-rightInv : ∀ x → ⟦ ⟦ x ⇑⟧ ⇓⟧ ≡ x
-- 𝔹-rightInv zero = refl
-- 𝔹-rightInv (suc x) = {!!}
-- 𝔹-leftInv : ∀ x → ⟦ ⟦ x ⇓⟧ ⇑⟧ ≡ x
-- 𝔹-leftInv [] = refl
-- 𝔹-leftInv (x ∷ xs) = {!!}
-- 𝔹⇔ℕ : 𝔹 ⇔ ℕ
-- 𝔹⇔ℕ .fun = ⟦_⇓⟧
-- 𝔹⇔ℕ .inv = ⟦_⇑⟧
-- 𝔹⇔ℕ .rightInv x = {!!}
-- 𝔹⇔ℕ .leftInv = {!!}
|
The circlepath function is defined as $z + r \exp(2 \pi i x)$.
|
module Text.WebIDL.Types.Definition
import Generics.Derive
import Text.WebIDL.Types.Attribute
import Text.WebIDL.Types.Argument
import Text.WebIDL.Types.Identifier
import Text.WebIDL.Types.Member
import Text.WebIDL.Types.StringLit
import Text.WebIDL.Types.Type
%hide Language.Reflection.TT.Namespace
%language ElabReflection
||| CallbackRest ::
||| identifier = Type ( ArgumentList ) ;
public export
record Callback where
constructor MkCallback
attributes : ExtAttributeList
name : Identifier
type : IdlType
args : ArgumentList
%runElab derive "Callback" [Generic,Meta,Eq,Show,HasAttributes]
||| CallbackRestOrInterface ::
||| CallbackRest
||| interface identifier { CallbackInterfaceMembers } ;
public export
record CallbackInterface where
constructor MkCallbackInterface
attributes : ExtAttributeList
name : Identifier
members : CallbackInterfaceMembers
%runElab derive "CallbackInterface" [Generic,Meta,Eq,Show,HasAttributes]
||| Dictionary ::
||| dictionary identifier Inheritance { DictionaryMembers } ;
public export
record Dictionary where
constructor MkDictionary
attributes : ExtAttributeList
name : Identifier
inherits : Inheritance
members : DictionaryMembers
%runElab derive "Dictionary" [Generic,Meta,Eq,Show,HasAttributes]
||| Enum ::
||| enum identifier { EnumValueList } ;
|||
||| EnumValueList ::
||| string EnumValueListComma
|||
||| EnumValueListComma ::
||| , EnumValueListString
||| ε
|||
||| EnumValueListString ::
||| string EnumValueListComma
||| ε
public export
record Enum where
constructor MkEnum
attributes : ExtAttributeList
name : Identifier
values : List1 StringLit
%runElab derive "Enum" [Generic,Meta,Eq,Show,HasAttributes]
||| IncludesStatement ::
||| identifier includes identifier ;
public export
record Includes where
constructor MkIncludes
attributes : ExtAttributeList
name : Identifier
includes : Identifier
%runElab derive "Includes" [Generic,Meta,Eq,Show,HasAttributes]
||| InterfaceRest ::
||| identifier Inheritance { InterfaceMembers } ;
public export
record Interface where
constructor MkInterface
attributes : ExtAttributeList
name : Identifier
inherits : Inheritance
members : InterfaceMembers
%runElab derive "Interface" [Generic,Meta,Eq,Show,HasAttributes]
||| MixinRest ::
||| mixin identifier { MixinMembers } ;
public export
record Mixin where
constructor MkMixin
attributes : ExtAttributeList
name : Identifier
members : MixinMembers
%runElab derive "Mixin" [Generic,Meta,Eq,Show,HasAttributes]
||| Namespace ::
||| namespace identifier { NamespaceMembers } ;
public export
record Namespace where
constructor MkNamespace
attributes : ExtAttributeList
name : Identifier
members : NamespaceMembers
%runElab derive "Namespace" [Generic,Meta,Eq,Show,HasAttributes]
||| Typedef ::
||| typedef TypeWithExtendedAttributes identifier ;
public export
record Typedef where
constructor MkTypedef
attributes : ExtAttributeList
typeAttributes : ExtAttributeList
type : IdlType
name : Identifier
%runElab derive "Typedef" [Generic,Meta,Eq,Show,HasAttributes]
||| PartialDictionary ::
||| dictionary identifier { DictionaryMembers } ;
public export
record PDictionary where
constructor MkPDictionary
attributes : ExtAttributeList
name : Identifier
members : DictionaryMembers
%runElab derive "PDictionary" [Generic,Meta,Eq,Show,HasAttributes]
||| PartialInterfaceRest ::
||| identifier { PartialInterfaceMembers } ;
public export
record PInterface where
constructor MkPInterface
attributes : ExtAttributeList
name : Identifier
members : PartialInterfaceMembers
%runElab derive "PInterface" [Generic,Meta,Eq,Show,HasAttributes]
||| MixinRest ::
||| mixin identifier { MixinMembers } ;
public export
record PMixin where
constructor MkPMixin
attributes : ExtAttributeList
name : Identifier
members : MixinMembers
%runElab derive "PMixin" [Generic,Meta,Eq,Show,HasAttributes]
||| Namespace ::
||| namespace identifier { NamespaceMembers } ;
public export
record PNamespace where
constructor MkPNamespace
attributes : ExtAttributeList
name : Identifier
members : NamespaceMembers
%runElab derive "PNamespace" [Generic,Meta,Eq,Show,HasAttributes]
public export
DefTypes : List Type
DefTypes = [ Callback
, CallbackInterface
, Dictionary
, Enum
, Includes
, Interface
, Mixin
, Namespace
, Typedef
]
public export
PartTypes : List Type
PartTypes = [PDictionary, PInterface, PMixin, PNamespace]
||| Definition ::
||| CallbackOrInterfaceOrMixin
||| Namespace
||| Partial
||| Dictionary
||| Enum
||| Typedef
||| IncludesStatement
||| CallbackOrInterfaceOrMixin ::
||| callback CallbackRestOrInterface
||| interface InterfaceOrMixin
|||
||| InterfaceOrMixin ::
||| InterfaceRest
||| MixinRest
public export
Definition : Type
Definition = NS I DefTypes
public export
0 Definitions : Type
Definitions = NP List DefTypes
||| PartialDefinition ::
||| interface PartialInterfaceOrPartialMixin
||| PartialDictionary
||| Namespace
|||
||| PartialInterfaceOrPartialMixin ::
||| PartialInterfaceRest
||| MixinRest
public export
Part : Type
Part = NS I PartTypes
public export
accumNs : {ts : _} -> List (NS I ts) -> NP List ts
accumNs = foldl (\np,ns => hliftA2 (++) (toNP ns) np) hempty
public export
0 PartOrDef : Type
PartOrDef = NS I [Part,Definition]
public export
0 PartsAndDefs : Type
PartsAndDefs = NP List [Part,Definition]
public export
defs : PartsAndDefs -> Definitions
defs = accumNs . get Definition
--------------------------------------------------------------------------------
-- Domain
--------------------------------------------------------------------------------
update : Eq k => (b -> b) -> k -> (b -> k) -> List b -> List b
update f k bk = map (\b => if bk b == k then f b else b)
mergeDict : PDictionary -> Dictionary -> Dictionary
mergeDict d = record { members $= (++ d.members) }
mergeIface : PInterface -> Interface -> Interface
mergeIface i = record { members $= (++ map to i.members) }
where to : (a,b) -> (a, NS I [c,b])
to (x, y) = (x, inject y)
mergeMixin : PMixin -> Mixin -> Mixin
mergeMixin m = record { members $= (++ m.members) }
mergeNamespace : PNamespace -> Namespace -> Namespace
mergeNamespace n = record { members $= (++ n.members) }
public export
record Domain where
constructor MkDomain
domain : String
callbacks : List Callback
callbackInterfaces : List CallbackInterface
dictionaries : List Dictionary
enums : List Enum
includeStatements : List Includes
interfaces : List Interface
mixins : List Mixin
namespaces : List Namespace
typedefs : List Typedef
%runElab derive "Domain" [Generic,Meta,Eq,Show,HasAttributes]
applyPart : Domain -> Part -> Domain
applyPart d (Z v) =
record { dictionaries $= update (mergeDict v) v.name name } d
applyPart d (S $ Z v) =
record { interfaces $= update (mergeIface v) v.name name } d
applyPart d (S $ S $ Z v) =
record { mixins $= update (mergeMixin v) v.name name } d
applyPart d (S $ S $ S $ Z v) =
record { namespaces $= update (mergeNamespace v) v.name name } d
export
toDomains : List (String,PartsAndDefs) -> List Domain
toDomains ps =
let defs = map (\(s,pad) => fromNP s (defs pad)) ps
prts = concatMap (\(_,pad) => get Part pad) ps
in map (\d => foldl applyPart d prts) defs
where fromNP : String -> Definitions -> Domain
fromNP s [c,ci,d,e,ic,it,m,n,t] = MkDomain s c ci d e ic it m n t
|
{-# OPTIONS --without-K --safe #-}
open import Categories.Category
open import Categories.Category.Monoidal
module Categories.Category.Monoidal.Symmetric {o ℓ e} {C : Category o ℓ e} (M : Monoidal C) where
open import Level
open import Data.Product using (Σ; _,_)
open import Categories.Functor.Bifunctor
open import Categories.NaturalTransformation.NaturalIsomorphism
open import Categories.Morphism C
open import Categories.Category.Monoidal.Braided M
open Category C
open Commutation
private
variable
X Y Z : Obj
-- symmetric monoidal category
-- commutative braided monoidal category
record Symmetric : Set (levelOfTerm M) where
field
braided : Braided
module braided = Braided braided
open braided public
private
B : ∀ {X Y} → X ⊗₀ Y ⇒ Y ⊗₀ X
B {X} {Y} = braiding.⇒.η (X , Y)
field
commutative : B {X} {Y} ∘ B {Y} {X} ≈ id
braided-iso : X ⊗₀ Y ≅ Y ⊗₀ X
braided-iso = record
{ from = B
; to = B
; iso = record
{ isoˡ = commutative
; isoʳ = commutative
}
}
|
-- Andreas, 2013-10-21
-- Test case for CheckInternal extracted from The Agda standard library
-- Propositional (intensional) equality
module FunExt where
open import Common.Level
open import Common.Equality
Extensionality : (a b : Level) → Set _
Extensionality a b =
{A : Set a} {B : A → Set b} {f g : (x : A) → B x} →
(∀ x → f x ≡ g x) → f ≡ g
-- Functional extensionality implies a form of extensionality for
-- Π-types.
∀-extensionality :
∀ {a b} →
Extensionality a (lsuc b) →
{A : Set a} (B₁ B₂ : A → Set b) →
(∀ x → B₁ x ≡ B₂ x) → (∀ x → B₁ x) ≡ (∀ x → B₂ x)
∀-extensionality ext B₁ B₂ B₁≡B₂ with ext B₁≡B₂
∀-extensionality ext B .B B₁≡B₂ | refl = refl
|
#include "gateserver.h"
#include <boost/property_tree/ptree.hpp>
#include <boost/property_tree/xml_parser.hpp>
#include "mylogger.h"
#include "protos/login.pb.h"
#include "protos/server.pb.h"
#include "net_asio/timer.h"
#include "url.h"
static LoggerPtr logger(Logger::getLogger("gate"));
void onLogin(Cmd::Login::LoginRequest* msg, NetAsio::ConnectionPtr conn)
{
LOG4CXX_INFO(logger, "[" << boost::this_thread::get_id() << "] " << msg->GetTypeName() << ":\n" << msg->DebugString());
Cmd::Login::LoginReply reply;
reply.set_retcode(Cmd::Login::LoginReply::OK);
conn->send(reply);
GateServer::instance()->syncInfo2Login();
}
GateServer::GateServer() : NetApp("gateserver", user_msg_dispatcher_, server_msg_dispatcher_)
{
user_msg_dispatcher_.bind<Cmd::Login::LoginRequest, NetAsio::ConnectionPtr>(onLogin);
}
struct SyncInfo2LoginTimer : public NetAsio::Timer
{
SyncInfo2LoginTimer(boost::asio::io_service& ios)
: NetAsio::Timer(ios, 3000) {}
virtual void run()
{
GateServer::instance()->syncInfo2Login();
}
};
bool GateServer::init()
{
if (NetApp::init())
{
// TODO:临时为之
SyncInfo2LoginTimer* timer = new SyncInfo2LoginTimer(worker_pool_->get_io_service());
if (timer)
{
}
return true;
}
return false;
}
void GateServer::fini()
{
NetApp::fini();
}
void GateServer::syncInfo2Login()
{
LOG4CXX_INFO(logger, "同步状态到登录服务器");
// 发送到WorldManager
URL listen;
if (listen.parse(config_.get<std::string>("listen")))
{
Cmd::Server::SyncGateUserCount sync;
sync.set_gate_ip(listen.host);
sync.set_gate_port(listen.port);
sync.set_usercount(acceptor()->size());
connector()->sendTo(1, sync);
}
}
|
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-- Andreas, 2017-09-03, issue #2729.
-- Expect non-indexed or -primed variables when splitting.
-- {-# OPTIONS -v interaction.case:100 #-}
-- {-# OPTIONS -v tc.cover:40 #-}
data Size : Set where
↑ : Size → Size
data Nat : Size → Set where
zero : ∀ i → Nat (↑ i)
suc : ∀ i → Nat i → Nat (↑ i)
pred : ∀ i → Nat i → Nat i
pred i x = {!x!} -- C-c C-c
-- WRONG (agda-2.5.3):
-- pred .(↑ i₁) (zero i₁) = ?
-- pred .(↑ i₁) (suc i₁ x) = ?
-- EXPECTED (correct in agda-2.5.1.1):
-- pred .(↑ i) (zero i) = ?
-- pred .(↑ i) (suc i x) = ?
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{- Formal verification of authenticated append-only skiplists in Agda, version 1.0.
Copyright (c) 2020 Oracle and/or its affiliates.
Licensed under the Universal Permissive License v 1.0 as shown at https://opensource.oracle.com/licenses/upl
-}
open import Data.Empty
open import Data.Fin.Properties using (toℕ<n; toℕ-injective)
open import Data.Product
open import Data.Sum
open import Data.Nat
open import Data.Nat.Divisibility
open import Data.Nat.Properties
open import Data.Nat.Induction
open import Data.List renaming (map to List-map)
open import Data.List.Relation.Unary.Any
open import Data.List.Relation.Unary.All
import Relation.Binary.PropositionalEquality as Eq
open import Relation.Binary.Definitions
open Eq using (_≡_; refl; trans; sym; cong; cong-app; subst)
open Eq.≡-Reasoning using (begin_; _≡⟨⟩_; _∎)
open import Relation.Binary.PropositionalEquality renaming ( [_] to Reveal[_])
open import Relation.Binary.PropositionalEquality
open import Relation.Binary.HeterogeneousEquality
using (_≅_; ≅-to-≡; ≡-to-≅; _≇_)
renaming (cong to ≅-cong; refl to ≅-refl; cong₂ to ≅-cong₂)
open import Relation.Nullary
open import Relation.Binary.Core
open import Relation.Nullary.Negation using (contradiction; contraposition)
import Relation.Nullary using (¬_)
open import Function
-- This module defines the hop relation used by the original AAOSL due to Maniatis
-- and Baker, and proves various properties needed to establish it as a valid
-- DepRel, so that we can instantiate the asbtract model with it to demonstrate that
-- it is an instance of the class of AAOSLs for which we prove our properties.
module AAOSL.Hops where
open import AAOSL.Lemmas
open import Data.Nat.Even
-- The level of an index is 0 for index 0,
-- otherwise, it is one plus the number of times
-- that two divides said index.
--
-- lvlOf must be marked terminating because in one branch
-- we make recursive call on the quotient of the argument, which
-- is not obviously smaller than that argument
-- This is justified by proving that lvlOf is equal to lvlOfWF,
-- which uses well-founded recursion
{-# TERMINATING #-}
lvlOf : ℕ → ℕ
lvlOf 0 = 0
lvlOf (suc n) with even? (suc n)
...| no _ = 1
...| yes e = suc (lvlOf (quotient e))
-- level of an index with well-founded recursion
lvlOfWFHelp : (n : ℕ) → Acc _<_ n → ℕ
lvlOfWFHelp 0 p = 0
lvlOfWFHelp (suc n) (acc rs) with even? (suc n)
... | no _ = 1
... | yes (divides q eq) = suc (lvlOfWFHelp q (rs q (1+n=m*2⇒m<1+n q n eq)))
lvlOfWF : ℕ → ℕ
lvlOfWF n = lvlOfWFHelp n (<-wellFounded n)
-- When looking at an index in the form 2^k * d, the level of
-- said index is more easily defined.
lvlOf' : ∀{n} → Pow2 n → ℕ
lvlOf' zero = zero
lvlOf' (pos l _ _ _) = suc l
-------------------------------------------
-- Properties of lvlOf, lvlOfWF, and lvlOf'
lvlOf≡lvlOfWFHelp : (n : ℕ) (p : Acc _<_ n) → lvlOf n ≡ lvlOfWFHelp n p
lvlOf≡lvlOfWFHelp 0 p = refl
lvlOf≡lvlOfWFHelp (suc n) (acc rs) with even? (suc n)
... | no _ = refl
... | yes (divides q eq) =
cong suc (lvlOf≡lvlOfWFHelp q (rs q (1+n=m*2⇒m<1+n q n eq)))
lvlOf≡lvlOfWF : (n : ℕ) → lvlOf n ≡ lvlOfWF n
lvlOf≡lvlOfWF n = lvlOf≡lvlOfWFHelp n (<-wellFounded n)
lvlOf≡lvlOf' : ∀ n → lvlOf n ≡ lvlOf' (to n)
lvlOf≡lvlOf' n rewrite lvlOf≡lvlOfWF n = go n (<-wellFounded n)
where
go : (n : ℕ) (p : Acc _<_ n) → lvlOfWFHelp n p ≡ lvlOf' (to n)
go 0 p = refl
go (suc n) (acc rs) with even? (suc n)
... | no _ = refl
... | yes (divides q eq) with go q (rs q (1+n=m*2⇒m<1+n q n eq))
... | ih with to q
... | pos l d odd prf = cong suc ih
lvl≥2-even : ∀ {n} → 2 ≤ lvlOf n → Even n
lvl≥2-even {suc n} x
with 2 ∣? (suc n)
...| yes prf = prf
...| no prf = ⊥-elim ((≤⇒≯ x) (s≤s (s≤s z≤n)))
lvlOfodd≡1 : ∀ n → Odd n → lvlOf n ≡ 1
lvlOfodd≡1 0 nodd = ⊥-elim (nodd (divides zero refl))
lvlOfodd≡1 (suc n) nodd
with even? (suc n)
...| yes prf = ⊥-elim (nodd prf)
...| no prf = refl
-- We eventually need to 'undo' a level
lvlOf-undo : ∀{j}(e : Even (suc j)) → suc (lvlOf (quotient e)) ≡ lvlOf (suc j)
lvlOf-undo {j} e with even? (suc j)
...| no abs = ⊥-elim (abs e)
...| yes prf rewrite even-irrelevant e prf = refl
∣-cmp : ∀{t u n} → (suc t * u) ∣ n → (d : suc t ∣ n) → u ∣ (quotient d)
∣-cmp {t} {u} {n} (divides q1 e1) (divides q2 e2)
rewrite sym (*-assoc q1 (suc t) u)
| *-comm q1 (suc t)
| *-comm q2 (suc t)
| *-assoc (suc t) q1 u
= divides q1 (*-cancelˡ-≡ t (trans (sym e2) e1))
∣-0< : ∀{n t} → 0 < n → (d : suc t ∣ n) → 0 < quotient d
∣-0< hip (divides zero e) = ⊥-elim (<⇒≢ hip (sym e))
∣-0< hip (divides (suc q) e) = s≤s z≤n
lvlOf-mono : ∀{n} k → 0 < n → 2 ^ k ∣ n → k ≤ lvlOf n
lvlOf-mono zero hip prf = z≤n
lvlOf-mono {suc n} (suc k) hip prf
with even? (suc n)
...| no abs = ⊥-elim (abs (divides (quotient prf * (2 ^ k))
(trans (_∣_.equality prf)
(trans (cong ((quotient prf) *_) (sym (*-comm (2 ^ k) 2)))
(sym (*-assoc (quotient prf) (2 ^ k) 2))))))
...| yes prf' = s≤s (lvlOf-mono {quotient prf'} k (∣-0< hip prf') (∣-cmp prf prf'))
-- This property can be strenghtened to < if we ever need.
lvlOf'-mono : ∀{k} d → 0 < d → k ≤ lvlOf' (to (2 ^ k * d))
lvlOf'-mono {k} d 0<d
with to d
...| pos {d} kk dd odd eq
with (2 ^ (k + kk)) * dd ≟ (2 ^ k) * d
...| no xx = ⊥-elim (xx ( trans (cong (_* dd) (^-distribˡ-+-* 2 k kk))
(trans (*-assoc (2 ^ k) (2 ^ kk) dd)
(cong (λ x → (2 ^ k) * x) (sym eq)))))
...| yes xx
with to-reduce {(2 ^ k) * d} {k + kk} {dd} (sym xx) odd
...| xx1 = ≤-trans (≮⇒≥ (m+n≮m k kk))
(≤-trans (n≤1+n (k + kk)) -- TODO-1: easy to strengthen to <; omit this step
(≤-reflexive (sym (cong lvlOf' xx1))))
-- And a progress property about levels:
-- These will be much easier to reason about in terms of lvlOf'
-- as we can see in lvlOf-correct.
lvlOf-correct : ∀{l j} → l < lvlOf j → 2 ^ l ≤ j
lvlOf-correct {l} {j} prf
rewrite lvlOf≡lvlOf' j
with to j
...| zero = ⊥-elim (1+n≢0 (n≤0⇒n≡0 prf))
...| pos l' d odd refl = 2^kd-mono (≤-unstep2 prf) (0<odd odd)
-- lvlOf-prog states that if we have not reached 0, we landed somewhere
-- where we can hop again at the same level.
lvlOf-prog : ∀{l j} → 0 < j ∸ 2 ^ l → l < lvlOf j → l < lvlOf (j ∸ 2 ^ l)
lvlOf-prog {l} {j} hip l<lvl
rewrite lvlOf≡lvlOf' j | lvlOf≡lvlOf' (j ∸ 2 ^ l)
with to j
...| zero = ⊥-elim (1+n≰n (≤-trans l<lvl z≤n))
...| pos l₁ d₁ o₁ refl
rewrite 2^ld-2l l₁ l d₁ (≤-unstep2 l<lvl)
with l ≟ l₁
...| no l≢l₁ rewrite to-2^kd l (odd-2^kd-1 (l₁ ∸ l) d₁
(0<m-n (≤∧≢⇒< (≤-unstep2 l<lvl) l≢l₁))
(0<odd o₁))
= ≤-refl
...| yes refl rewrite n∸n≡0 l₁ | +-comm d₁ 0
with odd∸1-even o₁
...| divides q prf rewrite prf
| sym (*-assoc (2 ^ l₁) q 2)
| a*b*2-lemma (2 ^ l₁) q
= lvlOf'-mono {suc l₁} q (1≤m*n⇒0<n {m = 2 ^ suc l₁} hip)
lvlOf-no-overshoot : ∀ j l → suc l < lvlOf j → 0 < j ∸ 2 ^ l
lvlOf-no-overshoot j l hip
rewrite lvlOf≡lvlOf' j with to j
...| zero = ⊥-elim (1+n≰n (≤-trans (s≤s z≤n) hip))
...| pos k d o refl = 0<m-n {2 ^ k * d} {2 ^ l}
(<-≤-trans (2^-mono (≤-unstep2 hip))
(2^kd-mono {k} {k} ≤-refl (0<odd o)))
---------------------------
-- The AAOSL Structure --
---------------------------
-------------------------------
-- Hops
-- Encoding our hops into a relation. A value of type 'H l j i'
-- witnesses the existence of a hop from j to i at level l.
data H : ℕ → ℕ → ℕ → Set where
hz : ∀ x → H 0 (suc x) x
hs : ∀ {l x y z}
→ H l x y
→ H l y z
→ suc l < lvlOf x
→ H (suc l) x z
-----------------------------
-- Hop's universal properties
-- The universal property comes for free
h-univ : ∀{l j i} → H l j i → i < j
h-univ (hz x) = s≤s ≤-refl
h-univ (hs h h₁ _) = <-trans (h-univ h₁) (h-univ h)
-- It is easy to prove there are no hops from zero
h-from0-⊥ : ∀{l i} → H l 0 i → ⊥
h-from0-⊥ (hs h h₁ _) = h-from0-⊥ h
-- And it is easy to prove that i is a distance of 2 ^ l away
-- from j.
h-univ₂ : ∀{l i j} → H l j i → i ≡ j ∸ 2 ^ l
h-univ₂ (hz x) = refl
h-univ₂ (hs {l = l} {j} h₀ h₁ _)
rewrite h-univ₂ h₀
| h-univ₂ h₁
| +-comm (2 ^ l) 0
| sym (∸-+-assoc j (2 ^ l) (2 ^ l))
= refl
-- and vice versa.
h-univ₁ : ∀{l i j} → H l j i → j ≡ i + 2 ^ l
h-univ₁ (hz x) = sym (+-comm x 1)
h-univ₁ (hs {l = l} {z = i} h₀ h₁ _)
rewrite h-univ₁ h₀
| h-univ₁ h₁
| +-comm (2 ^ l) 0
= +-assoc i (2 ^ l) (2 ^ l)
--------------
-- H and lvlOf
-- A value of type H says something about the levels of their indices
h-lvl-src : ∀{l j i} → H l j i → l < lvlOf j
h-lvl-src (hz x) with even? (suc x)
...| no _ = s≤s z≤n
...| yes _ = s≤s z≤n
h-lvl-src (hs h₀ h₁ prf) = prf
h-lvl-tgt : ∀{l j i} → 0 < i → H l j i → l < lvlOf i
h-lvl-tgt prf h rewrite h-univ₂ h = lvlOf-prog prf (h-lvl-src h)
h-lvl-inj : ∀{l₁ l₂ j i} (h₁ : H l₁ j i)(h₂ : H l₂ j i) → l₁ ≡ l₂
h-lvl-inj {i = i} h₁ h₂
= 2^-injective (+-cancelˡ-≡ i (trans (sym (h-univ₁ h₁)) (h-univ₁ h₂)))
-- TODO-1: document reasons for this pragma and justify it
{-# TERMINATING #-}
h-lvl-half : ∀{l j i y l₁} → H l j y → H l y i → H l₁ j i → lvlOf y ≡ suc l
h-lvl-half w₀ w₁ (hz n) = ⊥-elim (1+n≰n (≤-<-trans (h-univ w₁) (h-univ w₀)))
h-lvl-half {l}{j}{i}{y} w₀ w₁ (hs {l = l₁} {y = y₁} sh₀ sh₁ x)
-- TODO-2: factor out a lemma to prove l₁ ≡ l and y₁ ≡ y (already exists?)
with l₁ ≟ l
...| no imp
with j ≟ i + (2 ^ l₁) + (2 ^ l₁) | j ≟ i + (2 ^ l) + (2 ^ l)
...| no imp1 | _ rewrite h-univ₁ sh₁ = ⊥-elim (imp1 (h-univ₁ sh₀))
...| yes _ | no imp1 rewrite h-univ₁ w₁ = ⊥-elim (imp1 (h-univ₁ w₀))
...| yes j₁ | yes j₂
with trans (sym j₂) j₁
...| xx5 rewrite +-assoc i (2 ^ l) (2 ^ l)
| +-assoc i (2 ^ l₁) (2 ^ l₁)
with +-cancelˡ-≡ i xx5
...| xx6 rewrite sym (+-identityʳ (2 ^ l))
| sym (+-identityʳ (2 ^ l₁))
| +-assoc (2 ^ l) 0 ((2 ^ l) + 0)
| +-assoc (2 ^ l₁) 0 ((2 ^ l₁) + 0)
| *-comm 2 (2 ^ l)
| *-comm 2 (2 ^ l₁)
= ⊥-elim (imp (sym (2^-injective {l} {l₁} (
sym (*2-injective (2 ^ l) (2 ^ l₁) xx6)))))
h-lvl-half {l = l}{j = j}{i = i}{y = y} w₀ w₁ (hs {l = l₁} {y = y₁} sh₀ sh₁ x)
| yes xx1 rewrite xx1
with y₁ ≟ y
...| no imp = ⊥-elim (imp (+-cancelʳ-≡ y₁ y (trans (sym (h-univ₁ sh₀)) (h-univ₁ w₀))))
...| yes y₁≡y rewrite y₁≡y
with w₀
...| hs {l = l-1} ssh₀ ssh₁ xx rewrite sym xx1
= h-lvl-half sh₀ sh₁ (hs sh₀ sh₁ x)
...| hz y = lvlOfodd≡1 y (even-suc-odd y (lvl≥2-even {suc y} x))
-- If a hop goes over an index, then the level of this index is strictly
-- less than the level of the hop. The '≤' is there because
-- l starts at zero.
--
-- For example, lvlOf 4 ≡ 3; the only hops that can go over 4 are
-- those with l of 3 or higher. In fact, there is one at l ≡ 2
-- from 4 to 0: H 2 4 0
h-lvl-mid : ∀{l j i} → (k : ℕ) → H l j i → i < k → k < j → lvlOf k ≤ l
h-lvl-mid k (hz x) i<k k<j = ⊥-elim (n≮n k (<-≤-trans k<j i<k))
h-lvl-mid {j = j} k (hs {l = l₀}{y = y} w₀ w₁ x) i<k k<j
with <-cmp k y
...| tri< k<y k≢y k≯y = ≤-step (h-lvl-mid k w₁ i<k k<y)
...| tri> k≮y k≢y k>y = ≤-step (h-lvl-mid k w₀ k>y k<j)
...| tri≈ k≮y k≡y k≯y rewrite k≡y = ≤-reflexive (h-lvl-half w₀ w₁ (hs {l = l₀}{y = y} w₀ w₁ x))
h-lvl-≤₁ : ∀{l₁ l₂ j i₁ i₂}
→ (h : H l₁ j i₁)(v : H l₂ j i₂)
→ i₂ < i₁
→ l₁ < l₂
h-lvl-≤₁ {l₁} {l₂} {j} {i₁} {i₂} h v i₂<i₁ =
let h-univ = h-univ₁ h
v-univ = h-univ₁ v
eqj = trans (sym v-univ) h-univ
in log-mono l₁ l₂ (n+p≡m+q∧n<m⇒q<p i₂<i₁ eqj)
h-lvl-≤₂ : ∀{l₁ l₂ j₁ j₂ i}
→ (h : H l₁ j₁ i)(v : H l₂ j₂ i)
→ j₁ < j₂
→ l₁ < l₂
h-lvl-≤₂ {l₁} {l₂} {j₁} {j₂} {i} h v j₂<j₁ =
let h-univ = h-univ₁ h
v-univ = h-univ₁ v
in log-mono l₁ l₂ (+-cancelˡ-< i (subst (i + (2 ^ l₁) <_) v-univ (subst (_< j₂) h-univ j₂<j₁)))
------------------------------
-- Correctness and Irrelevance
h-correct : ∀ j l → l < lvlOf j → H l j (j ∸ 2 ^ l)
h-correct (suc j) zero prf = hz j
h-correct (suc j) (suc l) prf
with h-correct (suc j) l
...| ind with 2 ∣? (suc j)
...| no _ = ⊥-elim (ss≰1 prf)
...| yes e with ind (≤-unstep prf)
...| res₀
with h-correct (suc j ∸ 2 ^ l) l
(lvlOf-prog {l} {suc j}
(lvlOf-no-overshoot (suc j) l (subst (suc l <_ ) (lvlOf-undo e) prf))
(subst (l <_) (lvlOf-undo e) (≤-unstep prf)))
...| res₁
rewrite +-comm (2 ^ l) 0
| ∸-+-assoc (suc j) (2 ^ l) (2 ^ l)
= hs res₀ res₁ (subst (suc l <_) (lvlOf-undo e) prf)
h-irrelevant : ∀{l i j}
→ (h₁ : H l j i)
→ (h₂ : H l j i)
→ h₁ ≡ h₂
h-irrelevant (hz x) (hz .x) = refl
h-irrelevant (hs {y = y} h₁ h₃ x) (hs {y = z} h₂ h₄ x₁)
rewrite ≤-irrelevant x x₁
with y ≟ z
...| no abs = ⊥-elim (abs (trans (h-univ₂ h₁) (sym (h-univ₂ h₂))))
...| yes refl = cong₂ (λ P Q → hs P Q x₁) (h-irrelevant h₁ h₂) (h-irrelevant h₃ h₄)
-------------------------------------------------------------------
-- The non-overlapping property is stated in terms
-- of subhops. The idea is that a hop is either separate
-- from another one, or is entirely contained within the larger one.
--
-- Entirely contained comes from _⊆Hop_
data _⊆Hop_ : ∀{l₁ i₁ j₁ l₂ i₂ j₂}
→ H l₁ j₁ i₁
→ H l₂ j₂ i₂
→ Set where
here : ∀{l i j}(h : H l j i) → h ⊆Hop h
left : ∀{l₁ i₁ j₁ l₂ i₂ w j₂ }
→ (h : H l₁ j₁ i₁)
→ (w₀ : H l₂ j₂ w)
→ (w₁ : H l₂ w i₂)
→ (p : suc l₂ < lvlOf j₂)
→ h ⊆Hop w₀
→ h ⊆Hop (hs w₀ w₁ p)
right : ∀{l₁ i₁ j₁ l₂ i₂ w j₂}
→ (h : H l₁ j₁ i₁)
→ (w₀ : H l₂ j₂ w)
→ (w₁ : H l₂ w i₂)
→ (p : suc l₂ < lvlOf j₂)
→ h ⊆Hop w₁
→ h ⊆Hop (hs w₀ w₁ p)
⊆Hop-refl : ∀{l₁ l₂ j i}
→ (h₁ : H l₁ j i)
→ (h₂ : H l₂ j i)
→ h₁ ⊆Hop h₂
⊆Hop-refl h₁ h₂ with h-lvl-inj h₁ h₂
...| refl rewrite h-irrelevant h₁ h₂ = here h₂
⊆Hop-univ : ∀{l₁ i₁ j₁ l₂ i₂ j₂}
→ (h1 : H l₁ j₁ i₁)
→ (h2 : H l₂ j₂ i₂)
→ h1 ⊆Hop h2
→ i₂ ≤ i₁ × j₁ ≤ j₂ × l₁ ≤ l₂
⊆Hop-univ h1 .h1 (here .h1) = ≤-refl , ≤-refl , ≤-refl
⊆Hop-univ h1 (hs w₀ w₁ p) (left h1 w₀ w₁ q hip)
with ⊆Hop-univ h1 w₀ hip
...| a , b , c = (≤-trans (<⇒≤ (h-univ w₁)) a) , b , ≤-step c
⊆Hop-univ h1 (hs w₀ w₁ p) (right h1 w₀ w₁ q hip)
with ⊆Hop-univ h1 w₁ hip
...| a , b , c = a , ≤-trans b (<⇒≤ (h-univ w₀)) , ≤-step c
⊆Hop-univ₁ : ∀{l₁ i₁ j₁ l₂ i₂ j₂}
→ (h1 : H l₁ j₁ i₁)
→ (h2 : H l₂ j₂ i₂)
→ h1 ⊆Hop h2
→ i₂ ≤ i₁
⊆Hop-univ₁ h1 h2 h1h2 = proj₁ (⊆Hop-univ h1 h2 h1h2)
⊆Hop-src-≤ : ∀{l₁ i₁ j₁ l₂ i₂ j₂}
→ (h1 : H l₁ j₁ i₁)
→ (h2 : H l₂ j₂ i₂)
→ h1 ⊆Hop h2
→ j₁ ≤ j₂
⊆Hop-src-≤ h1 h2 h1h2 = (proj₁ ∘ proj₂) (⊆Hop-univ h1 h2 h1h2)
-- If two hops are not strictly the same, then the level of
-- the smaller hop is strictly smaller than the level of
-- the bigger hop.
--
-- VERY IMPORTANT
⊆Hop-univ-lvl : ∀{l₁ i₁ j₁ l₂ i₂ j₂}
→ (h₁ : H l₁ j₁ i₁)
→ (h₂ : H l₂ j₂ i₂)
→ h₁ ⊆Hop h₂
→ j₁ < j₂
→ l₁ < l₂
⊆Hop-univ-lvl {l₁}{i₁}{j₁}{l₂}{i₂}{j₂} h₁ h₂ h₁⊆Hoph₂ j₁<j₂
= let r₁ : i₂ + (2 ^ l₁) ≤ i₁ + (2 ^ l₁)
r₁ = +-monoˡ-≤ (2 ^ l₁) (proj₁ (⊆Hop-univ h₁ h₂ h₁⊆Hoph₂))
r₂ : i₁ + (2 ^ l₁) < i₂ + (2 ^ l₂)
r₂ = subst₂ _<_ (h-univ₁ h₁) (h-univ₁ h₂) j₁<j₂
in log-mono l₁ l₂ ((+-cancelˡ-< i₂) (≤-<-trans r₁ r₂))
hz-⊆ : ∀{l j i k}
→ (v : H l j i)
→ i ≤ k
→ k < j
→ hz k ⊆Hop v
hz-⊆ (hz x) i<k k<j
rewrite ≤-antisym (≤-unstep2 k<j) i<k = here (hz x)
hz-⊆ {k = k} (hs {y = y} v v₁ x) i<k k<j
with k <? y
...| yes k<y = right (hz k) v v₁ x (hz-⊆ v₁ i<k k<y)
...| no k≮y = left (hz k) v v₁ x (hz-⊆ v (≮⇒≥ k≮y) k<j)
⊆Hop-inj₁ : ∀{l₁ l₂ j i₁ i₂}
→ (h : H l₁ j i₁)(v : H l₂ j i₂)
→ i₂ < i₁
→ h ⊆Hop v
⊆Hop-inj₁ {i₁ = i₁} h (hz x) prf
= ⊥-elim (n≮n i₁ (<-≤-trans (h-univ h) prf))
⊆Hop-inj₁ {l} {j = j} {i₁ = i₁} h (hs {l = l₁} {y = y} v v₁ x) prf
with y ≟ i₁
...| yes refl = left h v v₁ x (⊆Hop-refl h v)
...| no y≢i₁
with h-lvl-≤₁ h (hs v v₁ x) prf
...| sl≤sl₁
with h-univ₂ h | h-univ₂ v
...| prf1 | prf2
= let r : j ∸ (2 ^ l₁) ≤ j ∸ (2 ^ l)
r = ∸-monoʳ-≤ {m = 2 ^ l} {2 ^ l₁} j (^-mono l l₁ (≤-unstep2 sl≤sl₁))
in left h v v₁ x (⊆Hop-inj₁ h v (≤∧≢⇒< (subst₂ _≤_ (sym prf2) (sym prf1) r) y≢i₁))
⊆Hop-inj₂ : ∀{l₁ l₂ j₁ j₂ i}
→ (h : H l₁ j₁ i)(v : H l₂ j₂ i)
→ j₁ < j₂
→ h ⊆Hop v
⊆Hop-inj₂ h (hz x) prf
= ⊥-elim (n≮n _ (<-≤-trans prf (h-univ h)))
⊆Hop-inj₂ {l} {j₁ = j₁} {i = i} h (hs {l = l₁} {y = y} v v₁ x) prf
with y ≟ j₁
...| yes refl = right h v v₁ x (⊆Hop-refl h v₁)
...| no y≢j₁
with h-lvl-≤₂ h (hs v v₁ x) prf
...| sl≤sl₁
with h-univ₁ h | h-univ₁ v₁
...| prf1 | prf2
= let r : i + 2 ^ l ≤ i + 2 ^ l₁
r = +-monoʳ-≤ i (^-mono l l₁ (≤-unstep2 sl≤sl₁))
in right h v v₁ x (⊆Hop-inj₂ h v₁ (≤∧≢⇒< (subst₂ _≤_ (sym prf1) (sym prf2) r) (y≢j₁ ∘ sym)))
⊆Hop-inj₃ : ∀{l₁ l₂ j₁ j₂ i₁ i₂}
→ (h : H l₁ j₁ i₁)(v : H l₂ j₂ i₂)
→ i₁ ≡ i₂ → j₁ ≡ j₂ → h ⊆Hop v
⊆Hop-inj₃ h v refl refl with h-lvl-inj h v
...| refl rewrite h-irrelevant h v = here v
-- This datatype encodes all the possible hop situations. This makes is
-- much easier to structure proofs talking about two hops.
data HopStatus : ∀{l₁ i₁ j₁ l₂ i₂ j₂}
→ H l₁ j₁ i₁
→ H l₂ j₂ i₂
→ Set where
-- Same hop; we carry the proofs explicitly here to be able to control
-- when to perform the rewrites.
Same : ∀{l₁ i₁ j₁ l₂ i₂ j₂}(h₁ : H l₁ j₁ i₁)(h₂ : H l₂ j₂ i₂)
→ i₁ ≡ i₂
→ j₁ ≡ j₂
→ HopStatus h₁ h₂
-- h₂ h₁
-- ⌜⁻⁻⁻⁻⁻⁻⁻⌝ ⌜⁻⁻⁻⁻⁻⁻⁻⌝
-- | | | |
-- i₂ < j₂ ≤ i₁ < j₁
SepL : ∀{l₁ i₁ j₁ l₂ i₂ j₂}(h₁ : H l₁ j₁ i₁)(h₂ : H l₂ j₂ i₂)
→ j₂ ≤ i₁
→ HopStatus h₁ h₂
-- h₁ h₂
-- ⌜⁻⁻⁻⁻⁻⁻⁻⌝ ⌜⁻⁻⁻⁻⁻⁻⁻⌝
-- | | | |
-- i₁ < j₁ ≤ i₂ < j₂
SepR : ∀{l₁ i₁ j₁ l₂ i₂ j₂}(h₁ : H l₁ j₁ i₁)(h₂ : H l₂ j₂ i₂)
→ j₁ ≤ i₂
→ HopStatus h₁ h₂
-- h₂
-- ⌜⁻⁻⁻⁻⁻⁻⁻⁻⁻⁻⁻⁻⁻⁻⁻⁻⁻⁻⁻⁻⁻⁻⌝
-- ∣ ∣
-- ∣ h₁ ∣
-- ∣ ⌜⁻⁻⁻⁻⁻⁻⁻⌝ |
-- | | | |
-- i₂ ≤ i₁ ⋯ j₁ ≤ j₂
SubL : ∀{l₁ i₁ j₁ l₂ i₂ j₂}(h₁ : H l₁ j₁ i₁)(h₂ : H l₂ j₂ i₂)
→ i₂ < i₁ ⊎ j₁ < j₂ -- makes sure hops differ!
→ h₁ ⊆Hop h₂
→ HopStatus h₁ h₂
-- h₁
-- ⌜⁻⁻⁻⁻⁻⁻⁻⁻⁻⁻⁻⁻⁻⁻⁻⁻⁻⁻⁻⁻⁻⁻⌝
-- ∣ ∣
-- ∣ h₂ ∣
-- ∣ ⌜⁻⁻⁻⁻⁻⁻⁻⌝ |
-- | | | |
-- i₁ ≤ i₂ ⋯ j₂ < j₁
SubR : ∀{l₁ i₁ j₁ l₂ i₂ j₂}(h₁ : H l₁ j₁ i₁)(h₂ : H l₂ j₂ i₂)
→ i₁ < i₂ ⊎ j₂ < j₁ -- makes sure hops differ
→ h₂ ⊆Hop h₁
→ HopStatus h₁ h₂
-- Finally, we can prove our no-overlap property. As it turns out, it is
-- just a special case of general non-overlapping, and therefore, it is
-- defined as such.
mutual
-- Distinguish is used to understand the relation between two arbitrary hops.
-- It is used to perform the induction step on arbitrary hops. Note how
-- 'no-overlap' has a clause that impedes the hops from being equal.
distinguish : ∀{l₁ i₁ j₁ l₂ i₂ j₂}
→ (h₁ : H l₁ j₁ i₁)
→ (h₂ : H l₂ j₂ i₂)
→ HopStatus h₁ h₂
distinguish {l₁} {i₁} {j₁} {l₂} {i₂} {j₂} h1 h2
with <-cmp i₁ i₂
...| tri≈ i₁≮i₂ i₁≡i₂ i₂≮i₁
with <-cmp j₁ j₂
...| tri≈ j₁≮j₂ j₁≡j₂ j₂≮j₁ = Same h1 h2 i₁≡i₂ j₁≡j₂
...| tri< j₁<j₂ j₁≢j₂ j₂≮j₁ rewrite i₁≡i₂ = SubL h1 h2 (inj₂ j₁<j₂) (⊆Hop-inj₂ h1 h2 j₁<j₂)
...| tri> j₁≮j₂ j₁≢j₂ j₂<j₁ rewrite i₁≡i₂ = SubR h1 h2 (inj₂ j₂<j₁) (⊆Hop-inj₂ h2 h1 j₂<j₁)
distinguish {l₁} {i₁} {j₁} {l₂} {i₂} {j₂} h1 h2
| tri< i₁<i₂ i₁≢i₂ i₂≮i₁
with <-cmp j₁ j₂
...| tri≈ j₁≮j₂ j₁≡j₂ j₂≮j₁ rewrite j₁≡j₂ = SubR h1 h2 (inj₁ i₁<i₂) (⊆Hop-inj₁ h2 h1 i₁<i₂)
...| tri< j₁<j₂ j₁≢j₂ j₂≮j₁ with no-overlap h2 h1 i₁<i₂
...| inj₁ a = SepR h1 h2 a
...| inj₂ b = SubR h1 h2 (inj₁ i₁<i₂) b
distinguish {l₁} {i₁} {j₁} {l₂} {i₂} {j₂} h1 h2
| tri< i₁<i₂ i₁≢i₂ i₂≮i₁
| tri> j₁≮j₂ j₁≢j₂ j₂<j₁ with no-overlap h2 h1 i₁<i₂
...| inj₁ a = SepR h1 h2 a
...| inj₂ b = SubR h1 h2 (inj₁ i₁<i₂) b
distinguish {l₁} {i₁} {j₁} {l₂} {i₂} {j₂} h1 h2
| tri> i₁≮i₂ i₁≢i₂ i₂<i₁
with <-cmp j₁ j₂
...| tri≈ j₁≮j₂ j₁≡j₂ j₂≮j₁ rewrite j₁≡j₂ = SubL h1 h2 (inj₁ i₂<i₁) (⊆Hop-inj₁ h1 h2 i₂<i₁)
...| tri< j₁<j₂ j₁≢j₂ j₂≮j₁ with no-overlap h1 h2 i₂<i₁
...| inj₁ a = SepL h1 h2 a
...| inj₂ b = SubL h1 h2 (inj₁ i₂<i₁) b
distinguish {l₁} {i₁} {j₁} {l₂} {i₂} {j₂} h1 h2
| tri> i₁≮i₂ i₁≢i₂ i₂<i₁
| tri> j₁≮j₂ j₁≢j₂ j₂<j₁ with no-overlap h1 h2 i₂<i₁
...| inj₁ a = SepL h1 h2 a
...| inj₂ b = SubL h1 h2 (inj₁ i₂<i₁) b
no-overlap-< : ∀{l₁ i₁ j₁ l₂ i₂ j₂}
→ (h₁ : H l₁ j₁ i₁)
→ (h₂ : H l₂ j₂ i₂)
→ i₂ < i₁
→ i₁ < j₂
→ j₁ ≤ j₂
no-overlap-< h₁ h₂ prf hip with no-overlap h₁ h₂ prf
...| inj₁ imp = ⊥-elim (1+n≰n (≤-trans hip imp))
...| inj₂ res = ⊆Hop-src-≤ h₁ h₂ res
-- TODO-1: rename to nocross for consistency with paper
-- Non-overlapping is more general, as hops might be completely
-- separate and then, naturally won't overlap.
no-overlap : ∀{l₁ i₁ j₁ l₂ i₂ j₂}
→ (h₁ : H l₁ j₁ i₁)
→ (h₂ : H l₂ j₂ i₂)
→ i₂ < i₁ -- this ensures h₁ ≢ h₂.
→ (j₂ ≤ i₁) ⊎ (h₁ ⊆Hop h₂)
no-overlap h (hz x) prf = inj₁ prf
no-overlap {l₁} {i₁} {j₁} {l₂} {i₂} {j₂} h₁ (hs {y = y} v₀ v₁ v-ok) hip
with distinguish h₁ v₀
...| SepL _ _ prf = inj₁ prf
...| SubL _ _ case prf = inj₂ (left h₁ v₀ v₁ v-ok prf)
...| Same _ _ p1 p2 = inj₂ (left h₁ v₀ v₁ v-ok (⊆Hop-inj₃ h₁ v₀ p1 p2))
no-overlap {l₁} {i₁} {j₁} {l₂} {i₂} {j₂} h₁ (hs {y = y} v₀ v₁ v-ok) hip
| SepR _ _ j₁≤y
with distinguish h₁ v₁
...| SepL _ _ prf = ⊥-elim (<⇒≱ (h-univ h₁) (≤-trans j₁≤y prf))
...| SepR _ _ prf = ⊥-elim (n≮n i₂ (<-trans hip (<-≤-trans (h-univ h₁) prf)))
...| SubR _ _ (inj₁ i₁<i₂) prf = ⊥-elim (n≮n i₂ (<-trans hip i₁<i₂))
...| SubR _ _ (inj₂ y<j₁) prf = ⊥-elim (n≮n j₁ (≤-<-trans j₁≤y y<j₁))
...| SubL _ _ case prf = inj₂ (right h₁ v₀ v₁ v-ok prf)
...| Same _ _ p1 p2 = inj₂ (right h₁ v₀ v₁ v-ok (⊆Hop-inj₃ h₁ v₁ p1 p2))
no-overlap {l₁} {i₁} {j₁} {l₂} {i₂} {j₂} h₁ (hs {y = y} v₀ v₁ v-ok) hip
| SubR _ _ (inj₁ i₁<y) v₀⊆h₁
with distinguish h₁ v₁
...| SepL _ _ prf = ⊥-elim (n≮n i₁ (<-≤-trans i₁<y prf))
...| SepR _ _ prf = ⊥-elim (n≮n i₂ (<-≤-trans (<-trans hip (h-univ h₁)) prf))
...| SubR _ _ (inj₁ i₁<i₂) prf = ⊥-elim (n≮n i₂ (<-trans hip i₁<i₂))
...| SubR _ _ (inj₂ y<j₁) prf = ⊥-elim (≤⇒≯ (no-overlap-< h₁ v₁ hip i₁<y) y<j₁)
...| SubL _ _ case prf = inj₂ (right h₁ v₀ v₁ v-ok prf)
...| Same _ _ p1 p2 = inj₂ (right h₁ v₀ v₁ v-ok (⊆Hop-inj₃ h₁ v₁ p1 p2))
no-overlap {l₁} {i₁} {j₁} {l₂} {i₂} {j₂} h₁ (hs {y = y} v₀ v₁ v-ok) hip
-- Here is the nasty case. We have to argue why this is impossible
-- WITHOUT resorting to 'nov h₁ (hs v₀ v₁ v-ok)', otherwise this would
-- result in an infinite loop. Note how 'nov' doesn't pattern match
-- on any argument.
--
-- Here's what this looks like:
--
-- (hs v₀ v₁ v-ok)
-- ⌜⁻⁻⁻⁻⁻⁻⁻⁻⁻⁻⁻⁻⁻⁻⁻⁻⁻⁻⁻⁻⁻⁻⁻⌝
-- | h₁ |
-- | ⌜⁻⁻⁻⁻⁻⁻⁻⁻⁻⁻⁻⁻⁻⁻⁺⁻⁻⁻⁻⁻⁻⁻⌝
-- | ∣ | ∣
-- | v₁ ∣ v₀ | ∣
-- ⌜⁻⁻⁻⁻⁻⁻⁻⁻⁺⁻⁻⁻⁻⁻⁻⌜⁻⁻⁻⁻⁻⁻⁻⌝ |
-- | | | | |
-- i₂ < i₁ ≤ y ⋯ j₂ < j₁
--
-- We can pattern match on i₁ ≟ y
| SubR _ _ (inj₂ j₂<j₁) v₀⊆h₁
with i₁ ≟ y
-- And we quickly discover that if i≢y, we have a crossing between
-- v₁ and h₁, and that's impossible.
...| no i₁≢y = ⊥-elim (n≮n y (<-≤-trans (<-trans (h-univ v₀) j₂<j₁)
(no-overlap-< h₁ v₁ hip (≤∧≢⇒< (⊆Hop-univ₁ v₀ h₁ v₀⊆h₁) i₁≢y))))
-- The hard part really is when i₁ ≡ y, here's how this looks like:
--
-- (hs v₀ v₁ v-ok)
-- lvl l+1 ⌜⁻⁻⁻⁻⁻⁻⁻⁻⁻⁻⁻⁻⁻⁻⁻⁻⌝
-- | | h₁
-- | ⌜⁻⁻⁻⁻⁻⁻⁻⁺⁻⁻⁻⁻⁻⁻⁻⌝ lvl l₁
-- | ∣ | ∣
-- | v₁ ∣ v₀ | ∣
-- lvl l ⌜⁻⁻⁻⁻⁻⁻⁻⁻⁺⁻⁻⁻⁻⁻⁻⁻⌝ |
-- | | | |
-- i₂ < i₁ ⋯ j₂ < j₁
--
-- We must show that the composite hop (hs v₀ v₁ v-ok) is impossible to build
-- to show that the crossing doesn't happen.
--
-- Hence, we MUST reason about the levels of the indices and eliminate 'v-ok',
-- Which is possible with a bit of struggling about levels.
...| yes refl with h-lvl-tgt (≤-trans (s≤s z≤n) hip) v₀
...| l≤lvli₁ with ⊆Hop-univ-lvl _ _ v₀⊆h₁ j₂<j₁
...| l<l₁ with h-lvl-mid i₁ (hs v₀ v₁ v-ok) hip (h-univ v₀)
...| lvli₁≤l+1 with h-lvl-tgt (≤-trans (s≤s z≤n) hip) h₁
...| l₁≤lvli₁ rewrite ≤-antisym lvli₁≤l+1 l≤lvli₁
= ⊥-elim (n≮n _ (<-≤-trans l<l₁ (≤-unstep2 l₁≤lvli₁)))
|
3 + 3 Fe → 2 Sb + 3 FeS
|
[STATEMENT]
lemma emeasure_L[simp]: "emeasure (qbs_to_measure X) = (\<lambda>A. if A = {} \<or> A \<notin> sigma_Mx X then 0 else \<infinity>)"
[PROOF STATE]
proof (prove)
goal (1 subgoal):
1. emeasure (qbs_to_measure X) = (\<lambda>A. if A = {} \<or> A \<notin> sigma_Mx X then 0 else \<infinity>)
[PROOF STEP]
by(auto simp: emeasure_def)
|
/* roots/brent.c
*
* Copyright (C) 1996, 1997, 1998, 1999, 2000 Reid Priedhorsky, Brian Gough
*
* This program 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 2 of the License, or (at
* your option) any later version.
*
* This program 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 this program; if not, write to the Free Software
* Foundation, Inc., 51 Franklin Street, Fifth Floor, Boston, MA 02110-1301, USA.
*/
/* brent.c -- brent root finding algorithm */
#include <config.h>
#include <stddef.h>
#include <stdlib.h>
#include <stdio.h>
#include <math.h>
#include <float.h>
#include <gsl/gsl_math.h>
#include <gsl/gsl_errno.h>
#include <gsl/gsl_roots.h>
#include "roots.h"
typedef struct
{
double a, b, c, d, e;
double fa, fb, fc;
}
brent_state_t;
static int brent_init (void * vstate, gsl_function * f, double * root, double x_lower, double x_upper);
static int brent_iterate (void * vstate, gsl_function * f, double * root, double * x_lower, double * x_upper);
static int
brent_init (void * vstate, gsl_function * f, double * root, double x_lower, double x_upper)
{
brent_state_t * state = (brent_state_t *) vstate;
double f_lower, f_upper ;
*root = 0.5 * (x_lower + x_upper) ;
SAFE_FUNC_CALL (f, x_lower, &f_lower);
SAFE_FUNC_CALL (f, x_upper, &f_upper);
state->a = x_lower;
state->fa = f_lower;
state->b = x_upper;
state->fb = f_upper;
state->c = x_upper;
state->fc = f_upper;
state->d = x_upper - x_lower ;
state->e = x_upper - x_lower ;
if ((f_lower < 0.0 && f_upper < 0.0) || (f_lower > 0.0 && f_upper > 0.0))
{
GSL_ERROR ("endpoints do not straddle y=0", GSL_EINVAL);
}
return GSL_SUCCESS;
}
static int
brent_iterate (void * vstate, gsl_function * f, double * root, double * x_lower, double * x_upper)
{
brent_state_t * state = (brent_state_t *) vstate;
double tol, m;
int ac_equal = 0;
double a = state->a, b = state->b, c = state->c;
double fa = state->fa, fb = state->fb, fc = state->fc;
double d = state->d, e = state->e;
if ((fb < 0 && fc < 0) || (fb > 0 && fc > 0))
{
ac_equal = 1;
c = a;
fc = fa;
d = b - a;
e = b - a;
}
if (fabs (fc) < fabs (fb))
{
ac_equal = 1;
a = b;
b = c;
c = a;
fa = fb;
fb = fc;
fc = fa;
}
tol = 0.5 * GSL_DBL_EPSILON * fabs (b);
m = 0.5 * (c - b);
if (fb == 0)
{
*root = b;
*x_lower = b;
*x_upper = b;
return GSL_SUCCESS;
}
if (fabs (m) <= tol)
{
*root = b;
if (b < c)
{
*x_lower = b;
*x_upper = c;
}
else
{
*x_lower = c;
*x_upper = b;
}
return GSL_SUCCESS;
}
if (fabs (e) < tol || fabs (fa) <= fabs (fb))
{
d = m; /* use bisection */
e = m;
}
else
{
double p, q, r; /* use inverse cubic interpolation */
double s = fb / fa;
if (ac_equal)
{
p = 2 * m * s;
q = 1 - s;
}
else
{
q = fa / fc;
r = fb / fc;
p = s * (2 * m * q * (q - r) - (b - a) * (r - 1));
q = (q - 1) * (r - 1) * (s - 1);
}
if (p > 0)
{
q = -q;
}
else
{
p = -p;
}
if (2 * p < GSL_MIN (3 * m * q - fabs (tol * q), fabs (e * q)))
{
e = d;
d = p / q;
}
else
{
/* interpolation failed, fall back to bisection */
d = m;
e = m;
}
}
a = b;
fa = fb;
if (fabs (d) > tol)
{
b += d;
}
else
{
b += (m > 0 ? +tol : -tol);
}
SAFE_FUNC_CALL (f, b, &fb);
state->a = a ;
state->b = b ;
state->c = c ;
state->d = d ;
state->e = e ;
state->fa = fa ;
state->fb = fb ;
state->fc = fc ;
/* Update the best estimate of the root and bounds on each
iteration */
*root = b;
if ((fb < 0 && fc < 0) || (fb > 0 && fc > 0))
{
c = a;
}
if (b < c)
{
*x_lower = b;
*x_upper = c;
}
else
{
*x_lower = c;
*x_upper = b;
}
return GSL_SUCCESS ;
}
static const gsl_root_fsolver_type brent_type =
{"brent", /* name */
sizeof (brent_state_t),
&brent_init,
&brent_iterate};
const gsl_root_fsolver_type * gsl_root_fsolver_brent = &brent_type;
|
State Before: α : Type u
β : Type v
γ : Type w
ι : Sort x
inst✝² : Preorder α
inst✝¹ : Preorder β
inst✝ : Preorder γ
f : α → β → γ
s : Set α
t : Set β
a : α
b : β
h₀ : ∀ (b : β), Antitone (swap f b)
h₁ : ∀ (a : α), Monotone (f a)
⊢ BddBelow s → BddAbove t → BddAbove (Set.image2 f s t) State After: case intro.intro
α : Type u
β : Type v
γ : Type w
ι : Sort x
inst✝² : Preorder α
inst✝¹ : Preorder β
inst✝ : Preorder γ
f : α → β → γ
s : Set α
t : Set β
a✝ : α
b✝ : β
h₀ : ∀ (b : β), Antitone (swap f b)
h₁ : ∀ (a : α), Monotone (f a)
a : α
ha : a ∈ lowerBounds s
b : β
hb : b ∈ upperBounds t
⊢ BddAbove (Set.image2 f s t) Tactic: rintro ⟨a, ha⟩ ⟨b, hb⟩ State Before: case intro.intro
α : Type u
β : Type v
γ : Type w
ι : Sort x
inst✝² : Preorder α
inst✝¹ : Preorder β
inst✝ : Preorder γ
f : α → β → γ
s : Set α
t : Set β
a✝ : α
b✝ : β
h₀ : ∀ (b : β), Antitone (swap f b)
h₁ : ∀ (a : α), Monotone (f a)
a : α
ha : a ∈ lowerBounds s
b : β
hb : b ∈ upperBounds t
⊢ BddAbove (Set.image2 f s t) State After: no goals Tactic: exact ⟨f a b, mem_upperBounds_image2_of_mem_upperBounds_of_mem_upperBounds h₀ h₁ ha hb⟩
|
Insuring your home against financial catastrophe can involve a number of high-stakes decisions. From which carrier to use to how large a deductible to choose, the insurance industry has become endlessly complicated. Volcanic eruptions, riots and falling planes? Your standard policy definitely covers those unlikely events. Pit bull bites, sinkholes and mold infestations? Perhaps, perhaps not. Flooding? Definitely not.
Here you will find an overview of what homeowners insurance protects and what it doesn’t, and how you can make informed decisions about buying coverage.
What Type of Homeowners Insurance Should You Get?
If you have a mortgage, homeowners insurance isn’t just extra security and protection — it’s required. Find out the average cost of home insurance in your state by using the below map.
Source: National Association of Insurance Commissioners (NAIC).
is injured on your property.
The most popular insurance is an “all risk” policy, although people with newly constructed homes may qualify for comprehensive coverage. Whether you live in a mobile home, condo, single family home, or rent an apartment, you can find a policy to protect your home and/or property.
A bare-bones policy that doesn’t include liability coverage. It’s no longer available in most states, according to the Insurance Information Institute.
Also called a “broad form” policy. The coverage protects your home from 16 different types of perils (see above list).
A “special form” policy that covers the attached structures of your home as well as detached structures like a garage, cottage and fence. It also includes personal liability coverage. The name “All Risk” Policy is misleading as the policy doesn’t cover all risks. It does cover the aforementioned 16 perils and everything except the named exclusions in the policy, which include war, power failure, flood, vermin and nuclear disaster, among others.
This is an insurance policy for renters. It covers your personal belongings, though not the structure of the property you rent.
A “comprehensive form” policy that covers the structure of your home and offers broader coverage for personal property. You’re protected from the same perils as an HO-3 but don’t have to prove your belongings were damaged by a peril named in the policy. This form of coverage is usually for new or recently constructed homes.
A policy for condo and co-op owners that works similar to HO-2 coverage. It covers the structural parts of the building you own, plus your personal belongings.
This coverage is like an HO-2 policy, but it protects mobile homes.
A policy that typically covers the replacement cost — minus depreciation — of damage to an older home.
Homeowners insurance is complicated, and rate quotes hinge on numerous variables. Check out our guide on comparing homeowners insurance quotes.
Homeowners and renters insurance policies almost never cover floods, hurricanes, earthquakes and other natural hazards. Learn how to protect yourself from Mother Nature’s worst.
You can’t control the wrath of nature, but you can prepare for the worst. Here are steps to take to get ready for an emergency, and what to do if disaster strikes.
Homeowners coverage varies based on the type of policy you buy and where you live. The most bare-bones insurance policy is known as “dwelling fire,” also known as HO-1 coverage, and it insures against hazards that include fire, smoke, lightning and explosions.
No homeowners policy covers flood damage if it’s caused by rising water. That peril is covered through a separate flood insurance policy from the federal government.
You need a separate federal flood insurance policy.
This risk is covered by separate earthquake insurance.
If you live in Hurricane Alley, you might need a policy from your state’s windstorm plan.
Many carriers have begun to exclude mold damage from policies.
You might be able to buy a special rider for this peril.
Some insurers exclude such breeds as pit bulls, Akitas and Rottweilers.
You might be able to buy a special rider for this hazard, too.
Say FedEx delivers work-related package to your house, and the driver slips and falls. Your policy doesn’t cover liability.
If your roof is simply old rather than damaged, you pay for the repair.
Does Your Location Affect Your Coverage?
Yes, where you live affects your coverage and policy costs. If you live in California, earthquake coverage isn’t part of your standard policy. You’ll need separate coverage, which usually carries a high deductible. And if you live in Florida or another hurricane-prone state, you might need separate windstorm insurance.
Some of the language in your policy might seem ridiculously specific, but there’s a reason for the hairsplitting. “Civil unrest” is covered in standard policies, but “war” isn’t. And the extensive damage left by Hurricane Katrina led to many lawsuits over whether water damage was technically caused by a flood or a windstorm.
For most homeowners, the high-value part of the policy applies to the structure of your home. If an insured hazard hits your humble abode, your coverage will pay to fix or rebuild your house. Insurers typically cover not just your living quarters but also garages, fences, tool sheds and gazebos. In addition, insurance policies generally pay to replace personal belongings if they’re stolen or damaged in a fire, hurricane or other covered disaster.
Be sure to review your dwelling coverage occasionally because capital improvements and inflation can affect your home’s cost of replacement. Due to the so-called 80 percent rule, if your coverage limits fall below 80 percent of the full replacement cost of your home, your insurance company may reduce the amount it pays on a claim.
In a typical policy with $200,000 in coverage for the dwelling, the insurer would cover up to $100,000 to replace furniture, electronics and other personal items. Pricey items such as jewelry, furs and silverware typically are covered, but many insurers impose dollar limits if they’re stolen. If you have an extensive jewelry collection, though, you might consider paying for an endorsement, which specifies coverage under certain special circumstances or for itemized valuables.
Homeowners policies also include liability protection, which pays for legal costs and any court awards in the event of a lawsuit against the homeowner. Typically, this coverage is limited to $100,000. Finally, your policy might pay for your living costs if your home is rendered uninhabitable.
Pays to replace home and possessions minus a deduction for depreciation — so don’t expect to get $2,000 for that ancient laptop.
The most generous coverage that pays to rebuild your home no matter the cost, useful if you’re the victim of a disaster that causes a spike in the costs of labor and construction materials.
If you’ve lived in your house for years, creating an inventory might seem tedious. Start with your most recent purchases and then work backward to your older possessions. It’s better to have an incomplete list than no list at all.
Source: Adapted from the Insurance Information Institute.
Don’t feel like cataloging every article of clothing, book and knickknack in your home? Jack Hungelmann, author of Insurance for Dummies, suggests tallying the value of furniture, TVs and other major items, then doubling it. That should approximate the value of all your stuff.
The average user is able to develop “a complete picture inventory of a property and generate detailed reports” in about an hour or less by using this app, according to its iTunes Store description.
This app gives you the ability to quickly produce a complete estimate of your belongings. The free version has a 25-item limit, while the $4.99 version unlocks all of the app’s available features.
Doubling as an organizer and home inventory resource, MyStuff2 allows you to keep track of your personal possessions. The “Lite” version is free; the main version of the app is $4.99 and the “Pro” version is $8.99.
This app is described as the “perfect companion for your move.” It helps with creating a visual inventory of your belongings through photos, videos, tags and other features. The basic version is free and it’s $4.99 for the “Plus” version which has additional features.
Stuffanizer is a visual inventory app that allows users to set custom locations, create tags and take photos to keep track of their stuff — for $2.99.
One of the most important variables determining how much you’ll pay for coverage. Hurricane Alley and Tornado Alley are the most expensive places to insure a home — Florida, Louisiana, Texas and Oklahoma top the list of priciest markets. Idaho, Oregon and Utah are the cheapest states to buy homeowners coverage.
If your Florida abode is a wood-frame house with jalousie windows, you’ll pay more. If it’s a concrete-block structure with impact-resistant windows, you’ll pay less. Insurers use a catastrophe model to determine prices. Rate regulation is the duty of insurance commissioners in each state.
If State Farm, Allstate, Nationwide and other carriers are competing for customers in your area, rates are likely to fall. This is known as a soft market. In the opposite case, a hard market, rates tend to rise.
How much insurers pay for this form of reinsurance can affect rates. Catastrophe reinsurance indemnifies the insurer for losses in excess of a stipulated sum arising from a single catastrophic event or series of events.
What Kind of Discounts Might You Expect?
Bundling your home insurance with your car insurance (buying it from the same insurance carrier). These are known as multi-line discounts.
Having dead-bolts and/or a home security system.
Bringing your old wiring up to code.
What happens if you’re the victim of a burglary, fire or other insured claim? First, your policy requires you to mitigate the damage as soon as possible. So, for instance, if you have a water leak, turn off the water and immediately call a company that handles water damage. Next, contact your insurance company to report the damage. When you reach your company’s adjuster, be sure to get his name and cell phone number, along with your claim number. In most cases, your carrier will come through with a check to cover your claim, up to the limits of your policy.
Whether you store it in the cloud or in a filing cabinet, your policy spells out what your insurer owes you in case of a loss.
If you have a large collection of jewelry or art, buy an endorsement to insure it.
Say you have a $1,000 deductible and your $500 bike is stolen from your garage. In that case, you’d be wise to keep your insurer out of the matter.
Hit by a storm or significant water damage, it makes sense to file a claim. However, you may want to skip a small, non-weather claim, lest your insurer drop you when your policy comes up for renewal.
Insurers handle each type of claim a bit differently. In case of a burglary, you’ll need to provide a police report and an inventory of the items taken. In the event of a fire or weather-related claim that damages the structure of your house, you’ll need to hire a contractor, and your insurer ultimately will reimburse you for your costs.
Source: National Association of Insurance Commissioners. Cost adjusted to 2014 dollars.
Do I Need a Public Insurance Adjuster?
The typical insurance claim goes relatively smoothly, and the carrier pays up as promised. But in some cases, an insurer simply refuses to pay what you think you’re owed. If you’re faced with an unreasonable settlement offer, it might make sense to hire a public adjuster. Unlike an adjuster who’s employed by your insurance company, a public adjuster works for you and is paid by you.
Public adjusters say they know policies and insurers’ processes, and they can persuade insurers to pay up when you can’t. You’ll pay your public adjuster a percentage of the settlement he negotiates on your behalf. Public adjusters stress that they’re not insurance agents nor paid by carriers, which puts them firmly on the side of policyholders. The National Association of Public Insurance Adjusters offers a listing of its members.
Sources: National Association of Insurance Commissioners, based on average annual premiums for 2013; Insurance Information Institute.
If you own a car, you’re familiar with the difference between insurance and a warranty. If your car is stolen or damaged in a crash or natural disaster, that’s a matter for your insurance policy. If your transmission blows out, you’ll turn to your warranty for financial help. The concept is similar for real estate.
A home warranty is a service contract that helps offset expenses if the roof springs a leak or the air conditioning needs a repair. Home warranties are a type of insurance, but they’re designed to cushion the blow of the smaller costs of routine maintenance rather than some significant loss from a catastrophic event. Sellers often provide home warranties as a tool to market their homes. A lender will require you to have homeowners insurance but not a warranty.
An individual employed by an insurance company to evaluate losses and settle policyholder claims.
This computerized method of predicting claims combines long-term disaster trends with current demographics and building patterns. The result is a prediction of the potential cost of catastrophic losses for a given area.
Insurers typically can handle billions of dollars of losses, but to protect against major disasters such as Hurricane Andrew and Hurricane Katrina, they buy reinsurance that helps cover claims filed after large-scale catastrophes.
The amount of loss paid by the policyholder. It’s either a specific dollar amount or a percentage of the claim amount.
A cut-rate type of insurance policy that covers only the most basic risks.
This type of policy covers a structure and its contents but has a high deductible.
A written form attached to an insurance policy altering the policy’s coverage, terms or conditions.
A provision in an insurance policy disallowing coverage for certain risks.
Coverage for flood damage is available from the federal government under the National Flood Insurance Program. It is sold by licensed insurance agents.
A period when carriers are reluctant to sell coverage.
A numerical ranking based on a consumer’s credit history. Insurers say people with poor credit histories have proven more likely to file claims.
A specific risk or cause of loss covered by an insurance policy, such as fire or theft.
A person who negotiates with insurers on behalf of policyholders and receives a portion of a claims settlement.
The process by which states monitor insurance companies’ rate changes.
An environment of plentiful, low-cost coverage.
State-sponsored insurance pools sell hurricane coverage to people who can’t buy it in the voluntary market because of their high risk. Alabama, Florida, Louisiana, Mississippi, North Carolina, South Carolina and Texas offer these plans. Georgia and New York also have special windstorm pools for certain coastal communities.
|
*DECK SSPEV
SUBROUTINE SSPEV (A, N, E, V, LDV, WORK, JOB, INFO)
C***BEGIN PROLOGUE SSPEV
C***PURPOSE Compute the eigenvalues and, optionally, the eigenvectors
C of a real symmetric matrix stored in packed form.
C***LIBRARY SLATEC (EISPACK)
C***CATEGORY D4A1
C***TYPE SINGLE PRECISION (SSPEV-S)
C***KEYWORDS EIGENVALUES, EIGENVECTORS, EISPACK, PACKED, SYMMETRIC
C***AUTHOR Kahaner, D. K., (NBS)
C Moler, C. B., (U. of New Mexico)
C Stewart, G. W., (U. of Maryland)
C***DESCRIPTION
C
C Abstract
C SSPEV computes the eigenvalues and, optionally, the eigenvectors
C of a real symmetric matrix stored in packed form.
C
C Call Sequence Parameters-
C (The values of parameters marked with * (star) will be changed
C by SSPEV.)
C
C A* REAL(N*(N+1)/2)
C real symmetric packed input matrix. Contains upper
C triangle and diagonal of A, by column (elements
C 11, 12, 22, 13, 23, 33, ...).
C
C N INTEGER
C set by the user to
C the order of the matrix A.
C
C E* REAL(N)
C on return from SSPEV, E contains the eigenvalues of A.
C See also INFO below.
C
C V* REAL(LDV,N)
C on return from SSPEV, if the user has set JOB
C = 0 V is not referenced.
C = nonzero the N eigenvectors of A are stored in the
C first N columns of V. See also INFO below.
C
C LDV INTEGER
C set by the user to
C the leading dimension of the array V if JOB is also
C set nonzero. In that case, N must be .LE. LDV.
C If JOB is set to zero, LDV is not referenced.
C
C WORK* REAL(2N)
C temporary storage vector. Contents changed by SSPEV.
C
C JOB INTEGER
C set by the user to
C = 0 eigenvalues only to be calculated by SSPEV.
C Neither V nor LDV are referenced.
C = nonzero eigenvalues and vectors to be calculated.
C In this case, A & V must be distinct arrays.
C Also, if LDA .GT. LDV, SSPEV changes all the
C elements of A thru column N. If LDA < LDV,
C SSPEV changes all the elements of V through
C column N. If LDA=LDV, only A(I,J) and V(I,
C J) for I,J = 1,...,N are changed by SSPEV.
C
C INFO* INTEGER
C on return from SSPEV, the value of INFO is
C = 0 for normal return.
C = K if the eigenvalue iteration fails to converge.
C Eigenvalues and vectors 1 through K-1 are correct.
C
C
C Error Messages-
C No. 1 recoverable N is greater than LDV and JOB is nonzero
C No. 2 recoverable N is less than one
C
C***REFERENCES (NONE)
C***ROUTINES CALLED IMTQL2, TQLRAT, TRBAK3, TRED3, XERMSG
C***REVISION HISTORY (YYMMDD)
C 800808 DATE WRITTEN
C 890831 Modified array declarations. (WRB)
C 890831 REVISION DATE from Version 3.2
C 891214 Prologue converted to Version 4.0 format. (BAB)
C 900315 CALLs to XERROR changed to CALLs to XERMSG. (THJ)
C 900326 Removed duplicate information from DESCRIPTION section.
C (WRB)
C***END PROLOGUE SSPEV
INTEGER I,INFO,J,LDV,M,N
REAL A(*),E(*),V(LDV,*),WORK(*)
C***FIRST EXECUTABLE STATEMENT SSPEV
IF (N .GT. LDV) CALL XERMSG ('SLATEC', 'SSPEV', 'N .GT. LDV.',
+ 1, 1)
IF(N .GT. LDV) RETURN
IF (N .LT. 1) CALL XERMSG ('SLATEC', 'SSPEV', 'N .LT. 1', 2, 1)
IF(N .LT. 1) RETURN
C
C CHECK N=1 CASE
C
E(1) = A(1)
INFO = 0
IF(N .EQ. 1) RETURN
C
IF(JOB.NE.0) GO TO 20
C
C EIGENVALUES ONLY
C
CALL TRED3(N,1,A,E,WORK(1),WORK(N+1))
CALL TQLRAT(N,E,WORK(N+1),INFO)
RETURN
C
C EIGENVALUES AND EIGENVECTORS
C
20 CALL TRED3(N,1,A,E,WORK(1),WORK(1))
DO 30 I = 1, N
DO 25 J = 1, N
25 V(I,J) = 0.
30 V(I,I) = 1.
CALL IMTQL2(LDV,N,E,WORK,V,INFO)
M = N
IF(INFO .NE. 0) M = INFO - 1
CALL TRBAK3(LDV,N,1,A,M,V)
RETURN
END
|
\documentclass{article}
\title{Learning Incoherent Subspaces: Classification via Incoherent Dictionary Learning.}
\author{Daniele Barchiesi and Mark D. Plumbley}
%\affiliation{Centre for Digital Music\\
% Queen Mary University of London\\
% Mile End Road, London E1 4NS, UK}
%%%%%%%%%%%%%%%%%%%%%%PACKAGES%%%%%%%%%%%%%%%%%%%%%
\usepackage{amsmath}
\usepackage[english]{babel}
\usepackage[applemac]{inputenc}
\usepackage[T1]{fontenc}
\usepackage{amsfonts,amssymb,amsmath,amsthm,bm}
\usepackage[boxruled,linesnumbered]{algorithm2e}
\usepackage{subfigure,graphicx}
\usepackage{setspace}
\usepackage{hyperref}
\usepackage{enumitem}
\usepackage{graphicx}
\usepackage{epstopdf}
%%%%%%%%%%%%%%%%%%%%%DEFINITIONS%%%%%%%%%%%%%%%%%%
\input{definitions.tex}
\def \nComponents{L} %number of significant components of ICA feature transform
\def \fea{\Vector{x}} %vector of features
\def \Feas{\Matrix{X}} %matrix containing set of training features
\def \iFea{n} %component index of features vector
\def \nDim{N} %dimensionality of features vector
\def \newFea{\Vector{y}} %features vector after feature transform
\def \NewFeas{\Matrix{Y}}%matrix of training features after feature transform
\def \Dic{\Matrix{\Phi}} %dictionary learned from features
\def \nAto{K} %number of atoms in the dictionary
\def \iAto{k} %atom index
\def \atom{\Vector{\phi}}%atom in a dictionary
\def \nFea{M} %number of signals or features
\def \iFea{m} %signal index
\def \Coeff{\Matrix{A}} %matrix of sparse approximation coefficients
\def \coherence{\mu} %mutual coherence
\def \coeff{\Vector{\alpha}}%vector of sparse approximation coefficients
\def \nActiveAtoms{S} %number of active atoms
\def \orthmat{\Matrix{W}} %orthonormal matrix
\def \cat{c} %category
\def \cats{\Vector{c}} %vector of categories of training signals
\def \Cat{\mathcal{C}} %set of possible categories
\def \uniCat{C} %element of the set of possible categories
\def \iCat{p} %index of the elements in the set of possible categories
\def \nCat{P} %number of elements in the set of possible categories
\def \definition{\overset{\Label{def}}{=}} %definitions
\def \Gram{\Matrix{G}} %Gram matrix
\def \gram{g} %element of the Gram matrix
\def \admissibleDictionary{\Function{D}} %set of admissible dictionaries
\def \ambient{\Set{R}} %ambient
\def \ipr{\Acronym{ipr}} %Iterative projections and rotations
\def \ip{\Acronym{ip}} %Iterative projections
\def \nDimSub{Q}
\def \Spa{\Matrix{\Psi}}
\begin{document}
%\ninept
%
\maketitle
%
\begin{abstract}
In this article we present the supervised iterative projections and rotations (\Acronym{s-ipr}) algorithm, a method for learning discriminative incoherent subspaces from data. We derive \Acronym{s-ipr} as a supervised extension of our previously proposed iterative projections and rotations (\Acronym{ipr}) algorithm for incoherent dictionary learning, and we employ it to learn incoherent sub-spaces that model signals belonging to different classes. We test our method as a feature transform for supervised classification, first by visualising transformed features from a synthetic dataset and from the `iris' dataset, then by using the resulting features in a classification experiment. While the visualisation results are promising, we find that \Acronym{s-ipr} generally performs worse than traditional and state-of-the-art techniques for supervised dimensionality reduction in terms of the misclassification ratio.
\end{abstract}
%
%\begin{keywords}
%Feature transforms, sparse approximation, dictionary learning, supervised classification.
%\end{keywords}
%
\section{Introduction: Classification And Feature Transforms}\label{sec:intro}
Supervised classification is one of the classic problems in machine learning where a system is designed to discriminate the category of an observed signal, having previously observed representative examples from the considered classes \cite{Duda1973Pa}.
Typically, a classification algorithm consists of a training phase where class-specific models are learned from labelled samples, followed by a testing phase where unlabelled data are classified by comparison with the learned models. Both training and testing comprise various stages. Firstly, we observe a signal that measures a process of interest, such as the recording of a sound or image, or a log of the temperatures in a particular geographic area. Then, a set of features are extracted from the raw signals using signal processing techniques. This step is performed in order to reduce the dimensionality of the data and provide a new signal that allows generalisation among examples of the same class, while retaining enough information to discriminate between different classes.
Following the features extraction step, a feature transform can be employed to further reduce the dimensionality of the data and to enhance discrimination between classes. Thus classification benefits from feature transforms especially when features are not separable, that is, when it is not possible to optimise a simple function that maps features belonging to signals of a given class to the corresponding category.
A further dimensionalty reduction may be performed when dealing with high dimensional signals (such as audio or high resolution images) by fitting the parameters of global statistical distributions with features learned on portions of the signal. Models learned on different classes are finally compared using a distance metric to the model learned form an unlabelled signal, which is typically assigned to the nearest class.
\subsection{Traditional Algorithms For Feature Transform}\label{sec:aft}
Two of the main feature transform techniques include principal component analysis (\Acronym{pca}) \cite{Pearson1901On} and Fisher's linear discriminant analysis (\Acronym{lda}) \cite{Duda1973Pa}.
\subsubsection{\Acronym{pca}}
Let $\curlyb{\fea_{\iFea} \in \real^{\nDim}}_{\iFea=1}^{\nFea}$ be a set of vectors containing features extracted from $\nFea$ training signals. The goal of \Acronym{pca} is to learn an orthonormal set of basis functions $\curlyb{\atom_{\iAto} \in \ambient^{\nDim}}_{\iAto=1}^{\nDim}$ such that $ \norm{\atom_{\iAtom}}{2}=1$ and $\inner{\atom_{i}}{\atom_{j}}=0 \; \forall i\neq j$ that are placed along the columns of a so-called \emph{dictionary} $\Dic\in\real^{\nDim\times\nDim}$. The bases are optimised from the data to identify their principal components, that is, the sub-spaces that retain the maximum variance of the features.
To compute the dictionary, the eigenvalue decomposition of the outer product
\begin{equation}
\Matrix{X}\Transpose{\Matrix{X}} = \Matrix{Q}\Matrix{\Lambda}\Transpose{\Matrix{Q}}
\end{equation}
is first calculated. Then, the $\nComponents$ eigenvectors corresponding to the $\nComponents$ largest eigenvalues are selected from the matrix $\Matrix{Q}$, and scaled to unit $\ell_{2}$ norm to form the dictionary $\Dic$. A new set of transformed features $\newFea_{\Acronym{pca}} = \Dic\Transpose{\Dic}\fea$ is computed by projecting the data onto the sub-space spanned by the columns of $\Dic$ (that is, onto the $\nComponents$-dimensional principal sub-space). This operation reduces the dimensionality of the features by projecting them onto a linear subspace embedded in $\real^{\nDim}$. It is an unsupervised technique that does not exploit knowledge about the classes associated with the training set, but implicitly relies in the assumption that the principal component directions encode relevant differences between classes.
\subsubsection{\Acronym{lda}}
In contrast, \Acronym{lda} is a supervised method for feature transform whose objective is to explicitly maximise the separability of classes in the transformed domain.
Let $\obsSet_{\iUniCat}$ be a set indexing features extracted from data belonging to the $\iUniCat$-th category, let
\begin{equation}
\average{\fea}_{\iUniCat} \definition \frac{1}{\abs{\obsSet_{\iUniCat}}}\sum_{\iObs\in\obsSet_{\iUniCat}} \fea_{\iObs}
\end{equation}
be the $\iUniCat$-th class feature centroid, and $\average{\fea}\definition\sum_{\iObs=1}^{\nObs}\fea_{\iObs}$ the centroid of the features extracted from the entire training dataset. The between-classes scatter matrix
\begin{equation}
\Matrix{S_{b}} \definition \sum_{\iUniCat=1}^{\nUniCat}\abs{\obsSet_{\iUniCat}}\roundb{\average{\fea}_{\iUniCat}-\average{\fea}}\Transpose{\roundb{\average{\fea}_{\iUniCat}-\average{\fea}}}
\end{equation}
is defined to measure the mutual distances between the centroids of different classes, while the within-classes scatter matrix
\begin{equation}
\Matrix{S}_{w}\definition \sum_{\iUniCat=1}^{\nUniCat}\sum_{\iObs\in\obsSet_{\iUniCat}}\roundb{\fea_{\iObs}-\average{\fea}_{\iUniCat}}\Transpose{\roundb{\fea_{\iObs}-\average{\fea}_{\iUniCat}}}
\end{equation}
quantifies the distances between features belonging to the same class.
To maximise an objective function $\objective(\Matrix{W})\definition\frac{\abs{\Transpose{\Matrix{W}}\Matrix{S}_{b}\Matrix{W}}}{\abs{\Transpose{\Matrix{W}}\Matrix{S}_{w}\Matrix{W}}}$ that promotes features belonging to the same class to be near each other and far away from features belonging to other classes, the eigenvalue decomposition of the matrix
\begin{equation}
\pseudoinverse{\Matrix{S}_{w}}\Matrix{S}_{b} = \Matrix{Q}\Matrix{\Lambda}\Transpose{\Matrix{Q}}
\end{equation}
is computed, and the features $\fea$ are projected onto the space spanned by its $(\nUniCat-1)$ eigenvectors corresponding to the largest $(\nUniCat-1)$ eigenvalues.
\Acronym{lda} explicitly seeks to enhance the discriminative power of features by optimising the objective $\objective$.
\subsection{Supervised \Acronym{pca}}
Related works that extend \Acronym{pca} include the supervised \Acronym{pca} (\Acronym{s-pca}) proposed by Barshan et al. \cite{Barshan2011Su}. \Acronym{s-pca} is based on the theory of reproducing kernel Hilbert spaces (\Acronym{rkhs}) (that are spaces of functions which satisfy certain properties and map elements from an arbitrary set to the set of complex numbers) \cite{Aronszajn:1950}, and on the so-called Hilbert-Schmidt independence criterion (\Acronym{hsic})\cite{gretton2005measuring}. The \Acronym{hsic} is used to estimate the statistical dependence of two random variables based on the fact that this quantity is related to the correlation of functions belonging to their respective \Acronym{rkhs}. While \Acronym{hsic} is defined in terms of the probability density function of the two random variables, empirical estimates of \Acronym{hsic} can be obtained from finite sequences of their realisations. The empirical \Acronym{hsic} can be used in turn to construct an objective function that maximises the dependence between the two variables. Hence, this strategy is adopted within the context of classification to maximise the statistical dependence between a transformed feature $\newFea_{\Acronym{s-pca}}$ and its corresponding category $\cat$.
In practice, \Acronym{s-pca} differs from \Acronym{pca} in that it calculates the eigenvalue decomposition of a matrix $\Matrix{R}$ defined as follows:
\begin{equation}
\Matrix{R} \definition \Matrix{X}\Matrix{H}\Matrix{L}\Matrix{H}\Transpose{\Matrix{X}}
\end{equation}
were $\Matrix{H} \definition \Matrix{I} - \Vector{e}\Transpose{\Vector{e}}$ is a so-called \emph{centring} matrix\footnote{Here $\Vector{e}$ is a vector of ones.} and $\Matrix{L} \definition \Vector{\cat}\Transpose{\Vector{\cat}}$ is the kernel matrix of the class variable that is constructed by computing the outer product of the vectors resulting from assigning different numerical values to each category.
\subsection{Other related work}
The union of incoherent sub-spaces model proposed by Schnass and Vandergheynst \cite{Schnass2010A-} employes a very similar intuition to the one that inspired our proposed method, and models features belonging to different classes using incoherent subspaces. Other methods for supervised dimensionality reduction include metric learning algorithms \cite{xing2002distance}, sufficient dimensionality reduction \cite{li1991sliced} and Bair's supervised principal components \cite{Bair06predictionby}.
Manifold learning techniques are used to model nonlinear data and reviewed by Van Der Maaten et al. \cite{Van-Der-Maaten2009Di}. Finally, the sparse sub-space clustering technique developed by Elhamifar and Vidal \cite{Elhamifar2013Sp} that applies concepts and algorithm from the field of sparse approximation to tackle unsupervised clustering problems.
\subsection{Paper organisation}
The method proposed in this paper is aimed at learning discriminative sub-spaces that allow dimensionality reduction, while at the same time enhancing the separability between classes. It is derived from our previous work on learning incoherent dictionaries for sparse approximation \cite{Barchiesi2013Le}.
The incoherent dictionary learning problem will be introduced in Section \ref{sec:idl}, while Section \ref{sec:lis} will contain the main contribution of this paper consisting in learning incoherent subspaces for classification. Numerical experiments are presented in Section \ref{sec:ne}, and conclusions are drawn in Section \ref{sec:end}.
\section{Incoherent Dictionary Learning}\label{sec:idl}
A sparse approximation of a signal $\fea\in\real^{\nDim}$ is a linear combination of $\nAto\geq\nDim$ basis functions $\curlyb{\atom_{\iAto}\in\real^{\nDim}}_{\iAto=1}^{\nAto}$ called \emph{atoms} described by:
\begin{equation}
\fea \approx \approximant{\fea} = \sum_{\iAto=1}^{\nAto} \alpha_{\iAto}\atom_{\iAto}
\end{equation}
where the vector of coefficients $\coeff$ contains a \emph{small} number of non-zero components, corresponding to a small number of atoms actively contributing to the approximation $\approximant{\fea}$. Given a signal $\fea$ and a dictionary, various algorithms have been proposed to find a sparse approximation that minimises the residual error $\norm{\fea-\approximant{\fea}}{2}$\cite{Elad2010Sp}.
Dictionary learning aims at optimising a dictionary $\Dic$ for sparse approximation given a set of training data. It is an unsupervised technique that can be thought as being a generalisation of \Acronym{pca}, as both methods learn linear subspaces that minimise the approximation error of the signals. Dictionary learning, however, is generally more flexible than \Acronym{pca} because it can be employed to learn more general non-orthogonal over-complete dictionaries \cite{Rubinstein2010Di}.
\subsection{The incoherent dictionary learning problem}
Dictionaries for sparse approximation have important intrinsic properties that describe the relations between their atoms, like the mutual coherence $\mu(\Dic)=\underset{i\neq j}{\max}{\inner{\atom_{i}}{\atom_{j}}}$ that is defined as the maximum inner product between any two different atoms. The goal of incoherent dictionary learning is to learn atoms that are well adapted to sparsely approximate a set of training signals, and that are at the same time mutually incoherent \cite{Barchiesi2013Le}.
Given a set of $\nFea$ training signals contained in the columns of the matrix $\Feas \in \real^{\nDim\times\nFea}$ and a matrix $\Coeff\in\real^{\nAto\times\nFea}$ indicating the sparse approximation coefficients, the incoherent dictionary learning problem can be expressed as:
\begin{align}\label{eq:iprcost}
\optimal{\Dic} = \MinimiseST{\Dic}{\norm{\Feas-\Dic\Coeff}{\F}}{\coherence(\Dic) \leq \coherence_{0} \nonumber \\
&\norm{\coeff_{\iFea}}{0}\leq \nActiveAtoms \quad \forall \iFea}
\end{align}
where $\coherence_{0}$ is a fixed mutual coherence constraint, the $\ell_{0}$ pseudo-norm $\norm{\cdot}{0}$ counts the number of non-zero components of its argument and $\nActiveAtoms$ is a fixed number of active atoms. Algorithms for (incoherent) dictionary learning generally follow an alternate optimisation heuristic, iteratively updating $\Dic$ and $\Coeff$ until a stopping criterion is met. In the case of the iterative projections and rotations algorithm (\Acronym{ipr}) algorithm \cite{Barchiesi2013Le}, a dictionary de-correlation step is added after updating the dictionary in order to satisfy the mutual coherence constraint.
Given $\Feas$, fixed $\coherence_{0}$, $\nActiveAtoms$ and a stopping criterion (such as a maximum number of iterations), the optimisation of \eqref{eq:iprcost} is tackled by iteratively performing the following steps:
\begin{itemize}
\item\emph{Sparse coding}: fix $\Dic$ and compute the matrix $\Coeff$ using a suitable sparse approximation method.
\item\emph{Dictionary update}: fix $\Coeff$ and update $\Dic$ using a suitable method for dictionary learning.
\item\emph{Dictionary de-correlation}: given $\Feas$, $\Dic$ and $\Coeff$ update the dictionary $\Dic$ to reduce its mutual coherence under the level $\coherence_{0}$.
\end{itemize}
\subsection{The iterative projections and rotations algorithm}\label{sec:ipr}
The \Acronym{ipr} algorithm has been proposed in order to solve the dictionary de-correlation step, while ensuring that the updated dictionary provides a sparse approximation with low residual norm, as indicated by the objective function \eqref{eq:iprcost} \cite{Barchiesi2013Le}.
The \Acronym{ipr} algorithm requires the calculation of the Gram matrix $\Gram=\Transpose{\Dic}\Dic$ which contains the inner products between any two atoms in the dictionary. $\Gram$ is iteratively projected onto two constraint sets, namely the structural constraint set $\stcset$ and the spectral constraint set $\spcset$. The former is the set of symmetric square matrices with unit diagonal values and off-diagonal values with magnitude smaller or equal than $\coherence_{0}$:
\small
\begin{equation*}
\stcset \definition \curlyb{\stcmat \in \ambient^{\nAtoms \times \nAtoms} : \stcmat = \Transpose{\stcmat}, \stcel_{i,i}=1,\max_{i > j}|\stcel_{i,j}|\leq \coherence_{0}}.
\end{equation*}
\normalsize
The latter is the set of symmetric positive semidefinite square matrices with rank smaller than or equal to $\nDimensions$:
\begin{equation*}
\spcset \definition \curlyb{ \spcmat \in \ambient^{\nAtoms \times \nAtoms} : \spcmat = \Transpose{\spcmat}, \operatorname{eig}(\spcmat)\geq \Vector{0}, \operatorname{rank}(\spcmat)\leq \nDimensions}
\end{equation*}
where the operator $\operatorname{eig}(\cdot)$ returns the vector of eigenvalues of its argument.
Starting from the Gram matrix of an initial dictionary $\Dictionary$, the \Acronym{ipr} method iteratively performs the following operations.
\begin{itemize}
\item \emph{Projection onto the structural constraint set}. The projection $\stcmat = \Projection_{\stcset}(\Gram)$ can be obtained by:
\begin{enumerate}
\item setting $\stcel_{i,i} = 1$,
\item limiting the off-diagonal elements so that, for $i \neq j$,
\small
\begin{equation}\label{eq:pscs}
\stcel_{i,j} = \operatorname{Limit}({\gram}_{i,j},\coherence_{0}) = \left\{ \begin{array}{rl}
\gram_{i,j} & \text{if} \quad |\gram_{i,j}|\leq \coherence_{0} \\
\operatorname{sgn}(\gram_{i,j})\coherence_{0} & \text{if} \quad |\gram_{i,j}| > \coherence_{0}
\end{array} \right.
\end{equation}
\normalsize
\end{enumerate}
\item \emph{Projection onto the spectral constraint set and factorization}. The projection $\spcmat = \Projection_{\spcset}(\Gram)$ and subsequent factorisation are obtained by:
\begin{enumerate}
\item calculating the eigenvalue decomposition (\Acronym{evd}) $\Gram = \eigvecmat \eigvalmat \Transpose{\eigvecmat}$,
\item thresholding the eigenvalues by keeping only the $\nDimensions$ largest positive ones.
\begin{equation*}
\left[\operatorname{Thresh}(\eigvalmat,\nDimensions) \right]_{i,i} = \left\{ \begin{array}{rl}
\lambda_{i,i} & \text{if} \quad i \leq N \; \text{and} \; \lambda_{i,i}>0 \\
0 & \text{if} \quad i > N \; \text{or} \; \lambda_{i,i}\leq 0
\end{array}\right.
\end{equation*}
where the eigenvalues in $\eigvalmat$ are ordered from the largest to the smallest. Following this step, at most $\nDimensions$ eigenvalues of the Gram matrix are different from zero,
\item factorizing the projected Gram matrix into the product $\Gram=\Transpose{\Dic}\Dic$ by setting:
\begin{equation}
\Dic = \eigvalmat^{1/2}\Transpose{\eigvecmat}.
\end{equation}
\end{enumerate}
\item \emph{Dictionary rotation}. Rotate the dictionary $\Dic$ to align it to the training set by solving the problem:
\begin{equation}\label{eq:rot}
\optimal{\orthmat} = \Minimise{\orthmat \Transpose{\orthmat} = \Matrix{I}}{\norm{\Feas - \orthmat\Dic\Coeff}{\Label{F}}}.
\end{equation}
The optimal rotation matrix can be calculated by:
\begin{enumerate}
\item computing the sample covariance between the observed signals and their approximations $\CovMat \definition (\Dic\Coeff)\Transpose{\Feas}$,
\item calculating the \Acronym{svd} of the covariance $\CovMat = \Matrix{U}\Matrix{\Sigma}\Transpose{\Matrix{V}}$,
\item setting the optimal rotation matrix to $\optimal{\orthmat}=\Matrix{V}\Transpose{\Matrix{U}}$,
\item rotating the dictionary $\Dic \leftarrow \optimal{\orthmat}\Dic$.
\end{enumerate}
\end{itemize}
More details about the \Acronym{ipr} algorithm can be found in \cite{Barchiesi2013Le}, including details of its computational cost.
%The code of the \Acronym{ipr} method is illustrated in Algorithm \ref{algo:ipr}.
%
%\begin{algo}
%\KwIn{$\Feas, \Dictionary, \Coeff, \coherence_{0}, \nIter$}
%\KwOut{$\optimal{\Dictionary}$}
%$\iter\gets1$\;
%\While{$\iter \leq \nIter$ and $\coherence(\Dictionary)>\coherence_{0}$}{
% \tcp{Calculate Gram matrix}
% $\Gram \gets \Transpose{\Dictionary}\Dictionary$\;
% \tcp{Project ont structural c.s.}
% $\operatorname{diag(\Gram)} \gets \Vector{1}$\;
% $\Gram \gets \operatorname{Limit}(\Gram,\coherence_{0})$\;
% \tcp{Project Gram matrix onto spectral c.s. and factorize}
% $[\eigvecmat, \eigvalmat] \gets \Acronym{evd}(\Gram)$\;
% $\eigvalmat \gets \operatorname{Thresh}(\eigvalmat,\nDimensions)$\;
% $\Dictionary \gets \eigvalmat^{1/2}\Transpose{\eigvecmat}$\;
% \tcp{Rotate dictionary}
% $\CovMat \gets \Feas\Transpose{(\Dictionary\Coeff)}$\;
% $[\Matrix{U},\Matrix{\Sigma},\Matrix{V}] \gets \Acronym{svd}(\CovMat)$\label{algo:ipr:svd}\;
% $\OrthMat \gets \Matrix{V}\Transpose{\Matrix{U}}$\;
% $\Dictionary \gets \OrthMat\Dictionary$\;
% $\iter\gets\iter+1$\;
%}
%\caption{\label{algo:ipr}Iterative projections and rotations (\Acronym{ipr})}
%\end{algo}
\section{Learning Incoherent Subspaces}\label{sec:lis}
The \Acronym{ipr} algorithm learns a dictionary where all the atoms are mutually incoherent. Therefore, given any two disjoint sets $\Lambda\bigcap\Gamma=\emptyset$ that identify non-overlapping collections of atoms, the sub-dictionaries $\Dic_{\Lambda}, \Dic_{\Gamma}$ are also mutually incoherent.
Starting from this observation, the main intuition driving the development of a supervised \Acronym{ipr} (\Acronym{s-ipr}) algorithm for classification is to learn mutually incoherent sub-dictionaries that approximate features from different classes of signals. The sub-dictionaries are in turn used to define incoherent sub-spaces, and features are projected onto these sub-spaces yielding discriminative dimensionality reduction.
\subsection{The supervised \Acronym{ipr} algorithm}\label{sec:iprclass}
Let $\curlyb{\cat_{\iFea}\in\Cat}_{\iFea=1}^{\nFea},\; \Cat=\curlyb{\uniCat_{1},\uniCat_{2},\dots,\uniCat_{\nCat}}$ be a set of labels that identify the category of the vectors of features $\fea_{\iFea}$, whose elements belong to a set $\Cat$ of $\nCat$ possible categories. The columns of the matrix $\Feas_{\iCat}$ contain a selection of the features extracted from signals belonging to the $\iCat$-th category.
To learn incoherent sub-dictionaries from the entire set of features, we must first cluster the atoms to different classes\footnote{Note that the term \emph{cluster} implies that a this stage the algorithm needs to make an unsupervised decision, since there is no any a-priori reason to assign a given atom to any particular class.}, and then only proceed with their de-correlation if they are assigned to different categories (while allowing coherent atoms to approximate features from the same class). To this aim, we employ the matrix $\Coeff$ to measure the contribution of every atom to the approximation of features belonging to each class.
Let $\coeff_{\iCat}^{\iAto}$ indicate the $\iAto$-th row of the matrix $\Coeff_{\iCat}$ containing the coefficients that contribute to the approximation of $\Feas_{\iCat}$, and $\nDim_{\iCat}$ indicate the number of its elements. A coefficient $\gamma_{\iAto,\iCat}$ is defined as:
\begin{equation}
\gamma_{\iAto,\iCat} \definition \frac{1}{\nDim_{\iCat}}\norm{\coeff_{\iCat}^{\iAto}}{1},
\end{equation}
and every atom $\atom_{\iAtom}$ is associated with the category to which it maximally contributes $\optimal{\iCat}_{\iAto} = \underset{\iCat}{\arg\max}\curlyb{\gamma_{\iAto,\iCat}}$.
Grouping together atoms that have been assigned to the same class leads to a set of sub-dictionaries whose size and rank depends on the number of atoms for each class, and to their linear dependence. As a general heuristic, if features corresponding to different classes do not occupy the same sub-space (according to the active elements in $\Coeff$), a full-rank dictionary $\Dic$ with $\nAto \geq \nDim \gg \nCat$ ensures that $\optimal{\iCat}_{\iAto}$ identify $\nCat$ non-empty and disjoint sub-dictionaries $\curlyb{\Dic_{\iCat}}_{\iCat=1}^{\nCat}$.
Once the atoms have been clustered, the Gram matrix $\Gram$ is computed and iteratively projected as in the method described in Section \ref{sec:ipr}, with the difference that equation \eqref{eq:pscs} is modified in order to only constraint the mutual coherence between atoms assigned to different categories
\small
\begin{equation}\label{eq:pscs2}
\operatorname{Limit}({\gram}_{i,j},\coherence_{0},\optimal{\Vector{\iCat}}) = \left\{ \begin{array}{rl}
\gram_{i,j} & \text{if} \quad |\gram_{i,j}|\leq \coherence_{0} \, \text{or} \, \optimal{\iCat}_{i}=\optimal{\iCat}_{j} \\
\operatorname{sgn}(\gram_{i,j}) \coherence_{0} & \text{if} \quad |\gram_{i,j}| > \coherence_{0} \, \text{and} \, \optimal{\iCat}_{i}\neq\optimal{\iCat}_{j}
\end{array} \right.
\end{equation}
\normalsize
A further modification of the standard \Acronym{ipr} algorithm presented in \cite{Barchiesi2013Le} consists in the update of the Gram matrix, performed by computing its element-wise average with the projection $\stcmat = \Projection_{\stcset}(\Gram)$ (rather than by using the projection alone). This heuristic has led to improved empirical results by preventing $\Gram$ from changing too abruptly.
The complete supervised \Acronym{s-ipr} method is summarised in Algorithm \ref{algo:sipr}. Note that the mutual coherence $\coherence_{\optimal{\iCat}}(\Dic) = \underset{\optimal{\iCat}_{i}\neq\optimal{\iCat}_{j}}{\arg\max}\inner{\atom_{i}}{\atom_{j}}$ indicated in this algorithm measures the inner product between any two atoms assigned to different categories since atoms assigned to the same category are allowed to be mutually coherent.
\begin{algo}
\KwIn{$\Feas, \Dictionary, \Coeff, \coherence_{0}, \cats, \nIter$}
\KwOut{$\optimal{\Dictionary}$}
$\iter\gets1$\;
\tcp{Cluster atoms}
$\Coeff_{\iCat} \gets \squareb{\coeff_{j}} \forall j \in \uniCat_{\iCat}$\;
$\gamma_{\iAto,\iCat} \gets \norm{\coeff_{\iCat}^{\iAto}}{1}/\nDim_{\iCat}$\;
$\optimal{\iCat}_{\iAto} = \underset{\iCat}{\arg\max}\curlyb{\gamma_{\iAto,\iCat}}$\;
\While{$\iter \leq \nIter$ and $\coherence_{\optimal{\iCat}}(\Dictionary)>\coherence_{0}$}{
\tcp{Calculate Gram matrix}
$\Gram \gets \Transpose{\Dictionary}\Dictionary$\;
\tcp{Project onto structural c.s.}
$\operatorname{diag}(\stcmat) \gets \Vector{1}$\;
$\stcmat \gets \operatorname{Limit}(\Gram,\coherence_{0},\optimal{\Vector{\iCat}})$\;
$\Gram \gets \frac{1}{2}\Gram + \frac{1}{2}\stcmat$\;
\tcp{Project onto spectral c.s. and factorize}
$[\eigvecmat, \eigvalmat] \gets \Acronym{evd}(\Gram)$\;
$\eigvalmat \gets \operatorname{Thresh}(\eigvalmat,\nDimensions)$\;
$\Dictionary \gets \eigvalmat^{1/2}\Transpose{\eigvecmat}$\;
\tcp{Rotate dictionary}
$\CovMat \gets \Feas\Transpose{(\Dictionary\Coeff)}$\;
$[\Matrix{U},\Matrix{\Sigma},\Matrix{V}] \gets \Acronym{svd}(\CovMat)$\label{algo:ipr:svd}\;
$\OrthMat \gets \Matrix{V}\Transpose{\Matrix{U}}$\;
$\Dictionary \gets \OrthMat\Dictionary$\;
$\iter\gets\iter+1$\;
}
\caption{\label{algo:sipr}Supervised \Acronym{ipr}}
\end{algo}
\subsection{Classification via incoherent subspaces}\label{sec:class}
The \Acronym{s-ipr} algorithm allows to learn a set of sub-dictionaries $\curlyb{\Dic_{\iCat}}$ that contain mutually incoherent atoms. These cannot be directly used to define discriminative subspaces because, depending on $\nDim$ and on the rank of each sub-dictionary, atoms belonging to disjoint sub-dictionaries might span identical subspaces. Instead, we fix a rank $\nDimSub\leq\floor{\nDim/\nCat}$ and choose a collection of $\nDimSub$ linearly independent atoms from each sub-dictionary $\Dic_{\iCat}$, using the largest values of $\gamma_{\iAto,\iCat}$ to define a picking order. Thus, we obtain a set $\curlyb{\Spa_{\iCat}}_{\iCat=1}^{\nCat}$ of incoherent sub-spaces of rank $\nDimSub$ embedded in the space $\ambient^{\nDim}$, and use them to derive a feature transform for classification.
Each feature vector $\fea_{\iFea}$ that belongs to the class $\cat_{\iFea}$ is projected onto the relative subspace, yielding a set of transformed features $\curlyb{\newFea_{\iFea}}_{\iFea=1}^{\nFea}$.
\begin{equation}
\newFea_{\iFea} = \Spa_{\cat_{\iFea}}\pseudoinverse{\Spa_{\cat_{\iFea}}}\fea_{\iFea}
\end{equation}
where $\Spa^{\dagger}$ denotes the Moore-Penrose pseudo-inverse of the matrix $\Spa$ and needs to be used in place of the transposition operator because the columns of $\Spa$ are in general not orthogonal.
When an unlabelled signal is presented to the classifier, the corresponding vector of features $\fea$ is projected onto all the learned sub-spaces. Then, the nearest sub-space is chosen using an Euclidean distance measure, and the corresponding projection $\newFea$ used as the transformed feature.
\begin{align}
\optimal{\iCat} &= \underset{\iCat}{\arg\min}\norm{\fea-\Spa_{\iCat}\pseudoinverse{\Spa_{\iCat}}\fea}{2} \\
\newFea &= \Spa_{\optimal{\iCat}}\pseudoinverse{\Spa_{\optimal{\iCat}}}\fea
\end{align}
The subspace $\optimal{\iCat}$ can be directly used as an estimator of the category of the signal $\optimal{\cat}$. Alternatively, a simple \emph{k-neaerst neighbour} classifier can be employed on the transformed features, and a class can be inferred as:
\begin{equation}
\optimal{\cat} = \texttt{knn}(\newFea,\NewFeas,\cats)
\end{equation}
where $\NewFeas$ represents the matrix of training features after the transform stage. This latter approach is especially suitable when working with a large number of classes in a space of relatively small dimension, as in this case multiple classes might be assigned to the same subspace.
\section{Numerical Experiments}\label{sec:ne}
\subsection{Feature visualisation}\label{sec:visu}
To illustrate the \Acronym{s-ipr} algorithm for feature transform, we first run visualisation experiments depicting how different feature transform methods act on training and test data.
\subsubsection{Synthetic data}
\begin{figure}
\centering
\includegraphics[width=.6\textwidth]{./Code/Datasets/GetToyExampleDataset.pdf}
\caption{\label{fig:toy}Synthetic data generated along one-dimensional subspaces of $\real^{2}$.}
\end{figure}
Figure \ref{fig:toy} displays a total of $1500$ synthetic features in $\real^{2}$ belonging to $3$ different classes that we generated for this experiment. For each class, first we draw values distributed uniformly in the interval $\squareb{-1,1}$ and assign them to the first component of the features (the \emph{x} coordinate). Then, we add Gaussian noise with variance $0.1$ to the second component (the \emph{y} coordinate), and we rotate the resulting data by the angles $\theta_{0}=0$, $\theta_{1}=\pi/4$ and $\theta_{3}=\pi/2$ for the $3$ classes respectively. This way, features belonging to different classes are clustered along different one-dimensional sub-spaces of $\real^{2}$.
\begin{figure}
\includegraphics[width=\textwidth]{./Code/toyvisu.pdf}
\caption{\label{fig:toyNewFea}Feature transform applied to the synthetic data in Figure \ref{fig:toy}. Different colours correspond to different classes, `+' and `o' markers represent samples taken from the training and test set respectively.}
\end{figure}
Figure \ref{fig:toyNewFea} displays the result of the application of feature transforms to the data depicted in Figure \ref{fig:toy} using subspaces of dimension $1$ (with the exception of \Acronym{lda} that projects the data onto a space of dimension $\nUniCat-1=2$). To generate the plots, we divided the data into a training set (displayed using the `+' marker) and a test set (displayed using the `o' marker). Samples were drawn in random order from the dataset and assigned to either the training set or the test set, with the former containing $70\%$ of the total data and the latter containing the remaining $30\%$. Then, we applied feature transforms on the training set, thereby learning the transform operators, and applied them to the test set.
Starting from the top-left plot, we can observe that \Acronym{PCA} identified the direction $x=y$ as the one-dimensional subspace that contains most of the variance of the training set. However, given the type of dataset and the dimensionality reduction caused by \Acronym{pca}, features from all classes are overlapping, making this transform a poor choice for classification. Similar observations can be drawn from analysing the result of \Acronym{s-pca}, although this transform identifies the direction $y=0$ as the one that leads to statistical dependence between the value of the transformed features in the training set and the relative class. \Acronym{lda} does not introduce any dimensionality reduction in this case, as it projects the features onto a space of dimension $\nUniCat-1=2$, leaving the original features unaltered. However, in the \Acronym{lda} plot we can appreciate the separation between training set and test set that is difficult to notice in the other plots.
Finally, the plot at the right-bottom corner of Figure \ref{fig:toyNewFea} displays the results of the \Acronym{s-ipr} algorithm. In setting the parameters of \Acronym{s-ipr}, we chose a $2$ times over-complete dictionary, a number of active atoms equal to half the dimension of the data, and minimal mutual coherence. In the case considered here, this means $\nAto = 4$, $\nActiveAtoms = 1$ and $\coherence = \sqrt{(\nAto-\nDim)/\nDim(\nAto-1)} \approx 0.33$. As discussed in Section \ref{sec:iprclass}, \Acronym{s-ipr} does not project whole sets of features onto a unique sub-space, but rather learns one sub-space for each category, and projects features onto the nearest sub-space. The result depicted here shows that three directions were identified containing data from mostly one category each. Since the incoherent dictionary learning is designed to learn atoms with minimal mutual coherence, the angles between the directions of the sub-spaces learned by \Acronym{s-ipr} are approximately equal. Prior information regarding the directions of the data would allow to relax the parameter $\coherence$, and track more closely the directions of the three data classes.
\subsubsection{Iris dataset}
\begin{figure}
\centering
\includegraphics[width=0.6\textwidth]{./Code/Datasets/GetFisherIrisDataset.pdf}
\caption{\label{fig:fisher3} First three features of the `iris' dataset depicting measurements of sepal length, sepal width and petal length of three iris species.}
\end{figure}
Figure \ref{fig:fisher3} displays a subset of the `iris' dataset, a popular database that has been used extensively to test and benchmark classification algorithms. The original dataset contains measurements of the sepal length, sepal width, petal length and petal width of three species of iris, namely `setosa', `versicolor' and `virginica'. In this visualisation experiment we selected the first $3$ features to be able to depict the data using three dimensional scatter plots. From observing the distribution of the data in the feature space, we see that `setosa' is relatively separated from the other two classes, while the features relative to `virginica' and `versicolor' substantially overlap, with only a few exemplars of `virginica' being distinguishable due to large sepal length and petal length.
\begin{figure}
\includegraphics[width=\textwidth]{./Code/fishervisu.eps}
\caption{\label{fig:fisher3NewFea}Feature transform applied to the iris data in Figure \ref{fig:fisher3}. Different colours correspond to different classes, `+' and `o' markers represent samples taken from the training and test set respectively.}
\end{figure}
The results of feature transforms are depicted in Figure \ref{fig:fisher3NewFea}. This time, we learn $2$ dimensional subspaces from the $3$ dimensional data points and plot the transformed features, along with the learned planes. We observe that \Acronym{pca} identifies a direction along a diagonal axis that follows the distribution of features displayed in Figure \ref{fig:fisher3}. \Acronym{s-pca}
, on the other hand, projects the features onto a horizontal plane that slightly enhances the separation between `versicolor' and `virginica' samples. \Acronym{lda} results in a projection where features belonging to the same category are closely clustered together, but fails to separate the classes `versicolor' and `virginica'. Finally, the output of \Acronym{s-ipr} displays three distinct sub-spaces associated with the three classes. As in the other plots, the separation between `versicolor' and `virginica' is far from perfect, however features from the two classes are mostly projected onto the respective sub-spaces. Features belonging to the `setosa' category are mostly clustered together as a result of their projection onto the black subspace, however we can note a few test samples that have been associated by the algorithm to the blue sub-space.
\subsection{Classification}\label{sec:cla}
\begin{table}
\centering
\begin{tabular}{lccc}\textbf{Name} & \nDim & \nUniCat & \nFea \\
\hline
Iris & 4 & 3 & 150\\
Balance & 4 & 3 & 625\\
Parkinsons & 23 & 2 & 197 \\
Sonar & 60 & 2 & 208\\
USPS & 256 & 3 & 1405
\end{tabular}
\caption{\label{tab:datasets}Dataset used in the classification evaluation of feature transform algorithms. All the datasets can be downloaded from \href{http://archive.ics.uci.edu/ml/datasets.html}{http://archive.ics.uci.edu/ml/datasets.html}. Note that we only use a subset of the USPS dataset containing the digits $1$, $3$ and $8$.}
\end{table}
In the previous section, we have illustrated how the \Acronym{s-ipr} algorithm is able to learn incoherent sub-spaces that model the distribution of features belonging to different classes. Here we evaluate \Acronym{s-ipr} and the other feature transform algorithms in the context of supervised classification. To perform the classification, features are transformed using the methods already used for comparison in Section \ref{sec:visu} by learning a transform operator on the training set and applying it to the test set. We use a $5$-fold stratified cross-validation to classify all the features in a dataset during the test stage. This method produces $5$ independent classification problems with a ratio between the number of training and test samples equal to $8:2$. Once the features have been transformed, a $k$-nearest neighbour classifier with $k=5$ is used to estimate a class.
We employ the datasets detailed in Table \ref{tab:datasets}, and for each of them we evaluate the misclassification ratio, that is defined as the fraction of misclassified samples as a proportion of the total number of samples in the test set, averaged over the $5$ independent classification problems created by the stratified cross-validation protocol.
\begin{figure}
\centering
\subfigure{\includegraphics[width=.48\textwidth]{Code/Util/fisheriris.pdf}}
\subfigure{\includegraphics[width=.48\textwidth]{Code/Util/balance.pdf}}
\subfigure{\includegraphics[width=.48\textwidth]{Code/Util/parkinsons.pdf}}
\subfigure{\includegraphics[width=.48\textwidth]{Code/Util/sonar.pdf}}
%\subfigure{\includegraphics[width=.7\textwidth]{Code/Util/usps.pdf}}
\caption{\label{fig:class} Misclassification ratio as a function of the rank of the subspace employed during feature transforms for the datasets `iris',`balance',`Parkinsons' and `sonar'}
\end{figure}
Figure \ref{fig:class} displays for each dataset the misclassification ratio as a function of the rank of the subspace learned by the algorithms. In the plots `none' indicates that no feature transform was applied (hence resulting in a sub-space rank equal to the dimension of the original features). In general we can see that \Acronym{s-ipr} does not perform as well as the other techniques, and is only comparable at high ranks that do not achieve an overall better classification ratio. Starting from the `iris' dataset, \Acronym{lda} achieves the best performance followed by one-dimensional subspaces learned using \Acronym{pca}. Both \Acronym{s-pca} and \Acronym{s-ipr} work better when learning subspaces of high rank. Note that, at rank $\nDim=4$ all the methods are equivalent because they are not performing dimensionality reduction. The results relative to the balance dataset are similar, with again \Acronym{lda} achieving the best misclassification ratio. Although the results on the `Parkinsons' and `sonar' datasets present similar trends regarding \Acronym{s-ipr}, here \Acronym{lda} does not prove to be as successful as \Acronym{pca} and \Acronym{s-pca} in separating features belonging to different classes.
\section{Conclusion}\label{sec:end}
\subsection{Summary}
We have presented the \Acronym{s-ipr} algorithm for learning incoherent subspaces from data belonging to different categories. The encouraging experimental results obtained on the visualisation of the synthetic dataset and of a subset of features taken from the 'iris` dataset motivated us to test \Acronym{s-ipr} as a general method for feature transform to be used in classification problems. Unfortunately, we found that the performance of our proposed method on a group of datasets commonly used to benchmark classification algorithms is only competitive compared to traditional and state-of-the-art methods for feature transform at high sub-space ranks.
The negative results presented in Section \ref{sec:cla} do not imply that \Acronym{s-ipr} is completely unsuitable as a tool for modelling data for classification, but they rather open a few important areas of future research that should be investigate to better understand the strengths and limitations of the proposed method.
\subsection{Future work}
The main assumption made when using incoherent dictionary learning for classification is that high dimensional features are arranged onto lower-dimensional sub-spaces, and that features belonging to different classes can be modelled using different subspaces that are mutually incoherent. This assumption might be met by some datasets, but might not generally be satisfied by others. Understanding the general distribution of the features in a dataset might be a necessary first step to inform a subsequent choice of algorithm, so that \Acronym{s-ipr} can be used in cases where its premise about the feature distribution is valid. This same argument holds for the whole class of linear models that comprises the dictionary learning model. Indeed, many feature transform techniques have equivalent \emph{kernelized} versions to model non-linear data.
Other substantial improvements can be made on the algorithm itself. The present implementation of \Acronym{s-ipr} contains a fixed parameter $\coherence$ that promotes minimal mutual coherence between the sub-spaces used to approximate different data classes. Knowledge about the distribution of the features might lead to relaxing this parameter, learning sub-spaces that are closer to the true distribution of the features and in turn improving class separation. Moreover, different values of mutual coherence for different pairs of subspaces can be easily included in the optimisation, greatly enhancing the flexibility of \Acronym{s-ipr} as a modelling tool.
\bibliographystyle{IEEEbib}
\bibliography{bibliography.bib}
\end{document}
|
%/* ----------------------------------------------------------- */
%/* */
%/* ___ */
%/* |_| | |_/ SPEECH */
%/* | | | | \ RECOGNITION */
%/* ========= SOFTWARE */
%/* */
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%/* developed at: */
%/* */
%/* Speech Vision and Robotics group */
%/* Cambridge University Engineering Department */
%/* http://svr-www.eng.cam.ac.uk/ */
%/* */
%/* Entropic Cambridge Research Laboratory */
%/* (now part of Microsoft) */
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%/* Copyright: Microsoft Corporation */
%/* 1995-2000 Redmond, Washington USA */
%/* http://www.microsoft.com */
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%/* ** See the file License for the Conditions of Use ** */
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%/* */
%/* ----------------------------------------------------------- */
%
% HTKBook - Steve Young 1/12/97
%
\mychap{Speech Input/Output}{speechio}
Many tools need to input parameterised speech data and \HTK\ provides
a number of different methods for doing this:
\begin{itemize}
\item input from a previously encoded speech parameter file
\item input from a waveform file which is encoded as part of the
input processing
\item input from an audio device which is encoded as part of the
input processing.
\end{itemize}
For input from a waveform file, a large number of different file formats
are supported, including all of the commonly used CD-ROM formats.
Input/output for parameter files is limited to the standard \HTK\ file format
and the new Entropic Esignal format.
\sidepic{Tool.spio}{60}{}
All \HTK\ speech input\index{speech input} is controlled by configuration
parameters which give details of what processing operations to apply to each
input speech file or audio source. This chapter describes speech input/output
in \HTK. The general mechanisms are explained and the various configuration
parameters are defined. The facilities for signal pre-processing, linear
prediction-based processing, Fourier-based processing and vector quantisation
are presented and the supported file formats are given. Also described are the
facilities for augmenting the basic speech parameters with energy measures,
delta coefficients and acceleration (delta-delta) coefficients and for
splitting each parameter vector into multiple data streams to form
\textit{observations}. The chapter concludes with a brief description of the
tools \htool{HList} and \htool{HCopy} which are provided for viewing,
manipulating and encoding speech files.
\mysect{General Mechanism}{genio}
The facilities for speech input and output in \HTK\ are provided
by five distinct modules: \htool{HAudio}, \htool{HWave},
\htool{HParm}, \htool{HVQ} and \htool{HSigP}. The interconnections
between these modules are shown in Fig.~\href{f:Spmods}.
\index{speech input!general mechanism}
\sidefig{Spmods}{62}{Speech Input Subsystem}{2}{
Waveforms
are read from files using \htool{HWave}, or are input direct from
an audio device using \htool{HAudio}. In a few rare cases, such as
in the display tool \htool{HSLab}, only the speech waveform is needed.
However, in most cases the waveform is wanted in parameterised form and
the required encoding is performed by \htool{HParm}
using the signal processing operations defined in
\htool{HSigP}. The parameter vectors are output by \htool{HParm}
in the form of observations which are the basic units of data processed
by the \HTK\ recognition and training tools. An observation contains all
components of a raw parameter vector but it may be possibly split into
a number of independent parts. Each such part is regarded by a \HTK\ tool
as a statistically independent data stream. Also, an observation
may include VQ indices attached to each data stream. Alternatively,
VQ indices can be read directly from a parameter file in which case the
observation will contain only VQ indices.
}
Usually a \HTK\ tool will require a number of speech data files to be
specified on the command line. In the majority of cases, these
files will be required in parameterised form. Thus, the following example
invokes the \HTK\ embedded training tool \htool{HERest}
to re-estimate a set of models using the speech data
files \texttt{s1}, \texttt{s2}, \texttt{s3}, \ldots . These are
input via the library module \htool{HParm} and they
must be in exactly the form needed by the models.
\begin{verbatim}
HERest ... s1 s2 s3 s4 ...
\end{verbatim}
However, if the external form of the speech data files is not in the
required form, it will often be possible to convert them automatically during
the input process.
To do this, configuration parameter values are specified whose function
is to define exactly
how the conversion should be done.
The key idea is that there is a
\textit{source parameter kind} and \textit{target parameter kind}.
The source refers to the natural form of the data in
the external medium and the target refers to the form of the
data that is required internally by the \HTK\ tool.
The principle function of the speech
input subsystem is to convert the source parameter kind into the
required target parameter kind. \index{speech input!automatic conversion}
Parameter kinds consist of a base form to which one or more
qualifiers may be attached where each qualifier consists of
a single letter preceded by an underscore character.\index{qualifiers}
Some examples of parameter kinds are
\begin{varlist}
\fwitem{2cm}{WAVEFORM} simple waveform
\fwitem{2cm}{LPC} linear prediction coefficients
\fwitem{2cm}{LPC\_D\_E} LPC with energy and delta coefficients
\fwitem{2cm}{MFCC\_C} compressed mel-cepstral coefficients
\end{varlist}
\index{speech input!target kind}
The required source and target parameter kinds are specified
using the configuration parameters \texttt{SOURCEKIND}
\index{sourcekind@\texttt{SOURCEKIND}} and
\texttt{TARGETKIND}\index{targetkind@\texttt{TARGETKIND}}.
Thus, if the following configuration parameters were defined
\begin{verbatim}
SOURCEKIND = WAVEFORM
TARGETKIND = MFCC_E
\end{verbatim}
then the speech input subsystem would expect each input file to contain
a speech waveform and it would convert it to mel-frequency cepstral
coefficients with log energy appended.
The source need not be a waveform. For example, the configuration
parameters
\begin{verbatim}
SOURCEKIND = LPC
TARGETKIND = LPREFC
\end{verbatim}
would be used to read in files containing linear prediction coefficients
and convert them to reflection coefficients.
For convenience, a special parameter kind called
\texttt{ANON}\index{anon@\texttt{ANON}} is provided. When the source is
specified as \texttt{ANON} then the actual kind of the source is determined
from the input file. When \texttt{ANON} is used in the target kind, then it is
assumed to be identical to the source. For example, the effect of the
following configuration parameters
\begin{verbatim}
SOURCEKIND = ANON
TARGETKIND = ANON_D
\end{verbatim}
would simply be to add delta coefficients to whatever the source form
happened to be.
The source and target parameter kinds default to \texttt{ANON}
to indicate that by default
no input conversions are performed. Note, however, that where two or more
files are listed on the command line, the meaning of
\texttt{ANON} will not be re-interpreted from one file to the next. Thus, it
is a general rule, that any tool reading multiple source speech files requires
that all the files have the same parameter kind.
The conversions applied by \HTK's input subsystem can be complex and may
not always behave exactly as expected. There are two facilities that can
be used to help check and debug the set-up of the speech i/o
configuration parameters.
Firstly, the tool \htool{HList} simply displays speech data by listing it
on the terminal. However, since \htool{HList} uses the speech
input subsystem like
all \HTK\ tools, if a value for \texttt{TARGETKIND} is set, then
it will display the target
form rather than the source form. This is the simplest way to check the form of
the speech data that will actually be delivered to a \HTK\ tool.
\htool{HList} is described
in more detail in section~\ref{s:UseHList} below.
Secondly, trace output can be generated from the \htool{HParm} module
by setting the \texttt{TRACE} configuration file parameter. This is a
bit-string in which individual bits cover different parts of the
conversion processing. The details are given in the reference section.
To summarise, speech input in \HTK\ is controlled by configuration
parameters. The key parameters are \texttt{SOURCEKIND} and {\tt
TARGETKIND} which specify the source and target parameter kinds.
These determine the end-points of the required input conversion.
However, to properly specify the detailed steps in between, more
configuration parameters must be defined.
These are described in subsequent sections.
\mysect{Speech Signal Processing}{sigproc}
In this section, the basic mechanisms involved in transforming a
speech waveform into a sequence of parameter vectors will be
described. Throughout this section, it is assumed that the
\texttt{SOURCEKIND} is \texttt{WAVEFORM} and that data is being read from
a HTK format file via \htool{HWave}. Reading from different format
files is described below in section~\ref{s:waveform}.
Much of the
material in this section also applies to data read direct from an audio
device, the
additional features needed to deal with this latter case are
described later in section~\ref{s:audioio}.
\vspace{0.2cm}
\index{speech input!blocking}
The overall process is illustrated in Fig.~\href{f:Blocking}
which shows the sampled waveform being converted into a
sequence of parameter blocks. In general, \HTK\ regards
both waveform files and parameter files as being just
sample sequences, the only difference being that in the former
case the samples are 2-byte integers and in the latter they
are multi-component vectors. The sample rate of the input
waveform will normally be determined from the input file
itself. However, it can be set explicitly using the
configuration parameter \texttt{SOURCERATE}. The period
between each parameter vector determines the output sample
rate and it is set using the configuration parameter
\texttt{TARGETRATE}. The segment of waveform used to determine
each parameter vector is usually referred to as a window
and its size is set by the
configuration parameter \texttt{WINDOWSIZE}. Notice that the
window size and frame rate are independent. Normally,
the window size will be larger than the frame rate so that
successive windows overlap as illustrated in
Fig.~\href{f:Blocking}.
\index{sourcerate@\texttt{SOURCERATE}}
\index{targetrate@\texttt{TARGETRATE}}
\index{windowsize@\texttt{WINDOWSIZE}}
For example, a waveform sampled
at 16kHz
would be converted into 100 parameter vectors per
second using a 25 msec window by setting the following
configuration parameters.
\begin{verbatim}
SOURCERATE = 625
TARGETRATE = 100000
WINDOWSIZE = 250000
\end{verbatim}
Remember that all durations are specified in 100 nsec units\footnote{
The somewhat bizarre choice of 100nsec units originated in Version 1 of
\HTK\ when times were represented by integers and this unit was the best
compromise between precision and range. Times are now represented by
doubles and hence the constraints no longer apply. However, the need for backwards
compatibility means that 100nsec units have been retained. The names
\texttt{SOURCERATE} and \texttt{TARGETRATE} are also non-ideal,
\texttt{SOURCEPERIOD} and \texttt{TARGETPERIOD} would be better.
}.
\sidefig{Blocking}{50}{Speech Encoding Process}{2}{}
Independent of what parameter kind is required, there are some simple
pre-processing operations that can be applied prior to performing the actual
signal analysis.\index{speech input!pre-processing}
Firstly, the DC mean can be removed from the source waveform by setting the
Boolean configuration parameter
\texttt{ZMEANSOURCE}\index{zmeansource@\texttt{ZMEANSOURCE}} to true
(i.e.\ \texttt{T}). This is useful when\index{speech input!DC offset}
the original analogue-digital conversion has added a DC offset to the
signal. It is applied to each window individually so that it can be
used both when reading from a file and when using direct audio
input\footnote{ This method of applying a zero mean is different to
HTK Version 1.5 where the mean was calculated and subtracted from the
whole speech file in one operation. The configuration variable
\texttt{V1COMPAT} can be set to revert to this older behaviour.}.
Secondly, it is common practice to pre-emphasise
the signal by applying the first order difference equation
\hequation{
{s^{\prime}}_n = s_n - k\,s_{n-1}
}{preemp}
to the samples\index{speech input!pre-emphasis}
$\{s_n, n=1,N \}$ in each window. Here $k$ is the
pre-emphasis\index{pre-emphasis} coefficient which should be in the range
$0 \leq k < 1$. It is specified using the configuration
parameter \texttt{PREEMCOEF}\index{preemcoef@\texttt{PREEMCOEF}}.
Finally,
it is usually beneficial to taper the
samples in each window so that discontinuities at the window
edges are attenuated. This is done by setting the
Boolean configuration
parameter \texttt{USEHAMMING}\index{usehamming@\texttt{USEHAMMING}}
to true.
This applies the following transformation to the samples
$\{s_n, n=1,N\}$ in the window
\hequation{
{s^{\prime}}_n = \left\{ 0.54 - 0.46 \cos \left( \frac{2 \pi (n-1)}{N-1}
\right) \right\} s_n
}{ham}
When both pre-emphasis and Hamming windowing are enabled,
pre-emphasis is performed
first.\index{speech input!Hamming window function} \index{Hamming Window}
In practice, all three of the above are usually applied.
Hence, a configuration file will typically contain the
following
\begin{verbatim}
ZMEANSOURCE = T
USEHAMMING = T
PREEMCOEF = 0.97
\end{verbatim}
Certain types of artificially generated waveform data can cause numerical
overflows with some coding schemes. In such cases adding a small amount of
random noise to the waveform data solves the problem. The noise is added
to the samples using
\hequation{
{s^{\prime}}_n = s_n + q RND()
}{dither}
where $RND()$ is a uniformly distributed random value over the interval
$[-1.0, +1.0)$ and $q$ is the scaling factor. The amount of noise added
to the data ($q$) is set with the configuration parameter
\index{adddither@\texttt{ADDDITHER}}\texttt{ADDDITHER} (default value $0.0$).
A positive value causes the noise signal added to be the same every time
(ensuring that the same file always gives exactly the same results). With a
negative value the noise is random and the same file may produce slightly
different results in different trials.
One problem that can arise when processing speech waveform files obtained from
external sources, such as databases on CD-ROM, is that the
byte-order\index{byte-order} may be different to that used by the machine on
which \HTK\ is running. To deal with this problem, \htool{HWave} can perform
automatic byte-swapping in order to preserve proper byte order. \HTK\ assumes
by default that speech waveform data is encoded as a sequence of 2-byte
integers as is the case for most current speech databases\footnote{Many of the
more recent speech databases use compression. In these cases, the data may be
regarded as being logically encoded as a sequence of 2-byte integers even if
the actual storage uses a variable length encoding scheme.}.
If the source format is known, then \htool{HWave} will also make an assumption
about the byte order used to create speech files in that format. It then checks
the byte order of the machine that it is running on and automatically performs
byte-swapping if the order is different. For unknown formats, proper byte order
can be ensured by setting the configuration parameter
\texttt{BYTEORDER}\index{byteorder@\texttt{BYTEORDER}} to \texttt{VAX} if the
speech data was created on a little-endian machine such as a VAX or an IBM PC,
and to anything else (e.g. \texttt{NONVAX}) if the speech data was created on a
big-endian machine such as a SUN, HP or Macintosh machine. \index{speech
input!byte order}
The reading/writing of \HTK\ format waveform files can be further controlled
via the configuration parameters \texttt{NATURALREADORDER} and
\texttt{NATURALWRITEORDER}. The effect and default settings of these parameters
are described in section~\href{s:byteswap}.
\index{byte swapping}
Note that \texttt{BYTEORDER} should not be used when \texttt{NATURALREADORDER}
is set to true. Finally, note that \HTK\ can also byte-swap parameterised
files in a similar way provided that only the byte-order of each 4 byte float
requires inversion.
\mysect{Linear Prediction Analysis}{lpcanal}
In linear prediction (LP) \index{linear prediction} analysis, the
vocal tract transfer function
is modelled by an all-pole filter\index{all-pole filter} with transfer function\footnote{
Note that some textbooks define the denominator of equation~\ref{e:allpole}
as $1 - \sum_{i=1}^p a_i z^{-i}$ so that the filter coefficients are the
negatives of those computed by \HTK.}
\hequation{
H(z) = \frac{1}{\sum_{i=0}^p a_i z^{-i}}
}{allpole}
where $p$ is the number of poles and $a_0 \equiv 1$.
The filter coefficients $\{a_i \}$ are chosen to minimise
the mean square filter prediction error summed over the analysis
window. The \HTK\ module \htool{HSigP} uses the \textit{autocorrelation
method} to perform this optimisation as follows.
Given a window of speech samples $\{s_n, n=1,N \}$,
the first $p+1$ terms of the autocorrelation sequence are
calculated from
\hequation{
r_i = \sum_{j=1}^{N-i} s_j s_{j+i}
}{autoco}
where $i = 0,p$.
The filter coefficients are then computed recursively
using a set of auxiliary coefficients $\{k_i\}$ which can be
interpreted as the reflection coefficients of an equivalent
acoustic tube and the prediction error $E$ which is initially
equal to $r_0$. Let $\{k_j^{(i-1)} \}$ and $\{a_j^{(i-1)} \}$
be the reflection and filter coefficients for a filter of order
$i-1$, then a filter of order $i$ can be calculated in three steps.
Firstly, a new set of reflection coefficients\index{reflection coefficients} are calculated.
\hequation{
k_j^{(i)} = k_j^{(i-1)}
}{kupdate1}
for $j = 1,i-1$ and
\hequation{
k_i^{(i)} = \left\{ r_i +
\sum_{j=1}^{i-1} a_j^{(i-1)} r_{i-j} \right\} / E^{(i-1)}
}{kupdate2}
Secondly, the prediction energy is updated.
\hequation{
E^{(i)} = (1 - k_i^{(i)} k_i^{(i)} ) E^{(i-1)}
}{Eupdate}
Finally, new filter coefficients are computed
\hequation{
a_j^{(i)} = a_j^{(i-1)} - k_i^{(i)} a_{i-j}^{(i-1)}
}{aupdate1}
for $j = 1,i-1$ and
\hequation{
a_i^{(i)} = - k_i^{(i)}
}{aupdate2}
This process is repeated from $i=1$ through to the required filter order
$i=p$.
To effect the above transformation, the target parameter kind must
be set to either \texttt{LPC}\index{lpc@\texttt{LPC}} to obtain the LP filter parameters $\{a_i\}$ or
\texttt{LPREFC}\index{lprefc@\texttt{LPREFC}} to obtain the reflection coefficients $\{k_i \}$. The
required filter order must also be set using the configuration
parameter \texttt{LPCORDER}\index{lpcorder@\texttt{LPCORDER}}.
Thus, for example, the following configuration
settings would produce a target parameterisation
consisting of 12 reflection coefficients per vector.
\begin{verbatim}
TARGETKIND = LPREFC
LPCORDER = 12
\end{verbatim}
An alternative LPC-based parameterisation is obtained by setting the
target kind to \texttt{LPCEPSTRA}\index{lpcepstra@\texttt{LPCEPSTRA}} to generate linear prediction cepstra.
The cepstrum of a signal is computed by taking a Fourier (or similar)
transform of the log spectrum. In the case of linear
prediction cepstra\index{linear prediction!cepstra}, the
required spectrum is the linear prediction spectrum which can be obtained
from the Fourier transform of the filter coefficients. However, it can be shown
that the required cepstra can be more efficiently computed using
a simple recursion
\hequation{
c_n = -a_n - \frac{1}{n} \sum_{i=1}^{n-1} (n-i) a_i c_{n-i}
}{lpcepstra}
The number of cepstra generated need not be the same as the number of
filter coefficients, hence it is set by a separate configuration
parameter called \texttt{NUMCEPS}\index{numceps@\texttt{NUMCEPS}}.
The principal advantage of cepstral coefficients is that they are
generally decorrelated and this allows diagonal covariances
to be used in the HMMs. However, one minor problem with
them is that the higher order cepstra are numerically quite small and
this results in
a very wide range of variances when going from the low to high cepstral
coefficients\index{cepstral coefficients!liftering}.
\HTK\ does not have a problem with this but for pragmatic reasons such as
displaying model parameters, flooring variances, etc., it is convenient to re-scale
the cepstral coefficients to have similar magnitudes. This is done by
setting the configuration parameter \texttt{CEPLIFTER}\index{ceplifter@\texttt{CEPLIFTER}} to some value $L$ to
\textit{lifter} the cepstra according to the following formula
\hequation{
{c^{\prime}}_n = \left( 1 + \frac{L}{2} sin \frac{\pi n}{L}
\right) c_n
}{ceplifter}
As an example, the following configuration parameters would
use a 14'th order linear prediction analysis to
generate 12 liftered LP cepstra per target vector
\begin{verbatim}
TARGETKIND = LPCEPSTRA
LPCORDER = 14
NUMCEPS = 12
CEPLIFTER = 22
\end{verbatim}
These are typical of the values needed to generate a good front-end
parameterisation for a speech recogniser based on linear prediction.
\index{cepstral analysis!LPC based}\index{cepstral analysis!liftering coefficient}
Finally, note that the conversions supported by \HTK\ are not limited to
the case where the source is a waveform. \HTK\ can convert any
LP-based parameter into any other LP-based parameter.
\mysect{Filterbank Analysis}{fbankanal}
The human ear resolves frequencies non-linearly across the audio spectrum and
empirical evidence suggests that designing a front-end to operate in a similar
non-linear manner improves recognition performance. A popular alternative to
linear prediction based analysis is therefore filterbank analysis since this
provides a much more straightforward route to obtaining the desired non-linear
frequency resolution. However, filterbank amplitudes are highly correlated and
hence, the use of a cepstral transformation in this case is virtually mandatory
if the data is to be used in a HMM based recogniser with diagonal covariances.
\index{cepstral analysis!filter bank} \index{speech input!filter bank}
\HTK\ provides a simple Fourier transform based filterbank designed to
give approximately equal resolution on a mel-scale. Fig.~\href{f:melfbank}
illustrates the general form of this filterbank. As can be seen,
the filters used are triangular and they are equally spaced along the mel-scale
which is defined by
\hequation{
\mbox{Mel}(f) = 2595 \log_{10}(1 + \frac{f}{700})
}{melscale}
To implement this filterbank, the window of speech data is
transformed\index{mel scale} using a Fourier transform and the magnitude is
taken. The magnitude coefficients are then \textit{binned} by correlating them
with each triangular filter. Here binning means that each FFT magnitude
coefficient is multiplied by the corresponding filter gain and the results
accumulated. Thus, each bin holds a weighted sum representing the spectral
magnitude in that filterbank channel.\index{binning} As an alternative, the
Boolean configuration parameter
\texttt{USEPOWER}\index{usepower@\texttt{USEPOWER}} can be set true to use the
power rather than the magnitude of the Fourier transform in the binning
process. \index{cepstral analysis!power vs magnitude}
\centrefig{melfbank}{110}{Mel-Scale Filter Bank}
\index{speech input!bandpass filtering}
Normally the triangular filters are spread over the whole frequency range from
zero upto the Nyquist frequency. However, band-limiting is often useful to
reject unwanted frequencies or avoid allocating filters to frequency regions in
which there is no useful signal energy. For filterbank analysis only, lower
and upper frequency cut-offs can be set using the configuration parameters
\texttt{LOFREQ}\index{lofreq@\texttt{LOFREQ}} and
\texttt{HIFREQ}\index{hifreq@\texttt{HIFREQ}}. For example,
\begin{verbatim}
LOFREQ = 300
HIFREQ = 3400
\end{verbatim}
might be used for processing telephone speech. When low and high pass cut-offs
are set in this way, the specified number of filterbank channels are distributed
equally on the mel-scale across the resulting pass-band such that the lower cut-off
of the first filter is at \texttt{LOFREQ} and the upper cut-off of the last
filter is at \texttt{HIFREQ}.
If mel-scale filterbank parameters are required directly, then the target kind
should be set to \texttt{MELSPEC}\index{melspec@\texttt{MELSPEC}}.
Alternatively, log filterbank parameters can be generated by setting the target
kind to \texttt{FBANK}.
\mysect{Vocal Tract Length Normalisation}{vtln}
A simple speaker normalisation technique can be implemented by
modifying the filterbank analysis described in the previous section.
Vocal tract length normalisation (VTLN) aims to compensate for the
fact that speakers have vocal tracts of different sizes. VTLN can be
implemented by warping the frequency axis in the filterbank analysis.
In HTK simple linear frequency warping is supported. The warping
factor~$\alpha$ is controlled by the configuration variable
\texttt{WARPFREQ}\index{melspec@\texttt{WARPFREQ}}. Here values of
$\alpha < 1.0$ correspond to a compression of the frequency axis. As
the warping would lead to some filters being placed outside the
analysis frequency range, the simple linear warping function is
modified at the upper and lower boundaries. The result is that the
lower boundary frequency of the analysis
(\texttt{LOFREQ}\index{melspec@\texttt{LOFREQ}}) and the upper
boundary frequency (\texttt{HIFREQ}\index{melspec@\texttt{HIFREQ}})
are always mapped to themselves. The regions in which the warping
function deviates from the linear warping with factor~$\alpha$ are
controlled with the two configuration variables
(\texttt{WARPLCUTOFF}\index{melspec@\texttt{WARPLCUTOFF}}) and
(\texttt{WARPUCUTOFF}\index{melspec@\texttt{WARPUCUTOFF}}).
Figure~\href{f:vtlnpiecewise} shows the overall shape of the resulting
piece-wise linear warping functions.
\centrefig{vtlnpiecewise}{60}{Frequency Warping}
The warping factor~$\alpha$ can for example be found using a search
procedure that compares likelihoods at different warping factors. A
typical procedure would involve recognising an utterance with
$\alpha=1.0$ and then performing forced alignment of the hypothesis
for all warping factors in the range $0.8 - 1.2$. The factor that
gives the highest likelihood is selected as the final warping factor.
Instead of estimating a separate warping factor for each utterance,
large units can be used by for example estimating only one~$\alpha$
per speaker.
Vocal tract length normalisation can be applied in testing as well as
in training the acoustic models.
\mysect{Cepstral Features}{cepstrum}
Most often, however, cepstral parameters are required
and these are indicated by setting the target kind to \texttt{MFCC} standing
for Mel-Frequency Cepstral Coefficients (MFCCs). These are calculated from the
log filterbank amplitudes $\{m_j\}$ using the Discrete Cosine Transform
\hequation{
c_i = \sqrt{\frac{2}{N}} \sum_{j=1}^N m_j \cos \left( \frac{\pi i}{N}(j-0.5) \right)
}{dct}
where $N$ is the number of filterbank channels set by the configuration
parameter \texttt{NUMCHANS}\index{numchans@\texttt{NUMCHANS}}. The required
number of cepstral coefficients is set by
\texttt{NUMCEPS}\index{numceps@\texttt{NUMCEPS}} as in the linear prediction
case. Liftering can also be applied to MFCCs using the
\texttt{CEPLIFTER}\index{ceplifter@\texttt{CEPLIFTER}} configuration parameter
(see equation~\ref{e:ceplifter}).
MFCCs are the parameterisation of choice for many speech recognition applications.
They give good discrimination and lend themselves to a number of manipulations.
In particular, the effect of inserting a transmission channel on the input
speech is to multiply the speech spectrum by the channel transfer function.
In the log cepstral domain, this multiplication becomes a simple addition which
can be removed by subtracting the cepstral mean from all input vectors.
In practice, of course, the mean has to be estimated over a limited amount
of speech data so the subtraction will not be perfect. Nevertheless, this
simple technique is very effective in practice where it
compensates for long-term spectral effects such as those caused by different
microphones and audio channels. To perform this
so-called \textit{Cepstral Mean Normalisation} (CMN) in \HTK\, it is only necessary
to add the \texttt{\_Z}\index{qualifiers!aaaz@\texttt{\_Z}} qualifier to the
target parameter kind. The mean is estimated by computing the average of
each cepstral parameter across each input speech file. Since this cannot be done
with live audio, cepstral mean compensation is not supported for this case.
\index{cepstral mean normalisation}
In addition to the mean normalisation the variance of the data can be
normalised. For improved robustness both mean and variance of the data
should be calculated on a larger units (e.g.\ on all the data from a
speaker instead of just on a single utterance). To use
speaker-/cluster-based normalisation the mean and variance estimates
are computed offline before the actual recognition and stored in
separate files (two files per cluster). The configuration variables
\texttt{CMEANDIR}\index{numchans@\texttt{CMEANDIR}} and
\texttt{VARSCALEDIR}\index{numchans@\texttt{VARSCALEDIR}} point to the
directories where these files are stored. To find the actual filename
a second set of variables
(\texttt{CMEANMASK}\index{numchans@\texttt{CMEANMASK}} and
\texttt{VARSCALEMASK}\index{numchans@\texttt{VARSCALEMASK}}) has to be
specified. These masks are regular expressions in which you can use
the special characters \texttt{?}, \texttt{*} and \texttt{\%}. The
appropriate mask is matched against the filename of the file to be
recognised and the substring that was matched against the \texttt{\%}
characters is used as the filename of the normalisation file. An
example config setting is:
\begin{verbatim}
CMEANDIR = /data/eval01/plp/cmn
CMEANMASK = %%%%%%%%%%_*
VARSCALEDIR = /data/eval01/plp/cvn
VARSCALEMASK = %%%%%%%%%%_*
VARSCALEFN = /data/eval01/plp/globvar
\end{verbatim}
So, if the file \verb|sw1-4930-B_4930Bx-sw1_000126_000439.plp| is to be
recognised then the normalisation estimates would be loaded from the
following files:
\begin{verbatim}
/data/eval01/plp/cmn/sw1-4930-B
/data/eval01/plp/cvn/sw1-4930-B
\end{verbatim}
The file specified by
\texttt{VARSCALEFN}\index{numchans@\texttt{VARSCALEFN}} contains the
global target variance vector, i.e. the variance of the data is first
normalised to 1.0 based on the estimate in the appropriate file in
\texttt{VARSCALEDIR}\index{numchans@\texttt{VARSCALEDIR}} and then
scaled to the target variance given in
\texttt{VARSCALEFN}\index{numchans@\texttt{VARSCALEFN}}.
The format of the files is very simple and each of them just contains
one vector. Note that in the case of the cepstral mean only the static
coefficients will be normalised. A cmn file could for example look like:
\begin{verbatim}
<CEPSNORM> <PLP_0>
<MEAN> 13
-10.285290 -9.484871 -6.454639 ...
\end{verbatim}
The cepstral variance normalised always applies to the full
observation vector after all qualifiers like delta and acceleration
coefficients have been added, e.g.:
\begin{verbatim}
<CEPSNORM> <PLP_D_A_Z_0>
<VARIANCE> 39
33.543018 31.241779 36.076199 ...
\end{verbatim}
The global variance vector will always have the same number of
dimensions as the cvn vector, e.g.:
\begin{verbatim}
<VARSCALE> 39
2.974308e+01 4.143743e+01 3.819999e+01 ...
\end{verbatim}
These estimates can be generated using \htool{HCompV}. See the
reference section for details.
\mysect{Perceptual Linear Prediction}{plp}
An alternative to the Mel-Frequency Cepstral Coefficients is the use
of Perceptual Linear Prediction (PLP) coefficients.
As implemented in HTK the PLP feature extraction is based on the
standard mel-frequency filterbank (possibly warped). The mel
filterbank coefficients are weighted by an equal-loudness curve and
then compressed by taking the cubic root.\footnote{the degree of
compression can be controlled by setting the configuration parameter
\texttt{COMPRESSFACT}\index{enormalise@\texttt{COMPRESSFACT}} which
is the power to which the amplitudes are raised and defaults to
0.33)} From the resulting auditory spectrum LP coefficents are
estimated which are then converted to cepstral coefficents in the
normal way (see above).
\mysect{Energy Measures}{energy}
\index{speech input!energy measures}
To augment the spectral parameters derived from linear prediction or
mel-filterbank analysis, an energy term can be appended by including the
qualifier \texttt{\_E}\index{qualifiers!aaae@\texttt{\_E}} in the target kind.
The energy is computed as the log of the signal energy, that is, for speech
samples $\{s_n, n=1,N \}$
\hequation{
E = log \sum_{n=1}^N s_n^2
}{logenergy}
This log energy measure can be normalised to the range $-E_{min}..1.0$ by
setting the Boolean configuration parameter
\texttt{ENORMALISE}\index{enormalise@\texttt{ENORMALISE}} to true (default
setting). This
normalisation is implemented by subtracting the maximum value of $E$ in the
utterance and adding $1.0$.
Note that energy normalisation is incompatible with live audio
input and in such circumstances the configuration variable \texttt{ENORMALISE}
should be explicitly set false.
The lowest energy in the utterance can be clamped using the configuration
parameter
\texttt{SILFLOOR}\index{silfloor@\texttt{SILFLOOR}} which gives the ratio
between the maximum and minimum energies in the utterance in dB. Its default
value is 50dB.
Finally, the overall log energy can be arbitrarily scaled by the value of the
configuration parameter \texttt{ESCALE}\index{escale@\texttt{ESCALE}} whose
default is $0.1$. \index{silence floor}
When calculating energy for LPC-derived parameterisations, the default is to
use the zero-th delay autocorrelation coefficient ($r_0$). However, this means
that the energy is calculated after windowing and pre-emphasis. If the
configuration parameter \texttt{RAWENERGY}\index{rawenergy@\texttt{RAWENERGY}}
is set true, however, then energy is calculated separately before any windowing
or pre-emphasis regardless of the requested parameterisation\footnote{ In any
event, setting the compatibility variable \texttt{V1COMPAT} to true in
\htool{HPARM} will ensure that the calculation of energy is compatible with
that computed by the Version 1 tool \htool{HCode}. }.
In addition to, or in place of, the log energy, the qualifier
\texttt{\_O}\index{qualifiers!aaao@\texttt{\_O}} can be added to a target kind
to indicate that the 0'th cepstral parameter $C_0$ is to be appended. This
qualifier is only valid if the target kind is \texttt{MFCC}. Unlike earlier
versions of \HTK\, scaling factors set by the configuration variable
\texttt{ESCALE} are not applied to $C_0$\footnote{ Unless \texttt{V1COMPAT} is
set to true. }.
\mysect{Delta, Acceleration and Third Differential Coefficients}{delta}
\index{speech input!dynamic coefficents}
The performance of a speech recognition system can be greatly enhanced by
adding time derivatives to the basic static parameters. In \HTK, these are
indicated by attaching qualifiers to the basic parameter kind. The qualifier
\texttt{\_D} indicates that first order regression coefficients (referred to as
delta coefficients) are appended, the qualifier
\texttt{\_A}\index{qualifiers!aaaa@\texttt{\_A}} indicates that second order
regression coefficients (referred to as acceleration coefficients) and
the qualifier
\texttt{\_T}\index{qualifiers!aaaa@\texttt{\_T}} indicates that third order
regression coefficients (referred to as third differential coefficients) are
appended. The \texttt{\_A} qualifier cannot be used without also using the
\texttt{\_D}\index{qualifiers!aaad@\texttt{\_D}} qualifier. Similarly
the \texttt{\_T} qualifier cannot be used without also using the
\texttt{\_D} and \texttt{\_A} qualifiers.
The delta coefficients\index{delta coefficients} are computed using the
following regression formula\index{regression formula}
\hequation{
d_t = \frac{ \sum_{\theta =1}^\Theta \theta(c_{t+\theta} - c_{t-\theta}) }{
2 \sum_{\theta = 1}^\Theta \theta^2 }
}{deltas}
where $d_t$ is a delta coefficient at time $t$ computed in terms of the
corresponding static coefficients $c_{t-\Theta}$ to $c_{t+\Theta}$. The value
of $\Theta$ is set using the configuration parameter
\texttt{DELTAWINDOW}\index{deltawindow@\texttt{DELTAWINDOW}}. The same formula
is applied to the delta coefficients to obtain acceleration coefficients except
that in this case the window size is set by
\texttt{ACCWINDOW}\index{accwindow@\texttt{ACCWINDOW}}. Similarly
the third differentials use \texttt{THIRDWINDOW}. Since
equation~\ref{e:deltas} relies on past and future speech parameter values,
some modification is needed at the beginning and end of the speech. The
default behaviour is to replicate the first or last vector as needed to fill
the regression window.
In older version 1.5 of \HTK\ and earlier, this end-effect problem was solved
by using simple
first order differences at the start and end of the speech, that is
\begin{equation}
d_t = c_{t+1} - c_t,\;\;\; t<\Theta
\end{equation}
and
\begin{equation}
d_t = c_t - c_{t-1}, \;\;\; t \geq T-\Theta
\end{equation}
where $T$ is the length of the data file. If required, this older behaviour
can be restored by setting the configuration variable
\texttt{V1COMPAT}\index{v1compat@\texttt{V1COMPAT}}
to true in \htool{HParm}.
For some purposes, it is useful to use simple differences throughout. This
can be achieved by setting the configuration
variable \texttt{SIMPLEDIFFS}\index{simplediffs@\texttt{SIMPLEDIFFS}}
to true in \htool{HParm}. In this case, just the end-points of the delta window
are used, i.e.
\hequation{
d_t = \frac{ (c_{t+\Theta} - c_{t-\Theta}) }{
2 \Theta}
}{simdiffs}
\index{simple differences}
When delta and acceleration coefficients are requested, they are computed for
all static parameters including energy if present. In some applications, the
absolute energy is not useful but time derivatives of the energy may be. By
including the \texttt{\_E} qualifier together with the
\texttt{\_N}\index{qualifiers!aaan@\texttt{\_N}} qualifier, the absolute energy
is suppressed leaving just the delta and acceleration coefficients of the
energy.
\mysect{Storage of Parameter Files}{parmstore}
Whereas \HTK\ can handle waveform data in a variety of file formats,
all parameterised speech data is stored externally in either native
\HTK\ format data files or Entropic Esignal format files.
Entropic ESPS format is no longer supported directly, but input and output
filters can be used to convert ESPS to Esignal format on input and
Esignal to ESPS on output.
\subsection{\HTK\ Format Parameter Files}
\HTK\ format files consist of a contiguous sequence of \textit{samples}
preceded by a header. Each sample is a vector of either 2-byte integers or
4-byte floats. 2-byte integers are used for compressed forms as described
below and for vector quantised data as described later in
section~\ref{s:vquant}. \HTK\ format data files can also be used to store
speech waveforms as described in section~\ref{s:waveform}. \index{file
formats!HTK}
The \HTK\ file format header is 12 bytes long and contains the following data
\begin{tabbing}
++ \= +++++++++ \= \kill
\>\texttt{nSamples}\>-- number of samples in file (4-byte integer) \\
\>\texttt{sampPeriod}\>-- sample period in 100ns units (4-byte integer) \\
\>\texttt{sampSize}\>-- number of bytes per sample (2-byte integer) \\
\>\texttt{parmKind}\>-- a code indicating the sample kind (2-byte integer)
\end{tabbing}
The parameter kind\index{parameter kind} consists of a 6 bit
code representing the basic parameter kind plus additional bits for
each of the possible qualifiers\index{qualifiers}. The basic parameter kind codes are
\begin{tabbing}
++++\= +++ \= ++++++++ \= \kill
\>0 \> \texttt{WAVEFORM} \> sampled waveform \\
\>1 \> \texttt{LPC} \> linear prediction filter coefficients \\
\>2 \> \texttt{LPREFC} \> linear prediction reflection coefficients \\
\>3 \> \texttt{LPCEPSTRA} \> LPC cepstral coefficients \\
\>4 \> \texttt{LPDELCEP} \> LPC cepstra plus delta coefficients \\
\>5 \> \texttt{IREFC} \> LPC reflection coef in 16 bit integer format \\
\>6 \> \texttt{MFCC} \> mel-frequency cepstral coefficients \\
\>7 \> \texttt{FBANK} \> log mel-filter bank channel outputs \\
\>8 \> \texttt{MELSPEC} \> linear mel-filter bank channel outputs \\
\>9 \> \texttt{USER} \> user defined sample kind \\
\>10 \> \texttt{DISCRETE} \> vector quantised data \\
\>11 \> \texttt{PLP} \> PLP cepstral coefficients \\
\end{tabbing}
and the bit-encoding for the qualifiers (in octal) is
\begin{tabbing}
++++\= +++ \= ++++++++ \= \kill
\>\texttt{\_E} \> 000100 \> has energy \\
\>\texttt{\_N} \> 000200 \> absolute energy suppressed \\
\>\texttt{\_D} \> 000400 \> has delta coefficients \\
\>\texttt{\_A} \> 001000 \> has acceleration coefficients\\
\>\texttt{\_C} \> 002000 \> is compressed \\
\>\texttt{\_Z} \> 004000 \> has zero mean static coef. \\
\>\texttt{\_K} \> 010000 \> has CRC checksum \\
\>\texttt{\_O} \> 020000 \> has 0'th cepstral coef. \\
\>\texttt{\_V} \> 040000 \> has VQ data \\
\>\texttt{\_T} \> 100000 \> has third differential coef. \\
\end{tabbing}\index{qualifiers!codes}
The \texttt{\_A} qualifier can only be specified when \texttt{\_D}
is also specified.
The \texttt{\_N} qualifier is only valid when both energy and delta
coefficients are present.
The sample kind \texttt{LPDELCEP} is identical to \texttt{LPCEPSTRA\_D}
and is retained for compatibility with older versions of \HTK.
The \texttt{\_C}\index{qualifiers!aaac@\texttt{\_C}} and
\texttt{\_K}\index{qualifiers!aaak@\texttt{\_K}} only exist in external files. Compressed
files are always decompressed on loading and any attached CRC
is checked and removed. An external file can contain both an energy
term and a 0'th order cepstral coefficient. These may be retained
on loading but normally one or the other is discarded\footnote{
Some applications may require the 0'th order cepstral coefficient
in order to recover the filterbank coefficients from the cepstral
coefficients.}.
\putfig{HTKFormat}{130}{Parameter Vector Layout in \HTK\ Format Files}
All parameterised forms of \HTK\ data files consist of a sequence of vectors.
Each vector is organised as shown by the examples in Fig~\href{f:HTKFormat}
where various different qualified forms are listed. As can be seen, an energy
value if present immediately follows the base coefficients. If delta
coefficients are added, these follow the base coefficients and energy value.
Note that the base form \texttt{LPC} is used in this figure only as an example,
the same layout applies to all base sample kinds. If the 0'th order cepstral
coefficient is included as well as energy then it is inserted immediately
before the energy coefficient, otherwise it replaces it.
For external storage of speech parameter files, two compression methods are
provided. For LP coding only, the \texttt{IREFC} parameter kind exploits the
fact that the reflection coefficients are bounded by $\pm 1$ and hence they can
be stored as scaled integers such that $+1.0$ is stored as $32767$ and $-1.0$
is stored as $-32767$. For other types of parameterisation, a more general
compression facility indicated by the
\texttt{\_C}\index{qualifiers!aaac@\texttt{\_C}} qualifier is used.
\HTK\ compressed parameter files consist of a set of compressed parameter
vectors stored as shorts such that for parameter $x$
\begin{eqnarray}
x_{short} & = & A*x_{float}-B \nonumber
\end{eqnarray}
The coefficients $A$ and $B$ are defined as
\begin{eqnarray}
A & = & 2*I/(x_{max}-x_{min}) \nonumber\\
B & = & (x_{max}+x_{min})*I/(x_{max}-x_{min}) \nonumber
\end{eqnarray}
where $x_{max}$ is the maximum value of parameter $x$ in the whole file and
$x_{min}$ is the corresponding minimum. $I$ is the maximum range of a 2-byte
integer i.e.\ 32767. The values of $A$ and $B$ are stored as two floating
point vectors prepended to the start of the file immediately after the header.
When a \HTK\ tool writes out a speech file to external storage, no further
signal conversions are performed. Thus, for most purposes, the target
parameter kind specifies both the required internal representation and the form
of the written output, if any. However, there is a distinction in the way that
the external data is actually stored. Firstly, it can be compressed as
described above by setting the configuration parameter \texttt{SAVECOMPRESSED}
to true. If the target kind is \texttt{LPREFC} then this compression is
implemented by converting to \texttt{IREFC} otherwise the general compression
algorithm described above is used. Secondly, in order to avoid data corruption
problems, externally stored \HTK\ parameter files can have a cyclic redundancy
checksum appended. This is indicated by the qualifier
\texttt{\_K}\index{qualifiers!aaak@\texttt{\_K}} and it is generated by setting
the configuration parameter \texttt{SAVEWITHCRC} to true. The principle tool
which uses these output conversions is \htool{HCopy} (see
section~\ref{s:UseHCopy}).
\subsection{Esignal Format Parameter Files}
\index{file formats!Esignal}
The default for parameter files is native \HTK\ format. However, \HTK\ tools
also support the Entropic Esignal format for both input and output. Esignal
replaces the Entropic ESPS file format. To ensure compatibility Entropic
provides conversion programs from ESPS to ESIG and vice versa.
To indicate that a source file is in Esignal format the configuration
variable \texttt{SOURCEFORMAT}\index{sourceformat@\texttt{SOURCEFORMAT}}
should be set to \texttt{ESIG}. Alternatively,
\texttt{-F ESIG}\index{standard options!aaaf@\texttt{-F}} can be specified
as a command-line option.
To generate Esignal format output files, the configuration variable
\texttt{TARGETFORMAT} should be set to \texttt{ESIG} or the command line option
\texttt{-O ESIG} should be set.
ESIG files consist of three parts: a preamble, a sequence of field
specifications called the field list and a sequence of records. The preamble
and the field list together constitute the header. The preamble is purely
ASCII. Currently it consists of 6 information items that are all terminated
by a new line. The information in the preamble is the following:
\begin{tabbing}
++ \= +++++++++ \= \kill
\>\texttt{line 1}\>-- identification of the file format \\
\>\texttt{line 2}\>-- version of the file format\\
\>\texttt{line 3}\>-- architecture (ASCII, EDR1, EDR2, machine name)\\
\>\texttt{line 4}\>-- preamble size (48 bytes)\\
\>\texttt{line 5}\>-- total header size\\
\>\texttt{line 6}\>-- record size\\
\end{tabbing}
All ESIG files that are output by \HTK\ programs contain the following
global fields:
\begin{description}
\item[commandLine] the command-line used to generate the file;
\item[recordFreq] a double value that indicates the sample frequency
in Herz;
\item[startTime] a double value that indicates a time at which the first
sample is presumed to be starting;
\item[parmKind] a character string that indicates the full
type of parameters in the file, e.g: \texttt{MFCC\_E\_D}.
\item[source\_1] if the input file was an ESIG file this field includes the
header items in the input file.
\end{description}
After that there are field specifiers for the records. The first specifier
is for the basekind of the parameters, e.g: \texttt{MFCC}. Then for each
available qualifier there are additional specifiers. Possible specifiers are:
\begin{tabbing}
++++\= \kill
\>\texttt{zeroc} \\
\>\texttt{energy}\\
\>\texttt{delta}\\
\>\texttt{delta\_zeroc} \\
\>\texttt{delta\_energy}\\
\>\texttt{accs}\\
\>\texttt{accs\_zeroc} \\
\>\texttt{accs\_energy}\\
\end{tabbing}\index{qualifiers!ESIG field specifiers}
The data segments of the ESIG files have exactly the same format as the
the corresponding \HTK\ files. This format was described in the previous
section.
\HTK\ can only input parameter files that have a valid parameter kind as value
of the header field \texttt{parmKind}. If this field does not exist or if the
value of this field does not contain a valid parameter kind, the file is
rejected. After the header has been read the file is treated as an \HTK\ file.
\mysect{Waveform File Formats}{waveform}
For reading waveform data files, \HTK\ can support a variety of different
formats and these are all briefly described in this section. The default
speech file format is \HTK. If a different format is to be used, it can be
specified by setting the configuration parameter
\texttt{SOURCEFORMAT}\index{sourceformat@\texttt{SOURCEFORMAT}}. However,
since file formats need to be changed often, they can also be set individually
via the \texttt{-F}\index{standard options!aaaf@\texttt{-F}} command-line
option. This over-rides any setting of the \texttt{SOURCEFORMAT} configuration
parameter.
Similarly for the output of waveforms, the format can be set using either the
configuration parameter \texttt{TARGETFORMAT} or the \texttt{-O} command-line
option. However, for output only native \HTK\ format (\texttt{HTK}), Esignal
format (\texttt{ESIG}) and headerless (\texttt{NOHEAD}) waveform files are
supported.
The following sub-sections give a brief description of each of the waveform
file formats supported by \HTK.
\subsection{HTK File Format}
\index{file formats!HTK}
The \HTK\ file format for waveforms is identical to that described in
section~\ref{s:parmstore} above. It consists of a 12 byte header followed
by a sequence of 2 byte integer speech samples. For waveforms, the
\texttt{sampSize} field will be 2 and the \texttt{parmKind} field will be 0.
The \texttt{sampPeriod} field gives the sample period in 100ns units, hence for
example, it will have the value 1000 for speech files sampled at 10kHz and 625
for speech files sampled at 16kHz.
\subsection{Esignal File Format}
\index{file formats!Esignal}
The Esignal file format for waveforms is similar to that described in
section~\ref{s:parmstore} above with the following exceptions. When reading an
ESIG waveform file the \HTK\ programs only check whether the record length
equals 2 and whether the datatype of the only field in the data records is
\texttt{SHORT}. The data field that is created on output of a waveform is
called \texttt{WAVEFORM}.
\subsection{TIMIT File Format}
\index{file formats!TIMIT}
The TIMIT format has the same structure as the HTK format except that the
12-byte header contains the following
\begin{tabbing}
++ \= +++++++++ \= \kill
\>\texttt{hdrSize}\>-- number of bytes in header ie 12 (2-byte integer) \\
\>\texttt{version}\>-- version number (2-byte integer) \\
\>\texttt{numChannels}\>-- number of channels (2-byte integer) \\
\>\texttt{sampRate}\>-- sample rate (2-byte integer) \\
\>\texttt{nSamples}\>-- number of samples in file (4-byte integer)
\end{tabbing}
TIMIT format data is used only on the prototype TIMIT CD ROM.
\subsection{NIST File Format}
\index{file formats!NIST}
The NIST file format is also referred to as the Sphere file format.
A NIST header consists of ASCII text. It begins with a label of the
form \texttt{NISTxx} where xx is a version code followed by the number
of bytes in the header. The remainder of the header consists of
name value pairs of which \HTK\ decodes the following
\begin{tabbing}
++ \= +++++++++++++ \= \kill
\>\texttt{sample\_rate} \>-- sample rate in Hz \\
\>\texttt{sample\_n\_bytes} \>-- number of bytes in each sample \\
\>\texttt{sample\_count} \>-- number of samples in file \\
\>\texttt{sample\_byte\_format} \>-- byte order \\
\>\texttt{sample\_coding} \>-- speech coding eg pcm, $\mu$law, shortpack \\
\>\texttt{channels\_interleaved} \>-- for 2 channel data only
\end{tabbing}
The current NIST Sphere data format\index{NIST Sphere data format} subsumes a
variety of internal data organisations. HTK currently supports interleaved
$\mu$law used in Switchboard, Shortpack compression used in the original
version of WSJ0 and standard 16bit linear PCM as used in Resource Management,
TIMIT, etc. It does not currently support the Shorten compression format as
used in WSJ1 due to licensing restrictions. Hence, to read WSJ1, the files
must be converted using the NIST supplied decompression routines into standard
16 bit linear PCM. This is most conveniently done under UNIX by using the
decompression program as an input filter set via the environment variable
\texttt{HWAVEFILTER}\index{hwavefilter@\texttt{HWAVEFILTER}} (see
section~\ref{s:iopipes}).
For interleaved $\mu$law as used in Switchboard, the default is to add the two
channels together. The left channel only can be obtained by setting the
environment variable \texttt{STEREOMODE} to \texttt{LEFT} and the right channel
only can be obtained by setting the environment variable \texttt{STEREOMODE} to
\texttt{RIGHT}. \index{mu law encoded files }
\subsection{SCRIBE File Format}
\index{file formats!SCRIBE}
The SCRIBE format is a subset of the standard laid down by the European Esprit
Programme SAM Project. SCRIBE data files are headerless and therefore consist
of just a sequence of 16 bit sample values. \HTK\ assumes by default that the
sample rate is 20kHz. The configuration parameter \texttt{SOURCERATE} should
be set to over-ride this. The byte ordering assumed for SCRIBE data files is
\texttt{VAX} (little-endian).
\subsection{SDES1 File Format}
\index{file formats!Sound Designer(SDES1)}
The SDES1 format refers to the ``Sound Designer I'' format defined by
Digidesign Inc in 1985 for multimedia and general audo applications. It is
used for storing short monoaural sound samples. The SDES1 header is complex
(1336 bytes) since it allows for associated display window information to be
stored in it as well as providing facilities for specifying repeat loops. The
HTK input routine for this format just picks out the following information
\begin{tabbing}
++ \= +++++++++ \= \kill
\>\texttt{headerSize} \>-- size of header ie 1336 (2 byte integer) \\
\>(182 byte filler) \\
\>\texttt{fileSize} \>-- number of bytes of sampled data (4 byte integer)\\
\>(832 byte filler) \\
\>\texttt{sampRate} \>-- sample rate in Hz (4 byte integer) \\
\>\texttt{sampPeriod} \>-- sample period in microseconds (4 byte integer) \\
\>\texttt{sampSize} \>-- number of bits per sample ie 16 (2 byte integer)
\end{tabbing}
\subsection{AIFF File Format}
\index{file formats!Audio Interchange (AIFF)}
The AIFF format was defined by Apple Computer for storing monoaural and
multichannel sampled sounds. An AIFF file consists of a number of {\it
chunks}. A {\it Common} chunk contains the fundamental parameters of the sound
(sample rate, number of channels, etc) and a {\it Sound Data} chunk contains
sampled audio data. \HTK\ only partially supports AIFF since some of the
information in it is stored as floating point numbers. In particular, the
sample rate is stored in this form and to avoid portability problems,
\HTK\ ignores the given sample rate and assumes that it is 16kHz. If this
default rate is incorrect, then the true sample period should be
specified by setting the \texttt{SOURCERATE} configuration parameter.
Full details of the AIFF format are available from Apple Developer
Technical Support.
\subsection{SUNAU8 File Format}
\index{file formats!Sun audio (SUNAU8)}
The SUNAU8 format defines a subset of the ``.au'' and ``.snd'' audio file
format used by Sun and NeXT. An SUNAU8 speech data file consists of a header
followed by 8 bit $\mu$law encoded speech samples. The header is 28 bytes and
contains the following fields, each of which is 4 bytes
\begin{tabbing}
++ \= +++++++++ \= \kill
\>\texttt{magicNumber} \>-- magic number 0x2e736e64 \\
\>\texttt{dataLocation} \>-- offset to start of data \\
\>\texttt{dataSize} \>-- number of bytes of data \\
\>\texttt{dataFormat} \>-- data format code which is 1 for 8 bit $\mu$law \\
\>\texttt{sampRate} \>-- a sample rate code which is always 8012.821 Hz \\
\>\texttt{numChan} \>-- the number of channels \\
\>\texttt{info} \>-- arbitrary character string min length 4 bytes
\end{tabbing}
No default byte ordering is assumed for this format. If the data source is
known to be different to the machine being used, then the environment variable
\texttt{BYTEORDER} must be set appropriately. Note that when used on Sun Sparc
machines with 16 bit audio device the sampling rate of 8012.821Hz is not
supported and playback will be peformed at 8KHz.
\subsection{OGI File Format}
\index{file formats!OGI}
The OGI format is similar to TIMIT. The header contains the following
\begin{tabbing}
++ \= +++++++++ \= \kill
\>\texttt{hdrSize}\>-- number of bytes in header \\
\>\texttt{version}\>-- version number (2-byte integer) \\
\>\texttt{numChannels}\>-- number of channels (2-byte integer) \\
\>\texttt{sampRate}\>-- sample rate (2-byte integer) \\
\>\texttt{nSamples}\>-- number of samples in file (4-byte integer) \\
\>\texttt{lendian}\>-- used to test for byte swapping (4-byte integer)
\end{tabbing}
\subsection{WAV File Format}{}
\index{file formats!WAV}
The WAV file format is a subset of Microsoft's RIFF specification for the
storage of multimedia files. A RIFF file starts out with a file header followed
by a sequence of data ``chunks''. A WAV file is often just a RIFF file with a
single ``WAVE'' chunk which consists of two sub-chunks - a ``fmt'' chunk
specifying the data format and a ``data'' chunk containing the actual sample
data. The WAV file header contains the following
\begin{tabbing}
++ \= +++++++++ \= \kill
\>\texttt{'RIFF'}\>-- RIFF file identification (4 bytes) \\
\>\texttt{<length>}\>-- length field (4 bytes)\\
\>\texttt{'WAVE'}\>-- WAVE chunk identification (4 bytes) \\
\>\texttt{'fmt '}\>-- format sub-chunk identification (4 bytes) \\
\>\texttt{flength}\>-- length of format sub-chunk (4 byte integer) \\
\>\texttt{format}\>-- format specifier (2 byte integer) \\
\>\texttt{chans}\>-- number of channels (2 byte integer) \\
\>\texttt{sampsRate}\>-- sample rate in Hz (4 byte integer) \\
\>\texttt{bpsec}\>-- bytes per second (4 byte integer) \\
\>\texttt{bpsample}\>-- bytes per sample (2 byte integer) \\
\>\texttt{bpchan}\>-- bits per channel (2 byte integer) \\
\>\texttt{'data'}\>-- data sub-chunk identification (4 bytes) \\
\>\texttt{dlength}\>-- length of data sub-chunk (4 byte integer)
\end{tabbing}
Support is provided for 8-bit CCITT mu-law, 8-bit CCITT a-law, 8-bit PCM
linear and 16-bit PCM linear - all in stereo or mono (use of \texttt{STEREOMODE}
parameter as per NIST). The default byte ordering assumed for \texttt{WAV}
data files is \texttt{VAX} (little-endian).
\subsection{ALIEN and NOHEAD File Formats}
\index{file formats!ALIEN}
\index{file formats!NOHEAD}
\HTK\ tools can read speech waveform files with alien formats provided that
their overall structure is that of a header followed by data. This is done by
setting the format to \texttt{ALIEN} and setting the environment variable
\texttt{HEADERSIZE} to the number of bytes in the header. \HTK\ will then
attempt to infer the rest of the information it needs. However, if input is
from a pipe, then the number of samples expected must be set using the
environment variable \texttt{NSAMPLES}\index{nsamples@\texttt{NSAMPLES}}. The
sample rate of the source file is defined by the configuration parameter
\texttt{SOURCERATE} as described in section~\ref{s:sigproc}. If the file has
no header then the format \texttt{NOHEAD} may be specified instead of
\texttt{ALIEN}\index{alien@\texttt{ALIEN}} in which case
\texttt{HEADERSIZE}\index{headersize@\texttt{HEADERSIZE}} is assumed to be
zero.
\mysect{Direct Audio Input/Output}{audioio}
\index{speech input!direct audio}
Many \HTK\ tools, particularly recognition tools, can input speech waveform
data directly from an audio device. The basic mechanism for doing this is to
simply specify the \texttt{SOURCEKIND} as being
\texttt{HAUDIO}\index{haudio@\texttt{HAUDIO}} following which speech samples
will be read directly from the host computer's audio input device.
Note that for live audio input, the configuration variable
\texttt{ENORMALISE} should be set to false both during training and recognition. Energy normalisation cannot
be used with live audio input, and the default setting for this variable
is \texttt{TRUE}. When training models for live audio input, be sure to
set \texttt{ENORMALISE} to false. If you have existing models trained with
\texttt{ENORMALISE} set to true, you can retrain them using {\it single-pass
retraining} (see section~\ref{s:singlepass}).
When using direct audio input\index{direct audio input}, the input sampling
rate may be set explicitly using the configuration parameter
\texttt{SOURCERATE}, \index{sourcerate@\texttt{SOURCERATE}} otherwise
\HTK\ will assume that it has been set by some external means such as an
audio control panel. In the latter case, it must be possible for
\htool{HAudio} to obtain the sample rate from the audio driver
otherwise an error message will be generated.
Although the detailed control of audio hardware is typically machine dependent,
\HTK\ provides a number of Boolean configuration variables to request specific
input and output sources. These are indicated by the following table
\begin{center}\index{audio source}\index{audio output}
\begin{tabular}{|c|l|} \hline
Variable & Source/Sink \\ \hline
\texttt{LINEIN} & line input \\
\texttt{MICIN} & microphone input \\
\texttt{LINEOUT} & line output \\
\texttt{PHONESOUT} & headphones output \\
\texttt{SPEAKEROUT} & speaker output \\ \hline
\end{tabular}
\end{center}
\index{linein@\texttt{LINEIN}}
\index{micin@\texttt{MICIN}}
\index{lineout@\texttt{LINEOUT}}
\index{phonesout@\texttt{PHONESOUT}}
\index{speakerout@\texttt{SPEAKEROUT}}
The major complication in using direct audio is in starting and stopping the
input device. The simplest approach to this is for \HTK\ tools to take direct
control and, for example, enable the audio input for a fixed period determined
via a command line option. However, the \htool{HAudio}/\htool{HParm} modules
provides two more powerful built-in facilities for audio input control.
\index{direct audio input!silence detector!speech detector}
The first method of audio input control involves the use of an automatic
energy-based speech/silence detector which is enabled by setting the
configuration parameter
\texttt{USESILDET}\index{usesildet@\texttt{USESILDET}} to true. Note that
the speech/silence detector can also operate on waveform input files.
The automatic speech/silence detector uses a two level algorithm which first
classifies each frame of data as either speech or silence and then applies a
heuristic to determine the start and end of each utterance.\index{HParm!SILENERGY} \index{HParm!SPEECHTHRESH}The detector classifies each
frame as speech or silence based solely on the log energy of the signal. When
the energy value exceeds a threshold the frame is marked as speech otherwise as
silence. The threshold is made up of two components both of which can be set by
configuration variables. The first component represents the mean energy level
of silence and can be set explicitly via the configuration
parameter \texttt{SILENERGY}. However, it is more usual to take a measurement
from the environment directly. Setting the configuration parameter
\texttt{MEASURESIL} to true will cause the detector to calibrate its parameters
from the current acoustic environment just prior to sampling. The second
threshold component is the level above which frames are classified as speech
(\texttt{SPEECHTHRESH}) .
\index{HParm!SPCSEQCOUNT} \index{HParm!SPCGLCHCOUNT} \index{HParm!SILGLCHCOUNT}
Once each frame has been classified as speech or silence they are grouped into
windows consisting of \texttt{SPCSEQCOUNT} consecutive frames. When the number
of frames marked as silence within each window falls below a glitch count the
whole window is classed as speech. Two separate glitch counts are used, {\tt
SPCGLCHCOUNT} before speech onset is detected and {\tt SILGLCHCOUNT} whilst
searching for the end of the utterance. This allows the algorithm to take
account of the tendancy for the end of an utterance to be somewhat quieter than
the beginning.
\index{HParm!SILMARGIN} \index{HParm!SILSEQCOUNT}
Finally, a top level heuristic is used to determine the start and end of the
utterance. The heuristic defines the start of speech as the beginning of the
first window classified as speech. The actual start of the processed utterance
is \texttt{SILMARGIN} frames before the detected start of speech to ensure that
when the speech detector triggers slightly late the recognition accuracy is not
affected. Once the start of the utterance has been found the detector searches
for \texttt{SILSEQCOUNT} windows all classified as silence and sets the end of
speech to be the end of the last window classified as speech. Once again the
processed utterance is extended \texttt{SILMARGIN} frames to ensure that if the
silence detector has triggered slightly early the whole of the speech is still
available for further processing.
\centrefig{endpointer}{120}{Endpointer Parameters}
Fig~\href{f:endpointer} shows an example of the speech/silence detection
process. The waveform data is first classified as speech or silence at frame
and then at window level before finally the start and end of the utterance are
marked. In the example, audio input starts at point {\tt A} and is stopped
automatically at point {\tt H}. The start of speech, {\tt C}, occurs when a
window of \texttt{SPCSEQCOUNT} frames are classified as speech and the start of
the utterance occurs \texttt{SILMARGIN} frames earlier at {\tt B}. The period
of silence from {\tt D} to {\tt E} is not marked as the end of the utterance
because it is shorter than \texttt{SILSEQCOUNT}. However after point {\tt F}
no more windows are classified as speech (although a few frames are) and so
this is marked as the end of speech with the end of the utterance extended to
{\tt G}.
\index{direct audio input!signal control!keypress}
The second built-in mechanism for controlling audio input is by arranging for
a signal to be sent from some other process. Sending the signal for the first
time starts the audio device. If the speech detector is not enabled then
sampling starts immediately and is stopped by sending the signal a second
time. If automatic speech/silence detection is enabled, then the first signal
starts the detector. Sampling stops immediately when a second signal is
received or when silence is detected. The signal number is set using the
configuration parameter \texttt{AUDIOSIG}\index{audiosig@\texttt{AUDIOSIG}}.
Keypress control operates in a similar fashion and is enabled by setting the
configuration parameter \texttt{AUDIOSIG} to a negative number. In this mode
an initial keypress will be required to start sampling/speech detection and
a second keypress will stop sampling immediately.
Audio output\index{audio output} is also supported by \HTK. There are no
generic facilities for output and the precise behaviour will depend on the tool
used. It should be noted, however, that the audio input facilities provided by
\htool{HAudio} include provision for attaching a \textit{replay buffer} to an
audio input channel. This is typically used to store the last few seconds of
each input to a recognition tool in a circular buffer so that the last
utterance input can be replayed on demand.
\mysect{Multiple Input Streams}{streams}
\index{multiple streams}
As noted in section~\ref{s:genio}, \HTK\ tools regard the input observation
sequence as being divided into a number of independent \textit{data streams}.
For building continuous density HMM systems, this facility is of limited
use and by far the most common case is that of a single data stream.
However, when building tied-mixture systems or when
using vector quantisation, a more uniform coverage of the
acoustic space is obtained by separating energy, deltas, etc., into
separate streams.
This separation of parameter vectors into streams takes place at
the point where the vectors
are extracted from the converted input file or audio device and
transformed into an observation. The tools for HMM construction
and for recognition thus view the input data as a sequence of observations
but note that this is entirely internal to \HTK. Externally data is
always stored as a single sequence of parameter vectors.
When multiple streams\index{multiple streams!rules for} are required,
the division of the parameter vectors is performed automatically
based on the parameter kind. This works according
to the following rules.
\begin{description}
\item[1 stream] single parameter vector. This is the default case.
\item[2 streams] if the parameter vector contains energy terms, then
they are extracted and placed in stream 2. Stream 1 contains the
remaining static coefficients and their deltas and accelerations,
if any. Otherwise,
the parameter vector must have appended delta
coefficients and no appended acceleration coefficients.
The vector is then split so that the static
coefficients form stream 1 and
the corresponding delta coefficients form stream 2.
\item[3 streams]
if the parameter vector has acceleration coefficients, then vector
is split with static coefficients plus any energy in stream 1,
delta coefficients plus any delta energy in stream 2 and
acceleration coefficients plus any acceleration energy in stream 3.
Otherwise, the parameter vector must include log energy and
must have appended delta coefficients. The vector is then split
into three parts so that the static coefficients form stream
1, the delta coefficients form stream 2, and the log energy
and delta log energy are combined to form stream 3.
\item[4 streams]
the parameter vector must include log energy and
must have appended delta and acceleration coefficients.
The vector is split into 4 parts so that the static coefficients form stream
1, the delta coefficients form stream 2, the acceleration
coefficients form stream 3 and the log energy, delta energy
and acceleration energy are combined to form stream 4.
\end{description}
In all cases,
the static log energy can be
suppressed (via the \texttt{\_N}\index{qualifiers!aaan@\texttt{\_N}} qualifier).
If none of the above rules apply for some required number of
streams, then the parameter vector is simply incompatible with that
form of observation. For example, the parameter kind \texttt{LPC\_D\_A}
cannot be split into 2 streams, instead 3 streams should be used.
\index{energy suppression}
\putfig{streams}{100}{Example Stream Construction}
Fig.~\href{f:streams} illustrates the way that streams are constructed
for a number of common cases. As earlier, the choice of \texttt{LPC}
as the static coefficients is purely for illustration and the same
mechanism applies to all base parameter kinds.
As discussed further in the next section,
multiple data streams are often used with vector quantised data. In this
case, each VQ symbol per input sample is placed in a separate data stream.
\mysect{Vector Quantisation}{vquant}
Although \HTK\ was designed primarily for building continuous density HMM
systems, it also supports discrete density HMMs. Discrete HMMs are
particularly useful for modelling data which is naturally symbolic. They can
also be used with continuous signals such as speech by quantising each speech
vector to give a unique VQ symbol for each input frame. The \HTK\ module
\htool{HVQ} provides a basic facility for performing this vector
quantisation\index{vector quantisation}. The VQ table (or codebook) can be
constructed using the \HTK\ tool \htool{HQuant}.
When used with speech, the principle justification for using discrete HMMs is
the much reduced computation. However, the use of vector quantisation
introduces errors and it can lead to rather fragile systems. For this reason,
the use of continuous density systems is generally preferred. To facilitate
the use of continuous density systems when there are computational constraints,
\HTK\ also allows VQ to be used as the basis for pre-selecting a subset of
Gaussian\index{Gaussian pre-selection} components for evaluation at each time
frame.
\sidefig{VQUse}{65}{Using Vector Quantisation}{2}{ Fig.~\href{f:VQUse}
illustrates the different ways that VQ can be used in \HTK\ for a single data
stream. For multiple streams, the same principles are applied to each stream
individually. A converted speech waveform or file of parameter vectors can
have VQ indices attached simply by specifying the name of a VQ table using the
configuration parameter \texttt{VQTABLE}\index{vqtable@\texttt{VQTABLE}} and by
adding the \texttt{\_V} qualifier to the target kind. The effect of this is
that each \textit{observation} passed to a recogniser can include both a
conventional parameter vector and a VQ index. \index{vector quantisation!uses
of} \index{qualifiers!aaav@\texttt{\_V}} For continuous density HMM systems, a
possible use of this might be to preselect Gaussians for evaluation (but note
that \HTK\ does not currently support this facility).
When used with a discrete HMM system, the
continuous parameter vectors are ignored and only the VQ
indices are used.
For training and evaluating discrete HMMs, it is convenient to
store speech data in vector quantised form. This is done using
the tool \htool{HCopy} to read in and vector quantise each speech file.
Normally, \htool{HCopy} copies the target form directly into the
output file. However, if the configuration parameter \texttt{SAVEASVQ}
is set, then it will
store only the VQ indices and mark the kind of the newly created
file as \texttt{DISCRETE}. Discrete files created in this
way can be read
directly by \htool{HParm} and the VQ symbols passed directly to
a tool as indicated by the lower part of Fig.~\href{f:VQUse}.
}
\index{saveasvq@\texttt{SAVEASVQ}}
\index{discrete@\texttt{DISCRETE}}
\index{vector quantisation!distance metrics}
\htool{HVQ} supports three types of distance metric and two organisations of
VQ codebook. Each codebook consists of a collection of nodes where each
node has a mean vector and optionally a covariance matrix or diagonal
variance vector. The corresponding distance metric used for each of these
is simple Euclidean, full covariance Mahalanobis or diagonal covariance
Mahalanobis. The codebook nodes are arranged in
the form of a simple linear table
or as a binary tree. In the linear case, the input vector is compared with
every node in turn and the nearest determines the VQ index. In the binary
tree case, each non-terminal node has a left and a right daughter. Starting
with the top-most root node, the input is compared with the left and right
daughter node and the nearest is selected. This process is repeated until a
terminal node is reached. \index{vector quantisation!type of}
\index{vector quantisation!code book external format}\index{files!VQ codebook}
VQ Tables are stored externally in text files consisting of a header
followed by a sequence of node entries. The header consists of the
following information
\begin{tabbing}
++ \= +++++++ \= + \= \kill
\> \textit{magic}\> --\> a magic number usually the original parameter kind \\
\> \textit{type} \> --\> 0 = linear tree, 1 = binary tree \\
\> \textit{mode} \> --\> 1 = diagonal covariance Mahalanobis \\
\>\>\> 2 = full covariance Mahalanobis \\
\>\>\> 5 = Euclidean \\
\> \textit{numNodes} \> --\> total number of nodes in the codebook \\
\> \textit{numS}\> --\> number of independent data streams \\
\> \textit{sw1,sw2,...}\> --\> width of each data stream \\
\end{tabbing}
Every node has a unique integer identifier and consists of the
following
\begin{tabbing}
++ \= +++++++ \= + \= \kill
\> \textit{stream}\> --\>stream number for this node \\
\> \textit{vqidx}\> --\>VQ index for this node (0 if non-terminal) \\
\> \textit{nodeId}\> --\>integer id of this node \\
\> \textit{leftId}\> --\>integer id of left daughter node \\
\> \textit{rightId}\> --\>integer id of right daughter node \\
\> \textit{mean}\> --\>mean vector \\
\> \textit{cov}\> --\>diagonal variance or full covariance \\
\end{tabbing}
The inclusion of the optional variance vector or covariance matrix depends
on the mode in the header. If present they are stored in inverse form.
In a binary tree, the root id is always 1. In linear codebooks, the
left and right daughter node id's are ignored.
\mysect{Viewing Speech with \htool{HList}}{UseHList}
\index{speech input!monitoring}
As mentioned in section~\ref{s:genio}, the tool \htool{HList}\index{hlist@\htool{HList}} provides
a dual r\^{o}le in \HTK. Firstly, it can be used for examining the contents
of speech data files.
In general, \htool{HList} displays three types of information
\begin{enumerate}
\item \textit{source header}: requested using the \texttt{-h} option
\item \textit{target header}: requested using the \texttt{-t} option
\item \textit{target data}: printed by default. The begin and end samples of the
displayed data can be specified using the \texttt{-s} and \texttt{-e} options.
\end{enumerate}
When the default configuration parameters are used,
no conversions are applied and the target
data is identical to the contents of the file.
\index{files!listing contents}
As an example, suppose that the file called \texttt{timit.wav} holds speech
waveform data using the TIMIT format. The command
\begin{verbatim}
HList -h -e 49 -F TIMIT timit.wav
\end{verbatim}
would display the source header information and the first 50 samples of the
file. The output would look something like the following
\begin{list}{}{\setlength{\leftmargin}{-1cm}}
\item
\begin{verbatim}
----------------------------- Source: timit.wav ---------------------------
Sample Bytes: 2 Sample Kind: WAVEFORM
Num Comps: 1 Sample Period: 62.5 us
Num Samples: 31437 File Format: TIMIT
------------------------------ Samples: 0->49 -----------------------------
0: 8 -4 -1 0 -2 -1 -3 -2 0 0
10: -1 0 -1 -2 -1 1 0 -1 -2 1
20: -2 0 0 0 2 1 -2 2 1 0
30: 1 0 0 -1 4 2 0 -1 4 0
40: 2 2 1 -1 -1 1 1 2 1 1
------------------------------------ END ----------------------------------
\end{verbatim}
\end{list}
The source information confirms that the file contains \texttt{WAVEFORM}
data with 2 byte samples and 31437 samples in total. The sample period is
$62.5\mu s$ which corresponds to a 16kHz sample rate.
The displayed data is numerically small because it corresponds to leading silence.
Any part of the file could be viewed by suitable choice of the begin and end
sample indices. For example,
\begin{verbatim}
HList -s 5000 -e 5049 -F TIMIT timit.wav
\end{verbatim}
would display samples 5000 through to 5049.
The output might look like the following
\begin{list}{}{\setlength{\leftmargin}{-1cm}}
\item
\begin{verbatim}
---------------------------- Samples: 5000->5049 --------------------------
5000: 85 -116 -159 -252 23 99 69 92 79 -166
5010: -100 -123 -111 48 -19 15 111 41 -126 -304
5020: -189 91 162 255 80 -134 -174 -55 57 155
5030: 90 -1 33 154 68 -149 -70 91 165 240
5040: 297 50 13 72 187 189 193 244 198 128
------------------------------------ END ----------------------------------
\end{verbatim}
\end{list}
The second use of \htool{HList} is to check that input conversions
are being performed properly. Suppose that the above TIMIT format file is
part of a database to be used for training a recogniser and that mel-frequency
cepstra are to be used along with energy and the first differential coefficients.
Suitable configuration parameters needed to achieve this might be as follows
\begin{verbatim}
# Wave -> MFCC config file
SOURCEFORMAT = TIMIT # same as -F TIMIT
TARGETKIND = MFCC_E_D # MFCC + Energy + Deltas
TARGETRATE = 100000 # 10ms frame rate
WINDOWSIZE = 200000 # 20ms window
NUMCHANS = 24 # num filterbank chans
NUMCEPS = 8 # compute c1 to c8
\end{verbatim}
\htool{HList} can be used to check this. For example, typing
\begin{verbatim}
HList -C config -o -h -t -s 100 -e 104 -i 9 timit.wav
\end{verbatim}
will cause the waveform file to be converted, then the source header,
the target header and parameter vectors 100 through to 104 to be listed.
A typical output would be as follows
\begin{verbatim}
------------------------------ Source: timit.wav ---------------------------
Sample Bytes: 2 Sample Kind: WAVEFORM
Num Comps: 1 Sample Period: 62.5 us
Num Samples: 31437 File Format: TIMIT
------------------------------------ Target --------------------------------
Sample Bytes: 72 Sample Kind: MFCC_E_D
Num Comps: 18 Sample Period: 10000.0 us
Num Samples: 195 File Format: HTK
-------------------------- Observation Structure ---------------------------
x: MFCC-1 MFCC-2 MFCC-3 MFCC-4 MFCC-5 MFCC-6 MFCC-7 MFCC-8 E
Del-1 Del-2 Del-3 Del-4 Del-5 Del-6 Del-7 Del-8 DelE
------------------------------ Samples: 100->104 ---------------------------
100: 3.573 -19.729 -1.256 -6.646 -8.293 -15.601 -23.404 10.988 0.834
3.161 -1.913 0.573 -0.069 -4.935 2.309 -5.336 2.460 0.080
101: 3.372 -16.278 -4.683 -3.600 -11.030 -8.481 -21.210 10.472 0.777
0.608 -1.850 -0.903 -0.665 -2.603 -0.194 -2.331 2.180 0.069
102: 2.823 -15.624 -5.367 -4.450 -12.045 -15.939 -22.082 14.794 0.830
-0.051 0.633 -0.881 -0.067 -1.281 -0.410 1.312 1.021 0.005
103: 3.752 -17.135 -5.656 -6.114 -12.336 -15.115 -17.091 11.640 0.825
-0.002 -0.204 0.015 -0.525 -1.237 -1.039 1.515 1.007 0.015
104: 3.127 -16.135 -5.176 -5.727 -14.044 -14.333 -18.905 15.506 0.833
-0.034 -0.247 0.103 -0.223 -1.575 0.513 1.507 0.754 0.006
------------------------------------- END ----------------------------------
\end{verbatim}
The target header information shows that the converted data consists
of 195 parameter vectors, each vector having 18 components and being 72 bytes in
size. The structure of each parameter vector is displayed as a simple sequence
of floating-point numbers. The layout information described in
section~\ref{s:parmstore} can be used to interpret the data.
However, including the \texttt{-o} option, as in the example, causes \htool{HList}
to output a schematic of the observation structure. Thus, it can be seen that
the first row of each sample contains the static coefficients and the second
contains the delta
coefficients. The energy is in the final column.
The command line option \texttt{-i 9} controls the number of values displayed
per line and can be used to aid in the visual interpretation of the data.
Notice finally that the command line option \texttt{-F TIMIT} was not required
in this case because the source format was specified in the configuration file.
It should be stressed that when \htool{HList} displays parameterised data,
it does so in exactly the form that \textit{observations} are passed to
a \HTK\ tool. So, for example, if the above data was input to a system
built using 3 data streams, then this can be simulated by using the
command line option \texttt{-n} to set the number of streams. For example, typing
\begin{verbatim}
HList -C config -n 3 -o -s 100 -e 101 -i 9 timit.wav
\end{verbatim}
would result in the following output
\begin{verbatim}
------------------------ Observation Structure -----------------------
nTotal=18 nStatic=8 nDel=16 eSep=T
x.1: MFCC-1 MFCC-2 MFCC-3 MFCC-4 MFCC-5 MFCC-6 MFCC-7 MFCC-8
x.2: Del-1 Del-2 Del-3 Del-4 Del-5 Del-6 Del-7 Del-8
x.3: E DelE
-------------------------- Samples: 100->101 -------------------------
100.1: 3.573 -19.729 -1.256 -6.646 -8.293 -15.601 -23.404 10.988
100.2: 3.161 -1.913 0.573 -0.069 -4.935 2.309 -5.336 2.460
100.3: 0.834 0.080
101.1: 3.372 -16.278 -4.683 -3.600 -11.030 -8.481 -21.210 10.472
101.2: 0.608 -1.850 -0.903 -0.665 -2.603 -0.194 -2.331 2.180
101.3: 0.777 0.069
--------------------------------- END --------------------------------
\end{verbatim}
Notice that the data is identical to the previous case, but it has been
re-organised into separate streams.\index{observations!displaying structure of}
\mysect{Copying and Coding using \htool{HCopy}}{UseHCopy}
\index{files!copying}
\htool{HCopy}\index{hcopy@\htool{HCopy}} is a general-purpose tool
for copying and manipulating speech files.
The general form of invocation is
\begin{verbatim}
HCopy src tgt
\end{verbatim}
which will make a new copy called \texttt{tgt} of the file called \texttt{src}.
\htool{HCopy} can also concatenate several sources together as in
\begin{verbatim}
HCopy src1 + src2 + src3 tgt
\end{verbatim}
which concatenates the contents of \texttt{src1}, \texttt{src2} and \texttt{src3},
storing the results in the file \texttt{tgt}. As well as putting speech files
together, \htool{HCopy} can also take them apart. For example,
\begin{verbatim}
HCopy -s 100 -e -100 src tgt
\end{verbatim}
will extract samples 100 through to N-100 of the file \texttt{src} to the file
\texttt{tgt} where N is the total number of samples in the source file.
The range of samples to be copied can also be specified with reference to
a label file, and modifications made to the speech file can be tracked in a
copy of the label file. All of the various options provided by \htool{HCopy}
are given in the reference section and in total they provide a powerful
facility for manipulating speech data files.
However, the use of \htool{HCopy} extends beyond that of copying, chopping
and concatenating files. \htool{HCopy} reads in all files using the speech
input/output subsystem described in the preceding sections. Hence, by specifying
an appropriate configuration file, \htool{HCopy} is also a speech coding tool.
For example, if the configuration file \texttt{config} was set-up to convert
waveform data to MFCC coefficients, the command
\begin{verbatim}
HCopy -C config -s 100 -e -100 src.wav tgt.mfc
\end{verbatim}
would parameterise the file waveform file \texttt{src.wav}, excluding the
first and last 100 samples, and store the result in \texttt{tgt.mfc}.
\htool{HCopy} will process its arguments in pairs, and as with all \HTK\ tools,
argument lists can be written in a script file specified via the \texttt{-S}
option. When coding a large database, the separate invocation of \htool{HCopy}
for each file needing to be processed would incur a very large overhead. Hence,
it is better to create a file, \texttt{flist} say, containing a list of
all source and target files, as in for example,
\begin{verbatim}
src1.wav tgt1.mfc
src2.wav tgt2.mfc
src3.wav tgt3.mfc
src4.wav tgt4.mfc
etc
\end{verbatim}
and then invoke \htool{HCopy} by
\begin{verbatim}
HCopy -C config -s 100 -e -100 -S flist
\end{verbatim}
which would encode each file listed in \texttt{flist} in a single invocation.
Normally \htool{HCopy} makes a direct copy of the target speech data in the
output file. However, if the configuration
parameter \texttt{SAVECOMPRESSED}\index{savecompressed@\texttt{SAVECOMPRESSED}}
is set true then the output is saved in compressed
form and if the
configuration parameter \texttt{SAVEWITHCRC}\index{savewithcrc@\texttt{SAVEWITHCRC}}
is set true then a checksum is appended to the output
(see section~\ref{s:parmstore}). If the configuration
parameter \texttt{SAVEASVQ} is set true then only
VQ indices are saved and the kind of the target file is changed to
\texttt{DISCRETE}. For this to work, the target kind must have the
qualifier \texttt{\_V}
\index{qualifiers!aaav@\texttt{\_V}}
attached (see section~\ref{s:vquant}).
\index{compression}\index{check sums}
\index{files!compressing}\index{files!adding checksums}
\centrefig{coercions}{100}{Valid Parameter Kind Conversions}
\mysect{Version 1.5 Compatibility}{v1spcompat}
The redesign of the \HTK\ front-end in version 2 has introduced
a number of differences in parameter encoding. The main
changes are
\begin{enumerate}
\item Source waveform zero mean processing is now performed on a frame-by-frame
basis.
\item Delta coefficients use a modified form of regression rather than
simple differences at the start and end of the utterance.
\item Energy scaling is no longer applied to the zero'th MFCC coefficient.
\end{enumerate}
If a parameter encoding is required which is as close as possible
to the version 1.5 encoding, then the compatibility configuration
variable \texttt{V1COMPAT} should be set to true.
Note also in this context that the default values for the various
configuration values have been chosen to be consistent with the
defaults or recommended practice for version 1.5.
\mysect{Summary}{spiosum}
\index{speech input!summary of variables}
This section summarises the various file formats, parameter kinds, qualifiers
and configuration parameters used by \HTK. Table~\href{t:fileform} lists the
audio speech file formats which can be read by the \htool{HWave} module.
Table~\href{t:parmkinds} lists the basic parameter kinds supported by the
\htool{HParm} module and Fig.~\href{f:coercions} shows the various automatic
conversions that can be performed by appropriate choice of source and target
parameter kinds. Table~\href{t:qualifiers} lists the available qualifiers for
parameter kinds. The first 6 of these are used to describe the target kind.
The source kind may already have some of these, \htool{HParm} adds the rest as
needed. Note that \htool{HParm} can also delete qualifiers when converting
from source to target. The final two qualifiers in Table~\href{t:qualifiers}
are only used in external files to indicate compression and an attached
checksum. \htool{HParm} adds these qualifiers to the target form during output
and only in response to setting the configuration parameters
\texttt{SAVECOMPRESSED} and \texttt{SAVEWITHCRC}. Adding the
\texttt{\_C}\index{qualifiers!aaac@\texttt{\_C}} or
\texttt{\_K}\index{qualifiers!aaak@\texttt{\_K}} qualifiers to the target kind
simply causes an error. Finally, Tables \href{t:spiocparms1} and
\href{t:spiocparms2} lists all of the configuration parameters along with their
meaning and default values.
\begin{center}
\begin{tabular}{|p{2.6cm}|p{8.7cm}|} \hline
Name & Description \\ \hline
\texttt{\HTK} & The standard \HTK\ file format\\
\texttt{TIMIT} & As used in the original prototype TIMIT CD-ROM\\
\texttt{NIST} & The standard SPHERE format used by the US NIST\\
\texttt{SCRIBE} & Subset of the European SAM standard used in the SCRIBE CD-ROM\\
\texttt{SDES1} & The Sound Designer 1 format defined by Digidesign Inc. \\
\texttt{AIFF} & Audio interchange file format\\
\texttt{SUNAU8} & Subset of 8bit ".au" and ".snd" formats used by Sun and NeXT\\
\texttt{OGI} & Format used by Oregan Graduate Institute similar to TIMIT\\
\texttt{WAV} & Microsoft WAVE files used on PCs\\
\texttt{ESIG} & Entropic Esignal file format\\ \hline
\texttt{AUDIO} & Pseudo format to indicate direct audio input \\
\texttt{ALIEN} & Pseudo format to indicate unsupported file, the
alien header size must be set via the environment variable \texttt{HDSIZE} \\
\texttt{NOHEAD} & As for the ALIEN format but header size is zero \\ \hline
\end{tabular}
\tabcap{fileform}{Supported File Formats}
\end{center}
\begin{center}
\begin{tabular}{|p{2.6cm}|p{8.7cm}|} \hline
Kind & Meaning \\ \hline
\texttt{WAVEFORM} & scalar samples (usually raw speech data) \\
\texttt{LPC} & linear prediction coefficients \\
\texttt{LPREFC} & linear prediction reflection coefficients \\
\texttt{LPCEPSTRA} & LP derived cepstral coefficients \\
\texttt{LPDELCEP} & LP cepstra + delta coef (obsolete) \\
\texttt{IREFC} & LPREFC stored as 16bit (short) integers \\
\texttt{MFCC} & mel-frequency cepstral coefficients \\
\texttt{FBANK} & log filter-bank parameters \\
\texttt{MELSPEC} & linear filter-bank parameters \\
\texttt{USER} & user defined parameters \\
\texttt{DISCRETE} & vector quantised codebook symbols \\
\texttt{PLP} & perceptual linaer prediction coefficients \\
\texttt{ANON} & matches actual parameter kind \\ \hline
\end{tabular}
\tabcap{parmkinds}{Supported Parameter Kinds}
\end{center}
\begin{center}
\begin{tabular}{|p{2.6cm}|p{8.7cm}|} \hline
Qualifier & Meaning \\ \hline
\texttt{\_A} & Acceleration coefficients appended \\
\texttt{\_C} & External form is compressed\\
\texttt{\_D} & Delta coefficients appended \\
\texttt{\_E} & Log energy appended\\
\texttt{\_K} & External form has checksum appended\\
\texttt{\_N} & Absolute log energy suppressed \\
\texttt{\_T} & Third differential coefficients appended \\
\texttt{\_V} & VQ index appended\\
\texttt{\_Z} & Cepstral mean subtracted\\
\texttt{\_0} & Cepstral C0 coefficient appended\\ \hline
\end{tabular}
\tabcap{qualifiers}{Parameter Kind Qualifiers}
\end{center}\index{qualifiers!summary}
\begin{center}
\begin{tabular}{|p{1.2cm}|p{3.0cm}|p{1.3cm}|p{6.5cm}|} \hline
Module & Name & Default & Description \\ \hline
\htool{HAudio} & \texttt{LINEIN} & \texttt{T} & Select line input for audio\\
\htool{HAudio} & \texttt{MICIN} & \texttt{F} & Select microphone input for audio\\
\htool{HAudio} & \texttt{LINEOUT} & \texttt{T} & Select line output for audio\\
\htool{HAudio} & \texttt{SPEAKEROUT} & \texttt{F} & Select speaker output for audio\\
\htool{HAudio} & \texttt{PHONESOUT} & \texttt{T} & Select headphones output for audio\\
& \texttt{SOURCEKIND} & \texttt{ANON} & Parameter kind of source \\
& \texttt{SOURCEFORMAT} & \texttt{HTK} & File format of source \\
& \texttt{SOURCERATE} & \texttt{0.0} & Sample period of source in 100ns units \\
\htool{HWave} & \texttt{NSAMPLES} & & Num samples in alien file input via a pipe\\
\htool{HWave} & \texttt{HEADERSIZE} & & Size of header in an alien file\\
\htool{HWave} & \texttt{STEREOMODE} & & Select channel: \texttt{RIGHT} or \texttt{LEFT} \\
\htool{HWave} & \texttt{BYTEORDER} & & Define byte order \texttt{VAX} or other\\
& \texttt{NATURALREADORDER} & \texttt{F} & Enable natural read order for HTK files \\
& \texttt{NATURALWRITEORDER} & \texttt{F} & Enable natural write order for HTK files \\
& \texttt{TARGETKIND} & \texttt{ANON} & Parameter kind of target \\
& \texttt{TARGETFORMAT} & \texttt{HTK} & File format of target \\
& \texttt{TARGETRATE} & \texttt{0.0} & Sample period of target in 100ns units \\
\htool{HParm} & \texttt{SAVECOMPRESSED} & \texttt{F} & Save the output file in compressed form \\
\htool{HParm} & \texttt{SAVEWITHCRC} & \texttt{T} & Attach a checksum to output parameter file \\
\htool{HParm} & \texttt{ADDDITHER} & \texttt{0.0} & Level of noise added to input signal \\
\htool{HParm} & \texttt{ZMEANSOURCE} & \texttt{F} & Zero mean source waveform before analysis \\
\htool{HParm} & \texttt{WINDOWSIZE} & \texttt{256000.0} & Analysis window size in 100ns units \\
\htool{HParm} & \texttt{USEHAMMING} & \texttt{T} & Use a Hamming window \\
\htool{HParm} & \texttt{PREEMCOEF} & \texttt{0.97} & Set pre-emphasis coefficient \\
\htool{HParm} & \texttt{LPCORDER} & \texttt{12} & Order of LPC analysis \\
\htool{HParm} & \texttt{NUMCHANS} & \texttt{20} & Number of filterbank channels \\
\htool{HParm} & \texttt{LOFREQ} & \texttt{-1.0} & Low frequency cut-off in fbank analysis \\
\htool{HParm} & \texttt{HIFREQ} & \texttt{-1.0} & High frequency cut-off in fbank analysis \\
\htool{HParm} & \texttt{USEPOWER} & \texttt{F} & Use power not magnitude in fbank analysis \\
\htool{HParm} & \texttt{NUMCEPS} & \texttt{12} & Number of cepstral parameters \\
\htool{HParm} & \texttt{CEPLIFTER} & \texttt{22} & Cepstral liftering coefficient \\
\htool{HParm} & \texttt{ENORMALISE} & \texttt{T} & Normalise log energy \\
\htool{HParm} & \texttt{ESCALE} & \texttt{0.1} & Scale log energy \\
\htool{HParm} & \texttt{SILFLOOR} & \texttt{50.0} & Energy silence floor (dB) \\
\htool{HParm} & \texttt{DELTAWINDOW} & \texttt{2} & Delta window size \\
\htool{HParm} & \texttt{ACCWINDOW} & \texttt{2} & Acceleration window size \\
\htool{HParm} & \texttt{VQTABLE} & \texttt{NULL} & Name of VQ table \\
\htool{HParm} & \texttt{SAVEASVQ} & \texttt{F} & Save only the VQ indices \\
\htool{HParm} & \texttt{AUDIOSIG} & \texttt{0} & Audio signal number for remote control \\ \hline
\end{tabular}
\tabcap{spiocparms1}{Configuration Parameters}
\end{center}
\begin{center}
\begin{tabular}{|p{1.1cm}|p{2.6cm}|p{1.4cm}|p{6.5cm}|} \hline
Module & Name & Default & Description \\ \hline
\htool{HParm} & \texttt{USESILDET} & \texttt{F} & Enable speech/silence detector \\
\htool{HParm} & \texttt{MEASURESIL} & \texttt{T} & Measure background noise level prior to sampling \\
\htool{HParm} & \texttt{OUTSILWARN} & \texttt{T} & Print a warning message to {\tt stdout} before
measuring audio levels \\
\htool{HParm} & \texttt{SPEECHTHRESH} & \texttt{9.0} & Threshold for speech above silence level (dB) \\
\htool{HParm} & \texttt{SILENERGY} & \texttt{0.0} & Average background noise level (dB) \\
\htool{HParm} & \texttt{SPCSEQCOUNT} & \texttt{10} & Window over which speech/silence decision reached \\
\htool{HParm} & \texttt{SPCGLCHCOUNT} & \texttt{0} & Maximum number of frames marked as silence in window which is
classified as speech whilst expecting start of speech \\
\htool{HParm} & \texttt{SILSEQCOUNT} & \texttt{100} & Number of frames classified as silence needed to mark end of
utterance \\
\htool{HParm} & \texttt{SILGLCHCOUNT} & \texttt{2} & Maximum number of frames marked as silence in window which is
classified as speech whilst expecting silence \\
\htool{HParm} & \texttt{SILMARGIN} & \texttt{40} & Number of extra frames included before and after start and end of
speech marks from the speech/silence detector \\
\htool{HParm} & \texttt{V1COMPAT} & \texttt{F} & Set Version 1.5 compatibility mode \\
& \texttt{TRACE} & \texttt{0} & Trace setting\\ \hline
\end{tabular}
\tabcap{spiocparms2}{Configuration Parameters (cont)}
\end{center}
%%% Local Variables:
%%% mode: latex
%%% TeX-master: "htkbook"
%%% End:
|
subroutine ptrac1
!***********************************************************************
! Copyright, 2004, The Regents of the University of California.
! This program was prepared by the Regents of the University of
! California at Los Alamos National Laboratory (the University) under
! contract No. W-7405-ENG-36 with the U.S. Department of Energy (DOE).
! All rights in the program are reserved by the DOE and the University.
! Permission is granted to the public to copy and use this software
! without charge, provided that this Notice and any statement of
! authorship are reproduced on all copies. Neither the U.S. Government
! nor the University makes any warranty, express or implied, or
! assumes any liability or responsibility for the use of this software.
!***********************************************************************
!D1
!D1 PURPOSE
!D1
!D1 Perform initial setup functions for the streamline particle
!D1 tracking calculations.
!D1
!***********************************************************************
!D2
!D2 REVISION HISTORY
!D2
!D2 FEHM Version 2.0, SC-194
!D2
!D2 $Log: /pvcs.config/fehm90/src/ptrac1.f_a $
!D2
!D2 Rev 2.5 06 Jan 2004 10:43:06 pvcs
!D2 FEHM Version 2.21, STN 10086-2.21-00, Qualified October 2003
!D2
!D2 Rev 2.4 29 Jan 2003 09:12:28 pvcs
!D2 FEHM Version 2.20, STN 10086-2.20-00
!D2
!D2 Rev 2.3 14 Nov 2001 13:11:44 pvcs
!D2 FEHM Version 2.12, STN 10086-2.12-00
!D2
!D2 Rev 2.2 06 Jun 2001 13:36:12 pvcs
!D2 FEHM Version 2.11, STN 10086-2.11-00
!D2
!D2 Rev 2.2 06 Jun 2001 08:26:14 pvcs
!D2 Update for extended dispersion tensor model
!D2
!D2 Rev 2.1 30 Nov 2000 12:06:02 pvcs
!D2 FEHM Version 2.10, STN 10086-2.10-00
!D2
!D2 Rev 2.0 Fri May 07 14:44:28 1999 pvcs
!D2 FEHM Version 2.0, SC-194 (Fortran 90)
!D2
!***********************************************************************
!D3
!D3 REQUIREMENTS TRACEABILITY
!D3
!D3 2.3.6 Streamline particle-tracking module
!D3
!***********************************************************************
!D4
!D4 SPECIAL COMMENTS AND REFERENCES
!D4
!D4 Requirements from SDN: 10086-RD-2.20-00
!D4 SOFTWARE REQUIREMENTS DOCUMENT (RD) for the
!D4 FEHM Application Version 2.20
!D4
!***********************************************************************
use comai
use combi
use comci
use comdi
use comsptr
use compart
use comsk
use comxi, only : nmfil
use davidi
use comwt
implicit none
integer neqp1,n50
integer i3,ii1,ii2,i1,i33,ix,iy,iz
integer ip,is,i5,kbp,kbm,kb,i,np1
integer current_node
integer n_porosi0,itemp_col,itemp_node, kb2,flag_box,inp1
real*8 dx,dy,dz,ep,ep5,x60,x33
real*8 rprime, spacing
real*8 xcoordw, ycoordw, zcoordw, del_plus, del_minus
real*8 ps_print
real*8 s_print
integer connect_flag, upper_limit
integer iprcount, iprint
real*8 tol_c
parameter(tol_c=1.d-20)
integer position_in_string, final_position
integer n_written, iprops, jprops, ijkv_find
c......dec 4 01 s kelkar insert omr changes ...................
integer inode,iwsk
real*8 aread,aread_max
integer flag_sk
integer iomr_flag
integer idbg,kdbg
real*8 gotcord
integer npart_ist2
real*8 epsilonwt
c
c&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&
! Called from insptr now
! if(cliff_flag) call setup_cliffnodes
c&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&
epsilonwt=1.e-12
count_steps = 0
if (omr_flag) iboulist = 0
if (omr_flag) ipc_save = 0
! Added to read static arrays for calculations once they have been computed zvd 20-Apr-04
if (save_omr) then
inquire ( file = nmfil(23), exist = sptr_exists )
if (sptr_exists) then
! Read arrays, no need to recompute
call sptr_save (0)
end if
else
sptr_exists = .false.
end if
c!!!for debuggin 5/13/04
c do i=1,neq
c if((i).eq.1..or.ddy(i).eq.1..or.ddz(i).eq.1.) then
c write(*,*)'i,ddx(i),ddy(i),ddz(i);',i,ddx(i),ddy(i),ddz(i)
c endif
c enddo
c!!!!!!
c Initialize parameters
neqp1=neq+1
ep=1.e-7
ep5=.5*ep
c Initialize particle time arrays
ttt=0
if (irsttime .ne. 0) then
tt1 = rsttime
c ZVD - 10-Dec-09, time can't be less than the simulation start time
do i = 1, num_part
if (ttp1(i) .lt. tt1) ttp1(i) = tt1
end do
else
tt1= 0.d0
end if
tt = 0
! ZVD - 14-Oct-05, initialized in insptr (for particle restarts)
! ttp1=0
c Initialize omr arrays if data not read in
if (.not. sptr_exists .and. omr_flag) then
node_omr = 0
isave_omr = 0
end if
c 5-Nov-02 zvd Add default nsp and icns values (liquid only)
nsp = 1
icns = 1
if(.not.unstruct_sptr) then
c Determine setup arrays for structured grids
c Call routine to construct the connectivity array for
c structured grids
c !!!!!!!!!!!!!!!!!5/13/04 next two lines taken out for debugging
if (.not. sptr_exists) then
call struct_conn_array(iomr_flag)
c.. s kelkar sep 28 04, moved call to geom_array inside conn_array
c Call routine to determine geometric sizes of structured grid cells
c call struct_geom_array
endif
if(node_count.eq.0) omr_flag=.false.
else
c Setup for unstructured grids
call unstruct_arrays
end if
c Determine initial state of particles (positions, nodes)
if(ist.eq.2) then
call particle_patch(npart_ist2)
end if
c *******initial state *********
c Find cell location where particle starts
if(abs(ist).ge.1) then
call find_particle_cell(npart_ist2,ierr,iptty)
c..............................................................
c...Oct 15 2008 s kelkar if porosity<=0 move down the column
c routine wtsi_column sorts nodes in vertical columns
c wcol(node)=column# corresponding to the node
c n_col(column#)=# of nodes in the column
c col(kb,wcol(node))=node #s in that column
n_porosi0= 0
do i=1,num_part
if (ijkv(i) .ne. 0) then
if(ps(ijkv(i)).le.0.) then
if (ifree .ne. 0) then
c Only try to move particle if wtsi problem
n_porosi0 = n_porosi0 +1
if( n_porosi0.eq.1) then
if(.not.allocated(wcol)) then
call wtsi_column
endif
endif
inp1=ijkv(i)
itemp_col = wcol(inp1)
do kb2 = 1, n_col(itemp_col)
itemp_node=col(itemp_col,kb2)
if(inp1.eq.itemp_node) then
do kb = kb2+1, n_col(itemp_col)
itemp_node=col(itemp_col,kb)
if(ps(itemp_node).gt.0.) then
ijkv(i) = itemp_node
z1(i)=cord(itemp_node,3)-
& corn(itemp_node,3)
goto 96969
endif
enddo
c did not find a porosity>0 node in the column. Do a neighbor search
call tree_search_porosity(ijkv(i),5,flag_box)
if(flag_box.gt.0) then
ijkv(i)=flag_box
z1(i) = cord(flag_box,3)
else
write(ierr, 222) i, ijkv(i)
ijkv(i) = 0
istop(i) = 1
c write(ierr,*)"error in ptrac1. can't find"
c write(ierr,*)'neighbor with porosity>0. ',
c & 'STOP.'
c stop
endif
endif
enddo
96969 continue
else
c Just remove the particle
write(ierr, 223) ijkv(i), i
istop(i) = 1
ijkv(i) = 0
end if
endif
end if
enddo
222 format ("Error in ptrac1: can't find neighbor with porosity>0",
& ' for particle ', i8, 'at node ', i8)
223 format ('Error in ptrac1: Invalid particle start ',
& 'at 0 porosity node ', i8, ' for particle number ', i8)
c..............................................................
if (ist .eq. 2) then
do i = 1, num_part
part_id(i,1) = i
part_id(i,2) = ijkv(i)
end do
end if
end if
c zvd 06-21-07 Set x3,y3,z3 to initial particle location
do i = 1, num_part
if (abs(ijkv(i)) .ne. 0) then
x3(i) = x1(i) + corn(ijkv(i), 1)
y3(i) = y1(i) + corn(ijkv(i), 2)
z3(i) = z1(i) + corn(ijkv(i), 3)
c zvd 03-16-2010 Set x3, y3, z3 to initial particle location in insptr for ist = 1
else if (ist .ne. 1) then
x3(i) = x1(i)
y3(i) = y1(i)
z3(i) = z1(i)
end if
end do
c...for a quick fix of bouncing particles s kelkar 1/16/02
do np1=1,num_part
oldnode(np1)=ijkv(np1)
oldnode2(np1)=ijkv(np1)
oldnode_rand(np1)=ijkv(np1)
enddo
c......................................
c Output of particle information changed: BAR 6-15-99
c
if(iprto.ne.0) then
call output_info
end if
c Subroutine to initialize particle tracking transport parameters
c s kelkar may 28 09 moved call to fehmn.f where ptrac1 used to be called
c call init_sptr_params
c s kelkar may 20 09 moved call to fehmn.f where ptrac1 used to be called
c if (.not. compute_flow) then
c if (.not. sptr_exists) call load_omr_flux_array
c if(.not.random_flag) then
c if(allocated(sx)) deallocate(sx)
c if(allocated(istrw)) deallocate(istrw)
c end if
c end if
! Added to save static arrays for calculations once they have been computed zvd 20-Apr-04
c zvd 19-Nov-2010
! Moved to fehmn, needs to be called after call to load_omr_flux_array
! if (.not. sptr_exists .and. save_omr) call sptr_save (1)
return
contains
******************************************************************
******************************************************************
subroutine struct_conn_array(iomr_flag)
use comsk
use comai, only : ierr, iout, iptty
implicit none
integer ibou, iflag_boundry, iomr_flag,idum,irray0,idumm
c **** get irray *****
if(icnl.eq.0) then
upper_limit = 3
else
upper_limit = 2
end if
do i=1,neq
do i3=-3,3
ggg(i,i3)=0.
enddo
enddo
c ...dec 4 01, s kelkar nov 1 01, OMR stuff ...............................
node_count=0
c.................................................................
do i=1,neq
do i3=-3,3
c do not zero out irray(i,0) because it has particle capture
c information read in insptr
if(i3.ne.0) irray(i,i3)=0
enddo
c ...dec 4 01, s kelkar nov 1 01, OMR stuff ......................
iomr_count=0
iomr_flag=0
isave_omr=0
do i1=1,iomrmax
iomr_neighbour(i1)=0
enddo
c.................................................................
ii1=nelm(i)+1
ii2=nelm(i+1)
c.....dec 4, 01 s kelkar 9/21/01 find the max area connection at this node
c for flagging connection-areas that are zero (less than epsilon)
aread_max=0.
aread=0.
do i1=ii1,ii2
kb=nelm(i1)
if(kb.ne.i) then
c istrw is a pointer array for sx, corresponding to the connection
c i-i1. Generally (but not always) it is a scalar, the negative of the
c magnitude of (cross sectional area0/3. divided by the inter-nodal
c distance
iwsk=istrw(i1-neq-1)
aread=-sx(iwsk,1)
if(aread.gt.aread_max) aread_max=aread
endif
enddo
aread_max=aread_max*1.e-8
c...................................................
do i1=ii1,ii2
kb=nelm(i1)
if(kb.ne.i) then
c Do loop filters out any connections that aren't
c oriented only in the x, y, or z directions
connect_flag = 0
do i3=1,upper_limit
x33=cord(kb,i3)-cord(i,i3)
if(x33.gt. tol_c ) connect_flag = connect_flag + 1
if(x33.lt.-tol_c ) connect_flag = connect_flag + 1
end do
if(connect_flag.lt.2) then
c if connect_flag =1 then its a good connection, store the node
c number in the correct slot of irray
i33=0
do i3=1,upper_limit
x33=cord(kb,i3)-cord(i,i3)
if(x33.gt. tol_c) then
i33= i3
endif
if(x33.lt.-tol_c) then
i33=-i3
endif
enddo
c s kelkar, may 25,04, check for -ve porosity at kb, these are treated
c as no flow connections. Flag these with -ve sign
! Use rock matirx pososity here to account for nodes that have been
! eliminated using negative porosities
if(ps(kb).gt.0.) then
irray(i,i33)=+kb
else
irray(i,i33)=-kb
endif
endif
if(connect_flag.gt.1) then
c ........... dec 4 01 s kelkar 9/21/01..........................
c if connect_flag > 1 then the connection is not lined up
c with any of the axis. If this connection has a nonzero area
c then it signals change from a structured to unstructured
c part of the grid. Howver, zero area connections that are
c not lined up with axis can also occure
c in a structured part of the grid, ie a diagonal in a square.
c these have to be filtered out.
iwsk=istrw(i1-neq-1)
aread=-sx(iwsk,1)
if(aread.gt.aread_max) then
c ...s kelkar nov 1 01, OMR stuff ...............................
c here is how the information is stored:
c node_count counts the number of nodes that have at least one bad
c connection with any one of its neighbours, and the node number of
c is stored in node_omr(node_count).
c for each such node i, the # of its neighbours that have a bad
c connection with it are counted in iomr_count, and the pointers
c (i1) to the neighbour
c node numbers are stored temporarily in iomr_neighbour(iomr_count)
c When the loop ovver the neighbours 'kb' is finished, then for each
c node i with a bad connection, the subroutine 'subomr' is called,
c which sorts out the faces on which bad connections occur.
c komr_count(k) is a temporary counter for the number of neighbours
c of 'i' on a particular face (k). and the pointers (i1) to the node
c numbers of such neighbours are saved in
c isave_omr(face#,komr_count(k))
if(iomr_flag.eq.0) then
iomr_flag=1
node_count=node_count+1
if(node_count.gt.omr_nodes) then
write (ierr, 1001) node_count, omr_nodes
if (iptty .ne. 0)
. write (iptty, 1001) node_count, omr_nodes
call exit_ptrac1
endif
1001 format ( 'ERROR in PTRAC1: count ', i8,
& ' greater than number of omr nodes ',
& i8, /, 'STOPPING')
node_omr(node_count)=i
endif
iomr_count=iomr_count+1
if(iomr_count.gt.iomrmax) then
write(ierr,*)'iomr_count.gt.iomrmax in ptrac1'
write(ierr,*)'increase dimension of ',
. 'iomr_neighbour'
write(ierr,*)'STOPPING'
if (iptty .ne. 0) then
write(iptty,*)'iomr_count.gt.iomrmax in ptrac1'
write(iptty,*)'increase dimension of ',
. 'iomr_neighbour'
write(iptty,*)'STOPPING'
end if
call exit_ptrac1
endif
c NOTE: saving the pointer i1 rather than the node # kb because
c i1 can be directly used as a pointer to sx and a_axy arrays
c and kb can be retrieved from nelm(i1)
iomr_neighbour(iomr_count)=i1
c close(87)
endif
endif
c.................................................................
endif
enddo
if(iomr_flag.eq.1) then
c OMR node can not be specified as a well-capture node. At this stage
c this could have only come from insptr- sptr input file must be
c modified
irray0=irray(i,0)
if(irray0.eq.-i.or.irray0.eq.-(i+1000000)
1 .or.(irray0.lt.-(10000000).and.irray0.gt.-(100000000))
2 )then
write(ierr,*)'OMR node can not specified as a '
write(ierr,*)'well-capture node.sptr input file'
write(ierr,*)'must be modified. check keyword cpatur'
write(ierr,*)'SUBROUTINE struct_conn_array (ptrac1)'
write(ierr,*)'Node Number=',i
write(ierr,*)'STOP'
call exit_ptrac1
endif
if(irray0.lt.-100000000) then
idumm=-(irray0+100000000)
irray0=-200000000-idumm
elseif(irray(i,0).ne.-(i+2000000)) then
irray0 = 0
endif
irray(i,0) = irray0
c omr node allowed to be a spring node
c....s kelkar Jan 27march 10, 04, 3D ORM stuff............
c irray(i,0) = +i : regular, interior node, not a source/sink
c irray(i,0) = -i-2000 : regular node on a external boundary
c irray(i,0) = -i : regular interior node that is a sink/source
c but not explicitly specified in sptr macro
c -100000000 < irray(i,0) < -10000000 : regular interior node that is
c specified as a sink/source in the sptr macro
c in this case -(irray(i,0)+10000000) is the pointer
c for the storage location in well_radius for this node
c -200000000 < irray(i,0) < -100000000 : non-OMR cliff node
c irray(i,0) < -200000000 : OMR cliff node
c if the node is a cliff node, but has a specified boundary
c outflow at it, remove cliff tag and mark as a regular
c boundry node in load_omr_flux_array
c = -(i+2000000) : spring node
c = -(i+1000000) : well-capture node on external bound
c similar to =-i case but with half space solution
c irray(i,0) = 0 : OMR node not on boundary
c irray(i,0) = -i-1000 : OMR node on a external boundary
c
c flag OMR nodes that are on the exterior boundary, and save the
c exterior faces in iboulist. Also set irray(i,0)=-i-1000 for
c OMR nodes on a external boundary
call boundary_nodes(i,+1)
c...................................................
else
c flag non-OMR nodes that are on the exterior boundary, and save the
c exterior faces in iboulist.
call boundary_nodes(i,0)
irray0=irray(i,0)
if(irray0.ne.-i.and.irray0.ne.-(i+1000000).and.
1 irray0.ne.-(i+2000000).and.irray0.ne.-(i+2000).
2 and.irray0.gt.-10000000) then
irray(i,0) = i
endif
endif
c ...dec 4 01 s kelkar nov 1 01, OMR stuff ..........................
if(iomr_flag.eq.1) call subomr2(i)
c.................................................................
c... s kelkar sep 28, 04 reduce storage by calling geom_array here
c and changin the dim of iomr_save to (la,k)
c calculate del_plus, del_minus- the distances to the
c control volume boundaries, for use in ddx_corn_array
call struct_geom_array(i)
enddo
c set up the ddx,ddy,ddz and corn arrays
call ddx_corn_array
c
do i=1,neq
do i3=-3,3
ggg(i,i3)=0.
enddo
enddo
write(iout, 1003) omr_nodes, node_count
if (iptty .ne. 0) write(iptty, 1003) omr_nodes, node_count
1003 format ('Number of OMR nodes set to ', i8, /,
. 'Actual number of OMR nodes ', i8)
c **** got irray *****
return
end subroutine struct_conn_array
******************************************************************
******************************************************************
subroutine output_info
implicit none
real*8 sptr_time
character*200 sptr_heading, sptr_prop_values
c Sets up output of particle tracking info
c Only used for regular output with transient particle start times,
c trans_flag is used for minimal output options
pstart_out = .true.
c Robinson added minimal write option 3-11-02
if(iprto.lt.0) then
if (xyz_flag) then
c ZVD modified minimal write option to include coordinates 12-08-2005
if (iprto .eq. -1) then
write(isptr2,100)
else
write (isptr2) 'XYZ'
end if
do np1 = 1, num_part
current_node = ijkv(np1)
xcoordw = x1(np1) + corn(current_node,1)
ycoordw = y1(np1) + corn(current_node,2)
zcoordw = z1(np1) + corn(current_node,3)
sptr_time = ttp1(np1)
if (iprto .eq. -1) then
write(isptr2,105) part_id(np1,1),sptr_time,
& current_node,xcoordw, ycoordw, zcoordw
else
write (isptr2) part_id(np1,1),sptr_time,current_node,
& xcoordw, ycoordw, zcoordw
end if
end do
else if (ip_flag .or. trans_flag) then
c ZVD added option to write initial position to abbreviated output file
if (iprto .eq. -1) then
if (ip_flag) then
write(isptr2, 110) ''
else
write(isptr2, 110) 'TRA : '
end if
else
if (trans_flag) write (isptr2) 'TRA'
end if
do np1 = 1, num_part
c ZVD added option for transient where particle start time is saved but
c initial node is set to 0
c ZVD 07-Feb-2011 negative of starting node is now output
c (this way particles that have been excluded can be distinguished from
c particles that have a delayed start time)
if (trans_flag) then
c current_node = 0
current_node = -ijkv(np1)
else
current_node = ijkv(np1)
end if
sptr_time = ttp1(np1)
if (iprto .eq. -1) then
write(isptr2,105) part_id(np1,1),sptr_time,
& current_node
else
write (isptr2) part_id(np1,1),sptr_time,current_node
end if
end do
end if
100 format ('XYZ : Part_no time_days cell_leaving',
& ' X Y Z')
105 format(1x,i8,1x,g21.14,1x,i8,3(1x,g16.9))
110 format (a, 'Part_no time_days cell_leaving')
elseif(iprto.eq.1) then
pstart_out = .false.
do iprint = 1, 200
sptr_heading(iprint:iprint) = ' '
end do
sptr_prop_values = ''
sptr_heading(1:58) =
2 ' particle_number x(m) y(m) z(m) time(days)'
c 2 ' particle_number x y z time '
position_in_string = 60
c Determine how many property columns are being written
n_written = 0
do iprops = 1, nsptrprops
if(write_prop(iprops).ne.0) then
n_written = n_written + 1
end if
end do
c Find which one is written next, write it to string
do iprops = 1, n_written
inner: do jprops = 1, nsptrprops
if(write_prop(jprops).eq.iprops) then
if(jprops.eq.1) then
sptr_heading(position_in_string:position_in_string+9)
2 = 'porosity '
position_in_string = position_in_string + 10
exit inner
c Fluid saturation
elseif(jprops.eq.2) then
sptr_heading(position_in_string:position_in_string+11)
2 = 'saturation '
position_in_string = position_in_string + 12
exit inner
c Permeability
elseif(jprops.eq.3) then
sptr_heading(position_in_string:position_in_string+13)
2 = 'permeability '
position_in_string = position_in_string + 14
exit inner
c Rock density
elseif(jprops.eq.4) then
sptr_heading(position_in_string:position_in_string+13)
2 = 'rock_density '
position_in_string = position_in_string + 14
exit inner
c Pressure
elseif(jprops.eq.5) then
sptr_heading(position_in_string:position_in_string+9)
2 = 'pressure '
position_in_string = position_in_string + 10
exit inner
c Temperature
elseif(jprops.eq.6) then
sptr_heading(position_in_string:position_in_string+12)
2 = 'temperature '
position_in_string = position_in_string + 13
exit inner
c Zone number
elseif(jprops.eq.7) then
sptr_heading(position_in_string:position_in_string+5)
2 = 'zone '
position_in_string = position_in_string + 6
exit inner
c Particle ID
elseif(jprops.eq.8) then
sptr_heading(position_in_string:position_in_string+3)
2 = 'ID '
position_in_string = position_in_string + 4
exit inner
end if
end if
end do inner
end do
sptr_heading(position_in_string:position_in_string+17)
2 = 'old_node new_node'
final_position = position_in_string+17
write(isptr2,'(a)') trim(sptr_heading)
do np1 = 1, num_part
current_node = ijkv(np1)
xcoordw = x1(np1) + corn(current_node,1)
ycoordw = y1(np1) + corn(current_node,2)
zcoordw = z1(np1) + corn(current_node,3)
sptr_prop = 0.
if(current_node.eq.0) then
iprcount = 0
do iprint = 1, nsptrprops
if(write_prop(iprint).ne.0) then
iprcount = iprcount + 1
sptr_prop(write_prop(iprint)) = 1.d-30
end if
end do
c ps_print = 1.e-30
c s_print = 1.e-30
else
iprcount = 0
c Porosity
if(write_prop(1).ne.0) then
iprcount = iprcount + 1
sptr_prop(write_prop(1)) = ps_trac(current_node)
end if
c Fluid saturation
if(write_prop(2).ne.0) then
iprcount = iprcount + 1
if (irdof .ne. 13 .or. ifree .ne. 0) then
sptr_prop(write_prop(2)) = s(current_node)
else
sptr_prop(write_prop(2)) = 1.0d0
end if
end if
c Permeability
if(write_prop(3).ne.0) then
iprcount = iprcount + 1
sptr_prop(write_prop(3)) = 1.d-6*pnx(current_node)
end if
c Rock density
if(write_prop(4).ne.0) then
iprcount = iprcount + 1
sptr_prop(write_prop(4)) = denr(current_node)
end if
c Pressure
if(write_prop(5).ne.0) then
iprcount = iprcount + 1
sptr_prop(write_prop(5)) = phi(current_node)
end if
c Temperature
if(write_prop(6).ne.0) then
iprcount = iprcount + 1
sptr_prop(write_prop(6)) = t(current_node)
end if
c Zone number
if(write_prop(7).ne.0) then
iprcount = iprcount + 1
sptr_prop(write_prop(7)) = izonef(current_node)
end if
c Particle ID
if(write_prop(8).ne.0) then
iprcount = iprcount + 1
sptr_prop(write_prop(8)) = part_id(np1, 2)
end if
position_in_string = 1
do i = 1, iprcount
if (i .eq. write_prop(7) .or. i .eq. write_prop(8))
& then
write (sptr_prop_values(position_in_string:
& position_in_string+9), '(i8,2x)')
& int(sptr_prop(i))
position_in_string = position_in_string+10
else
write (sptr_prop_values(position_in_string:
& position_in_string+17), '(g16.9,2x)')
& sptr_prop(i)
position_in_string = position_in_string+18
end if
end do
write (sptr_prop_values(position_in_string:
& position_in_string+17), '(2(i8,2x))')
& current_node, current_node
sptr_time = ttp1(np1)
c Don't output if particle time is greater than starting time
c if (sptr_time .gt. days) sptr_time = days
if (sptr_time .le. days) then
pstart_out(np1) = .true.
position_in_string = len_trim(sptr_prop_values)
write(isptr2,8001) part_id(np1,1), xcoordw, ycoordw,
2 zcoordw, sptr_time,
3 sptr_prop_values(1: position_in_string)
end if
end if
end do
end if
8001 format(1x, i8, 3(1x,g16.9), 1x, g21.14, 200a)
return
end subroutine output_info
******************************************************************
******************************************************************
subroutine struct_geom_array(i)
use comsptr
use comsk
implicit none
integer i,j,kb,kb_omr,jab
real*8 gotcord, delkb,del
c NOTE: using ggg as scratch storage for saving del+ and del-
c temporarily
do j=-3,3
delkb=0.
if(j.ne.0) then
jab=abs(j)
kb= abs (irray(i, j))
if((irray(i,0).eq.0).or.(irray(i,0).eq.-(i+1000)).or.
1 (irray(i,0).lt.-200000000)) then
kb_omr=0
call getcord(i,kb_omr,j,gotcord)
del = 0.5*abs((gotcord-cord(i,jab)))
c on return from gotcord, kb_omr =0 only for boundary nodes
c use expected symmetry of ggg to set ggg(kb_omr,-j)
c Also if del(i,j) and del(kb,-j)
c are not equal, then use the greater of the two- this way we may
c increase overlap, but we reduce chances of holes.
c for omr nodes, kb can be 0 either bcs its a boundry node or bcs
c grid refinement has a missing node there. In the case of a
c missing node, return value of kb_omr can be a legitimate node.
c in that case, use symmetry of ggg and irray. ggg(kb_omr could be
c nonzero thru this process even if kb_omr lt i. Hence when i
c becomes value of kb_omr, that del is recalculated and the
c greater value is used.
ggg(i,j)=del
if(irray(i,j).eq.0 .and. kb_omr .gt. 0) then
if (ps(kb_omr).gt.0.) then
irray(i,j)=+kb_omr
else
irray(i,j)=-kb_omr
endif
endif
c if(kb_omr.gt.0) then
c delkb=ggg(kb_omr,-j)
c if(delkb.gt.del) del=delkb
c ggg(kb_omr,-j)=del
c ggg(i,j)=del
c irray(i,j)=kb_omr
c irray(kb_omr,-j)=i
c else
c ggg(i,j)=del
c endif
else
c elseif(ggg(i,j).eq.0.) then
c non-OMR node.
c there is no need to recalculate ggg if it is already nonzero
kb_omr=abs(kb)
call getcord(i,kb_omr,j,gotcord)
del = 0.5*abs((gotcord-cord(i,jab)))
ggg(i,j)=del
c if (kb_omr.gt.0) ggg(kb_omr,-j)=del
endif
endif
enddo
return
end subroutine struct_geom_array
c******************************************************************
c Subroutine is empty - not yet implemented
subroutine unstruct_arrays
return
end subroutine unstruct_arrays
******************************************************************
******************************************************************
subroutine particle_patch(npart_ist2)
implicit none
integer npart_ist2
c *******initial state section *********
if(nx.eq.1) then
dx=0
else
dx=xdim/(1.e-20+nx-1)
endif
if(ny.eq.1) then
dy=0
else
dy=ydim/(1.e-20+ny-1)
endif
if(icnl.ne.0) nz = 1
if(nz.eq.1) then
dz=0
else
dz=zdim/(1.e-20+nz-1)
endif
do ix=1,nx
do iy=1,ny
do iz=1,nz
ip=ix+(iy-1)*nx+(iz-1)*nx*ny
x1(ip)=x10+(ix-1)*dx
y1(ip)=y10+(iy-1)*dy
z1(ip)=z10+(iz-1)*dz
enddo
enddo
enddo
npart_ist2=ip
c....dec 4 01, s kelkar sep 20 2001
if(ist.eq.3) then
do ip=1,num_part
inode=ijkv(ip)
x1(ip)=cord(inode,1)
y1(ip)=cord(inode,2)
z1(ip)=cord(inode,3)
enddo
endif
c.....................................
return
end subroutine particle_patch
******************************************************************
subroutine find_particle_cell(npart_ist2,ierr,iptty)
implicit none
integer flag_box,npart_ist2,ijkv_last,ierr,iptty
integer nout,save_out_face(6)
c *** fudged initial
xo=x1
yo=y1
zo=z1
c If the search algorithm starts at node 1 for all nodes,
c it can't get inside a locally structured part of the grid
c from outside of it. The following search gets each particle
c close or even at its starting node, so as long as that
c coordinate is inside a locally structured part of the
c grid and inside the model domain, the subsequent search
c should work. BAR 3-9-00
c
if(abs(ist).eq.1) then
! If we have read in the corresponding nodes use them otherwise search
if (.not. sptr_snode) then
c Particles could be anywhere, do search on each one
do i1 = 1, num_part
call near3(xo(i1),yo(i1),zo(i1),ijkv(i1),0)
end do
end if
elseif(ist.eq.2) then
c Particles are in a cluster, do search on first
c one only, use that as starting location
call near3(xo(1),yo(1),zo(1),ijkv_find,0)
ijkv(1)=ijkv_find
c..s kelkar Jan 21,05 replacing with tree-search
c *** n50 must exceed max flow path length/smallest nodal separation
c n50=100000
c
c
c do is=1,n50
c ijkvss=ijkv
c x1=xo-corn(ijkv,1)
c y1=yo-corn(ijkv,2)
c z1=zo-corn(ijkv,3)
c ddxv=ddx(ijkv)
c ddyv=ddy(ijkv)
c ddzv=ddz(ijkv)
c
c ijkvs=ijkv
c where(x1/ddxv.gt.1.)
c ijkv=irray(ijkv, 1)
c endwhere
c where(x1/ddxv.lt.0.)
c ijkv=irray(ijkv,-1)
c endwhere
c where(ijkv.eq.0) ijkv=ijkvs
c
c ijkvs=ijkv
c where(y1/ddyv.gt.1.)
c ijkv=irray(ijkv, 2)
c endwhere
c where(y1/ddyv.lt.0.)
c ijkv=irray(ijkv,-2)
c endwhere
c where(ijkv.eq.0) ijkv=ijkvs
c
c ijkvs=ijkv
c where(z1/ddzv.gt.1.)
c ijkv=irray(ijkv, 3)
c endwhere
c where(z1/ddzv.lt.0.)
c ijkv=irray(ijkv,-3)
c endwhere
c where(ijkv.eq.0) ijkv=ijkvs
c
c i5=0
c do i1=1,num_part
c if(ijkv(i1).ne.ijkvss(i1)) i5=1
c enddo
c if(i5.eq.0) go to 201
c
c enddo
c 201 continue
c........................................................
ijkv_last=ijkv_find
do is=2,npart_ist2
call tree_search(ijkv_last,20,ierr,iptty,
$ xo(is),yo(is),zo(is),flag_box,nout,save_out_face)
if(flag_box.gt.0) then
ijkv(is)=flag_box
ijkv_last=flag_box
else
call tree_search(ijkv(1),20,ierr,iptty,
$ xo(is),yo(is),zo(is),flag_box,nout,save_out_face)
if(flag_box.gt.0) then
ijkv(is)=flag_box
ijkv_last=flag_box
else
c tree-search failed, do a global search
call near3(xo(is),yo(is),zo(is),flag_box,0)
if(flag_box.gt.0) then
ijkv(is)=flag_box
ijkv_last=flag_box
else
write(ierr,*)'Error in find_particle_cell. Ist=2 '
write(ierr,*)'Cant find the CC for particle #', is
write(ierr,*)'xo,yo,zo=',xo(is),yo(is),zo(is)
write(ierr,*)'STOP'
if (iptty. ne. 0) then
write(iptty,*)'Error in find_particle_cell. Ist=2 '
write(iptty,*)'Cant find the CC for particle #', is
write(iptty,*)'xo,yo,zo=',xo(is),yo(is),zo(is)
write(iptty,*)'STOP'
end if
call exit_ptrac1
end if
endif
end if
enddo
end if
ijkvs=ijkv
c***
c s kelkar aug 29, 06
c for water table nodes, if S(ijkv(np1))>Smin then set the initial
c position below ddz*S, if S<Smin then move the particle vertically
c downward until a node with S>Smin is encountered.
c zvd added to time loop (ptrac3), to move particle down to wt
c when it starts to move aug 27, 2007
c if (ifree.ne.0) then
c if(deltawt.gt.epsilonwt) then
c call wtsi_ptrac1_init
c endif
c endif
c****
x1=xo-corn(ijkv,1)
y1=yo-corn(ijkv,2)
z1=zo-corn(ijkv,3)
c ********at final relative initial state *********
ddxv=ddx(ijkv)
ddyv=ddy(ijkv)
ddzv=ddz(ijkv)
c **** is the initial state valid? *****
cc where((ijkv.lt.1).or.(ijkv.gt.neq)) ijkv=0
c where((x1.gt.ddxv).or.(x1.lt.0.)) ijkv=0
c where((y1.gt.ddyv).or.(y1.lt.0.)) ijkv=0
c if(icnl.eq.0) then
c where((z1.gt.ddzv).or.(z1.lt.0.)) ijkv=0
c end if
do is=1,num_part
c Check to see if particle should be excluded if out side the model domain
if (exclude_particle) then
if(x1(is).gt.ddxv(is) .or. x1(is).lt.0. .or.
& y1(is).gt.ddyv(is) .or. y1(is).lt.0. .or.
& ps(ijkv(is)) .le. 0.) then
istop(is) = 1
ijkv(is) = 0
end if
if (icnl.eq.0) then
if(z1(is).gt.ddzv(is) .or. z1(is).lt.0.) then
istop(is) = 1
ijkv(is) = 0
end if
end if
else
if(x1(is).gt.ddxv(is)) x1(is)=ddxv(is)
if(x1(is).lt.0.) x1(is)=0.
if(y1(is).gt.ddyv(is)) y1(is)=ddyv(is)
if(y1(is).lt.0.) y1(is)=0.
if(icnl.eq.0) then
if(z1(is).gt.ddzv(is)) z1(is)=ddzv(is)
if(z1(is).lt.0.) z1(is)=0.
end if
end if
enddo
c....dec4 01 s kelkar nov 11 01 commented out next 3 lines to
c allow omr nodes as initial locations
c do np1 = 1, num_part
c if(irray(ijkv(np1),0).lt.0) ijkv(np1) = 0
c end do
c...................................
c *** print initial state ******
c ***** stop if initial state of any particle is invalid: ijkv=0 ****
istop=0
do i1=1,num_part
if(ijkv(i1).eq.0) then
write(ierr, 224) i1
! call exit_ptrac1
istop (i1)=1
end if
enddo
224 format ('Error in ptrac1: Initial state of particle is invalid',
& ' for particle number ', i8)
c **** set istop=1 if point out of domain****
c where(ijkv.eq.0) istop=1
c *** move initial points off element boundaries***
ddxv=ddx(ijkv)
ddyv=ddy(ijkv)
ddzv=ddz(ijkv)
where(x1.eq.0. ) x1=ddxv*ep
where(x1.eq.ddxv) x1=ddxv*(1.-ep)
where(y1.eq.0. ) y1=ddyv*ep
where(y1.eq.ddyv) y1=ddyv*(1.-ep)
if(icnl.eq.0) then
where(z1.eq.0. ) z1=ddzv*ep
where(z1.eq.ddzv) z1=ddzv*(1.-ep)
end if
return
end subroutine find_particle_cell
c******************************************************************
end subroutine ptrac1
c***********************************************************************
subroutine subomr2(i)
c s kelkar 11 jul 05
c this is a modified version (and simplified) of 'subomr'
c each connected node with a 'bad' connection, from array
c iomr_neighbour(:) is counted and stored for every direction
c that the connection if off-axis in the array isave_omr(:,:)
c connections are not classified as type I or II (that is done
c in subomr, but no longer needed. In the current version of ptrac1,
c that classification leads to wrong ddx,ddy,ddz values)
use comai
use combi
use comci
use comdi
use comflow
use comsptr
use comsk
use compart
implicit none
integer i,i1,k,ia,kb,ikb,ipos
real*8 epsilon,xkb,xia,d
epsilon=1.e-8
do i1=-3,3
komr_count(i1)=0
enddo
do i1=1,iomr_count
c NOTE: in iomr_neighbour,saved ikb rather than the node # kb
c because ikb can be directly used as a pointer to sx and a_axy
c arrays and kb can be retrieved from nelm(ikb)
ikb=iomr_neighbour(i1)
kb=nelm(ikb)
do k=1,3
c ia=abs(irray(i,k))
d=cord(kb,k)-cord(i,k)
if(d.gt.epsilon) then
komr_count(k)=komr_count(k)+1
if(komr_count(k).gt.komrmax) then
write(ierr,*)'komr_count(k).gt.komrmax in ',
& 'subomr2'
write(ierr,*)'change dimension in comomr.'
write(ierr,*)'i,iomr_count',i,iomr_count
write(ierr,*)'STOP'
if (iptty .ne. 0) then
write(iptty,*)'komr_count(k).gt.komrmax in ',
& 'subomr2'
write(iptty,*)'change dimension in comomr.'
write(iptty,*)'i,iomr_count',i,iomr_count
write(iptty,*)'STOP'
end if
call exit_ptrac1
endif
isave_omr(k,komr_count(k))=ikb
endif
c ia=abs(irray(i,-k))
d=cord(kb,k)-cord(i,k)
if(d.lt.-(epsilon)) then
komr_count(-k)=komr_count(-k)+1
if(komr_count(-k).gt.komrmax) then
write(ierr,*)'komr_count(k).gt.komrmax in ',
& 'subomr2'
write(ierr,*)'change dimension in comomr.'
write(ierr,*)'STOP'
if (iptty .ne. 0) then
write(iptty,*)'komr_count(k).gt.komrmax in ',
& 'subomr2'
write(iptty,*)'change dimension in comomr.'
write(iptty,*)'STOP'
end if
call exit_ptrac1
endif
isave_omr(-k,komr_count(-k))=ikb
endif
enddo
enddo
return
end
c.......................................................................
subroutine getcord(i,kb,la,gotcord)
c s kelkar, modified Feb 10,05
use comai, only : ierr, iptty
use combi
use comsptr
use comsk
implicit none
integer i,kb,k,l,la,j,jk,jkb,jkbmax,ibou,j1,j2,jkc
integer itempf, jtemp(200), jtemp_count, lasign,jjjj
integer i_augment,i1,i2,iii,jkd, jkbb,jkb1,jkb2,i_dir
integer l_perp,irray0
real*8 gotcord,dmax,d, djtemp, djkb,dtemp1,dtemp2
l=abs(la)
lasign=isign(1,la)
if(kb.gt.0) then
gotcord=cord(kb,l)
elseif(kb.eq.0) then
irray0=irray(i,0)
if((irray0.eq.i).or.(irray0.eq.-(i+1000000)).or.
1 (irray0.eq.-(i+2000000)).or.irray0.eq.-(i+2000).or.
2 (irray0.lt.-100000000.and.irray0.gt.-200000000))
3 then
c i is a non-OMR boundary node, a well-capture node on boundary
c or a spring node on a boundaryor a cliff node.
c Set gotcord = cord of i
gotcord=cord(i,l)
kb=0
c......s kelkar 1/27/04 3-D stuff................
elseif(irray(i,0).eq.-(i+1000).or.
1 (irray(i,0).lt.-200000000)) then
c handle OMR nodes on exterior boundaries, including omr-cliff nodes
itempf = 0
do ibou=1,6
if(iboulist(i,ibou).eq.la) then
c node i is an OMR node on an exterior boundary, with the exterior in
c the la direction. set gotcord =cord of i
if (irray(i, la) .lt. 0) then
kb = irray(i, la)
gotcord = cord (abs(kb), l)
else
gotcord=cord(i,l)
kb=0
end if
itempf = 1
endif
enddo
endif
c.................................................
if(irray(i,0).eq.0.or.((irray(i,0).eq.(-i-1000).or.
1 (irray(i,0).lt.-200000000)).and.
2 itempf.eq.0)) then
c we have either an non-boundary omr node
c , or a boundary OMR node but with boundary oriented in a plane
c different from that given by la. Define a
c fictious control volume face
c find the node furtherest from i in the la direction
c form the list of nodes with a bad connection with i in the
c la direction (these are saved in isave_omr)
dmax=0.
jtemp_count=0
do j=0,komrmax,1
if(j.eq.0) then
jkb=abs(irray(i,la))
else
jkb=0
jk=isave_omr(la,j)
if(jk.gt.0) jkb=nelm(jk)
endif
if(jkb.gt.0) then
d=abs(cord(i,l)-cord(jkb,l))
if(d.gt.dmax) then
c saving jkb in jtemp if a search is needed below over
c the neighbours of jkb to avoind creating holes in the mesh.
jtemp_count=jtemp_count+1
jtemp(jtemp_count)=jkb
dmax=d
jkbmax=jkb
endif
endif
enddo
if(jkbmax.gt.0 ) then
gotcord=cord(jkbmax,l)
kb=jkbmax
else
write(ierr,*)'STOP. getcord found jkbmax=0'
write(ierr,*)'i,kb,la=',i,kb,la
if (iptty .ne. 0) then
write(iptty,*)'STOP. getcord found jkbmax=0'
write(iptty,*)'i,kb,la=',i,kb,la
end if
call exit_ptrac1
endif
c now check for the rare situation when
c the node i lies on the side of a rectangle which has the central
c node missing due to refinement on all sides,
c and if so, getcord is set equal to the node on the other side
c of the squarerectangle, thus creating cc's that overlap in the
c middle of this rectangle, but avoid creating holes in the model
c this situation can only arise if the grid on the 'la' side is
c one level coarse compared to the grid on the '-la' side.
c the search for a node needs to be only over the neighbours of the
c nodes stored in jtemp(1:jtemp_count)
jkc=jkbmax
j1=nelm(jkc)+1
j2=nelm(jkc+1)
do j=j1,j2
jkb=nelm(j)
if(jkb.ne.i) then
djkb=abs(cord(i,l)-cord(jkb,l))
if(djkb.eq.dmax*2.) then
c the node jkb is at the expected distance from i along la axix
c check if the other 2 coordinates match
do jjjj=1,3
if(jjjj.ne.l) then
if(cord(i,jjjj).ne.cord(jkb,jjjj)) goto 91911
endif
enddo
c the coordinates match,
c find the normal axis to the coordinate plane formed the nodes
c i,jkc. note
c that i and jkb lie along the la coordinate axis
do jjjj=1,3
if(jjjj.ne.l) then
if(cord(i,jjjj).eq.cord(jkc,jjjj)) then
i_dir=jjjj
goto 91913
endif
endif
enddo
c i and jkc not in a coordinate plane, continue with other
c neighbours of jkc
goto 91911
91913 continue
c now check if the nodes i, jkc and jkb form
c a closed figure with another neighbour of i in the plane
c corrosponding to the coordinate i_dir. need to search only
c only the nodes on the la side of i, ie isave_omr(la,iii)
do iii=1,komrmax
jk=isave_omr(la,iii)
if(jk.gt.0) then
jkd=nelm(jk)
if(jkd.ne.jkc) then
c jkd is already ne jkb, and ne i
if(cord(i,i_dir).eq.cord(jkd,i_dir)) then
c found a neighbour of i in the plane of i-jkc-jkb. See if it is
c connected to jkb, forming a closed figure
jkb1=nelm(jkb)+1
jkb2=nelm(jkb+1)
do jkbb=jkb1,jkb2
if(jkd.eq.nelm(jkbb)) then
c closed figure is formed. By construction, we know that jkb and
c i are on the opposite sides of the line jkc-jkd: this is because
c jkc is the neighbour of i that is furthest from i in the la
c direction, and jkb is twice as far. Now check if
c jkc and jkd are on the opposite sides of the line i-jkb. Note that
c the line i-jkb is parallel to the la axis. First find l_perp,
c the axis normal to la(and l) and i_dir
if(l.eq.1) then
if(i_dir.eq.2) then
l_perp=3
elseif(i_dir.eq.3) then
l_perp=2
else
write(iptty,*)' Error in gotcord. l=i_dir=1'
stop
endif
elseif(l.eq.2) then
if(i_dir.eq.3) then
l_perp=1
elseif(i_dir.eq.1) then
l_perp=3
else
write(iptty,*)' Error in gotcord. l=i_dir=2'
stop
endif
elseif(l.eq.3) then
if(i_dir.eq.2) then
l_perp=1
elseif(i_dir.eq.1) then
l_perp=2
else
write(iptty,*)' Error in gotcord. l=i_dir=3'
stop
endif
endif
dtemp1=cord(i,l_perp)-cord(jkd,l_perp)
dtemp2=cord(i,l_perp)-cord(jkc,l_perp)
if((dtemp1*dtemp2).lt.0.) then
c set getchord equal to cord of node jkb
c and exit the search loop. jkb is the neighbour of a neighbour,
c and not in the original neighbour list for i in nelm. So
c save jkb as the la-neighbour of i in irray(i,la)
c irray(i,la)=jkb
c irray(jkb,-la)=i
gotcord=cord(jkb,l)
kb=jkb
goto 91912
endif
endif
enddo
endif
endif
endif
enddo
endif
endif
91911 continue
enddo
91912 continue
endif
endif
return
end
c..................................................................
subroutine flag_boundry(i,i1,i2,i3,iflag_boundry)
use combi
use comdi, only : ps
use comsk
implicit none
integer i,i1,i2,i3,iflag_boundry,i3ab,i3sign,kb,ksign
integer k
real*8 dist
c....s kelkar feb 3, 04, 3D ORM stuff............
c flag OMR nodes that are on the exterior boundary. The subroutine
c flag_boundry checks orientations for missing nodes, and if they are
c present, then checks if any other connetcions exist on the same side
c of the axis. If not, then it is a boundry node.
i3ab=iabs(i3)
i3sign=isign(1,i3)
do k=i1,i2
kb=nelm(k)
if (ps(kb) .gt. 0.d0) then
dist=cord(kb,i3ab)-cord(i,i3ab)
ksign=dsign(1.d0,dist)
if(ksign.eq.i3sign .and. abs(dist).gt.1.e-20) then
c found a neighbour node on the same side as the missing
c node. So i is not a boundry node. set flag and return
iflag_boundry = 0
goto 9999
endif
end if
enddo
c did not find any neighbour nodes on the same side as the
c missing node, so node i must be on an exterior boundry.
c set flag
iflag_boundry= +1
9999 continue
return
end subroutine flag_boundry
c...................................................................
subroutine exit_ptrac1
stop
return
end subroutine exit_ptrac1
c...........................................................
subroutine ddx_corn_array
c set up ddx, ddy,ddz and corn arrays
c NOTE: using ggg as scratch storage for saving del+ and del-
c temperorl
use comai, only : neq, isptr9
use combi, only : cord
use comsptr
use comsk
implicit none
integer i,j
real*8 del_plus,del_minus,dtemp(3)
do i=1,neq
do j=1,3
del_plus=ggg(i,j)
del_minus=ggg(i,-j)
dtemp(j)=del_plus+del_minus
corn(i,j)=cord(i,j)-del_minus
enddo
ddx(i)=dtemp(1)
ddy(i)=dtemp(2)
ddz(i)=dtemp(3)
enddo
c s kelkar sep 12 05 volume output for plumecalc
if(sptrx_flag) then
call sptr_volume_out
endif
return
end subroutine ddx_corn_array
c....................................................................
subroutine boundary_nodes(i,omrflag)
c....s kelkar April 4, 2005
c flag nodes that are on the exterior boundary. The subroutine
c flag_boundry checks orientations for missing nodes, and if they are
c present, then checks if any other connetcions exist on the same side
c of the axis. If not, then it is a boundry node.
c
c iboulist(i,7 )=# of boundary faces for node i (max 6)
c iboulist(i,1:6)= codes for boundary faces of node i
c irray(i,0) = +i : regular node, not a source/sink
c irray(i,0) = -i-2000 : regular node on a external boundary
c irray(i,0) = -i : regular interior node that is a sink/source
c but not explicitly specified in sptr macro
c -100000000 < irray(i,0) < -10000000 : regular interior node that is
c specified as a sink/source in the sptr macro
c in this case -(irray(i,0)+10000000) is the pointer
c for the storage location in well_radius for this node
c = -(i+2000000) : spring node
c = -(i+1000000) : well-capture node on external bound
c simillar to =-i case but with half space solution
c irray(i,0) = 0 : OMR node not on boundary
c irray(i,0) = -i-1000 : OMR node on a external boundary
c -200000000 < irray(i,0) < -100000000 : non-OMR cliff node
c irray(i,0) < -200000000 : OMR cliff node
c
use comai, only : ierr, iptty
use combi, only : nelm
use comsk
use comsptr
implicit none
integer i,ii1,ii2,i3,iflag_boundry,omrflag
integer ibou,upper_limit
ibou=0
upper_limit=3
ii1=nelm(i)+1
ii2=nelm(i+1)
do i3=1,upper_limit
iflag_boundry=0
if(irray(i,+i3).le.0) then
call flag_boundry(i,ii1,ii2,+i3,iflag_boundry)
endif
if(iflag_boundry.eq.1) then
if(irray(i,0).gt.-100000000) then
if(omrflag.eq.1) then
irray(i,0)=-i-1000
elseif(irray(i,0).ne.-(i+1000000).and.irray(i,0).ne.
$ -(i+2000000)) then
irray(i,0)=-i-2000
endif
endif
ibou=ibou+1
if(ibou.gt.6) then
write(ierr, 1002)
if (iptty .ne. 0) write(iptty, 1002)
call exit_ptrac1
endif
if (omr_flag) iboulist(i,ibou)=i3
endif
iflag_boundry=0
if(irray(i,-i3).le.0) then
call flag_boundry(i,ii1,ii2,-i3,iflag_boundry)
endif
if(iflag_boundry.eq.1) then
if(irray(i,0).gt.-100000000) then
if(omrflag.eq.1) then
irray(i,0)=-i-1000
elseif(irray(i,0).ne.-(i+1000000).and.irray(i,0).ne.
$ -(i+2000000)) then
irray(i,0)=-i-2000
endif
endif
ibou=ibou+1
if(ibou.gt.6) then
write(ierr, 1002)
if (iptty .ne. 0) write(iptty, 1002)
call exit_ptrac1
endif
if (omr_flag) iboulist(i,ibou)=-i3
endif
enddo
1002 format ('Error in stuc_conn_array, ibou > 6: STOPPING')
c save the number of faces on the boundary
if (omr_flag) iboulist(i,7)=ibou
return
end subroutine boundary_nodes
c...................................................
subroutine subomr(i)
c s kelkar 1 onv 0
c determin which faces the omr nodes lie
use comai
use combi
use comci
use comdi
use comflow
use comsptr
use comsk
use compart
implicit none
integer i,i1,k,ia,kb,ikb,ipos
real*8 epsilon,xkb,xia,d
epsilon=1.e-8
do i1=-3,3
komr_count(i1)=0
enddo
do i1=1,iomr_count
c NOTE: in iomr_neighbour,saved ikb rather than the node # kb
c because ikb can be directly used as a pointer to sx and a_axy
c arrays and kb can be retrieved from nelm(ikb)
ikb=iomr_neighbour(i1)
kb=nelm(ikb)
c now figure out the geometry stuff
c Look for type II connection, ie, kb is on a face defined
c without a central node, going from higher to lower level omr.
c we need to look at only those values of l for which irray(i,l)=0
c a connection node is missing. Use the sign of the difference
c in the l coordinate as an indicator
do k=1,3
ia=abs(irray(i,k))
if(ia.eq.0) then
d=cord(kb,k)-cord(i,k)
if(d.gt.epsilon) then
komr_count(k)=komr_count(k)+1
if(komr_count(k).gt.komrmax) then
write(ierr,*)'komr_count(k).gt.komrmax in ',
& 'subomr'
write(ierr,*)'change dimension in comomr.'
write(ierr,*)'i,iomr_count',i,iomr_count
write(ierr,*)'STOP'
if (iptty .ne. 0) then
write(iptty,*)'komr_count(k).gt.komrmax in ',
& 'subomr'
write(iptty,*)'change dimension in comomr.'
write(iptty,*)'i,iomr_count',i,iomr_count
write(iptty,*)'STOP'
end if
call exit_ptrac1
endif
isave_omr(k,komr_count(k))=ikb
! commenting out the next goto 9191 to allow node kb to be included
! as a neighbour on multiple sides of i
c goto 9191
endif
endif
ia=abs(irray(i,-k))
if(ia.eq.0) then
d=cord(kb,k)-cord(i,k)
if(d.lt.-(epsilon)) then
komr_count(-k)=komr_count(-k)+1
if(komr_count(-k).gt.komrmax) then
write(ierr,*)'komr_count(k).gt.komrmax in ',
& 'subomr'
write(ierr,*)'change dimension in comomr.'
write(ierr,*)'STOP'
if (iptty .ne. 0) then
write(iptty,*)'komr_count(k).gt.komrmax in ',
& 'subomr'
write(iptty,*)'change dimension in comomr.'
write(iptty,*)'STOP'
end if
call exit_ptrac1
endif
isave_omr(-k,komr_count(-k))=ikb
c goto 9191
endif
endif
enddo
c check if its a type-I connection, ie if kb is on a face defined
c by a central node, going from lower to higher level omr.
c check distance from each normal plane to kb to see if it is in the
c plane
c
do k=1,3
xkb=cord(kb,k)
ia=abs(irray(i,k))
if(ia.gt.0) then
xia=cord(ia,k)
d=abs(xkb-xia)
if(d.lt.epsilon) then
komr_count(k)=komr_count(k)+1
if(komr_count(k).gt.komrmax) then
write(ierr,*)'komr_count(k).gt.komrmax in ',
& 'subomr'
write(ierr,*)'change dimension in comomr.'
write(ierr,*)'STOP'
if (iptty .ne. 0) then
write(iptty,*)'komr_count(k).gt.komrmax in ',
& 'subomr'
write(iptty,*)'change dimension in comomr.'
write(iptty,*)'STOP'
end if
call exit_ptrac1
endif
isave_omr(k,komr_count(k))=ikb
c goto 9191
endif
endif
ia=abs(irray(i,-k))
if(ia.gt.0) then
xia=cord(ia,k)
d=abs(xkb-xia)
if(d.lt.-(epsilon)) then
komr_count(-k)=komr_count(-k)+1
if(komr_count(-k).gt.komrmax) then
write(ierr,*)'komr_count(-k).gt.komrmax in ',
& 'subomr'
write(ierr,*)'change dimension in comomr.'
write(ierr,*)'STOP'
if (iptty .ne. 0) then
write(iptty,*)'komr_count(k).gt.komrmax in ',
& 'subomr'
write(iptty,*)'change dimension in comomr.'
write(iptty,*)'STOP'
end if
call exit_ptrac1
endif
isave_omr(-k,komr_count(-k))=ikb
c goto 9191
endif
endif
enddo
9191 continue
enddo
return
end
c.......................................................................
subroutine wtsi_ptrac1_init
c s kelkar aug 29, 05
c for water table nodes, if S(ijkv(np1))>Smin then set the initial
c position below ddz*S, if S<Smin then move the particle vertically
c downward until a node with S>Smin is encountered.
use comai, only : days
use comdi, only : izone_free_nodes,s
use comsptr
use comsk
implicit none
integer i,inp1,newnode
real*8 xp,yp,zp,dumm,zwt
do i=1,num_part
inp1=ijkv(i)
c move only if it is time for the particle to move
if (izone_free_nodes(inp1).gt.1 .and. ttp1(i) .le. days) then
zp=zo(i)-corn(inp1,3)
zwt=ddz(inp1)*s(inp1)
if(zp.gt.zwt) then
xp=xo(i)-corn(inp1,1)
yp=yo(i)-corn(inp1,2)
newnode=inp1
call wtsi_find_water(inp1,i,xp,yp,zp,newnode)
if (newnode .ne. 0) then
call wtsi_displace_node(inp1,i,xp,yp,zp,newnode)
ijkv(i)=newnode
xo(i)=xp+corn(newnode,1)
yo(i)=yp+corn(newnode,2)
zo(i)=zp+corn(newnode,3)
end if
endif
endif
enddo
end subroutine wtsi_ptrac1_init
c....................................................................
subroutine wtsi_find_water(inp1,np1,xp,yp,zp,newnode)
c s kelkar aug 30, 05
c If newnode has irreducible water
c saturation, flagged by izone_free_nodes(inp1).gt.1 then
c search vertically downward to find a node with flowing water.
use comai, only : ierr, iptty
use comdi
use comsptr
use comsk
implicit none
integer inp1,np1,newnode,j,nextnode, node_flag, nodetemp
integer node_previous,ibou,i1
real*8 xp,yp,zp
real*8 epsilon
epsilon= 1.e-10
if (izone_free_nodes(newnode).ge.3.or.s(newnode).lt.smin) then
node_previous=newnode
do j=1,1000000
node_flag=irray(node_previous,0)
if(node_flag.eq.0) then
c nextnode is interior OMR, do OMR check
c xc and yc are updated wrt nodetemp in wtsi_neighbour and also
c zc is set on the +3 boundary of nodetemp.
call wtsi_neighbour2(inp1,np1,node_previous,
1 xp,yp,zp, nodetemp)
if(izone_free_nodes(nodetemp).le.2.and.s(nodetemp)
1 .ge.smin) then
c found a valid flowing node, return
newnode=nodetemp
goto 99999
else
node_previous = nodetemp
endif
elseif(node_flag.eq.-(1000+node_previous)) then
c node_previous is a boundary OMR node. check boundary faces,
c stored in iboulist(), to see if the -3 plane is a
c boundary plane. if so, particle is exiting the model- return.
c If it is not, find the new nearest node.
ibou=iboulist(node_previous,7)
do i1=1,ibou
if(-3.eq.iboulist(node_previous,i1)) then
c particle exited the model
newnode=0
write(ierr,*)'ptrac1:find_water_table. cant find'
write(ierr,*)'water table. Check initial particle'
write(ierr,*)'locations. np1=',np1
write(ierr,*)'exit_ptrac1. stop.'
call exit_ptrac1
c goto 99999
endif
enddo
c node_previous not on -3 boundary. In wtsi_neighbour,
c xc and yc are updated wrt nodetemp in wtsi_neighbour and also
c zc is set on the +3 boundary of nodetemp.
call wtsi_neighbour2(inp1,np1,node_previous,
1 xp,yp,zp, nodetemp)
if(izone_free_nodes(nodetemp).le.2.and.s(nodetemp)
1 .ge.smin) then
c found a valid flowing node, return
newnode=nodetemp
goto 99999
else
node_previous = nodetemp
endif
else
c regular interior node
nextnode=irray(node_previous,-3)
if(nextnode.gt.0) then
c wtsi_neighbour has not been called, so need to update xc,yc,zc
c change xc,yc to refere to nextnode and zc slightly below
c + 3 face of nextnode.
xp=xp+corn(node_previous,1)-corn(nextnode,1)
yp=yp+corn(node_previous,2)-corn(nextnode,2)
zp=(1.-epsilon)*ddz(nextnode)
If(izone_free_nodes(nextnode).le.2.and.s(nextnode)
1 .ge.smin) then
c found a valid flowing node, return
newnode=nextnode
goto 99999
else
c nextnode doesnt have flowing water,continue checking -3 neighbour
node_previous = nextnode
endif
elseif(nextnode.eq.0) then
c bottom node- particle has to exit
newnode=0
else
write(iptty,*)'error wtsi_newnode.nextnode=',nextnode
write(iptty,*)'STOP'
write(ierr,*)'error wtsi_newnode.nextnode=',nextnode
write(ierr,*)'STOP'
stop
endif
endif
enddo
else
c found a valid flowing node, return
endif
99999 continue
return
end subroutine wtsi_find_water
c...........................................................
subroutine wtsi_neighbour2(inp1,np1,node_previous,xc,yc,zc,
1 nodetemp)
use comai, only : ierr, iptty
use comdi
use comsptr
use comsk
implicit none
integer inp1,np1,j, nodetemp, node_previous
real*8 xc,yc,zc,epsilon,zcc
epsilon=1.e-8
c an Internal OMR node, check for neighbours on -3 side
c to begin ,place zcc slightly below -3 face of node_previous
zcc=-epsilon*ddz(node_previous)
call nearest_node(node_previous,np1,-3,xc,yc,zcc, nodetemp)
if(nodetemp.le.0) then
write(ierr,*)'Hole in the model. node=',node_previous
write(ierr,*)'stop in wtsi_neighbour2'
if (iptty .ne. 0) then
write(iptty,*)'Hole in the model. node=',node_previous
write(iptty,*)'stop in wtsi_neighbour2'
endif
call update_exit(-inp1,np1,-100,nodetemp,
$ 0.,xc,yc,zc)
c stop
end if
c change xc,yc to refere to nodetemp
c place zc at +3 face of nodetemp
xc=xc+corn(node_previous,1)-corn(nodetemp,1)
yc=yc+corn(node_previous,2)-corn(nodetemp,2)
zc=(1.-epsilon)*ddz(nodetemp)
return
end subroutine wtsi_neighbour2
c...........................................................
subroutine wtsi_newlocation2(nodeabove,np1,nodebelow,zc)
use comdi
use comsptr
use comsk
implicit none
integer nodeabove,np1,nodebelow,j
real*8 dzw,zc, epsilon
epsilon=1.e-12
dzw=s(nodebelow)*ddz(nodebelow)
if((dzw+corn(nodebelow,3)).lt.(ddz(nodeabove)+corn(nodeabove,3)))
1 then
c at this point zc is assumed to be wrt nodebelow already.
if(zc.gt.dzw) then
zc=dzw*(1.-deltawt)
endif
endif
return
end subroutine wtsi_newlocation2
c...........................................................
subroutine wtsi_displace_node(inp1,np1,xp,yp,zp,newnode)
c s kelkar aug 30, 05
c displace particle location vertically downward to deltawt
c meters below the water table and find the new node
use comai, only : ierr, iptty
use comdi
use comsptr
use comsk
implicit none
integer inp1,np1,newnode,j,nextnode, node_flag, nodetemp
integer node_previous,ibou,i1
real*8 xp,yp,zp,zwt,zptemp
real*8 epsilon
epsilon= 1.e-10
c place the particle deltawt(m) below the water table. note that
c xp,yp,zp are wrt newnode
zwt=ddz(newnode)*s(newnode)
zp=zwt*(1-epsilon)-abs(deltawt)
if (zp.lt.0.) then
node_previous=newnode
do j=1,1000000
node_flag=irray(node_previous,0)
if(node_flag.eq.0) then
c nextnode is interior OMR, do OMR check
c xp,yp,zp are updated wrt nodetemp in wtsi_neighbour3
c zp might be far below corn of node_previous, in which
c case nearest_node might not find nodetemp in level_max
c iterations, so use zptemp to place the particle
c just below corn(node_previous,3)
zptemp=-epsilon
call wtsi_neighbour3(inp1,np1,node_previous,
1 xp,yp,zptemp, nodetemp)
c recalculate zp wrt nodetemp
zp=zp+corn(node_previous,3)-corn(nodetemp,3)
if(zp.gt.0.) then
c found the new CC, return
newnode=nodetemp
goto 99999
else
node_previous = nodetemp
endif
elseif(node_flag.eq.-(1000+node_previous)) then
c node_previous is a boundary OMR node. check boundary faces,
c stored in iboulist(), to see if the -3 plane is a
c boundary plane. if so, particle is exiting the model- return.
c If it is not, find the new nearest node.
ibou=iboulist(node_previous,7)
do i1=1,ibou
if(-3.eq.iboulist(node_previous,i1)) then
c particle exited the model
newnode=0
write(ierr,*)'ptrac1:find_water_table. cant find'
write(ierr,*)'water table. Check initial particle'
write(ierr,*)'locations. np1=',np1
write(ierr,*)'exit_ptrac1. stop.'
call exit_ptrac1
c goto 99999
endif
enddo
c node_previous not on -3 boundary. In wtsi_neighbour,
c xc and yc are updated wrt nodetemp and also
c zc is set on the +3 boundary of nodetemp.
zptemp=-epsilon
call wtsi_neighbour3(inp1,np1,node_previous,
1 xp,yp,zptemp, nodetemp)
c recalculate zp wrt nodetemp
zp=zp+corn(node_previous,3)-corn(nodetemp,3)
if(zp.gt.0.) then
c found the new CC, return
newnode=nodetemp
goto 99999
else
node_previous = nodetemp
endif
else
c regular interior node
nextnode=irray(node_previous,-3)
if(nextnode.gt.0) then
c wtsi_neighbour has not been called, so need to update xc,yc,zc
c change xc,yc to refere to nextnode and zc slightly below
c + 3 face of nextnode.
xp=xp+corn(node_previous,1)-corn(nextnode,1)
yp=yp+corn(node_previous,2)-corn(nextnode,2)
zp=zp+corn(node_previous,3)-corn(nextnode,3)
If(zp.gt.0.) then
c found the new CC, return
newnode=nextnode
goto 99999
else
c not in CC of nextnode ,continue checking -3 neighbour
node_previous = nextnode
endif
elseif(nextnode.eq.0) then
c bottom node- particle displacement outside the model
c place the particle in the last CC
newnode=node_previous
else
write(iptty,*)'error wtsi_displace_node.nextnode=',
1 nextnode
write(iptty,*)'STOP'
write(ierr,*)'error wtsi_displace_node.nextnode=',
1 nextnode
write(ierr,*)'STOP'
stop
endif
endif
enddo
else
c found the new CC, return
endif
99999 continue
return
end subroutine wtsi_displace_node
c...........................................................
subroutine wtsi_neighbour3(inp1,np1,node_previous,xc,yc,zc,
1 nodetemp)
use comai, only : ierr, iptty
use comdi
use comsptr
use comsk
implicit none
integer inp1,np1,j, nodetemp, node_previous
real*8 xc,yc,zc,epsilon,zcc
epsilon=1.e-8
c an Internal OMR node, check for neighbours on -3 side
call nearest_node(node_previous,np1,-3,xc,yc,zc, nodetemp)
if(nodetemp.le.0) then
write(ierr,*)'Hole in the model. node=',node_previous
write(ierr,*)'stop in wtsi_neighbour'
write(iptty,*)'Hole in the model. node=',node_previous
write(iptty,*)'stop in wtsi_neighbour'
stop
endif
c change xc,yc,zc to refere to nodetemp
xc=xc+corn(node_previous,1)-corn(nodetemp,1)
yc=yc+corn(node_previous,2)-corn(nodetemp,2)
zc=zc+corn(node_previous,3)-corn(nodetemp,3)
return
end subroutine wtsi_neighbour3
c...........................................................
subroutine sptr_volume_out
c sep 13 05 s kelkar
c store volumes for plumecalc
use comai, only : iw, neq, neq_primary, isptr9
use combi
use comsptr
use comsk
use comxi, only : cform
implicit none
logical opened
integer i1, i1flag, i, ncoef, max_con
real*8, allocatable :: sx1temp(:)
allocate (sx1temp(neq))
do i1=1,neq
i1flag=irray(i1,0)
if(i1flag.eq.0.or.i1flag.eq.-(i1+1000).or.
1 (i1flag.lt.-200000000)) then
c omr node- compute the volume of the approximate brick shape
sx1temp(i1)=ddx(i1)*ddy(i1)*ddz(i1)
else
sx1temp(i1)=sx1(i1)
endif
enddo
ncoef=0
max_con=0
if(cform(26).eq.'formatted') then
c formatted
write(isptr9, '(5i10)' ) iw, neq_primary,
& nelm(neq_primary+1), ncoef, max_con
write(isptr9, '(5(1pe20.10))')
& (sx1temp(i), i = 1, neq_primary)
write(isptr9, '(5i10)' )
& (nelm(i), i = 1, nelm(neq_primary+1))
elseif(cform(26).eq.'unformatted') then
c unformatted
write(isptr9) iw, neq_primary, nelm(neq_primary+1),
& ncoef, max_con
write(isptr9)
& (sx1temp(i), i = 1, neq_primary)
write(isptr9)
& (nelm(i), i = 1, nelm(neq_primary+1) )
endif
if(allocated(sx1temp)) deallocate(sx1temp)
close(isptr9)
return
end subroutine sptr_volume_out
******************************************************************
subroutine init_sptr_params
use comai, only : neq
use combi, only : izonef
use comci, only : rolf
use comdi, only : denr, diffmfl, ifree, itrc, ps_trac, s
use compart, only : aperture, kd, matrix_por, secondary
use comsptr
use davidi, only : irdof
implicit none
integer i, np1
real*8 denominator, rprime, spacing
c Subroutine to initialize particle tracking parameters
if(nzbtc.gt.0) then
c zvd - initialized in allocmem and used in insptr, don't reset here
c izonebtc = 0
end if
cHari 01-Nov-06 include colloid diversity model (tprpflag=11)
do i = 1, neq
if(tprpflag(itrc(i)).eq.1.or.tprpflag(itrc(i)).eq.2.or.
2 tprpflag(itrc(i)).eq.11) then
c Compute denominator, make sure no divide by 0
if (irdof .ne. 13 .or. ifree .ne. 0) then
denominator = rolf(i)*s(i)*ps_trac(i)
else
denominator = rolf(i)*ps_trac(i)
endif
denominator = max(1.d-30, denominator)
omega_partial(i) = 1.+kd(itrc(i),1)*
2 denr(i)/denominator
elseif(tprpflag(itrc(i)).eq.3.or.tprpflag(itrc(i)).eq.4) then
if (irdof .ne. 13 .or. ifree .ne. 0) then
denominator = rolf(i)*s(i)*matrix_por(itrc(i))
else
denominator = rolf(i)*matrix_por(itrc(i))
endif
denominator = max(1.d-30, denominator)
rprime = 1.+kd(itrc(i),1)*
2 denr(i)/denominator
c aperture = 2 * b in Sudicky and Frind solution
c spacing = 2 * B in Sudicky and Frind solution
if(aperture(itrc(i)).lt.0.) then
if (irdof .ne. 13 .or. ifree .ne. 0) then
omega_partial(i) = (matrix_por(itrc(i))*s(i))**2*
2 diffmfl(1,itrc(i))*rprime
else
omega_partial(i) = (matrix_por(itrc(i)))**2*
2 diffmfl(1,itrc(i))*rprime
endif
sigma_partial(i) = aperture(itrc(i))
else
if (secondary(itrc(i)) .ne. 0.) then
spacing = secondary(itrc(i))
else
spacing = abs(aperture(itrc(i)))/max(1.d-30,
& ps_trac(i))
end if
if (irdof .ne. 13 .or. ifree .ne. 0) then
omega_partial(i) = s(i)*matrix_por(itrc(i))*
2 sqrt(rprime*diffmfl(1,itrc(i)))/(0.5*
3 abs(aperture(itrc(i))))
else
omega_partial(i) = matrix_por(itrc(i))*
2 sqrt(rprime*diffmfl(1,itrc(i)))/(0.5*
3 abs(aperture(itrc(i))))
endif
sigma_partial(i) = sqrt(rprime/diffmfl(1,itrc(i)))*0.5*
3 (spacing-abs(aperture(itrc(i))))
end if
end if
end do
return
end subroutine init_sptr_params
******************************************************************
|
(* This Isabelle theory is produced using the TIP tool offered at the following website:
https://github.com/tip-org/tools
This file was originally provided as part of TIP benchmark at the following website:
https://github.com/tip-org/benchmarks
Yutaka Nagashima at CIIRC, CTU changed the TIP output theory file slightly
to make it compatible with Isabelle2017.*)
theory TIP_list_concat_map_bind
imports "../../Test_Base"
begin
datatype 'a list = nil2 | cons2 "'a" "'a list"
fun x :: "'a list => 'a list => 'a list" where
"x (nil2) y2 = y2"
| "x (cons2 z2 xs) y2 = cons2 z2 (x xs y2)"
fun y :: "'a list => ('a => 'b list) => 'b list" where
"y (nil2) y2 = nil2"
| "y (cons2 z2 xs) y2 = x (y2 z2) (y xs y2)"
fun map :: "('a => 'b) => 'a list => 'b list" where
"map f (nil2) = nil2"
| "map f (cons2 y2 xs) = cons2 (f y2) (map f xs)"
fun concat :: "('a list) list => 'a list" where
"concat (nil2) = nil2"
| "concat (cons2 y2 xs) = x y2 (concat xs)"
theorem property0 :
"((concat (map f xs)) = (y xs f))"
oops
end
|
library(GenomicRanges)
library(bumphunter)
library(RColorBrewer)
load('/dcl01/lieber/ajaffe/lab/brain-epigenomics/bumphunting/BSobj_bsseqSmooth_Neuron_minCov_3.Rdata')
load("/dcl01/lieber/ajaffe/lab/brain-epigenomics/rdas/DMR/DMR_objects.rda")
load("/dcl01/lieber/ajaffe/lab/brain-epigenomics/bumphunting/rda/limma_Neuron_CpGs_minCov_3_ageInfo_dmrs.Rdata")
# load HARs
HARs = openxlsx::read.xlsx('/dcl01/lieber/ajaffe/lab/brain-epigenomics/rdas/HARs_hg19_Doan_Walsh_Table_S1.xlsx')
hars = makeGRangesFromDataFrame(HARs, keep.extra.columns=T)
length(hars) # 2737
# Identify all CpG clusters in the genome
gr <- granges(BSobj)
cl = clusterMaker( chr = as.character(seqnames(gr)),
pos = start(gr), maxGap = 1000)
gr.clusters = split(gr, cl)
gr.clusters = unlist(range(gr.clusters))
df.clusters = as.data.frame(gr.clusters)
# Find overlaps with DMRS in all three models
dmrs = split(dmrs, dmrs$k6cluster_label)
names(dmrs) = c("Gr1","Gr2","Gr3","Gr4","Gr5","Gr6")
oo = lapply(dmrs, function(x) findOverlaps(x, makeGRangesFromDataFrame(DMR$Interaction)))
dmrs = lapply(oo, function(x) DMR$Interaction[subjectHits(x),])
DMRgr = lapply(c(DMR, dmrs), function(x) makeGRangesFromDataFrame(x[which(x$fwer<=0.05),], keep.extra.columns=T))
oo = lapply(DMRgr, function(x) findOverlaps(gr.clusters, x))
lapply(oo, function(x) length(unique(queryHits(x))))
harOverlap = findOverlaps(hars, gr.clusters)
df.clusters$regionID = paste0(df.clusters$seqnames,":",df.clusters$start,"-",df.clusters$end)
df.clusters$rnum = 1:length(gr.clusters)
df.clusters$CellType = ifelse(df.clusters$rnum %in% queryHits(oo$CellType), "CellType","no")
df.clusters$Age = ifelse(df.clusters$rnum %in% queryHits(oo$Age), "Age","no")
df.clusters$Interaction = ifelse(df.clusters$rnum %in% queryHits(oo$Interaction), "Interaction","no")
df.clusters$Gr1 = ifelse(df.clusters$rnum %in% queryHits(oo$Gr1), "Gr1","no")
df.clusters$Gr2 = ifelse(df.clusters$rnum %in% queryHits(oo$Gr2), "Gr2","no")
df.clusters$Gr3 = ifelse(df.clusters$rnum %in% queryHits(oo$Gr3), "Gr3","no")
df.clusters$Gr4 = ifelse(df.clusters$rnum %in% queryHits(oo$Gr4), "Gr4","no")
df.clusters$Gr5 = ifelse(df.clusters$rnum %in% queryHits(oo$Gr5), "Gr5","no")
df.clusters$Gr6 = ifelse(df.clusters$rnum %in% queryHits(oo$Gr6), "Gr6","no")
df.clusters$HARs = ifelse(df.clusters$rnum %in% subjectHits(harOverlap), "HAR","no")
## make contingency tables
tables = list()
for (i in 1:length(names(DMRgr))) {
tables[[i]] = data.frame(YesHAR = c(nrow(df.clusters[df.clusters$HARs=="HAR" & df.clusters[,colnames(df.clusters)==names(DMRgr)[i]]==names(DMRgr)[i],]),
nrow(df.clusters[df.clusters$HARs=="HAR" & df.clusters[,colnames(df.clusters)==names(DMRgr)[i]]=="no",])),
NoHAR = c(nrow(df.clusters[df.clusters$HARs=="no" & df.clusters[,colnames(df.clusters)==names(DMRgr)[i]]==names(DMRgr)[i],]),
nrow(df.clusters[df.clusters$HARs=="no" & df.clusters[,colnames(df.clusters)==names(DMRgr)[i]]=="no",])),
row.names = c("YesDMR","NoDMR"))
}
names(tables) = names(DMRgr)
fisher = lapply(tables, fisher.test)
df = do.call(rbind, Map(cbind, lapply(fisher, function(x) data.frame(OR = x$estimate,
upper = x$conf.int[2],
lower = x$conf.int[1],
pval = x$p.value)), Model = as.list(names(fisher))))
df$fdr = p.adjust(df$pval, method= "fdr")
## test for enrichment of conserved, evolutionarily dated enhancers
noonan = openxlsx::read.xlsx('/dcl01/lieber/ajaffe/lab/brain-epigenomics/rdas/Emera_Noonan_supptable_1.xlsx')
x = GRanges(noonan[,1])
mcols(x) = noonan[,2:3]
noonan = x
length(noonan) # 30526
noonan = c(list(enhancers = noonan), as.list(split(noonan, noonan$Phylogenetic.age.assignment)))
noonanOv = lapply(noonan, function(x) findOverlaps(x, gr.clusters))
hits = lapply(noonanOv, function(x) ifelse(df.clusters$rnum %in% subjectHits(x), "yes","no"))
for (i in 1:length(hits)) { df.clusters[,17+i] = hits[[i]] }
colnames(df.clusters)[18:28] = names(hits)
## make contingency tables
tables = list()
for (i in 1:length(names(DMRgr))) {
tables[[i]] = list(enhancers = data.frame(YesHit = c(nrow(df.clusters[df.clusters$enhancers=="yes" & df.clusters[,colnames(df.clusters)==names(DMRgr)[i]]==names(DMRgr)[i],]),
nrow(df.clusters[df.clusters$enhancers=="yes" & df.clusters[,colnames(df.clusters)==names(DMRgr)[i]]=="no",])),
NoHit = c(nrow(df.clusters[df.clusters$enhancers=="no" & df.clusters[,colnames(df.clusters)==names(DMRgr)[i]]==names(DMRgr)[i],]),
nrow(df.clusters[df.clusters$enhancers=="no" & df.clusters[,colnames(df.clusters)==names(DMRgr)[i]]=="no",])),
row.names = c("YesDMR","NoDMR")),
Amniota = data.frame(YesHit = c(nrow(df.clusters[df.clusters$Amniota=="yes" & df.clusters[,colnames(df.clusters)==names(DMRgr)[i]]==names(DMRgr)[i],]),
nrow(df.clusters[df.clusters$Amniota=="yes" & df.clusters[,colnames(df.clusters)==names(DMRgr)[i]]=="no",])),
NoHit = c(nrow(df.clusters[df.clusters$Amniota=="no" & df.clusters[,colnames(df.clusters)==names(DMRgr)[i]]==names(DMRgr)[i],]),
nrow(df.clusters[df.clusters$Amniota=="no" & df.clusters[,colnames(df.clusters)==names(DMRgr)[i]]=="no",])),
row.names = c("YesDMR","NoDMR")),
Eutheria = data.frame(YesHit = c(nrow(df.clusters[df.clusters$Eutheria=="yes" & df.clusters[,colnames(df.clusters)==names(DMRgr)[i]]==names(DMRgr)[i],]),
nrow(df.clusters[df.clusters$Eutheria=="yes" & df.clusters[,colnames(df.clusters)==names(DMRgr)[i]]=="no",])),
NoHit = c(nrow(df.clusters[df.clusters$Eutheria=="no" & df.clusters[,colnames(df.clusters)==names(DMRgr)[i]]==names(DMRgr)[i],]),
nrow(df.clusters[df.clusters$Eutheria=="no" & df.clusters[,colnames(df.clusters)==names(DMRgr)[i]]=="no",])),
row.names = c("YesDMR","NoDMR")),
Gnathostomata = data.frame(YesHit = c(nrow(df.clusters[df.clusters$Gnathostomata=="yes" & df.clusters[,colnames(df.clusters)==names(DMRgr)[i]]==names(DMRgr)[i],]),
nrow(df.clusters[df.clusters$Gnathostomata=="yes" & df.clusters[,colnames(df.clusters)==names(DMRgr)[i]]=="no",])),
NoHit = c(nrow(df.clusters[df.clusters$Gnathostomata=="no" & df.clusters[,colnames(df.clusters)==names(DMRgr)[i]]==names(DMRgr)[i],]),
nrow(df.clusters[df.clusters$Gnathostomata=="no" & df.clusters[,colnames(df.clusters)==names(DMRgr)[i]]=="no",])),
row.names = c("YesDMR","NoDMR")),
Human = data.frame(YesHit = c(nrow(df.clusters[df.clusters$Human=="yes" & df.clusters[,colnames(df.clusters)==names(DMRgr)[i]]==names(DMRgr)[i],]),
nrow(df.clusters[df.clusters$Human=="yes" & df.clusters[,colnames(df.clusters)==names(DMRgr)[i]]=="no",])),
NoHit = c(nrow(df.clusters[df.clusters$Human=="no" & df.clusters[,colnames(df.clusters)==names(DMRgr)[i]]==names(DMRgr)[i],]),
nrow(df.clusters[df.clusters$Human=="no" & df.clusters[,colnames(df.clusters)==names(DMRgr)[i]]=="no",])),
row.names = c("YesDMR","NoDMR")),
Mammalia = data.frame(YesHit = c(nrow(df.clusters[df.clusters$Mammalia=="yes" & df.clusters[,colnames(df.clusters)==names(DMRgr)[i]]==names(DMRgr)[i],]),
nrow(df.clusters[df.clusters$Mammalia=="yes" & df.clusters[,colnames(df.clusters)==names(DMRgr)[i]]=="no",])),
NoHit = c(nrow(df.clusters[df.clusters$Mammalia=="no" & df.clusters[,colnames(df.clusters)==names(DMRgr)[i]]==names(DMRgr)[i],]),
nrow(df.clusters[df.clusters$Mammalia=="no" & df.clusters[,colnames(df.clusters)==names(DMRgr)[i]]=="no",])),
row.names = c("YesDMR","NoDMR")),
"No age assignment" = data.frame(YesHit = c(nrow(df.clusters[df.clusters$"No age assignment"=="yes" & df.clusters[,colnames(df.clusters)==names(DMRgr)[i]]==names(DMRgr)[i],]),
nrow(df.clusters[df.clusters$"No age assignment"=="yes" & df.clusters[,colnames(df.clusters)==names(DMRgr)[i]]=="no",])),
NoHit = c(nrow(df.clusters[df.clusters$"No age assignment"=="no" & df.clusters[,colnames(df.clusters)==names(DMRgr)[i]]==names(DMRgr)[i],]),
nrow(df.clusters[df.clusters$"No age assignment"=="no" & df.clusters[,colnames(df.clusters)==names(DMRgr)[i]]=="no",])),
row.names = c("YesDMR","NoDMR")),
Primate = data.frame(YesHit = c(nrow(df.clusters[df.clusters$Primate=="yes" & df.clusters[,colnames(df.clusters)==names(DMRgr)[i]]==names(DMRgr)[i],]),
nrow(df.clusters[df.clusters$Primate=="yes" & df.clusters[,colnames(df.clusters)==names(DMRgr)[i]]=="no",])),
NoHit = c(nrow(df.clusters[df.clusters$Primate=="no" & df.clusters[,colnames(df.clusters)==names(DMRgr)[i]]==names(DMRgr)[i],]),
nrow(df.clusters[df.clusters$Primate=="no" & df.clusters[,colnames(df.clusters)==names(DMRgr)[i]]=="no",])),
row.names = c("YesDMR","NoDMR")),
Tetrapoda = data.frame(YesHit = c(nrow(df.clusters[df.clusters$Tetrapoda=="yes" & df.clusters[,colnames(df.clusters)==names(DMRgr)[i]]==names(DMRgr)[i],]),
nrow(df.clusters[df.clusters$Tetrapoda=="yes" & df.clusters[,colnames(df.clusters)==names(DMRgr)[i]]=="no",])),
NoHit = c(nrow(df.clusters[df.clusters$Tetrapoda=="no" & df.clusters[,colnames(df.clusters)==names(DMRgr)[i]]==names(DMRgr)[i],]),
nrow(df.clusters[df.clusters$Tetrapoda=="no" & df.clusters[,colnames(df.clusters)==names(DMRgr)[i]]=="no",])),
row.names = c("YesDMR","NoDMR")),
Theria = data.frame(YesHit = c(nrow(df.clusters[df.clusters$Theria=="yes" & df.clusters[,colnames(df.clusters)==names(DMRgr)[i]]==names(DMRgr)[i],]),
nrow(df.clusters[df.clusters$Theria=="yes" & df.clusters[,colnames(df.clusters)==names(DMRgr)[i]]=="no",])),
NoHit = c(nrow(df.clusters[df.clusters$Theria=="no" & df.clusters[,colnames(df.clusters)==names(DMRgr)[i]]==names(DMRgr)[i],]),
nrow(df.clusters[df.clusters$Theria=="no" & df.clusters[,colnames(df.clusters)==names(DMRgr)[i]]=="no",])),
row.names = c("YesDMR","NoDMR")),
Vertebrata = data.frame(YesHit = c(nrow(df.clusters[df.clusters$Vertebrata=="yes" & df.clusters[,colnames(df.clusters)==names(DMRgr)[i]]==names(DMRgr)[i],]),
nrow(df.clusters[df.clusters$Vertebrata=="yes" & df.clusters[,colnames(df.clusters)==names(DMRgr)[i]]=="no",])),
NoHit = c(nrow(df.clusters[df.clusters$Vertebrata=="no" & df.clusters[,colnames(df.clusters)==names(DMRgr)[i]]==names(DMRgr)[i],]),
nrow(df.clusters[df.clusters$Vertebrata=="no" & df.clusters[,colnames(df.clusters)==names(DMRgr)[i]]=="no",])),
row.names = c("YesDMR","NoDMR")))
}
names(tables) = names(DMRgr)
fisher = lapply(tables, function(x) lapply(x, fisher.test))
res = do.call(rbind, Map(cbind, lapply(fisher, function(y)
do.call(rbind, Map(cbind, lapply(y, function(x) data.frame(OR = x$estimate,
upper = x$conf.int[2],
lower = x$conf.int[1],
pval = x$p.value)), Feature = as.list(names(y))))),
Model = as.list(names(fisher))))
res$fdr = p.adjust(res$pval, method= "fdr")
res$Feature = gsub("enhancers","All Enhancers", res$Feature)
res$Feature = gsub("assignment","assigned", res$Feature)
res$Feature = factor(res$Feature, levels=c("All Enhancers","No age assigned","Vertebrata","Gnathostomata","Tetrapoda",
"Amniota","Mammalia","Theria","Eutheria","Primate","Human"))
res = rbind(res, cbind(df[,1:4], Feature = "HAR", df[,5:6]))
rownames(res) = NULL
write.csv(res, quote=F, file = "/dcl01/lieber/ajaffe/lab/brain-epigenomics/rdas/DMR/HAR_conservedEnhancers_fisher.results.csv")
## Plot results
pdf("/dcl01/lieber/ajaffe/lab/brain-epigenomics/DMR/figures/HARs_DMR_oddsRatios.pdf", height = 4)
ggplot(df[which(df$Model %in% c("CellType", "Age", "Interaction")),], aes(Model, OR, fill = Model)) +
geom_col() + theme_classic() +
ylab("Odds Ratio") +
theme(axis.text.x = element_text(angle = 45, hjust = 1)) +
xlab("") + geom_hline(yintercept=1, linetype="dotted") +
ggtitle("Enrichment for HARs") +
theme(title = element_text(size = 20)) + theme(text = element_text(size = 20), legend.title=element_blank()) + theme(legend.position="bottom")
ggplot(df[which(df$Model %in% c("Gr1", "Gr2", "Gr3","Gr4","Gr5","Gr6")),], aes(Model, OR, fill = Model)) +
geom_col() + scale_fill_brewer(8, palette="Dark2") +
theme_classic() + geom_hline(yintercept=1, linetype="dotted") +
ylab("Odds Ratio") +
xlab("") + theme(axis.text.x = element_text(angle = 45, hjust = 1)) +
ggtitle("Enrichment for HARs") +
theme(title = element_text(size = 20)) + theme(text = element_text(size = 20), legend.title=element_blank()) + theme(legend.position="bottom")
dev.off()
pdf("/dcl01/lieber/ajaffe/lab/brain-epigenomics/DMR/figures/conservedEnhancers_DMR_oddsRatios.pdf", width = 16, height = 4)
ggplot(res[which(res$Model %in% c("CellType", "Age", "Interaction") & res$fdr<=0.05),], aes(Model,OR, fill = Model)) +
geom_col() + theme_classic() +
facet_grid(. ~ Feature) +
ylab("Odds Ratio") +
xlab("") + geom_hline(yintercept=1, linetype="dotted") +
theme(axis.text.x = element_text(angle = 45, hjust = 1)) +
ggtitle("Enrichment for Enhancers") +
theme(title = element_text(size = 20)) + theme(text = element_text(size = 20), legend.title=element_blank()) + theme(legend.position="bottom")
ggplot(res[which(res$Model %in% c("Gr1", "Gr2", "Gr3","Gr4","Gr5","Gr6") & res$fdr<=0.05),], aes(Model,OR, fill = Model)) +
geom_col() + scale_fill_brewer(8, palette="Dark2") +
theme_classic() + geom_hline(yintercept=1, linetype="dotted") +
facet_grid(. ~ Feature) +
ylab("Odds Ratio") +
xlab("") + theme(axis.text.x = element_text(angle = 45, hjust = 1)) +
ggtitle("Enrichment for Enhancers") +
theme(title = element_text(size = 20)) + theme(text = element_text(size = 20), legend.title=element_blank()) + theme(legend.position="bottom")
dev.off()
df = read.csv("../Desktop/BAMS/HAR_conservedEnhancers_fisher.results.csv")
df$sig = ifelse(df$fdr<=0.05, "Significant", "Not Significant")
df$sig = factor(df$sig, levels = c("Significant", "Not Significant"))
df = df[which(df$Model %in% c("Gr1", "Gr2", "Gr3","Gr4","Gr5","Gr6")),]
df$Model = factor(df$Model, levels = c("Gr1", "Gr2", "Gr3","Gr4","Gr5","Gr6"))
df$Feature = factor(df$Feature, levels=c("HAR","All Enhancers","No age assigned","Vertebrata","Gnathostomata",
"Tetrapoda", "Amniota","Mammalia","Theria","Eutheria","Primate","Human"))
pdf("./brain-epigenomics/DMR/figures/conservedEnhancers_DMR_oddsRatios_dotplot.pdf",
height = 5.5, width = 9.5)
ggplot(data = df[-which(df$Feature %in% c("HAR", "Primate", "Human")),],
aes(x = Model, y = log(OR), col = Model, shape = sig)) +
geom_point() + geom_pointrange(aes(ymin = log(lower), ymax = log(upper))) +
theme_bw() + facet_wrap(. ~ Feature) +
xlab("") + ylab("log(OR)") +
geom_hline(yintercept = 0, lty = 2) +
scale_shape_manual(values = c(16, 1)) +
scale_size_manual(values = c(2, 1)) +
scale_colour_brewer(8, palette="Dark2") +
ggtitle("Enrichment for Enhancers") +
theme(axis.ticks.x = element_blank(),
title = element_text(size = 20),
text = element_text(size = 20),
legend.title = element_blank(),
legend.position = "bottom") +
guides(col = FALSE)
dev.off()
df$Model = gsub("Gr1", "Gr1\n(G-N+)", df$Model)
df$Model = gsub("Gr2", "Gr2\n(G0N+)", df$Model)
df$Model = gsub("Gr3", "Gr3\n(G0N-)", df$Model)
df$Model = gsub("Gr4", "Gr4\n(G+N0)", df$Model)
df$Model = gsub("Gr5", "Gr5\n(G+N-)", df$Model)
df$Model = gsub("Gr6", "Gr6\n(G-N0)", df$Model)
df$Model = factor(df$Model, levels = c("Gr1\n(G-N+)","Gr2\n(G0N+)","Gr3\n(G0N-)",
"Gr4\n(G+N0)","Gr5\n(G+N-)","Gr6\n(G-N0)"))
pdf("./brain-epigenomics/DMR/figures/HARs_DMR_oddsRatios_dotplot.pdf",
height = 2.5, width = 6)
ggplot(data = df[which(df$Feature=="HAR"),],
aes(x = Model, y = log(OR), col = Model, shape = sig, size = sig)) +
geom_point() + geom_pointrange(aes(ymin = log(lower), ymax = log(upper))) +
theme_bw() +
xlab("") + ylab("log(OR)") +
geom_hline(yintercept = 0, lty = 2) +
scale_shape_manual(values = c(16, 1)) +
scale_size_manual(values = c(1, 2)) +
scale_colour_brewer(8, palette="Dark2") +
ggtitle("Enrichment for HARs") +
theme(axis.ticks.x = element_blank(),
title = element_text(size = 20),
text = element_text(size = 20)) +
guides(col = FALSE, shape = FALSE, size = FALSE)
dev.off()
|
# I arranged the working directory into "input" (all input files went here)
# and "output" - for submission files.
# For completeness I will past all the code in this one file. This includes:
# 1. Reading Data
# 2. Preparing Data
# 3. Building the Model.
# 4. Wrigin Submission File.
###############################################################################
################################### SOURCES ###################################
###############################################################################
setwd("~/GitHub/Data/mlsp-2014-mri")
getwd()
data_txt = read.csv("breadwrapper.txt")
data = read.csv("bf_study.csv")
data
# Used libraries
library(verification)
library(DWD)
###############################################################################
################################## LOAD DATA ##################################
###############################################################################
trainFNC = read.csv(file='Train/train_FNC.csv', head=TRUE, sep=",")
trainSBM = read.csv(file='Train/train_SBM.csv', head=TRUE, sep=",")
trainLAB = read.csv(file='Train/train_labels.csv', head=TRUE, sep=",")
testFNC = read.csv(file='Test/test_FNC.csv', head=TRUE, sep=",")
testSBM = read.csv(file='Test/test_SBM.csv', head=TRUE, sep=",")
# myExample <- read.csv(file.path(projectTree, "input/submission_example.csv"), as.is=T, header=T, sep=",")
###############################################################################
################################## PREP DATA ##################################
###############################################################################
myTrain = rbind(t(trainFNC[,-1]), t(trainSBM[,-1]))
colnames(myTrain) = trainLAB$Class
myTest = rbind(t(testFNC[,-1]), t(testSBM[,-1]))
colnames(myTest) = testFNC$Id
###############################################################################
############################## CROSS-VALIDATION ###############################
###############################################################################
# This part is optional and was used to select the values of C constraint.
# (This runs 100 itterations of 10-fold cross validation
ROCS = list()
Cs = c(1, 5, 10, 50, 100, 300, 500, 1000)
for(Cind in 1:length(Cs)) {
C = Cs[Cind]
tmpRocs = numeric()
for(i in 1:100) {
trainInds1 = sample(which(colnames(myTrain)==0), 42)
trainInds2 = sample(which(colnames(myTrain)==1), 36)
trainInds = c(trainInds1, trainInds2)
theTrain = myTrain[,trainInds]
theTest = myTrain[,-trainInds]
myFit = kdwd(t(myTrain), colnames(myTrain), C=C)
testScores = t(myFit@w[[1]]) %*% theTest
testScores = 1 - ((testScores - min(testScores)) / max(testScores - min(testScores)))
tmpRocs[i] = roc.area(as.numeric(colnames(theTest)), testScores)$A
print(i)
}
ROCS[[Cind]] = tmpRocs
}
###############################################################################
################################## FIT MODEL ##################################
###############################################################################
myFit = kdwd(t(myTrain), colnames(myTrain), C=300)
# Get scores for training data (meaningless for now).
scores = t(myFit@w[[1]]) %*% myTrain
scores = 1 - ((scores - min(scores)) / max(scores - min(scores)))
# Check ROC area. (meaningless, because of possible overfitting)
roc.area(as.numeric(colnames(myTrain)), scores)
###############################################################################
################################ WRITE SCORES #################################
###############################################################################
testScores = t(myFit@w[[1]]) %*% myTest
testScores = 1 - ((testScores - min(testScores)) / max(testScores - min(testScores)))
myExample$Probability = as.numeric(testScores)
write.csv(myExample, file=file.path(projectTree, "output/submission.csv"), row.names=F)
|
Mike Carey returned to the title for a single issue between Denise Mina and Andy Diggle 's runs on the title , and also wrote the well @-@ received Hellblazer graphic novel All His Engines about a strange illness sweeping the globe .
|
Any other trail is good if you like long XC rides.
The trail is fast with technical sections filled with rocks and roots with lots of JUMPS and some single track sections the trail is 1.9 miles long.
Any other trail if you like XC fun.
Echo Ridge was developed as a cross-country ski area by the U.S. Forest Service. It has 18 miles of trail cut into the hillsides. There are a number of options available to the rider depending on their experience.
One of my favorites is to ride from the Echo Valley Ski Area up the Forest Service road to the Echo Ridge Parking lot. This insures that you have earned a great downhill run through Bergman Gulch.
From the Ridge parking lot, head out to Grand Junction. At the GJ kiosk, look to your right. Between the two trails is a fun single track which takes the rider to Chaos Corner. From there head on down Alley Oop and do that loop which will bring you back to Chaos Corner. Then head up Windsinger to Grand Junction. From there head up Ridge View or go out Morning Glory to 5-Corners. Hit Little Critter. Keep your eye open for the single track taking off to the right after you have ridden most of Little Critter. This will dump you at North Junction at the beginning of the 1.9 mile Gulch. It is fast, sometimes technical and definitely dangerous if you hit a huge water bar going to fast. Your ride ends back at Echo Valley.
There are close to 20 plus miles of trails, old jammer (logging) roads, fire roads and xc ski trails open to Mountain Bikers. Pick up a map at the Chelan Ranger District located next to the Caravel Resort.
The Gulch rocks, if you have a shuttle to the top. Echo Ridge XC trails are good, plenty of different trails to ride. Plenty of jumps on the gulch and lots of different spurs off the main trail, look for them.
|
Antimony is incompatible with strong acids , halogenated acids , and oxidizers ; when exposed to newly formed hydrogen it may form stibine ( SbH3 ) .
|
# Fidelity estimation: Minimax Method
We show how to use the code for minimax method using some examples.
## Example 1: Cluster state and Pauli measurements
To demonstrate how to use the minimax method to estimate fidelity, we consider a 3-qubit linear cluster state as our target state.\
We will focus on Pauli measurements.
The minimax method constructs an estimator when the target state, the measurement settings, and the confidence level is specified. One we construct the estimator, we can repeatedly reuse the estimator for the chosen settings.
We outline the basic steps that we will go through in this example - from specifying the settings to computing the estimate.
1. Create a YAML file describing the settings required to construct the estimator.
2. Construct the estimator using the specified settings.
3. Create a CSV file containing the measurement outcomes.
4. Compute the fidelity estimate using the CSV file and the constructed estimator.
We remark that step 2 is usually computation intensive. Nevertheless, once the estimator has been constructed, the fidelity estimates (step 4) can be obtained almost instantaneously.
Let's look at each step in more detail.
## Step 1: Create a YAML file with the measurement settings.
[YAML](https://en.wikipedia.org/wiki/YAML) is a markup language that is human-readable, and can be parsed by a computer. The combination of these attributes makes it a good medium to specify the settings to the code.
We allow for different ways to specify the settings in the YAML file.\
For example, one could specify the target state as a list in the YAML file, or provide a path to a `.npy` file containing the numpy array for the target state.\
Since Pauli measurements and some special states like the stabilizer states are commonly used, we have provided a special interface to conveniently specify these settings.\
We will be using the latter interface in this demo for convenience. For details on all available formats to specify the settings, we encourage the reader to refer to the documentation of the code.
We create the following settings file for the cluster state. We specify those Pauli operators $P$ that have a non-zero weight $\text{Tr}(P \rho)$.
-------------------------------------------------
### cluster_state_settings.yaml
```
target:
- cluster: 3
POVM_list:
- pauli: [IZX, XIX, XZI, YXY, YYZ, ZXZ, ZYY]
R_list: 100
confidence_level: 0.95
```
-------------------------------------------------
Let's take a closer look at the settings.
- `target` refers to the target state. We can conveniently provide a linear cluster state using the syntax: `- cluster: nq`, where `nq` is the number of qubits.\
We have therefore specified a 3-qubit cluster state.
- `POVM_list` is a list of POVMs that will measured in order to estimate the fidelity.\
Pauli measurements can be specified in a few different ways, but here we use the most obvious one: list the Pauli operators that you want to measure.\
The default measurement is projection on each eigenvector of the Pauli operator, but if collective measurement on eigenspace with $+1$ and $-1$ eigenvalue is required, you can include the keyword `subspace` after listing all the Pauli operators.
- `R_list` corresponds to the number of outcomes recorded for each POVM.\
We want 100 outcomes for each Pauli measurement, so we simply write 100.\
If something more specific is required, write a list of outcomes, one for each Pauli measurement.
- `confidence_level` should be a number between 0.75 and 1, and it determines the confidence level of the computed risk.
----------
> **It is important to adhere to the syntax specified in the documentation when creating the YAML file.**\
The code is expected to throw an error when incorrect syntax is used. However, there could be some cases that slip past the sanity checks, and the code may end up constructing an estimator that was not intended by the user!
## Step 2: Construct the estimator
In order to construct the estimator, we use the function ```construct_fidelity_estimator``` included in ```handle_fidelity_estimation.py``` module.
The syntax for this function is pretty straightforward:
----------------------
```
construct_fidelity_estimator(yaml_filename, estimator_filename,
yaml_file_dir = './yaml_files', estimator_dir = './estimator_files')
```
----------------------
A closer look at the options:
- ```yaml_filename``` refers to the name of the YAML settings file.
- ```estimator_filename``` refers to the name of the (JSON) file to which the constructed estimator is saved.
- ```yaml_file_dir``` specifies the directory in which the YAML settings file is stored.\
This is an optional argument, and if nothing is specified, the code assumes that the YAML file lives in a sub-directory named `yaml_files` of the current directory.
- ```estimator_dir``` specifies the directory where the constructed estimator is saved.\
As before, this is an optional argument, and the default location is assumed to be a sub-directory named `estimator_files` of the current directory.
------------------
> We save the estimator because the same estimator can be re-used later for the same settings.\
The estimator is saved as a JSON file. These files are internally handled by the functions in the module, and need not be edited manually by the user.
------------------
Following the default options, we have created a subdirectory called `yaml_files` and placed `cluster_state_settings.yaml` YAML file there. Let us now construct the estimator.
> **It can take anywhere from a few minutes to many hours to compute the estimator depending on the settings that were specified.**\
If the dimension of the system is large or many measurement settings are begin used, please consider running the code on a workstation or a cluster.
The following code is expected to run in about 4 minutes on a laptop, though the actual time may vary depending on the hardware and the OS.
```python
import project_root # adds the root directory of the project to Python Path
from handle_fidelity_estimation import construct_fidelity_estimator
construct_fidelity_estimator(yaml_filename = 'cluster_state_settings.yaml',\
estimator_filename = 'cluster_state_estimator.json')
```
Optimization complete.
> 1. Note that `construct_fidelity_estimator` prints the progress of optimization by default.\
If you wish to turn this off, supply `print_progress = False` as an additional argument to the function.
2. If an estimator file already exists, `construct_fidelity_estimator` function will throw an error. You can delete the existing estimator, move it to a different directory, or use another name to save the estimator in that case.
## Step 3: Generate measurement outcomes
We generate the measurement outcomes separately and store them in a CSV file.\
Note that these outcomes are generated using the state obtained by applying 10% depolarizing noise to the target state.
> In practice, these outcomes will come from experiments.
The CSV file looks as follows:
---------------------
### cluster_state_outcomes.csv
| | | | | |
| --- | - | - | --- | - |
| IZX | 7 | 0 | ... | 7 |
| XIX | 7 | 7 | ... | 0 |
| . | . | . | . | . |
| ZYY | 5 | 3 | ... | 0 |
---------------------
The first column contains the labels of the Pauli measurements performed.\
Corresponding to each Pauli operator, we store the measurement outcomes in the same row as the Pauli operator.\
Outcome $i$ points to the eigenvector $\vert i \rangle$ that was observed upon measurement.
> 1. **It is important that the order of eigenvectors used for outcomes matches the POVM that was specified for constructing the estimator.**\
We use the following convention for the eigenvectors: $\vert+++\rangle$, $\vert++-\rangle$, $\vert+-+\rangle$, ..., $\vert---\rangle$.\
Basically, we use the binary expansion of numbers from $0$ to $2^{n_q} - 1$, where $n_q$ are the number of qubits, with $0$ replaced by $+$ and $1$ replaced by $-$.
2. **It is important that the outcomes for Pauli operators are listed in the same order as what we used for constructing the estimator.**\
That is, we must have outcomes for IZXZ, XIX, ..., ZYY in that order in the CSV file.
Note for any Pauli operator $P = X_1 \dotsb X_{n_q}$, a $+$ at the $i$th qubit location means that we are looking at the $+1$ eigenvector of $X_i$, where $X_i \in \{I, X, Y, Z\}$.
> In practice, steps 2 & 3 can occur in any order.
## Step 4: Compute the fidelity estimate
Let's use the estimator that we constructed in step 2 and the outcomes generated in step 3 to compute the fidelity estimate.\
This task is handled by `compute_fidelity_estimate_risk` function in `handle_fidelity_estimation.py` module.
This function takes the following form.
----------------
```
compute_fidelity_estimate_risk(outcomes, estimator_filename, estimator_dir = './estimator_files')
```
----------------
The options accept the following formats:
- `outcomes` can be one of the following:
1. A list of outcomes for each POVM measurement.
2. Path to a YAML file containing a list of outcomes for each POVM measurement.
3. - Path to a CSV file, or
- A dictionary:\
`{'csv_file_path': Path to CSV file, 'entries': 'row'/'column', 'start': (row index, column index)}`\
where `row` (`column`) is used if data is stored in rows (columns),\
and `start` denotes the index of the cell where the data starts (we start the row and column at index 0).
- `estimator_filename` is the name of the estimator file that we constructed previously.
- `estimator_dir` refers to the directory in which the estimator file has been saved.
We refer the reader to the documentation of the code which elaborates these options further.
As we can see from the CSV file outline in step 3, the data starts at the first row and the second column.\
The first column describes the data, but is not actually a part of it. Therefore, we set `start = (0, 1)`.\
As noted earlier, we label the rows and columns starting from 0, following Python convention.\
Also, it is clear that the data is stored row-wise, so we set `entries = 'row'`.
Note that we have saved the `cluster_state_outcomes.csv` file in a subdirectory called `outcome_files`.\
Using this, we compute the estimate as follows.
```python
import project_root # adds the root directory of the project to Python Path
from handle_fidelity_estimation import compute_fidelity_estimate_risk
compute_fidelity_estimate_risk(outcomes = {'csv_file_path': './outcome_files/cluster_state_outcomes.csv',\
'entries': 'row', 'start': (0, 1)},\
estimator_filename = 'cluster_state_estimator.json')
```
Fidelity estimate: 0.925
Risk: 0.086
We can see that the estimate $\widehat{F} \approx 0.925$ is close to the actual fidelity $F = 0.9125$.\
The risk can be reduced by increasing the number of shots and/or the Pauli measurements performed.
--------------------
--------------------
## Example 2: The Bell State and Randomized Pauli Measurement scheme
Suppose that our target state $\rho$ is the two-qubit Bell state
\begin{align}
\rho &= \vert \psi \rangle \langle \psi \vert \\
\text{where} \quad \vert \psi \rangle &= \frac{1}{\sqrt{2}} \left(\vert 00 \rangle + \vert 11 \rangle\right)
\end{align}
Observe that $\vert \psi \rangle$ is a stabilizer state that is generated by the stabilizers $XX$ and $ZZ$.
To compute the fidelty, we use the minimax optimal measurement scheme for stabilizer states. This amounts to sampling uniformly from the stabilizer group elements (except the identity) and recording their measurement outcome ($\pm 1).
Let's compute the estimator given by the minimax method for such a setting.
We know that
\begin{equation}
R = \left\lceil 2\frac{\ln\left(2/\epsilon\right)}{\left|\ln\left(1 - \left(\frac{d}{d - 1}\right)^2 \widehat{R}_*^2\right)\right|} \right\rceil
\end{equation}
outcomes are sufficient to achieve a risk $\widehat{\mathcal{R}}_* \in (0, 0.5)$ with a confidence level of $1 - \epsilon \in (0.75, 1)$.
As before, we break down the process of constructing an estimator & computing an estimate into four steps:
1. Create a YAML file describing the settings to construct the estimator and the risk.
2. Construct the estimator for the specified settings.
3. Store the outcomes in a YAML file. Convert outcomes to indices in case they are eigenvalues.
4. Use the outcomes and constructed estimator to compute the fidelity estimate (and the risk).
## Step 1: Create the YAML file containing the settings
The YAML file looks as follows.
### bell_state_settings.yml
```
target:
- stabilizer: [XX, ZZ]
POVM_list:
- pauli: [RPM]
R_list: 1657
confidence_level: 0.95
```
We describe the a couple of above options in more detail:
- The general syntax for specifying a target stabilizer state is `- stabilizer: list of stabilizer generators`. Note that we can include a sign in front of the Pauli operator if necessary.\
For example, we specify a stabilizer state above with $XX$ and $ZZ$ as the stabilizer generators. We could as well have used $XX$ and $-YY$ as the stabilizer generators.
- We have included a shortcut to specify the Randomized Pauli Measurement (RPM) scheme described in section II.E. of the PRA submission. The syntax is always `- pauli: [RPM]` for specifying this measurement scheme.\
For stabilizer states, this amounts to randomly sampling the stabilizer group (excluding the identity) and recording the eigenvalues of outcomes.
Note that we use a confidence level of $95\%$ and a risk of $\widehat{\mathcal{R}}_* = 0.05$ to obtain $R = 1657$.
## Step 2: Construct the estimator using the YAML settings file
As before, we use the function ```construct_fidelity_estimator``` in ```handle_fidelity_estimation.py``` module to construct the estimator.
We have placed `bell_state_settings.yaml` settings file in the `yaml_files` subdirectory.
> The estimator for the RPM measurement scheme is constructed efficiently. It should take at most a few minutes, if not seconds, to construct the estimator.
```python
import project_root # adds the root directory of the project to Python Path
from handle_fidelity_estimation import construct_fidelity_estimator
construct_fidelity_estimator(yaml_filename = 'bell_state_settings.yaml',\
estimator_filename = 'bell_state_estimator.json')
```
Optimization complete
You can check that there is a subdirectory called `estimator_files` (if it wasn't already there), and you can find the file `bell_state_estimator.json` there.
## Step 3: Create a YAML file with the measurement outcomes
We created some outcomes beforehand to test the estimator. For this purpose, we added $10\%$ depolarizing noise to the target state $\rho$, and then performed the Pauli measurements as prescibed by Randomized Pauli Measurement (RPM) scheme.
Note that for the RPM scheme, only the *number* of $+1$ and $-1$ eigenvalues are important. It doesn't matter which Pauli measurement gave a $+1$ outcome or a $-1$ outcome.\
Before we supply the outcomes to the estimator, we need to convert $+1 \to 0$ and $-1 \to 1$. The reason is that the estimator works by referring to the POVM elements and we used $\{E_+, E_-\}$ as the POVM when constructing the estimator, in that order.\
Because the outcomes are going be just $0$ and $1$, we put the outcomes in a list inside a YAML file.
> For the sake of demonstration, this time we choose to save our outcomes in a YAML file instead of a CSV file.\
A CSV file can be used if that's preferred.
The YAML file containing the outcomes looks as follows.
### bell_state_measurement_outcomes.yaml
```
outcomes:
- [0, 0, ...]
```
Note that there must be exactly $R = 1657$ measurement outcomes, because the estimator was constructed for this case.
> The syntax used in the YAML file is important for ensuring proper parsing of the file.\
The code documentation can be referred for details.
We use these outcomes to compute the fidelity estimate.
## Step 4: Compute the fidelity estimate
We now supply the outcomes to the `compute_fidelity_estimate_risk` function in `handle_fidelity_estimation.py` module.
We have saved the `bell_state_measurement_outcomes.yml` file in a subdirectory called `outcome_files`, and we use this to compute the estimate.
```python
import project_root # adds the root directory of the project to Python Path
from handle_fidelity_estimation import compute_fidelity_estimate_risk
compute_fidelity_estimate_risk(outcomes = './outcome_files/bell_state_measurement_outcomes.yaml',\
estimator_filename = 'bell_state_estimator.json')
```
Fidelity estimate: 0.933
Risk: 0.05
Observe that the risk is very close to the value of $0.05$ (only a rounded value is displayed) that we chose to determine the number of outcomes $R = 1657$.
We can also see that the fidelity estimate $\widehat{F} \approx 0.933$ is close to the actual fidelity $F = 0.925$, and within the specified risk of $0.05$.
This estimate can be found in Table II of PRA submission.
--------------------
--------------------
# Epilogue
Other formats are supported by the YAML settings file. You can directly supply lists to it or give a path to a `.npy` file which contains the array describing your target state or POVMs. Please read the documentation to see all the available options.
Note that the code can be run directly from the commandline. This is especially helpful if one needs to run the code on a cluster or even on a workstation. Please refer the documentation for details on how to use this functionality.
|
[STATEMENT]
lemma (in group) compl_fam_empty[simp]: "compl_fam S {}"
[PROOF STATE]
proof (prove)
goal (1 subgoal):
1. compl_fam S {}
[PROOF STEP]
unfolding compl_fam_def
[PROOF STATE]
proof (prove)
goal (1 subgoal):
1. \<forall>i\<in>{}. complementary (S i) (IDirProds G S ({} - {i}))
[PROOF STEP]
by simp
|
-- Prueba mediante encadenamiento de ecuaciones
-- ============================================
-- Sean a, b y c números reales. Demostrar que
-- (a * b) * c = b * (a * c)
import data.real.basic
variables (a b c : ℝ)
-- 1ª demostración
example : (a * b) * c = b * (a * c) :=
begin
rw mul_comm a b,
rw mul_assoc,
end
-- 2ª demostración
example : (a * b) * c = b * (a * c) :=
begin
calc (a * b) * c = (b * a) * c : by rw mul_comm a b
... = b * (a * c) : by rw mul_assoc,
end
-- 3ª demostración
example : (a * b) * c = b * (a * c) :=
by linarith
-- 4ª demostración
example : (a * b) * c = b * (a * c) :=
by finish
-- 5ª demostración
example : (a * b) * c = b * (a * c) :=
by ring
|
-- 2012-10-20 Andreas
module Issue721b where
data Bool : Set where
false true : Bool
record Foo (b : Bool) : Set where
field
_*_ : Bool → Bool → Bool
data _≡_ {A : Set} (x : A) : A → Set where
refl : x ≡ x
test : (F : Foo false) → let open Foo F in (x : Bool) → _*_ x ≡ (λ x → x)
test F x = x
where open Foo F
-- Don't want to see any anonymous module
|
Require Import CodeDeps.
Require Import Ident.
Local Open Scope Z_scope.
Definition _bit := 1%positive.
Definition _bits := 2%positive.
Definition _g := 3%positive.
Definition _i := 4%positive.
Definition _intid := 5%positive.
Definition _pending := 6%positive.
Definition _rec := 7%positive.
Definition _rec_rvic_state := 8%positive.
Definition _ret := 9%positive.
Definition _rvic := 10%positive.
Definition _t'1 := 11%positive.
Definition rvic_clear_pending_body :=
(Ssequence
(Scall (Some _t'1)
(Evar _get_rvic_pending_bits (Tfunction
(Tcons
(tptr Tvoid)
Tnil) (tptr Tvoid) cc_default))
((Etempvar _rvic (tptr Tvoid)) :: nil))
(Scall None
(Evar _rvic_clear_flag (Tfunction
(Tcons tulong (Tcons (tptr Tvoid) Tnil)) tvoid
cc_default))
((Etempvar _intid tulong) :: (Etempvar _t'1 (tptr Tvoid)) :: nil)))
.
Definition f_rvic_clear_pending := {|
fn_return := tvoid;
fn_callconv := cc_default;
fn_params := ((_rvic, (tptr Tvoid)) ::
(_intid, tulong) :: nil);
fn_vars := nil;
fn_temps := ((_t'1, (tptr Tvoid)) :: nil);
fn_body := rvic_clear_pending_body
|}.
|
Formal statement is: lemma frontier_subset_closed: "closed S \<Longrightarrow> frontier S \<subseteq> S" Informal statement is: If $S$ is a closed set, then the frontier of $S$ is a subset of $S$.
|
% SEGMENT_GRAPH Given a sparse, square graph of edges weights segment the nodes
% of the graph into connected sub-components using the greedy merge-based
% method of "Graph Based Image Segmentation".
%
% C = segment_graph(A)
% C = segment_graph(A,'ParameterName',ParameterValue, ...)
%
% Inputs:
% A #A by #A sparse, square matrix of edge weights
% Optional:
% 'Threshold' followed by "C" threshold to use (paper writes that this
% roughly corresponds to minimum size, though it's really just adding a
% weight of size/C to components. In any case, increasing this will tend
% to produce larger segments.
% 'MinSize' followed by the minimum size of an output component. This
% constraint is enforced as a _post process_.
% Output:
% C #A by 1 list of component ids
%
% Example:
% [V,F] = load_mesh('~/Dropbox/models/Cosmic blobs/Model9.off');
% A = adjacency_dihedral_angle_matrix(V,F);
% [AI,AJ,AV] = find(A);
% A = sparse(AI,AJ,exp(abs(pi-abs(AV-pi))),size(A,1),size(A,2));
% L = -(A - diag(sum(A,2)));
% C = segment_graph(L,'Threshold',500,'MinSize',20);
% tsurf(F,V,'CData',C);
% colormap(cbrewer('Set1',(max(C))));
% view(2);
% axis equal;
|
[STATEMENT]
lemma trnl\<^sub>\<epsilon>_eq:
assumes "ide u" and "ide v"
and "src f = trg v" and "src g = trg u"
and "\<nu> \<in> hom v (g \<star> u)"
shows "trnl\<^sub>\<epsilon> u \<nu> = (\<epsilon> \<star> u) \<cdot> \<a>\<^sup>-\<^sup>1[f, g, u] \<cdot> (f \<star> \<nu>)"
[PROOF STATE]
proof (prove)
goal (1 subgoal):
1. trnl\<^sub>\<epsilon> u \<nu> = (\<epsilon> \<star> u) \<cdot> \<a>\<^sup>-\<^sup>1[f, g, u] \<cdot> (f \<star> \<nu>)
[PROOF STEP]
using assms trnl\<^sub>\<epsilon>_def antipar strict_lunit comp_cod_arr hcomp_obj_arr
[PROOF STATE]
proof (prove)
using this:
ide u
ide v
src f = trg v
src g = trg u
\<nu> \<in> hom v (g \<star> u)
trnl\<^sub>\<epsilon> ?u ?\<nu> \<equiv> \<l>[?u] \<cdot> (\<epsilon> \<star> ?u) \<cdot> \<a>\<^sup>-\<^sup>1[f, g, ?u] \<cdot> (f \<star> ?\<nu>)
trg g = src f
src g = trg f
ide ?f \<Longrightarrow> \<l>[?f] = ?f
\<lbrakk>arr ?f; cod ?f = ?b\<rbrakk> \<Longrightarrow> ?b \<cdot> ?f = ?f
\<lbrakk>obj ?b; arr ?f; ?b = trg ?f\<rbrakk> \<Longrightarrow> ?b \<star> ?f = ?f
goal (1 subgoal):
1. trnl\<^sub>\<epsilon> u \<nu> = (\<epsilon> \<star> u) \<cdot> \<a>\<^sup>-\<^sup>1[f, g, u] \<cdot> (f \<star> \<nu>)
[PROOF STEP]
by auto
|
{-# OPTIONS --cubical --no-import-sorts #-}
open import Bundles
module Properties.ConstructiveField {ℓ ℓ'} (F : ConstructiveField {ℓ} {ℓ'}) where
open import Agda.Primitive renaming (_⊔_ to ℓ-max; lsuc to ℓ-suc; lzero to ℓ-zero)
private
variable
ℓ'' : Level
open import Cubical.Foundations.Everything renaming (_⁻¹ to _⁻¹ᵖ; assoc to ∙-assoc)
open import Cubical.Data.Sum.Base renaming (_⊎_ to infixr 4 _⊎_)
open import Cubical.Data.Sigma.Base renaming (_×_ to infixr 4 _×_)
open import Cubical.Data.Empty renaming (elim to ⊥-elim) -- `⊥` and `elim`
open import Function.Base using (it) -- instance search
open import MoreLogic
open MoreLogic.Reasoning
import MoreAlgebra
-- Lemma 4.1.6.
-- For a constructive field (F, 0, 1, +, ·, #), the following hold.
-- 1. 1 # 0.
-- 2. Addition + is #-compatible in the sense that for all x, y, z : F
-- x # y ⇔ x + z # y + z.
-- 3. Multiplication · is #-extensional in the sense that for all w, x, y, z : F
-- w · x # y · z ⇒ w # y ∨ x # z.
open ConstructiveField F
open import Cubical.Structures.Ring
R = (makeRing 0f 1f _+_ _·_ -_ is-set +-assoc +-rid +-rinv +-comm ·-assoc ·-rid ·-lid ·-rdist-+ ·-ldist-+)
open Cubical.Structures.Ring.Theory R
open MoreAlgebra.Properties.Ring R
-- Lemma 4.1.6.1
1f#0f : 1f # 0f
1f#0f with ·-identity 1f
1f#0f | 1·1≡1 , _ = fst (·-inv-back _ _ 1·1≡1)
-- Lemma 4.1.6.2
-- For #-compatibility of +, suppose x # y, that is, (x +z) −z # (y +z) −z.
-- Then #-extensionality gives (x + z # y + z) ∨ (−z # −z), where the latter case is excluded by irreflexivity of #.
+-#-compatible : ∀(x y z : Carrier) → x # y → x + z # y + z
+-#-compatible x y z x#y with
let P = transport (λ i → a+b-b≡a x z i # a+b-b≡a y z i ) x#y
in +-#-extensional _ _ _ _ P
... | inl x+z#y+z = x+z#y+z
... | inr -z#-z = ⊥-elim (#-irrefl _ -z#-z)
-- The other direction is similar.
+-#-compatible-inv : ∀(x y z : Carrier) → x + z # y + z → x # y
+-#-compatible-inv _ _ _ x+z#y+z with +-#-extensional _ _ _ _ x+z#y+z
... | inl x#y = x#y
... | inr z#z = ⊥-elim (#-irrefl _ z#z)
-- Lemma 4.1.6.3
·-#-extensional-case1 : ∀(w x y z : Carrier) → w · x # y · z → w · x # w · z → x # z
·-#-extensional-case1 w x y z w·x#y·z w·x#w·z =
let
instance -- this allows to use ⁻¹ᶠ without an instance argument
w·[z-x]#0f =
( w · x # w · z ⇒⟨ +-#-compatible _ _ (- (w · x)) ⟩
w · x - w · x # w · z - w · x ⇒⟨ transport (λ i → (fst (+-inv (w · x)) i) # a·b-a·c≡a·[b-c] w z x i) ⟩
0f # w · (z - x) ⇒⟨ #-sym _ _ ⟩
w · (z - x) # 0f ◼) w·x#w·z
in ( w · (z - x) # 0f ⇒⟨ (λ _ → ·-rinv (w · (z - x)) it ) ⟩ -- NOTE: "plugging in" the instance did not work, ∴ `it`
w · (z - x) · (w · (z - x)) ⁻¹ᶠ ≡ 1f ⇒⟨ transport (λ i → ·-comm w (z - x) i · (w · (z - x)) ⁻¹ᶠ ≡ 1f) ⟩
(z - x) · w · (w · (z - x)) ⁻¹ᶠ ≡ 1f ⇒⟨ transport (λ i → ·-assoc (z - x) w ((w · (z - x)) ⁻¹ᶠ) (~ i) ≡ 1f) ⟩
(z - x) · (w · (w · (z - x)) ⁻¹ᶠ) ≡ 1f ⇒⟨ fst ∘ (·-inv-back _ _) ⟩
z - x # 0f ⇒⟨ +-#-compatible _ _ x ⟩
(z - x) + x # 0f + x ⇒⟨ transport (λ i → +-assoc z (- x) x (~ i) # snd (+-identity x) i) ⟩
z + (- x + x) # x ⇒⟨ transport (λ i → z + snd (+-inv x) i # x) ⟩
z + 0f # x ⇒⟨ transport (λ i → fst (+-identity z) i # x) ⟩
z # x ⇒⟨ #-sym _ _ ⟩
x # z ◼) it -- conceptually we would plug `w·[z-x]#0f` in, but this breaks the very first step
·-#-extensional : ∀(w x y z : Carrier) → w · x # y · z → (w # y) ⊎ (x # z)
·-#-extensional w x y z w·x#y·z with #-cotrans _ _ w·x#y·z (w · z)
... | inl w·x#w·z = inr (·-#-extensional-case1 w x y z w·x#y·z w·x#w·z) -- first case
... | inr w·z#y·z = let z·w≡z·y = (transport (λ i → ·-comm w z i # ·-comm y z i) w·z#y·z)
in inl (·-#-extensional-case1 _ _ _ _ z·w≡z·y z·w≡z·y) -- second case reduced to first case
|
" Behind the Crooked Cross " is rarely played live as Hanneman hates the track , though King has always wanted to play it " because it 's got a cool intro " despite it not being his favorite song . King said " that 's fine " when speaking of the situation , noting " there are songs that he wants to play that I always shoot down . " " Ghosts of War " isn 't King 's favorite song either , which he attests " everybody always wants to hear " performed live . He confessed ; " I like the ending , you know , I like the big heavy part and I always say , ‘ Let 's put the heavy ending at the end of " Chemical Warfare " and just do the last half . ’ But I could never make that fly . "
|
/* test.c
*
* Copyright (C) 2018 Patrick Alken
*
* This program 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.
*
* This program 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 this program; if not, write to the Free Software
* Foundation, Inc., 51 Franklin Street, Fifth Floor, Boston, MA 02110-1301, USA.
*/
#include <config.h>
#include <stdlib.h>
#include <math.h>
#include <assert.h>
#include <gsl/gsl_math.h>
#include <gsl/gsl_bst.h>
#include <gsl/gsl_rng.h>
#include <gsl/gsl_sort.h>
#include <gsl/gsl_test.h>
enum array_order
{
ORD_RANDOM = 0, /* random order */
ORD_ASCENDING, /* ascending order */
ORD_DESCENDING, /* descending order */
ORD_BALANCED, /* balanced tree order */
ORD_ZIGZAG, /* zig-zag order */
ORD_ASCENDING_SHIFTED, /* ascending from middle, then beginning */
ORD_END_NODUP, /* end of no-duplicate ordering */
ORD_RANDOM_DUP /* random order with duplicates */
};
/* fill array[] with random integers in [lower,upper] with duplicates allowed */
static void
random_integers(const size_t n, const int lower, const int upper,
int array[], gsl_rng * r)
{
size_t i;
for (i = 0; i < n; ++i)
array[i] = (int) ((upper - lower) * gsl_rng_uniform(r) + lower);
}
/* fills array[] with a random permutation of the integers between 0 and n - 1 */
static void
random_permuted_integers (const size_t n, int array[], gsl_rng * r)
{
size_t i;
for (i = 0; i < n; i++)
array[i] = i;
for (i = 0; i < n; i++)
{
size_t j = i + (unsigned) (gsl_rng_uniform(r) * (n - i));
int t = array[j];
array[j] = array[i];
array[i] = t;
}
}
static int
compare_ints(const void *pa, const void *pb, void *params)
{
const int *a = pa;
const int *b = pb;
(void) params;
return (*a < *b) ? -1 : (*a > *b);
}
/* Generates a list of integers that produce a balanced tree when
inserted in order into a binary tree in the usual way.
|min| and |max| inclusively bound the values to be inserted.
Output is deposited starting at |*array|. */
static void
gen_balanced_tree (const int min, const int max, int **array)
{
int i;
if (min > max)
return;
i = (min + max + 1) / 2;
*(*array)++ = i;
gen_balanced_tree (min, i - 1, array);
gen_balanced_tree (i + 1, max, array);
}
/* generates a permutation of the integers |0| to |n - 1| */
static void
gen_int_array (const size_t n, const enum array_order order, int array[], gsl_rng * r)
{
size_t i;
switch (order)
{
case ORD_RANDOM:
random_permuted_integers (n, array, r);
break;
case ORD_ASCENDING:
for (i = 0; i < n; i++)
array[i] = i;
break;
case ORD_DESCENDING:
for (i = 0; i < n; i++)
array[i] = n - i - 1;
break;
case ORD_BALANCED:
gen_balanced_tree (0, n - 1, &array);
break;
case ORD_ZIGZAG:
for (i = 0; i < n; i++)
{
if (i % 2 == 0)
array[i] = i / 2;
else
array[i] = n - i / 2 - 1;
}
break;
case ORD_ASCENDING_SHIFTED:
for (i = 0; i < n; i++)
{
array[i] = i + n / 2;
if ((size_t) array[i] >= n)
array[i] -= n;
}
break;
case ORD_RANDOM_DUP:
random_integers(n, -10, 10, array, r);
break;
default:
assert (0);
}
}
static void
check_traverser(const size_t n, const enum array_order order, gsl_bst_trav * trav, int data,
const char *desc, const gsl_bst_workspace * w)
{
int *prev, *cur, *next;
prev = gsl_bst_trav_prev(trav);
if (prev != NULL)
{
gsl_test(*prev > data, "bst %s[n=%zu,order=%d] %s traverser ahead of %d, but should be ahead of %d",
gsl_bst_name(w), n, order, desc, *prev, data);
}
gsl_bst_trav_next(trav);
cur = gsl_bst_trav_cur(trav);
gsl_test(*cur != data, "bst %s[n=%zu,order=%d] %s traverser at %d, but should be at %d",
gsl_bst_name(w), n, order, desc, *cur, data);
next = gsl_bst_trav_next(trav);
if (next != NULL)
{
gsl_test(*next < data, "bst %s[n=%zu,order=%d] %s traverser behind %d, but should be behind %d",
gsl_bst_name(w), n, order, desc, *next, data);
}
gsl_bst_trav_prev(trav);
}
static void
test_bst_int(const size_t n, const gsl_bst_type * T, const enum array_order order, gsl_rng * r)
{
int *data = malloc(n * sizeof(int));
int *data_delete = malloc(n * sizeof(int));
int *sorted_data = malloc(n * sizeof(int));
gsl_bst_workspace * w = gsl_bst_alloc(T, NULL, compare_ints, NULL);
gsl_bst_trav trav;
int *p;
int i;
size_t nodes;
/* generate data to be inserted in tree */
gen_int_array(n, order, data, r);
for (i = 0; i < (int) n; ++i)
sorted_data[i] = data[i];
gsl_sort_int(sorted_data, 1, n);
if (order != ORD_RANDOM_DUP)
{
/* generate random order to delete data from tree */
gen_int_array(n, ORD_RANDOM, data_delete, r);
}
else
{
for (i = 0; i < (int) n; ++i)
data_delete[i] = sorted_data[i];
}
/* insert data */
for (i = 0; i < (int) n; ++i)
{
p = gsl_bst_insert(&data[i], w);
gsl_test(p != NULL, "bst_int %s[n=%zu,order=%d] insert i=%d", gsl_bst_name(w), n, order, i);
}
if (order != ORD_RANDOM_DUP)
{
nodes = gsl_bst_nodes(w);
gsl_test(nodes != n, "bst_int %s[n=%zu,order=%d] after insertion count = %zu/%zu",
gsl_bst_name(w), n, order, nodes, n);
}
/* test data was inserted and can be found */
for (i = 0; i < (int) n; ++i)
{
p = gsl_bst_find(&data[i], w);
gsl_test(*p != data[i], "bst_int %s[n=%zu,order=%d] find [%d,%d]",
gsl_bst_name(w), n, order, *p, data[i]);
p = gsl_bst_trav_find(&data[i], &trav, w);
gsl_test(p == NULL, "bst_int %s[n=%zu,order=%d] trav_find unable to find item %d",
gsl_bst_name(w), n, order, data[i]);
check_traverser(n, order, &trav, data[i], "post-insertion", w);
}
/* traverse tree in-order */
p = gsl_bst_trav_first(&trav, w);
i = 0;
while (p != NULL)
{
int *q = gsl_bst_trav_cur(&trav);
gsl_test(*p != sorted_data[i], "bst_int %s[n=%zu,order=%d] traverse i=%d [%d,%d]",
gsl_bst_name(w), n, order, i, *p, sorted_data[i]);
gsl_test(*p != *q, "bst_int %s[n=%zu,order=%d] traverse cur i=%d [%d,%d]",
gsl_bst_name(w), n, order, i, *p, *q);
p = gsl_bst_trav_next(&trav);
++i;
}
gsl_test(i != (int) n, "bst_int %s[n=%zu,order=%d] traverse number=%d",
gsl_bst_name(w), n, order, i);
/* traverse tree in reverse order */
p = gsl_bst_trav_last(&trav, w);
i = n - 1;
while (p != NULL)
{
int *q = gsl_bst_trav_cur(&trav);
gsl_test(*p != sorted_data[i], "bst_int %s[n=%zu,order=%d] traverse reverse i=%d [%d,%d]",
gsl_bst_name(w), n, order, i, *p, sorted_data[i]);
gsl_test(*p != *q, "bst_int %s[n=%zu,order=%d] traverse reverse cur i=%d [%d,%d]",
gsl_bst_name(w), n, order, i, *p, *q);
p = gsl_bst_trav_prev(&trav);
--i;
}
gsl_test(i != -1, "bst_int %s[n=%zu,order=%d] traverse reverse number=%d",
gsl_bst_name(w), n, order, i);
/* test traversal during tree modifications */
for (i = 0; i < (int) n; ++i)
{
gsl_bst_trav x, y, z;
gsl_bst_trav_find(&data[i], &x, w);
check_traverser(n, order, &x, data[i], "pre-deletion", w);
if (data[i] == data_delete[i])
continue;
p = gsl_bst_remove(&data_delete[i], w);
gsl_test(*p != data_delete[i], "bst_int %s[n=%zu,order=%d] remove i=%d [%d,%d]",
gsl_bst_name(w), n, order, i, *p, data_delete[i]);
p = gsl_bst_trav_copy(&y, &x);
gsl_test(*p != data[i], "bst_int %s[n=%zu,order=%d] copy i=%d [%d,%d]",
gsl_bst_name(w), n, order, i, *p, data[i]);
/* re-insert item */
p = gsl_bst_trav_insert(&data_delete[i], &z, w);
check_traverser(n, order, &x, data[i], "post-deletion", w);
check_traverser(n, order, &y, data[i], "copied", w);
check_traverser(n, order, &z, data_delete[i], "insertion", w);
#if 0
/* delete again */
gsl_bst_remove(&data[i], w);
#endif
}
/* emmpty tree */
gsl_bst_empty(w);
nodes = gsl_bst_nodes(w);
gsl_test(nodes != 0, "bst_int %s[n=%zu,order=%d] empty count = %zu",
gsl_bst_name(w), n, order, nodes);
gsl_bst_free(w);
free(data);
free(data_delete);
free(sorted_data);
}
static void
test_bst(const gsl_bst_type * T, gsl_rng * r)
{
enum array_order order;
for (order = 0; order < ORD_END_NODUP; ++order)
{
test_bst_int(50, T, order, r);
test_bst_int(100, T, order, r);
test_bst_int(500, T, order, r);
}
}
int
main(void)
{
gsl_rng * r = gsl_rng_alloc(gsl_rng_default);
test_bst(gsl_bst_avl, r);
test_bst(gsl_bst_rb, r);
gsl_rng_free(r);
exit (gsl_test_summary());
}
|
PROGRAM STEPFOR
INTEGER I
C This will print all even numbers from -10 to +10, inclusive.
DO 10 I = -10, 10, 2
WRITE (*,*) I
10 CONTINUE
STOP
END
|
With the Samsung Galaxy S9+, you get all the great features of the S9, including the shiny glass-and-metal design, the immersive Infinity Display, the stereo speakers, and the dual-aperture camera with 960fps slow-motion and AR Emoji. On top of this, you also get a larger screen, a bigger battery, and a second camera at the back enabling 2X lossless zoom and bokeh effect for your portraits. All of these goodies are backed up by a top-notch specs sheet making this one of the most powerful Android phones around.
3GLEB delivers the Samsung Galaxy S8 Plus to any location in Lebanon via Aramex.
|
module Protocol.Hex
import Data.Bits
import Data.List
-- Those three imports are for compatibility and should be removed after release of 0.6.0
import Data.DPair
import Data.Nat
import Data.Fin
%default total
hexDigit : Bits64 -> Char
hexDigit 0 = '0'
hexDigit 1 = '1'
hexDigit 2 = '2'
hexDigit 3 = '3'
hexDigit 4 = '4'
hexDigit 5 = '5'
hexDigit 6 = '6'
hexDigit 7 = '7'
hexDigit 8 = '8'
hexDigit 9 = '9'
hexDigit 10 = 'a'
hexDigit 11 = 'b'
hexDigit 12 = 'c'
hexDigit 13 = 'd'
hexDigit 14 = 'e'
hexDigit 15 = 'f'
hexDigit _ = 'X' -- TMP HACK: Ideally we'd have a bounds proof, generated below
-- `i4` is to be replaced with a `4` literal after release of 0.6.0
namespace Old
export
i4 : Subset Nat (`LT` 64)
i4 = Element (the Nat 4) %search
namespace New
export
i4 : Fin 64
i4 = 4
||| Convert a Bits64 value into a list of (lower case) hexadecimal characters
export
asHex : Bits64 -> String
asHex 0 = "0"
asHex n = pack $ asHex' n []
where
asHex' : Bits64 -> List Char -> List Char
asHex' 0 hex = hex
asHex' n hex = asHex' (assert_smaller n (n `shiftR` i4)) (hexDigit (n .&. 0xf) :: hex)
export
leftPad : Char -> Nat -> String -> String
leftPad paddingChar padToLength str =
if length str < padToLength
then pack (List.replicate (minus padToLength (length str)) paddingChar) ++ str
else str
export
fromHexDigit : Char -> Maybe Int
fromHexDigit '0' = Just 0
fromHexDigit '1' = Just 1
fromHexDigit '2' = Just 2
fromHexDigit '3' = Just 3
fromHexDigit '4' = Just 4
fromHexDigit '5' = Just 5
fromHexDigit '6' = Just 6
fromHexDigit '7' = Just 7
fromHexDigit '8' = Just 8
fromHexDigit '9' = Just 9
fromHexDigit 'a' = Just 10
fromHexDigit 'b' = Just 11
fromHexDigit 'c' = Just 12
fromHexDigit 'd' = Just 13
fromHexDigit 'e' = Just 14
fromHexDigit 'f' = Just 15
fromHexDigit _ = Nothing
export
fromHexChars : List Char -> Maybe Integer
fromHexChars = fromHexChars' 1
where
fromHexChars' : Integer -> List Char -> Maybe Integer
fromHexChars' _ [] = Just 0
fromHexChars' m (d :: ds)
= do digit <- fromHexDigit (toLower d)
digits <- fromHexChars' (m*16) ds
pure $ cast digit * m + digits
export
fromHex : String -> Maybe Integer
fromHex = fromHexChars . unpack
|
If $f$ is convex on the interval $[x,y]$, then for all $t \in [0,1]$, we have $f((1-t)x + ty) \leq (1-t)f(x) + tf(y)$.
|
<a href="https://colab.research.google.com/github/jhmartel/fp/blob/master/_notebooks/2022-02-22-Positronium_Part1.ipynb" target="_parent"></a>
# Positronium Part I.
> "Weber potential, positronium, two-body problem. "
- toc: false
- branch: master
- badges: false
- comments: true
- author: JHM
- categories: [weber, positronium, two-body]
Today we begin the study of Weber's potential in the isolated two-body system consisting of an electron and positron pair $e^-$ and $e^+$.
We assume the particles $e^\pm$ have equal mass $m=m_{e^{\pm}}$. The reduced mass is concentrated at the centre-of-mass $\mu=m/2$.
Weber's force is attractive between the pair $e^\pm$ at all distances.
The particles $e^\pm$ do not indefinitely spiral inwards. Simulations indicate that the radial distance between $e^\pm$ stays strictly bounded between two upper and lower limits $$0 < r_{lower} \leq r \leq r_{upper} < + \infty .$$ This is rigorously proved in [Weber-Clemente, 1990](https://www.ifi.unicamp.br/~assis/Int-J-Theor-Phys-V30-p537-545(1991).pdf).
If the electron is indivisible particle, then the above two-body problem models a pair $e^-$ and $e^+$ of isolated electron and positron.
*But do the particles $e^\pm$ ever 'collide' and annihalate?*
In the standard physics textbooks, it seems well known that annihalation between $e^\pm$ occurs and two gamma rays are ejected in opposite directions when $e^\pm$. conserving momentum, etc., and converting *all* their mass into energy. Thus it's determined that two gamma rays of energy $0.511 keV$ are released, where Einstein's formula $E=m_ec^2$ is applied, where $m_e$ is the reduced mass. [ref] The annihalation of $e^\pm$ is apparently an experimental test of the validity of Einstein's "mass-energy" hypothesis.
But what does Weber's potential say about the annihalation of $e^+$ and $e^-$ ?
If we know the centre of mass has zero net force, then we can replace the positions $r_1$, $r_2$ of the particles by their relative distance $r_{12}$ from the centre of mass. This yields $$r_1=R +\frac{m_2}{m_1+m_2} r_{12}$$ and $$r_2=R -\frac{m_1}{m_1+m_2} r_{12}.$$
Applying Newton's Second Law that $F_{21}=-F_{12}$ yields the following equation for $r_{12}''$: $$ \mu . r_{12}'' = F_{21},$$ where $\mu$ is the reduced mass of the system, namely $\mu=\frac{m_1 m_2}{m_1+m_2}=0.5$.
In the following equations we use numpy.odeint to solve Weber equations of motion of the relative distance $r_{12}$. Therefore we have reduced the two-body problem to a one-body problem. This is a standard reduction.
Given the solution for $r_{12}$, how do we reconstruct the paths/positions of the particles $r_1$, $r_2$ ? Answer: via the relation $r_1=R+\frac{m_2}{m_1+m_2} r_{12}$ and $r_2=R-\frac{m_1}{m_1 + m_2}r_{12}$.
Now the relative distance $r_{12}$ is a type of radial distance, and if $r, \omega$ is spherical coordinates, then we have $$r'^2=|v|^2= x'^2+y'^2+z'^2=(r')^2+r^2 (\theta')^2. $$ The above formula is the usual $|v|^2=v_r^2+v_t^2$, and the tangent velocity $v_t$ satisfies $v_t=r\theta'$, where $\theta'$ is the angular velocity.
The conservation of angular momentum says that the angular moment $L=\mu r\times v$, where $v$ is the linear velocity of $r$, is constant along the motion. Moreover one has $$|L|=\mu r^2 \theta'.$$ Thus we find the formula $$\theta'=\frac{|L|}{\mu r^2}.$$ This implies $$T=\frac{\mu}{2}v^2= \frac{\mu}{2}[(r')^2+\frac{|r\times v|^2}{r^{2}}]$$ represents the kinetic energy of the system.
The conservation of energy says $T+U$ is *constant* along trajectories.
```python
#collapse
# Here we define basic functions.
def cross(v1, v2):
x1, y1, z1 = v1
x2, y2, z2 = v2
return [y1*z2 - z1*y2, -(x1*z2 - z1*x2), x1*y2 - y1*x2 ]
def rho(rel_position):
x,y,z = rel_position
return (x*x+y*y+z*z)**0.5
def dot(vector1, vector2):
x1, y1, z1 = vector1
x2, y2, z2 = vector2
return x1*x2+y1*y2+z1*z2
def rdot(position, vector):
return dot(position, vector)/rho(position)
def norm(rel_velocity):
return rho(rel_velocity)
mu=0.5 ## reduced mass of the system. We assume m1 and m2 are equal, hence mu=1/2.
c=1.0 ## speed of light constant in Weber's potential
# Define the angular momentum
def AngMom(rel_position, rel_velocity):
return cross(rel_position, rel_velocity)
def L(rel_position, rel_velocity):
return norm(cross(rel_position, rel_velocity))
# Linear Kinetic Energy
def T(rel_position, rel_velocity):
vt = norm(cross(rel_position, rel_velocity))
# next formula decomposes v^2=(vr)^2+(vt)^2, where vt=r*θ'=|L|/(mu*r)
return (mu/2)*(rdot(rel_position, rel_velocity)**2) + (mu/2)*(rho(rel_position)**-2)*(vt**2)
## Weber Potential Energy
## Negative sign given -1=q1*q2
def U(rel_position, rel_velocity):
x,y,z = rel_position
vx,vy,vz = rel_velocity
rdot=dot(rel_position, rel_velocity)/rho(rel_position)
return -(1/rho(rel_position))*(1-(rdot*rdot)/2)
```
```python
## import the basic packages
import numpy as np
from scipy.integrate import odeint, solve_ivp
import matplotlib.pyplot as plt
from mpl_toolkits.mplot3d import Axes3D
## Integrating two-body isolated system of oppositely charged particles
## i.e. positron+electron pair.
## The product of the charges q1*q2 is factor in Weber's force law, and appears twice
## in the formula of Newton's F=ma.
def weber(t, rel_state):
x, y, z, vx, vy, vz = rel_state
r=(x*x + y*y + z*z)**0.5
rdot=(x*vx+y*vy+z*vz)/r
A=(-1)*r**-2 ## minus sign from q1*q2
B=1-(rdot*rdot)/2
C=(mu+((c*c*r)**-1))**-1 ## +plus instead of -minus.
dxdt = vx
dydt = vy
dzdt = vz
dvxdt = (x/r)*A*B*C
dvydt = (y/r)*A*B*C
dvzdt = (z/r)*A*B*C
return [dxdt, dydt, dzdt, dvxdt, dvydt, dvzdt]
t_span = (0.0, 100.0)
t = np.arange(0.0, 100.0, 0.1)
y1=[2.0,0,0,] # initial relative position
v1=[-0.4, 0.4, 0] # initial relative velocity
result = odeint(weber, y1+v1, t, tfirst=True) #here odeint solves the weber equations of motion relative y1+v1 for t.
Energy=T(y1,v1) + U(y1,v1)
print('The initial total energy T+U is equal to:', Energy)
print('The initial angular momentum is equal to', norm(AngMom(y1,v1)))
fig = plt.figure()
ax = fig.add_subplot(1, 2, 1, projection='3d')
ax.plot(result[:, 0],
result[:, 1],
result[:, 2])
ax.set_title("position")
ax = fig.add_subplot(1, 2, 2, projection='3d')
ax.plot(result[:, 3],
result[:, 4],
result[:, 5])
ax.set_title("velocity")
```
What does the above plot demonstrate?
It reveals a precession motion around the centre of mass. This is not predicted by Coulomb's force, which bounds the trajectories to elliptical orbits like Newton's Law of Gravitation.
The force is central, therefore we have conservation of angular momentum, and implies the system is constrained to a plane, namely orthogonal to the angular moment of the system.
Moreover the system satisfies a conservation of linear momentum, namely the sum $T+U$ is constant.
* Problem: Verify Assis-Clemente's 1990 formula for the lower and upper limits of the relative distance along the orbits.
```python
#collapse-output
import matplotlib.pyplot
import pylab
r_list=[]
for j in range(1000):
sample_position=[result[j,0], result[j,1], result[j,2]]
sample_velocity=[result[j,3], result[j,4], result[j,5]]
r_list.append(
(
int(j), rho(sample_position) )
)
prelistr = list(zip(*r_list))
pylab.scatter(list(prelistr[0]),list(prelistr[1]))
pylab.xlabel('time')
pylab.ylabel('rho')
pylab.title('Solutions have Upper and Lower Limits')
pylab.show()
#TU_list=[]
#for j in range(1000):
# sample_position=[result[j,0], result[j,1], result[j,2]]
# sample_velocity=[result[j,3], result[j,4], result[j,5]]
# TU_list.append(
# (rho(sample_position),T(sample_position,sample_velocity)+U(sample_position, sample_velocity)))
#prelist1 = list(zip(*TU_list))
#pylab.scatter(list(prelist1[0]),list(prelist1[1]))
#pylab.xlabel('distance r')
#pylab.ylabel('T+U')
#pylab.title('')
#pylab.show()
# The plot below demonstrates the conservation of angular momentum.
# Note that rdot is directly equal to the sample_velocity. I.e. there is
# no need to define rdot=v.hatr/r. This was error.
#A_list=[]
#for j in range(1000):
# sample_position=[result[j,0], result[j,1], result[j,2]]
# sample_velocity=[result[j,3], result[j,4], result[j,5]]
# A_list.append(
# (rho(sample_position), norm(cross(sample_position, sample_velocity)) ) )
#prelist2 = list(zip(*A_list))
#pylab.scatter(list(prelist2[0]),list(prelist2[1]))
#pylab.xlabel('rho')
#pylab.ylabel('Angular Momentum')
#pylab.title('Conservation of Angular Momentum')
#pylab.show()
#rho_list=[]
#for j in range(180):
# rho_list.append(
# (int(j), rho([result[j,0], result[j,1], result[j,2]]),
# )
# )
```
```python
#collapse
from sympy import *
t=symbols('t')
m=symbols('m')
c=symbols('c')
r=Function('r')(t)
P=Function('P')(r,t)
F=Function('F')(r,t)
U=-(r**-1)*(1-((r.diff(t))**2)*(2*c*c)**-1)
F=(-1)*(U.diff(t))*((r.diff(t))**-1)
pprint(simplify(U))
print()
pprint(simplify(F)) ## symbolic computation of the Force law.
```
2
⎛d ⎞
⎜──(r(t))⎟
2 ⎝dt ⎠
- c + ───────────
2
──────────────────
2
c ⋅r(t)
2
⎛d ⎞
2 ⎜──(r(t))⎟
2 d ⎝dt ⎠
- c - r(t)⋅───(r(t)) + ───────────
2 2
dt
───────────────────────────────────
2 2
c ⋅r (t)
|
c anirec - program to calculate receiver-function response of
c a stack of anisotropic layers to a plane wave incident from below
c CAN BE USED IN GRIDSEARCH OVER CC AND BAZ - SEE COMMENTED LINES FOR LOOP
c cannibalized from aniprop.f 11/18/95
c xf77 -o anirec_osc -fast -native -O5 anirec.f /data/d4/park/Plotxy/plotlib /data/d4/park/Ritz/eislib /data/d4/park/Ritz/jlib
c f77 -o anirec -fast -native -O5 anirec.f /data/d4/park/Ritz/eislib /data/d4/park/Ritz/jlib
c for hexagonally symmetric media
c reads fast axis orientation, constants A,B,C,D,E from file animodel
c calculate quadratic eigenvalue problem based on the Christoffel matrix
c see appendix of P. Shearer's thesis
c
c read model, phase velocity of incident wave, P, SV, or SH
c
c calc the eigenvector decomps for the layers
c loop over frequency, calc reflection/transmission matrices
c calc 3-comp transfer fct response at surface
c find distortion of reference wavelet
c modified to run with NA search by HY 2005
c outputs are in N-E-Z, R-T-Z or SV-SH-P coordinate systems
c note jeff's T is 180 off of that of sac.
c minor fix by HY 2006. add out_rot in the input
c out_rot = 0: output traces in N-E-Z
c out_rot = 1: R-T-Z
c out_rot = 2: P-SV-SH. But not computed here. just rotate to
c rtz and the is rotated by fs_traces_anirec elsewehere.
c this version has t-comp flipped to keep up the sac convention.
subroutine anirec(ntr, nsamp,dt,synth_cart,synth_cart2,
c synth_cart3,thick,rho1,alpha, beta, pct_a,
c pct_b,trend,plunge,nlay,
c per,shiftp,shifts,baz0,slow,out_rot,ipulse)
implicit real*8 (a-h,o-z)
implicit integer*4 (i-n)
include 'params.h'
c integer maxlay, maxtr, maxsamp
c parameter (maxlay =15, maxtr=200,maxsamp=2000)
c integer ntr, nsamp, nlay
ccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccc
c raysum parameters passed in
real synth_cart(3,maxsamp,maxtr), synth_cart2(3,maxsamp,maxtr)
real synth_cart3(3,maxsamp,maxtr)
real thick(maxlay),rho1(maxlay),alpha(maxlay),beta(maxlay)
real pct_a(maxlay),pct_b(maxlay),trend(maxlay),plunge(maxlay)
real strike(maxlay),dip(maxlay),dt
real baz0(maxtr), slow(maxtr), width, shiftp,shifts,per
integer out_rot,pulse
ccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccc
character*80 name,title,ylabel(3),name2
c character *4 chead(158)
c character *3 wf(3)
complex*16 pp,u0,ee,z1,z0,xnu,e1,e2,zla,xl
complex*16 rt,tt,rt0,trc,pfac,u,resp,zz
real*4 cc4,zz4,frqq,amn,amx,baseline,dat4,a,dat44
common/data/a(40000)
c common/header/ahead(158)
common/stfff/w(3,101),t(3,3),ttl(3,3),s(3,3),stl(3,3),
x r(3,3),x(3),y(3)
common/model/z(100),dz(100),rho(101),vp(101),vs(101),vp2(101),
x vp4(101),vs2(101),vss(101)
common/model2/xmu(101),xla(101),xmu2(101),xla2(101),xla4(101)
common/propag/xnu(6,101),xl(6,100),pfac(6,3),u(3,6)
common/mstff/qq(6,6),wr(6),wi(6),zr(6,6),zi(6,6),iv(6),fv(6)
common/pstff/pp(3),u0(3),ee(6,6,101),e1(6,6),e2(6,6),zla(6)
common/rstff/rt(3,3,101),tt(3,3,101),rt0(3,3),trc(3,3)
common/nstff/cc4(8200),zz4(8200),dat4(8200,3,3),ccc(8200)
common/nstfff/dat44(8200,3,3)
common/disper/resp(3,3,2050),frqq(2050)
common/disper2/roota(101),rootb(101),jtrval(101),kroots(2050)
common/evanes/ievan(10000)
c dimension iah(158)
c equivalence (iah,ahead),(chead,ahead)
data eps/1.d-6/,tol/1.d-3/
c we reduce the condition numbers of matrices by
c normalizing physical quantities to make them dimensionless
c Ill use the normal-mode normalizations
c which are a little peculiar for the crust, but what the hey!
rbar=5.515d3
ren=1.075190645d-3
radi=6.371d6
vbar=ren*radi
con=rbar*vbar**2
z1=dcmplx(1.d0,0.d0)
z0=dcmplx(0.d0,0.d0)
! write(*,*)ntr, nsamp,dt,nlay
! do i=1,nlay
! write(*,*)'layer',i,'-----------------------'
! write(*,*)thick(i),rho1(i),alpha(i), beta(i)
! write(*,*)pct_a(i),pct_b(i),trend(i),plunge(i)
! enddo
! write(*,*)baz0(1),slow(1)
! write(*,*)shift,out_rot,ipulse,per
! write(*,*)'------------------------------------------'
c Notes on the angle conventions for w-hat, the axis of symmetry:
c
c In the anisotropic reflectivity code, subroutine matget *assumes* a
c coordinate system in which z is down (anti-vertical), x is the radial
c direction, and y is anti-transverse. Therefore, the position angles
c theta,phi for w-hat are tilt relative to down, and azimuth defined as a
c rotation from x towards y. This rotation is CCW if viewed from below,
c and CW if viewed from above. Since w-hat and -(w-hat) define the same
c axis of symmetry, the position angles *also* can be defined as
c theta=(tilt from vertical) and phi=(rotation from anti-x (anti-radial)
c towards anti-y (transverse)). Viewed from above, this phi rotation is
c CW, and defines the strike of w-hat relative to the arrival azimuth of
c the wave.
c
c In order to compute seismograms for a variety of back-azimuths, we
c assume that the default is a wave approaching from the north, so that
c radial is south and transverse is east. Due to this orientation, the
c synthetic code interprets the layered model as having w-hat position
c angles defined as theta=(tilt from vertical) phi=(strike CW from N).
c For an event at back-azimuth psi (CW from N), routine matget rotates w-hat
c from geographic coordinates to ray-based coordinates before computing
c reflectivity matrices. If a wave arrives at back-azimuth psi, the strike
c of the axis of symmetry w-hat relative to its arrival azimuth is
c phi'=phi-psi. The code performs this rotation with this code in
c subroutine matget, for w-hat azimuth "az":
c
c caz=dcosd(az)
c saz=-dsind(az) ! sin(-az)
c do n=1,nlp
c ww(3)=w(3,n)
c ww(1)=w(1,n)*caz-w(2,n)*saz
c ww(2)=w(1,n)*saz+w(2,n)*caz
c ...
c
c In this manner, the axes of symmetry of the model, saved in array w(.,.),
c are never modified.
c in the driver code, "az" is the variable "baz" for back azimuth
c
c default baz = 0
c baz=0.
c set up for 10 sps data
c dt=0.1
c nfrq must be .le.npad/2
npad=2048!2048
nfrq=512!1024
dur=npad*dt
df=1./dur
frqmax=nfrq*df
if (dur < nsamp * dt + shiftp) then
write(*,*) 'HAHAHA IN ANIREC.F'
write(*,*) 'dur =',dur , ' < ', 'nsamp * dt + shiftp=',
&nsamp * dt + shiftp
endif
if (dur < nsamp * dt + shifts) then
write(*,*) 'HAHAHA IN ANIREC.F'
write(*,*) 'dur =',dur , ' < ', 'nsamp * dt + shifts=',
&nsamp * dt + shifts
endif
c print *,'dt,df,duration of record,max frequency:',dt,df,dur,frqmax
c print *,'NOTE: cosine^2 taper will be applied up to fmax'
c print *,'input model? (space-return: animodel)'
c read(5,102) name
c 102 format(a)
c if(name(1:1).eq.' ') then
c open(7,file='animodel',form='formatted')
c else
c open(7,file=name,form='formatted')
c endif
c read(7,102) title
c print *,title
c read(7,*) nl
nl = nlay-1
c read in theta,phi in degrees - polar coords of fast axis
nlp=nl+1
nlm=nl-1
z(1) = thick(1)
do i=2,nlp
z(i) = z(i-1) + thick(i)
enddo
c note the theta phi passed in are in radian
c also dip needs a 90 plus
do i=1,nlp
phi = trend(i) !baz
theta = pi/2.+plunge(i) !tilt
w(1,i)=dble(sin(theta )*cos(phi))
w(2,i)=dble(sin(theta)*sin(phi))
w(3,i)=dble(cos(theta))
c print *,(w(j,i),j=1,3)
c read depth to ith interface, vp (m/sec), pk-to-pk cos(2th) relative P pert
c pk-to-pk cos(4th) relative P pert, v_s, pk-to-pk cos(2th) relative S pert
c density (kg/m**3)
c read(7,*) z(i),vp(i),vp2(i),vp4(i),vs(i),vs2(i),rho(i)
vp(i) = alpha(i)
vp2(i) = pct_a(i) /100.
vp4(i) = 0.
vs(i) = beta(i)
vs2(i) = pct_b(i) /100.
rho(i) = rho1(i)
c recall that we interpret fractional values of b,c,e
c as peak-to-peak relative velocity perts.
c therefore, e=0.02 is 2% pert to mu from slowest to fastest
xmu(i)=rho(i)*vs(i)**2/con
xmu2(i)=vs2(i)*xmu(i)
xla(i)=rho(i)*vp(i)**2/con
xla2(i)=vp2(i)*xla(i)
xla4(i)=vp4(i)*xla(i)
vs(i)=vs(i)/vbar
vp(i)=vp(i)/vbar
rho(i)=rho(i)/rbar
z(i)=z(i)/radi
end do
do i=2,nl
dz(i)=z(i)-z(i-1)
end do
dz(1)=z(1)
c print the organ-pipe mode count for 1Hz
c the lowest layer (nl+1) is taken as evanescent region.
sdelay=0.
pdelay=0.
do i=1,nl
sdelay=sdelay+(dz(i)/vs(i))/ren
pdelay=pdelay+(dz(i)/vp(i))/ren
end do
c print *, sdelay,pdelay
c search for cmin, cmax is halfspace velocity
cmin=vs(1)
vss(1)=vs(1)
do i=2,nlp
if(cmin.gt.vs(i)) cmin=vs(i)
vss(i)=vs(i)
end do
900 csmin=vs(nlp)*vbar/1000.
cpmin=vp(nlp)*vbar/1000.
c source; passed in from the main body
c ipulse=1
c per=width
c print *, 'ipulse is ', ipulse, ' per is ', per
t1=0.
t2=nsamp * dt
npts=t2/dt+1
nst=t1/dt+1
c iah(80) = npts !number of points
c source pulse
npul=per/dt
do i=1,npad
zz4(i)=0.d0
end do
fac=2.d0*pi/per
if(ipulse.eq.1) then
do i=1,npul
time=i*dt
zz4(i)=(dsin(fac*time/2.d0))**2
end do
elseif(ipulse.eq.2) then
do i=1,npul
time=i*dt
zz4(i)=dsin(fac*time)*(dsin(fac*time/2.d0))**2
end do
elseif(ipulse.eq.3) then
do i=1,160
xx=0.05*(i-80)
zz4(i)=xx*cos(2.*pi*xx/sqrt(1.+xx**2))*exp(-0.4*xx**2)
end do
elseif(ipulse.eq.5) then
do i=1,npul*2
xx=0.01*(i-npul)
zz4(i)=xx*cos(2.*pi*xx/sqrt(1.+xx**2))*exp(-0.4*xx**2)
end do
else
do i=1,160
xx=0.05*(i-80)
zz4(i)=cos(5.*pi*xx/sqrt(1.+4*xx**2))*exp(-4.0*xx**2)
end do
endif
c print *,(zz4(i),i=1,npul*2)
c pause
call refft(zz4,npad,1,1)
c zero the DC and Nyquist
c ick switches the sign of y and z components to transverse & vertical
zz4(1)=0.
zz4(2)=0.
ccadd loop here for slowness/baz
do intra = 1, ntr
cc = 1./(slow(intra) * 1000.)
baz = baz0(intra)
cs=cos(baz )
sn=sin(baz )
if(cc.le.0.d0) go to 950
c non-dimensionalize cc
cc=cc*1000./vbar
c calc the eigenvector decomps for the layers
c need to identify upgoing P,SV,SH in the halfspace
call matget(nl,cc,baz * 180./pi)
c loop over frequency, calc reflection/transmission matrices
c calc 3-comp transfer fct response at surface
do jf=1,nfrq
om=2.d0*pi*jf*df/ren
frqq(jf)=jf*df
call respget(nl,om,cc,resp(1,1,jf))
end do
cccc=cc*vbar/1000.
c lets run a pulse thru these functions
c s(t)=cos(2pi*t/T)*sin^2(2pi*t/2T) for two oscillations of the cos
c OR
c s(t)=sin(2pi*t/2T)**2 for 1/2 oscillation of the sin
c OR bbpulse
c s(t)=t*cos(2.*pi*t/sqrt(1.+t**2))*exp(-0.4*t**2)
c let T=1 and 2 sec
c version for user-supplied cc and baz
c print *,
c x 'wavelet print: 1-onesidedpulse 2-oscillation 3-bbpulse 4-hfbb'
c read(5,*) ipulse
c version for gridsearch cc and baz
c ipulse=3
c
c another version for gridsearch
c ipulse=1
c end gridsearch lines
c if(ipulse.le.2) then
c print *,'enter wavelet period in seconds'
c read(5,*) per
c endif
c t1=0.
c dur=npad*dt
c t1=0.
c t2=dur
c version for usersupplied cc and baz
c 960 print *,'tstart, duration to plot? currently:',t1,t2
c read(5,*) t1,t2
c if(t1.lt.0.0.or.t2.le.0.) go to 900
c version for gridsearch cc and baz
c note S wave is aligned to start at shift sec.
c t1=0.
c t2=nsamp * dt
c end versions
c npts=t2/dt+1
c nst=t1/dt+1
c we start at dt, with duration 2T -- ONE CYCLE
do iprint=1,3
ick=1
baseline=0.
do k=1,3
cc4(1)=0.
cc4(2)=0.
if(k.gt.1) ick=-1
do jf=1,nfrq
zz=ick*dcmplx(dble(zz4(2*jf+1)),dble(zz4(2*jf+2)))
zz=zz*resp(k,iprint,jf)
cc4(2*jf+1)=dreal(zz)
cc4(2*jf+2)=dimag(zz)
end do
do jf=2*nfrq+3,npad
cc4(jf)=0.
end do
call refft(cc4,npad,-1,-1)
amx=cc4(1)
amn=cc4(1)
do i=1,npts
amx=amax1(amx,cc4(i+nst))
amn=amin1(amn,cc4(i+nst))
end do
baseline=baseline-amn
do i=1,npad
dat4(i,k,iprint)=cc4(i)+baseline
dat44(i,k,iprint)=cc4(i)
end do
baseline=baseline+amx
end do
end do
cc4(1)=nst*dt
cc4(2)=dt
c wf(1) = '.P'
c wf(2) = '.SV'
c wf(3) = '.SH'
cwe need SV and SH only
do ity=1,3
c use organ-pipe count to correct for traveltime of Swave thru stack
c tdelay=sdelay-shift.
c use organ-pipe count to correct for traveltime of Pwave thru stack
c tdelay=pdelay-4.
! write(*,*)sdelay,pdelay,shift
if (ity .ne.1) then
tdelay = sdelay-shifts!shift!sdelay!-t2/2.
!write(*,*)'sdel',sdelay
!if (tdelay.lt.0) stop "shift too large"
c tdelay =sdelay-per
else
tdelay= pdelay-shiftp!shift!pdelay!-t2/2.
!write(*,*)'pdel',pdelay
!if (tdelay.lt.0) stop "shift too large"
c tdelay =sdelay-per
endif
c if (ity .eq. 1) tdelay=tdelay*0.5
nst0=tdelay/dt
if(nst0.le.0) nst0=1
c print *, t2,tdelay
c switching the sign on T HY jan -11 for rtz
if (ity .eq. 1) then
do i=1,npts
synth_cart(3,i,intra)=dat44(nst0+i-nst,3,ity)
if (out_rot .ne. 0) then
synth_cart(1,i,intra)=dat44(nst0+i-nst,1,ity)
synth_cart(2,i,intra)=-dat44(nst0+i-nst,2,ity)
else
synth_cart(1,i,intra)=-sn*dat44(nst0+i,2,ity)
x -cs*dat44(nst0+i,1,ity)
synth_cart(2,i,intra)= cs*dat44(nst0+i,2,ity)
x -sn*dat44(nst0+i,1,ity)
endif
end do
end if
if (ity .eq. 2) then
amx = dat44(1,3,2)
amn = dat44(1,3,2)
do i=1,npts!per/dt*2.5!for the main pulse only
synth_cart2(3,i,intra)=dat44(nst0+i-nst,3,ity)
if (out_rot .ne. 0) then
synth_cart2(1,i,intra)=dat44(nst0+i-nst,1,ity)
synth_cart2(2,i,intra)=-dat44(nst0+i-nst,2,ity)
else
synth_cart2(1,i,intra)=-sn*dat44(nst0+i,2,ity)
x -cs*dat44(nst0+i,1,ity)
synth_cart2(2,i,intra)= cs*dat44(nst0+i,2,ity)
x -sn*dat44(nst0+i,1,ity)
endif
end do
end if
if (ity .eq. 3) then
do i=1,npts !per/dt*2.5!for the main pulse only
synth_cart3(3,i,intra)=dat44(nst0+i-nst,3,ity)
if (out_rot .ne. 0) then
synth_cart3(1,i,intra)=dat44(nst0+i-nst,1,ity)
synth_cart3(2,i,intra)=-dat44(nst0+i-nst,2,ity)
else
synth_cart3(1,i,intra)=-sn*dat44(nst0+i,2,ity)
x -cs*dat44(nst0+i,1,ity)
synth_cart3(2,i,intra)= cs*dat44(nst0+i,2,ity)
x -sn*dat44(nst0+i,1,ity)
endif
end do
cc if ((abs(amx) .lt. abs(amn)) ) then
cc do i=1,npts
cc synth_cart3(3,i,intra)=-synth_cart3(3,i,intra)
cc synth_cart3(2,i,intra)=-synth_cart3(2,i,intra)
cc synth_cart3(1,i,intra)=-synth_cart3(1,i,intra)
cc end do
cc end if
end if
end do
103 format(a,i2,a,i3,a)
c version for user-supplied cc and baz
end do !ntr
c end version for user-supplied cc and baz
c version for grid search over cc and baz
c end do
c end do
c end version for grid search over cc and baz
950 continue
101 format(80a)
c stop
end
subroutine matget(nl,cc,az)
c SPECIAL VERSION: az rotates the w-hat vector by -az degrees
c returns stress-displacement vectors for a stack of anisotropic layers
c P waves may be evanescent, but S waves are oscillatory in the stack
c the weirdness seen in the surface wave code should not appear in
c a receiver function code
c however, the iev parameter is retained to avoid leaving timebombs
implicit real*8 (a-h,o-z)
implicit integer*4 (i-n)
complex*16 pp,u0,ee,pw,uw,pu,z1,z0,xnu,eye,e1,e2,zla,rtm
complex*16 pfac,u,xl
common/stfff/w(3,101),t(3,3),ttl(3,3),s(3,3),stl(3,3),
x r(3,3),x(3),y(3)
common/model/z(100),dz(100),rho(101),vp(101),vs(101),vp2(101),
x vp4(101),vs2(101),vss(101)
common/model2/xmu(101),xla(101),xmu2(101),xla2(101),xla4(101)
common/propag/xnu(6,101),xl(6,100),pfac(6,3),u(3,6)
common/rrt/rtm(6,6,100)
common/mstff/qq(6,6),wr(6),wi(6),zr(6,6),zi(6,6),iv(6),fv(6)
common/pstff/pp(3),u0(3),ee(6,6,101),e1(6,6),e2(6,6),zla(6)
common/qstff/qi(6,6),xr(6),xi(6),yr(6),yi(6),ips(3)
dimension ww(3)
data pi/3.14159265358979d0/,eps/1.d-6/,tol/1.d-7/
c set iev=1 ** should be superfluous, save for now
c toggle to iev=0 if there is a purely propagating wave in the top layer n=1
iev=1
z1=dcmplx(1.d0,0.d0)
z0=dcmplx(0.d0,0.d0)
eye=dcmplx(0.d0,1.d0)
rbar=5.515d3
ren=1.075190645d-3
radi=6.371d6
vbar=radi*ren
con=rbar*radi*radi*ren*ren
nlp=nl+1
nlm=nl-1
c first calculate vertical wavenumbers and propagating waves for each layer
c requires an eigenvector problem be solved
c in general, the evanescent vertical wavenumbers have nonzero real parts
c complex exponential fct is used to avoid endless branching
c horizontal slowness p_x
px=1.d0/cc
caz=dble(cos(az* pi / 180.0))
saz=-dble(sin(az* pi / 180.0)) ! sin(-az)
do n=1,nlp
ww(3)=w(3,n)
ww(1)=w(1,n)*caz-w(2,n)*saz
ww(2)=w(1,n)*saz+w(2,n)*caz
a=xla(n)
b=xla2(n)
c=xla4(n)
d=xmu(n)
e=xmu2(n)
c print *,'a,b,c,d,e',a,b,c,d,e
fact=8.d0*ww(1)*ww(1)*c+2.d0*e
facs=16.d0*ww(1)*ww(3)*c
facr=8.d0*ww(3)*ww(3)*c+2.d0*e
c print *,'a,b,c,d,e',a,b,c,d,e
c print *,'w(.,n),fact,facs,facr',(ww(l),l=1,3),fact,facs,facr
do i=1,3
c first the what-0-what tensor
do j=1,3
t(j,i)=fact*ww(j)*ww(i)
s(j,i)=facs*ww(j)*ww(i)
r(j,i)=facr*ww(j)*ww(i)
end do
c next the identity tensor - correct an error on 7/6/95
t(i,i)=t(i,i)+d+e*(2.d0*ww(1)*ww(1)-1.d0)
s(i,i)=s(i,i)+4.d0*e*ww(1)*ww(3)
r(i,i)=r(i,i)+d+e*(2.d0*ww(3)*ww(3)-1.d0)
end do
c print 101,(ww(i),i=1,3)
c print 101,fact,facs,facr
c print *,'t,s,r'
c print 101,((t(i,j),j=1,3),i=1,3)
c print 101,((s(i,j),j=1,3),i=1,3)
c print 101,((r(i,j),j=1,3),i=1,3)
fac=b-4.d0*c-2.d0*e
c next the what-0-xhat and what-0-zhat tensors
do i=1,3
t(1,i)=t(1,i)+fac*ww(1)*ww(i)
t(i,1)=t(i,1)+fac*ww(1)*ww(i)
s(1,i)=s(1,i)+fac*ww(3)*ww(i)
s(i,1)=s(i,1)+fac*ww(3)*ww(i)
s(3,i)=s(3,i)+fac*ww(1)*ww(i)
s(i,3)=s(i,3)+fac*ww(1)*ww(i)
r(3,i)=r(3,i)+fac*ww(3)*ww(i)
r(i,3)=r(i,3)+fac*ww(3)*ww(i)
end do
fac=a-b+c-d+e
c finally the xhat-0-xhat, zhat-0-zhat, xhat-0-zhat, zhat-0-xhat tensors
t(1,1)=t(1,1)+fac
s(3,1)=s(3,1)+fac
s(1,3)=s(1,3)+fac
r(3,3)=r(3,3)+fac
c mult by horizontal slowness and calc the modified T-matrix
do i=1,3
do j=1,3
t(j,i)=t(j,i)*px*px
s(j,i)=s(j,i)*px
end do
t(i,i)=t(i,i)-rho(n)
end do
c calculate R**(-1).S, R**(-1).T, using routine solve
nn=3
do i=1,3
do j=1,3
y(j)=s(j,i)
end do
call solve(nn,r,x,y)
do j=1,3
stl(j,i)=x(j)
end do
nn=-3
end do
do i=1,3
do j=1,3
y(j)=t(j,i)
end do
call solve(nn,r,x,y)
do j=1,3
ttl(j,i)=x(j)
end do
end do
c fill the 6x6 Q-matrix
do i=1,3
do j=1,3
qq(j,i)=-stl(j,i)
qq(j,i+3)=-ttl(j,i)
qq(j+3,i)=0.d0
qq(j+3,i+3)=0.d0
end do
qq(i+3,i)=1.d0
end do
c solve eigenvalue problem for polarization vectors and vertical slownesses
c matrix system is nonsymmetric real valued
c solution from the eispack guide
call balanc(6,6,qq,is1,is2,fv)
call elmhes(6,6,is1,is2,qq,iv)
call eltran(6,6,is1,is2,qq,iv,zr)
call hqr2(6,6,is1,is2,qq,wr,wi,zr,ierr)
if(ierr.ne.0) then
print *, ierr,' error!'
stop
endif
call balbak(6,6,is1,is2,fv,6,zr)
c print *,'for layer',n
c print *, 'for phase velocity',cc,' the vertical slownesses are'
c print 101,(wr(i),wi(i),i=1,6)
c pause
101 format(6g12.4)
c eigenvector unpacking, see EISPACK guide, page 88
c bad eigenvector order is flagged by wi(i)>0. for odd i
iflag=0
do i=1,6
if(wi(i).eq.0.d0) then
if(n.eq.1) iev=0
do j=1,6
zi(j,i)=0.d0
end do
elseif(wi(i).gt.0.d0) then
c bad eigenvector order is flagged by wi(i)>0 for even i
if((i/2)*2.eq.i) then
iflag=iflag+1
iv(iflag)=i
endif
do j=1,6
zi(j,i)=zr(j,i+1)
end do
else
do j=1,6
zi(j,i)=-zi(j,i-1)
zr(j,i)=zr(j,i-1)
end do
endif
c normalize by the last three indices
sum=0.d0
do j=4,6
sum=sum+zr(j,i)**2+zi(j,i)**2
end do
sum=dsqrt(sum)
do j=1,6
zr(j,i)=zr(j,i)/sum
zi(j,i)=zi(j,i)/sum
end do
end do
c assemble the stress-displacement vectors
c calculate the traction components, with i removed
pp(1)=dcmplx(px,0.d0)
pp(2)=z0
do k=1,6
pp(3)=dcmplx(wr(k),wi(k))
do i=1,3
u0(i)=dcmplx(zr(i+3,k),zi(i+3,k))
end do
pu=z0
pw=z0
uw=z0
abcde=a-b+c-2.d0*d+2.d0*e
bce=b-4.d0*c-4.d0*e
de=d-e
do i=1,3
pu=pu+pp(i)*u0(i)
pw=pw+pp(i)*ww(i)
uw=uw+u0(i)*ww(i)
end do
do i=1,3
e1(i,k)=u0(i)
e1(i+3,k)=ww(i)*(pu*ww(3)*bce+8.d0*pw*uw*ww(3)*c
x +2.d0*(pw*u0(3)+uw*pp(3))*e)
e1(i+3,k)=e1(i+3,k)+pp(i)*(u0(3)*de+2.d0*uw*ww(3)*e)
e1(i+3,k)=e1(i+3,k)+u0(i)*(pp(3)*de+2.d0*pw*ww(3)*e)
end do
e1(6,k)=e1(6,k)+pu*abcde+pw*uw*bce
c almost lastly, mult traction by i
do i=1,3
e1(i+3,k)=eye*e1(i+3,k)
end do
end do
c reorder into upgoing and downgoing waves
c we use the exp(-i*omega*t) convention with z increasing downward
c so downgoing oscillatory waves have p_z>0, k_z real
c downgoing evanescent waves have Im(p_z)>0
c if the axis of symmetry is tilted, there are cases where a pair of
c near-horizontal plane waves will be both upgoing or both downgoing
c since Chen's algorithm depends on a 3,3 split, we must adopt a kluge
c similarly, there are cases where the EISPACK routines dont return
c the vertical wavenumbers in ordered pairs, but mix them up a bit
c this seems to cause problems, so a fix is necessary
c
c first, test for bad eigenvector order, switch k-1->k+1, k->k-1, k+1->k
c worst case is iflag=2, real,imag1+,imag1-,imag2+,imag2-,real
if(iflag.gt.0) then
do i=1,iflag
k=iv(i)
wrr=wr(k-1)
wii=wi(k-1)
wr(k-1)=wr(k)
wi(k-1)=wi(k)
wr(k)=wr(k+1)
wi(k)=wi(k+1)
wr(k+1)=wrr
wi(k+1)=wii
do j=1,6
pu=e1(j,k-1)
e1(j,k-1)=e1(j,k)
e1(j,k)=e1(j,k+1)
e1(j,k+1)=pu
end do
end do
endif
c second, divide into upgoing and downgoing waves
isum=0
do k=1,6
iv(k)=0
if(wi(k).eq.0.d0.and.wr(k).gt.0) iv(k)=1
if(wi(k).gt.0.d0) iv(k)=1
isum=isum+iv(k)
end do
c if up and downgoing cohorts are not equal, switch the sense of the
c pure-oscillatory wave with smallest wavenumber
140 continue
if(isum.ne.3) then
wr0=0.d0
do k=1,6
wr0=dmax1(wr0,dabs(wr(k)))
end do
do k=1,6
if(wi(k).eq.0.d0) then
if(dabs(wr(k)).lt.wr0) then
wr0=dabs(wr(k))
kk=k
endif
endif
end do
if(iv(kk).eq.0) then
iv(kk)=1
else
iv(kk)=0
endif
c check that we have equal up/down cohorts
isum=0
do k=1,6
isum=isum+iv(k)
end do
go to 140
endif
jdown=1
jup=4
c print *,'for layer',n,' the vert wavenums are (0=up,1=dn)'
do k=1,6
if(iv(k).eq.1) then
ki=jdown
jdown=jdown+1
else
ki=jup
jup=jup+1
endif
do i=1,6
ee(i,ki,n)=e1(i,k)
end do
c incorporate the factor of i into the stored vertical slowness
xnu(ki,n)=dcmplx(-wi(k),wr(k))
end do
1008 format(a,2g15.6,a,2g15.6)
end do
c now, must identify which upgoing waves in the halfspace are P,SV,SH
c crud, this goes back to array ee
c 3: SH is y-motion
c 2: SV is (-sqrt((1/vs)**2-p_x**2),0,-p_x) ! recall that z points down
c 1: P is (p_x,0,-sqrt((1/vp)**2-p_x**2)
c so we branch on size of u_y, and relative sign of u_x and u_z
c print *,'in the halfspace:'
c do i=4,6
c print *,'for i*k_z=',xnu(i,nlp),', the disp-stress vector is'
c do j=1,6
c xi(j)=dimag(ee(j,i,nlp))
c xr(j)=dreal(ee(j,i,nlp))
c end do
c print 101,(xr(j),j=1,6),(xi(j),j=1,6)
c end do
do i=4,6
ips(i-3)=3
if(zabs(ee(2,i,nlp)).lt.dsqrt(tol)) then ! not SH
test=dreal(ee(1,i,nlp))/dreal(ee(3,i,nlp))
if(test.gt.0.d0) then
ips(i-3)=2
else
ips(i-3)=1
endif
endif
end do
c print *,'wave prints:',(ips(i),i=1,3)
return
end
subroutine matget_old(nl,cc)
c returns stress-displacement vectors for a stack of anisotropic layers
c P waves may be evanescent, but S waves are oscillatory in the stack
c the weirdness seen in the surface wave code should not appear in
c a receiver function code
c however, the iev parameter is retained to avoid leaving timebombs
implicit real*8 (a-h,o-z)
implicit integer*4 (i-n)
complex*16 pp,u0,ee,pw,uw,pu,z1,z0,zz,xnu,eye,e1,e2,zla,rtm
complex*16 pfac,u,xl
common/stfff/w(3,101),t(3,3),ttl(3,3),s(3,3),stl(3,3),
x r(3,3),x(3),y(3)
common/model/z(100),dz(100),rho(101),vp(101),vs(101),vp2(101),
x vp4(101),vs2(101),vss(101)
common/model2/xmu(101),xla(101),xmu2(101),xla2(101),xla4(101)
common/propag/xnu(6,101),xl(6,100),pfac(6,3),u(3,6)
common/defect/idfct(4,101),adf(2,101)
common/rrt/rtm(6,6,100)
common/mstff/qq(6,6),wr(6),wi(6),zr(6,6),zi(6,6),iv(6),fv(6)
common/pstff/pp(3),u0(3),ee(6,6,101),e1(6,6),e2(6,6),zla(6)
common/qstff/qi(6,6),xr(6),xi(6),yr(6),yi(6),ips(3)
data pi/3.14159265358979d0/,eps/1.d-6/,tol/1.d-7/
c set iev=1 ** should be superfluous, but save for now
c toggle to iev=0 if there is a purely propagating wave in the top layer n=1
iev=1
z1=dcmplx(1.d0,0.d0)
z0=dcmplx(0.d0,0.d0)
eye=dcmplx(0.d0,1.d0)
rbar=5.515d3
ren=1.075190645d-3
radi=6.371d6
vbar=radi*ren
con=rbar*radi*radi*ren*ren
nlp=nl+1
nlm=nl-1
c first calculate vertical wavenumbers and propagating waves for each layer
c requires an eigenvector problem be solved
c in general, the evanescent vertical wavenumbers have nonzero real parts
c complex exponential fct is used to avoid endless branching
c horizontal slowness p_x
px=1.d0/cc
do n=1,nlp
a=xla(n)
b=xla2(n)
c=xla4(n)
d=xmu(n)
e=xmu2(n)
c print *,'a,b,c,d,e',a,b,c,d,e
fact=8.d0*w(1,n)*w(1,n)*c+2.d0*e
facs=16.d0*w(1,n)*w(3,n)*c
facr=8.d0*w(3,n)*w(3,n)*c+2.d0*e
do i=1,3
c first the what-0-what tensor
do j=1,3
t(j,i)=fact*w(j,n)*w(i,n)
s(j,i)=facs*w(j,n)*w(i,n)
r(j,i)=facr*w(j,n)*w(i,n)
end do
c next the identity tensor - correct an error on 7/6/95
t(i,i)=t(i,i)+d+e*(2.d0*w(1,n)*w(1,n)-1.d0)
s(i,i)=s(i,i)+4.d0*e*w(1,n)*w(3,n)
r(i,i)=r(i,i)+d+e*(2.d0*w(3,n)*w(3,n)-1.d0)
end do
c print 101,(w(i,n),i=1,3)
c print 101,fact,facs,facr
c print *,'t,s,r'
c print 101,((t(i,j),j=1,3),i=1,3)
c print 101,((s(i,j),j=1,3),i=1,3)
c print 101,((r(i,j),j=1,3),i=1,3)
fac=b-4.d0*c-2.d0*e
c next the what-0-xhat and what-0-zhat tensors
do i=1,3
t(1,i)=t(1,i)+fac*w(1,n)*w(i,n)
t(i,1)=t(i,1)+fac*w(1,n)*w(i,n)
s(1,i)=s(1,i)+fac*w(3,n)*w(i,n)
s(i,1)=s(i,1)+fac*w(3,n)*w(i,n)
s(3,i)=s(3,i)+fac*w(1,n)*w(i,n)
s(i,3)=s(i,3)+fac*w(1,n)*w(i,n)
r(3,i)=r(3,i)+fac*w(3,n)*w(i,n)
r(i,3)=r(i,3)+fac*w(3,n)*w(i,n)
end do
fac=a-b+c-d+e
c finally the xhat-0-xhat, zhat-0-zhat, xhat-0-zhat, zhat-0-xhat tensors
t(1,1)=t(1,1)+fac
s(3,1)=s(3,1)+fac
s(1,3)=s(1,3)+fac
r(3,3)=r(3,3)+fac
c mult by horizontal slowness and calc the modified T-matrix
do i=1,3
do j=1,3
t(j,i)=t(j,i)*px*px
s(j,i)=s(j,i)*px
end do
t(i,i)=t(i,i)-rho(n)
end do
c calculate R**(-1).S, R**(-1).T, using routine solve
nn=3
do i=1,3
do j=1,3
y(j)=s(j,i)
end do
call solve(nn,r,x,y)
do j=1,3
stl(j,i)=x(j)
end do
nn=-3
end do
do i=1,3
do j=1,3
y(j)=t(j,i)
end do
call solve(nn,r,x,y)
do j=1,3
ttl(j,i)=x(j)
end do
end do
c fill the 6x6 Q-matrix
do i=1,3
do j=1,3
qq(j,i)=-stl(j,i)
qq(j,i+3)=-ttl(j,i)
qq(j+3,i)=0.d0
qq(j+3,i+3)=0.d0
end do
qq(i+3,i)=1.d0
end do
c solve eigenvalue problem for polarization vectors and vertical slownesses
c matrix system is nonsymmetric real valued
c solution from the eispack guide
call balanc(6,6,qq,is1,is2,fv)
call elmhes(6,6,is1,is2,qq,iv)
call eltran(6,6,is1,is2,qq,iv,zr)
call hqr2(6,6,is1,is2,qq,wr,wi,zr,ierr)
if(ierr.ne.0) then
print *, ierr,' error!'
stop
endif
call balbak(6,6,is1,is2,fv,6,zr)
c print *,'for layer',n
c print *, 'for phase velocity',cc,' the vertical slownesses are'
c print 101,(wr(i),wi(i),i=1,6)
c pause
101 format(6g12.4)
c eigenvector unpacking, see EISPACK guide, page 88
c bad eigenvector order is flagged by wi(i)>0. for odd i
iflag=0
do i=1,6
if(wi(i).eq.0.d0) then
if(n.eq.1) iev=0
do j=1,6
zi(j,i)=0.d0
end do
elseif(wi(i).gt.0.d0) then
c bad eigenvector order is flagged by wi(i)>0 for even i
if((i/2)*2.eq.i) then
iflag=iflag+1
iv(iflag)=i
endif
do j=1,6
zi(j,i)=zr(j,i+1)
end do
else
do j=1,6
zi(j,i)=-zi(j,i-1)
zr(j,i)=zr(j,i-1)
end do
endif
c normalize by the last three indices
sum=0.d0
do j=4,6
sum=sum+zr(j,i)**2+zi(j,i)**2
end do
sum=dsqrt(sum)
do j=1,6
zr(j,i)=zr(j,i)/sum
zi(j,i)=zi(j,i)/sum
end do
end do
c assemble the stress-displacement vectors
c calculate the traction components, with i removed
pp(1)=dcmplx(px,0.d0)
pp(2)=z0
do k=1,6
pp(3)=dcmplx(wr(k),wi(k))
do i=1,3
u0(i)=dcmplx(zr(i+3,k),zi(i+3,k))
end do
pu=z0
pw=z0
uw=z0
abcde=a-b+c-2.d0*d+2.d0*e
bce=b-4.d0*c-4.d0*e
de=d-e
do i=1,3
pu=pu+pp(i)*u0(i)
pw=pw+pp(i)*w(i,n)
uw=uw+u0(i)*w(i,n)
end do
do i=1,3
e1(i,k)=u0(i)
e1(i+3,k)=w(i,n)*(pu*w(3,n)*bce+8.d0*pw*uw*w(3,n)*c
x +2.d0*(pw*u0(3)+uw*pp(3))*e)
e1(i+3,k)=e1(i+3,k)+pp(i)*(u0(3)*de+2.d0*uw*w(3,n)*e)
e1(i+3,k)=e1(i+3,k)+u0(i)*(pp(3)*de+2.d0*pw*w(3,n)*e)
end do
e1(6,k)=e1(6,k)+pu*abcde+pw*uw*bce
c almost lastly, mult traction by i
do i=1,3
e1(i+3,k)=eye*e1(i+3,k)
end do
end do
c reorder into upgoing and downgoing waves
c we use the exp(-i*omega*t) convention with z increasing downward
c so downgoing oscillatory waves have p_z>0, k_z real
c downgoing evanescent waves have Im(p_z)>0
c if the axis of symmetry is tilted, there are cases where a pair of
c near-horizontal plane waves will be both upgoing or both downgoing
c since Chen's algorithm depends on a 3,3 split, we must adopt a kluge
c similarly, there are cases where the EISPACK routines dont return
c the vertical wavenumbers in ordered pairs, but mix them up a bit
c this seems to cause problems, so a fix is necessary
c
c first, test for bad eigenvector order, switch k-1->k+1, k->k-1, k+1->k
c worst case is iflag=2, real,imag1+,imag1-,imag2+,imag2-,real
if(iflag.gt.0) then
do i=1,iflag
k=iv(i)
wrr=wr(k-1)
wii=wi(k-1)
wr(k-1)=wr(k)
wi(k-1)=wi(k)
wr(k)=wr(k+1)
wi(k)=wi(k+1)
wr(k+1)=wrr
wi(k+1)=wii
do j=1,6
pu=e1(j,k-1)
e1(j,k-1)=e1(j,k)
e1(j,k)=e1(j,k+1)
e1(j,k+1)=pu
end do
end do
endif
c second, divide into upgoing and downgoing waves
isum=0
do k=1,6
iv(k)=0
if(wi(k).eq.0.d0.and.wr(k).gt.0) iv(k)=1
if(wi(k).gt.0.d0) iv(k)=1
isum=isum+iv(k)
end do
c if up and downgoing cohorts are not equal, switch the sense of the
c pure-oscillatory wave with smallest wavenumber
140 continue
if(isum.ne.3) then
wr0=0.d0
do k=1,6
wr0=dmax1(wr0,dabs(wr(k)))
end do
do k=1,6
if(wi(k).eq.0.d0) then
if(dabs(wr(k)).lt.wr0) then
wr0=dabs(wr(k))
kk=k
endif
endif
end do
if(iv(kk).eq.0) then
iv(kk)=1
else
iv(kk)=0
endif
c check that we have equal up/down cohorts
isum=0
do k=1,6
isum=isum+iv(k)
end do
go to 140
endif
jdown=1
jup=4
c print *,'for layer',n,' the vert wavenums are (0=up,1=dn)'
1001 format(i2,2g15.6)
do k=1,6
if(iv(k).eq.1) then
ki=jdown
jdown=jdown+1
else
ki=jup
jup=jup+1
endif
do i=1,6
ee(i,ki,n)=e1(i,k)
end do
c incorporate the factor of i into the stored vertical slowness
xnu(ki,n)=dcmplx(-wi(k),wr(k))
end do
c OK, here's where we check whether two downgoing stress-disp vectors
c are nearly parallel - we check the dotproducts of displacement components
do i=1,4
idfct(i,n)=0
adf((i+1)/2,n)=0.d0
end do
do i=1,2
do j=i+1,3
r1=0.d0
r2=0.d0
zz=z0
do k=1,3
r1=r1+zabs(ee(k,i,n))**2
r2=r2+zabs(ee(k,j,n))**2
zz=zz+ee(k,j,n)*conjg(ee(k,i,n))
end do
qqq=1.d0-zabs(zz)/dsqrt(r1*r2)
if(qqq.lt.tol) then
ccc=cc*vbar
idfct(1,n)=i
idfct(2,n)=j
c print 1008,'vert slownesses',xnu(i,n),' and',xnu(j,n)
c we average eigenvalues (vert slownesses)
c and solve for eigenvector in subroutine defective
xnu(i,n)=(xnu(i,n)+xnu(j,n))/2.d0
xnu(j,n)=xnu(i,n)
c calculate the extravector for defective repeated eigenvalue
call defective(i,j,n,adf(1,n),a,b,c,d,e,px)
c print *,i,j,n,ccc,qqq,adf(1,n)
endif
end do
end do
1008 format(a,2g15.6,a,2g15.6)
c OK, here's where we check whether two upgoing stress-disp vectors
c are nearly parallel - we check the dotproducts of displacement components
do i=4,5
do j=i+1,6
r1=0.d0
r2=0.d0
zz=z0
do k=1,3
r1=r1+zabs(ee(k,i,n))**2
r2=r2+zabs(ee(k,j,n))**2
zz=zz+ee(k,j,n)*conjg(ee(k,i,n))
end do
qqq=1.d0-zabs(zz)/dsqrt(r1*r2)
if(qqq.lt.tol) then
ccc=cc*vbar
idfct(3,n)=i
idfct(4,n)=j
c print 1008,'vert slownesses',xnu(i,n),' and',xnu(j,n)
c we average the eigenvalues
xnu(i,n)=(xnu(i,n)+xnu(j,n))/2.d0
xnu(j,n)=xnu(i,n)
c calculate the extravector for defective repeated eigenvalue
c as well as coefficient (adf) of linear (z-z0) term
call defective(i,j,n,adf(2,n),a,b,c,d,e,px)
c print *,i,j,n,ccc,qqq,adf(2,n)
endif
end do
end do
end do
c now, must identify which upgoing waves in the halfspace are P,SV,SH
c crud, this goes back to array ee
c 3: SH is y-motion
c 2: SV is (-sqrt((1/vs)**2-p_x**2),0,-p_x) ! recall that z points down
c 1: P is (p_x,0,-sqrt((1/vp)**2-p_x**2)
c so we branch on size of u_y, and relative sign of u_x and u_z
print *,'in the halfspace:'
do i=4,6
print *,'for i*k_z=',xnu(i,nlp),', the disp-stress vector is'
do j=1,6
xi(j)=dimag(ee(j,i,nlp))
xr(j)=dreal(ee(j,i,nlp))
end do
print 101,(xr(j),j=1,6),(xi(j),j=1,6)
end do
do i=4,6
ips(i-3)=3
if(zabs(ee(2,i,nlp)).lt.dsqrt(tol)) then ! not SH
test=dreal(ee(1,i,nlp))/dreal(ee(3,i,nlp))
if(test.gt.0.d0) then
ips(i-3)=2
else
ips(i-3)=1
endif
endif
end do
print *,'wave prints:',(ips(i),i=1,3)
return
end
subroutine respget(nl,om,cc,resp)
c returns surface response for a stack of anisotropic layers
c incident p,sv,sh waves with freq om and phase velocity cc
c iev=1 if the waves are evanescent in the top layer, iev=0 otherwise
implicit real*8 (a-h,o-z)
implicit integer*4 (i-n)
complex*16 pp,u0,ee,z1,z0,xnu,eye,e1,e2,zla,rtm
complex*16 rt,tt,rt0,trc,xl,pfac,u,co,resp,ur
common/stfff/w(3,101),t(3,3),ttl(3,3),s(3,3),stl(3,3),
x r(3,3),x(3),y(3)
common/model/z(100),dz(100),rho(101),vp(101),vs(101),vp2(101),
x vp4(101),vs2(101),vss(101)
common/model2/xmu(101),xla(101),xmu2(101),xla2(101),xla4(101)
common/propag/xnu(6,101),xl(6,100),pfac(6,3),u(3,6)
common/defect/idfct(4,101),adf(2,101)
common/rrt/rtm(6,6,100)
common/mstff/qq(6,6),wr(6),wi(6),zr(6,6),zi(6,6),iv(6),fv(6)
common/pstff/pp(3),u0(3),ee(6,6,101),e1(6,6),e2(6,6),zla(6)
common/pstf2/co(6,101),ur(3)
common/qstff/qi(6,6),xr(6),xi(6),yr(6),yi(6),ips(3)
common/rstff/rt(3,3,101),tt(3,3,101),rt0(3,3),trc(3,3)
dimension resp(3,3)
data pi/3.14159265358979d0/,eps/1.d-6/,tol/1.d-7/
c set iev=1
c toggle to iev=0 if there is a purely propagating wave in the top layer n=1
iev=1
z1=dcmplx(1.d0,0.d0)
z0=dcmplx(0.d0,0.d0)
eye=dcmplx(0.d0,1.d0)
rbar=5.515d3
ren=1.075190645d-3
radi=6.371d6
vbar=radi*ren
con=rbar*radi*radi*ren*ren
nlp=nl+1
nlm=nl-1
c first calculate vertical wavenumbers and propagating waves for each layer
c an eigenvector problem was solved in prior subroutine
c results stored in array ee(6,6,101)
c in general, the evanescent vertical wavenumbers have nonzero real parts
c complex exponential fct is used to avoid endless branching
c
c calculate modified R/T coefficients
c first calc the propagation factors
c note that for dipping fast axes the upgoing and downgoing wavenumbers are
c independent, so we must calc all to be safe
do n=1,nl
do k=1,3
xl(k,n)=zexp(om*xnu(k,n)*dz(n)) ! downgoing
xl(k+3,n)=zexp(-om*xnu(k+3,n)*dz(n)) ! upgoing
end do
end do
c do i=1,6
c print 1002,xnu(i,3),xl(i,3)
c end do
1002 format('i*k_z:',2g15.6,', propfac is',2g15.6)
c calculate modified R/T coefficients at each interface
do n=1,nl
c rearrange to e1: waves approaching and e2: waves leaving an interface
do k=1,3
do i=1,6
e1(i,k)=ee(i,k,n+1)
e2(i,k)=ee(i,k,n)
e1(i,k+3)=-ee(i,k+3,n)
e2(i,k+3)=-ee(i,k+3,n+1)
end do
zla(k)=xl(k,n)
if(n.lt.nl) then
zla(k+3)=xl(k+3,n+1)
else
c reference the upcoming wave amplitude to the top of halfspace
c therefore no propagation factor, not valid for evanescent waves in halfspace
c in surface wave code this is zero, so that upgoing evanescent waves vanish
zla(k+3)=1.d0
endif
end do
c mult the columns of e2
do k=1,6
do i=1,6
e2(i,k)=e2(i,k)*zla(k)
end do
end do
c the possibility of defective matrices must be contemplated here
c k=1,2,3 columns are downgoing in nth layer
c k=4,5,6 columns are upgoing in (n+1)th layer
c the vector e2(.,k1) has already been multiplied by exponential factor zla
if(idfct(1,n).ne.0) then
k1=idfct(1,n)
k2=idfct(2,n)
do i=1,6
e2(i,k2)=e2(i,k2)+adf(1,n)*dz(n)*e2(i,k1)
end do
endif
c the sign change on dz is for upgoing waves
if(idfct(3,n+1).ne.0) then
k1=idfct(3,n+1)
k2=idfct(4,n+1)
do i=1,6
e2(i,k2)=e2(i,k2)-adf(2,n+1)*dz(n+1)*e2(i,k1)
end do
endif
c in order to use csolve to invert e1, must separate into real/imag parts
c its clumsy, but im lazy
c we calc e1^{-1}\cdot e2\cdot \Gamma one column at a time
do k=1,6
do i=1,6
qq(i,k)=dreal(e1(i,k))
qi(i,k)=dimag(e1(i,k))
end do
end do
nn=6
do k=1,6
do i=1,6
yr(i)=dreal(e2(i,k))
yi(i)=dimag(e2(i,k))
end do
call csolve(nn,qq,qi,xr,xi,yr,yi)
nn=-6
do i=1,6
rtm(i,k,n)=dcmplx(xr(i),xi(i))
end do
end do
end do
c calc R_ud at the free surface
c note that first two factors in Chen (20) dont collapse
c mult by inv-matrix one column at a time
do k=1,3
do i=1,3
rt0(i,k)=ee(i+3,k+3,1)*xl(k+3,1)
s(i,k)=dreal(ee(i+3,k,1))
t(i,k)=dimag(ee(i+3,k,1))
end do
end do
c the possibility of defective matrices must be contemplated here
c these waves are upgoing in 1st layer
c the sign change on dz is for upgoing waves, and xl(k1,1)=xl(k2,1)
if(idfct(3,1).ne.0) then
k1=idfct(3,1)
k2=idfct(4,1)-3
do i=1,3
rt0(i,k2)=rt0(i,k2)-adf(2,1)*dz(1)*ee(i+3,k1,1)*xl(k1,1)
end do
endif
nn=3
do k=1,3
do i=1,3
yr(i)=dreal(rt0(i,k))
yi(i)=dimag(rt0(i,k))
end do
call csolve(nn,s,t,xr,xi,yr,yi)
nn=-3
do i=1,3
rt0(i,k)=-dcmplx(xr(i),xi(i))
end do
end do
c recursive calc of generalized R/T coefs:
c in contrast to the surface-wave code, we start from the top layer and
c iterate down to the halfspace
c we also uses submatrices of generalized R/T matrix in different order
do n=1,nl
c first the generalized upward-transmission coef:
do k=1,3
do i=1,3
trc(i,k)=z0
if(n.gt.1) then
do j=1,3
trc(i,k)=trc(i,k)-rtm(i+3,j,n)*rt(j,k,n-1)
end do
else
c use free-surface reflection matrix in top layer (interface "zero")
do j=1,3
trc(i,k)=trc(i,k)-rtm(i+3,j,n)*rt0(j,k)
end do
endif
end do
trc(k,k)=trc(k,k)+z1
end do
do k=1,3
do i=1,3
s(i,k)=dreal(trc(i,k))
t(i,k)=dimag(trc(i,k))
end do
end do
nn=3
do k=1,3
do i=1,3
yr(i)=dreal(rtm(i+3,k+3,n))
yi(i)=dimag(rtm(i+3,k+3,n))
end do
call csolve(nn,s,t,xr,xi,yr,yi)
nn=-3
do i=1,3
tt(i,k,n)=dcmplx(xr(i),xi(i))
end do
end do
c next the generalized reflection coef:
do k=1,3
do i=1,3
trc(i,k)=z0
if(n.gt.1) then
do j=1,3
trc(i,k)=trc(i,k)+rt(i,j,n-1)*tt(j,k,n)
end do
else
c use free-surface reflection matrix in top layer (interface "zero")
do j=1,3
trc(i,k)=trc(i,k)+rt0(i,j)*tt(j,k,n)
end do
endif
end do
end do
do k=1,3
do i=1,3
rt(i,k,n)=rtm(i,k+3,n)
do j=1,3
rt(i,k,n)=rt(i,k,n)+rtm(i,j,n)*trc(j,k)
end do
end do
end do
end do
1001 format(6f14.6)
c print *,'free-surface reflection'
c print 1001,((rt0(i,j),j=1,3),i=1,3)
c do n=1,nl
c print *,'interface',n
c print 1001,((rt(i,j,n),j=1,3),i=1,3)
c print 1001,((tt(i,j,n),j=1,3),i=1,3)
c end do
c using the p,sv,sh identification, we propagate upward to the surface,
c calculate the wave coefs in the top layer, then the particle displacement
do iup=1,3
do i=1,3
co(i+3,nlp)=z0
end do
co(iup+3,nlp)=z1
c from upgoing coefs in the n+1 layer, calculate
c upgoing coefs in the nth layer, downgoing coefs in the n+1 layer
do n=nl,1,-1
do i=1,3
co(i+3,n)=z0
co(i,n+1)=z0
do j=1,3
co(i+3,n)=co(i+3,n)+tt(i,j,n)*co(j+3,n+1)
co(i,n+1)=co(i,n+1)+rt(i,j,n)*co(j+3,n+1)
end do
end do
end do
c then downgoing coefs in the top layer:
do i=1,3
co(i,1)=z0
do j=1,3
co(i,1)=co(i,1)+rt0(i,j)*co(j+3,1)
end do
end do
c print *,'upgoing coefs'
c print 1001,((co(j+3,n),j=1,3),n=1,nlp)
c print *,'downgoing coefs'
c print 1001,((co(j,n),j=1,3),n=1,nlp)
c calc the surface displacement
h1=0.d0
h2=z(1)
do i=1,3
ur(i)=z0
do k=1,3
ur(i)=ur(i)+co(k,1)*ee(i,k,1)
x +co(k+3,1)*ee(i,k+3,1)*(zexp(om*xnu(k+3,1)*(-h2)))
end do
c check for the xtra terms associated with defective matrices
if(idfct(1,1).ne.0) then
ii=idfct(1,1)
jj=idfct(2,1)
ur(i)=ur(i)+co(jj,1)*adf(1,1)*ee(i,ii,1)*(-h1)
endif
if(idfct(3,1).ne.0) then
ii=idfct(3,1)
jj=idfct(4,1)
ur(i)=ur(i)
x +co(jj,1)*adf(2,1)*ee(i,ii,1)*(-h2)*(zexp(om*xnu(ii,1)*(-h2)))
endif
end do
c copy the surface displacement into the response matrix
do i=1,3
resp(i,ips(iup))=ur(i)
end do
end do
return
end
subroutine defective(i,j,n,adf,a,b,c,d,e,px)
c kluge for dealing with nearly defective propagator matrices
c in which the eigenvectors,
c which represent the particle motion of upgoing and downgoing waves
c become nearly parallel.
c in this case the solution for system of ODEs is
c a_1 \bf_1 e^xnu*(z-z0) + a_2*(\bf_2 + adf*(z-z0)*\bf_1)e^xnu*(z-z0)
c
implicit real*8 (a-h,o-z)
implicit integer*4 (i-n)
complex*16 pp,u0,ee,z1,z0,znu,xnu,e1,e2,zla,xl,u,pfac,eye
complex*16 zq1,zq2,u1,u2,zq3,xee
common/stfff/w(3,101),t(3,3),ttl(3,3),s(3,3),stl(3,3),
x r(3,3),x(3),y(3)
common/propag/xnu(6,101),xl(6,100),pfac(6,3),u(3,6)
common/defect1/zq1(3,3),zq2(3,3),u1(3),u2(3),zq3(2,2),xee(3)
common/defect2/edr(6),edi(6),qdr(5,5),qdi(5,5),ydr(6),ydi(6)
common/defect3/q1r(3,3),q1i(3,3),q2r(3,3),q2i(3,3),fv2(3),fv3(3)
common/mstff/qq(6,6),wr(6),wi(6),zr(6,6),zi(6,6),iv(6),fv(6)
common/pstff/pp(3),u0(3),ee(6,6,101),e1(6,6),e2(6,6),zla(6)
common/qstff/qi(6,6),xr(6),xi(6),yr(6),yi(6),ips(3)
z1=dcmplx(1.d0,0.d0)
z0=dcmplx(0.d0,0.d0)
eye=dcmplx(0.d0,1.d0)
c for the extravector, need to solve system of equations
c based on original 6x6 Q matrix
c the plane-wave solutions generalize to the form
c u0*e^{i*nu*(z-z0)} and u1*e^{i*nu*(z-z0)} + adf* u0*(z-z0)*e^{i*nu*(z-z0)}
c u1 is the solution to
c (\bTtil + nu*\bStil + nu^2*\bI).u1=i*adf*(\bStil + 2*nu*\bI).u0
c in practice, we absorb the adf factor into u1, then normalize
c (\bTtil + nu*\bStil + nu^2*\bI).(u1/adf)=i*(\bStil + 2*nu*\bI).u0
c since nu is the known eigenvalue of u0, the solution is easier
c form the matrices on either side
znu=-eye*xnu(i,n)
do ii=1,3
do jj=1,3
zq1(jj,ii)=dcmplx(ttl(jj,ii),0.d0)+znu*stl(jj,ii)
zq2(jj,ii)=dcmplx(stl(jj,ii),0.d0)
end do
zq1(ii,ii)=zq1(ii,ii)+znu*znu
zq2(ii,ii)=zq2(ii,ii)+2.d0*znu
end do
c we wish to find the eigenvector of the near-defective matrix
c in the region where its eigenvectors are numerically unstable
c we explicitly calculate the eigenvector with smallest right-eigenvalue of
c (\bTtil + nu*\bStil + nu^2*\bI)=zq1
c copy into real, imag matrices
do ii=1,3
do jj=1,3
q1r(jj,ii)=dreal(zq1(jj,ii))
q1i(jj,ii)=dimag(zq1(jj,ii))
end do
end do
c into eispack
call cbal(3,3,q1r,q1i,low,igh,fv)
call corth(3,3,low,igh,q1r,q1i,fv2,fv3)
call comqr2(3,3,low,igh,fv2,fv3,q1r,q1i,wr,wi,q2r,q2i,ierr)
if(ierr.ne.0) go to 400
call cbabk2(3,3,low,igh,fv,3,q2r,q2i)
amn=wr(1)**2+wi(1)**2
ij=1
do ii=2,3
amm=wr(ii)**2+wi(ii)**2
if(amm.lt.amn) then
ij=ii
amn=amm
endif
end do
sum=0.d0
do ii=1,3
u0(ii)=dcmplx(q2r(ii,ij),q2i(ii,ij))
sum=sum+zabs(u0(ii))**2
end do
sum=dsqrt(sum)
do ii=1,3
u0(ii)=u0(ii)/sum
end do
c assemble the ith stress-displacement vector
c calculate the traction components, with i removed
pp(1)=dcmplx(px,0.d0)
pp(2)=z0
pp(3)=znu
pu=z0
pw=z0
uw=z0
abcde=a-b+c-2.d0*d+2.d0*e
bce=b-4.d0*c-4.d0*e
de=d-e
do ii=1,3
pu=pu+pp(ii)*u0(ii)
pw=pw+pp(ii)*w(ii,n)
uw=uw+u0(ii)*w(ii,n)
end do
do ii=1,3
ee(ii,i,n)= u0(ii)
ee(ii+3,i,n)=w(ii,n)*(pu*w(3,n)*bce+8.d0*pw*uw*w(3,n)*c
x +2.d0*(pw*u0(3)+uw*pp(3))*e)
ee(ii+3,i,n)=ee(ii+3,i,n)+pp(ii)*(u0(3)*de+2.d0*uw*w(3,n)*e)
ee(ii+3,i,n)=ee(ii+3,i,n)+u0(ii)*(pp(3)*de+2.d0*pw*w(3,n)*e)
end do
ee(6,i,n)=ee(6,i,n)+pu*abcde+pw*uw*bce
c almost lastly, mult traction by i
do ii=1,3
ee(ii+3,i,n)=eye*ee(ii+3,i,n)
end do
c extract u0 from ee(*,i,n) use it to calculate the additional traction terms
c and store in ee(*,j,n)
c additional traction terms involve gradient of (z-z0)
c so can be calculated from standard formulas with \bk=zhat
c we dont multiply by i
pp(1)=z0
pp(2)=z0
pp(3)=z1
pu=z0
pw=z0
uw=z0
abcde=a-b+c-2.d0*d+2.d0*e
bce=b-4.d0*c-4.d0*e
de=d-e
do ii=1,3
u0(ii)=ee(ii,i,n)
pu=pu+pp(ii)*u0(ii)
pw=pw+pp(ii)*w(ii,n)
uw=uw+u0(ii)*w(ii,n)
end do
do ii=1,3
xee(ii)=w(ii,n)*(pu*w(3,n)*bce+8.d0*pw*uw*w(3,n)*c
x +2.d0*(pw*u0(3)+uw*pp(3))*e)
xee(ii)=xee(ii)+pp(ii)*(u0(3)*de+2.d0*uw*w(3,n)*e)
xee(ii)=xee(ii)+u0(ii)*(pp(3)*de+2.d0*pw*w(3,n)*e)
end do
xee(3)=xee(3)+pu*abcde+pw*uw*bce
c extract u0 from ee(*,i,n), mult by i*(\bStil + 2*nu*\bI), replace in u0
do ii=1,3
u0(ii)=z0
do jj=1,3
u0(ii)=u0(ii)+zq2(ii,jj)*ee(jj,i,n)
end do
u0(ii)=eye*u0(ii)
end do
1002 format(3(2g14.6,3x))
c for znu NOT an eigenvalue,
c but rather the average of closely-space eigenvalues
c in this case, zq1 is nonsingular, and we just solve for u1
do ii=1,3
yr(ii)=dreal(u0(ii))
yi(ii)=dimag(u0(ii))
do jj=1,3
q1r(jj,ii)=dreal(zq1(jj,ii))
q1i(jj,ii)=dimag(zq1(jj,ii))
end do
end do
call csolve(3,q1r,q1i,xr,xi,yr,yi)
do ii=1,3
u1(ii)=dcmplx(xr(ii),xi(ii))
end do
c End, different tactic
c
c normalize
sum=0.d0
do ii=1,3
sum=sum+zabs(u1(ii))**2
end do
sum=dsqrt(sum)
do ii=1,3
u1(ii)=u1(ii)/sum
end do
c adf is the normalization constant
adf=1.d0/sum
c calculate the traction
c and place the new stress-displacement vector in column j
c pp is the wavenumber vector, and first two components are already in place
pp(1)=dcmplx(px,0.d0)
pp(2)=z0
pp(3)=znu
pu=z0
pw=z0
uw=z0
abcde=a-b+c-2.d0*d+2.d0*e
bce=b-4.d0*c-4.d0*e
de=d-e
do ii=1,3
pu=pu+pp(ii)*u1(ii)
pw=pw+pp(ii)*w(ii,n)
uw=uw+u1(ii)*w(ii,n)
end do
do ii=1,3
ee(ii,j,n)=u1(ii)
ee(ii+3,j,n)=w(ii,n)*(pu*w(3,n)*bce+8.d0*pw*uw*w(3,n)*c
x +2.d0*(pw*u1(3)+uw*pp(3))*e)
ee(ii+3,j,n)=ee(ii+3,j,n)+pp(ii)*(u1(3)*de+2.d0*uw*w(3,n)*e)
ee(ii+3,j,n)=ee(ii+3,j,n)+u1(ii)*(pp(3)*de+2.d0*pw*w(3,n)*e)
end do
ee(6,j,n)=ee(6,j,n)+pu*abcde+pw*uw*bce
c almost lastly, mult traction by i
c and add extra traction from (z-z0) term (not mult by i)
c TEST - mult xee by zero, see if it is important --- it IS important
do ii=1,3
ee(ii+3,j,n)=eye*ee(ii+3,j,n)+adf*xee(ii)
end do
return
400 print *,'eispack error'
stop
end
|
From Undecidability.L Require Export Util.L_facts.
(* **** Closure calculus *)
Inductive Comp : Type :=
| CompVar (x:nat)
| CompApp (s : Comp) (t : Comp) : Comp
| CompClos (s : term) (A : list Comp) : Comp.
Coercion CompApp : Comp >-> Funclass.
Inductive lamComp : Comp -> Prop := lambdaComp s A: lamComp (CompClos (lam s) A).
Inductive validComp : Comp -> Prop :=
| validCompApp s t : validComp s -> validComp t -> validComp (s t)
| validCompClos (s : term) (A : list Comp) :
(forall a, a el A -> validComp a) -> (forall a, a el A -> lamComp a) -> bound (length A) s -> validComp (CompClos s A).
Hint Constructors Comp lamComp validComp : core.
Definition validEnv A := forall a, a el A -> validComp a (*/\ lamComp a)*).
Definition validEnv' A := forall a, a el A -> closed a.
Hint Unfold validEnv validEnv' : core.
Lemma validEnv_cons a A : validEnv (a::A) <-> ((validComp a) /\ validEnv A).
Proof.
unfold validEnv. simpl. split. auto. intros [? ?] a' [eq|el']; subst;auto.
Qed.
Lemma validEnv'_cons a A : validEnv' (a::A) <-> (closed a /\ validEnv' A).
Proof.
unfold validEnv'. simpl. intuition. now subst.
Qed.
Ltac inv_validComp :=
match goal with
| H : validComp (CompApp _ _) |- _ => inv H
| H : validComp (CompClos _ _) |- _ => inv H
end.
Definition Comp_ind_deep'
(P : Comp -> Prop)
(Pl : list Comp -> Prop)
(IHVar : forall x : nat, P (CompVar x))
(IHApp : forall s : Comp, P s -> forall t : Comp, P t -> P (s t))
(IHClos : forall (s : term) (A : list Comp),
Pl A -> P (CompClos s A))
(IHNil : Pl nil)
(IHCons : forall (a:Comp) (A : list Comp),
P a -> Pl A -> Pl (a::A))
(x:Comp) : P x :=
(fix f c : P c:=
match c with
| CompVar x => IHVar x
| CompApp s t => IHApp s (f s) t (f t)
| CompClos s A => IHClos s A
((fix g A : Pl A :=
match A with
[] => IHNil
| a::A => IHCons a A (f a) (g A)
end) A)
end) x
.
Definition Comp_ind_deep
(P : Comp -> Prop)
(IHVar : forall x : nat, P (CompVar x))
(IHApp : forall s : Comp, P s -> forall t : Comp, P t -> P (s t))
(IHClos : forall (s : term) (A : list Comp),
(forall a, a el A -> P a) -> P (CompClos s A)) : forall x, P x.
Proof.
apply Comp_ind_deep' with (Pl:=fun A => (forall a, a el A -> P a));auto.
intros. inv H1;auto.
Qed.
(*
Lemma subst_comm s x1 u1 x2 u2 : closed u1 -> closed u2 -> x1 <> x2 -> subst (subst s x1 u1) x2 u2 = subst (subst s x2 u2) x1 u1.
Proof with try (congruence||auto).
intros cl1 cl2 neq. revert x1 x2 neq;induction s;simpl;intros.
-decide (n=x1); decide (n=x2); try rewrite cl1;try rewrite cl2;subst;simpl...
+decide (x1=x1)...
+decide (x2=x2)...
+decide (n=x2);decide (n=x1)...
-rewrite IHs1,IHs2...
-rewrite IHs...
Qed.
*) (*
Lemma subst_twice s x u1 u2 : closed u1 -> subst (subst s x u1) x u2 = subst s x u1.
Proof with try (congruence||auto).
intros cl. revert x;induction s;simpl;intros.
-decide (n=x);subst. now rewrite cl. simpl. decide (n=x);subst;congruence.
-rewrite IHs1,IHs2...
-rewrite IHs...
Qed.*)
(*
Lemma subst_free a s k u y: closed a -> subst s k u = s -> subst (subst s y a) k u = subst s y a.
Proof.
intros ca eq. revert y k u eq. induction s;simpl;intros.
-decide (n=y). now rewrite ca. apply eq.
-simpl in eq. inversion eq. rewrite H0, H1, IHs1, IHs2;auto.
-f_equal. simpl in eq. inversion eq. rewrite !H0. now rewrite IHs.
Qed.*)
(*
Lemma bound_ge k s m: bound k s -> m >= k -> bound m s.
Proof.
intros. decide (m=k);subst.
-auto.
-eapply bound_gt;eauto. lia.
Qed.
*)
(*
Lemma bound_subst' x s a y: bound x s -> closed a -> bound x (subst s y a).
Proof.
intros dcl cl. revert y. induction dcl;simpl;intros.
-decide (n=y);subst.
+eapply bound_ge. now apply closed_dcl. lia.
+now constructor.
-now constructor.
-now constructor.
Qed.
*)
(*
Fixpoint substList' (s:term) (x:nat) (A: list term): term :=
match A with
| nil => s
| a::A => substList' (subst s x a) (S x) A
end.*)
Fixpoint substList (s:term) (x:nat) (A: list term): term :=
match s with
| var n => if Dec (x>n) then var n else nth (n-x) A (var n)
| app s t => app (substList s x A) (substList t x A)
| lam s => lam (substList s (S x) A)
end.
Fixpoint deClos (s:Comp) : term :=
match s with
| CompVar x => var x
| CompApp s t => app (deClos s) (deClos t)
| CompClos s A => substList s 0 (map deClos A)
end.
(* Reduction *)
Reserved Notation "s '>[(' l ')]' t" (at level 50, format "s '>[(' l ')]' t").
Declare Scope LClos.
Inductive CPow : nat -> Comp -> Comp -> Prop :=
| CPowRefl (s:Comp) : s >[(0)] s
| CPowTrans (s t u:Comp) i j : s >[(i)] t -> t >[(j)] u -> s >[(i+j)] u
| CPowAppL (s s' t :Comp) l: s >[(l)] s' -> (s t) >[(l)] (s' t)
| CPowAppR (s t t':Comp) l: t >[(l)] t' -> (s t) >[(l)] (s t')
| CPowApp (s t:term) (A:list Comp) :
CompClos (app s t) A >[(0)] (CompClos s A) (CompClos t A)
| CPowVar (x:nat) (A:list Comp):
CompClos (var x) A >[(0)] nth x A (CompVar x)
| CPowVal (s t:term) (A B:list Comp):
lambda t -> (CompClos (lam s) A) (CompClos t B) >[(1)] (CompClos s ((CompClos t B)::A))
where "s '>[(' l ')]' t" := (CPow l s t) : LClos.
Open Scope LClos.
Ltac inv_CompStep :=
match goal with
| H : (CompApp _ _) >(_) CompClos _ _ |- _ => inv H
| H : (CompClos _ _) >(_) CompApp _ _ |- _ => inv H
end.
Hint Constructors CPow : core.
Lemma CPow_congL n s s' t :
s >[(n)] s' -> s t >[(n)] s' t.
Proof.
induction 1;eauto.
Qed.
Lemma CPow_congR n (s t t' : Comp) :
t >[(n)] t' -> s t >[(n)] s t'.
Proof.
induction 1;eauto.
Qed.
Lemma CPow_trans s t u i j k : s >[(i)] t -> t >[(j)] u -> i + j = k -> s >[(k)] u.
Proof.
intros. subst. eauto.
Qed.
Instance CPow'_App_properR n:
Proper (eq ==> (CPow n) ==> (CPow n)) CompApp.
Proof.
intros ? ? -> ? ? ?. now eapply CPow_congR.
Qed.
(*
Definition CStar s t:= exists k , CPow k s t .
Notation "s '>[]*' t" := (CStar s t) (at level 50) : L.los.
Instance rStar'_PreOrder : PreOrder CStar.
Proof.
constructor; hnf.
-now eexists.
-eapply star_trans.
Qed.
Lemma rStar'_trans_l s s' t :
s >[]* s' -> s t >[]* s' t.
Proof.
induction 1; eauto using star.
Qed.
Lemma rStar'_trans_r (s t t' : Comp):
t >[]* t' -> s t >[]* s t'.
Proof.
induction 1; eauto using star.
Qed.
Instance rStar'_App_proper :
Proper ((star CStep) ==> (star CStep) ==> (star CStep)) CompApp.
Proof.
cbv. intros s s' A t t' B. etransitivity.
apply rStar'_trans_l, A. apply rStar'_trans_r, B.
Qed.
Instance CStep_star_subrelation : subrelation CStep (star CStep).
Proof.
intros s t st. eauto using star.
Qed.
*)
(* Properties of step-indexed version *)
(*
Notation "x '>[]^' n y" := (ARS.pow CStep n x y) (at level 50) : L.cope.
Lemma CStep_Lam n: forall (s t u:Comp), lamComp u -> (ARS.pow CStep n (s t) u) ->
exists m1 m2 (s' t':Comp),(m1 < n /\ ARS.pow CStep m1 s s' /\ lamComp s')
/\ (m2 < n /\ ARS.pow CStep m2 t t' /\ lamComp t').
Proof with repeat intuition;try now reflexivity.
induction n;intros ? ? ? lu R.
-inv R. inv lu.
-destruct R as [u' [R R']]. inv R.
+apply IHn in R'... decompose [ex and] R'. exists (S x), x0, x1, x2... change (S x) with (1+x). apply pow_add;simpl. exists s';intuition. eexists;simpl...
+apply IHn in R'... decompose [ex and] R'. exists x, (S x0), x1, x2... change (S x0) with (1+x0). apply pow_add;simpl. exists t';intuition. eexists;simpl...
+inv H2. eexists 0,0,_,_...
Qed.
Lemma CStep_Lam' (s t u:Comp) : lamComp u -> (s t) >[]* u ->
exists (s' t':Comp),( s >[]* s' /\ lamComp s')
/\ (t >[]* t' /\ lamComp t').
Proof with repeat intuition;try now reflexivity.
intros lu R. apply star_pow in R. destruct R as [n R]. revert s t u lu R. induction n;intros.
-inv R. inv lu.
-destruct R as [u' [R R']]. inv R.
+apply IHn in R'... decompose [ex and] R'. exists x, x0... eauto using star.
+apply IHn in R'... decompose [ex and] R'. exists x, x0... eauto using star.
+inv H2. eexists _,_...
Qed.
*)
Lemma substList_bound x s A: bound x s -> substList s x A = s.
Proof.
revert x;induction s;intros;simpl.
-inv H. decide (x>n);tauto.
-inv H. now rewrite IHs1,IHs2.
-inv H. rewrite IHs;auto.
Qed.
Lemma substList_closed s A x: closed s -> substList s x A = s.
Proof.
intros. apply substList_bound. destruct x. now apply closed_dcl. eapply bound_gt;[rewrite <- closed_dcl|];auto. lia.
Qed.
Lemma substList_var' y x A: y >= x -> substList (var y) x A = nth (y-x) A (var y).
Proof.
intros ge. simpl. decide (x>y). lia. auto.
Qed.
Lemma substList_var y A: substList (var y) 0 A = nth y A (var y).
Proof.
rewrite substList_var'. f_equal. lia. lia.
Qed.
Lemma substList_is_bound y A s: validEnv' A -> bound (y+|A|) (s) -> bound y (substList s y A).
Proof.
intros vA. revert y. induction s;intros y dA.
-apply closed_k_bound. intros k u ge. simpl. decide (y>n).
+simpl. destruct (Nat.eqb_spec n k). lia. auto.
+inv dA. assert (n-y<|A|) by lia. now rewrite (vA _ (nth_In A #n H)).
-inv dA. simpl. constructor;auto.
-simpl. constructor. apply IHs. now inv dA.
Qed.
Lemma substList_closed' A s: validEnv' A -> bound (|A|) (s) -> closed (substList s 0 A).
Proof.
intros. rewrite closed_dcl. apply substList_is_bound;auto.
Qed.
Lemma deClos_valComp a: validComp a -> closed (deClos a).
Proof.
intros va. induction va;simpl.
-now apply app_closed.
-apply substList_closed'. intros a ain. rewrite in_map_iff in ain. destruct ain as [a' [eq a'in]];subst. now apply H0. now rewrite map_length.
Qed.
Lemma deClos_validEnv A : validEnv A -> validEnv' (map deClos A).
Proof.
intros vA. induction A;simpl.
-unfold validEnv'. simpl. tauto.
-rewrite validEnv'_cons. apply validEnv_cons in vA as [ca cA]. split;auto. apply deClos_valComp; auto.
Qed.
Hint Resolve deClos_validEnv : core.
Lemma subst_substList x s t A: validEnv' A -> subst (substList s (S x) A) x t = substList s x (t::A).
Proof.
revert x;induction s;simpl;intros x cl.
-decide (S x > n);simpl. decide (x>n); destruct (Nat.eqb_spec n x);try lia;try tauto. subst. now rewrite minus_diag. decide (x>n). lia. destruct (n-x) eqn: eq. lia. assert (n2=n-S x) by lia. subst n2. destruct (nth_in_or_default (n-S x) A #n).
+ apply cl in i. now rewrite i.
+rewrite e. simpl. destruct (Nat.eqb_spec n x). lia. auto.
-now rewrite IHs1,IHs2.
-now rewrite IHs.
Qed.
Lemma validComp_step s t l: validComp s -> s >[(l)] t -> validComp t.
Proof with repeat (subst || firstorder).
intros vs R. induction R;repeat inv_validComp...
-inv H3. constructor...
-inv H3. apply H1. apply nth_In. lia.
-inv H8. constructor;auto;intros a [?|?];subst;auto.
Qed.
Hint Resolve validComp_step : core.
(*
Lemma deClos_correct''' s t : validComp s -> s >(0) t -> deClos s = deClos t.
Proof with repeat (cbn in * || eauto || congruence || lia || subst).
intros cs R. remember 0 as n eqn:eq in R. revert eq. induction R;intros ?;repeat inv_validComp...
-destruct i... rewrite IHR1,IHR2...
-destruct IHR...
-rewrite IHR...
-simpl. rewrite <- minus_n_O. rewrite <-map_nth with (f:=deClos)...
Qed.
Lemma deClos_correct'' s t : validComp s -> s >(1) t -> deClos s = deClos t \/ deClos s ≻ deClos t.
Proof with repeat (cbn in * || eauto || congruence || lia || subst).
intros cs R. remember 1 as n eqn:eq in R. revert eq. induction R;intros ?;repeat inv_validComp...
-destruct i...
+destruct IHR2... apply deClos_correct''' in R1... left... aply deClos_correct''' in R1... right...
right... split;eauto. destruct IHR. auto. left... right...
-destruct IHR. auto. left... right...
-left...
-left. simpl. rewrite <- minus_n_O. rewrite <-map_nth with (f:=deClos)...
-right. inv H. simpl. rewrite <-subst_substList...
Qed.*)
Lemma deClos_correct l s t : validComp s -> s >[(l)] t -> deClos s >(l) deClos t.
Proof with repeat (cbn in * || eauto 10 using star || congruence || lia || subst).
intros cs R.
induction R...
-eapply pow_trans;eauto.
-inv cs;apply pow_step_congL...
-inv cs;apply pow_step_congR...
-rewrite <- minus_n_O. rewrite <-map_nth with (f:=deClos)...
-inv H. inv cs. inv H1. eexists;split... rewrite <- subst_substList...
Qed.
(*
(* relation that tries to capture that two closures 'reduce' to one another *)
Reserved Notation "s '=[]>' t" (at level 70).
Inductive reduceC : Comp -> Comp -> Prop :=
| redC s t: deClos s >* deClos t -> s =[]> t
where "s '=[]>' t" := (reduceC s t).
Hint Constructors reduceC.
Lemma reduceC_if s t : s =[]> t -> deClos s >* deClos t.
Proof.
now inversion 1.
Qed.
(* ** Properties of the extended reduction relation *)
Instance reduceC_PreOrder : PreOrder reduceC.
Proof.
constructor;repeat intros;constructor.
-reflexivity.
-inv H. inv H0. now rewrite H1.
Qed.
Instance reduceC_App_proper :
Proper (reduceC ==> reduceC ==> reduceC) CompApp.
Proof.
cbv. intros s s' A t t' B. constructor. simpl. apply star_step_app_proper.
-now inv A.
-now inv B.
Qed.
Lemma CStep_reduceC l s t: validComp s -> s >(l) t -> s =[]> t.
Proof.
intros. constructor. eapply deClos_correct;eauto.
Qed.
(* relation that tries to capture that two closures 'are the same' *)
Reserved Notation "s '=[]=' t" (at level 70).
Inductive equivC : Comp -> Comp -> Prop :=
| eqC s t: deClos s == deClos t -> s =[]= t
where "s '=[]=' t" := (equivC s t).
Hint Constructors equivC.
Lemma equivC_if s t : s =[]= t -> deClos s == deClos t.
Proof.
now inversion 1.
Qed.
(* ** Properties of the equivalence relation *)
Instance equivC_Equivalence : Equivalence equivC.
Proof.
constructor;repeat intros;constructor.
-reflexivity.
-inv H. now rewrite H0.
-inv H0. inv H. now rewrite H0.
Qed.
Instance equivC_App_proper :
Proper (equivC ==> equivC ==> equivC) CompApp.
Proof.
cbv. intros s s' A t t' B. constructor. simpl. apply equiv_app_proper.
-now inv A.
-now inv B.
Qed.
Lemma CStep_equivC s t: validComp s -> s >[]> t -> s =[]= t.
intros vs R. induction R;repeat inv_validComp.
-now rewrite IHR.
-now rewrite IHR.
-constructor. reflexivity.
-constructor. simpl. rewrite <- minus_n_O. rewrite <-map_nth with (f:= deClos). reflexivity.
-constructor. rewrite deClos_correct'. reflexivity. auto. auto.
Qed.
Lemma starC_equivC s t :
validComp s -> s >[]* t -> s =[]= t.
Proof.
intros vs R. induction R.
-reflexivity.
-rewrite <-IHR.
+eauto using CStep_equivC.
+eauto using validComp_step.
Qed.
*)
Lemma substList_nil s x: substList s x [] = s.
Proof.
revert x. induction s;intros;simpl.
-decide (x>n). reflexivity. now destruct(n-x).
-congruence.
-congruence.
Qed.
(*
Lemma equivC_deClos s : s =[]> CompClos (deClos s) [].
Proof.
constructor. simpl. induction s;simpl.
-now destruct x.
-rewrite IHs1 at 1. rewrite IHs2 at 1. reflexivity.
-now rewrite substList_nil.
Qed.
*)
(*
Goal uniform_confluent CStep.
Proof with try (congruence||(now (subst;tauto))||(now (right;eauto))||(now (right;eauto;eexists;eauto))).
intros s. induction s;intros.
-inv H.
-inv H;inv H0...
+destruct (IHs1 _ _ H4 H3) as [?|[? [? ?]]]...
+destruct (IHs2 _ _ H4 H3) as [?|[? [? ?]]]...
+inv H4; now inv H3.
+inv H3; now inv H4.
-inv H; inv H0...
Qed.*)
(*
Lemma lamComp_noStep l s t : lamComp s -> ~ s>(S l)t.
Proof.
intros H R. remember (S l). revert Heqn. revert H. induction R;intros;try congruence.
destruct i. inv H. inv R.lia. .
Qed.
*)
Lemma validComp_closed s: closed s -> validComp (CompClos s []).
Proof.
intros cs. constructor;simpl;try tauto. now apply closed_dcl.
Qed.
(*
Lemma lamComp_star s t : lamComp s -> s >[]* t -> s = t.
Proof.
intros H R. induction R. auto. now apply lamComp_noStep in H0.
Qed.
Lemma validComp_star s t: validComp s -> s >[]* t -> validComp t.
Proof.
intros vs R. induction R; eauto using validComp_step.
Qed.
*)
(*
Lemma deClos_lam p s: (λ s) = deClos p -> exists t, lamComp t /\ deClos t = (lam s) /\ p >[]* t.
Proof.
revert s. apply Comp_ind_deep with (x:=p);clear p;simpl.
-congruence.
-congruence.
-intros p A IH s eq. destruct p; simpl in eq.
+rewrite <- minus_n_O in eq. change (var n) with (deClos (CompVar n)) in eq. rewrite map_nth in eq. apply IH in eq as [t [? [? R]]]. exists t;repeat split;auto. now rewrite CStepVar. destruct (nth_in_or_default n A (CompVar n)).
*auto.
*rewrite e in eq. simpl in eq. congruence.
+inv eq.
+exists (CompClos (lam p) A). simpl. repeat split;auto. reflexivity.
Qed.
Fixpoint normComp' s A:=
match s with
| app s t => (normComp' s A) (normComp' t A)
| var x => CompClos (var x) A (*nth x A (CompVar x)*)
| lam s => CompClos (lam s) A
end.
Fixpoint normComp s :=
match s with
| CompApp s t => (normComp s) (normComp t)
| CompClos s A => normComp' s A
| s => s
end.
Lemma normComp'_deClos s A: deClos (CompClos s A) = deClos (normComp' s A).
Proof.
induction s;simpl.
-rewrite <- minus_n_O. reflexivity.
-simpl in *. congruence.
-simpl in *. congruence.
Qed.
Lemma normComp_deClos s: deClos s = deClos (normComp s).
Proof.
induction s;simpl.
-auto.
-congruence.
-rewrite <- normComp'_deClos. reflexivity.
Qed.
Lemma normComp'_star s A: CompClos s A >[]* normComp' s A.
Proof.
induction s;simpl;eauto using star.
-rewrite CStepApp. now rewrite IHs1,IHs2.
Qed.
Lemma normComp_star s: s >[]* normComp s.
Proof.
induction s;simpl.
-reflexivity.
-now rewrite <- IHs1,<-IHs2.
-apply normComp'_star.
Qed.
Lemma normComp'_idem s A:normComp (normComp' s A)=normComp' s A.
Proof.
induction s;simpl; congruence.
Qed.
Lemma normComp_idem s: normComp (normComp s)=normComp s.
Proof.
induction s;simpl.
-reflexivity.
-congruence.
-apply normComp'_idem.
Qed.
Lemma normComp'_valid s A: validComp (CompClos s A) -> validComp (normComp' s A).
Proof.
intros vA. induction s;simpl.
-auto.
-inv vA. inv H3. auto.
-auto.
Qed.
Lemma normComp_valid s: validComp s -> validComp (normComp s).
Proof.
intros vs. induction s;simpl.
-auto.
-inv vs. auto.
-apply normComp'_valid. auto.
Qed.
Lemma CompStep_correct2' s t : normComp s = s -> validComp s -> deClos s ≻ t -> exists t', t = deClos t' /\ s >[]* t'.
Proof.
intros nc vs. revert t. induction vs as [s1 s2|]; intros t R.
-simpl in R. inv R;simpl in nc.
+destruct (deClos_lam H0) as [t'[lt' [eq R]]].
destruct (deClos_lam H1) as [u [lu [equ Ru]]].
inv lt'.
exists (CompClos s0 (u::A)). simpl;split.
*rewrite equ. rewrite <-subst_substList. simpl in eq. congruence. apply deClos_validEnv. apply validComp_star in R;auto. inv R. auto.
*rewrite R, Ru. inv lu. rewrite <- CStepVal. reflexivity. auto.
+apply IHvs2 in H2 as [u [? R]]. exists (s1 u). split; simpl. congruence. now rewrite R. congruence.
+apply IHvs1 in H2 as [u [? R]]. exists (u s2). split; simpl. congruence. now rewrite R. congruence.
-destruct s;simpl in nc.
+simpl in R. rewrite <- minus_n_O in R. change (var n) with (deClos (CompVar n)) in R. rewrite map_nth in R. apply H0 in R. destruct R as [t' [? ?]].
*eexists. split. eauto. now rewrite CStepVar.
*apply nth_In. now inv H2.
*destruct (nth_in_or_default n A (CompVar n)). apply H1 in i. inv i. now simpl. rewrite e. reflexivity.
+inv nc.
+simpl in R. inv R.
Qed.
Lemma CompStep_correct2 s t : validComp s -> deClos s ≻ t -> exists t', t = deClos t' /\ s >[]* t'.
Proof.
intros vs R. rewrite normComp_deClos in R. destruct (CompStep_correct2' (normComp_idem s) (normComp_valid vs) R) as [t' [eq R']]. exists t'. split. auto. now rewrite normComp_star.
Qed.
Close Scope L.los.*)
|
#ifndef QUBUS_QTL_KERNEL_HELPERS_HPP
#define QUBUS_QTL_KERNEL_HELPERS_HPP
#include <boost/hana/for_each.hpp>
#include <boost/hana/functional/apply.hpp>
#include <boost/hana/range.hpp>
#include <boost/hana/transform.hpp>
#include <boost/hana/tuple.hpp>
#include <boost/hana/type.hpp>
#include <boost/hana/unpack.hpp>
#include <qubus/util/function_traits.hpp>
#include <functional>
#include <type_traits>
#include <utility>
namespace qubus
{
namespace qtl
{
template <typename Kernel>
struct get_kernel_arg_type_t
{
template <typename Index>
constexpr auto operator()(Index index) const
{
return boost::hana::type_c<util::arg_type<Kernel, Index::value>>;
}
};
template <typename Kernel>
constexpr auto get_kernel_arg_type = get_kernel_arg_type_t<Kernel>{};
struct instantiate_t
{
template <typename Type>
auto operator()(Type type) const
{
using value_type = typename Type::type;
return value_type{};
}
};
constexpr auto instantiate = instantiate_t{};
template <typename Kernel>
constexpr auto instantiate_kernel_args()
{
constexpr std::size_t kernel_arity = util::function_traits<Kernel>::arity;
constexpr auto arg_types = boost::hana::transform(
boost::hana::to_tuple(boost::hana::range_c<std::size_t, 0, kernel_arity>),
get_kernel_arg_type<Kernel>);
return boost::hana::transform(arg_types, instantiate);
}
}
}
#endif
|
module _ where
open import Common.Prelude renaming (_+_ to _+N_)
open import Common.Integer
diff : Nat → Nat → Integer
diff a zero = pos a
diff zero (suc b) = negsuc b
diff (suc a) (suc b) = diff a b
_+_ : Integer → Integer → Integer
pos a + pos b = pos (a +N b)
pos a + negsuc b = diff a (suc b)
negsuc a + pos b = diff b (suc a)
negsuc a + negsuc b = negsuc (suc a +N b)
printInt : Integer → IO Unit
printInt n = putStrLn (intToString n)
main : IO Unit
main = printInt (pos 42 + pos 58) ,,
printInt (pos 42 + negsuc 141) ,,
printInt (pos 42 + negsuc 31) ,,
printInt (negsuc 42 + pos 143) ,,
printInt (negsuc 42 + pos 33) ,,
printInt (negsuc 42 + negsuc 56)
|
[STATEMENT]
lemma Union_in_lattice: "[|M \<subseteq> L; lattice L|] ==> \<Union>M \<in> L"
[PROOF STATE]
proof (prove)
goal (1 subgoal):
1. \<lbrakk>M \<subseteq> L; lattice L\<rbrakk> \<Longrightarrow> \<Union> M \<in> L
[PROOF STEP]
by (simp add: lattice_def)
|
Require Import Crypto.Arithmetic.PrimeFieldTheorems.
Require Import Crypto.Specific.solinas64_2e382m105_7limbs.Synthesis.
(* TODO : change this to field once field isomorphism happens *)
Definition carry :
{ carry : feBW_loose -> feBW_tight
| forall a, phiBW_tight (carry a) = (phiBW_loose a) }.
Proof.
Set Ltac Profiling.
Time synthesize_carry ().
Show Ltac Profile.
Time Defined.
Print Assumptions carry.
|
MODULE GWFLPFMODULE
INTEGER, SAVE, POINTER ::ILPFCB,IWDFLG,IWETIT,IHDWET
INTEGER, SAVE, POINTER ::ISFAC,ICONCV,ITHFLG,NOCVCO,NOVFC
REAL, SAVE, POINTER ::WETFCT
INTEGER, SAVE, POINTER, DIMENSION(:) ::LAYTYP
INTEGER, SAVE, POINTER, DIMENSION(:) ::LAYAVG
REAL, SAVE, POINTER, DIMENSION(:) ::CHANI
INTEGER, SAVE, POINTER, DIMENSION(:) ::LAYVKA
INTEGER, SAVE, POINTER, DIMENSION(:) ::LAYWET
INTEGER, SAVE, POINTER, DIMENSION(:) ::LAYSTRT
INTEGER, SAVE, POINTER, DIMENSION(:,:) ::LAYFLG
REAL, SAVE, POINTER, DIMENSION(:,:,:) ::VKA
REAL, SAVE, POINTER, DIMENSION(:,:,:) ::VKCB
REAL, SAVE, POINTER, DIMENSION(:,:,:) ::SC1
REAL, SAVE, POINTER, DIMENSION(:,:,:) ::SC2
REAL, SAVE, POINTER, DIMENSION(:,:,:) ::HANI
REAL, SAVE, POINTER, DIMENSION(:,:,:) ::WETDRY
REAL, SAVE, POINTER, DIMENSION(:,:,:) ::HK
TYPE GWFLPFTYPE
INTEGER, POINTER ::ILPFCB,IWDFLG,IWETIT,IHDWET
INTEGER, POINTER ::ISFAC,ICONCV,ITHFLG,NOCVCO,NOVFC
REAL, POINTER ::WETFCT
INTEGER, POINTER, DIMENSION(:) ::LAYTYP
INTEGER, POINTER, DIMENSION(:) ::LAYAVG
REAL, POINTER, DIMENSION(:) ::CHANI
INTEGER, POINTER, DIMENSION(:) ::LAYVKA
INTEGER, POINTER, DIMENSION(:) ::LAYWET
INTEGER, POINTER, DIMENSION(:) ::LAYSTRT
INTEGER, POINTER, DIMENSION(:,:) ::LAYFLG
REAL, POINTER, DIMENSION(:,:,:) ::VKA
REAL, POINTER, DIMENSION(:,:,:) ::VKCB
REAL, POINTER, DIMENSION(:,:,:) ::SC1
REAL, POINTER, DIMENSION(:,:,:) ::SC2
REAL, POINTER, DIMENSION(:,:,:) ::HANI
REAL, POINTER, DIMENSION(:,:,:) ::WETDRY
REAL, POINTER, DIMENSION(:,:,:) ::HK
END TYPE
TYPE(GWFLPFTYPE) GWFLPFDAT(10)
END MODULE GWFLPFMODULE
SUBROUTINE GWF2LPF7AR(IN,IGRID)
C ******************************************************************
C ALLOCATE AND READ DATA FOR LAYER PROPERTY FLOW PACKAGE
C ******************************************************************
C
C SPECIFICATIONS:
C ------------------------------------------------------------------
USE GLOBAL, ONLY:NCOL,NROW,NLAY,ITRSS,LAYHDT,LAYHDS,LAYCBD,
1 NCNFBD,IBOUND,BUFF,BOTM,NBOTM,DELR,DELC,IOUT
USE GWFBASMODULE,ONLY:HDRY
USE GWFLPFMODULE,ONLY:ILPFCB,IWDFLG,IWETIT,IHDWET,
1 ISFAC,ICONCV,ITHFLG,NOCVCO,NOVFC,WETFCT,
2 LAYTYP,LAYAVG,CHANI,LAYVKA,LAYWET,LAYSTRT,
3 LAYFLG,VKA,VKCB,SC1,SC2,HANI,WETDRY,HK
C
CHARACTER*14 LAYPRN(5),AVGNAM(3),TYPNAM(2),VKANAM(2),WETNAM(2),
1 HANNAM
DATA AVGNAM/' HARMONIC',' LOGARITHMIC',' LOG-ARITH'/
DATA TYPNAM/' CONFINED',' CONVERTIBLE'/
DATA VKANAM/' VERTICAL K',' ANISOTROPY'/
DATA WETNAM/' NON-WETTABLE',' WETTABLE'/
DATA HANNAM/' VARIABLE'/
CHARACTER*200 LINE
CHARACTER*24 ANAME(9),STOTXT
CHARACTER*4 PTYP
C
DATA ANAME(1) /' HYD. COND. ALONG ROWS'/
DATA ANAME(2) /' HORIZ. ANI. (COL./ROW)'/
DATA ANAME(3) /' VERTICAL HYD. COND.'/
DATA ANAME(4) /' HORIZ. TO VERTICAL ANI.'/
DATA ANAME(5) /'QUASI3D VERT. HYD. COND.'/
DATA ANAME(6) /' SPECIFIC STORAGE'/
DATA ANAME(7) /' SPECIFIC YIELD'/
DATA ANAME(8) /' WETDRY PARAMETER'/
DATA ANAME(9) /' STORAGE COEFFICIENT'/
C ------------------------------------------------------------------
C1------Allocate scalar data.
ALLOCATE(ILPFCB,IWDFLG,IWETIT,IHDWET)
ALLOCATE(ISFAC,ICONCV,ITHFLG,NOCVCO,NOVFC)
ALLOCATE(WETFCT)
ZERO=0.
C
C2------IDENTIFY PACKAGE
WRITE(IOUT,1) IN
1 FORMAT(1X,/1X,'LPF -- LAYER-PROPERTY FLOW PACKAGE, VERSION 7',
1', 5/2/2005',/,9X,'INPUT READ FROM UNIT ',I4)
C
C3------READ COMMENTS AND ITEM 1.
CALL URDCOM(IN,IOUT,LINE)
LLOC=1
CALL URWORD(LINE,LLOC,ISTART,ISTOP,2,ILPFCB,R,IOUT,IN)
CALL URWORD(LINE,LLOC,ISTART,ISTOP,3,I,HDRY,IOUT,IN)
CALL URWORD(LINE,LLOC,ISTART,ISTOP,2,NPLPF,R,IOUT,IN)
C
C3A-----WRITE ITEM 1
IF(ILPFCB.LT.0) WRITE(IOUT,8)
8 FORMAT(1X,'CONSTANT-HEAD CELL-BY-CELL FLOWS WILL BE PRINTED',
1 ' WHEN ICBCFL IS NOT 0')
IF(ILPFCB.GT.0) WRITE(IOUT,9) ILPFCB
9 FORMAT(1X,'CELL-BY-CELL FLOWS WILL BE SAVED ON UNIT ',I4)
WRITE(IOUT,11) HDRY
11 FORMAT(1X,'HEAD AT CELLS THAT CONVERT TO DRY=',1PG13.5)
IF(NPLPF.GT.0) THEN
WRITE(IOUT,15) NPLPF
15 FORMAT(1X,I5,' Named Parameters ')
ELSE
NPLPF=0
WRITE(IOUT,'(A)') ' No named parameters'
END IF
C
C3B-----GET OPTIONS.
ISFAC=0
ICONCV=0
ITHFLG=0
NOCVCO=0
NOVFC=0
NOPCHK=0
STOTXT=ANAME(6)
20 CALL URWORD(LINE,LLOC,ISTART,ISTOP,1,I,R,IOUT,IN)
IF(LINE(ISTART:ISTOP).EQ.'STORAGECOEFFICIENT') THEN
ISFAC=1
STOTXT=ANAME(9)
WRITE(IOUT,21)
21 FORMAT(1X,'STORAGECOEFFICIENT OPTION:',/,
1 1X,'Read storage coefficient rather than specific storage')
ELSE IF(LINE(ISTART:ISTOP).EQ.'CONSTANTCV') THEN
ICONCV=1
WRITE(IOUT,23)
23 FORMAT(1X,'CONSTANTCV OPTION:',/,1X,'Constant vertical',
1 ' conductance for convertible layers')
ELSE IF(LINE(ISTART:ISTOP).EQ.'THICKSTRT') THEN
ITHFLG=1
WRITE(IOUT,25)
25 FORMAT(1X,'THICKSTRT OPTION:',/,1X,'Negative LAYTYP indicates',
1 ' confined layer with thickness computed from STRT-BOT')
ELSE IF(LINE(ISTART:ISTOP).EQ.'NOCVCORRECTION') THEN
NOCVCO=1
WRITE(IOUT,27)
27 FORMAT(1X,'NOCVCORRECTION OPTION:',/,1X,
1 'Do not adjust vertical conductance when applying',
2 ' the vertical flow correction')
ELSE IF(LINE(ISTART:ISTOP).EQ.'NOVFC') THEN
NOVFC=1
NOCVCO=1
WRITE(IOUT,29)
29 FORMAT(1X,'NOVFC OPTION:',/,1X,
1 'Do not apply the vertical flow correction')
ELSE IF(LINE(ISTART:ISTOP).EQ.'NOPARCHECK') THEN
NOPCHK=1
WRITE(IOUT,30)
30 FORMAT(1X,'NOPARCHECK OPTION:',/,1X,
1 'For data defined by parameters, do not check to see if ',
2 'parameters define data at all cells')
END IF
IF(LLOC.LT.200) GO TO 20
C
C4------ALLOCATE AND READ LAYTYP, LAYAVG, CHANI, LAYVKA, LAYWET, LAYSTRT.
ALLOCATE(LAYTYP(NLAY))
ALLOCATE(LAYAVG(NLAY))
ALLOCATE(CHANI(NLAY))
ALLOCATE(LAYVKA(NLAY))
ALLOCATE(LAYWET(NLAY))
ALLOCATE(LAYSTRT(NLAY))
READ(IN,*) (LAYTYP(K),K=1,NLAY)
READ(IN,*) (LAYAVG(K),K=1,NLAY)
READ(IN,*) (CHANI(K),K=1,NLAY)
READ(IN,*) (LAYVKA(K),K=1,NLAY)
READ(IN,*) (LAYWET(K),K=1,NLAY)
C
C4A-----PRINT A TABLE OF VALUES FOR LAYTYP, LAYAVG, CHANI, LAYVKA, LAYWET.
WRITE(IOUT,47)
47 FORMAT(1X,/3X,'LAYER FLAGS:',/1X,
1 'LAYER LAYTYP LAYAVG CHANI ',
2 ' LAYVKA LAYWET',/1X,75('-'))
DO 50 K=1,NLAY
WRITE(IOUT,48) K,LAYTYP(K),LAYAVG(K),CHANI(K),LAYVKA(K),LAYWET(K)
48 FORMAT(1X,I4,2I14,1PE14.3,2I14)
C
C4A1----SET GLOBAL HEAD-DEPENDENT TRANSMISSIVITY AND STORAGE FLAGS.
IF (LAYTYP(K).NE.0) THEN
LAYHDT(K)=1
LAYHDS(K)=1
ELSE
LAYHDT(K)=0
LAYHDS(K)=0
ENDIF
50 CONTINUE
C
C4A2----SET LAYSTRT AND RESET LAYTYP IF THICKSTRT OPTION IS ACTIVE.
DO 60 K=1,NLAY
LAYSTRT(K)=0
IF(LAYTYP(K).LT.0 .AND. ITHFLG.NE.0) THEN
LAYSTRT(K)=1
LAYTYP(K)=0
LAYHDT(K)=0
LAYHDS(K)=0
WRITE(IOUT,57) K
57 FORMAT(1X,'Layer',I5,
1 ' is confined because LAYTYP<0 and THICKSTRT option is active')
END IF
60 CONTINUE
C
C4B-----BASED ON LAYTYP, LAYAVG, CHANI, LAYWET, COUNT THE NUMBER OF EACH
C4B-----TYPE OF 2-D ARRAY; CHECK VALUES FOR CONSISTENCY; AND SETUP
C4B-----POINTERS IN LAYTYP, CHANI, AND LAYWET FOR CONVENIENT ACCESS
C4B-----TO SC2, HANI, and WETDRY. PRINT INTERPRETED VALUES OF FLAGS.
NCNVRT=0
NHANI=0
NWETD=0
WRITE(IOUT,67)
67 FORMAT(1X,/3X,'INTERPRETATION OF LAYER FLAGS:',/1X,
1 ' INTERBLOCK HORIZONTAL',
2 ' DATA IN',/1X,
3 ' LAYER TYPE TRANSMISSIVITY ANISOTROPY',
4 ' ARRAY VKA WETTABILITY',/1X,
5 'LAYER (LAYTYP) (LAYAVG) (CHANI)',
6 ' (LAYVKA) (LAYWET)',/1X,75('-'))
DO 100 K=1,NLAY
IF(LAYTYP(K).NE.0) THEN
NCNVRT=NCNVRT+1
LAYTYP(K)=NCNVRT
END IF
IF(CHANI(K).LE.ZERO) THEN
NHANI=NHANI+1
CHANI(K)=-NHANI
END IF
IF(LAYWET(K).NE.0) THEN
IF(LAYTYP(K).EQ.0) THEN
WRITE(IOUT,*)
1 ' LAYWET is not 0 and LAYTYP is 0 for layer:',K
WRITE(IOUT,*) ' LAYWET must be 0 if LAYTYP is 0'
CALL USTOP(' ')
ELSE
NWETD=NWETD+1
LAYWET(K)=NWETD
END IF
END IF
IF(LAYAVG(K).LT.0 .OR. LAYAVG(K).GT.2) THEN
WRITE(IOUT,74) LAYAVG(K)
74 FORMAT(1X,I8,
1 ' IS AN INVALID LAYAVG VALUE -- MUST BE 0, 1, or 2')
CALL USTOP(' ')
END IF
LAYPRN(1)=TYPNAM(1)
IF(LAYTYP(K).NE.0) LAYPRN(1)=TYPNAM(2)
LAYPRN(2)=AVGNAM(LAYAVG(K)+1)
IF(CHANI(K).LE.0) THEN
LAYPRN(3)=HANNAM
ELSE
WRITE(LAYPRN(3),'(1PE14.3)') CHANI(K)
END IF
LAYPRN(4)=VKANAM(1)
IF(LAYVKA(K).NE.0) LAYPRN(4)=VKANAM(2)
LAYPRN(5)=WETNAM(1)
IF(LAYWET(K).NE.0) LAYPRN(5)=WETNAM(2)
WRITE(IOUT,78) K,(LAYPRN(I),I=1,5)
78 FORMAT(1X,I4,5A)
100 CONTINUE
C
C4C-----PRINT WETTING INFORMATION.
IF(NWETD.EQ.0) THEN
WRITE(IOUT,13)
13 FORMAT(1X,/,1X,'WETTING CAPABILITY IS NOT ACTIVE IN ANY LAYER')
IWDFLG=0
ELSE
WRITE(IOUT,12) NWETD
12 FORMAT(1X,/,1X,'WETTING CAPABILITY IS ACTIVE IN',I4,' LAYERS')
IWDFLG=1
READ(IN,*) WETFCT,IWETIT,IHDWET
IF(IWETIT.LE.0) IWETIT=1
WRITE(IOUT,*) ' WETTING FACTOR=',WETFCT
WRITE(IOUT,*) ' WETTING ITERATION INTERVAL=',IWETIT
WRITE(IOUT,*) ' IHDWET=',IHDWET
END IF
C
C5------ALLOCATE MEMORY FOR ARRAYS.
ALLOCATE(LAYFLG(6,NLAY))
ALLOCATE(HK(NCOL,NROW,NLAY))
ALLOCATE(VKA(NCOL,NROW,NLAY))
IF(NCNFBD.GT.0) THEN
ALLOCATE(VKCB(NCOL,NROW,NCNFBD))
ELSE
ALLOCATE(VKCB(1,1,1))
END IF
IF(ITRSS.NE.0) THEN
ALLOCATE(SC1(NCOL,NROW,NLAY))
ELSE
ALLOCATE(SC1(1,1,1))
END IF
IF(ITRSS.NE.0 .AND. NCNVRT.GT.0) THEN
ALLOCATE(SC2(NCOL,NROW,NCNVRT))
ELSE
ALLOCATE(SC2(1,1,1))
END IF
IF(NHANI.GT.0) THEN
ALLOCATE(HANI(NCOL,NROW,NHANI))
ELSE
ALLOCATE(HANI(1,1,1))
END IF
IF(NWETD.GT.0) THEN
ALLOCATE(WETDRY(NCOL,NROW,NWETD))
ELSE
ALLOCATE(WETDRY(1,1,1))
END IF
C
C6------READ PARAMETER DEFINITIONS
NPHK=0
NPVKCB=0
NPVK=0
NPVANI=0
NPSS=0
NPSY=0
NPHANI=0
IF(NPLPF.GT.0) THEN
WRITE(IOUT,115)
115 FORMAT(/,' PARAMETERS DEFINED IN THE LPF PACKAGE')
DO 120 K=1,NPLPF
CALL UPARARRRP(IN,IOUT,N,1,PTYP,1,0,-1)
C Note that NPHK and the other NP variables in
C this group are used only as flags, not counts
IF(PTYP.EQ.'HK') THEN
NPHK=1
ELSE IF(PTYP.EQ.'HANI') THEN
C6A-----WHEN A HANI PARAMETER IS USED, THEN ALL HORIZONTAL ANISOTROPY
C6A-----MUST BE DEFINED USING PARAMETERS. ENSURE THAT ALL CHANI <= 0
DO 118 I = 1, NLAY
IF (CHANI(I).GT.0.0) THEN
WRITE(IOUT,117)
117 FORMAT(/,
&' ERROR: WHEN A HANI PARAMETER IS USED, CHANI FOR ALL LAYERS',/,
&' MUST BE LESS THAN OR EQUAL TO 0.0 -- STOP EXECUTION',
&' (GWF2LPF7AR)')
CALL USTOP(' ')
ENDIF
118 CONTINUE
NPHANI=1
ELSE IF(PTYP.EQ.'VKCB') THEN
NPVKCB=1
ELSE IF(PTYP.EQ.'VK') THEN
NPVK=1
CALL SGWF2LPF7CK(IOUT,N,'VK ')
ELSE IF(PTYP.EQ.'VANI') THEN
NPVANI=1
CALL SGWF2LPF7CK(IOUT,N,'VANI')
ELSE IF(PTYP.EQ.'SS') THEN
NPSS=1
ELSE IF(PTYP.EQ.'SY') THEN
NPSY=1
ELSE
WRITE(IOUT,*) ' Invalid parameter type for LPF Package'
CALL USTOP(' ')
END IF
120 CONTINUE
END IF
C
C7------DEFINE DATA FOR EACH LAYER -- VIA READING OR NAMED PARAMETERS.
DO 200 K=1,NLAY
KK=K
C
C7A-----DEFINE HORIZONTAL HYDRAULIC CONDUCTIVITY (HK)
IF(NPHK.EQ.0) THEN
CALL U2DREL(HK(:,:,KK),ANAME(1),NROW,NCOL,KK,IN,IOUT)
ELSE
READ(IN,*) LAYFLG(1,K)
WRITE(IOUT,121) ANAME(1),K,LAYFLG(1,K)
121 FORMAT(1X,/1X,A,' FOR LAYER',I4,
1 ' WILL BE DEFINED BY PARAMETERS',/1X,'(PRINT FLAG=',I4,')')
CALL UPARARRSUB1(HK(:,:,KK),NCOL,NROW,KK,'HK',
1 IOUT,ANAME(1),LAYFLG(1,KK))
IF(NOPCHK.EQ.0) CALL UPARARRCK(BUFF,IBOUND,IOUT,K,NCOL,NLAY,
1 NROW,'HK ')
END IF
C
C7B-----READ HORIZONTAL ANISOTROPY IF CHANI IS NON-ZERO
IF(CHANI(K).LE.ZERO) THEN
KHANI=-CHANI(K)
IF(NPHANI.EQ.0) THEN
CALL U2DREL(HANI(:,:,KHANI),ANAME(2),NROW,NCOL,KK,IN,IOUT)
ELSE
READ(IN,*) LAYFLG(6,K)
WRITE(IOUT,121) ANAME(2),K,LAYFLG(6,K)
CALL UPARARRSUB1(HANI(:,:,KHANI),NCOL,NROW,KK,'HANI',
1 IOUT,ANAME(2),LAYFLG(6,KK))
IF(NOPCHK.EQ.0) CALL UPARARRCK(BUFF,IBOUND,IOUT,K,NCOL,
1 NLAY,NROW,'HANI')
END IF
END IF
C
C7C-----DEFINE VERTICAL HYDRAULIC CONDUCTIVITY OR HORIZONTAL TO VERTICAL
C7C-----ANISOTROPY (VKA).
IANAME=3
PTYP='VK'
IF(LAYVKA(K).NE.0) THEN
IANAME=4
PTYP='VANI'
END IF
IF(NPVK.EQ.0 .AND. NPVANI.EQ.0) THEN
CALL U2DREL(VKA(:,:,KK),ANAME(IANAME),NROW,NCOL,KK,IN,IOUT)
ELSE
READ(IN,*) LAYFLG(2,K)
WRITE(IOUT,121) ANAME(IANAME),K,LAYFLG(2,K)
CALL UPARARRSUB1(VKA(:,:,KK),NCOL,NROW,KK,PTYP,IOUT,
1 ANAME(IANAME),LAYFLG(2,KK))
IF(NOPCHK.EQ.0) CALL UPARARRCK(BUFF,IBOUND,IOUT,K,NCOL,NLAY,
1 NROW,PTYP)
END IF
C
C7D-----DEFINE SPECIFIC STORAGE OR STORAGE COEFFICIENT IN ARRAY SC1 IF TRANSIENT.
IF(ITRSS.NE.0) THEN
IF(NPSS.EQ.0) THEN
CALL U2DREL(SC1(:,:,KK),STOTXT,NROW,NCOL,KK,IN,IOUT)
ELSE
READ(IN,*) LAYFLG(3,K)
WRITE(IOUT,121) STOTXT,K,LAYFLG(3,K)
CALL UPARARRSUB1(SC1(:,:,KK),NCOL,NROW,KK,'SS',
1 IOUT,STOTXT,LAYFLG(3,KK))
IF(NOPCHK.EQ.0) CALL UPARARRCK(BUFF,IBOUND,IOUT,K,NCOL,
1 NLAY,NROW,'SS ')
END IF
IF(ISFAC.EQ.0) THEN
CALL SGWF2LPF7SC(SC1(:,:,KK),KK,1)
ELSE
CALL SGWF2LPF7SC(SC1(:,:,KK),KK,0)
END IF
END IF
C
C7E-----DEFINE SPECIFIC YIELD IN ARRAY SC2 IF TRANSIENT AND LAYER IS
C7E-----IS CONVERTIBLE.
IF(LAYTYP(K).NE.0) THEN
IF(ITRSS.NE.0) THEN
IF(NPSY.EQ.0) THEN
CALL U2DREL(SC2(:,:,LAYTYP(K)),ANAME(7),NROW,NCOL,KK,IN,
1 IOUT)
ELSE
READ(IN,*) LAYFLG(4,K)
WRITE(IOUT,121) ANAME(7),K,LAYFLG(4,K)
CALL UPARARRSUB1(SC2(:,:,LAYTYP(K)),NCOL,
1 NROW,KK,'SY',IOUT,ANAME(7),LAYFLG(4,KK))
IF(NOPCHK.EQ.0) CALL UPARARRCK(BUFF,IBOUND,IOUT,K,
1 NCOL,NLAY,NROW,'SY ')
END IF
CALL SGWF2LPF7SC(SC2(:,:,LAYTYP(K)),KK,0)
END IF
END IF
C
C7F-----READ CONFINING BED VERTICAL HYDRAULIC CONDUCTIVITY (VKCB)
IF(LAYCBD(K).NE.0) THEN
IF(NPVKCB.EQ.0) THEN
CALL U2DREL(VKCB(:,:,LAYCBD(K)),ANAME(5),NROW,NCOL,KK,IN,
1 IOUT)
ELSE
READ(IN,*) LAYFLG(5,K)
WRITE(IOUT,121) ANAME(5),K,LAYFLG(5,K)
CALL UPARARRSUB1(VKCB(:,:,LAYCBD(K)),NCOL,NROW,KK,
1 'VKCB',IOUT,ANAME(5),LAYFLG(5,KK))
IF(NOPCHK.EQ.0) CALL UPARARRCK(BUFF,IBOUND,IOUT,K,NCOL,
1 NLAY,NROW,'VKCB')
END IF
END IF
C
C7G-----READ WETDRY CODES IF WETTING CAPABILITY HAS BEEN INVOKED
C7G-----(LAYWET NOT 0).
IF(LAYWET(K).NE.0) THEN
CALL U2DREL(WETDRY(:,:,LAYWET(K)),ANAME(8),NROW,NCOL,KK,IN,
1 IOUT)
END IF
200 CONTINUE
C
C8------PREPARE AND CHECK LPF DATA.
CALL SGWF2LPF7N()
C
C9------RETURN
CALL GWF2LPF7PSV(IGRID)
RETURN
END
SUBROUTINE GWF2LPF7AD(KPER,IGRID)
C ******************************************************************
C SET HOLD TO BOTM WHENEVER A WETTABLE CELL IS DRY
C ******************************************************************
C
C SPECIFICATIONS:
C ------------------------------------------------------------------
USE GLOBAL, ONLY:NCOL,NROW,NLAY,ISSFLG,IBOUND,HOLD,BOTM,LBOTM
USE GWFLPFMODULE,ONLY:LAYWET,WETDRY
C ------------------------------------------------------------------
C
CALL SGWF2LPF7PNT(IGRID)
ISS=ISSFLG(KPER)
C
C1------RETURN IF STEADY STATE.
IF(ISS.NE.0) RETURN
C
C2------LOOP THROUGH ALL LAYERS TO SET HOLD=BOT IF A WETTABLE CELL IS DRY
ZERO=0.
DO 100 K=1,NLAY
C
C2A-----SKIP LAYERS THAT CANNOT CONVERT BETWEEN WET AND DRY
IF(LAYWET(K).EQ.0) GO TO 100
DO 90 I=1,NROW
DO 90 J=1,NCOL
C
C2B-----SKIP CELLS THAT ARE CURRENTLY WET OR ARE NOT WETTABLE
IF(IBOUND(J,I,K).NE.0) GO TO 90
IF(WETDRY(J,I,LAYWET(K)).EQ.ZERO) GO TO 90
C
C2C-----SET HOLD=BOT
HOLD(J,I,K)=BOTM(J,I,LBOTM(K))
90 CONTINUE
100 CONTINUE
C
C3-----RETURN
RETURN
END
SUBROUTINE GWF2LPF7FM(KITER,KSTP,KPER,IGRID)
C ******************************************************************
C ADD LEAKAGE CORRECTION AND STORAGE TO HCOF AND RHS, AND CALCULATE
C CONDUCTANCE AS REQUIRED.
C ******************************************************************
C
C SPECIFICATIONS:
C ------------------------------------------------------------------
USE GLOBAL, ONLY:NCOL,NROW,NLAY,IBOUND,BOTM,NBOTM,DELR,DELC,
1 LBOTM,CV,HNEW,RHS,HCOF,HOLD,ISSFLG,IOUT
USE GWFBASMODULE,ONLY:DELT
USE GWFLPFMODULE,ONLY:LAYTYP,SC1,SC2,NOVFC
C ------------------------------------------------------------------
C
C1------SET POINTERS TO DATA, GET STEADY-STATE FLAG FOR STRESS PERIOD,
C1------DEFINE CONSTANT.
CALL SGWF2LPF7PNT(IGRID)
ISS=ISSFLG(KPER)
ONE=1.
C
C2------FOR EACH LAYER: IF CONVERTIBLE, CALCULATE CONDUCTANCES.
DO 100 K=1,NLAY
KK=K
IF(LAYTYP(K).NE.0)
1 CALL SGWF2LPF7HCOND(KK,KITER,KSTP,KPER)
100 CONTINUE
DO 101 K=1,NLAY
KK=K
IF(K.NE.NLAY) THEN
IF(LAYTYP(K).NE.0 .OR. LAYTYP(K+1).NE.0)
1 CALL SGWF2LPF7VCOND(KK)
END IF
101 CONTINUE
C
C3------IF THE STRESS PERIOD IS TRANSIENT, ADD STORAGE TO HCOF AND RHS
IF(ISS.EQ.0) THEN
TLED=ONE/DELT
DO 200 K=1,NLAY
C
C4------SEE IF THIS LAYER IS CONVERTIBLE OR NON-CONVERTIBLE.
IF(LAYTYP(K).EQ.0) THEN
C5------NON-CONVERTIBLE LAYER, SO USE PRIMARY STORAGE
DO 140 I=1,NROW
DO 140 J=1,NCOL
IF(IBOUND(J,I,K).LE.0) GO TO 140
RHO=SC1(J,I,K)*TLED
HCOF(J,I,K)=HCOF(J,I,K)-RHO
RHS(J,I,K)=RHS(J,I,K)-RHO*HOLD(J,I,K)
140 CONTINUE
ELSE
C
C6------A CONVERTIBLE LAYER, SO CHECK OLD AND NEW HEADS TO DETERMINE
C6------WHEN TO USE PRIMARY AND SECONDARY STORAGE
DO 180 I=1,NROW
DO 180 J=1,NCOL
C
C6A-----IF THE CELL IS EXTERNAL THEN SKIP IT.
IF(IBOUND(J,I,K).LE.0) GO TO 180
TP=BOTM(J,I,LBOTM(K)-1)
RHO2=SC2(J,I,LAYTYP(K))*TLED
RHO1=SC1(J,I,K)*TLED
C
C6B-----FIND STORAGE FACTOR AT START OF TIME STEP.
SOLD=RHO2
IF(HOLD(J,I,K).GT.TP) SOLD=RHO1
C
C6C-----FIND STORAGE FACTOR AT END OF TIME STEP.
HTMP=HNEW(J,I,K)
SNEW=RHO2
IF(HTMP.GT.TP) SNEW=RHO1
C
C6D-----ADD STORAGE TERMS TO RHS AND HCOF.
HCOF(J,I,K)=HCOF(J,I,K)-SNEW
RHS(J,I,K)=RHS(J,I,K) - SOLD*(HOLD(J,I,K)-TP) - SNEW*TP
C
180 CONTINUE
END IF
C
200 CONTINUE
END IF
C
C7------FOR EACH LAYER DETERMINE IF CORRECTION TERMS ARE NEEDED FOR
C7------FLOW DOWN INTO PARTIALLY SATURATED LAYERS.
IF(NOVFC.EQ.0) THEN
DO 300 K=1,NLAY
C
C8------SEE IF CORRECTION IS NEEDED FOR LEAKAGE FROM ABOVE.
IF(LAYTYP(K).NE.0 .AND. K.NE.1) THEN
C
C8A-----FOR EACH CELL MAKE THE CORRECTION IF NEEDED.
DO 220 I=1,NROW
DO 220 J=1,NCOL
C
C8B-----IF THE CELL IS EXTERNAL(IBOUND<=0) THEN SKIP IT.
IF(IBOUND(J,I,K).LE.0) GO TO 220
HTMP=HNEW(J,I,K)
C
C8C-----IF HEAD IS ABOVE TOP THEN CORRECTION NOT NEEDED
TOP=BOTM(J,I,LBOTM(K)-1)
IF(HTMP.GE.TOP) GO TO 220
C
C8D-----WITH HEAD BELOW TOP ADD CORRECTION TERMS TO RHS.
RHS(J,I,K)=RHS(J,I,K) + CV(J,I,K-1)*(TOP-HTMP)
220 CONTINUE
END IF
C
C9------SEE IF THIS LAYER MAY NEED CORRECTION FOR LEAKAGE TO BELOW.
IF(K.EQ.NLAY) GO TO 300
IF(LAYTYP(K+1).NE.0) THEN
C
C9A-----FOR EACH CELL MAKE THE CORRECTION IF NEEDED.
DO 280 I=1,NROW
DO 280 J=1,NCOL
C
C9B-----IF CELL IS EXTERNAL (IBOUND<=0) THEN SKIP IT.
IF(IBOUND(J,I,K).LE.0) GO TO 280
C
C9C-----IF HEAD IN THE LOWER CELL IS LESS THAN TOP ADD CORRECTION
C9C-----TERM TO RHS.
HTMP=HNEW(J,I,K+1)
TOP=BOTM(J,I,LBOTM(K+1)-1)
IF(HTMP.LT.TOP) RHS(J,I,K)=RHS(J,I,K)- CV(J,I,K)*(TOP-HTMP)
280 CONTINUE
END IF
C
300 CONTINUE
END IF
C
C10-----RETURN
RETURN
END
SUBROUTINE SGWF2LPF7N()
C ******************************************************************
C INITIALIZE AND CHECK LPF DATA
C ******************************************************************
C
C SPECIFICATIONS:
C ------------------------------------------------------------------
USE GLOBAL, ONLY:NCOL,NROW,NLAY,IBOUND,HNEW,LAYCBD,CV,
1 BOTM,NBOTM,DELR,DELC,IOUT
USE GWFBASMODULE,ONLY:HNOFLO
USE GWFLPFMODULE,ONLY:LAYWET,WETDRY,HK,VKCB,LAYTYP,VKA
C ------------------------------------------------------------------
C
C1------DEFINE CONSTANTS.
ZERO=0.
HCNV=HNOFLO
C
C2------INSURE THAT EACH ACTIVE CELL HAS AT LEAST ONE NON-ZERO
C2------TRANSMISSIVE PARAMETER.
DO 60 K=1,NLAY
IF(LAYWET(K).NE.0) THEN
C
C3------WETTING IS ACTIVE.
DO 40 I=1,NROW
DO 40 J=1,NCOL
IF(IBOUND(J,I,K).EQ.0 .AND. WETDRY(J,I,LAYWET(K)).EQ.ZERO)
1 GO TO 40
C
C3A-----CHECK HORIZONTAL HYDRAULIC CONDUCTIVITY (HK).
IF(HK(J,I,K).NE.ZERO) GO TO 40
C
C3B-----CHECK VERTICAL HYDRAULIC CONDUCTIVITY AND CONFINING BED
C3B-----VERTICAL HYDRAULIC CONDUCTIVITY.
IF(NLAY.GT.1) THEN
IF(VKA(J,I,K).NE.ZERO) THEN
IF(K.NE.NLAY) THEN
IF (VKA(J,I,K+1).NE.ZERO) THEN
IF(LAYCBD(K).NE.0) THEN
IF(VKCB(J,I,LAYCBD(K)).NE.ZERO) GO TO 40
ELSE
GO TO 40
END IF
END IF
END IF
IF(K.NE.1) THEN
IF (VKA(J,I,K-1).NE.ZERO) THEN
IF (LAYCBD(K-1).NE.0) THEN
IF(VKCB(J,I,LAYCBD(K-1)).NE.ZERO) GO TO 40
ELSE
GO TO 40
END IF
ENDIF
END IF
END IF
END IF
C
C3C-----ALL TRANSMISSIVE TERMS ARE ALL 0, SO CONVERT CELL TO NO FLOW.
IBOUND(J,I,K)=0
HNEW(J,I,K)=HCNV
WETDRY(J,I,LAYWET(K))=ZERO
WRITE(IOUT,43) K,I,J
40 CONTINUE
C
ELSE
C
C4------WETTING IS INACTIVE
DO 50 I=1,NROW
DO 50 J=1,NCOL
IF(IBOUND(J,I,K).EQ.0) GO TO 50
C
C4A-----CHECK HORIZONTAL HYDRAULIC CONDUCTIVITY (HK).
IF(HK(J,I,K).NE.ZERO) GO TO 50
C
C4B-----CHECK VERTICAL HYDRAULIC CONDUCTIVITY AND CONFINING BED
C4B-----VERTICAL HYDRAULIC CONDUCTIVITY.
IF(NLAY.GT.1) THEN
IF(VKA(J,I,K).NE.ZERO) THEN
IF(K.NE.NLAY) THEN
IF (VKA(J,I,K+1).NE.ZERO) THEN
IF(LAYCBD(K).NE.0) THEN
IF(VKCB(J,I,LAYCBD(K)).NE.ZERO) GO TO 50
ELSE
GO TO 50
END IF
END IF
END IF
IF(K.NE.1) THEN
IF (VKA(J,I,K-1).NE.ZERO) THEN
IF (LAYCBD(K-1).NE.0) THEN
IF(VKCB(J,I,LAYCBD(K-1)).NE.ZERO) GO TO 50
ELSE
GO TO 50
END IF
ENDIF
END IF
END IF
END IF
C
C4C-----ALL TRANSMISSIVE TERMS ARE 0, SO CONVERT CELL TO NO FLOW.
IBOUND(J,I,K)=0
HNEW(J,I,K)=HCNV
WRITE(IOUT,43) K,I,J
43 FORMAT(1X,'NODE (LAYER,ROW,COL) ',I3,2(1X,I5),
1 ' ELIMINATED BECAUSE ALL HYDRAULIC',/,
2 ' CONDUCTIVITIES TO NODE ARE 0')
50 CONTINUE
END IF
60 CONTINUE
C
C5------CALCULATE HOR. CONDUCTANCE(CR AND CC) FOR CONSTANT T LAYERS.
DO 70 K=1,NLAY
KK=K
IF(LAYTYP(K).EQ.0) CALL SGWF2LPF7HCOND(KK,0,0,0)
70 CONTINUE
C
C6------CALCULATE VERTICAL CONDUCTANCE BETWEEN CONFINED LAYERS.
IF(NLAY.GT.1) THEN
DO 10 K=1,NLAY-1
KK=K
IF(LAYTYP(K).EQ.0 .AND. LAYTYP(K+1).EQ.0)
1 CALL SGWF2LPF7VCOND(KK)
10 CONTINUE
END IF
C
C7------RETURN.
RETURN
END
SUBROUTINE GWF2LPF7BDADJ(KSTP,KPER,IDIR,IBDRET,
1 IC1,IC2,IR1,IR2,IL1,IL2,IGRID)
C ******************************************************************
C COMPUTE FLOW BETWEEN ADJACENT CELLS IN A SUBREGION OF THE GRID
C ******************************************************************
C
C SPECIFICATIONS:
C ------------------------------------------------------------------
USE GLOBAL, ONLY:NCOL,NROW,NLAY,IBOUND,HNEW,BUFF,CR,CC,CV,
1 BOTM,LBOTM,IOUT
USE GWFBASMODULE,ONLY:ICBCFL,DELT,PERTIM,TOTIM,ICHFLG
USE GWFLPFMODULE,ONLY:ILPFCB,LAYTYP,NOVFC
C
CHARACTER*16 TEXT(3)
DOUBLE PRECISION HD
C
DATA TEXT(1),TEXT(2),TEXT(3)
1 /'FLOW RIGHT FACE ','FLOW FRONT FACE ','FLOW LOWER FACE '/
C ------------------------------------------------------------------
C
CALL SGWF2LPF7PNT(IGRID)
C
C1------IF CELL-BY-CELL FLOWS WILL BE SAVED IN A FILE, SET FLAG IBD.
C1------RETURN IF FLOWS ARE NOT BEING SAVED OR RETURNED.
ZERO=0.
IBD=0
IF(ILPFCB.GT.0) IBD=ICBCFL
IF(IBD.EQ.0 .AND. IBDRET.EQ.0) RETURN
C
C2------SET THE SUBREGION EQUAL TO THE ENTIRE GRID IF VALUES ARE BEING
C2------SAVED IN A FILE.
IF(IBD.NE.0) THEN
K1=1
K2=NLAY
I1=1
I2=NROW
J1=1
J2=NCOL
END IF
C
C3------TEST FOR DIRECTION OF CALCULATION; IF NOT ACROSS COLUMNS, GO TO
C3------STEP 4. IF ONLY 1 COLUMN, RETURN.
IF(IDIR.NE.1) GO TO 405
IF(NCOL.EQ.1) RETURN
C
C3A-----CALCULATE FLOW ACROSS COLUMNS (THROUGH RIGHT FACE). IF NOT
C3A-----SAVING IN A FILE, SET THE SUBREGION. CLEAR THE BUFFER.
IF(IBD.EQ.0) THEN
K1=IL1
K2=IL2
I1=IR1
I2=IR2
J1=IC1-1
IF(J1.LT.1) J1=1
J2=IC2
END IF
DO 310 K=K1,K2
DO 310 I=I1,I2
DO 310 J=J1,J2
BUFF(J,I,K)=ZERO
310 CONTINUE
C
C3B-----FOR EACH CELL CALCULATE FLOW THRU RIGHT FACE & STORE IN BUFFER.
IF(J2.EQ.NCOL) J2=J2-1
DO 400 K=K1,K2
DO 400 I=I1,I2
DO 400 J=J1,J2
IF(ICHFLG.EQ.0) THEN
IF((IBOUND(J,I,K).LE.0) .AND. (IBOUND(J+1,I,K).LE.0)) GO TO 400
ELSE
IF((IBOUND(J,I,K).EQ.0) .OR. (IBOUND(J+1,I,K).EQ.0)) GO TO 400
END IF
HDIFF=HNEW(J,I,K)-HNEW(J+1,I,K)
BUFF(J,I,K)=HDIFF*CR(J,I,K)
400 CONTINUE
C
C3C-----RECORD CONTENTS OF BUFFER AND RETURN.
IF(IBD.EQ.1)
1 CALL UBUDSV(KSTP,KPER,TEXT(1),ILPFCB,BUFF,NCOL,NROW,NLAY,IOUT)
IF(IBD.EQ.2) CALL UBDSV1(KSTP,KPER,TEXT(1),ILPFCB,BUFF,NCOL,NROW,
1 NLAY,IOUT,DELT,PERTIM,TOTIM,IBOUND)
RETURN
C
C4------TEST FOR DIRECTION OF CALCULATION; IF NOT ACROSS ROWS, GO TO
C4------STEP 5. IF ONLY 1 ROW, RETURN.
405 IF(IDIR.NE.2) GO TO 505
IF(NROW.EQ.1) RETURN
C
C4A-----CALCULATE FLOW ACROSS ROWS (THROUGH FRONT FACE). IF NOT SAVING
C4A-----IN A FILE, SET THE SUBREGION. CLEAR THE BUFFER.
IF(IBD.EQ.0) THEN
K1=IL1
K2=IL2
I1=IR1-1
IF(I1.LT.1) I1=1
I2=IR2
J1=IC1
J2=IC2
END IF
DO 410 K=K1,K2
DO 410 I=I1,I2
DO 410 J=J1,J2
BUFF(J,I,K)=ZERO
410 CONTINUE
C
C4B-----FOR EACH CELL CALCULATE FLOW THRU FRONT FACE & STORE IN BUFFER.
IF(I2.EQ.NROW) I2=I2-1
DO 500 K=K1,K2
DO 500 I=I1,I2
DO 500 J=J1,J2
IF(ICHFLG.EQ.0) THEN
IF((IBOUND(J,I,K).LE.0) .AND. (IBOUND(J,I+1,K).LE.0)) GO TO 500
ELSE
IF((IBOUND(J,I,K).EQ.0) .OR. (IBOUND(J,I+1,K).EQ.0)) GO TO 500
END IF
HDIFF=HNEW(J,I,K)-HNEW(J,I+1,K)
BUFF(J,I,K)=HDIFF*CC(J,I,K)
500 CONTINUE
C
C4C-----RECORD CONTENTS OF BUFFER AND RETURN.
IF(IBD.EQ.1)
1 CALL UBUDSV(KSTP,KPER,TEXT(2),ILPFCB,BUFF,NCOL,NROW,NLAY,IOUT)
IF(IBD.EQ.2) CALL UBDSV1(KSTP,KPER,TEXT(2),ILPFCB,BUFF,NCOL,NROW,
1 NLAY,IOUT,DELT,PERTIM,TOTIM,IBOUND)
RETURN
C
C5------DIRECTION OF CALCULATION IS ACROSS LAYERS BY ELIMINATION. IF
C5------ONLY 1 LAYER, RETURN.
505 IF(NLAY.EQ.1) RETURN
C
C5A-----CALCULATE FLOW ACROSS LAYERS (THROUGH LOWER FACE). IF NOT
C5A-----SAVING IN A FILE, SET THE SUBREGION. CLEAR THE BUFFER.
IF(IBD.EQ.0) THEN
K1=IL1-1
IF(K1.LT.1) K1=1
K2=IL2
I1=IR1
I2=IR2
J1=IC1
J2=IC2
END IF
DO 510 K=K1,K2
DO 510 I=I1,I2
DO 510 J=J1,J2
BUFF(J,I,K)=ZERO
510 CONTINUE
C
C5B-----FOR EACH CELL CALCULATE FLOW THRU LOWER FACE & STORE IN BUFFER.
IF(K2.EQ.NLAY) K2=K2-1
DO 600 K=1,K2
IF(K.LT.K1) GO TO 600
DO 590 I=I1,I2
DO 590 J=J1,J2
IF(ICHFLG.EQ.0) THEN
IF((IBOUND(J,I,K).LE.0) .AND. (IBOUND(J,I,K+1).LE.0)) GO TO 590
ELSE
IF((IBOUND(J,I,K).EQ.0) .OR. (IBOUND(J,I,K+1).EQ.0)) GO TO 590
END IF
HD=HNEW(J,I,K+1)
IF(NOVFC.NE.0 .OR. LAYTYP(K+1).EQ.0) GO TO 580
TMP=HD
TOP=BOTM(J,I,LBOTM(K+1)-1)
IF(TMP.LT.TOP) HD=TOP
580 HDIFF=HNEW(J,I,K)-HD
BUFF(J,I,K)=HDIFF*CV(J,I,K)
590 CONTINUE
600 CONTINUE
C
C5C-----RECORD CONTENTS OF BUFFER AND RETURN.
IF(IBD.EQ.1)
1 CALL UBUDSV(KSTP,KPER,TEXT(3),ILPFCB,BUFF,NCOL,NROW,NLAY,IOUT)
IF(IBD.EQ.2) CALL UBDSV1(KSTP,KPER,TEXT(3),ILPFCB,BUFF,NCOL,NROW,
1 NLAY,IOUT,DELT,PERTIM,TOTIM,IBOUND)
RETURN
END
SUBROUTINE GWF2LPF7BDS(KSTP,KPER,IGRID)
C ******************************************************************
C COMPUTE STORAGE BUDGET FLOW TERM FOR LPF.
C ******************************************************************
C
C SPECIFICATIONS:
C ------------------------------------------------------------------
USE GLOBAL, ONLY:NCOL,NROW,NLAY,ISSFLG,IBOUND,HNEW,HOLD,
1 BUFF,BOTM,LBOTM,IOUT
USE GWFBASMODULE,ONLY:MSUM,ICBCFL,VBVL,VBNM,DELT,PERTIM,TOTIM
USE GWFLPFMODULE,ONLY:ILPFCB,LAYTYP,SC1,SC2
CHARACTER*16 TEXT
DOUBLE PRECISION STOIN,STOUT,SSTRG
C
DATA TEXT /' STORAGE'/
C ------------------------------------------------------------------
C
CALL SGWF2LPF7PNT(IGRID)
C
C1------INITIALIZE BUDGET ACCUMULATORS AND 1/DELT.
ISS=ISSFLG(KPER)
ZERO=0.
STOIN=ZERO
STOUT=ZERO
C2------IF STEADY STATE, STORAGE TERM IS ZERO
IF(ISS.NE.0) GOTO 400
ONE=1.
TLED=ONE/DELT
C
C3------IF CELL-BY-CELL FLOWS WILL BE SAVED, SET FLAG IBD.
IBD=0
IF(ILPFCB.GT.0) IBD=ICBCFL
C
C4------CLEAR BUFFER.
DO 210 K=1,NLAY
DO 210 I=1,NROW
DO 210 J=1,NCOL
BUFF(J,I,K)=ZERO
210 CONTINUE
C
C5------LOOP THROUGH EVERY CELL IN THE GRID.
KT=0
DO 300 K=1,NLAY
LC=LAYTYP(K)
IF(LC.NE.0) KT=KT+1
DO 300 I=1,NROW
DO 300 J=1,NCOL
C
C6------SKIP NO-FLOW AND CONSTANT-HEAD CELLS.
IF(IBOUND(J,I,K).LE.0) GO TO 300
HSING=HNEW(J,I,K)
C
C7-----CHECK LAYER TYPE TO SEE IF ONE STORAGE CAPACITY OR TWO.
IF(LC.EQ.0) GO TO 285
C
C7A----TWO STORAGE CAPACITIES.
TP=BOTM(J,I,LBOTM(K)-1)
RHO2=SC2(J,I,KT)*TLED
RHO1=SC1(J,I,K)*TLED
SOLD=RHO2
IF(HOLD(J,I,K).GT.TP) SOLD=RHO1
SNEW=RHO2
IF(HSING.GT.TP) SNEW=RHO1
STRG=SOLD*(HOLD(J,I,K)-TP) + SNEW*TP - SNEW*HSING
GO TO 288
C
C7B----ONE STORAGE CAPACITY.
285 RHO=SC1(J,I,K)*TLED
STRG=RHO*HOLD(J,I,K) - RHO*HSING
C
C8-----STORE CELL-BY-CELL FLOW IN BUFFER AND ADD TO ACCUMULATORS.
288 BUFF(J,I,K)=STRG
SSTRG=STRG
IF(STRG.LT.ZERO) THEN
STOUT=STOUT-SSTRG
ELSE
STOIN=STOIN+SSTRG
END IF
C
300 CONTINUE
C
C9-----IF IBD FLAG IS SET RECORD THE CONTENTS OF THE BUFFER.
IF(IBD.EQ.1) CALL UBUDSV(KSTP,KPER,TEXT,
1 ILPFCB,BUFF,NCOL,NROW,NLAY,IOUT)
IF(IBD.EQ.2) CALL UBDSV1(KSTP,KPER,TEXT,ILPFCB,
1 BUFF,NCOL,NROW,NLAY,IOUT,DELT,PERTIM,TOTIM,IBOUND)
C
C10-----ADD TOTAL RATES AND VOLUMES TO VBVL & PUT TITLE IN VBNM.
400 CONTINUE
SIN=STOIN
SOUT=STOUT
VBVL(1,MSUM)=VBVL(1,MSUM)+SIN*DELT
VBVL(2,MSUM)=VBVL(2,MSUM)+SOUT*DELT
VBVL(3,MSUM)=SIN
VBVL(4,MSUM)=SOUT
VBNM(MSUM)=TEXT
MSUM=MSUM+1
C
C11----RETURN.
RETURN
END
SUBROUTINE GWF2LPF7BDCH(KSTP,KPER,IGRID)
C ******************************************************************
C COMPUTE FLOW FROM CONSTANT-HEAD CELLS
C ******************************************************************
C
C SPECIFICATIONS:
C ------------------------------------------------------------------
USE GLOBAL, ONLY:NCOL,NROW,NLAY,IBOUND,HNEW,BUFF,CR,CC,CV,
1 BOTM,LBOTM,IOUT
USE GWFBASMODULE,ONLY:MSUM,VBVL,VBNM,DELT,PERTIM,TOTIM,ICBCFL,
1 ICHFLG
USE GWFLPFMODULE,ONLY:ILPFCB,LAYTYP,NOVFC
CHARACTER*16 TEXT
DOUBLE PRECISION HD,CHIN,CHOUT,XX1,XX2,XX3,XX4,XX5,XX6
C
DATA TEXT /' CONSTANT HEAD'/
C ------------------------------------------------------------------
CALL SGWF2LPF7PNT(IGRID)
C
C1------SET IBD TO INDICATE IF CELL-BY-CELL BUDGET VALUES WILL BE SAVED.
IBD=0
IF(ILPFCB.LT.0 .AND. ICBCFL.NE.0) IBD=-1
IF(ILPFCB.GT.0) IBD=ICBCFL
C
C2------CLEAR BUDGET ACCUMULATORS.
ZERO=0.
CHIN=ZERO
CHOUT=ZERO
IBDLBL=0
C
C3------CLEAR BUFFER.
DO 5 K=1,NLAY
DO 5 I=1,NROW
DO 5 J=1,NCOL
BUFF(J,I,K)=ZERO
5 CONTINUE
C
C3A-----IF SAVING CELL-BY-CELL FLOW IN A LIST, COUNT CONSTANT-HEAD
C3A-----CELLS AND WRITE HEADER RECORDS.
IF(IBD.EQ.2) THEN
NCH=0
DO 7 K=1,NLAY
DO 7 I=1,NROW
DO 7 J=1,NCOL
IF(IBOUND(J,I,K).LT.0) NCH=NCH+1
7 CONTINUE
CALL UBDSV2(KSTP,KPER,TEXT,ILPFCB,NCOL,NROW,NLAY,
1 NCH,IOUT,DELT,PERTIM,TOTIM,IBOUND)
END IF
C
C4------LOOP THROUGH EACH CELL AND CALCULATE FLOW INTO MODEL FROM EACH
C4------CONSTANT-HEAD CELL.
DO 200 K=1,NLAY
DO 200 I=1,NROW
DO 200 J=1,NCOL
C
C5------IF CELL IS NOT CONSTANT HEAD SKIP IT & GO ON TO NEXT CELL.
IF (IBOUND(J,I,K).GE.0)GO TO 200
C
C6------CLEAR VALUES FOR FLOW RATE THROUGH EACH FACE OF CELL.
X1=ZERO
X2=ZERO
X3=ZERO
X4=ZERO
X5=ZERO
X6=ZERO
CHCH1=ZERO
CHCH2=ZERO
CHCH3=ZERO
CHCH4=ZERO
CHCH5=ZERO
CHCH6=ZERO
C
C7------CALCULATE FLOW THROUGH THE LEFT FACE.
C7------COMMENTS A-C APPEAR ONLY IN THE SECTION HEADED BY COMMENT 7,
C7------BUT THEY APPLY IN A SIMILAR MANNER TO SECTIONS 8-12.
C
C7A-----IF THERE IS NO FLOW TO CALCULATE THROUGH THIS FACE, THEN GO ON
C7A-----TO NEXT FACE. NO FLOW OCCURS AT THE EDGE OF THE GRID, TO AN
C7A-----ADJACENT NO-FLOW CELL, OR TO AN ADJACENT CONSTANT-HEAD CELL.
IF(J.EQ.1) GO TO 30
IF(IBOUND(J-1,I,K).EQ.0) GO TO 30
IF(IBOUND(J-1,I,K).LT.0 .AND. ICHFLG.EQ.0) GO TO 30
C
C7B-----CALCULATE FLOW THROUGH THIS FACE INTO THE ADJACENT CELL.
HDIFF=HNEW(J,I,K)-HNEW(J-1,I,K)
CHCH1=HDIFF*CR(J-1,I,K)
IF(IBOUND(J-1,I,K).LT.0) GO TO 30
X1=CHCH1
XX1=X1
C
C7C-----ACCUMULATE POSITIVE AND NEGATIVE FLOW.
IF(X1.LT.ZERO) THEN
CHOUT=CHOUT-XX1
ELSE
CHIN=CHIN+XX1
END IF
C
C8------CALCULATE FLOW THROUGH THE RIGHT FACE.
30 IF(J.EQ.NCOL) GO TO 60
IF(IBOUND(J+1,I,K).EQ.0) GO TO 60
IF(IBOUND(J+1,I,K).LT.0 .AND. ICHFLG.EQ.0) GO TO 60
HDIFF=HNEW(J,I,K)-HNEW(J+1,I,K)
CHCH2=HDIFF*CR(J,I,K)
IF(IBOUND(J+1,I,K).LT.0) GO TO 60
X2=CHCH2
XX2=X2
IF(X2.LT.ZERO) THEN
CHOUT=CHOUT-XX2
ELSE
CHIN=CHIN+XX2
END IF
C
C9------CALCULATE FLOW THROUGH THE BACK FACE.
60 IF(I.EQ.1) GO TO 90
IF (IBOUND(J,I-1,K).EQ.0) GO TO 90
IF (IBOUND(J,I-1,K).LT.0 .AND. ICHFLG.EQ.0) GO TO 90
HDIFF=HNEW(J,I,K)-HNEW(J,I-1,K)
CHCH3=HDIFF*CC(J,I-1,K)
IF(IBOUND(J,I-1,K).LT.0) GO TO 90
X3=CHCH3
XX3=X3
IF(X3.LT.ZERO) THEN
CHOUT=CHOUT-XX3
ELSE
CHIN=CHIN+XX3
END IF
C
C10-----CALCULATE FLOW THROUGH THE FRONT FACE.
90 IF(I.EQ.NROW) GO TO 120
IF(IBOUND(J,I+1,K).EQ.0) GO TO 120
IF(IBOUND(J,I+1,K).LT.0 .AND. ICHFLG.EQ.0) GO TO 120
HDIFF=HNEW(J,I,K)-HNEW(J,I+1,K)
CHCH4=HDIFF*CC(J,I,K)
IF(IBOUND(J,I+1,K).LT.0) GO TO 120
X4=CHCH4
XX4=X4
IF(X4.LT.ZERO) THEN
CHOUT=CHOUT-XX4
ELSE
CHIN=CHIN+XX4
END IF
C
C11-----CALCULATE FLOW THROUGH THE UPPER FACE.
120 IF(K.EQ.1) GO TO 150
IF (IBOUND(J,I,K-1).EQ.0) GO TO 150
IF (IBOUND(J,I,K-1).LT.0 .AND. ICHFLG.EQ.0) GO TO 150
HD=HNEW(J,I,K)
IF(NOVFC.NE.0 .OR. LAYTYP(K).EQ.0) GO TO 122
TMP=HD
TOP=BOTM(J,I,LBOTM(K)-1)
IF(TMP.LT.TOP) HD=TOP
122 HDIFF=HD-HNEW(J,I,K-1)
CHCH5=HDIFF*CV(J,I,K-1)
IF(IBOUND(J,I,K-1).LT.0) GO TO 150
X5=CHCH5
XX5=X5
IF(X5.LT.ZERO) THEN
CHOUT=CHOUT-XX5
ELSE
CHIN=CHIN+XX5
END IF
C
C12-----CALCULATE FLOW THROUGH THE LOWER FACE.
150 IF(K.EQ.NLAY) GO TO 180
IF(IBOUND(J,I,K+1).EQ.0) GO TO 180
IF(IBOUND(J,I,K+1).LT.0 .AND. ICHFLG.EQ.0) GO TO 180
HD=HNEW(J,I,K+1)
IF(NOVFC.NE.0 .OR. LAYTYP(K+1).EQ.0) GO TO 152
TMP=HD
TOP=BOTM(J,I,LBOTM(K+1)-1)
IF(TMP.LT.TOP) HD=TOP
152 HDIFF=HNEW(J,I,K)-HD
CHCH6=HDIFF*CV(J,I,K)
IF(IBOUND(J,I,K+1).LT.0) GO TO 180
X6=CHCH6
XX6=X6
IF(X6.LT.ZERO) THEN
CHOUT=CHOUT-XX6
ELSE
CHIN=CHIN+XX6
END IF
C
C13-----SUM THE FLOWS THROUGH SIX FACES OF CONSTANT HEAD CELL, AND
C13-----STORE SUM IN BUFFER.
180 RATE=CHCH1+CHCH2+CHCH3+CHCH4+CHCH5+CHCH6
BUFF(J,I,K)=RATE
C
C14-----PRINT THE FLOW FOR THE CELL IF REQUESTED.
IF(IBD.LT.0) THEN
IF(IBDLBL.EQ.0) WRITE(IOUT,899) TEXT,KPER,KSTP
899 FORMAT(1X,/1X,A,' PERIOD ',I4,' STEP ',I3)
WRITE(IOUT,900) K,I,J,RATE
900 FORMAT(1X,'LAYER ',I3,' ROW ',I5,' COL ',I5,
1 ' RATE ',1PG15.6)
IBDLBL=1
END IF
C
C15-----IF SAVING CELL-BY-CELL FLOW IN LIST, WRITE FLOW FOR CELL.
IF(IBD.EQ.2) CALL UBDSVA(ILPFCB,NCOL,NROW,J,I,K,RATE,IBOUND,NLAY)
200 CONTINUE
C
C16-----IF SAVING CELL-BY-CELL FLOW IN 3-D ARRAY, WRITE THE ARRAY.
IF(IBD.EQ.1) CALL UBUDSV(KSTP,KPER,TEXT,
1 ILPFCB,BUFF,NCOL,NROW,NLAY,IOUT)
C
C17-----SAVE TOTAL CONSTANT HEAD FLOWS AND VOLUMES IN VBVL TABLE
C17-----FOR INCLUSION IN BUDGET. PUT LABELS IN VBNM TABLE.
CIN=CHIN
COUT=CHOUT
VBVL(1,MSUM)=VBVL(1,MSUM)+CIN*DELT
VBVL(2,MSUM)=VBVL(2,MSUM)+COUT*DELT
VBVL(3,MSUM)=CIN
VBVL(4,MSUM)=COUT
VBNM(MSUM)=TEXT
MSUM=MSUM+1
C
C18-----RETURN.
RETURN
END
SUBROUTINE SGWF2LPF7SC(SC,K,ISPST)
C ******************************************************************
C COMPUTE STORAGE CAPACITY
C ******************************************************************
C
C SPECIFICATIONS:
C ------------------------------------------------------------------
USE GLOBAL, ONLY:NCOL,NROW,DELR,DELC,BOTM,LBOTM,LAYCBD
C
DIMENSION SC(NCOL,NROW)
C ------------------------------------------------------------------
C
C1------MULTIPLY SPECIFIC STORAGE BY THICKNESS, DELR, AND DELC TO GET
C1------CONFINED STORAGE CAPACITY.
IF(ISPST.NE.0) THEN
DO 80 I=1,NROW
DO 80 J=1,NCOL
THICK=BOTM(J,I,LBOTM(K)-1)-BOTM(J,I,LBOTM(K))
SC(J,I)=SC(J,I)*THICK*DELR(J)*DELC(I)
80 CONTINUE
ELSE
C
C2------MULTIPLY SPECIFIC YIELD BY DELR AND DELC TO GET UNCONFINED
C2------STORAGE CAPACITY(SC2).
DO 85 I=1,NROW
DO 85 J=1,NCOL
SC(J,I)=SC(J,I)*DELR(J)*DELC(I)
85 CONTINUE
END IF
C
RETURN
END
SUBROUTINE SGWF2LPF7HCOND(K,KITER,KSTP,KPER)
C ******************************************************************
C COMPUTE HORIZONTAL BRANCH CONDUCTANCE FOR ONE LAYER.
C ******************************************************************
C
C SPECIFICATIONS:
C ------------------------------------------------------------------
USE GLOBAL, ONLY:IOUT,NCOL,NROW,IBOUND,HNEW,BOTM,NBOTM,
1 LBOTM,CC,STRT
USE GWFBASMODULE,ONLY:HDRY
USE GWFLPFMODULE,ONLY:LAYWET,IWETIT,LAYTYP,LAYAVG,LAYSTRT
C
CHARACTER*3 ACNVRT
DIMENSION ICNVRT(5),JCNVRT(5),ACNVRT(5)
C
C ------------------------------------------------------------------
C1------INITIALIZE DATA.
ZERO=0.
NCNVRT=0
IHDCNV=0
C
C2------IF LAYER IS WETTABLE CONVERT DRY CELLS TO WET WHEN APPROPRIATE.
ITFLG=1
IF(LAYWET(K).NE.0) ITFLG=MOD(KITER,IWETIT)
IF(ITFLG.EQ.0) CALL SGWF2LPF7WET(K,KITER,KSTP,KPER,
2 IHDCNV,NCNVRT,ICNVRT,JCNVRT,ACNVRT)
C
C3------LOOP THROUGH EACH CELL, AND CALCULATE SATURATED THICKNESS.
DO 200 I=1,NROW
DO 200 J=1,NCOL
C
C3A-----SET SATURATED THICKNESS=0. FOR DRY CELLS.
IF(IBOUND(J,I,K).EQ.0) THEN
CC(J,I,K)=ZERO
ELSE
C
C3B-----CALCULATE SATURATED THICKNESS FOR A WET CELL.
BBOT=BOTM(J,I,LBOTM(K))
IF(LAYSTRT(K).NE.0) THEN
TTOP=STRT(J,I,K)
IF(BBOT.GT.TTOP) THEN
WRITE(IOUT,33) K,I,J
33 FORMAT(1X,/1X,'Negative cell thickness at (layer,row,col)',
1 I4,',',I5,',',I5)
WRITE(IOUT,34) TTOP,BBOT
34 FORMAT(1X,'Initial head, bottom elevation:',1P,2G13.5)
CALL USTOP(' ')
END IF
ELSE
TTOP=BOTM(J,I,LBOTM(K)-1)
IF(BBOT.GT.TTOP) THEN
WRITE(IOUT,35) K,I,J
35 FORMAT(1X,/1X,'Negative cell thickness at (layer,row,col)',
1 I4,',',I5,',',I5)
WRITE(IOUT,36) TTOP,BBOT
36 FORMAT(1X,'Top elevation, bottom elevation:',1P,2G13.5)
CALL USTOP(' ')
END IF
END IF
IF(LAYTYP(K).NE.0) THEN
HHD=HNEW(J,I,K)
IF(HHD.LT.TTOP) TTOP=HHD
END IF
THCK=TTOP-BBOT
CC(J,I,K)=THCK
C
C
C3C-----WHEN SATURATED THICKNESS <= 0, PRINT A MESSAGE AND SET
C3C-----HNEW=HDRY, SATURATED THICKNESS=0.0, AND IBOUND=0.
IF(THCK.LE.ZERO) THEN
CALL SGWF2LPF7WDMSG(1,NCNVRT,ICNVRT,JCNVRT,ACNVRT,IHDCNV,
1 IOUT,KITER,J,I,K,KSTP,KPER,NCOL,NROW)
HNEW(J,I,K)=HDRY
CC(J,I,K)=ZERO
IF(IBOUND(J,I,K).LT.0) THEN
WRITE(IOUT,151)
151 FORMAT(1X,/1X,'CONSTANT-HEAD CELL WENT DRY',
1 ' -- SIMULATION ABORTED')
WRITE(IOUT,*) TTOP, BBOT, THCK
WRITE(IOUT,152) K,I,J,KITER,KSTP,KPER
152 FORMAT(1X,'LAYER=',I3,' ROW=',I5,' COLUMN=',I5,
1 ' ITERATION=',I3,' TIME STEP=',I3,' STRESS PERIOD=',I4)
CALL USTOP(' ')
END IF
IBOUND(J,I,K)=0
END IF
END IF
200 CONTINUE
C
C4------PRINT ANY REMAINING CELL CONVERSIONS NOT YET PRINTED.
CALL SGWF2LPF7WDMSG(0,NCNVRT,ICNVRT,JCNVRT,ACNVRT,IHDCNV,
1 IOUT,KITER,J,I,K,KSTP,KPER,NCOL,NROW)
C
C5------CHANGE IBOUND VALUE FOR CELLS THAT CONVERTED TO WET THIS
C5------ITERATION FROM 30000 to 1.
IF(LAYWET(K).NE.0) THEN
DO 205 I=1,NROW
DO 205 J=1,NCOL
IF(IBOUND(J,I,K).EQ.30000) IBOUND(J,I,K)=1
205 CONTINUE
END IF
C
C6------COMPUTE HORIZONTAL BRANCH CONDUCTANCES FROM CELL HYDRAULIC
C6------CONDUCTIVITY, SATURATED THICKNESS, AND GRID DIMENSIONS.
IF(LAYAVG(K).EQ.0) THEN
CALL SGWF2LPF7HHARM(K)
ELSE IF(LAYAVG(K).EQ.1) THEN
CALL SGWF2LPF7HLOG(K)
ELSE
CALL SGWF2LPF7HUNCNF(K)
END IF
C
C7------RETURN.
RETURN
END
SUBROUTINE SGWF2LPF7WET(K,KITER,KSTP,KPER,IHDCNV,NCNVRT,
1 ICNVRT,JCNVRT,ACNVRT)
C
C ******************************************************************
C CONVERT DRY CELLS TO WET.
C ******************************************************************
C
C SPECIFICATIONS:
C ------------------------------------------------------------------
USE GLOBAL, ONLY:IOUT,NCOL,NROW,NLAY,HNEW,IBOUND,BOTM,LBOTM
USE GWFLPFMODULE, ONLY:LAYTYP,CHANI,LAYVKA,LAYWET,WETDRY,
1 WETFCT,IHDWET
C
CHARACTER*3 ACNVRT
DIMENSION ICNVRT(5),JCNVRT(5),ACNVRT(5)
C ------------------------------------------------------------------
C
C1------LOOP THROUGH ALL CELLS.
ZERO=0.0
DO 100 I=1,NROW
DO 100 J=1,NCOL
C
C2------IF CELL IS DRY AND IF IT IS WETTABLE, CONTINUE CHECKING TO SEE
C2------IF IT SHOULD BECOME WET.
IF(IBOUND(J,I,K).EQ.0 .AND. WETDRY(J,I,LAYWET(K)).NE.ZERO) THEN
C
C3------CALCULATE WETTING ELEVATION.
WD=WETDRY(J,I,LAYWET(K))
IF(WD.LT.ZERO) WD=-WD
TURNON=BOTM(J,I,LBOTM(K))+WD
C
C4------CHECK HEAD IN CELL BELOW TO SEE IF WETTING ELEVATION HAS BEEN
C4------REACHED.
IF(K.NE.NLAY) THEN
HTMP=HNEW(J,I,K+1)
IF(IBOUND(J,I,K+1).GT.0 .AND. HTMP.GE.TURNON) GO TO 50
END IF
C
C5------CHECK HEAD IN ADJACENT HORIZONTAL CELLS TO SEE IF WETTING
C5------ELEVATION HAS BEEN REACHED.
IF(WETDRY(J,I,LAYWET(K)).GT.ZERO) THEN
IF(J.NE.1) THEN
HTMP=HNEW(J-1,I,K)
IF(IBOUND(J-1,I,K).GT.0 .AND. IBOUND(J-1,I,K).NE.30000.
1 AND. HTMP.GE.TURNON) GO TO 50
END IF
IF(J.NE.NCOL) THEN
HTMP=HNEW(J+1,I,K)
IF(IBOUND(J+1,I,K).GT.0 .AND. HTMP.GE.TURNON) GO TO 50
END IF
IF(I.NE.1) THEN
HTMP=HNEW(J,I-1,K)
IF(IBOUND(J,I-1,K).GT.0 .AND. IBOUND(J,I-1,K).NE.30000.
1 AND. HTMP.GE.TURNON) GO TO 50
END IF
IF(I.NE.NROW) THEN
HTMP=HNEW(J,I+1,K)
IF(IBOUND(J,I+1,K).GT.0 .AND. HTMP.GE.TURNON) GO TO 50
END IF
END IF
C
C6------WETTING ELEVATION HAS NOT BEEN REACHED, SO CELL REMAINS DRY.
GO TO 100
C
C7------CELL BECOMES WET. PRINT MESSAGE, SET INITIAL HEAD, AND SET
C7------IBOUND.
50 CALL SGWF2LPF7WDMSG(2,NCNVRT,ICNVRT,JCNVRT,ACNVRT,IHDCNV,
1 IOUT,KITER,J,I,K,KSTP,KPER,NCOL,NROW)
C
C7A-----USE EQUATION 3A IF IHDWET=0; USE EQUATION 3B IF IHDWET IS NOT 0.
IF(IHDWET.EQ.0) THEN
HNEW(J,I,K)=BOTM(J,I,LBOTM(K))+
1 WETFCT*(HTMP-BOTM(J,I,LBOTM(K)))
ELSE
HNEW(J,I,K)=BOTM(J,I,LBOTM(K))+WETFCT*WD
END IF
IBOUND(J,I,K)=30000
END IF
C
C8------END OF LOOP FOR ALL CELLS IN LAYER.
100 CONTINUE
C
C9------RETURN.
RETURN
END
SUBROUTINE SGWF2LPF7WDMSG(ICODE,NCNVRT,ICNVRT,JCNVRT,ACNVRT,
1 IHDCNV,IOUT,KITER,J,I,K,KSTP,KPER,NCOL,NROW)
C ******************************************************************
C PRINT MESSAGE WHEN CELLS CONVERT BETWEEN WET AND DRY.
C ******************************************************************
C
C SPECIFICATIONS:
C ------------------------------------------------------------------
CHARACTER*3 ACNVRT
DIMENSION ICNVRT(5),JCNVRT(5),ACNVRT(5)
C ------------------------------------------------------------------
C
C1------KEEP TRACK OF CELL CONVERSIONS.
IF(ICODE.GT.0) THEN
NCNVRT=NCNVRT+1
ICNVRT(NCNVRT)=I
JCNVRT(NCNVRT)=J
IF(ICODE.EQ.1) THEN
ACNVRT(NCNVRT)='DRY'
ELSE
ACNVRT(NCNVRT)='WET'
END IF
END IF
C
C2------PRINT A LINE OF DATA IF 5 CONVERSIONS HAVE OCCURRED OR IF ICODE
C2------INDICATES THAT A PARTIAL LINE SHOULD BE PRINTED.
IF(NCNVRT.EQ.5 .OR. (ICODE.EQ.0 .AND. NCNVRT.GT.0)) THEN
IF(IHDCNV.EQ.0) WRITE(IOUT,17) KITER,K,KSTP,KPER
17 FORMAT(1X,/1X,'CELL CONVERSIONS FOR ITER.=',I3,' LAYER=',
1 I3,' STEP=',I3,' PERIOD=',I4,' (ROW,COL)')
IHDCNV=1
IF (NROW.LE.999 .AND. NCOL.LE.999) THEN
WRITE(IOUT,18) (ACNVRT(L),ICNVRT(L),JCNVRT(L),L=1,NCNVRT)
18 FORMAT(1X,3X,5(A,'(',I3,',',I3,') '))
ELSE
WRITE(IOUT,19) (ACNVRT(L),ICNVRT(L),JCNVRT(L),L=1,NCNVRT)
19 FORMAT(1X,2X,5(A,'(',I5,',',I5,')'))
ENDIF
NCNVRT=0
END IF
C
C3------RETURN.
RETURN
END
SUBROUTINE SGWF2LPF7HHARM(K)
C ******************************************************************
C COMPUTE HORIZONTAL BRANCH CONDUCTANCE USING HARMONIC MEAN OF BLOCK
C CONDUCTANCES (DISTANCE WEIGHTED HARMONIC MEAN OF TRANSMISSIVITY).
C CELL THICKNESS IS IN CC UPON ENTRY.
C ******************************************************************
C
C SPECIFICATIONS:
C ------------------------------------------------------------------
USE GLOBAL, ONLY:NCOL,NROW,IBOUND,CR,CC,DELR,DELC
USE GWFLPFMODULE,ONLY:HK,CHANI,HANI
C ------------------------------------------------------------------
C
ZERO=0.
TWO=2.
C
C1------FOR EACH CELL CALCULATE BRANCH CONDUCTANCES FROM THAT CELL
C1------TO THE ONE ON THE RIGHT AND THE ONE IN FRONT.
DO 100 I=1,NROW
DO 100 J=1,NCOL
C
C2------IF CELL IS DRY OR HK=0., SET CONDUCTANCE EQUAL TO 0 AND GO ON
C2------TO NEXT CELL.
IF(IBOUND(J,I,K).EQ.0 .OR. HK(J,I,K).EQ.ZERO) THEN
CR(J,I,K)=ZERO
CC(J,I,K)=ZERO
ELSE
C
C3------CELL IS WET -- CALCULATE TRANSMISSIVITY OF CELL.
T1=HK(J,I,K)*CC(J,I,K)
C3A-----IF THIS IS NOT THE LAST COLUMN (RIGHTMOST), CALCULATE
C3A-----BRANCH CONDUCTANCE IN THE ROW DIRECTION (CR) TO THE RIGHT.
IF(J.NE.NCOL) THEN
IF(IBOUND(J+1,I,K).NE.0) THEN
T2=HK(J+1,I,K)*CC(J+1,I,K)
CR(J,I,K)=TWO*T2*T1*DELC(I)/(T1*DELR(J+1)+T2*DELR(J))
ELSE
CR(J,I,K)=ZERO
END IF
ELSE
C3B-----IF THIS IS THE LAST COLUMN, SET BRANCH CONDUCTANCE=0.
CR(J,I,K)=ZERO
END IF
C
C3C-----IF THIS IS NOT THE LAST ROW (FRONTMOST) THEN CALCULATE
C3C-----BRANCH CONDUCTANCE IN THE COLUMN DIRECTION (CC) TO THE FRONT.
IF(I.NE.NROW) THEN
IF(IBOUND(J,I+1,K).NE.0) THEN
T2=HK(J,I+1,K)*CC(J,I+1,K)
IF(CHANI(K).LE.ZERO) THEN
KHANI=-CHANI(K)
T1=T1*HANI(J,I,KHANI)
T2=T2*HANI(J,I+1,KHANI)
ELSE
T1=T1*CHANI(K)
T2=T2*CHANI(K)
END IF
CC(J,I,K)=TWO*T2*T1*DELR(J)/(T1*DELC(I+1)+T2*DELC(I))
ELSE
C3D-----IF THIS IS THE LAST ROW, SET BRANCH CONDUCTANCE=0.
CC(J,I,K)=ZERO
END IF
ELSE
CC(J,I,K)=ZERO
END IF
END IF
100 CONTINUE
C
C4------RETURN
RETURN
END
SUBROUTINE SGWF2LPF7HLOG(K)
C ******************************************************************
C-----COMPUTE HORIZONTAL CONDUCTANCE USING LOGARITHMIC MEAN
C-----TRANSMISSIVITY -- ACTIVATED BY LAYAVG=1
C-----CELL SATURATED THICKNESS IS IN CC.
C ******************************************************************
C
C SPECIFICATIONS:
C ------------------------------------------------------------------
USE GLOBAL, ONLY:NCOL,NROW,IBOUND,CR,CC,DELR,DELC
USE GWFLPFMODULE,ONLY:HK,CHANI,HANI
C ------------------------------------------------------------------
C
ZERO=0.
TWO=2.
HALF=0.5
FRAC1=1.005
FRAC2=0.995
C
C1------FOR EACH CELL CALCULATE BRANCH CONDUCTANCES FROM THAT CELL
C1------TO THE ONE ON THE RIGHT AND THE ONE IN FRONT.
DO 100 I=1,NROW
DO 100 J=1,NCOL
C
C2------IF CELL IS DRY OR HK=0., SET CONDUCTANCE EQUAL TO 0 AND GO ON
C2------TO NEXT CELL.
IF(IBOUND(J,I,K).EQ.0 .OR. HK(J,I,K).EQ.ZERO) THEN
CR(J,I,K)=ZERO
CC(J,I,K)=ZERO
ELSE
C
C3------CELL IS WET -- CALCULATE TRANSMISSIVITY OF CELL.
T1=HK(J,I,K)*CC(J,I,K)
C3A-----IF THIS IS NOT THE LAST COLUMN(RIGHTMOST) THEN CALCULATE
C3A-----BRANCH CONDUCTANCE IN THE ROW DIRECTION (CR) TO THE RIGHT.
IF(J.NE.NCOL) THEN
IF(IBOUND(J+1,I,K).NE.0) THEN
C3A1----LOGARITHMIC MEAN INTERBLOCK TRANSMISSIVITY
T2=HK(J+1,I,K)*CC(J+1,I,K)
RATIO=T2/T1
IF(RATIO.GT.FRAC1 .OR. RATIO.LT.FRAC2) THEN
T=(T2-T1)/LOG(RATIO)
ELSE
T=HALF*(T1+T2)
END IF
CR(J,I,K)=TWO*DELC(I)*T/(DELR(J+1)+DELR(J))
ELSE
CR(J,I,K)=ZERO
END IF
ELSE
CR(J,I,K)=ZERO
END IF
C
C3B-----IF THIS IS NOT THE LAST ROW (FRONTMOST) THEN CALCULATE
C3B-----BRANCH CONDUCTANCE IN THE COLUMN DIRECTION (CC) TO THE FRONT.
IF(I.NE.NROW) THEN
IF(IBOUND(J,I+1,K).NE.0) THEN
T2=HK(J,I+1,K)*CC(J,I+1,K)
IF(CHANI(K).LE.ZERO) THEN
KHANI=-CHANI(K)
T1=T1*HANI(J,I,KHANI)
T2=T2*HANI(J,I+1,KHANI)
ELSE
T1=T1*CHANI(K)
T2=T2*CHANI(K)
END IF
RATIO=T2/T1
IF(RATIO.GT.FRAC1 .OR. RATIO.LT.FRAC2) THEN
T=(T2-T1)/LOG(RATIO)
ELSE
T=HALF*(T1+T2)
END IF
CC(J,I,K)=TWO*DELR(J)*T/(DELC(I+1)+DELC(I))
ELSE
CC(J,I,K)=ZERO
END IF
ELSE
CC(J,I,K)=ZERO
END IF
END IF
100 CONTINUE
C
C4------RETURN
RETURN
END
SUBROUTINE SGWF2LPF7HUNCNF(K)
C ******************************************************************
C-----COMPUTE HORIZONTAL CONDUCTANCE USING ARITHMETIC MEAN SATURATED
C-----THICKNESS AND LOGARITHMIC MEAN HYDRAULIC CONDUCTIVITY.
C-----CELL SATURATED THICKNESS IS IN CC.
C-----ACTIVATED BY LAYAVG=2
C ******************************************************************
C
C SPECIFICATIONS:
C ------------------------------------------------------------------
USE GLOBAL, ONLY:NCOL,NROW,IBOUND,CR,CC,DELR,DELC
USE GWFLPFMODULE,ONLY:HK,CHANI,HANI
C ------------------------------------------------------------------
C
ZERO=0.
HALF=0.5
FRAC1=1.005
FRAC2=0.995
C
C1------FOR EACH CELL CALCULATE BRANCH CONDUCTANCES FROM THAT CELL
C1------TO THE ONE ON THE RIGHT AND THE ONE IN FRONT.
DO 100 I=1,NROW
DO 100 J=1,NCOL
C
C2------IF CELL IS DRY OR HK=0., SET CONDUCTANCE EQUAL TO 0 AND GO ON
C2------TO NEXT CELL.
IF(IBOUND(J,I,K).EQ.0 .OR. HK(J,I,K).EQ.ZERO) THEN
CR(J,I,K)=ZERO
CC(J,I,K)=ZERO
ELSE
C
C3------CELL IS WET -- CALCULATE TRANSMISSIVITY OF CELL.
HYC1=HK(J,I,K)
C3A-----IF THIS IS NOT THE LAST COLUMN(RIGHTMOST) THEN CALCULATE
C3A-----BRANCH CONDUCTANCE IN THE ROW DIRECTION (CR) TO THE RIGHT.
IF(J.NE.NCOL) THEN
IF(IBOUND(J+1,I,K).NE.0) THEN
C3A1----LOGARITHMIC MEAN HYDRAULIC CONDUCTIVITY
HYC2=HK(J+1,I,K)
RATIO=HYC2/HYC1
IF(RATIO.GT.FRAC1 .OR. RATIO.LT.FRAC2) THEN
HYC=(HYC2-HYC1)/LOG(RATIO)
ELSE
HYC=HALF*(HYC1+HYC2)
END IF
C3A2----MULTIPLY LOGARITHMIC K BY ARITHMETIC SATURATED THICKNESS.
CR(J,I,K)=DELC(I)*HYC*(CC(J,I,K)+CC(J+1,I,K))/
1 (DELR(J+1)+DELR(J))
ELSE
CR(J,I,K)=ZERO
END IF
ELSE
CR(J,I,K)=ZERO
END IF
C
C3B-----IF THIS IS NOT THE LAST ROW (FRONTMOST) THEN CALCULATE
C3B-----BRANCH CONDUCTANCE IN THE COLUMN DIRECTION (CC) TO THE FRONT.
IF(I.NE.NROW) THEN
IF(IBOUND(J,I+1,K).NE.0) THEN
C3B1----LOGARITHMIC MEAN HYDRAULIC CONDUCTIVITY
HYC2=HK(J,I+1,K)
IF(CHANI(K).LE.ZERO) THEN
KHANI=-CHANI(K)
HYC1=HYC1*HANI(J,I,KHANI)
HYC2=HYC2*HANI(J,I+1,KHANI)
ELSE
HYC1=HYC1*CHANI(K)
HYC2=HYC2*CHANI(K)
END IF
RATIO=HYC2/HYC1
IF(RATIO.GT.FRAC1 .OR. RATIO.LT.FRAC2) THEN
HYC=(HYC2-HYC1)/LOG(RATIO)
ELSE
HYC=HALF*(HYC1+HYC2)
END IF
C3B2----MULTIPLY LOGARITHMIC K BY ARITHMETIC SATURATED THICKNESS.
CC(J,I,K)=DELR(J)*HYC*(CC(J,I,K)+CC(J,I+1,K))/
1 (DELC(I+1)+DELC(I))
ELSE
CC(J,I,K)=ZERO
END IF
ELSE
CC(J,I,K)=ZERO
END IF
END IF
100 CONTINUE
C
C4------RETURN.
RETURN
END
SUBROUTINE SGWF2LPF7VCOND(K)
C ******************************************************************
C COMPUTE VERTICAL BRANCH CONDUCTANCE BETWEEN A LAYER AND THE NEXT
C LOWER LAYER FROM VERTICAL HYDRAULIC CONDUCTIVITY.
C ******************************************************************
C
C SPECIFICATIONS:
C ------------------------------------------------------------------
USE GLOBAL, ONLY:NCOL,NROW,NLAY,IBOUND,HNEW,CV,DELR,DELC,
1 BOTM,LBOTM,LAYCBD,IOUT,STRT
USE GWFLPFMODULE, ONLY:LAYTYP,LAYAVG,CHANI,LAYVKA,LAYWET,
1 HK,VKA,VKCB,NOCVCO,ICONCV,LAYSTRT
C
DOUBLE PRECISION BBOT,TTOP,HHD
C ------------------------------------------------------------------
C
IF(K.EQ.NLAY) RETURN
ZERO=0.
HALF=0.5
C
C1------LOOP THROUGH ALL CELLS IN THE LAYER.
DO 100 I=1,NROW
DO 100 J=1,NCOL
CV(J,I,K)=ZERO
IF(IBOUND(J,I,K).NE.0 .AND. IBOUND(J,I,K+1).NE.0) THEN
C
C2------CALCULATE VERTICAL HYDRAULIC CONDUCTIVITY FOR CELL.
IF(LAYVKA(K).EQ.0) THEN
HYC1=VKA(J,I,K)
ELSE
HYC1=HK(J,I,K)/VKA(J,I,K)
END IF
IF(HYC1.GT.ZERO) THEN
C3------CALCULATE VERTICAL HYDRAULIC CONDUCTIVITY FOR CELL BELOW.
IF(LAYVKA(K+1).EQ.0) THEN
HYC2=VKA(J,I,K+1)
ELSE
HYC2=(HK(J,I,K+1)/VKA(J,I,K+1))
END IF
IF(HYC2.GT.ZERO) THEN
C
C4------CALCULATE INVERSE LEAKANCE FOR CELL. ICONCV FLAG PREVENTS
C4------CV FROM BEING HEAD DEPENDENT.
BBOT=BOTM(J,I,LBOTM(K))
TTOP=BOTM(J,I,LBOTM(K)-1)
IF(LAYSTRT(K).NE.0) TTOP=STRT(J,I,K)
IF(LAYTYP(K).NE.0 .AND. ICONCV.EQ.0) THEN
HHD=HNEW(J,I,K)
IF(HHD.LT.TTOP) TTOP=HHD
END IF
BOVK1=(TTOP-BBOT)*HALF/HYC1
C
C5------CALCULATE INVERSE LEAKANCE FOR CELL BELOW.
BBOT=BOTM(J,I,LBOTM(K+1))
TTOP=BOTM(J,I,LBOTM(K+1)-1)
IF(LAYSTRT(K+1).NE.0) TTOP=STRT(J,I,K+1)
B=(TTOP-BBOT)*HALF
C
C5A-----IF CELL BELOW IS NOT SATURATED, DO NOT INCLUDE ITS CONDUCTANCE
C5A-----IN THE VERTICAL CONDUCTANCE CALULATION, EXCEPT THAT THE NOCVCO
C5A-----AND ICONCV FLAGS TURN OFF THIS CORRECTION.
IF(LAYTYP(K+1).NE.0
1 .AND.NOCVCO.EQ.0 .AND. ICONCV.EQ.0) THEN
HHD=HNEW(J,I,K+1)
IF(HHD.LT.TTOP) B=ZERO
END IF
BOVK2=B/HYC2
C
C6------CALCULATE VERTICAL HYDRAULIC CONDUCTIVITY FOR CONFINING BED.
IF(LAYCBD(K).NE.0) THEN
IF(VKCB(J,I,LAYCBD(K)).GT.ZERO) THEN
C
C7------CALCULATE INVERSE LEAKANCE FOR CONFINING BED.
B=BOTM(J,I,LBOTM(K))-BOTM(J,I,LBOTM(K)+1)
IF(B.LT.ZERO) THEN
WRITE(IOUT,45) K,I,J
45 FORMAT(1X,/1X,
1 'Negative confining bed thickness below cell (Layer,row,col)',
2 I4,',',I5,',',I5)
WRITE(IOUT,46) BOTM(J,I,LBOTM(K)),BOTM(J,I,LBOTM(K)+1)
46 FORMAT(1X,'Top elevation, bottom elevation:',1P,2G13.5)
CALL USTOP(' ')
END IF
CBBOVK=B/VKCB(J,I,LAYCBD(K))
CV(J,I,K)=DELR(J)*DELC(I)/(BOVK1+CBBOVK+BOVK2)
END IF
ELSE
CV(J,I,K)=DELR(J)*DELC(I)/(BOVK1+BOVK2)
END IF
END IF
END IF
END IF
100 CONTINUE
C
C8------RETURN.
RETURN
END
SUBROUTINE SGWF2LPF7CK(IOUT,NP,PTYP)
C ******************************************************************
C CHECK THAT JUST-DEFINED PARAMETER OF TYPE 'VK' OR 'VANI' IS USED
C CONSISTENTLY WITH LAYVKA ENTRIES FOR LAYERS LISTED IN CLUSTERS FOR
C THE PARAMETER
C ******************************************************************
C
C SPECIFICATIONS:
C ------------------------------------------------------------------
USE GWFLPFMODULE, ONLY:LAYTYP,LAYAVG,CHANI,LAYVKA,LAYWET
USE PARAMMODULE
C
CHARACTER*4 PTYP
C ------------------------------------------------------------------
C
C1------LOOP THROUGH THE CLUSTERS FOR THIS PARAMETER.
DO 10 ICL = IPLOC(1,NP),IPLOC(2,NP)
LAY = IPCLST(1,ICL)
LV = LAYVKA(LAY)
IF (PTYP.EQ.'VK ' .AND. LV.NE.0) THEN
WRITE (IOUT,590) LAY,LV,LAY,PARNAM(NP),'VK'
590 FORMAT(/,
&1X,'LAYVKA entered for layer ',i3,' is: ',i3,'; however,',
&' layer ',i3,' is',/,' listed in a cluster for parameter "',a,
&'" of type ',a,' and')
WRITE (IOUT,600)
600 FORMAT(
&1X,'parameters of type VK can apply only to layers for which',
&/,' LAYVKA is specified as zero -- STOP EXECUTION (SGWF2LPF7CK)')
CALL USTOP(' ')
ELSEIF (PTYP.EQ.'VANI' .AND. LV.EQ.0) THEN
WRITE (IOUT,590) LAY,LV,LAY,PARNAM(NP),'VANI'
WRITE (IOUT,610)
610 FORMAT(
&1X,'parameters of type VANI can apply only to layers for which',/,
&' LAYVKA is not specified as zero -- STOP EXECUTION',
&' (SGWF2LPF7CK)')
CALL USTOP(' ')
ENDIF
10 CONTINUE
C
C2------Return.
RETURN
END
SUBROUTINE GWF2LPF7DA(IGRID)
C Deallocate LPF DATA
USE GWFLPFMODULE
C
DEALLOCATE(GWFLPFDAT(IGRID)%ILPFCB)
DEALLOCATE(GWFLPFDAT(IGRID)%IWDFLG)
DEALLOCATE(GWFLPFDAT(IGRID)%IWETIT)
DEALLOCATE(GWFLPFDAT(IGRID)%IHDWET)
DEALLOCATE(GWFLPFDAT(IGRID)%ISFAC)
DEALLOCATE(GWFLPFDAT(IGRID)%ICONCV)
DEALLOCATE(GWFLPFDAT(IGRID)%ITHFLG)
DEALLOCATE(GWFLPFDAT(IGRID)%NOCVCO)
DEALLOCATE(GWFLPFDAT(IGRID)%NOVFC)
DEALLOCATE(GWFLPFDAT(IGRID)%WETFCT)
DEALLOCATE(GWFLPFDAT(IGRID)%LAYTYP)
DEALLOCATE(GWFLPFDAT(IGRID)%LAYAVG)
DEALLOCATE(GWFLPFDAT(IGRID)%CHANI)
DEALLOCATE(GWFLPFDAT(IGRID)%LAYVKA)
DEALLOCATE(GWFLPFDAT(IGRID)%LAYWET)
DEALLOCATE(GWFLPFDAT(IGRID)%LAYSTRT)
DEALLOCATE(GWFLPFDAT(IGRID)%LAYFLG)
DEALLOCATE(GWFLPFDAT(IGRID)%VKA)
DEALLOCATE(GWFLPFDAT(IGRID)%VKCB)
DEALLOCATE(GWFLPFDAT(IGRID)%SC1)
DEALLOCATE(GWFLPFDAT(IGRID)%SC2)
DEALLOCATE(GWFLPFDAT(IGRID)%HANI)
DEALLOCATE(GWFLPFDAT(IGRID)%WETDRY)
DEALLOCATE(GWFLPFDAT(IGRID)%HK)
C
RETURN
END
SUBROUTINE SGWF2LPF7PNT(IGRID)
C Point to LPF data for a grid.
USE GWFLPFMODULE
C
ILPFCB=>GWFLPFDAT(IGRID)%ILPFCB
IWDFLG=>GWFLPFDAT(IGRID)%IWDFLG
IWETIT=>GWFLPFDAT(IGRID)%IWETIT
IHDWET=>GWFLPFDAT(IGRID)%IHDWET
ISFAC=>GWFLPFDAT(IGRID)%ISFAC
ICONCV=>GWFLPFDAT(IGRID)%ICONCV
ITHFLG=>GWFLPFDAT(IGRID)%ITHFLG
NOCVCO=>GWFLPFDAT(IGRID)%NOCVCO
NOVFC=>GWFLPFDAT(IGRID)%NOVFC
WETFCT=>GWFLPFDAT(IGRID)%WETFCT
LAYTYP=>GWFLPFDAT(IGRID)%LAYTYP
LAYAVG=>GWFLPFDAT(IGRID)%LAYAVG
CHANI=>GWFLPFDAT(IGRID)%CHANI
LAYVKA=>GWFLPFDAT(IGRID)%LAYVKA
LAYWET=>GWFLPFDAT(IGRID)%LAYWET
LAYSTRT=>GWFLPFDAT(IGRID)%LAYSTRT
LAYFLG=>GWFLPFDAT(IGRID)%LAYFLG
VKA=>GWFLPFDAT(IGRID)%VKA
VKCB=>GWFLPFDAT(IGRID)%VKCB
SC1=>GWFLPFDAT(IGRID)%SC1
SC2=>GWFLPFDAT(IGRID)%SC2
HANI=>GWFLPFDAT(IGRID)%HANI
WETDRY=>GWFLPFDAT(IGRID)%WETDRY
HK=>GWFLPFDAT(IGRID)%HK
C
RETURN
END
SUBROUTINE GWF2LPF7PSV(IGRID)
C Save LPF data for a grid.
USE GWFLPFMODULE
C
GWFLPFDAT(IGRID)%ILPFCB=>ILPFCB
GWFLPFDAT(IGRID)%IWDFLG=>IWDFLG
GWFLPFDAT(IGRID)%IWETIT=>IWETIT
GWFLPFDAT(IGRID)%IHDWET=>IHDWET
GWFLPFDAT(IGRID)%ISFAC=>ISFAC
GWFLPFDAT(IGRID)%ICONCV=>ICONCV
GWFLPFDAT(IGRID)%ITHFLG=>ITHFLG
GWFLPFDAT(IGRID)%NOCVCO=>NOCVCO
GWFLPFDAT(IGRID)%NOVFC=>NOVFC
GWFLPFDAT(IGRID)%WETFCT=>WETFCT
GWFLPFDAT(IGRID)%LAYTYP=>LAYTYP
GWFLPFDAT(IGRID)%LAYAVG=>LAYAVG
GWFLPFDAT(IGRID)%CHANI=>CHANI
GWFLPFDAT(IGRID)%LAYVKA=>LAYVKA
GWFLPFDAT(IGRID)%LAYWET=>LAYWET
GWFLPFDAT(IGRID)%LAYSTRT=>LAYSTRT
GWFLPFDAT(IGRID)%LAYFLG=>LAYFLG
GWFLPFDAT(IGRID)%VKA=>VKA
GWFLPFDAT(IGRID)%VKCB=>VKCB
GWFLPFDAT(IGRID)%SC1=>SC1
GWFLPFDAT(IGRID)%SC2=>SC2
GWFLPFDAT(IGRID)%HANI=>HANI
GWFLPFDAT(IGRID)%WETDRY=>WETDRY
GWFLPFDAT(IGRID)%HK=>HK
C
RETURN
END
|
\hypertarget{naomi-and-ruth}{%
\subsection{Naomi and Ruth}\label{naomi-and-ruth}}
\hypertarget{section}{%
\section{1}\label{section}}
\bibverse{1} In the time when the judges ruled, there was once a famine
in the land. A man from Bethlehem in Judah took his wife and two sons to
live in the territory of Moab. \bibverse{2} His name was Elimelech and
his wife's was Naomi, and his two sons were Mahlon and Chilion. They
were Ephrathites from Bethlehem in Judah. After they had been living in
Moab for some time, \bibverse{3} Elimelech died, and Naomi was left with
her two sons, \bibverse{4} who married Moabite women named Orpah and
Ruth. After they had lived there about ten years, \bibverse{5} Mahlon
and Chilion both died, and Naomi was left alone, without husband or
sons.
\bibverse{6} So she set out with her daughters-in-law to return from the
land of Moab, for she had heard that the Lord had remembered his people
and given them food. \bibverse{7} As they were setting out together on
the journey to Judah, \bibverse{8} Naomi said to her daughters-in-law,
``Go, return both of you to the home of your mother. May the Lord be
kind to you as you have been kind to the dead and to me. \bibverse{9}
The Lord grant that each of you may find peace and happiness in the
house of a new husband.''
Then she kissed them; but they began to weep aloud \bibverse{10} and
said to her, ``No, we will return with you to your people.''
\bibverse{11} But Naomi said, ``Go back, my daughters; why should you go
with me? Can I still bear sons who might become your husbands?
\bibverse{12} Go back, my daughters, go your own way, because I am too
old to have a husband. Even if I should say, `I have hope,' even if I
should have a husband tonight and should bear sons, \bibverse{13} would
you wait for them until they were grown up? Would you remain single for
them? No, my daughters! My heart grieves for you, for the Lord has sent
me adversity.'' \bibverse{14} Then they again wept aloud, and Orpah
kissed her mother-in-law goodbye, but Ruth stayed with her.
\bibverse{15} ``Look,'' said Naomi, ``your sister-in-law is going back
to her own people and to her own gods; go along with her!''
\bibverse{16} But Ruth answered, ``Do not urge me to leave you or to go
back. I will go where you go, and I will stay wherever you stay. Your
people will be my people, and your God my God; \bibverse{17} I will die
where you die, and be buried there. May the Lord bring a curse upon me,
if anything but death separate you and me.'' \bibverse{18} When Naomi
saw that Ruth was determined to go with her, she ceased urging her to
return.
\bibverse{19} So they journeyed on until they came to Bethlehem. Their
arrival stirred the whole town, and the women said, ``Can this be
Naomi?''
\bibverse{20} ``Do not call me Naomi,'' she said to them, ``call me
Mara+ 1.20 In Hebrew ``Naomi'' means ``pleasant,'' and ``Mara'' means
``bitter.'', for the Almighty has given me a bitter lot. \bibverse{21} I
had plenty when I left, but the Lord has brought me back empty handed.
Why should you call me Naomi, now that the Lord has afflicted me, and
the Almighty has brought misfortune on me?'' \bibverse{22} So Naomi and
Ruth, her Moabite daughter-in-law, returned from Moab. They reached
Bethlehem at the beginning of the barley harvest.
\hypertarget{in-the-fields-of-boaz}{%
\subsection{In the Fields of Boaz}\label{in-the-fields-of-boaz}}
\hypertarget{section-1}{%
\section{2}\label{section-1}}
\bibverse{1} Now Naomi was related through her husband to a very wealthy
man of the family of Elimelech named Boaz. \bibverse{2} Ruth the Moabite
said to Naomi, ``Let me now go into the fields and gather leftover grain
behind anyone who will allow me.''
``Go, my daughter.'' she replied.
\bibverse{3} So she went to glean in the field after the reapers. As it
happened, she was in that part of the field which belonged to Boaz, who
was of the family of Elimelech. \bibverse{4} When Boaz came from
Bethlehem and said to the reapers, ``The Lord be with you,'' they
answered him, ``May the Lord bless you.''
\bibverse{5} ``Whose girl is this?'' Boaz asked his servant who had
charge of the reapers. \bibverse{6} The servant who had charge of the
reapers replied, ``It is the Moabite girl who came back with Naomi from
the territory of Moab. \bibverse{7} She asked to be allowed to glean and
gather sheaves after the reapers. So she came and has continued to work
until now and she has not rested a moment in the field.''
\bibverse{8} Then Boaz said to Ruth, ``Listen, my daughter. Do not go to
glean in another field nor leave this place, but stay here with my
girls. \bibverse{9} Watch where the men are reaping and follow the
gleaners. I have told the young men not to trouble you. When you are
thirsty, go to the jars and drink of that which the young men have
drawn.''
\bibverse{10} Then she bowed low and said to him, ``Why are you so kind
to me, to take interest in me when I am just a foreigner?''
\bibverse{11} Boaz replied, ``I have heard what you have done for your
mother-in-law since the death of your husband, and how you left your
father and mother and your native land to come to a people that you did
not know before. \bibverse{12} May the Lord repay you for what you have
done, and may you be fully rewarded by the God of Israel, under whose
wings you have come to take refuge.'' \bibverse{13} Then she said, ``I
trust I may please you, my lord, for you have comforted me and spoken
kindly to your servant, although I am not really equal to one of your
own servants.''
\bibverse{14} At mealtime Boaz said to Ruth, ``Come here and eat some of
the food and dip your piece of bread in the vinegar.'' So she sat beside
the reapers, and he passed her some roasted grain. She ate until she was
satisfied and had some left. \bibverse{15} When she rose to glean, Boaz
gave this order to his young men: ``Let her glean even amongst the
sheaves and do not disturb her. \bibverse{16} Also pull out some for her
from the bundles and leave for her to glean, and do not find fault with
her.''
\bibverse{17} So she gleaned in the field until evening, then beat out
what she had gleaned. It was about a bushel of barley. \bibverse{18}
Then she took it up and went into the town and showed her mother-in-law
what she had gleaned. She also brought out and gave her that which she
had left from her meal after she had had enough.
\bibverse{19} ``Where did you glean today, and where did you work?''
asked her mother-in-law. ``A blessing on him who took notice of you!''
So she told her mother-in-law where she had worked. ``The name of the
man with whom I worked today,'' she said, ``is Boaz.''
\bibverse{20} Naomi said to her daughter-in-law, ``May the blessing of
the Lord rest on this man who has not ceased to show his loving-kindness
to the living and to the dead. The man,'' she added, ``is a near
relation of ours.''
\bibverse{21} ``He told me,'' Ruth said, ``that I must keep near his
young men until they have completed all his harvest.''
\bibverse{22} Naomi said to Ruth, ``It is best, my daughter, that you
should go out with his girls because you might not be as safe in another
field.'' \bibverse{23} So she gleaned with the girls of Boaz until the
end of the barley and wheat harvest; but she lived with her
mother-in-law.
\hypertarget{night-and-morning}{%
\subsection{Night and morning}\label{night-and-morning}}
\hypertarget{section-2}{%
\section{3}\label{section-2}}
\bibverse{1} One day, Naomi said to Ruth, ``My daughter, should I not
seek to secure a home for you where you will be happy and prosperous?
\bibverse{2} Is not Boaz, with whose girls you have been, a relative of
ours? \bibverse{3} Tonight he is going to winnow barley on the
threshing-floor. So bathe and anoint yourself and put on your best
clothes and go down to the threshing-floor. But do not make yourself
known to the man until he has finished eating and drinking. \bibverse{4}
Then when he lies down, mark the place where he lies. Go in, uncover his
feet, lie down, and then he will tell you what to do.''
\bibverse{5} ``I will do as you say.'' Ruth said to her.
\bibverse{6} So she went down to the threshing-floor and did just as her
mother-in-law told her. \bibverse{7} When Boaz had finished eating and
drinking and was in a happy mood, he went to lie down at the end of the
heap of grain. Then Ruth came quietly and uncovered his feet and lay
down. \bibverse{8} At midnight the man was startled and turned over, and
there was a woman lying at his feet! \bibverse{9} ``Who are you?'' he
said. ``I am Ruth your servant.'' she answered, ``Spread your cloak over
your servant, for you are a near relative.'' \bibverse{10} He said,
``May you be blest by the Lord, my daughter. You have shown me greater
favour now than at first, for you have not followed young men, whether
poor or rich. \bibverse{11} My daughter, have no fear; I will do for you
all that you ask; for the whole town knows that you are a virtuous
woman. \bibverse{12} Now it is true that I am a near relative, but there
is another man nearer than I. \bibverse{13} Stay here tonight, and then
in the morning, if he will perform for you the duty of a kinsman, well,
let him do it. But if he will not perform for you the duty of a kinsman,
then as surely as the Lord lives, I will do it for you. Lie down until
morning.''
\bibverse{14} So she lay at his feet until morning, but rose before
anyone could recognise her, for Boaz said, ``No one must know that a
woman came to the threshing-floor.'' \bibverse{15} He also said, ``Bring
the cloak which you have on and hold it.'' So she held it while he
poured into it six measures of barley and laid it on her shoulders. Then
he went into the city.
\bibverse{16} When Ruth came to her mother-in-law, Naomi asked, ``Is it
you, my daughter?'' Then Ruth told Naomi all that the man had done for
her. \bibverse{17} ``He gave me these six measures of barley,'' she
said, ``for he said I should not go to my mother-in-law empty-handed.''
\bibverse{18} ``Wait quietly, my daughter.'' Naomi said, ``Until you
know how the affair will turn out, for the man will not rest unless he
settles it all today.''
\hypertarget{section-3}{%
\section{4}\label{section-3}}
\bibverse{1} Then Boaz went up to the gate and sat down. Just then the
near kinsman of whom Boaz had spoken came along. Boaz said, ``Hello,
So-and-so (calling him by name), come here and sit down.'' So he stopped
and sat down. \bibverse{2} Boaz also took ten of the town elders and
said, ``Sit down here.'' So they sat down.
\bibverse{3} Then he said to the near relative, ``Naomi, who has come
back from the country of Moab, is offering for sale the piece of land
which belonged to our relative Elimelech, \bibverse{4} and I thought
that I would lay the matter before you, suggesting that you buy it in
the presence of these men who sit here and of the elders of my people.
If you will buy it and so keep it in the possession of the family, do
so; but if not; then tell me, so that I may know; for no one but you has
the right to buy it, and I am next to you.'' ``I will buy it.'' he said.
\bibverse{5} Then Boaz said, ``On the day you buy the field from Naomi,
you must also marry Ruth the Moabite, the widow of the dead, in order to
preserve the name of the dead in connection with his inheritance.''
\bibverse{6} ``I cannot buy it for myself without spoiling my own
inheritance,'' the near relative said. ``You take my right of buying it
as a relative, because I cannot do so.''
\bibverse{7} Now this used to be the custom in Israel: to make valid
anything relating to a matter of redemption or exchange, a man drew off
his sandal and gave it to the other man; and this was the way contracts
were attested in Israel. \bibverse{8} So when the near relative said to
Boaz, ``Buy it for yourself,'' Boaz drew off the man's sandal.
\bibverse{9} Then Boaz said to the elders and to all the people, ``You
are witnesses at this time that I have bought all that was Elimelech's
and all that was Chilion's and Mahlon's from Naomi. \bibverse{10}
Moreover I have secured Ruth the Moabite, the wife of Mahlon, to be my
wife, in order to perpetuate the name of the dead in connection with his
inheritance, so that his name will not disappear from amongst his
relatives and from the household where he lived. You are witnesses this
day.''
\bibverse{11} Then all the people who were at the gate and the elders
said, ``We are witnesses. May the Lord make the woman who is coming into
your house like Rachel and Leah, who together built the house of Israel.
May you do well in Ephrata, and become famous in Bethlehem.
\bibverse{12} From the children whom the Lord will give you by this
young woman may your household become like the household of Perez, whom
Tamar bore to Judah.''
\bibverse{13} So Boaz married Ruth, and she became his wife; and the
Lord gave to her a son. \bibverse{14} Then the women said to Naomi,
``Blessed be the Lord who has not left you at this time without a near
relative, and may his name be famous in Israel. \bibverse{15} This child
will restore your vigour and nourish you in your old age; for your
daughter-in-law who loves you, who is worth more to you than seven sons,
has borne a son to Boaz!''
\bibverse{16} So Naomi took the child in her arms and cared for him as
if he was her own. \bibverse{17} The women of the neighbourhood gave him
a name, saying, ``A son is born to Naomi!'' They named him Obed; he
became the father of Jesse, who was the father of David. \#\# Genealogy
\bibverse{18} This is the genealogy of Perez: Perez was the father of
Hezron, \bibverse{19} Hezron of Ram, Ram of Amminadab, \bibverse{20}
Amminidab of Nashon, Nashon of Salmon, \bibverse{21} Salmon of Boaz,
Boaz of Obed, \bibverse{22} Obed of Jesse, Jesse of David.
|
{-# OPTIONS --without-K --safe #-}
module Dodo.Binary.Intersection where
-- Stdlib imports
open import Level using (Level; _⊔_)
open import Data.Product as P
open import Data.Product using (_×_; _,_; swap; proj₁; proj₂)
open import Relation.Binary using (REL)
-- Local imports
open import Dodo.Binary.Equality
-- # Definitions
infixl 30 _∩₂_
_∩₂_ : {a b ℓ₁ ℓ₂ : Level} {A : Set a} {B : Set b}
→ REL A B ℓ₁
→ REL A B ℓ₂
→ REL A B (ℓ₁ ⊔ ℓ₂)
_∩₂_ P Q x y = P x y × Q x y
-- # Properties
module _ {a b ℓ : Level} {A : Set a} {B : Set b} {R : REL A B ℓ} where
∩₂-idem : (R ∩₂ R) ⇔₂ R
∩₂-idem = ⇔: ⊆-proof ⊇-proof
where
⊆-proof : (R ∩₂ R) ⊆₂' R
⊆-proof _ _ = proj₁
⊇-proof : R ⊆₂' (R ∩₂ R)
⊇-proof _ _ Rxy = (Rxy , Rxy)
module _ {a b ℓ₁ ℓ₂ : Level} {A : Set a} {B : Set b}
{P : REL A B ℓ₁} {Q : REL A B ℓ₂} where
∩₂-comm : (P ∩₂ Q) ⇔₂ (Q ∩₂ P)
∩₂-comm = ⇔: (λ _ _ → swap) (λ _ _ → swap)
module _ {a b ℓ₁ ℓ₂ ℓ₃ : Level} {A : Set a} {B : Set b}
{P : REL A B ℓ₁} {Q : REL A B ℓ₂} {R : REL A B ℓ₃} where
∩₂-assoc : P ∩₂ (Q ∩₂ R) ⇔₂ (P ∩₂ Q) ∩₂ R
∩₂-assoc = ⇔: ⊆-proof ⊇-proof
where
⊆-proof : P ∩₂ (Q ∩₂ R) ⊆₂' (P ∩₂ Q) ∩₂ R
⊆-proof _ _ (Pxy , (Qxy , Rxy)) = ((Pxy , Qxy) , Rxy)
⊇-proof : (P ∩₂ Q) ∩₂ R ⊆₂' P ∩₂ (Q ∩₂ R)
⊇-proof _ _ ((Pxy , Qxy) , Rxy) = (Pxy , (Qxy , Rxy))
-- # Operations
-- ## Operations: ⊆₂
module _ {a b ℓ₁ ℓ₂ ℓ₃ : Level} {A : Set a} {B : Set b}
{P : REL A B ℓ₁} {Q : REL A B ℓ₂} {R : REL A B ℓ₃} where
∩₂-combine-⊆₂ : P ⊆₂ Q → P ⊆₂ R → P ⊆₂ (Q ∩₂ R)
∩₂-combine-⊆₂ (⊆: P⊆Q) (⊆: P⊆R) = ⊆: (λ x y Pxy → (P⊆Q x y Pxy , P⊆R x y Pxy))
module _ {a b ℓ₁ ℓ₂ : Level} {A : Set a} {B : Set b}
{P : REL A B ℓ₁} {Q : REL A B ℓ₂} where
∩₂-introˡ-⊆₂ : (P ∩₂ Q) ⊆₂ Q
∩₂-introˡ-⊆₂ = ⊆: λ _ _ → proj₂
∩₂-introʳ-⊆₂ : (P ∩₂ Q) ⊆₂ P
∩₂-introʳ-⊆₂ = ⊆: λ _ _ → proj₁
module _ {a b ℓ₁ ℓ₂ ℓ₃ : Level} {A : Set a} {B : Set b}
{P : REL A B ℓ₁} {Q : REL A B ℓ₂} {R : REL A B ℓ₃} where
∩₂-elimˡ-⊆₂ : P ⊆₂ (Q ∩₂ R) → P ⊆₂ R
∩₂-elimˡ-⊆₂ (⊆: P⊆[Q∩R]) = ⊆: (λ x y Pxy → proj₂ (P⊆[Q∩R] x y Pxy))
∩₂-elimʳ-⊆₂ : P ⊆₂ (Q ∩₂ R) → P ⊆₂ Q
∩₂-elimʳ-⊆₂ (⊆: P⊆[Q∩R]) = ⊆: (λ x y Pxy → proj₁ (P⊆[Q∩R] x y Pxy))
module _ {a b ℓ₁ ℓ₂ ℓ₃ : Level} {A : Set a} {B : Set b}
{P : REL A B ℓ₁} {Q : REL A B ℓ₂} {R : REL A B ℓ₃} where
∩₂-substˡ-⊆₂ : P ⊆₂ Q → (P ∩₂ R) ⊆₂ (Q ∩₂ R)
∩₂-substˡ-⊆₂ (⊆: P⊆Q) = ⊆: (λ x y → P.map₁ (P⊆Q x y))
∩₂-substʳ-⊆₂ : P ⊆₂ Q → (R ∩₂ P) ⊆₂ (R ∩₂ Q)
∩₂-substʳ-⊆₂ (⊆: P⊆Q) = ⊆: (λ x y → P.map₂ (P⊆Q x y))
-- ## Operations: ⇔₂
module _ {a b ℓ₁ ℓ₂ ℓ₃ : Level} {A : Set a} {B : Set b}
{P : REL A B ℓ₁} {Q : REL A B ℓ₂} {R : REL A B ℓ₃} where
∩₂-substˡ : P ⇔₂ Q → (P ∩₂ R) ⇔₂ (Q ∩₂ R)
∩₂-substˡ = ⇔₂-compose ∩₂-substˡ-⊆₂ ∩₂-substˡ-⊆₂
∩₂-substʳ : P ⇔₂ Q → (R ∩₂ P) ⇔₂ (R ∩₂ Q)
∩₂-substʳ = ⇔₂-compose ∩₂-substʳ-⊆₂ ∩₂-substʳ-⊆₂
|
If $f$ is continuous on the closed segment $[a,b]$, then the contour integral of $f$ along the line segment $[a,b]$ is equal to the sum of the contour integrals of $f$ along the line segments $[a,c]$ and $[c,b]$.
|
Formal statement is: lemma orthogonal_self: "orthogonal x x \<longleftrightarrow> x = 0" Informal statement is: Two vectors are orthogonal if and only if they are equal to zero.
|
module ReadNum
import Data.Strings
%default total
readNumber : HasIO io => io (Maybe Nat)
readNumber = do n <- getLine
if all isDigit (unpack n)
then pure $ Just (stringToNatOrZ n)
else pure Nothing
readNumbers : HasIO io => io (Maybe (Nat, Nat))
readNumbers = do m <- readNumber
case m of
Nothing => pure Nothing
Just mm => do n <- readNumber
case n of
Nothing => pure Nothing
Just nn => pure $ Just (mm, nn)
-- pattern-matched destucturing
readNumbersImproved : HasIO io => io (Maybe (Nat, Nat))
readNumbersImproved = do Just m <- readNumber | Nothing => pure Nothing
Just n <- readNumber | Nothing => pure Nothing
pure $ Just (m, n)
|
Formal statement is: lemma asymp_equiv_sym: "f \<sim>[F] g \<longleftrightarrow> g \<sim>[F] f" Informal statement is: If $f$ is asymptotically equivalent to $g$, then $g$ is asymptotically equivalent to $f$.
|
program main
use mpi
DOUBLE PRECISION f(10,2)
integer code,np,world,ierr
do i=1,10
f(i,1)=1
f(i,2)=2
enddo
call mpi_init(ierr)
call mpi_comm_dup(mpi_comm_world,world,ierr)
call mpi_comm_rank(world,code,ierr)
call mpi_comm_size(world,np,ierr)
do i=1,3
call update()
if(code.eq.0)then
print*,'global f',f
endif
enddo
call mpi_comm_free(world,ierr)
call mpi_finalize(ierr)
contains
subroutine update()
integer i,j,istr,iend,avgtask,extra,currtasknum
double precision, allocatable :: kf(:,:)
character*8 name
!parallel
avgtask=10/np
extra=mod(10,np)
!print*,'extra',extra
if(code .lt. extra)then
istr=(avgtask+1)*code+1
iend=(avgtask+1)*(code+1)
currtasknum=avgtask+1
else
istr=avgtask*code+1+extra
iend=avgtask*(code+1)+extra
currtasknum=avgtask
endif
print*,'code,istr,iend',code,',',istr,',',iend
do i=istr,iend
f(i,1)=f(i,1)+10*code
f(i,2)=f(i,2)+10*code
enddo
if(code.eq.0)then
allocate(kf(10,2))
endif
call mpi_gather(f(istr,1),2*currtasknum,mpi_integer,kf(istr,1),2*currtasknum,mpi_integer,0,world,ierr)
call mpi_gather(f(istr,2),2*currtasknum,mpi_integer,kf(istr,2),2*currtasknum,mpi_integer,0,world,ierr)
if(code.eq.0)then
f=kf
! print*,'dddddddd',f
deallocate(kf)
endif
end subroutine update
end program
|
module TTImp.Elab.Prim
import Core.TT
%default covering
export
checkPrim : FC -> Constant -> (Term vars, Term vars)
checkPrim fc (I i) = (PrimVal fc (I i), PrimVal fc IntType)
checkPrim fc (BI i) = (PrimVal fc (BI i), PrimVal fc IntegerType)
checkPrim fc (B8 i) = (PrimVal fc (B8 i), PrimVal fc Bits8Type)
checkPrim fc (B16 i) = (PrimVal fc (B16 i), PrimVal fc Bits16Type)
checkPrim fc (B32 i) = (PrimVal fc (B32 i), PrimVal fc Bits32Type)
checkPrim fc (B64 i) = (PrimVal fc (B64 i), PrimVal fc Bits64Type)
checkPrim fc (Str s) = (PrimVal fc (Str s), PrimVal fc StringType)
checkPrim fc (Ch c) = (PrimVal fc (Ch c), PrimVal fc CharType)
checkPrim fc (Db d) = (PrimVal fc (Db d), PrimVal fc DoubleType)
checkPrim fc WorldVal = (PrimVal fc WorldVal, PrimVal fc WorldType)
checkPrim fc IntType = (PrimVal fc IntType, TType fc)
checkPrim fc IntegerType = (PrimVal fc IntegerType, TType fc)
checkPrim fc Bits8Type = (PrimVal fc Bits8Type, TType fc)
checkPrim fc Bits16Type = (PrimVal fc Bits16Type, TType fc)
checkPrim fc Bits32Type = (PrimVal fc Bits32Type, TType fc)
checkPrim fc Bits64Type = (PrimVal fc Bits64Type, TType fc)
checkPrim fc StringType = (PrimVal fc StringType, TType fc)
checkPrim fc CharType = (PrimVal fc CharType, TType fc)
checkPrim fc DoubleType = (PrimVal fc DoubleType, TType fc)
checkPrim fc WorldType = (PrimVal fc WorldType, TType fc)
|
{-# OPTIONS --without-K --safe #-}
open import Definition.Typed.EqualityRelation
module Definition.LogicalRelation {{eqrel : EqRelSet}} where
open EqRelSet {{...}}
open import Definition.Untyped as U
open import Definition.Untyped.Properties
open import Definition.Typed.Properties
open import Definition.Typed
open import Definition.Typed.Weakening
open import Tools.Product
import Tools.PropositionalEquality as PE
-- The different cases of the logical relation are spread out through out
-- this file. This is due to them having different dependencies.
-- We will refer to expressions that satisfies the logical relation as reducible.
-- Reducibility of Neutrals:
-- Neutral type
record _⊩ne_ (Γ : Con Term) (A : Term) : Set where
constructor ne
field
K : Term
D : Γ ⊢ A :⇒*: K
neK : Neutral K
K≡K : Γ ⊢ K ~ K ∷ U
-- Neutral type equality
record _⊩ne_≡_/_ (Γ : Con Term) (A B : Term) ([A] : Γ ⊩ne A) : Set where
constructor ne₌
open _⊩ne_ [A]
field
M : Term
D′ : Γ ⊢ B :⇒*: M
neM : Neutral M
K≡M : Γ ⊢ K ~ M ∷ U
-- Neutral term in WHNF
record _⊩neNf_∷_ (Γ : Con Term) (k A : Term) : Set where
inductive
constructor neNfₜ
field
neK : Neutral k
⊢k : Γ ⊢ k ∷ A
k≡k : Γ ⊢ k ~ k ∷ A
-- Neutral term
record _⊩ne_∷_/_ (Γ : Con Term) (t A : Term) ([A] : Γ ⊩ne A) : Set where
inductive
constructor neₜ
open _⊩ne_ [A]
field
k : Term
d : Γ ⊢ t :⇒*: k ∷ K
nf : Γ ⊩neNf k ∷ K
-- Neutral term equality in WHNF
record _⊩neNf_≡_∷_ (Γ : Con Term) (k m A : Term) : Set where
inductive
constructor neNfₜ₌
field
neK : Neutral k
neM : Neutral m
k≡m : Γ ⊢ k ~ m ∷ A
-- Neutral term equality
record _⊩ne_≡_∷_/_ (Γ : Con Term) (t u A : Term) ([A] : Γ ⊩ne A) : Set where
constructor neₜ₌
open _⊩ne_ [A]
field
k m : Term
d : Γ ⊢ t :⇒*: k ∷ K
d′ : Γ ⊢ u :⇒*: m ∷ K
nf : Γ ⊩neNf k ≡ m ∷ K
-- Reducibility of natural numbers:
-- Natural number type
_⊩ℕ_ : (Γ : Con Term) (A : Term) → Set
Γ ⊩ℕ A = Γ ⊢ A :⇒*: ℕ
-- Natural number type equality
_⊩ℕ_≡_ : (Γ : Con Term) (A B : Term) → Set
Γ ⊩ℕ A ≡ B = Γ ⊢ B ⇒* ℕ
mutual
-- Natural number term
record _⊩ℕ_∷ℕ (Γ : Con Term) (t : Term) : Set where
inductive
constructor ℕₜ
field
n : Term
d : Γ ⊢ t :⇒*: n ∷ ℕ
n≡n : Γ ⊢ n ≅ n ∷ ℕ
prop : Natural-prop Γ n
-- WHNF property of natural number terms
data Natural-prop (Γ : Con Term) : (n : Term) → Set where
sucᵣ : ∀ {n} → Γ ⊩ℕ n ∷ℕ → Natural-prop Γ (suc n)
zeroᵣ : Natural-prop Γ zero
ne : ∀ {n} → Γ ⊩neNf n ∷ ℕ → Natural-prop Γ n
mutual
-- Natural number term equality
record _⊩ℕ_≡_∷ℕ (Γ : Con Term) (t u : Term) : Set where
inductive
constructor ℕₜ₌
field
k k′ : Term
d : Γ ⊢ t :⇒*: k ∷ ℕ
d′ : Γ ⊢ u :⇒*: k′ ∷ ℕ
k≡k′ : Γ ⊢ k ≅ k′ ∷ ℕ
prop : [Natural]-prop Γ k k′
-- WHNF property of Natural number term equality
data [Natural]-prop (Γ : Con Term) : (n n′ : Term) → Set where
sucᵣ : ∀ {n n′} → Γ ⊩ℕ n ≡ n′ ∷ℕ → [Natural]-prop Γ (suc n) (suc n′)
zeroᵣ : [Natural]-prop Γ zero zero
ne : ∀ {n n′} → Γ ⊩neNf n ≡ n′ ∷ ℕ → [Natural]-prop Γ n n′
-- Natural extraction from term WHNF property
natural : ∀ {Γ n} → Natural-prop Γ n → Natural n
natural (sucᵣ x) = sucₙ
natural zeroᵣ = zeroₙ
natural (ne (neNfₜ neK ⊢k k≡k)) = ne neK
-- Natural extraction from term equality WHNF property
split : ∀ {Γ a b} → [Natural]-prop Γ a b → Natural a × Natural b
split (sucᵣ x) = sucₙ , sucₙ
split zeroᵣ = zeroₙ , zeroₙ
split (ne (neNfₜ₌ neK neM k≡m)) = ne neK , ne neM
-- Reducibility of Empty
-- Empty type
_⊩Empty_ : (Γ : Con Term) (A : Term) → Set
Γ ⊩Empty A = Γ ⊢ A :⇒*: Empty
-- Empty type equality
_⊩Empty_≡_ : (Γ : Con Term) (A B : Term) → Set
Γ ⊩Empty A ≡ B = Γ ⊢ B ⇒* Empty
-- WHNF property of absurd terms
data Empty-prop (Γ : Con Term) : (n : Term) → Set where
ne : ∀ {n} → Γ ⊩neNf n ∷ Empty → Empty-prop Γ n
-- Empty term
record _⊩Empty_∷Empty (Γ : Con Term) (t : Term) : Set where
inductive
constructor Emptyₜ
field
n : Term
d : Γ ⊢ t :⇒*: n ∷ Empty
n≡n : Γ ⊢ n ≅ n ∷ Empty
prop : Empty-prop Γ n
data [Empty]-prop (Γ : Con Term) : (n n′ : Term) → Set where
ne : ∀ {n n′} → Γ ⊩neNf n ≡ n′ ∷ Empty → [Empty]-prop Γ n n′
-- Empty term equality
record _⊩Empty_≡_∷Empty (Γ : Con Term) (t u : Term) : Set where
inductive
constructor Emptyₜ₌
field
k k′ : Term
d : Γ ⊢ t :⇒*: k ∷ Empty
d′ : Γ ⊢ u :⇒*: k′ ∷ Empty
k≡k′ : Γ ⊢ k ≅ k′ ∷ Empty
prop : [Empty]-prop Γ k k′
empty : ∀ {Γ n} → Empty-prop Γ n → Neutral n
empty (ne (neNfₜ neK _ _)) = neK
esplit : ∀ {Γ a b} → [Empty]-prop Γ a b → Neutral a × Neutral b
esplit (ne (neNfₜ₌ neK neM k≡m)) = neK , neM
-- Reducibility of Unit
-- Unit type
_⊩Unit_ : (Γ : Con Term) (A : Term) → Set
Γ ⊩Unit A = Γ ⊢ A :⇒*: Unit
-- Unit type equality
_⊩Unit_≡_ : (Γ : Con Term) (A B : Term) → Set
Γ ⊩Unit A ≡ B = Γ ⊢ B ⇒* Unit
record _⊩Unit_∷Unit (Γ : Con Term) (t : Term) : Set where
inductive
constructor Unitₜ
field
n : Term
d : Γ ⊢ t :⇒*: n ∷ Unit
prop : Whnf n
-- Unit term equality
record _⊩Unit_≡_∷Unit (Γ : Con Term) (t u : Term) : Set where
constructor Unitₜ₌
field
⊢t : Γ ⊢ t ∷ Unit
⊢u : Γ ⊢ u ∷ Unit
-- Type levels
data TypeLevel : Set where
⁰ : TypeLevel
¹ : TypeLevel
data _<_ : (i j : TypeLevel) → Set where
0<1 : ⁰ < ¹
-- Logical relation
-- Exported interface
record LogRelKit : Set₁ where
constructor Kit
field
_⊩U : (Γ : Con Term) → Set
_⊩B⟨_⟩_ : (Γ : Con Term) (W : BindingType) → Term → Set
_⊩_ : (Γ : Con Term) → Term → Set
_⊩_≡_/_ : (Γ : Con Term) (A B : Term) → Γ ⊩ A → Set
_⊩_∷_/_ : (Γ : Con Term) (t A : Term) → Γ ⊩ A → Set
_⊩_≡_∷_/_ : (Γ : Con Term) (t u A : Term) → Γ ⊩ A → Set
module LogRel (l : TypeLevel) (rec : ∀ {l′} → l′ < l → LogRelKit) where
-- Reducibility of Universe:
-- Universe type
record _⊩¹U (Γ : Con Term) : Set where
constructor Uᵣ
field
l′ : TypeLevel
l< : l′ < l
⊢Γ : ⊢ Γ
-- Universe type equality
_⊩¹U≡_ : (Γ : Con Term) (B : Term) → Set
Γ ⊩¹U≡ B = B PE.≡ U -- Note lack of reduction
-- Universe term
record _⊩¹U_∷U/_ {l′} (Γ : Con Term) (t : Term) (l< : l′ < l) : Set where
constructor Uₜ
open LogRelKit (rec l<)
field
A : Term
d : Γ ⊢ t :⇒*: A ∷ U
typeA : Type A
A≡A : Γ ⊢ A ≅ A ∷ U
[t] : Γ ⊩ t
-- Universe term equality
record _⊩¹U_≡_∷U/_ {l′} (Γ : Con Term) (t u : Term) (l< : l′ < l) : Set where
constructor Uₜ₌
open LogRelKit (rec l<)
field
A B : Term
d : Γ ⊢ t :⇒*: A ∷ U
d′ : Γ ⊢ u :⇒*: B ∷ U
typeA : Type A
typeB : Type B
A≡B : Γ ⊢ A ≅ B ∷ U
[t] : Γ ⊩ t
[u] : Γ ⊩ u
[t≡u] : Γ ⊩ t ≡ u / [t]
mutual
-- Reducibility of Binding types (Π, Σ):
-- B-type
record _⊩¹B⟨_⟩_ (Γ : Con Term) (W : BindingType) (A : Term) : Set where
inductive
constructor Bᵣ
field
F : Term
G : Term
D : Γ ⊢ A :⇒*: ⟦ W ⟧ F ▹ G
⊢F : Γ ⊢ F
⊢G : Γ ∙ F ⊢ G
A≡A : Γ ⊢ ⟦ W ⟧ F ▹ G ≅ ⟦ W ⟧ F ▹ G
[F] : ∀ {ρ Δ} → ρ ∷ Δ ⊆ Γ → ⊢ Δ → Δ ⊩¹ U.wk ρ F
[G] : ∀ {ρ Δ a}
→ ([ρ] : ρ ∷ Δ ⊆ Γ) (⊢Δ : ⊢ Δ)
→ Δ ⊩¹ a ∷ U.wk ρ F / [F] [ρ] ⊢Δ
→ Δ ⊩¹ U.wk (lift ρ) G [ a ]
G-ext : ∀ {ρ Δ a b}
→ ([ρ] : ρ ∷ Δ ⊆ Γ) (⊢Δ : ⊢ Δ)
→ ([a] : Δ ⊩¹ a ∷ U.wk ρ F / [F] [ρ] ⊢Δ)
→ ([b] : Δ ⊩¹ b ∷ U.wk ρ F / [F] [ρ] ⊢Δ)
→ Δ ⊩¹ a ≡ b ∷ U.wk ρ F / [F] [ρ] ⊢Δ
→ Δ ⊩¹ U.wk (lift ρ) G [ a ] ≡ U.wk (lift ρ) G [ b ] / [G] [ρ] ⊢Δ [a]
-- B-type equality
record _⊩¹B⟨_⟩_≡_/_ (Γ : Con Term) (W : BindingType) (A B : Term) ([A] : Γ ⊩¹B⟨ W ⟩ A) : Set where
inductive
constructor B₌
open _⊩¹B⟨_⟩_ [A]
field
F′ : Term
G′ : Term
D′ : Γ ⊢ B ⇒* ⟦ W ⟧ F′ ▹ G′
A≡B : Γ ⊢ ⟦ W ⟧ F ▹ G ≅ ⟦ W ⟧ F′ ▹ G′
[F≡F′] : ∀ {ρ Δ}
→ ([ρ] : ρ ∷ Δ ⊆ Γ) (⊢Δ : ⊢ Δ)
→ Δ ⊩¹ U.wk ρ F ≡ U.wk ρ F′ / [F] [ρ] ⊢Δ
[G≡G′] : ∀ {ρ Δ a}
→ ([ρ] : ρ ∷ Δ ⊆ Γ) (⊢Δ : ⊢ Δ)
→ ([a] : Δ ⊩¹ a ∷ U.wk ρ F / [F] [ρ] ⊢Δ)
→ Δ ⊩¹ U.wk (lift ρ) G [ a ] ≡ U.wk (lift ρ) G′ [ a ] / [G] [ρ] ⊢Δ [a]
-- Term reducibility of Π-type
_⊩¹Π_∷_/_ : (Γ : Con Term) (t A : Term) ([A] : Γ ⊩¹B⟨ BΠ ⟩ A) → Set
Γ ⊩¹Π t ∷ A / Bᵣ F G D ⊢F ⊢G A≡A [F] [G] G-ext =
∃ λ f → Γ ⊢ t :⇒*: f ∷ Π F ▹ G
× Function f
× Γ ⊢ f ≅ f ∷ Π F ▹ G
× (∀ {ρ Δ a b}
([ρ] : ρ ∷ Δ ⊆ Γ) (⊢Δ : ⊢ Δ)
([a] : Δ ⊩¹ a ∷ U.wk ρ F / [F] [ρ] ⊢Δ)
([b] : Δ ⊩¹ b ∷ U.wk ρ F / [F] [ρ] ⊢Δ)
([a≡b] : Δ ⊩¹ a ≡ b ∷ U.wk ρ F / [F] [ρ] ⊢Δ)
→ Δ ⊩¹ U.wk ρ f ∘ a ≡ U.wk ρ f ∘ b ∷ U.wk (lift ρ) G [ a ] / [G] [ρ] ⊢Δ [a])
× (∀ {ρ Δ a} → ([ρ] : ρ ∷ Δ ⊆ Γ) (⊢Δ : ⊢ Δ)
→ ([a] : Δ ⊩¹ a ∷ U.wk ρ F / [F] [ρ] ⊢Δ)
→ Δ ⊩¹ U.wk ρ f ∘ a ∷ U.wk (lift ρ) G [ a ] / [G] [ρ] ⊢Δ [a])
{- NOTE(WN): Last 2 fields could be refactored to a single forall.
But touching this definition is painful, so only do it
if you have to change it anyway. -}
-- Issue: Agda complains about record use not being strictly positive.
-- Therefore we have to use ×
-- Term equality of Π-type
_⊩¹Π_≡_∷_/_ : (Γ : Con Term) (t u A : Term) ([A] : Γ ⊩¹B⟨ BΠ ⟩ A) → Set
Γ ⊩¹Π t ≡ u ∷ A / [A]@(Bᵣ F G D ⊢F ⊢G A≡A [F] [G] G-ext) =
∃₂ λ f g → Γ ⊢ t :⇒*: f ∷ Π F ▹ G
× Γ ⊢ u :⇒*: g ∷ Π F ▹ G
× Function f
× Function g
× Γ ⊢ f ≅ g ∷ Π F ▹ G
× Γ ⊩¹Π t ∷ A / [A]
× Γ ⊩¹Π u ∷ A / [A]
× (∀ {ρ Δ a} ([ρ] : ρ ∷ Δ ⊆ Γ) (⊢Δ : ⊢ Δ)
([a] : Δ ⊩¹ a ∷ U.wk ρ F / [F] [ρ] ⊢Δ)
→ Δ ⊩¹ U.wk ρ f ∘ a ≡ U.wk ρ g ∘ a ∷ U.wk (lift ρ) G [ a ] / [G] [ρ] ⊢Δ [a])
-- Issue: Same as above.
-- Term reducibility of Σ-type
_⊩¹Σ_∷_/_ : (Γ : Con Term) (t A : Term) ([A] : Γ ⊩¹B⟨ BΣ ⟩ A) → Set
Γ ⊩¹Σ t ∷ A / [A]@(Bᵣ F G D ⊢F ⊢G A≡A [F] [G] G-ext) =
∃ λ p → Γ ⊢ t :⇒*: p ∷ Σ F ▹ G
× Product p
× Γ ⊢ p ≅ p ∷ Σ F ▹ G
× (Σ (Γ ⊩¹ fst p ∷ U.wk id F / [F] id (wf ⊢F)) λ [fst]
→ Γ ⊩¹ snd p ∷ U.wk (lift id) G [ fst p ] / [G] id (wf ⊢F) [fst])
-- Term equality of Σ-type
_⊩¹Σ_≡_∷_/_ : (Γ : Con Term) (t u A : Term) ([A] : Γ ⊩¹B⟨ BΣ ⟩ A) → Set
Γ ⊩¹Σ t ≡ u ∷ A / [A]@(Bᵣ F G D ⊢F ⊢G A≡A [F] [G] G-ext) =
∃₂ λ p r → Γ ⊢ t :⇒*: p ∷ Σ F ▹ G
× Γ ⊢ u :⇒*: r ∷ Σ F ▹ G
× Product p
× Product r
× Γ ⊢ p ≅ r ∷ Σ F ▹ G
× Γ ⊩¹Σ t ∷ A / [A]
× Γ ⊩¹Σ u ∷ A / [A]
× (Σ (Γ ⊩¹ fst p ∷ U.wk id F / [F] id (wf ⊢F)) λ [fstp]
→ Γ ⊩¹ fst r ∷ U.wk id F / [F] id (wf ⊢F)
× Γ ⊩¹ fst p ≡ fst r ∷ U.wk id F / [F] id (wf ⊢F)
× Γ ⊩¹ snd p ≡ snd r ∷ U.wk (lift id) G [ fst p ] / [G] id (wf ⊢F) [fstp])
-- Logical relation definition
data _⊩¹_ (Γ : Con Term) : Term → Set where
Uᵣ : Γ ⊩¹U → Γ ⊩¹ U
ℕᵣ : ∀ {A} → Γ ⊩ℕ A → Γ ⊩¹ A
Emptyᵣ : ∀ {A} → Γ ⊩Empty A → Γ ⊩¹ A
Unitᵣ : ∀ {A} → Γ ⊩Unit A → Γ ⊩¹ A
ne : ∀ {A} → Γ ⊩ne A → Γ ⊩¹ A
Bᵣ : ∀ {A} W → Γ ⊩¹B⟨ W ⟩ A → Γ ⊩¹ A
emb : ∀ {A l′} (l< : l′ < l) (let open LogRelKit (rec l<))
([A] : Γ ⊩ A) → Γ ⊩¹ A
_⊩¹_≡_/_ : (Γ : Con Term) (A B : Term) → Γ ⊩¹ A → Set
Γ ⊩¹ A ≡ B / Uᵣ UA = Γ ⊩¹U≡ B
Γ ⊩¹ A ≡ B / ℕᵣ D = Γ ⊩ℕ A ≡ B
Γ ⊩¹ A ≡ B / Emptyᵣ D = Γ ⊩Empty A ≡ B
Γ ⊩¹ A ≡ B / Unitᵣ D = Γ ⊩Unit A ≡ B
Γ ⊩¹ A ≡ B / ne neA = Γ ⊩ne A ≡ B / neA
Γ ⊩¹ A ≡ B / Bᵣ W BA = Γ ⊩¹B⟨ W ⟩ A ≡ B / BA
Γ ⊩¹ A ≡ B / emb l< [A] = Γ ⊩ A ≡ B / [A]
where open LogRelKit (rec l<)
_⊩¹_∷_/_ : (Γ : Con Term) (t A : Term) → Γ ⊩¹ A → Set
Γ ⊩¹ t ∷ .U / Uᵣ (Uᵣ l′ l< ⊢Γ) = Γ ⊩¹U t ∷U/ l<
Γ ⊩¹ t ∷ A / ℕᵣ D = Γ ⊩ℕ t ∷ℕ
Γ ⊩¹ t ∷ A / Emptyᵣ D = Γ ⊩Empty t ∷Empty
Γ ⊩¹ t ∷ A / Unitᵣ D = Γ ⊩Unit t ∷Unit
Γ ⊩¹ t ∷ A / ne neA = Γ ⊩ne t ∷ A / neA
Γ ⊩¹ t ∷ A / Bᵣ BΠ ΠA = Γ ⊩¹Π t ∷ A / ΠA
Γ ⊩¹ t ∷ A / Bᵣ BΣ ΣA = Γ ⊩¹Σ t ∷ A / ΣA
Γ ⊩¹ t ∷ A / emb l< [A] = Γ ⊩ t ∷ A / [A]
where open LogRelKit (rec l<)
_⊩¹_≡_∷_/_ : (Γ : Con Term) (t u A : Term) → Γ ⊩¹ A → Set
Γ ⊩¹ t ≡ u ∷ .U / Uᵣ (Uᵣ l′ l< ⊢Γ) = Γ ⊩¹U t ≡ u ∷U/ l<
Γ ⊩¹ t ≡ u ∷ A / ℕᵣ D = Γ ⊩ℕ t ≡ u ∷ℕ
Γ ⊩¹ t ≡ u ∷ A / Emptyᵣ D = Γ ⊩Empty t ≡ u ∷Empty
Γ ⊩¹ t ≡ u ∷ A / Unitᵣ D = Γ ⊩Unit t ≡ u ∷Unit
Γ ⊩¹ t ≡ u ∷ A / ne neA = Γ ⊩ne t ≡ u ∷ A / neA
Γ ⊩¹ t ≡ u ∷ A / Bᵣ BΠ ΠA = Γ ⊩¹Π t ≡ u ∷ A / ΠA
Γ ⊩¹ t ≡ u ∷ A / Bᵣ BΣ ΣA = Γ ⊩¹Σ t ≡ u ∷ A / ΣA
Γ ⊩¹ t ≡ u ∷ A / emb l< [A] = Γ ⊩ t ≡ u ∷ A / [A]
where open LogRelKit (rec l<)
kit : LogRelKit
kit = Kit _⊩¹U _⊩¹B⟨_⟩_
_⊩¹_ _⊩¹_≡_/_ _⊩¹_∷_/_ _⊩¹_≡_∷_/_
open LogRel public using (Uᵣ; ℕᵣ; Emptyᵣ; Unitᵣ; ne; Bᵣ; B₌; emb; Uₜ; Uₜ₌)
-- Patterns for the non-records of Π
pattern Πₜ f d funcF f≡f [f] [f]₁ = f , d , funcF , f≡f , [f] , [f]₁
pattern Πₜ₌ f g d d′ funcF funcG f≡g [f] [g] [f≡g] = f , g , d , d′ , funcF , funcG , f≡g , [f] , [g] , [f≡g]
pattern Σₜ p d pProd p≅p [fst] [snd] = p , d , pProd , p≅p , ([fst] , [snd])
pattern Σₜ₌ p r d d′ pProd rProd p≅r [t] [u] [fstp] [fstr] [fst≡] [snd≡] = p , r , d , d′ , pProd , rProd , p≅r , [t] , [u] , ([fstp] , [fstr] , [fst≡] , [snd≡])
pattern Uᵣ′ a b c = Uᵣ (Uᵣ a b c)
pattern ne′ a b c d = ne (ne a b c d)
pattern Bᵣ′ W a b c d e f g h i = Bᵣ W (Bᵣ a b c d e f g h i)
pattern Πᵣ′ a b c d e f g h i = Bᵣ′ BΠ a b c d e f g h i
pattern Σᵣ′ a b c d e f g h i = Bᵣ′ BΣ a b c d e f g h i
logRelRec : ∀ l {l′} → l′ < l → LogRelKit
logRelRec ⁰ = λ ()
logRelRec ¹ 0<1 = LogRel.kit ⁰ (λ ())
kit : ∀ (i : TypeLevel) → LogRelKit
kit l = LogRel.kit l (logRelRec l)
-- a bit of repetition in "kit ¹" definition, would work better with Fin 2 for
-- TypeLevel because you could recurse.
_⊩′⟨_⟩U : (Γ : Con Term) (l : TypeLevel) → Set
Γ ⊩′⟨ l ⟩U = Γ ⊩U where open LogRelKit (kit l)
_⊩′⟨_⟩B⟨_⟩_ : (Γ : Con Term) (l : TypeLevel) (W : BindingType) → Term → Set
Γ ⊩′⟨ l ⟩B⟨ W ⟩ A = Γ ⊩B⟨ W ⟩ A where open LogRelKit (kit l)
_⊩⟨_⟩_ : (Γ : Con Term) (l : TypeLevel) → Term → Set
Γ ⊩⟨ l ⟩ A = Γ ⊩ A where open LogRelKit (kit l)
_⊩⟨_⟩_≡_/_ : (Γ : Con Term) (l : TypeLevel) (A B : Term) → Γ ⊩⟨ l ⟩ A → Set
Γ ⊩⟨ l ⟩ A ≡ B / [A] = Γ ⊩ A ≡ B / [A] where open LogRelKit (kit l)
_⊩⟨_⟩_∷_/_ : (Γ : Con Term) (l : TypeLevel) (t A : Term) → Γ ⊩⟨ l ⟩ A → Set
Γ ⊩⟨ l ⟩ t ∷ A / [A] = Γ ⊩ t ∷ A / [A] where open LogRelKit (kit l)
_⊩⟨_⟩_≡_∷_/_ : (Γ : Con Term) (l : TypeLevel) (t u A : Term) → Γ ⊩⟨ l ⟩ A → Set
Γ ⊩⟨ l ⟩ t ≡ u ∷ A / [A] = Γ ⊩ t ≡ u ∷ A / [A] where open LogRelKit (kit l)
|
Formal statement is: lemma lim_cnj: "((\<lambda>x. cnj(f x)) \<longlongrightarrow> cnj l) F \<longleftrightarrow> (f \<longlongrightarrow> l) F" Informal statement is: The limit of the complex conjugate of a function is the complex conjugate of the limit of the function.
|
module Y2016.M12.D14.Exercise where
import Data.Complex
-- import available via 1HaskellADay git repository
import Data.Matrix
{--
I'm thinking about quantum computation. IBM has released a 5-qubit computer
for public experimentation. So, let's experiment.
One way to go about that is to dive right in, so, yes, if you wish: dive
right in.
Another approach is to comprehend the maths behind quantum computation.
So, let's look at that.
I was going to bewail that Shor's prime factors algorithm needs 7 qubits to
work, but, NEWSFLASH! IBM has added Shor's algorithm to their API, so ...
CANCEL BEWAILMENT.
*ahem*
Moving on.
First, let's look at qubits. Qubits are 'bra-ket'ted numbers (ket numbers)
with the representation
|0> = | 1 | or |1> = | 0 |
| 0 | | 1 |
OOH! MATRICES!
exercise 1. Represent ket0 and ket1 states as matrices in Haskell
--}
type Qubit = Matrix (Complex Float) --- where Float is 1 or 0 but okay
ket0, ket1 :: Qubit
ket0 = undefined
ket1 = undefined
{--
It MAY be helpful to have a show-instance of a qubit that abbreviates the
complex number to something more presentable. Your choice.
A qubit state is most-times in a super-position of |0> or |1> and we represent
that as
|ψ> = α|0> + β|1>
And we KNOW that |α|² + |β|² = 1
YAY! Okay. Whatever.
So, we have a qubit at |0>-state and we want to flip it to |1>-state, or vice
versa. How do we do that?
We put it through a Pauli X gate
The Pauli X operator is = | 0 1 |
| 1 0 |
That is to say, zero goes to 1 and 1 goes to zero.
excercise 2: represent the Pauli X, Y, and Z operators
--}
type PauliOperator = Matrix (Complex Float)
pauliX, pauliY, pauliZ :: PauliOperator
pauliX = undefined
pauliY = undefined
pauliZ = undefined
-- exercise 3: rotate the qubits ket0 and ket1 through the pauliX operator
-- (figure out what that means). The intended result is:
-- X|0> = |1> and X|1> = |0>
-- what are your results?
rotate :: PauliOperator -> Qubit -> Qubit
rotate p q = undefined
|
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