text
stringlengths 0
3.34M
|
|---|
open import Agda.Builtin.Bool
data D : Set where
c : Bool → D
f : @0 D → Bool
f (c true) = true
f (c false) = false
|
import Data.Vect
f1 : (n : Nat) -> Vect n Nat -> Vect (S n) Nat
f1 a b = a :: b
f2 : (n : Nat) -> case n of
k => Vect k Nat -> Vect (S k) Nat
f2 a b = a :: b
f3 : (n : Nat) -> case n of
Z => Vect Z Nat -> Vect (S Z) Nat
(S k) => Vect (S k) Nat -> Vect (S (S k)) Nat
f3 Z b = a :: b
f3 (S a) b = a :: b
f4 : (n : (Nat, Nat)) -> case n of
(x, y) => Vect x Nat -> Vect (S x) Nat
f4 (_, a) b = a :: b
|
(* Title: HOL/Auth/n_germanSimp_lemma_on_inv__44.thy
Author: Yongjian Li and Kaiqiang Duan, State Key Lab of Computer Science, Institute of Software, Chinese Academy of Sciences
Copyright 2016 State Key Lab of Computer Science, Institute of Software, Chinese Academy of Sciences
*)
header{*The n_germanSimp Protocol Case Study*}
theory n_germanSimp_lemma_on_inv__44 imports n_germanSimp_base
begin
section{*All lemmas on causal relation between inv__44 and some rule r*}
lemma n_RecvReqSVsinv__44:
assumes a1: "(\<exists> i. i\<le>N\<and>r=n_RecvReqS N i)" and
a2: "(\<exists> p__Inv4. p__Inv4\<le>N\<and>f=inv__44 p__Inv4)"
shows "invHoldForRule s f r (invariants N)" (is "?P1 s \<or> ?P2 s \<or> ?P3 s")
proof -
from a1 obtain i where a1:"i\<le>N\<and>r=n_RecvReqS N i" apply fastforce done
from a2 obtain p__Inv4 where a2:"p__Inv4\<le>N\<and>f=inv__44 p__Inv4" apply fastforce done
have "(i=p__Inv4)\<or>(i~=p__Inv4)" apply (cut_tac a1 a2, auto) done
moreover {
assume b1: "(i=p__Inv4)"
have "?P3 s"
apply (cut_tac a1 a2 b1, simp, rule_tac x="(neg (andForm (andForm (andForm (eqn (IVar (Field (Para (Ident ''Chan2'') p__Inv4) ''Cmd'')) (Const Empty)) (eqn (IVar (Para (Ident ''ShrSet'') p__Inv4)) (Const true))) (eqn (IVar (Field (Para (Ident ''Cache'') p__Inv4) ''State'')) (Const I))) (eqn (IVar (Ident ''CurCmd'')) (Const Empty))))" in exI, auto) done
then have "invHoldForRule s f r (invariants N)" by auto
}
moreover {
assume b1: "(i~=p__Inv4)"
have "?P3 s"
apply (cut_tac a1 a2 b1, simp, rule_tac x="(neg (andForm (andForm (andForm (eqn (IVar (Field (Para (Ident ''Chan2'') p__Inv4) ''Cmd'')) (Const Empty)) (eqn (IVar (Para (Ident ''ShrSet'') p__Inv4)) (Const true))) (eqn (IVar (Field (Para (Ident ''Cache'') p__Inv4) ''State'')) (Const I))) (eqn (IVar (Ident ''CurCmd'')) (Const Empty))))" in exI, auto) done
then have "invHoldForRule s f r (invariants N)" by auto
}
ultimately show "invHoldForRule s f r (invariants N)" by satx
qed
lemma n_RecvReqE__part__0Vsinv__44:
assumes a1: "(\<exists> i. i\<le>N\<and>r=n_RecvReqE__part__0 N i)" and
a2: "(\<exists> p__Inv4. p__Inv4\<le>N\<and>f=inv__44 p__Inv4)"
shows "invHoldForRule s f r (invariants N)" (is "?P1 s \<or> ?P2 s \<or> ?P3 s")
proof -
from a1 obtain i where a1:"i\<le>N\<and>r=n_RecvReqE__part__0 N i" apply fastforce done
from a2 obtain p__Inv4 where a2:"p__Inv4\<le>N\<and>f=inv__44 p__Inv4" apply fastforce done
have "(i=p__Inv4)\<or>(i~=p__Inv4)" apply (cut_tac a1 a2, auto) done
moreover {
assume b1: "(i=p__Inv4)"
have "?P1 s"
proof(cut_tac a1 a2 b1, auto) qed
then have "invHoldForRule s f r (invariants N)" by auto
}
moreover {
assume b1: "(i~=p__Inv4)"
have "?P1 s"
proof(cut_tac a1 a2 b1, auto) qed
then have "invHoldForRule s f r (invariants N)" by auto
}
ultimately show "invHoldForRule s f r (invariants N)" by satx
qed
lemma n_RecvReqE__part__1Vsinv__44:
assumes a1: "(\<exists> i. i\<le>N\<and>r=n_RecvReqE__part__1 N i)" and
a2: "(\<exists> p__Inv4. p__Inv4\<le>N\<and>f=inv__44 p__Inv4)"
shows "invHoldForRule s f r (invariants N)" (is "?P1 s \<or> ?P2 s \<or> ?P3 s")
proof -
from a1 obtain i where a1:"i\<le>N\<and>r=n_RecvReqE__part__1 N i" apply fastforce done
from a2 obtain p__Inv4 where a2:"p__Inv4\<le>N\<and>f=inv__44 p__Inv4" apply fastforce done
have "(i=p__Inv4)\<or>(i~=p__Inv4)" apply (cut_tac a1 a2, auto) done
moreover {
assume b1: "(i=p__Inv4)"
have "?P1 s"
proof(cut_tac a1 a2 b1, auto) qed
then have "invHoldForRule s f r (invariants N)" by auto
}
moreover {
assume b1: "(i~=p__Inv4)"
have "?P1 s"
proof(cut_tac a1 a2 b1, auto) qed
then have "invHoldForRule s f r (invariants N)" by auto
}
ultimately show "invHoldForRule s f r (invariants N)" by satx
qed
lemma n_SendInv__part__0Vsinv__44:
assumes a1: "(\<exists> i. i\<le>N\<and>r=n_SendInv__part__0 i)" and
a2: "(\<exists> p__Inv4. p__Inv4\<le>N\<and>f=inv__44 p__Inv4)"
shows "invHoldForRule s f r (invariants N)" (is "?P1 s \<or> ?P2 s \<or> ?P3 s")
proof -
from a1 obtain i where a1:"i\<le>N\<and>r=n_SendInv__part__0 i" apply fastforce done
from a2 obtain p__Inv4 where a2:"p__Inv4\<le>N\<and>f=inv__44 p__Inv4" apply fastforce done
have "(i=p__Inv4)\<or>(i~=p__Inv4)" apply (cut_tac a1 a2, auto) done
moreover {
assume b1: "(i=p__Inv4)"
have "?P1 s"
proof(cut_tac a1 a2 b1, auto) qed
then have "invHoldForRule s f r (invariants N)" by auto
}
moreover {
assume b1: "(i~=p__Inv4)"
have "?P2 s"
proof(cut_tac a1 a2 b1, auto) qed
then have "invHoldForRule s f r (invariants N)" by auto
}
ultimately show "invHoldForRule s f r (invariants N)" by satx
qed
lemma n_SendInv__part__1Vsinv__44:
assumes a1: "(\<exists> i. i\<le>N\<and>r=n_SendInv__part__1 i)" and
a2: "(\<exists> p__Inv4. p__Inv4\<le>N\<and>f=inv__44 p__Inv4)"
shows "invHoldForRule s f r (invariants N)" (is "?P1 s \<or> ?P2 s \<or> ?P3 s")
proof -
from a1 obtain i where a1:"i\<le>N\<and>r=n_SendInv__part__1 i" apply fastforce done
from a2 obtain p__Inv4 where a2:"p__Inv4\<le>N\<and>f=inv__44 p__Inv4" apply fastforce done
have "(i=p__Inv4)\<or>(i~=p__Inv4)" apply (cut_tac a1 a2, auto) done
moreover {
assume b1: "(i=p__Inv4)"
have "?P1 s"
proof(cut_tac a1 a2 b1, auto) qed
then have "invHoldForRule s f r (invariants N)" by auto
}
moreover {
assume b1: "(i~=p__Inv4)"
have "?P2 s"
proof(cut_tac a1 a2 b1, auto) qed
then have "invHoldForRule s f r (invariants N)" by auto
}
ultimately show "invHoldForRule s f r (invariants N)" by satx
qed
lemma n_SendInvAckVsinv__44:
assumes a1: "(\<exists> i. i\<le>N\<and>r=n_SendInvAck i)" and
a2: "(\<exists> p__Inv4. p__Inv4\<le>N\<and>f=inv__44 p__Inv4)"
shows "invHoldForRule s f r (invariants N)" (is "?P1 s \<or> ?P2 s \<or> ?P3 s")
proof -
from a1 obtain i where a1:"i\<le>N\<and>r=n_SendInvAck i" apply fastforce done
from a2 obtain p__Inv4 where a2:"p__Inv4\<le>N\<and>f=inv__44 p__Inv4" apply fastforce done
have "(i=p__Inv4)\<or>(i~=p__Inv4)" apply (cut_tac a1 a2, auto) done
moreover {
assume b1: "(i=p__Inv4)"
have "?P3 s"
apply (cut_tac a1 a2 b1, simp, rule_tac x="(neg (andForm (andForm (eqn (IVar (Ident ''CurCmd'')) (Const ReqS)) (eqn (IVar (Ident ''ExGntd'')) (Const false))) (eqn (IVar (Field (Para (Ident ''Chan2'') p__Inv4) ''Cmd'')) (Const Inv))))" in exI, auto) done
then have "invHoldForRule s f r (invariants N)" by auto
}
moreover {
assume b1: "(i~=p__Inv4)"
have "?P2 s"
proof(cut_tac a1 a2 b1, auto) qed
then have "invHoldForRule s f r (invariants N)" by auto
}
ultimately show "invHoldForRule s f r (invariants N)" by satx
qed
lemma n_RecvInvAckVsinv__44:
assumes a1: "(\<exists> i. i\<le>N\<and>r=n_RecvInvAck i)" and
a2: "(\<exists> p__Inv4. p__Inv4\<le>N\<and>f=inv__44 p__Inv4)"
shows "invHoldForRule s f r (invariants N)" (is "?P1 s \<or> ?P2 s \<or> ?P3 s")
proof -
from a1 obtain i where a1:"i\<le>N\<and>r=n_RecvInvAck i" apply fastforce done
from a2 obtain p__Inv4 where a2:"p__Inv4\<le>N\<and>f=inv__44 p__Inv4" apply fastforce done
have "(i=p__Inv4)\<or>(i~=p__Inv4)" apply (cut_tac a1 a2, auto) done
moreover {
assume b1: "(i=p__Inv4)"
have "((formEval (eqn (IVar (Ident ''ExGntd'')) (Const true)) s))\<or>((formEval (neg (eqn (IVar (Ident ''ExGntd'')) (Const true))) s))" by auto
moreover {
assume c1: "((formEval (eqn (IVar (Ident ''ExGntd'')) (Const true)) s))"
have "?P1 s"
proof(cut_tac a1 a2 b1 c1, auto) qed
then have "invHoldForRule s f r (invariants N)" by auto
}
moreover {
assume c1: "((formEval (neg (eqn (IVar (Ident ''ExGntd'')) (Const true))) s))"
have "?P1 s"
proof(cut_tac a1 a2 b1 c1, auto) qed
then have "invHoldForRule s f r (invariants N)" by auto
}
ultimately have "invHoldForRule s f r (invariants N)" by satx
}
moreover {
assume b1: "(i~=p__Inv4)"
have "((formEval (eqn (IVar (Ident ''ExGntd'')) (Const true)) s))\<or>((formEval (neg (eqn (IVar (Ident ''ExGntd'')) (Const true))) s))" by auto
moreover {
assume c1: "((formEval (eqn (IVar (Ident ''ExGntd'')) (Const true)) s))"
have "?P3 s"
apply (cut_tac a1 a2 b1 c1, simp, rule_tac x="(neg (andForm (andForm (eqn (IVar (Para (Ident ''ShrSet'') p__Inv4)) (Const true)) (eqn (IVar (Ident ''CurCmd'')) (Const ReqS))) (eqn (IVar (Field (Para (Ident ''Chan3'') i) ''Cmd'')) (Const InvAck))))" in exI, auto) done
then have "invHoldForRule s f r (invariants N)" by auto
}
moreover {
assume c1: "((formEval (neg (eqn (IVar (Ident ''ExGntd'')) (Const true))) s))"
have "?P2 s"
proof(cut_tac a1 a2 b1 c1, auto) qed
then have "invHoldForRule s f r (invariants N)" by auto
}
ultimately have "invHoldForRule s f r (invariants N)" by satx
}
ultimately show "invHoldForRule s f r (invariants N)" by satx
qed
lemma n_SendGntSVsinv__44:
assumes a1: "(\<exists> i. i\<le>N\<and>r=n_SendGntS i)" and
a2: "(\<exists> p__Inv4. p__Inv4\<le>N\<and>f=inv__44 p__Inv4)"
shows "invHoldForRule s f r (invariants N)" (is "?P1 s \<or> ?P2 s \<or> ?P3 s")
proof -
from a1 obtain i where a1:"i\<le>N\<and>r=n_SendGntS i" apply fastforce done
from a2 obtain p__Inv4 where a2:"p__Inv4\<le>N\<and>f=inv__44 p__Inv4" apply fastforce done
have "(i=p__Inv4)\<or>(i~=p__Inv4)" apply (cut_tac a1 a2, auto) done
moreover {
assume b1: "(i=p__Inv4)"
have "?P1 s"
proof(cut_tac a1 a2 b1, auto) qed
then have "invHoldForRule s f r (invariants N)" by auto
}
moreover {
assume b1: "(i~=p__Inv4)"
have "?P1 s"
proof(cut_tac a1 a2 b1, auto) qed
then have "invHoldForRule s f r (invariants N)" by auto
}
ultimately show "invHoldForRule s f r (invariants N)" by satx
qed
lemma n_SendGntEVsinv__44:
assumes a1: "(\<exists> i. i\<le>N\<and>r=n_SendGntE N i)" and
a2: "(\<exists> p__Inv4. p__Inv4\<le>N\<and>f=inv__44 p__Inv4)"
shows "invHoldForRule s f r (invariants N)" (is "?P1 s \<or> ?P2 s \<or> ?P3 s")
proof -
from a1 obtain i where a1:"i\<le>N\<and>r=n_SendGntE N i" apply fastforce done
from a2 obtain p__Inv4 where a2:"p__Inv4\<le>N\<and>f=inv__44 p__Inv4" apply fastforce done
have "(i=p__Inv4)\<or>(i~=p__Inv4)" apply (cut_tac a1 a2, auto) done
moreover {
assume b1: "(i=p__Inv4)"
have "?P1 s"
proof(cut_tac a1 a2 b1, auto) qed
then have "invHoldForRule s f r (invariants N)" by auto
}
moreover {
assume b1: "(i~=p__Inv4)"
have "?P1 s"
proof(cut_tac a1 a2 b1, auto) qed
then have "invHoldForRule s f r (invariants N)" by auto
}
ultimately show "invHoldForRule s f r (invariants N)" by satx
qed
lemma n_RecvGntSVsinv__44:
assumes a1: "(\<exists> i. i\<le>N\<and>r=n_RecvGntS i)" and
a2: "(\<exists> p__Inv4. p__Inv4\<le>N\<and>f=inv__44 p__Inv4)"
shows "invHoldForRule s f r (invariants N)" (is "?P1 s \<or> ?P2 s \<or> ?P3 s")
proof -
from a1 obtain i where a1:"i\<le>N\<and>r=n_RecvGntS i" apply fastforce done
from a2 obtain p__Inv4 where a2:"p__Inv4\<le>N\<and>f=inv__44 p__Inv4" apply fastforce done
have "(i=p__Inv4)\<or>(i~=p__Inv4)" apply (cut_tac a1 a2, auto) done
moreover {
assume b1: "(i=p__Inv4)"
have "?P1 s"
proof(cut_tac a1 a2 b1, auto) qed
then have "invHoldForRule s f r (invariants N)" by auto
}
moreover {
assume b1: "(i~=p__Inv4)"
have "?P2 s"
proof(cut_tac a1 a2 b1, auto) qed
then have "invHoldForRule s f r (invariants N)" by auto
}
ultimately show "invHoldForRule s f r (invariants N)" by satx
qed
lemma n_RecvGntEVsinv__44:
assumes a1: "(\<exists> i. i\<le>N\<and>r=n_RecvGntE i)" and
a2: "(\<exists> p__Inv4. p__Inv4\<le>N\<and>f=inv__44 p__Inv4)"
shows "invHoldForRule s f r (invariants N)" (is "?P1 s \<or> ?P2 s \<or> ?P3 s")
proof -
from a1 obtain i where a1:"i\<le>N\<and>r=n_RecvGntE i" apply fastforce done
from a2 obtain p__Inv4 where a2:"p__Inv4\<le>N\<and>f=inv__44 p__Inv4" apply fastforce done
have "(i=p__Inv4)\<or>(i~=p__Inv4)" apply (cut_tac a1 a2, auto) done
moreover {
assume b1: "(i=p__Inv4)"
have "?P1 s"
proof(cut_tac a1 a2 b1, auto) qed
then have "invHoldForRule s f r (invariants N)" by auto
}
moreover {
assume b1: "(i~=p__Inv4)"
have "?P2 s"
proof(cut_tac a1 a2 b1, auto) qed
then have "invHoldForRule s f r (invariants N)" by auto
}
ultimately show "invHoldForRule s f r (invariants N)" by satx
qed
lemma n_StoreVsinv__44:
assumes a1: "\<exists> i d. i\<le>N\<and>d\<le>N\<and>r=n_Store i d" and
a2: "(\<exists> p__Inv4. p__Inv4\<le>N\<and>f=inv__44 p__Inv4)"
shows "invHoldForRule s f r (invariants N)"
apply (rule noEffectOnRule, cut_tac a1 a2, auto) done
end
|
/-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes
! This file was ported from Lean 3 source module data.polynomial.splits
! leanprover-community/mathlib commit f694c7dead66f5d4c80f446c796a5aad14707f0e
! Please do not edit these lines, except to modify the commit id
! if you have ported upstream changes.
-/
import Mathbin.Data.List.Prime
import Mathbin.Data.Polynomial.FieldDivision
import Mathbin.Data.Polynomial.Lifts
/-!
# Split polynomials
A polynomial `f : K[X]` splits over a field extension `L` of `K` if it is zero or all of its
irreducible factors over `L` have degree `1`.
## Main definitions
* `polynomial.splits i f`: A predicate on a homomorphism `i : K →+* L` from a commutative ring to a
field and a polynomial `f` saying that `f.map i` is zero or all of its irreducible factors over
`L` have degree `1`.
## Main statements
* `lift_of_splits`: If `K` and `L` are field extensions of a field `F` and for some finite subset
`S` of `K`, the minimal polynomial of every `x ∈ K` splits as a polynomial with coefficients in
`L`, then `algebra.adjoin F S` embeds into `L`.
-/
noncomputable section
open Classical BigOperators Polynomial
universe u v w
variable {F : Type u} {K : Type v} {L : Type w}
namespace Polynomial
open Polynomial
section Splits
section CommRing
variable [CommRing K] [Field L] [Field F]
variable (i : K →+* L)
#print Polynomial.Splits /-
/-- A polynomial `splits` iff it is zero or all of its irreducible factors have `degree` 1. -/
def Splits (f : K[X]) : Prop :=
f.map i = 0 ∨ ∀ {g : L[X]}, Irreducible g → g ∣ f.map i → degree g = 1
#align polynomial.splits Polynomial.Splits
-/
#print Polynomial.splits_zero /-
@[simp]
theorem splits_zero : Splits i (0 : K[X]) :=
Or.inl (Polynomial.map_zero i)
#align polynomial.splits_zero Polynomial.splits_zero
-/
/- warning: polynomial.splits_of_map_eq_C -> Polynomial.splits_of_map_eq_C is a dubious translation:
lean 3 declaration is
forall {K : Type.{u1}} {L : Type.{u2}} [_inst_1 : CommRing.{u1} K] [_inst_2 : Field.{u2} L] (i : RingHom.{u1, u2} K L (NonAssocRing.toNonAssocSemiring.{u1} K (Ring.toNonAssocRing.{u1} K (CommRing.toRing.{u1} K _inst_1))) (NonAssocRing.toNonAssocSemiring.{u2} L (Ring.toNonAssocRing.{u2} L (DivisionRing.toRing.{u2} L (Field.toDivisionRing.{u2} L _inst_2))))) {f : Polynomial.{u1} K (Ring.toSemiring.{u1} K (CommRing.toRing.{u1} K _inst_1))} {a : L}, (Eq.{succ u2} (Polynomial.{u2} L (Ring.toSemiring.{u2} L (DivisionRing.toRing.{u2} L (Field.toDivisionRing.{u2} L _inst_2)))) (Polynomial.map.{u1, u2} K L (Ring.toSemiring.{u1} K (CommRing.toRing.{u1} K _inst_1)) (Ring.toSemiring.{u2} L (DivisionRing.toRing.{u2} L (Field.toDivisionRing.{u2} L _inst_2))) i f) (coeFn.{succ u2, succ u2} (RingHom.{u2, u2} L (Polynomial.{u2} L (Ring.toSemiring.{u2} L (DivisionRing.toRing.{u2} L (Field.toDivisionRing.{u2} L _inst_2)))) (Semiring.toNonAssocSemiring.{u2} L (Ring.toSemiring.{u2} L (DivisionRing.toRing.{u2} L (Field.toDivisionRing.{u2} L _inst_2)))) (Semiring.toNonAssocSemiring.{u2} (Polynomial.{u2} L (Ring.toSemiring.{u2} L (DivisionRing.toRing.{u2} L (Field.toDivisionRing.{u2} L _inst_2)))) (Polynomial.semiring.{u2} L (Ring.toSemiring.{u2} L (DivisionRing.toRing.{u2} L (Field.toDivisionRing.{u2} L _inst_2)))))) (fun (_x : RingHom.{u2, u2} L (Polynomial.{u2} L (Ring.toSemiring.{u2} L (DivisionRing.toRing.{u2} L (Field.toDivisionRing.{u2} L _inst_2)))) (Semiring.toNonAssocSemiring.{u2} L (Ring.toSemiring.{u2} L (DivisionRing.toRing.{u2} L (Field.toDivisionRing.{u2} L _inst_2)))) (Semiring.toNonAssocSemiring.{u2} (Polynomial.{u2} L (Ring.toSemiring.{u2} L (DivisionRing.toRing.{u2} L (Field.toDivisionRing.{u2} L _inst_2)))) (Polynomial.semiring.{u2} L (Ring.toSemiring.{u2} L (DivisionRing.toRing.{u2} L (Field.toDivisionRing.{u2} L _inst_2)))))) => L -> (Polynomial.{u2} L (Ring.toSemiring.{u2} L (DivisionRing.toRing.{u2} L (Field.toDivisionRing.{u2} L _inst_2))))) (RingHom.hasCoeToFun.{u2, u2} L (Polynomial.{u2} L (Ring.toSemiring.{u2} L (DivisionRing.toRing.{u2} L (Field.toDivisionRing.{u2} L _inst_2)))) (Semiring.toNonAssocSemiring.{u2} L (Ring.toSemiring.{u2} L (DivisionRing.toRing.{u2} L (Field.toDivisionRing.{u2} L _inst_2)))) (Semiring.toNonAssocSemiring.{u2} (Polynomial.{u2} L (Ring.toSemiring.{u2} L (DivisionRing.toRing.{u2} L (Field.toDivisionRing.{u2} L _inst_2)))) (Polynomial.semiring.{u2} L (Ring.toSemiring.{u2} L (DivisionRing.toRing.{u2} L (Field.toDivisionRing.{u2} L _inst_2)))))) (Polynomial.C.{u2} L (Ring.toSemiring.{u2} L (DivisionRing.toRing.{u2} L (Field.toDivisionRing.{u2} L _inst_2)))) a)) -> (Polynomial.Splits.{u1, u2} K L _inst_1 _inst_2 i f)
but is expected to have type
forall {K : Type.{u1}} {L : Type.{u2}} [_inst_1 : CommRing.{u1} K] [_inst_2 : Field.{u2} L] (i : RingHom.{u1, u2} K L (NonAssocRing.toNonAssocSemiring.{u1} K (Ring.toNonAssocRing.{u1} K (CommRing.toRing.{u1} K _inst_1))) (NonAssocRing.toNonAssocSemiring.{u2} L (Ring.toNonAssocRing.{u2} L (DivisionRing.toRing.{u2} L (Field.toDivisionRing.{u2} L _inst_2))))) {f : Polynomial.{u1} K (Ring.toSemiring.{u1} K (CommRing.toRing.{u1} K _inst_1))} {a : L}, (Eq.{succ u2} (Polynomial.{u2} L (DivisionSemiring.toSemiring.{u2} L (Semifield.toDivisionSemiring.{u2} L (Field.toSemifield.{u2} L _inst_2)))) (Polynomial.map.{u1, u2} K L (Ring.toSemiring.{u1} K (CommRing.toRing.{u1} K _inst_1)) (DivisionSemiring.toSemiring.{u2} L (Semifield.toDivisionSemiring.{u2} L (Field.toSemifield.{u2} L _inst_2))) i f) (FunLike.coe.{succ u2, succ u2, succ u2} (RingHom.{u2, u2} L (Polynomial.{u2} L (DivisionSemiring.toSemiring.{u2} L (Semifield.toDivisionSemiring.{u2} L (Field.toSemifield.{u2} L _inst_2)))) (Semiring.toNonAssocSemiring.{u2} L (DivisionSemiring.toSemiring.{u2} L (Semifield.toDivisionSemiring.{u2} L (Field.toSemifield.{u2} L _inst_2)))) (Semiring.toNonAssocSemiring.{u2} (Polynomial.{u2} L (DivisionSemiring.toSemiring.{u2} L (Semifield.toDivisionSemiring.{u2} L (Field.toSemifield.{u2} L _inst_2)))) (Polynomial.semiring.{u2} L (DivisionSemiring.toSemiring.{u2} L (Semifield.toDivisionSemiring.{u2} L (Field.toSemifield.{u2} L _inst_2)))))) L (fun (_x : L) => (fun ([email protected]._hyg.2391 : L) => Polynomial.{u2} L (DivisionSemiring.toSemiring.{u2} L (Semifield.toDivisionSemiring.{u2} L (Field.toSemifield.{u2} L _inst_2)))) _x) (MulHomClass.toFunLike.{u2, u2, u2} (RingHom.{u2, u2} L (Polynomial.{u2} L (DivisionSemiring.toSemiring.{u2} L (Semifield.toDivisionSemiring.{u2} L (Field.toSemifield.{u2} L _inst_2)))) (Semiring.toNonAssocSemiring.{u2} L (DivisionSemiring.toSemiring.{u2} L (Semifield.toDivisionSemiring.{u2} L (Field.toSemifield.{u2} L _inst_2)))) (Semiring.toNonAssocSemiring.{u2} (Polynomial.{u2} L (DivisionSemiring.toSemiring.{u2} L (Semifield.toDivisionSemiring.{u2} L (Field.toSemifield.{u2} L _inst_2)))) (Polynomial.semiring.{u2} L (DivisionSemiring.toSemiring.{u2} L (Semifield.toDivisionSemiring.{u2} L (Field.toSemifield.{u2} L _inst_2)))))) L (Polynomial.{u2} L (DivisionSemiring.toSemiring.{u2} L (Semifield.toDivisionSemiring.{u2} L (Field.toSemifield.{u2} L _inst_2)))) (NonUnitalNonAssocSemiring.toMul.{u2} L (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} L (Semiring.toNonAssocSemiring.{u2} L (DivisionSemiring.toSemiring.{u2} L (Semifield.toDivisionSemiring.{u2} L (Field.toSemifield.{u2} L _inst_2)))))) (NonUnitalNonAssocSemiring.toMul.{u2} (Polynomial.{u2} L (DivisionSemiring.toSemiring.{u2} L (Semifield.toDivisionSemiring.{u2} L (Field.toSemifield.{u2} L _inst_2)))) (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} (Polynomial.{u2} L (DivisionSemiring.toSemiring.{u2} L (Semifield.toDivisionSemiring.{u2} L (Field.toSemifield.{u2} L _inst_2)))) (Semiring.toNonAssocSemiring.{u2} (Polynomial.{u2} L (DivisionSemiring.toSemiring.{u2} L (Semifield.toDivisionSemiring.{u2} L (Field.toSemifield.{u2} L _inst_2)))) (Polynomial.semiring.{u2} L (DivisionSemiring.toSemiring.{u2} L (Semifield.toDivisionSemiring.{u2} L (Field.toSemifield.{u2} L _inst_2))))))) (NonUnitalRingHomClass.toMulHomClass.{u2, u2, u2} (RingHom.{u2, u2} L (Polynomial.{u2} L (DivisionSemiring.toSemiring.{u2} L (Semifield.toDivisionSemiring.{u2} L (Field.toSemifield.{u2} L _inst_2)))) (Semiring.toNonAssocSemiring.{u2} L (DivisionSemiring.toSemiring.{u2} L (Semifield.toDivisionSemiring.{u2} L (Field.toSemifield.{u2} L _inst_2)))) (Semiring.toNonAssocSemiring.{u2} (Polynomial.{u2} L (DivisionSemiring.toSemiring.{u2} L (Semifield.toDivisionSemiring.{u2} L (Field.toSemifield.{u2} L _inst_2)))) (Polynomial.semiring.{u2} L (DivisionSemiring.toSemiring.{u2} L (Semifield.toDivisionSemiring.{u2} L (Field.toSemifield.{u2} L _inst_2)))))) L (Polynomial.{u2} L (DivisionSemiring.toSemiring.{u2} L (Semifield.toDivisionSemiring.{u2} L (Field.toSemifield.{u2} L _inst_2)))) (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} L (Semiring.toNonAssocSemiring.{u2} L (DivisionSemiring.toSemiring.{u2} L (Semifield.toDivisionSemiring.{u2} L (Field.toSemifield.{u2} L _inst_2))))) (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} (Polynomial.{u2} L (DivisionSemiring.toSemiring.{u2} L (Semifield.toDivisionSemiring.{u2} L (Field.toSemifield.{u2} L _inst_2)))) (Semiring.toNonAssocSemiring.{u2} (Polynomial.{u2} L (DivisionSemiring.toSemiring.{u2} L (Semifield.toDivisionSemiring.{u2} L (Field.toSemifield.{u2} L _inst_2)))) (Polynomial.semiring.{u2} L (DivisionSemiring.toSemiring.{u2} L (Semifield.toDivisionSemiring.{u2} L (Field.toSemifield.{u2} L _inst_2)))))) (RingHomClass.toNonUnitalRingHomClass.{u2, u2, u2} (RingHom.{u2, u2} L (Polynomial.{u2} L (DivisionSemiring.toSemiring.{u2} L (Semifield.toDivisionSemiring.{u2} L (Field.toSemifield.{u2} L _inst_2)))) (Semiring.toNonAssocSemiring.{u2} L (DivisionSemiring.toSemiring.{u2} L (Semifield.toDivisionSemiring.{u2} L (Field.toSemifield.{u2} L _inst_2)))) (Semiring.toNonAssocSemiring.{u2} (Polynomial.{u2} L (DivisionSemiring.toSemiring.{u2} L (Semifield.toDivisionSemiring.{u2} L (Field.toSemifield.{u2} L _inst_2)))) (Polynomial.semiring.{u2} L (DivisionSemiring.toSemiring.{u2} L (Semifield.toDivisionSemiring.{u2} L (Field.toSemifield.{u2} L _inst_2)))))) L (Polynomial.{u2} L (DivisionSemiring.toSemiring.{u2} L (Semifield.toDivisionSemiring.{u2} L (Field.toSemifield.{u2} L _inst_2)))) (Semiring.toNonAssocSemiring.{u2} L (DivisionSemiring.toSemiring.{u2} L (Semifield.toDivisionSemiring.{u2} L (Field.toSemifield.{u2} L _inst_2)))) (Semiring.toNonAssocSemiring.{u2} (Polynomial.{u2} L (DivisionSemiring.toSemiring.{u2} L (Semifield.toDivisionSemiring.{u2} L (Field.toSemifield.{u2} L _inst_2)))) (Polynomial.semiring.{u2} L (DivisionSemiring.toSemiring.{u2} L (Semifield.toDivisionSemiring.{u2} L (Field.toSemifield.{u2} L _inst_2))))) (RingHom.instRingHomClassRingHom.{u2, u2} L (Polynomial.{u2} L (DivisionSemiring.toSemiring.{u2} L (Semifield.toDivisionSemiring.{u2} L (Field.toSemifield.{u2} L _inst_2)))) (Semiring.toNonAssocSemiring.{u2} L (DivisionSemiring.toSemiring.{u2} L (Semifield.toDivisionSemiring.{u2} L (Field.toSemifield.{u2} L _inst_2)))) (Semiring.toNonAssocSemiring.{u2} (Polynomial.{u2} L (DivisionSemiring.toSemiring.{u2} L (Semifield.toDivisionSemiring.{u2} L (Field.toSemifield.{u2} L _inst_2)))) (Polynomial.semiring.{u2} L (DivisionSemiring.toSemiring.{u2} L (Semifield.toDivisionSemiring.{u2} L (Field.toSemifield.{u2} L _inst_2))))))))) (Polynomial.C.{u2} L (DivisionSemiring.toSemiring.{u2} L (Semifield.toDivisionSemiring.{u2} L (Field.toSemifield.{u2} L _inst_2)))) a)) -> (Polynomial.Splits.{u1, u2} K L _inst_1 _inst_2 i f)
Case conversion may be inaccurate. Consider using '#align polynomial.splits_of_map_eq_C Polynomial.splits_of_map_eq_Cₓ'. -/
theorem splits_of_map_eq_C {f : K[X]} {a : L} (h : f.map i = C a) : Splits i f :=
if ha : a = 0 then Or.inl (h.trans (ha.symm ▸ C_0))
else
Or.inr fun g hg ⟨p, hp⟩ =>
absurd hg.1 <|
Classical.not_not.2 <|
isUnit_iff_degree_eq_zero.2 <| by
have := congr_arg degree hp
rw [h, degree_C ha, degree_mul, @eq_comm (WithBot ℕ) 0, Nat.WithBot.add_eq_zero_iff] at
this
exact this.1
#align polynomial.splits_of_map_eq_C Polynomial.splits_of_map_eq_C
/- warning: polynomial.splits_C -> Polynomial.splits_C is a dubious translation:
lean 3 declaration is
forall {K : Type.{u1}} {L : Type.{u2}} [_inst_1 : CommRing.{u1} K] [_inst_2 : Field.{u2} L] (i : RingHom.{u1, u2} K L (NonAssocRing.toNonAssocSemiring.{u1} K (Ring.toNonAssocRing.{u1} K (CommRing.toRing.{u1} K _inst_1))) (NonAssocRing.toNonAssocSemiring.{u2} L (Ring.toNonAssocRing.{u2} L (DivisionRing.toRing.{u2} L (Field.toDivisionRing.{u2} L _inst_2))))) (a : K), Polynomial.Splits.{u1, u2} K L _inst_1 _inst_2 i (coeFn.{succ u1, succ u1} (RingHom.{u1, u1} K (Polynomial.{u1} K (Ring.toSemiring.{u1} K (CommRing.toRing.{u1} K _inst_1))) (Semiring.toNonAssocSemiring.{u1} K (Ring.toSemiring.{u1} K (CommRing.toRing.{u1} K _inst_1))) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} K (Ring.toSemiring.{u1} K (CommRing.toRing.{u1} K _inst_1))) (Polynomial.semiring.{u1} K (Ring.toSemiring.{u1} K (CommRing.toRing.{u1} K _inst_1))))) (fun (_x : RingHom.{u1, u1} K (Polynomial.{u1} K (Ring.toSemiring.{u1} K (CommRing.toRing.{u1} K _inst_1))) (Semiring.toNonAssocSemiring.{u1} K (Ring.toSemiring.{u1} K (CommRing.toRing.{u1} K _inst_1))) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} K (Ring.toSemiring.{u1} K (CommRing.toRing.{u1} K _inst_1))) (Polynomial.semiring.{u1} K (Ring.toSemiring.{u1} K (CommRing.toRing.{u1} K _inst_1))))) => K -> (Polynomial.{u1} K (Ring.toSemiring.{u1} K (CommRing.toRing.{u1} K _inst_1)))) (RingHom.hasCoeToFun.{u1, u1} K (Polynomial.{u1} K (Ring.toSemiring.{u1} K (CommRing.toRing.{u1} K _inst_1))) (Semiring.toNonAssocSemiring.{u1} K (Ring.toSemiring.{u1} K (CommRing.toRing.{u1} K _inst_1))) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} K (Ring.toSemiring.{u1} K (CommRing.toRing.{u1} K _inst_1))) (Polynomial.semiring.{u1} K (Ring.toSemiring.{u1} K (CommRing.toRing.{u1} K _inst_1))))) (Polynomial.C.{u1} K (Ring.toSemiring.{u1} K (CommRing.toRing.{u1} K _inst_1))) a)
but is expected to have type
forall {K : Type.{u1}} {L : Type.{u2}} [_inst_1 : CommRing.{u1} K] [_inst_2 : Field.{u2} L] (i : RingHom.{u1, u2} K L (NonAssocRing.toNonAssocSemiring.{u1} K (Ring.toNonAssocRing.{u1} K (CommRing.toRing.{u1} K _inst_1))) (NonAssocRing.toNonAssocSemiring.{u2} L (Ring.toNonAssocRing.{u2} L (DivisionRing.toRing.{u2} L (Field.toDivisionRing.{u2} L _inst_2))))) (a : K), Polynomial.Splits.{u1, u2} K L _inst_1 _inst_2 i (FunLike.coe.{succ u1, succ u1, succ u1} (RingHom.{u1, u1} K (Polynomial.{u1} K (Ring.toSemiring.{u1} K (CommRing.toRing.{u1} K _inst_1))) (Semiring.toNonAssocSemiring.{u1} K (Ring.toSemiring.{u1} K (CommRing.toRing.{u1} K _inst_1))) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} K (Ring.toSemiring.{u1} K (CommRing.toRing.{u1} K _inst_1))) (Polynomial.semiring.{u1} K (Ring.toSemiring.{u1} K (CommRing.toRing.{u1} K _inst_1))))) K (fun (_x : K) => (fun ([email protected]._hyg.2391 : K) => Polynomial.{u1} K (Ring.toSemiring.{u1} K (CommRing.toRing.{u1} K _inst_1))) _x) (MulHomClass.toFunLike.{u1, u1, u1} (RingHom.{u1, u1} K (Polynomial.{u1} K (Ring.toSemiring.{u1} K (CommRing.toRing.{u1} K _inst_1))) (Semiring.toNonAssocSemiring.{u1} K (Ring.toSemiring.{u1} K (CommRing.toRing.{u1} K _inst_1))) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} K (Ring.toSemiring.{u1} K (CommRing.toRing.{u1} K _inst_1))) (Polynomial.semiring.{u1} K (Ring.toSemiring.{u1} K (CommRing.toRing.{u1} K _inst_1))))) K (Polynomial.{u1} K (Ring.toSemiring.{u1} K (CommRing.toRing.{u1} K _inst_1))) (NonUnitalNonAssocSemiring.toMul.{u1} K (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} K (Semiring.toNonAssocSemiring.{u1} K (Ring.toSemiring.{u1} K (CommRing.toRing.{u1} K _inst_1))))) (NonUnitalNonAssocSemiring.toMul.{u1} (Polynomial.{u1} K (Ring.toSemiring.{u1} K (CommRing.toRing.{u1} K _inst_1))) (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} (Polynomial.{u1} K (Ring.toSemiring.{u1} K (CommRing.toRing.{u1} K _inst_1))) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} K (Ring.toSemiring.{u1} K (CommRing.toRing.{u1} K _inst_1))) (Polynomial.semiring.{u1} K (Ring.toSemiring.{u1} K (CommRing.toRing.{u1} K _inst_1)))))) (NonUnitalRingHomClass.toMulHomClass.{u1, u1, u1} (RingHom.{u1, u1} K (Polynomial.{u1} K (Ring.toSemiring.{u1} K (CommRing.toRing.{u1} K _inst_1))) (Semiring.toNonAssocSemiring.{u1} K (Ring.toSemiring.{u1} K (CommRing.toRing.{u1} K _inst_1))) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} K (Ring.toSemiring.{u1} K (CommRing.toRing.{u1} K _inst_1))) (Polynomial.semiring.{u1} K (Ring.toSemiring.{u1} K (CommRing.toRing.{u1} K _inst_1))))) K (Polynomial.{u1} K (Ring.toSemiring.{u1} K (CommRing.toRing.{u1} K _inst_1))) (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} K (Semiring.toNonAssocSemiring.{u1} K (Ring.toSemiring.{u1} K (CommRing.toRing.{u1} K _inst_1)))) (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} (Polynomial.{u1} K (Ring.toSemiring.{u1} K (CommRing.toRing.{u1} K _inst_1))) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} K (Ring.toSemiring.{u1} K (CommRing.toRing.{u1} K _inst_1))) (Polynomial.semiring.{u1} K (Ring.toSemiring.{u1} K (CommRing.toRing.{u1} K _inst_1))))) (RingHomClass.toNonUnitalRingHomClass.{u1, u1, u1} (RingHom.{u1, u1} K (Polynomial.{u1} K (Ring.toSemiring.{u1} K (CommRing.toRing.{u1} K _inst_1))) (Semiring.toNonAssocSemiring.{u1} K (Ring.toSemiring.{u1} K (CommRing.toRing.{u1} K _inst_1))) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} K (Ring.toSemiring.{u1} K (CommRing.toRing.{u1} K _inst_1))) (Polynomial.semiring.{u1} K (Ring.toSemiring.{u1} K (CommRing.toRing.{u1} K _inst_1))))) K (Polynomial.{u1} K (Ring.toSemiring.{u1} K (CommRing.toRing.{u1} K _inst_1))) (Semiring.toNonAssocSemiring.{u1} K (Ring.toSemiring.{u1} K (CommRing.toRing.{u1} K _inst_1))) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} K (Ring.toSemiring.{u1} K (CommRing.toRing.{u1} K _inst_1))) (Polynomial.semiring.{u1} K (Ring.toSemiring.{u1} K (CommRing.toRing.{u1} K _inst_1)))) (RingHom.instRingHomClassRingHom.{u1, u1} K (Polynomial.{u1} K (Ring.toSemiring.{u1} K (CommRing.toRing.{u1} K _inst_1))) (Semiring.toNonAssocSemiring.{u1} K (Ring.toSemiring.{u1} K (CommRing.toRing.{u1} K _inst_1))) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} K (Ring.toSemiring.{u1} K (CommRing.toRing.{u1} K _inst_1))) (Polynomial.semiring.{u1} K (Ring.toSemiring.{u1} K (CommRing.toRing.{u1} K _inst_1)))))))) (Polynomial.C.{u1} K (Ring.toSemiring.{u1} K (CommRing.toRing.{u1} K _inst_1))) a)
Case conversion may be inaccurate. Consider using '#align polynomial.splits_C Polynomial.splits_Cₓ'. -/
@[simp]
theorem splits_C (a : K) : Splits i (C a) :=
splits_of_map_eq_C i (map_C i)
#align polynomial.splits_C Polynomial.splits_C
#print Polynomial.splits_of_map_degree_eq_one /-
theorem splits_of_map_degree_eq_one {f : K[X]} (hf : degree (f.map i) = 1) : Splits i f :=
Or.inr fun g hg ⟨p, hp⟩ => by
have := congr_arg degree hp <;>
simp [Nat.WithBot.add_eq_one_iff, hf, @eq_comm (WithBot ℕ) 1,
mt is_unit_iff_degree_eq_zero.2 hg.1] at this <;>
clear _fun_match <;>
tauto
#align polynomial.splits_of_map_degree_eq_one Polynomial.splits_of_map_degree_eq_one
-/
#print Polynomial.splits_of_degree_le_one /-
theorem splits_of_degree_le_one {f : K[X]} (hf : degree f ≤ 1) : Splits i f :=
if hif : degree (f.map i) ≤ 0 then splits_of_map_eq_C i (degree_le_zero_iff.mp hif)
else by
push_neg at hif
rw [← Order.succ_le_iff, ← WithBot.coe_zero, WithBot.succ_coe, Nat.succ_eq_succ] at hif
exact splits_of_map_degree_eq_one i (le_antisymm ((degree_map_le i _).trans hf) hif)
#align polynomial.splits_of_degree_le_one Polynomial.splits_of_degree_le_one
-/
#print Polynomial.splits_of_degree_eq_one /-
theorem splits_of_degree_eq_one {f : K[X]} (hf : degree f = 1) : Splits i f :=
splits_of_degree_le_one i hf.le
#align polynomial.splits_of_degree_eq_one Polynomial.splits_of_degree_eq_one
-/
#print Polynomial.splits_of_natDegree_le_one /-
theorem splits_of_natDegree_le_one {f : K[X]} (hf : natDegree f ≤ 1) : Splits i f :=
splits_of_degree_le_one i (degree_le_of_natDegree_le hf)
#align polynomial.splits_of_nat_degree_le_one Polynomial.splits_of_natDegree_le_one
-/
#print Polynomial.splits_of_natDegree_eq_one /-
theorem splits_of_natDegree_eq_one {f : K[X]} (hf : natDegree f = 1) : Splits i f :=
splits_of_natDegree_le_one i (le_of_eq hf)
#align polynomial.splits_of_nat_degree_eq_one Polynomial.splits_of_natDegree_eq_one
-/
#print Polynomial.splits_mul /-
theorem splits_mul {f g : K[X]} (hf : Splits i f) (hg : Splits i g) : Splits i (f * g) :=
if h : (f * g).map i = 0 then Or.inl h
else
Or.inr fun p hp hpf =>
((PrincipalIdealRing.irreducible_iff_prime.1 hp).2.2 _ _
(show p ∣ map i f * map i g by convert hpf <;> rw [Polynomial.map_mul])).elim
(hf.resolve_left (fun hf => by simpa [hf] using h) hp)
(hg.resolve_left (fun hg => by simpa [hg] using h) hp)
#align polynomial.splits_mul Polynomial.splits_mul
-/
#print Polynomial.splits_of_splits_mul' /-
theorem splits_of_splits_mul' {f g : K[X]} (hfg : (f * g).map i ≠ 0) (h : Splits i (f * g)) :
Splits i f ∧ Splits i g :=
⟨Or.inr fun g hgi hg =>
Or.resolve_left h hfg hgi (by rw [Polynomial.map_mul] <;> exact hg.trans (dvd_mul_right _ _)),
Or.inr fun g hgi hg =>
Or.resolve_left h hfg hgi (by rw [Polynomial.map_mul] <;> exact hg.trans (dvd_mul_left _ _))⟩
#align polynomial.splits_of_splits_mul' Polynomial.splits_of_splits_mul'
-/
#print Polynomial.splits_map_iff /-
theorem splits_map_iff (j : L →+* F) {f : K[X]} : Splits j (f.map i) ↔ Splits (j.comp i) f := by
simp [splits, Polynomial.map_map]
#align polynomial.splits_map_iff Polynomial.splits_map_iff
-/
#print Polynomial.splits_one /-
theorem splits_one : Splits i 1 :=
splits_C i 1
#align polynomial.splits_one Polynomial.splits_one
-/
/- warning: polynomial.splits_of_is_unit -> Polynomial.splits_of_isUnit is a dubious translation:
lean 3 declaration is
forall {K : Type.{u1}} {L : Type.{u2}} [_inst_1 : CommRing.{u1} K] [_inst_2 : Field.{u2} L] (i : RingHom.{u1, u2} K L (NonAssocRing.toNonAssocSemiring.{u1} K (Ring.toNonAssocRing.{u1} K (CommRing.toRing.{u1} K _inst_1))) (NonAssocRing.toNonAssocSemiring.{u2} L (Ring.toNonAssocRing.{u2} L (DivisionRing.toRing.{u2} L (Field.toDivisionRing.{u2} L _inst_2))))) [_inst_4 : IsDomain.{u1} K (Ring.toSemiring.{u1} K (CommRing.toRing.{u1} K _inst_1))] {u : Polynomial.{u1} K (Ring.toSemiring.{u1} K (CommRing.toRing.{u1} K _inst_1))}, (IsUnit.{u1} (Polynomial.{u1} K (Ring.toSemiring.{u1} K (CommRing.toRing.{u1} K _inst_1))) (Ring.toMonoid.{u1} (Polynomial.{u1} K (Ring.toSemiring.{u1} K (CommRing.toRing.{u1} K _inst_1))) (Polynomial.ring.{u1} K (CommRing.toRing.{u1} K _inst_1))) u) -> (Polynomial.Splits.{u1, u2} K L _inst_1 _inst_2 i u)
but is expected to have type
forall {K : Type.{u1}} {L : Type.{u2}} [_inst_1 : CommRing.{u1} K] [_inst_2 : Field.{u2} L] (i : RingHom.{u1, u2} K L (NonAssocRing.toNonAssocSemiring.{u1} K (Ring.toNonAssocRing.{u1} K (CommRing.toRing.{u1} K _inst_1))) (NonAssocRing.toNonAssocSemiring.{u2} L (Ring.toNonAssocRing.{u2} L (DivisionRing.toRing.{u2} L (Field.toDivisionRing.{u2} L _inst_2))))) [_inst_4 : IsDomain.{u1} K (Ring.toSemiring.{u1} K (CommRing.toRing.{u1} K _inst_1))] {u : Polynomial.{u1} K (Ring.toSemiring.{u1} K (CommRing.toRing.{u1} K _inst_1))}, (IsUnit.{u1} (Polynomial.{u1} K (Ring.toSemiring.{u1} K (CommRing.toRing.{u1} K _inst_1))) (MonoidWithZero.toMonoid.{u1} (Polynomial.{u1} K (Ring.toSemiring.{u1} K (CommRing.toRing.{u1} K _inst_1))) (Semiring.toMonoidWithZero.{u1} (Polynomial.{u1} K (Ring.toSemiring.{u1} K (CommRing.toRing.{u1} K _inst_1))) (Polynomial.semiring.{u1} K (Ring.toSemiring.{u1} K (CommRing.toRing.{u1} K _inst_1))))) u) -> (Polynomial.Splits.{u1, u2} K L _inst_1 _inst_2 i u)
Case conversion may be inaccurate. Consider using '#align polynomial.splits_of_is_unit Polynomial.splits_of_isUnitₓ'. -/
theorem splits_of_isUnit [IsDomain K] {u : K[X]} (hu : IsUnit u) : u.Splits i :=
(isUnit_iff.mp hu).choose_spec.2 ▸ splits_C _ _
#align polynomial.splits_of_is_unit Polynomial.splits_of_isUnit
/- warning: polynomial.splits_X_sub_C -> Polynomial.splits_X_sub_C is a dubious translation:
lean 3 declaration is
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but is expected to have type
forall {K : Type.{u1}} {L : Type.{u2}} [_inst_1 : CommRing.{u1} K] [_inst_2 : Field.{u2} L] (i : RingHom.{u1, u2} K L (NonAssocRing.toNonAssocSemiring.{u1} K (Ring.toNonAssocRing.{u1} K (CommRing.toRing.{u1} K _inst_1))) (NonAssocRing.toNonAssocSemiring.{u2} L (Ring.toNonAssocRing.{u2} L (DivisionRing.toRing.{u2} L (Field.toDivisionRing.{u2} L _inst_2))))) {x : K}, Polynomial.Splits.{u1, u2} K L _inst_1 _inst_2 i (HSub.hSub.{u1, u1, u1} (Polynomial.{u1} K (Ring.toSemiring.{u1} K (CommRing.toRing.{u1} K _inst_1))) ((fun ([email protected]._hyg.2391 : K) => Polynomial.{u1} K (Ring.toSemiring.{u1} K (CommRing.toRing.{u1} K _inst_1))) x) (Polynomial.{u1} K (Ring.toSemiring.{u1} K (CommRing.toRing.{u1} K _inst_1))) (instHSub.{u1} (Polynomial.{u1} K (Ring.toSemiring.{u1} K (CommRing.toRing.{u1} K _inst_1))) (Polynomial.sub.{u1} K (CommRing.toRing.{u1} K _inst_1))) (Polynomial.X.{u1} K (Ring.toSemiring.{u1} K (CommRing.toRing.{u1} K _inst_1))) (FunLike.coe.{succ u1, succ u1, succ u1} (RingHom.{u1, u1} K (Polynomial.{u1} K (Ring.toSemiring.{u1} K (CommRing.toRing.{u1} K _inst_1))) (Semiring.toNonAssocSemiring.{u1} K (Ring.toSemiring.{u1} K (CommRing.toRing.{u1} K _inst_1))) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} K (Ring.toSemiring.{u1} K (CommRing.toRing.{u1} K _inst_1))) (Polynomial.semiring.{u1} K (Ring.toSemiring.{u1} K (CommRing.toRing.{u1} K _inst_1))))) K (fun (_x : K) => (fun ([email protected]._hyg.2391 : K) => Polynomial.{u1} K (Ring.toSemiring.{u1} K (CommRing.toRing.{u1} K _inst_1))) _x) (MulHomClass.toFunLike.{u1, u1, u1} (RingHom.{u1, u1} K (Polynomial.{u1} K (Ring.toSemiring.{u1} K (CommRing.toRing.{u1} K _inst_1))) (Semiring.toNonAssocSemiring.{u1} K (Ring.toSemiring.{u1} K (CommRing.toRing.{u1} K _inst_1))) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} K (Ring.toSemiring.{u1} K (CommRing.toRing.{u1} K _inst_1))) (Polynomial.semiring.{u1} K (Ring.toSemiring.{u1} K (CommRing.toRing.{u1} K _inst_1))))) K (Polynomial.{u1} K (Ring.toSemiring.{u1} K (CommRing.toRing.{u1} K _inst_1))) (NonUnitalNonAssocSemiring.toMul.{u1} K (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} K (Semiring.toNonAssocSemiring.{u1} K (Ring.toSemiring.{u1} K (CommRing.toRing.{u1} K _inst_1))))) (NonUnitalNonAssocSemiring.toMul.{u1} (Polynomial.{u1} K (Ring.toSemiring.{u1} K (CommRing.toRing.{u1} K _inst_1))) (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} (Polynomial.{u1} K (Ring.toSemiring.{u1} K (CommRing.toRing.{u1} K _inst_1))) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} K (Ring.toSemiring.{u1} K (CommRing.toRing.{u1} K _inst_1))) (Polynomial.semiring.{u1} K (Ring.toSemiring.{u1} K (CommRing.toRing.{u1} K _inst_1)))))) (NonUnitalRingHomClass.toMulHomClass.{u1, u1, u1} (RingHom.{u1, u1} K (Polynomial.{u1} K (Ring.toSemiring.{u1} K (CommRing.toRing.{u1} K _inst_1))) (Semiring.toNonAssocSemiring.{u1} K (Ring.toSemiring.{u1} K (CommRing.toRing.{u1} K _inst_1))) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} K (Ring.toSemiring.{u1} K (CommRing.toRing.{u1} K _inst_1))) (Polynomial.semiring.{u1} K (Ring.toSemiring.{u1} K (CommRing.toRing.{u1} K _inst_1))))) K (Polynomial.{u1} K (Ring.toSemiring.{u1} K (CommRing.toRing.{u1} K _inst_1))) (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} K (Semiring.toNonAssocSemiring.{u1} K (Ring.toSemiring.{u1} K (CommRing.toRing.{u1} K _inst_1)))) (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} (Polynomial.{u1} K (Ring.toSemiring.{u1} K (CommRing.toRing.{u1} K _inst_1))) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} K (Ring.toSemiring.{u1} K (CommRing.toRing.{u1} K _inst_1))) (Polynomial.semiring.{u1} K (Ring.toSemiring.{u1} K (CommRing.toRing.{u1} K _inst_1))))) (RingHomClass.toNonUnitalRingHomClass.{u1, u1, u1} (RingHom.{u1, u1} K (Polynomial.{u1} K (Ring.toSemiring.{u1} K (CommRing.toRing.{u1} K _inst_1))) (Semiring.toNonAssocSemiring.{u1} K (Ring.toSemiring.{u1} K (CommRing.toRing.{u1} K _inst_1))) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} K (Ring.toSemiring.{u1} K (CommRing.toRing.{u1} K _inst_1))) (Polynomial.semiring.{u1} K (Ring.toSemiring.{u1} K (CommRing.toRing.{u1} K _inst_1))))) K (Polynomial.{u1} K (Ring.toSemiring.{u1} K (CommRing.toRing.{u1} K _inst_1))) (Semiring.toNonAssocSemiring.{u1} K (Ring.toSemiring.{u1} K (CommRing.toRing.{u1} K _inst_1))) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} K (Ring.toSemiring.{u1} K (CommRing.toRing.{u1} K _inst_1))) (Polynomial.semiring.{u1} K (Ring.toSemiring.{u1} K (CommRing.toRing.{u1} K _inst_1)))) (RingHom.instRingHomClassRingHom.{u1, u1} K (Polynomial.{u1} K (Ring.toSemiring.{u1} K (CommRing.toRing.{u1} K _inst_1))) (Semiring.toNonAssocSemiring.{u1} K (Ring.toSemiring.{u1} K (CommRing.toRing.{u1} K _inst_1))) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} K (Ring.toSemiring.{u1} K (CommRing.toRing.{u1} K _inst_1))) (Polynomial.semiring.{u1} K (Ring.toSemiring.{u1} K (CommRing.toRing.{u1} K _inst_1)))))))) (Polynomial.C.{u1} K (Ring.toSemiring.{u1} K (CommRing.toRing.{u1} K _inst_1))) x))
Case conversion may be inaccurate. Consider using '#align polynomial.splits_X_sub_C Polynomial.splits_X_sub_Cₓ'. -/
theorem splits_X_sub_C {x : K} : (X - C x).Splits i :=
splits_of_degree_le_one _ <| degree_X_sub_C_le _
#align polynomial.splits_X_sub_C Polynomial.splits_X_sub_C
#print Polynomial.splits_X /-
theorem splits_X : X.Splits i :=
splits_of_degree_le_one _ degree_X_le
#align polynomial.splits_X Polynomial.splits_X
-/
#print Polynomial.splits_prod /-
theorem splits_prod {ι : Type u} {s : ι → K[X]} {t : Finset ι} :
(∀ j ∈ t, (s j).Splits i) → (∏ x in t, s x).Splits i :=
by
refine' Finset.induction_on t (fun _ => splits_one i) fun a t hat ih ht => _
rw [Finset.forall_mem_insert] at ht; rw [Finset.prod_insert hat]
exact splits_mul i ht.1 (ih ht.2)
#align polynomial.splits_prod Polynomial.splits_prod
-/
/- warning: polynomial.splits_pow -> Polynomial.splits_pow is a dubious translation:
lean 3 declaration is
forall {K : Type.{u1}} {L : Type.{u2}} [_inst_1 : CommRing.{u1} K] [_inst_2 : Field.{u2} L] (i : RingHom.{u1, u2} K L (NonAssocRing.toNonAssocSemiring.{u1} K (Ring.toNonAssocRing.{u1} K (CommRing.toRing.{u1} K _inst_1))) (NonAssocRing.toNonAssocSemiring.{u2} L (Ring.toNonAssocRing.{u2} L (DivisionRing.toRing.{u2} L (Field.toDivisionRing.{u2} L _inst_2))))) {f : Polynomial.{u1} K (Ring.toSemiring.{u1} K (CommRing.toRing.{u1} K _inst_1))}, (Polynomial.Splits.{u1, u2} K L _inst_1 _inst_2 i f) -> (forall (n : Nat), Polynomial.Splits.{u1, u2} K L _inst_1 _inst_2 i (HPow.hPow.{u1, 0, u1} (Polynomial.{u1} K (Ring.toSemiring.{u1} K (CommRing.toRing.{u1} K _inst_1))) Nat (Polynomial.{u1} K (Ring.toSemiring.{u1} K (CommRing.toRing.{u1} K _inst_1))) (instHPow.{u1, 0} (Polynomial.{u1} K (Ring.toSemiring.{u1} K (CommRing.toRing.{u1} K _inst_1))) Nat (Monoid.Pow.{u1} (Polynomial.{u1} K (Ring.toSemiring.{u1} K (CommRing.toRing.{u1} K _inst_1))) (Ring.toMonoid.{u1} (Polynomial.{u1} K (Ring.toSemiring.{u1} K (CommRing.toRing.{u1} K _inst_1))) (Polynomial.ring.{u1} K (CommRing.toRing.{u1} K _inst_1))))) f n))
but is expected to have type
forall {K : Type.{u1}} {L : Type.{u2}} [_inst_1 : CommRing.{u1} K] [_inst_2 : Field.{u2} L] (i : RingHom.{u1, u2} K L (NonAssocRing.toNonAssocSemiring.{u1} K (Ring.toNonAssocRing.{u1} K (CommRing.toRing.{u1} K _inst_1))) (NonAssocRing.toNonAssocSemiring.{u2} L (Ring.toNonAssocRing.{u2} L (DivisionRing.toRing.{u2} L (Field.toDivisionRing.{u2} L _inst_2))))) {f : Polynomial.{u1} K (Ring.toSemiring.{u1} K (CommRing.toRing.{u1} K _inst_1))}, (Polynomial.Splits.{u1, u2} K L _inst_1 _inst_2 i f) -> (forall (n : Nat), Polynomial.Splits.{u1, u2} K L _inst_1 _inst_2 i (HPow.hPow.{u1, 0, u1} (Polynomial.{u1} K (Ring.toSemiring.{u1} K (CommRing.toRing.{u1} K _inst_1))) Nat (Polynomial.{u1} K (Ring.toSemiring.{u1} K (CommRing.toRing.{u1} K _inst_1))) (instHPow.{u1, 0} (Polynomial.{u1} K (Ring.toSemiring.{u1} K (CommRing.toRing.{u1} K _inst_1))) Nat (Monoid.Pow.{u1} (Polynomial.{u1} K (Ring.toSemiring.{u1} K (CommRing.toRing.{u1} K _inst_1))) (MonoidWithZero.toMonoid.{u1} (Polynomial.{u1} K (Ring.toSemiring.{u1} K (CommRing.toRing.{u1} K _inst_1))) (Semiring.toMonoidWithZero.{u1} (Polynomial.{u1} K (Ring.toSemiring.{u1} K (CommRing.toRing.{u1} K _inst_1))) (Polynomial.semiring.{u1} K (Ring.toSemiring.{u1} K (CommRing.toRing.{u1} K _inst_1))))))) f n))
Case conversion may be inaccurate. Consider using '#align polynomial.splits_pow Polynomial.splits_powₓ'. -/
theorem splits_pow {f : K[X]} (hf : f.Splits i) (n : ℕ) : (f ^ n).Splits i :=
by
rw [← Finset.card_range n, ← Finset.prod_const]
exact splits_prod i fun j hj => hf
#align polynomial.splits_pow Polynomial.splits_pow
/- warning: polynomial.splits_X_pow -> Polynomial.splits_X_pow is a dubious translation:
lean 3 declaration is
forall {K : Type.{u1}} {L : Type.{u2}} [_inst_1 : CommRing.{u1} K] [_inst_2 : Field.{u2} L] (i : RingHom.{u1, u2} K L (NonAssocRing.toNonAssocSemiring.{u1} K (Ring.toNonAssocRing.{u1} K (CommRing.toRing.{u1} K _inst_1))) (NonAssocRing.toNonAssocSemiring.{u2} L (Ring.toNonAssocRing.{u2} L (DivisionRing.toRing.{u2} L (Field.toDivisionRing.{u2} L _inst_2))))) (n : Nat), Polynomial.Splits.{u1, u2} K L _inst_1 _inst_2 i (HPow.hPow.{u1, 0, u1} (Polynomial.{u1} K (Ring.toSemiring.{u1} K (CommRing.toRing.{u1} K _inst_1))) Nat (Polynomial.{u1} K (Ring.toSemiring.{u1} K (CommRing.toRing.{u1} K _inst_1))) (instHPow.{u1, 0} (Polynomial.{u1} K (Ring.toSemiring.{u1} K (CommRing.toRing.{u1} K _inst_1))) Nat (Monoid.Pow.{u1} (Polynomial.{u1} K (Ring.toSemiring.{u1} K (CommRing.toRing.{u1} K _inst_1))) (Ring.toMonoid.{u1} (Polynomial.{u1} K (Ring.toSemiring.{u1} K (CommRing.toRing.{u1} K _inst_1))) (Polynomial.ring.{u1} K (CommRing.toRing.{u1} K _inst_1))))) (Polynomial.X.{u1} K (Ring.toSemiring.{u1} K (CommRing.toRing.{u1} K _inst_1))) n)
but is expected to have type
forall {K : Type.{u1}} {L : Type.{u2}} [_inst_1 : CommRing.{u1} K] [_inst_2 : Field.{u2} L] (i : RingHom.{u1, u2} K L (NonAssocRing.toNonAssocSemiring.{u1} K (Ring.toNonAssocRing.{u1} K (CommRing.toRing.{u1} K _inst_1))) (NonAssocRing.toNonAssocSemiring.{u2} L (Ring.toNonAssocRing.{u2} L (DivisionRing.toRing.{u2} L (Field.toDivisionRing.{u2} L _inst_2))))) (n : Nat), Polynomial.Splits.{u1, u2} K L _inst_1 _inst_2 i (HPow.hPow.{u1, 0, u1} (Polynomial.{u1} K (Ring.toSemiring.{u1} K (CommRing.toRing.{u1} K _inst_1))) Nat (Polynomial.{u1} K (Ring.toSemiring.{u1} K (CommRing.toRing.{u1} K _inst_1))) (instHPow.{u1, 0} (Polynomial.{u1} K (Ring.toSemiring.{u1} K (CommRing.toRing.{u1} K _inst_1))) Nat (Monoid.Pow.{u1} (Polynomial.{u1} K (Ring.toSemiring.{u1} K (CommRing.toRing.{u1} K _inst_1))) (MonoidWithZero.toMonoid.{u1} (Polynomial.{u1} K (Ring.toSemiring.{u1} K (CommRing.toRing.{u1} K _inst_1))) (Semiring.toMonoidWithZero.{u1} (Polynomial.{u1} K (Ring.toSemiring.{u1} K (CommRing.toRing.{u1} K _inst_1))) (Polynomial.semiring.{u1} K (Ring.toSemiring.{u1} K (CommRing.toRing.{u1} K _inst_1))))))) (Polynomial.X.{u1} K (Ring.toSemiring.{u1} K (CommRing.toRing.{u1} K _inst_1))) n)
Case conversion may be inaccurate. Consider using '#align polynomial.splits_X_pow Polynomial.splits_X_powₓ'. -/
theorem splits_X_pow (n : ℕ) : (X ^ n).Splits i :=
splits_pow i (splits_X i) n
#align polynomial.splits_X_pow Polynomial.splits_X_pow
#print Polynomial.splits_id_iff_splits /-
theorem splits_id_iff_splits {f : K[X]} : (f.map i).Splits (RingHom.id L) ↔ f.Splits i := by
rw [splits_map_iff, RingHom.id_comp]
#align polynomial.splits_id_iff_splits Polynomial.splits_id_iff_splits
-/
/- warning: polynomial.exists_root_of_splits' -> Polynomial.exists_root_of_splits' is a dubious translation:
lean 3 declaration is
forall {K : Type.{u1}} {L : Type.{u2}} [_inst_1 : CommRing.{u1} K] [_inst_2 : Field.{u2} L] (i : RingHom.{u1, u2} K L (NonAssocRing.toNonAssocSemiring.{u1} K (Ring.toNonAssocRing.{u1} K (CommRing.toRing.{u1} K _inst_1))) (NonAssocRing.toNonAssocSemiring.{u2} L (Ring.toNonAssocRing.{u2} L (DivisionRing.toRing.{u2} L (Field.toDivisionRing.{u2} L _inst_2))))) {f : Polynomial.{u1} K (Ring.toSemiring.{u1} K (CommRing.toRing.{u1} K _inst_1))}, (Polynomial.Splits.{u1, u2} K L _inst_1 _inst_2 i f) -> (Ne.{1} (WithBot.{0} Nat) (Polynomial.degree.{u2} L (Ring.toSemiring.{u2} L (DivisionRing.toRing.{u2} L (Field.toDivisionRing.{u2} L _inst_2))) (Polynomial.map.{u1, u2} K L (Ring.toSemiring.{u1} K (CommRing.toRing.{u1} K _inst_1)) (Ring.toSemiring.{u2} L (DivisionRing.toRing.{u2} L (Field.toDivisionRing.{u2} L _inst_2))) i f)) (OfNat.ofNat.{0} (WithBot.{0} Nat) 0 (OfNat.mk.{0} (WithBot.{0} Nat) 0 (Zero.zero.{0} (WithBot.{0} Nat) (WithBot.hasZero.{0} Nat Nat.hasZero))))) -> (Exists.{succ u2} L (fun (x : L) => Eq.{succ u2} L (Polynomial.eval₂.{u1, u2} K L (Ring.toSemiring.{u1} K (CommRing.toRing.{u1} K _inst_1)) (Ring.toSemiring.{u2} L (DivisionRing.toRing.{u2} L (Field.toDivisionRing.{u2} L _inst_2))) i x f) (OfNat.ofNat.{u2} L 0 (OfNat.mk.{u2} L 0 (Zero.zero.{u2} L (MulZeroClass.toHasZero.{u2} L (NonUnitalNonAssocSemiring.toMulZeroClass.{u2} L (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u2} L (NonAssocRing.toNonUnitalNonAssocRing.{u2} L (Ring.toNonAssocRing.{u2} L (DivisionRing.toRing.{u2} L (Field.toDivisionRing.{u2} L _inst_2))))))))))))
but is expected to have type
forall {K : Type.{u1}} {L : Type.{u2}} [_inst_1 : CommRing.{u1} K] [_inst_2 : Field.{u2} L] (i : RingHom.{u1, u2} K L (NonAssocRing.toNonAssocSemiring.{u1} K (Ring.toNonAssocRing.{u1} K (CommRing.toRing.{u1} K _inst_1))) (NonAssocRing.toNonAssocSemiring.{u2} L (Ring.toNonAssocRing.{u2} L (DivisionRing.toRing.{u2} L (Field.toDivisionRing.{u2} L _inst_2))))) {f : Polynomial.{u1} K (Ring.toSemiring.{u1} K (CommRing.toRing.{u1} K _inst_1))}, (Polynomial.Splits.{u1, u2} K L _inst_1 _inst_2 i f) -> (Ne.{1} (WithBot.{0} Nat) (Polynomial.degree.{u2} L (DivisionSemiring.toSemiring.{u2} L (Semifield.toDivisionSemiring.{u2} L (Field.toSemifield.{u2} L _inst_2))) (Polynomial.map.{u1, u2} K L (Ring.toSemiring.{u1} K (CommRing.toRing.{u1} K _inst_1)) (DivisionSemiring.toSemiring.{u2} L (Semifield.toDivisionSemiring.{u2} L (Field.toSemifield.{u2} L _inst_2))) i f)) (OfNat.ofNat.{0} (WithBot.{0} Nat) 0 (Zero.toOfNat0.{0} (WithBot.{0} Nat) (WithBot.zero.{0} Nat (LinearOrderedCommMonoidWithZero.toZero.{0} Nat Nat.linearOrderedCommMonoidWithZero))))) -> (Exists.{succ u2} L (fun (x : L) => Eq.{succ u2} L (Polynomial.eval₂.{u1, u2} K L (Ring.toSemiring.{u1} K (CommRing.toRing.{u1} K _inst_1)) (DivisionSemiring.toSemiring.{u2} L (Semifield.toDivisionSemiring.{u2} L (Field.toSemifield.{u2} L _inst_2))) i x f) (OfNat.ofNat.{u2} L 0 (Zero.toOfNat0.{u2} L (CommMonoidWithZero.toZero.{u2} L (CommGroupWithZero.toCommMonoidWithZero.{u2} L (Semifield.toCommGroupWithZero.{u2} L (Field.toSemifield.{u2} L _inst_2))))))))
Case conversion may be inaccurate. Consider using '#align polynomial.exists_root_of_splits' Polynomial.exists_root_of_splits'ₓ'. -/
theorem exists_root_of_splits' {f : K[X]} (hs : Splits i f) (hf0 : degree (f.map i) ≠ 0) :
∃ x, eval₂ i x f = 0 :=
if hf0' : f.map i = 0 then by simp [eval₂_eq_eval_map, hf0']
else
let ⟨g, hg⟩ :=
WfDvdMonoid.exists_irreducible_factor
(show ¬IsUnit (f.map i) from mt isUnit_iff_degree_eq_zero.1 hf0) hf0'
let ⟨x, hx⟩ := exists_root_of_degree_eq_one (hs.resolve_left hf0' hg.1 hg.2)
let ⟨i, hi⟩ := hg.2
⟨x, by rw [← eval_map, hi, eval_mul, show _ = _ from hx, MulZeroClass.zero_mul]⟩
#align polynomial.exists_root_of_splits' Polynomial.exists_root_of_splits'
#print Polynomial.roots_ne_zero_of_splits' /-
theorem roots_ne_zero_of_splits' {f : K[X]} (hs : Splits i f) (hf0 : natDegree (f.map i) ≠ 0) :
(f.map i).roots ≠ 0 :=
let ⟨x, hx⟩ := exists_root_of_splits' i hs fun h => hf0 <| natDegree_eq_of_degree_eq_some h
fun h => by
rw [← eval_map] at hx
cases h.subst ((mem_roots _).2 hx)
exact ne_zero_of_nat_degree_gt (Nat.pos_of_ne_zero hf0)
#align polynomial.roots_ne_zero_of_splits' Polynomial.roots_ne_zero_of_splits'
-/
#print Polynomial.rootOfSplits' /-
/-- Pick a root of a polynomial that splits. See `root_of_splits` for polynomials over a field
which has simpler assumptions. -/
def rootOfSplits' {f : K[X]} (hf : f.Splits i) (hfd : (f.map i).degree ≠ 0) : L :=
Classical.choose <| exists_root_of_splits' i hf hfd
#align polynomial.root_of_splits' Polynomial.rootOfSplits'
-/
/- warning: polynomial.map_root_of_splits' -> Polynomial.map_rootOfSplits' is a dubious translation:
lean 3 declaration is
forall {K : Type.{u1}} {L : Type.{u2}} [_inst_1 : CommRing.{u1} K] [_inst_2 : Field.{u2} L] (i : RingHom.{u1, u2} K L (NonAssocRing.toNonAssocSemiring.{u1} K (Ring.toNonAssocRing.{u1} K (CommRing.toRing.{u1} K _inst_1))) (NonAssocRing.toNonAssocSemiring.{u2} L (Ring.toNonAssocRing.{u2} L (DivisionRing.toRing.{u2} L (Field.toDivisionRing.{u2} L _inst_2))))) {f : Polynomial.{u1} K (Ring.toSemiring.{u1} K (CommRing.toRing.{u1} K _inst_1))} (hf : Polynomial.Splits.{u1, u2} K L _inst_1 _inst_2 i f) (hfd : Ne.{1} (WithBot.{0} Nat) (Polynomial.degree.{u2} L (Ring.toSemiring.{u2} L (DivisionRing.toRing.{u2} L (Field.toDivisionRing.{u2} L _inst_2))) (Polynomial.map.{u1, u2} K L (Ring.toSemiring.{u1} K (CommRing.toRing.{u1} K _inst_1)) (Ring.toSemiring.{u2} L (DivisionRing.toRing.{u2} L (Field.toDivisionRing.{u2} L _inst_2))) i f)) (OfNat.ofNat.{0} (WithBot.{0} Nat) 0 (OfNat.mk.{0} (WithBot.{0} Nat) 0 (Zero.zero.{0} (WithBot.{0} Nat) (WithBot.hasZero.{0} Nat Nat.hasZero))))), Eq.{succ u2} L (Polynomial.eval₂.{u1, u2} K L (Ring.toSemiring.{u1} K (CommRing.toRing.{u1} K _inst_1)) (Ring.toSemiring.{u2} L (DivisionRing.toRing.{u2} L (Field.toDivisionRing.{u2} L _inst_2))) i (Polynomial.rootOfSplits'.{u1, u2} K L _inst_1 _inst_2 i f hf hfd) f) (OfNat.ofNat.{u2} L 0 (OfNat.mk.{u2} L 0 (Zero.zero.{u2} L (MulZeroClass.toHasZero.{u2} L (NonUnitalNonAssocSemiring.toMulZeroClass.{u2} L (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u2} L (NonAssocRing.toNonUnitalNonAssocRing.{u2} L (Ring.toNonAssocRing.{u2} L (DivisionRing.toRing.{u2} L (Field.toDivisionRing.{u2} L _inst_2))))))))))
but is expected to have type
forall {K : Type.{u1}} {L : Type.{u2}} [_inst_1 : CommRing.{u1} K] [_inst_2 : Field.{u2} L] (i : RingHom.{u1, u2} K L (NonAssocRing.toNonAssocSemiring.{u1} K (Ring.toNonAssocRing.{u1} K (CommRing.toRing.{u1} K _inst_1))) (NonAssocRing.toNonAssocSemiring.{u2} L (Ring.toNonAssocRing.{u2} L (DivisionRing.toRing.{u2} L (Field.toDivisionRing.{u2} L _inst_2))))) {f : Polynomial.{u1} K (Ring.toSemiring.{u1} K (CommRing.toRing.{u1} K _inst_1))} (hf : Polynomial.Splits.{u1, u2} K L _inst_1 _inst_2 i f) (hfd : Ne.{1} (WithBot.{0} Nat) (Polynomial.degree.{u2} L (DivisionSemiring.toSemiring.{u2} L (Semifield.toDivisionSemiring.{u2} L (Field.toSemifield.{u2} L _inst_2))) (Polynomial.map.{u1, u2} K L (Ring.toSemiring.{u1} K (CommRing.toRing.{u1} K _inst_1)) (DivisionSemiring.toSemiring.{u2} L (Semifield.toDivisionSemiring.{u2} L (Field.toSemifield.{u2} L _inst_2))) i f)) (OfNat.ofNat.{0} (WithBot.{0} Nat) 0 (Zero.toOfNat0.{0} (WithBot.{0} Nat) (WithBot.zero.{0} Nat (LinearOrderedCommMonoidWithZero.toZero.{0} Nat Nat.linearOrderedCommMonoidWithZero))))), Eq.{succ u2} L (Polynomial.eval₂.{u1, u2} K L (Ring.toSemiring.{u1} K (CommRing.toRing.{u1} K _inst_1)) (DivisionSemiring.toSemiring.{u2} L (Semifield.toDivisionSemiring.{u2} L (Field.toSemifield.{u2} L _inst_2))) i (Polynomial.rootOfSplits'.{u1, u2} K L _inst_1 _inst_2 i f hf hfd) f) (OfNat.ofNat.{u2} L 0 (Zero.toOfNat0.{u2} L (CommMonoidWithZero.toZero.{u2} L (CommGroupWithZero.toCommMonoidWithZero.{u2} L (Semifield.toCommGroupWithZero.{u2} L (Field.toSemifield.{u2} L _inst_2))))))
Case conversion may be inaccurate. Consider using '#align polynomial.map_root_of_splits' Polynomial.map_rootOfSplits'ₓ'. -/
theorem map_rootOfSplits' {f : K[X]} (hf : f.Splits i) (hfd) :
f.eval₂ i (rootOfSplits' i hf hfd) = 0 :=
Classical.choose_spec <| exists_root_of_splits' i hf hfd
#align polynomial.map_root_of_splits' Polynomial.map_rootOfSplits'
/- warning: polynomial.nat_degree_eq_card_roots' -> Polynomial.natDegree_eq_card_roots' is a dubious translation:
lean 3 declaration is
forall {K : Type.{u1}} {L : Type.{u2}} [_inst_1 : CommRing.{u1} K] [_inst_2 : Field.{u2} L] {p : Polynomial.{u1} K (Ring.toSemiring.{u1} K (CommRing.toRing.{u1} K _inst_1))} {i : RingHom.{u1, u2} K L (NonAssocRing.toNonAssocSemiring.{u1} K (Ring.toNonAssocRing.{u1} K (CommRing.toRing.{u1} K _inst_1))) (NonAssocRing.toNonAssocSemiring.{u2} L (Ring.toNonAssocRing.{u2} L (DivisionRing.toRing.{u2} L (Field.toDivisionRing.{u2} L _inst_2))))}, (Polynomial.Splits.{u1, u2} K L _inst_1 _inst_2 i p) -> (Eq.{1} Nat (Polynomial.natDegree.{u2} L (Ring.toSemiring.{u2} L (DivisionRing.toRing.{u2} L (Field.toDivisionRing.{u2} L _inst_2))) (Polynomial.map.{u1, u2} K L (Ring.toSemiring.{u1} K (CommRing.toRing.{u1} K _inst_1)) (Ring.toSemiring.{u2} L (DivisionRing.toRing.{u2} L (Field.toDivisionRing.{u2} L _inst_2))) i p)) (coeFn.{succ u2, succ u2} (AddMonoidHom.{u2, 0} (Multiset.{u2} L) Nat (AddMonoid.toAddZeroClass.{u2} (Multiset.{u2} L) (AddRightCancelMonoid.toAddMonoid.{u2} (Multiset.{u2} L) (AddCancelMonoid.toAddRightCancelMonoid.{u2} (Multiset.{u2} L) (AddCancelCommMonoid.toAddCancelMonoid.{u2} (Multiset.{u2} L) (OrderedCancelAddCommMonoid.toCancelAddCommMonoid.{u2} (Multiset.{u2} L) (Multiset.orderedCancelAddCommMonoid.{u2} L)))))) (AddMonoid.toAddZeroClass.{0} Nat Nat.addMonoid)) (fun (_x : AddMonoidHom.{u2, 0} (Multiset.{u2} L) Nat (AddMonoid.toAddZeroClass.{u2} (Multiset.{u2} L) (AddRightCancelMonoid.toAddMonoid.{u2} (Multiset.{u2} L) (AddCancelMonoid.toAddRightCancelMonoid.{u2} (Multiset.{u2} L) (AddCancelCommMonoid.toAddCancelMonoid.{u2} (Multiset.{u2} L) (OrderedCancelAddCommMonoid.toCancelAddCommMonoid.{u2} (Multiset.{u2} L) (Multiset.orderedCancelAddCommMonoid.{u2} L)))))) (AddMonoid.toAddZeroClass.{0} Nat Nat.addMonoid)) => (Multiset.{u2} L) -> Nat) (AddMonoidHom.hasCoeToFun.{u2, 0} (Multiset.{u2} L) Nat (AddMonoid.toAddZeroClass.{u2} (Multiset.{u2} L) (AddRightCancelMonoid.toAddMonoid.{u2} (Multiset.{u2} L) (AddCancelMonoid.toAddRightCancelMonoid.{u2} (Multiset.{u2} L) (AddCancelCommMonoid.toAddCancelMonoid.{u2} (Multiset.{u2} L) (OrderedCancelAddCommMonoid.toCancelAddCommMonoid.{u2} (Multiset.{u2} L) (Multiset.orderedCancelAddCommMonoid.{u2} L)))))) (AddMonoid.toAddZeroClass.{0} Nat Nat.addMonoid)) (Multiset.card.{u2} L) (Polynomial.roots.{u2} L (EuclideanDomain.toCommRing.{u2} L (Field.toEuclideanDomain.{u2} L _inst_2)) (Field.isDomain.{u2} L _inst_2) (Polynomial.map.{u1, u2} K L (Ring.toSemiring.{u1} K (CommRing.toRing.{u1} K _inst_1)) (Ring.toSemiring.{u2} L (DivisionRing.toRing.{u2} L (Field.toDivisionRing.{u2} L _inst_2))) i p))))
but is expected to have type
forall {K : Type.{u1}} {L : Type.{u2}} [_inst_1 : CommRing.{u1} K] [_inst_2 : Field.{u2} L] {p : Polynomial.{u1} K (Ring.toSemiring.{u1} K (CommRing.toRing.{u1} K _inst_1))} {i : RingHom.{u1, u2} K L (NonAssocRing.toNonAssocSemiring.{u1} K (Ring.toNonAssocRing.{u1} K (CommRing.toRing.{u1} K _inst_1))) (NonAssocRing.toNonAssocSemiring.{u2} L (Ring.toNonAssocRing.{u2} L (DivisionRing.toRing.{u2} L (Field.toDivisionRing.{u2} L _inst_2))))}, (Polynomial.Splits.{u1, u2} K L _inst_1 _inst_2 i p) -> (Eq.{1} Nat (Polynomial.natDegree.{u2} L (DivisionSemiring.toSemiring.{u2} L (Semifield.toDivisionSemiring.{u2} L (Field.toSemifield.{u2} L _inst_2))) (Polynomial.map.{u1, u2} K L (Ring.toSemiring.{u1} K (CommRing.toRing.{u1} K _inst_1)) (DivisionSemiring.toSemiring.{u2} L (Semifield.toDivisionSemiring.{u2} L (Field.toSemifield.{u2} L _inst_2))) i p)) (FunLike.coe.{succ u2, succ u2, 1} (AddMonoidHom.{u2, 0} (Multiset.{u2} L) Nat (AddMonoid.toAddZeroClass.{u2} (Multiset.{u2} L) (AddRightCancelMonoid.toAddMonoid.{u2} (Multiset.{u2} L) (AddCancelMonoid.toAddRightCancelMonoid.{u2} (Multiset.{u2} L) (AddCancelCommMonoid.toAddCancelMonoid.{u2} (Multiset.{u2} L) (OrderedCancelAddCommMonoid.toCancelAddCommMonoid.{u2} (Multiset.{u2} L) (Multiset.instOrderedCancelAddCommMonoidMultiset.{u2} L)))))) (AddMonoid.toAddZeroClass.{0} Nat Nat.addMonoid)) (Multiset.{u2} L) (fun (_x : Multiset.{u2} L) => (fun ([email protected]._hyg.403 : Multiset.{u2} L) => Nat) _x) (AddHomClass.toFunLike.{u2, u2, 0} (AddMonoidHom.{u2, 0} (Multiset.{u2} L) Nat (AddMonoid.toAddZeroClass.{u2} (Multiset.{u2} L) (AddRightCancelMonoid.toAddMonoid.{u2} (Multiset.{u2} L) (AddCancelMonoid.toAddRightCancelMonoid.{u2} (Multiset.{u2} L) (AddCancelCommMonoid.toAddCancelMonoid.{u2} (Multiset.{u2} L) (OrderedCancelAddCommMonoid.toCancelAddCommMonoid.{u2} (Multiset.{u2} L) (Multiset.instOrderedCancelAddCommMonoidMultiset.{u2} L)))))) (AddMonoid.toAddZeroClass.{0} Nat Nat.addMonoid)) (Multiset.{u2} L) Nat (AddZeroClass.toAdd.{u2} (Multiset.{u2} L) (AddMonoid.toAddZeroClass.{u2} (Multiset.{u2} L) (AddRightCancelMonoid.toAddMonoid.{u2} (Multiset.{u2} L) (AddCancelMonoid.toAddRightCancelMonoid.{u2} (Multiset.{u2} L) (AddCancelCommMonoid.toAddCancelMonoid.{u2} (Multiset.{u2} L) (OrderedCancelAddCommMonoid.toCancelAddCommMonoid.{u2} (Multiset.{u2} L) (Multiset.instOrderedCancelAddCommMonoidMultiset.{u2} L))))))) (AddZeroClass.toAdd.{0} Nat (AddMonoid.toAddZeroClass.{0} Nat Nat.addMonoid)) (AddMonoidHomClass.toAddHomClass.{u2, u2, 0} (AddMonoidHom.{u2, 0} (Multiset.{u2} L) Nat (AddMonoid.toAddZeroClass.{u2} (Multiset.{u2} L) (AddRightCancelMonoid.toAddMonoid.{u2} (Multiset.{u2} L) (AddCancelMonoid.toAddRightCancelMonoid.{u2} (Multiset.{u2} L) (AddCancelCommMonoid.toAddCancelMonoid.{u2} (Multiset.{u2} L) (OrderedCancelAddCommMonoid.toCancelAddCommMonoid.{u2} (Multiset.{u2} L) (Multiset.instOrderedCancelAddCommMonoidMultiset.{u2} L)))))) (AddMonoid.toAddZeroClass.{0} Nat Nat.addMonoid)) (Multiset.{u2} L) Nat (AddMonoid.toAddZeroClass.{u2} (Multiset.{u2} L) (AddRightCancelMonoid.toAddMonoid.{u2} (Multiset.{u2} L) (AddCancelMonoid.toAddRightCancelMonoid.{u2} (Multiset.{u2} L) (AddCancelCommMonoid.toAddCancelMonoid.{u2} (Multiset.{u2} L) (OrderedCancelAddCommMonoid.toCancelAddCommMonoid.{u2} (Multiset.{u2} L) (Multiset.instOrderedCancelAddCommMonoidMultiset.{u2} L)))))) (AddMonoid.toAddZeroClass.{0} Nat Nat.addMonoid) (AddMonoidHom.addMonoidHomClass.{u2, 0} (Multiset.{u2} L) Nat (AddMonoid.toAddZeroClass.{u2} (Multiset.{u2} L) (AddRightCancelMonoid.toAddMonoid.{u2} (Multiset.{u2} L) (AddCancelMonoid.toAddRightCancelMonoid.{u2} (Multiset.{u2} L) (AddCancelCommMonoid.toAddCancelMonoid.{u2} (Multiset.{u2} L) (OrderedCancelAddCommMonoid.toCancelAddCommMonoid.{u2} (Multiset.{u2} L) (Multiset.instOrderedCancelAddCommMonoidMultiset.{u2} L)))))) (AddMonoid.toAddZeroClass.{0} Nat Nat.addMonoid)))) (Multiset.card.{u2} L) (Polynomial.roots.{u2} L (EuclideanDomain.toCommRing.{u2} L (Field.toEuclideanDomain.{u2} L _inst_2)) (Field.isDomain.{u2} L _inst_2) (Polynomial.map.{u1, u2} K L (Ring.toSemiring.{u1} K (CommRing.toRing.{u1} K _inst_1)) (DivisionSemiring.toSemiring.{u2} L (Semifield.toDivisionSemiring.{u2} L (Field.toSemifield.{u2} L _inst_2))) i p))))
Case conversion may be inaccurate. Consider using '#align polynomial.nat_degree_eq_card_roots' Polynomial.natDegree_eq_card_roots'ₓ'. -/
theorem natDegree_eq_card_roots' {p : K[X]} {i : K →+* L} (hsplit : Splits i p) :
(p.map i).natDegree = (p.map i).roots.card :=
by
by_cases hp : p.map i = 0
· rw [hp, nat_degree_zero, roots_zero, Multiset.card_zero]
obtain ⟨q, he, hd, hr⟩ := exists_prod_multiset_X_sub_C_mul (p.map i)
rw [← splits_id_iff_splits, ← he] at hsplit
rw [← he] at hp
have hq : q ≠ 0 := fun h => hp (by rw [h, MulZeroClass.mul_zero])
rw [← hd, add_right_eq_self]
by_contra
have h' : (map (RingHom.id L) q).natDegree ≠ 0 := by simp [h]
have := roots_ne_zero_of_splits' (RingHom.id L) (splits_of_splits_mul' _ _ hsplit).2 h'
· rw [map_id] at this
exact this hr
· rw [map_id]
exact mul_ne_zero monic_prod_multiset_X_sub_C.ne_zero hq
#align polynomial.nat_degree_eq_card_roots' Polynomial.natDegree_eq_card_roots'
/- warning: polynomial.degree_eq_card_roots' -> Polynomial.degree_eq_card_roots' is a dubious translation:
lean 3 declaration is
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but is expected to have type
forall {K : Type.{u1}} {L : Type.{u2}} [_inst_1 : CommRing.{u1} K] [_inst_2 : Field.{u2} L] {p : Polynomial.{u1} K (Ring.toSemiring.{u1} K (CommRing.toRing.{u1} K _inst_1))} {i : RingHom.{u1, u2} K L (NonAssocRing.toNonAssocSemiring.{u1} K (Ring.toNonAssocRing.{u1} K (CommRing.toRing.{u1} K _inst_1))) (NonAssocRing.toNonAssocSemiring.{u2} L (Ring.toNonAssocRing.{u2} L (DivisionRing.toRing.{u2} L (Field.toDivisionRing.{u2} L _inst_2))))}, (Ne.{succ u2} (Polynomial.{u2} L (DivisionSemiring.toSemiring.{u2} L (Semifield.toDivisionSemiring.{u2} L (Field.toSemifield.{u2} L _inst_2)))) (Polynomial.map.{u1, u2} K L (Ring.toSemiring.{u1} K (CommRing.toRing.{u1} K _inst_1)) (DivisionSemiring.toSemiring.{u2} L (Semifield.toDivisionSemiring.{u2} L (Field.toSemifield.{u2} L _inst_2))) i p) (OfNat.ofNat.{u2} (Polynomial.{u2} L (DivisionSemiring.toSemiring.{u2} L (Semifield.toDivisionSemiring.{u2} L (Field.toSemifield.{u2} L _inst_2)))) 0 (Zero.toOfNat0.{u2} (Polynomial.{u2} L (DivisionSemiring.toSemiring.{u2} L (Semifield.toDivisionSemiring.{u2} L (Field.toSemifield.{u2} L _inst_2)))) (Polynomial.zero.{u2} L (DivisionSemiring.toSemiring.{u2} L (Semifield.toDivisionSemiring.{u2} L (Field.toSemifield.{u2} L _inst_2))))))) -> (Polynomial.Splits.{u1, u2} K L _inst_1 _inst_2 i p) -> (Eq.{1} (WithBot.{0} Nat) (Polynomial.degree.{u2} L (DivisionSemiring.toSemiring.{u2} L (Semifield.toDivisionSemiring.{u2} L (Field.toSemifield.{u2} L _inst_2))) (Polynomial.map.{u1, u2} K L (Ring.toSemiring.{u1} K (CommRing.toRing.{u1} K _inst_1)) (DivisionSemiring.toSemiring.{u2} L (Semifield.toDivisionSemiring.{u2} L (Field.toSemifield.{u2} L _inst_2))) i p)) (Nat.cast.{0} (WithBot.{0} Nat) (Semiring.toNatCast.{0} (WithBot.{0} Nat) (OrderedSemiring.toSemiring.{0} (WithBot.{0} Nat) (OrderedCommSemiring.toOrderedSemiring.{0} (WithBot.{0} Nat) (WithBot.orderedCommSemiring.{0} Nat (fun (a : Nat) (b : Nat) => instDecidableEqNat a b) Nat.canonicallyOrderedCommSemiring Nat.nontrivial)))) (FunLike.coe.{succ u2, succ u2, 1} (AddMonoidHom.{u2, 0} (Multiset.{u2} L) Nat (AddMonoid.toAddZeroClass.{u2} (Multiset.{u2} L) (AddRightCancelMonoid.toAddMonoid.{u2} (Multiset.{u2} L) (AddCancelMonoid.toAddRightCancelMonoid.{u2} (Multiset.{u2} L) (AddCancelCommMonoid.toAddCancelMonoid.{u2} (Multiset.{u2} L) (OrderedCancelAddCommMonoid.toCancelAddCommMonoid.{u2} (Multiset.{u2} L) (Multiset.instOrderedCancelAddCommMonoidMultiset.{u2} L)))))) (AddMonoid.toAddZeroClass.{0} Nat Nat.addMonoid)) (Multiset.{u2} L) (fun (_x : Multiset.{u2} L) => (fun ([email protected]._hyg.403 : Multiset.{u2} L) => Nat) _x) (AddHomClass.toFunLike.{u2, u2, 0} (AddMonoidHom.{u2, 0} (Multiset.{u2} L) Nat (AddMonoid.toAddZeroClass.{u2} (Multiset.{u2} L) (AddRightCancelMonoid.toAddMonoid.{u2} (Multiset.{u2} L) (AddCancelMonoid.toAddRightCancelMonoid.{u2} (Multiset.{u2} L) (AddCancelCommMonoid.toAddCancelMonoid.{u2} (Multiset.{u2} L) (OrderedCancelAddCommMonoid.toCancelAddCommMonoid.{u2} (Multiset.{u2} L) (Multiset.instOrderedCancelAddCommMonoidMultiset.{u2} L)))))) (AddMonoid.toAddZeroClass.{0} Nat Nat.addMonoid)) (Multiset.{u2} L) Nat (AddZeroClass.toAdd.{u2} (Multiset.{u2} L) (AddMonoid.toAddZeroClass.{u2} (Multiset.{u2} L) (AddRightCancelMonoid.toAddMonoid.{u2} (Multiset.{u2} L) (AddCancelMonoid.toAddRightCancelMonoid.{u2} (Multiset.{u2} L) (AddCancelCommMonoid.toAddCancelMonoid.{u2} (Multiset.{u2} L) (OrderedCancelAddCommMonoid.toCancelAddCommMonoid.{u2} (Multiset.{u2} L) (Multiset.instOrderedCancelAddCommMonoidMultiset.{u2} L))))))) (AddZeroClass.toAdd.{0} Nat (AddMonoid.toAddZeroClass.{0} Nat Nat.addMonoid)) (AddMonoidHomClass.toAddHomClass.{u2, u2, 0} (AddMonoidHom.{u2, 0} (Multiset.{u2} L) Nat (AddMonoid.toAddZeroClass.{u2} (Multiset.{u2} L) (AddRightCancelMonoid.toAddMonoid.{u2} (Multiset.{u2} L) (AddCancelMonoid.toAddRightCancelMonoid.{u2} (Multiset.{u2} L) (AddCancelCommMonoid.toAddCancelMonoid.{u2} (Multiset.{u2} L) (OrderedCancelAddCommMonoid.toCancelAddCommMonoid.{u2} (Multiset.{u2} L) (Multiset.instOrderedCancelAddCommMonoidMultiset.{u2} L)))))) (AddMonoid.toAddZeroClass.{0} Nat Nat.addMonoid)) (Multiset.{u2} L) Nat (AddMonoid.toAddZeroClass.{u2} (Multiset.{u2} L) (AddRightCancelMonoid.toAddMonoid.{u2} (Multiset.{u2} L) (AddCancelMonoid.toAddRightCancelMonoid.{u2} (Multiset.{u2} L) (AddCancelCommMonoid.toAddCancelMonoid.{u2} (Multiset.{u2} L) (OrderedCancelAddCommMonoid.toCancelAddCommMonoid.{u2} (Multiset.{u2} L) (Multiset.instOrderedCancelAddCommMonoidMultiset.{u2} L)))))) (AddMonoid.toAddZeroClass.{0} Nat Nat.addMonoid) (AddMonoidHom.addMonoidHomClass.{u2, 0} (Multiset.{u2} L) Nat (AddMonoid.toAddZeroClass.{u2} (Multiset.{u2} L) (AddRightCancelMonoid.toAddMonoid.{u2} (Multiset.{u2} L) (AddCancelMonoid.toAddRightCancelMonoid.{u2} (Multiset.{u2} L) (AddCancelCommMonoid.toAddCancelMonoid.{u2} (Multiset.{u2} L) (OrderedCancelAddCommMonoid.toCancelAddCommMonoid.{u2} (Multiset.{u2} L) (Multiset.instOrderedCancelAddCommMonoidMultiset.{u2} L)))))) (AddMonoid.toAddZeroClass.{0} Nat Nat.addMonoid)))) (Multiset.card.{u2} L) (Polynomial.roots.{u2} L (EuclideanDomain.toCommRing.{u2} L (Field.toEuclideanDomain.{u2} L _inst_2)) (Field.isDomain.{u2} L _inst_2) (Polynomial.map.{u1, u2} K L (Ring.toSemiring.{u1} K (CommRing.toRing.{u1} K _inst_1)) (DivisionSemiring.toSemiring.{u2} L (Semifield.toDivisionSemiring.{u2} L (Field.toSemifield.{u2} L _inst_2))) i p)))))
Case conversion may be inaccurate. Consider using '#align polynomial.degree_eq_card_roots' Polynomial.degree_eq_card_roots'ₓ'. -/
theorem degree_eq_card_roots' {p : K[X]} {i : K →+* L} (p_ne_zero : p.map i ≠ 0)
(hsplit : Splits i p) : (p.map i).degree = (p.map i).roots.card := by
rw [degree_eq_nat_degree p_ne_zero, nat_degree_eq_card_roots' hsplit]
#align polynomial.degree_eq_card_roots' Polynomial.degree_eq_card_roots'
end CommRing
variable [Field K] [Field L] [Field F]
variable (i : K →+* L)
/- warning: polynomial.splits_iff -> Polynomial.splits_iff is a dubious translation:
lean 3 declaration is
forall {K : Type.{u1}} {L : Type.{u2}} [_inst_1 : Field.{u1} K] [_inst_2 : Field.{u2} L] (i : RingHom.{u1, u2} K L (NonAssocRing.toNonAssocSemiring.{u1} K (Ring.toNonAssocRing.{u1} K (DivisionRing.toRing.{u1} K (Field.toDivisionRing.{u1} K _inst_1)))) (NonAssocRing.toNonAssocSemiring.{u2} L (Ring.toNonAssocRing.{u2} L (DivisionRing.toRing.{u2} L (Field.toDivisionRing.{u2} L _inst_2))))) (f : Polynomial.{u1} K (Ring.toSemiring.{u1} K (DivisionRing.toRing.{u1} K (Field.toDivisionRing.{u1} K _inst_1)))), Iff (Polynomial.Splits.{u1, u2} K L (EuclideanDomain.toCommRing.{u1} K (Field.toEuclideanDomain.{u1} K _inst_1)) _inst_2 i f) (Or (Eq.{succ u1} (Polynomial.{u1} K (Ring.toSemiring.{u1} K (DivisionRing.toRing.{u1} K (Field.toDivisionRing.{u1} K _inst_1)))) f (OfNat.ofNat.{u1} (Polynomial.{u1} K (Ring.toSemiring.{u1} K (DivisionRing.toRing.{u1} K (Field.toDivisionRing.{u1} K _inst_1)))) 0 (OfNat.mk.{u1} (Polynomial.{u1} K (Ring.toSemiring.{u1} K (DivisionRing.toRing.{u1} K (Field.toDivisionRing.{u1} K _inst_1)))) 0 (Zero.zero.{u1} (Polynomial.{u1} K (Ring.toSemiring.{u1} K (DivisionRing.toRing.{u1} K (Field.toDivisionRing.{u1} K _inst_1)))) (Polynomial.zero.{u1} K (Ring.toSemiring.{u1} K (DivisionRing.toRing.{u1} K (Field.toDivisionRing.{u1} K _inst_1)))))))) (forall {g : Polynomial.{u2} L (Ring.toSemiring.{u2} L (DivisionRing.toRing.{u2} L (Field.toDivisionRing.{u2} L _inst_2)))}, (Irreducible.{u2} (Polynomial.{u2} L (Ring.toSemiring.{u2} L (DivisionRing.toRing.{u2} L (Field.toDivisionRing.{u2} L _inst_2)))) (Ring.toMonoid.{u2} (Polynomial.{u2} L (Ring.toSemiring.{u2} L (DivisionRing.toRing.{u2} L (Field.toDivisionRing.{u2} L _inst_2)))) (Polynomial.ring.{u2} L (DivisionRing.toRing.{u2} L (Field.toDivisionRing.{u2} L _inst_2)))) g) -> (Dvd.Dvd.{u2} (Polynomial.{u2} L (Ring.toSemiring.{u2} L (DivisionRing.toRing.{u2} L (Field.toDivisionRing.{u2} L _inst_2)))) (semigroupDvd.{u2} (Polynomial.{u2} L (Ring.toSemiring.{u2} L (DivisionRing.toRing.{u2} L (Field.toDivisionRing.{u2} L _inst_2)))) (SemigroupWithZero.toSemigroup.{u2} (Polynomial.{u2} L (Ring.toSemiring.{u2} L (DivisionRing.toRing.{u2} L (Field.toDivisionRing.{u2} L _inst_2)))) (NonUnitalSemiring.toSemigroupWithZero.{u2} (Polynomial.{u2} L (Ring.toSemiring.{u2} L (DivisionRing.toRing.{u2} L (Field.toDivisionRing.{u2} L _inst_2)))) (NonUnitalRing.toNonUnitalSemiring.{u2} (Polynomial.{u2} L (Ring.toSemiring.{u2} L (DivisionRing.toRing.{u2} L (Field.toDivisionRing.{u2} L _inst_2)))) (NonUnitalCommRing.toNonUnitalRing.{u2} (Polynomial.{u2} L (Ring.toSemiring.{u2} L (DivisionRing.toRing.{u2} L (Field.toDivisionRing.{u2} L _inst_2)))) (CommRing.toNonUnitalCommRing.{u2} (Polynomial.{u2} L (Ring.toSemiring.{u2} L (DivisionRing.toRing.{u2} L (Field.toDivisionRing.{u2} L _inst_2)))) (Polynomial.commRing.{u2} L (EuclideanDomain.toCommRing.{u2} L (Field.toEuclideanDomain.{u2} L _inst_2))))))))) g (Polynomial.map.{u1, u2} K L (Ring.toSemiring.{u1} K (DivisionRing.toRing.{u1} K (Field.toDivisionRing.{u1} K _inst_1))) (Ring.toSemiring.{u2} L (DivisionRing.toRing.{u2} L (Field.toDivisionRing.{u2} L _inst_2))) i f)) -> (Eq.{1} (WithBot.{0} Nat) (Polynomial.degree.{u2} L (Ring.toSemiring.{u2} L (DivisionRing.toRing.{u2} L (Field.toDivisionRing.{u2} L _inst_2))) g) (OfNat.ofNat.{0} (WithBot.{0} Nat) 1 (OfNat.mk.{0} (WithBot.{0} Nat) 1 (One.one.{0} (WithBot.{0} Nat) (WithBot.hasOne.{0} Nat Nat.hasOne)))))))
but is expected to have type
forall {K : Type.{u1}} {L : Type.{u2}} [_inst_1 : Field.{u1} K] [_inst_2 : Field.{u2} L] (i : RingHom.{u1, u2} K L (NonAssocRing.toNonAssocSemiring.{u1} K (Ring.toNonAssocRing.{u1} K (DivisionRing.toRing.{u1} K (Field.toDivisionRing.{u1} K _inst_1)))) (NonAssocRing.toNonAssocSemiring.{u2} L (Ring.toNonAssocRing.{u2} L (DivisionRing.toRing.{u2} L (Field.toDivisionRing.{u2} L _inst_2))))) (f : Polynomial.{u1} K (DivisionSemiring.toSemiring.{u1} K (Semifield.toDivisionSemiring.{u1} K (Field.toSemifield.{u1} K _inst_1)))), Iff (Polynomial.Splits.{u1, u2} K L (EuclideanDomain.toCommRing.{u1} K (Field.toEuclideanDomain.{u1} K _inst_1)) _inst_2 i f) (Or (Eq.{succ u1} (Polynomial.{u1} K (DivisionSemiring.toSemiring.{u1} K (Semifield.toDivisionSemiring.{u1} K (Field.toSemifield.{u1} K _inst_1)))) f (OfNat.ofNat.{u1} (Polynomial.{u1} K (DivisionSemiring.toSemiring.{u1} K (Semifield.toDivisionSemiring.{u1} K (Field.toSemifield.{u1} K _inst_1)))) 0 (Zero.toOfNat0.{u1} (Polynomial.{u1} K (DivisionSemiring.toSemiring.{u1} K (Semifield.toDivisionSemiring.{u1} K (Field.toSemifield.{u1} K _inst_1)))) (Polynomial.zero.{u1} K (DivisionSemiring.toSemiring.{u1} K (Semifield.toDivisionSemiring.{u1} K (Field.toSemifield.{u1} K _inst_1))))))) (forall {g : Polynomial.{u2} L (DivisionSemiring.toSemiring.{u2} L (Semifield.toDivisionSemiring.{u2} L (Field.toSemifield.{u2} L _inst_2)))}, (Irreducible.{u2} (Polynomial.{u2} L (DivisionSemiring.toSemiring.{u2} L (Semifield.toDivisionSemiring.{u2} L (Field.toSemifield.{u2} L _inst_2)))) (MonoidWithZero.toMonoid.{u2} (Polynomial.{u2} L (DivisionSemiring.toSemiring.{u2} L (Semifield.toDivisionSemiring.{u2} L (Field.toSemifield.{u2} L _inst_2)))) (Semiring.toMonoidWithZero.{u2} (Polynomial.{u2} L (DivisionSemiring.toSemiring.{u2} L (Semifield.toDivisionSemiring.{u2} L (Field.toSemifield.{u2} L _inst_2)))) (Polynomial.semiring.{u2} L (DivisionSemiring.toSemiring.{u2} L (Semifield.toDivisionSemiring.{u2} L (Field.toSemifield.{u2} L _inst_2)))))) g) -> (Dvd.dvd.{u2} (Polynomial.{u2} L (DivisionSemiring.toSemiring.{u2} L (Semifield.toDivisionSemiring.{u2} L (Field.toSemifield.{u2} L _inst_2)))) (semigroupDvd.{u2} (Polynomial.{u2} L (DivisionSemiring.toSemiring.{u2} L (Semifield.toDivisionSemiring.{u2} L (Field.toSemifield.{u2} L _inst_2)))) (SemigroupWithZero.toSemigroup.{u2} (Polynomial.{u2} L (DivisionSemiring.toSemiring.{u2} L (Semifield.toDivisionSemiring.{u2} L (Field.toSemifield.{u2} L _inst_2)))) (NonUnitalSemiring.toSemigroupWithZero.{u2} (Polynomial.{u2} L (DivisionSemiring.toSemiring.{u2} L (Semifield.toDivisionSemiring.{u2} L (Field.toSemifield.{u2} L _inst_2)))) (NonUnitalRing.toNonUnitalSemiring.{u2} (Polynomial.{u2} L (DivisionSemiring.toSemiring.{u2} L (Semifield.toDivisionSemiring.{u2} L (Field.toSemifield.{u2} L _inst_2)))) (NonUnitalCommRing.toNonUnitalRing.{u2} (Polynomial.{u2} L (DivisionSemiring.toSemiring.{u2} L (Semifield.toDivisionSemiring.{u2} L (Field.toSemifield.{u2} L _inst_2)))) (CommRing.toNonUnitalCommRing.{u2} (Polynomial.{u2} L (DivisionSemiring.toSemiring.{u2} L (Semifield.toDivisionSemiring.{u2} L (Field.toSemifield.{u2} L _inst_2)))) (EuclideanDomain.toCommRing.{u2} (Polynomial.{u2} L (DivisionSemiring.toSemiring.{u2} L (Semifield.toDivisionSemiring.{u2} L (Field.toSemifield.{u2} L _inst_2)))) (Polynomial.instEuclideanDomainPolynomialToSemiringToDivisionSemiringToSemifield.{u2} L _inst_2)))))))) g (Polynomial.map.{u1, u2} K L (DivisionSemiring.toSemiring.{u1} K (Semifield.toDivisionSemiring.{u1} K (Field.toSemifield.{u1} K _inst_1))) (DivisionSemiring.toSemiring.{u2} L (Semifield.toDivisionSemiring.{u2} L (Field.toSemifield.{u2} L _inst_2))) i f)) -> (Eq.{1} (WithBot.{0} Nat) (Polynomial.degree.{u2} L (DivisionSemiring.toSemiring.{u2} L (Semifield.toDivisionSemiring.{u2} L (Field.toSemifield.{u2} L _inst_2))) g) (OfNat.ofNat.{0} (WithBot.{0} Nat) 1 (One.toOfNat1.{0} (WithBot.{0} Nat) (WithBot.one.{0} Nat (CanonicallyOrderedCommSemiring.toOne.{0} Nat Nat.canonicallyOrderedCommSemiring)))))))
Case conversion may be inaccurate. Consider using '#align polynomial.splits_iff Polynomial.splits_iffₓ'. -/
/-- This lemma is for polynomials over a field. -/
theorem splits_iff (f : K[X]) :
Splits i f ↔ f = 0 ∨ ∀ {g : L[X]}, Irreducible g → g ∣ f.map i → degree g = 1 := by
rw [splits, map_eq_zero]
#align polynomial.splits_iff Polynomial.splits_iff
/- warning: polynomial.splits.def -> Polynomial.Splits.def is a dubious translation:
lean 3 declaration is
forall {K : Type.{u1}} {L : Type.{u2}} [_inst_1 : Field.{u1} K] [_inst_2 : Field.{u2} L] {i : RingHom.{u1, u2} K L (NonAssocRing.toNonAssocSemiring.{u1} K (Ring.toNonAssocRing.{u1} K (DivisionRing.toRing.{u1} K (Field.toDivisionRing.{u1} K _inst_1)))) (NonAssocRing.toNonAssocSemiring.{u2} L (Ring.toNonAssocRing.{u2} L (DivisionRing.toRing.{u2} L (Field.toDivisionRing.{u2} L _inst_2))))} {f : Polynomial.{u1} K (Ring.toSemiring.{u1} K (DivisionRing.toRing.{u1} K (Field.toDivisionRing.{u1} K _inst_1)))}, (Polynomial.Splits.{u1, u2} K L (EuclideanDomain.toCommRing.{u1} K (Field.toEuclideanDomain.{u1} K _inst_1)) _inst_2 i f) -> (Or (Eq.{succ u1} (Polynomial.{u1} K (Ring.toSemiring.{u1} K (DivisionRing.toRing.{u1} K (Field.toDivisionRing.{u1} K _inst_1)))) f (OfNat.ofNat.{u1} (Polynomial.{u1} K (Ring.toSemiring.{u1} K (DivisionRing.toRing.{u1} K (Field.toDivisionRing.{u1} K _inst_1)))) 0 (OfNat.mk.{u1} (Polynomial.{u1} K (Ring.toSemiring.{u1} K (DivisionRing.toRing.{u1} K (Field.toDivisionRing.{u1} K _inst_1)))) 0 (Zero.zero.{u1} (Polynomial.{u1} K (Ring.toSemiring.{u1} K (DivisionRing.toRing.{u1} K (Field.toDivisionRing.{u1} K _inst_1)))) (Polynomial.zero.{u1} K (Ring.toSemiring.{u1} K (DivisionRing.toRing.{u1} K (Field.toDivisionRing.{u1} K _inst_1)))))))) (forall {g : Polynomial.{u2} L (Ring.toSemiring.{u2} L (DivisionRing.toRing.{u2} L (Field.toDivisionRing.{u2} L _inst_2)))}, (Irreducible.{u2} (Polynomial.{u2} L (Ring.toSemiring.{u2} L (DivisionRing.toRing.{u2} L (Field.toDivisionRing.{u2} L _inst_2)))) (Ring.toMonoid.{u2} (Polynomial.{u2} L (Ring.toSemiring.{u2} L (DivisionRing.toRing.{u2} L (Field.toDivisionRing.{u2} L _inst_2)))) (Polynomial.ring.{u2} L (DivisionRing.toRing.{u2} L (Field.toDivisionRing.{u2} L _inst_2)))) g) -> (Dvd.Dvd.{u2} (Polynomial.{u2} L (Ring.toSemiring.{u2} L (DivisionRing.toRing.{u2} L (Field.toDivisionRing.{u2} L _inst_2)))) (semigroupDvd.{u2} (Polynomial.{u2} L (Ring.toSemiring.{u2} L (DivisionRing.toRing.{u2} L (Field.toDivisionRing.{u2} L _inst_2)))) (SemigroupWithZero.toSemigroup.{u2} (Polynomial.{u2} L (Ring.toSemiring.{u2} L (DivisionRing.toRing.{u2} L (Field.toDivisionRing.{u2} L _inst_2)))) (NonUnitalSemiring.toSemigroupWithZero.{u2} (Polynomial.{u2} L (Ring.toSemiring.{u2} L (DivisionRing.toRing.{u2} L (Field.toDivisionRing.{u2} L _inst_2)))) (NonUnitalRing.toNonUnitalSemiring.{u2} (Polynomial.{u2} L (Ring.toSemiring.{u2} L (DivisionRing.toRing.{u2} L (Field.toDivisionRing.{u2} L _inst_2)))) (NonUnitalCommRing.toNonUnitalRing.{u2} (Polynomial.{u2} L (Ring.toSemiring.{u2} L (DivisionRing.toRing.{u2} L (Field.toDivisionRing.{u2} L _inst_2)))) (CommRing.toNonUnitalCommRing.{u2} (Polynomial.{u2} L (Ring.toSemiring.{u2} L (DivisionRing.toRing.{u2} L (Field.toDivisionRing.{u2} L _inst_2)))) (Polynomial.commRing.{u2} L (EuclideanDomain.toCommRing.{u2} L (Field.toEuclideanDomain.{u2} L _inst_2))))))))) g (Polynomial.map.{u1, u2} K L (Ring.toSemiring.{u1} K (DivisionRing.toRing.{u1} K (Field.toDivisionRing.{u1} K _inst_1))) (Ring.toSemiring.{u2} L (DivisionRing.toRing.{u2} L (Field.toDivisionRing.{u2} L _inst_2))) i f)) -> (Eq.{1} (WithBot.{0} Nat) (Polynomial.degree.{u2} L (Ring.toSemiring.{u2} L (DivisionRing.toRing.{u2} L (Field.toDivisionRing.{u2} L _inst_2))) g) (OfNat.ofNat.{0} (WithBot.{0} Nat) 1 (OfNat.mk.{0} (WithBot.{0} Nat) 1 (One.one.{0} (WithBot.{0} Nat) (WithBot.hasOne.{0} Nat Nat.hasOne)))))))
but is expected to have type
forall {K : Type.{u1}} {L : Type.{u2}} [_inst_1 : Field.{u1} K] [_inst_2 : Field.{u2} L] {i : RingHom.{u1, u2} K L (NonAssocRing.toNonAssocSemiring.{u1} K (Ring.toNonAssocRing.{u1} K (DivisionRing.toRing.{u1} K (Field.toDivisionRing.{u1} K _inst_1)))) (NonAssocRing.toNonAssocSemiring.{u2} L (Ring.toNonAssocRing.{u2} L (DivisionRing.toRing.{u2} L (Field.toDivisionRing.{u2} L _inst_2))))} {f : Polynomial.{u1} K (DivisionSemiring.toSemiring.{u1} K (Semifield.toDivisionSemiring.{u1} K (Field.toSemifield.{u1} K _inst_1)))}, (Polynomial.Splits.{u1, u2} K L (EuclideanDomain.toCommRing.{u1} K (Field.toEuclideanDomain.{u1} K _inst_1)) _inst_2 i f) -> (Or (Eq.{succ u1} (Polynomial.{u1} K (DivisionSemiring.toSemiring.{u1} K (Semifield.toDivisionSemiring.{u1} K (Field.toSemifield.{u1} K _inst_1)))) f (OfNat.ofNat.{u1} (Polynomial.{u1} K (DivisionSemiring.toSemiring.{u1} K (Semifield.toDivisionSemiring.{u1} K (Field.toSemifield.{u1} K _inst_1)))) 0 (Zero.toOfNat0.{u1} (Polynomial.{u1} K (DivisionSemiring.toSemiring.{u1} K (Semifield.toDivisionSemiring.{u1} K (Field.toSemifield.{u1} K _inst_1)))) (Polynomial.zero.{u1} K (DivisionSemiring.toSemiring.{u1} K (Semifield.toDivisionSemiring.{u1} K (Field.toSemifield.{u1} K _inst_1))))))) (forall {g : Polynomial.{u2} L (DivisionSemiring.toSemiring.{u2} L (Semifield.toDivisionSemiring.{u2} L (Field.toSemifield.{u2} L _inst_2)))}, (Irreducible.{u2} (Polynomial.{u2} L (DivisionSemiring.toSemiring.{u2} L (Semifield.toDivisionSemiring.{u2} L (Field.toSemifield.{u2} L _inst_2)))) (MonoidWithZero.toMonoid.{u2} (Polynomial.{u2} L (DivisionSemiring.toSemiring.{u2} L (Semifield.toDivisionSemiring.{u2} L (Field.toSemifield.{u2} L _inst_2)))) (Semiring.toMonoidWithZero.{u2} (Polynomial.{u2} L (DivisionSemiring.toSemiring.{u2} L (Semifield.toDivisionSemiring.{u2} L (Field.toSemifield.{u2} L _inst_2)))) (Polynomial.semiring.{u2} L (DivisionSemiring.toSemiring.{u2} L (Semifield.toDivisionSemiring.{u2} L (Field.toSemifield.{u2} L _inst_2)))))) g) -> (Dvd.dvd.{u2} (Polynomial.{u2} L (DivisionSemiring.toSemiring.{u2} L (Semifield.toDivisionSemiring.{u2} L (Field.toSemifield.{u2} L _inst_2)))) (semigroupDvd.{u2} (Polynomial.{u2} L (DivisionSemiring.toSemiring.{u2} L (Semifield.toDivisionSemiring.{u2} L (Field.toSemifield.{u2} L _inst_2)))) (SemigroupWithZero.toSemigroup.{u2} (Polynomial.{u2} L (DivisionSemiring.toSemiring.{u2} L (Semifield.toDivisionSemiring.{u2} L (Field.toSemifield.{u2} L _inst_2)))) (NonUnitalSemiring.toSemigroupWithZero.{u2} (Polynomial.{u2} L (DivisionSemiring.toSemiring.{u2} L (Semifield.toDivisionSemiring.{u2} L (Field.toSemifield.{u2} L _inst_2)))) (NonUnitalRing.toNonUnitalSemiring.{u2} (Polynomial.{u2} L (DivisionSemiring.toSemiring.{u2} L (Semifield.toDivisionSemiring.{u2} L (Field.toSemifield.{u2} L _inst_2)))) (NonUnitalCommRing.toNonUnitalRing.{u2} (Polynomial.{u2} L (DivisionSemiring.toSemiring.{u2} L (Semifield.toDivisionSemiring.{u2} L (Field.toSemifield.{u2} L _inst_2)))) (CommRing.toNonUnitalCommRing.{u2} (Polynomial.{u2} L (DivisionSemiring.toSemiring.{u2} L (Semifield.toDivisionSemiring.{u2} L (Field.toSemifield.{u2} L _inst_2)))) (EuclideanDomain.toCommRing.{u2} (Polynomial.{u2} L (DivisionSemiring.toSemiring.{u2} L (Semifield.toDivisionSemiring.{u2} L (Field.toSemifield.{u2} L _inst_2)))) (Polynomial.instEuclideanDomainPolynomialToSemiringToDivisionSemiringToSemifield.{u2} L _inst_2)))))))) g (Polynomial.map.{u1, u2} K L (DivisionSemiring.toSemiring.{u1} K (Semifield.toDivisionSemiring.{u1} K (Field.toSemifield.{u1} K _inst_1))) (DivisionSemiring.toSemiring.{u2} L (Semifield.toDivisionSemiring.{u2} L (Field.toSemifield.{u2} L _inst_2))) i f)) -> (Eq.{1} (WithBot.{0} Nat) (Polynomial.degree.{u2} L (DivisionSemiring.toSemiring.{u2} L (Semifield.toDivisionSemiring.{u2} L (Field.toSemifield.{u2} L _inst_2))) g) (OfNat.ofNat.{0} (WithBot.{0} Nat) 1 (One.toOfNat1.{0} (WithBot.{0} Nat) (WithBot.one.{0} Nat (CanonicallyOrderedCommSemiring.toOne.{0} Nat Nat.canonicallyOrderedCommSemiring)))))))
Case conversion may be inaccurate. Consider using '#align polynomial.splits.def Polynomial.Splits.defₓ'. -/
/-- This lemma is for polynomials over a field. -/
theorem Splits.def {i : K →+* L} {f : K[X]} (h : Splits i f) :
f = 0 ∨ ∀ {g : L[X]}, Irreducible g → g ∣ f.map i → degree g = 1 :=
(splits_iff i f).mp h
#align polynomial.splits.def Polynomial.Splits.def
#print Polynomial.splits_of_splits_mul /-
theorem splits_of_splits_mul {f g : K[X]} (hfg : f * g ≠ 0) (h : Splits i (f * g)) :
Splits i f ∧ Splits i g :=
splits_of_splits_mul' i (map_ne_zero hfg) h
#align polynomial.splits_of_splits_mul Polynomial.splits_of_splits_mul
-/
#print Polynomial.splits_of_splits_of_dvd /-
theorem splits_of_splits_of_dvd {f g : K[X]} (hf0 : f ≠ 0) (hf : Splits i f) (hgf : g ∣ f) :
Splits i g := by
obtain ⟨f, rfl⟩ := hgf
exact (splits_of_splits_mul i hf0 hf).1
#align polynomial.splits_of_splits_of_dvd Polynomial.splits_of_splits_of_dvd
-/
/- warning: polynomial.splits_of_splits_gcd_left -> Polynomial.splits_of_splits_gcd_left is a dubious translation:
lean 3 declaration is
forall {K : Type.{u1}} {L : Type.{u2}} [_inst_1 : Field.{u1} K] [_inst_2 : Field.{u2} L] (i : RingHom.{u1, u2} K L (NonAssocRing.toNonAssocSemiring.{u1} K (Ring.toNonAssocRing.{u1} K (DivisionRing.toRing.{u1} K (Field.toDivisionRing.{u1} K _inst_1)))) (NonAssocRing.toNonAssocSemiring.{u2} L (Ring.toNonAssocRing.{u2} L (DivisionRing.toRing.{u2} L (Field.toDivisionRing.{u2} L _inst_2))))) {f : Polynomial.{u1} K (Ring.toSemiring.{u1} K (DivisionRing.toRing.{u1} K (Field.toDivisionRing.{u1} K _inst_1)))} {g : Polynomial.{u1} K (Ring.toSemiring.{u1} K (DivisionRing.toRing.{u1} K (Field.toDivisionRing.{u1} K _inst_1)))}, (Ne.{succ u1} (Polynomial.{u1} K (Ring.toSemiring.{u1} K (DivisionRing.toRing.{u1} K (Field.toDivisionRing.{u1} K _inst_1)))) f (OfNat.ofNat.{u1} (Polynomial.{u1} K (Ring.toSemiring.{u1} K (DivisionRing.toRing.{u1} K (Field.toDivisionRing.{u1} K _inst_1)))) 0 (OfNat.mk.{u1} (Polynomial.{u1} K (Ring.toSemiring.{u1} K (DivisionRing.toRing.{u1} K (Field.toDivisionRing.{u1} K _inst_1)))) 0 (Zero.zero.{u1} (Polynomial.{u1} K (Ring.toSemiring.{u1} K (DivisionRing.toRing.{u1} K (Field.toDivisionRing.{u1} K _inst_1)))) (Polynomial.zero.{u1} K (Ring.toSemiring.{u1} K (DivisionRing.toRing.{u1} K (Field.toDivisionRing.{u1} K _inst_1)))))))) -> (Polynomial.Splits.{u1, u2} K L (EuclideanDomain.toCommRing.{u1} K (Field.toEuclideanDomain.{u1} K _inst_1)) _inst_2 i f) -> (Polynomial.Splits.{u1, u2} K L (EuclideanDomain.toCommRing.{u1} K (Field.toEuclideanDomain.{u1} K _inst_1)) _inst_2 i (EuclideanDomain.gcd.{u1} (Polynomial.{u1} K (Ring.toSemiring.{u1} K (CommRing.toRing.{u1} K (EuclideanDomain.toCommRing.{u1} K (Field.toEuclideanDomain.{u1} K _inst_1))))) (Polynomial.euclideanDomain.{u1} K _inst_1) (fun (a : Polynomial.{u1} K (Ring.toSemiring.{u1} K (CommRing.toRing.{u1} K (EuclideanDomain.toCommRing.{u1} K (Field.toEuclideanDomain.{u1} K _inst_1))))) (b : Polynomial.{u1} K (Ring.toSemiring.{u1} K (CommRing.toRing.{u1} K (EuclideanDomain.toCommRing.{u1} K (Field.toEuclideanDomain.{u1} K _inst_1))))) => Classical.propDecidable (Eq.{succ u1} (Polynomial.{u1} K (Ring.toSemiring.{u1} K (CommRing.toRing.{u1} K (EuclideanDomain.toCommRing.{u1} K (Field.toEuclideanDomain.{u1} K _inst_1))))) a b)) f g))
but is expected to have type
forall {K : Type.{u1}} {L : Type.{u2}} [_inst_1 : Field.{u1} K] [_inst_2 : Field.{u2} L] (i : RingHom.{u1, u2} K L (NonAssocRing.toNonAssocSemiring.{u1} K (Ring.toNonAssocRing.{u1} K (DivisionRing.toRing.{u1} K (Field.toDivisionRing.{u1} K _inst_1)))) (NonAssocRing.toNonAssocSemiring.{u2} L (Ring.toNonAssocRing.{u2} L (DivisionRing.toRing.{u2} L (Field.toDivisionRing.{u2} L _inst_2))))) {f : Polynomial.{u1} K (DivisionSemiring.toSemiring.{u1} K (Semifield.toDivisionSemiring.{u1} K (Field.toSemifield.{u1} K _inst_1)))} {g : Polynomial.{u1} K (DivisionSemiring.toSemiring.{u1} K (Semifield.toDivisionSemiring.{u1} K (Field.toSemifield.{u1} K _inst_1)))}, (Ne.{succ u1} (Polynomial.{u1} K (DivisionSemiring.toSemiring.{u1} K (Semifield.toDivisionSemiring.{u1} K (Field.toSemifield.{u1} K _inst_1)))) f (OfNat.ofNat.{u1} (Polynomial.{u1} K (DivisionSemiring.toSemiring.{u1} K (Semifield.toDivisionSemiring.{u1} K (Field.toSemifield.{u1} K _inst_1)))) 0 (Zero.toOfNat0.{u1} (Polynomial.{u1} K (DivisionSemiring.toSemiring.{u1} K (Semifield.toDivisionSemiring.{u1} K (Field.toSemifield.{u1} K _inst_1)))) (Polynomial.zero.{u1} K (DivisionSemiring.toSemiring.{u1} K (Semifield.toDivisionSemiring.{u1} K (Field.toSemifield.{u1} K _inst_1))))))) -> (Polynomial.Splits.{u1, u2} K L (EuclideanDomain.toCommRing.{u1} K (Field.toEuclideanDomain.{u1} K _inst_1)) _inst_2 i f) -> (Polynomial.Splits.{u1, u2} K L (EuclideanDomain.toCommRing.{u1} K (Field.toEuclideanDomain.{u1} K _inst_1)) _inst_2 i (EuclideanDomain.gcd.{u1} (Polynomial.{u1} K (Ring.toSemiring.{u1} K (CommRing.toRing.{u1} K (EuclideanDomain.toCommRing.{u1} K (Field.toEuclideanDomain.{u1} K _inst_1))))) (Polynomial.instEuclideanDomainPolynomialToSemiringToDivisionSemiringToSemifield.{u1} K _inst_1) (fun (a : Polynomial.{u1} K (Ring.toSemiring.{u1} K (CommRing.toRing.{u1} K (EuclideanDomain.toCommRing.{u1} K (Field.toEuclideanDomain.{u1} K _inst_1))))) (b : Polynomial.{u1} K (Ring.toSemiring.{u1} K (CommRing.toRing.{u1} K (EuclideanDomain.toCommRing.{u1} K (Field.toEuclideanDomain.{u1} K _inst_1))))) => Classical.propDecidable (Eq.{succ u1} (Polynomial.{u1} K (Ring.toSemiring.{u1} K (CommRing.toRing.{u1} K (EuclideanDomain.toCommRing.{u1} K (Field.toEuclideanDomain.{u1} K _inst_1))))) a b)) f g))
Case conversion may be inaccurate. Consider using '#align polynomial.splits_of_splits_gcd_left Polynomial.splits_of_splits_gcd_leftₓ'. -/
theorem splits_of_splits_gcd_left {f g : K[X]} (hf0 : f ≠ 0) (hf : Splits i f) :
Splits i (EuclideanDomain.gcd f g) :=
Polynomial.splits_of_splits_of_dvd i hf0 hf (EuclideanDomain.gcd_dvd_left f g)
#align polynomial.splits_of_splits_gcd_left Polynomial.splits_of_splits_gcd_left
/- warning: polynomial.splits_of_splits_gcd_right -> Polynomial.splits_of_splits_gcd_right is a dubious translation:
lean 3 declaration is
forall {K : Type.{u1}} {L : Type.{u2}} [_inst_1 : Field.{u1} K] [_inst_2 : Field.{u2} L] (i : RingHom.{u1, u2} K L (NonAssocRing.toNonAssocSemiring.{u1} K (Ring.toNonAssocRing.{u1} K (DivisionRing.toRing.{u1} K (Field.toDivisionRing.{u1} K _inst_1)))) (NonAssocRing.toNonAssocSemiring.{u2} L (Ring.toNonAssocRing.{u2} L (DivisionRing.toRing.{u2} L (Field.toDivisionRing.{u2} L _inst_2))))) {f : Polynomial.{u1} K (Ring.toSemiring.{u1} K (DivisionRing.toRing.{u1} K (Field.toDivisionRing.{u1} K _inst_1)))} {g : Polynomial.{u1} K (Ring.toSemiring.{u1} K (DivisionRing.toRing.{u1} K (Field.toDivisionRing.{u1} K _inst_1)))}, (Ne.{succ u1} (Polynomial.{u1} K (Ring.toSemiring.{u1} K (DivisionRing.toRing.{u1} K (Field.toDivisionRing.{u1} K _inst_1)))) g (OfNat.ofNat.{u1} (Polynomial.{u1} K (Ring.toSemiring.{u1} K (DivisionRing.toRing.{u1} K (Field.toDivisionRing.{u1} K _inst_1)))) 0 (OfNat.mk.{u1} (Polynomial.{u1} K (Ring.toSemiring.{u1} K (DivisionRing.toRing.{u1} K (Field.toDivisionRing.{u1} K _inst_1)))) 0 (Zero.zero.{u1} (Polynomial.{u1} K (Ring.toSemiring.{u1} K (DivisionRing.toRing.{u1} K (Field.toDivisionRing.{u1} K _inst_1)))) (Polynomial.zero.{u1} K (Ring.toSemiring.{u1} K (DivisionRing.toRing.{u1} K (Field.toDivisionRing.{u1} K _inst_1)))))))) -> (Polynomial.Splits.{u1, u2} K L (EuclideanDomain.toCommRing.{u1} K (Field.toEuclideanDomain.{u1} K _inst_1)) _inst_2 i g) -> (Polynomial.Splits.{u1, u2} K L (EuclideanDomain.toCommRing.{u1} K (Field.toEuclideanDomain.{u1} K _inst_1)) _inst_2 i (EuclideanDomain.gcd.{u1} (Polynomial.{u1} K (Ring.toSemiring.{u1} K (CommRing.toRing.{u1} K (EuclideanDomain.toCommRing.{u1} K (Field.toEuclideanDomain.{u1} K _inst_1))))) (Polynomial.euclideanDomain.{u1} K _inst_1) (fun (a : Polynomial.{u1} K (Ring.toSemiring.{u1} K (CommRing.toRing.{u1} K (EuclideanDomain.toCommRing.{u1} K (Field.toEuclideanDomain.{u1} K _inst_1))))) (b : Polynomial.{u1} K (Ring.toSemiring.{u1} K (CommRing.toRing.{u1} K (EuclideanDomain.toCommRing.{u1} K (Field.toEuclideanDomain.{u1} K _inst_1))))) => Classical.propDecidable (Eq.{succ u1} (Polynomial.{u1} K (Ring.toSemiring.{u1} K (CommRing.toRing.{u1} K (EuclideanDomain.toCommRing.{u1} K (Field.toEuclideanDomain.{u1} K _inst_1))))) a b)) f g))
but is expected to have type
forall {K : Type.{u1}} {L : Type.{u2}} [_inst_1 : Field.{u1} K] [_inst_2 : Field.{u2} L] (i : RingHom.{u1, u2} K L (NonAssocRing.toNonAssocSemiring.{u1} K (Ring.toNonAssocRing.{u1} K (DivisionRing.toRing.{u1} K (Field.toDivisionRing.{u1} K _inst_1)))) (NonAssocRing.toNonAssocSemiring.{u2} L (Ring.toNonAssocRing.{u2} L (DivisionRing.toRing.{u2} L (Field.toDivisionRing.{u2} L _inst_2))))) {f : Polynomial.{u1} K (DivisionSemiring.toSemiring.{u1} K (Semifield.toDivisionSemiring.{u1} K (Field.toSemifield.{u1} K _inst_1)))} {g : Polynomial.{u1} K (DivisionSemiring.toSemiring.{u1} K (Semifield.toDivisionSemiring.{u1} K (Field.toSemifield.{u1} K _inst_1)))}, (Ne.{succ u1} (Polynomial.{u1} K (DivisionSemiring.toSemiring.{u1} K (Semifield.toDivisionSemiring.{u1} K (Field.toSemifield.{u1} K _inst_1)))) g (OfNat.ofNat.{u1} (Polynomial.{u1} K (DivisionSemiring.toSemiring.{u1} K (Semifield.toDivisionSemiring.{u1} K (Field.toSemifield.{u1} K _inst_1)))) 0 (Zero.toOfNat0.{u1} (Polynomial.{u1} K (DivisionSemiring.toSemiring.{u1} K (Semifield.toDivisionSemiring.{u1} K (Field.toSemifield.{u1} K _inst_1)))) (Polynomial.zero.{u1} K (DivisionSemiring.toSemiring.{u1} K (Semifield.toDivisionSemiring.{u1} K (Field.toSemifield.{u1} K _inst_1))))))) -> (Polynomial.Splits.{u1, u2} K L (EuclideanDomain.toCommRing.{u1} K (Field.toEuclideanDomain.{u1} K _inst_1)) _inst_2 i g) -> (Polynomial.Splits.{u1, u2} K L (EuclideanDomain.toCommRing.{u1} K (Field.toEuclideanDomain.{u1} K _inst_1)) _inst_2 i (EuclideanDomain.gcd.{u1} (Polynomial.{u1} K (Ring.toSemiring.{u1} K (CommRing.toRing.{u1} K (EuclideanDomain.toCommRing.{u1} K (Field.toEuclideanDomain.{u1} K _inst_1))))) (Polynomial.instEuclideanDomainPolynomialToSemiringToDivisionSemiringToSemifield.{u1} K _inst_1) (fun (a : Polynomial.{u1} K (Ring.toSemiring.{u1} K (CommRing.toRing.{u1} K (EuclideanDomain.toCommRing.{u1} K (Field.toEuclideanDomain.{u1} K _inst_1))))) (b : Polynomial.{u1} K (Ring.toSemiring.{u1} K (CommRing.toRing.{u1} K (EuclideanDomain.toCommRing.{u1} K (Field.toEuclideanDomain.{u1} K _inst_1))))) => Classical.propDecidable (Eq.{succ u1} (Polynomial.{u1} K (Ring.toSemiring.{u1} K (CommRing.toRing.{u1} K (EuclideanDomain.toCommRing.{u1} K (Field.toEuclideanDomain.{u1} K _inst_1))))) a b)) f g))
Case conversion may be inaccurate. Consider using '#align polynomial.splits_of_splits_gcd_right Polynomial.splits_of_splits_gcd_rightₓ'. -/
theorem splits_of_splits_gcd_right {f g : K[X]} (hg0 : g ≠ 0) (hg : Splits i g) :
Splits i (EuclideanDomain.gcd f g) :=
Polynomial.splits_of_splits_of_dvd i hg0 hg (EuclideanDomain.gcd_dvd_right f g)
#align polynomial.splits_of_splits_gcd_right Polynomial.splits_of_splits_gcd_right
#print Polynomial.splits_mul_iff /-
theorem splits_mul_iff {f g : K[X]} (hf : f ≠ 0) (hg : g ≠ 0) :
(f * g).Splits i ↔ f.Splits i ∧ g.Splits i :=
⟨splits_of_splits_mul i (mul_ne_zero hf hg), fun ⟨hfs, hgs⟩ => splits_mul i hfs hgs⟩
#align polynomial.splits_mul_iff Polynomial.splits_mul_iff
-/
#print Polynomial.splits_prod_iff /-
theorem splits_prod_iff {ι : Type u} {s : ι → K[X]} {t : Finset ι} :
(∀ j ∈ t, s j ≠ 0) → ((∏ x in t, s x).Splits i ↔ ∀ j ∈ t, (s j).Splits i) :=
by
refine'
Finset.induction_on t (fun _ => ⟨fun _ _ h => h.elim, fun _ => splits_one i⟩)
fun a t hat ih ht => _
rw [Finset.forall_mem_insert] at ht⊢
rw [Finset.prod_insert hat, splits_mul_iff i ht.1 (Finset.prod_ne_zero_iff.2 ht.2), ih ht.2]
#align polynomial.splits_prod_iff Polynomial.splits_prod_iff
-/
/- warning: polynomial.degree_eq_one_of_irreducible_of_splits -> Polynomial.degree_eq_one_of_irreducible_of_splits is a dubious translation:
lean 3 declaration is
forall {K : Type.{u1}} [_inst_1 : Field.{u1} K] {p : Polynomial.{u1} K (Ring.toSemiring.{u1} K (DivisionRing.toRing.{u1} K (Field.toDivisionRing.{u1} K _inst_1)))}, (Irreducible.{u1} (Polynomial.{u1} K (Ring.toSemiring.{u1} K (DivisionRing.toRing.{u1} K (Field.toDivisionRing.{u1} K _inst_1)))) (Ring.toMonoid.{u1} (Polynomial.{u1} K (Ring.toSemiring.{u1} K (DivisionRing.toRing.{u1} K (Field.toDivisionRing.{u1} K _inst_1)))) (Polynomial.ring.{u1} K (DivisionRing.toRing.{u1} K (Field.toDivisionRing.{u1} K _inst_1)))) p) -> (Polynomial.Splits.{u1, u1} K K (EuclideanDomain.toCommRing.{u1} K (Field.toEuclideanDomain.{u1} K _inst_1)) _inst_1 (RingHom.id.{u1} K (NonAssocRing.toNonAssocSemiring.{u1} K (Ring.toNonAssocRing.{u1} K (DivisionRing.toRing.{u1} K (Field.toDivisionRing.{u1} K _inst_1))))) p) -> (Eq.{1} (WithBot.{0} Nat) (Polynomial.degree.{u1} K (Ring.toSemiring.{u1} K (DivisionRing.toRing.{u1} K (Field.toDivisionRing.{u1} K _inst_1))) p) (OfNat.ofNat.{0} (WithBot.{0} Nat) 1 (OfNat.mk.{0} (WithBot.{0} Nat) 1 (One.one.{0} (WithBot.{0} Nat) (WithBot.hasOne.{0} Nat Nat.hasOne)))))
but is expected to have type
forall {K : Type.{u1}} [_inst_1 : Field.{u1} K] {p : Polynomial.{u1} K (DivisionSemiring.toSemiring.{u1} K (Semifield.toDivisionSemiring.{u1} K (Field.toSemifield.{u1} K _inst_1)))}, (Irreducible.{u1} (Polynomial.{u1} K (DivisionSemiring.toSemiring.{u1} K (Semifield.toDivisionSemiring.{u1} K (Field.toSemifield.{u1} K _inst_1)))) (MonoidWithZero.toMonoid.{u1} (Polynomial.{u1} K (DivisionSemiring.toSemiring.{u1} K (Semifield.toDivisionSemiring.{u1} K (Field.toSemifield.{u1} K _inst_1)))) (Semiring.toMonoidWithZero.{u1} (Polynomial.{u1} K (DivisionSemiring.toSemiring.{u1} K (Semifield.toDivisionSemiring.{u1} K (Field.toSemifield.{u1} K _inst_1)))) (Polynomial.semiring.{u1} K (DivisionSemiring.toSemiring.{u1} K (Semifield.toDivisionSemiring.{u1} K (Field.toSemifield.{u1} K _inst_1)))))) p) -> (Polynomial.Splits.{u1, u1} K K (EuclideanDomain.toCommRing.{u1} K (Field.toEuclideanDomain.{u1} K _inst_1)) _inst_1 (RingHom.id.{u1} K (NonAssocRing.toNonAssocSemiring.{u1} K (Ring.toNonAssocRing.{u1} K (DivisionRing.toRing.{u1} K (Field.toDivisionRing.{u1} K _inst_1))))) p) -> (Eq.{1} (WithBot.{0} Nat) (Polynomial.degree.{u1} K (DivisionSemiring.toSemiring.{u1} K (Semifield.toDivisionSemiring.{u1} K (Field.toSemifield.{u1} K _inst_1))) p) (OfNat.ofNat.{0} (WithBot.{0} Nat) 1 (One.toOfNat1.{0} (WithBot.{0} Nat) (WithBot.one.{0} Nat (CanonicallyOrderedCommSemiring.toOne.{0} Nat Nat.canonicallyOrderedCommSemiring)))))
Case conversion may be inaccurate. Consider using '#align polynomial.degree_eq_one_of_irreducible_of_splits Polynomial.degree_eq_one_of_irreducible_of_splitsₓ'. -/
theorem degree_eq_one_of_irreducible_of_splits {p : K[X]} (hp : Irreducible p)
(hp_splits : Splits (RingHom.id K) p) : p.degree = 1 :=
by
rcases hp_splits with ⟨⟩
· exfalso
simp_all
· apply hp_splits hp
simp
#align polynomial.degree_eq_one_of_irreducible_of_splits Polynomial.degree_eq_one_of_irreducible_of_splits
/- warning: polynomial.exists_root_of_splits -> Polynomial.exists_root_of_splits is a dubious translation:
lean 3 declaration is
forall {K : Type.{u1}} {L : Type.{u2}} [_inst_1 : Field.{u1} K] [_inst_2 : Field.{u2} L] (i : RingHom.{u1, u2} K L (NonAssocRing.toNonAssocSemiring.{u1} K (Ring.toNonAssocRing.{u1} K (DivisionRing.toRing.{u1} K (Field.toDivisionRing.{u1} K _inst_1)))) (NonAssocRing.toNonAssocSemiring.{u2} L (Ring.toNonAssocRing.{u2} L (DivisionRing.toRing.{u2} L (Field.toDivisionRing.{u2} L _inst_2))))) {f : Polynomial.{u1} K (Ring.toSemiring.{u1} K (DivisionRing.toRing.{u1} K (Field.toDivisionRing.{u1} K _inst_1)))}, (Polynomial.Splits.{u1, u2} K L (EuclideanDomain.toCommRing.{u1} K (Field.toEuclideanDomain.{u1} K _inst_1)) _inst_2 i f) -> (Ne.{1} (WithBot.{0} Nat) (Polynomial.degree.{u1} K (Ring.toSemiring.{u1} K (DivisionRing.toRing.{u1} K (Field.toDivisionRing.{u1} K _inst_1))) f) (OfNat.ofNat.{0} (WithBot.{0} Nat) 0 (OfNat.mk.{0} (WithBot.{0} Nat) 0 (Zero.zero.{0} (WithBot.{0} Nat) (WithBot.hasZero.{0} Nat Nat.hasZero))))) -> (Exists.{succ u2} L (fun (x : L) => Eq.{succ u2} L (Polynomial.eval₂.{u1, u2} K L (Ring.toSemiring.{u1} K (DivisionRing.toRing.{u1} K (Field.toDivisionRing.{u1} K _inst_1))) (Ring.toSemiring.{u2} L (DivisionRing.toRing.{u2} L (Field.toDivisionRing.{u2} L _inst_2))) i x f) (OfNat.ofNat.{u2} L 0 (OfNat.mk.{u2} L 0 (Zero.zero.{u2} L (MulZeroClass.toHasZero.{u2} L (NonUnitalNonAssocSemiring.toMulZeroClass.{u2} L (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u2} L (NonAssocRing.toNonUnitalNonAssocRing.{u2} L (Ring.toNonAssocRing.{u2} L (DivisionRing.toRing.{u2} L (Field.toDivisionRing.{u2} L _inst_2))))))))))))
but is expected to have type
forall {K : Type.{u1}} {L : Type.{u2}} [_inst_1 : Field.{u1} K] [_inst_2 : Field.{u2} L] (i : RingHom.{u1, u2} K L (NonAssocRing.toNonAssocSemiring.{u1} K (Ring.toNonAssocRing.{u1} K (DivisionRing.toRing.{u1} K (Field.toDivisionRing.{u1} K _inst_1)))) (NonAssocRing.toNonAssocSemiring.{u2} L (Ring.toNonAssocRing.{u2} L (DivisionRing.toRing.{u2} L (Field.toDivisionRing.{u2} L _inst_2))))) {f : Polynomial.{u1} K (DivisionSemiring.toSemiring.{u1} K (Semifield.toDivisionSemiring.{u1} K (Field.toSemifield.{u1} K _inst_1)))}, (Polynomial.Splits.{u1, u2} K L (EuclideanDomain.toCommRing.{u1} K (Field.toEuclideanDomain.{u1} K _inst_1)) _inst_2 i f) -> (Ne.{1} (WithBot.{0} Nat) (Polynomial.degree.{u1} K (DivisionSemiring.toSemiring.{u1} K (Semifield.toDivisionSemiring.{u1} K (Field.toSemifield.{u1} K _inst_1))) f) (OfNat.ofNat.{0} (WithBot.{0} Nat) 0 (Zero.toOfNat0.{0} (WithBot.{0} Nat) (WithBot.zero.{0} Nat (LinearOrderedCommMonoidWithZero.toZero.{0} Nat Nat.linearOrderedCommMonoidWithZero))))) -> (Exists.{succ u2} L (fun (x : L) => Eq.{succ u2} L (Polynomial.eval₂.{u1, u2} K L (DivisionSemiring.toSemiring.{u1} K (Semifield.toDivisionSemiring.{u1} K (Field.toSemifield.{u1} K _inst_1))) (DivisionSemiring.toSemiring.{u2} L (Semifield.toDivisionSemiring.{u2} L (Field.toSemifield.{u2} L _inst_2))) i x f) (OfNat.ofNat.{u2} L 0 (Zero.toOfNat0.{u2} L (CommMonoidWithZero.toZero.{u2} L (CommGroupWithZero.toCommMonoidWithZero.{u2} L (Semifield.toCommGroupWithZero.{u2} L (Field.toSemifield.{u2} L _inst_2))))))))
Case conversion may be inaccurate. Consider using '#align polynomial.exists_root_of_splits Polynomial.exists_root_of_splitsₓ'. -/
theorem exists_root_of_splits {f : K[X]} (hs : Splits i f) (hf0 : degree f ≠ 0) :
∃ x, eval₂ i x f = 0 :=
exists_root_of_splits' i hs ((f.degree_map i).symm ▸ hf0)
#align polynomial.exists_root_of_splits Polynomial.exists_root_of_splits
#print Polynomial.roots_ne_zero_of_splits /-
theorem roots_ne_zero_of_splits {f : K[X]} (hs : Splits i f) (hf0 : natDegree f ≠ 0) :
(f.map i).roots ≠ 0 :=
roots_ne_zero_of_splits' i hs (ne_of_eq_of_ne (natDegree_map i) hf0)
#align polynomial.roots_ne_zero_of_splits Polynomial.roots_ne_zero_of_splits
-/
#print Polynomial.rootOfSplits /-
/-- Pick a root of a polynomial that splits. This version is for polynomials over a field and has
simpler assumptions. -/
def rootOfSplits {f : K[X]} (hf : f.Splits i) (hfd : f.degree ≠ 0) : L :=
rootOfSplits' i hf ((f.degree_map i).symm ▸ hfd)
#align polynomial.root_of_splits Polynomial.rootOfSplits
-/
#print Polynomial.rootOfSplits'_eq_rootOfSplits /-
/-- `root_of_splits'` is definitionally equal to `root_of_splits`. -/
theorem rootOfSplits'_eq_rootOfSplits {f : K[X]} (hf : f.Splits i) (hfd) :
rootOfSplits' i hf hfd = rootOfSplits i hf (f.degree_map i ▸ hfd) :=
rfl
#align polynomial.root_of_splits'_eq_root_of_splits Polynomial.rootOfSplits'_eq_rootOfSplits
-/
/- warning: polynomial.map_root_of_splits -> Polynomial.map_rootOfSplits is a dubious translation:
lean 3 declaration is
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Case conversion may be inaccurate. Consider using '#align polynomial.map_root_of_splits Polynomial.map_rootOfSplitsₓ'. -/
theorem map_rootOfSplits {f : K[X]} (hf : f.Splits i) (hfd) :
f.eval₂ i (rootOfSplits i hf hfd) = 0 :=
map_rootOfSplits' i hf (ne_of_eq_of_ne (degree_map f i) hfd)
#align polynomial.map_root_of_splits Polynomial.map_rootOfSplits
/- warning: polynomial.nat_degree_eq_card_roots -> Polynomial.natDegree_eq_card_roots is a dubious translation:
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Case conversion may be inaccurate. Consider using '#align polynomial.nat_degree_eq_card_roots Polynomial.natDegree_eq_card_rootsₓ'. -/
theorem natDegree_eq_card_roots {p : K[X]} {i : K →+* L} (hsplit : Splits i p) :
p.natDegree = (p.map i).roots.card :=
(natDegree_map i).symm.trans <| natDegree_eq_card_roots' hsplit
#align polynomial.nat_degree_eq_card_roots Polynomial.natDegree_eq_card_roots
/- warning: polynomial.degree_eq_card_roots -> Polynomial.degree_eq_card_roots is a dubious translation:
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(AddCancelCommMonoid.toAddCancelMonoid.{u2} (Multiset.{u2} L) (OrderedCancelAddCommMonoid.toCancelAddCommMonoid.{u2} (Multiset.{u2} L) (Multiset.instOrderedCancelAddCommMonoidMultiset.{u2} L)))))) (AddMonoid.toAddZeroClass.{0} Nat Nat.addMonoid)) (Multiset.{u2} L) (fun (_x : Multiset.{u2} L) => (fun ([email protected]._hyg.403 : Multiset.{u2} L) => Nat) _x) (AddHomClass.toFunLike.{u2, u2, 0} (AddMonoidHom.{u2, 0} (Multiset.{u2} L) Nat (AddMonoid.toAddZeroClass.{u2} (Multiset.{u2} L) (AddRightCancelMonoid.toAddMonoid.{u2} (Multiset.{u2} L) (AddCancelMonoid.toAddRightCancelMonoid.{u2} (Multiset.{u2} L) (AddCancelCommMonoid.toAddCancelMonoid.{u2} (Multiset.{u2} L) (OrderedCancelAddCommMonoid.toCancelAddCommMonoid.{u2} (Multiset.{u2} L) (Multiset.instOrderedCancelAddCommMonoidMultiset.{u2} L)))))) (AddMonoid.toAddZeroClass.{0} Nat Nat.addMonoid)) (Multiset.{u2} L) Nat (AddZeroClass.toAdd.{u2} (Multiset.{u2} L) (AddMonoid.toAddZeroClass.{u2} (Multiset.{u2} L) (AddRightCancelMonoid.toAddMonoid.{u2} (Multiset.{u2} L) (AddCancelMonoid.toAddRightCancelMonoid.{u2} (Multiset.{u2} L) (AddCancelCommMonoid.toAddCancelMonoid.{u2} (Multiset.{u2} L) (OrderedCancelAddCommMonoid.toCancelAddCommMonoid.{u2} (Multiset.{u2} L) (Multiset.instOrderedCancelAddCommMonoidMultiset.{u2} L))))))) (AddZeroClass.toAdd.{0} Nat (AddMonoid.toAddZeroClass.{0} Nat Nat.addMonoid)) (AddMonoidHomClass.toAddHomClass.{u2, u2, 0} (AddMonoidHom.{u2, 0} (Multiset.{u2} L) Nat (AddMonoid.toAddZeroClass.{u2} (Multiset.{u2} L) (AddRightCancelMonoid.toAddMonoid.{u2} (Multiset.{u2} L) (AddCancelMonoid.toAddRightCancelMonoid.{u2} (Multiset.{u2} L) (AddCancelCommMonoid.toAddCancelMonoid.{u2} (Multiset.{u2} L) (OrderedCancelAddCommMonoid.toCancelAddCommMonoid.{u2} (Multiset.{u2} L) (Multiset.instOrderedCancelAddCommMonoidMultiset.{u2} L)))))) (AddMonoid.toAddZeroClass.{0} Nat Nat.addMonoid)) (Multiset.{u2} L) Nat (AddMonoid.toAddZeroClass.{u2} (Multiset.{u2} L) (AddRightCancelMonoid.toAddMonoid.{u2} (Multiset.{u2} L) (AddCancelMonoid.toAddRightCancelMonoid.{u2} (Multiset.{u2} L) (AddCancelCommMonoid.toAddCancelMonoid.{u2} (Multiset.{u2} L) (OrderedCancelAddCommMonoid.toCancelAddCommMonoid.{u2} (Multiset.{u2} L) (Multiset.instOrderedCancelAddCommMonoidMultiset.{u2} L)))))) (AddMonoid.toAddZeroClass.{0} Nat Nat.addMonoid) (AddMonoidHom.addMonoidHomClass.{u2, 0} (Multiset.{u2} L) Nat (AddMonoid.toAddZeroClass.{u2} (Multiset.{u2} L) (AddRightCancelMonoid.toAddMonoid.{u2} (Multiset.{u2} L) (AddCancelMonoid.toAddRightCancelMonoid.{u2} (Multiset.{u2} L) (AddCancelCommMonoid.toAddCancelMonoid.{u2} (Multiset.{u2} L) (OrderedCancelAddCommMonoid.toCancelAddCommMonoid.{u2} (Multiset.{u2} L) (Multiset.instOrderedCancelAddCommMonoidMultiset.{u2} L)))))) (AddMonoid.toAddZeroClass.{0} Nat Nat.addMonoid)))) (Multiset.card.{u2} L) (Polynomial.roots.{u2} L (EuclideanDomain.toCommRing.{u2} L (Field.toEuclideanDomain.{u2} L _inst_2)) (Field.isDomain.{u2} L _inst_2) (Polynomial.map.{u1, u2} K L (DivisionSemiring.toSemiring.{u1} K (Semifield.toDivisionSemiring.{u1} K (Field.toSemifield.{u1} K _inst_1))) (DivisionSemiring.toSemiring.{u2} L (Semifield.toDivisionSemiring.{u2} L (Field.toSemifield.{u2} L _inst_2))) i p)))))
Case conversion may be inaccurate. Consider using '#align polynomial.degree_eq_card_roots Polynomial.degree_eq_card_rootsₓ'. -/
theorem degree_eq_card_roots {p : K[X]} {i : K →+* L} (p_ne_zero : p ≠ 0) (hsplit : Splits i p) :
p.degree = (p.map i).roots.card := by
rw [degree_eq_nat_degree p_ne_zero, nat_degree_eq_card_roots hsplit]
#align polynomial.degree_eq_card_roots Polynomial.degree_eq_card_roots
/- warning: polynomial.roots_map -> Polynomial.roots_map is a dubious translation:
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but is expected to have type
forall {K : Type.{u1}} {L : Type.{u2}} [_inst_1 : Field.{u1} K] [_inst_2 : Field.{u2} L] (i : RingHom.{u1, u2} K L (NonAssocRing.toNonAssocSemiring.{u1} K (Ring.toNonAssocRing.{u1} K (DivisionRing.toRing.{u1} K (Field.toDivisionRing.{u1} K _inst_1)))) (NonAssocRing.toNonAssocSemiring.{u2} L (Ring.toNonAssocRing.{u2} L (DivisionRing.toRing.{u2} L (Field.toDivisionRing.{u2} L _inst_2))))) {f : Polynomial.{u1} K (DivisionSemiring.toSemiring.{u1} K (Semifield.toDivisionSemiring.{u1} K (Field.toSemifield.{u1} K _inst_1)))}, (Polynomial.Splits.{u1, u1} K K (EuclideanDomain.toCommRing.{u1} K (Field.toEuclideanDomain.{u1} K _inst_1)) _inst_1 (RingHom.id.{u1} K (NonAssocRing.toNonAssocSemiring.{u1} K (Ring.toNonAssocRing.{u1} K (DivisionRing.toRing.{u1} K (Field.toDivisionRing.{u1} K _inst_1))))) f) -> (Eq.{succ u2} (Multiset.{u2} L) (Polynomial.roots.{u2} L (EuclideanDomain.toCommRing.{u2} L (Field.toEuclideanDomain.{u2} L _inst_2)) (EuclideanDomain.instIsDomainToSemiringToRingToCommRing.{u2} L (Field.toEuclideanDomain.{u2} L _inst_2)) (Polynomial.map.{u1, u2} K L (DivisionSemiring.toSemiring.{u1} K (Semifield.toDivisionSemiring.{u1} K (Field.toSemifield.{u1} K _inst_1))) (DivisionSemiring.toSemiring.{u2} L (Semifield.toDivisionSemiring.{u2} L (Field.toSemifield.{u2} L _inst_2))) i f)) (Multiset.map.{u1, u2} K L (FunLike.coe.{max (succ u1) (succ u2), succ u1, succ u2} (RingHom.{u1, u2} K L (NonAssocRing.toNonAssocSemiring.{u1} K (Ring.toNonAssocRing.{u1} K (DivisionRing.toRing.{u1} K (Field.toDivisionRing.{u1} K _inst_1)))) (NonAssocRing.toNonAssocSemiring.{u2} L (Ring.toNonAssocRing.{u2} L (DivisionRing.toRing.{u2} L (Field.toDivisionRing.{u2} L _inst_2))))) K (fun (_x : K) => (fun ([email protected]._hyg.2391 : K) => L) _x) (MulHomClass.toFunLike.{max u1 u2, u1, u2} (RingHom.{u1, u2} K L (NonAssocRing.toNonAssocSemiring.{u1} K (Ring.toNonAssocRing.{u1} K (DivisionRing.toRing.{u1} K (Field.toDivisionRing.{u1} K _inst_1)))) (NonAssocRing.toNonAssocSemiring.{u2} L (Ring.toNonAssocRing.{u2} L (DivisionRing.toRing.{u2} L (Field.toDivisionRing.{u2} L _inst_2))))) K L (NonUnitalNonAssocSemiring.toMul.{u1} K (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} K (NonAssocRing.toNonAssocSemiring.{u1} K (Ring.toNonAssocRing.{u1} K (DivisionRing.toRing.{u1} K (Field.toDivisionRing.{u1} K _inst_1)))))) (NonUnitalNonAssocSemiring.toMul.{u2} L (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} L (NonAssocRing.toNonAssocSemiring.{u2} L (Ring.toNonAssocRing.{u2} L (DivisionRing.toRing.{u2} L (Field.toDivisionRing.{u2} L _inst_2)))))) (NonUnitalRingHomClass.toMulHomClass.{max u1 u2, u1, u2} (RingHom.{u1, u2} K L (NonAssocRing.toNonAssocSemiring.{u1} K (Ring.toNonAssocRing.{u1} K (DivisionRing.toRing.{u1} K (Field.toDivisionRing.{u1} K _inst_1)))) (NonAssocRing.toNonAssocSemiring.{u2} L (Ring.toNonAssocRing.{u2} L (DivisionRing.toRing.{u2} L (Field.toDivisionRing.{u2} L _inst_2))))) K L (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} K (NonAssocRing.toNonAssocSemiring.{u1} K (Ring.toNonAssocRing.{u1} K (DivisionRing.toRing.{u1} K (Field.toDivisionRing.{u1} K _inst_1))))) (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} L (NonAssocRing.toNonAssocSemiring.{u2} L (Ring.toNonAssocRing.{u2} L (DivisionRing.toRing.{u2} L (Field.toDivisionRing.{u2} L _inst_2))))) (RingHomClass.toNonUnitalRingHomClass.{max u1 u2, u1, u2} (RingHom.{u1, u2} K L (NonAssocRing.toNonAssocSemiring.{u1} K (Ring.toNonAssocRing.{u1} K (DivisionRing.toRing.{u1} K (Field.toDivisionRing.{u1} K _inst_1)))) (NonAssocRing.toNonAssocSemiring.{u2} L (Ring.toNonAssocRing.{u2} L (DivisionRing.toRing.{u2} L (Field.toDivisionRing.{u2} L _inst_2))))) K L (NonAssocRing.toNonAssocSemiring.{u1} K (Ring.toNonAssocRing.{u1} K (DivisionRing.toRing.{u1} K (Field.toDivisionRing.{u1} K _inst_1)))) (NonAssocRing.toNonAssocSemiring.{u2} L (Ring.toNonAssocRing.{u2} L (DivisionRing.toRing.{u2} L (Field.toDivisionRing.{u2} L _inst_2)))) (RingHom.instRingHomClassRingHom.{u1, u2} K L (NonAssocRing.toNonAssocSemiring.{u1} K (Ring.toNonAssocRing.{u1} K (DivisionRing.toRing.{u1} K (Field.toDivisionRing.{u1} K _inst_1)))) (NonAssocRing.toNonAssocSemiring.{u2} L (Ring.toNonAssocRing.{u2} L (DivisionRing.toRing.{u2} L (Field.toDivisionRing.{u2} L _inst_2)))))))) i) (Polynomial.roots.{u1} K (EuclideanDomain.toCommRing.{u1} K (Field.toEuclideanDomain.{u1} K _inst_1)) (EuclideanDomain.instIsDomainToSemiringToRingToCommRing.{u1} K (Field.toEuclideanDomain.{u1} K _inst_1)) f)))
Case conversion may be inaccurate. Consider using '#align polynomial.roots_map Polynomial.roots_mapₓ'. -/
theorem roots_map {f : K[X]} (hf : f.Splits <| RingHom.id K) : (f.map i).roots = f.roots.map i :=
(roots_map_of_injective_of_card_eq_natDegree i.Injective <|
by
convert(nat_degree_eq_card_roots hf).symm
rw [map_id]).symm
#align polynomial.roots_map Polynomial.roots_map
#print Polynomial.image_rootSet /-
theorem image_rootSet [Algebra F K] [Algebra F L] {p : F[X]} (h : p.Splits (algebraMap F K))
(f : K →ₐ[F] L) : f '' p.rootSet K = p.rootSet L := by
classical rw [root_set, ← Finset.coe_image, ← Multiset.toFinset_map, ← f.coe_to_ring_hom, ←
roots_map (↑f) ((splits_id_iff_splits (algebraMap F K)).mpr h), map_map, f.comp_algebra_map, ←
root_set]
#align polynomial.image_root_set Polynomial.image_rootSet
-/
/- warning: polynomial.adjoin_root_set_eq_range -> Polynomial.adjoin_rootSet_eq_range is a dubious translation:
lean 3 declaration is
forall {F : Type.{u1}} {K : Type.{u2}} {L : Type.{u3}} [_inst_1 : Field.{u2} K] [_inst_2 : Field.{u3} L] [_inst_3 : Field.{u1} F] [_inst_4 : Algebra.{u1, u2} F K (Semifield.toCommSemiring.{u1} F (Field.toSemifield.{u1} F _inst_3)) (Ring.toSemiring.{u2} K (DivisionRing.toRing.{u2} K (Field.toDivisionRing.{u2} K _inst_1)))] [_inst_5 : Algebra.{u1, u3} F L (Semifield.toCommSemiring.{u1} F (Field.toSemifield.{u1} F _inst_3)) (Ring.toSemiring.{u3} L (DivisionRing.toRing.{u3} L (Field.toDivisionRing.{u3} L _inst_2)))] {p : Polynomial.{u1} F (Ring.toSemiring.{u1} F (DivisionRing.toRing.{u1} F (Field.toDivisionRing.{u1} F _inst_3)))}, (Polynomial.Splits.{u1, u2} F K (EuclideanDomain.toCommRing.{u1} F (Field.toEuclideanDomain.{u1} F _inst_3)) _inst_1 (algebraMap.{u1, u2} F K (Semifield.toCommSemiring.{u1} F (Field.toSemifield.{u1} F _inst_3)) (Ring.toSemiring.{u2} K (DivisionRing.toRing.{u2} K (Field.toDivisionRing.{u2} K _inst_1))) _inst_4) p) -> (forall (f : AlgHom.{u1, u2, u3} F K L (Semifield.toCommSemiring.{u1} F (Field.toSemifield.{u1} F _inst_3)) (Ring.toSemiring.{u2} K (DivisionRing.toRing.{u2} K (Field.toDivisionRing.{u2} K _inst_1))) (Ring.toSemiring.{u3} L (DivisionRing.toRing.{u3} L (Field.toDivisionRing.{u3} L _inst_2))) _inst_4 _inst_5), Iff (Eq.{succ u3} (Subalgebra.{u1, u3} F L (Semifield.toCommSemiring.{u1} F (Field.toSemifield.{u1} F _inst_3)) (Ring.toSemiring.{u3} L (DivisionRing.toRing.{u3} L (Field.toDivisionRing.{u3} L _inst_2))) _inst_5) (Algebra.adjoin.{u1, u3} F L (Semifield.toCommSemiring.{u1} F (Field.toSemifield.{u1} F _inst_3)) (Ring.toSemiring.{u3} L (DivisionRing.toRing.{u3} L (Field.toDivisionRing.{u3} L _inst_2))) _inst_5 (Polynomial.rootSet.{u1, u3} F (EuclideanDomain.toCommRing.{u1} F (Field.toEuclideanDomain.{u1} F _inst_3)) p L (EuclideanDomain.toCommRing.{u3} L (Field.toEuclideanDomain.{u3} L _inst_2)) (Field.isDomain.{u3} L _inst_2) _inst_5)) (AlgHom.range.{u1, u2, u3} F K L (Semifield.toCommSemiring.{u1} F (Field.toSemifield.{u1} F _inst_3)) (Ring.toSemiring.{u2} K (DivisionRing.toRing.{u2} K (Field.toDivisionRing.{u2} K _inst_1))) _inst_4 (Ring.toSemiring.{u3} L (DivisionRing.toRing.{u3} L (Field.toDivisionRing.{u3} L _inst_2))) _inst_5 f)) (Eq.{succ u2} (Subalgebra.{u1, u2} F K (Semifield.toCommSemiring.{u1} F (Field.toSemifield.{u1} F _inst_3)) (Ring.toSemiring.{u2} K (DivisionRing.toRing.{u2} K (Field.toDivisionRing.{u2} K _inst_1))) _inst_4) (Algebra.adjoin.{u1, u2} F K (Semifield.toCommSemiring.{u1} F (Field.toSemifield.{u1} F _inst_3)) (Ring.toSemiring.{u2} K (DivisionRing.toRing.{u2} K (Field.toDivisionRing.{u2} K _inst_1))) _inst_4 (Polynomial.rootSet.{u1, u2} F (EuclideanDomain.toCommRing.{u1} F (Field.toEuclideanDomain.{u1} F _inst_3)) p K (EuclideanDomain.toCommRing.{u2} K (Field.toEuclideanDomain.{u2} K _inst_1)) (Field.isDomain.{u2} K _inst_1) _inst_4)) (Top.top.{u2} (Subalgebra.{u1, u2} F K (Semifield.toCommSemiring.{u1} F (Field.toSemifield.{u1} F _inst_3)) (Ring.toSemiring.{u2} K (DivisionRing.toRing.{u2} K (Field.toDivisionRing.{u2} K _inst_1))) _inst_4) (CompleteLattice.toHasTop.{u2} (Subalgebra.{u1, u2} F K (Semifield.toCommSemiring.{u1} F (Field.toSemifield.{u1} F _inst_3)) (Ring.toSemiring.{u2} K (DivisionRing.toRing.{u2} K (Field.toDivisionRing.{u2} K _inst_1))) _inst_4) (Algebra.Subalgebra.completeLattice.{u1, u2} F K (Semifield.toCommSemiring.{u1} F (Field.toSemifield.{u1} F _inst_3)) (Ring.toSemiring.{u2} K (DivisionRing.toRing.{u2} K (Field.toDivisionRing.{u2} K _inst_1))) _inst_4)))))
but is expected to have type
forall {F : Type.{u1}} {K : Type.{u2}} {L : Type.{u3}} [_inst_1 : Field.{u2} K] [_inst_2 : Field.{u3} L] [_inst_3 : Field.{u1} F] [_inst_4 : Algebra.{u1, u2} F K (Semifield.toCommSemiring.{u1} F (Field.toSemifield.{u1} F _inst_3)) (DivisionSemiring.toSemiring.{u2} K (Semifield.toDivisionSemiring.{u2} K (Field.toSemifield.{u2} K _inst_1)))] [_inst_5 : Algebra.{u1, u3} F L (Semifield.toCommSemiring.{u1} F (Field.toSemifield.{u1} F _inst_3)) (DivisionSemiring.toSemiring.{u3} L (Semifield.toDivisionSemiring.{u3} L (Field.toSemifield.{u3} L _inst_2)))] {p : Polynomial.{u1} F (DivisionSemiring.toSemiring.{u1} F (Semifield.toDivisionSemiring.{u1} F (Field.toSemifield.{u1} F _inst_3)))}, (Polynomial.Splits.{u1, u2} F K (EuclideanDomain.toCommRing.{u1} F (Field.toEuclideanDomain.{u1} F _inst_3)) _inst_1 (algebraMap.{u1, u2} F K (Semifield.toCommSemiring.{u1} F (Field.toSemifield.{u1} F _inst_3)) (DivisionSemiring.toSemiring.{u2} K (Semifield.toDivisionSemiring.{u2} K (Field.toSemifield.{u2} K _inst_1))) _inst_4) p) -> (forall (f : AlgHom.{u1, u2, u3} F K L (Semifield.toCommSemiring.{u1} F (Field.toSemifield.{u1} F _inst_3)) (DivisionSemiring.toSemiring.{u2} K (Semifield.toDivisionSemiring.{u2} K (Field.toSemifield.{u2} K _inst_1))) (DivisionSemiring.toSemiring.{u3} L (Semifield.toDivisionSemiring.{u3} L (Field.toSemifield.{u3} L _inst_2))) _inst_4 _inst_5), Iff (Eq.{succ u3} (Subalgebra.{u1, u3} F L (Semifield.toCommSemiring.{u1} F (Field.toSemifield.{u1} F _inst_3)) (DivisionSemiring.toSemiring.{u3} L (Semifield.toDivisionSemiring.{u3} L (Field.toSemifield.{u3} L _inst_2))) _inst_5) (Algebra.adjoin.{u1, u3} F L (Semifield.toCommSemiring.{u1} F (Field.toSemifield.{u1} F _inst_3)) (DivisionSemiring.toSemiring.{u3} L (Semifield.toDivisionSemiring.{u3} L (Field.toSemifield.{u3} L _inst_2))) _inst_5 (Polynomial.rootSet.{u1, u3} F (EuclideanDomain.toCommRing.{u1} F (Field.toEuclideanDomain.{u1} F _inst_3)) p L (EuclideanDomain.toCommRing.{u3} L (Field.toEuclideanDomain.{u3} L _inst_2)) (Field.isDomain.{u3} L _inst_2) _inst_5)) (AlgHom.range.{u1, u2, u3} F K L (Semifield.toCommSemiring.{u1} F (Field.toSemifield.{u1} F _inst_3)) (DivisionSemiring.toSemiring.{u2} K (Semifield.toDivisionSemiring.{u2} K (Field.toSemifield.{u2} K _inst_1))) _inst_4 (DivisionSemiring.toSemiring.{u3} L (Semifield.toDivisionSemiring.{u3} L (Field.toSemifield.{u3} L _inst_2))) _inst_5 f)) (Eq.{succ u2} (Subalgebra.{u1, u2} F K (Semifield.toCommSemiring.{u1} F (Field.toSemifield.{u1} F _inst_3)) (DivisionSemiring.toSemiring.{u2} K (Semifield.toDivisionSemiring.{u2} K (Field.toSemifield.{u2} K _inst_1))) _inst_4) (Algebra.adjoin.{u1, u2} F K (Semifield.toCommSemiring.{u1} F (Field.toSemifield.{u1} F _inst_3)) (DivisionSemiring.toSemiring.{u2} K (Semifield.toDivisionSemiring.{u2} K (Field.toSemifield.{u2} K _inst_1))) _inst_4 (Polynomial.rootSet.{u1, u2} F (EuclideanDomain.toCommRing.{u1} F (Field.toEuclideanDomain.{u1} F _inst_3)) p K (EuclideanDomain.toCommRing.{u2} K (Field.toEuclideanDomain.{u2} K _inst_1)) (Field.isDomain.{u2} K _inst_1) _inst_4)) (Top.top.{u2} (Subalgebra.{u1, u2} F K (Semifield.toCommSemiring.{u1} F (Field.toSemifield.{u1} F _inst_3)) (DivisionSemiring.toSemiring.{u2} K (Semifield.toDivisionSemiring.{u2} K (Field.toSemifield.{u2} K _inst_1))) _inst_4) (CompleteLattice.toTop.{u2} (Subalgebra.{u1, u2} F K (Semifield.toCommSemiring.{u1} F (Field.toSemifield.{u1} F _inst_3)) (DivisionSemiring.toSemiring.{u2} K (Semifield.toDivisionSemiring.{u2} K (Field.toSemifield.{u2} K _inst_1))) _inst_4) (Algebra.instCompleteLatticeSubalgebra.{u1, u2} F K (Semifield.toCommSemiring.{u1} F (Field.toSemifield.{u1} F _inst_3)) (DivisionSemiring.toSemiring.{u2} K (Semifield.toDivisionSemiring.{u2} K (Field.toSemifield.{u2} K _inst_1))) _inst_4)))))
Case conversion may be inaccurate. Consider using '#align polynomial.adjoin_root_set_eq_range Polynomial.adjoin_rootSet_eq_rangeₓ'. -/
theorem adjoin_rootSet_eq_range [Algebra F K] [Algebra F L] {p : F[X]}
(h : p.Splits (algebraMap F K)) (f : K →ₐ[F] L) :
Algebra.adjoin F (p.rootSet L) = f.range ↔ Algebra.adjoin F (p.rootSet K) = ⊤ :=
by
rw [← image_root_set h f, Algebra.adjoin_image, ← Algebra.map_top]
exact (Subalgebra.map_injective f.to_ring_hom.injective).eq_iff
#align polynomial.adjoin_root_set_eq_range Polynomial.adjoin_rootSet_eq_range
/- warning: polynomial.eq_prod_roots_of_splits -> Polynomial.eq_prod_roots_of_splits is a dubious translation:
lean 3 declaration is
forall {K : Type.{u1}} {L : Type.{u2}} [_inst_1 : Field.{u1} K] [_inst_2 : Field.{u2} L] {p : Polynomial.{u1} K (Ring.toSemiring.{u1} K (DivisionRing.toRing.{u1} K (Field.toDivisionRing.{u1} K _inst_1)))} {i : RingHom.{u1, u2} K L (NonAssocRing.toNonAssocSemiring.{u1} K (Ring.toNonAssocRing.{u1} K (DivisionRing.toRing.{u1} K (Field.toDivisionRing.{u1} K _inst_1)))) (NonAssocRing.toNonAssocSemiring.{u2} L (Ring.toNonAssocRing.{u2} L (DivisionRing.toRing.{u2} L (Field.toDivisionRing.{u2} L _inst_2))))}, (Polynomial.Splits.{u1, u2} K L (EuclideanDomain.toCommRing.{u1} K (Field.toEuclideanDomain.{u1} K _inst_1)) _inst_2 i p) -> (Eq.{succ u2} (Polynomial.{u2} L (Ring.toSemiring.{u2} L (DivisionRing.toRing.{u2} L (Field.toDivisionRing.{u2} L _inst_2)))) (Polynomial.map.{u1, u2} K L (Ring.toSemiring.{u1} K (DivisionRing.toRing.{u1} K (Field.toDivisionRing.{u1} K _inst_1))) (Ring.toSemiring.{u2} L (DivisionRing.toRing.{u2} L (Field.toDivisionRing.{u2} L _inst_2))) i p) (HMul.hMul.{u2, u2, u2} 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but is expected to have type
forall {K : Type.{u1}} {L : Type.{u2}} [_inst_1 : Field.{u1} K] [_inst_2 : Field.{u2} L] {p : Polynomial.{u1} K (DivisionSemiring.toSemiring.{u1} K (Semifield.toDivisionSemiring.{u1} K (Field.toSemifield.{u1} K _inst_1)))} {i : RingHom.{u1, u2} K L (NonAssocRing.toNonAssocSemiring.{u1} K (Ring.toNonAssocRing.{u1} K (DivisionRing.toRing.{u1} K (Field.toDivisionRing.{u1} K _inst_1)))) (NonAssocRing.toNonAssocSemiring.{u2} L (Ring.toNonAssocRing.{u2} L (DivisionRing.toRing.{u2} L (Field.toDivisionRing.{u2} L _inst_2))))}, (Polynomial.Splits.{u1, u2} K L (EuclideanDomain.toCommRing.{u1} K (Field.toEuclideanDomain.{u1} K _inst_1)) _inst_2 i p) -> (Eq.{succ u2} (Polynomial.{u2} L (DivisionSemiring.toSemiring.{u2} L (Semifield.toDivisionSemiring.{u2} L (Field.toSemifield.{u2} L _inst_2)))) (Polynomial.map.{u1, u2} K L (DivisionSemiring.toSemiring.{u1} K (Semifield.toDivisionSemiring.{u1} K (Field.toSemifield.{u1} K _inst_1))) (DivisionSemiring.toSemiring.{u2} L (Semifield.toDivisionSemiring.{u2} L (Field.toSemifield.{u2} L _inst_2))) i p) (HMul.hMul.{u2, u2, u2} ((fun ([email protected]._hyg.2391 : (fun ([email protected]._hyg.2391 : K) => L) (Polynomial.leadingCoeff.{u1} K (DivisionSemiring.toSemiring.{u1} K (Semifield.toDivisionSemiring.{u1} K (Field.toSemifield.{u1} K _inst_1))) p)) => Polynomial.{u2} ((fun ([email protected]._hyg.2391 : K) => L) (Polynomial.leadingCoeff.{u1} K (DivisionSemiring.toSemiring.{u1} K (Semifield.toDivisionSemiring.{u1} K (Field.toSemifield.{u1} K _inst_1))) p)) (DivisionSemiring.toSemiring.{u2} ((fun ([email protected]._hyg.2391 : K) => L) (Polynomial.leadingCoeff.{u1} K (DivisionSemiring.toSemiring.{u1} K (Semifield.toDivisionSemiring.{u1} K (Field.toSemifield.{u1} K _inst_1))) p)) (Semifield.toDivisionSemiring.{u2} ((fun ([email protected]._hyg.2391 : K) => L) (Polynomial.leadingCoeff.{u1} K (DivisionSemiring.toSemiring.{u1} K (Semifield.toDivisionSemiring.{u1} K (Field.toSemifield.{u1} K _inst_1))) p)) (Field.toSemifield.{u2} ((fun ([email protected]._hyg.2391 : K) => L) (Polynomial.leadingCoeff.{u1} K (DivisionSemiring.toSemiring.{u1} K (Semifield.toDivisionSemiring.{u1} K (Field.toSemifield.{u1} K _inst_1))) p)) _inst_2)))) (FunLike.coe.{max (succ u1) (succ u2), succ u1, succ u2} (RingHom.{u1, u2} K L (NonAssocRing.toNonAssocSemiring.{u1} K (Ring.toNonAssocRing.{u1} K (DivisionRing.toRing.{u1} K (Field.toDivisionRing.{u1} K _inst_1)))) (NonAssocRing.toNonAssocSemiring.{u2} L (Ring.toNonAssocRing.{u2} L (DivisionRing.toRing.{u2} L (Field.toDivisionRing.{u2} L _inst_2))))) K (fun (a : K) => (fun ([email protected]._hyg.2391 : K) => L) a) (MulHomClass.toFunLike.{max u1 u2, u1, u2} (RingHom.{u1, u2} K L (NonAssocRing.toNonAssocSemiring.{u1} K (Ring.toNonAssocRing.{u1} K (DivisionRing.toRing.{u1} K (Field.toDivisionRing.{u1} K _inst_1)))) (NonAssocRing.toNonAssocSemiring.{u2} L (Ring.toNonAssocRing.{u2} L (DivisionRing.toRing.{u2} L (Field.toDivisionRing.{u2} L _inst_2))))) K L (NonUnitalNonAssocSemiring.toMul.{u1} K (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} K (NonAssocRing.toNonAssocSemiring.{u1} K (Ring.toNonAssocRing.{u1} K (DivisionRing.toRing.{u1} K (Field.toDivisionRing.{u1} K _inst_1)))))) (NonUnitalNonAssocSemiring.toMul.{u2} L (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} L (NonAssocRing.toNonAssocSemiring.{u2} L (Ring.toNonAssocRing.{u2} L (DivisionRing.toRing.{u2} L (Field.toDivisionRing.{u2} L _inst_2)))))) (NonUnitalRingHomClass.toMulHomClass.{max u1 u2, u1, u2} (RingHom.{u1, u2} K L (NonAssocRing.toNonAssocSemiring.{u1} K (Ring.toNonAssocRing.{u1} K (DivisionRing.toRing.{u1} K (Field.toDivisionRing.{u1} K _inst_1)))) (NonAssocRing.toNonAssocSemiring.{u2} L (Ring.toNonAssocRing.{u2} L (DivisionRing.toRing.{u2} L (Field.toDivisionRing.{u2} L _inst_2))))) K L (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} K (NonAssocRing.toNonAssocSemiring.{u1} K (Ring.toNonAssocRing.{u1} K (DivisionRing.toRing.{u1} K (Field.toDivisionRing.{u1} K _inst_1))))) (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} L (NonAssocRing.toNonAssocSemiring.{u2} L (Ring.toNonAssocRing.{u2} L (DivisionRing.toRing.{u2} L (Field.toDivisionRing.{u2} L _inst_2))))) (RingHomClass.toNonUnitalRingHomClass.{max u1 u2, u1, u2} (RingHom.{u1, u2} K L (NonAssocRing.toNonAssocSemiring.{u1} K (Ring.toNonAssocRing.{u1} K (DivisionRing.toRing.{u1} K (Field.toDivisionRing.{u1} K _inst_1)))) (NonAssocRing.toNonAssocSemiring.{u2} L (Ring.toNonAssocRing.{u2} L (DivisionRing.toRing.{u2} L (Field.toDivisionRing.{u2} L _inst_2))))) K L (NonAssocRing.toNonAssocSemiring.{u1} K (Ring.toNonAssocRing.{u1} K (DivisionRing.toRing.{u1} K (Field.toDivisionRing.{u1} K _inst_1)))) (NonAssocRing.toNonAssocSemiring.{u2} L (Ring.toNonAssocRing.{u2} L (DivisionRing.toRing.{u2} L (Field.toDivisionRing.{u2} L _inst_2)))) (RingHom.instRingHomClassRingHom.{u1, u2} K L (NonAssocRing.toNonAssocSemiring.{u1} K (Ring.toNonAssocRing.{u1} K (DivisionRing.toRing.{u1} K (Field.toDivisionRing.{u1} K _inst_1)))) (NonAssocRing.toNonAssocSemiring.{u2} L (Ring.toNonAssocRing.{u2} L (DivisionRing.toRing.{u2} L (Field.toDivisionRing.{u2} L _inst_2)))))))) i (Polynomial.leadingCoeff.{u1} K (DivisionSemiring.toSemiring.{u1} K (Semifield.toDivisionSemiring.{u1} K (Field.toSemifield.{u1} K _inst_1))) p))) (Polynomial.{u2} L (DivisionSemiring.toSemiring.{u2} L (Semifield.toDivisionSemiring.{u2} L (Field.toSemifield.{u2} L _inst_2)))) ((fun ([email protected]._hyg.2391 : (fun ([email protected]._hyg.2391 : K) => L) (Polynomial.leadingCoeff.{u1} K (DivisionSemiring.toSemiring.{u1} K (Semifield.toDivisionSemiring.{u1} K (Field.toSemifield.{u1} K _inst_1))) p)) => Polynomial.{u2} ((fun ([email protected]._hyg.2391 : K) => L) (Polynomial.leadingCoeff.{u1} K (DivisionSemiring.toSemiring.{u1} K (Semifield.toDivisionSemiring.{u1} K (Field.toSemifield.{u1} K _inst_1))) p)) (DivisionSemiring.toSemiring.{u2} ((fun ([email protected]._hyg.2391 : K) => L) 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(DivisionRing.toRing.{u1} K (Field.toDivisionRing.{u1} K _inst_1)))) (NonAssocRing.toNonAssocSemiring.{u2} L (Ring.toNonAssocRing.{u2} L (DivisionRing.toRing.{u2} L (Field.toDivisionRing.{u2} L _inst_2))))) K L (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} K (NonAssocRing.toNonAssocSemiring.{u1} K (Ring.toNonAssocRing.{u1} K (DivisionRing.toRing.{u1} K (Field.toDivisionRing.{u1} K _inst_1))))) (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} L (NonAssocRing.toNonAssocSemiring.{u2} L (Ring.toNonAssocRing.{u2} L (DivisionRing.toRing.{u2} L (Field.toDivisionRing.{u2} L _inst_2))))) (RingHomClass.toNonUnitalRingHomClass.{max u1 u2, u1, u2} (RingHom.{u1, u2} K L (NonAssocRing.toNonAssocSemiring.{u1} K (Ring.toNonAssocRing.{u1} K (DivisionRing.toRing.{u1} K (Field.toDivisionRing.{u1} K _inst_1)))) (NonAssocRing.toNonAssocSemiring.{u2} L (Ring.toNonAssocRing.{u2} L (DivisionRing.toRing.{u2} L (Field.toDivisionRing.{u2} L _inst_2))))) K L (NonAssocRing.toNonAssocSemiring.{u1} K (Ring.toNonAssocRing.{u1} K (DivisionRing.toRing.{u1} K (Field.toDivisionRing.{u1} K _inst_1)))) (NonAssocRing.toNonAssocSemiring.{u2} L (Ring.toNonAssocRing.{u2} L (DivisionRing.toRing.{u2} L (Field.toDivisionRing.{u2} L _inst_2)))) (RingHom.instRingHomClassRingHom.{u1, u2} K L (NonAssocRing.toNonAssocSemiring.{u1} K (Ring.toNonAssocRing.{u1} K (DivisionRing.toRing.{u1} K (Field.toDivisionRing.{u1} K _inst_1)))) (NonAssocRing.toNonAssocSemiring.{u2} L (Ring.toNonAssocRing.{u2} L (DivisionRing.toRing.{u2} L (Field.toDivisionRing.{u2} L _inst_2)))))))) i (Polynomial.leadingCoeff.{u1} K (DivisionSemiring.toSemiring.{u1} K (Semifield.toDivisionSemiring.{u1} K (Field.toSemifield.{u1} K _inst_1))) p))) (instHMul.{u2} ((fun ([email protected]._hyg.2391 : (fun ([email protected]._hyg.2391 : K) => L) (Polynomial.leadingCoeff.{u1} K (DivisionSemiring.toSemiring.{u1} K (Semifield.toDivisionSemiring.{u1} K (Field.toSemifield.{u1} K _inst_1))) p)) => Polynomial.{u2} ((fun ([email protected]._hyg.2391 : K) => L) (Polynomial.leadingCoeff.{u1} K (DivisionSemiring.toSemiring.{u1} K (Semifield.toDivisionSemiring.{u1} K (Field.toSemifield.{u1} K _inst_1))) p)) (DivisionSemiring.toSemiring.{u2} ((fun ([email protected]._hyg.2391 : K) => L) (Polynomial.leadingCoeff.{u1} K (DivisionSemiring.toSemiring.{u1} K (Semifield.toDivisionSemiring.{u1} K (Field.toSemifield.{u1} K _inst_1))) p)) (Semifield.toDivisionSemiring.{u2} ((fun ([email protected]._hyg.2391 : K) => L) (Polynomial.leadingCoeff.{u1} K (DivisionSemiring.toSemiring.{u1} K (Semifield.toDivisionSemiring.{u1} K (Field.toSemifield.{u1} K _inst_1))) p)) (Field.toSemifield.{u2} ((fun ([email protected]._hyg.2391 : K) => L) (Polynomial.leadingCoeff.{u1} K (DivisionSemiring.toSemiring.{u1} K (Semifield.toDivisionSemiring.{u1} K (Field.toSemifield.{u1} K _inst_1))) p)) _inst_2)))) (FunLike.coe.{max (succ u1) (succ u2), succ u1, succ u2} (RingHom.{u1, u2} K L (NonAssocRing.toNonAssocSemiring.{u1} K 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(Ring.toNonAssocRing.{u1} K (DivisionRing.toRing.{u1} K (Field.toDivisionRing.{u1} K _inst_1)))) (NonAssocRing.toNonAssocSemiring.{u2} L (Ring.toNonAssocRing.{u2} L (DivisionRing.toRing.{u2} L (Field.toDivisionRing.{u2} L _inst_2))))) K L (NonAssocRing.toNonAssocSemiring.{u1} K (Ring.toNonAssocRing.{u1} K (DivisionRing.toRing.{u1} K (Field.toDivisionRing.{u1} K _inst_1)))) (NonAssocRing.toNonAssocSemiring.{u2} L (Ring.toNonAssocRing.{u2} L (DivisionRing.toRing.{u2} L (Field.toDivisionRing.{u2} L _inst_2)))) (RingHom.instRingHomClassRingHom.{u1, u2} K L (NonAssocRing.toNonAssocSemiring.{u1} K (Ring.toNonAssocRing.{u1} K (DivisionRing.toRing.{u1} K (Field.toDivisionRing.{u1} K _inst_1)))) (NonAssocRing.toNonAssocSemiring.{u2} L (Ring.toNonAssocRing.{u2} L (DivisionRing.toRing.{u2} L (Field.toDivisionRing.{u2} L _inst_2)))))))) i (Polynomial.leadingCoeff.{u1} K (DivisionSemiring.toSemiring.{u1} K (Semifield.toDivisionSemiring.{u1} K (Field.toSemifield.{u1} K _inst_1))) p))) (Polynomial.mul'.{u2} ((fun ([email protected]._hyg.2391 : K) => L) (Polynomial.leadingCoeff.{u1} K (DivisionSemiring.toSemiring.{u1} K (Semifield.toDivisionSemiring.{u1} K (Field.toSemifield.{u1} K _inst_1))) p)) (DivisionSemiring.toSemiring.{u2} ((fun ([email protected]._hyg.2391 : K) => L) (Polynomial.leadingCoeff.{u1} K (DivisionSemiring.toSemiring.{u1} K (Semifield.toDivisionSemiring.{u1} K (Field.toSemifield.{u1} K _inst_1))) p)) (Semifield.toDivisionSemiring.{u2} ((fun ([email protected]._hyg.2391 : K) => L) (Polynomial.leadingCoeff.{u1} K (DivisionSemiring.toSemiring.{u1} K (Semifield.toDivisionSemiring.{u1} K (Field.toSemifield.{u1} K _inst_1))) p)) (Field.toSemifield.{u2} ((fun ([email protected]._hyg.2391 : K) => L) (Polynomial.leadingCoeff.{u1} K (DivisionSemiring.toSemiring.{u1} K (Semifield.toDivisionSemiring.{u1} K (Field.toSemifield.{u1} K _inst_1))) p)) _inst_2))))) (FunLike.coe.{succ u2, succ u2, succ u2} (RingHom.{u2, u2} ((fun ([email protected]._hyg.2391 : K) => L) (Polynomial.leadingCoeff.{u1} K (DivisionSemiring.toSemiring.{u1} K (Semifield.toDivisionSemiring.{u1} K (Field.toSemifield.{u1} K _inst_1))) p)) (Polynomial.{u2} ((fun ([email protected]._hyg.2391 : K) => L) (Polynomial.leadingCoeff.{u1} K (DivisionSemiring.toSemiring.{u1} K (Semifield.toDivisionSemiring.{u1} K (Field.toSemifield.{u1} K _inst_1))) p)) (DivisionSemiring.toSemiring.{u2} ((fun ([email protected]._hyg.2391 : K) => L) (Polynomial.leadingCoeff.{u1} K (DivisionSemiring.toSemiring.{u1} K (Semifield.toDivisionSemiring.{u1} K (Field.toSemifield.{u1} K _inst_1))) p)) (Semifield.toDivisionSemiring.{u2} ((fun ([email protected]._hyg.2391 : K) => L) (Polynomial.leadingCoeff.{u1} K (DivisionSemiring.toSemiring.{u1} K (Semifield.toDivisionSemiring.{u1} K (Field.toSemifield.{u1} K _inst_1))) p)) (Field.toSemifield.{u2} ((fun ([email protected]._hyg.2391 : K) => L) (Polynomial.leadingCoeff.{u1} K (DivisionSemiring.toSemiring.{u1} K (Semifield.toDivisionSemiring.{u1} K (Field.toSemifield.{u1} K _inst_1))) p)) _inst_2)))) (Semiring.toNonAssocSemiring.{u2} ((fun ([email protected]._hyg.2391 : K) => L) (Polynomial.leadingCoeff.{u1} K (DivisionSemiring.toSemiring.{u1} K (Semifield.toDivisionSemiring.{u1} K (Field.toSemifield.{u1} K _inst_1))) p)) (DivisionSemiring.toSemiring.{u2} ((fun ([email protected]._hyg.2391 : K) => L) (Polynomial.leadingCoeff.{u1} K (DivisionSemiring.toSemiring.{u1} K (Semifield.toDivisionSemiring.{u1} K (Field.toSemifield.{u1} K _inst_1))) p)) (Semifield.toDivisionSemiring.{u2} ((fun ([email protected]._hyg.2391 : K) => L) (Polynomial.leadingCoeff.{u1} K (DivisionSemiring.toSemiring.{u1} K (Semifield.toDivisionSemiring.{u1} K (Field.toSemifield.{u1} K _inst_1))) p)) (Field.toSemifield.{u2} ((fun ([email protected]._hyg.2391 : K) => L) (Polynomial.leadingCoeff.{u1} K (DivisionSemiring.toSemiring.{u1} K (Semifield.toDivisionSemiring.{u1} K (Field.toSemifield.{u1} K _inst_1))) p)) _inst_2)))) (Semiring.toNonAssocSemiring.{u2} (Polynomial.{u2} ((fun ([email protected]._hyg.2391 : K) => L) (Polynomial.leadingCoeff.{u1} K (DivisionSemiring.toSemiring.{u1} K (Semifield.toDivisionSemiring.{u1} K (Field.toSemifield.{u1} K _inst_1))) p)) (DivisionSemiring.toSemiring.{u2} ((fun ([email protected]._hyg.2391 : K) => L) (Polynomial.leadingCoeff.{u1} K (DivisionSemiring.toSemiring.{u1} K (Semifield.toDivisionSemiring.{u1} K (Field.toSemifield.{u1} K _inst_1))) p)) (Semifield.toDivisionSemiring.{u2} ((fun ([email protected]._hyg.2391 : K) => L) (Polynomial.leadingCoeff.{u1} K (DivisionSemiring.toSemiring.{u1} K (Semifield.toDivisionSemiring.{u1} K (Field.toSemifield.{u1} K _inst_1))) p)) (Field.toSemifield.{u2} ((fun ([email protected]._hyg.2391 : K) => L) (Polynomial.leadingCoeff.{u1} K (DivisionSemiring.toSemiring.{u1} K (Semifield.toDivisionSemiring.{u1} K (Field.toSemifield.{u1} K _inst_1))) p)) _inst_2)))) (Polynomial.semiring.{u2} ((fun ([email protected]._hyg.2391 : K) => L) (Polynomial.leadingCoeff.{u1} K (DivisionSemiring.toSemiring.{u1} K (Semifield.toDivisionSemiring.{u1} K (Field.toSemifield.{u1} K _inst_1))) p)) (DivisionSemiring.toSemiring.{u2} ((fun ([email protected]._hyg.2391 : K) => L) (Polynomial.leadingCoeff.{u1} K (DivisionSemiring.toSemiring.{u1} K (Semifield.toDivisionSemiring.{u1} K (Field.toSemifield.{u1} K _inst_1))) p)) (Semifield.toDivisionSemiring.{u2} ((fun ([email protected]._hyg.2391 : K) => L) (Polynomial.leadingCoeff.{u1} K (DivisionSemiring.toSemiring.{u1} K (Semifield.toDivisionSemiring.{u1} K (Field.toSemifield.{u1} K _inst_1))) p)) (Field.toSemifield.{u2} ((fun ([email protected]._hyg.2391 : K) => L) (Polynomial.leadingCoeff.{u1} K (DivisionSemiring.toSemiring.{u1} K (Semifield.toDivisionSemiring.{u1} K (Field.toSemifield.{u1} K _inst_1))) p)) _inst_2)))))) ((fun ([email protected]._hyg.2391 : K) => L) (Polynomial.leadingCoeff.{u1} K (DivisionSemiring.toSemiring.{u1} K (Semifield.toDivisionSemiring.{u1} K (Field.toSemifield.{u1} K _inst_1))) p)) (fun (_x : (fun ([email protected]._hyg.2391 : K) => L) (Polynomial.leadingCoeff.{u1} K (DivisionSemiring.toSemiring.{u1} K (Semifield.toDivisionSemiring.{u1} K (Field.toSemifield.{u1} K _inst_1))) p)) => (fun ([email protected]._hyg.2391 : (fun ([email protected]._hyg.2391 : K) => L) (Polynomial.leadingCoeff.{u1} K (DivisionSemiring.toSemiring.{u1} K (Semifield.toDivisionSemiring.{u1} K (Field.toSemifield.{u1} K _inst_1))) p)) => Polynomial.{u2} ((fun ([email protected]._hyg.2391 : K) => L) (Polynomial.leadingCoeff.{u1} K (DivisionSemiring.toSemiring.{u1} K (Semifield.toDivisionSemiring.{u1} K (Field.toSemifield.{u1} K _inst_1))) p)) (DivisionSemiring.toSemiring.{u2} ((fun ([email protected]._hyg.2391 : K) => L) (Polynomial.leadingCoeff.{u1} K (DivisionSemiring.toSemiring.{u1} K (Semifield.toDivisionSemiring.{u1} K (Field.toSemifield.{u1} K _inst_1))) p)) (Semifield.toDivisionSemiring.{u2} ((fun ([email protected]._hyg.2391 : K) => L) (Polynomial.leadingCoeff.{u1} K (DivisionSemiring.toSemiring.{u1} K (Semifield.toDivisionSemiring.{u1} K (Field.toSemifield.{u1} K _inst_1))) p)) (Field.toSemifield.{u2} ((fun ([email protected]._hyg.2391 : K) => L) (Polynomial.leadingCoeff.{u1} K (DivisionSemiring.toSemiring.{u1} K (Semifield.toDivisionSemiring.{u1} K (Field.toSemifield.{u1} K _inst_1))) p)) _inst_2)))) _x) (MulHomClass.toFunLike.{u2, u2, u2} (RingHom.{u2, u2} ((fun ([email protected]._hyg.2391 : K) => L) (Polynomial.leadingCoeff.{u1} K (DivisionSemiring.toSemiring.{u1} K (Semifield.toDivisionSemiring.{u1} K (Field.toSemifield.{u1} K _inst_1))) p)) (Polynomial.{u2} ((fun ([email protected]._hyg.2391 : K) => L) (Polynomial.leadingCoeff.{u1} K (DivisionSemiring.toSemiring.{u1} K (Semifield.toDivisionSemiring.{u1} K (Field.toSemifield.{u1} K _inst_1))) p)) (DivisionSemiring.toSemiring.{u2} ((fun ([email protected]._hyg.2391 : K) => L) (Polynomial.leadingCoeff.{u1} K (DivisionSemiring.toSemiring.{u1} K (Semifield.toDivisionSemiring.{u1} K (Field.toSemifield.{u1} K _inst_1))) p)) (Semifield.toDivisionSemiring.{u2} ((fun ([email protected]._hyg.2391 : K) => L) (Polynomial.leadingCoeff.{u1} K (DivisionSemiring.toSemiring.{u1} K (Semifield.toDivisionSemiring.{u1} K (Field.toSemifield.{u1} K _inst_1))) p)) (Field.toSemifield.{u2} ((fun ([email protected]._hyg.2391 : K) => L) (Polynomial.leadingCoeff.{u1} K (DivisionSemiring.toSemiring.{u1} K (Semifield.toDivisionSemiring.{u1} K (Field.toSemifield.{u1} K _inst_1))) p)) _inst_2)))) (Semiring.toNonAssocSemiring.{u2} ((fun ([email protected]._hyg.2391 : K) => L) (Polynomial.leadingCoeff.{u1} K (DivisionSemiring.toSemiring.{u1} K (Semifield.toDivisionSemiring.{u1} K (Field.toSemifield.{u1} K _inst_1))) p)) (DivisionSemiring.toSemiring.{u2} ((fun ([email protected]._hyg.2391 : K) => L) (Polynomial.leadingCoeff.{u1} K (DivisionSemiring.toSemiring.{u1} K (Semifield.toDivisionSemiring.{u1} K (Field.toSemifield.{u1} K _inst_1))) p)) (Semifield.toDivisionSemiring.{u2} ((fun ([email protected]._hyg.2391 : K) => L) (Polynomial.leadingCoeff.{u1} K (DivisionSemiring.toSemiring.{u1} K (Semifield.toDivisionSemiring.{u1} K (Field.toSemifield.{u1} K _inst_1))) p)) (Field.toSemifield.{u2} ((fun ([email protected]._hyg.2391 : K) => L) (Polynomial.leadingCoeff.{u1} K (DivisionSemiring.toSemiring.{u1} K (Semifield.toDivisionSemiring.{u1} K (Field.toSemifield.{u1} K _inst_1))) p)) _inst_2)))) (Semiring.toNonAssocSemiring.{u2} (Polynomial.{u2} ((fun ([email protected]._hyg.2391 : K) => L) (Polynomial.leadingCoeff.{u1} K (DivisionSemiring.toSemiring.{u1} K (Semifield.toDivisionSemiring.{u1} K (Field.toSemifield.{u1} K _inst_1))) p)) (DivisionSemiring.toSemiring.{u2} ((fun ([email protected]._hyg.2391 : K) => L) (Polynomial.leadingCoeff.{u1} K (DivisionSemiring.toSemiring.{u1} K (Semifield.toDivisionSemiring.{u1} K (Field.toSemifield.{u1} K _inst_1))) p)) (Semifield.toDivisionSemiring.{u2} ((fun ([email protected]._hyg.2391 : K) => L) (Polynomial.leadingCoeff.{u1} K (DivisionSemiring.toSemiring.{u1} K (Semifield.toDivisionSemiring.{u1} K (Field.toSemifield.{u1} K _inst_1))) p)) (Field.toSemifield.{u2} ((fun ([email protected]._hyg.2391 : K) => L) (Polynomial.leadingCoeff.{u1} K (DivisionSemiring.toSemiring.{u1} K (Semifield.toDivisionSemiring.{u1} K (Field.toSemifield.{u1} K _inst_1))) p)) _inst_2)))) (Polynomial.semiring.{u2} ((fun ([email protected]._hyg.2391 : K) => L) (Polynomial.leadingCoeff.{u1} K (DivisionSemiring.toSemiring.{u1} K (Semifield.toDivisionSemiring.{u1} K (Field.toSemifield.{u1} K _inst_1))) p)) (DivisionSemiring.toSemiring.{u2} ((fun ([email protected]._hyg.2391 : K) => L) (Polynomial.leadingCoeff.{u1} K (DivisionSemiring.toSemiring.{u1} K (Semifield.toDivisionSemiring.{u1} K (Field.toSemifield.{u1} K _inst_1))) p)) (Semifield.toDivisionSemiring.{u2} ((fun ([email protected]._hyg.2391 : K) => L) (Polynomial.leadingCoeff.{u1} K (DivisionSemiring.toSemiring.{u1} K (Semifield.toDivisionSemiring.{u1} K (Field.toSemifield.{u1} K _inst_1))) p)) (Field.toSemifield.{u2} ((fun ([email protected]._hyg.2391 : K) => L) (Polynomial.leadingCoeff.{u1} K (DivisionSemiring.toSemiring.{u1} K (Semifield.toDivisionSemiring.{u1} K (Field.toSemifield.{u1} K _inst_1))) p)) _inst_2)))))) ((fun ([email protected]._hyg.2391 : K) => L) (Polynomial.leadingCoeff.{u1} K (DivisionSemiring.toSemiring.{u1} K (Semifield.toDivisionSemiring.{u1} K (Field.toSemifield.{u1} K _inst_1))) p)) (Polynomial.{u2} ((fun ([email protected]._hyg.2391 : K) => L) (Polynomial.leadingCoeff.{u1} K (DivisionSemiring.toSemiring.{u1} K (Semifield.toDivisionSemiring.{u1} K (Field.toSemifield.{u1} K _inst_1))) p)) (DivisionSemiring.toSemiring.{u2} ((fun ([email protected]._hyg.2391 : K) => L) (Polynomial.leadingCoeff.{u1} K (DivisionSemiring.toSemiring.{u1} K (Semifield.toDivisionSemiring.{u1} K (Field.toSemifield.{u1} K _inst_1))) p)) (Semifield.toDivisionSemiring.{u2} ((fun ([email protected]._hyg.2391 : K) => L) (Polynomial.leadingCoeff.{u1} K (DivisionSemiring.toSemiring.{u1} K (Semifield.toDivisionSemiring.{u1} K (Field.toSemifield.{u1} K _inst_1))) p)) (Field.toSemifield.{u2} ((fun ([email protected]._hyg.2391 : K) => L) (Polynomial.leadingCoeff.{u1} K (DivisionSemiring.toSemiring.{u1} K (Semifield.toDivisionSemiring.{u1} K (Field.toSemifield.{u1} K _inst_1))) p)) _inst_2)))) (NonUnitalNonAssocSemiring.toMul.{u2} ((fun ([email protected]._hyg.2391 : K) => L) (Polynomial.leadingCoeff.{u1} K (DivisionSemiring.toSemiring.{u1} K (Semifield.toDivisionSemiring.{u1} K (Field.toSemifield.{u1} K _inst_1))) p)) (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} ((fun ([email protected]._hyg.2391 : K) => L) (Polynomial.leadingCoeff.{u1} K (DivisionSemiring.toSemiring.{u1} K (Semifield.toDivisionSemiring.{u1} K (Field.toSemifield.{u1} K _inst_1))) p)) (Semiring.toNonAssocSemiring.{u2} ((fun ([email protected]._hyg.2391 : K) => L) (Polynomial.leadingCoeff.{u1} K (DivisionSemiring.toSemiring.{u1} K (Semifield.toDivisionSemiring.{u1} K (Field.toSemifield.{u1} K _inst_1))) p)) (DivisionSemiring.toSemiring.{u2} ((fun ([email protected]._hyg.2391 : K) => L) (Polynomial.leadingCoeff.{u1} K (DivisionSemiring.toSemiring.{u1} K (Semifield.toDivisionSemiring.{u1} K (Field.toSemifield.{u1} K _inst_1))) p)) (Semifield.toDivisionSemiring.{u2} ((fun ([email protected]._hyg.2391 : K) => L) (Polynomial.leadingCoeff.{u1} K (DivisionSemiring.toSemiring.{u1} K (Semifield.toDivisionSemiring.{u1} K (Field.toSemifield.{u1} K _inst_1))) p)) (Field.toSemifield.{u2} ((fun ([email protected]._hyg.2391 : K) => L) (Polynomial.leadingCoeff.{u1} K (DivisionSemiring.toSemiring.{u1} K (Semifield.toDivisionSemiring.{u1} K (Field.toSemifield.{u1} K _inst_1))) p)) _inst_2)))))) (NonUnitalNonAssocSemiring.toMul.{u2} (Polynomial.{u2} ((fun ([email protected]._hyg.2391 : K) => L) (Polynomial.leadingCoeff.{u1} K (DivisionSemiring.toSemiring.{u1} K (Semifield.toDivisionSemiring.{u1} K (Field.toSemifield.{u1} K _inst_1))) p)) (DivisionSemiring.toSemiring.{u2} ((fun ([email protected]._hyg.2391 : K) => L) (Polynomial.leadingCoeff.{u1} K (DivisionSemiring.toSemiring.{u1} K (Semifield.toDivisionSemiring.{u1} K (Field.toSemifield.{u1} K _inst_1))) p)) (Semifield.toDivisionSemiring.{u2} ((fun ([email protected]._hyg.2391 : K) => L) (Polynomial.leadingCoeff.{u1} K (DivisionSemiring.toSemiring.{u1} K (Semifield.toDivisionSemiring.{u1} K (Field.toSemifield.{u1} K _inst_1))) p)) (Field.toSemifield.{u2} ((fun ([email protected]._hyg.2391 : K) => L) (Polynomial.leadingCoeff.{u1} K (DivisionSemiring.toSemiring.{u1} K (Semifield.toDivisionSemiring.{u1} K (Field.toSemifield.{u1} K _inst_1))) p)) _inst_2)))) (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} (Polynomial.{u2} ((fun ([email protected]._hyg.2391 : K) => L) (Polynomial.leadingCoeff.{u1} K (DivisionSemiring.toSemiring.{u1} K (Semifield.toDivisionSemiring.{u1} K (Field.toSemifield.{u1} K _inst_1))) p)) (DivisionSemiring.toSemiring.{u2} ((fun ([email protected]._hyg.2391 : K) => L) (Polynomial.leadingCoeff.{u1} K (DivisionSemiring.toSemiring.{u1} K (Semifield.toDivisionSemiring.{u1} K (Field.toSemifield.{u1} K _inst_1))) p)) (Semifield.toDivisionSemiring.{u2} ((fun ([email protected]._hyg.2391 : K) => L) (Polynomial.leadingCoeff.{u1} K (DivisionSemiring.toSemiring.{u1} K (Semifield.toDivisionSemiring.{u1} K (Field.toSemifield.{u1} K _inst_1))) p)) (Field.toSemifield.{u2} ((fun ([email protected]._hyg.2391 : K) => L) (Polynomial.leadingCoeff.{u1} K (DivisionSemiring.toSemiring.{u1} K (Semifield.toDivisionSemiring.{u1} K (Field.toSemifield.{u1} K _inst_1))) p)) _inst_2)))) (Semiring.toNonAssocSemiring.{u2} (Polynomial.{u2} ((fun ([email protected]._hyg.2391 : K) => L) (Polynomial.leadingCoeff.{u1} K (DivisionSemiring.toSemiring.{u1} K (Semifield.toDivisionSemiring.{u1} K (Field.toSemifield.{u1} K _inst_1))) p)) (DivisionSemiring.toSemiring.{u2} ((fun ([email protected]._hyg.2391 : K) => L) (Polynomial.leadingCoeff.{u1} K (DivisionSemiring.toSemiring.{u1} K (Semifield.toDivisionSemiring.{u1} K (Field.toSemifield.{u1} K _inst_1))) p)) (Semifield.toDivisionSemiring.{u2} ((fun ([email protected]._hyg.2391 : K) => L) (Polynomial.leadingCoeff.{u1} K (DivisionSemiring.toSemiring.{u1} K (Semifield.toDivisionSemiring.{u1} K (Field.toSemifield.{u1} K _inst_1))) p)) (Field.toSemifield.{u2} ((fun ([email protected]._hyg.2391 : K) => L) (Polynomial.leadingCoeff.{u1} K (DivisionSemiring.toSemiring.{u1} K (Semifield.toDivisionSemiring.{u1} K (Field.toSemifield.{u1} K _inst_1))) p)) _inst_2)))) (Polynomial.semiring.{u2} ((fun ([email protected]._hyg.2391 : K) => L) (Polynomial.leadingCoeff.{u1} K (DivisionSemiring.toSemiring.{u1} K (Semifield.toDivisionSemiring.{u1} K (Field.toSemifield.{u1} K _inst_1))) p)) (DivisionSemiring.toSemiring.{u2} ((fun ([email protected]._hyg.2391 : K) => L) (Polynomial.leadingCoeff.{u1} K (DivisionSemiring.toSemiring.{u1} K (Semifield.toDivisionSemiring.{u1} K (Field.toSemifield.{u1} K _inst_1))) p)) (Semifield.toDivisionSemiring.{u2} ((fun ([email protected]._hyg.2391 : K) => L) (Polynomial.leadingCoeff.{u1} K (DivisionSemiring.toSemiring.{u1} K (Semifield.toDivisionSemiring.{u1} K (Field.toSemifield.{u1} K _inst_1))) p)) (Field.toSemifield.{u2} ((fun ([email protected]._hyg.2391 : K) => L) (Polynomial.leadingCoeff.{u1} K (DivisionSemiring.toSemiring.{u1} K (Semifield.toDivisionSemiring.{u1} K (Field.toSemifield.{u1} K _inst_1))) p)) _inst_2))))))) (NonUnitalRingHomClass.toMulHomClass.{u2, u2, u2} (RingHom.{u2, u2} ((fun ([email protected]._hyg.2391 : K) => L) (Polynomial.leadingCoeff.{u1} K (DivisionSemiring.toSemiring.{u1} K (Semifield.toDivisionSemiring.{u1} K (Field.toSemifield.{u1} K _inst_1))) p)) (Polynomial.{u2} ((fun ([email protected]._hyg.2391 : K) => L) (Polynomial.leadingCoeff.{u1} K (DivisionSemiring.toSemiring.{u1} K (Semifield.toDivisionSemiring.{u1} K (Field.toSemifield.{u1} K _inst_1))) p)) (DivisionSemiring.toSemiring.{u2} ((fun ([email protected]._hyg.2391 : K) => L) (Polynomial.leadingCoeff.{u1} K (DivisionSemiring.toSemiring.{u1} K (Semifield.toDivisionSemiring.{u1} K (Field.toSemifield.{u1} K _inst_1))) p)) (Semifield.toDivisionSemiring.{u2} ((fun ([email protected]._hyg.2391 : K) => L) (Polynomial.leadingCoeff.{u1} K (DivisionSemiring.toSemiring.{u1} K (Semifield.toDivisionSemiring.{u1} K (Field.toSemifield.{u1} K _inst_1))) p)) (Field.toSemifield.{u2} ((fun ([email protected]._hyg.2391 : K) => L) (Polynomial.leadingCoeff.{u1} K (DivisionSemiring.toSemiring.{u1} K (Semifield.toDivisionSemiring.{u1} K (Field.toSemifield.{u1} K _inst_1))) p)) _inst_2)))) (Semiring.toNonAssocSemiring.{u2} ((fun ([email protected]._hyg.2391 : K) => L) (Polynomial.leadingCoeff.{u1} K (DivisionSemiring.toSemiring.{u1} K (Semifield.toDivisionSemiring.{u1} K (Field.toSemifield.{u1} K _inst_1))) p)) (DivisionSemiring.toSemiring.{u2} ((fun ([email protected]._hyg.2391 : K) => L) (Polynomial.leadingCoeff.{u1} K (DivisionSemiring.toSemiring.{u1} K (Semifield.toDivisionSemiring.{u1} K (Field.toSemifield.{u1} K _inst_1))) p)) (Semifield.toDivisionSemiring.{u2} ((fun ([email protected]._hyg.2391 : K) => L) (Polynomial.leadingCoeff.{u1} K (DivisionSemiring.toSemiring.{u1} K (Semifield.toDivisionSemiring.{u1} K (Field.toSemifield.{u1} K _inst_1))) p)) (Field.toSemifield.{u2} ((fun ([email protected]._hyg.2391 : K) => L) (Polynomial.leadingCoeff.{u1} K (DivisionSemiring.toSemiring.{u1} K (Semifield.toDivisionSemiring.{u1} K (Field.toSemifield.{u1} K _inst_1))) p)) _inst_2)))) (Semiring.toNonAssocSemiring.{u2} (Polynomial.{u2} ((fun ([email protected]._hyg.2391 : K) => L) (Polynomial.leadingCoeff.{u1} K (DivisionSemiring.toSemiring.{u1} K (Semifield.toDivisionSemiring.{u1} K (Field.toSemifield.{u1} K _inst_1))) p)) (DivisionSemiring.toSemiring.{u2} ((fun ([email protected]._hyg.2391 : K) => L) (Polynomial.leadingCoeff.{u1} K (DivisionSemiring.toSemiring.{u1} K (Semifield.toDivisionSemiring.{u1} K (Field.toSemifield.{u1} K _inst_1))) p)) (Semifield.toDivisionSemiring.{u2} ((fun ([email protected]._hyg.2391 : K) => L) (Polynomial.leadingCoeff.{u1} K (DivisionSemiring.toSemiring.{u1} K (Semifield.toDivisionSemiring.{u1} K (Field.toSemifield.{u1} K _inst_1))) p)) (Field.toSemifield.{u2} ((fun ([email protected]._hyg.2391 : K) => L) (Polynomial.leadingCoeff.{u1} K (DivisionSemiring.toSemiring.{u1} K (Semifield.toDivisionSemiring.{u1} K (Field.toSemifield.{u1} K _inst_1))) p)) _inst_2)))) (Polynomial.semiring.{u2} ((fun ([email protected]._hyg.2391 : K) => L) (Polynomial.leadingCoeff.{u1} K (DivisionSemiring.toSemiring.{u1} K (Semifield.toDivisionSemiring.{u1} K (Field.toSemifield.{u1} K _inst_1))) p)) (DivisionSemiring.toSemiring.{u2} ((fun ([email protected]._hyg.2391 : K) => L) (Polynomial.leadingCoeff.{u1} K (DivisionSemiring.toSemiring.{u1} K (Semifield.toDivisionSemiring.{u1} K (Field.toSemifield.{u1} K _inst_1))) p)) (Semifield.toDivisionSemiring.{u2} ((fun ([email protected]._hyg.2391 : K) => L) (Polynomial.leadingCoeff.{u1} K (DivisionSemiring.toSemiring.{u1} K (Semifield.toDivisionSemiring.{u1} K (Field.toSemifield.{u1} K _inst_1))) p)) (Field.toSemifield.{u2} ((fun ([email protected]._hyg.2391 : K) => L) (Polynomial.leadingCoeff.{u1} K (DivisionSemiring.toSemiring.{u1} K (Semifield.toDivisionSemiring.{u1} K (Field.toSemifield.{u1} K _inst_1))) p)) _inst_2)))))) ((fun ([email protected]._hyg.2391 : K) => L) (Polynomial.leadingCoeff.{u1} K (DivisionSemiring.toSemiring.{u1} K (Semifield.toDivisionSemiring.{u1} K (Field.toSemifield.{u1} K _inst_1))) p)) (Polynomial.{u2} ((fun ([email protected]._hyg.2391 : K) => L) (Polynomial.leadingCoeff.{u1} K (DivisionSemiring.toSemiring.{u1} K (Semifield.toDivisionSemiring.{u1} K (Field.toSemifield.{u1} K _inst_1))) p)) (DivisionSemiring.toSemiring.{u2} ((fun ([email protected]._hyg.2391 : K) => L) (Polynomial.leadingCoeff.{u1} K (DivisionSemiring.toSemiring.{u1} K (Semifield.toDivisionSemiring.{u1} K (Field.toSemifield.{u1} K _inst_1))) p)) (Semifield.toDivisionSemiring.{u2} ((fun ([email protected]._hyg.2391 : K) => L) (Polynomial.leadingCoeff.{u1} K (DivisionSemiring.toSemiring.{u1} K (Semifield.toDivisionSemiring.{u1} K (Field.toSemifield.{u1} K _inst_1))) p)) (Field.toSemifield.{u2} ((fun ([email protected]._hyg.2391 : K) => L) (Polynomial.leadingCoeff.{u1} K (DivisionSemiring.toSemiring.{u1} K (Semifield.toDivisionSemiring.{u1} K (Field.toSemifield.{u1} K _inst_1))) p)) _inst_2)))) (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} ((fun ([email protected]._hyg.2391 : K) => L) (Polynomial.leadingCoeff.{u1} K (DivisionSemiring.toSemiring.{u1} K (Semifield.toDivisionSemiring.{u1} K (Field.toSemifield.{u1} K _inst_1))) p)) (Semiring.toNonAssocSemiring.{u2} ((fun ([email protected]._hyg.2391 : K) => L) (Polynomial.leadingCoeff.{u1} K (DivisionSemiring.toSemiring.{u1} K (Semifield.toDivisionSemiring.{u1} K (Field.toSemifield.{u1} K _inst_1))) p)) (DivisionSemiring.toSemiring.{u2} ((fun ([email protected]._hyg.2391 : K) => L) (Polynomial.leadingCoeff.{u1} K (DivisionSemiring.toSemiring.{u1} K (Semifield.toDivisionSemiring.{u1} K (Field.toSemifield.{u1} K _inst_1))) p)) (Semifield.toDivisionSemiring.{u2} ((fun ([email protected]._hyg.2391 : K) => L) (Polynomial.leadingCoeff.{u1} K (DivisionSemiring.toSemiring.{u1} K (Semifield.toDivisionSemiring.{u1} K (Field.toSemifield.{u1} K _inst_1))) p)) (Field.toSemifield.{u2} ((fun ([email protected]._hyg.2391 : K) => L) (Polynomial.leadingCoeff.{u1} K (DivisionSemiring.toSemiring.{u1} K (Semifield.toDivisionSemiring.{u1} K (Field.toSemifield.{u1} K _inst_1))) p)) _inst_2))))) (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} (Polynomial.{u2} ((fun ([email protected]._hyg.2391 : K) => L) (Polynomial.leadingCoeff.{u1} K (DivisionSemiring.toSemiring.{u1} K (Semifield.toDivisionSemiring.{u1} K (Field.toSemifield.{u1} K _inst_1))) p)) (DivisionSemiring.toSemiring.{u2} ((fun ([email protected]._hyg.2391 : K) => L) (Polynomial.leadingCoeff.{u1} K (DivisionSemiring.toSemiring.{u1} K (Semifield.toDivisionSemiring.{u1} K (Field.toSemifield.{u1} K _inst_1))) p)) (Semifield.toDivisionSemiring.{u2} ((fun ([email protected]._hyg.2391 : K) => L) (Polynomial.leadingCoeff.{u1} K (DivisionSemiring.toSemiring.{u1} K (Semifield.toDivisionSemiring.{u1} K (Field.toSemifield.{u1} K _inst_1))) p)) (Field.toSemifield.{u2} ((fun ([email protected]._hyg.2391 : K) => L) (Polynomial.leadingCoeff.{u1} K (DivisionSemiring.toSemiring.{u1} K (Semifield.toDivisionSemiring.{u1} K (Field.toSemifield.{u1} K _inst_1))) p)) _inst_2)))) (Semiring.toNonAssocSemiring.{u2} (Polynomial.{u2} ((fun ([email protected]._hyg.2391 : K) => L) (Polynomial.leadingCoeff.{u1} K (DivisionSemiring.toSemiring.{u1} K (Semifield.toDivisionSemiring.{u1} K (Field.toSemifield.{u1} K _inst_1))) p)) (DivisionSemiring.toSemiring.{u2} ((fun ([email protected]._hyg.2391 : K) => L) (Polynomial.leadingCoeff.{u1} K (DivisionSemiring.toSemiring.{u1} K (Semifield.toDivisionSemiring.{u1} K (Field.toSemifield.{u1} K _inst_1))) p)) (Semifield.toDivisionSemiring.{u2} ((fun ([email protected]._hyg.2391 : K) => L) (Polynomial.leadingCoeff.{u1} K (DivisionSemiring.toSemiring.{u1} K (Semifield.toDivisionSemiring.{u1} K (Field.toSemifield.{u1} K _inst_1))) p)) (Field.toSemifield.{u2} ((fun ([email protected]._hyg.2391 : K) => L) (Polynomial.leadingCoeff.{u1} K (DivisionSemiring.toSemiring.{u1} K (Semifield.toDivisionSemiring.{u1} K (Field.toSemifield.{u1} K _inst_1))) p)) _inst_2)))) (Polynomial.semiring.{u2} ((fun ([email protected]._hyg.2391 : K) => L) (Polynomial.leadingCoeff.{u1} K (DivisionSemiring.toSemiring.{u1} K (Semifield.toDivisionSemiring.{u1} K (Field.toSemifield.{u1} K _inst_1))) p)) (DivisionSemiring.toSemiring.{u2} ((fun ([email protected]._hyg.2391 : K) => L) (Polynomial.leadingCoeff.{u1} K (DivisionSemiring.toSemiring.{u1} K (Semifield.toDivisionSemiring.{u1} K (Field.toSemifield.{u1} K _inst_1))) p)) (Semifield.toDivisionSemiring.{u2} ((fun ([email protected]._hyg.2391 : K) => L) (Polynomial.leadingCoeff.{u1} K (DivisionSemiring.toSemiring.{u1} K (Semifield.toDivisionSemiring.{u1} K (Field.toSemifield.{u1} K _inst_1))) p)) (Field.toSemifield.{u2} ((fun ([email protected]._hyg.2391 : K) => L) (Polynomial.leadingCoeff.{u1} K (DivisionSemiring.toSemiring.{u1} K (Semifield.toDivisionSemiring.{u1} K (Field.toSemifield.{u1} K _inst_1))) p)) _inst_2)))))) (RingHomClass.toNonUnitalRingHomClass.{u2, u2, u2} (RingHom.{u2, u2} ((fun ([email protected]._hyg.2391 : K) => L) (Polynomial.leadingCoeff.{u1} K (DivisionSemiring.toSemiring.{u1} K (Semifield.toDivisionSemiring.{u1} K (Field.toSemifield.{u1} K _inst_1))) p)) (Polynomial.{u2} ((fun ([email protected]._hyg.2391 : K) => L) (Polynomial.leadingCoeff.{u1} K (DivisionSemiring.toSemiring.{u1} K (Semifield.toDivisionSemiring.{u1} K (Field.toSemifield.{u1} K _inst_1))) p)) (DivisionSemiring.toSemiring.{u2} ((fun ([email protected]._hyg.2391 : K) => L) (Polynomial.leadingCoeff.{u1} K (DivisionSemiring.toSemiring.{u1} K (Semifield.toDivisionSemiring.{u1} K (Field.toSemifield.{u1} K _inst_1))) p)) (Semifield.toDivisionSemiring.{u2} ((fun ([email protected]._hyg.2391 : K) => L) (Polynomial.leadingCoeff.{u1} K (DivisionSemiring.toSemiring.{u1} K (Semifield.toDivisionSemiring.{u1} K (Field.toSemifield.{u1} K _inst_1))) p)) (Field.toSemifield.{u2} ((fun ([email protected]._hyg.2391 : K) => L) (Polynomial.leadingCoeff.{u1} K (DivisionSemiring.toSemiring.{u1} K (Semifield.toDivisionSemiring.{u1} K (Field.toSemifield.{u1} K _inst_1))) p)) _inst_2)))) (Semiring.toNonAssocSemiring.{u2} ((fun ([email protected]._hyg.2391 : K) => L) (Polynomial.leadingCoeff.{u1} K (DivisionSemiring.toSemiring.{u1} K (Semifield.toDivisionSemiring.{u1} K (Field.toSemifield.{u1} K _inst_1))) p)) (DivisionSemiring.toSemiring.{u2} ((fun ([email protected]._hyg.2391 : K) => L) (Polynomial.leadingCoeff.{u1} K (DivisionSemiring.toSemiring.{u1} K (Semifield.toDivisionSemiring.{u1} K (Field.toSemifield.{u1} K _inst_1))) p)) (Semifield.toDivisionSemiring.{u2} ((fun ([email protected]._hyg.2391 : K) => L) (Polynomial.leadingCoeff.{u1} K (DivisionSemiring.toSemiring.{u1} K (Semifield.toDivisionSemiring.{u1} K (Field.toSemifield.{u1} K _inst_1))) p)) (Field.toSemifield.{u2} ((fun ([email protected]._hyg.2391 : K) => L) (Polynomial.leadingCoeff.{u1} K (DivisionSemiring.toSemiring.{u1} K (Semifield.toDivisionSemiring.{u1} K (Field.toSemifield.{u1} K _inst_1))) p)) _inst_2)))) (Semiring.toNonAssocSemiring.{u2} (Polynomial.{u2} ((fun ([email protected]._hyg.2391 : K) => L) (Polynomial.leadingCoeff.{u1} K (DivisionSemiring.toSemiring.{u1} K (Semifield.toDivisionSemiring.{u1} K (Field.toSemifield.{u1} K _inst_1))) p)) (DivisionSemiring.toSemiring.{u2} ((fun ([email protected]._hyg.2391 : K) => L) (Polynomial.leadingCoeff.{u1} K (DivisionSemiring.toSemiring.{u1} K (Semifield.toDivisionSemiring.{u1} K (Field.toSemifield.{u1} K _inst_1))) p)) (Semifield.toDivisionSemiring.{u2} ((fun ([email protected]._hyg.2391 : K) => L) (Polynomial.leadingCoeff.{u1} K (DivisionSemiring.toSemiring.{u1} K (Semifield.toDivisionSemiring.{u1} K (Field.toSemifield.{u1} K _inst_1))) p)) (Field.toSemifield.{u2} ((fun ([email protected]._hyg.2391 : K) => L) (Polynomial.leadingCoeff.{u1} K (DivisionSemiring.toSemiring.{u1} K (Semifield.toDivisionSemiring.{u1} K (Field.toSemifield.{u1} K _inst_1))) p)) _inst_2)))) (Polynomial.semiring.{u2} ((fun ([email protected]._hyg.2391 : K) => L) (Polynomial.leadingCoeff.{u1} K (DivisionSemiring.toSemiring.{u1} K (Semifield.toDivisionSemiring.{u1} K (Field.toSemifield.{u1} K _inst_1))) p)) (DivisionSemiring.toSemiring.{u2} ((fun ([email protected]._hyg.2391 : K) => L) (Polynomial.leadingCoeff.{u1} K (DivisionSemiring.toSemiring.{u1} K (Semifield.toDivisionSemiring.{u1} K (Field.toSemifield.{u1} K _inst_1))) p)) (Semifield.toDivisionSemiring.{u2} ((fun ([email protected]._hyg.2391 : K) => L) (Polynomial.leadingCoeff.{u1} K (DivisionSemiring.toSemiring.{u1} K (Semifield.toDivisionSemiring.{u1} K (Field.toSemifield.{u1} K _inst_1))) p)) (Field.toSemifield.{u2} ((fun ([email protected]._hyg.2391 : K) => L) (Polynomial.leadingCoeff.{u1} K (DivisionSemiring.toSemiring.{u1} K (Semifield.toDivisionSemiring.{u1} K (Field.toSemifield.{u1} K _inst_1))) p)) _inst_2)))))) ((fun ([email protected]._hyg.2391 : K) => L) (Polynomial.leadingCoeff.{u1} K (DivisionSemiring.toSemiring.{u1} K (Semifield.toDivisionSemiring.{u1} K (Field.toSemifield.{u1} K _inst_1))) p)) (Polynomial.{u2} ((fun ([email protected]._hyg.2391 : K) => L) (Polynomial.leadingCoeff.{u1} K (DivisionSemiring.toSemiring.{u1} K (Semifield.toDivisionSemiring.{u1} K (Field.toSemifield.{u1} K _inst_1))) p)) (DivisionSemiring.toSemiring.{u2} ((fun ([email protected]._hyg.2391 : K) => L) (Polynomial.leadingCoeff.{u1} K (DivisionSemiring.toSemiring.{u1} K (Semifield.toDivisionSemiring.{u1} K (Field.toSemifield.{u1} K _inst_1))) p)) (Semifield.toDivisionSemiring.{u2} ((fun ([email protected]._hyg.2391 : K) => L) (Polynomial.leadingCoeff.{u1} K (DivisionSemiring.toSemiring.{u1} K (Semifield.toDivisionSemiring.{u1} K (Field.toSemifield.{u1} K _inst_1))) p)) (Field.toSemifield.{u2} ((fun ([email protected]._hyg.2391 : K) => L) (Polynomial.leadingCoeff.{u1} K (DivisionSemiring.toSemiring.{u1} K (Semifield.toDivisionSemiring.{u1} K (Field.toSemifield.{u1} K _inst_1))) p)) _inst_2)))) (Semiring.toNonAssocSemiring.{u2} ((fun ([email protected]._hyg.2391 : K) => L) (Polynomial.leadingCoeff.{u1} K (DivisionSemiring.toSemiring.{u1} K (Semifield.toDivisionSemiring.{u1} K (Field.toSemifield.{u1} K _inst_1))) p)) (DivisionSemiring.toSemiring.{u2} ((fun ([email protected]._hyg.2391 : K) => L) (Polynomial.leadingCoeff.{u1} K (DivisionSemiring.toSemiring.{u1} K (Semifield.toDivisionSemiring.{u1} K (Field.toSemifield.{u1} K _inst_1))) p)) (Semifield.toDivisionSemiring.{u2} ((fun ([email protected]._hyg.2391 : K) => L) (Polynomial.leadingCoeff.{u1} K (DivisionSemiring.toSemiring.{u1} K (Semifield.toDivisionSemiring.{u1} K (Field.toSemifield.{u1} K _inst_1))) p)) (Field.toSemifield.{u2} ((fun ([email protected]._hyg.2391 : K) => L) (Polynomial.leadingCoeff.{u1} K (DivisionSemiring.toSemiring.{u1} K (Semifield.toDivisionSemiring.{u1} K (Field.toSemifield.{u1} K _inst_1))) p)) _inst_2)))) (Semiring.toNonAssocSemiring.{u2} (Polynomial.{u2} ((fun ([email protected]._hyg.2391 : K) => L) (Polynomial.leadingCoeff.{u1} K (DivisionSemiring.toSemiring.{u1} K (Semifield.toDivisionSemiring.{u1} K (Field.toSemifield.{u1} K _inst_1))) p)) (DivisionSemiring.toSemiring.{u2} ((fun ([email protected]._hyg.2391 : K) => L) (Polynomial.leadingCoeff.{u1} K (DivisionSemiring.toSemiring.{u1} K (Semifield.toDivisionSemiring.{u1} K (Field.toSemifield.{u1} K _inst_1))) p)) (Semifield.toDivisionSemiring.{u2} ((fun ([email protected]._hyg.2391 : K) => L) (Polynomial.leadingCoeff.{u1} K (DivisionSemiring.toSemiring.{u1} K (Semifield.toDivisionSemiring.{u1} K (Field.toSemifield.{u1} K _inst_1))) p)) (Field.toSemifield.{u2} ((fun ([email protected]._hyg.2391 : K) => L) (Polynomial.leadingCoeff.{u1} K (DivisionSemiring.toSemiring.{u1} K (Semifield.toDivisionSemiring.{u1} K (Field.toSemifield.{u1} K _inst_1))) p)) _inst_2)))) (Polynomial.semiring.{u2} ((fun ([email protected]._hyg.2391 : K) => L) (Polynomial.leadingCoeff.{u1} K (DivisionSemiring.toSemiring.{u1} K (Semifield.toDivisionSemiring.{u1} K (Field.toSemifield.{u1} K _inst_1))) p)) (DivisionSemiring.toSemiring.{u2} ((fun ([email protected]._hyg.2391 : K) => L) (Polynomial.leadingCoeff.{u1} K (DivisionSemiring.toSemiring.{u1} K (Semifield.toDivisionSemiring.{u1} K (Field.toSemifield.{u1} K _inst_1))) p)) (Semifield.toDivisionSemiring.{u2} ((fun ([email protected]._hyg.2391 : K) => L) (Polynomial.leadingCoeff.{u1} K (DivisionSemiring.toSemiring.{u1} K (Semifield.toDivisionSemiring.{u1} K (Field.toSemifield.{u1} K _inst_1))) p)) (Field.toSemifield.{u2} ((fun ([email protected]._hyg.2391 : K) => L) (Polynomial.leadingCoeff.{u1} K (DivisionSemiring.toSemiring.{u1} K (Semifield.toDivisionSemiring.{u1} K (Field.toSemifield.{u1} K _inst_1))) p)) _inst_2))))) (RingHom.instRingHomClassRingHom.{u2, u2} ((fun ([email protected]._hyg.2391 : K) => L) (Polynomial.leadingCoeff.{u1} K (DivisionSemiring.toSemiring.{u1} K (Semifield.toDivisionSemiring.{u1} K (Field.toSemifield.{u1} K _inst_1))) p)) (Polynomial.{u2} ((fun ([email protected]._hyg.2391 : K) => L) (Polynomial.leadingCoeff.{u1} K (DivisionSemiring.toSemiring.{u1} K (Semifield.toDivisionSemiring.{u1} K (Field.toSemifield.{u1} K _inst_1))) p)) (DivisionSemiring.toSemiring.{u2} ((fun ([email protected]._hyg.2391 : K) => L) (Polynomial.leadingCoeff.{u1} K (DivisionSemiring.toSemiring.{u1} K (Semifield.toDivisionSemiring.{u1} K (Field.toSemifield.{u1} K _inst_1))) p)) (Semifield.toDivisionSemiring.{u2} ((fun ([email protected]._hyg.2391 : K) => L) (Polynomial.leadingCoeff.{u1} K (DivisionSemiring.toSemiring.{u1} K (Semifield.toDivisionSemiring.{u1} K (Field.toSemifield.{u1} K _inst_1))) p)) (Field.toSemifield.{u2} ((fun ([email protected]._hyg.2391 : K) => L) (Polynomial.leadingCoeff.{u1} K (DivisionSemiring.toSemiring.{u1} K (Semifield.toDivisionSemiring.{u1} K (Field.toSemifield.{u1} K _inst_1))) p)) _inst_2)))) (Semiring.toNonAssocSemiring.{u2} ((fun ([email protected]._hyg.2391 : K) => L) (Polynomial.leadingCoeff.{u1} K (DivisionSemiring.toSemiring.{u1} K (Semifield.toDivisionSemiring.{u1} K (Field.toSemifield.{u1} K _inst_1))) p)) (DivisionSemiring.toSemiring.{u2} ((fun ([email protected]._hyg.2391 : K) => L) (Polynomial.leadingCoeff.{u1} K (DivisionSemiring.toSemiring.{u1} K (Semifield.toDivisionSemiring.{u1} K (Field.toSemifield.{u1} K _inst_1))) p)) (Semifield.toDivisionSemiring.{u2} ((fun ([email protected]._hyg.2391 : K) => L) (Polynomial.leadingCoeff.{u1} K (DivisionSemiring.toSemiring.{u1} K (Semifield.toDivisionSemiring.{u1} K (Field.toSemifield.{u1} K _inst_1))) p)) (Field.toSemifield.{u2} ((fun ([email protected]._hyg.2391 : K) => L) (Polynomial.leadingCoeff.{u1} K (DivisionSemiring.toSemiring.{u1} K (Semifield.toDivisionSemiring.{u1} K (Field.toSemifield.{u1} K _inst_1))) p)) _inst_2)))) (Semiring.toNonAssocSemiring.{u2} (Polynomial.{u2} ((fun ([email protected]._hyg.2391 : K) => L) (Polynomial.leadingCoeff.{u1} K (DivisionSemiring.toSemiring.{u1} K (Semifield.toDivisionSemiring.{u1} K (Field.toSemifield.{u1} K _inst_1))) p)) (DivisionSemiring.toSemiring.{u2} ((fun ([email protected]._hyg.2391 : K) => L) (Polynomial.leadingCoeff.{u1} K (DivisionSemiring.toSemiring.{u1} K (Semifield.toDivisionSemiring.{u1} K (Field.toSemifield.{u1} K _inst_1))) p)) (Semifield.toDivisionSemiring.{u2} ((fun ([email protected]._hyg.2391 : K) => L) (Polynomial.leadingCoeff.{u1} K (DivisionSemiring.toSemiring.{u1} K (Semifield.toDivisionSemiring.{u1} K (Field.toSemifield.{u1} K _inst_1))) p)) (Field.toSemifield.{u2} ((fun 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Case conversion may be inaccurate. Consider using '#align polynomial.eq_prod_roots_of_splits Polynomial.eq_prod_roots_of_splitsₓ'. -/
theorem eq_prod_roots_of_splits {p : K[X]} {i : K →+* L} (hsplit : Splits i p) :
p.map i = C (i p.leadingCoeff) * ((p.map i).roots.map fun a => X - C a).Prod :=
by
rw [← leading_coeff_map]; symm
apply C_leading_coeff_mul_prod_multiset_X_sub_C
rw [nat_degree_map]; exact (nat_degree_eq_card_roots hsplit).symm
#align polynomial.eq_prod_roots_of_splits Polynomial.eq_prod_roots_of_splits
/- warning: polynomial.eq_prod_roots_of_splits_id -> Polynomial.eq_prod_roots_of_splits_id is a dubious translation:
lean 3 declaration is
forall {K : Type.{u1}} [_inst_1 : Field.{u1} K] {p : Polynomial.{u1} K (Ring.toSemiring.{u1} K (DivisionRing.toRing.{u1} K (Field.toDivisionRing.{u1} K _inst_1)))}, (Polynomial.Splits.{u1, u1} K K (EuclideanDomain.toCommRing.{u1} K (Field.toEuclideanDomain.{u1} K _inst_1)) _inst_1 (RingHom.id.{u1} K (NonAssocRing.toNonAssocSemiring.{u1} K (Ring.toNonAssocRing.{u1} K (DivisionRing.toRing.{u1} K (Field.toDivisionRing.{u1} K _inst_1))))) p) -> (Eq.{succ u1} (Polynomial.{u1} K (Ring.toSemiring.{u1} K (DivisionRing.toRing.{u1} K (Field.toDivisionRing.{u1} K _inst_1)))) p (HMul.hMul.{u1, u1, u1} (Polynomial.{u1} K (Ring.toSemiring.{u1} K (DivisionRing.toRing.{u1} K (Field.toDivisionRing.{u1} K _inst_1)))) (Polynomial.{u1} K (Ring.toSemiring.{u1} K (DivisionRing.toRing.{u1} K (Field.toDivisionRing.{u1} K _inst_1)))) (Polynomial.{u1} K (Ring.toSemiring.{u1} K (DivisionRing.toRing.{u1} K (Field.toDivisionRing.{u1} K _inst_1)))) (instHMul.{u1} (Polynomial.{u1} K (Ring.toSemiring.{u1} K (DivisionRing.toRing.{u1} K (Field.toDivisionRing.{u1} K _inst_1)))) (Polynomial.mul'.{u1} K (Ring.toSemiring.{u1} K (DivisionRing.toRing.{u1} K (Field.toDivisionRing.{u1} K _inst_1))))) (coeFn.{succ u1, succ u1} (RingHom.{u1, u1} K (Polynomial.{u1} K (Ring.toSemiring.{u1} K (DivisionRing.toRing.{u1} K (Field.toDivisionRing.{u1} K _inst_1)))) (Semiring.toNonAssocSemiring.{u1} K (Ring.toSemiring.{u1} K (DivisionRing.toRing.{u1} K (Field.toDivisionRing.{u1} K _inst_1)))) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} K (Ring.toSemiring.{u1} K (DivisionRing.toRing.{u1} K (Field.toDivisionRing.{u1} K _inst_1)))) (Polynomial.semiring.{u1} K (Ring.toSemiring.{u1} K (DivisionRing.toRing.{u1} K (Field.toDivisionRing.{u1} K _inst_1)))))) (fun (_x : RingHom.{u1, u1} K (Polynomial.{u1} K (Ring.toSemiring.{u1} K (DivisionRing.toRing.{u1} K (Field.toDivisionRing.{u1} K _inst_1)))) (Semiring.toNonAssocSemiring.{u1} K (Ring.toSemiring.{u1} K (DivisionRing.toRing.{u1} K (Field.toDivisionRing.{u1} K _inst_1)))) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} K (Ring.toSemiring.{u1} K (DivisionRing.toRing.{u1} K (Field.toDivisionRing.{u1} K _inst_1)))) (Polynomial.semiring.{u1} K (Ring.toSemiring.{u1} K (DivisionRing.toRing.{u1} K (Field.toDivisionRing.{u1} K _inst_1)))))) => K -> (Polynomial.{u1} K (Ring.toSemiring.{u1} K (DivisionRing.toRing.{u1} K (Field.toDivisionRing.{u1} K _inst_1))))) (RingHom.hasCoeToFun.{u1, u1} K (Polynomial.{u1} K (Ring.toSemiring.{u1} K (DivisionRing.toRing.{u1} K (Field.toDivisionRing.{u1} K _inst_1)))) (Semiring.toNonAssocSemiring.{u1} K (Ring.toSemiring.{u1} K (DivisionRing.toRing.{u1} K (Field.toDivisionRing.{u1} K _inst_1)))) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} K (Ring.toSemiring.{u1} K (DivisionRing.toRing.{u1} K (Field.toDivisionRing.{u1} K _inst_1)))) (Polynomial.semiring.{u1} K (Ring.toSemiring.{u1} K (DivisionRing.toRing.{u1} K (Field.toDivisionRing.{u1} K _inst_1)))))) (Polynomial.C.{u1} K (Ring.toSemiring.{u1} K (DivisionRing.toRing.{u1} K (Field.toDivisionRing.{u1} K _inst_1)))) (Polynomial.leadingCoeff.{u1} K (Ring.toSemiring.{u1} K (DivisionRing.toRing.{u1} K (Field.toDivisionRing.{u1} K _inst_1))) p)) (Multiset.prod.{u1} (Polynomial.{u1} K (Ring.toSemiring.{u1} K (DivisionRing.toRing.{u1} K (Field.toDivisionRing.{u1} K _inst_1)))) (CommRing.toCommMonoid.{u1} (Polynomial.{u1} K (Ring.toSemiring.{u1} K (DivisionRing.toRing.{u1} K (Field.toDivisionRing.{u1} K _inst_1)))) (Polynomial.commRing.{u1} K (EuclideanDomain.toCommRing.{u1} K (Field.toEuclideanDomain.{u1} K _inst_1)))) (Multiset.map.{u1, u1} K (Polynomial.{u1} K (Ring.toSemiring.{u1} K (DivisionRing.toRing.{u1} K (Field.toDivisionRing.{u1} K _inst_1)))) (fun (a : K) => HSub.hSub.{u1, u1, u1} (Polynomial.{u1} K (Ring.toSemiring.{u1} K (DivisionRing.toRing.{u1} K (Field.toDivisionRing.{u1} K _inst_1)))) (Polynomial.{u1} K (Ring.toSemiring.{u1} K (DivisionRing.toRing.{u1} K (Field.toDivisionRing.{u1} K _inst_1)))) (Polynomial.{u1} K (Ring.toSemiring.{u1} K (DivisionRing.toRing.{u1} K (Field.toDivisionRing.{u1} K _inst_1)))) (instHSub.{u1} (Polynomial.{u1} K (Ring.toSemiring.{u1} K (DivisionRing.toRing.{u1} K (Field.toDivisionRing.{u1} K _inst_1)))) (Polynomial.sub.{u1} K (DivisionRing.toRing.{u1} K (Field.toDivisionRing.{u1} K _inst_1)))) (Polynomial.X.{u1} K (Ring.toSemiring.{u1} K (DivisionRing.toRing.{u1} K (Field.toDivisionRing.{u1} K _inst_1)))) (coeFn.{succ u1, succ u1} (RingHom.{u1, u1} K (Polynomial.{u1} K (Ring.toSemiring.{u1} K (DivisionRing.toRing.{u1} K (Field.toDivisionRing.{u1} K _inst_1)))) (Semiring.toNonAssocSemiring.{u1} K (Ring.toSemiring.{u1} K (DivisionRing.toRing.{u1} K (Field.toDivisionRing.{u1} K _inst_1)))) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} K (Ring.toSemiring.{u1} K (DivisionRing.toRing.{u1} K (Field.toDivisionRing.{u1} K _inst_1)))) (Polynomial.semiring.{u1} K (Ring.toSemiring.{u1} K (DivisionRing.toRing.{u1} K (Field.toDivisionRing.{u1} K _inst_1)))))) (fun (_x : RingHom.{u1, u1} K (Polynomial.{u1} K (Ring.toSemiring.{u1} K (DivisionRing.toRing.{u1} K (Field.toDivisionRing.{u1} K _inst_1)))) (Semiring.toNonAssocSemiring.{u1} K (Ring.toSemiring.{u1} K (DivisionRing.toRing.{u1} K (Field.toDivisionRing.{u1} K _inst_1)))) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} K (Ring.toSemiring.{u1} K (DivisionRing.toRing.{u1} K (Field.toDivisionRing.{u1} K _inst_1)))) (Polynomial.semiring.{u1} K (Ring.toSemiring.{u1} K (DivisionRing.toRing.{u1} K (Field.toDivisionRing.{u1} K _inst_1)))))) => K -> (Polynomial.{u1} K (Ring.toSemiring.{u1} K (DivisionRing.toRing.{u1} K (Field.toDivisionRing.{u1} K _inst_1))))) (RingHom.hasCoeToFun.{u1, u1} K (Polynomial.{u1} K (Ring.toSemiring.{u1} K (DivisionRing.toRing.{u1} K (Field.toDivisionRing.{u1} K _inst_1)))) (Semiring.toNonAssocSemiring.{u1} K (Ring.toSemiring.{u1} K (DivisionRing.toRing.{u1} K (Field.toDivisionRing.{u1} K _inst_1)))) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} K (Ring.toSemiring.{u1} K (DivisionRing.toRing.{u1} K (Field.toDivisionRing.{u1} K _inst_1)))) (Polynomial.semiring.{u1} K (Ring.toSemiring.{u1} K (DivisionRing.toRing.{u1} K (Field.toDivisionRing.{u1} K _inst_1)))))) (Polynomial.C.{u1} K (Ring.toSemiring.{u1} K (DivisionRing.toRing.{u1} K (Field.toDivisionRing.{u1} K _inst_1)))) a)) (Polynomial.roots.{u1} K (EuclideanDomain.toCommRing.{u1} K (Field.toEuclideanDomain.{u1} K _inst_1)) (Field.isDomain.{u1} K _inst_1) p)))))
but is expected to have type
forall {K : Type.{u1}} [_inst_1 : Field.{u1} K] {p : Polynomial.{u1} K (DivisionSemiring.toSemiring.{u1} K (Semifield.toDivisionSemiring.{u1} K (Field.toSemifield.{u1} K _inst_1)))}, (Polynomial.Splits.{u1, u1} K K (EuclideanDomain.toCommRing.{u1} K (Field.toEuclideanDomain.{u1} K _inst_1)) _inst_1 (RingHom.id.{u1} K (NonAssocRing.toNonAssocSemiring.{u1} K (Ring.toNonAssocRing.{u1} K (DivisionRing.toRing.{u1} K (Field.toDivisionRing.{u1} K _inst_1))))) p) -> (Eq.{succ u1} (Polynomial.{u1} K (DivisionSemiring.toSemiring.{u1} K (Semifield.toDivisionSemiring.{u1} K (Field.toSemifield.{u1} K _inst_1)))) p (HMul.hMul.{u1, u1, u1} ((fun ([email protected]._hyg.2391 : K) => Polynomial.{u1} K (DivisionSemiring.toSemiring.{u1} K (Semifield.toDivisionSemiring.{u1} K (Field.toSemifield.{u1} K _inst_1)))) (Polynomial.leadingCoeff.{u1} K (DivisionSemiring.toSemiring.{u1} K (Semifield.toDivisionSemiring.{u1} K (Field.toSemifield.{u1} K _inst_1))) p)) (Polynomial.{u1} K (DivisionSemiring.toSemiring.{u1} K (Semifield.toDivisionSemiring.{u1} K (Field.toSemifield.{u1} K _inst_1)))) ((fun ([email protected]._hyg.2391 : K) => Polynomial.{u1} K (DivisionSemiring.toSemiring.{u1} K (Semifield.toDivisionSemiring.{u1} K (Field.toSemifield.{u1} K _inst_1)))) (Polynomial.leadingCoeff.{u1} K (DivisionSemiring.toSemiring.{u1} K (Semifield.toDivisionSemiring.{u1} K (Field.toSemifield.{u1} K _inst_1))) p)) (instHMul.{u1} ((fun ([email protected]._hyg.2391 : K) => Polynomial.{u1} K (DivisionSemiring.toSemiring.{u1} K (Semifield.toDivisionSemiring.{u1} K (Field.toSemifield.{u1} K _inst_1)))) (Polynomial.leadingCoeff.{u1} K (DivisionSemiring.toSemiring.{u1} K (Semifield.toDivisionSemiring.{u1} K (Field.toSemifield.{u1} K _inst_1))) p)) (Polynomial.mul'.{u1} K (DivisionSemiring.toSemiring.{u1} K (Semifield.toDivisionSemiring.{u1} K (Field.toSemifield.{u1} K _inst_1))))) (FunLike.coe.{succ u1, succ u1, succ u1} (RingHom.{u1, u1} K (Polynomial.{u1} K (DivisionSemiring.toSemiring.{u1} K (Semifield.toDivisionSemiring.{u1} K (Field.toSemifield.{u1} K _inst_1)))) (Semiring.toNonAssocSemiring.{u1} K (DivisionSemiring.toSemiring.{u1} K (Semifield.toDivisionSemiring.{u1} K (Field.toSemifield.{u1} K _inst_1)))) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} K (DivisionSemiring.toSemiring.{u1} K (Semifield.toDivisionSemiring.{u1} K (Field.toSemifield.{u1} K _inst_1)))) (Polynomial.semiring.{u1} K (DivisionSemiring.toSemiring.{u1} K (Semifield.toDivisionSemiring.{u1} K (Field.toSemifield.{u1} K _inst_1)))))) K (fun (_x : K) => (fun ([email protected]._hyg.2391 : K) => Polynomial.{u1} K (DivisionSemiring.toSemiring.{u1} K (Semifield.toDivisionSemiring.{u1} K (Field.toSemifield.{u1} K _inst_1)))) _x) (MulHomClass.toFunLike.{u1, u1, u1} (RingHom.{u1, u1} K (Polynomial.{u1} K (DivisionSemiring.toSemiring.{u1} K (Semifield.toDivisionSemiring.{u1} K (Field.toSemifield.{u1} K _inst_1)))) (Semiring.toNonAssocSemiring.{u1} K (DivisionSemiring.toSemiring.{u1} K (Semifield.toDivisionSemiring.{u1} K (Field.toSemifield.{u1} K _inst_1)))) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} K (DivisionSemiring.toSemiring.{u1} K (Semifield.toDivisionSemiring.{u1} K (Field.toSemifield.{u1} K _inst_1)))) (Polynomial.semiring.{u1} K (DivisionSemiring.toSemiring.{u1} K (Semifield.toDivisionSemiring.{u1} K (Field.toSemifield.{u1} K _inst_1)))))) K (Polynomial.{u1} K (DivisionSemiring.toSemiring.{u1} K (Semifield.toDivisionSemiring.{u1} K (Field.toSemifield.{u1} K _inst_1)))) (NonUnitalNonAssocSemiring.toMul.{u1} K (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} K (Semiring.toNonAssocSemiring.{u1} K (DivisionSemiring.toSemiring.{u1} K (Semifield.toDivisionSemiring.{u1} K (Field.toSemifield.{u1} K _inst_1)))))) (NonUnitalNonAssocSemiring.toMul.{u1} (Polynomial.{u1} K (DivisionSemiring.toSemiring.{u1} K (Semifield.toDivisionSemiring.{u1} K (Field.toSemifield.{u1} K _inst_1)))) (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} (Polynomial.{u1} K (DivisionSemiring.toSemiring.{u1} K (Semifield.toDivisionSemiring.{u1} K (Field.toSemifield.{u1} K _inst_1)))) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} K (DivisionSemiring.toSemiring.{u1} K (Semifield.toDivisionSemiring.{u1} K (Field.toSemifield.{u1} K _inst_1)))) (Polynomial.semiring.{u1} K (DivisionSemiring.toSemiring.{u1} K (Semifield.toDivisionSemiring.{u1} K (Field.toSemifield.{u1} K _inst_1))))))) (NonUnitalRingHomClass.toMulHomClass.{u1, u1, u1} (RingHom.{u1, u1} K (Polynomial.{u1} K (DivisionSemiring.toSemiring.{u1} K (Semifield.toDivisionSemiring.{u1} K (Field.toSemifield.{u1} K _inst_1)))) (Semiring.toNonAssocSemiring.{u1} K (DivisionSemiring.toSemiring.{u1} K (Semifield.toDivisionSemiring.{u1} K (Field.toSemifield.{u1} K _inst_1)))) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} K (DivisionSemiring.toSemiring.{u1} K (Semifield.toDivisionSemiring.{u1} K (Field.toSemifield.{u1} K _inst_1)))) (Polynomial.semiring.{u1} K (DivisionSemiring.toSemiring.{u1} K (Semifield.toDivisionSemiring.{u1} K (Field.toSemifield.{u1} K _inst_1)))))) K (Polynomial.{u1} K (DivisionSemiring.toSemiring.{u1} K (Semifield.toDivisionSemiring.{u1} K (Field.toSemifield.{u1} K _inst_1)))) (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} K (Semiring.toNonAssocSemiring.{u1} K (DivisionSemiring.toSemiring.{u1} K (Semifield.toDivisionSemiring.{u1} K (Field.toSemifield.{u1} K _inst_1))))) (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} (Polynomial.{u1} K (DivisionSemiring.toSemiring.{u1} K (Semifield.toDivisionSemiring.{u1} K (Field.toSemifield.{u1} K _inst_1)))) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} K (DivisionSemiring.toSemiring.{u1} K (Semifield.toDivisionSemiring.{u1} K (Field.toSemifield.{u1} K _inst_1)))) (Polynomial.semiring.{u1} K (DivisionSemiring.toSemiring.{u1} K (Semifield.toDivisionSemiring.{u1} K (Field.toSemifield.{u1} K _inst_1)))))) (RingHomClass.toNonUnitalRingHomClass.{u1, u1, u1} (RingHom.{u1, u1} K (Polynomial.{u1} K (DivisionSemiring.toSemiring.{u1} K (Semifield.toDivisionSemiring.{u1} K (Field.toSemifield.{u1} K _inst_1)))) (Semiring.toNonAssocSemiring.{u1} K (DivisionSemiring.toSemiring.{u1} K (Semifield.toDivisionSemiring.{u1} K (Field.toSemifield.{u1} K _inst_1)))) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} K (DivisionSemiring.toSemiring.{u1} K (Semifield.toDivisionSemiring.{u1} K (Field.toSemifield.{u1} K _inst_1)))) (Polynomial.semiring.{u1} K (DivisionSemiring.toSemiring.{u1} K (Semifield.toDivisionSemiring.{u1} K (Field.toSemifield.{u1} K _inst_1)))))) K (Polynomial.{u1} K (DivisionSemiring.toSemiring.{u1} K (Semifield.toDivisionSemiring.{u1} K (Field.toSemifield.{u1} K _inst_1)))) (Semiring.toNonAssocSemiring.{u1} K (DivisionSemiring.toSemiring.{u1} K (Semifield.toDivisionSemiring.{u1} K (Field.toSemifield.{u1} K _inst_1)))) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} K (DivisionSemiring.toSemiring.{u1} K (Semifield.toDivisionSemiring.{u1} K (Field.toSemifield.{u1} K _inst_1)))) (Polynomial.semiring.{u1} K (DivisionSemiring.toSemiring.{u1} K (Semifield.toDivisionSemiring.{u1} K (Field.toSemifield.{u1} K _inst_1))))) (RingHom.instRingHomClassRingHom.{u1, u1} K (Polynomial.{u1} K (DivisionSemiring.toSemiring.{u1} K (Semifield.toDivisionSemiring.{u1} K (Field.toSemifield.{u1} K _inst_1)))) (Semiring.toNonAssocSemiring.{u1} K (DivisionSemiring.toSemiring.{u1} K (Semifield.toDivisionSemiring.{u1} K (Field.toSemifield.{u1} K _inst_1)))) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} K (DivisionSemiring.toSemiring.{u1} K (Semifield.toDivisionSemiring.{u1} K (Field.toSemifield.{u1} K _inst_1)))) (Polynomial.semiring.{u1} K (DivisionSemiring.toSemiring.{u1} K (Semifield.toDivisionSemiring.{u1} K (Field.toSemifield.{u1} K _inst_1))))))))) (Polynomial.C.{u1} K (DivisionSemiring.toSemiring.{u1} K (Semifield.toDivisionSemiring.{u1} K (Field.toSemifield.{u1} K _inst_1)))) (Polynomial.leadingCoeff.{u1} K (DivisionSemiring.toSemiring.{u1} K (Semifield.toDivisionSemiring.{u1} K (Field.toSemifield.{u1} K _inst_1))) p)) (Multiset.prod.{u1} (Polynomial.{u1} K (DivisionSemiring.toSemiring.{u1} K (Semifield.toDivisionSemiring.{u1} K (Field.toSemifield.{u1} K _inst_1)))) (CommRing.toCommMonoid.{u1} (Polynomial.{u1} K (DivisionSemiring.toSemiring.{u1} K (Semifield.toDivisionSemiring.{u1} K (Field.toSemifield.{u1} K _inst_1)))) (EuclideanDomain.toCommRing.{u1} (Polynomial.{u1} K (DivisionSemiring.toSemiring.{u1} K (Semifield.toDivisionSemiring.{u1} K (Field.toSemifield.{u1} K _inst_1)))) (Polynomial.instEuclideanDomainPolynomialToSemiringToDivisionSemiringToSemifield.{u1} K _inst_1))) (Multiset.map.{u1, u1} K (Polynomial.{u1} K (DivisionSemiring.toSemiring.{u1} K (Semifield.toDivisionSemiring.{u1} K (Field.toSemifield.{u1} K _inst_1)))) (fun (a : K) => HSub.hSub.{u1, u1, u1} (Polynomial.{u1} K (DivisionSemiring.toSemiring.{u1} K (Semifield.toDivisionSemiring.{u1} K (Field.toSemifield.{u1} K _inst_1)))) ((fun ([email protected]._hyg.2391 : K) => Polynomial.{u1} K (DivisionSemiring.toSemiring.{u1} K (Semifield.toDivisionSemiring.{u1} K (Field.toSemifield.{u1} K _inst_1)))) a) (Polynomial.{u1} K (DivisionSemiring.toSemiring.{u1} K (Semifield.toDivisionSemiring.{u1} K (Field.toSemifield.{u1} K _inst_1)))) (instHSub.{u1} (Polynomial.{u1} K (DivisionSemiring.toSemiring.{u1} K (Semifield.toDivisionSemiring.{u1} K (Field.toSemifield.{u1} K _inst_1)))) (Polynomial.sub.{u1} K (DivisionRing.toRing.{u1} K (Field.toDivisionRing.{u1} K _inst_1)))) (Polynomial.X.{u1} K (DivisionSemiring.toSemiring.{u1} K (Semifield.toDivisionSemiring.{u1} K (Field.toSemifield.{u1} K _inst_1)))) (FunLike.coe.{succ u1, succ u1, succ u1} (RingHom.{u1, u1} K (Polynomial.{u1} K (DivisionSemiring.toSemiring.{u1} K (Semifield.toDivisionSemiring.{u1} K (Field.toSemifield.{u1} K _inst_1)))) (Semiring.toNonAssocSemiring.{u1} K (DivisionSemiring.toSemiring.{u1} K (Semifield.toDivisionSemiring.{u1} K (Field.toSemifield.{u1} K _inst_1)))) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} K (DivisionSemiring.toSemiring.{u1} K (Semifield.toDivisionSemiring.{u1} K (Field.toSemifield.{u1} K _inst_1)))) (Polynomial.semiring.{u1} K (DivisionSemiring.toSemiring.{u1} K (Semifield.toDivisionSemiring.{u1} K (Field.toSemifield.{u1} K _inst_1)))))) K (fun (_x : K) => (fun ([email protected]._hyg.2391 : K) => Polynomial.{u1} K (DivisionSemiring.toSemiring.{u1} K (Semifield.toDivisionSemiring.{u1} K (Field.toSemifield.{u1} K _inst_1)))) _x) (MulHomClass.toFunLike.{u1, u1, u1} (RingHom.{u1, u1} K (Polynomial.{u1} K (DivisionSemiring.toSemiring.{u1} K (Semifield.toDivisionSemiring.{u1} K (Field.toSemifield.{u1} K _inst_1)))) (Semiring.toNonAssocSemiring.{u1} K (DivisionSemiring.toSemiring.{u1} K (Semifield.toDivisionSemiring.{u1} K (Field.toSemifield.{u1} K _inst_1)))) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} K 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(Field.toSemifield.{u1} K _inst_1))))) (RingHom.instRingHomClassRingHom.{u1, u1} K (Polynomial.{u1} K (DivisionSemiring.toSemiring.{u1} K (Semifield.toDivisionSemiring.{u1} K (Field.toSemifield.{u1} K _inst_1)))) (Semiring.toNonAssocSemiring.{u1} K (DivisionSemiring.toSemiring.{u1} K (Semifield.toDivisionSemiring.{u1} K (Field.toSemifield.{u1} K _inst_1)))) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} K (DivisionSemiring.toSemiring.{u1} K (Semifield.toDivisionSemiring.{u1} K (Field.toSemifield.{u1} K _inst_1)))) (Polynomial.semiring.{u1} K (DivisionSemiring.toSemiring.{u1} K (Semifield.toDivisionSemiring.{u1} K (Field.toSemifield.{u1} K _inst_1))))))))) (Polynomial.C.{u1} K (DivisionSemiring.toSemiring.{u1} K (Semifield.toDivisionSemiring.{u1} K (Field.toSemifield.{u1} K _inst_1)))) a)) (Polynomial.roots.{u1} K (EuclideanDomain.toCommRing.{u1} K (Field.toEuclideanDomain.{u1} K _inst_1)) (EuclideanDomain.instIsDomainToSemiringToRingToCommRing.{u1} K (Field.toEuclideanDomain.{u1} K _inst_1)) p)))))
Case conversion may be inaccurate. Consider using '#align polynomial.eq_prod_roots_of_splits_id Polynomial.eq_prod_roots_of_splits_idₓ'. -/
theorem eq_prod_roots_of_splits_id {p : K[X]} (hsplit : Splits (RingHom.id K) p) :
p = C p.leadingCoeff * (p.roots.map fun a => X - C a).Prod := by
simpa using eq_prod_roots_of_splits hsplit
#align polynomial.eq_prod_roots_of_splits_id Polynomial.eq_prod_roots_of_splits_id
/- warning: polynomial.eq_prod_roots_of_monic_of_splits_id -> Polynomial.eq_prod_roots_of_monic_of_splits_id is a dubious translation:
lean 3 declaration is
forall {K : Type.{u1}} [_inst_1 : Field.{u1} K] {p : Polynomial.{u1} K (Ring.toSemiring.{u1} K (DivisionRing.toRing.{u1} K (Field.toDivisionRing.{u1} K _inst_1)))}, (Polynomial.Monic.{u1} K (Ring.toSemiring.{u1} K (DivisionRing.toRing.{u1} K (Field.toDivisionRing.{u1} K _inst_1))) p) -> (Polynomial.Splits.{u1, u1} K K (EuclideanDomain.toCommRing.{u1} K (Field.toEuclideanDomain.{u1} K _inst_1)) _inst_1 (RingHom.id.{u1} K (NonAssocRing.toNonAssocSemiring.{u1} K (Ring.toNonAssocRing.{u1} K (DivisionRing.toRing.{u1} K (Field.toDivisionRing.{u1} K _inst_1))))) p) -> (Eq.{succ u1} (Polynomial.{u1} K (Ring.toSemiring.{u1} K (DivisionRing.toRing.{u1} K (Field.toDivisionRing.{u1} K _inst_1)))) p (Multiset.prod.{u1} (Polynomial.{u1} K (Ring.toSemiring.{u1} K (DivisionRing.toRing.{u1} K (Field.toDivisionRing.{u1} K _inst_1)))) (CommRing.toCommMonoid.{u1} (Polynomial.{u1} K (Ring.toSemiring.{u1} K (DivisionRing.toRing.{u1} K (Field.toDivisionRing.{u1} K _inst_1)))) (Polynomial.commRing.{u1} K (EuclideanDomain.toCommRing.{u1} K (Field.toEuclideanDomain.{u1} K _inst_1)))) (Multiset.map.{u1, u1} K (Polynomial.{u1} K (Ring.toSemiring.{u1} K (DivisionRing.toRing.{u1} K (Field.toDivisionRing.{u1} K _inst_1)))) (fun (a : K) => HSub.hSub.{u1, u1, u1} (Polynomial.{u1} K (Ring.toSemiring.{u1} K (DivisionRing.toRing.{u1} K (Field.toDivisionRing.{u1} K _inst_1)))) (Polynomial.{u1} K (Ring.toSemiring.{u1} K (DivisionRing.toRing.{u1} K (Field.toDivisionRing.{u1} K _inst_1)))) (Polynomial.{u1} K (Ring.toSemiring.{u1} K (DivisionRing.toRing.{u1} K (Field.toDivisionRing.{u1} K _inst_1)))) (instHSub.{u1} (Polynomial.{u1} K (Ring.toSemiring.{u1} K (DivisionRing.toRing.{u1} K (Field.toDivisionRing.{u1} K _inst_1)))) (Polynomial.sub.{u1} K (DivisionRing.toRing.{u1} K (Field.toDivisionRing.{u1} K _inst_1)))) (Polynomial.X.{u1} K (Ring.toSemiring.{u1} K (DivisionRing.toRing.{u1} K (Field.toDivisionRing.{u1} K _inst_1)))) (coeFn.{succ u1, succ u1} (RingHom.{u1, u1} K (Polynomial.{u1} K (Ring.toSemiring.{u1} K (DivisionRing.toRing.{u1} K (Field.toDivisionRing.{u1} K _inst_1)))) (Semiring.toNonAssocSemiring.{u1} K (Ring.toSemiring.{u1} K (DivisionRing.toRing.{u1} K (Field.toDivisionRing.{u1} K _inst_1)))) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} K (Ring.toSemiring.{u1} K (DivisionRing.toRing.{u1} K (Field.toDivisionRing.{u1} K _inst_1)))) (Polynomial.semiring.{u1} K (Ring.toSemiring.{u1} K (DivisionRing.toRing.{u1} K (Field.toDivisionRing.{u1} K _inst_1)))))) (fun (_x : RingHom.{u1, u1} K (Polynomial.{u1} K (Ring.toSemiring.{u1} K (DivisionRing.toRing.{u1} K (Field.toDivisionRing.{u1} K _inst_1)))) (Semiring.toNonAssocSemiring.{u1} K (Ring.toSemiring.{u1} K (DivisionRing.toRing.{u1} K (Field.toDivisionRing.{u1} K _inst_1)))) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} K (Ring.toSemiring.{u1} K (DivisionRing.toRing.{u1} K (Field.toDivisionRing.{u1} K _inst_1)))) (Polynomial.semiring.{u1} K (Ring.toSemiring.{u1} K (DivisionRing.toRing.{u1} K (Field.toDivisionRing.{u1} K _inst_1)))))) => K -> (Polynomial.{u1} K (Ring.toSemiring.{u1} K (DivisionRing.toRing.{u1} K (Field.toDivisionRing.{u1} K _inst_1))))) (RingHom.hasCoeToFun.{u1, u1} K (Polynomial.{u1} K (Ring.toSemiring.{u1} K (DivisionRing.toRing.{u1} K (Field.toDivisionRing.{u1} K _inst_1)))) (Semiring.toNonAssocSemiring.{u1} K (Ring.toSemiring.{u1} K (DivisionRing.toRing.{u1} K (Field.toDivisionRing.{u1} K _inst_1)))) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} K (Ring.toSemiring.{u1} K (DivisionRing.toRing.{u1} K (Field.toDivisionRing.{u1} K _inst_1)))) (Polynomial.semiring.{u1} K (Ring.toSemiring.{u1} K (DivisionRing.toRing.{u1} K (Field.toDivisionRing.{u1} K _inst_1)))))) (Polynomial.C.{u1} K (Ring.toSemiring.{u1} K (DivisionRing.toRing.{u1} K (Field.toDivisionRing.{u1} K _inst_1)))) a)) (Polynomial.roots.{u1} K (EuclideanDomain.toCommRing.{u1} K (Field.toEuclideanDomain.{u1} K _inst_1)) (Field.isDomain.{u1} K _inst_1) p))))
but is expected to have type
forall {K : Type.{u1}} [_inst_1 : Field.{u1} K] {p : Polynomial.{u1} K (DivisionSemiring.toSemiring.{u1} K (Semifield.toDivisionSemiring.{u1} K (Field.toSemifield.{u1} K _inst_1)))}, (Polynomial.Monic.{u1} K (DivisionSemiring.toSemiring.{u1} K (Semifield.toDivisionSemiring.{u1} K (Field.toSemifield.{u1} K _inst_1))) p) -> (Polynomial.Splits.{u1, u1} K K (EuclideanDomain.toCommRing.{u1} K (Field.toEuclideanDomain.{u1} K _inst_1)) _inst_1 (RingHom.id.{u1} K (NonAssocRing.toNonAssocSemiring.{u1} K (Ring.toNonAssocRing.{u1} K (DivisionRing.toRing.{u1} K (Field.toDivisionRing.{u1} K _inst_1))))) p) -> (Eq.{succ u1} (Polynomial.{u1} K (DivisionSemiring.toSemiring.{u1} K (Semifield.toDivisionSemiring.{u1} K (Field.toSemifield.{u1} K _inst_1)))) p (Multiset.prod.{u1} (Polynomial.{u1} K (DivisionSemiring.toSemiring.{u1} K (Semifield.toDivisionSemiring.{u1} K (Field.toSemifield.{u1} K _inst_1)))) (CommRing.toCommMonoid.{u1} (Polynomial.{u1} K (DivisionSemiring.toSemiring.{u1} K 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K (Field.toSemifield.{u1} K _inst_1)))) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} K (DivisionSemiring.toSemiring.{u1} K (Semifield.toDivisionSemiring.{u1} K (Field.toSemifield.{u1} K _inst_1)))) (Polynomial.semiring.{u1} K (DivisionSemiring.toSemiring.{u1} K (Semifield.toDivisionSemiring.{u1} K (Field.toSemifield.{u1} K _inst_1)))))) (RingHomClass.toNonUnitalRingHomClass.{u1, u1, u1} (RingHom.{u1, u1} K (Polynomial.{u1} K (DivisionSemiring.toSemiring.{u1} K (Semifield.toDivisionSemiring.{u1} K (Field.toSemifield.{u1} K _inst_1)))) (Semiring.toNonAssocSemiring.{u1} K (DivisionSemiring.toSemiring.{u1} K (Semifield.toDivisionSemiring.{u1} K (Field.toSemifield.{u1} K _inst_1)))) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} K (DivisionSemiring.toSemiring.{u1} K (Semifield.toDivisionSemiring.{u1} K (Field.toSemifield.{u1} K _inst_1)))) (Polynomial.semiring.{u1} K (DivisionSemiring.toSemiring.{u1} K (Semifield.toDivisionSemiring.{u1} K (Field.toSemifield.{u1} K _inst_1)))))) K (Polynomial.{u1} K (DivisionSemiring.toSemiring.{u1} K (Semifield.toDivisionSemiring.{u1} K (Field.toSemifield.{u1} K _inst_1)))) (Semiring.toNonAssocSemiring.{u1} K (DivisionSemiring.toSemiring.{u1} K (Semifield.toDivisionSemiring.{u1} K (Field.toSemifield.{u1} K _inst_1)))) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} K (DivisionSemiring.toSemiring.{u1} K (Semifield.toDivisionSemiring.{u1} K (Field.toSemifield.{u1} K _inst_1)))) (Polynomial.semiring.{u1} K (DivisionSemiring.toSemiring.{u1} K (Semifield.toDivisionSemiring.{u1} K (Field.toSemifield.{u1} K _inst_1))))) (RingHom.instRingHomClassRingHom.{u1, u1} K (Polynomial.{u1} K (DivisionSemiring.toSemiring.{u1} K (Semifield.toDivisionSemiring.{u1} K (Field.toSemifield.{u1} K _inst_1)))) (Semiring.toNonAssocSemiring.{u1} K (DivisionSemiring.toSemiring.{u1} K (Semifield.toDivisionSemiring.{u1} K (Field.toSemifield.{u1} K _inst_1)))) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} K (DivisionSemiring.toSemiring.{u1} K (Semifield.toDivisionSemiring.{u1} K (Field.toSemifield.{u1} K _inst_1)))) (Polynomial.semiring.{u1} K (DivisionSemiring.toSemiring.{u1} K (Semifield.toDivisionSemiring.{u1} K (Field.toSemifield.{u1} K _inst_1))))))))) (Polynomial.C.{u1} K (DivisionSemiring.toSemiring.{u1} K (Semifield.toDivisionSemiring.{u1} K (Field.toSemifield.{u1} K _inst_1)))) a)) (Polynomial.roots.{u1} K (EuclideanDomain.toCommRing.{u1} K (Field.toEuclideanDomain.{u1} K _inst_1)) (EuclideanDomain.instIsDomainToSemiringToRingToCommRing.{u1} K (Field.toEuclideanDomain.{u1} K _inst_1)) p))))
Case conversion may be inaccurate. Consider using '#align polynomial.eq_prod_roots_of_monic_of_splits_id Polynomial.eq_prod_roots_of_monic_of_splits_idₓ'. -/
theorem eq_prod_roots_of_monic_of_splits_id {p : K[X]} (m : Monic p)
(hsplit : Splits (RingHom.id K) p) : p = (p.roots.map fun a => X - C a).Prod :=
by
convert eq_prod_roots_of_splits_id hsplit
simp [m]
#align polynomial.eq_prod_roots_of_monic_of_splits_id Polynomial.eq_prod_roots_of_monic_of_splits_id
/- warning: polynomial.eq_X_sub_C_of_splits_of_single_root -> Polynomial.eq_X_sub_C_of_splits_of_single_root is a dubious translation:
lean 3 declaration is
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but is expected to have type
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(DivisionSemiring.toSemiring.{u1} K (Semifield.toDivisionSemiring.{u1} K (Field.toSemifield.{u1} K _inst_1)))) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} K (DivisionSemiring.toSemiring.{u1} K (Semifield.toDivisionSemiring.{u1} K (Field.toSemifield.{u1} K _inst_1)))) (Polynomial.semiring.{u1} K (DivisionSemiring.toSemiring.{u1} K (Semifield.toDivisionSemiring.{u1} K (Field.toSemifield.{u1} K _inst_1))))))))) (Polynomial.C.{u1} K (DivisionSemiring.toSemiring.{u1} K (Semifield.toDivisionSemiring.{u1} K (Field.toSemifield.{u1} K _inst_1)))) (Polynomial.leadingCoeff.{u1} K (DivisionSemiring.toSemiring.{u1} K (Semifield.toDivisionSemiring.{u1} K (Field.toSemifield.{u1} K _inst_1))) h)) (HSub.hSub.{u1, u1, u1} (Polynomial.{u1} K (DivisionSemiring.toSemiring.{u1} K (Semifield.toDivisionSemiring.{u1} K (Field.toSemifield.{u1} K _inst_1)))) ((fun ([email protected]._hyg.2391 : K) => Polynomial.{u1} K (DivisionSemiring.toSemiring.{u1} K (Semifield.toDivisionSemiring.{u1} K 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(DivisionSemiring.toSemiring.{u1} K (Semifield.toDivisionSemiring.{u1} K (Field.toSemifield.{u1} K _inst_1))))))) (NonUnitalRingHomClass.toMulHomClass.{u1, u1, u1} (RingHom.{u1, u1} K (Polynomial.{u1} K (DivisionSemiring.toSemiring.{u1} K (Semifield.toDivisionSemiring.{u1} K (Field.toSemifield.{u1} K _inst_1)))) (Semiring.toNonAssocSemiring.{u1} K (DivisionSemiring.toSemiring.{u1} K (Semifield.toDivisionSemiring.{u1} K (Field.toSemifield.{u1} K _inst_1)))) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} K (DivisionSemiring.toSemiring.{u1} K (Semifield.toDivisionSemiring.{u1} K (Field.toSemifield.{u1} K _inst_1)))) (Polynomial.semiring.{u1} K (DivisionSemiring.toSemiring.{u1} K (Semifield.toDivisionSemiring.{u1} K (Field.toSemifield.{u1} K _inst_1)))))) K (Polynomial.{u1} K (DivisionSemiring.toSemiring.{u1} K (Semifield.toDivisionSemiring.{u1} K (Field.toSemifield.{u1} K _inst_1)))) (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} K (Semiring.toNonAssocSemiring.{u1} K (DivisionSemiring.toSemiring.{u1} K (Semifield.toDivisionSemiring.{u1} K (Field.toSemifield.{u1} K _inst_1))))) (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} (Polynomial.{u1} K (DivisionSemiring.toSemiring.{u1} K (Semifield.toDivisionSemiring.{u1} K (Field.toSemifield.{u1} K _inst_1)))) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} K (DivisionSemiring.toSemiring.{u1} K (Semifield.toDivisionSemiring.{u1} K (Field.toSemifield.{u1} K _inst_1)))) (Polynomial.semiring.{u1} K (DivisionSemiring.toSemiring.{u1} K (Semifield.toDivisionSemiring.{u1} K (Field.toSemifield.{u1} K _inst_1)))))) (RingHomClass.toNonUnitalRingHomClass.{u1, u1, u1} (RingHom.{u1, u1} K (Polynomial.{u1} K (DivisionSemiring.toSemiring.{u1} K (Semifield.toDivisionSemiring.{u1} K (Field.toSemifield.{u1} K _inst_1)))) (Semiring.toNonAssocSemiring.{u1} K (DivisionSemiring.toSemiring.{u1} K (Semifield.toDivisionSemiring.{u1} K (Field.toSemifield.{u1} K _inst_1)))) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} K (DivisionSemiring.toSemiring.{u1} K (Semifield.toDivisionSemiring.{u1} K (Field.toSemifield.{u1} K _inst_1)))) (Polynomial.semiring.{u1} K (DivisionSemiring.toSemiring.{u1} K (Semifield.toDivisionSemiring.{u1} K (Field.toSemifield.{u1} K _inst_1)))))) K (Polynomial.{u1} K (DivisionSemiring.toSemiring.{u1} K (Semifield.toDivisionSemiring.{u1} K (Field.toSemifield.{u1} K _inst_1)))) (Semiring.toNonAssocSemiring.{u1} K (DivisionSemiring.toSemiring.{u1} K (Semifield.toDivisionSemiring.{u1} K (Field.toSemifield.{u1} K _inst_1)))) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} K (DivisionSemiring.toSemiring.{u1} K (Semifield.toDivisionSemiring.{u1} K (Field.toSemifield.{u1} K _inst_1)))) (Polynomial.semiring.{u1} K (DivisionSemiring.toSemiring.{u1} K (Semifield.toDivisionSemiring.{u1} K (Field.toSemifield.{u1} K _inst_1))))) (RingHom.instRingHomClassRingHom.{u1, u1} K (Polynomial.{u1} K (DivisionSemiring.toSemiring.{u1} K (Semifield.toDivisionSemiring.{u1} K (Field.toSemifield.{u1} K _inst_1)))) (Semiring.toNonAssocSemiring.{u1} K (DivisionSemiring.toSemiring.{u1} K (Semifield.toDivisionSemiring.{u1} K (Field.toSemifield.{u1} K _inst_1)))) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} K (DivisionSemiring.toSemiring.{u1} K (Semifield.toDivisionSemiring.{u1} K (Field.toSemifield.{u1} K _inst_1)))) (Polynomial.semiring.{u1} K (DivisionSemiring.toSemiring.{u1} K (Semifield.toDivisionSemiring.{u1} K (Field.toSemifield.{u1} K _inst_1))))))))) (Polynomial.C.{u1} K (DivisionSemiring.toSemiring.{u1} K (Semifield.toDivisionSemiring.{u1} K (Field.toSemifield.{u1} K _inst_1)))) x))))
Case conversion may be inaccurate. Consider using '#align polynomial.eq_X_sub_C_of_splits_of_single_root Polynomial.eq_X_sub_C_of_splits_of_single_rootₓ'. -/
theorem eq_X_sub_C_of_splits_of_single_root {x : K} {h : K[X]} (h_splits : Splits i h)
(h_roots : (h.map i).roots = {i x}) : h = C h.leadingCoeff * (X - C x) :=
by
apply Polynomial.map_injective _ i.injective
rw [eq_prod_roots_of_splits h_splits, h_roots]
simp
#align polynomial.eq_X_sub_C_of_splits_of_single_root Polynomial.eq_X_sub_C_of_splits_of_single_root
/- warning: polynomial.mem_lift_of_splits_of_roots_mem_range -> Polynomial.mem_lift_of_splits_of_roots_mem_range is a dubious translation:
lean 3 declaration is
forall {K : Type.{u1}} [_inst_1 : Field.{u1} K] (R : Type.{u2}) [_inst_4 : CommRing.{u2} R] [_inst_5 : Algebra.{u2, u1} R K (CommRing.toCommSemiring.{u2} R _inst_4) (Ring.toSemiring.{u1} K (DivisionRing.toRing.{u1} K (Field.toDivisionRing.{u1} K _inst_1)))] {f : Polynomial.{u1} K (Ring.toSemiring.{u1} K (DivisionRing.toRing.{u1} K (Field.toDivisionRing.{u1} K _inst_1)))}, (Polynomial.Splits.{u1, u1} K K (EuclideanDomain.toCommRing.{u1} K (Field.toEuclideanDomain.{u1} K _inst_1)) _inst_1 (RingHom.id.{u1} K (NonAssocRing.toNonAssocSemiring.{u1} K (Ring.toNonAssocRing.{u1} K (DivisionRing.toRing.{u1} K (Field.toDivisionRing.{u1} K _inst_1))))) f) -> (Polynomial.Monic.{u1} K (Ring.toSemiring.{u1} K (DivisionRing.toRing.{u1} K (Field.toDivisionRing.{u1} K _inst_1))) f) -> (forall (a : K), (Membership.Mem.{u1, u1} K (Multiset.{u1} K) (Multiset.hasMem.{u1} K) a (Polynomial.roots.{u1} K (EuclideanDomain.toCommRing.{u1} K (Field.toEuclideanDomain.{u1} K _inst_1)) (Field.isDomain.{u1} K _inst_1) f)) -> (Membership.Mem.{u1, u1} K (Subring.{u1} K (DivisionRing.toRing.{u1} K (Field.toDivisionRing.{u1} K _inst_1))) (SetLike.hasMem.{u1, u1} (Subring.{u1} K (DivisionRing.toRing.{u1} K (Field.toDivisionRing.{u1} K _inst_1))) K (Subring.setLike.{u1} K (DivisionRing.toRing.{u1} K (Field.toDivisionRing.{u1} K _inst_1)))) a (RingHom.range.{u2, u1} R K (CommRing.toRing.{u2} R _inst_4) (DivisionRing.toRing.{u1} K (Field.toDivisionRing.{u1} K _inst_1)) (algebraMap.{u2, u1} R K (CommRing.toCommSemiring.{u2} R _inst_4) (Ring.toSemiring.{u1} K (DivisionRing.toRing.{u1} K (Field.toDivisionRing.{u1} K _inst_1))) _inst_5)))) -> (Membership.Mem.{u1, u1} (Polynomial.{u1} K (Ring.toSemiring.{u1} K (DivisionRing.toRing.{u1} K (Field.toDivisionRing.{u1} K _inst_1)))) (Subsemiring.{u1} (Polynomial.{u1} K (Ring.toSemiring.{u1} K (DivisionRing.toRing.{u1} K (Field.toDivisionRing.{u1} K _inst_1)))) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} K (Ring.toSemiring.{u1} K (DivisionRing.toRing.{u1} K (Field.toDivisionRing.{u1} K _inst_1)))) (Polynomial.semiring.{u1} K (Ring.toSemiring.{u1} K (DivisionRing.toRing.{u1} K (Field.toDivisionRing.{u1} K _inst_1)))))) (SetLike.hasMem.{u1, u1} (Subsemiring.{u1} (Polynomial.{u1} K (Ring.toSemiring.{u1} K (DivisionRing.toRing.{u1} K (Field.toDivisionRing.{u1} K _inst_1)))) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} K (Ring.toSemiring.{u1} K (DivisionRing.toRing.{u1} K (Field.toDivisionRing.{u1} K _inst_1)))) (Polynomial.semiring.{u1} K (Ring.toSemiring.{u1} K (DivisionRing.toRing.{u1} K (Field.toDivisionRing.{u1} K _inst_1)))))) (Polynomial.{u1} K (Ring.toSemiring.{u1} K (DivisionRing.toRing.{u1} K (Field.toDivisionRing.{u1} K _inst_1)))) (Subsemiring.setLike.{u1} (Polynomial.{u1} K (Ring.toSemiring.{u1} K (DivisionRing.toRing.{u1} K (Field.toDivisionRing.{u1} K _inst_1)))) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} K (Ring.toSemiring.{u1} K (DivisionRing.toRing.{u1} K (Field.toDivisionRing.{u1} K _inst_1)))) (Polynomial.semiring.{u1} K (Ring.toSemiring.{u1} K (DivisionRing.toRing.{u1} K (Field.toDivisionRing.{u1} K _inst_1))))))) f (Polynomial.lifts.{u2, u1} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_4)) K (Ring.toSemiring.{u1} K (DivisionRing.toRing.{u1} K (Field.toDivisionRing.{u1} K _inst_1))) (algebraMap.{u2, u1} R K (CommRing.toCommSemiring.{u2} R _inst_4) (Ring.toSemiring.{u1} K (DivisionRing.toRing.{u1} K (Field.toDivisionRing.{u1} K _inst_1))) _inst_5)))
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Case conversion may be inaccurate. Consider using '#align polynomial.mem_lift_of_splits_of_roots_mem_range Polynomial.mem_lift_of_splits_of_roots_mem_rangeₓ'. -/
theorem mem_lift_of_splits_of_roots_mem_range (R : Type _) [CommRing R] [Algebra R K] {f : K[X]}
(hs : f.Splits (RingHom.id K)) (hm : f.Monic) (hr : ∀ a ∈ f.roots, a ∈ (algebraMap R K).range) :
f ∈ Polynomial.lifts (algebraMap R K) :=
by
rw [eq_prod_roots_of_monic_of_splits_id hm hs, lifts_iff_lifts_ring]
refine' Subring.multiset_prod_mem _ _ fun P hP => _
obtain ⟨b, hb, rfl⟩ := Multiset.mem_map.1 hP
exact Subring.sub_mem _ (X_mem_lifts _) (C'_mem_lifts (hr _ hb))
#align polynomial.mem_lift_of_splits_of_roots_mem_range Polynomial.mem_lift_of_splits_of_roots_mem_range
section UFD
attribute [local instance] PrincipalIdealRing.to_uniqueFactorizationMonoid
-- mathport name: «expr ~ᵤ »
local infixl:50 " ~ᵤ " => Associated
open UniqueFactorizationMonoid Associates
/- warning: polynomial.splits_of_exists_multiset -> Polynomial.splits_of_exists_multiset is a dubious translation:
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but is expected to have type
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protected]._hyg.2391 : (fun ([email protected]._hyg.2391 : K) => L) (Polynomial.leadingCoeff.{u1} K (DivisionSemiring.toSemiring.{u1} K (Semifield.toDivisionSemiring.{u1} K (Field.toSemifield.{u1} K _inst_1))) f)) => Polynomial.{u2} ((fun ([email protected]._hyg.2391 : K) => L) (Polynomial.leadingCoeff.{u1} K (DivisionSemiring.toSemiring.{u1} K (Semifield.toDivisionSemiring.{u1} K (Field.toSemifield.{u1} K _inst_1))) f)) (DivisionSemiring.toSemiring.{u2} ((fun ([email protected]._hyg.2391 : K) => L) (Polynomial.leadingCoeff.{u1} K (DivisionSemiring.toSemiring.{u1} K (Semifield.toDivisionSemiring.{u1} K (Field.toSemifield.{u1} K _inst_1))) f)) (Semifield.toDivisionSemiring.{u2} ((fun ([email protected]._hyg.2391 : K) => L) (Polynomial.leadingCoeff.{u1} K (DivisionSemiring.toSemiring.{u1} K (Semifield.toDivisionSemiring.{u1} K (Field.toSemifield.{u1} K _inst_1))) f)) (Field.toSemifield.{u2} ((fun ([email protected]._hyg.2391 : K) => L) (Polynomial.leadingCoeff.{u1} K (DivisionSemiring.toSemiring.{u1} K (Semifield.toDivisionSemiring.{u1} K (Field.toSemifield.{u1} K _inst_1))) f)) _inst_2)))) (FunLike.coe.{max (succ u1) (succ u2), succ u1, succ u2} (RingHom.{u1, u2} K L (NonAssocRing.toNonAssocSemiring.{u1} K (Ring.toNonAssocRing.{u1} K (DivisionRing.toRing.{u1} K (Field.toDivisionRing.{u1} K _inst_1)))) (NonAssocRing.toNonAssocSemiring.{u2} L (Ring.toNonAssocRing.{u2} L (DivisionRing.toRing.{u2} L (Field.toDivisionRing.{u2} L _inst_2))))) K (fun (a : K) => (fun ([email protected]._hyg.2391 : K) => L) a) (MulHomClass.toFunLike.{max u1 u2, u1, u2} (RingHom.{u1, u2} K L (NonAssocRing.toNonAssocSemiring.{u1} K (Ring.toNonAssocRing.{u1} K (DivisionRing.toRing.{u1} K (Field.toDivisionRing.{u1} K _inst_1)))) (NonAssocRing.toNonAssocSemiring.{u2} L (Ring.toNonAssocRing.{u2} L (DivisionRing.toRing.{u2} L (Field.toDivisionRing.{u2} L _inst_2))))) K L (NonUnitalNonAssocSemiring.toMul.{u1} K (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} K (NonAssocRing.toNonAssocSemiring.{u1} K (Ring.toNonAssocRing.{u1} K (DivisionRing.toRing.{u1} K (Field.toDivisionRing.{u1} K _inst_1)))))) (NonUnitalNonAssocSemiring.toMul.{u2} L (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} L (NonAssocRing.toNonAssocSemiring.{u2} L (Ring.toNonAssocRing.{u2} L (DivisionRing.toRing.{u2} L (Field.toDivisionRing.{u2} L _inst_2)))))) (NonUnitalRingHomClass.toMulHomClass.{max u1 u2, u1, u2} (RingHom.{u1, u2} K L (NonAssocRing.toNonAssocSemiring.{u1} K (Ring.toNonAssocRing.{u1} K (DivisionRing.toRing.{u1} K (Field.toDivisionRing.{u1} K _inst_1)))) (NonAssocRing.toNonAssocSemiring.{u2} L (Ring.toNonAssocRing.{u2} L (DivisionRing.toRing.{u2} L (Field.toDivisionRing.{u2} L _inst_2))))) K L (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} K (NonAssocRing.toNonAssocSemiring.{u1} K (Ring.toNonAssocRing.{u1} K (DivisionRing.toRing.{u1} K (Field.toDivisionRing.{u1} K _inst_1))))) (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} L (NonAssocRing.toNonAssocSemiring.{u2} L (Ring.toNonAssocRing.{u2} L (DivisionRing.toRing.{u2} L (Field.toDivisionRing.{u2} L _inst_2))))) (RingHomClass.toNonUnitalRingHomClass.{max u1 u2, u1, u2} (RingHom.{u1, u2} K L (NonAssocRing.toNonAssocSemiring.{u1} K (Ring.toNonAssocRing.{u1} K (DivisionRing.toRing.{u1} K (Field.toDivisionRing.{u1} K _inst_1)))) (NonAssocRing.toNonAssocSemiring.{u2} L (Ring.toNonAssocRing.{u2} L (DivisionRing.toRing.{u2} L (Field.toDivisionRing.{u2} L _inst_2))))) K L (NonAssocRing.toNonAssocSemiring.{u1} K (Ring.toNonAssocRing.{u1} K (DivisionRing.toRing.{u1} K (Field.toDivisionRing.{u1} K _inst_1)))) (NonAssocRing.toNonAssocSemiring.{u2} L (Ring.toNonAssocRing.{u2} L (DivisionRing.toRing.{u2} L (Field.toDivisionRing.{u2} L _inst_2)))) (RingHom.instRingHomClassRingHom.{u1, u2} K L (NonAssocRing.toNonAssocSemiring.{u1} K (Ring.toNonAssocRing.{u1} K (DivisionRing.toRing.{u1} K (Field.toDivisionRing.{u1} K _inst_1)))) (NonAssocRing.toNonAssocSemiring.{u2} L (Ring.toNonAssocRing.{u2} L (DivisionRing.toRing.{u2} L (Field.toDivisionRing.{u2} L _inst_2)))))))) i (Polynomial.leadingCoeff.{u1} K (DivisionSemiring.toSemiring.{u1} K (Semifield.toDivisionSemiring.{u1} K (Field.toSemifield.{u1} K _inst_1))) f))) (Polynomial.{u2} L (DivisionSemiring.toSemiring.{u2} L (Semifield.toDivisionSemiring.{u2} L (Field.toSemifield.{u2} L _inst_2)))) ((fun ([email protected]._hyg.2391 : (fun ([email protected]._hyg.2391 : K) => L) (Polynomial.leadingCoeff.{u1} K (DivisionSemiring.toSemiring.{u1} K (Semifield.toDivisionSemiring.{u1} K (Field.toSemifield.{u1} K _inst_1))) f)) => Polynomial.{u2} ((fun ([email protected]._hyg.2391 : K) => L) (Polynomial.leadingCoeff.{u1} K (DivisionSemiring.toSemiring.{u1} K (Semifield.toDivisionSemiring.{u1} K (Field.toSemifield.{u1} K _inst_1))) f)) (DivisionSemiring.toSemiring.{u2} ((fun ([email protected]._hyg.2391 : K) => L) (Polynomial.leadingCoeff.{u1} K (DivisionSemiring.toSemiring.{u1} K 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(RingHom.{u1, u2} K L (NonAssocRing.toNonAssocSemiring.{u1} K (Ring.toNonAssocRing.{u1} K (DivisionRing.toRing.{u1} K (Field.toDivisionRing.{u1} K _inst_1)))) (NonAssocRing.toNonAssocSemiring.{u2} L (Ring.toNonAssocRing.{u2} L (DivisionRing.toRing.{u2} L (Field.toDivisionRing.{u2} L _inst_2))))) K L (NonUnitalNonAssocSemiring.toMul.{u1} K (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} K (NonAssocRing.toNonAssocSemiring.{u1} K (Ring.toNonAssocRing.{u1} K (DivisionRing.toRing.{u1} K (Field.toDivisionRing.{u1} K _inst_1)))))) (NonUnitalNonAssocSemiring.toMul.{u2} L (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} L (NonAssocRing.toNonAssocSemiring.{u2} L (Ring.toNonAssocRing.{u2} L (DivisionRing.toRing.{u2} L (Field.toDivisionRing.{u2} L _inst_2)))))) (NonUnitalRingHomClass.toMulHomClass.{max u1 u2, u1, u2} (RingHom.{u1, u2} K L (NonAssocRing.toNonAssocSemiring.{u1} K (Ring.toNonAssocRing.{u1} K (DivisionRing.toRing.{u1} K (Field.toDivisionRing.{u1} K _inst_1)))) (NonAssocRing.toNonAssocSemiring.{u2} L (Ring.toNonAssocRing.{u2} L (DivisionRing.toRing.{u2} L (Field.toDivisionRing.{u2} L _inst_2))))) K L (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} K (NonAssocRing.toNonAssocSemiring.{u1} K (Ring.toNonAssocRing.{u1} K (DivisionRing.toRing.{u1} K (Field.toDivisionRing.{u1} K _inst_1))))) (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} L (NonAssocRing.toNonAssocSemiring.{u2} L (Ring.toNonAssocRing.{u2} L (DivisionRing.toRing.{u2} L (Field.toDivisionRing.{u2} L _inst_2))))) (RingHomClass.toNonUnitalRingHomClass.{max u1 u2, u1, u2} (RingHom.{u1, u2} K L (NonAssocRing.toNonAssocSemiring.{u1} K (Ring.toNonAssocRing.{u1} K (DivisionRing.toRing.{u1} K (Field.toDivisionRing.{u1} K _inst_1)))) (NonAssocRing.toNonAssocSemiring.{u2} L (Ring.toNonAssocRing.{u2} L (DivisionRing.toRing.{u2} L (Field.toDivisionRing.{u2} L _inst_2))))) K L (NonAssocRing.toNonAssocSemiring.{u1} K (Ring.toNonAssocRing.{u1} K (DivisionRing.toRing.{u1} K (Field.toDivisionRing.{u1} K _inst_1)))) (NonAssocRing.toNonAssocSemiring.{u2} L (Ring.toNonAssocRing.{u2} L (DivisionRing.toRing.{u2} L (Field.toDivisionRing.{u2} L _inst_2)))) (RingHom.instRingHomClassRingHom.{u1, u2} K L (NonAssocRing.toNonAssocSemiring.{u1} K (Ring.toNonAssocRing.{u1} K (DivisionRing.toRing.{u1} K (Field.toDivisionRing.{u1} K _inst_1)))) (NonAssocRing.toNonAssocSemiring.{u2} L (Ring.toNonAssocRing.{u2} L (DivisionRing.toRing.{u2} L (Field.toDivisionRing.{u2} L _inst_2)))))))) i (Polynomial.leadingCoeff.{u1} K (DivisionSemiring.toSemiring.{u1} K (Semifield.toDivisionSemiring.{u1} K (Field.toSemifield.{u1} K _inst_1))) f))) (instHMul.{u2} ((fun ([email protected]._hyg.2391 : (fun ([email protected]._hyg.2391 : K) => L) (Polynomial.leadingCoeff.{u1} K (DivisionSemiring.toSemiring.{u1} K (Semifield.toDivisionSemiring.{u1} K (Field.toSemifield.{u1} K _inst_1))) f)) => Polynomial.{u2} ((fun ([email protected]._hyg.2391 : K) => L) (Polynomial.leadingCoeff.{u1} K (DivisionSemiring.toSemiring.{u1} K (Semifield.toDivisionSemiring.{u1} K (Field.toSemifield.{u1} K _inst_1))) f)) (DivisionSemiring.toSemiring.{u2} ((fun ([email protected]._hyg.2391 : K) => L) (Polynomial.leadingCoeff.{u1} K (DivisionSemiring.toSemiring.{u1} K (Semifield.toDivisionSemiring.{u1} K (Field.toSemifield.{u1} K _inst_1))) f)) (Semifield.toDivisionSemiring.{u2} ((fun ([email protected]._hyg.2391 : K) => L) (Polynomial.leadingCoeff.{u1} K (DivisionSemiring.toSemiring.{u1} K (Semifield.toDivisionSemiring.{u1} K (Field.toSemifield.{u1} K _inst_1))) f)) (Field.toSemifield.{u2} ((fun ([email protected]._hyg.2391 : K) => L) (Polynomial.leadingCoeff.{u1} K (DivisionSemiring.toSemiring.{u1} K (Semifield.toDivisionSemiring.{u1} K (Field.toSemifield.{u1} K _inst_1))) f)) _inst_2)))) (FunLike.coe.{max (succ u1) (succ u2), succ u1, succ u2} (RingHom.{u1, u2} K L (NonAssocRing.toNonAssocSemiring.{u1} K (Ring.toNonAssocRing.{u1} K (DivisionRing.toRing.{u1} K (Field.toDivisionRing.{u1} K _inst_1)))) (NonAssocRing.toNonAssocSemiring.{u2} L (Ring.toNonAssocRing.{u2} L (DivisionRing.toRing.{u2} L (Field.toDivisionRing.{u2} L _inst_2))))) K (fun (a : K) => (fun ([email protected]._hyg.2391 : K) => L) a) (MulHomClass.toFunLike.{max u1 u2, u1, u2} (RingHom.{u1, u2} K L (NonAssocRing.toNonAssocSemiring.{u1} K (Ring.toNonAssocRing.{u1} K (DivisionRing.toRing.{u1} K (Field.toDivisionRing.{u1} K _inst_1)))) (NonAssocRing.toNonAssocSemiring.{u2} L (Ring.toNonAssocRing.{u2} L (DivisionRing.toRing.{u2} L (Field.toDivisionRing.{u2} L _inst_2))))) K L (NonUnitalNonAssocSemiring.toMul.{u1} K (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} K (NonAssocRing.toNonAssocSemiring.{u1} K (Ring.toNonAssocRing.{u1} K (DivisionRing.toRing.{u1} K (Field.toDivisionRing.{u1} K _inst_1)))))) (NonUnitalNonAssocSemiring.toMul.{u2} L (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} L (NonAssocRing.toNonAssocSemiring.{u2} L (Ring.toNonAssocRing.{u2} L (DivisionRing.toRing.{u2} L (Field.toDivisionRing.{u2} L _inst_2)))))) (NonUnitalRingHomClass.toMulHomClass.{max u1 u2, u1, u2} (RingHom.{u1, u2} K L (NonAssocRing.toNonAssocSemiring.{u1} K (Ring.toNonAssocRing.{u1} K (DivisionRing.toRing.{u1} K (Field.toDivisionRing.{u1} K _inst_1)))) (NonAssocRing.toNonAssocSemiring.{u2} L (Ring.toNonAssocRing.{u2} L (DivisionRing.toRing.{u2} L (Field.toDivisionRing.{u2} L _inst_2))))) K L (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} K (NonAssocRing.toNonAssocSemiring.{u1} K (Ring.toNonAssocRing.{u1} K (DivisionRing.toRing.{u1} K (Field.toDivisionRing.{u1} K _inst_1))))) (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} L (NonAssocRing.toNonAssocSemiring.{u2} L (Ring.toNonAssocRing.{u2} L (DivisionRing.toRing.{u2} L (Field.toDivisionRing.{u2} L _inst_2))))) (RingHomClass.toNonUnitalRingHomClass.{max u1 u2, u1, u2} (RingHom.{u1, u2} K L (NonAssocRing.toNonAssocSemiring.{u1} K (Ring.toNonAssocRing.{u1} K (DivisionRing.toRing.{u1} K (Field.toDivisionRing.{u1} K _inst_1)))) (NonAssocRing.toNonAssocSemiring.{u2} L (Ring.toNonAssocRing.{u2} L (DivisionRing.toRing.{u2} L (Field.toDivisionRing.{u2} L _inst_2))))) K L (NonAssocRing.toNonAssocSemiring.{u1} K (Ring.toNonAssocRing.{u1} K (DivisionRing.toRing.{u1} K (Field.toDivisionRing.{u1} K _inst_1)))) (NonAssocRing.toNonAssocSemiring.{u2} L (Ring.toNonAssocRing.{u2} L (DivisionRing.toRing.{u2} L (Field.toDivisionRing.{u2} L _inst_2)))) (RingHom.instRingHomClassRingHom.{u1, u2} K L (NonAssocRing.toNonAssocSemiring.{u1} K (Ring.toNonAssocRing.{u1} K (DivisionRing.toRing.{u1} K (Field.toDivisionRing.{u1} K _inst_1)))) (NonAssocRing.toNonAssocSemiring.{u2} L (Ring.toNonAssocRing.{u2} L (DivisionRing.toRing.{u2} L (Field.toDivisionRing.{u2} L _inst_2)))))))) i (Polynomial.leadingCoeff.{u1} K (DivisionSemiring.toSemiring.{u1} K (Semifield.toDivisionSemiring.{u1} K (Field.toSemifield.{u1} K _inst_1))) f))) (Polynomial.mul'.{u2} ((fun ([email protected]._hyg.2391 : K) => L) (Polynomial.leadingCoeff.{u1} K (DivisionSemiring.toSemiring.{u1} K (Semifield.toDivisionSemiring.{u1} K (Field.toSemifield.{u1} K _inst_1))) f)) (DivisionSemiring.toSemiring.{u2} ((fun ([email protected]._hyg.2391 : K) => L) (Polynomial.leadingCoeff.{u1} K (DivisionSemiring.toSemiring.{u1} K (Semifield.toDivisionSemiring.{u1} K (Field.toSemifield.{u1} K _inst_1))) f)) (Semifield.toDivisionSemiring.{u2} ((fun ([email protected]._hyg.2391 : K) => L) (Polynomial.leadingCoeff.{u1} K (DivisionSemiring.toSemiring.{u1} K (Semifield.toDivisionSemiring.{u1} K (Field.toSemifield.{u1} K _inst_1))) f)) (Field.toSemifield.{u2} ((fun ([email protected]._hyg.2391 : K) => L) (Polynomial.leadingCoeff.{u1} K (DivisionSemiring.toSemiring.{u1} K (Semifield.toDivisionSemiring.{u1} K (Field.toSemifield.{u1} K _inst_1))) f)) _inst_2))))) (FunLike.coe.{succ u2, succ u2, succ u2} (RingHom.{u2, u2} ((fun ([email protected]._hyg.2391 : K) => L) (Polynomial.leadingCoeff.{u1} K (DivisionSemiring.toSemiring.{u1} K 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(Semiring.toNonAssocSemiring.{u2} ((fun ([email protected]._hyg.2391 : K) => L) (Polynomial.leadingCoeff.{u1} K (DivisionSemiring.toSemiring.{u1} K (Semifield.toDivisionSemiring.{u1} K (Field.toSemifield.{u1} K _inst_1))) f)) (DivisionSemiring.toSemiring.{u2} ((fun ([email protected]._hyg.2391 : K) => L) (Polynomial.leadingCoeff.{u1} K (DivisionSemiring.toSemiring.{u1} K (Semifield.toDivisionSemiring.{u1} K (Field.toSemifield.{u1} K _inst_1))) f)) (Semifield.toDivisionSemiring.{u2} ((fun ([email protected]._hyg.2391 : K) => L) (Polynomial.leadingCoeff.{u1} K (DivisionSemiring.toSemiring.{u1} K (Semifield.toDivisionSemiring.{u1} K (Field.toSemifield.{u1} K _inst_1))) f)) (Field.toSemifield.{u2} ((fun ([email protected]._hyg.2391 : K) => L) (Polynomial.leadingCoeff.{u1} K (DivisionSemiring.toSemiring.{u1} K (Semifield.toDivisionSemiring.{u1} K (Field.toSemifield.{u1} K _inst_1))) f)) _inst_2)))) (Semiring.toNonAssocSemiring.{u2} (Polynomial.{u2} ((fun ([email protected]._hyg.2391 : K) => L) (Polynomial.leadingCoeff.{u1} K (DivisionSemiring.toSemiring.{u1} K (Semifield.toDivisionSemiring.{u1} K (Field.toSemifield.{u1} K _inst_1))) f)) (DivisionSemiring.toSemiring.{u2} ((fun ([email protected]._hyg.2391 : K) => L) (Polynomial.leadingCoeff.{u1} K (DivisionSemiring.toSemiring.{u1} K (Semifield.toDivisionSemiring.{u1} K (Field.toSemifield.{u1} K _inst_1))) f)) (Semifield.toDivisionSemiring.{u2} ((fun ([email protected]._hyg.2391 : K) => L) (Polynomial.leadingCoeff.{u1} K (DivisionSemiring.toSemiring.{u1} K (Semifield.toDivisionSemiring.{u1} K (Field.toSemifield.{u1} K _inst_1))) f)) (Field.toSemifield.{u2} ((fun ([email protected]._hyg.2391 : K) => L) (Polynomial.leadingCoeff.{u1} K (DivisionSemiring.toSemiring.{u1} K (Semifield.toDivisionSemiring.{u1} K (Field.toSemifield.{u1} K _inst_1))) f)) _inst_2)))) (Polynomial.semiring.{u2} ((fun ([email protected]._hyg.2391 : K) => L) (Polynomial.leadingCoeff.{u1} K (DivisionSemiring.toSemiring.{u1} K (Semifield.toDivisionSemiring.{u1} K (Field.toSemifield.{u1} K _inst_1))) f)) (DivisionSemiring.toSemiring.{u2} ((fun ([email protected]._hyg.2391 : K) => L) (Polynomial.leadingCoeff.{u1} K (DivisionSemiring.toSemiring.{u1} K (Semifield.toDivisionSemiring.{u1} K (Field.toSemifield.{u1} K _inst_1))) f)) (Semifield.toDivisionSemiring.{u2} ((fun ([email protected]._hyg.2391 : K) => L) (Polynomial.leadingCoeff.{u1} K (DivisionSemiring.toSemiring.{u1} K (Semifield.toDivisionSemiring.{u1} K (Field.toSemifield.{u1} K _inst_1))) f)) (Field.toSemifield.{u2} ((fun ([email protected]._hyg.2391 : K) => L) (Polynomial.leadingCoeff.{u1} K (DivisionSemiring.toSemiring.{u1} K (Semifield.toDivisionSemiring.{u1} K (Field.toSemifield.{u1} K _inst_1))) f)) _inst_2)))))) ((fun ([email protected]._hyg.2391 : K) => L) (Polynomial.leadingCoeff.{u1} K (DivisionSemiring.toSemiring.{u1} K (Semifield.toDivisionSemiring.{u1} K (Field.toSemifield.{u1} K _inst_1))) f)) (fun (_x : (fun ([email protected]._hyg.2391 : K) => L) (Polynomial.leadingCoeff.{u1} K (DivisionSemiring.toSemiring.{u1} K (Semifield.toDivisionSemiring.{u1} K (Field.toSemifield.{u1} K _inst_1))) f)) => (fun ([email protected]._hyg.2391 : (fun ([email protected]._hyg.2391 : K) => L) (Polynomial.leadingCoeff.{u1} K (DivisionSemiring.toSemiring.{u1} K (Semifield.toDivisionSemiring.{u1} K (Field.toSemifield.{u1} K _inst_1))) f)) => Polynomial.{u2} ((fun ([email protected]._hyg.2391 : K) => L) (Polynomial.leadingCoeff.{u1} K (DivisionSemiring.toSemiring.{u1} K (Semifield.toDivisionSemiring.{u1} K (Field.toSemifield.{u1} K _inst_1))) f)) (DivisionSemiring.toSemiring.{u2} ((fun ([email protected]._hyg.2391 : K) => L) (Polynomial.leadingCoeff.{u1} K (DivisionSemiring.toSemiring.{u1} K (Semifield.toDivisionSemiring.{u1} K (Field.toSemifield.{u1} K _inst_1))) f)) (Semifield.toDivisionSemiring.{u2} ((fun ([email protected]._hyg.2391 : K) => L) (Polynomial.leadingCoeff.{u1} K (DivisionSemiring.toSemiring.{u1} K (Semifield.toDivisionSemiring.{u1} K (Field.toSemifield.{u1} K _inst_1))) f)) (Field.toSemifield.{u2} ((fun ([email protected]._hyg.2391 : K) => L) (Polynomial.leadingCoeff.{u1} K (DivisionSemiring.toSemiring.{u1} K (Semifield.toDivisionSemiring.{u1} K (Field.toSemifield.{u1} K _inst_1))) f)) _inst_2)))) _x) (MulHomClass.toFunLike.{u2, u2, u2} (RingHom.{u2, u2} ((fun ([email protected]._hyg.2391 : K) => L) (Polynomial.leadingCoeff.{u1} K (DivisionSemiring.toSemiring.{u1} K (Semifield.toDivisionSemiring.{u1} K (Field.toSemifield.{u1} K _inst_1))) f)) (Polynomial.{u2} ((fun ([email protected]._hyg.2391 : K) => L) (Polynomial.leadingCoeff.{u1} K (DivisionSemiring.toSemiring.{u1} K (Semifield.toDivisionSemiring.{u1} K (Field.toSemifield.{u1} K _inst_1))) f)) (DivisionSemiring.toSemiring.{u2} ((fun ([email protected]._hyg.2391 : K) => L) (Polynomial.leadingCoeff.{u1} K (DivisionSemiring.toSemiring.{u1} K (Semifield.toDivisionSemiring.{u1} K (Field.toSemifield.{u1} K _inst_1))) f)) (Semifield.toDivisionSemiring.{u2} ((fun ([email protected]._hyg.2391 : K) => L) (Polynomial.leadingCoeff.{u1} K (DivisionSemiring.toSemiring.{u1} K (Semifield.toDivisionSemiring.{u1} K (Field.toSemifield.{u1} K _inst_1))) f)) (Field.toSemifield.{u2} ((fun ([email protected]._hyg.2391 : K) => L) (Polynomial.leadingCoeff.{u1} K (DivisionSemiring.toSemiring.{u1} K (Semifield.toDivisionSemiring.{u1} K (Field.toSemifield.{u1} K _inst_1))) f)) _inst_2)))) (Semiring.toNonAssocSemiring.{u2} ((fun ([email protected]._hyg.2391 : K) => L) (Polynomial.leadingCoeff.{u1} K (DivisionSemiring.toSemiring.{u1} K (Semifield.toDivisionSemiring.{u1} K (Field.toSemifield.{u1} K _inst_1))) f)) (DivisionSemiring.toSemiring.{u2} ((fun ([email protected]._hyg.2391 : K) => L) (Polynomial.leadingCoeff.{u1} K (DivisionSemiring.toSemiring.{u1} K (Semifield.toDivisionSemiring.{u1} K (Field.toSemifield.{u1} K _inst_1))) f)) (Semifield.toDivisionSemiring.{u2} ((fun ([email protected]._hyg.2391 : K) => L) (Polynomial.leadingCoeff.{u1} K (DivisionSemiring.toSemiring.{u1} K (Semifield.toDivisionSemiring.{u1} K (Field.toSemifield.{u1} K _inst_1))) f)) (Field.toSemifield.{u2} ((fun ([email protected]._hyg.2391 : K) => L) (Polynomial.leadingCoeff.{u1} K (DivisionSemiring.toSemiring.{u1} K (Semifield.toDivisionSemiring.{u1} K (Field.toSemifield.{u1} K _inst_1))) f)) _inst_2)))) (Semiring.toNonAssocSemiring.{u2} (Polynomial.{u2} ((fun ([email protected]._hyg.2391 : K) => L) (Polynomial.leadingCoeff.{u1} K (DivisionSemiring.toSemiring.{u1} K (Semifield.toDivisionSemiring.{u1} K (Field.toSemifield.{u1} K _inst_1))) f)) (DivisionSemiring.toSemiring.{u2} ((fun ([email protected]._hyg.2391 : K) => L) (Polynomial.leadingCoeff.{u1} K (DivisionSemiring.toSemiring.{u1} K (Semifield.toDivisionSemiring.{u1} K (Field.toSemifield.{u1} K _inst_1))) f)) (Semifield.toDivisionSemiring.{u2} ((fun ([email protected]._hyg.2391 : K) => L) (Polynomial.leadingCoeff.{u1} K (DivisionSemiring.toSemiring.{u1} K (Semifield.toDivisionSemiring.{u1} K (Field.toSemifield.{u1} K _inst_1))) f)) (Field.toSemifield.{u2} ((fun ([email protected]._hyg.2391 : K) => L) (Polynomial.leadingCoeff.{u1} K (DivisionSemiring.toSemiring.{u1} K (Semifield.toDivisionSemiring.{u1} K (Field.toSemifield.{u1} K _inst_1))) f)) _inst_2)))) (Polynomial.semiring.{u2} ((fun ([email protected]._hyg.2391 : K) => L) (Polynomial.leadingCoeff.{u1} K (DivisionSemiring.toSemiring.{u1} K (Semifield.toDivisionSemiring.{u1} K (Field.toSemifield.{u1} K _inst_1))) f)) (DivisionSemiring.toSemiring.{u2} ((fun ([email protected]._hyg.2391 : K) => L) (Polynomial.leadingCoeff.{u1} K (DivisionSemiring.toSemiring.{u1} K (Semifield.toDivisionSemiring.{u1} K (Field.toSemifield.{u1} K _inst_1))) f)) (Semifield.toDivisionSemiring.{u2} ((fun ([email protected]._hyg.2391 : K) => L) (Polynomial.leadingCoeff.{u1} K (DivisionSemiring.toSemiring.{u1} K (Semifield.toDivisionSemiring.{u1} K (Field.toSemifield.{u1} K _inst_1))) f)) (Field.toSemifield.{u2} ((fun ([email protected]._hyg.2391 : K) => L) (Polynomial.leadingCoeff.{u1} K (DivisionSemiring.toSemiring.{u1} K (Semifield.toDivisionSemiring.{u1} K (Field.toSemifield.{u1} K _inst_1))) f)) _inst_2)))))) ((fun ([email protected]._hyg.2391 : K) => L) (Polynomial.leadingCoeff.{u1} K (DivisionSemiring.toSemiring.{u1} K (Semifield.toDivisionSemiring.{u1} K (Field.toSemifield.{u1} K _inst_1))) f)) (Polynomial.{u2} ((fun ([email protected]._hyg.2391 : K) => L) (Polynomial.leadingCoeff.{u1} K (DivisionSemiring.toSemiring.{u1} K (Semifield.toDivisionSemiring.{u1} K (Field.toSemifield.{u1} K _inst_1))) f)) (DivisionSemiring.toSemiring.{u2} ((fun ([email protected]._hyg.2391 : K) => L) (Polynomial.leadingCoeff.{u1} K (DivisionSemiring.toSemiring.{u1} K (Semifield.toDivisionSemiring.{u1} K (Field.toSemifield.{u1} K _inst_1))) f)) (Semifield.toDivisionSemiring.{u2} ((fun ([email protected]._hyg.2391 : K) => L) (Polynomial.leadingCoeff.{u1} K (DivisionSemiring.toSemiring.{u1} K (Semifield.toDivisionSemiring.{u1} K (Field.toSemifield.{u1} K _inst_1))) f)) (Field.toSemifield.{u2} ((fun ([email protected]._hyg.2391 : K) => L) (Polynomial.leadingCoeff.{u1} K (DivisionSemiring.toSemiring.{u1} K (Semifield.toDivisionSemiring.{u1} K (Field.toSemifield.{u1} K _inst_1))) f)) _inst_2)))) (NonUnitalNonAssocSemiring.toMul.{u2} ((fun ([email protected]._hyg.2391 : K) => L) (Polynomial.leadingCoeff.{u1} K (DivisionSemiring.toSemiring.{u1} K (Semifield.toDivisionSemiring.{u1} K (Field.toSemifield.{u1} K _inst_1))) f)) (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} ((fun ([email protected]._hyg.2391 : K) => L) (Polynomial.leadingCoeff.{u1} K (DivisionSemiring.toSemiring.{u1} K (Semifield.toDivisionSemiring.{u1} K (Field.toSemifield.{u1} K _inst_1))) f)) (Semiring.toNonAssocSemiring.{u2} ((fun ([email protected]._hyg.2391 : K) => L) (Polynomial.leadingCoeff.{u1} K (DivisionSemiring.toSemiring.{u1} K (Semifield.toDivisionSemiring.{u1} K (Field.toSemifield.{u1} K _inst_1))) f)) (DivisionSemiring.toSemiring.{u2} ((fun ([email protected]._hyg.2391 : K) => L) (Polynomial.leadingCoeff.{u1} K (DivisionSemiring.toSemiring.{u1} K (Semifield.toDivisionSemiring.{u1} K (Field.toSemifield.{u1} K _inst_1))) f)) (Semifield.toDivisionSemiring.{u2} ((fun ([email protected]._hyg.2391 : K) => L) (Polynomial.leadingCoeff.{u1} K (DivisionSemiring.toSemiring.{u1} K (Semifield.toDivisionSemiring.{u1} K (Field.toSemifield.{u1} K _inst_1))) f)) (Field.toSemifield.{u2} ((fun ([email protected]._hyg.2391 : K) => L) (Polynomial.leadingCoeff.{u1} K (DivisionSemiring.toSemiring.{u1} K (Semifield.toDivisionSemiring.{u1} K (Field.toSemifield.{u1} K _inst_1))) f)) _inst_2)))))) (NonUnitalNonAssocSemiring.toMul.{u2} (Polynomial.{u2} ((fun ([email protected]._hyg.2391 : K) => L) (Polynomial.leadingCoeff.{u1} K (DivisionSemiring.toSemiring.{u1} K (Semifield.toDivisionSemiring.{u1} K (Field.toSemifield.{u1} K _inst_1))) f)) (DivisionSemiring.toSemiring.{u2} ((fun ([email protected]._hyg.2391 : K) => L) (Polynomial.leadingCoeff.{u1} K (DivisionSemiring.toSemiring.{u1} K (Semifield.toDivisionSemiring.{u1} K (Field.toSemifield.{u1} K _inst_1))) f)) (Semifield.toDivisionSemiring.{u2} ((fun ([email protected]._hyg.2391 : K) => L) (Polynomial.leadingCoeff.{u1} K (DivisionSemiring.toSemiring.{u1} K (Semifield.toDivisionSemiring.{u1} K (Field.toSemifield.{u1} K _inst_1))) f)) (Field.toSemifield.{u2} ((fun ([email protected]._hyg.2391 : K) => L) (Polynomial.leadingCoeff.{u1} K (DivisionSemiring.toSemiring.{u1} K (Semifield.toDivisionSemiring.{u1} K (Field.toSemifield.{u1} K _inst_1))) f)) _inst_2)))) (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} (Polynomial.{u2} ((fun ([email protected]._hyg.2391 : K) => L) (Polynomial.leadingCoeff.{u1} K (DivisionSemiring.toSemiring.{u1} K (Semifield.toDivisionSemiring.{u1} K (Field.toSemifield.{u1} K _inst_1))) f)) (DivisionSemiring.toSemiring.{u2} ((fun ([email protected]._hyg.2391 : K) => L) (Polynomial.leadingCoeff.{u1} K (DivisionSemiring.toSemiring.{u1} K (Semifield.toDivisionSemiring.{u1} K (Field.toSemifield.{u1} K _inst_1))) f)) (Semifield.toDivisionSemiring.{u2} ((fun ([email protected]._hyg.2391 : K) => L) (Polynomial.leadingCoeff.{u1} K (DivisionSemiring.toSemiring.{u1} K (Semifield.toDivisionSemiring.{u1} K (Field.toSemifield.{u1} K _inst_1))) f)) (Field.toSemifield.{u2} ((fun ([email protected]._hyg.2391 : K) => L) (Polynomial.leadingCoeff.{u1} K (DivisionSemiring.toSemiring.{u1} K (Semifield.toDivisionSemiring.{u1} K (Field.toSemifield.{u1} K _inst_1))) f)) _inst_2)))) (Semiring.toNonAssocSemiring.{u2} (Polynomial.{u2} ((fun ([email protected]._hyg.2391 : K) => L) (Polynomial.leadingCoeff.{u1} K (DivisionSemiring.toSemiring.{u1} K (Semifield.toDivisionSemiring.{u1} K (Field.toSemifield.{u1} K _inst_1))) f)) (DivisionSemiring.toSemiring.{u2} ((fun ([email protected]._hyg.2391 : K) => L) (Polynomial.leadingCoeff.{u1} K (DivisionSemiring.toSemiring.{u1} K (Semifield.toDivisionSemiring.{u1} K (Field.toSemifield.{u1} K _inst_1))) f)) (Semifield.toDivisionSemiring.{u2} ((fun ([email protected]._hyg.2391 : K) => L) (Polynomial.leadingCoeff.{u1} K (DivisionSemiring.toSemiring.{u1} K (Semifield.toDivisionSemiring.{u1} K (Field.toSemifield.{u1} K _inst_1))) f)) (Field.toSemifield.{u2} ((fun ([email protected]._hyg.2391 : K) => L) (Polynomial.leadingCoeff.{u1} K (DivisionSemiring.toSemiring.{u1} K (Semifield.toDivisionSemiring.{u1} K (Field.toSemifield.{u1} K _inst_1))) f)) _inst_2)))) (Polynomial.semiring.{u2} ((fun ([email protected]._hyg.2391 : K) => L) (Polynomial.leadingCoeff.{u1} K (DivisionSemiring.toSemiring.{u1} K (Semifield.toDivisionSemiring.{u1} K (Field.toSemifield.{u1} K _inst_1))) f)) (DivisionSemiring.toSemiring.{u2} ((fun ([email protected]._hyg.2391 : K) => L) (Polynomial.leadingCoeff.{u1} K (DivisionSemiring.toSemiring.{u1} K (Semifield.toDivisionSemiring.{u1} K (Field.toSemifield.{u1} K _inst_1))) f)) (Semifield.toDivisionSemiring.{u2} ((fun ([email protected]._hyg.2391 : K) => L) (Polynomial.leadingCoeff.{u1} K (DivisionSemiring.toSemiring.{u1} K (Semifield.toDivisionSemiring.{u1} K (Field.toSemifield.{u1} K _inst_1))) f)) (Field.toSemifield.{u2} ((fun ([email protected]._hyg.2391 : K) => L) (Polynomial.leadingCoeff.{u1} K (DivisionSemiring.toSemiring.{u1} K (Semifield.toDivisionSemiring.{u1} K (Field.toSemifield.{u1} K _inst_1))) f)) _inst_2))))))) (NonUnitalRingHomClass.toMulHomClass.{u2, u2, u2} (RingHom.{u2, u2} ((fun ([email protected]._hyg.2391 : K) => L) (Polynomial.leadingCoeff.{u1} K (DivisionSemiring.toSemiring.{u1} K (Semifield.toDivisionSemiring.{u1} K (Field.toSemifield.{u1} K _inst_1))) f)) (Polynomial.{u2} ((fun ([email protected]._hyg.2391 : K) => L) (Polynomial.leadingCoeff.{u1} K (DivisionSemiring.toSemiring.{u1} K (Semifield.toDivisionSemiring.{u1} K (Field.toSemifield.{u1} K _inst_1))) f)) (DivisionSemiring.toSemiring.{u2} ((fun ([email protected]._hyg.2391 : K) => L) (Polynomial.leadingCoeff.{u1} K (DivisionSemiring.toSemiring.{u1} K (Semifield.toDivisionSemiring.{u1} K (Field.toSemifield.{u1} K _inst_1))) f)) (Semifield.toDivisionSemiring.{u2} ((fun ([email protected]._hyg.2391 : K) => L) (Polynomial.leadingCoeff.{u1} K (DivisionSemiring.toSemiring.{u1} K (Semifield.toDivisionSemiring.{u1} K (Field.toSemifield.{u1} K _inst_1))) f)) (Field.toSemifield.{u2} ((fun ([email protected]._hyg.2391 : K) => L) (Polynomial.leadingCoeff.{u1} K (DivisionSemiring.toSemiring.{u1} K (Semifield.toDivisionSemiring.{u1} K (Field.toSemifield.{u1} K _inst_1))) f)) _inst_2)))) (Semiring.toNonAssocSemiring.{u2} ((fun ([email protected]._hyg.2391 : K) => L) (Polynomial.leadingCoeff.{u1} K (DivisionSemiring.toSemiring.{u1} K (Semifield.toDivisionSemiring.{u1} K (Field.toSemifield.{u1} K _inst_1))) f)) (DivisionSemiring.toSemiring.{u2} ((fun ([email protected]._hyg.2391 : K) => L) (Polynomial.leadingCoeff.{u1} K (DivisionSemiring.toSemiring.{u1} K (Semifield.toDivisionSemiring.{u1} K (Field.toSemifield.{u1} K _inst_1))) f)) (Semifield.toDivisionSemiring.{u2} ((fun ([email protected]._hyg.2391 : K) => L) (Polynomial.leadingCoeff.{u1} K (DivisionSemiring.toSemiring.{u1} K (Semifield.toDivisionSemiring.{u1} K (Field.toSemifield.{u1} K _inst_1))) f)) (Field.toSemifield.{u2} ((fun ([email protected]._hyg.2391 : K) => L) (Polynomial.leadingCoeff.{u1} K (DivisionSemiring.toSemiring.{u1} K (Semifield.toDivisionSemiring.{u1} K (Field.toSemifield.{u1} K _inst_1))) f)) _inst_2)))) (Semiring.toNonAssocSemiring.{u2} (Polynomial.{u2} ((fun ([email protected]._hyg.2391 : K) => L) (Polynomial.leadingCoeff.{u1} K (DivisionSemiring.toSemiring.{u1} K (Semifield.toDivisionSemiring.{u1} K (Field.toSemifield.{u1} K _inst_1))) f)) (DivisionSemiring.toSemiring.{u2} ((fun ([email protected]._hyg.2391 : K) => L) (Polynomial.leadingCoeff.{u1} K (DivisionSemiring.toSemiring.{u1} K (Semifield.toDivisionSemiring.{u1} K (Field.toSemifield.{u1} K _inst_1))) f)) (Semifield.toDivisionSemiring.{u2} ((fun ([email protected]._hyg.2391 : K) => L) (Polynomial.leadingCoeff.{u1} K (DivisionSemiring.toSemiring.{u1} K (Semifield.toDivisionSemiring.{u1} K (Field.toSemifield.{u1} K _inst_1))) f)) (Field.toSemifield.{u2} ((fun ([email protected]._hyg.2391 : K) => L) (Polynomial.leadingCoeff.{u1} K (DivisionSemiring.toSemiring.{u1} K (Semifield.toDivisionSemiring.{u1} K (Field.toSemifield.{u1} K _inst_1))) f)) _inst_2)))) (Polynomial.semiring.{u2} ((fun ([email protected]._hyg.2391 : K) => L) (Polynomial.leadingCoeff.{u1} K (DivisionSemiring.toSemiring.{u1} K (Semifield.toDivisionSemiring.{u1} K (Field.toSemifield.{u1} K _inst_1))) f)) (DivisionSemiring.toSemiring.{u2} ((fun ([email protected]._hyg.2391 : K) => L) (Polynomial.leadingCoeff.{u1} K (DivisionSemiring.toSemiring.{u1} K (Semifield.toDivisionSemiring.{u1} K (Field.toSemifield.{u1} K _inst_1))) f)) (Semifield.toDivisionSemiring.{u2} ((fun ([email protected]._hyg.2391 : K) => L) (Polynomial.leadingCoeff.{u1} K (DivisionSemiring.toSemiring.{u1} K (Semifield.toDivisionSemiring.{u1} K (Field.toSemifield.{u1} K _inst_1))) f)) (Field.toSemifield.{u2} ((fun ([email protected]._hyg.2391 : K) => L) (Polynomial.leadingCoeff.{u1} K (DivisionSemiring.toSemiring.{u1} K (Semifield.toDivisionSemiring.{u1} K (Field.toSemifield.{u1} K _inst_1))) f)) _inst_2)))))) ((fun ([email protected]._hyg.2391 : K) => L) (Polynomial.leadingCoeff.{u1} K (DivisionSemiring.toSemiring.{u1} K (Semifield.toDivisionSemiring.{u1} K (Field.toSemifield.{u1} K _inst_1))) f)) (Polynomial.{u2} ((fun ([email protected]._hyg.2391 : K) => L) (Polynomial.leadingCoeff.{u1} K (DivisionSemiring.toSemiring.{u1} K (Semifield.toDivisionSemiring.{u1} K (Field.toSemifield.{u1} K _inst_1))) f)) (DivisionSemiring.toSemiring.{u2} ((fun ([email protected]._hyg.2391 : K) => L) (Polynomial.leadingCoeff.{u1} K (DivisionSemiring.toSemiring.{u1} K (Semifield.toDivisionSemiring.{u1} K (Field.toSemifield.{u1} K _inst_1))) f)) (Semifield.toDivisionSemiring.{u2} ((fun ([email protected]._hyg.2391 : K) => L) (Polynomial.leadingCoeff.{u1} K (DivisionSemiring.toSemiring.{u1} K (Semifield.toDivisionSemiring.{u1} K (Field.toSemifield.{u1} K _inst_1))) f)) (Field.toSemifield.{u2} ((fun ([email protected]._hyg.2391 : K) => L) (Polynomial.leadingCoeff.{u1} K (DivisionSemiring.toSemiring.{u1} K (Semifield.toDivisionSemiring.{u1} K (Field.toSemifield.{u1} K _inst_1))) f)) _inst_2)))) (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} ((fun ([email protected]._hyg.2391 : K) => L) (Polynomial.leadingCoeff.{u1} K (DivisionSemiring.toSemiring.{u1} K (Semifield.toDivisionSemiring.{u1} K (Field.toSemifield.{u1} K _inst_1))) f)) (Semiring.toNonAssocSemiring.{u2} ((fun ([email protected]._hyg.2391 : K) => L) (Polynomial.leadingCoeff.{u1} K (DivisionSemiring.toSemiring.{u1} K (Semifield.toDivisionSemiring.{u1} K (Field.toSemifield.{u1} K _inst_1))) f)) (DivisionSemiring.toSemiring.{u2} ((fun ([email protected]._hyg.2391 : K) => L) (Polynomial.leadingCoeff.{u1} K (DivisionSemiring.toSemiring.{u1} K (Semifield.toDivisionSemiring.{u1} K (Field.toSemifield.{u1} K _inst_1))) f)) (Semifield.toDivisionSemiring.{u2} ((fun ([email protected]._hyg.2391 : K) => L) (Polynomial.leadingCoeff.{u1} K (DivisionSemiring.toSemiring.{u1} K (Semifield.toDivisionSemiring.{u1} K (Field.toSemifield.{u1} K _inst_1))) f)) (Field.toSemifield.{u2} ((fun ([email protected]._hyg.2391 : K) => L) (Polynomial.leadingCoeff.{u1} K (DivisionSemiring.toSemiring.{u1} K (Semifield.toDivisionSemiring.{u1} K (Field.toSemifield.{u1} K _inst_1))) f)) _inst_2))))) (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} (Polynomial.{u2} ((fun ([email protected]._hyg.2391 : K) => L) (Polynomial.leadingCoeff.{u1} K (DivisionSemiring.toSemiring.{u1} K (Semifield.toDivisionSemiring.{u1} K (Field.toSemifield.{u1} K _inst_1))) f)) (DivisionSemiring.toSemiring.{u2} ((fun ([email protected]._hyg.2391 : K) => L) (Polynomial.leadingCoeff.{u1} K (DivisionSemiring.toSemiring.{u1} K (Semifield.toDivisionSemiring.{u1} K (Field.toSemifield.{u1} K _inst_1))) f)) (Semifield.toDivisionSemiring.{u2} ((fun ([email protected]._hyg.2391 : K) => L) (Polynomial.leadingCoeff.{u1} K (DivisionSemiring.toSemiring.{u1} K (Semifield.toDivisionSemiring.{u1} K (Field.toSemifield.{u1} K _inst_1))) f)) (Field.toSemifield.{u2} ((fun ([email protected]._hyg.2391 : K) => L) (Polynomial.leadingCoeff.{u1} K (DivisionSemiring.toSemiring.{u1} K (Semifield.toDivisionSemiring.{u1} K (Field.toSemifield.{u1} K _inst_1))) f)) _inst_2)))) (Semiring.toNonAssocSemiring.{u2} (Polynomial.{u2} ((fun ([email protected]._hyg.2391 : K) => L) (Polynomial.leadingCoeff.{u1} K (DivisionSemiring.toSemiring.{u1} K (Semifield.toDivisionSemiring.{u1} K (Field.toSemifield.{u1} K _inst_1))) f)) (DivisionSemiring.toSemiring.{u2} ((fun ([email protected]._hyg.2391 : K) => L) (Polynomial.leadingCoeff.{u1} K (DivisionSemiring.toSemiring.{u1} K (Semifield.toDivisionSemiring.{u1} K (Field.toSemifield.{u1} K _inst_1))) f)) (Semifield.toDivisionSemiring.{u2} ((fun ([email protected]._hyg.2391 : K) => L) (Polynomial.leadingCoeff.{u1} K (DivisionSemiring.toSemiring.{u1} K (Semifield.toDivisionSemiring.{u1} K (Field.toSemifield.{u1} K _inst_1))) f)) (Field.toSemifield.{u2} ((fun ([email protected]._hyg.2391 : K) => L) (Polynomial.leadingCoeff.{u1} K (DivisionSemiring.toSemiring.{u1} K (Semifield.toDivisionSemiring.{u1} K (Field.toSemifield.{u1} K _inst_1))) f)) _inst_2)))) (Polynomial.semiring.{u2} ((fun ([email protected]._hyg.2391 : K) => L) (Polynomial.leadingCoeff.{u1} K (DivisionSemiring.toSemiring.{u1} K (Semifield.toDivisionSemiring.{u1} K (Field.toSemifield.{u1} K _inst_1))) f)) (DivisionSemiring.toSemiring.{u2} ((fun ([email protected]._hyg.2391 : K) => L) (Polynomial.leadingCoeff.{u1} K (DivisionSemiring.toSemiring.{u1} K (Semifield.toDivisionSemiring.{u1} K (Field.toSemifield.{u1} K _inst_1))) f)) (Semifield.toDivisionSemiring.{u2} ((fun ([email protected]._hyg.2391 : K) => L) (Polynomial.leadingCoeff.{u1} K (DivisionSemiring.toSemiring.{u1} K (Semifield.toDivisionSemiring.{u1} K (Field.toSemifield.{u1} K _inst_1))) f)) (Field.toSemifield.{u2} ((fun ([email protected]._hyg.2391 : K) => L) (Polynomial.leadingCoeff.{u1} K (DivisionSemiring.toSemiring.{u1} K (Semifield.toDivisionSemiring.{u1} K (Field.toSemifield.{u1} K _inst_1))) f)) _inst_2)))))) (RingHomClass.toNonUnitalRingHomClass.{u2, u2, u2} (RingHom.{u2, u2} ((fun ([email protected]._hyg.2391 : K) => L) (Polynomial.leadingCoeff.{u1} K (DivisionSemiring.toSemiring.{u1} K (Semifield.toDivisionSemiring.{u1} K (Field.toSemifield.{u1} K _inst_1))) f)) (Polynomial.{u2} ((fun ([email protected]._hyg.2391 : K) => L) (Polynomial.leadingCoeff.{u1} K (DivisionSemiring.toSemiring.{u1} K (Semifield.toDivisionSemiring.{u1} K (Field.toSemifield.{u1} K _inst_1))) f)) (DivisionSemiring.toSemiring.{u2} ((fun ([email protected]._hyg.2391 : K) => L) (Polynomial.leadingCoeff.{u1} K (DivisionSemiring.toSemiring.{u1} K (Semifield.toDivisionSemiring.{u1} K (Field.toSemifield.{u1} K _inst_1))) f)) (Semifield.toDivisionSemiring.{u2} ((fun ([email protected]._hyg.2391 : K) => L) (Polynomial.leadingCoeff.{u1} K (DivisionSemiring.toSemiring.{u1} K (Semifield.toDivisionSemiring.{u1} K (Field.toSemifield.{u1} K _inst_1))) f)) (Field.toSemifield.{u2} ((fun ([email protected]._hyg.2391 : K) => L) (Polynomial.leadingCoeff.{u1} K (DivisionSemiring.toSemiring.{u1} K (Semifield.toDivisionSemiring.{u1} K (Field.toSemifield.{u1} K _inst_1))) f)) _inst_2)))) (Semiring.toNonAssocSemiring.{u2} ((fun ([email protected]._hyg.2391 : K) => L) (Polynomial.leadingCoeff.{u1} K (DivisionSemiring.toSemiring.{u1} K (Semifield.toDivisionSemiring.{u1} K (Field.toSemifield.{u1} K _inst_1))) f)) (DivisionSemiring.toSemiring.{u2} ((fun ([email protected]._hyg.2391 : K) => L) (Polynomial.leadingCoeff.{u1} K (DivisionSemiring.toSemiring.{u1} K (Semifield.toDivisionSemiring.{u1} K (Field.toSemifield.{u1} K _inst_1))) f)) (Semifield.toDivisionSemiring.{u2} ((fun ([email protected]._hyg.2391 : K) => L) (Polynomial.leadingCoeff.{u1} K (DivisionSemiring.toSemiring.{u1} K (Semifield.toDivisionSemiring.{u1} K (Field.toSemifield.{u1} K _inst_1))) f)) (Field.toSemifield.{u2} ((fun ([email protected]._hyg.2391 : K) => L) (Polynomial.leadingCoeff.{u1} K (DivisionSemiring.toSemiring.{u1} K (Semifield.toDivisionSemiring.{u1} K (Field.toSemifield.{u1} K _inst_1))) f)) _inst_2)))) (Semiring.toNonAssocSemiring.{u2} (Polynomial.{u2} ((fun ([email protected]._hyg.2391 : K) => L) (Polynomial.leadingCoeff.{u1} K (DivisionSemiring.toSemiring.{u1} K (Semifield.toDivisionSemiring.{u1} K (Field.toSemifield.{u1} K _inst_1))) f)) (DivisionSemiring.toSemiring.{u2} ((fun ([email protected]._hyg.2391 : K) => L) (Polynomial.leadingCoeff.{u1} K (DivisionSemiring.toSemiring.{u1} K (Semifield.toDivisionSemiring.{u1} K (Field.toSemifield.{u1} K _inst_1))) f)) (Semifield.toDivisionSemiring.{u2} ((fun ([email protected]._hyg.2391 : K) => L) (Polynomial.leadingCoeff.{u1} K (DivisionSemiring.toSemiring.{u1} K (Semifield.toDivisionSemiring.{u1} K (Field.toSemifield.{u1} K _inst_1))) f)) (Field.toSemifield.{u2} ((fun ([email protected]._hyg.2391 : K) => L) (Polynomial.leadingCoeff.{u1} K (DivisionSemiring.toSemiring.{u1} K (Semifield.toDivisionSemiring.{u1} K (Field.toSemifield.{u1} K _inst_1))) f)) _inst_2)))) (Polynomial.semiring.{u2} ((fun ([email protected]._hyg.2391 : K) => L) (Polynomial.leadingCoeff.{u1} K (DivisionSemiring.toSemiring.{u1} K (Semifield.toDivisionSemiring.{u1} K (Field.toSemifield.{u1} K _inst_1))) f)) (DivisionSemiring.toSemiring.{u2} ((fun ([email protected]._hyg.2391 : K) => L) (Polynomial.leadingCoeff.{u1} K (DivisionSemiring.toSemiring.{u1} K (Semifield.toDivisionSemiring.{u1} K (Field.toSemifield.{u1} K _inst_1))) f)) (Semifield.toDivisionSemiring.{u2} ((fun ([email protected]._hyg.2391 : K) => L) (Polynomial.leadingCoeff.{u1} K (DivisionSemiring.toSemiring.{u1} K (Semifield.toDivisionSemiring.{u1} K (Field.toSemifield.{u1} K _inst_1))) f)) (Field.toSemifield.{u2} ((fun ([email protected]._hyg.2391 : K) => L) (Polynomial.leadingCoeff.{u1} K (DivisionSemiring.toSemiring.{u1} K (Semifield.toDivisionSemiring.{u1} K (Field.toSemifield.{u1} K _inst_1))) f)) _inst_2)))))) ((fun ([email protected]._hyg.2391 : K) => L) (Polynomial.leadingCoeff.{u1} K (DivisionSemiring.toSemiring.{u1} K (Semifield.toDivisionSemiring.{u1} K (Field.toSemifield.{u1} K _inst_1))) f)) (Polynomial.{u2} ((fun ([email protected]._hyg.2391 : K) => L) (Polynomial.leadingCoeff.{u1} K (DivisionSemiring.toSemiring.{u1} K (Semifield.toDivisionSemiring.{u1} K (Field.toSemifield.{u1} K _inst_1))) f)) (DivisionSemiring.toSemiring.{u2} ((fun ([email protected]._hyg.2391 : K) => L) (Polynomial.leadingCoeff.{u1} K (DivisionSemiring.toSemiring.{u1} K (Semifield.toDivisionSemiring.{u1} K (Field.toSemifield.{u1} K _inst_1))) f)) (Semifield.toDivisionSemiring.{u2} ((fun ([email protected]._hyg.2391 : K) => L) (Polynomial.leadingCoeff.{u1} K (DivisionSemiring.toSemiring.{u1} K (Semifield.toDivisionSemiring.{u1} K (Field.toSemifield.{u1} K _inst_1))) f)) (Field.toSemifield.{u2} ((fun ([email protected]._hyg.2391 : K) => L) (Polynomial.leadingCoeff.{u1} K (DivisionSemiring.toSemiring.{u1} K (Semifield.toDivisionSemiring.{u1} K (Field.toSemifield.{u1} K _inst_1))) f)) _inst_2)))) (Semiring.toNonAssocSemiring.{u2} ((fun ([email protected]._hyg.2391 : K) => L) (Polynomial.leadingCoeff.{u1} K (DivisionSemiring.toSemiring.{u1} K (Semifield.toDivisionSemiring.{u1} K (Field.toSemifield.{u1} K _inst_1))) f)) (DivisionSemiring.toSemiring.{u2} ((fun ([email protected]._hyg.2391 : K) => L) (Polynomial.leadingCoeff.{u1} K (DivisionSemiring.toSemiring.{u1} K (Semifield.toDivisionSemiring.{u1} K (Field.toSemifield.{u1} K _inst_1))) f)) (Semifield.toDivisionSemiring.{u2} ((fun ([email protected]._hyg.2391 : K) => L) (Polynomial.leadingCoeff.{u1} K (DivisionSemiring.toSemiring.{u1} K (Semifield.toDivisionSemiring.{u1} K (Field.toSemifield.{u1} K _inst_1))) f)) (Field.toSemifield.{u2} ((fun ([email protected]._hyg.2391 : K) => L) (Polynomial.leadingCoeff.{u1} K (DivisionSemiring.toSemiring.{u1} K (Semifield.toDivisionSemiring.{u1} K (Field.toSemifield.{u1} K _inst_1))) f)) _inst_2)))) (Semiring.toNonAssocSemiring.{u2} (Polynomial.{u2} ((fun ([email protected]._hyg.2391 : K) => L) (Polynomial.leadingCoeff.{u1} K (DivisionSemiring.toSemiring.{u1} K (Semifield.toDivisionSemiring.{u1} K (Field.toSemifield.{u1} K _inst_1))) f)) (DivisionSemiring.toSemiring.{u2} ((fun ([email protected]._hyg.2391 : K) => L) (Polynomial.leadingCoeff.{u1} K (DivisionSemiring.toSemiring.{u1} K (Semifield.toDivisionSemiring.{u1} K (Field.toSemifield.{u1} K _inst_1))) f)) (Semifield.toDivisionSemiring.{u2} ((fun ([email protected]._hyg.2391 : K) => L) (Polynomial.leadingCoeff.{u1} K (DivisionSemiring.toSemiring.{u1} K (Semifield.toDivisionSemiring.{u1} K (Field.toSemifield.{u1} K _inst_1))) f)) (Field.toSemifield.{u2} ((fun ([email protected]._hyg.2391 : K) => L) (Polynomial.leadingCoeff.{u1} K (DivisionSemiring.toSemiring.{u1} K (Semifield.toDivisionSemiring.{u1} K (Field.toSemifield.{u1} K _inst_1))) f)) _inst_2)))) (Polynomial.semiring.{u2} ((fun ([email protected]._hyg.2391 : K) => L) (Polynomial.leadingCoeff.{u1} K (DivisionSemiring.toSemiring.{u1} K (Semifield.toDivisionSemiring.{u1} K (Field.toSemifield.{u1} K _inst_1))) f)) (DivisionSemiring.toSemiring.{u2} ((fun ([email protected]._hyg.2391 : K) => L) (Polynomial.leadingCoeff.{u1} K (DivisionSemiring.toSemiring.{u1} K (Semifield.toDivisionSemiring.{u1} K (Field.toSemifield.{u1} K _inst_1))) f)) (Semifield.toDivisionSemiring.{u2} ((fun ([email protected]._hyg.2391 : K) => L) (Polynomial.leadingCoeff.{u1} K (DivisionSemiring.toSemiring.{u1} K (Semifield.toDivisionSemiring.{u1} K (Field.toSemifield.{u1} K _inst_1))) f)) (Field.toSemifield.{u2} ((fun ([email protected]._hyg.2391 : K) => L) (Polynomial.leadingCoeff.{u1} K (DivisionSemiring.toSemiring.{u1} K (Semifield.toDivisionSemiring.{u1} K (Field.toSemifield.{u1} K _inst_1))) f)) _inst_2))))) (RingHom.instRingHomClassRingHom.{u2, u2} ((fun ([email protected]._hyg.2391 : K) => L) (Polynomial.leadingCoeff.{u1} K (DivisionSemiring.toSemiring.{u1} K (Semifield.toDivisionSemiring.{u1} K (Field.toSemifield.{u1} K _inst_1))) f)) (Polynomial.{u2} ((fun ([email protected]._hyg.2391 : K) => L) (Polynomial.leadingCoeff.{u1} K (DivisionSemiring.toSemiring.{u1} K (Semifield.toDivisionSemiring.{u1} K (Field.toSemifield.{u1} K _inst_1))) f)) (DivisionSemiring.toSemiring.{u2} ((fun ([email protected]._hyg.2391 : K) => L) (Polynomial.leadingCoeff.{u1} K (DivisionSemiring.toSemiring.{u1} K (Semifield.toDivisionSemiring.{u1} K (Field.toSemifield.{u1} K _inst_1))) f)) (Semifield.toDivisionSemiring.{u2} ((fun ([email protected]._hyg.2391 : K) => L) (Polynomial.leadingCoeff.{u1} K (DivisionSemiring.toSemiring.{u1} K (Semifield.toDivisionSemiring.{u1} K (Field.toSemifield.{u1} K _inst_1))) f)) (Field.toSemifield.{u2} ((fun ([email protected]._hyg.2391 : K) => L) (Polynomial.leadingCoeff.{u1} K (DivisionSemiring.toSemiring.{u1} K (Semifield.toDivisionSemiring.{u1} K (Field.toSemifield.{u1} K _inst_1))) f)) _inst_2)))) (Semiring.toNonAssocSemiring.{u2} ((fun ([email protected]._hyg.2391 : K) => L) (Polynomial.leadingCoeff.{u1} K (DivisionSemiring.toSemiring.{u1} K (Semifield.toDivisionSemiring.{u1} K (Field.toSemifield.{u1} K _inst_1))) f)) (DivisionSemiring.toSemiring.{u2} ((fun ([email protected]._hyg.2391 : K) => L) (Polynomial.leadingCoeff.{u1} K (DivisionSemiring.toSemiring.{u1} K (Semifield.toDivisionSemiring.{u1} K (Field.toSemifield.{u1} K _inst_1))) f)) (Semifield.toDivisionSemiring.{u2} ((fun ([email protected]._hyg.2391 : K) => L) (Polynomial.leadingCoeff.{u1} K (DivisionSemiring.toSemiring.{u1} K (Semifield.toDivisionSemiring.{u1} K (Field.toSemifield.{u1} K _inst_1))) f)) (Field.toSemifield.{u2} ((fun ([email protected]._hyg.2391 : K) => L) (Polynomial.leadingCoeff.{u1} K (DivisionSemiring.toSemiring.{u1} K (Semifield.toDivisionSemiring.{u1} K (Field.toSemifield.{u1} K _inst_1))) f)) _inst_2)))) (Semiring.toNonAssocSemiring.{u2} (Polynomial.{u2} ((fun ([email protected]._hyg.2391 : K) => L) (Polynomial.leadingCoeff.{u1} K (DivisionSemiring.toSemiring.{u1} K (Semifield.toDivisionSemiring.{u1} K (Field.toSemifield.{u1} K _inst_1))) f)) (DivisionSemiring.toSemiring.{u2} ((fun ([email protected]._hyg.2391 : K) => L) (Polynomial.leadingCoeff.{u1} K (DivisionSemiring.toSemiring.{u1} K (Semifield.toDivisionSemiring.{u1} K (Field.toSemifield.{u1} K _inst_1))) f)) (Semifield.toDivisionSemiring.{u2} ((fun ([email protected]._hyg.2391 : K) => L) (Polynomial.leadingCoeff.{u1} K (DivisionSemiring.toSemiring.{u1} K (Semifield.toDivisionSemiring.{u1} K (Field.toSemifield.{u1} K _inst_1))) f)) (Field.toSemifield.{u2} ((fun 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(Semiring.toNonAssocSemiring.{u2} (Polynomial.{u2} L (DivisionSemiring.toSemiring.{u2} L (Semifield.toDivisionSemiring.{u2} L (Field.toSemifield.{u2} L _inst_2)))) (Polynomial.semiring.{u2} L (DivisionSemiring.toSemiring.{u2} L (Semifield.toDivisionSemiring.{u2} L (Field.toSemifield.{u2} L _inst_2)))))) L (Polynomial.{u2} L (DivisionSemiring.toSemiring.{u2} L (Semifield.toDivisionSemiring.{u2} L (Field.toSemifield.{u2} L _inst_2)))) (Semiring.toNonAssocSemiring.{u2} L (DivisionSemiring.toSemiring.{u2} L (Semifield.toDivisionSemiring.{u2} L (Field.toSemifield.{u2} L _inst_2)))) (Semiring.toNonAssocSemiring.{u2} (Polynomial.{u2} L (DivisionSemiring.toSemiring.{u2} L (Semifield.toDivisionSemiring.{u2} L (Field.toSemifield.{u2} L _inst_2)))) (Polynomial.semiring.{u2} L (DivisionSemiring.toSemiring.{u2} L (Semifield.toDivisionSemiring.{u2} L (Field.toSemifield.{u2} L _inst_2))))) (RingHom.instRingHomClassRingHom.{u2, u2} L (Polynomial.{u2} L (DivisionSemiring.toSemiring.{u2} L (Semifield.toDivisionSemiring.{u2} L (Field.toSemifield.{u2} L _inst_2)))) (Semiring.toNonAssocSemiring.{u2} L (DivisionSemiring.toSemiring.{u2} L (Semifield.toDivisionSemiring.{u2} L (Field.toSemifield.{u2} L _inst_2)))) (Semiring.toNonAssocSemiring.{u2} (Polynomial.{u2} L (DivisionSemiring.toSemiring.{u2} L (Semifield.toDivisionSemiring.{u2} L (Field.toSemifield.{u2} L _inst_2)))) (Polynomial.semiring.{u2} L (DivisionSemiring.toSemiring.{u2} L (Semifield.toDivisionSemiring.{u2} L (Field.toSemifield.{u2} L _inst_2))))))))) (Polynomial.C.{u2} L (DivisionSemiring.toSemiring.{u2} L (Semifield.toDivisionSemiring.{u2} L (Field.toSemifield.{u2} L _inst_2)))) a)) s)))) -> (Polynomial.Splits.{u1, u2} K L (EuclideanDomain.toCommRing.{u1} K (Field.toEuclideanDomain.{u1} K _inst_1)) _inst_2 i f)
Case conversion may be inaccurate. Consider using '#align polynomial.splits_of_exists_multiset Polynomial.splits_of_exists_multisetₓ'. -/
theorem splits_of_exists_multiset {f : K[X]} {s : Multiset L}
(hs : f.map i = C (i f.leadingCoeff) * (s.map fun a : L => X - C a).Prod) : Splits i f :=
if hf0 : f = 0 then hf0.symm ▸ splits_zero i
else
Or.inr fun p hp hdp => by
rw [irreducible_iff_prime] at hp
rw [hs, ← Multiset.prod_toList] at hdp
obtain hd | hd := hp.2.2 _ _ hdp
· refine' (hp.2.1 <| isUnit_of_dvd_unit hd _).elim
exact is_unit_C.2 ((leading_coeff_ne_zero.2 hf0).IsUnit.map i)
· obtain ⟨q, hq, hd⟩ := hp.dvd_prod_iff.1 hd
obtain ⟨a, ha, rfl⟩ := Multiset.mem_map.1 (Multiset.mem_toList.1 hq)
rw [degree_eq_degree_of_associated ((hp.dvd_prime_iff_associated <| prime_X_sub_C a).1 hd)]
exact degree_X_sub_C a
#align polynomial.splits_of_exists_multiset Polynomial.splits_of_exists_multiset
#print Polynomial.splits_of_splits_id /-
theorem splits_of_splits_id {f : K[X]} : Splits (RingHom.id K) f → Splits i f :=
UniqueFactorizationMonoid.induction_on_prime f (fun _ => splits_zero _)
(fun _ hu _ => splits_of_degree_le_one _ ((isUnit_iff_degree_eq_zero.1 hu).symm ▸ by decide))
fun a p ha0 hp ih hfi =>
splits_mul _
(splits_of_degree_eq_one _
((splits_of_splits_mul _ (mul_ne_zero hp.1 ha0) hfi).1.def.resolve_left hp.1 hp.Irreducible
(by rw [map_id])))
(ih (splits_of_splits_mul _ (mul_ne_zero hp.1 ha0) hfi).2)
#align polynomial.splits_of_splits_id Polynomial.splits_of_splits_id
-/
end UFD
/- warning: polynomial.splits_iff_exists_multiset -> Polynomial.splits_iff_exists_multiset is a dubious translation:
lean 3 declaration is
forall {K : Type.{u1}} {L : Type.{u2}} [_inst_1 : Field.{u1} K] [_inst_2 : Field.{u2} L] (i : RingHom.{u1, u2} K L (NonAssocRing.toNonAssocSemiring.{u1} K (Ring.toNonAssocRing.{u1} K (DivisionRing.toRing.{u1} K (Field.toDivisionRing.{u1} K _inst_1)))) (NonAssocRing.toNonAssocSemiring.{u2} L (Ring.toNonAssocRing.{u2} L (DivisionRing.toRing.{u2} L (Field.toDivisionRing.{u2} L _inst_2))))) {f : Polynomial.{u1} K (Ring.toSemiring.{u1} K (DivisionRing.toRing.{u1} K (Field.toDivisionRing.{u1} K _inst_1)))}, Iff (Polynomial.Splits.{u1, u2} K L (EuclideanDomain.toCommRing.{u1} K (Field.toEuclideanDomain.{u1} K _inst_1)) _inst_2 i f) (Exists.{succ u2} (Multiset.{u2} L) (fun (s : Multiset.{u2} L) => Eq.{succ u2} (Polynomial.{u2} L (Ring.toSemiring.{u2} L (DivisionRing.toRing.{u2} L (Field.toDivisionRing.{u2} L _inst_2)))) (Polynomial.map.{u1, u2} K L (Ring.toSemiring.{u1} K (DivisionRing.toRing.{u1} K (Field.toDivisionRing.{u1} K _inst_1))) (Ring.toSemiring.{u2} L (DivisionRing.toRing.{u2} L (Field.toDivisionRing.{u2} L _inst_2))) i f) (HMul.hMul.{u2, u2, u2} (Polynomial.{u2} L (Ring.toSemiring.{u2} L (DivisionRing.toRing.{u2} L (Field.toDivisionRing.{u2} L _inst_2)))) (Polynomial.{u2} L (Ring.toSemiring.{u2} L (DivisionRing.toRing.{u2} L (Field.toDivisionRing.{u2} L _inst_2)))) (Polynomial.{u2} L (Ring.toSemiring.{u2} L (DivisionRing.toRing.{u2} L (Field.toDivisionRing.{u2} L _inst_2)))) (instHMul.{u2} (Polynomial.{u2} L (Ring.toSemiring.{u2} L (DivisionRing.toRing.{u2} L (Field.toDivisionRing.{u2} L _inst_2)))) (Polynomial.mul'.{u2} L (Ring.toSemiring.{u2} L (DivisionRing.toRing.{u2} L (Field.toDivisionRing.{u2} L _inst_2))))) (coeFn.{succ u2, succ u2} (RingHom.{u2, u2} L (Polynomial.{u2} L (Ring.toSemiring.{u2} L (DivisionRing.toRing.{u2} L (Field.toDivisionRing.{u2} L _inst_2)))) (Semiring.toNonAssocSemiring.{u2} L (Ring.toSemiring.{u2} L (DivisionRing.toRing.{u2} L (Field.toDivisionRing.{u2} L _inst_2)))) (Semiring.toNonAssocSemiring.{u2} (Polynomial.{u2} L (Ring.toSemiring.{u2} L (DivisionRing.toRing.{u2} L (Field.toDivisionRing.{u2} L _inst_2)))) (Polynomial.semiring.{u2} L (Ring.toSemiring.{u2} L (DivisionRing.toRing.{u2} L (Field.toDivisionRing.{u2} L _inst_2)))))) (fun (_x : RingHom.{u2, u2} L (Polynomial.{u2} L (Ring.toSemiring.{u2} L (DivisionRing.toRing.{u2} L (Field.toDivisionRing.{u2} L _inst_2)))) (Semiring.toNonAssocSemiring.{u2} L (Ring.toSemiring.{u2} L (DivisionRing.toRing.{u2} L (Field.toDivisionRing.{u2} L _inst_2)))) (Semiring.toNonAssocSemiring.{u2} (Polynomial.{u2} L (Ring.toSemiring.{u2} L (DivisionRing.toRing.{u2} L (Field.toDivisionRing.{u2} L _inst_2)))) (Polynomial.semiring.{u2} L (Ring.toSemiring.{u2} L (DivisionRing.toRing.{u2} L (Field.toDivisionRing.{u2} L _inst_2)))))) => L -> (Polynomial.{u2} L (Ring.toSemiring.{u2} L (DivisionRing.toRing.{u2} L (Field.toDivisionRing.{u2} L _inst_2))))) (RingHom.hasCoeToFun.{u2, u2} L (Polynomial.{u2} L (Ring.toSemiring.{u2} L (DivisionRing.toRing.{u2} L (Field.toDivisionRing.{u2} L _inst_2)))) (Semiring.toNonAssocSemiring.{u2} L (Ring.toSemiring.{u2} L (DivisionRing.toRing.{u2} L (Field.toDivisionRing.{u2} L _inst_2)))) (Semiring.toNonAssocSemiring.{u2} (Polynomial.{u2} L (Ring.toSemiring.{u2} L (DivisionRing.toRing.{u2} L (Field.toDivisionRing.{u2} L _inst_2)))) (Polynomial.semiring.{u2} L (Ring.toSemiring.{u2} L (DivisionRing.toRing.{u2} L (Field.toDivisionRing.{u2} L _inst_2)))))) (Polynomial.C.{u2} L (Ring.toSemiring.{u2} L (DivisionRing.toRing.{u2} L (Field.toDivisionRing.{u2} L _inst_2)))) (coeFn.{max (succ u1) (succ u2), max (succ u1) (succ u2)} (RingHom.{u1, u2} K L (NonAssocRing.toNonAssocSemiring.{u1} K (Ring.toNonAssocRing.{u1} K (DivisionRing.toRing.{u1} K (Field.toDivisionRing.{u1} K _inst_1)))) (NonAssocRing.toNonAssocSemiring.{u2} L (Ring.toNonAssocRing.{u2} L (DivisionRing.toRing.{u2} L (Field.toDivisionRing.{u2} L _inst_2))))) (fun (_x : RingHom.{u1, u2} K L (NonAssocRing.toNonAssocSemiring.{u1} K (Ring.toNonAssocRing.{u1} K (DivisionRing.toRing.{u1} K (Field.toDivisionRing.{u1} K _inst_1)))) (NonAssocRing.toNonAssocSemiring.{u2} L (Ring.toNonAssocRing.{u2} L (DivisionRing.toRing.{u2} L (Field.toDivisionRing.{u2} L _inst_2))))) => K -> L) (RingHom.hasCoeToFun.{u1, u2} K L (NonAssocRing.toNonAssocSemiring.{u1} K (Ring.toNonAssocRing.{u1} K (DivisionRing.toRing.{u1} K (Field.toDivisionRing.{u1} K _inst_1)))) (NonAssocRing.toNonAssocSemiring.{u2} L (Ring.toNonAssocRing.{u2} L (DivisionRing.toRing.{u2} L (Field.toDivisionRing.{u2} L _inst_2))))) i (Polynomial.leadingCoeff.{u1} K (Ring.toSemiring.{u1} K (DivisionRing.toRing.{u1} K (Field.toDivisionRing.{u1} K _inst_1))) f))) (Multiset.prod.{u2} (Polynomial.{u2} L (Ring.toSemiring.{u2} L (DivisionRing.toRing.{u2} L (Field.toDivisionRing.{u2} L _inst_2)))) (CommRing.toCommMonoid.{u2} (Polynomial.{u2} L (Ring.toSemiring.{u2} L (DivisionRing.toRing.{u2} L (Field.toDivisionRing.{u2} L _inst_2)))) (Polynomial.commRing.{u2} L (EuclideanDomain.toCommRing.{u2} L (Field.toEuclideanDomain.{u2} L _inst_2)))) (Multiset.map.{u2, u2} L (Polynomial.{u2} L (Ring.toSemiring.{u2} L (DivisionRing.toRing.{u2} L (Field.toDivisionRing.{u2} L _inst_2)))) (fun (a : L) => HSub.hSub.{u2, u2, u2} (Polynomial.{u2} L (Ring.toSemiring.{u2} L (DivisionRing.toRing.{u2} L (Field.toDivisionRing.{u2} L _inst_2)))) (Polynomial.{u2} L (Ring.toSemiring.{u2} L (DivisionRing.toRing.{u2} L (Field.toDivisionRing.{u2} L _inst_2)))) (Polynomial.{u2} L (Ring.toSemiring.{u2} L (DivisionRing.toRing.{u2} L (Field.toDivisionRing.{u2} L _inst_2)))) (instHSub.{u2} (Polynomial.{u2} L (Ring.toSemiring.{u2} L (DivisionRing.toRing.{u2} L (Field.toDivisionRing.{u2} L _inst_2)))) (Polynomial.sub.{u2} L (DivisionRing.toRing.{u2} L (Field.toDivisionRing.{u2} L _inst_2)))) (Polynomial.X.{u2} L (Ring.toSemiring.{u2} L (DivisionRing.toRing.{u2} L (Field.toDivisionRing.{u2} L _inst_2)))) (coeFn.{succ u2, succ u2} (RingHom.{u2, u2} L (Polynomial.{u2} L (Ring.toSemiring.{u2} L (DivisionRing.toRing.{u2} L (Field.toDivisionRing.{u2} L _inst_2)))) (Semiring.toNonAssocSemiring.{u2} L (Ring.toSemiring.{u2} L (DivisionRing.toRing.{u2} L (Field.toDivisionRing.{u2} L _inst_2)))) (Semiring.toNonAssocSemiring.{u2} (Polynomial.{u2} L (Ring.toSemiring.{u2} L (DivisionRing.toRing.{u2} L (Field.toDivisionRing.{u2} L _inst_2)))) (Polynomial.semiring.{u2} L (Ring.toSemiring.{u2} L (DivisionRing.toRing.{u2} L (Field.toDivisionRing.{u2} L _inst_2)))))) (fun (_x : RingHom.{u2, u2} L (Polynomial.{u2} L (Ring.toSemiring.{u2} L (DivisionRing.toRing.{u2} L (Field.toDivisionRing.{u2} L _inst_2)))) (Semiring.toNonAssocSemiring.{u2} L (Ring.toSemiring.{u2} L (DivisionRing.toRing.{u2} L (Field.toDivisionRing.{u2} L _inst_2)))) (Semiring.toNonAssocSemiring.{u2} (Polynomial.{u2} L (Ring.toSemiring.{u2} L (DivisionRing.toRing.{u2} L (Field.toDivisionRing.{u2} L _inst_2)))) (Polynomial.semiring.{u2} L (Ring.toSemiring.{u2} L (DivisionRing.toRing.{u2} L (Field.toDivisionRing.{u2} L _inst_2)))))) => L -> (Polynomial.{u2} L (Ring.toSemiring.{u2} L (DivisionRing.toRing.{u2} L (Field.toDivisionRing.{u2} L _inst_2))))) (RingHom.hasCoeToFun.{u2, u2} L (Polynomial.{u2} L (Ring.toSemiring.{u2} L (DivisionRing.toRing.{u2} L (Field.toDivisionRing.{u2} L _inst_2)))) (Semiring.toNonAssocSemiring.{u2} L (Ring.toSemiring.{u2} L (DivisionRing.toRing.{u2} L (Field.toDivisionRing.{u2} L _inst_2)))) (Semiring.toNonAssocSemiring.{u2} (Polynomial.{u2} L (Ring.toSemiring.{u2} L (DivisionRing.toRing.{u2} L (Field.toDivisionRing.{u2} L _inst_2)))) (Polynomial.semiring.{u2} L (Ring.toSemiring.{u2} L (DivisionRing.toRing.{u2} L (Field.toDivisionRing.{u2} L _inst_2)))))) (Polynomial.C.{u2} L (Ring.toSemiring.{u2} L (DivisionRing.toRing.{u2} L (Field.toDivisionRing.{u2} L _inst_2)))) a)) s)))))
but is expected to have type
forall {K : Type.{u1}} {L : Type.{u2}} [_inst_1 : Field.{u1} K] [_inst_2 : Field.{u2} L] (i : RingHom.{u1, u2} K L (NonAssocRing.toNonAssocSemiring.{u1} K (Ring.toNonAssocRing.{u1} K (DivisionRing.toRing.{u1} K (Field.toDivisionRing.{u1} K _inst_1)))) (NonAssocRing.toNonAssocSemiring.{u2} L (Ring.toNonAssocRing.{u2} L (DivisionRing.toRing.{u2} L (Field.toDivisionRing.{u2} L _inst_2))))) {f : Polynomial.{u1} K (DivisionSemiring.toSemiring.{u1} K (Semifield.toDivisionSemiring.{u1} K (Field.toSemifield.{u1} K _inst_1)))}, Iff (Polynomial.Splits.{u1, u2} K L (EuclideanDomain.toCommRing.{u1} K (Field.toEuclideanDomain.{u1} K _inst_1)) _inst_2 i f) (Exists.{succ u2} (Multiset.{u2} L) (fun (s : Multiset.{u2} L) => Eq.{succ u2} (Polynomial.{u2} L (DivisionSemiring.toSemiring.{u2} L (Semifield.toDivisionSemiring.{u2} L (Field.toSemifield.{u2} L _inst_2)))) (Polynomial.map.{u1, u2} K L (DivisionSemiring.toSemiring.{u1} K (Semifield.toDivisionSemiring.{u1} K (Field.toSemifield.{u1} K _inst_1))) (DivisionSemiring.toSemiring.{u2} L (Semifield.toDivisionSemiring.{u2} L (Field.toSemifield.{u2} L _inst_2))) i f) (HMul.hMul.{u2, u2, u2} ((fun ([email protected]._hyg.2391 : (fun ([email protected]._hyg.2391 : K) => L) (Polynomial.leadingCoeff.{u1} K (DivisionSemiring.toSemiring.{u1} K (Semifield.toDivisionSemiring.{u1} K (Field.toSemifield.{u1} K _inst_1))) f)) => Polynomial.{u2} ((fun ([email protected]._hyg.2391 : K) => L) (Polynomial.leadingCoeff.{u1} K (DivisionSemiring.toSemiring.{u1} K (Semifield.toDivisionSemiring.{u1} K (Field.toSemifield.{u1} K _inst_1))) f)) (DivisionSemiring.toSemiring.{u2} ((fun ([email protected]._hyg.2391 : K) => L) (Polynomial.leadingCoeff.{u1} K (DivisionSemiring.toSemiring.{u1} K (Semifield.toDivisionSemiring.{u1} K (Field.toSemifield.{u1} K _inst_1))) f)) (Semifield.toDivisionSemiring.{u2} ((fun ([email protected]._hyg.2391 : K) => L) (Polynomial.leadingCoeff.{u1} K (DivisionSemiring.toSemiring.{u1} K (Semifield.toDivisionSemiring.{u1} K (Field.toSemifield.{u1} K _inst_1))) f)) (Field.toSemifield.{u2} ((fun ([email protected]._hyg.2391 : K) => L) (Polynomial.leadingCoeff.{u1} K (DivisionSemiring.toSemiring.{u1} K (Semifield.toDivisionSemiring.{u1} K (Field.toSemifield.{u1} K _inst_1))) f)) _inst_2)))) (FunLike.coe.{max (succ u1) (succ u2), succ u1, succ u2} (RingHom.{u1, u2} K L (NonAssocRing.toNonAssocSemiring.{u1} K (Ring.toNonAssocRing.{u1} K (DivisionRing.toRing.{u1} K (Field.toDivisionRing.{u1} K _inst_1)))) (NonAssocRing.toNonAssocSemiring.{u2} L (Ring.toNonAssocRing.{u2} L (DivisionRing.toRing.{u2} L (Field.toDivisionRing.{u2} L _inst_2))))) K (fun (a : K) => (fun ([email protected]._hyg.2391 : K) => L) a) (MulHomClass.toFunLike.{max u1 u2, u1, u2} (RingHom.{u1, u2} K L (NonAssocRing.toNonAssocSemiring.{u1} K (Ring.toNonAssocRing.{u1} K (DivisionRing.toRing.{u1} K (Field.toDivisionRing.{u1} K _inst_1)))) (NonAssocRing.toNonAssocSemiring.{u2} L (Ring.toNonAssocRing.{u2} L (DivisionRing.toRing.{u2} L (Field.toDivisionRing.{u2} L _inst_2))))) K L (NonUnitalNonAssocSemiring.toMul.{u1} K (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} K (NonAssocRing.toNonAssocSemiring.{u1} K (Ring.toNonAssocRing.{u1} K (DivisionRing.toRing.{u1} K (Field.toDivisionRing.{u1} K _inst_1)))))) (NonUnitalNonAssocSemiring.toMul.{u2} L (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} L (NonAssocRing.toNonAssocSemiring.{u2} L (Ring.toNonAssocRing.{u2} L (DivisionRing.toRing.{u2} L (Field.toDivisionRing.{u2} L _inst_2)))))) (NonUnitalRingHomClass.toMulHomClass.{max u1 u2, u1, u2} (RingHom.{u1, u2} K L (NonAssocRing.toNonAssocSemiring.{u1} K (Ring.toNonAssocRing.{u1} K (DivisionRing.toRing.{u1} K (Field.toDivisionRing.{u1} K _inst_1)))) (NonAssocRing.toNonAssocSemiring.{u2} L (Ring.toNonAssocRing.{u2} L (DivisionRing.toRing.{u2} L (Field.toDivisionRing.{u2} L _inst_2))))) K L (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} K (NonAssocRing.toNonAssocSemiring.{u1} K (Ring.toNonAssocRing.{u1} K (DivisionRing.toRing.{u1} K (Field.toDivisionRing.{u1} K _inst_1))))) (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} L (NonAssocRing.toNonAssocSemiring.{u2} L (Ring.toNonAssocRing.{u2} L (DivisionRing.toRing.{u2} L (Field.toDivisionRing.{u2} L _inst_2))))) (RingHomClass.toNonUnitalRingHomClass.{max u1 u2, u1, u2} (RingHom.{u1, u2} K L (NonAssocRing.toNonAssocSemiring.{u1} K (Ring.toNonAssocRing.{u1} K (DivisionRing.toRing.{u1} K (Field.toDivisionRing.{u1} K _inst_1)))) (NonAssocRing.toNonAssocSemiring.{u2} L (Ring.toNonAssocRing.{u2} L (DivisionRing.toRing.{u2} L (Field.toDivisionRing.{u2} L _inst_2))))) K L (NonAssocRing.toNonAssocSemiring.{u1} K (Ring.toNonAssocRing.{u1} K (DivisionRing.toRing.{u1} K (Field.toDivisionRing.{u1} K _inst_1)))) (NonAssocRing.toNonAssocSemiring.{u2} L (Ring.toNonAssocRing.{u2} L (DivisionRing.toRing.{u2} L (Field.toDivisionRing.{u2} L _inst_2)))) (RingHom.instRingHomClassRingHom.{u1, u2} K L (NonAssocRing.toNonAssocSemiring.{u1} K (Ring.toNonAssocRing.{u1} K (DivisionRing.toRing.{u1} K (Field.toDivisionRing.{u1} K _inst_1)))) (NonAssocRing.toNonAssocSemiring.{u2} L (Ring.toNonAssocRing.{u2} L (DivisionRing.toRing.{u2} L (Field.toDivisionRing.{u2} L _inst_2)))))))) i (Polynomial.leadingCoeff.{u1} K (DivisionSemiring.toSemiring.{u1} K (Semifield.toDivisionSemiring.{u1} K (Field.toSemifield.{u1} K _inst_1))) f))) (Polynomial.{u2} L (DivisionSemiring.toSemiring.{u2} L (Semifield.toDivisionSemiring.{u2} L (Field.toSemifield.{u2} L _inst_2)))) ((fun ([email protected]._hyg.2391 : (fun ([email protected]._hyg.2391 : K) => L) (Polynomial.leadingCoeff.{u1} K (DivisionSemiring.toSemiring.{u1} K (Semifield.toDivisionSemiring.{u1} K (Field.toSemifield.{u1} K _inst_1))) f)) => Polynomial.{u2} ((fun ([email protected]._hyg.2391 : K) => L) (Polynomial.leadingCoeff.{u1} K (DivisionSemiring.toSemiring.{u1} K (Semifield.toDivisionSemiring.{u1} K (Field.toSemifield.{u1} K _inst_1))) f)) (DivisionSemiring.toSemiring.{u2} ((fun ([email protected]._hyg.2391 : K) => L) (Polynomial.leadingCoeff.{u1} K (DivisionSemiring.toSemiring.{u1} K (Semifield.toDivisionSemiring.{u1} K (Field.toSemifield.{u1} K _inst_1))) f)) (Semifield.toDivisionSemiring.{u2} ((fun ([email protected]._hyg.2391 : K) => L) (Polynomial.leadingCoeff.{u1} K (DivisionSemiring.toSemiring.{u1} K (Semifield.toDivisionSemiring.{u1} K (Field.toSemifield.{u1} K _inst_1))) f)) (Field.toSemifield.{u2} ((fun ([email protected]._hyg.2391 : K) => L) (Polynomial.leadingCoeff.{u1} K (DivisionSemiring.toSemiring.{u1} K (Semifield.toDivisionSemiring.{u1} K (Field.toSemifield.{u1} K _inst_1))) f)) _inst_2)))) (FunLike.coe.{max (succ u1) (succ u2), succ u1, succ u2} (RingHom.{u1, u2} K L (NonAssocRing.toNonAssocSemiring.{u1} K (Ring.toNonAssocRing.{u1} K (DivisionRing.toRing.{u1} K (Field.toDivisionRing.{u1} K _inst_1)))) (NonAssocRing.toNonAssocSemiring.{u2} L (Ring.toNonAssocRing.{u2} L (DivisionRing.toRing.{u2} L (Field.toDivisionRing.{u2} L _inst_2))))) K 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(Ring.toNonAssocRing.{u1} K (DivisionRing.toRing.{u1} K (Field.toDivisionRing.{u1} K _inst_1)))) (NonAssocRing.toNonAssocSemiring.{u2} L (Ring.toNonAssocRing.{u2} L (DivisionRing.toRing.{u2} L (Field.toDivisionRing.{u2} L _inst_2))))) K L (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} K (NonAssocRing.toNonAssocSemiring.{u1} K (Ring.toNonAssocRing.{u1} K (DivisionRing.toRing.{u1} K (Field.toDivisionRing.{u1} K _inst_1))))) (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} L (NonAssocRing.toNonAssocSemiring.{u2} L (Ring.toNonAssocRing.{u2} L (DivisionRing.toRing.{u2} L (Field.toDivisionRing.{u2} L _inst_2))))) (RingHomClass.toNonUnitalRingHomClass.{max u1 u2, u1, u2} (RingHom.{u1, u2} K L (NonAssocRing.toNonAssocSemiring.{u1} K (Ring.toNonAssocRing.{u1} K (DivisionRing.toRing.{u1} K (Field.toDivisionRing.{u1} K _inst_1)))) (NonAssocRing.toNonAssocSemiring.{u2} L (Ring.toNonAssocRing.{u2} L (DivisionRing.toRing.{u2} L (Field.toDivisionRing.{u2} L _inst_2))))) K L (NonAssocRing.toNonAssocSemiring.{u1} K (Ring.toNonAssocRing.{u1} K (DivisionRing.toRing.{u1} K (Field.toDivisionRing.{u1} K _inst_1)))) (NonAssocRing.toNonAssocSemiring.{u2} L (Ring.toNonAssocRing.{u2} L (DivisionRing.toRing.{u2} L (Field.toDivisionRing.{u2} L _inst_2)))) (RingHom.instRingHomClassRingHom.{u1, u2} K L (NonAssocRing.toNonAssocSemiring.{u1} K (Ring.toNonAssocRing.{u1} K (DivisionRing.toRing.{u1} K (Field.toDivisionRing.{u1} K _inst_1)))) (NonAssocRing.toNonAssocSemiring.{u2} L (Ring.toNonAssocRing.{u2} L (DivisionRing.toRing.{u2} L (Field.toDivisionRing.{u2} L _inst_2)))))))) i (Polynomial.leadingCoeff.{u1} K (DivisionSemiring.toSemiring.{u1} K (Semifield.toDivisionSemiring.{u1} K (Field.toSemifield.{u1} K _inst_1))) f))) (instHMul.{u2} ((fun ([email protected]._hyg.2391 : (fun ([email protected]._hyg.2391 : K) => L) (Polynomial.leadingCoeff.{u1} K (DivisionSemiring.toSemiring.{u1} K (Semifield.toDivisionSemiring.{u1} K (Field.toSemifield.{u1} K _inst_1))) f)) => 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f)) (Semifield.toDivisionSemiring.{u2} ((fun ([email protected]._hyg.2391 : K) => L) (Polynomial.leadingCoeff.{u1} K (DivisionSemiring.toSemiring.{u1} K (Semifield.toDivisionSemiring.{u1} K (Field.toSemifield.{u1} K _inst_1))) f)) (Field.toSemifield.{u2} ((fun ([email protected]._hyg.2391 : K) => L) (Polynomial.leadingCoeff.{u1} K (DivisionSemiring.toSemiring.{u1} K (Semifield.toDivisionSemiring.{u1} K (Field.toSemifield.{u1} K _inst_1))) f)) _inst_2)))) (Polynomial.semiring.{u2} ((fun ([email protected]._hyg.2391 : K) => L) (Polynomial.leadingCoeff.{u1} K (DivisionSemiring.toSemiring.{u1} K (Semifield.toDivisionSemiring.{u1} K (Field.toSemifield.{u1} K _inst_1))) f)) (DivisionSemiring.toSemiring.{u2} ((fun ([email protected]._hyg.2391 : K) => L) (Polynomial.leadingCoeff.{u1} K (DivisionSemiring.toSemiring.{u1} K (Semifield.toDivisionSemiring.{u1} K (Field.toSemifield.{u1} K _inst_1))) f)) (Semifield.toDivisionSemiring.{u2} ((fun ([email protected]._hyg.2391 : K) => L) (Polynomial.leadingCoeff.{u1} K (DivisionSemiring.toSemiring.{u1} K (Semifield.toDivisionSemiring.{u1} K (Field.toSemifield.{u1} K _inst_1))) f)) (Field.toSemifield.{u2} ((fun ([email protected]._hyg.2391 : K) => L) (Polynomial.leadingCoeff.{u1} K (DivisionSemiring.toSemiring.{u1} K (Semifield.toDivisionSemiring.{u1} K (Field.toSemifield.{u1} K _inst_1))) f)) _inst_2)))))) ((fun ([email protected]._hyg.2391 : K) => L) (Polynomial.leadingCoeff.{u1} K (DivisionSemiring.toSemiring.{u1} K (Semifield.toDivisionSemiring.{u1} K (Field.toSemifield.{u1} K _inst_1))) f)) (Polynomial.{u2} ((fun ([email protected]._hyg.2391 : K) => L) (Polynomial.leadingCoeff.{u1} K (DivisionSemiring.toSemiring.{u1} K (Semifield.toDivisionSemiring.{u1} K (Field.toSemifield.{u1} K _inst_1))) f)) (DivisionSemiring.toSemiring.{u2} ((fun ([email protected]._hyg.2391 : K) => L) (Polynomial.leadingCoeff.{u1} K (DivisionSemiring.toSemiring.{u1} K (Semifield.toDivisionSemiring.{u1} K (Field.toSemifield.{u1} K _inst_1))) f)) (Semifield.toDivisionSemiring.{u2} ((fun ([email protected]._hyg.2391 : K) => L) (Polynomial.leadingCoeff.{u1} K (DivisionSemiring.toSemiring.{u1} K (Semifield.toDivisionSemiring.{u1} K (Field.toSemifield.{u1} K _inst_1))) f)) (Field.toSemifield.{u2} ((fun ([email protected]._hyg.2391 : K) => L) (Polynomial.leadingCoeff.{u1} K (DivisionSemiring.toSemiring.{u1} K (Semifield.toDivisionSemiring.{u1} K (Field.toSemifield.{u1} K _inst_1))) f)) _inst_2)))) (NonUnitalNonAssocSemiring.toMul.{u2} ((fun ([email protected]._hyg.2391 : K) => L) (Polynomial.leadingCoeff.{u1} K (DivisionSemiring.toSemiring.{u1} K (Semifield.toDivisionSemiring.{u1} K (Field.toSemifield.{u1} K _inst_1))) f)) (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} ((fun ([email protected]._hyg.2391 : K) => L) (Polynomial.leadingCoeff.{u1} K (DivisionSemiring.toSemiring.{u1} K (Semifield.toDivisionSemiring.{u1} K (Field.toSemifield.{u1} K _inst_1))) f)) (Semiring.toNonAssocSemiring.{u2} ((fun ([email protected]._hyg.2391 : K) => L) (Polynomial.leadingCoeff.{u1} K (DivisionSemiring.toSemiring.{u1} K (Semifield.toDivisionSemiring.{u1} K (Field.toSemifield.{u1} K _inst_1))) f)) (DivisionSemiring.toSemiring.{u2} ((fun ([email protected]._hyg.2391 : K) => L) (Polynomial.leadingCoeff.{u1} K (DivisionSemiring.toSemiring.{u1} K (Semifield.toDivisionSemiring.{u1} K (Field.toSemifield.{u1} K _inst_1))) f)) (Semifield.toDivisionSemiring.{u2} ((fun ([email protected]._hyg.2391 : K) => L) (Polynomial.leadingCoeff.{u1} K (DivisionSemiring.toSemiring.{u1} K (Semifield.toDivisionSemiring.{u1} K (Field.toSemifield.{u1} K _inst_1))) f)) (Field.toSemifield.{u2} ((fun ([email protected]._hyg.2391 : K) => L) (Polynomial.leadingCoeff.{u1} K (DivisionSemiring.toSemiring.{u1} K (Semifield.toDivisionSemiring.{u1} K (Field.toSemifield.{u1} K _inst_1))) f)) _inst_2)))))) (NonUnitalNonAssocSemiring.toMul.{u2} (Polynomial.{u2} ((fun ([email protected]._hyg.2391 : K) => L) (Polynomial.leadingCoeff.{u1} K (DivisionSemiring.toSemiring.{u1} K (Semifield.toDivisionSemiring.{u1} K (Field.toSemifield.{u1} K _inst_1))) f)) (DivisionSemiring.toSemiring.{u2} ((fun ([email protected]._hyg.2391 : K) => L) (Polynomial.leadingCoeff.{u1} K (DivisionSemiring.toSemiring.{u1} K (Semifield.toDivisionSemiring.{u1} K (Field.toSemifield.{u1} K _inst_1))) f)) (Semifield.toDivisionSemiring.{u2} ((fun ([email protected]._hyg.2391 : K) => L) (Polynomial.leadingCoeff.{u1} K (DivisionSemiring.toSemiring.{u1} K (Semifield.toDivisionSemiring.{u1} K (Field.toSemifield.{u1} K _inst_1))) f)) (Field.toSemifield.{u2} ((fun ([email protected]._hyg.2391 : K) => L) (Polynomial.leadingCoeff.{u1} K (DivisionSemiring.toSemiring.{u1} K (Semifield.toDivisionSemiring.{u1} K (Field.toSemifield.{u1} K _inst_1))) f)) _inst_2)))) (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} (Polynomial.{u2} ((fun ([email protected]._hyg.2391 : K) => L) (Polynomial.leadingCoeff.{u1} K (DivisionSemiring.toSemiring.{u1} K (Semifield.toDivisionSemiring.{u1} K (Field.toSemifield.{u1} K _inst_1))) f)) (DivisionSemiring.toSemiring.{u2} ((fun ([email protected]._hyg.2391 : K) => L) (Polynomial.leadingCoeff.{u1} K (DivisionSemiring.toSemiring.{u1} K (Semifield.toDivisionSemiring.{u1} K (Field.toSemifield.{u1} K _inst_1))) f)) (Semifield.toDivisionSemiring.{u2} ((fun ([email protected]._hyg.2391 : K) => L) (Polynomial.leadingCoeff.{u1} K (DivisionSemiring.toSemiring.{u1} K (Semifield.toDivisionSemiring.{u1} K (Field.toSemifield.{u1} K _inst_1))) f)) (Field.toSemifield.{u2} ((fun ([email protected]._hyg.2391 : K) => L) (Polynomial.leadingCoeff.{u1} K (DivisionSemiring.toSemiring.{u1} K (Semifield.toDivisionSemiring.{u1} K (Field.toSemifield.{u1} K _inst_1))) f)) _inst_2)))) (Semiring.toNonAssocSemiring.{u2} (Polynomial.{u2} ((fun ([email protected]._hyg.2391 : K) => L) (Polynomial.leadingCoeff.{u1} K (DivisionSemiring.toSemiring.{u1} K (Semifield.toDivisionSemiring.{u1} K (Field.toSemifield.{u1} K _inst_1))) f)) (DivisionSemiring.toSemiring.{u2} ((fun ([email protected]._hyg.2391 : K) => L) (Polynomial.leadingCoeff.{u1} K (DivisionSemiring.toSemiring.{u1} K (Semifield.toDivisionSemiring.{u1} K (Field.toSemifield.{u1} K _inst_1))) f)) (Semifield.toDivisionSemiring.{u2} ((fun ([email protected]._hyg.2391 : K) => L) (Polynomial.leadingCoeff.{u1} K (DivisionSemiring.toSemiring.{u1} K (Semifield.toDivisionSemiring.{u1} K (Field.toSemifield.{u1} K _inst_1))) f)) (Field.toSemifield.{u2} ((fun ([email protected]._hyg.2391 : K) => L) (Polynomial.leadingCoeff.{u1} K (DivisionSemiring.toSemiring.{u1} K (Semifield.toDivisionSemiring.{u1} K (Field.toSemifield.{u1} K _inst_1))) f)) _inst_2)))) (Polynomial.semiring.{u2} ((fun ([email protected]._hyg.2391 : K) => L) (Polynomial.leadingCoeff.{u1} K (DivisionSemiring.toSemiring.{u1} K (Semifield.toDivisionSemiring.{u1} K (Field.toSemifield.{u1} K _inst_1))) f)) (DivisionSemiring.toSemiring.{u2} ((fun ([email protected]._hyg.2391 : K) => L) (Polynomial.leadingCoeff.{u1} K (DivisionSemiring.toSemiring.{u1} K (Semifield.toDivisionSemiring.{u1} K (Field.toSemifield.{u1} K _inst_1))) f)) (Semifield.toDivisionSemiring.{u2} ((fun ([email protected]._hyg.2391 : K) => L) (Polynomial.leadingCoeff.{u1} K (DivisionSemiring.toSemiring.{u1} K (Semifield.toDivisionSemiring.{u1} K (Field.toSemifield.{u1} K _inst_1))) f)) (Field.toSemifield.{u2} ((fun ([email protected]._hyg.2391 : K) => L) (Polynomial.leadingCoeff.{u1} K (DivisionSemiring.toSemiring.{u1} K (Semifield.toDivisionSemiring.{u1} K (Field.toSemifield.{u1} K _inst_1))) f)) _inst_2))))))) (NonUnitalRingHomClass.toMulHomClass.{u2, u2, u2} (RingHom.{u2, u2} ((fun ([email protected]._hyg.2391 : K) => L) (Polynomial.leadingCoeff.{u1} K (DivisionSemiring.toSemiring.{u1} K (Semifield.toDivisionSemiring.{u1} K (Field.toSemifield.{u1} K _inst_1))) f)) (Polynomial.{u2} ((fun ([email protected]._hyg.2391 : K) => L) (Polynomial.leadingCoeff.{u1} K (DivisionSemiring.toSemiring.{u1} K (Semifield.toDivisionSemiring.{u1} K (Field.toSemifield.{u1} K _inst_1))) f)) (DivisionSemiring.toSemiring.{u2} ((fun ([email protected]._hyg.2391 : K) => L) (Polynomial.leadingCoeff.{u1} K (DivisionSemiring.toSemiring.{u1} K (Semifield.toDivisionSemiring.{u1} K (Field.toSemifield.{u1} K _inst_1))) f)) (Semifield.toDivisionSemiring.{u2} ((fun ([email protected]._hyg.2391 : K) => L) (Polynomial.leadingCoeff.{u1} K (DivisionSemiring.toSemiring.{u1} K (Semifield.toDivisionSemiring.{u1} K (Field.toSemifield.{u1} K _inst_1))) f)) (Field.toSemifield.{u2} ((fun ([email protected]._hyg.2391 : K) => L) (Polynomial.leadingCoeff.{u1} K (DivisionSemiring.toSemiring.{u1} K (Semifield.toDivisionSemiring.{u1} K (Field.toSemifield.{u1} K _inst_1))) f)) _inst_2)))) (Semiring.toNonAssocSemiring.{u2} ((fun ([email protected]._hyg.2391 : K) => L) (Polynomial.leadingCoeff.{u1} K (DivisionSemiring.toSemiring.{u1} K (Semifield.toDivisionSemiring.{u1} K (Field.toSemifield.{u1} K _inst_1))) f)) (DivisionSemiring.toSemiring.{u2} ((fun ([email protected]._hyg.2391 : K) => L) (Polynomial.leadingCoeff.{u1} K (DivisionSemiring.toSemiring.{u1} K (Semifield.toDivisionSemiring.{u1} K (Field.toSemifield.{u1} K _inst_1))) f)) (Semifield.toDivisionSemiring.{u2} ((fun ([email protected]._hyg.2391 : K) => L) (Polynomial.leadingCoeff.{u1} K (DivisionSemiring.toSemiring.{u1} K (Semifield.toDivisionSemiring.{u1} K (Field.toSemifield.{u1} K _inst_1))) f)) (Field.toSemifield.{u2} ((fun ([email protected]._hyg.2391 : K) => L) (Polynomial.leadingCoeff.{u1} K (DivisionSemiring.toSemiring.{u1} K (Semifield.toDivisionSemiring.{u1} K (Field.toSemifield.{u1} K _inst_1))) f)) _inst_2)))) (Semiring.toNonAssocSemiring.{u2} (Polynomial.{u2} ((fun ([email protected]._hyg.2391 : K) => L) (Polynomial.leadingCoeff.{u1} K (DivisionSemiring.toSemiring.{u1} K (Semifield.toDivisionSemiring.{u1} K (Field.toSemifield.{u1} K _inst_1))) f)) (DivisionSemiring.toSemiring.{u2} ((fun ([email protected]._hyg.2391 : K) => L) (Polynomial.leadingCoeff.{u1} K (DivisionSemiring.toSemiring.{u1} K (Semifield.toDivisionSemiring.{u1} K (Field.toSemifield.{u1} K _inst_1))) f)) (Semifield.toDivisionSemiring.{u2} ((fun ([email protected]._hyg.2391 : K) => L) (Polynomial.leadingCoeff.{u1} K (DivisionSemiring.toSemiring.{u1} K (Semifield.toDivisionSemiring.{u1} K (Field.toSemifield.{u1} K _inst_1))) f)) (Field.toSemifield.{u2} ((fun ([email protected]._hyg.2391 : K) => L) (Polynomial.leadingCoeff.{u1} K (DivisionSemiring.toSemiring.{u1} K (Semifield.toDivisionSemiring.{u1} K (Field.toSemifield.{u1} K _inst_1))) f)) _inst_2)))) (Polynomial.semiring.{u2} ((fun ([email protected]._hyg.2391 : K) => L) (Polynomial.leadingCoeff.{u1} K (DivisionSemiring.toSemiring.{u1} K (Semifield.toDivisionSemiring.{u1} K (Field.toSemifield.{u1} K _inst_1))) f)) (DivisionSemiring.toSemiring.{u2} ((fun ([email protected]._hyg.2391 : K) => L) (Polynomial.leadingCoeff.{u1} K (DivisionSemiring.toSemiring.{u1} K (Semifield.toDivisionSemiring.{u1} K (Field.toSemifield.{u1} K _inst_1))) f)) (Semifield.toDivisionSemiring.{u2} ((fun ([email protected]._hyg.2391 : K) => L) (Polynomial.leadingCoeff.{u1} K (DivisionSemiring.toSemiring.{u1} K (Semifield.toDivisionSemiring.{u1} K (Field.toSemifield.{u1} K _inst_1))) f)) (Field.toSemifield.{u2} ((fun ([email protected]._hyg.2391 : K) => L) (Polynomial.leadingCoeff.{u1} K (DivisionSemiring.toSemiring.{u1} K (Semifield.toDivisionSemiring.{u1} K (Field.toSemifield.{u1} K _inst_1))) f)) _inst_2)))))) ((fun ([email protected]._hyg.2391 : K) => L) (Polynomial.leadingCoeff.{u1} K (DivisionSemiring.toSemiring.{u1} K (Semifield.toDivisionSemiring.{u1} K (Field.toSemifield.{u1} K _inst_1))) f)) (Polynomial.{u2} ((fun ([email protected]._hyg.2391 : K) => L) (Polynomial.leadingCoeff.{u1} K (DivisionSemiring.toSemiring.{u1} K (Semifield.toDivisionSemiring.{u1} K (Field.toSemifield.{u1} K _inst_1))) f)) (DivisionSemiring.toSemiring.{u2} ((fun ([email protected]._hyg.2391 : K) => L) (Polynomial.leadingCoeff.{u1} K (DivisionSemiring.toSemiring.{u1} K (Semifield.toDivisionSemiring.{u1} K (Field.toSemifield.{u1} K _inst_1))) f)) (Semifield.toDivisionSemiring.{u2} ((fun ([email protected]._hyg.2391 : K) => L) (Polynomial.leadingCoeff.{u1} K (DivisionSemiring.toSemiring.{u1} K (Semifield.toDivisionSemiring.{u1} K (Field.toSemifield.{u1} K _inst_1))) f)) (Field.toSemifield.{u2} ((fun ([email protected]._hyg.2391 : K) => L) (Polynomial.leadingCoeff.{u1} K (DivisionSemiring.toSemiring.{u1} K (Semifield.toDivisionSemiring.{u1} K (Field.toSemifield.{u1} K _inst_1))) f)) _inst_2)))) (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} ((fun ([email protected]._hyg.2391 : K) => L) (Polynomial.leadingCoeff.{u1} K (DivisionSemiring.toSemiring.{u1} K (Semifield.toDivisionSemiring.{u1} K (Field.toSemifield.{u1} K _inst_1))) f)) (Semiring.toNonAssocSemiring.{u2} ((fun ([email protected]._hyg.2391 : K) => L) (Polynomial.leadingCoeff.{u1} K (DivisionSemiring.toSemiring.{u1} K (Semifield.toDivisionSemiring.{u1} K (Field.toSemifield.{u1} K _inst_1))) f)) (DivisionSemiring.toSemiring.{u2} ((fun ([email protected]._hyg.2391 : K) => L) (Polynomial.leadingCoeff.{u1} K (DivisionSemiring.toSemiring.{u1} K (Semifield.toDivisionSemiring.{u1} K (Field.toSemifield.{u1} K _inst_1))) f)) (Semifield.toDivisionSemiring.{u2} ((fun ([email protected]._hyg.2391 : K) => L) (Polynomial.leadingCoeff.{u1} K (DivisionSemiring.toSemiring.{u1} K (Semifield.toDivisionSemiring.{u1} K (Field.toSemifield.{u1} K _inst_1))) f)) (Field.toSemifield.{u2} ((fun ([email protected]._hyg.2391 : K) => L) (Polynomial.leadingCoeff.{u1} K (DivisionSemiring.toSemiring.{u1} K (Semifield.toDivisionSemiring.{u1} K (Field.toSemifield.{u1} K _inst_1))) f)) _inst_2))))) (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} (Polynomial.{u2} ((fun ([email protected]._hyg.2391 : K) => L) (Polynomial.leadingCoeff.{u1} K (DivisionSemiring.toSemiring.{u1} K (Semifield.toDivisionSemiring.{u1} K (Field.toSemifield.{u1} K _inst_1))) f)) (DivisionSemiring.toSemiring.{u2} ((fun ([email protected]._hyg.2391 : K) => L) (Polynomial.leadingCoeff.{u1} K (DivisionSemiring.toSemiring.{u1} K (Semifield.toDivisionSemiring.{u1} K (Field.toSemifield.{u1} K _inst_1))) f)) (Semifield.toDivisionSemiring.{u2} ((fun ([email protected]._hyg.2391 : K) => L) (Polynomial.leadingCoeff.{u1} K (DivisionSemiring.toSemiring.{u1} K (Semifield.toDivisionSemiring.{u1} K (Field.toSemifield.{u1} K _inst_1))) f)) (Field.toSemifield.{u2} ((fun ([email protected]._hyg.2391 : K) => L) (Polynomial.leadingCoeff.{u1} K (DivisionSemiring.toSemiring.{u1} K (Semifield.toDivisionSemiring.{u1} K (Field.toSemifield.{u1} K _inst_1))) f)) _inst_2)))) (Semiring.toNonAssocSemiring.{u2} (Polynomial.{u2} ((fun ([email protected]._hyg.2391 : K) => L) (Polynomial.leadingCoeff.{u1} K (DivisionSemiring.toSemiring.{u1} K (Semifield.toDivisionSemiring.{u1} K (Field.toSemifield.{u1} K _inst_1))) f)) (DivisionSemiring.toSemiring.{u2} ((fun ([email protected]._hyg.2391 : K) => L) (Polynomial.leadingCoeff.{u1} K (DivisionSemiring.toSemiring.{u1} K (Semifield.toDivisionSemiring.{u1} K (Field.toSemifield.{u1} K _inst_1))) f)) (Semifield.toDivisionSemiring.{u2} ((fun ([email protected]._hyg.2391 : K) => L) (Polynomial.leadingCoeff.{u1} K (DivisionSemiring.toSemiring.{u1} K (Semifield.toDivisionSemiring.{u1} K (Field.toSemifield.{u1} K _inst_1))) f)) (Field.toSemifield.{u2} ((fun ([email protected]._hyg.2391 : K) => L) (Polynomial.leadingCoeff.{u1} K (DivisionSemiring.toSemiring.{u1} K (Semifield.toDivisionSemiring.{u1} K (Field.toSemifield.{u1} K _inst_1))) f)) _inst_2)))) (Polynomial.semiring.{u2} ((fun ([email protected]._hyg.2391 : K) => L) (Polynomial.leadingCoeff.{u1} K (DivisionSemiring.toSemiring.{u1} K (Semifield.toDivisionSemiring.{u1} K (Field.toSemifield.{u1} K _inst_1))) f)) (DivisionSemiring.toSemiring.{u2} ((fun ([email protected]._hyg.2391 : K) => L) (Polynomial.leadingCoeff.{u1} K (DivisionSemiring.toSemiring.{u1} K (Semifield.toDivisionSemiring.{u1} K (Field.toSemifield.{u1} K _inst_1))) f)) (Semifield.toDivisionSemiring.{u2} ((fun ([email protected]._hyg.2391 : K) => L) (Polynomial.leadingCoeff.{u1} K (DivisionSemiring.toSemiring.{u1} K (Semifield.toDivisionSemiring.{u1} K (Field.toSemifield.{u1} K _inst_1))) f)) (Field.toSemifield.{u2} ((fun ([email protected]._hyg.2391 : K) => L) (Polynomial.leadingCoeff.{u1} K (DivisionSemiring.toSemiring.{u1} K (Semifield.toDivisionSemiring.{u1} K (Field.toSemifield.{u1} K _inst_1))) f)) _inst_2)))))) (RingHomClass.toNonUnitalRingHomClass.{u2, u2, u2} (RingHom.{u2, u2} ((fun ([email protected]._hyg.2391 : K) => L) (Polynomial.leadingCoeff.{u1} K (DivisionSemiring.toSemiring.{u1} K (Semifield.toDivisionSemiring.{u1} K (Field.toSemifield.{u1} K _inst_1))) f)) (Polynomial.{u2} ((fun ([email protected]._hyg.2391 : K) => L) (Polynomial.leadingCoeff.{u1} K (DivisionSemiring.toSemiring.{u1} K (Semifield.toDivisionSemiring.{u1} K (Field.toSemifield.{u1} K _inst_1))) f)) (DivisionSemiring.toSemiring.{u2} ((fun ([email protected]._hyg.2391 : K) => L) (Polynomial.leadingCoeff.{u1} K (DivisionSemiring.toSemiring.{u1} K (Semifield.toDivisionSemiring.{u1} K (Field.toSemifield.{u1} K _inst_1))) f)) (Semifield.toDivisionSemiring.{u2} ((fun ([email protected]._hyg.2391 : K) => L) (Polynomial.leadingCoeff.{u1} K (DivisionSemiring.toSemiring.{u1} K (Semifield.toDivisionSemiring.{u1} K (Field.toSemifield.{u1} K _inst_1))) f)) (Field.toSemifield.{u2} ((fun ([email protected]._hyg.2391 : K) => L) (Polynomial.leadingCoeff.{u1} K (DivisionSemiring.toSemiring.{u1} K (Semifield.toDivisionSemiring.{u1} K (Field.toSemifield.{u1} K _inst_1))) f)) _inst_2)))) (Semiring.toNonAssocSemiring.{u2} ((fun ([email protected]._hyg.2391 : K) => L) (Polynomial.leadingCoeff.{u1} K (DivisionSemiring.toSemiring.{u1} K (Semifield.toDivisionSemiring.{u1} K (Field.toSemifield.{u1} K _inst_1))) f)) 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(DivisionSemiring.toSemiring.{u2} L (Semifield.toDivisionSemiring.{u2} L (Field.toSemifield.{u2} L _inst_2)))) (Polynomial.semiring.{u2} L (DivisionSemiring.toSemiring.{u2} L (Semifield.toDivisionSemiring.{u2} L (Field.toSemifield.{u2} L _inst_2))))) (RingHom.instRingHomClassRingHom.{u2, u2} L (Polynomial.{u2} L (DivisionSemiring.toSemiring.{u2} L (Semifield.toDivisionSemiring.{u2} L (Field.toSemifield.{u2} L _inst_2)))) (Semiring.toNonAssocSemiring.{u2} L (DivisionSemiring.toSemiring.{u2} L (Semifield.toDivisionSemiring.{u2} L (Field.toSemifield.{u2} L _inst_2)))) (Semiring.toNonAssocSemiring.{u2} (Polynomial.{u2} L (DivisionSemiring.toSemiring.{u2} L (Semifield.toDivisionSemiring.{u2} L (Field.toSemifield.{u2} L _inst_2)))) (Polynomial.semiring.{u2} L (DivisionSemiring.toSemiring.{u2} L (Semifield.toDivisionSemiring.{u2} L (Field.toSemifield.{u2} L _inst_2))))))))) (Polynomial.C.{u2} L (DivisionSemiring.toSemiring.{u2} L (Semifield.toDivisionSemiring.{u2} L (Field.toSemifield.{u2} L _inst_2)))) a)) s)))))
Case conversion may be inaccurate. Consider using '#align polynomial.splits_iff_exists_multiset Polynomial.splits_iff_exists_multisetₓ'. -/
theorem splits_iff_exists_multiset {f : K[X]} :
Splits i f ↔
∃ s : Multiset L, f.map i = C (i f.leadingCoeff) * (s.map fun a : L => X - C a).Prod :=
⟨fun hf => ⟨(f.map i).roots, eq_prod_roots_of_splits hf⟩, fun ⟨s, hs⟩ =>
splits_of_exists_multiset i hs⟩
#align polynomial.splits_iff_exists_multiset Polynomial.splits_iff_exists_multiset
#print Polynomial.splits_comp_of_splits /-
theorem splits_comp_of_splits (j : L →+* F) {f : K[X]} (h : Splits i f) : Splits (j.comp i) f :=
by
change i with (RingHom.id _).comp i at h
rw [← splits_map_iff]
rw [← splits_map_iff i] at h
exact splits_of_splits_id _ h
#align polynomial.splits_comp_of_splits Polynomial.splits_comp_of_splits
-/
/- warning: polynomial.splits_iff_card_roots -> Polynomial.splits_iff_card_roots is a dubious translation:
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but is expected to have type
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Case conversion may be inaccurate. Consider using '#align polynomial.splits_iff_card_roots Polynomial.splits_iff_card_rootsₓ'. -/
/-- A polynomial splits if and only if it has as many roots as its degree. -/
theorem splits_iff_card_roots {p : K[X]} : Splits (RingHom.id K) p ↔ p.roots.card = p.natDegree :=
by
constructor
· intro H
rw [nat_degree_eq_card_roots H, map_id]
· intro hroots
rw [splits_iff_exists_multiset (RingHom.id K)]
use p.roots
simp only [RingHom.id_apply, map_id]
exact (C_leading_coeff_mul_prod_multiset_X_sub_C hroots).symm
#align polynomial.splits_iff_card_roots Polynomial.splits_iff_card_roots
#print Polynomial.aeval_root_derivative_of_splits /-
theorem aeval_root_derivative_of_splits [Algebra K L] {P : K[X]} (hmo : P.Monic)
(hP : P.Splits (algebraMap K L)) {r : L} (hr : r ∈ (P.map (algebraMap K L)).roots) :
aeval r P.derivative = (((P.map <| algebraMap K L).roots.eraseₓ r).map fun a => r - a).Prod :=
by
replace hmo := hmo.map (algebraMap K L)
replace hP := (splits_id_iff_splits (algebraMap K L)).2 hP
rw [aeval_def, ← eval_map, ← derivative_map]
nth_rw 1 [eq_prod_roots_of_monic_of_splits_id hmo hP]
rw [eval_multiset_prod_X_sub_C_derivative hr]
#align polynomial.aeval_root_derivative_of_splits Polynomial.aeval_root_derivative_of_splits
-/
/- warning: polynomial.prod_roots_eq_coeff_zero_of_monic_of_split -> Polynomial.prod_roots_eq_coeff_zero_of_monic_of_split is a dubious translation:
lean 3 declaration is
forall {K : Type.{u1}} [_inst_1 : Field.{u1} K] {P : Polynomial.{u1} K (Ring.toSemiring.{u1} K (DivisionRing.toRing.{u1} K (Field.toDivisionRing.{u1} K _inst_1)))}, (Polynomial.Monic.{u1} K (Ring.toSemiring.{u1} K (DivisionRing.toRing.{u1} K (Field.toDivisionRing.{u1} K _inst_1))) P) -> (Polynomial.Splits.{u1, u1} K K (EuclideanDomain.toCommRing.{u1} K (Field.toEuclideanDomain.{u1} K _inst_1)) _inst_1 (RingHom.id.{u1} K (NonAssocRing.toNonAssocSemiring.{u1} K (Ring.toNonAssocRing.{u1} K (DivisionRing.toRing.{u1} K (Field.toDivisionRing.{u1} K _inst_1))))) P) -> (Eq.{succ u1} K (Polynomial.coeff.{u1} K (Ring.toSemiring.{u1} K (DivisionRing.toRing.{u1} K (Field.toDivisionRing.{u1} K _inst_1))) P (OfNat.ofNat.{0} Nat 0 (OfNat.mk.{0} Nat 0 (Zero.zero.{0} Nat Nat.hasZero)))) (HMul.hMul.{u1, u1, u1} K K K (instHMul.{u1} K (Distrib.toHasMul.{u1} K (Ring.toDistrib.{u1} K (DivisionRing.toRing.{u1} K (Field.toDivisionRing.{u1} K _inst_1))))) (HPow.hPow.{u1, 0, u1} K Nat K (instHPow.{u1, 0} K Nat (Monoid.Pow.{u1} K (Ring.toMonoid.{u1} K (DivisionRing.toRing.{u1} K (Field.toDivisionRing.{u1} K _inst_1))))) (Neg.neg.{u1} K (SubNegMonoid.toHasNeg.{u1} K (AddGroup.toSubNegMonoid.{u1} K (AddGroupWithOne.toAddGroup.{u1} K (AddCommGroupWithOne.toAddGroupWithOne.{u1} K (Ring.toAddCommGroupWithOne.{u1} K (DivisionRing.toRing.{u1} K (Field.toDivisionRing.{u1} K _inst_1))))))) (OfNat.ofNat.{u1} K 1 (OfNat.mk.{u1} K 1 (One.one.{u1} K (AddMonoidWithOne.toOne.{u1} K (AddGroupWithOne.toAddMonoidWithOne.{u1} K (AddCommGroupWithOne.toAddGroupWithOne.{u1} K (Ring.toAddCommGroupWithOne.{u1} K (DivisionRing.toRing.{u1} K (Field.toDivisionRing.{u1} K _inst_1)))))))))) (Polynomial.natDegree.{u1} K (Ring.toSemiring.{u1} K (DivisionRing.toRing.{u1} K (Field.toDivisionRing.{u1} K _inst_1))) P)) (Multiset.prod.{u1} K (CommRing.toCommMonoid.{u1} K (EuclideanDomain.toCommRing.{u1} K (Field.toEuclideanDomain.{u1} K _inst_1))) (Polynomial.roots.{u1} K (EuclideanDomain.toCommRing.{u1} K (Field.toEuclideanDomain.{u1} K _inst_1)) (Field.isDomain.{u1} K _inst_1) P))))
but is expected to have type
forall {K : Type.{u1}} [_inst_1 : Field.{u1} K] {P : Polynomial.{u1} K (DivisionSemiring.toSemiring.{u1} K (Semifield.toDivisionSemiring.{u1} K (Field.toSemifield.{u1} K _inst_1)))}, (Polynomial.Monic.{u1} K (DivisionSemiring.toSemiring.{u1} K (Semifield.toDivisionSemiring.{u1} K (Field.toSemifield.{u1} K _inst_1))) P) -> (Polynomial.Splits.{u1, u1} K K (EuclideanDomain.toCommRing.{u1} K (Field.toEuclideanDomain.{u1} K _inst_1)) _inst_1 (RingHom.id.{u1} K (NonAssocRing.toNonAssocSemiring.{u1} K (Ring.toNonAssocRing.{u1} K (DivisionRing.toRing.{u1} K (Field.toDivisionRing.{u1} K _inst_1))))) P) -> (Eq.{succ u1} K (Polynomial.coeff.{u1} K (DivisionSemiring.toSemiring.{u1} K (Semifield.toDivisionSemiring.{u1} K (Field.toSemifield.{u1} K _inst_1))) P (OfNat.ofNat.{0} Nat 0 (instOfNatNat 0))) (HMul.hMul.{u1, u1, u1} K K K (instHMul.{u1} K (NonUnitalNonAssocRing.toMul.{u1} K (NonAssocRing.toNonUnitalNonAssocRing.{u1} K (Ring.toNonAssocRing.{u1} K (DivisionRing.toRing.{u1} K (Field.toDivisionRing.{u1} K _inst_1)))))) (HPow.hPow.{u1, 0, u1} K Nat K (instHPow.{u1, 0} K Nat (Monoid.Pow.{u1} K (MonoidWithZero.toMonoid.{u1} K (Semiring.toMonoidWithZero.{u1} K (DivisionSemiring.toSemiring.{u1} K (Semifield.toDivisionSemiring.{u1} K (Field.toSemifield.{u1} K _inst_1))))))) (Neg.neg.{u1} K (Ring.toNeg.{u1} K (DivisionRing.toRing.{u1} K (Field.toDivisionRing.{u1} K _inst_1))) (OfNat.ofNat.{u1} K 1 (One.toOfNat1.{u1} K (NonAssocRing.toOne.{u1} K (Ring.toNonAssocRing.{u1} K (DivisionRing.toRing.{u1} K (Field.toDivisionRing.{u1} K _inst_1))))))) (Polynomial.natDegree.{u1} K (DivisionSemiring.toSemiring.{u1} K (Semifield.toDivisionSemiring.{u1} K (Field.toSemifield.{u1} K _inst_1))) P)) (Multiset.prod.{u1} K (CommRing.toCommMonoid.{u1} K (EuclideanDomain.toCommRing.{u1} K (Field.toEuclideanDomain.{u1} K _inst_1))) (Polynomial.roots.{u1} K (EuclideanDomain.toCommRing.{u1} K (Field.toEuclideanDomain.{u1} K _inst_1)) (EuclideanDomain.instIsDomainToSemiringToRingToCommRing.{u1} K (Field.toEuclideanDomain.{u1} K _inst_1)) P))))
Case conversion may be inaccurate. Consider using '#align polynomial.prod_roots_eq_coeff_zero_of_monic_of_split Polynomial.prod_roots_eq_coeff_zero_of_monic_of_splitₓ'. -/
/-- If `P` is a monic polynomial that splits, then `coeff P 0` equals the product of the roots. -/
theorem prod_roots_eq_coeff_zero_of_monic_of_split {P : K[X]} (hmo : P.Monic)
(hP : P.Splits (RingHom.id K)) : coeff P 0 = (-1) ^ P.natDegree * P.roots.Prod :=
by
nth_rw 1 [eq_prod_roots_of_monic_of_splits_id hmo hP]
rw [coeff_zero_eq_eval_zero, eval_multiset_prod, Multiset.map_map]
simp_rw [Function.comp_apply, eval_sub, eval_X, zero_sub, eval_C]
conv_lhs =>
congr
congr
ext
rw [neg_eq_neg_one_mul]
rw [Multiset.prod_map_mul, Multiset.map_const, Multiset.prod_replicate, Multiset.map_id',
splits_iff_card_roots.1 hP]
#align polynomial.prod_roots_eq_coeff_zero_of_monic_of_split Polynomial.prod_roots_eq_coeff_zero_of_monic_of_split
/- warning: polynomial.sum_roots_eq_next_coeff_of_monic_of_split -> Polynomial.sum_roots_eq_nextCoeff_of_monic_of_split is a dubious translation:
lean 3 declaration is
forall {K : Type.{u1}} [_inst_1 : Field.{u1} K] {P : Polynomial.{u1} K (Ring.toSemiring.{u1} K (DivisionRing.toRing.{u1} K (Field.toDivisionRing.{u1} K _inst_1)))}, (Polynomial.Monic.{u1} K (Ring.toSemiring.{u1} K (DivisionRing.toRing.{u1} K (Field.toDivisionRing.{u1} K _inst_1))) P) -> (Polynomial.Splits.{u1, u1} K K (EuclideanDomain.toCommRing.{u1} K (Field.toEuclideanDomain.{u1} K _inst_1)) _inst_1 (RingHom.id.{u1} K (NonAssocRing.toNonAssocSemiring.{u1} K (Ring.toNonAssocRing.{u1} K (DivisionRing.toRing.{u1} K (Field.toDivisionRing.{u1} K _inst_1))))) P) -> (Eq.{succ u1} K (Polynomial.nextCoeff.{u1} K (Ring.toSemiring.{u1} K (DivisionRing.toRing.{u1} K (Field.toDivisionRing.{u1} K _inst_1))) P) (Neg.neg.{u1} K (SubNegMonoid.toHasNeg.{u1} K (AddGroup.toSubNegMonoid.{u1} K (AddGroupWithOne.toAddGroup.{u1} K (AddCommGroupWithOne.toAddGroupWithOne.{u1} K (Ring.toAddCommGroupWithOne.{u1} K (DivisionRing.toRing.{u1} K (Field.toDivisionRing.{u1} K _inst_1))))))) (Multiset.sum.{u1} K (AddCommGroup.toAddCommMonoid.{u1} K (NonUnitalNonAssocRing.toAddCommGroup.{u1} K (NonAssocRing.toNonUnitalNonAssocRing.{u1} K (Ring.toNonAssocRing.{u1} K (DivisionRing.toRing.{u1} K (Field.toDivisionRing.{u1} K _inst_1)))))) (Polynomial.roots.{u1} K (EuclideanDomain.toCommRing.{u1} K (Field.toEuclideanDomain.{u1} K _inst_1)) (Field.isDomain.{u1} K _inst_1) P))))
but is expected to have type
forall {K : Type.{u1}} [_inst_1 : Field.{u1} K] {P : Polynomial.{u1} K (DivisionSemiring.toSemiring.{u1} K (Semifield.toDivisionSemiring.{u1} K (Field.toSemifield.{u1} K _inst_1)))}, (Polynomial.Monic.{u1} K (DivisionSemiring.toSemiring.{u1} K (Semifield.toDivisionSemiring.{u1} K (Field.toSemifield.{u1} K _inst_1))) P) -> (Polynomial.Splits.{u1, u1} K K (EuclideanDomain.toCommRing.{u1} K (Field.toEuclideanDomain.{u1} K _inst_1)) _inst_1 (RingHom.id.{u1} K (NonAssocRing.toNonAssocSemiring.{u1} K (Ring.toNonAssocRing.{u1} K (DivisionRing.toRing.{u1} K (Field.toDivisionRing.{u1} K _inst_1))))) P) -> (Eq.{succ u1} K (Polynomial.nextCoeff.{u1} K (DivisionSemiring.toSemiring.{u1} K (Semifield.toDivisionSemiring.{u1} K (Field.toSemifield.{u1} K _inst_1))) P) (Neg.neg.{u1} K (Ring.toNeg.{u1} K (DivisionRing.toRing.{u1} K (Field.toDivisionRing.{u1} K _inst_1))) (Multiset.sum.{u1} K (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} K (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u1} K (NonAssocRing.toNonUnitalNonAssocRing.{u1} K (Ring.toNonAssocRing.{u1} K (DivisionRing.toRing.{u1} K (Field.toDivisionRing.{u1} K _inst_1)))))) (Polynomial.roots.{u1} K (EuclideanDomain.toCommRing.{u1} K (Field.toEuclideanDomain.{u1} K _inst_1)) (EuclideanDomain.instIsDomainToSemiringToRingToCommRing.{u1} K (Field.toEuclideanDomain.{u1} K _inst_1)) P))))
Case conversion may be inaccurate. Consider using '#align polynomial.sum_roots_eq_next_coeff_of_monic_of_split Polynomial.sum_roots_eq_nextCoeff_of_monic_of_splitₓ'. -/
/-- If `P` is a monic polynomial that splits, then `P.next_coeff` equals the sum of the roots. -/
theorem sum_roots_eq_nextCoeff_of_monic_of_split {P : K[X]} (hmo : P.Monic)
(hP : P.Splits (RingHom.id K)) : P.nextCoeff = -P.roots.Sum :=
by
nth_rw 1 [eq_prod_roots_of_monic_of_splits_id hmo hP]
rw [monic.next_coeff_multiset_prod _ _ fun a ha => _]
· simp_rw [next_coeff_X_sub_C, Multiset.sum_map_neg']
· exact monic_X_sub_C a
#align polynomial.sum_roots_eq_next_coeff_of_monic_of_split Polynomial.sum_roots_eq_nextCoeff_of_monic_of_split
end Splits
end Polynomial
|
1
-- @@stderr --
dtrace: invalid probe specifier sysinfo:genunix:read: probe description :sysinfo:genunix:read does not match any probes
|
function engine = dv_unrolled_dbn_inf_engine(bnet, T, varargin)
% JTREE_UNROLLED_DBN_INF_ENGINE Unroll the DBN for T time-slices and apply jtree to the resulting static net
% engine = jtree_unrolled_dbn_inf_engine(bnet, T, ...)
%
% The following optional arguments can be specified in the form of name/value pairs:
% [default value in brackets]
%
% useC - 1 means use jtree_C_inf_engine instead of jtree_inf_engine [0]
% constrained - 1 means we constrain ourselves to eliminate slice t before t+1 [1]
%
% e.g., engine = jtree_unrolled_inf_engine(bnet, 'useC', 1);
% set default params
N = length(bnet.intra);
useC = 0;
constrained = 1;
if nargin >= 3
args = varargin;
nargs = length(args);
if isstr(args{1})
for i=1:2:nargs
switch args{i},
case 'useC', useC = args{i+1};
case 'constrained', constrained = args{i+1};
otherwise,
error(['invalid argument name ' args{i}]);
end
end
else
error(['invalid argument name ' args{1}]);
end
end
bnet2 = hodbn_to_bnet(bnet, T);
ss = length(bnet.intra);
engine.ss = ss;
% If constrained_order = 1 we constrain ourselves to eliminate slice t before t+1.
% This prevents cliques containing nodes from far-apart time-slices.
if constrained
stages = num2cell(unroll_set(1:ss, ss, T), 1);
else
stages = { 1:length(bnet2.dag) };
end
if useC
%jengine = jtree_C_inf_engine(bnet2, 'stages', stages);
%function is not implemented
assert(0)
else
jengine = stab_cond_gauss_inf_engine(bnet2);
end
engine.unrolled_engine = jengine;
% we don't inherit from jtree_inf_engine, because that would only store bnet2,
% and we would lose access to the DBN-specific fields like intra/inter
engine.nslices = T;
engine = class(engine, 'stable_ho_inf_engine', inf_engine(bnet));
|
#
abstract type Equation end
#----------------------------------------------------------------------
export Diffusion
#----------------------------------------------------------------------
struct Diffusion{T,U} <: Equation # {T,U,D,K} # type, dimension, (bdfK order)
fld::Field{T}
ν ::Array{T} # viscosity
f ::Array{T} # forcing
rhs::Array{T} # RHS
tstep::TimeStepper{T,U}
msh::Mesh{T} # mesh
end
#--------------------------------------#
function Diffusion(bc::Array{Char,1},msh::Mesh
;Ti=0.,Tf=0.,dt=0.,k=3)
fld = Field(bc,msh)
ν = zero(fld.u)
f = zero(fld.u)
rhs = zero(fld.u)
tstep = TimeStepper(Ti,Tf,dt,k)
return Diffusion(fld
,ν,f,rhs
,tstep
,msh)
end
#----------------------------------------------------------------------
function opLHS(u::Array,dfn::Diffusion)
@unpack fld, msh, ν = dfn
@unpack bdfB = dfn.tstep
lhs = hlmz(u,ν,bdfB[1],msh)
lhs .= gatherScatter(lhs,msh)
lhs .= mask(lhs,fld.M)
return lhs
end
function opPrecond(u::Array,dfn::Diffusion)
return u
end
function makeRHS!(dfn::Diffusion)
@unpack fld, rhs, ν, f, msh = dfn
@unpack bdfA, bdfB = dfn.tstep
rhs .= mass(f ,msh) # forcing
rhs .-= ν .* lapl(fld.ub,msh) # boundary data
for i=1:length(fld.uh) # histories
rhs .-= bdfB[1+i] .* mass(fld.uh[i],msh)
end
rhs .= mask(rhs,fld.M)
rhs .= gatherScatter(rhs,msh)
return
end
function solve!(dfn::Diffusion)
@unpack rhs, msh, fld = dfn
@unpack u,ub = fld
opL(u) = opLHS(u,dfn)
opP(u) = opPrecond(u,dfn)
pcg!(u,rhs,opL;opM=opP,mult=msh.mult,ifv=false)
u .+= ub
return
end
#----------------------------------------------------------------------
export evolve!
#----------------------------------------------------------------------
function evolve!(dfn::Diffusion
,setBC! =fixU!
,setForcing! =fixU!
,setVisc! =fixU!)
@unpack fld, f, ν, msh = dfn
@unpack time, bdfA, bdfB, istep, dt = dfn.tstep
updateHist!(fld)
Zygote.ignore() do
updateHist!(time)
istep .+= 1
time[1] += dt[1]
bdfExtK!(bdfA,bdfB,time)
end
setBC!(fld.ub,msh.x,msh.y,time[1])
setForcing!(f,msh.x,msh.y,time[1])
setVisc!(ν ,msh.x,msh.y,time[1])
makeRHS!(dfn)
solve!(dfn)
return
end
#----------------------------------------------------------------------
export simulate!
#----------------------------------------------------------------------
function simulate!(dfn::Diffusion,callback!::Function
,setIC! =fixU!
,setBC! =fixU!
,setForcing! =fixU!
,setVisc! =fixU!)
@unpack fld, msh = dfn
@unpack time, istep, dt, Tf = dfn.tstep
setIC!(fld.u,msh.x,msh.y,time[1])
Zygote.ignore() do
callback!(dfn)
end
while time[1] <= Tf[1]
evolve!(dfn,setBC!,setForcing!,setVisc!)
Zygote.ignore() do
callback!(dfn)
end
if(time[1] < 1e-12) break end
end
return
end
#----------------------------------------------------------------------
#
|
theory conditions_relativized
imports conditions_negative
begin
(****************** Relativized order and equality relations ****************)
definition subset_in::\<open>'p \<sigma> \<Rightarrow> 'p \<sigma> \<Rightarrow> 'p \<sigma> \<Rightarrow> bool\<close> ("_\<preceq>\<^sub>__")
where \<open>A \<preceq>\<^sub>U B \<equiv> \<forall>x. U x \<longrightarrow> (A x \<longrightarrow> B x)\<close>
definition subset_out::\<open>'p \<sigma> \<Rightarrow> 'p \<sigma> \<Rightarrow> 'p \<sigma> \<Rightarrow> bool\<close> ("_\<preceq>\<^sup>__")
where \<open>A \<preceq>\<^sup>U B \<equiv> \<forall>x. \<not>U x \<longrightarrow> (A x \<longrightarrow> B x)\<close>
definition setequ_in::\<open>'p \<sigma> \<Rightarrow> 'p \<sigma> \<Rightarrow> 'p \<sigma> \<Rightarrow> bool\<close> ("_\<approx>\<^sub>__")
where \<open>A \<approx>\<^sub>U B \<equiv> \<forall>x. U x \<longrightarrow> (A x \<longleftrightarrow> B x)\<close>
definition setequ_out::\<open>'p \<sigma> \<Rightarrow> 'p \<sigma> \<Rightarrow> 'p \<sigma> \<Rightarrow> bool\<close> ("_\<approx>\<^sup>__")
where \<open>A \<approx>\<^sup>U B \<equiv> \<forall>x. \<not>U x \<longrightarrow> (A x \<longleftrightarrow> B x)\<close>
declare subset_in_def[order] subset_out_def[order] setequ_in_def[order] setequ_out_def[order]
lemma subset_in_out: "(let U=C in (A \<preceq>\<^sub>U B)) = (let U=\<^bold>\<midarrow>C in (A \<preceq>\<^sup>U B))" by (simp add: compl_def subset_in_def subset_out_def)
lemma setequ_in_out: "(let U=C in (A \<approx>\<^sub>U B)) = (let U=\<^bold>\<midarrow>C in (A \<approx>\<^sup>U B))" by (simp add: compl_def setequ_in_def setequ_out_def)
lemma subset_in_char: "(A \<preceq>\<^sub>U B) = (U \<^bold>\<and> A \<preceq> U \<^bold>\<and> B)" unfolding order conn by blast
lemma subset_out_char: "(A \<preceq>\<^sup>U B) = (U \<^bold>\<or> A \<preceq> U \<^bold>\<or> B)" unfolding order conn by blast
lemma setequ_in_char: "(A \<approx>\<^sub>U B) = (U \<^bold>\<and> A \<approx> U \<^bold>\<and> B)" unfolding order conn by blast
lemma setequ_out_char: "(A \<approx>\<^sup>U B) = (U \<^bold>\<or> A \<approx> U \<^bold>\<or> B)" unfolding order conn by blast
(**Relativization cannot be meaningfully applied to conditions (n)NORM or (n)DNRM.*)
lemma "NORM \<phi> = (let U = \<^bold>\<top> in ((\<phi> \<^bold>\<bottom>) \<approx>\<^sub>U \<^bold>\<bottom>))" by (simp add: NORM_def setequ_def setequ_in_def top_def)
lemma "(let U = \<^bold>\<bottom> in ((\<phi> \<^bold>\<bottom>) \<approx>\<^sub>U \<^bold>\<bottom>))" by (simp add: bottom_def setequ_in_def)
(*Relativization ('in' resp. 'out') leaves (n)EXPN/(n)CNTR unchanged or trivializes them.*)
lemma "EXPN \<phi> = (\<forall>A. A \<preceq>\<^sub>A \<phi> A)" by (simp add: EXPN_def subset_def subset_in_def)
lemma "CNTR \<phi> = (\<forall>A. (\<phi> A) \<preceq>\<^sup>A A)" by (metis (mono_tags, lifting) CNTR_def subset_def subset_out_def)
lemma "\<forall>A. A \<preceq>\<^sup>A \<phi> A" by (simp add: subset_out_def)
lemma "\<forall>A. (\<phi> A) \<preceq>\<^sub>A A" by (simp add: subset_in_def)
(****************** Relativized ADDI variants ****************)
definition ADDIr::"('w \<sigma> \<Rightarrow> 'w \<sigma>) \<Rightarrow> bool" ("ADDIr")
where "ADDIr \<phi> \<equiv> \<forall>A B. let U = (A \<^bold>\<or> B) in (\<phi>(A \<^bold>\<or> B) \<approx>\<^sup>U (\<phi> A) \<^bold>\<or> (\<phi> B))"
definition ADDIr_a::"('w \<sigma> \<Rightarrow> 'w \<sigma>) \<Rightarrow> bool" ("ADDIr\<^sup>a")
where "ADDIr\<^sup>a \<phi> \<equiv> \<forall>A B. let U = (A \<^bold>\<or> B) in (\<phi>(A \<^bold>\<or> B) \<preceq>\<^sup>U (\<phi> A) \<^bold>\<or> (\<phi> B))"
definition ADDIr_b::"('w \<sigma> \<Rightarrow> 'w \<sigma>) \<Rightarrow> bool" ("ADDIr\<^sup>b")
where "ADDIr\<^sup>b \<phi> \<equiv> \<forall>A B. let U = (A \<^bold>\<or> B) in ((\<phi> A) \<^bold>\<or> (\<phi> B) \<preceq>\<^sup>U \<phi>(A \<^bold>\<or> B))"
declare ADDIr_def[cond] ADDIr_a_def[cond] ADDIr_b_def[cond]
lemma ADDIr_char: "ADDIr \<phi> = (ADDIr\<^sup>a \<phi> \<and> ADDIr\<^sup>b \<phi>)" unfolding cond by (meson setequ_char setequ_out_char subset_out_char)
lemma ADDIr_a_impl: "ADDI\<^sup>a \<phi> \<longrightarrow> ADDIr\<^sup>a \<phi>" by (simp add: ADDI_a_def ADDIr_a_def subset_def subset_out_def)
lemma ADDIr_a_equ: "EXPN \<phi> \<Longrightarrow> ADDIr\<^sup>a \<phi> = ADDI\<^sup>a \<phi>" unfolding cond by (smt (verit, del_insts) join_def subset_def subset_out_def)
lemma ADDIr_a_equ':"nEXPN \<phi> \<Longrightarrow> ADDIr\<^sup>a \<phi> = ADDI\<^sup>a \<phi>" unfolding cond by (smt (verit, ccfv_threshold) compl_def subset_def subset_out_def)
lemma ADDIr_b_impl: "ADDI\<^sup>b \<phi> \<longrightarrow> ADDIr\<^sup>b \<phi>" by (simp add: ADDI_b_def ADDIr_b_def subset_def subset_out_def)
lemma "nEXPN \<phi> \<Longrightarrow> ADDIr\<^sup>b \<phi> \<longrightarrow> ADDI\<^sup>b \<phi>" nitpick oops
lemma ADDIr_b_equ: "EXPN \<phi> \<Longrightarrow> ADDIr\<^sup>b \<phi> = ADDI\<^sup>b \<phi>" unfolding cond by (smt (z3) subset_def subset_out_def)
(****************** Relativized MULT variants ****************)
definition MULTr::"('w \<sigma> \<Rightarrow> 'w \<sigma>) \<Rightarrow> bool" ("MULTr")
where "MULTr \<phi> \<equiv> \<forall>A B. let U = (A \<^bold>\<and> B) in (\<phi>(A \<^bold>\<and> B) \<approx>\<^sub>U (\<phi> A) \<^bold>\<and> (\<phi> B))"
definition MULTr_a::"('w \<sigma> \<Rightarrow> 'w \<sigma>) \<Rightarrow> bool" ("MULTr\<^sup>a")
where "MULTr\<^sup>a \<phi> \<equiv> \<forall>A B. let U = (A \<^bold>\<and> B) in (\<phi>(A \<^bold>\<and> B) \<preceq>\<^sub>U (\<phi> A) \<^bold>\<and> (\<phi> B))"
definition MULTr_b::"('w \<sigma> \<Rightarrow> 'w \<sigma>) \<Rightarrow> bool" ("MULTr\<^sup>b")
where "MULTr\<^sup>b \<phi> \<equiv> \<forall>A B. let U = (A \<^bold>\<and> B) in ((\<phi> A) \<^bold>\<and> (\<phi> B) \<preceq>\<^sub>U \<phi>(A \<^bold>\<and> B))"
declare MULTr_def[cond] MULTr_a_def[cond] MULTr_b_def[cond]
lemma MULTr_char: "MULTr \<phi> = (MULTr\<^sup>a \<phi> \<and> MULTr\<^sup>b \<phi>)" unfolding cond by (meson setequ_char setequ_in_char subset_in_char)
lemma MULTr_a_impl: "MULT\<^sup>a \<phi> \<longrightarrow> MULTr\<^sup>a \<phi>" by (simp add: MULT_a_def MULTr_a_def subset_def subset_in_def)
lemma "nCNTR \<phi> \<Longrightarrow> MULTr\<^sup>a \<phi> \<longrightarrow> MULT\<^sup>a \<phi>" nitpick oops
lemma MULTr_a_equ: "CNTR \<phi> \<Longrightarrow> MULTr\<^sup>a \<phi> = MULT\<^sup>a \<phi>" unfolding cond by (smt (verit, del_insts) subset_def subset_in_def)
lemma MULTr_b_impl: "MULT\<^sup>b \<phi> \<longrightarrow> MULTr\<^sup>b \<phi>" by (simp add: MULT_b_def MULTr_b_def subset_def subset_in_def)
lemma "MULTr\<^sup>b \<phi> \<longrightarrow> MULT\<^sup>b \<phi>" nitpick oops
lemma MULTr_b_equ: "CNTR \<phi> \<Longrightarrow> MULTr\<^sup>b \<phi> = MULT\<^sup>b \<phi>" unfolding cond by (smt (verit, del_insts) meet_def subset_def subset_in_def)
lemma MULTr_b_equ':"nCNTR \<phi> \<Longrightarrow> MULTr\<^sup>b \<phi> = MULT\<^sup>b \<phi>" unfolding cond by (smt (z3) compl_def subset_def subset_in_def)
(**** Weak variants of monotonicity ***)
definition MONOw1::"('w \<sigma> \<Rightarrow> 'w \<sigma>) \<Rightarrow> bool" ("MONOw\<^sup>1")
where "MONOw\<^sup>1 \<phi> \<equiv> \<forall>A B. A \<preceq> B \<longrightarrow> (\<phi> A) \<preceq> B \<^bold>\<or> (\<phi> B)"
definition MONOw2::"('w \<sigma> \<Rightarrow> 'w \<sigma>) \<Rightarrow> bool" ("MONOw\<^sup>2")
where "MONOw\<^sup>2 \<phi> \<equiv> \<forall>A B. A \<preceq> B \<longrightarrow> A \<^bold>\<and> (\<phi> A) \<preceq> (\<phi> B)"
declare MONOw1_def[cond] MONOw2_def[cond]
lemma MONOw1_ADDIr_b: "MONOw\<^sup>1 \<phi> = ADDIr\<^sup>b \<phi>" proof -
have l2r: "MONOw\<^sup>1 \<phi> \<longrightarrow> ADDIr\<^sup>b \<phi>" unfolding cond subset_out_char by (metis (mono_tags, opaque_lifting) L7 join_def subset_def)
have r2l: "ADDIr\<^sup>b \<phi> \<longrightarrow> MONOw\<^sup>1 \<phi>" unfolding cond subset_out_char by (metis (full_types) L9 join_def setequ_ext subset_def)
show ?thesis using l2r r2l by blast
qed
lemma MONOw2_MULTr_a: "MONOw\<^sup>2 \<phi> = MULTr\<^sup>a \<phi>" proof -
have l2r: "MONOw\<^sup>2 \<phi> \<longrightarrow> MULTr\<^sup>a \<phi>" unfolding cond subset_in_char by (meson L4 L5 L8 L9)
have r2l:"MULTr\<^sup>a \<phi> \<longrightarrow> MONOw\<^sup>2 \<phi>" unfolding cond subset_in_char by (metis BA_distr1 L2 L5 L6 L9 setequ_ext)
show ?thesis using l2r r2l by blast
qed
lemma MONOw1_impl: "MONO \<phi> \<longrightarrow> MONOw\<^sup>1 \<phi>" by (simp add: ADDIr_b_impl MONO_ADDIb MONOw1_ADDIr_b)
lemma "MONOw\<^sup>1 \<phi> \<longrightarrow> MONO \<phi>" nitpick oops
lemma MONOw2_impl: "MONO \<phi> \<longrightarrow> MONOw\<^sup>2 \<phi>" by (simp add: MONO_MULTa MONOw2_MULTr_a MULTr_a_impl)
lemma "MONOw\<^sup>2 \<phi> \<longrightarrow> MONO \<phi>" nitpick oops
(** We have in fact that (n)CNTR (resp. (n)EXPN) implies MONOw\<^sup>1/ADDIr\<^sup>b (resp. MONOw\<^sup>2/MULTr\<^sup>a) *)
lemma CNTR_MONOw1_impl: "CNTR \<phi> \<longrightarrow> MONOw\<^sup>1 \<phi>" by (metis CNTR_def L3 MONOw1_def subset_char1)
lemma nCNTR_MONOw1_impl: "nCNTR \<phi> \<longrightarrow> MONOw\<^sup>1 \<phi>" by (smt (verit, ccfv_threshold) MONOw1_def compl_def join_def nCNTR_def subset_def)
lemma EXPN_MONOw2_impl: "EXPN \<phi> \<longrightarrow> MONOw\<^sup>2 \<phi>" by (metis EXPN_def L4 MONOw2_def subset_char1)
lemma nEXPN_MONOw2_impl: "nEXPN \<phi> \<longrightarrow> MONOw\<^sup>2 \<phi>" by (smt (verit) MONOw2_def compl_def meet_def nEXPN_def subset_def)
(****************** Relativized nADDI variants ****************)
definition nADDIr::"('w \<sigma> \<Rightarrow> 'w \<sigma>) \<Rightarrow> bool" ("nADDIr")
where "nADDIr \<phi> \<equiv> \<forall>A B. let U = (A \<^bold>\<or> B) in (\<phi>(A \<^bold>\<or> B) \<approx>\<^sup>U (\<phi> A) \<^bold>\<and> (\<phi> B))"
definition nADDIr_a::"('w \<sigma> \<Rightarrow> 'w \<sigma>) \<Rightarrow> bool" ("nADDIr\<^sup>a")
where "nADDIr\<^sup>a \<phi> \<equiv> \<forall>A B. let U = (A \<^bold>\<or> B) in ((\<phi> A) \<^bold>\<and> (\<phi> B) \<preceq>\<^sup>U \<phi>(A \<^bold>\<or> B))"
definition nADDIr_b::"('w \<sigma> \<Rightarrow> 'w \<sigma>) \<Rightarrow> bool" ("nADDIr\<^sup>b")
where "nADDIr\<^sup>b \<phi> \<equiv> \<forall>A B. let U = (A \<^bold>\<or> B) in (\<phi>(A \<^bold>\<or> B) \<preceq>\<^sup>U (\<phi> A) \<^bold>\<and> (\<phi> B))"
declare nADDIr_def[cond] nADDIr_a_def[cond] nADDIr_b_def[cond]
lemma nADDIr_char: "nADDIr \<phi> = (nADDIr\<^sup>a \<phi> \<and> nADDIr\<^sup>b \<phi>)" unfolding cond by (meson setequ_char setequ_out_char subset_out_char)
lemma nADDIr_a_impl: "nADDI\<^sup>a \<phi> \<longrightarrow> nADDIr\<^sup>a \<phi>" unfolding cond by (simp add: subset_def subset_out_def)
lemma "nADDIr\<^sup>a \<phi> \<longrightarrow> nADDI\<^sup>a \<phi>" nitpick oops
lemma nADDIr_a_equ: "EXPN \<phi> \<Longrightarrow> nADDIr\<^sup>a \<phi> = nADDI\<^sup>a \<phi>" unfolding cond by (smt (z3) subset_def subset_out_def)
lemma nADDIr_a_equ':"nEXPN \<phi> \<Longrightarrow> nADDIr\<^sup>a \<phi> = nADDI\<^sup>a \<phi>" unfolding cond by (smt (z3) compl_def join_def meet_def subset_def subset_out_def)
lemma nADDIr_b_impl: "nADDI\<^sup>b \<phi> \<longrightarrow> nADDIr\<^sup>b \<phi>" by (simp add: nADDI_b_def nADDIr_b_def subset_def subset_out_def)
lemma "EXPN \<phi> \<Longrightarrow> nADDIr\<^sup>b \<phi> \<longrightarrow> nADDI\<^sup>b \<phi>" nitpick oops
lemma nADDIr_b_equ: "nEXPN \<phi> \<Longrightarrow> nADDIr\<^sup>b \<phi> = nADDI\<^sup>b \<phi>" unfolding cond by (smt (z3) compl_def subset_def subset_out_def)
(****************** Relativized nMULT variants ****************)
definition nMULTr::"('w \<sigma> \<Rightarrow> 'w \<sigma>) \<Rightarrow> bool" ("nMULTr")
where "nMULTr \<phi> \<equiv> \<forall>A B. let U = (A \<^bold>\<and> B) in (\<phi>(A \<^bold>\<and> B) \<approx>\<^sub>U (\<phi> A) \<^bold>\<or> (\<phi> B))"
definition nMULTr_a::"('w \<sigma> \<Rightarrow> 'w \<sigma>) \<Rightarrow> bool" ("nMULTr\<^sup>a")
where "nMULTr\<^sup>a \<phi> \<equiv> \<forall>A B. let U = (A \<^bold>\<and> B) in ((\<phi> A) \<^bold>\<or> (\<phi> B) \<preceq>\<^sub>U \<phi>(A \<^bold>\<and> B))"
definition nMULTr_b::"('w \<sigma> \<Rightarrow> 'w \<sigma>) \<Rightarrow> bool" ("nMULTr\<^sup>b")
where "nMULTr\<^sup>b \<phi> \<equiv> \<forall>A B. let U = (A \<^bold>\<and> B) in (\<phi>(A \<^bold>\<and> B) \<preceq>\<^sub>U (\<phi> A) \<^bold>\<or> (\<phi> B))"
declare nMULTr_def[cond] nMULTr_a_def[cond] nMULTr_b_def[cond]
lemma nMULTr_char: "nMULTr \<phi> = (nMULTr\<^sup>a \<phi> \<and> nMULTr\<^sup>b \<phi>)" unfolding cond by (meson setequ_char setequ_in_char subset_in_char)
lemma nMULTr_a_impl: "nMULT\<^sup>a \<phi> \<longrightarrow> nMULTr\<^sup>a \<phi>" by (simp add: nMULT_a_def nMULTr_a_def subset_def subset_in_def)
lemma "CNTR \<phi> \<Longrightarrow> nMULTr\<^sup>a \<phi> \<longrightarrow> nMULT\<^sup>a \<phi>" nitpick oops
lemma nMULTr_a_equ: "nCNTR \<phi> \<Longrightarrow> nMULTr\<^sup>a \<phi> = nMULT\<^sup>a \<phi>" unfolding cond by (smt (z3) compl_def subset_def subset_in_def)
lemma nMULTr_b_impl: "nMULT\<^sup>b \<phi> \<longrightarrow> nMULTr\<^sup>b \<phi>" by (simp add: nMULT_b_def nMULTr_b_def subset_def subset_in_def)
lemma "nMULTr\<^sup>b \<phi> \<longrightarrow> nMULT\<^sup>b \<phi>" nitpick oops
lemma nMULTr_b_equ: "CNTR \<phi> \<Longrightarrow> nMULTr\<^sup>b \<phi> = nMULT\<^sup>b \<phi>" unfolding cond by (smt (z3) compl_def join_def meet_def subset_def subset_in_def)
lemma nMULTr_b_equ':"nCNTR \<phi> \<Longrightarrow> nMULTr\<^sup>b \<phi> = nMULT\<^sup>b \<phi>" unfolding cond by (smt (z3) compl_def join_def meet_def subset_def subset_in_def)
(**** Weak variants of antitonicity ***)
definition ANTIw1::"('w \<sigma> \<Rightarrow> 'w \<sigma>) \<Rightarrow> bool" ("ANTIw\<^sup>1")
where "ANTIw\<^sup>1 \<phi> \<equiv> \<forall>A B. A \<preceq> B \<longrightarrow> (\<phi> B) \<preceq> B \<^bold>\<or> (\<phi> A)"
definition ANTIw2::"('w \<sigma> \<Rightarrow> 'w \<sigma>) \<Rightarrow> bool" ("ANTIw\<^sup>2")
where "ANTIw\<^sup>2 \<phi> \<equiv> \<forall>A B. A \<preceq> B \<longrightarrow> A \<^bold>\<and> (\<phi> B) \<preceq> (\<phi> A)"
declare ANTIw1_def[cond] ANTIw2_def[cond]
lemma ANTIw1_nADDIr_b: "ANTIw\<^sup>1 \<phi> = nADDIr\<^sup>b \<phi>" proof -
have l2r: "ANTIw\<^sup>1 \<phi> \<longrightarrow> nADDIr\<^sup>b \<phi>" unfolding cond subset_out_char by (smt (verit, ccfv_SIG) BA_distr2 L8 join_def setequ_ext subset_def)
have r2l: "nADDIr\<^sup>b \<phi> \<longrightarrow> ANTIw\<^sup>1 \<phi>" unfolding cond subset_out_def by (metis (full_types) L9 join_def meet_def setequ_ext subset_def)
show ?thesis using l2r r2l by blast
qed
lemma ANTIw2_nMULTr_a: "ANTIw\<^sup>2 \<phi> = nMULTr\<^sup>a \<phi>" proof -
have l2r: "ANTIw\<^sup>2 \<phi> \<longrightarrow> nMULTr\<^sup>a \<phi>" unfolding cond subset_in_char by (metis BA_distr1 L3 L4 L5 L7 L8 setequ_ext)
have r2l: "nMULTr\<^sup>a \<phi> \<longrightarrow> ANTIw\<^sup>2 \<phi>" unfolding cond subset_in_def by (metis (full_types) L10 join_def meet_def setequ_ext subset_def)
show ?thesis using l2r r2l by blast
qed
lemma "ANTI \<phi> \<longrightarrow> ANTIw\<^sup>1 \<phi>" by (simp add: ANTI_nADDIb ANTIw1_nADDIr_b nADDIr_b_impl)
lemma "ANTIw\<^sup>1 \<phi> \<longrightarrow> ANTI \<phi>" nitpick oops
lemma "ANTI \<phi> \<longrightarrow> ANTIw\<^sup>2 \<phi>" by (simp add: ANTI_nMULTa ANTIw2_nMULTr_a nMULTr_a_impl)
lemma "ANTIw\<^sup>2 \<phi> \<longrightarrow> ANTI \<phi>" nitpick oops
(** We have in fact that (n)CNTR (resp. (n)EXPN) implies ANTIw\<^sup>1/nADDIr\<^sup>b (resp. ANTIw\<^sup>2/nMULTr\<^sup>a) *)
lemma CNTR_ANTIw1_impl: "CNTR \<phi> \<longrightarrow> ANTIw\<^sup>1 \<phi>" unfolding cond using L3 subset_char1 by blast
lemma nCNTR_ANTIw1_impl: "nCNTR \<phi> \<longrightarrow> ANTIw\<^sup>1 \<phi>" unfolding cond by (metis (full_types) compl_def join_def subset_def)
lemma EXPN_ANTIw2_impl: "EXPN \<phi> \<longrightarrow> ANTIw\<^sup>2 \<phi>" unfolding cond using L4 subset_char1 by blast
lemma nEXPN_ANTIw2_impl: "nEXPN \<phi> \<longrightarrow> ANTIw\<^sup>2 \<phi>" unfolding cond by (metis (full_types) compl_def meet_def subset_def)
(****************** Dual interrelations ****************)
lemma ADDIr_dual1: "ADDIr\<^sup>a \<phi> = MULTr\<^sup>b \<phi>\<^sup>d" unfolding cond subset_in_char subset_out_char by (smt (z3) BA_cp BA_deMorgan1 BA_dn op_dual_def setequ_ext)
lemma ADDIr_dual2: "ADDIr\<^sup>b \<phi> = MULTr\<^sup>a \<phi>\<^sup>d" unfolding cond subset_in_char subset_out_char by (smt (verit, ccfv_threshold) BA_cp BA_deMorgan1 BA_dn op_dual_def setequ_ext)
lemma ADDIr_dual: "ADDIr \<phi> = MULTr \<phi>\<^sup>d" using ADDIr_char ADDIr_dual1 ADDIr_dual2 MULTr_char by blast
lemma nADDIr_dual1: "nADDIr\<^sup>a \<phi> = nMULTr\<^sup>b \<phi>\<^sup>d" unfolding cond subset_in_char subset_out_char by (smt (verit, del_insts) BA_cp BA_deMorgan1 BA_dn op_dual_def setequ_ext)
lemma nADDIr_dual2: "nADDIr\<^sup>b \<phi> = nMULTr\<^sup>a \<phi>\<^sup>d" by (smt (z3) BA_deMorgan1 BA_dn compl_def nADDIr_b_def nMULTr_a_def op_dual_def setequ_ext subset_in_def subset_out_def)
lemma nADDIr_dual: "nADDIr \<phi> = nMULTr \<phi>\<^sup>d" using nADDIr_char nADDIr_dual1 nADDIr_dual2 nMULTr_char by blast
(****************** Complement interrelations ****************)
lemma ADDIr_a_cmpl: "ADDIr\<^sup>a \<phi> = nADDIr\<^sup>a \<phi>\<^sup>c" unfolding cond by (smt (verit, del_insts) BA_deMorgan1 compl_def setequ_ext subset_out_def svfun_compl_def)
lemma ADDIr_b_cmpl: "ADDIr\<^sup>b \<phi> = nADDIr\<^sup>b \<phi>\<^sup>c" unfolding cond by (smt (verit, del_insts) BA_deMorgan1 compl_def setequ_ext subset_out_def svfun_compl_def)
lemma ADDIr_cmpl: "ADDIr \<phi> = nADDIr \<phi>\<^sup>c" by (simp add: ADDIr_a_cmpl ADDIr_b_cmpl ADDIr_char nADDIr_char)
lemma MULTr_a_cmpl: "MULTr\<^sup>a \<phi> = nMULTr\<^sup>a \<phi>\<^sup>c" unfolding cond by (smt (verit, del_insts) BA_deMorgan2 compl_def setequ_ext subset_in_def svfun_compl_def)
lemma MULTr_b_cmpl: "MULTr\<^sup>b \<phi> = nMULTr\<^sup>b \<phi>\<^sup>c" unfolding cond by (smt (verit, ccfv_threshold) BA_deMorgan2 compl_def setequ_ext subset_in_def svfun_compl_def)
lemma MULTr_cmpl: "MULTr \<phi> = nMULTr \<phi>\<^sup>c" by (simp add: MULTr_a_cmpl MULTr_b_cmpl MULTr_char nMULTr_char)
(****************** Fixed-point interrelations ****************)
lemma EXPN_fp: "EXPN \<phi> = EXPN \<phi>\<^sup>f\<^sup>p" by (simp add: EXPN_def dimpl_def op_fixpoint_def subset_def)
lemma EXPN_fpc: "EXPN \<phi> = nEXPN \<phi>\<^sup>f\<^sup>p\<^sup>c" using EXPN_fp nEXPN_CNTR_compl by blast
lemma CNTR_fp: "CNTR \<phi> = nCNTR \<phi>\<^sup>f\<^sup>p" by (metis EXPN_CNTR_dual1 EXPN_fp dual_compl_char2 dual_invol nCNTR_EXPN_compl ofp_comm_dc1 sfun_compl_invol)
lemma CNTR_fpc: "CNTR \<phi> = CNTR \<phi>\<^sup>f\<^sup>p\<^sup>c" by (metis CNTR_fp nCNTR_EXPN_compl ofp_comm_compl ofp_invol)
lemma nNORM_fp: "NORM \<phi> = nNORM \<phi>\<^sup>f\<^sup>p" by (metis NORM_def fixpoints_def fp_rel nNORM_def)
lemma NORM_fpc: "NORM \<phi> = NORM \<phi>\<^sup>f\<^sup>p\<^sup>c" by (simp add: NORM_def bottom_def ofp_fixpoint_compl_def sdiff_def)
lemma DNRM_fp: "DNRM \<phi> = DNRM \<phi>\<^sup>f\<^sup>p" by (simp add: DNRM_def dimpl_def op_fixpoint_def top_def)
lemma DNRM_fpc: "DNRM \<phi> = nDNRM \<phi>\<^sup>f\<^sup>p\<^sup>c" using DNRM_fp nDNRM_DNRM_compl by blast
lemma ADDIr_a_fpc: "ADDIr\<^sup>a \<phi> = ADDIr\<^sup>a \<phi>\<^sup>f\<^sup>p\<^sup>c" unfolding cond subset_out_def by (simp add: join_def ofp_fixpoint_compl_def sdiff_def)
lemma ADDIr_a_fp: "ADDIr\<^sup>a \<phi> = nADDIr\<^sup>a \<phi>\<^sup>f\<^sup>p" by (metis ADDIr_a_cmpl ADDIr_a_fpc sfun_compl_invol)
lemma ADDIr_b_fpc: "ADDIr\<^sup>b \<phi> = ADDIr\<^sup>b \<phi>\<^sup>f\<^sup>p\<^sup>c" unfolding cond subset_out_def by (simp add: join_def ofp_fixpoint_compl_def sdiff_def)
lemma ADDIr_b_fp: "ADDIr\<^sup>b \<phi> = nADDIr\<^sup>b \<phi>\<^sup>f\<^sup>p" by (metis ADDIr_b_cmpl ADDIr_b_fpc sfun_compl_invol)
lemma MULTr_a_fp: "MULTr\<^sup>a \<phi> = MULTr\<^sup>a \<phi>\<^sup>f\<^sup>p" unfolding cond subset_in_def by (simp add: dimpl_def meet_def op_fixpoint_def)
lemma MULTr_a_fpc: "MULTr\<^sup>a \<phi> = nMULTr\<^sup>a \<phi>\<^sup>f\<^sup>p\<^sup>c" using MULTr_a_cmpl MULTr_a_fp by blast
lemma MULTr_b_fp: "MULTr\<^sup>b \<phi> = MULTr\<^sup>b \<phi>\<^sup>f\<^sup>p" unfolding cond subset_in_def by (simp add: dimpl_def meet_def op_fixpoint_def)
lemma MULTr_b_fpc: "MULTr\<^sup>b \<phi> = nMULTr\<^sup>b \<phi>\<^sup>f\<^sup>p\<^sup>c" using MULTr_b_cmpl MULTr_b_fp by blast
(****************** Relativized IDEM variants ****************)
definition IDEMr_a::"('w \<sigma> \<Rightarrow> 'w \<sigma>) \<Rightarrow> bool" ("IDEMr\<^sup>a")
where "IDEMr\<^sup>a \<phi> \<equiv> \<forall>A. \<phi>(A \<^bold>\<or> \<phi> A) \<preceq>\<^sup>A (\<phi> A)"
definition IDEMr_b::"('w \<sigma> \<Rightarrow> 'w \<sigma>) \<Rightarrow> bool" ("IDEMr\<^sup>b")
where "IDEMr\<^sup>b \<phi> \<equiv> \<forall>A. (\<phi> A) \<preceq>\<^sub>A \<phi>(A \<^bold>\<and> \<phi> A)"
declare IDEMr_a_def[cond] IDEMr_b_def[cond]
(****************** Relativized nIDEM variants ****************)
definition nIDEMr_a::"('w \<sigma> \<Rightarrow> 'w \<sigma>) \<Rightarrow> bool" ("nIDEMr\<^sup>a")
where "nIDEMr\<^sup>a \<phi> \<equiv> \<forall>A. (\<phi> A) \<preceq>\<^sup>A \<phi>(A \<^bold>\<or> \<^bold>\<midarrow>(\<phi> A))"
definition nIDEMr_b::"('w \<sigma> \<Rightarrow> 'w \<sigma>) \<Rightarrow> bool" ("nIDEMr\<^sup>b")
where "nIDEMr\<^sup>b \<phi> \<equiv> \<forall>A. \<phi>(A \<^bold>\<and> \<^bold>\<midarrow>(\<phi> A)) \<preceq>\<^sub>A (\<phi> A)"
declare nIDEMr_a_def[cond] nIDEMr_b_def[cond]
(****************** Complement interrelations ****************)
lemma IDEMr_a_cmpl: "IDEMr\<^sup>a \<phi> = nIDEMr\<^sup>a \<phi>\<^sup>c" unfolding cond subset_in_def subset_out_def by (metis compl_def sfun_compl_invol svfun_compl_def)
lemma IDEMr_b_cmpl: "IDEMr\<^sup>b \<phi> = nIDEMr\<^sup>b \<phi>\<^sup>c" unfolding cond subset_in_def subset_out_def by (metis compl_def sfun_compl_invol svfun_compl_def)
(****************** Dual interrelation ****************)
lemma IDEMr_dual: "IDEMr\<^sup>a \<phi> = IDEMr\<^sup>b \<phi>\<^sup>d" unfolding cond subset_in_def subset_out_def op_dual_def by (metis (mono_tags, opaque_lifting) BA_dn compl_def diff_char1 diff_char2 impl_char setequ_ext)
lemma nIDEMr_dual: "nIDEMr\<^sup>a \<phi> = nIDEMr\<^sup>b \<phi>\<^sup>d" by (metis IDEMr_dual IDEMr_a_cmpl IDEMr_b_cmpl dual_compl_char1 dual_compl_char2 sfun_compl_invol)
(****************** Fixed-point interrelations ****************)
lemma IDEMr_a_fp: "IDEMr\<^sup>a \<phi> = nIDEMr\<^sup>a \<phi>\<^sup>f\<^sup>p" proof -
have l2r: "IDEMr\<^sup>a \<phi> \<longrightarrow> nIDEMr\<^sup>a \<phi>\<^sup>f\<^sup>p"
unfolding cond subset_out_def op_fixpoint_def conn order apply simp (*by metis*) sorry (*fix proof reconstruction in kernel*)
have r2l: "nIDEMr\<^sup>a \<phi>\<^sup>f\<^sup>p \<longrightarrow> IDEMr\<^sup>a \<phi>"
unfolding cond subset_out_def op_fixpoint_def conn order apply simp (*by metis*) sorry (*fix proof reconstruction in kernel*)
from l2r r2l show ?thesis by blast
qed
lemma IDEMr_a_fpc: "IDEMr\<^sup>a \<phi> = IDEMr\<^sup>a \<phi>\<^sup>f\<^sup>p\<^sup>c" using IDEMr_a_fp by (metis IDEMr_a_cmpl sfun_compl_invol)
lemma IDEMr_b_fp: "IDEMr\<^sup>b \<phi> = IDEMr\<^sup>b \<phi>\<^sup>f\<^sup>p" proof -
have l2r: "IDEMr\<^sup>b \<phi> \<longrightarrow> IDEMr\<^sup>b \<phi>\<^sup>f\<^sup>p"
unfolding cond subset_in_def op_fixpoint_def conn order apply simp (*by metis*) sorry (*fix proof reconstruction in kernel*)
have r2l: "IDEMr\<^sup>b \<phi>\<^sup>f\<^sup>p \<longrightarrow> IDEMr\<^sup>b \<phi>"
unfolding cond subset_in_def op_fixpoint_def conn order apply simp (*by metis*) sorry (*fix proof reconstruction in kernel*)
from l2r r2l show ?thesis by blast
qed
lemma IDEMr_b_fpc: "IDEMr\<^sup>b \<phi> = nIDEMr\<^sup>b \<phi>\<^sup>f\<^sup>p\<^sup>c" using IDEMr_b_fp IDEMr_b_cmpl by blast
(***************************************************)
(*** Verifying original border axioms by Zarycki ***)
(***************************************************)
(*The original border condition B1' is equivalent to the conjuntion of nMULTr and CNTR*)
abbreviation "B1' \<phi> \<equiv> \<forall>A B. \<phi>(A \<^bold>\<and> B) \<approx> (A \<^bold>\<and> \<phi> B) \<^bold>\<or> (\<phi> A \<^bold>\<and> B)"
lemma "B1' \<phi> = (nMULTr \<phi> \<and> CNTR \<phi>)" proof -
have l2ra: "B1' \<phi> \<longrightarrow> nMULTr \<phi>" unfolding cond by (smt (z3) join_def meet_def setequ_ext setequ_in_def)
have l2rb: "B1' \<phi> \<longrightarrow> CNTR \<phi>" unfolding cond by (metis L2 L4 L5 L7 L9 setequ_ext)
have r2l: "(nMULTr \<phi> \<and> CNTR \<phi>) \<longrightarrow> B1' \<phi>" unfolding cond by (smt (z3) L10 join_def meet_def setequ_def setequ_in_char)
from l2ra l2rb r2l show ?thesis by blast
qed
(*Modulo conditions nMULTr and CNTR the border condition B4 is equivalent to nIDEMr\<^sup>b*)
abbreviation "B4 \<phi> \<equiv> \<forall>A. \<phi>(\<^bold>\<midarrow>\<phi>(\<^bold>\<midarrow>A)) \<preceq> A"
lemma "nMULTr \<phi> \<Longrightarrow> CNTR \<phi> \<Longrightarrow> B4 \<phi> = nIDEMr\<^sup>b \<phi>" proof -
assume a1: "nMULTr \<phi>" and a2: "CNTR \<phi>"
have l2r: "nMULTr\<^sup>b \<phi> \<Longrightarrow> B4 \<phi> \<longrightarrow> nIDEMr\<^sup>b \<phi>" unfolding cond subset_in_char subset_def by (metis BA_deMorgan1 BA_dn compl_def meet_def setequ_ext)
have r2l: "nMULTr\<^sup>a \<phi> \<Longrightarrow> CNTR \<phi> \<Longrightarrow> nIDEMr\<^sup>b \<phi> \<longrightarrow> B4 \<phi>" unfolding cond by (smt (verit) compl_def join_def meet_def subset_def subset_in_def)
from l2r r2l show ?thesis using a1 a2 nMULTr_char by blast
qed
end |
In animals , amino acids are obtained through the consumption of foods containing protein . Ingested proteins are then broken down into amino acids through digestion , which typically involves denaturation of the protein through exposure to acid and hydrolysis by enzymes called proteases . Some ingested amino acids are used for protein biosynthesis , while others are converted to glucose through gluconeogenesis , or fed into the citric acid cycle . This use of protein as a fuel is particularly important under starvation conditions as it allows the body 's own proteins to be used to support life , particularly those found in muscle . Amino acids are also an important dietary source of nitrogen .
|
#!/usr/bin/R
# compute descriptive statistics, correlation, and regression from raw.txt
# required pacakges
# - dplyr
# - Hmisc
# library
library("dplyr")
library("Hmisc")
# improt raw .txt to data frame
raw_data <- read.delim("raw_data-caffeine.txt")
# convert data to numerical matrix
df <- as.matrix(as.data.frame(lapply(raw_data, as.numeric)))
# output file - descriptive stats
sink("descriptive_stats-caffeine.txt", split=TRUE, append=FALSE)
# descriptive stats
summary(raw_data)
# output file - correlation and regression
sink("correlation-caffeine.txt", split=TRUE, append=FALSE)
# correlations (r) and significance (p)
rcorr(df, type=c("pearson"))
|
/-
Copyright (c) 2021 Eric Wieser. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Eric Wieser
! This file was ported from Lean 3 source module ring_theory.subsemiring.pointwise
! leanprover-community/mathlib commit 59694bd07f0a39c5beccba34bd9f413a160782bf
! Please do not edit these lines, except to modify the commit id
! if you have ported upstream changes.
-/
import Mathlib.Algebra.GroupRingAction.Basic
import Mathlib.RingTheory.Subsemiring.Basic
import Mathlib.GroupTheory.Submonoid.Pointwise
import Mathlib.Data.Set.Pointwise.Basic
/-! # Pointwise instances on `Subsemiring`s
This file provides the action `Subsemiring.PointwiseMulAction` which matches the action of
`MulActionSet`.
This actions is available in the `Pointwise` locale.
## Implementation notes
This file is almost identical to `GroupTheory/Submonoid/Pointwise.lean`. Where possible, try to
keep them in sync.
-/
open Set
variable {M R : Type _}
namespace Subsemiring
section Monoid
variable [Monoid M] [Semiring R] [MulSemiringAction M R]
/-- The action on a subsemiring corresponding to applying the action to every element.
This is available as an instance in the `pointwise` locale. -/
protected def pointwiseMulAction : MulAction M (Subsemiring R)
where
smul a S := S.map (MulSemiringAction.toRingHom _ _ a)
one_smul S := (congr_arg (fun f => S.map f) (RingHom.ext <| one_smul M)).trans S.map_id
mul_smul _a₁ _a₂ S :=
(congr_arg (fun f => S.map f) (RingHom.ext <| mul_smul _ _)).trans (S.map_map _ _).symm
#align subsemiring.pointwise_mul_action Subsemiring.pointwiseMulAction
scoped[Pointwise] attribute [instance] Subsemiring.pointwiseMulAction
open Pointwise
theorem pointwise_smul_def {a : M} (S : Subsemiring R) :
a • S = S.map (MulSemiringAction.toRingHom _ _ a) :=
rfl
#align subsemiring.pointwise_smul_def Subsemiring.pointwise_smul_def
@[simp]
theorem coe_pointwise_smul (m : M) (S : Subsemiring R) : ↑(m • S) = m • (S : Set R) :=
rfl
#align subsemiring.coe_pointwise_smul Subsemiring.coe_pointwise_smul
@[simp]
theorem pointwise_smul_toAddSubmonoid (m : M) (S : Subsemiring R) :
(m • S).toAddSubmonoid = m • S.toAddSubmonoid :=
rfl
#align subsemiring.pointwise_smul_to_add_submonoid Subsemiring.pointwise_smul_toAddSubmonoid
theorem smul_mem_pointwise_smul (m : M) (r : R) (S : Subsemiring R) : r ∈ S → m • r ∈ m • S :=
(Set.smul_mem_smul_set : _ → _ ∈ m • (S : Set R))
#align subsemiring.smul_mem_pointwise_smul Subsemiring.smul_mem_pointwise_smul
theorem mem_smul_pointwise_iff_exists (m : M) (r : R) (S : Subsemiring R) :
r ∈ m • S ↔ ∃ s : R, s ∈ S ∧ m • s = r :=
(Set.mem_smul_set : r ∈ m • (S : Set R) ↔ _)
#align subsemiring.mem_smul_pointwise_iff_exists Subsemiring.mem_smul_pointwise_iff_exists
@[simp]
theorem smul_bot (a : M) : a • (⊥ : Subsemiring R) = ⊥ :=
map_bot _
#align subsemiring.smul_bot Subsemiring.smul_bot
theorem smul_sup (a : M) (S T : Subsemiring R) : a • (S ⊔ T) = a • S ⊔ a • T :=
map_sup _ _ _
#align subsemiring.smul_sup Subsemiring.smul_sup
theorem smul_closure (a : M) (s : Set R) : a • closure s = closure (a • s) :=
RingHom.map_closureS _ _
#align subsemiring.smul_closure Subsemiring.smul_closure
instance pointwise_central_scalar [MulSemiringAction Mᵐᵒᵖ R] [IsCentralScalar M R] :
IsCentralScalar M (Subsemiring R) :=
⟨fun _a S => (congr_arg fun f => S.map f) <| RingHom.ext <| op_smul_eq_smul _⟩
#align subsemiring.pointwise_central_scalar Subsemiring.pointwise_central_scalar
end Monoid
section Group
variable [Group M] [Semiring R] [MulSemiringAction M R]
open Pointwise
@[simp]
theorem smul_mem_pointwise_smul_iff {a : M} {S : Subsemiring R} {x : R} : a • x ∈ a • S ↔ x ∈ S :=
smul_mem_smul_set_iff
#align subsemiring.smul_mem_pointwise_smul_iff Subsemiring.smul_mem_pointwise_smul_iff
theorem mem_pointwise_smul_iff_inv_smul_mem {a : M} {S : Subsemiring R} {x : R} :
x ∈ a • S ↔ a⁻¹ • x ∈ S :=
mem_smul_set_iff_inv_smul_mem
#align subsemiring.mem_pointwise_smul_iff_inv_smul_mem Subsemiring.mem_pointwise_smul_iff_inv_smul_mem
theorem mem_inv_pointwise_smul_iff {a : M} {S : Subsemiring R} {x : R} : x ∈ a⁻¹ • S ↔ a • x ∈ S :=
mem_inv_smul_set_iff
#align subsemiring.mem_inv_pointwise_smul_iff Subsemiring.mem_inv_pointwise_smul_iff
@[simp]
theorem pointwise_smul_le_pointwise_smul_iff {a : M} {S T : Subsemiring R} :
a • S ≤ a • T ↔ S ≤ T :=
set_smul_subset_set_smul_iff
#align subsemiring.pointwise_smul_le_pointwise_smul_iff Subsemiring.pointwise_smul_le_pointwise_smul_iff
theorem pointwise_smul_subset_iff {a : M} {S T : Subsemiring R} : a • S ≤ T ↔ S ≤ a⁻¹ • T :=
set_smul_subset_iff
#align subsemiring.pointwise_smul_subset_iff Subsemiring.pointwise_smul_subset_iff
theorem subset_pointwise_smul_iff {a : M} {S T : Subsemiring R} : S ≤ a • T ↔ a⁻¹ • S ≤ T :=
subset_set_smul_iff
#align subsemiring.subset_pointwise_smul_iff Subsemiring.subset_pointwise_smul_iff
/-! TODO: add `equiv_smul` like we have for subgroup. -/
end Group
section GroupWithZero
variable [GroupWithZero M] [Semiring R] [MulSemiringAction M R]
open Pointwise
@[simp]
theorem smul_mem_pointwise_smul_iff₀ {a : M} (ha : a ≠ 0) (S : Subsemiring R) (x : R) :
a • x ∈ a • S ↔ x ∈ S :=
smul_mem_smul_set_iff₀ ha (S : Set R) x
#align subsemiring.smul_mem_pointwise_smul_iff₀ Subsemiring.smul_mem_pointwise_smul_iff₀
theorem mem_pointwise_smul_iff_inv_smul_mem₀ {a : M} (ha : a ≠ 0) (S : Subsemiring R) (x : R) :
x ∈ a • S ↔ a⁻¹ • x ∈ S :=
mem_smul_set_iff_inv_smul_mem₀ ha (S : Set R) x
#align subsemiring.mem_pointwise_smul_iff_inv_smul_mem₀ Subsemiring.mem_pointwise_smul_iff_inv_smul_mem₀
theorem mem_inv_pointwise_smul_iff₀ {a : M} (ha : a ≠ 0) (S : Subsemiring R) (x : R) :
x ∈ a⁻¹ • S ↔ a • x ∈ S :=
mem_inv_smul_set_iff₀ ha (S : Set R) x
#align subsemiring.mem_inv_pointwise_smul_iff₀ Subsemiring.mem_inv_pointwise_smul_iff₀
@[simp]
theorem pointwise_smul_le_pointwise_smul_iff₀ {a : M} (ha : a ≠ 0) {S T : Subsemiring R} :
a • S ≤ a • T ↔ S ≤ T :=
set_smul_subset_set_smul_iff₀ ha
#align subsemiring.pointwise_smul_le_pointwise_smul_iff₀ Subsemiring.pointwise_smul_le_pointwise_smul_iff₀
theorem pointwise_smul_le_iff₀ {a : M} (ha : a ≠ 0) {S T : Subsemiring R} :
a • S ≤ T ↔ S ≤ a⁻¹ • T :=
set_smul_subset_iff₀ ha
#align subsemiring.pointwise_smul_le_iff₀ Subsemiring.pointwise_smul_le_iff₀
theorem le_pointwise_smul_iff₀ {a : M} (ha : a ≠ 0) {S T : Subsemiring R} :
S ≤ a • T ↔ a⁻¹ • S ≤ T :=
subset_set_smul_iff₀ ha
#align subsemiring.le_pointwise_smul_iff₀ Subsemiring.le_pointwise_smul_iff₀
end GroupWithZero
end Subsemiring
|
## Tests involing building packages and whatnot
build_tests_dir = joinpath(@__DIR__, "build_tests")
libfoo_products = [
LibraryProduct("libfoo", :libfoo),
ExecutableProduct("fooifier", :fooifier),
]
libfoo_make_script = raw"""
cd ${WORKSPACE}/srcdir/libfoo
make install
install_license ${WORKSPACE}/srcdir/libfoo/LICENSE.md
"""
libfoo_cmake_script = raw"""
mkdir ${WORKSPACE}/srcdir/libfoo/build && cd ${WORKSPACE}/srcdir/libfoo/build
cmake -DCMAKE_INSTALL_PREFIX=${prefix} -DCMAKE_TOOLCHAIN_FILE=${CMAKE_TARGET_TOOLCHAIN} ..
make install
install_license ${WORKSPACE}/srcdir/libfoo/LICENSE.md
"""
libfoo_meson_script = raw"""
mkdir ${WORKSPACE}/srcdir/libfoo/build && cd ${WORKSPACE}/srcdir/libfoo/build
meson .. -Dprefix=${prefix} --cross-file="${MESON_TARGET_TOOLCHAIN}"
ninja install -v
# grumble grumble meson! Why do you go to all the trouble to build it properly
# in `build`, then screw it up when you `install` it?! Silly willy.
if [[ ${target} == *apple* ]]; then
install_name_tool ${prefix}/bin/fooifier -change ${prefix}/lib/libfoo.0.dylib @rpath/libfoo.0.dylib
fi
install_license ${WORKSPACE}/srcdir/libfoo/LICENSE.md
"""
@testset "Building libfoo" begin
# Test building with both `make` and `cmake`, using directory and git repository
for script in (libfoo_make_script, libfoo_cmake_script, libfoo_meson_script)
# Do build within a separate temporary directory
mktempdir() do build_path
# Create local git repository of `libfoo` sources
git_path = joinpath(build_path, "libfoo.git")
mkpath(git_path)
# Copy files in, commit them. This is the commit we will build.
repo = LibGit2.init(git_path)
LibGit2.commit(repo, "Initial empty commit")
libfoo_src_dir = joinpath(build_tests_dir, "libfoo")
run(`cp -r $(libfoo_src_dir)/$(readdir(libfoo_src_dir)) $(git_path)/`)
for file in readdir(git_path)
LibGit2.add!(repo, file)
end
commit = LibGit2.commit(repo, "Add libfoo files")
# Add another commit to ensure that the git checkout is getting the right commit.
open(joinpath(git_path, "Makefile"), "w") do io
println(io, "THIS WILL BREAK EVERYTHING")
end
LibGit2.add!(repo, "Makefile")
LibGit2.commit(repo, "Break Makefile")
for source in (build_tests_dir, git_path => bytes2hex(LibGit2.raw(LibGit2.GitHash(commit))))
build_output_meta = autobuild(
build_path,
"libfoo",
v"1.0.0",
# Copy in the libfoo sources
[source],
# Use the particular build script we're interested in
script,
# Build for this platform
[platform],
# The products we expect to be build
libfoo_products,
# No depenedencies
[];
# Don't do audit passes
skip_audit=true,
# Make one verbose for the coverage. We do it all for the coverage, Morty.
verbose=true,
)
@test haskey(build_output_meta, platform)
tarball_path, tarball_hash = build_output_meta[platform][1:2]
# Ensure the build products were created
@test isfile(tarball_path)
# Ensure that the file contains what we expect
contents = list_tarball_files(tarball_path)
@test "bin/fooifier$(exeext(platform))" in contents
@test "lib/libfoo.$(dlext(platform))" in contents
# Unpack it somewhere else
@test verify(tarball_path, tarball_hash)
testdir = joinpath(build_path, "testdir")
mkpath(testdir)
unpack(tarball_path, testdir)
# Ensure we can use it
prefix = Prefix(testdir)
fooifier_path = joinpath(bindir(prefix), "fooifier$(exeext(platform))")
libfoo_path = first(filter(f -> isfile(f), joinpath.(libdirs(prefix), "libfoo.$(dlext(platform))")))
# We know that foo(a, b) returns 2*a^2 - b
result = 2*2.2^2 - 1.1
# Test that we can invoke fooifier
@test !success(`$fooifier_path`)
@test success(`$fooifier_path 1.5 2.0`)
@test parse(Float64,readchomp(`$fooifier_path 2.2 1.1`)) ≈ result
# Test that we can dlopen() libfoo and invoke it directly
libfoo = Libdl.dlopen_e(libfoo_path)
@test libfoo != C_NULL
foo = Libdl.dlsym_e(libfoo, :foo)
@test foo != C_NULL
@test ccall(foo, Cdouble, (Cdouble, Cdouble), 2.2, 1.1) ≈ result
Libdl.dlclose(libfoo)
end
end
end
end
shards_to_test = expand_cxxstring_abis(expand_gfortran_versions(platform))
if lowercase(get(ENV, "BINARYBUILDER_FULL_SHARD_TEST", "false")) == "true"
@info("Beginning full shard test... (this can take a while)")
shards_to_test = supported_platforms()
else
shards_to_test = [platform]
end
# Expand to all platforms
shards_to_test = expand_cxxstring_abis(expand_gfortran_versions(shards_to_test))
# Perform a sanity test on each and every shard.
@testset "Shard testsuites" begin
mktempdir() do build_path
products = Product[
ExecutableProduct("hello_world_c", :hello_world_c),
ExecutableProduct("hello_world_cxx", :hello_world_cxx),
ExecutableProduct("hello_world_fortran", :hello_world_fortran),
ExecutableProduct("hello_world_go", :hello_world_go),
ExecutableProduct("hello_world_rust", :hello_world_rust),
]
build_output_meta = autobuild(
build_path,
"testsuite",
v"1.0.0",
# No sources
[],
# Build the test suite, install the binaries into our prefix's `bin`
raw"""
# Build testsuite
make -j${nproc} -sC /usr/share/testsuite install
# Install fake license just to silence the warning
install_license /usr/share/licenses/libuv/LICENSE
""",
# Build for ALL the platforms
shards_to_test,
products,
# No dependencies
[];
# We need to be able to build go and rust and whatnot
compilers=[:c, :go, :rust],
)
# Test that we built everything (I'm not entirely sure how I expect
# this to fail without some kind of error being thrown earlier on,
# to be honest I just like seeing lots of large green numbers.)
@test length(keys(shards_to_test)) == length(keys(build_output_meta))
# Extract our platform's build, run the hello_world tests:
output_meta = select_platform(build_output_meta, platform)
@test output_meta != nothing
tarball_path, tarball_hash = output_meta[1:2]
# Ensure the build products were created
@test isfile(tarball_path)
# Unpack it somewhere else
@test verify(tarball_path, tarball_hash)
testdir = joinpath(build_path, "testdir")
mkdir(testdir)
unpack(tarball_path, testdir)
prefix = Prefix(testdir)
for product in products
hw_path = locate(product, prefix)
@test hw_path !== nothing && isfile(hw_path)
with_libgfortran() do
@test strip(String(read(`$hw_path`))) == "Hello, World!"
end
end
end
end
@testset "Dependency Specification" begin
mktempdir() do build_path
@test_logs (:error, r"BadDependency_jll") (:error, r"WorseDependency_jll") match_mode=:any begin
@test_throws ErrorException autobuild(
build_path,
"baddeps",
v"1.0.0",
# No sources
[],
"true",
[platform],
[ExecutableProduct("foo", :foo)],
# Three dependencies; one good, two bad
[
"Zlib_jll",
# We hope nobody will ever register something named this
"BadDependency_jll",
"WorseDependency_jll",
]
)
end
end
end
|
-- @@stderr --
dtrace: failed to compile script test/unittest/speculation/err.NegativeBufSize.d: line 27: failed to set option 'bufsize' to '-72': Invalid value for specified option
|
{-# OPTIONS --safe #-}
module Cubical.Algebra.Ring.Base where
open import Cubical.Foundations.Prelude
open import Cubical.Foundations.Equiv
open import Cubical.Foundations.Equiv.HalfAdjoint
open import Cubical.Foundations.Function
open import Cubical.Foundations.HLevels
open import Cubical.Foundations.Isomorphism
open import Cubical.Foundations.Univalence
open import Cubical.Foundations.Transport
open import Cubical.Foundations.SIP
open import Cubical.Data.Sigma
open import Cubical.Algebra.Semigroup
open import Cubical.Algebra.Monoid
open import Cubical.Algebra.Group
open import Cubical.Algebra.AbGroup
open import Cubical.Displayed.Base
open import Cubical.Displayed.Auto
open import Cubical.Displayed.Record
open import Cubical.Displayed.Universe
open import Cubical.Reflection.RecordEquiv
open Iso
private
variable
ℓ ℓ' : Level
record IsRing {R : Type ℓ}
(0r 1r : R) (_+_ _·_ : R → R → R) (-_ : R → R) : Type ℓ where
constructor isring
field
+IsAbGroup : IsAbGroup 0r _+_ -_
·IsMonoid : IsMonoid 1r _·_
dist : (x y z : R) → (x · (y + z) ≡ (x · y) + (x · z))
× ((x + y) · z ≡ (x · z) + (y · z))
-- This is in the Agda stdlib, but it's redundant
-- zero : (x : R) → (x · 0r ≡ 0r) × (0r · x ≡ 0r)
open IsAbGroup +IsAbGroup public
renaming
( assoc to +Assoc
; identity to +Identity
; lid to +Lid
; rid to +Rid
; inverse to +Inv
; invl to +Linv
; invr to +Rinv
; comm to +Comm
; isSemigroup to +IsSemigroup
; isMonoid to +IsMonoid
; isGroup to +IsGroup )
open IsMonoid ·IsMonoid public
renaming
( assoc to ·Assoc
; identity to ·Identity
; lid to ·Lid
; rid to ·Rid
; isSemigroup to ·IsSemigroup )
hiding
( is-set ) -- We only want to export one proof of this
·Rdist+ : (x y z : R) → x · (y + z) ≡ (x · y) + (x · z)
·Rdist+ x y z = dist x y z .fst
·Ldist+ : (x y z : R) → (x + y) · z ≡ (x · z) + (y · z)
·Ldist+ x y z = dist x y z .snd
record RingStr (A : Type ℓ) : Type (ℓ-suc ℓ) where
constructor ringstr
field
0r : A
1r : A
_+_ : A → A → A
_·_ : A → A → A
-_ : A → A
isRing : IsRing 0r 1r _+_ _·_ -_
infix 8 -_
infixl 7 _·_
infixl 6 _+_
open IsRing isRing public
Ring : ∀ ℓ → Type (ℓ-suc ℓ)
Ring ℓ = TypeWithStr ℓ RingStr
isSetRing : (R : Ring ℓ) → isSet ⟨ R ⟩
isSetRing R = R .snd .RingStr.isRing .IsRing.·IsMonoid .IsMonoid.isSemigroup .IsSemigroup.is-set
makeIsRing : {R : Type ℓ} {0r 1r : R} {_+_ _·_ : R → R → R} { -_ : R → R}
(is-setR : isSet R)
(+-assoc : (x y z : R) → x + (y + z) ≡ (x + y) + z)
(+-rid : (x : R) → x + 0r ≡ x)
(+-rinv : (x : R) → x + (- x) ≡ 0r)
(+-comm : (x y : R) → x + y ≡ y + x)
(r+-assoc : (x y z : R) → x · (y · z) ≡ (x · y) · z)
(·-rid : (x : R) → x · 1r ≡ x)
(·-lid : (x : R) → 1r · x ≡ x)
(·-rdist-+ : (x y z : R) → x · (y + z) ≡ (x · y) + (x · z))
(·-ldist-+ : (x y z : R) → (x + y) · z ≡ (x · z) + (y · z))
→ IsRing 0r 1r _+_ _·_ -_
makeIsRing is-setR assoc +-rid +-rinv +-comm ·-assoc ·-rid ·-lid ·-rdist-+ ·-ldist-+ =
isring (makeIsAbGroup is-setR assoc +-rid +-rinv +-comm)
(makeIsMonoid is-setR ·-assoc ·-rid ·-lid)
λ x y z → ·-rdist-+ x y z , ·-ldist-+ x y z
makeRing : {R : Type ℓ} (0r 1r : R) (_+_ _·_ : R → R → R) (-_ : R → R)
(is-setR : isSet R)
(+-assoc : (x y z : R) → x + (y + z) ≡ (x + y) + z)
(+-rid : (x : R) → x + 0r ≡ x)
(+-rinv : (x : R) → x + (- x) ≡ 0r)
(+-comm : (x y : R) → x + y ≡ y + x)
(+-assoc : (x y z : R) → x · (y · z) ≡ (x · y) · z)
(·-rid : (x : R) → x · 1r ≡ x)
(·-lid : (x : R) → 1r · x ≡ x)
(·-rdist-+ : (x y z : R) → x · (y + z) ≡ (x · y) + (x · z))
(·-ldist-+ : (x y z : R) → (x + y) · z ≡ (x · z) + (y · z))
→ Ring ℓ
makeRing 0r 1r _+_ _·_ -_ is-setR assoc +-rid +-rinv +-comm ·-assoc ·-rid ·-lid ·-rdist-+ ·-ldist-+ =
_ , ringstr 0r 1r _+_ _·_ -_
(makeIsRing is-setR assoc +-rid +-rinv +-comm
·-assoc ·-rid ·-lid ·-rdist-+ ·-ldist-+ )
record IsRingHom {A : Type ℓ} {B : Type ℓ'} (R : RingStr A) (f : A → B) (S : RingStr B)
: Type (ℓ-max ℓ ℓ')
where
-- Shorter qualified names
private
module R = RingStr R
module S = RingStr S
field
pres0 : f R.0r ≡ S.0r
pres1 : f R.1r ≡ S.1r
pres+ : (x y : A) → f (x R.+ y) ≡ f x S.+ f y
pres· : (x y : A) → f (x R.· y) ≡ f x S.· f y
pres- : (x : A) → f (R.- x) ≡ S.- (f x)
unquoteDecl IsRingHomIsoΣ = declareRecordIsoΣ IsRingHomIsoΣ (quote IsRingHom)
RingHom : (R : Ring ℓ) (S : Ring ℓ') → Type (ℓ-max ℓ ℓ')
RingHom R S = Σ[ f ∈ (⟨ R ⟩ → ⟨ S ⟩) ] IsRingHom (R .snd) f (S .snd)
IsRingEquiv : {A : Type ℓ} {B : Type ℓ'} (M : RingStr A) (e : A ≃ B) (N : RingStr B)
→ Type (ℓ-max ℓ ℓ')
IsRingEquiv M e N = IsRingHom M (e .fst) N
RingEquiv : (R : Ring ℓ) (S : Ring ℓ') → Type (ℓ-max ℓ ℓ')
RingEquiv R S = Σ[ e ∈ (⟨ R ⟩ ≃ ⟨ S ⟩) ] IsRingEquiv (R .snd) e (S .snd)
_$_ : {R S : Ring ℓ} → (φ : RingHom R S) → (x : ⟨ R ⟩) → ⟨ S ⟩
φ $ x = φ .fst x
RingEquiv→RingHom : {A B : Ring ℓ} → RingEquiv A B → RingHom A B
RingEquiv→RingHom (e , eIsHom) = e .fst , eIsHom
isPropIsRing : {R : Type ℓ} (0r 1r : R) (_+_ _·_ : R → R → R) (-_ : R → R)
→ isProp (IsRing 0r 1r _+_ _·_ -_)
isPropIsRing 0r 1r _+_ _·_ -_ (isring RG RM RD) (isring SG SM SD) =
λ i → isring (isPropIsAbGroup _ _ _ RG SG i)
(isPropIsMonoid _ _ RM SM i)
(isPropDistr RD SD i)
where
isSetR : isSet _
isSetR = RM .IsMonoid.isSemigroup .IsSemigroup.is-set
isPropDistr : isProp ((x y z : _) → ((x · (y + z)) ≡ ((x · y) + (x · z)))
× (((x + y) · z) ≡ ((x · z) + (y · z))))
isPropDistr = isPropΠ3 λ _ _ _ → isProp× (isSetR _ _) (isSetR _ _)
isPropIsRingHom : {A : Type ℓ} {B : Type ℓ'} (R : RingStr A) (f : A → B) (S : RingStr B)
→ isProp (IsRingHom R f S)
isPropIsRingHom R f S = isOfHLevelRetractFromIso 1 IsRingHomIsoΣ
(isProp×4 (isSetRing (_ , S) _ _)
(isSetRing (_ , S) _ _)
(isPropΠ2 λ _ _ → isSetRing (_ , S) _ _)
(isPropΠ2 λ _ _ → isSetRing (_ , S) _ _)
(isPropΠ λ _ → isSetRing (_ , S) _ _))
isSetRingHom : (R : Ring ℓ) (S : Ring ℓ') → isSet (RingHom R S)
isSetRingHom R S = isSetΣSndProp (isSetΠ (λ _ → isSetRing S)) (λ f → isPropIsRingHom (snd R) f (snd S))
RingHomPathP : (R S T : Ring ℓ) (p : S ≡ T) (φ : RingHom R S) (ψ : RingHom R T)
→ PathP (λ i → R .fst → p i .fst) (φ .fst) (ψ .fst)
→ PathP (λ i → RingHom R (p i)) φ ψ
RingHomPathP R S T p φ ψ q = ΣPathP (q , isProp→PathP (λ _ → isPropIsRingHom _ _ _) _ _)
RingHom≡ : {R S : Ring ℓ} {φ ψ : RingHom R S} → fst φ ≡ fst ψ → φ ≡ ψ
RingHom≡ = Σ≡Prop λ f → isPropIsRingHom _ f _
𝒮ᴰ-Ring : DUARel (𝒮-Univ ℓ) RingStr ℓ
𝒮ᴰ-Ring =
𝒮ᴰ-Record (𝒮-Univ _) IsRingEquiv
(fields:
data[ 0r ∣ null ∣ pres0 ]
data[ 1r ∣ null ∣ pres1 ]
data[ _+_ ∣ bin ∣ pres+ ]
data[ _·_ ∣ bin ∣ pres· ]
data[ -_ ∣ un ∣ pres- ]
prop[ isRing ∣ (λ _ _ → isPropIsRing _ _ _ _ _) ])
where
open RingStr
open IsRingHom
-- faster with some sharing
null = autoDUARel (𝒮-Univ _) (λ A → A)
un = autoDUARel (𝒮-Univ _) (λ A → A → A)
bin = autoDUARel (𝒮-Univ _) (λ A → A → A → A)
RingPath : (R S : Ring ℓ) → RingEquiv R S ≃ (R ≡ S)
RingPath = ∫ 𝒮ᴰ-Ring .UARel.ua
-- Rings have an abelian group and a monoid
Ring→AbGroup : Ring ℓ → AbGroup ℓ
Ring→AbGroup (A , ringstr _ _ _ _ _ R) = A , abgroupstr _ _ _ (IsRing.+IsAbGroup R)
Ring→Group : Ring ℓ → Group ℓ
Ring→Group = AbGroup→Group ∘ Ring→AbGroup
Ring→AddMonoid : Ring ℓ → Monoid ℓ
Ring→AddMonoid = Group→Monoid ∘ Ring→Group
Ring→MultMonoid : Ring ℓ → Monoid ℓ
Ring→MultMonoid (A , ringstr _ _ _ _ _ R) = monoid _ _ _ (IsRing.·IsMonoid R)
-- Smart constructor for ring homomorphisms
-- that infers the other equations from pres1, pres+, and pres·
module _ {R : Ring ℓ} {S : Ring ℓ'} {f : ⟨ R ⟩ → ⟨ S ⟩} where
private
module R = RingStr (R .snd)
module S = RingStr (S .snd)
module _
(p1 : f R.1r ≡ S.1r)
(p+ : (x y : ⟨ R ⟩) → f (x R.+ y) ≡ f x S.+ f y)
(p· : (x y : ⟨ R ⟩) → f (x R.· y) ≡ f x S.· f y)
where
open IsRingHom
private
isGHom : IsGroupHom (Ring→Group R .snd) f (Ring→Group S .snd)
isGHom = makeIsGroupHom p+
makeIsRingHom : IsRingHom (R .snd) f (S .snd)
makeIsRingHom .pres0 = isGHom .IsGroupHom.pres1
makeIsRingHom .pres1 = p1
makeIsRingHom .pres+ = p+
makeIsRingHom .pres· = p·
makeIsRingHom .pres- = isGHom .IsGroupHom.presinv
|
#################################################
###### Integers and Floating-Point Numbers ######
#################################################
# Julia's primitive numeric types:
# Integer types
# Type Signed? Number of bits Smallest value Largest value
# Int8 ✓ 8 -2^7 2^7 - 1
# UInt8 8 0 2^8 - 1
# Int16 ✓ 16 -2^15 2^15 - 1
# UInt16 16 0 2^16 - 1
# Int32 ✓ 32 -2^31 2^31 - 1
# UInt32 32 0 2^32 - 1
# Int64 ✓ 64 -2^63 2^63 - 1
# UInt64 64 0 2^64 - 1
# Int128 ✓ 128 -2^127 2^127 - 1
# UInt128 128 0 2^128 - 1
# Bool N/A 8 false (0) true (1) <- implemented as an 8-bit primitive type
# Floating-point types:
# Type Precision Number of bits
# Float16 half 16
# Float32 single 32
# Float64 double 64
typeof(12) # Int64
# Unsigned integers are input and output using the 0x prefix and hexadecimal (base 16) digits 0-9a-f
# (the capitalized digits A-F also work for input). The size of the unsigned value is determined
# by the number of hex digits used:
x = 0x1
# 0x01
typeof(x) # UInt8
x = 0x123 # UInt16
x = 0x1234567 # UInt32
##################################
###### Arithmetic Operators ######
##################################
# The following arithmetic operators are supported on all primitive numeric types:
# Expression Name Description
# +x unary plus the identity operation
# -x unary minus maps values to their additive inverses
# x + y binary plus performs addition
# x - y binary minus performs subtraction
# x * y times performs multiplication
# x / y divide performs division
# x ÷ y integer divide x / y, truncated to an integer
# x \ y inverse divide equivalent to y / x
# x ^ y power raises x to the yth power
# x % y remainder equivalent to rem(x,y)
###############################
###### Boolean Operators ######
###############################
# Expression Name
# !x negation
# x && y short-circuiting and
# x || y short-circuiting or
###############################
###### Bitwise Operators ######
###############################
# Expression Name
# ~x bitwise not
# x & y bitwise and
# x | y bitwise or
# x ⊻ y bitwise xor (exclusive or)
# x >>> y logical shift right
# x >> y arithmetic shift right
# x << y logical/arithmetic shift left
#################################
###### Numeric Comparisons ######
#################################
# Operator Name
# == equality
# !=, ≠ inequality
# < less than
# <=, ≤ less than or equal to
# > greater than
# >=, ≥ greater than or equal to
################################
###### Updating operators ######
################################
# += -= *= /= \= ÷= %= ^= &= |= ⊻= >>>= >>= <<=
#######################
###### Functions ######
#######################
function g(x,y)
return x + y
end
function hypot(x,y)
x = abs(x)
y = abs(y)
if x > y
r = y/x
return x*sqrt(1+r*r)
end
if y == 0
return zero(x)
end
r = x/y
return y*sqrt(1+r*r)
end
function fact(n::Int)
n >= 0 || error("n must be non-negative")
n == 0 && return 1
n * fact(n-1)
end
#################################
###### Anonymous Functions ######
#################################
x -> x + x
function (x)
x + x
end
|
{-# OPTIONS --without-K --safe #-}
open import Categories.Category
open import Categories.Functor hiding (id)
-- Cocone over a Functor F (from shape category J into category C)
module Categories.Diagram.Cocone
{o ℓ e} {o′ ℓ′ e′} {C : Category o ℓ e} {J : Category o′ ℓ′ e′} (F : Functor J C) where
open Category C
open Functor F
open import Level
record Coapex (N : Obj) : Set (o ⊔ ℓ ⊔ e ⊔ o′ ⊔ ℓ′) where
field
ψ : (X : Category.Obj J) → F₀ X ⇒ N
commute : ∀ {X Y} (f : J [ X , Y ]) → ψ Y ∘ F₁ f ≈ ψ X
record Cocone : Set (o ⊔ ℓ ⊔ e ⊔ o′ ⊔ ℓ′) where
field
{N} : Obj
coapex : Coapex N
open Coapex coapex public
open Coapex
open Cocone
record Cocone⇒ (c c′ : Cocone) : Set (ℓ ⊔ e ⊔ o′) where
field
arr : N c ⇒ N c′
commute : ∀ {X} → arr ∘ ψ c X ≈ ψ c′ X
open Cocone⇒
|
import numpy as np
from . import edgefit
from . import imagefit
from .models import simulate
def analyze(qpi, r0, method="edge", model="projection", edgekw={}, imagekw={},
ret_center=False, ret_pha_offset=False, ret_qpi=False):
"""Determine refractive index and radius of a spherical object
Parameters
----------
qpi: qpimage.QPImage
Quantitative phase image data
r0: float
Approximate radius of the sphere [m]
method: str
The method used to determine the refractive index
can either be "edge" (determine the radius from the
edge detected in the phase image) or "image" (perform
a 2D phase image fit).
model: str
The light-scattering model used by `method`. If
`method` is "edge", only "projection" is allowed.
If `method` is "image", `model` can be one of
"mie", "projection", "rytov", or "rytov-sc".
edgekw: dict
Keyword arguments for tuning the edge detection algorithm,
see :func:`qpsphere.edgefit.contour_canny`.
imagekw: dict
Keyword arguments for tuning the image fitting algorithm,
see :func:`qpsphere.imagefit.alg.match_phase`
ret_center: bool
If True, return the center coordinate of the sphere.
ret_pha_offset: bool
If True, return the phase image background offset.
ret_qpi: bool
If True, return the modeled data as a :class:`qpimage.QPImage`.
Returns
-------
n: float
Computed refractive index
r: float
Computed radius [m]
c: tuple of floats
Only returned if `ret_center` is True;
Center position of the sphere [px]
pha_offset: float
Only returned if `ret_pha_offset` is True;
Phase image background offset
qpi_sim: qpimage.QPImage
Only returned if `ret_qpi` is True;
Modeled data
Notes
-----
If `method` is "image", then the "edge" method is used
as a first step to estimate initial parameters for radius,
refractive index, and position of the sphere using `edgekw`.
If this behavior is not desired, please make use of the
method :func:`qpsphere.imagefit.analyze`.
"""
if method == "edge":
if model != "projection":
raise ValueError("`method='edge'` requires `model='projection'`!")
n, r, c = edgefit.analyze(qpi=qpi,
r0=r0,
edgekw=edgekw,
ret_center=True,
ret_edge=False,
)
res = [n, r]
if ret_center:
res.append(c)
if ret_pha_offset:
res.append(0)
if ret_qpi:
qpi_sim = simulate(radius=r,
sphere_index=n,
medium_index=qpi["medium index"],
wavelength=qpi["wavelength"],
grid_size=qpi.shape,
model="projection",
pixel_size=qpi["pixel size"],
center=c)
res.append(qpi_sim)
elif method == "image":
try:
n0, r0, c0 = edgefit.analyze(qpi=qpi,
r0=r0,
edgekw=edgekw,
ret_center=True,
ret_edge=False,
)
except (edgefit.EdgeDetectionError,
edgefit.RadiusExceedsImageSizeError):
# proceed with best guess
c0 = np.array(qpi.shape) / 2
n0 = qpi["medium index"] + np.sign(np.sum(qpi.pha)) * .01
res = imagefit.analyze(qpi=qpi,
model=model,
n0=n0,
r0=r0,
c0=c0,
imagekw=imagekw,
ret_center=ret_center,
ret_pha_offset=ret_pha_offset,
ret_qpi=ret_qpi
)
else:
raise NotImplementedError("`method` must be 'edge' or 'image'!")
return res
def bg_phase_mask_from_sim(sim, radial_clearance=1.1):
"""Return the background phase mask of a qpsphere simulation
Parameters
----------
sim: qpimage.QPImage
Quantitative phase data simulated with qpsphere;
The simulation keyword arguments "sim center", "sim radius",
and "pixel size" must be present in `sim.meta`.
radial_clearance: float
Multiplicator to the fitted radius of the sphere; modifies
the size of the mask; set to "1" to use the radius determined
by :func:`qpsphere.analyze`.
The circular area containing the phase object is set to
`False` in the output `mask` image.
Returns
-------
mask: boolean 2d np.ndarray
The mask is `True` for background regions and `False` for
object regions.
"""
# Mask values around the object
cx, cy = sim["sim center"]
radius = sim["sim radius"]
px_um = sim["pixel size"]
x = np.arange(sim.shape[0]).reshape(-1, 1)
y = np.arange(sim.shape[1]).reshape(1, -1)
rsq = (x - cx)**2 + (y - cy)**2
mask = rsq > (radius/px_um * radial_clearance)**2
return mask
def bg_phase_mask_for_qpi(qpi, r0, method="edge", model="projection",
edgekw={}, imagekw={}, radial_clearance=1.1):
"""Determine the background phase mask for a spherical phase object
The position and radius of the phase object are determined with
:func:`analyze`, to which the corresponding keyword arguments
are passed. A binary mask is created from the simulation results
via :func:`bg_phase_mask_from_sim`.
Parameters
----------
qpi: qpimage.QPImage
Quantitative phase image data
r0: float
Approximate radius of the sphere [m]
method: str
The method used to determine the refractive index
can either be "edge" (determine the radius from the
edge detected in the phase image) or "image" (perform
a 2D phase image fit).
model: str
The light-scattering model used by `method`. If
`method` is "edge", only "projection" is allowed.
If `method` is "image", `model` can be one of
"mie", "projection", "rytov", or "rytov-sc".
edgekw: dict
Keyword arguments for tuning the edge detection algorithm,
see :func:`qpsphere.edgefit.contour_canny`.
imagekw: dict
Keyword arguments for tuning the image fitting algorithm,
see :func:`qpsphere.imagefit.alg.match_phase`
radial_clearance: float
Multiplicator to the fitted radius of the sphere; modifies
the size of the mask; set to "1" to use the radius determined
by :func:`qpsphere.analyze`.
The circular area containing the phase object is set to
`False` in the output `mask` image.
Returns
-------
mask: boolean 2d np.ndarray
The mask is `True` for background regions and `False` for
object regions.
"""
# fit sphere
_, _, sim = analyze(qpi=qpi,
r0=r0,
method=method,
model=model,
edgekw=edgekw,
imagekw=imagekw,
ret_qpi=True)
# determine mask
mask = bg_phase_mask_from_sim(sim=sim,
radial_clearance=radial_clearance)
return mask
|
-- @@stderr --
dtrace: failed to compile script test/unittest/types/err.D_DECL_ENCONST.badeval.d: [D_DECL_ENCONST] line 18: enumerator 'TAG' must be assigned to an integral constant expression
|
State Before: 𝕜 : Type u_1
inst✝⁴ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝³ : SeminormedAddCommGroup E
inst✝² : NormedSpace 𝕜 E
F : Type ?u.27970
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace 𝕜 F
⊢ ‖inclusionInDoubleDual 𝕜 E‖ ≤ 1 State After: 𝕜 : Type u_1
inst✝⁴ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝³ : SeminormedAddCommGroup E
inst✝² : NormedSpace 𝕜 E
F : Type ?u.27970
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace 𝕜 F
⊢ ‖ContinuousLinearMap.id 𝕜 (Dual 𝕜 E)‖ ≤ 1 Tactic: rw [inclusionInDoubleDual_norm_eq] State Before: 𝕜 : Type u_1
inst✝⁴ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝³ : SeminormedAddCommGroup E
inst✝² : NormedSpace 𝕜 E
F : Type ?u.27970
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace 𝕜 F
⊢ ‖ContinuousLinearMap.id 𝕜 (Dual 𝕜 E)‖ ≤ 1 State After: no goals Tactic: exact ContinuousLinearMap.norm_id_le |
[STATEMENT]
lemma ladder_shift_j_length:
"length (ladder_shift_j d L) = length L"
[PROOF STATE]
proof (prove)
goal (1 subgoal):
1. length (ladder_shift_j d L) = length L
[PROOF STEP]
by (induct L, auto) |
Set Warnings "-notation-overridden,-parsing".
Require Import Tweetnacl_verif.init_tweetnacl.
Require Import Tweetnacl_verif.spec_A.
Require Import Tweetnacl.Libs.Export.
Require Import Tweetnacl.ListsOp.Export.
Require Import Tweetnacl.Low.A.
Open Scope Z.
Definition Gprog : funspecs :=
ltac:(with_library prog [A_spec]).
Import Low.
Lemma body_A: semax_body Vprog Gprog f_A A_spec.
Proof.
start_function.
unfold nm_overlap_array_sep_3, nm_overlap_array_sep_3' in *.
assert(HA: Zlength (A a b) = 16). rewrite A_Zlength ; omega.
assert(HmA: Zlength (mVI64 (A a b)) = 16). rewrite ?Zlength_map //.
assert(Forall (fun x : ℤ => amin + bmin < x < amax + bmax) (A a b)). apply A_bound_Zlength_lt ; trivial ; omega.
assert(Htkdp: tkdp 16 (mVI64 (A a b)) o = mVI64 (A a b)). rewrite -HmA tkdp_all ; trivial ; rewrite ?Zlength_map; omega.
assert(Haux1: forall i, 0 <= i < 16 -> exists aux1, Vlong aux1 = Znth i (mVI64 a) Vundef). intros; erewrite (Znth_map Int64.zero) ; [eexists ; reflexivity | rewrite Zlength_map; omega].
assert(Haux2: forall i, 0 <= i < 16 -> exists aux2, Vlong aux2 = Znth i (mVI64 b) Vundef). intros; erewrite (Znth_map Int64.zero) ; [eexists ; reflexivity | rewrite Zlength_map; omega].
assert(Haux3: forall i, 0 <= i < 16 -> exists aux3, Vlong aux3 = Znth i (tkdp i (mVI64 (A a b)) (mVI64 a)) Vundef).
intros ; rewrite /tkdp -?map_firstn -?map_skipn -?map_app ;
erewrite (Znth_map Int64.zero);[eexists ; reflexivity | rewrite ?Zlength_map];
change (firstn (nat_of_Z i) (A a b) ++ skipn (nat_of_Z i) a) with (tkdp i (A a b) a);
rewrite tkdp_Zlength HA ; omega.
assert(Haux4: forall i, 0 <= i < 16 -> exists aux4, Vlong aux4 = Znth i (tkdp i (mVI64 (A a b)) (mVI64 b)) Vundef).
intros ; rewrite /tkdp -?map_firstn -?map_skipn -?map_app ;
erewrite (Znth_map Int64.zero);[eexists ; reflexivity | rewrite ?Zlength_map];
change (firstn (nat_of_Z i) (A a b) ++ skipn (nat_of_Z i) a) with (tkdp i (A a b) a);
rewrite tkdp_Zlength HA ; omega.
flatten ; Intros.
1: subst o v_a.
2: subst o v_b.
3: subst b v_b.
4: subst o a v_o v_a.
1: forward_for_simple_bound 16 (A_Inv sho sha shb v_o v_o v_b (mVI64 a) a amin amax b bmin bmax 0);
[ unfold nm_overlap_array_sep_3' ; entailer!| |
rewrite Htkdp; forward; unfold nm_overlap_array_sep_3' ; entailer!].
2: forward_for_simple_bound 16 (A_Inv sho sha shb v_o v_a v_o (mVI64 b) a amin amax b bmin bmax 1);
[ unfold nm_overlap_array_sep_3' ; entailer! | |
rewrite Htkdp; forward; unfold nm_overlap_array_sep_3' ; entailer!].
3: forward_for_simple_bound 16 (A_Inv sho sha shb v_o v_a v_a o a amin amax a amin amax 2);
[ unfold nm_overlap_array_sep_3' ; entailer!| |
rewrite Htkdp; forward; unfold nm_overlap_array_sep_3' ; entailer!].
4: forward_for_simple_bound 16 (A_Inv sho sha shb v_b v_b v_b (mVI64 b) b bmin bmax b bmin bmax 3);
[ unfold nm_overlap_array_sep_3' ; entailer!| |
rewrite Htkdp; forward; unfold nm_overlap_array_sep_3' ; entailer!].
5: forward_for_simple_bound 16 (A_Inv sho sha shb v_o v_a v_b o a amin amax b bmin bmax 4);
[ unfold nm_overlap_array_sep_3' ; entailer!| |
rewrite Htkdp; forward; unfold nm_overlap_array_sep_3' ; entailer!].
all: unfold nm_overlap_array_sep_3' ; simpl ; Intros.
all: specialize Haux1 with i ; destruct (Haux1 H7) as [aux1 HHaux1].
all: specialize Haux2 with i ; destruct (Haux2 H7) as [aux2 HHaux2].
all: specialize Haux3 with i ; destruct (Haux3 H7) as [aux3 HHaux3].
all: specialize Haux4 with i ; destruct (Haux4 H7) as [aux4 HHaux4].
all: forward.
3,4,5,6,9,10: rewrite -HHaux1.
1,2,9,10: rewrite -HHaux3.
1,3,5,7,9: entailer!.
all: forward.
7,8: rewrite -HHaux1.
1,2,9,10: rewrite -HHaux2.
5,6: rewrite -HHaux3.
7,8: rewrite -HHaux4.
1,3,5,7,9: entailer!.
all: forward.
all: entailer!.
all: rewrite map_map in HHaux1.
all: rewrite map_map in HHaux2.
all: rewrite (Znth_map 0) in HHaux1; [ | omega ].
all: rewrite (Znth_map 0) in HHaux2; [ | omega ].
all: rewrite Znth_tkdp in HHaux3 ; [ | omega].
all: rewrite Znth_tkdp in HHaux4 ; [ | omega].
all: rewrite map_map in HHaux4.
all: rewrite map_map in HHaux3.
all: try (rewrite (Znth_map 0) in HHaux3; [ | omega ]).
all: try (rewrite (Znth_map 0) in HHaux4; [ | omega ]).
1,3,5,7,9: inversion HHaux1.
1,2,3,4,5: inversion HHaux2.
1,2,3,4,5: inversion HHaux3.
1,2,3,4,5: inversion HHaux4.
all: try assert(-2^62 < (Znth i a 0) < 2 ^ 62) by (solve_bounds_by_values_ H).
all: try assert(-2^62 < (Znth i b 0) < 2 ^ 62) by (solve_bounds_by_values_ H0).
all: try assert((-2^62) + (-2^62) <= Znth i a 0 + Znth i b 0 <= 2^62 + 2^62) by omega.
all: try assert((-2^62) + (-2^62) <= Znth i a 0 + Znth i a 0 <= 2^62 + 2^62) by omega.
all: try assert((-2^62) + (-2^62) <= Znth i b 0 + Znth i b 0 <= 2^62 + 2^62) by omega.
1,2,3,4,5: rewrite ?Int64.signed_repr ; solve_bounds_by_values.
all: unfold nm_overlap_array_sep_3' ; simpl ; data_atify ; cancel ; replace_cancel.
all: unfold A.
(* postcond |-- loop invariant *)
all: clean_context_from_VST.
all: rewrite /tkdp -?map_firstn -?map_skipn -?map_app in HHaux3.
all: rewrite /tkdp -?map_firstn -?map_skipn -?map_app in HHaux4.
all: inv HHaux1 ; inv HHaux2 ; inv HHaux3 ; inv HHaux4.
all: rewrite add64_repr /nat_of_Z.
all: rewrite ?Znth_nth; try omega.
all: rewrite <- ZsubList_nth_Zlength ; try omega.
all: rewrite /tkdp ?simple_S_i ; try omega.
all: rewrite /A in HA, HmA.
all: rewrite (upd_Znth_app_step_Zlength _ _ _ Vundef); try omega.
all: f_equal ; rewrite map_map (Znth_map 0) ?Znth_nth ; try reflexivity.
all: omega.
Qed.
Close Scope Z. |
CCHHAAPPTTEERR 11
FFRRAANNZZ LLIISSPP
11..11.. FRANZ LISPwas algebraic manipulation, artificial
intelligence, and programming languages at the Univer-
sity of California at Berkeley. Its roots are in a
PDP-11 Lisp system which originally came from Harvard.
As it grew it adopted features of Maclisp and Lisp
Machine Lisp. Substantial compatibility with other
Lisp dialects (Interlisp, UCILisp, CMULisp) is
achieved by means of support packages and compiler
switches. The heart of FRANZ LISP is written almost
entirely in the programming language C. Of course, it
has been greatly extended by additions written in
Lisp. A small part is written in the assembly lan-
guage for the current host machines, VAXen and a cou-
ple of flavors of 68000. Because FRANZ LISP is writ-
ten in C, it is relatively portable and easy to com-
prehend.
FRANZ LISP is capable of running large lisp pro-
grams in a timesharing environment, has facilities for
arrays and user defined structures, has a user con-
trolled reader with character and word macro capabil-
ities, and can interact directly with compiled Lisp,
C, Fortran, and Pascal code.
This document is a reference manual for the FRANZ
LISP system. It is not a Lisp primer or introduction
to the language. Some parts will be of interest pri-
marily to those maintaining FRANZ LISP at their com-
puter site. There is an additional document entitled
_T_h_e _F_r_a_n_z _L_i_s_p _S_y_s_t_e_m_, _b_y _J_o_h_n _F_o_d_e_r_a_r_o_, _w_h_i_c_h _p_a_r_-
_t_i_a_l_l_y _d_e_s_c_r_i_b_e_s _t_h_e _s_y_s_t_e_m _i_m_p_l_e_m_e_n_t_a_t_i_o_n_. _F_R_A_N_Z
_L_I_S_P_, _a_s _d_e_l_i_v_e_r_e_d _b_y _B_e_r_k_e_l_e_y_, _i_n_c_l_u_d_e_s _a_l_l _s_o_u_r_c_e
_c_o_d_e _a_n_d _m_a_c_h_i_n_e _r_e_a_d_a_b_l_e _v_e_r_s_i_o_n _o_f _t_h_i_s _m_a_n_u_a_l _a_n_d
_s_y_s_t_e_m _d_o_c_u_m_e_n_t_. _T_h_e _s_y_s_t_e_m _d_o_c_u_m_e_n_t _i_s _i_n _a _s_i_n_g_l_e
_f_i_l_e _n_a_m_e_d _"_f_r_a_n_z_._n_" _i_n _t_h_e _"_d_o_c_" _s_u_b_d_i_r_e_c_t_o_r_y_.
____________________
is rumored that this name has something to do with Franz
Liszt [F_rants List] (1811-1886) a Hungarian composer and
keyboard virtuoso. These allegations have never been
proven.
FFRRAANNZZ LLIISSPP 11--11
FFRRAANNZZ LLIISSPP 11--22
This document is divided into four Movements. In
the first one we will attempt to describe the language
of FRANZ LISP precisely and completely as it now
stands (Opus 38.69, June 1983). In the second Move-
ment we will look at the reader, function types,
arrays and exception handling. In the third Movement
we will look at several large support packages written
to help the FRANZ LISP user, namely the trace package,
compiler, fixit and stepping package. Finally the
fourth movement contains an index into the other
movements. In the rest of this chapter we shall exam-
ine the data types of FRANZ LISP. The conventions
used in the description of the FRANZ LISP functions
will be given in S1.3 -- it is very important that
these conventions are understood.
11..22.. DDaattaa TTyyppeess FRANZ LISP has fourteen data types.
In this section we shall look in detail at each type
and if a type is divisible we shall look inside it.
There is a Lisp function _t_y_p_e which will return the
type name of a lisp object. This is the official
FRANZ LISP name for that type and we will use this
name and this name only in the manual to avoid confus-
ing the reader. The types are listed in terms of
importance rather than alphabetically.
11..22..00.. lliissppvvaall This is the name we use to describe
any Lisp object. The function _t_y_p_e will never
return `lispval'.
11..22..11.. ssyymmbbooll This object corresponds to a variable
in most other programming languages. It may have a
value or may be `unbound'. A symbol may be _l_a_m_b_d_a
_b_o_u_n_d meaning that its current value is stored away
somewhere and the symbol is given a new value for
the duration of a certain context. When the Lisp
processor leaves that context, the symbol's cur-
rent value is thrown away and its old value is
restored.
A symbol may also have a _f_u_n_c_t_i_o_n _b_i_n_d_i_n_g. This
function binding is static; it cannot be lambda
bound. Whenever the symbol is used in the func-
tional position of a Lisp expression the function
binding of the symbol is examined (see Chapter 4
for more details on evaluation).
A symbol may also have a _p_r_o_p_e_r_t_y _l_i_s_t, another
static data structure. The property list consists
Printed: October 16, 1993
FFRRAANNZZ LLIISSPP 11--33
of a list of an even number of elements, considered
to be grouped as pairs. The first element of the
pair is the _i_n_d_i_c_a_t_o_r the second the _v_a_l_u_e of that
indicator.
Each symbol has a print name _(_p_n_a_m_e_) which is how
this symbol is accessed from input and referred to
on (printed) output.
A symbol also has a hashlink used to link symbols
together in the oblist -- this field is inaccessi-
ble to the lisp user.
Symbols are created by the reader and by the func-
tions _c_o_n_c_a_t, _m_a_k_n_a_m and their derivatives. Most
symbols live on FRANZ LISP's sole _o_b_l_i_s_t, and
therefore two symbols with the same print name are
usually the exact same object (they are _e_q). Sym-
bols which are not on the oblist are said to be
_u_n_i_n_t_e_r_n_e_d_. The function _m_a_k_n_a_m creates uninterned
symbols while _c_o_n_c_a_t creates _i_n_t_e_r_n_e_d ones.
+-------------+-----------+-----------+---------------------+
|Subpart name | Get value | Set value | Type |
| | | | |
+-------------+-----------+-----------+---------------------+
| value | eval | set | lispval |
| | | setq | |
+-------------+-----------+-----------+---------------------+
| property | plist | setplist | list or nil |
| list | get | putprop | |
| | | defprop | |
+-------------+-----------+-----------+---------------------+
| function | getd | putd | array, binary, list |
| binding | | def | or nil |
+-------------+-----------+-----------+---------------------+
| print name | get_pname | | string |
+-------------+-----------+-----------+---------------------+
| hash link | | | |
+-------------+-----------+-----------+---------------------+
11..22..22.. lliisstt A list cell has two parts, called the
car and cdr. List cells are created by the func-
tion _c_o_n_s.
Printed: October 16, 1993
FFRRAANNZZ LLIISSPP 11--44
+-------------+-----------+-----------+---------+
|Subpart name | Get value | Set value | Type |
| | | | |
+-------------+-----------+-----------+---------+
| car | car | rplaca | lispval |
+-------------+-----------+-----------+---------+
| cdr | cdr | rplacd | lispval |
+-------------+-----------+-----------+---------+
11..22..33.. bbiinnaarryy This type acts as a function header
for machine coded functions. It has two parts, a
pointer to the start of the function and a symbol
whose print name describes the argument _d_i_s_c_i_p_l_i_n_e.
The discipline (if _l_a_m_b_d_a, _m_a_c_r_o or _n_l_a_m_b_d_a) deter-
mines whether the arguments to this function will
be evaluated by the caller before this function is
called. If the discipline is a string (specifi-
cally "_s_u_b_r_o_u_t_i_n_e", "_f_u_n_c_t_i_o_n", "_i_n_t_e_g_e_r_-_f_u_n_c_t_i_o_n",
"_r_e_a_l_-_f_u_n_c_t_i_o_n", "_c_-_f_u_n_c_t_i_o_n", "_d_o_u_b_l_e_-_c_-_f_u_n_c_t_i_o_n",
or "_v_e_c_t_o_r_-_c_-_f_u_n_c_t_i_o_n" ) then this function is a
foreign subroutine or function (see S8.5 for more
details on this). Although the type of the _e_n_t_r_y
field of a binary type object is usually ssttrriinngg or
ootthheerr, the object pointed to is actually a sequence
of machine instructions.
Objects of type binary are created by _m_f_u_n_c_t_i_o_n_,
_c_f_a_s_l_, and _g_e_t_a_d_d_r_e_s_s_.
+-------------+-----------+-----------+------------------+
|Subpart name | Get value | Set value | Type |
| | | | |
+-------------+-----------+-----------+------------------+
| entry | getentry | | string or fixnum |
+-------------+-----------+-----------+------------------+
| discipline | getdisc | putdisc | symbol or fixnum |
+-------------+-----------+-----------+------------------+
11..22..44.. ffiixxnnuumm A fixnum is an integer constant in
the range -2to 2Small fixnums (-1024 to 1023) are
stored in a special table so they needn't be allo-
cated each time one is needed. In principle, the
range for fixnums is machine dependent, although
all current implementations for franz have this
range.
Printed: October 16, 1993
FFRRAANNZZ LLIISSPP 11--55
11..22..55.. fflloonnuumm A flonum is a double precision real
number. On the VAX, the range is +-2.9x10to
+-1.7x10There are approximately sixteen decimal
digits of precision. Other machines may have other
ranges.
11..22..66.. bbiiggnnuumm A bignum is an integer of potentially
unbounded size. When integer arithmetic exceeds
the limits of fixnums mentioned above, the calcula-
tion is automatically done with bignums. Should
calculation with bignums give a result which can be
represented as a fixnum, then the fixnum represen-
tation will be used.(f current algorithms for inte-
ger arithmetic operations will return (in certain
cases) a result between +-2and 2as a bignum
although this could be represented as a fixnum.
This contraction is known as _i_n_t_e_g_e_r _n_o_r_m_a_l_i_z_a_t_i_o_n.
Many Lisp functions assume that integers are nor-
malized. Bignums are composed of a sequence of
lliisstt cells and a cell known as an ssddoott.. The user
should consider a bbiiggnnuumm structure indivisible and
use functions such as _h_a_i_p_a_r_t, and _b_i_g_n_u_m_-_l_e_f_t_s_h_i_f_t
to extract parts of it.
11..22..77.. ssttrriinngg A string is a null terminated
sequence of characters. Most functions of symbols
which operate on the symbol's print name will also
work on strings. The default reader syntax is set
so that a sequence of characters surrounded by dou-
ble quotes is a string.
11..22..88.. ppoorrtt A port is a structure which the system
I/O routines can reference to transfer data between
the Lisp system and external media. Unlike other
Lisp objects there are a very limited number of
ports (20). Ports are allocated by _i_n_f_i_l_e and _o_u_t_-
_f_i_l_e and deallocated by _c_l_o_s_e and _r_e_s_e_t_i_o. The
_p_r_i_n_t function prints a port as a percent sign fol-
lowed by the name of the file it is connected to
(if the port was opened by _f_i_l_e_o_p_e_n_, _i_n_f_i_l_e_, _o_r
_o_u_t_f_i_l_e). During initialization, FRANZ LISP binds
the symbol ppiippoorrtt to a port attached to the stan-
dard input stream. This port prints as %$stdin.
There are ports connected to the standard output
and error streams, which print as %$stdout and
%$stderr. This is discussed in more detail at the
beginning of Chapter 5.
Printed: October 16, 1993
FFRRAANNZZ LLIISSPP 11--66
11..22..99.. vveeccttoorr Vectors are indexed sequences of
data. They can be used to implement a notion of
user-defined types via their associated property
list. They make hhuunnkkss (see below) logically unnec-
essary, although hunks are very efficiently garbage
collected. There is a second kind of vector,
called an immediate-vector, which stores binary
data. The name that the function _t_y_p_e returns for
immediate-vectors is vveeccttoorrii. Immediate-vectors
could be used to implement strings and block-flonum
arrays, for example. Vectors are discussed in
chapter 9. The functions _n_e_w_-_v_e_c_t_o_r, and _v_e_c_t_o_r,
can be used to create vectors.
+-------------+-----------+-----------+---------+
|Subpart name | Get value | Set value | Type |
| | | | |
+-------------+-----------+-----------+---------+
| datum[_i] | vref | vset | lispval |
+-------------+-----------+-----------+---------+
| property | vprop | vsetprop | lispval |
| | | vputprop | |
+-------------+-----------+-----------+---------+
| size | vsize | - | fixnum |
+-------------+-----------+-----------+---------+
11..22..1100.. aarrrraayy Arrays are rather complicated types
and are fully described in Chapter 9. An array
consists of a block of contiguous data, a function
to access that data, and auxiliary fields for use
by the accessing function. Since an array's
accessing function is created by the user, an array
can have any form the user chooses (e.g. n-
dimensional, triangular, or hash table).
Arrays are created by the function _m_a_r_r_a_y.
Printed: October 16, 1993
FFRRAANNZZ LLIISSPP 11--77
+----------------+-----------+-----------+---------------------+
| Subpart name | Get value | Set value | Type |
| | | | |
+----------------+-----------+-----------+---------------------+
|access function | getaccess | putaccess | binary, list |
| | | | or symbol |
+----------------+-----------+-----------+---------------------+
| auxiliary | getaux | putaux | lispval |
+----------------+-----------+-----------+---------------------+
| data | arrayref | replace | block of contiguous |
| | | set | lispval |
+----------------+-----------+-----------+---------------------+
| length | getlength | putlength | fixnum |
+----------------+-----------+-----------+---------------------+
| delta | getdelta | putdelta | fixnum |
+----------------+-----------+-----------+---------------------+
11..22..1111.. vvaalluuee A value cell contains a pointer to a
lispval. This type is used mainly by arrays of
general lisp objects. Value cells are created with
the _p_t_r function. A value cell containing a
pointer to the symbol `foo' is printed as
`(ptr to)foo'
11..22..1122.. hhuunnkk A hunk is a vector of from 1 to 128
lispvals. Once a hunk is created (by _h_u_n_k or
_m_a_k_h_u_n_k) it cannot grow or shrink. The access time
for an element of a hunk is slower than a list cell
element but faster than an array. Hunks are really
only allocated in sizes which are powers of two,
but can appear to the user to be any size in the 1
to 128 range. Users of hunks must realize that
_(_n_o_t _(_a_t_o_m _'_l_i_s_p_v_a_l_)_) will return true if _l_i_s_p_v_a_l
is a hunk. Most lisp systems do not have a direct
test for a list cell and instead use the above test
and assume that a true result means _l_i_s_p_v_a_l is a
list cell. In FRANZ LISP you can use _d_t_p_r to check
for a list cell. Although hunks are not list
cells, you can still access the first two hunk ele-
ments with _c_d_r and _c_a_r and you can access any hunk
element with _c_x_r.(f a hunk, the function _c_d_r refer-
ences the first element and _c_a_r the second.
You can set the value of the first two elements of
a hunk with _r_p_l_a_c_d and _r_p_l_a_c_a and you can set the
value of any element of the hunk with _r_p_l_a_c_x. A
hunk is printed by printing its contents surrounded
by { and }. However a hunk cannot be read in in
this way in the standard lisp system. It is easy
Printed: October 16, 1993
FFRRAANNZZ LLIISSPP 11--88
to write a reader macro to do this if desired.
11..22..1133.. ootthheerr Occasionally, you can obtain a
pointer to storage not allocated by the lisp sys-
tem. One example of this is the entry field of
those FRANZ LISP functions written in C. Such
objects are classified as of type ootthheerr. Foreign
functions which call malloc to allocate their own
space, may also inadvertantly create such objects.
The garbage collector is supposed to ignore such
objects.
11..33.. DDooccuummeennttaattiioonn The conventions used in the follow-
ing chapters were designed to give a great deal of
information in a brief space. The first line of a
function description contains the function name in
bboolldd ffaaccee and then lists the arguments, if any. The
arguments all have names which begin with a letter or
letters and an underscore. The letter(s) gives the
allowable type(s) for that argument according to this
table.
Printed: October 16, 1993
FFRRAANNZZ LLIISSPP 11--99
+-------+----------------------------------------------+
|Letter | Allowable type(s) |
| | |
+-------+----------------------------------------------+
|g | any type |
+-------+----------------------------------------------+
|s | symbol (although nil may not be allowed) |
+-------+----------------------------------------------+
|t | string |
+-------+----------------------------------------------+
|l | list (although nil may be allowed) |
+-------+----------------------------------------------+
|n | number (fixnum, flonum, bignum) |
+-------+----------------------------------------------+
|i | integer (fixnum, bignum) |
+-------+----------------------------------------------+
|x | fixnum |
+-------+----------------------------------------------+
|b | bignum |
+-------+----------------------------------------------+
|f | flonum |
+-------+----------------------------------------------+
|u | function type (either binary or lambda body) |
+-------+----------------------------------------------+
|y | binary |
+-------+----------------------------------------------+
|v | vector |
+-------+----------------------------------------------+
|V | vectori |
+-------+----------------------------------------------+
|a | array |
+-------+----------------------------------------------+
|e | value |
+-------+----------------------------------------------+
|p | port (or nil) |
+-------+----------------------------------------------+
|h | hunk |
+-------+----------------------------------------------+
In the first line of a function description, those
arguments preceded by a quote mark are evaluated (usu-
ally before the function is called). The quoting con-
vention is used so that we can give a name to the
result of evaluating the argument and we can describe
the allowable types. If an argument is not quoted it
does not mean that that argument will not be evalu-
ated, but rather that if it is evaluated, the time at
which it is evaluated will be specifically mentioned
in the function description. Optional arguments are
surrounded by square brackets. An ellipsis (...)
means zero or more occurrences of an argument of the
directly preceding type.
Printed: October 16, 1993
|
[STATEMENT]
lemma mult_lres_sub_assoc:
"x * (y / z) \<le> (x * y) / z"
[PROOF STATE]
proof (prove)
goal (1 subgoal):
1. x * (y / z) \<le> x * y / z
[PROOF STEP]
by (meson dual_order.trans lres_galois mult_right_isotone lres_inverse lres_mult_sub_lres_lres) |
lemma sets_restrict_space_cong: "sets M = sets N \<Longrightarrow> sets (restrict_space M \<Omega>) = sets (restrict_space N \<Omega>)" |
import Data.Vect
import Decidable.Equality
myExactLength : {m : _} -> (len : Nat) -> (input : Vect m a) -> Maybe (Vect len a)
myExactLength len input = case decEq m len of
Yes Refl => Just input
No contra => Nothing |
If $f$ and $g$ are continuous real-valued functions on a topological space $X$, then the function $x \mapsto \max(f(x), g(x))$ is continuous. |
The information contained in JoeBobBriggs.com is for general information purposes only. The information is provided by JoeBobBriggs.com and while we endeavour to keep the information up to date and correct, we make no representations or warranties of any kind, express or implied, about the completeness, accuracy, reliability, suitability or availability with respect to JoeBobBriggs.com or the information, products, services, or related graphics contained on JoeBobBriggs.com for any purpose. Any reliance you place on such information is therefore strictly at your own risk.
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lemma sphere_empty [simp]: "r < 0 \<Longrightarrow> sphere a r = {}" for a :: "'a::metric_space" |
If $f$ is holomorphic on an open set $M$ and $f(w) \neq 0$ and $f(w) \neq a$ for all $w \in M - \{z\}$, then either $f$ or $1/f$ is bounded on a punctured neighborhood of $z$. |
In 2014 , Fey was recognized by Elle Magazine during The Women in Hollywood Awards , honoring women for their outstanding achievements in film , spanning all aspects of the motion picture industry , including acting , directing , and producing .
|
module GDAL.Complex (
Complex(..)
, module C
) where
import Data.Complex as C hiding (Complex(..))
import qualified Data.Complex as C
newtype Complex a =
Complex { unComplex :: C.Complex a }
deriving (Eq)
instance Show a => Show (Complex a) where
show (Complex c) = show c
|
[STATEMENT]
lemma eff_Done[simp]: "eff Done s i = s"
[PROOF STATE]
proof (prove)
goal (1 subgoal):
1. eff Done s i = s
[PROOF STEP]
unfolding eff_def
[PROOF STATE]
proof (prove)
goal (1 subgoal):
1. snd (snd (wt_cont_eff Done s i)) = s
[PROOF STEP]
by simp |
State Before: α : Type u_2
β : Type u_3
ι : Type ?u.400065
mα : MeasurableSpace α
mβ : MeasurableSpace β
κ : { x // x ∈ kernel α β }
E : Type u_1
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
inst✝ : CompleteSpace E
f : β → E
g : α → β
a : α
hg : Measurable g
hf : StronglyMeasurable f
⊢ (∫ (x : β), f x ∂↑(deterministic g hg) a) = f (g a) State After: no goals Tactic: rw [kernel.deterministic_apply, integral_dirac' _ _ hf] |
lemma continuous_on_mult_right: fixes c::"'a::real_normed_algebra" shows "continuous_on s f \<Longrightarrow> continuous_on s (\<lambda>x. f x * c)" |
State Before: α : Type u_1
β : Type ?u.842481
inst✝ : DecidableEq α
l : List α
x : α
hl : Nodup l
⊢ formPerm l ^ length l = 1 State After: case H
α : Type u_1
β : Type ?u.842481
inst✝ : DecidableEq α
l : List α
x✝ : α
hl : Nodup l
x : α
⊢ ↑(formPerm l ^ length l) x = ↑1 x Tactic: ext x State Before: case H
α : Type u_1
β : Type ?u.842481
inst✝ : DecidableEq α
l : List α
x✝ : α
hl : Nodup l
x : α
⊢ ↑(formPerm l ^ length l) x = ↑1 x State After: case pos
α : Type u_1
β : Type ?u.842481
inst✝ : DecidableEq α
l : List α
x✝ : α
hl : Nodup l
x : α
hx : x ∈ l
⊢ ↑(formPerm l ^ length l) x = ↑1 x
case neg
α : Type u_1
β : Type ?u.842481
inst✝ : DecidableEq α
l : List α
x✝ : α
hl : Nodup l
x : α
hx : ¬x ∈ l
⊢ ↑(formPerm l ^ length l) x = ↑1 x Tactic: by_cases hx : x ∈ l State Before: case pos
α : Type u_1
β : Type ?u.842481
inst✝ : DecidableEq α
l : List α
x✝ : α
hl : Nodup l
x : α
hx : x ∈ l
⊢ ↑(formPerm l ^ length l) x = ↑1 x State After: case pos.intro.intro
α : Type u_1
β : Type ?u.842481
inst✝ : DecidableEq α
l : List α
x : α
hl : Nodup l
k : ℕ
hk : k < length l
hx : nthLe l k hk ∈ l
⊢ ↑(formPerm l ^ length l) (nthLe l k hk) = ↑1 (nthLe l k hk) Tactic: obtain ⟨k, hk, rfl⟩ := nthLe_of_mem hx State Before: case pos.intro.intro
α : Type u_1
β : Type ?u.842481
inst✝ : DecidableEq α
l : List α
x : α
hl : Nodup l
k : ℕ
hk : k < length l
hx : nthLe l k hk ∈ l
⊢ ↑(formPerm l ^ length l) (nthLe l k hk) = ↑1 (nthLe l k hk) State After: no goals Tactic: simp [formPerm_pow_apply_nthLe _ hl, Nat.mod_eq_of_lt hk] State Before: case neg
α : Type u_1
β : Type ?u.842481
inst✝ : DecidableEq α
l : List α
x✝ : α
hl : Nodup l
x : α
hx : ¬x ∈ l
⊢ ↑(formPerm l ^ length l) x = ↑1 x State After: case neg
α : Type u_1
β : Type ?u.842481
inst✝ : DecidableEq α
l : List α
x✝ : α
hl : Nodup l
x : α
hx : ¬x ∈ l
this : ¬x ∈ {x | ↑(formPerm l ^ length l) x ≠ x}
⊢ ↑(formPerm l ^ length l) x = ↑1 x Tactic: have : x ∉ { x | (l.formPerm ^ l.length) x ≠ x } := by
intro H
refine' hx _
replace H := set_support_zpow_subset l.formPerm l.length H
simpa using support_formPerm_le' _ H State Before: case neg
α : Type u_1
β : Type ?u.842481
inst✝ : DecidableEq α
l : List α
x✝ : α
hl : Nodup l
x : α
hx : ¬x ∈ l
this : ¬x ∈ {x | ↑(formPerm l ^ length l) x ≠ x}
⊢ ↑(formPerm l ^ length l) x = ↑1 x State After: no goals Tactic: simpa using this State Before: α : Type u_1
β : Type ?u.842481
inst✝ : DecidableEq α
l : List α
x✝ : α
hl : Nodup l
x : α
hx : ¬x ∈ l
⊢ ¬x ∈ {x | ↑(formPerm l ^ length l) x ≠ x} State After: α : Type u_1
β : Type ?u.842481
inst✝ : DecidableEq α
l : List α
x✝ : α
hl : Nodup l
x : α
hx : ¬x ∈ l
H : x ∈ {x | ↑(formPerm l ^ length l) x ≠ x}
⊢ False Tactic: intro H State Before: α : Type u_1
β : Type ?u.842481
inst✝ : DecidableEq α
l : List α
x✝ : α
hl : Nodup l
x : α
hx : ¬x ∈ l
H : x ∈ {x | ↑(formPerm l ^ length l) x ≠ x}
⊢ False State After: α : Type u_1
β : Type ?u.842481
inst✝ : DecidableEq α
l : List α
x✝ : α
hl : Nodup l
x : α
hx : ¬x ∈ l
H : x ∈ {x | ↑(formPerm l ^ length l) x ≠ x}
⊢ x ∈ l Tactic: refine' hx _ State Before: α : Type u_1
β : Type ?u.842481
inst✝ : DecidableEq α
l : List α
x✝ : α
hl : Nodup l
x : α
hx : ¬x ∈ l
H : x ∈ {x | ↑(formPerm l ^ length l) x ≠ x}
⊢ x ∈ l State After: α : Type u_1
β : Type ?u.842481
inst✝ : DecidableEq α
l : List α
x✝ : α
hl : Nodup l
x : α
hx : ¬x ∈ l
H : x ∈ {x | ↑(formPerm l) x ≠ x}
⊢ x ∈ l Tactic: replace H := set_support_zpow_subset l.formPerm l.length H State Before: α : Type u_1
β : Type ?u.842481
inst✝ : DecidableEq α
l : List α
x✝ : α
hl : Nodup l
x : α
hx : ¬x ∈ l
H : x ∈ {x | ↑(formPerm l) x ≠ x}
⊢ x ∈ l State After: no goals Tactic: simpa using support_formPerm_le' _ H |
module Issue2858-nbe where
open import Agda.Builtin.List
data Ty : Set where
α : Ty
_↝_ : Ty → Ty → Ty
variable
σ τ : Ty
Γ Δ Θ : List Ty
Scoped : Set₁
Scoped = Ty → List Ty → Set
data Var : Scoped where
z : Var σ (σ ∷ Γ)
s : Var σ Γ → Var σ (τ ∷ Γ)
record Ren (Γ Δ : List Ty) : Set where
field lookup : ∀ {σ} → Var σ Γ → Var σ Δ
open Ren public
bind : Ren Γ Δ → Ren (σ ∷ Γ) (σ ∷ Δ)
lookup (bind ρ) z = z
lookup (bind ρ) (s v) = s (lookup ρ v)
refl : Ren Γ Γ
lookup refl v = v
step : Ren Γ (σ ∷ Γ)
lookup step v = s v
_∘_ : Ren Δ Θ → Ren Γ Δ → Ren Γ Θ
lookup (ρ′ ∘ ρ) v = lookup ρ′ (lookup ρ v)
interleaved mutual
data Syn : Scoped
data Chk : Scoped
th^Syn : Ren Γ Δ → Syn σ Γ → Syn σ Δ
th^Chk : Ren Γ Δ → Chk σ Γ → Chk σ Δ
-- variable rule
constructor
var : Var σ Γ → Syn σ Γ
th^Syn ρ (var v) = var (lookup ρ v)
-- change of direction rules
constructor
emb : Syn σ Γ → Chk σ Γ
cut : Chk σ Γ → Syn σ Γ
th^Chk ρ (emb t) = emb (th^Syn ρ t)
th^Syn ρ (cut c) = cut (th^Chk ρ c)
-- function introduction and elimination
constructor
app : Syn (σ ↝ τ) Γ → Chk σ Γ → Syn τ Γ
lam : Chk τ (σ ∷ Γ) → Chk (σ ↝ τ) Γ
th^Syn ρ (app f t) = app (th^Syn ρ f) (th^Chk ρ t)
th^Chk ρ (lam b) = lam (th^Chk (bind ρ) b)
-- Model construction
Val : Scoped
Val α Γ = Syn α Γ
Val (σ ↝ τ) Γ = ∀ {Δ} → Ren Γ Δ → Val σ Δ → Val τ Δ
th^Val : Ren Γ Δ → Val σ Γ → Val σ Δ
th^Val {σ = α} ρ t = th^Syn ρ t
th^Val {σ = σ ↝ τ} ρ t = λ ρ′ → t (ρ′ ∘ ρ)
interleaved mutual
reify : ∀ σ → Val σ Γ → Chk σ Γ
reflect : ∀ σ → Syn σ Γ → Val σ Γ
-- base case
reify α t = emb t
reflect α t = t
-- arrow case
reify (σ ↝ τ) t = lam (reify τ (t step (reflect σ (var z))))
reflect (σ ↝ τ) t = λ ρ v → reflect τ (app (th^Syn ρ t) (reify σ v))
record Env (Γ Δ : List Ty) : Set where
field lookup : ∀ {σ} → Var σ Γ → Val σ Δ
open Env public
th^Env : Ren Δ Θ → Env Γ Δ → Env Γ Θ
lookup (th^Env ρ vs) v = th^Val ρ (lookup vs v)
placeholders : Env Γ Γ
lookup placeholders v = reflect _ (var v)
extend : Env Γ Δ → Val σ Δ → Env (σ ∷ Γ) Δ
lookup (extend ρ t) z = t
lookup (extend ρ t) (s v) = lookup ρ v
interleaved mutual
eval^Syn : Env Γ Δ → Syn σ Γ → Val σ Δ
eval^Chk : Env Γ Δ → Chk σ Γ → Val σ Δ
-- variable
eval^Syn vs (var v) = lookup vs v
-- change of direction
eval^Syn vs (cut t) = eval^Chk vs t
eval^Chk vs (emb t) = eval^Syn vs t
-- function introduction & elimination
eval^Syn vs (app f t) = eval^Syn vs f refl (eval^Chk vs t)
eval^Chk vs (lam b) = λ ρ v → eval^Chk (extend (th^Env ρ vs) v) b
interleaved mutual
norm^Syn : Syn σ Γ → Chk σ Γ
norm^Chk : Chk σ Γ → Chk σ Γ
norm^Syn t = norm^Chk (emb t)
norm^Chk t = reify _ (eval^Chk placeholders t)
|
/-
Copyright (c) 2018 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Kenny Lau, Johan Commelin, Mario Carneiro, Kevin Buzzard
-/
import group_theory.submonoid.basic
import algebra.big_operators.basic
import deprecated.group
/-!
# Unbundled submonoids (deprecated)
This file is deprecated, and is no longer imported by anything in mathlib other than other
deprecated files, and test files. You should not need to import it.
This file defines unbundled multiplicative and additive submonoids. Instead of using this file,
please use `submonoid G` and `add_submonoid A`, defined in `group_theory.submonoid.basic`.
## Main definitions
`is_add_submonoid (S : set M)` : the predicate that `S` is the underlying subset of an additive
submonoid of `M`. The bundled variant `add_submonoid M` should be used in preference to this.
`is_submonoid (S : set M)` : the predicate that `S` is the underlying subset of a submonoid
of `M`. The bundled variant `submonoid M` should be used in preference to this.
## Tags
submonoid, submonoids, is_submonoid
-/
open_locale big_operators
variables {M : Type*} [monoid M] {s : set M}
variables {A : Type*} [add_monoid A] {t : set A}
/-- `s` is an additive submonoid: a set containing 0 and closed under addition.
Note that this structure is deprecated, and the bundled variant `add_submonoid A` should be
preferred. -/
structure is_add_submonoid (s : set A) : Prop :=
(zero_mem : (0:A) ∈ s)
(add_mem {a b} : a ∈ s → b ∈ s → a + b ∈ s)
/-- `s` is a submonoid: a set containing 1 and closed under multiplication.
Note that this structure is deprecated, and the bundled variant `submonoid M` should be
preferred. -/
@[to_additive]
structure is_submonoid (s : set M) : Prop :=
(one_mem : (1:M) ∈ s)
(mul_mem {a b} : a ∈ s → b ∈ s → a * b ∈ s)
lemma additive.is_add_submonoid
{s : set M} : ∀ (is : is_submonoid s), @is_add_submonoid (additive M) _ s
| ⟨h₁, h₂⟩ := ⟨h₁, @h₂⟩
theorem additive.is_add_submonoid_iff
{s : set M} : @is_add_submonoid (additive M) _ s ↔ is_submonoid s :=
⟨λ ⟨h₁, h₂⟩, ⟨h₁, @h₂⟩, additive.is_add_submonoid⟩
lemma multiplicative.is_submonoid
{s : set A} : ∀ (is : is_add_submonoid s), @is_submonoid (multiplicative A) _ s
| ⟨h₁, h₂⟩ := ⟨h₁, @h₂⟩
theorem multiplicative.is_submonoid_iff
{s : set A} : @is_submonoid (multiplicative A) _ s ↔ is_add_submonoid s :=
⟨λ ⟨h₁, h₂⟩, ⟨h₁, @h₂⟩, multiplicative.is_submonoid⟩
/-- The intersection of two submonoids of a monoid `M` is a submonoid of `M`. -/
@[to_additive "The intersection of two `add_submonoid`s of an `add_monoid` `M` is
an `add_submonoid` of M."]
lemma is_submonoid.inter {s₁ s₂ : set M} (is₁ : is_submonoid s₁) (is₂ : is_submonoid s₂) :
is_submonoid (s₁ ∩ s₂) :=
{ one_mem := ⟨is₁.one_mem, is₂.one_mem⟩,
mul_mem := λ x y hx hy,
⟨is₁.mul_mem hx.1 hy.1, is₂.mul_mem hx.2 hy.2⟩ }
/-- The intersection of an indexed set of submonoids of a monoid `M` is a submonoid of `M`. -/
@[to_additive "The intersection of an indexed set of `add_submonoid`s of an `add_monoid` `M` is
an `add_submonoid` of `M`."]
lemma is_submonoid.Inter {ι : Sort*} {s : ι → set M} (h : ∀ y : ι, is_submonoid (s y)) :
is_submonoid (set.Inter s) :=
{ one_mem := set.mem_Inter.2 $ λ y, (h y).one_mem,
mul_mem := λ x₁ x₂ h₁ h₂, set.mem_Inter.2 $
λ y, (h y).mul_mem (set.mem_Inter.1 h₁ y) (set.mem_Inter.1 h₂ y) }
/-- The union of an indexed, directed, nonempty set of submonoids of a monoid `M` is a submonoid
of `M`. -/
@[to_additive "The union of an indexed, directed, nonempty set
of `add_submonoid`s of an `add_monoid` `M` is an `add_submonoid` of `M`. "]
lemma is_submonoid_Union_of_directed {ι : Type*} [hι : nonempty ι]
{s : ι → set M} (hs : ∀ i, is_submonoid (s i))
(directed : ∀ i j, ∃ k, s i ⊆ s k ∧ s j ⊆ s k) :
is_submonoid (⋃i, s i) :=
{ one_mem := let ⟨i⟩ := hι in set.mem_Union.2 ⟨i, (hs i).one_mem⟩,
mul_mem := λ a b ha hb,
let ⟨i, hi⟩ := set.mem_Union.1 ha in
let ⟨j, hj⟩ := set.mem_Union.1 hb in
let ⟨k, hk⟩ := directed i j in
set.mem_Union.2 ⟨k, (hs k).mul_mem (hk.1 hi) (hk.2 hj)⟩ }
section powers
/-- The set of natural number powers `1, x, x², ...` of an element `x` of a monoid. -/
@[to_additive multiples
"The set of natural number multiples `0, x, 2x, ...` of an element `x` of an `add_monoid`."]
def powers (x : M) : set M := {y | ∃ n:ℕ, x^n = y}
/-- 1 is in the set of natural number powers of an element of a monoid. -/
@[to_additive "0 is in the set of natural number multiples of an element of an `add_monoid`."]
lemma powers.one_mem {x : M} : (1 : M) ∈ powers x := ⟨0, pow_zero _⟩
/-- An element of a monoid is in the set of that element's natural number powers. -/
@[to_additive
"An element of an `add_monoid` is in the set of that element's natural number multiples."]
lemma powers.self_mem {x : M} : x ∈ powers x := ⟨1, pow_one _⟩
/-- The set of natural number powers of an element of a monoid is closed under multiplication. -/
@[to_additive
"The set of natural number multiples of an element of an `add_monoid` is closed under addition."]
lemma powers.mul_mem {x y z : M} : (y ∈ powers x) → (z ∈ powers x) → (y * z ∈ powers x) :=
λ ⟨n₁, h₁⟩ ⟨n₂, h₂⟩, ⟨n₁ + n₂, by simp only [pow_add, *]⟩
/-- The set of natural number powers of an element of a monoid `M` is a submonoid of `M`. -/
@[to_additive "The set of natural number multiples of an element of
an `add_monoid` `M` is an `add_submonoid` of `M`."]
lemma powers.is_submonoid (x : M) : is_submonoid (powers x) :=
{ one_mem := powers.one_mem,
mul_mem := λ y z, powers.mul_mem }
/-- A monoid is a submonoid of itself. -/
@[to_additive "An `add_monoid` is an `add_submonoid` of itself."]
lemma univ.is_submonoid : is_submonoid (@set.univ M) := by split; simp
/-- The preimage of a submonoid under a monoid hom is a submonoid of the domain. -/
@[to_additive "The preimage of an `add_submonoid` under an `add_monoid` hom is
an `add_submonoid` of the domain."]
lemma is_submonoid.preimage {N : Type*} [monoid N] {f : M → N} (hf : is_monoid_hom f)
{s : set N} (hs : is_submonoid s) : is_submonoid (f ⁻¹' s) :=
{ one_mem := show f 1 ∈ s, by rw is_monoid_hom.map_one hf; exact hs.one_mem,
mul_mem := λ a b (ha : f a ∈ s) (hb : f b ∈ s),
show f (a * b) ∈ s, by rw is_monoid_hom.map_mul hf; exact hs.mul_mem ha hb }
/-- The image of a submonoid under a monoid hom is a submonoid of the codomain. -/
@[to_additive "The image of an `add_submonoid` under an `add_monoid`
hom is an `add_submonoid` of the codomain."]
lemma is_submonoid.image {γ : Type*} [monoid γ] {f : M → γ} (hf : is_monoid_hom f)
{s : set M} (hs : is_submonoid s) : is_submonoid (f '' s) :=
{ one_mem := ⟨1, hs.one_mem, hf.map_one⟩,
mul_mem := λ a b ⟨x, hx⟩ ⟨y, hy⟩, ⟨x * y, hs.mul_mem hx.1 hy.1,
by rw [hf.map_mul, hx.2, hy.2]⟩ }
/-- The image of a monoid hom is a submonoid of the codomain. -/
@[to_additive "The image of an `add_monoid` hom is an `add_submonoid`
of the codomain."]
lemma range.is_submonoid {γ : Type*} [monoid γ] {f : M → γ} (hf : is_monoid_hom f) :
is_submonoid (set.range f) :=
by { rw ← set.image_univ, exact univ.is_submonoid.image hf }
/-- Submonoids are closed under natural powers. -/
@[to_additive is_add_submonoid.smul_mem
"An `add_submonoid` is closed under multiplication by naturals."]
lemma is_submonoid.pow_mem {a : M} (hs : is_submonoid s) (h : a ∈ s) : ∀ {n : ℕ}, a ^ n ∈ s
| 0 := by { rw pow_zero, exact hs.one_mem }
| (n + 1) := by { rw pow_succ, exact hs.mul_mem h is_submonoid.pow_mem }
/-- The set of natural number powers of an element of a submonoid is a subset of the submonoid. -/
@[to_additive is_add_submonoid.multiples_subset "The set of natural number multiples of an element
of an `add_submonoid` is a subset of the `add_submonoid`."]
lemma is_submonoid.power_subset {a : M} (hs : is_submonoid s) (h : a ∈ s) : powers a ⊆ s :=
assume x ⟨n, hx⟩, hx ▸ hs.pow_mem h
end powers
namespace is_submonoid
/-- The product of a list of elements of a submonoid is an element of the submonoid. -/
@[to_additive "The sum of a list of elements of an `add_submonoid` is an element of the
`add_submonoid`."]
lemma list_prod_mem (hs : is_submonoid s) : ∀{l : list M}, (∀x∈l, x ∈ s) → l.prod ∈ s
| [] h := hs.one_mem
| (a::l) h :=
suffices a * l.prod ∈ s, by simpa,
have a ∈ s ∧ (∀x∈l, x ∈ s), by simpa using h,
hs.mul_mem this.1 (list_prod_mem this.2)
/-- The product of a multiset of elements of a submonoid of a `comm_monoid` is an element of
the submonoid. -/
@[to_additive "The sum of a multiset of elements of an `add_submonoid` of an `add_comm_monoid`
is an element of the `add_submonoid`. "]
lemma multiset_prod_mem {M} [comm_monoid M] {s : set M} (hs : is_submonoid s) (m : multiset M) :
(∀a∈m, a ∈ s) → m.prod ∈ s :=
begin
refine quotient.induction_on m (assume l hl, _),
rw [multiset.quot_mk_to_coe, multiset.coe_prod],
exact list_prod_mem hs hl
end
/-- The product of elements of a submonoid of a `comm_monoid` indexed by a `finset` is an element
of the submonoid. -/
@[to_additive "The sum of elements of an `add_submonoid` of an `add_comm_monoid` indexed by
a `finset` is an element of the `add_submonoid`."]
lemma finset_prod_mem {M A} [comm_monoid M] {s : set M} (hs : is_submonoid s) (f : A → M) :
∀(t : finset A), (∀b∈t, f b ∈ s) → ∏ b in t, f b ∈ s
| ⟨m, hm⟩ _ := multiset_prod_mem hs _ (by simpa)
end is_submonoid
namespace add_monoid
/-- The inductively defined membership predicate for the submonoid generated by a subset of a
monoid. -/
inductive in_closure (s : set A) : A → Prop
| basic {a : A} : a ∈ s → in_closure a
| zero : in_closure 0
| add {a b : A} : in_closure a → in_closure b → in_closure (a + b)
end add_monoid
namespace monoid
/-- The inductively defined membership predicate for the `submonoid` generated by a subset of an
monoid. -/
@[to_additive]
inductive in_closure (s : set M) : M → Prop
| basic {a : M} : a ∈ s → in_closure a
| one : in_closure 1
| mul {a b : M} : in_closure a → in_closure b → in_closure (a * b)
/-- The inductively defined submonoid generated by a subset of a monoid. -/
@[to_additive "The inductively defined `add_submonoid` genrated by a subset of an `add_monoid`."]
def closure (s : set M) : set M := {a | in_closure s a }
@[to_additive]
lemma closure.is_submonoid (s : set M) : is_submonoid (closure s) :=
{ one_mem := in_closure.one, mul_mem := assume a b, in_closure.mul }
/-- A subset of a monoid is contained in the submonoid it generates. -/
@[to_additive "A subset of an `add_monoid` is contained in the `add_submonoid` it generates."]
theorem subset_closure {s : set M} : s ⊆ closure s :=
assume a, in_closure.basic
/-- The submonoid generated by a set is contained in any submonoid that contains the set. -/
@[to_additive "The `add_submonoid` generated by a set is contained in any `add_submonoid` that
contains the set."]
theorem closure_subset {s t : set M} (ht : is_submonoid t) (h : s ⊆ t) : closure s ⊆ t :=
assume a ha, by induction ha; simp [h _, *, is_submonoid.one_mem, is_submonoid.mul_mem]
/-- Given subsets `t` and `s` of a monoid `M`, if `s ⊆ t`, the submonoid of `M` generated by `s` is
contained in the submonoid generated by `t`. -/
@[to_additive "Given subsets `t` and `s` of an `add_monoid M`, if `s ⊆ t`, the `add_submonoid`
of `M` generated by `s` is contained in the `add_submonoid` generated by `t`."]
theorem closure_mono {s t : set M} (h : s ⊆ t) : closure s ⊆ closure t :=
closure_subset (closure.is_submonoid t) $ set.subset.trans h subset_closure
/-- The submonoid generated by an element of a monoid equals the set of natural number powers of
the element. -/
@[to_additive "The `add_submonoid` generated by an element of an `add_monoid` equals the set of
natural number multiples of the element."]
theorem closure_singleton {x : M} : closure ({x} : set M) = powers x :=
set.eq_of_subset_of_subset (closure_subset (powers.is_submonoid x) $ set.singleton_subset_iff.2 $
powers.self_mem) $ is_submonoid.power_subset (closure.is_submonoid _) $
set.singleton_subset_iff.1 $ subset_closure
/-- The image under a monoid hom of the submonoid generated by a set equals the submonoid generated
by the image of the set under the monoid hom. -/
@[to_additive "The image under an `add_monoid` hom of the `add_submonoid` generated by a set equals
the `add_submonoid` generated by the image of the set under the `add_monoid` hom."]
lemma image_closure {A : Type*} [monoid A] {f : M → A} (hf : is_monoid_hom f) (s : set M) :
f '' closure s = closure (f '' s) :=
le_antisymm
begin
rintros _ ⟨x, hx, rfl⟩,
apply in_closure.rec_on hx; intros,
{ solve_by_elim [subset_closure, set.mem_image_of_mem] },
{ rw [hf.map_one], apply is_submonoid.one_mem (closure.is_submonoid (f '' s))},
{ rw [hf.map_mul], solve_by_elim [(closure.is_submonoid _).mul_mem] }
end
(closure_subset (is_submonoid.image hf (closure.is_submonoid _)) $
set.image_subset _ subset_closure)
/-- Given an element `a` of the submonoid of a monoid `M` generated by a set `s`, there exists
a list of elements of `s` whose product is `a`. -/
@[to_additive "Given an element `a` of the `add_submonoid` of an `add_monoid M` generated by
a set `s`, there exists a list of elements of `s` whose sum is `a`."]
theorem exists_list_of_mem_closure {s : set M} {a : M} (h : a ∈ closure s) :
(∃l:list M, (∀x∈l, x ∈ s) ∧ l.prod = a) :=
begin
induction h,
case in_closure.basic : a ha { existsi ([a]), simp [ha] },
case in_closure.one { existsi ([]), simp },
case in_closure.mul : a b _ _ ha hb
{ rcases ha with ⟨la, ha, eqa⟩,
rcases hb with ⟨lb, hb, eqb⟩,
existsi (la ++ lb),
simp [eqa.symm, eqb.symm, or_imp_distrib],
exact assume a, ⟨ha a, hb a⟩ }
end
/-- Given sets `s, t` of a commutative monoid `M`, `x ∈ M` is in the submonoid of `M` generated by
`s ∪ t` iff there exists an element of the submonoid generated by `s` and an element of the
submonoid generated by `t` whose product is `x`. -/
@[to_additive "Given sets `s, t` of a commutative `add_monoid M`, `x ∈ M` is in the `add_submonoid`
of `M` generated by `s ∪ t` iff there exists an element of the `add_submonoid` generated by `s`
and an element of the `add_submonoid` generated by `t` whose sum is `x`."]
theorem mem_closure_union_iff {M : Type*} [comm_monoid M] {s t : set M} {x : M} :
x ∈ closure (s ∪ t) ↔ ∃ y ∈ closure s, ∃ z ∈ closure t, y * z = x :=
⟨λ hx, let ⟨L, HL1, HL2⟩ := exists_list_of_mem_closure hx in HL2 ▸
list.rec_on L (λ _, ⟨1, (closure.is_submonoid _).one_mem, 1,
(closure.is_submonoid _).one_mem, mul_one _⟩)
(λ hd tl ih HL1, let ⟨y, hy, z, hz, hyzx⟩ := ih (list.forall_mem_of_forall_mem_cons HL1) in
or.cases_on (HL1 hd $ list.mem_cons_self _ _)
(λ hs, ⟨hd * y, (closure.is_submonoid _).mul_mem (subset_closure hs) hy, z, hz,
by rw [mul_assoc, list.prod_cons, ← hyzx]; refl⟩)
(λ ht, ⟨y, hy, z * hd, (closure.is_submonoid _).mul_mem hz (subset_closure ht),
by rw [← mul_assoc, list.prod_cons, ← hyzx, mul_comm hd]; refl⟩)) HL1,
λ ⟨y, hy, z, hz, hyzx⟩, hyzx ▸ (closure.is_submonoid _).mul_mem
(closure_mono (set.subset_union_left _ _) hy)
(closure_mono (set.subset_union_right _ _) hz)⟩
end monoid
/-- Create a bundled submonoid from a set `s` and `[is_submonoid s]`. -/
@[to_additive "Create a bundled additive submonoid from a set `s` and `[is_add_submonoid s]`."]
def submonoid.of {s : set M} (h : is_submonoid s) : submonoid M := ⟨s, h.2, h.1⟩
@[to_additive]
lemma submonoid.is_submonoid (S : submonoid M) : is_submonoid (S : set M) := ⟨S.3, S.2⟩
|
\chapter{March}
\section{Perceptron algorithm} \index{Perceptron algorithm}
Suppose we have input data $(x_1, y_1), (x_2, y_2), ..., (x_n, y_n) \in \mathbb{R}^p \times \{-1, 1\}$,
and if the data points are separable, the perceptron algorithm works as following:
\begin{minted}[frame=lines, framesep=2mm,tabsize=4]{cpp}
w = 0
while some (x, y) is misclassified:
w = w + yx
\end{minted}
\begin{remark}
In the separable case, perceptron algorithm guarantees to converge.
\end{remark}
\myheader{Multi-class perceptron}
\begin{minted}[frame=lines, framesep=2mm,tabsize=4]{cpp}
w_1 = w_2 = ... = w_k = 0
while some (x, y) is misclassified:
for correct label y: w_y = w_y + x
for incorrect label y*: w_(y*) = w_(y*) - x
\end{minted}
\section{Kernel function} \index{Kernel function}
Following the perceptron algorithm, suppose $\phi$ is a function that maps $x$ to another feature space, such as $\phi(x) = (1, x_1, x_2, ..., x_1^2, x_2^2,..., x_1 x_2,...)$, which is a quadratic embedding.
In this case we can also run perceptron algorithm in the new feature space.
\begin{minted}[frame=lines, framesep=2mm,tabsize=4]{cpp}
w = 0
while y*(w * \phi(x)) < 0:
w = w + y\phi(x)
\end{minted}
A problem is that every time we need to calculate $\phi(x)$, which may be of high dimensions. To solve this problem, we observe that in fact we don't need to access $\phi(x)$ at all to make a decision, instead we
can write $w$ as following:
\myequ{
w = a_1 \phi(x_1) + a_2 \phi(x_2) + ... + a_n \phi(x_n)
}
then $w\cdot\phi(x)$ is a weighted sum of $\phi(x)\cdot\phi(x_i)$. In addition, we also observe that
\myequ{
\phi(x) \cdot \phi(z) = (1 + x\cdot z)^2
}
That is, we don't need to calculate $\phi(x)$.
\myheader{kernel function} From above we know that we don't care about the embedding $\phi(x)$, we only
care about the similarity between a pair of data points. Therefore, the kernel function is defined as following:
\vspace{0.5cm}
\begin{definition}[Kernel function]
A function $k$: $\mathbb{R}^p \times \mathbb{R}^p \rightarrow \mathbb{R}$ is a valid kernel if it corresponds to some embedding, that is, there exists $\phi$ defined on $\mathbb{R}^p$ such that
\myequ{
k(x,z) = \phi(x) \cdot \phi(z)
}
\end{definition}
This is equivalent to require that for any finite subset $\{x_1, x_2, ..., x_m\} \subset \mathbb{R}^p$,
the $m \times m$ similarity matrix
\myequ{
K_{ij} = k(x_i, x_j)
}
is \textit{positive semidefinite}. Proof:
\myequ{
Z^T K Z = Z^T (X^T X) Z = (XZ)^T (XZ) \geq 0
}
\myheader{RBF kernel}
RBF kernel or Gaussian kernel is defined as
\myequ{
k(x,z) = e^{-||x-z||^2 / 2\sigma^2}
}
\myheader{string kernel}
For each substring $s$, we define feature:
\myequ{
\phi_s(x) &= \# \text{ of times substring $s$ appears in $x$} \\
\phi(x) &= (\phi_s(x): \text{ all strings } s)
}
\section{$k$-means Clustering}\index{$k$-means Clustering}
\myheader{$k$-means} Minimize average squared distance between points and their nearest representatives.
The input is data points $x_1, x_2, ..., x_n$, and integer $k$, and the output is centers
$\mu_1, \mu_2, ..., \mu_k$.
\myheader{Lloyd's $k$-means algorithm}
\begin{minted}[frame=lines, framesep=2mm,tabsize=4]{cpp}
Initialize centers u_1, u_2, ... u_k in some manner.
Repeat until convergence:
assign each point to its nearest center
update each u_j to the mean of points assigned to it
\end{minted}
\myheader{How to initialize centers?} $k$-means++: start with extra centers, then prune later.
\begin{lstlisting}[
style=liststy,
]
Pick a data point x at random as the first center
Let $C = \{x\}$
Repeat until the desired number of centers is attained:
pick a data point $x$ at random from the following distributions:
$Pr(x) \propto dist(x, C)^2$, where $dist(x, C) = min_{z\in C}||x-z||$
Add $x$ to $C$
\end{lstlisting}
\myheader{Streaming and online computation} If there are too much data to fit in memory, or the data
is continuously collected, we have to update the model gradually.
\myheader{The good and the bad} Good: fast and easy, effective in quantization. Bad: geared towards data in which the clusters are spherical, and of roughly the same results.
\section{Mixtures of Gaussians} \index{Mixtures of Gaussians}
Each of $k$ clusters is specified by a Guassian distribution $P_j = N(\mu_i, \sum_i)$ and a mixing weight
$\pi_j$. The overall distribution is a mixture of all Gaussians:
\myequ{
Pr(x) = \pi_1 P_1(x) + \pi_2 P_2(x) + ... + \pi_k P_k(x)
}
We need to determine all the parameters including $\pi, \mu, \sum$. We apply \textbf{EM} algorithm to solve this problem
(see Figure~\ref{fig:em_mar}).
\begin{figure}[H]
\centering{
\includegraphics[width=0.9\textwidth]{./images/mar/gmm_em.PNG}
}
\caption{EM algorithm for GMM clustering.}
\label{fig:em_mar}
\end{figure}
\section{Hierarchical clustering} \index{Hierarchical clustering}
Clustering is of multi-scale, and often there is no single right answer. Hierarchical
clustering avoids these problems.
\begin{lstlisting}[
style=liststy,
]
Start with each point on its own
Repeat until there is just one cluster:
Merge the two clusters with the $closest$ pair of points
Discard singleton clusters
\end{lstlisting}
\myheader{Linkage method} The problem is how we measure the distance
between two cluster of points.
\begin{enumerate}
\item Single linkage: $dist(C, C') = min_{x \in C, x' \in C'} ||x - x'||$
\item Complete linkage: $dist(C, C') = max_{x \in C, x' \in C'} ||x - x'||$
\item Average linkage:
\begin{enumerate}
\item average pairwise distance between all pair of points in the two clusters
\item distance between cluster centers
\item Ward's method
\end{enumerate}
\end{enumerate}
\section{Boosting}
\myheader{Adaboost} See Figure~\ref{fig:adaboost_mar}.
\begin{figure}[H]
\centering{
\includegraphics[width=0.9\textwidth]{./images/mar/adaboost.PNG}
}
\caption{Adaboost algorithms.}
\label{fig:adaboost_mar}
\end{figure}
\myheader{Bagging} See Figure~\ref{fig:bagging_mar}
\begin{figure}[H]
\centering{
\includegraphics[width=0.9\textwidth]{./images/mar/bagging.PNG}
}
\caption{Bagging.}
\label{fig:bagging_mar}
\end{figure}
\myheader{Random forest} See Figure~\ref{fig:random_forest_mar}
\begin{figure}[H]
\centering{
\includegraphics[width=0.9\textwidth]{./images/mar/random_forest.PNG}
}
\caption{Random forests.}
\label{fig:random_forest_mar}
\end{figure}
\section{Informative projections} \index{Informative projections}
\myheader{Project to multiple directions} Suppose we want to project $x \in \mathbb{R}^p$
into the $k$-dimensional subspace spanned by $u_1, u_2, ..., u_k \in \mathbb{R}^p$,
and suppose all $u_i$ are orthonormal (each has length one, and they are perpendicular to
each other). Then the projection is:
\myequ{
(x \cdot u_1)u_1 + (x \cdot u_2)u_2 + ... + (x \cdot u_k)u_k = UU^Tx
}
\myheader{Best single direction} Suppose we want to map our data $x_1, x_2, ..., x_n \in
\mathbb{R}^p$ into just one dimension $x \mapsto u \cdot x$, what is the best direction $u$.
The best direction $u$ should be the one that maximize the variance after projection. Let $X$
be the data matrix, where each column is a data point, and $\sum$ be the covariance matrix of $X$.
Suppose the mean of $X$ is $\mu \in \mathbb{R}^p$, then
\myequ{
\mathbb{E}(u^T X) & = u^T \mathbb{E}(X) = u^T \mu \\
var(u^T X) & = \mathbb{E}(u^T X - u^T \mu) = \mathbb{E} (u^T (X - \mu) (X - \mu)^T u) \\
& = u^T \mathbb{E}(X - \mu)(X - \mu)^T u = u^T \sum u
}
\begin{remark}
$u^T\sum u$ is maximized by setting $u$ to the first \textbf{eigenvector} of $\sum$. The maximum
value is the corresponding eigenvalue.
\end{remark}
\myheader{Best $k$-dimensional projection} Let $\sum$ be the $p\times p$ covariance matrix of $X$.
and $\lambda_1 \geq \lambda_2 \geq ... \geq \lambda_p$ are the eigenvalues, and $u_1, u_2, ...
u_p$ are the corresponding eigenvectors. Then the best $k$-dimensional projection directions are
$u_1, u_2, ..., u_k$.
\myheader{Spectral decomposition} See Figure~\ref{fig:spec_decomp_mar}.
\begin{figure}[H]
\centering{
\includegraphics[width=0.9\textwidth]{./images/mar/spectral_decomp.PNG}
}
\caption{Spectral decomposition.}
\label{fig:spec_decomp_mar}
\end{figure}
\myheader{Singular value decomposition (SVD)} See Figure~\ref{fig:svd_mar}.
Where $u_i, \sigma_i, v_i$ comes from? We know that:
\begin{itemize}
\item $Mv_i = \sigma_i u_i, M^T u_i = \sigma_i v_i$
\item $M M^T u_i = \sigma_i^2 u_i, M^T M v_i = \sigma_i^2 v_i$
\end{itemize}
Therefore, $v_i$ is the eigenvectors of $M^T M$, and $u_i$ is the eigenvectors
of $M M^T$. $MM^T$ and $M^TM$ has the same eigenvalues $\sigma_i^2$. Note
that all $\sigma_i$ are \textbf{non-negative}.
\begin{figure}[H]
\centering{
\includegraphics[width=0.9\textwidth]{./images/mar/SVD.PNG}
}
\caption{Singular value decomposition.}
\label{fig:svd_mar}
\end{figure}
\section{Positive definite matrix }
\vspace{0.5cm}
\begin{definition}[Positive definite matrix]
A square $p \times p$ symmetric matrix $A$ is positive definite if for all
nonzero $x \in \mathbb{R}^p$,
\myequ{
x^T A x > 0
}
\end{definition}
\myheader{Properties of positive definite matrix}
\begin{enumerate}
\item The $r \times r$ submatrix $A_r$ (start from top left element) is also positive
semidefinite.
\item The $p$ eigenvalues of A $\lambda_1, \lambda_2, ..., \lambda_p$ are positive.
Conversely, if all the eigenvalues of a matrix $B$ are
positive, then $B$ is positive definite.
\item There exist a unique decomposition of $A = LL^T$, where $L$ is a lower triangular
matrix. This is called \textit{Cholesky Decomposition}.
\item There exists a unique decomposition of $A = VDV^T$.
\end{enumerate}
\section{Beyond projections} \index{Beyond projections}
Sometimes data in a high-dimensional space $\mathbb{R}^p$ in fact
lies close to a $k$-dimensional manifold, for $k\ll p$.
\myheader{ISOMAP algorithm}
Given data $x_1, x_2, ..., x_n$,
\begin{enumerate}
\item estimate \textit{geodesic distances} between the data points, that is, distance along the manifold.
\item embed these points in Euclidean space so as to match these distances.
\end{enumerate}
\myheader{Geodesic distances}
To estimate geodesic distances:
\begin{enumerate}
\item Construct neighborhood graph, connect nodes whenever two nodes are close together.
\item Compute distance in this graph (shortest-path algorithm).
\end{enumerate}
\myheader{Distance-preserving embeddings} Problem definition:
\begin{lstlisting}[
style=liststy,]
Input:an $n\times n$ matrix of pairwise distances $D$, where $D_ij$ is the distance between
points $i$ and $j$.
Output: an embedding $z_1, z_2, ..., z_n \in \mathbb{R}^k$ that realizes these distances as closely as
possible.
\end{lstlisting}
\myheader{Gram matrix}
Gram matrix on a set of vectors $z_1, z_2, ..., z_n$ is the matrix $B$ where
$B_{ij} = z_i \cdot z_j$.
\myheader{Classical multidimensional scaling} See Figure~\ref{fig:cms_mar}
\begin{figure}[H]
\centering{
\includegraphics[width=0.9\textwidth]{./images/mar/cms.PNG}
}
\caption{Classical multidimensional scaling.}
\label{fig:cms_mar}
\end{figure}
\section{Jacobian matrix} \index{Jacobian matrix}
In vector calculus, the Jacobian matrix is the matrix of all first-order partial derivatives of a vector-valued function.
\myequ{
J = \frac{d \mathbf{f}}{d \mathbf{x}} &= \begin{bmatrix}
\frac{\partial \mathbf{f}}{\partial x_1} &
\frac{\partial \mathbf{f}}{\partial x_2} &
\dots & \frac{\partial \mathbf{f}}{\partial x_n}
\end{bmatrix} \\
& = \begin{bmatrix}
\frac{\partial f_1}{\partial x_1} &
\dots &
\frac{\partial f_1}{\partial x_n} \\
\vdots & \ddots & \vdots \\
\frac{\partial f_m}{\partial x_1} &
\dots &
\frac{\partial f_m}{\partial x_n}
\end{bmatrix}
}
\section{Gaussian-Newton algorithm} \index{Gaussian-Newton algorithm}
The Gaussian-Newton algorithm is used to solve non-linear least square problems. It is a modification
of Newton's method for finding a minimum of a function. Unlike Newton's method, the Gaussian-Newton algorithm
can only be used to minimize a sum of squared function values, but it has the advantage that the second
derivatives, which can be challenging to compute, are not required.
Given $m$ functions $r = (r_1, r_2, \dots, r_m)$ (often called residuals) of $n$ variables $\mathbf{\beta} = (\beta_1, \dots, \beta_n)$ with $m \geq n$, the Gauss–Newton algorithm iteratively finds the value of the variables which minimizes the sum of squares:
\myequ{
S(\beta) = \sum\limits_{i=1}^{m} r_i^2(\beta)
}
Start with an initial guess $\mathbf{\beta}^{(0)}$, the method proceeds by the iterations:
\myequ{
\mathbf{\beta}^{(s+1)} = \mathbf{\beta}^{(s)} - (J_r^T J_r)^{-1} J_r^T r(\mathbf{\beta}^{(s)})
}
where
\myequ{
(J_r)_{ij} = \frac{\partial r_i(\beta^{(s)})}{\partial \beta_j}
}
If $m = n$, the iteration simplifies to
\myequ{
\mathbf{\beta}^{(s+1)} = \mathbf{\beta}^{(s)} - (J_r)^{-1} r(\mathbf{\beta}^{(s)})
}
\section{Functional programming: introductions} \index{Functional programming: introductions}
\myheader{Why FP?}
We want software to be \textit{readable, reusable, modifiable, predictable and checkable}. Functional
programming could satisfy these requirements.
There is no assignment, mutation, or loop in FP.
\section{$\lambda$-calculus}\index{$\lambda$-calculus}
Lambda calculus (also written as $\lambda$-calculus) is a formal system in mathematical logic for expressing computation
based on function abstraction and application using variable binding and substitution.
\myheader{Syntax}
Three kinds of expressions:
\begin{lstlisting}[
style=liststy,]
e ::= x
| $\lambda$ x. e
| $\lambda$ e_1 e_2
\end{lstlisting}
called \textit{variables}, \textit{functions}($\lambda$-abstraction), and \textit{application}.
Functions $\lambda x. e$ takes $x$ as an input and output $e$.
Application associates to the left: $x y z$ means $(x y) z$. However, abstraction extends as far right
as possible:
$\lambda x. x \lambda y. x y z \Rightarrow \lambda x. (x \lambda y. ((x y) z))$
Identity function: $I = \lambda x. x$
\myheader{Scope of identifier (variable)}
Scope of a variable is the part of program where the variable is \textit{accessible}.
$\lambda x. E$ binds variable $x$ in $E$:
\begin{enumerate}
\item $x$ is the newly introduced varialbe.
\item $E$ is the scope of $x$.
\item $x$ is bound in $\lambda x. E$
\end{enumerate}
$y$ is \textit{free} if it occurs not bound in $E$:
\begin{enumerate}
\item $Free(x) = \{x\}$
\item $Free(E_1 E_2) = Free(E_1) \cup Free(E_2)$
\item $Free(\lambda x. E) = Free(E) - \{x\}$
\end{enumerate}
$\alpha$-renaming:
\begin{enumerate}
\item Allows bound variable names to be changed.
\item $\lambda x. x = \lambda y. y$
\end{enumerate}
substitution:
\begin{enumerate}
\item $[E'/x]E$: use $E'$ to substitute all $x$ bounded in $E$
\item $[y (\lambda x. x)/x] \lambda y. (\lambda x. x) y x
\equiv$
$[y (\lambda v. v)/x] \lambda y. (\lambda u. u) z x $ (renaming)
$\equiv \lambda z. (\lambda u. u) z (y (\lambda v. v))$ (substitution)
\end{enumerate}
$\beta$-reduction:
\begin{enumerate}
\item $\beta$-reduction captures the idea of function application.
\item $(\lambda x. e) e' \rightarrow [e'/x]e$
\item $((\lambda n. n\times2) 7) \rightarrow 7\times 2$
\end{enumerate}
local variable:
\begin{lstlisting}[
style=liststy,]
let x = $e_1$ in $e_2$
$\equiv (\lambda x. e_2) e_1$
\end{lstlisting}
boolean:
\begin{lstlisting}[
style=liststy,]
true $\equiv$ $\lambda x. \lambda y. x$
false $\equiv$ $\lambda x. \lambda y. y$
if $E_1$ then $E_2$ else $E_3$ $\equiv$ $E_1 E_2 E_3$
if true then u else v $\rightarrow$ $(\lambda x. \lambda y. x) u\ v \rightarrow (\lambda y. u) \rightarrow u$
\end{lstlisting}
not:
\begin{lstlisting}[
style=liststy,]
function takes b:
return function x, y:
return (if b then y else x)
not $\equiv \lambda b. (\lambda x. \lambda y. \ b\ y\ x)$
not true $\rightarrow \lambda x. \lambda y. \ true \ y \ x \rightarrow \lambda x. \lambda y. \ y \rightarrow false$
\end{lstlisting}
or:
\begin{lstlisting}[
style=liststy,]
function takes $b_1, b_2$
return function takes x, y:
return (if $b_1$ then x else (if $b_2$ then x else y))
or $\equiv \lambda b_1 . \lambda b_2 . (\lambda x. \lambda y.\ b_1\ x (b_2 \ x \ y))$
\end{lstlisting}
records:
\begin{lstlisting}[
style=liststy,]
pair = function takes a bool
return the left or right element
mkpair $e_1 \ e_2 \equiv \lambda b. \ b\ e_1\ e_2$
fst p $\equiv$ p true
snd p $\equiv$ p false
\end{lstlisting}
natural numbers:
\begin{lstlisting}[
style=liststy,]
natural number: iterate a number of times over some function
$n$ = function that takes function $f$, starting value $s$
returns: $f$ applied to $s$ $n$ times
$0 \equiv \lambda f. \lambda s. \ s$
$1 \equiv \lambda f. \lambda s. \ f \ s$
$2 \equiv \lambda f. \lambda s. \ f (f \ s)$
(n f s) = apply f to s n times
\end{lstlisting}
operations on natural numbers:
\begin{lstlisting}[
style=liststy,]
iszero n $\Leftrightarrow$ n ($\lambda$ b. false) true
iszero $\Leftrightarrow$ $\lambda $ n. ($\lambda$ b. false) true
succ n $\Leftrightarrow$ $\lambda$f. $\lambda$s. f (n f s)
add a b $\Leftrightarrow$ a succ b
multi a b $\Leftrightarrow$ a (add b) 0
\end{lstlisting}
\section{Haskell: basics}\index{Haskell: basics}
Haskell is a standardized, general-purpose purely functional programming language, with non-strict semantics and strong static typing.
\myheader{GHC system}
\begin{lstlisting}[
style=liststy,
language=Haskell,]
:load foo.hs
:type expression
:info variable
\end{lstlisting}
\myheader{Basic types}
\begin{haskellcode}
32 :: Integer
4.2 :: Double
'a' :: Char
True :: Bool
-- function types
pos :: Integer -> Bool
pos x = (x > 0)
-- multiple argument function types
-- function takes args of A1, A2, A3, gives out B
A1 -> A2 -> A3 -> B
-- tuples, elements do not have to be of the same type
(A1, ..., An)
-- pattern matching extracts values from tuple
pat :: (Int, Int, Int) -> Int
pat (x, y, z) = x * (y + z)
-- Lists, elements have to be of the same type
[1, 3, 5, 7]
-- construct lists
'a' : ['b', 'c'] = ['a', 'b', 'c']
cons2 x y zs = x : y : zs
-- type
-- Not a new type, just shorthand
type XY = (Double, Double)
type Circle = (Double, Double, Double)
-- data creates new types
data CircleT = Circle (Double, Double, Double)
data Shape =
| Rectangle Side Side
| Ellipse Radius Radius
| RtTriangle Side Side
| Polygon [Vertex]
type Side = Double
type Radius = Double
type Vertex = (Double, Double)
\end{haskellcode}
\myheader{Input and output}
\begin{haskellcode}
-- action: value describing an effect on world
IO a -- type of an action that returns an a
-- takes input string, return action that writes string to stdout
putStr :: String -> IO ()
-- only one way to execute action: make it the value of name main
main :: IO ()
main = putStr "hello world\n"
-- do many actions with 'do'
do putStr "Hello"
putStr "World"
putStr "\n"
-- input
getLine :: IO String
main:: IO()
main = do putStr "What is your name?"
n <- getLine -- assignment
putStrLn ("Happy New Year " ++ n)
\end{haskellcode}
\section{Haskell: higher-order functions}\index{Haskell: higher-order functions}
In all functional languages, functions are first-class values, meaning, that they can be treated just as you would any other data. That is, you can pass functions around to in any manner that you can pass any other data around in.
\myheader{Functions are data}
\begin{haskellcode}
plus1 :: Int -> Int
plus1 x = x + 1
minus1 :: Int -> Int
minus1 x = x - 1
funp :: (Int -> Int, Int -> Int)
funp = (plus1, minus1)
\end{haskellcode}
\myheader{Take functions as input and output}
\begin{haskellcode}
-- functions as input
doTwice :: (t -> t) -> t -> t
doTwice f x = f (f x)
-- functions as output
plusn :: Int -> (Int -> Int)
plusn n = f
where f x = x + n
-- partially apply functions
plus :: Int -> Int -> Int
plus a b = a + b
plus5 :: Int -> Int
plus5 = plus 5 -- plus5 1000 outputs 1005
\end{haskellcode}
\myheader{Anonymous functions}
We will see many situations where a particular function is only used once, and hence,
there is no need to explicitly name it. Haskell provides a mechanism to create such anonymous functions.
\begin{haskellcode}
\x -> x + 1
(\x -> x + 1) 100 -- 101
doTwice (\x -> x + 1) 100 -- 102
\end{haskellcode}
\myheader{Infix operations}
Haskell allows you to use any function as an infix operator,
simply by wrapping it inside backticks.
To further improve readability, Haskell allows you to use partially applied infix operators, ie infix operators with only a single argument. These are called \textit{sections}.
\begin{haskellcode}
2 `plus` 4 -- 6
doTwice (+1) 0 -- 2
doTwice (1+) 0 -- 2
doTwice (1:) [2..5] -- [1, 1, 2, 3, 4, 5]
\end{haskellcode}
\myheader{Polymorphism}
\textit{doTwice} is polymorphic in that it works with different kinds of values,
e.g. functions that increment integers and concatenate strings. This is vital for \textit{abstraction}.
Polymorphic functions which can operate on different kinds values are often associated with polymorphic
data structures which can contain different kinds of values.
These are also represented by types containing type variables.
\begin{haskellcode}
foo :: a -> (a -> b) -> b
foo x f = f x
x |> f = f x
do1 :: (a -> b) -> a -> b
do1 f x = x |> f
do2 :: (a -> a) -> a -> a
do2 f x = x |> f |> f
\end{haskellcode}
\myheader{Bottling Computation Patterns With Polymorphic Higher-Order Functions}
\begin{haskellcode}
toUpperString :: String -> String
toUpperString [] = []
toUpperString (c:cs) = toUpper c : toUpperString cs
-- map
map :: (a -> b) -> [a] -> [b]
map f [] = []
map f (x:xs) = (f x) : (map f xs)
toUpperString = map toUpper
-- foldr
foldr op base [] = base
foldr op base (x:xs) = x `op` (foldr op base xs)
-- foldr on actions
fuseActions :: [IO ()] -> IO ()
fuseActions [] = return ()
fuseActions (a1:acts) = do a1
fuseActions acts
fuseActions :: [IO ()] -> IO ()
fuseActions = foldr (>>) (return ())
\end{haskellcode}
\section{Higher-order programming} \index{Higher-order programming}
\myheader{Recursive types}
\begin{haskellcode}
data Shape = Rectangle Double Double
| Polygon [(Double, Double)]
data IntTree = ILeaf Int
| INode IntTree IntTree
\end{haskellcode}
\myheader{Parameterized types}
\begin{haskellcode}
-- a is a type
data List a = Empty
| OneAndMore a (List a)
data Tree a = Leaf a
| Node (Tree a) (Tree a)
type IntList = List Int
type CharList = List Char
type DoubleList = List Double
\end{haskellcode}
\myheader{Kinds}
In the area of mathematical logic and computer science known as type theory,
a kind is the type of a type constructor or, less commonly, the type of a higher-order type operator.
The \textit{Tree a} corresponds to trees of values of type a.
If a is the type parameter, then what is Tree ?
A function that takes a type a as input and returns a type Tree a as output!
But wait, if List is a function then what is its type? A \textit{kind} is the type of a type.
\begin{haskellcode}
:kind Int -- *
:kind Char -- *
:kind Bool -- * -> *
:kind (->) -- * -> * -> *
\end{haskellcode}
\begin{enumerate}
\item *: the kind of all data types seen as nullary type constructors,
and also called proper types in this context.
\item * -> * : the kind of a unary type constructor, e.g. list type constructor.
\item * -> * -> * : the kind of a binary type constructor (via currying), e.g. of
a pair type constructor and also that of a function type constructor.
\item (* -> *) -> *: the kind of a higher-order type operator from unary type constructors to proper types.
\end{enumerate}
\section{Haskell: typeclass}\index{Haskell: typeclass}
We want the operator such as $+$ to work for a bunch of different data types
such as integers, doubles. So what should be the type of $+$?
\begin{haskellcode}
-- too anemic
(+) :: Integer -> Integer -> Integer
-- too aggressive, it doesn’t make sense to add two functions
(+) :: a -> a -> a
\end{haskellcode}
Haskell solves this problem with an insanely slick mechanism called typeclasses.
\myheader{Qualified types}
\begin{haskellcode}
-- truth
(+) :: (Num a) => a -> a -> a
\end{haskellcode}
We call the above a qualified type. Read it as, $+$ takes in two a values and returns an a value
for any type a that is a Num or is an instance of Num.
The name Num can be thought of as a predicate over types.
Some types satisfy the Num predicate. Examples include Integer, Double etc, and any values of those types can be passed to +.
Other types do not satisfy the predicate. Examples include Char, String, functions etc, and
so values of those types cannot be passed to +.
\myheader{Typeclass}
A \textit{typeclass} is a collection of operations (functions) that must exist for the underlying type.
A typeclass defines some behavior (like comparing for equality, comparing for ordering, enumeration) and then types that can behave in that way are made instances of that typeclass. The behavior of typeclasses is achieved by defining functions or just type declarations that we then implement. So when we say that a type is an instance of a typeclass, we mean that we can use the functions that the typeclass defines with that type.
\begin{haskellcode}
-- typeclass Eq
class Eq a where
(==) :: a -> a -> Bool
(/=) :: a -> a -> Bool
-- typeclass show
class Show a where
show :: a -> String
\end{haskellcode}
A type a can be an instance of Eq as long as there are two functions
that determine if two a values are respectively equal or disequal.
Similarly, the typeclass Show captures the requirements that make a particular datatype be viewable,
\myheader{Automatic derivation}
Haskell allows us automatically derive functions for certain key type classes, namely those in the standard library.
\begin{haskellcode}
data Showable = A' | B' | C' deriving (Eq, Show)
class (Eq a, Show a) => Num a where
(+) :: a -> a -> a
(*) :: a -> a -> a
(-) :: a -> a -> a
negate :: a -> a
abs :: a -> a
signum :: a -> a
fromInteger :: Integer -> a
-- A type T can only be deemed an instance of Num if
-- 1. The type is also an instance of Eq and Show, and
-- 2. There are functions for adding, multiplying, etc
\end{haskellcode}
\myheader{Explicit signatures}
While Haskell is pretty good about inferring types in general,
there are cases when the use of type classes requires explicit annotations.
\begin{haskellcode}
-- Read is a build typeclass, parse a string and turn it into an a
read:: (Read a) => String -> a
-- read "2" -- error: it doesn’t know what to convert the string to
-- (read "2") :: Int -- correct
-- (read "2") :: Double -- correct
\end{haskellcode}
\section{Haskell: monads}\index{Haskell: monads}
The function programming community divides into two camps:
\begin{enumerate}
\item "Pure" languages, such as Haskell, are based directly upon the
mathematical notion of a function as a mapping from arguments to
results.
\item "Impure" languages, such as ML, are based upon the extension of this notion with a range of possible effects, such as exceptions and assignments.
\end{enumerate}
Pure languages are easier to reason about and may benefit from lazy
evaluation, while impure languages may be more efficient
and can lead to shorter programs.
\subsection{Abstracting programming patterns}
Monads are an example of the idea of abstracting out a common
programming pattern as a definition.
\begin{haskellcode}
inc :: [Int] -> [Int]
inc [] = []
inc (n:ns) = n+1 : inc ns
sqr :: [Int] -> [Int]
sqr [] = []
sqr (n:ns) = n^2 : sqr ns
-- above functions have the same programming pattern,
-- namely mapping the empty list to itself, and a non-empty
-- list to some function applied to each element in the list
-- abstract this pattern gives us the map function
map :: (a -> b) -> [a] -> [b]
map f [] = []
map f (x:xs) = f x : map f xs
inc = map (+1)
sqr = map (^2)
\end{haskellcode}
\myheader{Maybe}
The \texttt{Maybe} type encapsulates an optional value.
A value of type \texttt{Maybe a} either
contains a value of type \texttt{a} (represented as \text{Just a}),
or it is empty
(represented as \texttt{Nothing}).
Using \texttt{Maybe} is a good way to deal with errors or
exceptional cases without resorting to drastic measures such as error.
\begin{haskellcode}
data Maybe a = Just a | Nothing
foo :: (a -> b) -> Maybe a -> Maybe b
foo f z = case z of
Just x -> Just (f x)
Nothing -> Nothing
\end{haskellcode}
\myheader{Functor typeclass}
Functor typeclass is basically for things that can be mapped over.
\begin{haskellcode}
class Functor f where
fmap :: (a -> b) -> f a -> f b
-- map is just a map that works on lists
instance Functor [] where
fmap = map
-- Maybe is also a functor
instance Functor Maybe where
fmap f (Just x) = Just (f x)
fmap f Nothing = Nothing
-- make Tree type instance of Functor
instance Functor Tree where
fmap f EmptyTree = EmptyTree
fmap f (Node x left right) = Node (f x)
(fmap f left) (fmap f right)
-- IO is an instance of Functor
instance Functor IO where
fmap f action = do
results <- action
return (f result)
-- <-: bind that result to a name
result <- getLine
-- return: a function that makes an I/O action that doesn't do
-- anything but only presents something as its result
\end{haskellcode}
\myheader{Generalize map to many arguments}
We can generalize map to many arguments.
\begin{haskellcode}
lift1 :: (a -> b) -> [a] -> [b]
lift2 :: (a1 -> a2 -> b) -> [a1] -> [a2] -> [b]
lift3 :: (a1 -> a2 -> a3 -> b) -> [a1] -> [a2] -> [a3] -> [b]
\end{haskellcode}
There is a typeclass called \texttt{Applicative} that corresponds to
the type constructors that you can \texttt{lift2} or \texttt{lift3} over.
\begin{haskellcode}
liftA :: Applicative t => (a -> b) -> t a -> t b
liftA2 :: Applicative t
=> (a1 -> a2 -> b)
-> t a1
-> t a2
-> t b
liftA3 :: Applicative t
=> (a1 -> a2 -> a3 -> b)
-> t a1
-> t a2
-> t a3
-> t b
\end{haskellcode}
\myheader{Sequencing operator}
Here we introduce a new sequencing operator that we write as
\texttt{>>=}, and read as \textit{then}.
\begin{haskellcode}
(>>=) :: Maybe a -> (a -> Maybe b) -> Maybe b
m >>= f = case m of
Nothing -> Nothing
Just x -> f x
-- evaluate each of the expression m1, m2, ..., mn in turn,
-- and combine their results x1, x2, ..., xn by applying
-- the function f
m1 >>= \x1 ->
m2 >>= \x2 ->
...
mn >>= \xn ->
f x1 x2 ... xn
\end{haskellcode}
\section{To add} |
-- --------------------------------------------------------------- [ Day04.idr ]
-- Module : Data.Advent.Day04
-- Description : My solution to the Day 4 puzzle of the 2016 Advent of Code.
-- Copyright : Copyright (c) 2016, Eric Bailey
-- License : MIT
-- Link : http://adventofcode.com/2016/day/4
-- --------------------------------------------------------------------- [ EOH ]
||| Day 4: Security Through Obscurity
module Data.Advent.Day04
import public Data.SortedMap
import public Lightyear
import public Lightyear.Char
import public Lightyear.Strings
import public Lightyear.StringFile
%access export
-- NOTE: This is not awesome, but the maps here will be small...
private
inc : k -> SortedMap k Nat -> SortedMap k Nat
inc k m =
case lookup k m of
Nothing => insert k 1 m
Just v => insert k (S v) $ delete k m
-- ----------------------------------------------------------------- [ Parsers ]
encryptedName : Parser (List Char)
encryptedName = concat <$> some (some letter <* token "-")
sectorId : Parser Integer
sectorId = integer
checksum : Parser String
checksum = quoted' '[' ']'
room : Parser (List Char, Integer, String)
room = liftA2 MkPair encryptedName $ liftA2 MkPair sectorId checksum
-- ------------------------------------------------------------------- [ Logic ]
implementation [day04] Ord (Char, Nat) where
compare (a,x) (b,y) = let o = compare y x in
if EQ /= o
then o
else compare a b
computeChecksum : List Char -> String
computeChecksum = pack . map Basics.fst .
take 5 . sort @{day04} . toList .
foldr inc empty
isReal : (List Char, Integer, String) -> Bool
isReal (cs, _, cksum) = cksum == computeChecksum cs
main' : Show a => Parser (List (List Char, Integer, String)) ->
(List (List Char, Integer, String) -> a) -> IO ()
main' p f =
either putStrLn (printLn . f)
!(run $ parseFile (const show) (const id) p "input/day04.txt")
-- ---------------------------------------------------------------- [ Part One ]
namespace PartOne
main : IO ()
main = main' (some room) $ foldr go 0
where
go r@(_,sid,_) sum = if isReal r then sum + sid else sum
-- ---------------------------------------------------------------- [ Part Two ]
namespace PartTwo
roomToMessage : (List Char, Integer, String) -> String
roomToMessage (cs, sid, _) =
let az = cycle ['a' .. 'z']
n = toNat sid `mod` 26 in
pack $ map (\c => index n (drop (toNat c `minus` 97) az)) cs
main : IO ()
main = main' (filter isReal <$> some room) $ \rs =>
fromMaybe (-404) $ (\(_,sid,_) => sid) <$>
find (("northpoleobjectstorage" ==) . roomToMessage) rs
-- -------------------------------------------------------------------- [ Main ]
namespace Main
main : IO ()
main = putStr "Part One: " *> PartOne.main *>
putStr "Part Two: " *> PartTwo.main
-- --------------------------------------------------------------------- [ EOF ]
|
State Before: α : Type u_1
β : Type ?u.34335
m : MeasurableSpace α
μ✝ ν✝ μ ν : Measure α
⊢ withDensity ν (rnDeriv μ ν) ≤ μ State After: case pos
α : Type u_1
β : Type ?u.34335
m : MeasurableSpace α
μ✝ ν✝ μ ν : Measure α
hl : HaveLebesgueDecomposition μ ν
⊢ withDensity ν (rnDeriv μ ν) ≤ μ
case neg
α : Type u_1
β : Type ?u.34335
m : MeasurableSpace α
μ✝ ν✝ μ ν : Measure α
hl : ¬HaveLebesgueDecomposition μ ν
⊢ withDensity ν (rnDeriv μ ν) ≤ μ Tactic: by_cases hl : HaveLebesgueDecomposition μ ν State Before: case pos
α : Type u_1
β : Type ?u.34335
m : MeasurableSpace α
μ✝ ν✝ μ ν : Measure α
hl : HaveLebesgueDecomposition μ ν
⊢ withDensity ν (rnDeriv μ ν) ≤ μ State After: case pos.intro
α : Type u_1
β : Type ?u.34335
m : MeasurableSpace α
μ✝ ν✝ μ ν : Measure α
hl : HaveLebesgueDecomposition μ ν
left✝ : singularPart μ ν ⟂ₘ ν
h : μ = singularPart μ ν + withDensity ν (rnDeriv μ ν)
⊢ withDensity ν (rnDeriv μ ν) ≤ μ Tactic: cases' (haveLebesgueDecomposition_spec μ ν).2 with _ h State Before: case pos.intro
α : Type u_1
β : Type ?u.34335
m : MeasurableSpace α
μ✝ ν✝ μ ν : Measure α
hl : HaveLebesgueDecomposition μ ν
left✝ : singularPart μ ν ⟂ₘ ν
h : μ = singularPart μ ν + withDensity ν (rnDeriv μ ν)
⊢ withDensity ν (rnDeriv μ ν) ≤ μ State After: case pos.intro
α : Type u_1
β : Type ?u.34335
m : MeasurableSpace α
μ✝ ν✝ μ ν : Measure α
hl : HaveLebesgueDecomposition μ ν
left✝ : singularPart μ ν ⟂ₘ ν
h : μ = singularPart μ ν + withDensity ν (rnDeriv μ ν)
⊢ withDensity ν (rnDeriv μ ν) ≤ singularPart μ ν + withDensity ν (rnDeriv μ ν) Tactic: conv_rhs => rw [h] State Before: case pos.intro
α : Type u_1
β : Type ?u.34335
m : MeasurableSpace α
μ✝ ν✝ μ ν : Measure α
hl : HaveLebesgueDecomposition μ ν
left✝ : singularPart μ ν ⟂ₘ ν
h : μ = singularPart μ ν + withDensity ν (rnDeriv μ ν)
⊢ withDensity ν (rnDeriv μ ν) ≤ singularPart μ ν + withDensity ν (rnDeriv μ ν) State After: no goals Tactic: exact Measure.le_add_left le_rfl State Before: case neg
α : Type u_1
β : Type ?u.34335
m : MeasurableSpace α
μ✝ ν✝ μ ν : Measure α
hl : ¬HaveLebesgueDecomposition μ ν
⊢ withDensity ν (rnDeriv μ ν) ≤ μ State After: case neg
α : Type u_1
β : Type ?u.34335
m : MeasurableSpace α
μ✝ ν✝ μ ν : Measure α
hl : ¬HaveLebesgueDecomposition μ ν
⊢ 0 ≤ μ Tactic: rw [rnDeriv, dif_neg hl, withDensity_zero] State Before: case neg
α : Type u_1
β : Type ?u.34335
m : MeasurableSpace α
μ✝ ν✝ μ ν : Measure α
hl : ¬HaveLebesgueDecomposition μ ν
⊢ 0 ≤ μ State After: no goals Tactic: exact Measure.zero_le μ |
function ConstantMagneticPotentialDifference(;name, V_m=1.0)
val = V_m
@named two_port_elementary = TwoPortElementary()
@unpack port_p, port_n = two_port_elementary
@parameters V_m
@variables Phi(t)
eqs = [
V_m ~ port_p.V_m - port_n.V_m,
Phi ~ port_p.Phi,
0 ~ port_p.Phi + port_n.Phi,
]
extend(ODESystem(eqs, t, [Phi], [V_m], systems=[port_p, port_n], defaults=Dict(V_m => val), name=name), two_port_elementary)
end
function ConstantMagneticFlux(;name, Phi=1.0)
val = Phi
@named two_port_elementary = TwoPortElementary()
@unpack port_p, port_n = two_port_elementary
@parameters Phi
@variables V_m(t)
eqs = [
V_m ~ port_p.V_m - port_n.V_m,
Phi ~ port_p.Phi,
0 ~ port_p.Phi + port_n.Phi,
]
extend(ODESystem(eqs, t, [V_m], [Phi], systems=[port_p, port_n], defaults=Dict(Phi => val), name=name), two_port_elementary)
end
|
Formal statement is: lemma LIMSEQ_linear: "X \<longlonglongrightarrow> x \<Longrightarrow> l > 0 \<Longrightarrow> (\<lambda> n. X (n * l)) \<longlonglongrightarrow> x" Informal statement is: If $X$ converges to $x$, then $X(n \cdot l)$ converges to $x$ for any $l > 0$. |
function newfp = slchangefilepart(fp, varargin)
%SLCHANGEFILEPART Changes some parts of the file path
%
% $ Syntax $
% - newfp = slchangefilepart(fp, partname1, part1, ...)
%
% $ Description $
% - newfp = slchangefilepart(fp, partname1, part1, ...) changes the
% specified part of a path to a new value to form a new path.
% Please refer to slfilepart for part names
%
% $ Remarks $
% - If you specify the name, then you should be specify title and ext.
%
% $ History $
% - Created by Dahua Lin, on Aug 12nd, 2006
%
%% Main body
if isempty(varargin)
newfp = fp;
else
opts.parent = '';
opts.name = '';
opts.title = '';
opts.ext = '';
opts = slparseprops(opts, varargin{:});
if ~isempty(opts.ext)
if opts.ext(1) ~= '.'
error('sltoolbox:invalidarg', ...
'The extension string should start with a dot . ');
end
end
[p.parent, p.title, p.ext] = fileparts(fp);
if isempty(opts.name)
p = updatefields(p, opts, {'parent', 'title', 'ext'});
newfp = fullfile(p.parent, [p.title, p.ext]);
else
if ~isempty(opts.title) || ~isempty(opts.ext)
error('sltoolbox:invalidarg', ...
'When name is specified, title and ext should not be');
end
p = updatefields(p, opts, {'parent', 'name'});
newfp = fullfile(p.parent, p.name);
end
end
%% Auxiliary functions
function S = updatefields(S, newS, fns)
nf = length(fns);
for i = 1 : nf
f = fns{i};
if ~isempty(newS.(f))
S.(f) = newS.(f);
end
end
|
lemma mult_nat_right_at_top: "c > 0 \<Longrightarrow> filterlim (\<lambda>x. x * c) at_top sequentially" for c :: nat |
Formal statement is: lemma tendsto_mult_one: fixes f g :: "_ \<Rightarrow> 'b::topological_monoid_mult" shows "(f \<longlongrightarrow> 1) F \<Longrightarrow> (g \<longlongrightarrow> 1) F \<Longrightarrow> ((\<lambda>x. f x * g x) \<longlongrightarrow> 1) F" Informal statement is: If $f$ and $g$ converge to $1$, then $f \cdot g$ converges to $1$. |
MODULE myjunk
TYPE POINT
REAL :: X, Y
CONTAINS
PROCEDURE, PASS :: LENGTH => POINT_LENGTH
END TYPE POINT
CONTAINS
REAL FUNCTION POINT_LENGTH(A,B)
CLASS(POINT), INTENT(IN) :: A, B
POINT_LENGTH = SQRT((A%X-B%X)**2+(A%Y-B%Y)**2)
END FUNCTION POINT_LENGTH
END MODULE
|
module Minecraft.Base.PreClassic.Cobblestone
import public Minecraft.Core.Block.GenericData
import public Minecraft.Core.Entity.Pickup.GenericData
import public Minecraft.Core.Item.GenericData
%default total
public export
record Block where
constructor MkBlock
{auto base : GenericBlockData}
public export
record Item where
constructor MkItem
{auto base : GenericItemData}
public export
record ItemEntity where
constructor MkItemEntity
{auto base : GenericItemEntityData Cobblestone.Item}
|
The broadcast of Sister Wives came at a time that polygamy and multiple marriages were a prevalent topic in American pop culture . Big Love , the hit HBO series about fictional Utah polygamist Bill Henrickson , his three sister wives , and their struggle to gain acceptance in society , had already been on the air for several years . In early September 2010 , the drama series Lone Star , about a con man on the verge of entering into multiple marriages , premiered on Fox but was quickly canceled after two episodes , and when Sister Wives debuted , actress Katherine Heigl was in the process of developing a film about Carolyn Jessop , a woman who fled from a polygamist sect .
|
(*
Title: Psi-calculi
Author/Maintainer: Jesper Bengtson ([email protected]), 2012
*)
theory Bisim_Struct_Cong
imports Bisim_Pres Sim_Struct_Cong Structural_Congruence
begin
context env begin
lemma bisimParComm:
fixes \<Psi> :: 'b
and P :: "('a, 'b, 'c) psi"
and Q :: "('a, 'b, 'c) psi"
shows "\<Psi> \<rhd> P \<parallel> Q \<sim> Q \<parallel> P"
proof -
let ?X = "{((\<Psi>::'b), \<lparr>\<nu>*xvec\<rparr>((P::('a, 'b, 'c) psi) \<parallel> Q), \<lparr>\<nu>*xvec\<rparr>(Q \<parallel> P)) | xvec \<Psi> P Q. xvec \<sharp>* \<Psi>}"
have "eqvt ?X"
by(force simp add: eqvt_def pt_fresh_star_bij[OF pt_name_inst, OF at_name_inst] eqvts)
have "(\<Psi>, P \<parallel> Q, Q \<parallel> P) \<in> ?X"
apply auto by(rule_tac x="[]" in exI) auto
thus ?thesis
proof(coinduct rule: bisimWeakCoinduct)
case(cStatEq \<Psi> PQ QP)
from \<open>(\<Psi>, PQ, QP) \<in> ?X\<close>
obtain xvec P Q where PFrQ: "PQ = \<lparr>\<nu>*xvec\<rparr>(P \<parallel> Q)" and QFrP: "QP = \<lparr>\<nu>*xvec\<rparr>(Q \<parallel> P)" and "xvec \<sharp>* \<Psi>"
by auto
obtain A\<^sub>P \<Psi>\<^sub>P where FrP: "extractFrame P = \<langle>A\<^sub>P, \<Psi>\<^sub>P\<rangle>" and "A\<^sub>P \<sharp>* \<Psi>" and "A\<^sub>P \<sharp>* Q"
by(rule_tac C="(\<Psi>, Q)" in freshFrame) auto
obtain A\<^sub>Q \<Psi>\<^sub>Q where FrQ: "extractFrame Q = \<langle>A\<^sub>Q, \<Psi>\<^sub>Q\<rangle>" and "A\<^sub>Q \<sharp>* \<Psi>" and "A\<^sub>Q \<sharp>* A\<^sub>P" and "A\<^sub>Q \<sharp>* \<Psi>\<^sub>P"
by(rule_tac C="(\<Psi>, A\<^sub>P, \<Psi>\<^sub>P)" in freshFrame) auto
from FrQ \<open>A\<^sub>Q \<sharp>* A\<^sub>P\<close> \<open>A\<^sub>P \<sharp>* Q\<close> have "A\<^sub>P \<sharp>* \<Psi>\<^sub>Q" by(force dest: extractFrameFreshChain)
have "\<langle>(xvec@A\<^sub>P@A\<^sub>Q), \<Psi> \<otimes> \<Psi>\<^sub>P \<otimes> \<Psi>\<^sub>Q\<rangle> \<simeq>\<^sub>F \<langle>(xvec@A\<^sub>Q@A\<^sub>P), \<Psi> \<otimes> \<Psi>\<^sub>Q \<otimes> \<Psi>\<^sub>P\<rangle>"
by(simp add: frameChainAppend)
(metis frameResChainPres frameResChainComm frameNilStatEq compositionSym Associativity Commutativity FrameStatEqTrans)
with FrP FrQ PFrQ QFrP \<open>A\<^sub>P \<sharp>* \<Psi>\<^sub>Q\<close> \<open>A\<^sub>Q \<sharp>* \<Psi>\<^sub>P\<close> \<open>A\<^sub>Q \<sharp>* A\<^sub>P\<close> \<open>xvec \<sharp>* \<Psi>\<close> \<open>A\<^sub>P \<sharp>* \<Psi>\<close> \<open>A\<^sub>Q \<sharp>* \<Psi>\<close>
show ?case by(auto simp add: frameChainAppend)
next
case(cSim \<Psi> PQ QP)
from \<open>(\<Psi>, PQ, QP) \<in> ?X\<close>
obtain xvec P Q where PFrQ: "PQ = \<lparr>\<nu>*xvec\<rparr>(P \<parallel> Q)" and QFrP: "QP = \<lparr>\<nu>*xvec\<rparr>(Q \<parallel> P)"
and "xvec \<sharp>* \<Psi>"
by auto
moreover have "\<Psi> \<rhd> \<lparr>\<nu>*xvec\<rparr>(P \<parallel> Q) \<leadsto>[?X] \<lparr>\<nu>*xvec\<rparr>(Q \<parallel> P)"
proof -
have "\<Psi> \<rhd> P \<parallel> Q \<leadsto>[?X] Q \<parallel> P"
proof -
note \<open>eqvt ?X\<close>
moreover have "\<And>\<Psi> P Q. (\<Psi>, P \<parallel> Q, Q \<parallel> P) \<in> ?X"
apply auto by(rule_tac x="[]" in exI) auto
moreover have "\<And>\<Psi> P Q xvec. \<lbrakk>(\<Psi>, P, Q) \<in> ?X; xvec \<sharp>* \<Psi>\<rbrakk> \<Longrightarrow> (\<Psi>, \<lparr>\<nu>*xvec\<rparr>P, \<lparr>\<nu>*xvec\<rparr>Q) \<in> ?X"
apply(induct xvec, auto)
by(rule_tac x="xvec@xveca" in exI) (auto simp add: resChainAppend)
ultimately show ?thesis by(rule simParComm)
qed
moreover note \<open>eqvt ?X\<close> \<open>xvec \<sharp>* \<Psi>\<close>
moreover have "\<And>\<Psi> P Q x. \<lbrakk>(\<Psi>, P, Q) \<in> ?X; x \<sharp> \<Psi>\<rbrakk> \<Longrightarrow> (\<Psi>, \<lparr>\<nu>x\<rparr>P, \<lparr>\<nu>x\<rparr>Q) \<in> ?X"
apply auto
by(rule_tac x="x#xvec" in exI) auto
ultimately show ?thesis by(rule resChainPres)
qed
ultimately show ?case by simp
next
case(cExt \<Psi> PQ QP \<Psi>')
from \<open>(\<Psi>, PQ, QP) \<in> ?X\<close>
obtain xvec P Q where PFrQ: "PQ = \<lparr>\<nu>*xvec\<rparr>(P \<parallel> Q)" and QFrP: "QP = \<lparr>\<nu>*xvec\<rparr>(Q \<parallel> P)"
and "xvec \<sharp>* \<Psi>"
by auto
obtain p where "(p \<bullet> xvec) \<sharp>* \<Psi>"
and "(p \<bullet> xvec) \<sharp>* P"
and "(p \<bullet> xvec) \<sharp>* Q"
and "(p \<bullet> xvec) \<sharp>* \<Psi>'"
and S: "(set p) \<subseteq> (set xvec) \<times> (set(p \<bullet> xvec))" and "distinctPerm p"
by(rule_tac c="(\<Psi>, P, Q, \<Psi>')" in name_list_avoiding) auto
from \<open>(p \<bullet> xvec) \<sharp>* P\<close> \<open>(p \<bullet> xvec) \<sharp>* Q\<close> S have "\<lparr>\<nu>*xvec\<rparr>(P \<parallel> Q) = \<lparr>\<nu>*(p \<bullet> xvec)\<rparr>(p \<bullet> (P \<parallel> Q))"
by(subst resChainAlpha) auto
hence PQAlpha: "\<lparr>\<nu>*xvec\<rparr>(P \<parallel> Q) = \<lparr>\<nu>*(p \<bullet> xvec)\<rparr>((p \<bullet> P) \<parallel> (p \<bullet> Q))"
by(simp add: eqvts)
from \<open>(p \<bullet> xvec) \<sharp>* P\<close> \<open>(p \<bullet> xvec) \<sharp>* Q\<close> S have "\<lparr>\<nu>*xvec\<rparr>(Q \<parallel> P) = \<lparr>\<nu>*(p \<bullet> xvec)\<rparr>(p \<bullet> (Q \<parallel> P))"
by(subst resChainAlpha) auto
hence QPAlpha: "\<lparr>\<nu>*xvec\<rparr>(Q \<parallel> P) = \<lparr>\<nu>*(p \<bullet> xvec)\<rparr>((p \<bullet> Q) \<parallel> (p \<bullet> P))"
by(simp add: eqvts)
from \<open>(p \<bullet> xvec) \<sharp>* \<Psi>\<close> \<open>(p \<bullet> xvec) \<sharp>* \<Psi>'\<close> have "(\<Psi> \<otimes> \<Psi>', \<lparr>\<nu>*(p \<bullet> xvec)\<rparr>((p \<bullet> P) \<parallel> (p \<bullet> Q)), \<lparr>\<nu>*(p \<bullet> xvec)\<rparr>((p \<bullet> Q) \<parallel> (p \<bullet> P))) \<in> ?X"
by auto
with PFrQ QFrP PQAlpha QPAlpha show ?case by simp
next
case(cSym \<Psi> PR QR)
thus ?case by blast
qed
qed
lemma bisimResComm:
fixes x :: name
and \<Psi> :: 'b
and y :: name
and P :: "('a, 'b, 'c) psi"
shows "\<Psi> \<rhd> \<lparr>\<nu>x\<rparr>(\<lparr>\<nu>y\<rparr>P) \<sim> \<lparr>\<nu>y\<rparr>(\<lparr>\<nu>x\<rparr>P)"
proof(cases "x=y")
case True
thus ?thesis by(blast intro: bisimReflexive)
next
case False
{
fix x::name and y::name and P::"('a, 'b, 'c) psi"
assume "x \<sharp> \<Psi>" and "y \<sharp> \<Psi>"
let ?X = "{((\<Psi>::'b), \<lparr>\<nu>x\<rparr>(\<lparr>\<nu>y\<rparr>(P::('a, 'b, 'c) psi)), \<lparr>\<nu>y\<rparr>(\<lparr>\<nu>x\<rparr>P)) | \<Psi> x y P. x \<sharp> \<Psi> \<and> y \<sharp> \<Psi>}"
from \<open>x \<sharp> \<Psi>\<close> \<open>y \<sharp> \<Psi>\<close> have "(\<Psi>, \<lparr>\<nu>x\<rparr>(\<lparr>\<nu>y\<rparr>P), \<lparr>\<nu>y\<rparr>(\<lparr>\<nu>x\<rparr>P)) \<in> ?X" by auto
hence "\<Psi> \<rhd> \<lparr>\<nu>x\<rparr>(\<lparr>\<nu>y\<rparr>P) \<sim> \<lparr>\<nu>y\<rparr>(\<lparr>\<nu>x\<rparr>P)"
proof(coinduct rule: bisimCoinduct)
case(cStatEq \<Psi> xyP yxP)
from \<open>(\<Psi>, xyP, yxP) \<in> ?X\<close> obtain x y P where "x \<sharp> \<Psi>" and "y \<sharp> \<Psi>" and "xyP = \<lparr>\<nu>x\<rparr>(\<lparr>\<nu>y\<rparr>P)" and "yxP = \<lparr>\<nu>y\<rparr>(\<lparr>\<nu>x\<rparr>P)" by auto
moreover obtain A\<^sub>P \<Psi>\<^sub>P where "extractFrame P = \<langle>A\<^sub>P, \<Psi>\<^sub>P\<rangle>" and "A\<^sub>P \<sharp>* \<Psi>" and "x \<sharp> A\<^sub>P" and "y \<sharp> A\<^sub>P"
by(rule_tac C="(x, y, \<Psi>)" in freshFrame) auto
ultimately show ?case by(force intro: frameResComm FrameStatEqTrans)
next
case(cSim \<Psi> xyP yxP)
from \<open>(\<Psi>, xyP, yxP) \<in> ?X\<close> obtain x y P where "x \<sharp> \<Psi>" and "y \<sharp> \<Psi>" and "xyP = \<lparr>\<nu>x\<rparr>(\<lparr>\<nu>y\<rparr>P)" and "yxP = \<lparr>\<nu>y\<rparr>(\<lparr>\<nu>x\<rparr>P)" by auto
note \<open>x \<sharp> \<Psi>\<close> \<open>y \<sharp> \<Psi>\<close>
moreover have "eqvt ?X" by(force simp add: eqvt_def pt_fresh_bij[OF pt_name_inst, OF at_name_inst])
hence "eqvt(?X \<union> bisim)" by auto
moreover have "\<And>\<Psi> P. (\<Psi>, P, P) \<in> ?X \<union> bisim" by(blast intro: bisimReflexive)
moreover have "\<And>\<Psi> x y P. \<lbrakk>x \<sharp> \<Psi>; y \<sharp> \<Psi>\<rbrakk> \<Longrightarrow> (\<Psi>, \<lparr>\<nu>x\<rparr>(\<lparr>\<nu>y\<rparr>P), \<lparr>\<nu>y\<rparr>(\<lparr>\<nu>x\<rparr>P)) \<in> ?X \<union> bisim" by auto
ultimately have "\<Psi> \<rhd> \<lparr>\<nu>x\<rparr>(\<lparr>\<nu>y\<rparr>P) \<leadsto>[(?X \<union> bisim)] \<lparr>\<nu>y\<rparr>(\<lparr>\<nu>x\<rparr>P)" by(rule resComm)
with \<open>xyP = \<lparr>\<nu>x\<rparr>(\<lparr>\<nu>y\<rparr>P)\<close> \<open>yxP = \<lparr>\<nu>y\<rparr>(\<lparr>\<nu>x\<rparr>P)\<close> show ?case
by simp
next
case(cExt \<Psi> xyP yxP \<Psi>')
from \<open>(\<Psi>, xyP, yxP) \<in> ?X\<close> obtain x y P where "x \<sharp> \<Psi>" and "y \<sharp> \<Psi>" and xyPeq: "xyP = \<lparr>\<nu>x\<rparr>(\<lparr>\<nu>y\<rparr>P)" and yxPeq: "yxP = \<lparr>\<nu>y\<rparr>(\<lparr>\<nu>x\<rparr>P)" by auto
show ?case
proof(case_tac "x=y")
assume "x = y"
with xyPeq yxPeq show ?case
by(blast intro: bisimReflexive)
next
assume "x \<noteq> y"
obtain x' where "x' \<sharp> \<Psi>" and "x' \<sharp> \<Psi>'" and "x' \<noteq> x" and "x' \<noteq> y" and "x' \<sharp> P" by(generate_fresh "name") (auto simp add: fresh_prod)
obtain y' where "y' \<sharp> \<Psi>" and "y' \<sharp> \<Psi>'" and "y' \<noteq> x" and "x' \<noteq> y'" and "y' \<noteq> y" and "y' \<sharp> P" by(generate_fresh "name") (auto simp add: fresh_prod)
with xyPeq \<open>y' \<sharp> P\<close> \<open>x' \<sharp> P\<close> \<open>x \<noteq> y\<close> \<open>x' \<noteq> y\<close> \<open>y' \<noteq> x\<close> have "\<lparr>\<nu>x\<rparr>(\<lparr>\<nu>y\<rparr>P) = \<lparr>\<nu>x'\<rparr>(\<lparr>\<nu>y'\<rparr>([(x, x')] \<bullet> [(y, y')] \<bullet> P))"
apply(subst alphaRes[of x']) apply(simp add: abs_fresh) by(subst alphaRes[of y' _ y]) (auto simp add: eqvts calc_atm)
moreover with yxPeq \<open>y' \<sharp> P\<close> \<open>x' \<sharp> P\<close> \<open>x \<noteq> y\<close> \<open>x' \<noteq> y\<close> \<open>y' \<noteq> x\<close> \<open>x' \<noteq> y'\<close> have "\<lparr>\<nu>y\<rparr>(\<lparr>\<nu>x\<rparr>P) = \<lparr>\<nu>y'\<rparr>(\<lparr>\<nu>x'\<rparr>([(y, y')] \<bullet> [(x, x')] \<bullet> P))"
apply(subst alphaRes[of y']) apply(simp add: abs_fresh) by(subst alphaRes[of x' _ x]) (auto simp add: eqvts calc_atm)
with \<open>x \<noteq> y\<close> \<open>x' \<noteq> y\<close> \<open>y' \<noteq> y\<close> \<open>x' \<noteq> x\<close> \<open>y' \<noteq> x\<close> \<open>x' \<noteq> y'\<close> have "\<lparr>\<nu>y\<rparr>(\<lparr>\<nu>x\<rparr>P) = \<lparr>\<nu>y'\<rparr>(\<lparr>\<nu>x'\<rparr>([(x, x')] \<bullet> [(y, y')] \<bullet> P))"
by(subst perm_compose) (simp add: calc_atm)
moreover from \<open>x' \<sharp> \<Psi>\<close> \<open>x' \<sharp> \<Psi>'\<close> \<open>y' \<sharp> \<Psi>\<close> \<open>y' \<sharp> \<Psi>'\<close> have "(\<Psi> \<otimes> \<Psi>', \<lparr>\<nu>x'\<rparr>(\<lparr>\<nu>y'\<rparr>([(x, x')] \<bullet> [(y, y')] \<bullet> P)), \<lparr>\<nu>y'\<rparr>(\<lparr>\<nu>x'\<rparr>([(x, x')] \<bullet> [(y, y')] \<bullet> P))) \<in> ?X"
by auto
ultimately show ?case using xyPeq yxPeq by simp
qed
next
case(cSym \<Psi> xyP yxP)
thus ?case by auto
qed
}
moreover obtain x'::name where "x' \<sharp> \<Psi>" and "x' \<sharp> P" and "x' \<noteq> x" and "x' \<noteq> y"
by(generate_fresh "name") auto
moreover obtain y'::name where "y' \<sharp> \<Psi>" and "y' \<sharp> P" and "y' \<noteq> x" and "y' \<noteq> y" and "y' \<noteq> x'"
by(generate_fresh "name") auto
ultimately have "\<Psi> \<rhd> \<lparr>\<nu>x'\<rparr>(\<lparr>\<nu>y'\<rparr>([(y, y'), (x, x')] \<bullet> P)) \<sim> \<lparr>\<nu>y'\<rparr>(\<lparr>\<nu>x'\<rparr>([(y, y'), (x, x')] \<bullet> P))" by auto
thus ?thesis using \<open>x' \<sharp> P\<close> \<open>x' \<noteq> x\<close> \<open>x' \<noteq> y\<close> \<open>y' \<sharp> P\<close> \<open>y' \<noteq> x\<close> \<open>y' \<noteq> y\<close> \<open>y' \<noteq> x'\<close> \<open>x \<noteq> y\<close>
apply(subst alphaRes[where x=x and y=x' and P=P], auto)
apply(subst alphaRes[where x=y and y=y' and P=P], auto)
apply(subst alphaRes[where x=x and y=x' and P="\<lparr>\<nu>y'\<rparr>([(y, y')] \<bullet> P)"], auto simp add: abs_fresh fresh_left)
apply(subst alphaRes[where x=y and y=y' and P="\<lparr>\<nu>x'\<rparr>([(x, x')] \<bullet> P)"], auto simp add: abs_fresh fresh_left)
by(subst perm_compose) (simp add: eqvts calc_atm)
qed
lemma bisimResComm':
fixes x :: name
and \<Psi> :: 'b
and xvec :: "name list"
and P :: "('a, 'b, 'c) psi"
assumes "x \<sharp> \<Psi>"
and "xvec \<sharp>* \<Psi>"
shows "\<Psi> \<rhd> \<lparr>\<nu>x\<rparr>(\<lparr>\<nu>*xvec\<rparr>P) \<sim> \<lparr>\<nu>*xvec\<rparr>(\<lparr>\<nu>x\<rparr>P)"
using assms
by(induct xvec) (auto intro: bisimResComm bisimReflexive bisimResPres bisimTransitive)
lemma bisimScopeExt:
fixes x :: name
and \<Psi> :: 'b
and P :: "('a, 'b, 'c) psi"
and Q :: "('a, 'b, 'c) psi"
assumes "x \<sharp> P"
shows "\<Psi> \<rhd> \<lparr>\<nu>x\<rparr>(P \<parallel> Q) \<sim> P \<parallel> \<lparr>\<nu>x\<rparr>Q"
proof -
{
fix x::name and Q :: "('a, 'b, 'c) psi"
assume "x \<sharp> \<Psi>" and "x \<sharp> P"
let ?X1 = "{((\<Psi>::'b), \<lparr>\<nu>*xvec\<rparr>(\<lparr>\<nu>x\<rparr>((P::('a, 'b, 'c) psi) \<parallel> Q)), \<lparr>\<nu>*xvec\<rparr>(P \<parallel> \<lparr>\<nu>x\<rparr>Q)) | \<Psi> xvec x P Q. x \<sharp> \<Psi> \<and> x \<sharp> P \<and> xvec \<sharp>* \<Psi>}"
let ?X2 = "{((\<Psi>::'b), \<lparr>\<nu>*xvec\<rparr>((P::('a, 'b, 'c) psi) \<parallel> \<lparr>\<nu>x\<rparr>Q), \<lparr>\<nu>*xvec\<rparr>(\<lparr>\<nu>x\<rparr>(P \<parallel> Q))) | \<Psi> xvec x P Q. x \<sharp> \<Psi> \<and> x \<sharp> P \<and> xvec \<sharp>* \<Psi>}"
let ?X = "?X1 \<union> ?X2"
from \<open>x \<sharp> \<Psi>\<close> \<open>x \<sharp> P\<close> have "(\<Psi>, \<lparr>\<nu>x\<rparr>(P \<parallel> Q), P \<parallel> \<lparr>\<nu>x\<rparr>Q) \<in> ?X"
by(auto, rule_tac x="[]" in exI) (auto simp add: fresh_list_nil)
moreover have "eqvt ?X"
by(rule eqvtUnion)
(fastforce simp add: eqvt_def eqvts pt_fresh_star_bij[OF pt_name_inst, OF at_name_inst] pt_fresh_bij[OF pt_name_inst, OF at_name_inst])+
ultimately have "\<Psi> \<rhd> \<lparr>\<nu>x\<rparr>(P \<parallel> Q) \<sim> P \<parallel> \<lparr>\<nu>x\<rparr>Q"
proof(coinduct rule: transitiveCoinduct)
case(cStatEq \<Psi> R T)
show ?case
proof(case_tac "(\<Psi>, R, T) \<in> ?X1")
assume "(\<Psi>, R, T) \<in> ?X1"
then obtain xvec x P Q where "R = \<lparr>\<nu>*xvec\<rparr>(\<lparr>\<nu>x\<rparr>(P \<parallel> Q))" and "T = \<lparr>\<nu>*xvec\<rparr>(P \<parallel> \<lparr>\<nu>x\<rparr>Q)" and "xvec \<sharp>* \<Psi>" and "x \<sharp> P" and "x \<sharp> \<Psi>"
by auto
moreover obtain A\<^sub>P \<Psi>\<^sub>P where FrP: "extractFrame P = \<langle>A\<^sub>P, \<Psi>\<^sub>P\<rangle>" and "A\<^sub>P \<sharp>* \<Psi>" and "x \<sharp> A\<^sub>P" and "A\<^sub>P \<sharp>* Q"
by(rule_tac C="(\<Psi>, x, Q)" in freshFrame) auto
moreover obtain A\<^sub>Q \<Psi>\<^sub>Q where FrQ: "extractFrame Q = \<langle>A\<^sub>Q, \<Psi>\<^sub>Q\<rangle>" and "A\<^sub>Q \<sharp>* \<Psi>" and "x \<sharp> A\<^sub>Q" and "A\<^sub>Q \<sharp>* A\<^sub>P" and "A\<^sub>Q \<sharp>* \<Psi>\<^sub>P"
by(rule_tac C="(\<Psi>, x, A\<^sub>P, \<Psi>\<^sub>P)" in freshFrame) auto
moreover from FrQ \<open>A\<^sub>P \<sharp>* Q\<close> \<open>A\<^sub>Q \<sharp>* A\<^sub>P\<close> have "A\<^sub>P \<sharp>* \<Psi>\<^sub>Q"
by(drule_tac extractFrameFreshChain) auto
moreover from \<open>x \<sharp> P\<close> \<open>x \<sharp> A\<^sub>P\<close> FrP have "x \<sharp> \<Psi>\<^sub>P" by(drule_tac extractFrameFresh) auto
ultimately show ?case
by(force simp add: frameChainAppend intro: frameResComm' FrameStatEqTrans frameResChainPres)
next
assume "(\<Psi>, R, T) \<notin> ?X1"
with \<open>(\<Psi>, R, T) \<in> ?X\<close> have "(\<Psi>, R, T) \<in> ?X2" by blast
then obtain xvec x P Q where "T = \<lparr>\<nu>*xvec\<rparr>(\<lparr>\<nu>x\<rparr>(P \<parallel> Q))" and "R = \<lparr>\<nu>*xvec\<rparr>(P \<parallel> \<lparr>\<nu>x\<rparr>Q)" and "xvec \<sharp>* \<Psi>" and "x \<sharp> P" and "x \<sharp> \<Psi>"
by auto
moreover obtain A\<^sub>P \<Psi>\<^sub>P where FrP: "extractFrame P = \<langle>A\<^sub>P, \<Psi>\<^sub>P\<rangle>" and "A\<^sub>P \<sharp>* \<Psi>" and "x \<sharp> A\<^sub>P" and "A\<^sub>P \<sharp>* Q"
by(rule_tac C="(\<Psi>, x, Q)" in freshFrame) auto
moreover obtain A\<^sub>Q \<Psi>\<^sub>Q where FrQ: "extractFrame Q = \<langle>A\<^sub>Q, \<Psi>\<^sub>Q\<rangle>" and "A\<^sub>Q \<sharp>* \<Psi>" and "x \<sharp> A\<^sub>Q" and "A\<^sub>Q \<sharp>* A\<^sub>P" and "A\<^sub>Q \<sharp>* \<Psi>\<^sub>P"
by(rule_tac C="(\<Psi>, x, A\<^sub>P, \<Psi>\<^sub>P)" in freshFrame) auto
moreover from FrQ \<open>A\<^sub>P \<sharp>* Q\<close> \<open>A\<^sub>Q \<sharp>* A\<^sub>P\<close> have "A\<^sub>P \<sharp>* \<Psi>\<^sub>Q"
by(drule_tac extractFrameFreshChain) auto
moreover from \<open>x \<sharp> P\<close> \<open>x \<sharp> A\<^sub>P\<close> FrP have "x \<sharp> \<Psi>\<^sub>P" by(drule_tac extractFrameFresh) auto
ultimately show ?case
apply auto
by(force simp add: frameChainAppend intro: frameResComm' FrameStatEqTrans frameResChainPres FrameStatEqSym)
qed
next
case(cSim \<Psi> R T)
let ?Y = "{(\<Psi>, P, Q) | \<Psi> P P' Q' Q. \<Psi> \<rhd> P \<sim> P' \<and> ((\<Psi>, P', Q') \<in> ?X \<or> \<Psi> \<rhd> P' \<sim> Q') \<and> \<Psi> \<rhd> Q' \<sim> Q}"
from \<open>eqvt ?X\<close> have "eqvt ?Y" by blast
have C1: "\<And>\<Psi> R T y. \<lbrakk>(\<Psi>, R, T) \<in> ?Y; (y::name) \<sharp> \<Psi>\<rbrakk> \<Longrightarrow> (\<Psi>, \<lparr>\<nu>y\<rparr>R, \<lparr>\<nu>y\<rparr>T) \<in> ?Y"
proof -
fix \<Psi> R T y
assume "(\<Psi>, R, T) \<in> ?Y"
then obtain R' T' where "\<Psi> \<rhd> R \<sim> R'" and "(\<Psi>, R', T') \<in> (?X \<union> bisim)" and "\<Psi> \<rhd> T' \<sim> T" by fastforce
assume "(y::name) \<sharp> \<Psi>"
show "(\<Psi>, \<lparr>\<nu>y\<rparr>R, \<lparr>\<nu>y\<rparr>T) \<in> ?Y"
proof(case_tac "(\<Psi>, R', T') \<in> ?X")
assume "(\<Psi>, R', T') \<in> ?X"
show ?thesis
proof(case_tac "(\<Psi>, R', T') \<in> ?X1")
assume "(\<Psi>, R', T') \<in> ?X1"
then obtain xvec x P Q where R'eq: "R' = \<lparr>\<nu>*xvec\<rparr>(\<lparr>\<nu>x\<rparr>(P \<parallel> Q))" and T'eq: "T' = \<lparr>\<nu>*xvec\<rparr>(P \<parallel> \<lparr>\<nu>x\<rparr>Q)"
and "xvec \<sharp>* \<Psi>" and "x \<sharp> P" and "x \<sharp> \<Psi>"
by auto
from \<open>\<Psi> \<rhd> R \<sim> R'\<close> \<open>y \<sharp> \<Psi>\<close> have "\<Psi> \<rhd> \<lparr>\<nu>y\<rparr>R \<sim> \<lparr>\<nu>y\<rparr>R'" by(rule bisimResPres)
moreover from \<open>xvec \<sharp>* \<Psi>\<close> \<open>y \<sharp> \<Psi>\<close> \<open>x \<sharp> P\<close> \<open>x \<sharp> \<Psi>\<close> have "(\<Psi>, \<lparr>\<nu>*(y#xvec)\<rparr>\<lparr>\<nu>x\<rparr>(P \<parallel> Q), \<lparr>\<nu>*(y#xvec)\<rparr>(P \<parallel> \<lparr>\<nu>x\<rparr>Q)) \<in> ?X1"
by(force simp del: resChain.simps)
with R'eq T'eq have "(\<Psi>, \<lparr>\<nu>y\<rparr>R', \<lparr>\<nu>y\<rparr>T') \<in> ?X \<union> bisim" by simp
moreover from \<open>\<Psi> \<rhd> T' \<sim> T\<close> \<open>y \<sharp> \<Psi>\<close> have "\<Psi> \<rhd> \<lparr>\<nu>y\<rparr>T' \<sim> \<lparr>\<nu>y\<rparr>T" by(rule bisimResPres)
ultimately show ?thesis by blast
next
assume "(\<Psi>, R', T') \<notin> ?X1"
with \<open>(\<Psi>, R', T') \<in> ?X\<close> have "(\<Psi>, R', T') \<in> ?X2" by blast
then obtain xvec x P Q where T'eq: "T' = \<lparr>\<nu>*xvec\<rparr>(\<lparr>\<nu>x\<rparr>(P \<parallel> Q))" and R'eq: "R' = \<lparr>\<nu>*xvec\<rparr>(P \<parallel> \<lparr>\<nu>x\<rparr>Q)" and "xvec \<sharp>* \<Psi>" and "x \<sharp> P" and "x \<sharp> \<Psi>"
by auto
from \<open>\<Psi> \<rhd> R \<sim> R'\<close> \<open>y \<sharp> \<Psi>\<close> have "\<Psi> \<rhd> \<lparr>\<nu>y\<rparr>R \<sim> \<lparr>\<nu>y\<rparr>R'" by(rule bisimResPres)
moreover from \<open>xvec \<sharp>* \<Psi>\<close> \<open>y \<sharp> \<Psi>\<close> \<open>x \<sharp> P\<close> \<open>x \<sharp> \<Psi>\<close> have "(\<Psi>, \<lparr>\<nu>*(y#xvec)\<rparr>(P \<parallel> \<lparr>\<nu>x\<rparr>Q), \<lparr>\<nu>*(y#xvec)\<rparr>(\<lparr>\<nu>x\<rparr>(P \<parallel> Q))) \<in> ?X2"
by(force simp del: resChain.simps)
with R'eq T'eq have "(\<Psi>, \<lparr>\<nu>y\<rparr>R', \<lparr>\<nu>y\<rparr>T') \<in> ?X \<union> bisim" by simp
moreover from \<open>\<Psi> \<rhd> T' \<sim> T\<close> \<open>y \<sharp> \<Psi>\<close> have "\<Psi> \<rhd> \<lparr>\<nu>y\<rparr>T' \<sim> \<lparr>\<nu>y\<rparr>T" by(rule bisimResPres)
ultimately show ?thesis by blast
qed
next
assume "(\<Psi>, R', T') \<notin> ?X"
with \<open>(\<Psi>, R', T') \<in> ?X \<union> bisim\<close> have "\<Psi> \<rhd> R' \<sim> T'" by blast
with \<open>\<Psi> \<rhd> R \<sim> R'\<close> \<open>\<Psi> \<rhd> T' \<sim> T\<close> \<open>y \<sharp> \<Psi>\<close> show ?thesis
by(blast dest: bisimResPres)
qed
qed
show ?case
proof(case_tac "(\<Psi>, R, T) \<in> ?X1")
assume "(\<Psi>, R, T) \<in> ?X1"
then obtain xvec x P Q where Req: "R = \<lparr>\<nu>*xvec\<rparr>(\<lparr>\<nu>x\<rparr>(P \<parallel> Q))" and Teq: "T = \<lparr>\<nu>*xvec\<rparr>(P \<parallel> \<lparr>\<nu>x\<rparr>Q)" and "xvec \<sharp>* \<Psi>" and "x \<sharp> P" and "x \<sharp> \<Psi>"
by auto
have "\<Psi> \<rhd> \<lparr>\<nu>*xvec\<rparr>(\<lparr>\<nu>x\<rparr>(P \<parallel> Q)) \<leadsto>[?Y] \<lparr>\<nu>*xvec\<rparr>(P \<parallel> \<lparr>\<nu>x\<rparr>Q)"
proof -
have "\<Psi> \<rhd> \<lparr>\<nu>x\<rparr>(P \<parallel> Q) \<leadsto>[?Y] P \<parallel> \<lparr>\<nu>x\<rparr>Q"
proof -
note \<open>x \<sharp> P\<close> \<open>x \<sharp> \<Psi>\<close> \<open>eqvt ?Y\<close>
moreover have "\<And>\<Psi> P. (\<Psi>, P, P) \<in> ?Y" by(blast intro: bisimReflexive)
moreover have "\<And>x \<Psi> P Q xvec. \<lbrakk>x \<sharp> \<Psi>; x \<sharp> P; xvec \<sharp>* \<Psi>\<rbrakk> \<Longrightarrow> (\<Psi>, \<lparr>\<nu>x\<rparr>(\<lparr>\<nu>*xvec\<rparr>(P \<parallel> Q)), \<lparr>\<nu>*xvec\<rparr>(P \<parallel> \<lparr>\<nu>x\<rparr>Q)) \<in> ?Y"
proof -
fix x \<Psi> P Q xvec
assume "(x::name) \<sharp> (\<Psi>::'b)" and "x \<sharp> (P::('a, 'b, 'c) psi)" and "(xvec::name list) \<sharp>* \<Psi>"
from \<open>x \<sharp> \<Psi>\<close> \<open>xvec \<sharp>* \<Psi>\<close> have "\<Psi> \<rhd> \<lparr>\<nu>x\<rparr>(\<lparr>\<nu>*xvec\<rparr>(P \<parallel> Q)) \<sim> \<lparr>\<nu>*xvec\<rparr>(\<lparr>\<nu>x\<rparr>(P \<parallel> Q))"
by(rule bisimResComm')
moreover from \<open>xvec \<sharp>* \<Psi>\<close> \<open>x \<sharp> \<Psi>\<close> \<open>x \<sharp> P\<close> have "(\<Psi>, \<lparr>\<nu>*xvec\<rparr>(\<lparr>\<nu>x\<rparr>(P \<parallel> Q)), \<lparr>\<nu>*xvec\<rparr>(P \<parallel> \<lparr>\<nu>x\<rparr>Q)) \<in> ?X \<union> bisim"
by blast
ultimately show "(\<Psi>, \<lparr>\<nu>x\<rparr>(\<lparr>\<nu>*xvec\<rparr>(P \<parallel> Q)), \<lparr>\<nu>*xvec\<rparr>(P \<parallel> \<lparr>\<nu>x\<rparr>Q)) \<in> ?Y"
by(blast intro: bisimReflexive)
qed
moreover have "\<And>\<Psi> xvec P x. \<lbrakk>x \<sharp> \<Psi>; xvec \<sharp>* \<Psi>\<rbrakk> \<Longrightarrow> (\<Psi>, \<lparr>\<nu>x\<rparr>(\<lparr>\<nu>*xvec\<rparr>P), \<lparr>\<nu>*xvec\<rparr>(\<lparr>\<nu>x\<rparr>P)) \<in> ?Y"
by(blast intro: bisimResComm' bisimReflexive)
ultimately show ?thesis by(rule scopeExtLeft)
qed
thus ?thesis using \<open>eqvt ?Y\<close> \<open>xvec \<sharp>* \<Psi>\<close> C1
by(rule resChainPres)
qed
with Req Teq show ?case by simp
next
assume "(\<Psi>, R, T) \<notin> ?X1"
with \<open>(\<Psi>, R, T) \<in> ?X\<close> have "(\<Psi>, R, T) \<in> ?X2" by blast
then obtain xvec x P Q where Teq: "T = \<lparr>\<nu>*xvec\<rparr>(\<lparr>\<nu>x\<rparr>(P \<parallel> Q))" and Req: "R = \<lparr>\<nu>*xvec\<rparr>(P \<parallel> \<lparr>\<nu>x\<rparr>Q)" and "xvec \<sharp>* \<Psi>" and "x \<sharp> P" and "x \<sharp> \<Psi>"
by auto
have "\<Psi> \<rhd> \<lparr>\<nu>*xvec\<rparr>(P \<parallel> \<lparr>\<nu>x\<rparr>Q) \<leadsto>[?Y] \<lparr>\<nu>*xvec\<rparr>(\<lparr>\<nu>x\<rparr>(P \<parallel> Q))"
proof -
have "\<Psi> \<rhd> P \<parallel> \<lparr>\<nu>x\<rparr>Q \<leadsto>[?Y] \<lparr>\<nu>x\<rparr>(P \<parallel> Q)"
proof -
note \<open>x \<sharp> P\<close> \<open>x \<sharp> \<Psi>\<close> \<open>eqvt ?Y\<close>
moreover have "\<And>\<Psi> P. (\<Psi>, P, P) \<in> ?Y" by(blast intro: bisimReflexive)
moreover have "\<And>x \<Psi> P Q xvec. \<lbrakk>x \<sharp> \<Psi>; x \<sharp> P; xvec \<sharp>* \<Psi>\<rbrakk> \<Longrightarrow> (\<Psi>, \<lparr>\<nu>*xvec\<rparr>(P \<parallel> \<lparr>\<nu>x\<rparr>Q), \<lparr>\<nu>x\<rparr>(\<lparr>\<nu>*xvec\<rparr>(P \<parallel> Q))) \<in> ?Y"
proof -
fix x \<Psi> P Q xvec
assume "(x::name) \<sharp> (\<Psi>::'b)" and "x \<sharp> (P::('a, 'b, 'c) psi)" and "(xvec::name list) \<sharp>* \<Psi>"
from \<open>xvec \<sharp>* \<Psi>\<close> \<open>x \<sharp> \<Psi>\<close> \<open>x \<sharp> P\<close> have "(\<Psi>, \<lparr>\<nu>*xvec\<rparr>(P \<parallel> \<lparr>\<nu>x\<rparr>Q), \<lparr>\<nu>*xvec\<rparr>(\<lparr>\<nu>x\<rparr>(P \<parallel> Q))) \<in> ?X \<union> bisim"
by blast
moreover from \<open>x \<sharp> \<Psi>\<close> \<open>xvec \<sharp>* \<Psi>\<close> have "\<Psi> \<rhd> \<lparr>\<nu>*xvec\<rparr>(\<lparr>\<nu>x\<rparr>(P \<parallel> Q)) \<sim> \<lparr>\<nu>x\<rparr>(\<lparr>\<nu>*xvec\<rparr>(P \<parallel> Q))"
by(blast intro: bisimResComm' bisimE)
ultimately show "(\<Psi>, \<lparr>\<nu>*xvec\<rparr>(P \<parallel> \<lparr>\<nu>x\<rparr>Q), \<lparr>\<nu>x\<rparr>(\<lparr>\<nu>*xvec\<rparr>(P \<parallel> Q))) \<in> ?Y"
by(blast intro: bisimReflexive)
qed
ultimately show ?thesis by(rule scopeExtRight)
qed
thus ?thesis using \<open>eqvt ?Y\<close> \<open>xvec \<sharp>* \<Psi>\<close> C1
by(rule resChainPres)
qed
with Req Teq show ?case by simp
qed
next
case(cExt \<Psi> R T \<Psi>')
show ?case
proof(case_tac "(\<Psi>, R, T) \<in> ?X1")
assume "(\<Psi>, R, T) \<in> ?X1"
then obtain xvec x P Q where Req: "R = \<lparr>\<nu>*xvec\<rparr>(\<lparr>\<nu>x\<rparr>(P \<parallel> Q))" and Teq: "T = \<lparr>\<nu>*xvec\<rparr>(P \<parallel> \<lparr>\<nu>x\<rparr>Q)" and "xvec \<sharp>* \<Psi>" and "x \<sharp> P" and "x \<sharp> \<Psi>"
by auto
obtain y::name where "y \<sharp> P" and "y \<sharp> Q" and "y \<sharp> xvec" and "y \<sharp> \<Psi>" and "y \<sharp> \<Psi>'"
by(generate_fresh "name", auto simp add: fresh_prod)
obtain p where "(p \<bullet> xvec) \<sharp>* \<Psi>" and "(p \<bullet> xvec) \<sharp>* P" and "(p \<bullet> xvec) \<sharp>* Q" and "(p \<bullet> xvec) \<sharp>* \<Psi>'"
and "x \<sharp> (p \<bullet> xvec)" and "y \<sharp> (p \<bullet> xvec)"
and S: "(set p) \<subseteq> (set xvec) \<times> (set(p \<bullet> xvec))" and "distinctPerm p"
by(rule_tac c="(\<Psi>, P, Q, x, y, \<Psi>')" in name_list_avoiding) auto
from \<open>y \<sharp> P\<close> have "(p \<bullet> y) \<sharp> (p \<bullet> P)" by(simp add: pt_fresh_bij[OF pt_name_inst, OF at_name_inst])
with S \<open>y \<sharp> xvec\<close> \<open>y \<sharp> (p \<bullet> xvec)\<close> have "y \<sharp> (p \<bullet> P)" by simp
with \<open>(p \<bullet> xvec) \<sharp>* \<Psi>\<close> \<open>y \<sharp> \<Psi>\<close> \<open>(p \<bullet> xvec) \<sharp>* \<Psi>'\<close> \<open>y \<sharp> \<Psi>'\<close>
have "(\<Psi> \<otimes> \<Psi>', \<lparr>\<nu>*(p \<bullet> xvec)\<rparr>(\<lparr>\<nu>y\<rparr>((p \<bullet> P) \<parallel> (p \<bullet> [(x, y)] \<bullet> Q))), \<lparr>\<nu>*(p \<bullet> xvec)\<rparr>((p \<bullet> P) \<parallel> (\<lparr>\<nu>y\<rparr>(p \<bullet> [(x, y)] \<bullet> Q)))) \<in> ?X"
by auto
moreover from Req \<open>(p \<bullet> xvec) \<sharp>* P\<close> \<open>(p \<bullet> xvec) \<sharp>* Q\<close> \<open>y \<sharp> xvec\<close> \<open>y \<sharp> (p \<bullet> xvec)\<close> \<open>x \<sharp> (p \<bullet> xvec)\<close> \<open>y \<sharp> P\<close> \<open>y \<sharp> Q\<close> \<open>x \<sharp> P\<close> S
have "R = \<lparr>\<nu>*(p \<bullet> xvec)\<rparr>(\<lparr>\<nu>y\<rparr>((p \<bullet> P) \<parallel> (p \<bullet> [(x, y)] \<bullet> Q)))"
apply(erule_tac rev_mp)
apply(subst alphaRes[of y])
apply(clarsimp simp add: eqvts)
apply(subst resChainAlpha[of p])
by(auto simp add: eqvts)
moreover from Teq \<open>(p \<bullet> xvec) \<sharp>* P\<close> \<open>(p \<bullet> xvec) \<sharp>* Q\<close> \<open>y \<sharp> xvec\<close> \<open>y \<sharp> (p \<bullet> xvec)\<close> \<open>x \<sharp> (p \<bullet> xvec)\<close> \<open>y \<sharp> P\<close> \<open>y \<sharp> Q\<close> \<open>x \<sharp> P\<close> S
have "T = \<lparr>\<nu>*(p \<bullet> xvec)\<rparr>((p \<bullet> P) \<parallel> \<lparr>\<nu>y\<rparr>(p \<bullet> [(x, y)] \<bullet> Q))"
apply(erule_tac rev_mp)
apply(subst alphaRes[of y])
apply(clarsimp simp add: eqvts)
apply(subst resChainAlpha[of p])
by(auto simp add: eqvts)
ultimately show ?case
by blast
next
assume "(\<Psi>, R, T) \<notin> ?X1"
with \<open>(\<Psi>, R, T) \<in> ?X\<close> have "(\<Psi>, R, T) \<in> ?X2" by blast
then obtain xvec x P Q where Teq: "T = \<lparr>\<nu>*xvec\<rparr>(\<lparr>\<nu>x\<rparr>(P \<parallel> Q))" and Req: "R = \<lparr>\<nu>*xvec\<rparr>(P \<parallel> \<lparr>\<nu>x\<rparr>Q)" and "xvec \<sharp>* \<Psi>" and "x \<sharp> P" and "x \<sharp> \<Psi>"
by auto
obtain y::name where "y \<sharp> P" and "y \<sharp> Q" and "y \<sharp> xvec" and "y \<sharp> \<Psi>" and "y \<sharp> \<Psi>'"
by(generate_fresh "name", auto simp add: fresh_prod)
obtain p where "(p \<bullet> xvec) \<sharp>* \<Psi>" and "(p \<bullet> xvec) \<sharp>* P" and "(p \<bullet> xvec) \<sharp>* Q" and "(p \<bullet> xvec) \<sharp>* \<Psi>'"
and "x \<sharp> (p \<bullet> xvec)" and "y \<sharp> (p \<bullet> xvec)"
and S: "(set p) \<subseteq> (set xvec) \<times> (set(p \<bullet> xvec))" and "distinctPerm p"
by(rule_tac c="(\<Psi>, P, Q, x, y, \<Psi>')" in name_list_avoiding) auto
from \<open>y \<sharp> P\<close> have "(p \<bullet> y) \<sharp> (p \<bullet> P)" by(simp add: pt_fresh_bij[OF pt_name_inst, OF at_name_inst])
with S \<open>y \<sharp> xvec\<close> \<open>y \<sharp> (p \<bullet> xvec)\<close> have "y \<sharp> (p \<bullet> P)" by simp
with \<open>(p \<bullet> xvec) \<sharp>* \<Psi>\<close> \<open>y \<sharp> \<Psi>\<close> \<open>(p \<bullet> xvec) \<sharp>* \<Psi>'\<close> \<open>y \<sharp> \<Psi>'\<close>
have "(\<Psi> \<otimes> \<Psi>', \<lparr>\<nu>*(p \<bullet> xvec)\<rparr>((p \<bullet> P) \<parallel> \<lparr>\<nu>y\<rparr>(p \<bullet> [(x, y)] \<bullet> Q)), \<lparr>\<nu>*(p \<bullet> xvec)\<rparr>(\<lparr>\<nu>y\<rparr>((p \<bullet> P) \<parallel> (p \<bullet> [(x, y)] \<bullet> Q)))) \<in> ?X2"
by auto
moreover from Teq \<open>(p \<bullet> xvec) \<sharp>* P\<close> \<open>(p \<bullet> xvec) \<sharp>* Q\<close> \<open>y \<sharp> xvec\<close> \<open>y \<sharp> (p \<bullet> xvec)\<close> \<open>x \<sharp> (p \<bullet> xvec)\<close> \<open>y \<sharp> P\<close> \<open>y \<sharp> Q\<close> \<open>x \<sharp> P\<close> S
have "T = \<lparr>\<nu>*(p \<bullet> xvec)\<rparr>(\<lparr>\<nu>y\<rparr>((p \<bullet> P) \<parallel> (p \<bullet> [(x, y)] \<bullet> Q)))"
apply(erule_tac rev_mp)
apply(subst alphaRes[of y])
apply(clarsimp simp add: eqvts)
apply(subst resChainAlpha[of p])
by(auto simp add: eqvts)
moreover from Req \<open>(p \<bullet> xvec) \<sharp>* P\<close> \<open>(p \<bullet> xvec) \<sharp>* Q\<close> \<open>y \<sharp> xvec\<close> \<open>y \<sharp> (p \<bullet> xvec)\<close> \<open>x \<sharp> (p \<bullet> xvec)\<close> \<open>y \<sharp> P\<close> \<open>y \<sharp> Q\<close> \<open>x \<sharp> P\<close> S
have "R = \<lparr>\<nu>*(p \<bullet> xvec)\<rparr>((p \<bullet> P) \<parallel> \<lparr>\<nu>y\<rparr>(p \<bullet> [(x, y)] \<bullet> Q))"
apply(erule_tac rev_mp)
apply(subst alphaRes[of y])
apply(clarsimp simp add: eqvts)
apply(subst resChainAlpha[of p])
by(auto simp add: eqvts)
ultimately show ?case
by blast
qed
next
case(cSym \<Psi> P Q)
thus ?case
by(blast dest: bisimE)
qed
}
moreover obtain y::name where "y \<sharp> \<Psi>" and "y \<sharp> P" "y \<sharp> Q"
by(generate_fresh "name") auto
ultimately have "\<Psi> \<rhd> \<lparr>\<nu>y\<rparr>(P \<parallel> ([(x, y)] \<bullet> Q)) \<sim> P \<parallel> \<lparr>\<nu>y\<rparr>([(x, y)] \<bullet> Q)" by auto
thus ?thesis using assms \<open>y \<sharp> P\<close> \<open>y \<sharp> Q\<close>
apply(subst alphaRes[where x=x and y=y and P=Q], auto)
by(subst alphaRes[where x=x and y=y and P="P \<parallel> Q"]) auto
qed
lemma bisimScopeExtChain:
fixes xvec :: "name list"
and \<Psi> :: 'b
and P :: "('a, 'b, 'c) psi"
and Q :: "('a, 'b, 'c) psi"
assumes "xvec \<sharp>* \<Psi>"
and "xvec \<sharp>* P"
shows "\<Psi> \<rhd> \<lparr>\<nu>*xvec\<rparr>(P \<parallel> Q) \<sim> P \<parallel> (\<lparr>\<nu>*xvec\<rparr>Q)"
using assms
by(induct xvec) (auto intro: bisimScopeExt bisimReflexive bisimTransitive bisimResPres)
lemma bisimParAssoc:
fixes \<Psi> :: 'b
and P :: "('a, 'b, 'c) psi"
and Q :: "('a, 'b, 'c) psi"
and R :: "('a, 'b, 'c) psi"
shows "\<Psi> \<rhd> (P \<parallel> Q) \<parallel> R \<sim> P \<parallel> (Q \<parallel> R)"
proof -
let ?X = "{(\<Psi>, \<lparr>\<nu>*xvec\<rparr>((P \<parallel> Q) \<parallel> R), \<lparr>\<nu>*xvec\<rparr>(P \<parallel> (Q \<parallel> R))) | \<Psi> xvec P Q R. xvec \<sharp>* \<Psi>}"
let ?Y = "{(\<Psi>, P, Q) | \<Psi> P P' Q' Q. \<Psi> \<rhd> P \<sim> P' \<and> (\<Psi>, P', Q') \<in> ?X \<and> \<Psi> \<rhd> Q' \<sim> Q}"
have "(\<Psi>, (P \<parallel> Q) \<parallel> R, P \<parallel> (Q \<parallel> R)) \<in> ?X"
by(auto, rule_tac x="[]" in exI) auto
moreover have "eqvt ?X" by(force simp add: eqvt_def simp add: pt_fresh_star_bij[OF pt_name_inst, OF at_name_inst] eqvts)
ultimately show ?thesis
proof(coinduct rule: weakTransitiveCoinduct')
case(cStatEq \<Psi> PQR PQR')
from \<open>(\<Psi>, PQR, PQR') \<in> ?X\<close> obtain xvec P Q R where "xvec \<sharp>* \<Psi>" and "PQR = \<lparr>\<nu>*xvec\<rparr>((P \<parallel> Q) \<parallel> R)" and "PQR' = \<lparr>\<nu>*xvec\<rparr>(P \<parallel> (Q \<parallel> R))"
by auto
moreover obtain A\<^sub>P \<Psi>\<^sub>P where FrP: "extractFrame P = \<langle>A\<^sub>P, \<Psi>\<^sub>P\<rangle>" and "A\<^sub>P \<sharp>* \<Psi>" and "A\<^sub>P \<sharp>* Q" and "A\<^sub>P \<sharp>* R"
by(rule_tac C="(\<Psi>, Q, R)" in freshFrame) auto
moreover obtain A\<^sub>Q \<Psi>\<^sub>Q where FrQ: "extractFrame Q = \<langle>A\<^sub>Q, \<Psi>\<^sub>Q\<rangle>" and "A\<^sub>Q \<sharp>* \<Psi>" and "A\<^sub>Q \<sharp>* A\<^sub>P" and "A\<^sub>Q \<sharp>* \<Psi>\<^sub>P" and "A\<^sub>Q \<sharp>* R"
by(rule_tac C="(\<Psi>, A\<^sub>P, \<Psi>\<^sub>P, R)" in freshFrame) auto
moreover obtain A\<^sub>R \<Psi>\<^sub>R where FrR: "extractFrame R = \<langle>A\<^sub>R, \<Psi>\<^sub>R\<rangle>" and "A\<^sub>R \<sharp>* \<Psi>" and "A\<^sub>R \<sharp>* A\<^sub>P" and "A\<^sub>R \<sharp>* \<Psi>\<^sub>P" and "A\<^sub>R \<sharp>* A\<^sub>Q" and "A\<^sub>R \<sharp>* \<Psi>\<^sub>Q"
by(rule_tac C="(\<Psi>, A\<^sub>P, \<Psi>\<^sub>P, A\<^sub>Q, \<Psi>\<^sub>Q)" in freshFrame) auto
moreover from FrQ \<open>A\<^sub>P \<sharp>* Q\<close> \<open>A\<^sub>Q \<sharp>* A\<^sub>P\<close> have "A\<^sub>P \<sharp>* \<Psi>\<^sub>Q"
by(drule_tac extractFrameFreshChain) auto
moreover from FrR \<open>A\<^sub>P \<sharp>* R\<close> \<open>A\<^sub>R \<sharp>* A\<^sub>P\<close> have "A\<^sub>P \<sharp>* \<Psi>\<^sub>R"
by(drule_tac extractFrameFreshChain) auto
moreover from FrR \<open>A\<^sub>Q \<sharp>* R\<close> \<open>A\<^sub>R \<sharp>* A\<^sub>Q\<close> have "A\<^sub>Q \<sharp>* \<Psi>\<^sub>R"
by(drule_tac extractFrameFreshChain) auto
ultimately show ?case using freshCompChain
by auto (metis frameChainAppend compositionSym Associativity frameNilStatEq frameResChainPres)
next
case(cSim \<Psi> T S)
from \<open>(\<Psi>, T, S) \<in> ?X\<close> obtain xvec P Q R where "xvec \<sharp>* \<Psi>" and TEq: "T = \<lparr>\<nu>*xvec\<rparr>((P \<parallel> Q) \<parallel> R)"
and SEq: "S = \<lparr>\<nu>*xvec\<rparr>(P \<parallel> (Q \<parallel> R))"
by auto
from \<open>eqvt ?X\<close>have "eqvt ?Y" by blast
have C1: "\<And>\<Psi> T S yvec. \<lbrakk>(\<Psi>, T, S) \<in> ?Y; yvec \<sharp>* \<Psi>\<rbrakk> \<Longrightarrow> (\<Psi>, \<lparr>\<nu>*yvec\<rparr>T, \<lparr>\<nu>*yvec\<rparr>S) \<in> ?Y"
proof -
fix \<Psi> T S yvec
assume "(\<Psi>, T, S) \<in> ?Y"
then obtain T' S' where "\<Psi> \<rhd> T \<sim> T'" and "(\<Psi>, T', S') \<in> ?X" and "\<Psi> \<rhd> S' \<sim> S" by fastforce
assume "(yvec::name list) \<sharp>* \<Psi>"
from \<open>(\<Psi>, T', S') \<in> ?X\<close> obtain xvec P Q R where T'eq: "T' = \<lparr>\<nu>*xvec\<rparr>((P \<parallel> Q) \<parallel> R)" and S'eq: "S' = \<lparr>\<nu>*xvec\<rparr>(P \<parallel> (Q \<parallel> R))"
and "xvec \<sharp>* \<Psi>"
by auto
from \<open>\<Psi> \<rhd> T \<sim> T'\<close> \<open>yvec \<sharp>* \<Psi>\<close> have "\<Psi> \<rhd> \<lparr>\<nu>*yvec\<rparr>T \<sim> \<lparr>\<nu>*yvec\<rparr>T'" by(rule bisimResChainPres)
moreover from \<open>xvec \<sharp>* \<Psi>\<close> \<open>yvec \<sharp>* \<Psi>\<close> have "(\<Psi>, \<lparr>\<nu>*(yvec@xvec)\<rparr>((P \<parallel> Q) \<parallel> R), \<lparr>\<nu>*(yvec@xvec)\<rparr>(P \<parallel> (Q \<parallel> R))) \<in> ?X"
by force
with T'eq S'eq have "(\<Psi>, \<lparr>\<nu>*yvec\<rparr>T', \<lparr>\<nu>*yvec\<rparr>S') \<in> ?X" by(simp add: resChainAppend)
moreover from \<open>\<Psi> \<rhd> S' \<sim> S\<close> \<open>yvec \<sharp>* \<Psi>\<close> have "\<Psi> \<rhd> \<lparr>\<nu>*yvec\<rparr>S' \<sim> \<lparr>\<nu>*yvec\<rparr>S" by(rule bisimResChainPres)
ultimately show "(\<Psi>, \<lparr>\<nu>*yvec\<rparr>T, \<lparr>\<nu>*yvec\<rparr>S) \<in> ?Y" by blast
qed
have C2: "\<And>\<Psi> T S y. \<lbrakk>(\<Psi>, T, S) \<in> ?Y; y \<sharp> \<Psi>\<rbrakk> \<Longrightarrow> (\<Psi>, \<lparr>\<nu>y\<rparr>T, \<lparr>\<nu>y\<rparr>S) \<in> ?Y"
by(drule_tac yvec2="[y]" in C1) auto
have "\<Psi> \<rhd> \<lparr>\<nu>*xvec\<rparr>((P \<parallel> Q) \<parallel> R) \<leadsto>[?Y] \<lparr>\<nu>*xvec\<rparr>(P \<parallel> (Q \<parallel> R))"
proof -
have "\<Psi> \<rhd> (P \<parallel> Q) \<parallel> R \<leadsto>[?Y] P \<parallel> (Q \<parallel> R)"
proof -
note \<open>eqvt ?Y\<close>
moreover have "\<And>\<Psi> P Q R. (\<Psi>, (P \<parallel> Q) \<parallel> R, P \<parallel> (Q \<parallel> R)) \<in> ?Y"
proof -
fix \<Psi> P Q R
have "(\<Psi>::'b, ((P::('a, 'b, 'c) psi) \<parallel> Q) \<parallel> R, P \<parallel> (Q \<parallel> R)) \<in> ?X"
by(auto, rule_tac x="[]" in exI) auto
thus "(\<Psi>, (P \<parallel> Q) \<parallel> R, P \<parallel> (Q \<parallel> R)) \<in> ?Y"
by(blast intro: bisimReflexive)
qed
moreover have "\<And>xvec \<Psi> P Q R. \<lbrakk>xvec \<sharp>* \<Psi>; xvec \<sharp>* P\<rbrakk> \<Longrightarrow> (\<Psi>, \<lparr>\<nu>*xvec\<rparr>((P \<parallel> Q) \<parallel> R), P \<parallel> (\<lparr>\<nu>*xvec\<rparr>(Q \<parallel> R))) \<in> ?Y"
proof -
fix xvec \<Psi> P Q R
assume "(xvec::name list) \<sharp>* (\<Psi>::'b)" and "xvec \<sharp>* (P::('a, 'b, 'c) psi)"
from \<open>xvec \<sharp>* \<Psi>\<close> have "(\<Psi>, \<lparr>\<nu>*xvec\<rparr>((P \<parallel> Q) \<parallel> R), \<lparr>\<nu>*xvec\<rparr>(P \<parallel> (Q \<parallel> R))) \<in> ?X" by blast
moreover from \<open>xvec \<sharp>* \<Psi>\<close> \<open>xvec \<sharp>* P\<close> have "\<Psi> \<rhd> \<lparr>\<nu>*xvec\<rparr>(P \<parallel> (Q \<parallel> R)) \<sim> P \<parallel> (\<lparr>\<nu>*xvec\<rparr>(Q \<parallel> R))"
by(rule bisimScopeExtChain)
ultimately show "(\<Psi>, \<lparr>\<nu>*xvec\<rparr>((P \<parallel> Q) \<parallel> R), P \<parallel> (\<lparr>\<nu>*xvec\<rparr>(Q \<parallel> R))) \<in> ?Y"
by(blast intro: bisimReflexive)
qed
moreover have "\<And>xvec \<Psi> P Q R. \<lbrakk>xvec \<sharp>* \<Psi>; xvec \<sharp>* R\<rbrakk> \<Longrightarrow> (\<Psi>, (\<lparr>\<nu>*xvec\<rparr>(P \<parallel> Q)) \<parallel> R, \<lparr>\<nu>*xvec\<rparr>(P \<parallel> (Q \<parallel> R))) \<in> ?Y"
proof -
fix xvec \<Psi> P Q R
assume "(xvec::name list) \<sharp>* (\<Psi>::'b)" and "xvec \<sharp>* (R::('a, 'b, 'c) psi)"
have "\<Psi> \<rhd> (\<lparr>\<nu>*xvec\<rparr>(P \<parallel> Q)) \<parallel> R \<sim> R \<parallel> (\<lparr>\<nu>*xvec\<rparr>(P \<parallel> Q))" by(rule bisimParComm)
moreover from \<open>xvec \<sharp>* \<Psi>\<close> \<open>xvec \<sharp>* R\<close> have "\<Psi> \<rhd> \<lparr>\<nu>*xvec\<rparr>(R \<parallel> (P \<parallel> Q)) \<sim> R \<parallel> (\<lparr>\<nu>*xvec\<rparr>(P \<parallel> Q))" by(rule bisimScopeExtChain)
hence "\<Psi> \<rhd> R \<parallel> (\<lparr>\<nu>*xvec\<rparr>(P \<parallel> Q)) \<sim> \<lparr>\<nu>*xvec\<rparr>(R \<parallel> (P \<parallel> Q))" by(rule bisimE)
moreover from \<open>xvec \<sharp>* \<Psi>\<close> have "\<Psi> \<rhd> \<lparr>\<nu>*xvec\<rparr>(R \<parallel> (P \<parallel> Q)) \<sim> \<lparr>\<nu>*xvec\<rparr>((P \<parallel> Q) \<parallel> R)"
by(metis bisimResChainPres bisimParComm)
moreover from \<open>xvec \<sharp>* \<Psi>\<close> have "(\<Psi>, \<lparr>\<nu>*xvec\<rparr>((P \<parallel> Q) \<parallel> R), \<lparr>\<nu>*xvec\<rparr>(P \<parallel> (Q \<parallel> R))) \<in> ?X" by blast
ultimately show "(\<Psi>, (\<lparr>\<nu>*xvec\<rparr>(P \<parallel> Q)) \<parallel> R, \<lparr>\<nu>*xvec\<rparr>(P \<parallel> (Q \<parallel> R))) \<in> ?Y" by(blast dest: bisimTransitive intro: bisimReflexive)
qed
ultimately show ?thesis using C1
by(rule parAssocLeft)
qed
thus ?thesis using \<open>eqvt ?Y\<close> \<open>xvec \<sharp>* \<Psi>\<close> C2
by(rule resChainPres)
qed
with TEq SEq show ?case by simp
next
case(cExt \<Psi> T S \<Psi>')
from \<open>(\<Psi>, T, S) \<in> ?X\<close> obtain xvec P Q R where "xvec \<sharp>* \<Psi>" and TEq: "T = \<lparr>\<nu>*xvec\<rparr>((P \<parallel> Q) \<parallel> R)"
and SEq: "S = \<lparr>\<nu>*xvec\<rparr>(P \<parallel> (Q \<parallel> R))"
by auto
obtain p where "(p \<bullet> xvec) \<sharp>* \<Psi>" and "(p \<bullet> xvec) \<sharp>* P" and "(p \<bullet> xvec) \<sharp>* Q" and "(p \<bullet> xvec) \<sharp>* R" and "(p \<bullet> xvec) \<sharp>* \<Psi>'"
and S: "(set p) \<subseteq> (set xvec) \<times> (set(p \<bullet> xvec))" and "distinctPerm p"
by(rule_tac c="(\<Psi>, P, Q, R, \<Psi>')" in name_list_avoiding) auto
from \<open>(p \<bullet> xvec) \<sharp>* \<Psi>\<close> \<open>(p \<bullet> xvec) \<sharp>* \<Psi>'\<close> have "(\<Psi> \<otimes> \<Psi>', \<lparr>\<nu>*(p \<bullet> xvec)\<rparr>(((p \<bullet> P) \<parallel> (p \<bullet> Q)) \<parallel> (p \<bullet> R)), \<lparr>\<nu>*(p \<bullet> xvec)\<rparr>((p \<bullet> P) \<parallel> ((p \<bullet> Q) \<parallel> (p \<bullet> R)))) \<in> ?X"
by auto
moreover from TEq \<open>(p \<bullet> xvec) \<sharp>* P\<close> \<open>(p \<bullet> xvec) \<sharp>* Q\<close> \<open>(p \<bullet> xvec) \<sharp>* R\<close> S have "T = \<lparr>\<nu>*(p \<bullet> xvec)\<rparr>(((p \<bullet> P) \<parallel> (p \<bullet> Q)) \<parallel> (p \<bullet> R))"
apply auto by(subst resChainAlpha[of p]) auto
moreover from SEq \<open>(p \<bullet> xvec) \<sharp>* P\<close> \<open>(p \<bullet> xvec) \<sharp>* Q\<close> \<open>(p \<bullet> xvec) \<sharp>* R\<close> S have "S = \<lparr>\<nu>*(p \<bullet> xvec)\<rparr>((p \<bullet> P) \<parallel> ((p \<bullet> Q) \<parallel> (p \<bullet> R)))"
apply auto by(subst resChainAlpha[of p]) auto
ultimately show ?case by simp
next
case(cSym \<Psi> T S)
from \<open>(\<Psi>, T, S) \<in> ?X\<close> obtain xvec P Q R where "xvec \<sharp>* \<Psi>" and TEq: "T = \<lparr>\<nu>*xvec\<rparr>((P \<parallel> Q) \<parallel> R)"
and SEq: "\<lparr>\<nu>*xvec\<rparr>(P \<parallel> (Q \<parallel> R)) = S"
by auto
from \<open>xvec \<sharp>* \<Psi>\<close> have "\<Psi> \<rhd> \<lparr>\<nu>*xvec\<rparr>(P \<parallel> (Q \<parallel> R)) \<sim> \<lparr>\<nu>*xvec\<rparr>((R \<parallel> Q) \<parallel> P)"
by(metis bisimParComm bisimParPres bisimTransitive bisimResChainPres)
moreover from \<open>xvec \<sharp>* \<Psi>\<close> have "(\<Psi>, \<lparr>\<nu>*xvec\<rparr>((R \<parallel> Q) \<parallel> P), \<lparr>\<nu>*xvec\<rparr>(R \<parallel> (Q \<parallel> P))) \<in> ?X" by blast
moreover from \<open>xvec \<sharp>* \<Psi>\<close> have "\<Psi> \<rhd> \<lparr>\<nu>*xvec\<rparr>(R \<parallel> (Q \<parallel> P)) \<sim> \<lparr>\<nu>*xvec\<rparr>((P \<parallel> Q) \<parallel> R)"
by(metis bisimParComm bisimParPres bisimTransitive bisimResChainPres)
ultimately show ?case using TEq SEq by(blast dest: bisimTransitive)
qed
qed
lemma bisimParNil:
fixes P :: "('a, 'b, 'c) psi"
shows "\<Psi> \<rhd> P \<parallel> \<zero> \<sim> P"
proof -
let ?X1 = "{(\<Psi>, P \<parallel> \<zero>, P) | \<Psi> P. True}"
let ?X2 = "{(\<Psi>, P, P \<parallel> \<zero>) | \<Psi> P. True}"
let ?X = "?X1 \<union> ?X2"
have "eqvt ?X" by(auto simp add: eqvt_def)
have "(\<Psi>, P \<parallel> \<zero>, P) \<in> ?X" by simp
thus ?thesis
proof(coinduct rule: bisimWeakCoinduct)
case(cStatEq \<Psi> Q R)
show ?case
proof(case_tac "(\<Psi>, Q, R) \<in> ?X1")
assume "(\<Psi>, Q, R) \<in> ?X1"
then obtain P where "Q = P \<parallel> \<zero>" and "R = P" by auto
moreover obtain A\<^sub>P \<Psi>\<^sub>P where "extractFrame P = \<langle>A\<^sub>P, \<Psi>\<^sub>P\<rangle>" and "A\<^sub>P \<sharp>* \<Psi>"
by(rule freshFrame)
ultimately show ?case
apply auto by(metis frameResChainPres frameNilStatEq Identity Associativity AssertionStatEqTrans Commutativity)
next
assume "(\<Psi>, Q, R) \<notin> ?X1"
with \<open>(\<Psi>, Q, R) \<in> ?X\<close> have "(\<Psi>, Q, R) \<in> ?X2" by blast
then obtain P where "Q = P" and "R = P \<parallel> \<zero>" by auto
moreover obtain A\<^sub>P \<Psi>\<^sub>P where "extractFrame P = \<langle>A\<^sub>P, \<Psi>\<^sub>P\<rangle>" and "A\<^sub>P \<sharp>* \<Psi>"
by(rule freshFrame)
ultimately show ?case
apply auto by(metis frameResChainPres frameNilStatEq Identity Associativity AssertionStatEqTrans AssertionStatEqSym Commutativity)
qed
next
case(cSim \<Psi> Q R)
thus ?case using \<open>eqvt ?X\<close>
by(auto intro: parNilLeft parNilRight)
next
case(cExt \<Psi> Q R \<Psi>')
thus ?case by auto
next
case(cSym \<Psi> Q R)
thus ?case by auto
qed
qed
lemma bisimResNil:
fixes x :: name
and \<Psi> :: 'b
shows "\<Psi> \<rhd> \<lparr>\<nu>x\<rparr>\<zero> \<sim> \<zero>"
proof -
{
fix x::name
assume "x \<sharp> \<Psi>"
have "\<Psi> \<rhd> \<lparr>\<nu>x\<rparr>\<zero> \<sim> \<zero>"
proof -
let ?X1 = "{(\<Psi>, \<lparr>\<nu>x\<rparr>\<zero>, \<zero>) | \<Psi> x. x \<sharp> \<Psi>}"
let ?X2 = "{(\<Psi>, \<zero>, \<lparr>\<nu>x\<rparr>\<zero>) | \<Psi> x. x \<sharp> \<Psi>}"
let ?X = "?X1 \<union> ?X2"
from \<open>x \<sharp> \<Psi>\<close> have "(\<Psi>, \<lparr>\<nu>x\<rparr>\<zero>, \<zero>) \<in> ?X" by auto
thus ?thesis
proof(coinduct rule: bisimWeakCoinduct)
case(cStatEq \<Psi> P Q)
thus ?case using freshComp by(force intro: frameResFresh FrameStatEqSym)
next
case(cSim \<Psi> P Q)
thus ?case
by(force intro: resNilLeft resNilRight)
next
case(cExt \<Psi> P Q \<Psi>')
obtain y where "y \<sharp> \<Psi>" and "y \<sharp> \<Psi>'" and "y \<noteq> x"
by(generate_fresh "name") (auto simp add: fresh_prod)
show ?case
proof(case_tac "(\<Psi>, P, Q) \<in> ?X1")
assume "(\<Psi>, P, Q) \<in> ?X1"
then obtain x where "P = \<lparr>\<nu>x\<rparr>\<zero>" and "Q = \<zero>" by auto
moreover have "\<lparr>\<nu>x\<rparr>\<zero> = \<lparr>\<nu>y\<rparr> \<zero>" by(subst alphaRes) auto
ultimately show ?case using \<open>y \<sharp> \<Psi>\<close> \<open>y \<sharp> \<Psi>'\<close> by auto
next
assume "(\<Psi>, P, Q) \<notin> ?X1"
with \<open>(\<Psi>, P, Q) \<in> ?X\<close> have "(\<Psi>, P, Q) \<in> ?X2" by auto
then obtain x where "Q = \<lparr>\<nu>x\<rparr>\<zero>" and "P = \<zero>" by auto
moreover have "\<lparr>\<nu>x\<rparr>\<zero> = \<lparr>\<nu>y\<rparr> \<zero>" by(subst alphaRes) auto
ultimately show ?case using \<open>y \<sharp> \<Psi>\<close> \<open>y \<sharp> \<Psi>'\<close> by auto
qed
next
case(cSym \<Psi> P Q)
thus ?case by auto
qed
qed
}
moreover obtain y::name where "y \<sharp> \<Psi>" by(generate_fresh "name") auto
ultimately have "\<Psi> \<rhd> \<lparr>\<nu>y\<rparr>\<zero> \<sim> \<zero>" by auto
thus ?thesis by(subst alphaRes[where x=x and y=y]) auto
qed
lemma bisimOutputPushRes:
fixes x :: name
and \<Psi> :: 'b
and M :: 'a
and N :: 'a
and P :: "('a, 'b, 'c) psi"
assumes "x \<sharp> M"
and "x \<sharp> N"
shows "\<Psi> \<rhd> \<lparr>\<nu>x\<rparr>(M\<langle>N\<rangle>.P) \<sim> M\<langle>N\<rangle>.\<lparr>\<nu>x\<rparr>P"
proof -
{
fix x::name and P::"('a, 'b, 'c) psi"
assume "x \<sharp> \<Psi>" and "x \<sharp> M" and "x \<sharp> N"
let ?X1 = "{(\<Psi>, \<lparr>\<nu>x\<rparr>(M\<langle>N\<rangle>.P), M\<langle>N\<rangle>.\<lparr>\<nu>x\<rparr>P) | \<Psi> x M N P. x \<sharp> \<Psi> \<and> x \<sharp> M \<and> x \<sharp> N}"
let ?X2 = "{(\<Psi>, M\<langle>N\<rangle>.\<lparr>\<nu>x\<rparr>P, \<lparr>\<nu>x\<rparr>(M\<langle>N\<rangle>.P)) | \<Psi> x M N P. x \<sharp> \<Psi> \<and> x \<sharp> M \<and> x \<sharp> N}"
let ?X = "?X1 \<union> ?X2"
have "eqvt ?X" by(rule_tac eqvtUnion) (force simp add: eqvt_def pt_fresh_bij[OF pt_name_inst, OF at_name_inst] eqvts)+
from \<open>x \<sharp> \<Psi>\<close> \<open>x \<sharp> M\<close> \<open>x \<sharp> N\<close> have "(\<Psi>, \<lparr>\<nu>x\<rparr>(M\<langle>N\<rangle>.P), M\<langle>N\<rangle>.\<lparr>\<nu>x\<rparr>P) \<in> ?X" by auto
hence "\<Psi> \<rhd> \<lparr>\<nu>x\<rparr>(M\<langle>N\<rangle>.P) \<sim> M\<langle>N\<rangle>.\<lparr>\<nu>x\<rparr>P"
proof(coinduct rule: bisimCoinduct)
case(cStatEq \<Psi> Q R)
thus ?case using freshComp by(force intro: frameResFresh FrameStatEqSym)
next
case(cSim \<Psi> Q R)
thus ?case using \<open>eqvt ?X\<close>
by(fastforce intro: outputPushResLeft outputPushResRight bisimReflexive)
next
case(cExt \<Psi> Q R \<Psi>')
show ?case
proof(case_tac "(\<Psi>, Q, R) \<in> ?X1")
assume "(\<Psi>, Q, R) \<in> ?X1"
then obtain x M N P where Qeq: "Q = \<lparr>\<nu>x\<rparr>(M\<langle>N\<rangle>.P)" and Req: "R = M\<langle>N\<rangle>.\<lparr>\<nu>x\<rparr>P" and "x \<sharp> \<Psi>" and "x \<sharp> M" and "x \<sharp> N" by auto
obtain y::name where "y \<sharp> \<Psi>" and "y \<sharp> \<Psi>'" and "y \<sharp> M" and "y \<sharp> N" and "y \<sharp> P"
by(generate_fresh "name") (auto simp add: fresh_prod)
moreover hence "(\<Psi> \<otimes> \<Psi>', \<lparr>\<nu>y\<rparr>(M\<langle>N\<rangle>.([(x, y)] \<bullet> P)), M\<langle>N\<rangle>.\<lparr>\<nu>y\<rparr>([(x, y)] \<bullet> P)) \<in> ?X" by auto
moreover from Qeq \<open>x \<sharp> M\<close> \<open>y \<sharp> M\<close> \<open>x \<sharp> N\<close> \<open>y \<sharp> N\<close> \<open>y \<sharp> P\<close> have "Q = \<lparr>\<nu>y\<rparr>(M\<langle>N\<rangle>.([(x, y)] \<bullet> P))"
apply auto by(subst alphaRes[of y]) (auto simp add: eqvts)
moreover from Req \<open>y \<sharp> P\<close> have "R = M\<langle>N\<rangle>.\<lparr>\<nu>y\<rparr>([(x, y)] \<bullet> P)"
apply auto by(subst alphaRes[of y]) (auto simp add: eqvts)
ultimately show ?case by blast
next
assume "(\<Psi>, Q, R) \<notin> ?X1"
with \<open>(\<Psi>, Q, R) \<in> ?X\<close> have "(\<Psi>, Q, R) \<in> ?X2" by blast
then obtain x M N P where Req: "R = \<lparr>\<nu>x\<rparr>(M\<langle>N\<rangle>.P)" and Qeq: "Q = M\<langle>N\<rangle>.\<lparr>\<nu>x\<rparr>P" and "x \<sharp> \<Psi>" and "x \<sharp> M" and "x \<sharp> N" by auto
obtain y::name where "y \<sharp> \<Psi>" and "y \<sharp> \<Psi>'" and "y \<sharp> M" and "y \<sharp> N" and "y \<sharp> P"
by(generate_fresh "name") (auto simp add: fresh_prod)
moreover hence "(\<Psi> \<otimes> \<Psi>', \<lparr>\<nu>y\<rparr>(M\<langle>N\<rangle>.([(x, y)] \<bullet> P)), M\<langle>N\<rangle>.\<lparr>\<nu>y\<rparr>([(x, y)] \<bullet> P)) \<in> ?X" by auto
moreover from Req \<open>x \<sharp> M\<close> \<open>y \<sharp> M\<close> \<open>x \<sharp> N\<close> \<open>y \<sharp> N\<close> \<open>y \<sharp> P\<close> have "R = \<lparr>\<nu>y\<rparr>(M\<langle>N\<rangle>.([(x, y)] \<bullet> P))"
apply auto by(subst alphaRes[of y]) (auto simp add: eqvts)
moreover from Qeq \<open>y \<sharp> P\<close> have "Q = M\<langle>N\<rangle>.\<lparr>\<nu>y\<rparr>([(x, y)] \<bullet> P)"
apply auto by(subst alphaRes[of y]) (auto simp add: eqvts)
ultimately show ?case by blast
qed
next
case(cSym \<Psi> R Q)
thus ?case by blast
qed
}
moreover obtain y::name where "y \<sharp> \<Psi>" and "y \<sharp> M" and "y \<sharp> N" "y \<sharp> P"
by(generate_fresh "name") auto
ultimately have "\<Psi> \<rhd> \<lparr>\<nu>y\<rparr>(M\<langle>N\<rangle>.([(x, y)] \<bullet> P)) \<sim> M\<langle>N\<rangle>.\<lparr>\<nu>y\<rparr>([(x, y)] \<bullet> P)" by auto
thus ?thesis using assms \<open>y \<sharp> P\<close> \<open>y \<sharp> M\<close> \<open>y \<sharp> N\<close>
apply(subst alphaRes[where x=x and y=y and P=P], auto)
by(subst alphaRes[where x=x and y=y and P="M\<langle>N\<rangle>.P"]) auto
qed
lemma bisimInputPushRes:
fixes x :: name
and \<Psi> :: 'b
and M :: 'a
and xvec :: "name list"
and N :: 'a
and P :: "('a, 'b, 'c) psi"
assumes "x \<sharp> M"
and "x \<sharp> xvec"
and "x \<sharp> N"
shows "\<Psi> \<rhd> \<lparr>\<nu>x\<rparr>(M\<lparr>\<lambda>*xvec N\<rparr>.P) \<sim> M\<lparr>\<lambda>*xvec N\<rparr>.\<lparr>\<nu>x\<rparr>P"
proof -
{
fix x::name and P::"('a, 'b, 'c) psi"
assume "x \<sharp> \<Psi>" and "x \<sharp> M" and "x \<sharp> N" and "x \<sharp> xvec"
let ?X1 = "{(\<Psi>, \<lparr>\<nu>x\<rparr>(M\<lparr>\<lambda>*xvec N\<rparr>.P), M\<lparr>\<lambda>*xvec N\<rparr>.\<lparr>\<nu>x\<rparr>P) | \<Psi> x M xvec N P. x \<sharp> \<Psi> \<and> x \<sharp> M \<and> x \<sharp> xvec \<and> x \<sharp> N}"
let ?X2 = "{(\<Psi>, M\<lparr>\<lambda>*xvec N\<rparr>.\<lparr>\<nu>x\<rparr>P, \<lparr>\<nu>x\<rparr>(M\<lparr>\<lambda>*xvec N\<rparr>.P)) | \<Psi> x M xvec N P. x \<sharp> \<Psi> \<and> x \<sharp> M \<and> x \<sharp> xvec \<and> x \<sharp> N}"
let ?X = "?X1 \<union> ?X2"
have "eqvt ?X" by(rule_tac eqvtUnion) (force simp add: eqvt_def pt_fresh_bij[OF pt_name_inst, OF at_name_inst] eqvts)+
from \<open>x \<sharp> \<Psi>\<close> \<open>x \<sharp> M\<close> \<open>x \<sharp> xvec\<close> \<open>x \<sharp> N\<close> have "(\<Psi>, \<lparr>\<nu>x\<rparr>(M\<lparr>\<lambda>*xvec N\<rparr>.P), M\<lparr>\<lambda>*xvec N\<rparr>.\<lparr>\<nu>x\<rparr>P) \<in> ?X" by blast
hence "\<Psi> \<rhd> \<lparr>\<nu>x\<rparr>(M\<lparr>\<lambda>*xvec N\<rparr>.P) \<sim> M\<lparr>\<lambda>*xvec N\<rparr>.\<lparr>\<nu>x\<rparr>P"
proof(coinduct rule: bisimCoinduct)
case(cStatEq \<Psi> Q R)
thus ?case using freshComp by(force intro: frameResFresh FrameStatEqSym)
next
case(cSim \<Psi> Q R)
thus ?case using \<open>eqvt ?X\<close>
by(fastforce intro: inputPushResLeft inputPushResRight bisimReflexive)
next
case(cExt \<Psi> Q R \<Psi>')
show ?case
proof(case_tac "(\<Psi>, Q, R) \<in> ?X1")
assume "(\<Psi>, Q, R) \<in> ?X1"
then obtain x M xvec N P where Qeq: "Q = \<lparr>\<nu>x\<rparr>(M\<lparr>\<lambda>*xvec N\<rparr>.P)" and Req: "R = M\<lparr>\<lambda>*xvec N\<rparr>.\<lparr>\<nu>x\<rparr>P" and "x \<sharp> \<Psi>"
and "x \<sharp> M" and "x \<sharp> xvec" and "x \<sharp> N" by auto
obtain y::name where "y \<sharp> \<Psi>" and "y \<sharp> \<Psi>'" and "y \<sharp> M" and "y \<sharp> N" and "y \<sharp> P" and "y \<sharp> xvec"
by(generate_fresh "name") (auto simp add: fresh_prod)
moreover hence "(\<Psi> \<otimes> \<Psi>', \<lparr>\<nu>y\<rparr>(M\<lparr>\<lambda>*xvec N\<rparr>.([(x, y)] \<bullet> P)), M\<lparr>\<lambda>*xvec N\<rparr>.\<lparr>\<nu>y\<rparr>([(x, y)] \<bullet> P)) \<in> ?X" by fastforce
moreover from Qeq \<open>x \<sharp> M\<close> \<open>y \<sharp> M\<close> \<open>x \<sharp> xvec\<close> \<open>y \<sharp> xvec\<close> \<open>x \<sharp> N\<close> \<open>y \<sharp> N\<close> \<open>y \<sharp> P\<close> have "Q = \<lparr>\<nu>y\<rparr>(M\<lparr>\<lambda>*xvec N\<rparr>.([(x, y)] \<bullet> P))"
apply auto by(subst alphaRes[of y]) (auto simp add: eqvts inputChainFresh)
moreover from Req \<open>y \<sharp> P\<close> have "R = M\<lparr>\<lambda>*xvec N \<rparr>.\<lparr>\<nu>y\<rparr>([(x, y)] \<bullet> P)"
apply auto by(subst alphaRes[of y]) (auto simp add: eqvts)
ultimately show ?case by blast
next
assume "(\<Psi>, Q, R) \<notin> ?X1"
with \<open>(\<Psi>, Q, R) \<in> ?X\<close> have "(\<Psi>, Q, R) \<in> ?X2" by blast
then obtain x M xvec N P where Req: "R = \<lparr>\<nu>x\<rparr>(M\<lparr>\<lambda>*xvec N\<rparr>.P)" and Qeq: "Q = M\<lparr>\<lambda>*xvec N\<rparr>.\<lparr>\<nu>x\<rparr>P" and "x \<sharp> \<Psi>"
and "x \<sharp> M" and "x \<sharp> xvec" and "x \<sharp> N" by auto
obtain y::name where "y \<sharp> \<Psi>" and "y \<sharp> \<Psi>'" and "y \<sharp> M" and "y \<sharp> N" and "y \<sharp> P" and "y \<sharp> xvec"
by(generate_fresh "name") (auto simp add: fresh_prod)
moreover hence "(\<Psi> \<otimes> \<Psi>', \<lparr>\<nu>y\<rparr>(M\<lparr>\<lambda>*xvec N\<rparr>.([(x, y)] \<bullet> P)), M\<lparr>\<lambda>*xvec N\<rparr>.\<lparr>\<nu>y\<rparr>([(x, y)] \<bullet> P)) \<in> ?X" by fastforce
moreover from Req \<open>x \<sharp> M\<close> \<open>y \<sharp> M\<close> \<open>x \<sharp> xvec\<close> \<open>y \<sharp> xvec\<close> \<open>x \<sharp> N\<close> \<open>y \<sharp> N\<close> \<open>y \<sharp> P\<close> have "R = \<lparr>\<nu>y\<rparr>(M\<lparr>\<lambda>*xvec N\<rparr>.([(x, y)] \<bullet> P))"
apply auto by(subst alphaRes[of y]) (auto simp add: eqvts inputChainFresh)
moreover from Qeq \<open>y \<sharp> P\<close> have "Q = M\<lparr>\<lambda>*xvec N \<rparr>.\<lparr>\<nu>y\<rparr>([(x, y)] \<bullet> P)"
apply auto by(subst alphaRes[of y]) (auto simp add: eqvts)
ultimately show ?case by blast
qed
next
case(cSym \<Psi> R Q)
thus ?case by blast
qed
}
moreover obtain y::name where "y \<sharp> \<Psi>" and "y \<sharp> M" and "y \<sharp> N" and "y \<sharp> P" and "y \<sharp> xvec"
by(generate_fresh "name") auto
ultimately have "\<Psi> \<rhd> \<lparr>\<nu>y\<rparr>(M\<lparr>\<lambda>*xvec N\<rparr>.([(x, y)] \<bullet> P)) \<sim> M\<lparr>\<lambda>*xvec N\<rparr>.\<lparr>\<nu>y\<rparr>([(x, y)] \<bullet> P)" by auto
thus ?thesis using assms \<open>y \<sharp> P\<close> \<open>y \<sharp> M\<close> \<open>y \<sharp> N\<close> \<open>y \<sharp> xvec\<close>
apply(subst alphaRes[where x=x and y=y and P=P], auto)
by(subst alphaRes[where x=x and y=y and P="M\<lparr>\<lambda>*xvec N\<rparr>.P"]) (auto simp add: inputChainFresh eqvts)
qed
lemma bisimCasePushRes:
fixes x :: name
and \<Psi> :: 'b
and Cs :: "('c \<times> ('a, 'b, 'c) psi) list"
assumes "x \<sharp> (map fst Cs)"
shows "\<Psi> \<rhd> \<lparr>\<nu>x\<rparr>(Cases Cs) \<sim> Cases(map (\<lambda>(\<phi>, P). (\<phi>, \<lparr>\<nu>x\<rparr>P)) Cs)"
proof -
{
fix x::name and Cs::"('c \<times> ('a, 'b, 'c) psi) list"
assume "x \<sharp> \<Psi>" and "x \<sharp> (map fst Cs)"
let ?X1 = "{(\<Psi>, \<lparr>\<nu>x\<rparr>(Cases Cs), Cases(map (\<lambda>(\<phi>, P). (\<phi>, \<lparr>\<nu>x\<rparr>P)) Cs)) | \<Psi> x Cs. x \<sharp> \<Psi> \<and> x \<sharp> (map fst Cs)}"
let ?X2 = "{(\<Psi>, Cases(map (\<lambda>(\<phi>, P). (\<phi>, \<lparr>\<nu>x\<rparr>P)) Cs), \<lparr>\<nu>x\<rparr>(Cases Cs)) | \<Psi> x Cs. x \<sharp> \<Psi> \<and> x \<sharp> (map fst Cs)}"
let ?X = "?X1 \<union> ?X2"
have "eqvt ?X" apply(rule_tac eqvtUnion)
apply(auto simp add: eqvt_def eqvts)
apply(rule_tac x="p \<bullet> x" in exI)
apply(rule_tac x="p \<bullet> Cs" in exI)
apply(perm_extend_simp)
apply(auto simp add: eqvts)
apply(simp add: pt_fresh_bij[OF pt_name_inst, OF at_name_inst])
apply(drule_tac pi=p in pt_fresh_bij1[OF pt_name_inst, OF at_name_inst])
apply(drule_tac pi=p in pt_fresh_bij1[OF pt_name_inst, OF at_name_inst])
apply(simp add: eqvts)
apply(perm_extend_simp)
apply(simp add: eqvts)
apply(rule_tac x="p \<bullet> x" in exI)
apply(rule_tac x="p \<bullet> Cs" in exI)
apply auto
apply(perm_extend_simp)
apply(simp add: pt_fresh_bij[OF pt_name_inst, OF at_name_inst])
apply(drule_tac pi=p in pt_fresh_bij1[OF pt_name_inst, OF at_name_inst])
apply(drule_tac pi=p in pt_fresh_bij1[OF pt_name_inst, OF at_name_inst])
apply(simp add: eqvts)
apply(perm_extend_simp)
by(simp add: eqvts)
from \<open>x \<sharp> \<Psi>\<close> \<open>x \<sharp> map fst Cs\<close> have "(\<Psi>, \<lparr>\<nu>x\<rparr>(Cases Cs), Cases(map (\<lambda>(\<phi>, P). (\<phi>, \<lparr>\<nu>x\<rparr>P)) Cs)) \<in> ?X" by auto
hence "\<Psi> \<rhd> \<lparr>\<nu>x\<rparr>(Cases Cs) \<sim> Cases(map (\<lambda>(\<phi>, P). (\<phi>, \<lparr>\<nu>x\<rparr>P)) Cs)"
proof(coinduct rule: bisimCoinduct)
case(cStatEq \<Psi> Q R)
thus ?case using freshComp by(force intro: frameResFresh FrameStatEqSym)
next
case(cSim \<Psi> Q R)
thus ?case using \<open>eqvt ?X\<close>
by(fastforce intro: casePushResLeft casePushResRight bisimReflexive)
next
case(cExt \<Psi> Q R \<Psi>')
show ?case
proof(case_tac "(\<Psi>, Q, R) \<in> ?X1")
assume "(\<Psi>, Q, R) \<in> ?X1"
then obtain x Cs where Qeq: "Q = \<lparr>\<nu>x\<rparr>(Cases Cs)" and Req: "R = Cases(map (\<lambda>(\<phi>, P). (\<phi>, \<lparr>\<nu>x\<rparr>P)) Cs)"
and "x \<sharp> \<Psi>" and "x \<sharp> (map fst Cs)" by blast
obtain y::name where "y \<sharp> \<Psi>" and "y \<sharp> \<Psi>'" and "y \<sharp> Cs"
by(generate_fresh "name") (auto simp add: fresh_prod)
from \<open>y \<sharp> Cs\<close> \<open>x \<sharp> (map fst Cs)\<close> have "y \<sharp> map fst ([(x, y)] \<bullet> Cs)" by(induct Cs) (auto simp add: fresh_list_cons fresh_list_nil)
moreover with \<open>y \<sharp> \<Psi>\<close> \<open>y \<sharp> \<Psi>'\<close> have "(\<Psi> \<otimes> \<Psi>', \<lparr>\<nu>y\<rparr>(Cases ([(x, y)] \<bullet> Cs)), Cases(map (\<lambda>(\<phi>, P). (\<phi>, \<lparr>\<nu>y\<rparr>P)) ([(x, y)] \<bullet> Cs))) \<in> ?X"
by auto
moreover from Qeq \<open>y \<sharp> Cs\<close> have "Q = \<lparr>\<nu>y\<rparr>(Cases([(x, y)] \<bullet> Cs))"
apply auto by(subst alphaRes[of y]) (auto simp add: eqvts)
moreover from Req \<open>y \<sharp> Cs\<close> \<open>x \<sharp> (map fst Cs)\<close> have "R = Cases(map (\<lambda>(\<phi>, P). (\<phi>, \<lparr>\<nu>y\<rparr>P)) ([(x, y)] \<bullet> Cs))"
by(induct Cs arbitrary: R) (auto simp add: fresh_list_cons fresh_prod alphaRes)
ultimately show ?case by blast
next
assume "(\<Psi>, Q, R) \<notin> ?X1"
with \<open>(\<Psi>, Q, R) \<in> ?X\<close> have "(\<Psi>, Q, R) \<in> ?X2" by blast
then obtain x Cs where Req: "R = \<lparr>\<nu>x\<rparr>(Cases Cs)" and Qeq: "Q = Cases(map (\<lambda>(\<phi>, P). (\<phi>, \<lparr>\<nu>x\<rparr>P)) Cs)"
and "x \<sharp> \<Psi>" and "x \<sharp> (map fst Cs)" by blast
obtain y::name where "y \<sharp> \<Psi>" and "y \<sharp> \<Psi>'" and "y \<sharp> Cs"
by(generate_fresh "name") (auto simp add: fresh_prod)
from \<open>y \<sharp> Cs\<close> \<open>x \<sharp> (map fst Cs)\<close> have "y \<sharp> map fst ([(x, y)] \<bullet> Cs)" by(induct Cs) (auto simp add: fresh_list_cons fresh_list_nil)
moreover with \<open>y \<sharp> \<Psi>\<close> \<open>y \<sharp> \<Psi>'\<close> have "(\<Psi> \<otimes> \<Psi>', \<lparr>\<nu>y\<rparr>(Cases ([(x, y)] \<bullet> Cs)), Cases(map (\<lambda>(\<phi>, P). (\<phi>, \<lparr>\<nu>y\<rparr>P)) ([(x, y)] \<bullet> Cs))) \<in> ?X"
by auto
moreover from Req \<open>y \<sharp> Cs\<close> have "R = \<lparr>\<nu>y\<rparr>(Cases([(x, y)] \<bullet> Cs))"
apply auto by(subst alphaRes[of y]) (auto simp add: eqvts)
moreover from Qeq \<open>y \<sharp> Cs\<close> \<open>x \<sharp> (map fst Cs)\<close> have "Q = Cases(map (\<lambda>(\<phi>, P). (\<phi>, \<lparr>\<nu>y\<rparr>P)) ([(x, y)] \<bullet> Cs))"
by(induct Cs arbitrary: Q) (auto simp add: fresh_list_cons fresh_prod alphaRes)
ultimately show ?case by blast
qed
next
case(cSym \<Psi> R Q)
thus ?case by blast
qed
}
moreover obtain y::name where "y \<sharp> \<Psi>" and "y \<sharp> Cs" by(generate_fresh "name") auto
moreover from \<open>x \<sharp> map fst Cs\<close> have "y \<sharp> map fst([(x, y)] \<bullet> Cs)"
by(induct Cs) (auto simp add: fresh_left calc_atm)
ultimately have "\<Psi> \<rhd> \<lparr>\<nu>y\<rparr>(Cases ([(x, y)] \<bullet> Cs)) \<sim> Cases(map (\<lambda>(\<phi>, P). (\<phi>, \<lparr>\<nu>y\<rparr>P)) ([(x, y)] \<bullet> Cs))"
by auto
moreover from \<open>y \<sharp> Cs\<close> have "\<lparr>\<nu>y\<rparr>(Cases ([(x, y)] \<bullet> Cs)) = \<lparr>\<nu>x\<rparr>(Cases Cs)"
by(simp add: alphaRes eqvts)
moreover from \<open>x \<sharp> map fst Cs\<close> \<open>y \<sharp> Cs\<close> have "Cases(map (\<lambda>(\<phi>, P). (\<phi>, \<lparr>\<nu>y\<rparr>P)) ([(x, y)] \<bullet> Cs)) = Cases(map (\<lambda>(\<phi>, P). (\<phi>, \<lparr>\<nu>x\<rparr>P)) Cs)"
by(induct Cs) (auto simp add: alphaRes)
ultimately show ?thesis by auto
qed
lemma bangExt:
fixes \<Psi> :: 'b
and P :: "('a, 'b, 'c) psi"
assumes "guarded P"
shows "\<Psi> \<rhd> !P \<sim> P \<parallel> !P"
proof -
let ?X = "{(\<Psi>, !P, P \<parallel> !P) | \<Psi> P. guarded P} \<union> {(\<Psi>, P \<parallel> !P, !P) | \<Psi> P. guarded P}"
from \<open>guarded P\<close> have "(\<Psi>, !P, P \<parallel> !P) \<in> ?X" by auto
thus ?thesis
proof(coinduct rule: bisimCoinduct)
case(cStatEq \<Psi> Q R)
from \<open>(\<Psi>, Q, R) \<in> ?X\<close> obtain P where Eq: "(Q = !P \<and> R = P \<parallel> !P) \<or> (Q = P \<parallel> !P \<and> R = !P)" and "guarded P"
by auto
obtain A\<^sub>P \<Psi>\<^sub>P where FrP: "extractFrame P = \<langle>A\<^sub>P, \<Psi>\<^sub>P\<rangle>" and "A\<^sub>P \<sharp>* \<Psi>" by(rule freshFrame)
from FrP \<open>guarded P\<close> have "\<Psi>\<^sub>P \<simeq> SBottom'" by(blast dest: guardedStatEq)
from \<open>\<Psi>\<^sub>P \<simeq> SBottom'\<close> have "\<Psi> \<otimes> SBottom' \<simeq> \<Psi> \<otimes> \<Psi>\<^sub>P \<otimes> SBottom'" by(metis Identity Composition AssertionStatEqTrans Commutativity AssertionStatEqSym)
hence "\<langle>A\<^sub>P, \<Psi> \<otimes> SBottom'\<rangle> \<simeq>\<^sub>F \<langle>A\<^sub>P, \<Psi> \<otimes> \<Psi>\<^sub>P \<otimes> SBottom'\<rangle>"
by(force intro: frameResChainPres)
moreover from \<open>A\<^sub>P \<sharp>* \<Psi>\<close> have "\<langle>\<epsilon>, \<Psi> \<otimes> SBottom'\<rangle> \<simeq>\<^sub>F \<langle>A\<^sub>P, \<Psi> \<otimes> SBottom'\<rangle>"
by(rule_tac FrameStatEqSym) (fastforce intro: frameResFreshChain)
ultimately show ?case using Eq \<open>A\<^sub>P \<sharp>* \<Psi>\<close> FrP
by auto (blast dest: FrameStatEqTrans FrameStatEqSym)+
next
case(cSim \<Psi> Q R)
thus ?case by(auto intro: bangExtLeft bangExtRight bisimReflexive)
next
case(cExt \<Psi> Q R)
thus ?case by auto
next
case(cSym \<Psi> Q R)
thus ?case by auto
qed
qed
lemma bisimParPresSym:
fixes \<Psi> :: 'b
and P :: "('a, 'b, 'c) psi"
and Q :: "('a, 'b, 'c) psi"
and R :: "('a, 'b, 'c) psi"
assumes "\<Psi> \<rhd> P \<sim> Q"
shows "\<Psi> \<rhd> R \<parallel> P \<sim> R \<parallel> Q"
using assms
by(metis bisimParComm bisimParPres bisimTransitive)
lemma bisimScopeExtSym:
fixes x :: name
and Q :: "('a, 'b, 'c) psi"
and P :: "('a, 'b, 'c) psi"
assumes "x \<sharp> \<Psi>"
and "x \<sharp> Q"
shows "\<Psi> \<rhd> \<lparr>\<nu>x\<rparr>(P \<parallel> Q) \<sim> (\<lparr>\<nu>x\<rparr>P) \<parallel> Q"
using assms
by(metis bisimScopeExt bisimTransitive bisimParComm bisimSymmetric bisimResPres)
lemma bisimScopeExtChainSym:
fixes xvec :: "name list"
and Q :: "('a, 'b, 'c) psi"
and P :: "('a, 'b, 'c) psi"
assumes "xvec \<sharp>* \<Psi>"
and "xvec \<sharp>* Q"
shows "\<Psi> \<rhd> \<lparr>\<nu>*xvec\<rparr>(P \<parallel> Q) \<sim> (\<lparr>\<nu>*xvec\<rparr>P) \<parallel> Q"
using assms
by(induct xvec) (auto intro: bisimScopeExtSym bisimReflexive bisimTransitive bisimResPres)
lemma bisimParPresAuxSym:
fixes \<Psi> :: 'b
and P :: "('a, 'b, 'c) psi"
and Q :: "('a, 'b, 'c) psi"
and R :: "('a, 'b, 'c) psi"
assumes "\<Psi> \<otimes> \<Psi>\<^sub>R \<rhd> P \<sim> Q"
and "extractFrame R = \<langle>A\<^sub>R, \<Psi>\<^sub>R\<rangle>"
and "A\<^sub>R \<sharp>* \<Psi>"
and "A\<^sub>R \<sharp>* P"
and "A\<^sub>R \<sharp>* Q"
shows "\<Psi> \<rhd> R \<parallel> P \<sim> R \<parallel> Q"
using assms
by(metis bisimParComm bisimParPresAux bisimTransitive)
lemma bangDerivative:
fixes \<Psi> :: 'b
and P :: "('a, 'b, 'c) psi"
and \<alpha> :: "'a action"
and P' :: "('a, 'b, 'c) psi"
assumes "\<Psi> \<rhd> !P \<longmapsto>\<alpha> \<prec> P'"
and "\<Psi> \<rhd> P \<sim> Q"
and "bn \<alpha> \<sharp>* \<Psi>"
and "bn \<alpha> \<sharp>* P"
and "bn \<alpha> \<sharp>* Q"
and "bn \<alpha> \<sharp>* subject \<alpha>"
and "guarded Q"
obtains Q' R T where "\<Psi> \<rhd> !Q \<longmapsto>\<alpha> \<prec> Q'" and "\<Psi> \<rhd> P' \<sim> R \<parallel> !P" and "\<Psi> \<rhd> Q' \<sim> T \<parallel> !Q" and "\<Psi> \<rhd> R \<sim> T"
and "((supp R)::name set) \<subseteq> supp P'" and "((supp T)::name set) \<subseteq> supp Q'"
proof -
from \<open>\<Psi> \<rhd> !P \<longmapsto>\<alpha> \<prec> P'\<close> have "guarded P" apply - by(ind_cases "\<Psi> \<rhd> !P \<longmapsto>\<alpha> \<prec> P'") (auto simp add: psi.inject)
assume "\<And>Q' R T. \<lbrakk>\<Psi> \<rhd> !Q \<longmapsto>\<alpha> \<prec> Q'; \<Psi> \<rhd> P' \<sim> R \<parallel> !P; \<Psi> \<rhd> Q' \<sim> T \<parallel> !Q; \<Psi> \<rhd> R \<sim> T; ((supp R)::name set) \<subseteq> supp P';
((supp T)::name set) \<subseteq> supp Q'\<rbrakk> \<Longrightarrow> thesis"
moreover from \<open>\<Psi> \<rhd> !P \<longmapsto>\<alpha> \<prec> P'\<close> \<open>bn \<alpha> \<sharp>* subject \<alpha>\<close> \<open>bn \<alpha> \<sharp>* \<Psi>\<close> \<open>bn \<alpha> \<sharp>* P\<close> \<open>bn \<alpha> \<sharp>* Q\<close> \<open>\<Psi> \<rhd> P \<sim> Q\<close> \<open>guarded Q\<close>
have "\<exists>Q' T R . \<Psi> \<rhd> !Q \<longmapsto>\<alpha> \<prec> Q' \<and> \<Psi> \<rhd> P' \<sim> R \<parallel> !P \<and> \<Psi> \<rhd> Q' \<sim> T \<parallel> !Q \<and> \<Psi> \<rhd> R \<sim> T \<and>
((supp R)::name set) \<subseteq> supp P' \<and> ((supp T)::name set) \<subseteq> supp Q'"
proof(nominal_induct avoiding: Q rule: bangInduct')
case(cAlpha \<alpha> P' p Q)
then obtain Q' T R where QTrans: "\<Psi> \<rhd> !Q \<longmapsto>\<alpha> \<prec> Q'" and "\<Psi> \<rhd> P' \<sim> R \<parallel> (P \<parallel> !P)" and "\<Psi> \<rhd> Q' \<sim> T \<parallel> !Q" and "\<Psi> \<rhd> R \<sim> T"
and suppR: "((supp R)::name set) \<subseteq> supp P'" and suppT: "((supp T)::name set) \<subseteq> supp Q'"
by blast
from QTrans have "distinct(bn \<alpha>)" by(rule boundOutputDistinct)
have S: "set p \<subseteq> set(bn \<alpha>) \<times> set(bn(p \<bullet> \<alpha>))" by fact
from QTrans \<open>bn(p \<bullet> \<alpha>) \<sharp>* Q\<close> \<open>bn(p \<bullet> \<alpha>) \<sharp>* \<alpha>\<close> \<open>bn \<alpha> \<sharp>* subject \<alpha>\<close> \<open>distinct(bn \<alpha>)\<close> have "bn(p \<bullet> \<alpha>) \<sharp>* Q'"
by(drule_tac freeFreshChainDerivative) simp+
with QTrans \<open>bn(p \<bullet> \<alpha>) \<sharp>* \<alpha>\<close> S \<open>bn \<alpha> \<sharp>* subject \<alpha>\<close> have "\<Psi> \<rhd> !Q \<longmapsto>(p \<bullet> \<alpha>) \<prec> (p \<bullet> Q')"
by(force simp add: residualAlpha)
moreover from \<open>\<Psi> \<rhd> P' \<sim> R \<parallel> (P \<parallel> !P)\<close> have "(p \<bullet> \<Psi>) \<rhd> (p \<bullet> P') \<sim> (p \<bullet> (R \<parallel> (P \<parallel> !P)))"
by(rule bisimClosed)
with \<open>bn \<alpha> \<sharp>* \<Psi>\<close> \<open>bn \<alpha> \<sharp>* P\<close> \<open>bn(p \<bullet> \<alpha>) \<sharp>* \<Psi>\<close> \<open>bn(p \<bullet> \<alpha>) \<sharp>* P\<close> S have "\<Psi> \<rhd> (p \<bullet> P') \<sim> (p \<bullet> R) \<parallel> (P \<parallel> !P)"
by(simp add: eqvts)
moreover from \<open>\<Psi> \<rhd> Q' \<sim> T \<parallel> !Q\<close> have "(p \<bullet> \<Psi>) \<rhd> (p \<bullet> Q') \<sim> (p \<bullet> (T \<parallel> !Q))"
by(rule bisimClosed)
with \<open>bn \<alpha> \<sharp>* \<Psi>\<close> \<open>bn \<alpha> \<sharp>* Q\<close> \<open>bn(p \<bullet> \<alpha>) \<sharp>* \<Psi>\<close> \<open>bn(p \<bullet> \<alpha>) \<sharp>* Q\<close> S have "\<Psi> \<rhd> (p \<bullet> Q') \<sim> (p \<bullet> T) \<parallel> !Q"
by(simp add: eqvts)
moreover from \<open>\<Psi> \<rhd> R \<sim> T\<close> have "(p \<bullet> \<Psi>) \<rhd> (p \<bullet> R) \<sim> (p \<bullet> T)"
by(rule bisimClosed)
with \<open>bn \<alpha> \<sharp>* \<Psi>\<close> \<open>bn(p \<bullet> \<alpha>) \<sharp>* \<Psi>\<close> S have "\<Psi> \<rhd> (p \<bullet> R) \<sim> (p \<bullet> T)"
by(simp add: eqvts)
moreover from suppR have "((supp(p \<bullet> R))::name set) \<subseteq> supp(p \<bullet> P')"
apply(erule_tac rev_mp)
by(subst subsetClosed[of p, symmetric]) (simp add: eqvts)
moreover from suppT have "((supp(p \<bullet> T))::name set) \<subseteq> supp(p \<bullet> Q')"
apply(erule_tac rev_mp)
by(subst subsetClosed[of p, symmetric]) (simp add: eqvts)
ultimately show ?case by blast
next
case(cPar1 \<alpha> P' Q)
from \<open>\<Psi> \<rhd> P \<sim> Q\<close> \<open>\<Psi> \<rhd> P \<longmapsto>\<alpha> \<prec> P'\<close> \<open>bn \<alpha> \<sharp>* \<Psi>\<close> \<open>bn \<alpha> \<sharp>* Q\<close>
obtain Q' where QTrans: "\<Psi> \<rhd> Q \<longmapsto>\<alpha> \<prec> Q'" and "\<Psi> \<rhd> P' \<sim> Q'"
by(blast dest: bisimE simE)
from QTrans have "\<Psi> \<otimes> SBottom' \<rhd> Q \<longmapsto>\<alpha> \<prec> Q'" by(metis statEqTransition Identity AssertionStatEqSym)
hence "\<Psi> \<rhd> Q \<parallel> !Q \<longmapsto>\<alpha> \<prec> (Q' \<parallel> !Q)" using \<open>bn \<alpha> \<sharp>* Q\<close> by(rule_tac Par1) (assumption | simp)+
hence "\<Psi> \<rhd> !Q \<longmapsto>\<alpha> \<prec> (Q' \<parallel> !Q)" using \<open>guarded Q\<close> by(rule Bang)
moreover from \<open>guarded P\<close> have "\<Psi> \<rhd> P' \<parallel> !P \<sim> P' \<parallel> (P \<parallel> !P)" by(metis bangExt bisimParPresSym)
moreover have "\<Psi> \<rhd> Q' \<parallel> !Q \<sim> Q' \<parallel> !Q" by(rule bisimReflexive)
ultimately show ?case using \<open>\<Psi> \<rhd> P' \<sim> Q'\<close> by(force simp add: psi.supp)
next
case(cPar2 \<alpha> P' Q)
then obtain Q' T R where QTrans: "\<Psi> \<rhd> !Q \<longmapsto>\<alpha> \<prec> Q'" and "\<Psi> \<rhd> P' \<sim> R \<parallel> !P" and "\<Psi> \<rhd> Q' \<sim> T \<parallel> !Q" and "\<Psi> \<rhd> R \<sim> T"
and suppR: "((supp R)::name set) \<subseteq> supp P'" and suppT: "((supp T)::name set) \<subseteq> supp Q'"
by blast
note QTrans
from \<open>\<Psi> \<rhd> P' \<sim> R \<parallel> !P\<close> have "\<Psi> \<rhd> P \<parallel> P' \<sim> R \<parallel> (P \<parallel> !P)"
by(metis bisimParPresSym bisimParComm bisimTransitive bisimParAssoc)
with QTrans show ?case using \<open>\<Psi> \<rhd> Q' \<sim> T \<parallel> !Q\<close> \<open>\<Psi> \<rhd> R \<sim> T\<close> suppR suppT
by(force simp add: psi.supp)
next
case(cComm1 M N P' K xvec P'' Q)
from \<open>\<Psi> \<rhd> P \<sim> Q\<close> have "\<Psi> \<rhd> Q \<leadsto>[bisim] P" by(metis bisimE)
with \<open>\<Psi> \<rhd> P \<longmapsto>M\<lparr>N\<rparr> \<prec> P'\<close> obtain Q' where QTrans: "\<Psi> \<rhd> Q \<longmapsto>M\<lparr>N\<rparr> \<prec> Q'" and "\<Psi> \<rhd> Q' \<sim> P'"
by(force dest: simE)
from QTrans have "\<Psi> \<otimes> SBottom' \<rhd> Q \<longmapsto>M\<lparr>N\<rparr> \<prec> Q'" by(metis statEqTransition Identity AssertionStatEqSym)
moreover obtain A\<^sub>Q \<Psi>\<^sub>Q where FrQ: "extractFrame Q = \<langle>A\<^sub>Q, \<Psi>\<^sub>Q\<rangle>" and "A\<^sub>Q \<sharp>* \<Psi>" and "A\<^sub>Q \<sharp>* Q" and "A\<^sub>Q \<sharp>* M"
by(rule_tac C="(\<Psi>, Q, M)" in freshFrame) auto
note FrQ
moreover from FrQ \<open>guarded Q\<close> have "\<Psi>\<^sub>Q \<simeq> SBottom'" by(blast dest: guardedStatEq)
from \<open>\<Psi> \<rhd> !P \<longmapsto>K\<lparr>\<nu>*xvec\<rparr>\<langle>N\<rangle> \<prec> P''\<close> \<open>xvec \<sharp>* K\<close> \<open>\<Psi> \<rhd> P \<sim> Q\<close> \<open>xvec \<sharp>* \<Psi>\<close> \<open>xvec \<sharp>* P\<close> \<open>xvec \<sharp>* Q\<close> \<open>guarded Q\<close>
obtain Q'' T R where QTrans': "\<Psi> \<rhd> !Q \<longmapsto>K\<lparr>\<nu>*xvec\<rparr>\<langle>N\<rangle> \<prec> Q''" and "\<Psi> \<rhd> P'' \<sim> R \<parallel> !P" and "\<Psi> \<rhd> Q'' \<sim> T \<parallel> !Q" and "\<Psi> \<rhd> R \<sim> T"
and suppR: "((supp R)::name set) \<subseteq> supp P''" and suppT: "((supp T)::name set) \<subseteq> supp Q''" using cComm1
by fastforce
from QTrans' \<open>\<Psi>\<^sub>Q \<simeq> SBottom'\<close> have "\<Psi> \<otimes> \<Psi>\<^sub>Q \<rhd> !Q \<longmapsto>K\<lparr>\<nu>*xvec\<rparr>\<langle>N\<rangle> \<prec> Q''"
by(metis statEqTransition Identity compositionSym AssertionStatEqSym)
moreover from \<open>\<Psi> \<turnstile> M \<leftrightarrow> K\<close> \<open>\<Psi>\<^sub>Q \<simeq> SBottom'\<close> have "\<Psi> \<otimes> \<Psi>\<^sub>Q \<otimes> SBottom' \<turnstile> M \<leftrightarrow> K" by(metis statEqEnt Identity compositionSym AssertionStatEqSym)
ultimately have "\<Psi> \<rhd> Q \<parallel> !Q \<longmapsto>\<tau> \<prec> (\<lparr>\<nu>*xvec\<rparr>(Q' \<parallel> Q''))" using \<open>A\<^sub>Q \<sharp>* \<Psi>\<close> \<open>A\<^sub>Q \<sharp>* Q\<close> \<open>A\<^sub>Q \<sharp>* M\<close> \<open>xvec \<sharp>* Q\<close>
by(rule_tac Comm1) (assumption | simp)+
hence "\<Psi> \<rhd> !Q \<longmapsto>\<tau> \<prec> (\<lparr>\<nu>*xvec\<rparr>(Q' \<parallel> Q''))" using \<open>guarded Q\<close> by(rule Bang)
moreover from \<open>\<Psi> \<rhd> P'' \<sim> R \<parallel> !P\<close> \<open>guarded P\<close> \<open>xvec \<sharp>* \<Psi>\<close> have "\<Psi> \<rhd> \<lparr>\<nu>*xvec\<rparr>(P' \<parallel> P'') \<sim> \<lparr>\<nu>*xvec\<rparr>((P' \<parallel> R) \<parallel> (P \<parallel> !P))"
by(metis bisimParPresSym bangExt bisimTransitive bisimParAssoc bisimSymmetric bisimResChainPres)
with \<open>xvec \<sharp>* \<Psi>\<close> \<open>xvec \<sharp>* P\<close> have "\<Psi> \<rhd> \<lparr>\<nu>*xvec\<rparr>(P' \<parallel> P'') \<sim> (\<lparr>\<nu>*xvec\<rparr>(P' \<parallel> R)) \<parallel> (P \<parallel> !P)"
by(metis bisimScopeExtChainSym bisimTransitive psiFreshVec)
moreover from \<open>\<Psi> \<rhd> Q'' \<sim> T \<parallel> !Q\<close> \<open>xvec \<sharp>* \<Psi>\<close> \<open>xvec \<sharp>* Q\<close> have "\<Psi> \<rhd> \<lparr>\<nu>*xvec\<rparr>(Q' \<parallel> Q'') \<sim> (\<lparr>\<nu>*xvec\<rparr>(Q' \<parallel> T)) \<parallel> !Q"
by(metis bisimParPresSym bisimTransitive bisimParAssoc bisimSymmetric bisimResChainPres bisimScopeExtChainSym psiFreshVec)
moreover from \<open>\<Psi> \<rhd> R \<sim> T\<close> \<open>\<Psi> \<rhd> Q' \<sim> P'\<close> \<open>xvec \<sharp>* \<Psi>\<close> have "\<Psi> \<rhd> \<lparr>\<nu>*xvec\<rparr>(P' \<parallel> R) \<sim> \<lparr>\<nu>*xvec\<rparr>(Q' \<parallel> T)"
by(metis bisimParPresSym bisimTransitive bisimResChainPres bisimParComm bisimE(4))
moreover from suppR have "((supp(\<lparr>\<nu>*xvec\<rparr>(P' \<parallel> R)))::name set) \<subseteq> supp((\<lparr>\<nu>*xvec\<rparr>(P' \<parallel> P'')))"
by(auto simp add: psi.supp resChainSupp)
moreover from suppT have "((supp(\<lparr>\<nu>*xvec\<rparr>(Q' \<parallel> T)))::name set) \<subseteq> supp((\<lparr>\<nu>*xvec\<rparr>(Q' \<parallel> Q'')))"
by(auto simp add: psi.supp resChainSupp)
ultimately show ?case by blast
next
case(cComm2 M xvec N P' K P'' Q)
from \<open>\<Psi> \<rhd> P \<sim> Q\<close> \<open>\<Psi> \<rhd> P \<longmapsto>M\<lparr>\<nu>*xvec\<rparr>\<langle>N\<rangle> \<prec> P'\<close> \<open>xvec \<sharp>* \<Psi>\<close> \<open>xvec \<sharp>* Q\<close>
obtain Q' where QTrans: "\<Psi> \<rhd> Q \<longmapsto>M\<lparr>\<nu>*xvec\<rparr>\<langle>N\<rangle> \<prec> Q'" and "\<Psi> \<rhd> P' \<sim> Q'"
by(metis bisimE simE bn.simps)
from QTrans have "\<Psi> \<otimes> SBottom' \<rhd> Q \<longmapsto>M\<lparr>\<nu>*xvec\<rparr>\<langle>N\<rangle> \<prec> Q'" by(metis statEqTransition Identity AssertionStatEqSym)
moreover obtain A\<^sub>Q \<Psi>\<^sub>Q where FrQ: "extractFrame Q = \<langle>A\<^sub>Q, \<Psi>\<^sub>Q\<rangle>" and "A\<^sub>Q \<sharp>* \<Psi>" and "A\<^sub>Q \<sharp>* Q" and "A\<^sub>Q \<sharp>* M"
by(rule_tac C="(\<Psi>, Q, M)" in freshFrame) auto
note FrQ
moreover from FrQ \<open>guarded Q\<close> have "\<Psi>\<^sub>Q \<simeq> SBottom'" by(blast dest: guardedStatEq)
from \<open>\<Psi> \<rhd> !P \<longmapsto>K\<lparr>N\<rparr> \<prec> P''\<close> \<open>\<Psi> \<rhd> P \<sim> Q\<close> \<open>guarded Q\<close>
obtain Q'' T R where QTrans': "\<Psi> \<rhd> !Q \<longmapsto>K\<lparr>N\<rparr> \<prec> Q''" and "\<Psi> \<rhd> P'' \<sim> R \<parallel> !P" and "\<Psi> \<rhd> Q'' \<sim> T \<parallel> !Q" and "\<Psi> \<rhd> R \<sim> T"
and suppR: "((supp R)::name set) \<subseteq> supp P''" and suppT: "((supp T)::name set) \<subseteq> supp Q''" using cComm2
by fastforce
from QTrans' \<open>\<Psi>\<^sub>Q \<simeq> SBottom'\<close> have "\<Psi> \<otimes> \<Psi>\<^sub>Q \<rhd> !Q \<longmapsto>K\<lparr>N\<rparr> \<prec> Q''"
by(metis statEqTransition Identity compositionSym AssertionStatEqSym)
moreover from \<open>\<Psi> \<turnstile> M \<leftrightarrow> K\<close> \<open>\<Psi>\<^sub>Q \<simeq> SBottom'\<close> have "\<Psi> \<otimes> \<Psi>\<^sub>Q \<otimes> SBottom' \<turnstile> M \<leftrightarrow> K" by(metis statEqEnt Identity compositionSym AssertionStatEqSym)
ultimately have "\<Psi> \<rhd> Q \<parallel> !Q \<longmapsto>\<tau> \<prec> (\<lparr>\<nu>*xvec\<rparr>(Q' \<parallel> Q''))" using \<open>A\<^sub>Q \<sharp>* \<Psi>\<close> \<open>A\<^sub>Q \<sharp>* Q\<close> \<open>A\<^sub>Q \<sharp>* M\<close> \<open>xvec \<sharp>* Q\<close>
by(rule_tac Comm2) (assumption | simp)+
hence "\<Psi> \<rhd> !Q \<longmapsto>\<tau> \<prec> (\<lparr>\<nu>*xvec\<rparr>(Q' \<parallel> Q''))" using \<open>guarded Q\<close> by(rule Bang)
moreover from \<open>\<Psi> \<rhd> P'' \<sim> R \<parallel> !P\<close> \<open>guarded P\<close> \<open>xvec \<sharp>* \<Psi>\<close> have "\<Psi> \<rhd> \<lparr>\<nu>*xvec\<rparr>(P' \<parallel> P'') \<sim> \<lparr>\<nu>*xvec\<rparr>((P' \<parallel> R) \<parallel> (P \<parallel> !P))"
by(metis bisimParPresSym bangExt bisimTransitive bisimParAssoc bisimSymmetric bisimResChainPres)
with \<open>xvec \<sharp>* \<Psi>\<close> \<open>xvec \<sharp>* P\<close> have "\<Psi> \<rhd> \<lparr>\<nu>*xvec\<rparr>(P' \<parallel> P'') \<sim> (\<lparr>\<nu>*xvec\<rparr>(P' \<parallel> R)) \<parallel> (P \<parallel> !P)"
by(metis bisimScopeExtChainSym bisimTransitive psiFreshVec)
moreover from \<open>\<Psi> \<rhd> Q'' \<sim> T \<parallel> !Q\<close> \<open>xvec \<sharp>* \<Psi>\<close> \<open>xvec \<sharp>* Q\<close> have "\<Psi> \<rhd> \<lparr>\<nu>*xvec\<rparr>(Q' \<parallel> Q'') \<sim> (\<lparr>\<nu>*xvec\<rparr>(Q' \<parallel> T)) \<parallel> !Q"
by(metis bisimParPresSym bisimTransitive bisimParAssoc bisimSymmetric bisimResChainPres bisimScopeExtChainSym psiFreshVec)
moreover from \<open>\<Psi> \<rhd> R \<sim> T\<close> \<open>\<Psi> \<rhd> P' \<sim> Q'\<close> \<open>xvec \<sharp>* \<Psi>\<close> have "\<Psi> \<rhd> \<lparr>\<nu>*xvec\<rparr>(P' \<parallel> R) \<sim> \<lparr>\<nu>*xvec\<rparr>(Q' \<parallel> T)"
by(metis bisimParPresSym bisimTransitive bisimResChainPres bisimParComm)
moreover from suppR have "((supp(\<lparr>\<nu>*xvec\<rparr>(P' \<parallel> R)))::name set) \<subseteq> supp((\<lparr>\<nu>*xvec\<rparr>(P' \<parallel> P'')))"
by(auto simp add: psi.supp resChainSupp)
moreover from suppT have "((supp(\<lparr>\<nu>*xvec\<rparr>(Q' \<parallel> T)))::name set) \<subseteq> supp((\<lparr>\<nu>*xvec\<rparr>(Q' \<parallel> Q'')))"
by(auto simp add: psi.supp resChainSupp)
ultimately show ?case by blast
next
case(cBang \<alpha> P' Q)
then obtain Q' T R where QTrans: "\<Psi> \<rhd> !Q \<longmapsto>\<alpha> \<prec> Q'" and "\<Psi> \<rhd> P' \<sim> R \<parallel> (P \<parallel> !P)" and "\<Psi> \<rhd> Q' \<sim> T \<parallel> !Q" and "\<Psi> \<rhd> R \<sim> T"
and suppR: "((supp R)::name set) \<subseteq> supp P'" and suppT: "((supp T)::name set) \<subseteq> supp Q'"
by blast
from \<open>\<Psi> \<rhd> P' \<sim> R \<parallel> (P \<parallel> !P)\<close> \<open>guarded P\<close> have "\<Psi> \<rhd> P' \<sim> R \<parallel> !P" by(metis bangExt bisimParPresSym bisimTransitive bisimSymmetric)
with QTrans show ?case using \<open>\<Psi> \<rhd> Q' \<sim> T \<parallel> !Q\<close> \<open>\<Psi> \<rhd> R \<sim> T\<close> suppR suppT
by blast
qed
ultimately show ?thesis by blast
qed
lemma structCongBisim:
fixes P :: "('a, 'b, 'c) psi"
and Q :: "('a, 'b, 'c) psi"
assumes "P \<equiv>\<^sub>s Q"
shows "P \<sim> Q"
using assms
by(induct rule: structCong.induct)
(auto intro: bisimReflexive bisimSymmetric bisimTransitive bisimParComm bisimParAssoc bisimParNil bisimResNil bisimResComm bisimScopeExt bisimCasePushRes bisimInputPushRes bisimOutputPushRes bangExt)
lemma bisimBangPres:
fixes \<Psi> :: 'b
and P :: "('a, 'b, 'c) psi"
and Q :: "('a, 'b, 'c) psi"
assumes "\<Psi> \<rhd> P \<sim> Q"
and "guarded P"
and "guarded Q"
shows "\<Psi> \<rhd> !P \<sim> !Q"
proof -
let ?X = "{(\<Psi>, R \<parallel> !P, R \<parallel> !Q) | \<Psi> P Q R. \<Psi> \<rhd> P \<sim> Q \<and> guarded P \<and> guarded Q}"
let ?Y = "{(\<Psi>, P, Q) | \<Psi> P P' Q' Q. \<Psi> \<rhd> P \<sim> P' \<and> (\<Psi>, P', Q') \<in> ?X \<and> \<Psi> \<rhd> Q' \<sim> Q}"
from assms have "(\<Psi>, \<zero> \<parallel> !P, \<zero> \<parallel> !Q) \<in> ?X" by(blast intro: bisimReflexive)
moreover have "eqvt ?X"
apply(auto simp add: eqvt_def)
apply(drule_tac p=p in bisimClosed)
by fastforce
ultimately have "\<Psi> \<rhd> \<zero> \<parallel> !P \<sim> \<zero> \<parallel> !Q"
proof(coinduct rule: weakTransitiveCoinduct)
case(cStatEq \<Psi> P Q)
thus ?case by auto
next
case(cSim \<Psi> RP RQ)
from \<open>(\<Psi>, RP, RQ) \<in> ?X\<close> obtain P Q R where "\<Psi> \<rhd> P \<sim> Q" and "guarded P" and "guarded Q"
and "RP = R \<parallel> !P" and "RQ = R \<parallel> !Q"
by auto
note \<open>\<Psi> \<rhd> P \<sim> Q\<close>
moreover from \<open>eqvt ?X\<close> have "eqvt ?Y" by blast
moreover note \<open>guarded P\<close> \<open>guarded Q\<close> bisimE(2) bisimE(3) bisimE(4) statEqBisim bisimClosed bisimParAssoc[THEN bisimSymmetric]
bisimParPres bisimParPresAuxSym bisimResChainPres bisimScopeExtChainSym bisimTransitive
moreover have "\<And>\<Psi> P Q R T. \<lbrakk>\<Psi> \<rhd> P \<sim> Q; (\<Psi>, Q, R) \<in> ?Y; \<Psi> \<rhd> R \<sim> T\<rbrakk> \<Longrightarrow> (\<Psi>, P, T) \<in> ?Y"
by auto (metis bisimTransitive)
moreover have "\<And>\<Psi> P Q R. \<lbrakk>\<Psi> \<rhd> P \<sim> Q; guarded P; guarded Q\<rbrakk> \<Longrightarrow> (\<Psi>, R \<parallel> !P, R \<parallel> !Q) \<in> ?Y" by(blast intro: bisimReflexive)
moreover have "\<And>\<Psi> P \<alpha> P' Q. \<lbrakk>\<Psi> \<rhd> !P \<longmapsto>\<alpha> \<prec> P'; \<Psi> \<rhd> P \<sim> Q; bn \<alpha> \<sharp>* \<Psi>; bn \<alpha> \<sharp>* P; bn \<alpha> \<sharp>* Q; guarded Q; bn \<alpha> \<sharp>* subject \<alpha>\<rbrakk> \<Longrightarrow>
\<exists>Q' R T. \<Psi> \<rhd> !Q \<longmapsto>\<alpha> \<prec> Q' \<and> \<Psi> \<rhd> P' \<sim> R \<parallel> !P \<and> \<Psi> \<rhd> Q' \<sim> T \<parallel> !Q \<and>
\<Psi> \<rhd> R \<sim> T \<and> ((supp R)::name set) \<subseteq> supp P' \<and>
((supp T)::name set) \<subseteq> supp Q'"
by(blast elim: bangDerivative)
ultimately have "\<Psi> \<rhd> R \<parallel> !P \<leadsto>[?Y] R \<parallel> !Q"
by(rule bangPres)
with \<open>RP = R \<parallel> !P\<close> \<open>RQ = R \<parallel> !Q\<close> show ?case
by blast
next
case(cExt \<Psi> RP RQ \<Psi>')
thus ?case by(blast dest: bisimE)
next
case(cSym \<Psi> RP RQ)
thus ?case by(blast dest: bisimE)
qed
thus ?thesis
by(metis bisimTransitive bisimParNil bisimSymmetric bisimParComm)
qed
end
end
|
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State Before: α : Type ?u.27674
M✝ : Type u
N : Type v
G : Type w
H : Type x
A : Type y
B : Type z
R : Type u₁
S : Type u₂
inst✝² : Monoid M✝
inst✝¹ : AddMonoid A
M : Type u_1
inst✝ : Monoid M
x : M
m n : ℕ
h : x ^ n = 1
⊢ x ^ m = x ^ (m % n) State After: α : Type ?u.27674
M✝ : Type u
N : Type v
G : Type w
H : Type x
A : Type y
B : Type z
R : Type u₁
S : Type u₂
inst✝² : Monoid M✝
inst✝¹ : AddMonoid A
M : Type u_1
inst✝ : Monoid M
x : M
m n : ℕ
h : x ^ n = 1
t : x ^ m = x ^ (n * (m / n) + m % n)
⊢ x ^ m = x ^ (m % n) Tactic: have t : x ^ m = x ^ (n * (m / n) + m % n) :=
congr_arg (fun a => x ^ a) ((Nat.add_comm _ _).trans (Nat.mod_add_div _ _)).symm State Before: α : Type ?u.27674
M✝ : Type u
N : Type v
G : Type w
H : Type x
A : Type y
B : Type z
R : Type u₁
S : Type u₂
inst✝² : Monoid M✝
inst✝¹ : AddMonoid A
M : Type u_1
inst✝ : Monoid M
x : M
m n : ℕ
h : x ^ n = 1
t : x ^ m = x ^ (n * (m / n) + m % n)
⊢ x ^ m = x ^ (m % n) State After: α : Type ?u.27674
M✝ : Type u
N : Type v
G : Type w
H : Type x
A : Type y
B : Type z
R : Type u₁
S : Type u₂
inst✝² : Monoid M✝
inst✝¹ : AddMonoid A
M : Type u_1
inst✝ : Monoid M
x : M
m n : ℕ
h : x ^ n = 1
t : x ^ m = x ^ (n * (m / n) + m % n)
⊢ x ^ m = x ^ (m % n) Tactic: dsimp at t State Before: α : Type ?u.27674
M✝ : Type u
N : Type v
G : Type w
H : Type x
A : Type y
B : Type z
R : Type u₁
S : Type u₂
inst✝² : Monoid M✝
inst✝¹ : AddMonoid A
M : Type u_1
inst✝ : Monoid M
x : M
m n : ℕ
h : x ^ n = 1
t : x ^ m = x ^ (n * (m / n) + m % n)
⊢ x ^ m = x ^ (m % n) State After: no goals Tactic: rw [t, pow_add, pow_mul, h, one_pow, one_mul] |
function [gal] = ft32gal(ft3)
% Convert volume from cubic feet to US liquid gallons.
% Chad Greene 2012
gal = ft3*7.4805194805; |
const UIntOrChar = Union{Unsigned, AbstractChar}
struct StaticString{N, T<:Unsigned} <: AbstractString
data::NTuple{N, T}
function StaticString{N, T}(t::NTuple{M, <:UIntOrChar}) where {N, T, M}
N == M || throw(DimensionMismatch(
"cannot construct StaticString{$N, $T} from input of length $M"))
new{N, T}(t)
end
end
StaticString{N, T}(cs::UIntOrChar...) where {N, T} = StaticString{N, T}(cs)
StaticString{N, T}(fs::StaticString{N, T}) where {N, T} = fs
StaticString{N, T}(s) where {N, T} = StaticString{N, T}(String(s)...)
_units_to_type(n::Int) = ifelse(n == 1, UInt8, ifelse(n==2, UInt16, UInt32))
macro static_str(s)
quote
StaticString{
length($(esc(s))),
_units_to_type(maximum(ncodeunits, $(esc(s))))
}($(esc(s)))
end
end
convert(::Type{StaticString{N, T}}, s::AbstractString) where {N, T} = StaticString{N, T}(s)
ncodeunits(s::StaticString{N, T}) where {T,N} = N
sizeof(s::StaticString) = sizeof(s.data)
length(s::StaticString) = ncodeunits(s)
length(S::Type{StaticString{N, T}}) where {N, T} = N
lastindex(s::StaticString{N, T}) where {N, T} = N
function iterate(s::StaticString{N, T}, i::Int = 1) where {N, T}
i > N && return nothing
return Char(s[i]), i+1
end
codeunit(::StaticString{N, T}) where {N, T} = T
@propagate_inbounds codeunit(s::StaticString, i::Integer) = s.data[i]
@propagate_inbounds getindex(s::StaticString, i::Integer)::Char = s.data[i]
isvalid(s::StaticString, i::Integer) = checkbounds(Bool, s, i)
|
function p=polelague(n)
% p=polegend(n)
% Almacena en las filas de la matriz p los coefs de los polinomios de Legendre
p(1,1)=1;
p(2,1:2)=[-1 1];
for k=2:n
p(k+1,1:k+1)=((2*(k-2)*[0 p(k,1:k)]+3*[0 p(k,1:k)]-[p(k,1:k) 0]-(k-1).^2*[0 0 p(k-1,1:k-1)]));
end
|
module ExVectDecEq
import Decidable.Equality
%default total
data Vect : Nat -> Type -> Type where
Nil : Vect Z a
(::) : a -> Vect k a -> Vect (S k) a
{-
interface DecEq ty where
decEq : (val1 : ty) -> (val2 : ty) -> Dec (val1 = val2)
data Dec : (prop : Type) -> Type where
Yes : (prf : prop) -> Dec prop
No : (contra ; prop -> Void) -> Dec prop
-}
headUnequal : DecEq a => { xs: Vect n a } -> { ys : Vect n a } ->
(contra : (x = y) -> Void) -> ((x :: xs) = (y :: ys)) -> Void
headUnequal contra Refl = contra Refl
tailUnequal : DecEq a => { xs : Vect n a } -> { ys : Vect n a } ->
(contra : (xs = ys) -> Void) -> ((x :: xs) = (y:: ys)) -> Void
tailUnequal contra Refl = contra Refl
DecEq a => DecEq (Vect n a) where
decEq [] [] = Yes Refl
decEq [] _ = No ?nilNotEqualToCons
decEq _ [] = No ?consNotEqualToNil
decEq (x :: xs) (y :: ys) = case decEq x y of
No contra => No (headUnequal contra)
Yes Refl => case decEq xs ys of
No contra => No (tailUnequal contra)
Yes Refl => Yes Refl
|
module Intro
sm : List Nat -> Nat
sm [] = 0
sm (x :: xs) = x + (sm xs)
fct : Nat -> Nat
fct Z = 1
fct (S k) = (S k) * (fct k)
fbp : Nat -> (Nat, Nat)
fbp Z = (1, 1)
fbp (S k) = (snd (fbp k), fst (fbp k) + snd (fbp k))
fib : Nat -> Nat
fib n = fst (fbp n)
public export
add : Nat -> Nat -> Nat
add Z j = j
add (S k) j = S (add k j)
mul : Nat -> Nat -> Nat
mul Z j = Z
mul (S k) j = add j (mul k j)
sub : (n: Nat) -> (m : Nat) -> (LTE m n) -> Nat
sub n Z LTEZero = n
sub (S right) (S left) (LTESucc x) = sub right left x
oneLTEFour : LTE 1 4
oneLTEFour = LTESucc LTEZero
fourMinusOne : Nat
fourMinusOne = sub 4 1 oneLTEFour
reflLTE : (n: Nat) -> LTE n n
reflLTE Z = LTEZero
reflLTE (S k) = LTESucc (reflLTE k)
sillyZero: Nat -> Nat
sillyZero n = sub n n (reflLTE n)
idNat : Nat -> Nat
idNat = \x => x
loop: Nat -> Nat
loop k = loop (S k)
|
Formal statement is: proposition Schwarz_reflection: assumes "open S" and cnjs: "cnj ` S \<subseteq> S" and holf: "f holomorphic_on (S \<inter> {z. 0 < Im z})" and contf: "continuous_on (S \<inter> {z. 0 \<le> Im z}) f" and f: "\<And>z. \<lbrakk>z \<in> S; z \<in> \<real>\<rbrakk> \<Longrightarrow> (f z) \<in> \<real>" shows "(\<lambda>z. if 0 \<le> Im z then f z else cnj(f(cnj z))) holomorphic_on S" Informal statement is: If $f$ is holomorphic on the upper half-plane and continuous on the real line, then the Schwarz reflection of $f$ is holomorphic on the whole plane. |
## these are differently named than SymPy or missing or ...
Base.abs(x::SymbolicObject) = sympy.Abs(x)
Base.abs2(x::SymbolicObject) = x * conj(x)
Base.max(x::Sym, a) = sympy.Max(x, a)
Base.min(x::Sym, a) = sympy.Min(x, a)
Base.cbrt(x::Sym) = x^(1//3)
Base.ceil(x::Sym) = sympy.ceiling(x)
## Trig
Base.asech(z::Sym) = log(sqrt(1/z-1)*sqrt(1/z+1) + 1/z)
Base.acsch(z::Sym) = log(sqrt(1+1/z^2) + 1/z) ## http://mathworld.wolfram.com/InverseHyperbolicCosecant.html
Base.atan(y::Sym, x) = sympy.atan2(y,x)
Base.sinc(x::Sym) = iszero(x) ? one(x) : sin(PI*x)/(PI*x)
cosc(x::Sym) = diff(sinc(x))
Base.sincos(x::Sym) = (sin(x), cos(x))
Base.sinpi(x::Sym) = sympy.sin(x*PI)
Base.cospi(x::Sym) = sympy.cos(x*PI)
degree_variants = (:sind, :cosd, :tand, :cotd, :secd, :cscd,
:asind, :acosd, :atand, :acotd, :asecd, :acscd)
for methvar in degree_variants
meth = Symbol(String(methvar)[1:end-1])
@eval begin
(Base.$methvar)(ex::SymbolicObject) = ($meth)((PI/180)*ex)
end
end
Base.rad2deg(x::Sym) = (x * 180) / PI
Base.deg2rad(x::Sym) = (x * PI) / 180
Base.hypot(x::Sym, y::Number) = hypot(promote(x,y)...)
Base.hypot(xs::Sym...) = sqrt(sum(abs(xᵢ)^2 for xᵢ ∈ xs))
## exponential
Base.log1p(x::Sym) = sympy.log(1 + x)
Base.log(x::Sym) = sympy.log(x)
Base.log(b::Number, x::Sym) = sympy.log(x, b)
Base.log2(x::SymbolicObject) = log(2,x)
Base.log10(x::SymbolicObject) = log(10,x)
## calculus.
## use a pair for limit x=>0
limit(x::SymbolicObject, xc::Pair, args...;kwargs...) = limit(x, xc[1], xc[2], args...;kwargs...)
## allow a function
limit(f::Function, x::Sym, c;kwargs...) = limit(Sym(f(x)), x, c; kwargs...)
function limit(f::Function, c;kwargs...)
@vars x
limit(f, x, c; kwargs...)
end
## This is type piracy and a bad idea
function Base.diff(f::Function, n::Integer=1)
@vars x
sympy.diff(f(x), x, n)
end
## integrate(ex,a,b)
function integrate(ex::SymbolicObject, a::Number, b::Number)
fs = free_symbols(ex)
if length(fs) !== 1
@warn "Need exactly on free symbol. Use `integrate(ex, (x, a, b))` instead"
return
end
integrate(ex, (fs[1], a, b))
end
function integrate(f::Function, a::Number, b::Number)
@vars x
sympy.integrate(f(x), (x, a, b))
end
function integrate(f::Function)
@syms x
sympy.integrate(f(x), x)
end
## Add interfaces for solve, nonlinsolve when vector of equations passed in
## An alternative to Eq(lhs, rhs) following Symbolics.jl
"""
lhs ~ rhs
Specify an equation.
Alternative syntax to `Eq(lhs, rhs)` or `lhs ⩵ rhs` (`\\Equal[tab]`) following `Symbolics.jl`.
"""
Base.:~(lhs::Number, rhs::SymbolicObject) = Eq(lhs, rhs)
Base.:~(lhs::SymbolicObject, rhs::Number) = Eq(lhs, rhs)
Base.:~(lhs::SymbolicObject, rhs::SymbolicObject) = Eq(lhs, rhs)
"""
solve
Use `solve` to solve algebraic equations.
Examples:
```julia
julia> using SymPy
julia> @syms x y a b c d
(x, y, a, b, c, d)
julia> solve(x^2 + 2x + 1, x) # [-1]
1-element Vector{Sym}:
-1
julia> solve(x^2 + 2a*x + a^2, x) # [-a]
1-element Vector{Sym}:
-a
julia> solve([a*x + b*y-3, c*x + b*y - 1], [x,y]) # Dict(y => (a - 3*c)/(b*(a - c)),x => 2/(a - c))
Dict{Any, Any} with 2 entries:
y => (a - 3*c)/(a*b - b*c)
x => 2/(a - c)
```
!!! note
A very nice example using `solve` is a [blog](https://newptcai.github.io/euclidean-plane-geometry-with-julia.html) entry on [Napolean's theorem](https://en.wikipedia.org/wiki/Napoleon%27s_theorem) by Xing Shi Cai.
"""
solve() = ()
solve(V::Vector{T}, args...; kwargs...) where {T <: SymbolicObject} =
sympy.solve(V, args...; kwargs...)
"""
nonlinsolve
Note: if passing variables in use a tuple (e.g., `(x,y)`) and *not* a vector (e.g., `[x,y]`).
"""
nonlinsolve(V::AbstractArray{T,N}, args...; kwargs...) where {T <: SymbolicObject, N} =
sympy.nonlinsolve(V, args...; kwargs...)
linsolve(V::AbstractArray{T,N}, args...; kwargs...) where {T <: SymbolicObject, N} =
sympy.linsolve(V, args...; kwargs...)
linsolve(Ts::Tuple, args...; kwargs...) where {T <: SymbolicObject} =
sympy.linsolve(Ts, args...; kwargs...)
## dsolve allowing initial condiation to be specified
"""
dsolve(eqn, var, args..,; ics=nothing, kwargs...)
Call `sympy.dsolve`.
The initial conditions are specified with a dictionary.
Example:
```jldoctest dsolve
julia> using SymPy
julia> @syms α, x, f(), g()
(α, x, f, g)
julia> ∂ = Differential(x)
Differential(x)
julia> eqn = ∂(f(x)) ~ α * x
d
──(f(x)) = x⋅α
dx
```
```julia
julia> dsolve(eqn)
2
x ⋅α
f(x) = C₁ + ────
2
```
```jldoctest dsolve
julia> dsolve(eqn(α=>2); ics=Dict(f(0)=>1)) |> print # fill in parameter, initial condition
Eq(f(x), x^2 + 1)
julia> eqn = ∂(∂(f(x))) ~ -f(x); print(eqn)
Eq(Derivative(f(x), (x, 2)), -f(x))
julia> dsolve(eqn)
f(x) = C₁⋅sin(x) + C₂⋅cos(x)
julia> dsolve(eqn; ics = Dict(f(0)=>1, ∂(f)(0) => -1))
f(x) = -sin(x) + cos(x)
julia> eqn = ∂(∂(f(x))) - f(x) - exp(x);
julia> dsolve(eqn, ics=Dict(f(0) => 1, f(1) => Sym(1//2))) |> print # not just 1//2
Eq(f(x), (x/2 + (-exp(2) - 2 + E)/(-2 + 2*exp(2)))*exp(x) + (-E + 3*exp(2))*exp(-x)/(-2 + 2*exp(2)))
```
Systems
```jldoctest dsolve
julia> @syms x() y() t g
(x, y, t, g)
julia> ∂ = Differential(t)
Differential(t)
julia> eqns = [∂(x(t)) ~ y(t), ∂(y(t)) ~ x(t)]
2-element Vector{Sym}:
Eq(Derivative(x(t), t), y(t))
Eq(Derivative(y(t), t), x(t))
julia> dsolve(eqns)
2-element Vector{Sym}:
Eq(x(t), -C1*exp(-t) + C2*exp(t))
Eq(y(t), C1*exp(-t) + C2*exp(t))
julia> dsolve(eqns, ics = Dict(x(0) => 1, y(0) => 2))
2-element Vector{Sym}:
Eq(x(t), 3*exp(t)/2 - exp(-t)/2)
Eq(y(t), 3*exp(t)/2 + exp(-t)/2)
julia> eqns = [∂(∂(x(t))) ~ 0, ∂(∂(y(t))) ~ -g]
2-element Vector{Sym}:
Eq(Derivative(x(t), (t, 2)), 0)
Eq(Derivative(y(t), (t, 2)), -g)
julia> dsolve(eqns) # can't solve for initial conditions though! (NotAlgebraic)
2-element Vector{Sym}:
x(t) = C₁ + C₂⋅t
Eq(y(t), C3 + C4*t - g*t^2/2)
julia> @syms t x() y()
(t, x, y)
julia> eq = (∂(x)(t) ~ x(t)*y(t)*sin(t), ∂(y)(t) ~ y(t)^2 * sin(t))
(Eq(Derivative(x(t), t), x(t)*y(t)*sin(t)), Eq(Derivative(y(t), t), y(t)^2*sin(t)))
```
```julia
julia> dsolve(eq) # returns a set to be `collect`ed:
PyObject {Eq(x(t), -exp(C1)/(C2*exp(C1) - cos(t))), Eq(y(t), -1/(C1 - cos(t)))}
```
```julia
julia> dsolve(eq) |> collect
2-element Vector{Any}:
Eq(x(t), -exp(C1)/(C2*exp(C1) - cos(t)))
Eq(y(t), -1/(C1 - cos(t)))
```
"""
function dsolve(eqn, args...;
ics::Union{Nothing, AbstractDict, Tuple}=nothing,
kwargs...)
if isa(ics, Tuple) # legacy
_dsolve(eqn, args...; ics=ics, kwargs...)
else
sympy.dsolve(eqn, args...; ics=ics, kwargs...)
end
end
rhs(x::SymbolicObject) = pycall_hasproperty(x, :rhs) ? x.rhs : x
lhs(x::SymbolicObject) = pycall_hasproperty(x, :lhs) ? x.lhs : x
export dsolve, rhs, lhs
## ---- deprecate ----
## used with ics=(u,0,1) style
function _dsolve(eqn::Sym, args...; ics=nothing, kwargs...)
Base.depwarn("Use of tuple(s), `(u, x₀, u₀)`, to specify initial conditions is deprecated. Use a dictionary: `ics=Dict(u(x₀) => u₀)`.", :_dsolve)
if isempty(args)
var = first(free_symbols(eqn))
else
var = first(args)
end
# var might be f(x) or x, we want `x`
if Introspection.classname(var) != "Symbol"
var = first(var.args)
end
## if we have one initial condition, can be passed in a (u,x0,y0) *or* ((u,x0,y0),)
## if more than oneq a tuple of tuples
if eltype(ics) <: Tuple
__dsolve(eqn, var, ics; kwargs...)
else
__dsolve(eqn, var, (ics,); kwargs...)
end
end
function __dsolve(eqn::Sym, var::Sym, ics; kwargs...)
if length(ics) == 0
throw(ArgumentError("""Some initial value specification is needed.
Specifying the function, as in `dsolve(ex, f(x))`, is deprecated.
Use `sympy.dsolve(ex, f(x); kwargs...)` directly for that underlying interface.
"""))
end
out = sympy.dsolve(eqn; kwargs...)
ord = sympy.ode_order(eqn, var)
## `out` may be an array of solutions. If so we do each one.
## we want to use an array for output only if needed
if !isa(out, Array)
return _solve_ivp(out, var, ics,ord)
else
output = Sym[]
for o in out
a = _solve_ivp(o, var, ics,ord)
a != nothing && push!(output, a)
end
return length(output) == 1 ? output[1] : output
end
end
## Helper.
## out is an equation in var with constants. Args are intial conditions
## Return `nothing` if initial condition is not satisfied (found by `solve`)
function _solve_ivp(out, var, args, o)
eqns = Sym[(diff(out.rhs(), var, f.n))(var=>x0) - y0 for (f, x0, y0) in args]
sols = solve(eqns, Sym["C$i" for i in 1:o], dict=true)
if length(sols) == 0
return nothing
end
## massage output
## Might have more than one solution, though unlikely. But if we substitute a variable
## for y0 we will get an array back from solve which may have length 1.
if isa(sols, Array)
if length(sols) == 1
sols = sols[1]
else
return [out([Pair(k,v) for (k,v) in sol]...) for sol in sols]
end
end
out([Pair(k,v) for (k,v) in sols]...)
end
## For System Of Ordinary Differential Equations
## may need to collect return values
# dsolve(eqs::Union{Array, Tuple}, args...; kwargs...) = sympy.dsolve(eqs, args...; kwargs...)
|
theory Hotel_Example_Small_Generator
imports Hotel_Example "~~/src/HOL/Library/Predicate_Compile_Alternative_Defs"
begin
ML_file "~~/src/HOL/Tools/Predicate_Compile/predicate_compile_quickcheck.ML"
declare Let_def[code_pred_inline]
lemma [code_pred_inline]: "insert == (%y A x. y = x | A x)"
by (auto simp add: insert_iff[unfolded mem_def] fun_eq_iff intro!: eq_reflection)
lemma [code_pred_inline]: "(op -) == (%A B x. A x \<and> \<not> B x)"
by (auto simp add: Diff_iff[unfolded mem_def] fun_eq_iff intro!: eq_reflection)
instantiation room :: small_lazy
begin
definition
"small_lazy i = Lazy_Sequence.single Room0"
instance ..
end
instantiation key :: small_lazy
begin
definition
"small_lazy i = Lazy_Sequence.append (Lazy_Sequence.single Key0) (Lazy_Sequence.append (Lazy_Sequence.single Key1) (Lazy_Sequence.append (Lazy_Sequence.single Key2) (Lazy_Sequence.single Key3)))"
instance ..
end
instantiation guest :: small_lazy
begin
definition
"small_lazy i = Lazy_Sequence.append (Lazy_Sequence.single Guest0) (Lazy_Sequence.single Guest1)"
instance ..
end
ML \<open>
val small_15_active = Attrib.setup_config_bool @{binding quickcheck_small_14_active} (K false);
val small_14_active = Attrib.setup_config_bool @{binding quickcheck_small_15_active} (K false);
\<close>
setup \<open>
Context.theory_map (Quickcheck.add_tester ("small_generators_depth_14",
(small_14_active, Predicate_Compile_Quickcheck.test_goals
(Predicate_Compile_Aux.Pos_Generator_DSeq, true, true, 14))))
#> Context.theory_map (Quickcheck.add_tester ("small_generators_depth_15",
(small_15_active, Predicate_Compile_Quickcheck.test_goals
(Predicate_Compile_Aux.Pos_Generator_DSeq, true, true, 15))))
\<close>
lemma
"hotel s ==> feels_safe s r ==> g \<in> isin s r ==> owns s r = Some g"
(*quickcheck[tester = small_generators_depth_14, finite_types = false, iterations = 1, size = 1, timeout = 1200.0, expect = no_counterexample]*)
quickcheck[tester = small_generators_depth_15, finite_types = false, iterations = 1, size = 1, timeout = 2400.0, expect = counterexample]
oops
end
|
Parameter A : Set.
Definition Eq : A -> A -> Prop :=
fun a => fun b => forall P : A -> Prop, P a <-> P b.
Lemma Eq_eq : forall x y, Eq x y <-> x = y.
Proof.
unfold Eq.
intros x y; split.
- intro E.
destruct (E (fun z => x = z)).
apply (H (eq_refl x)).
- intro E.
subst; split; auto.
Qed. |
# Managing Complexity with BondGraphTools
https://github.com/peter-cudmore/seminars/CellML-2019
Dr. Peter Cudmore.
Systems Biology Labratory,
The School of Chemical and Biomedical Engineering,
The University of Melbourne.
```python
## Make sympy print pretty text
import sympy as sp
sp.init_printing()
# Load the examples
from examples import *
# Import the source-code inspector
from inspect import getsourcelines
# Start the julia interpreter
from BondGraphTools.config import config
_ = config.start_julia()
# Parameters from Safaei et.al. 2018
blood = Fluid(density=0.004, viscosity=1050)
# Approximate parameters for 10cm worth of artery.
artery = Vessel(radius=0.03, thickness=0.005, youngs_modulus=0.16e6, length=0.1)
```
## The problem with big systems is that they're _big_...
<center> </center>
## Complex Physical Systems
A _complex physical system_ has:
* many parts or subsystems (High-dimensional).
* subsystems which are not all of the same (Heterogenenous).
* subsystems which are complicated (Nonlinear and/or Noisy).
* well defined boundaries between the subsystems (Network Topology).
* **subsystems interact via resource exchange (Conservation Laws).**
Examples include: Biochemical Networks, Ecosystems, Electrical Grids, Hydraulic networks, etc.
## Some Obvious Questions
Human Metabolism Map @ https://www.vmh.life
<center> </center>
1. Why?
2. Where do we get the topology from?
3. What are the dynamic features?
4. How do we paramaterise them?
5. What is a useful representation?
6. How do we manipulate and refine our model?
7. How should one manage complexity at scale?
## Why?
To predict, understand and control systemic phenomenon such as emergence, multiscale dynamics and long-range interactions.
To _rationally engineer_ systems in general and biological systems in particular.
## Where do we get network data?
For Systems Biology:
- Databases (Brenda, KEGG, BIGG, SABIO-RK, reactome)
- Model Repositories (Physiome Project, BioModels)
- Publications (supp. materials, images, tables)
- Collaborators.
More generally: connectivity maps.
## A Subtle Issue: Dynamics.
Recall that Complex Physical Systems have a _network topology_ of _conservative interactions_.
- As edges are 'resource exchange',
- _effort_ must be imposed to move resources from one node to another,
- which move at some corresponding _flow_ rate.
- Nodes must either store resources, or conservatively pass them along.
- The exceptions are the boundary conditions such as resource sources and sinks (including dissipation).
## A Major Issue: Parameterisation.
Ideally we would use parameters that:
- are physically meaningful,
- are able to be theoretically estimated,
- can be derived from underlying physics/chemistry,
- or that have been shown to be consistent across many experiemental conditions.
- and hence tabulated (or able to be derived from tablated data)
(but this is easier said than done)
## The Relevant Questions for Today.
- What is a useful representation of a complex physical system?
- What is a good way to manipulate models?
- How do we manage complexity and scale?
_Object Oriented Modelling via_ `BondGraphTools`.
## 'Energetic Systems' as an Object Oriented Programming.
Object Oriented Programming (OOP) is a software development paradigm that seeks to manage large, complicated projects by breaking problems into _data_ plus _methods_ that act on the data.
Three big ideas in OOP are:
1. _Inheritance_ or is-a relationships.
2. _Composition_ or has-a relationships.
3. _Encapsulation_ or infomation hiding.
This allows for _hierarchical_ and _modular_ design which reduces model complexity.
```python
import BondGraphTools
help(BondGraphTools)
```
Help on package BondGraphTools:
NAME
BondGraphTools
DESCRIPTION
BondGraphTools
==============
BondGraphTools is a python library for symbolic modelling
of energetic systems.
Package Documentation::
https://bondgraphtools.readthedocs.io/
Source::
https://github.com/BondGraphTools/BondGraphTools
Bug reports:
https://github.com/BondGraphTools/BondGraphTools/issues
Simple Example
--------------
Build and simulate a RLC driven RLC circuit::
import BondGraphTools as bgt
# Create a new model
model = bgt.new(name="RLC")
# Create components
# 1 Ohm Resistor
resistor = bgt.new("R", name="R1", value=1.0)
# 1 Henry Inductor
inductor = bgt.new("L", name="L1", value=1.0)
# 1 Farad Capacitor
capacitor = bgt.new("C", name="C1", value=1.0)
# Conservation Law
law = bgt.new("0") # Common voltage conservation law
# Connect the components
connect(law, resistor)
connect(law, capacitor)
connect(law, inductor)
# produce timeseries data
t, x = simulate(model, x0=[1,1], timespan=[0, 10])
Bugs
----
Please report any bugs `here <https://github.com/BondGraphTools/BondGraphTools/issues>`_,
or fork the repository and submit a pull request.
License
-------
Released under the Apache 2.0 License::
Copyright (C) 2018
Peter Cudmore <[email protected]>
PACKAGE CONTENTS
actions
algebra
atomic
base
component_manager
compound
config
exceptions
fileio
port_hamiltonian
reaction_builder
sim_tools
version
view
DATA
version = '0.3.7'
FILE
/Users/pete/Workspace/BondGraphTools/BondGraphTools/__init__.py
### Bond Graphs, Port Hamiltonians and BondGraphTools
_Bond graphs_ are a **graphical** framework for modelling energetic systems.
_Port Hamiltonians_ are a **geometric** framework for modelling energetic systems.
_BondGraphTools_ is a **programmatic** framework for modelling energetic systems.
## Goals of this talk.
In the remaining time i hope to convince you that:
- Object oriented modelling is suited to complex physical systems.
- `BondGraphTools` is a useful library for this purpose.
- Incorporating scripting into your work is worthwhile.
- `BondGraphTools` and `libCellML`
# Object Oriented Modelling for Energetic Systems
Inheritance, Composition and Encapsulation
## Inheritance
For networked systems, _inheritance_ means that for each node or subsystems have:
- conditions on the interals.
- a description of the interface.
## Inheritance
### Nodes are particular _energetic systems_
Each node is described by a set of differential-algebraic equations; the constitutive relations
$$\Phi(\dot{x},x,e,f) = 0.$$
## Inheritance
### Edges are constraints on port variables.
An edge represents how state is shared between systems.
## Inheritance
#### Example Node Subclasses:
- Resistive dissipation.
- Elastic defomation of vessel walls.
- Conservation of mass.
_This_ chemical reation or _that particular_ aeortic compliance are **instances** of a particular subclass.
## Anatomy of an Energetic System
Nodes can have _state_, represented by the variables $(\dot{x}, x)$
Examples with state:
- Charge accumulation.
- Chemical concentration.
- Elastic deformation.
## Anatomy of an Energetic System
Nodes can alternatively be stateless.
Examples without state:
- resistance / friction
- semiconductance
- elementary chemical reactions
## Anatomy of an Energetic System
Nodes can have _external ports_ (here labeled $[1]$ , $[2]$ and $[3]$) which provide an interface to the external environment.
Examples of ports:
- The poles of a dielectic membrane
- The open end of a vessel segment.
- The enzyme/substrate/cofactor mix of a biochemical reaction.
## Anatomy of an Energetic System
$\Phi$ relates _internal state_ to _external environment_.
The relation $\Phi$ may have parameters such as:
- Temperature and pressure
- Gibbs formation energy
- Vessel wall compliance
- Electrical resistance
## Composition
_Composition_ means that we can replace subgraphs with a single node and vice-versa.
_This means we both abstract parts of the model, or refine parts of the model as necessary!_
## Composition
Recall that each node is a set of DAE's
$$\Phi_j(\dot{x},x,e,f) = 0.$$
One can simply take the direct sum of the systems to produce a composite system
$$
\Phi_0 = \left(
\Phi_6,
\Phi_7,\Phi_8,
\Phi_{edges}\right)^T = 0
$$
## Compostion
For
$$
\Phi_0 = \left(
\Phi_6,
\Phi_7, \Phi_8
\Phi_{edges}\right)^T = 0
$$
the relation
$$\Phi_{edges} = (e^i_\alpha - e^j_\beta, f^i_\alpha + f^j_\beta,\ldots)$$
turns edges between node-ports pairs $(i,\alpha)$ and $(j,\beta)$ into constraints upon the composed system $\Phi_0$.
## Compositon
_Like joining pipe segments!_
- $e^i_\alpha - e^j_\beta = 0$ implies 'pressure'(effort) is identical at the join.
- $f^i_\alpha + f^j_\beta = 0$ implies the flow goes losslessly from one, to the other.
## Encapsulation
Encapsulation = Modularity!
- Model Sharing.
- Scalablity!
## Encapsulation
Encapsulation allows Model Comparison.
- In-place model swaps.
- Model re-use.
## Object Oriented Modelling and Energetic Systems
Energetic systems provide:
- _Inheritance_; an abstract base representation of energetic systems.
- _Composition_; a way to hierarchically compose systems of systems.
- _Encapsulation_; a framework inside which simplifications can occur.
# `BondGraphTools`
Modelling Object Oriented Physical System
## Energetic Modelling as Object Oriented Programming.
`BondGraphTools` provides the infrastructure to
- Represent complex physical systems as object oriented python code.
- Manipulate and organise models of complex phyiscal systems programmitcally.
- Algorithmically simplify the resulting models.
## What `BondGraphTools` is good for
(and why you should use it, or pinch ideas from it)
- automated model reduciton
- scripted model building
- formal (code) modelling of systems.
- tool integration
```python
from BondGraphTools import draw
segment1 = VesselSegmentA(
"Example_1", artery, blood
)
draw(segment1)
```
```python
source, _ = getsourcelines(VesselSegmentA)
for line in source:
print(line[:-1])
```
class VesselSegmentA(bgt.BondGraph):
"""A vascular vessel segment.
This class is an example of Vessel Segment A from:
Safaei, Sorous. Blanco, Pablo J. Müller, Lucas O. Hellevik, Leif R. and Hunter, Peter J.
Bond Graph Model of Cerebral Circulation: Toward Clinically Feasible Systemic Blood Flow Simulations
Frontiers in Physiology, 2018, volume 9, page 148
The vessel segment is of the $uv$ type that has components:
- pressure inlet $u_i$
- flow outlet $v_o$
- fluid interia $I$
- wall dissipation $R$
- and compliance $C$.
The linear resistance, compliance and inertance are computed as per Safaei et.al..
See Also: BondGraph
"""
def __init__(self, name, vessel, fluid):
"""
Args:
name (str): The name of this vessel segement
vessel (Vessel): The vessel material properties
fluid (Fluid): The fluid properties
"""
# Parameters
resistance = 8 * fluid.viscosity * vessel.length / (pi * vessel.radius**4)
inertance = 2 * pi * vessel.radius**3 / (vessel.thickness * vessel.youngs_modulus)
compliance = fluid.density * vessel.length / (pi* vessel.radius**2)
# Instantiating Components
R_component = bgt.new('R', value=resistance, name=f'R')
C_component = bgt.new('C', value=compliance, name=f'C')
I_component = bgt.new('I', value=inertance, name=f'I')
conserved_flow = bgt.new("1", name=f'1')
conserved_pressure = bgt.new("0", name=f'0')
u_in = bgt.new('SS', name=f'u_i')
v_out = bgt.new('SS', name=f'v_o')
bonds = [
(u_in, conserved_flow),
(conserved_flow, R_component),
(conserved_flow, I_component),
(conserved_flow, conserved_pressure),
(conserved_pressure, C_component),
(conserved_pressure, v_out)
]
# Build the BondGraph via the inherited initialise function
super().__init__(
name=name,
components=(R_component, C_component, I_component, conserved_flow, conserved_pressure, u_in, v_out)
)
# wire it up
for bond in bonds:
bgt.connect(*bond)
# expose the ports
bgt.expose(u_in, label="u_i")
bgt.expose(v_out, label="v_o")
## Scripting the construction of an artery model
```python
from BondGraphTools import new, add, connect
length = 1
current_length = 0
segment_counter = 0
artery_model = new()
inlet = new('Se', name='inlet', label='u_i')
add(artery_model, inlet)
outlet = bgt.new('Sf', name='outlet', label='v_o')
add(artery_model, outlet)
last_outlet = inlet
```
```python
while current_length < length:
# Add a new segment
this_segment = VesselSegmentA(
name=f"Segment_{{{segment_counter}}}",
vessel=artery, fluid=blood)
add(artery_model, this_segment)
current_inlet_port = (this_segment, 'u_i')
# and connect it to the previous one
connect(last_outlet, current_inlet_port)
# Update the counter variables
last_outlet = (this_segment, 'v_o')
current_length = current_length + artery.length
segment_counter = segment_counter + 1
connect(last_outlet, outlet)
```
## Automatically Generating Equations
```python
artery_model.constitutive_relations
```
## Algorithmic Substitution
```python
from BondGraphTools import (
BondGraph, expose, new, connect)
class Voigt_Model(BondGraph):
def __init__(self, name,
compliance, dissipation):
# ------ Define the Subcomponents
C = new("C", name='C', value=compliance)
R = new("R", name='R',value=dissipation)
law = new('1')
port = new("SS", name='SS')
# ------ Build the model
super().__init__(name=name,
components=(C, R, law, port)
)
# ------ Wire it up
connect(port, law)
connect(law, R)
connect(law, C)
# ----- Expose the port
expose(port, label="C_v")
```
```python
from BondGraphTools import swap
# simple iterator function
def next_segment(model):
i = 0
try:
while True:
yield model / f"Segment_{{{i}}}"
i += 1
except ValueError:
return StopIteration
# swap the components out for each segment
for segment in next_segment(artery_model):
C = segment / "C"
C_v = Voigt_Model('C_v',
C.params['C'],
0.001)
swap(C, C_v)
```
```python
print_tree(artery_model)
```
BG: BG1
|-BG: Segment_{10}
|--BG: C_v
|---SS: SS
|---1: 124
|---R: R
|---C: C
|--SS: v_o
|--SS: u_i
|--0: 0
|--1: 1
|--I: I
|--R: R
|-BG: Segment_{9}
|--BG: C_v
|---SS: SS
|---1: 123
|---R: R
|---C: C
|--SS: v_o
|--SS: u_i
|--0: 0
|--1: 1
|--I: I
|--R: R
|-BG: Segment_{8}
|--BG: C_v
|---SS: SS
|---1: 122
|---R: R
|---C: C
|--SS: v_o
|--SS: u_i
|--0: 0
|--1: 1
|--I: I
|--R: R
|-BG: Segment_{7}
|--BG: C_v
|---SS: SS
|---1: 121
|---R: R
|---C: C
|--SS: v_o
|--SS: u_i
|--0: 0
|--1: 1
|--I: I
|--R: R
|-BG: Segment_{6}
|--BG: C_v
|---SS: SS
|---1: 120
|---R: R
|---C: C
|--SS: v_o
|--SS: u_i
|--0: 0
|--1: 1
|--I: I
|--R: R
|-BG: Segment_{5}
|--BG: C_v
|---SS: SS
|---1: 119
|---R: R
|---C: C
|--SS: v_o
|--SS: u_i
|--0: 0
|--1: 1
|--I: I
|--R: R
|-BG: Segment_{4}
|--BG: C_v
|---SS: SS
|---1: 118
|---R: R
|---C: C
|--SS: v_o
|--SS: u_i
|--0: 0
|--1: 1
|--I: I
|--R: R
|-BG: Segment_{3}
|--BG: C_v
|---SS: SS
|---1: 117
|---R: R
|---C: C
|--SS: v_o
|--SS: u_i
|--0: 0
|--1: 1
|--I: I
|--R: R
|-BG: Segment_{2}
|--BG: C_v
|---SS: SS
|---1: 116
|---R: R
|---C: C
|--SS: v_o
|--SS: u_i
|--0: 0
|--1: 1
|--I: I
|--R: R
|-BG: Segment_{1}
|--BG: C_v
|---SS: SS
|---1: 115
|---R: R
|---C: C
|--SS: v_o
|--SS: u_i
|--0: 0
|--1: 1
|--I: I
|--R: R
|-BG: Segment_{0}
|--BG: C_v
|---SS: SS
|---1: 114
|---R: R
|---C: C
|--SS: v_o
|--SS: u_i
|--0: 0
|--1: 1
|--I: I
|--R: R
|-SS: outlet
|-SS: inlet
## Automated Model Building
```python
TCA_reactions = {
"Citrate synthase":
["acetyl-CoA + oxaloacetate + H2O = citrate + CoA-SH"],
"Aconitase":
["Citrate = cis-Aconitate + H2O", "cis-Aconitate + H2O = Isocitrate"],
"Isocitrate dehydrogenase":
["Isocitrate + NAD = Oxalosuccinate + NADH + H",
"Oxalosuccinate = a-Ketoglutarate + CO2" ],
"a-Ketoglutarate dehydrogenase":
["a-Ketoglutarate + NAD + CoA-SH = Succinyl-CoA + NADH + H + CO2"],
"Succinyl-CoA synthetase":
["Succinyl-CoA + ADP + Pi = Succinate + CoA-SH + ATP"],
"Succinate dehydrogenase":
["Succinate + Q = Fumarate + QH2"],
"Fumarase":
["Fumarate + H2O = L-Malate"],
"Malate dehydrogenase":
["L-Malate + NAD = Oxaloacetate + NADH + H"]
}
```
```python
from BondGraphTools.reaction_builder import (
Reaction_Network)
def TCA_Cycle():
reaction_net = Reaction_Network(
name="TCA_Cycle"
)
# loop through each enzyme
for enzyme in TCA_reactions:
for i, reaction in enumerate(TCA_reactions[enzyme]):
# add each reaction.
reaction_net.add_reaction(
reaction,
name=f"{enzyme} - {i}"
)
return reaction_net
```
```python
from BondGraphTools import draw
tca_model = TCA_Cycle().as_network_model()
draw(tca_model)
```
## ...from data to equations
```python
tca_model.constitutive_relations
```
```python
# Parameters
tca_model.params
```
{0: (C: acetyl-CoA, 'k'),
1: (C: acetyl-CoA, 'R'),
2: (C: acetyl-CoA, 'T'),
3: (C: oxaloacetate, 'k'),
4: (C: oxaloacetate, 'R'),
5: (C: oxaloacetate, 'T'),
6: (C: H2O, 'k'),
7: (C: H2O, 'R'),
8: (C: H2O, 'T'),
9: (C: citrate, 'k'),
10: (C: citrate, 'R'),
11: (C: citrate, 'T'),
12: (C: CoA-SH, 'k'),
13: (C: CoA-SH, 'R'),
14: (C: CoA-SH, 'T'),
15: (C: Citrate, 'k'),
16: (C: Citrate, 'R'),
17: (C: Citrate, 'T'),
18: (C: cis-Aconitate, 'k'),
19: (C: cis-Aconitate, 'R'),
20: (C: cis-Aconitate, 'T'),
21: (C: Isocitrate, 'k'),
22: (C: Isocitrate, 'R'),
23: (C: Isocitrate, 'T'),
24: (C: NAD, 'k'),
25: (C: NAD, 'R'),
26: (C: NAD, 'T'),
27: (C: Oxalosuccinate, 'k'),
28: (C: Oxalosuccinate, 'R'),
29: (C: Oxalosuccinate, 'T'),
30: (C: NADH, 'k'),
31: (C: NADH, 'R'),
32: (C: NADH, 'T'),
33: (C: H, 'k'),
34: (C: H, 'R'),
35: (C: H, 'T'),
36: (C: a-Ketoglutarate, 'k'),
37: (C: a-Ketoglutarate, 'R'),
38: (C: a-Ketoglutarate, 'T'),
39: (C: CO2, 'k'),
40: (C: CO2, 'R'),
41: (C: CO2, 'T'),
42: (C: Succinyl-CoA, 'k'),
43: (C: Succinyl-CoA, 'R'),
44: (C: Succinyl-CoA, 'T'),
45: (C: ADP, 'k'),
46: (C: ADP, 'R'),
47: (C: ADP, 'T'),
48: (C: Pi, 'k'),
49: (C: Pi, 'R'),
50: (C: Pi, 'T'),
51: (C: Succinate, 'k'),
52: (C: Succinate, 'R'),
53: (C: Succinate, 'T'),
54: (C: ATP, 'k'),
55: (C: ATP, 'R'),
56: (C: ATP, 'T'),
57: (C: Q, 'k'),
58: (C: Q, 'R'),
59: (C: Q, 'T'),
60: (C: Fumarate, 'k'),
61: (C: Fumarate, 'R'),
62: (C: Fumarate, 'T'),
63: (C: QH2, 'k'),
64: (C: QH2, 'R'),
65: (C: QH2, 'T'),
66: (C: L-Malate, 'k'),
67: (C: L-Malate, 'R'),
68: (C: L-Malate, 'T'),
69: (C: Oxaloacetate, 'k'),
70: (C: Oxaloacetate, 'R'),
71: (C: Oxaloacetate, 'T'),
72: (R: Citrate synthase - 0, 'r'),
73: (R: Citrate synthase - 0, 'R'),
74: (R: Citrate synthase - 0, 'T'),
75: (R: Aconitase - 0, 'r'),
76: (R: Aconitase - 0, 'R'),
77: (R: Aconitase - 0, 'T'),
78: (R: Aconitase - 1, 'r'),
79: (R: Aconitase - 1, 'R'),
80: (R: Aconitase - 1, 'T'),
81: (R: Isocitrate dehydrogenase - 0, 'r'),
82: (R: Isocitrate dehydrogenase - 0, 'R'),
83: (R: Isocitrate dehydrogenase - 0, 'T'),
84: (R: Isocitrate dehydrogenase - 1, 'r'),
85: (R: Isocitrate dehydrogenase - 1, 'R'),
86: (R: Isocitrate dehydrogenase - 1, 'T'),
87: (R: a-Ketoglutarate dehydrogenase - 0, 'r'),
88: (R: a-Ketoglutarate dehydrogenase - 0, 'R'),
89: (R: a-Ketoglutarate dehydrogenase - 0, 'T'),
90: (R: Succinyl-CoA synthetase - 0, 'r'),
91: (R: Succinyl-CoA synthetase - 0, 'R'),
92: (R: Succinyl-CoA synthetase - 0, 'T'),
93: (R: Succinate dehydrogenase - 0, 'r'),
94: (R: Succinate dehydrogenase - 0, 'R'),
95: (R: Succinate dehydrogenase - 0, 'T'),
96: (R: Fumarase - 0, 'r'),
97: (R: Fumarase - 0, 'R'),
98: (R: Fumarase - 0, 'T'),
99: (R: Malate dehydrogenase - 0, 'r'),
100: (R: Malate dehydrogenase - 0, 'R'),
101: (R: Malate dehydrogenase - 0, 'T')}
```python
from BondGraphTools import set_param, swap, new
# Set Parameters to 1
value = 1
for param in tca_model.params:
set_param(tca_model, param, value)
# Swap acetyl_CoA for a effort source
acetyl_CoA = tca_model / "C: acetyl-CoA"
flow_control = new("Se", value=None)
swap(acetyl_CoA, flow_control)
```
```python
tca_model.constitutive_relations
```
## Basic Simulation
```python
import numpy as np
x_dim = len(tca_model.state_vars)
x0 = np.exp(np.random.randn(x_dim))
from BondGraphTools import simulate
t, x = simulate(tca_model,
timespan=[0,1],
x0=x0,
control_vars=["t > 0.2 ? 2 : 0"])
```
```python
from matplotlib.pyplot import plot
_ = plot(t, x)
```
## BondGraphTools Development Philosophy
BondGraphTools ideals:
- working is better than right.
- ... it should just do the thing.
- ... it does what it says on the box.
- don't make the user fight the tools.
## What `BondGraphTools` does not do
- dimensional anaysis
- ensure realistic models
- graphical user interfaces
- parameter fitting
- ontologies, metadata...
## `BondGraphTools` in literature
Used in forthcoming work by:
- Prof. Peter Gawthrop (Physically Plausible Models)
- Michael Pan (Algorithmic Model Evalutition)
- PC (`BondGraphTools`)
## Current Status:
Currently version: 0.3.8
The next version 0.4 will include:
- Symbolics overhaul.
- Improved model reduction.
- Cleaner parameter handline.
- Observables.
# Thanks
- Andre and the CellML workshop organisers
- Prof. Peter Hunter and the ABI
- Prof. Edmund Crampin, Prof. Peter Gawthrop and Michael Pan & The Systems Biology Lab
<table >
<tr style="background-color:#FFFFFF;">
<td></td>
<td></td>
</tr>
</table>
## Please check out `BondGraphTools`
Docs: https://bondgraphtools.readthedocs.io/
GitHub: https://github.com/BondGraphTools
|
-- -------------------------------------------------------------- [ Common.idr ]
-- Module : Common.idr
-- Copyright : (c) Jan de Muijnck-Hughes
-- License : see LICENSE
-- --------------------------------------------------------------------- [ EOH ]
module Freya.Common
%access export
public export
data RTy = FUNC | USAB | RELI | PERF | SUPP
implementation Cast RTy String where
cast FUNC = "functional"
cast USAB = "usability"
cast RELI = "reliability"
cast PERF = "performance"
cast SUPP = "supportability"
implementation Show RTy where
show FUNC = "FUNC"
show USAB = "USAB"
show RELI = "RELI"
show PERF = "PERF"
show SUPP = "SUPP"
readRTy : String -> Maybe RTy
readRTy s =
case s of
"functional" => Just FUNC
"usability" => Just USAB
"reliability" => Just RELI
"performance" => Just PERF
"supportability" => Just SUPP
otherwise => Nothing
public export
data TTy = ADV | DIS | GEN
implementation Show TTy where
show ADV = "ADVANTAGE"
show DIS = "DISADVANTAGE"
show GEN = "GENERAL"
implementation Cast TTy String where
cast ADV = "advantage"
cast DIS = "disadvantage"
cast GEN = "general"
readTTy : String -> Maybe TTy
readTTy s =
case s of
"advantage" => Just ADV
"disadvantage" => Just DIS
"general" => Just GEN
otherwise => Nothing
public export
data MTy = STRUCT | DYN
implementation Show MTy where
show STRUCT = "STRUCT"
show DYN = "DYN"
implementation Cast MTy String where
cast STRUCT = "structure"
cast DYN = "dynamic"
readMTy : String -> Maybe MTy
readMTy s =
case s of
"structure" => Just STRUCT
"dynamic" => Just DYN
otherwise => Nothing
public export
data LTy = SPECIAL | IMPL | USES | LINK
implementation Show LTy where
show SPECIAL = "SPECIAL"
show IMPL = "IMPL"
show USES = "USES"
show LINK = "LINK"
public export
implementation Cast LTy String where
cast SPECIAL = "specialises"
cast IMPL = "implements"
cast USES = "requires"
cast LINK = "linked"
readLTy : String -> Maybe LTy
readLTy s =
case s of
"specialises" => Just SPECIAL
"implements" => Just IMPL
"requires" => Just USES
"linked" => Just LINK
otherwise => Nothing
-- --------------------------------------------------------------------- [ EOF ]
|
State Before: α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
f g : Perm α
l : List (Perm α)
h : f ∈ l
hl : List.Pairwise Disjoint l
⊢ ∀ (x : α), x ∈ support f → ↑f x = ↑(List.prod l) x State After: case nil
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
f g : Perm α
l : List (Perm α)
h✝ : f ∈ l
hl✝ : List.Pairwise Disjoint l
h : f ∈ []
hl : List.Pairwise Disjoint []
⊢ ∀ (x : α), x ∈ support f → ↑f x = ↑(List.prod []) x
case cons
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
f g : Perm α
l : List (Perm α)
h✝ : f ∈ l
hl✝ : List.Pairwise Disjoint l
hd : Perm α
tl : List (Perm α)
IH : f ∈ tl → List.Pairwise Disjoint tl → ∀ (x : α), x ∈ support f → ↑f x = ↑(List.prod tl) x
h : f ∈ hd :: tl
hl : List.Pairwise Disjoint (hd :: tl)
⊢ ∀ (x : α), x ∈ support f → ↑f x = ↑(List.prod (hd :: tl)) x Tactic: induction' l with hd tl IH State Before: case nil
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
f g : Perm α
l : List (Perm α)
h✝ : f ∈ l
hl✝ : List.Pairwise Disjoint l
h : f ∈ []
hl : List.Pairwise Disjoint []
⊢ ∀ (x : α), x ∈ support f → ↑f x = ↑(List.prod []) x State After: no goals Tactic: simp at h State Before: case cons
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
f g : Perm α
l : List (Perm α)
h✝ : f ∈ l
hl✝ : List.Pairwise Disjoint l
hd : Perm α
tl : List (Perm α)
IH : f ∈ tl → List.Pairwise Disjoint tl → ∀ (x : α), x ∈ support f → ↑f x = ↑(List.prod tl) x
h : f ∈ hd :: tl
hl : List.Pairwise Disjoint (hd :: tl)
⊢ ∀ (x : α), x ∈ support f → ↑f x = ↑(List.prod (hd :: tl)) x State After: case cons
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
f g : Perm α
l : List (Perm α)
h✝ : f ∈ l
hl✝ : List.Pairwise Disjoint l
hd : Perm α
tl : List (Perm α)
IH : f ∈ tl → List.Pairwise Disjoint tl → ∀ (x : α), x ∈ support f → ↑f x = ↑(List.prod tl) x
h : f ∈ hd :: tl
hl : List.Pairwise Disjoint (hd :: tl)
x : α
hx : x ∈ support f
⊢ ↑f x = ↑(List.prod (hd :: tl)) x Tactic: intro x hx State Before: case cons
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
f g : Perm α
l : List (Perm α)
h✝ : f ∈ l
hl✝ : List.Pairwise Disjoint l
hd : Perm α
tl : List (Perm α)
IH : f ∈ tl → List.Pairwise Disjoint tl → ∀ (x : α), x ∈ support f → ↑f x = ↑(List.prod tl) x
h : f ∈ hd :: tl
hl : List.Pairwise Disjoint (hd :: tl)
x : α
hx : x ∈ support f
⊢ ↑f x = ↑(List.prod (hd :: tl)) x State After: case cons
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
f g : Perm α
l : List (Perm α)
h✝ : f ∈ l
hl✝ : List.Pairwise Disjoint l
hd : Perm α
tl : List (Perm α)
IH : f ∈ tl → List.Pairwise Disjoint tl → ∀ (x : α), x ∈ support f → ↑f x = ↑(List.prod tl) x
h : f ∈ hd :: tl
hl : (∀ (a' : Perm α), a' ∈ tl → Disjoint hd a') ∧ List.Pairwise Disjoint tl
x : α
hx : x ∈ support f
⊢ ↑f x = ↑(List.prod (hd :: tl)) x Tactic: rw [List.pairwise_cons] at hl State Before: case cons
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
f g : Perm α
l : List (Perm α)
h✝ : f ∈ l
hl✝ : List.Pairwise Disjoint l
hd : Perm α
tl : List (Perm α)
IH : f ∈ tl → List.Pairwise Disjoint tl → ∀ (x : α), x ∈ support f → ↑f x = ↑(List.prod tl) x
h : f ∈ hd :: tl
hl : (∀ (a' : Perm α), a' ∈ tl → Disjoint hd a') ∧ List.Pairwise Disjoint tl
x : α
hx : x ∈ support f
⊢ ↑f x = ↑(List.prod (hd :: tl)) x State After: case cons
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
f g : Perm α
l : List (Perm α)
h✝ : f ∈ l
hl✝ : List.Pairwise Disjoint l
hd : Perm α
tl : List (Perm α)
IH : f ∈ tl → List.Pairwise Disjoint tl → ∀ (x : α), x ∈ support f → ↑f x = ↑(List.prod tl) x
h : f = hd ∨ f ∈ tl
hl : (∀ (a' : Perm α), a' ∈ tl → Disjoint hd a') ∧ List.Pairwise Disjoint tl
x : α
hx : x ∈ support f
⊢ ↑f x = ↑(List.prod (hd :: tl)) x Tactic: rw [List.mem_cons] at h State Before: case cons
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
f g : Perm α
l : List (Perm α)
h✝ : f ∈ l
hl✝ : List.Pairwise Disjoint l
hd : Perm α
tl : List (Perm α)
IH : f ∈ tl → List.Pairwise Disjoint tl → ∀ (x : α), x ∈ support f → ↑f x = ↑(List.prod tl) x
h : f = hd ∨ f ∈ tl
hl : (∀ (a' : Perm α), a' ∈ tl → Disjoint hd a') ∧ List.Pairwise Disjoint tl
x : α
hx : x ∈ support f
⊢ ↑f x = ↑(List.prod (hd :: tl)) x State After: case cons.inl
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
f g : Perm α
l : List (Perm α)
h : f ∈ l
hl✝ : List.Pairwise Disjoint l
tl : List (Perm α)
IH : f ∈ tl → List.Pairwise Disjoint tl → ∀ (x : α), x ∈ support f → ↑f x = ↑(List.prod tl) x
x : α
hx : x ∈ support f
hl : (∀ (a' : Perm α), a' ∈ tl → Disjoint f a') ∧ List.Pairwise Disjoint tl
⊢ ↑f x = ↑(List.prod (f :: tl)) x
case cons.inr
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
f g : Perm α
l : List (Perm α)
h✝ : f ∈ l
hl✝ : List.Pairwise Disjoint l
hd : Perm α
tl : List (Perm α)
IH : f ∈ tl → List.Pairwise Disjoint tl → ∀ (x : α), x ∈ support f → ↑f x = ↑(List.prod tl) x
hl : (∀ (a' : Perm α), a' ∈ tl → Disjoint hd a') ∧ List.Pairwise Disjoint tl
x : α
hx : x ∈ support f
h : f ∈ tl
⊢ ↑f x = ↑(List.prod (hd :: tl)) x Tactic: rcases h with (rfl | h) State Before: case cons.inl
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
f g : Perm α
l : List (Perm α)
h : f ∈ l
hl✝ : List.Pairwise Disjoint l
tl : List (Perm α)
IH : f ∈ tl → List.Pairwise Disjoint tl → ∀ (x : α), x ∈ support f → ↑f x = ↑(List.prod tl) x
x : α
hx : x ∈ support f
hl : (∀ (a' : Perm α), a' ∈ tl → Disjoint f a') ∧ List.Pairwise Disjoint tl
⊢ ↑f x = ↑(List.prod (f :: tl)) x State After: no goals Tactic: rw [List.prod_cons, mul_apply,
not_mem_support.mp ((disjoint_prod_right tl hl.left).mem_imp hx)] State Before: case cons.inr
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
f g : Perm α
l : List (Perm α)
h✝ : f ∈ l
hl✝ : List.Pairwise Disjoint l
hd : Perm α
tl : List (Perm α)
IH : f ∈ tl → List.Pairwise Disjoint tl → ∀ (x : α), x ∈ support f → ↑f x = ↑(List.prod tl) x
hl : (∀ (a' : Perm α), a' ∈ tl → Disjoint hd a') ∧ List.Pairwise Disjoint tl
x : α
hx : x ∈ support f
h : f ∈ tl
⊢ ↑f x = ↑(List.prod (hd :: tl)) x State After: case cons.inr
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
f g : Perm α
l : List (Perm α)
h✝ : f ∈ l
hl✝ : List.Pairwise Disjoint l
hd : Perm α
tl : List (Perm α)
IH : f ∈ tl → List.Pairwise Disjoint tl → ∀ (x : α), x ∈ support f → ↑f x = ↑(List.prod tl) x
hl : (∀ (a' : Perm α), a' ∈ tl → Disjoint hd a') ∧ List.Pairwise Disjoint tl
x : α
hx : x ∈ support f
h : f ∈ tl
⊢ ¬↑f x ∈ support hd Tactic: rw [List.prod_cons, mul_apply, ← IH h hl.right _ hx, eq_comm, ← not_mem_support] State Before: case cons.inr
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
f g : Perm α
l : List (Perm α)
h✝ : f ∈ l
hl✝ : List.Pairwise Disjoint l
hd : Perm α
tl : List (Perm α)
IH : f ∈ tl → List.Pairwise Disjoint tl → ∀ (x : α), x ∈ support f → ↑f x = ↑(List.prod tl) x
hl : (∀ (a' : Perm α), a' ∈ tl → Disjoint hd a') ∧ List.Pairwise Disjoint tl
x : α
hx : x ∈ support f
h : f ∈ tl
⊢ ¬↑f x ∈ support hd State After: case cons.inr
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
f g : Perm α
l : List (Perm α)
h✝ : f ∈ l
hl✝ : List.Pairwise Disjoint l
hd : Perm α
tl : List (Perm α)
IH : f ∈ tl → List.Pairwise Disjoint tl → ∀ (x : α), x ∈ support f → ↑f x = ↑(List.prod tl) x
hl : (∀ (a' : Perm α), a' ∈ tl → Disjoint hd a') ∧ List.Pairwise Disjoint tl
x : α
hx : x ∈ support f
h : f ∈ tl
⊢ ↑f x ∈ support f Tactic: refine' (hl.left _ h).symm.mem_imp _ State Before: case cons.inr
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
f g : Perm α
l : List (Perm α)
h✝ : f ∈ l
hl✝ : List.Pairwise Disjoint l
hd : Perm α
tl : List (Perm α)
IH : f ∈ tl → List.Pairwise Disjoint tl → ∀ (x : α), x ∈ support f → ↑f x = ↑(List.prod tl) x
hl : (∀ (a' : Perm α), a' ∈ tl → Disjoint hd a') ∧ List.Pairwise Disjoint tl
x : α
hx : x ∈ support f
h : f ∈ tl
⊢ ↑f x ∈ support f State After: no goals Tactic: simpa using hx |
State Before: α : Type u_2
β : Type ?u.56609
γ : Type ?u.56612
δ : Type ?u.56615
ι : Type u_1
R : Type ?u.56621
R' : Type ?u.56624
m : MeasurableSpace α
μ μ₁ μ₂ : Measure α
s✝ s₁ s₂ t : Set α
inst✝¹ : SemilatticeSup ι
inst✝ : Countable ι
s : ι → Set α
hm : Monotone s
⊢ Tendsto (↑↑μ ∘ s) atTop (𝓝 (↑↑μ (⋃ (n : ι), s n))) State After: α : Type u_2
β : Type ?u.56609
γ : Type ?u.56612
δ : Type ?u.56615
ι : Type u_1
R : Type ?u.56621
R' : Type ?u.56624
m : MeasurableSpace α
μ μ₁ μ₂ : Measure α
s✝ s₁ s₂ t : Set α
inst✝¹ : SemilatticeSup ι
inst✝ : Countable ι
s : ι → Set α
hm : Monotone s
⊢ Tendsto (↑↑μ ∘ s) atTop (𝓝 (⨆ (i : ι), ↑↑μ (s i))) Tactic: rw [measure_iUnion_eq_iSup (directed_of_sup hm)] State Before: α : Type u_2
β : Type ?u.56609
γ : Type ?u.56612
δ : Type ?u.56615
ι : Type u_1
R : Type ?u.56621
R' : Type ?u.56624
m : MeasurableSpace α
μ μ₁ μ₂ : Measure α
s✝ s₁ s₂ t : Set α
inst✝¹ : SemilatticeSup ι
inst✝ : Countable ι
s : ι → Set α
hm : Monotone s
⊢ Tendsto (↑↑μ ∘ s) atTop (𝓝 (⨆ (i : ι), ↑↑μ (s i))) State After: no goals Tactic: exact tendsto_atTop_iSup fun n m hnm => measure_mono <| hm hnm |
(* generated by Ott 0.32, locally-nameless lngen from: ../Qualitative.ott *)
Require Import Metalib.Metatheory.
Require Export Metalib.LibLNgen.
Require Export Qual.grade_sig.
Require Export Qual.sort_sig.
(** syntax *)
Definition tmvar : Set := var. (*r variables *)
Definition qualityvar : Set := atom. (*r qualities *)
Definition grade : Set := grade.
Definition sort : Set := sort.
Inductive tm : Set := (*r terms and types *)
| a_TyUnit : tm (*r unit type *)
| a_TmUnit : tm (*r unit term *)
| a_Pi (psi:grade) (A:tm) (B:tm) (*r dependent function type *)
| a_Abs (psi:grade) (A:tm) (a:tm) (*r function *)
| a_App (a:tm) (psi:grade) (b:tm) (*r function application *)
| a_Type (s:sort) (*r sort *)
| a_Var_b (_:nat) (*r variable *)
| a_Var_f (x:tmvar) (*r variable *)
| a_Sum (A1:tm) (A2:tm) (*r sum type *)
| a_Inj1 (a:tm) (*r injection into sum type *)
| a_Inj2 (a:tm) (*r injection into sum type *)
| a_Case (psi:grade) (a:tm) (b1:tm) (b2:tm) (*r case elimination of sum type *)
| a_WSigma (psi:grade) (A:tm) (B:tm) (*r dependent tuple type *)
| a_WPair (a:tm) (psi:grade) (b:tm) (*r tuple creation *)
| a_LetPair (psi:grade) (a:tm) (b:tm) (*r tuple elimination *)
| a_SSigma (psi:grade) (A:tm) (B:tm)
| a_SPair (a:tm) (psi:grade) (b:tm)
| a_Proj1 (psi:grade) (a:tm)
| a_Proj2 (psi:grade) (a:tm).
Definition econtext : Set := list ( atom * grade ).
Definition context : Set := list ( atom * (grade * tm) ).
(* EXPERIMENTAL *)
(** auxiliary functions on the new list types *)
(** library functions *)
(** subrules *)
(** arities *)
(** opening up abstractions *)
Fixpoint open_tm_wrt_tm_rec (k:nat) (a5:tm) (a_6:tm) {struct a_6}: tm :=
match a_6 with
| a_TyUnit => a_TyUnit
| a_TmUnit => a_TmUnit
| (a_Pi psi A B) => a_Pi psi (open_tm_wrt_tm_rec k a5 A) (open_tm_wrt_tm_rec (S k) a5 B)
| (a_Abs psi A a) => a_Abs psi (open_tm_wrt_tm_rec k a5 A) (open_tm_wrt_tm_rec (S k) a5 a)
| (a_App a psi b) => a_App (open_tm_wrt_tm_rec k a5 a) psi (open_tm_wrt_tm_rec k a5 b)
| (a_Type s) => a_Type s
| (a_Var_b nat) =>
match lt_eq_lt_dec nat k with
| inleft (left _) => a_Var_b nat
| inleft (right _) => a5
| inright _ => a_Var_b (nat - 1)
end
| (a_Var_f x) => a_Var_f x
| (a_Sum A1 A2) => a_Sum (open_tm_wrt_tm_rec k a5 A1) (open_tm_wrt_tm_rec k a5 A2)
| (a_Inj1 a) => a_Inj1 (open_tm_wrt_tm_rec k a5 a)
| (a_Inj2 a) => a_Inj2 (open_tm_wrt_tm_rec k a5 a)
| (a_Case psi a b1 b2) => a_Case psi (open_tm_wrt_tm_rec k a5 a) (open_tm_wrt_tm_rec k a5 b1) (open_tm_wrt_tm_rec k a5 b2)
| (a_WSigma psi A B) => a_WSigma psi (open_tm_wrt_tm_rec k a5 A) (open_tm_wrt_tm_rec (S k) a5 B)
| (a_WPair a psi b) => a_WPair (open_tm_wrt_tm_rec k a5 a) psi (open_tm_wrt_tm_rec k a5 b)
| (a_LetPair psi a b) => a_LetPair psi (open_tm_wrt_tm_rec k a5 a) (open_tm_wrt_tm_rec (S k) a5 b)
| (a_SSigma psi A B) => a_SSigma psi (open_tm_wrt_tm_rec k a5 A) (open_tm_wrt_tm_rec (S k) a5 B)
| (a_SPair a psi b) => a_SPair (open_tm_wrt_tm_rec k a5 a) psi (open_tm_wrt_tm_rec k a5 b)
| (a_Proj1 psi a) => a_Proj1 psi (open_tm_wrt_tm_rec k a5 a)
| (a_Proj2 psi a) => a_Proj2 psi (open_tm_wrt_tm_rec k a5 a)
end.
Definition open_tm_wrt_tm a5 a_6 := open_tm_wrt_tm_rec 0 a_6 a5.
(** terms are locally-closed pre-terms *)
(** definitions *)
(* defns LC_tm *)
Inductive lc_tm : tm -> Prop := (* defn lc_tm *)
| lc_a_TyUnit :
(lc_tm a_TyUnit)
| lc_a_TmUnit :
(lc_tm a_TmUnit)
| lc_a_Pi : forall (psi:grade) (A B:tm),
(lc_tm A) ->
( forall x , lc_tm ( open_tm_wrt_tm B (a_Var_f x) ) ) ->
(lc_tm (a_Pi psi A B))
| lc_a_Abs : forall (psi:grade) (A a:tm),
(lc_tm A) ->
( forall x , lc_tm ( open_tm_wrt_tm a (a_Var_f x) ) ) ->
(lc_tm (a_Abs psi A a))
| lc_a_App : forall (a:tm) (psi:grade) (b:tm),
(lc_tm a) ->
(lc_tm b) ->
(lc_tm (a_App a psi b))
| lc_a_Type : forall (s:sort),
(lc_tm (a_Type s))
| lc_a_Var_f : forall (x:tmvar),
(lc_tm (a_Var_f x))
| lc_a_Sum : forall (A1 A2:tm),
(lc_tm A1) ->
(lc_tm A2) ->
(lc_tm (a_Sum A1 A2))
| lc_a_Inj1 : forall (a:tm),
(lc_tm a) ->
(lc_tm (a_Inj1 a))
| lc_a_Inj2 : forall (a:tm),
(lc_tm a) ->
(lc_tm (a_Inj2 a))
| lc_a_Case : forall (psi:grade) (a b1 b2:tm),
(lc_tm a) ->
(lc_tm b1) ->
(lc_tm b2) ->
(lc_tm (a_Case psi a b1 b2))
| lc_a_WSigma : forall (psi:grade) (A B:tm),
(lc_tm A) ->
( forall x , lc_tm ( open_tm_wrt_tm B (a_Var_f x) ) ) ->
(lc_tm (a_WSigma psi A B))
| lc_a_WPair : forall (a:tm) (psi:grade) (b:tm),
(lc_tm a) ->
(lc_tm b) ->
(lc_tm (a_WPair a psi b))
| lc_a_LetPair : forall (psi:grade) (a b:tm),
(lc_tm a) ->
( forall x , lc_tm ( open_tm_wrt_tm b (a_Var_f x) ) ) ->
(lc_tm (a_LetPair psi a b))
| lc_a_SSigma : forall (psi:grade) (A B:tm),
(lc_tm A) ->
( forall x , lc_tm ( open_tm_wrt_tm B (a_Var_f x) ) ) ->
(lc_tm (a_SSigma psi A B))
| lc_a_SPair : forall (a:tm) (psi:grade) (b:tm),
(lc_tm a) ->
(lc_tm b) ->
(lc_tm (a_SPair a psi b))
| lc_a_Proj1 : forall (psi:grade) (a:tm),
(lc_tm a) ->
(lc_tm (a_Proj1 psi a))
| lc_a_Proj2 : forall (psi:grade) (a:tm),
(lc_tm a) ->
(lc_tm (a_Proj2 psi a)).
(** free variables *)
Fixpoint fv_tm_tm (a5:tm) : vars :=
match a5 with
| a_TyUnit => {}
| a_TmUnit => {}
| (a_Pi psi A B) => (fv_tm_tm A) \u (fv_tm_tm B)
| (a_Abs psi A a) => (fv_tm_tm A) \u (fv_tm_tm a)
| (a_App a psi b) => (fv_tm_tm a) \u (fv_tm_tm b)
| (a_Type s) => {}
| (a_Var_b nat) => {}
| (a_Var_f x) => {{x}}
| (a_Sum A1 A2) => (fv_tm_tm A1) \u (fv_tm_tm A2)
| (a_Inj1 a) => (fv_tm_tm a)
| (a_Inj2 a) => (fv_tm_tm a)
| (a_Case psi a b1 b2) => (fv_tm_tm a) \u (fv_tm_tm b1) \u (fv_tm_tm b2)
| (a_WSigma psi A B) => (fv_tm_tm A) \u (fv_tm_tm B)
| (a_WPair a psi b) => (fv_tm_tm a) \u (fv_tm_tm b)
| (a_LetPair psi a b) => (fv_tm_tm a) \u (fv_tm_tm b)
| (a_SSigma psi A B) => (fv_tm_tm A) \u (fv_tm_tm B)
| (a_SPair a psi b) => (fv_tm_tm a) \u (fv_tm_tm b)
| (a_Proj1 psi a) => (fv_tm_tm a)
| (a_Proj2 psi a) => (fv_tm_tm a)
end.
(** substitutions *)
Fixpoint subst_tm_tm (a5:tm) (x5:tmvar) (a_6:tm) {struct a_6} : tm :=
match a_6 with
| a_TyUnit => a_TyUnit
| a_TmUnit => a_TmUnit
| (a_Pi psi A B) => a_Pi psi (subst_tm_tm a5 x5 A) (subst_tm_tm a5 x5 B)
| (a_Abs psi A a) => a_Abs psi (subst_tm_tm a5 x5 A) (subst_tm_tm a5 x5 a)
| (a_App a psi b) => a_App (subst_tm_tm a5 x5 a) psi (subst_tm_tm a5 x5 b)
| (a_Type s) => a_Type s
| (a_Var_b nat) => a_Var_b nat
| (a_Var_f x) => (if eq_var x x5 then a5 else (a_Var_f x))
| (a_Sum A1 A2) => a_Sum (subst_tm_tm a5 x5 A1) (subst_tm_tm a5 x5 A2)
| (a_Inj1 a) => a_Inj1 (subst_tm_tm a5 x5 a)
| (a_Inj2 a) => a_Inj2 (subst_tm_tm a5 x5 a)
| (a_Case psi a b1 b2) => a_Case psi (subst_tm_tm a5 x5 a) (subst_tm_tm a5 x5 b1) (subst_tm_tm a5 x5 b2)
| (a_WSigma psi A B) => a_WSigma psi (subst_tm_tm a5 x5 A) (subst_tm_tm a5 x5 B)
| (a_WPair a psi b) => a_WPair (subst_tm_tm a5 x5 a) psi (subst_tm_tm a5 x5 b)
| (a_LetPair psi a b) => a_LetPair psi (subst_tm_tm a5 x5 a) (subst_tm_tm a5 x5 b)
| (a_SSigma psi A B) => a_SSigma psi (subst_tm_tm a5 x5 A) (subst_tm_tm a5 x5 B)
| (a_SPair a psi b) => a_SPair (subst_tm_tm a5 x5 a) psi (subst_tm_tm a5 x5 b)
| (a_Proj1 psi a) => a_Proj1 psi (subst_tm_tm a5 x5 a)
| (a_Proj2 psi a) => a_Proj2 psi (subst_tm_tm a5 x5 a)
end.
(* %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% *)
Local Open Scope grade_scope.
Definition labels : context -> econtext :=
map (fun '(u , s) => u).
Definition subst_ctx (a : tm) (x:var) : context -> context :=
map (fun '(g, A) => (g, (subst_tm_tm a x A))).
Definition join_ctx_l (psi : grade) : context -> context :=
map (fun '(g, A) => (psi * g, A)).
Definition join_ctx_r (psi : grade) : context -> context :=
map (fun '(g, A) => (g * psi, A)).
Definition meet_ctx_l (psi : grade) : context -> context :=
map (fun '(g, A) => (psi + g, A)).
Definition meet_ctx_r (psi : grade) : context -> context :=
map (fun '(g, A) => (g + psi, A)).
Fixpoint close_tm_wrt_tm_rec (n1 : nat) (x1 : tmvar) (a1 : tm) {struct a1} : tm :=
match a1 with
| a_TyUnit => a_TyUnit
| a_TmUnit => a_TmUnit
| a_Pi psi1 A1 B1 => a_Pi psi1 (close_tm_wrt_tm_rec n1 x1 A1) (close_tm_wrt_tm_rec (S n1) x1 B1)
| a_Abs psi1 A1 a2 => a_Abs psi1 (close_tm_wrt_tm_rec n1 x1 A1) (close_tm_wrt_tm_rec (S n1) x1 a2)
| a_App a2 psi1 b1 => a_App (close_tm_wrt_tm_rec n1 x1 a2) psi1 (close_tm_wrt_tm_rec n1 x1 b1)
| a_Type s => a_Type s
| a_Var_f x2 => if (x1 == x2) then (a_Var_b n1) else (a_Var_f x2)
| a_Var_b n2 => if (lt_ge_dec n2 n1) then (a_Var_b n2) else (a_Var_b (S n2))
| a_Sum A1 A2 => a_Sum (close_tm_wrt_tm_rec n1 x1 A1) (close_tm_wrt_tm_rec n1 x1 A2)
| a_Inj1 a2 => a_Inj1 (close_tm_wrt_tm_rec n1 x1 a2)
| a_Inj2 a2 => a_Inj2 (close_tm_wrt_tm_rec n1 x1 a2)
| a_Case psi a2 b1 b2 => a_Case psi (close_tm_wrt_tm_rec n1 x1 a2) (close_tm_wrt_tm_rec n1 x1 b1) (close_tm_wrt_tm_rec n1 x1 b2)
| a_WSigma psi1 A1 B1 => a_WSigma psi1 (close_tm_wrt_tm_rec n1 x1 A1) (close_tm_wrt_tm_rec (S n1) x1 B1)
| a_WPair a2 psi1 b1 => a_WPair (close_tm_wrt_tm_rec n1 x1 a2) psi1 (close_tm_wrt_tm_rec n1 x1 b1)
| a_LetPair psi1 a2 b1 => a_LetPair psi1 (close_tm_wrt_tm_rec n1 x1 a2) (close_tm_wrt_tm_rec (S n1) x1 b1)
| a_SSigma psi1 A1 B1 => a_SSigma psi1 (close_tm_wrt_tm_rec n1 x1 A1) (close_tm_wrt_tm_rec (S n1) x1 B1)
| a_SPair a2 psi1 b1 => a_SPair (close_tm_wrt_tm_rec n1 x1 a2) psi1 (close_tm_wrt_tm_rec n1 x1 b1)
| a_Proj1 psi1 a2 => a_Proj1 psi1 (close_tm_wrt_tm_rec n1 x1 a2)
| a_Proj2 psi1 a2 => a_Proj2 psi1 (close_tm_wrt_tm_rec n1 x1 a2)
end.
Definition close_tm_wrt_tm x1 a1 := close_tm_wrt_tm_rec 0 x1 a1.
(** definitions *)
(* defns JOp *)
Inductive Step : tm -> tm -> Prop := (* defn Step *)
| S_AppCong : forall (a:tm) (psi:grade) (b a':tm),
lc_tm b ->
Step a a' ->
Step (a_App a psi b) (a_App a' psi b)
| S_Beta : forall (psi:grade) (A a b:tm),
lc_tm A ->
lc_tm (a_Abs psi A a) ->
lc_tm b ->
Step (a_App ( (a_Abs psi A a) ) psi b) (open_tm_wrt_tm a b )
| S_CaseCong : forall (psi:grade) (a b1 b2 a':tm),
lc_tm b1 ->
lc_tm b2 ->
Step a a' ->
Step (a_Case psi a b1 b2) (a_Case psi a' b1 b2)
| S_Case1Beta : forall (psi:grade) (a b1 b2:tm),
lc_tm b2 ->
lc_tm b1 ->
lc_tm a ->
Step (a_Case psi ( (a_Inj1 a) ) b1 b2) (a_App b1 psi a)
| S_Case2Beta : forall (psi:grade) (a b1 b2:tm),
lc_tm b1 ->
lc_tm b2 ->
lc_tm a ->
Step (a_Case psi ( (a_Inj2 a) ) b1 b2) (a_App b2 psi a)
| S_Proj1Cong : forall (psi:grade) (a a':tm),
Step a a' ->
Step (a_Proj1 psi a) (a_Proj1 psi a')
| S_Proj2Cong : forall (psi:grade) (a a':tm),
Step a a' ->
Step (a_Proj2 psi a) (a_Proj2 psi a')
| S_Proj1Beta : forall (psi:grade) (a1 a2:tm),
lc_tm a2 ->
lc_tm a1 ->
Step (a_Proj1 psi (a_SPair a1 psi a2)) a1
| S_Proj2Beta : forall (psi:grade) (a1 a2:tm),
lc_tm a1 ->
lc_tm a2 ->
Step (a_Proj2 psi (a_SPair a1 psi a2)) a2
| S_LetPairCong : forall (psi:grade) (a b a':tm),
lc_tm (a_LetPair psi a b) ->
Step a a' ->
Step (a_LetPair psi a b) (a_LetPair psi a' b)
| S_LetPairBeta : forall (psi:grade) (a1 a2 b:tm),
lc_tm (a_LetPair psi (a_WPair a1 psi a2) b) ->
lc_tm a1 ->
lc_tm a2 ->
Step (a_LetPair psi (a_WPair a1 psi a2) b) (a_App (open_tm_wrt_tm b a1 ) q_Bot a2).
(* defns Jsub *)
Inductive P_sub : econtext -> econtext -> Prop := (* defn P_sub *)
| P_Empty :
P_sub nil nil
| P_Cons : forall (P1:econtext) (x:tmvar) (psi1:grade) (P2:econtext) (psi2:grade),
( psi1 <= psi2 ) ->
P_sub P1 P2 ->
~ AtomSetImpl.In x (dom P1 ) ->
~ AtomSetImpl.In x (dom P2 ) ->
P_sub ( ( x ~ psi1 ) ++ P1 ) ( ( x ~ psi2 ) ++ P2 ) .
(* defns Wsub *)
Inductive ctx_sub : context -> context -> Prop := (* defn ctx_sub *)
| CS_Empty :
ctx_sub nil nil
| CS_ConsTm : forall (W1:context) (x:tmvar) (psi1:grade) (A:tm) (W2:context) (psi2:grade),
( psi1 <= psi2 ) ->
ctx_sub W1 W2 ->
~ AtomSetImpl.In x (dom W1 ) ->
~ AtomSetImpl.In x (dom W2 ) ->
True ->
ctx_sub ( ( x ~( psi1 , A )) ++ W1 ) ( ( x ~( psi2 , A )) ++ W2 ) .
(* defns JGrade *)
Inductive CGrade : econtext -> grade -> grade -> tm -> Prop := (* defn CGrade *)
| CG_Leq : forall (P:econtext) (phi phi0:grade) (a:tm),
( phi0 <= phi ) ->
Grade P phi a ->
CGrade P phi phi0 a
| CG_Nleq : forall (P:econtext) (phi phi0:grade) (a:tm),
lc_tm a ->
not ( ( ( phi0 <= phi ) ) ) ->
uniq P ->
CGrade P phi phi0 a
with Grade : econtext -> grade -> tm -> Prop := (* defn Grade *)
| G_Type : forall (P:econtext) (psi:grade) (s:sort),
uniq P ->
Grade P psi (a_Type s)
| G_Var : forall (P:econtext) (psi:grade) (x:tmvar) (psi0:grade),
uniq P ->
( psi0 <= psi ) ->
binds x psi0 P ->
Grade P psi (a_Var_f x)
| G_Pi : forall (L:vars) (P:econtext) (psi psi0:grade) (A B:tm),
Grade P psi A ->
( forall x , x \notin L -> Grade ( ( x ~ psi ) ++ P ) psi ( open_tm_wrt_tm B (a_Var_f x) ) ) ->
Grade P psi (a_Pi psi0 A B)
| G_Abs : forall (L:vars) (P:econtext) (psi psi0:grade) (A b:tm),
( forall x , x \notin L -> Grade ( ( x ~ psi0 ) ++ P ) psi ( open_tm_wrt_tm b (a_Var_f x) ) ) ->
CGrade P psi q_Top A ->
Grade P psi (a_Abs psi0 A b)
| G_App : forall (P:econtext) (psi:grade) (b:tm) (psi0:grade) (a:tm),
Grade P psi b ->
CGrade P psi psi0 a ->
Grade P psi (a_App b psi0 a)
| G_WSigma : forall (L:vars) (P:econtext) (psi psi0:grade) (A B:tm),
Grade P psi A ->
( forall x , x \notin L -> Grade ( ( x ~ psi ) ++ P ) psi ( open_tm_wrt_tm B (a_Var_f x) ) ) ->
Grade P psi (a_WSigma psi0 A B)
| G_WPair : forall (P:econtext) (psi:grade) (a:tm) (psi0:grade) (b:tm),
CGrade P psi psi0 a ->
Grade P psi b ->
Grade P psi (a_WPair a psi0 b)
| G_LetPair : forall (L:vars) (P:econtext) (psi psi0:grade) (a c:tm),
Grade P psi a ->
( forall x , x \notin L -> Grade ( ( ( x ~ psi0 ) ++ P ) ) psi ( open_tm_wrt_tm c (a_Var_f x) ) ) ->
Grade P psi (a_LetPair psi0 a c)
| G_SSigma : forall (L:vars) (P:econtext) (psi psi0:grade) (A B:tm),
Grade P psi A ->
( forall x , x \notin L -> Grade ( ( x ~ psi ) ++ P ) psi ( open_tm_wrt_tm B (a_Var_f x) ) ) ->
Grade P psi (a_SSigma psi0 A B)
| G_SPair : forall (P:econtext) (psi:grade) (a:tm) (psi0:grade) (b:tm),
CGrade P psi psi0 a ->
Grade P psi b ->
Grade P psi (a_SPair a psi0 b)
| G_Proj1 : forall (P:econtext) (psi psi0:grade) (a:tm),
Grade P psi a ->
( psi0 <= psi ) ->
Grade P psi (a_Proj1 psi0 a)
| G_Proj2 : forall (P:econtext) (psi psi0:grade) (a:tm),
Grade P psi a ->
Grade P psi (a_Proj2 psi0 a)
| G_Sum : forall (P:econtext) (psi:grade) (A B:tm),
Grade P psi A ->
Grade P psi B ->
Grade P psi (a_Sum A B)
| G_Inj1 : forall (P:econtext) (psi:grade) (a1:tm),
Grade P psi a1 ->
Grade P psi (a_Inj1 a1)
| G_Inj2 : forall (P:econtext) (psi:grade) (a2:tm),
Grade P psi a2 ->
Grade P psi (a_Inj2 a2)
| G_Case : forall (P:econtext) (psi psi0:grade) (a b1 b2:tm),
Grade P psi a ->
Grade P psi b1 ->
Grade P psi b2 ->
( psi0 <= psi ) ->
Grade P psi (a_Case psi0 a b1 b2)
| G_TyUnit : forall (P:econtext) (psi:grade),
uniq P ->
Grade P psi a_TyUnit
| G_TmUnit : forall (P:econtext) (psi:grade),
uniq P ->
Grade P psi a_TmUnit.
(* defns JGEq *)
Inductive CEq : econtext -> grade -> grade -> tm -> tm -> Prop := (* defn CEq *)
| CEq_Leq : forall (P:econtext) (psi psi0:grade) (a1 a2:tm),
( psi0 <= psi ) ->
GEq P psi a1 a2 ->
CEq P psi psi0 a1 a2
| CEq_Nleq : forall (P:econtext) (psi psi0:grade) (a1 a2:tm),
lc_tm a1 ->
lc_tm a2 ->
not ( ( ( psi0 <= psi ) ) ) ->
uniq P ->
CEq P psi psi0 a1 a2
with GEq : econtext -> grade -> tm -> tm -> Prop := (* defn GEq *)
| GEq_Var : forall (P:econtext) (psi:grade) (x:tmvar) (psi0:grade),
uniq P ->
binds x psi0 P ->
( psi0 <= psi ) ->
GEq P psi (a_Var_f x) (a_Var_f x)
| GEq_Type : forall (P:econtext) (psi:grade) (s:sort),
uniq P ->
GEq P psi (a_Type s) (a_Type s)
| GEq_Pi : forall (L:vars) (P:econtext) (psi psi0:grade) (A1 B1 A2 B2:tm),
GEq P psi A1 A2 ->
( forall x , x \notin L -> GEq ( ( x ~ psi ) ++ P ) psi ( open_tm_wrt_tm B1 (a_Var_f x) ) ( open_tm_wrt_tm B2 (a_Var_f x) ) ) ->
GEq P psi (a_Pi psi0 A1 B1) (a_Pi psi0 A2 B2)
| GEq_Abs : forall (L:vars) (P:econtext) (psi psi0:grade) (A1 b1 A2 b2:tm),
( forall x , x \notin L -> GEq ( ( x ~ psi0 ) ++ P ) psi ( open_tm_wrt_tm b1 (a_Var_f x) ) ( open_tm_wrt_tm b2 (a_Var_f x) ) ) ->
CEq P psi q_Top A1 A2 ->
GEq P psi (a_Abs psi0 A1 b1) (a_Abs psi0 A2 b2)
| GEq_App : forall (P:econtext) (psi:grade) (b1:tm) (psi0:grade) (a1 b2 a2:tm),
GEq P psi b1 b2 ->
CEq P psi psi0 a1 a2 ->
GEq P psi (a_App b1 psi0 a1) (a_App b2 psi0 a2)
| GEq_WSigma : forall (L:vars) (P:econtext) (psi psi0:grade) (A1 B1 A2 B2:tm),
GEq P psi A1 A2 ->
( forall x , x \notin L -> GEq ( ( x ~ psi ) ++ P ) psi ( open_tm_wrt_tm B1 (a_Var_f x) ) ( open_tm_wrt_tm B2 (a_Var_f x) ) ) ->
GEq P psi (a_WSigma psi0 A1 B1) (a_WSigma psi0 A2 B2)
| GEq_WPair : forall (P:econtext) (psi:grade) (a1:tm) (psi0:grade) (b1 a2 b2:tm),
CEq P psi psi0 a1 a2 ->
GEq P psi b1 b2 ->
GEq P psi (a_WPair a1 psi0 b1) (a_WPair a2 psi0 b2)
| GEq_LetPair : forall (L:vars) (P:econtext) (psi psi0:grade) (a1 b1 a2 b2:tm),
GEq P psi a1 a2 ->
( forall x , x \notin L -> GEq ( ( x ~ psi0 ) ++ P ) psi ( open_tm_wrt_tm b1 (a_Var_f x) ) ( open_tm_wrt_tm b2 (a_Var_f x) ) ) ->
GEq P psi (a_LetPair psi0 a1 b1) (a_LetPair psi0 a2 b2)
| GEq_SSigma : forall (L:vars) (P:econtext) (psi psi0:grade) (A1 B1 A2 B2:tm),
GEq P psi A1 A2 ->
( forall x , x \notin L -> GEq ( ( x ~ psi ) ++ P ) psi ( open_tm_wrt_tm B1 (a_Var_f x) ) ( open_tm_wrt_tm B2 (a_Var_f x) ) ) ->
GEq P psi (a_SSigma psi0 A1 B1) (a_SSigma psi0 A2 B2)
| GEq_SPair : forall (P:econtext) (psi:grade) (a1:tm) (psi0:grade) (b1 a2 b2:tm),
CEq P psi psi0 a1 a2 ->
GEq P psi b1 b2 ->
GEq P psi (a_SPair a1 psi0 b1) (a_SPair a2 psi0 b2)
| GEq_Proj1 : forall (P:econtext) (psi psi0:grade) (a1 a2:tm),
GEq P psi a1 a2 ->
( psi0 <= psi ) ->
GEq P psi (a_Proj1 psi0 a1) (a_Proj1 psi0 a2)
| GEq_Proj2 : forall (P:econtext) (psi psi0:grade) (a1 a2:tm),
GEq P psi a1 a2 ->
GEq P psi (a_Proj2 psi0 a1) (a_Proj2 psi0 a2)
| GEq_Sum : forall (P:econtext) (psi:grade) (A1 A2 A1' A2':tm),
GEq P psi A1 A1' ->
GEq P psi A2 A2' ->
GEq P psi (a_Sum A1 A2) (a_Sum A1' A2')
| GEq_Inj1 : forall (P:econtext) (psi:grade) (a1 a1':tm),
GEq P psi a1 a1' ->
GEq P psi (a_Inj1 a1) (a_Inj1 a1')
| GEq_Inj2 : forall (P:econtext) (psi:grade) (a2 a2':tm),
GEq P psi a2 a2' ->
GEq P psi (a_Inj2 a2) (a_Inj2 a2')
| GEq_Case : forall (P:econtext) (psi psi0:grade) (a b1 b2 a' b1' b2':tm),
GEq P psi a a' ->
GEq P psi b1 b1' ->
GEq P psi b2 b2' ->
( psi0 <= psi ) ->
GEq P psi (a_Case psi0 a b1 b2) (a_Case psi0 a' b1' b2')
| GEq_TyUnit : forall (P:econtext) (psi:grade),
uniq P ->
GEq P psi a_TyUnit a_TyUnit
| GEq_TmUnit : forall (P:econtext) (psi:grade),
uniq P ->
GEq P psi a_TmUnit a_TmUnit.
(* defns JUnTyDefEq *)
Inductive CDefEq : econtext -> grade -> grade -> tm -> tm -> Prop := (* defn CDefEq *)
| CDefEq_Leq : forall (P:econtext) (phi phi0:grade) (a b:tm),
( phi0 <= phi ) ->
DefEq P phi a b ->
CDefEq P phi phi0 a b
| CDefEq_Nleq : forall (P:econtext) (phi phi0:grade) (a b:tm),
lc_tm a ->
lc_tm b ->
not ( ( ( phi0 <= phi ) ) ) ->
uniq P ->
CDefEq P phi phi0 a b
with DefEq : econtext -> grade -> tm -> tm -> Prop := (* defn DefEq *)
| Eq_Refl : forall (P:econtext) (psi:grade) (a:tm),
Grade P psi a ->
DefEq P psi a a
| Eq_Trans : forall (P:econtext) (psi:grade) (a c b:tm),
DefEq P psi a b ->
DefEq P psi b c ->
DefEq P psi a c
| Eq_Sym : forall (P:econtext) (psi:grade) (b a:tm),
DefEq P psi a b ->
DefEq P psi b a
| Eq_Beta : forall (P:econtext) (psi:grade) (a b:tm),
Grade P psi a ->
Step a b ->
Grade P psi b ->
DefEq P psi a b
| Eq_Pi : forall (L:vars) (P:econtext) (psi psi0:grade) (A1 B1 A2 B2:tm),
DefEq P psi A1 A2 ->
( forall x , x \notin L -> DefEq ( ( x ~ psi ) ++ P ) psi ( open_tm_wrt_tm B1 (a_Var_f x) ) ( open_tm_wrt_tm B2 (a_Var_f x) ) ) ->
DefEq P psi (a_Pi psi0 A1 B1) (a_Pi psi0 A2 B2)
| Eq_Abs : forall (L:vars) (P:econtext) (psi psi0:grade) (A1 b1 A2 b2:tm),
CDefEq P psi q_Top A1 A2 ->
( forall x , x \notin L -> DefEq ( ( x ~ psi0 ) ++ P ) psi ( open_tm_wrt_tm b1 (a_Var_f x) ) ( open_tm_wrt_tm b2 (a_Var_f x) ) ) ->
DefEq P psi (a_Abs psi0 A1 b1) (a_Abs psi0 A2 b2)
| Eq_App : forall (P:econtext) (psi:grade) (b1:tm) (psi0:grade) (a1 b2 a2:tm),
DefEq P psi b1 b2 ->
CDefEq P psi psi0 a1 a2 ->
DefEq P psi (a_App b1 psi0 a1) (a_App b2 psi0 a2)
| Eq_PiFst : forall (P:econtext) (psi:grade) (A1 A2:tm) (psi0:grade) (B1 B2:tm),
DefEq P psi (a_Pi psi0 A1 B1) (a_Pi psi0 A2 B2) ->
DefEq P psi A1 A2
| Eq_PiSnd : forall (P:econtext) (psi:grade) (B1 a1 B2 a2:tm) (psi0:grade) (A1 A2:tm),
DefEq P psi (a_Pi psi0 A1 B1) (a_Pi psi0 A2 B2) ->
DefEq P psi a1 a2 ->
DefEq P psi (open_tm_wrt_tm B1 a1 ) (open_tm_wrt_tm B2 a2 )
| Eq_WSigma : forall (L:vars) (P:econtext) (psi psi0:grade) (A1 B1 A2 B2:tm),
DefEq P psi A1 A2 ->
( forall x , x \notin L -> DefEq ( ( x ~ psi ) ++ P ) psi ( open_tm_wrt_tm B1 (a_Var_f x) ) ( open_tm_wrt_tm B2 (a_Var_f x) ) ) ->
DefEq P psi (a_WSigma psi0 A1 B1) (a_WSigma psi0 A2 B2)
| Eq_WSigmaFst : forall (P:econtext) (psi:grade) (A1 A2:tm) (psi0:grade) (B1 B2:tm),
DefEq P psi (a_WSigma psi0 A1 B1) (a_WSigma psi0 A2 B2) ->
DefEq P psi A1 A2
| Eq_WSigmaSnd : forall (P:econtext) (psi:grade) (B1 a B2:tm) (psi0:grade) (A1 A2:tm),
DefEq P psi (a_WSigma psi0 A1 B1) (a_WSigma psi0 A2 B2) ->
Grade P psi a ->
DefEq P psi (open_tm_wrt_tm B1 a ) (open_tm_wrt_tm B2 a )
| Eq_WPair : forall (P:econtext) (psi:grade) (a1:tm) (psi0:grade) (b1 a2 b2:tm),
CDefEq P psi psi0 a1 a2 ->
DefEq P psi b1 b2 ->
DefEq P psi (a_WPair a1 psi0 b1) (a_WPair a2 psi0 b2)
| Eq_LetPair : forall (L:vars) (P:econtext) (psi psi0:grade) (a1 b1 a2 b2:tm),
DefEq P psi a1 a2 ->
( forall x , x \notin L -> DefEq ( ( x ~ psi0 ) ++ P ) psi ( open_tm_wrt_tm b1 (a_Var_f x) ) ( open_tm_wrt_tm b2 (a_Var_f x) ) ) ->
DefEq P psi (a_LetPair psi0 a1 b1) (a_LetPair psi0 a2 b2)
| Eq_SSigma : forall (L:vars) (P:econtext) (psi psi0:grade) (A1 B1 A2 B2:tm),
DefEq P psi A1 A2 ->
( forall x , x \notin L -> DefEq ( ( x ~ psi ) ++ P ) psi ( open_tm_wrt_tm B1 (a_Var_f x) ) ( open_tm_wrt_tm B2 (a_Var_f x) ) ) ->
DefEq P psi (a_SSigma psi0 A1 B1) (a_SSigma psi0 A2 B2)
| Eq_SSigmaFst : forall (P:econtext) (psi:grade) (A1 A2:tm) (psi0:grade) (B1 B2:tm),
DefEq P psi (a_SSigma psi0 A1 B1) (a_SSigma psi0 A2 B2) ->
DefEq P psi A1 A2
| Eq_SSigmaSnd : forall (P:econtext) (psi:grade) (B1 a1 B2 a2:tm) (psi0:grade) (A1 A2:tm),
DefEq P psi (a_SSigma psi0 A1 B1) (a_SSigma psi0 A2 B2) ->
DefEq P psi a1 a2 ->
DefEq P psi (open_tm_wrt_tm B1 a1 ) (open_tm_wrt_tm B2 a2 )
| Eq_SPair : forall (P:econtext) (psi:grade) (a1:tm) (psi0:grade) (b1 a2 b2:tm),
CDefEq P psi psi0 a1 a2 ->
DefEq P psi b1 b2 ->
DefEq P psi (a_SPair a1 psi0 b1) (a_SPair a2 psi0 b2)
| Eq_Proj1 : forall (P:econtext) (psi psi0:grade) (a1 a2:tm),
DefEq P psi a1 a2 ->
( psi0 <= psi ) ->
DefEq P psi (a_Proj1 psi0 a1) (a_Proj1 psi0 a2)
| Eq_Proj2 : forall (P:econtext) (psi psi0:grade) (a1 a2:tm),
DefEq P psi a1 a2 ->
DefEq P psi (a_Proj2 psi0 a1) (a_Proj2 psi0 a2)
| Eq_Sum : forall (P:econtext) (psi:grade) (A1 A2 A1' A2':tm),
DefEq P psi A1 A1' ->
DefEq P psi A2 A2' ->
DefEq P psi (a_Sum A1 A2) (a_Sum A1' A2')
| Eq_SumFst : forall (P:econtext) (psi:grade) (A1 A1' A2 A2':tm),
DefEq P psi (a_Sum A1 A2) (a_Sum A1' A2') ->
DefEq P psi A1 A1'
| Eq_SumSnd : forall (P:econtext) (psi:grade) (A2 A2' A1 A1':tm),
DefEq P psi (a_Sum A1 A2) (a_Sum A1' A2') ->
DefEq P psi A2 A2'
| Eq_Inj1 : forall (P:econtext) (psi:grade) (a1 a1':tm),
DefEq P psi a1 a1' ->
DefEq P psi (a_Inj1 a1) (a_Inj1 a1')
| Eq_Inj2 : forall (P:econtext) (psi:grade) (a2 a2':tm),
DefEq P psi a2 a2' ->
DefEq P psi (a_Inj2 a2) (a_Inj2 a2')
| Eq_Case : forall (P:econtext) (psi psi0:grade) (a b1 b2 a' b1' b2':tm),
DefEq P psi a a' ->
DefEq P psi b1 b1' ->
DefEq P psi b2 b2' ->
( psi0 <= psi ) ->
DefEq P psi (a_Case psi0 a b1 b2) (a_Case psi0 a' b1' b2')
| Eq_TyUnit : forall (P:econtext) (psi:grade),
uniq P ->
DefEq P psi a_TyUnit a_TyUnit
| Eq_TmUnit : forall (P:econtext) (psi:grade),
uniq P ->
DefEq P psi a_TmUnit a_TmUnit
| Eq_SubstIrrel : forall (L:vars) (P:econtext) (phi:grade) (b1 a1 b2 a2:tm) (psi:grade),
lc_tm a1 ->
lc_tm a2 ->
True ->
True ->
( forall x , x \notin L -> DefEq ( ( x ~ psi ) ++ P ) phi ( open_tm_wrt_tm b1 (a_Var_f x) ) ( open_tm_wrt_tm b2 (a_Var_f x) ) ) ->
not ( ( ( psi <= phi ) ) ) ->
DefEq P phi (open_tm_wrt_tm b1 a1 ) (open_tm_wrt_tm b2 a2 ) .
(* defns JTyping *)
Inductive Typing : context -> grade -> tm -> tm -> Prop := (* defn Typing *)
| T_Type : forall (W:context) (psi:grade) (s1 s2:sort),
uniq W ->
( psi <= q_C ) ->
axiom s1 s2 ->
Typing W psi (a_Type s1) (a_Type s2)
| T_Conv : forall (W:context) (psi:grade) (a B A:tm) (s:sort),
Typing W psi a A ->
DefEq (labels (meet_ctx_l q_C W ) ) q_C A B ->
Typing (meet_ctx_l q_C W ) q_C B (a_Type s) ->
Typing W psi a B
| T_Var : forall (W:context) (psi:grade) (x:tmvar) (A:tm) (psi0:grade),
uniq W ->
( psi0 <= psi ) ->
binds x ( psi0 , A ) W ->
( psi <= q_C ) ->
Typing W psi (a_Var_f x) A
| T_Pi : forall (L:vars) (W:context) (psi psi0:grade) (A B:tm) (s3 s1 s2:sort),
Typing W psi A (a_Type s1) ->
( forall x , x \notin L -> Typing ( ( x ~( psi , A )) ++ W ) psi ( open_tm_wrt_tm B (a_Var_f x) ) (a_Type s2) ) ->
rule_pi s1 s2 s3 ->
Typing W psi (a_Pi psi0 A B) (a_Type s3)
| T_Abs : forall (L:vars) (W:context) (psi psi0:grade) (A b B:tm) (s:sort),
( forall x , x \notin L -> Typing ( ( x ~( (q_join psi0 psi ) , A )) ++ W ) psi ( open_tm_wrt_tm b (a_Var_f x) ) ( open_tm_wrt_tm B (a_Var_f x) ) ) ->
Typing (meet_ctx_l q_C W ) q_C ( (a_Pi psi0 A B) ) (a_Type s) ->
Typing W psi (a_Abs psi0 A b) (a_Pi psi0 A B)
| T_App : forall (W:context) (psi:grade) (b:tm) (psi0:grade) (a B A:tm),
Typing W psi b (a_Pi psi0 A B) ->
Typing W (q_join psi0 psi ) a A ->
( psi0 <= q_C ) ->
Typing W psi (a_App b psi0 a) (open_tm_wrt_tm B a )
| T_AppIrrel : forall (W:context) (psi:grade) (b:tm) (psi0:grade) (a B A:tm),
Typing W psi b (a_Pi psi0 A B) ->
Typing (meet_ctx_l q_C W ) q_C a A ->
( q_C < psi0 ) ->
Typing W psi (a_App b psi0 a) (open_tm_wrt_tm B a )
| T_WSigma : forall (L:vars) (W:context) (psi psi0:grade) (A B:tm) (s3 s1 s2:sort),
Typing W psi A (a_Type s1) ->
( forall x , x \notin L -> Typing ( ( x ~( psi , A )) ++ W ) psi ( open_tm_wrt_tm B (a_Var_f x) ) (a_Type s2) ) ->
rule_sig s1 s2 s3 ->
Typing W psi (a_WSigma psi0 A B) (a_Type s3)
| T_WPair : forall (W:context) (psi:grade) (a:tm) (psi0:grade) (b A B:tm) (s:sort),
Typing (meet_ctx_l q_C W ) q_C ( (a_WSigma psi0 A B) ) (a_Type s) ->
Typing W (q_join psi0 psi ) a A ->
Typing W psi b (open_tm_wrt_tm B a ) ->
( psi0 <= q_C ) ->
Typing W psi (a_WPair a psi0 b) (a_WSigma psi0 A B)
| T_WPairIrrel : forall (W:context) (psi:grade) (a:tm) (psi0:grade) (b A B:tm) (s:sort),
Typing (meet_ctx_l q_C W ) q_C ( (a_WSigma psi0 A B) ) (a_Type s) ->
Typing (meet_ctx_l q_C W ) q_C a A ->
( q_C < psi0 ) ->
Typing W psi b (open_tm_wrt_tm B a ) ->
Typing W psi (a_WPair a psi0 b) (a_WSigma psi0 A B)
| T_LetPair : forall (L:vars) (W:context) (psi psi0:grade) (a c C B A:tm) (s:sort),
( forall x , x \notin L ->
Typing ((x ~ (q_C, a_WSigma psi0 A B)) ++ meet_ctx_l q_C W) q_C (open_tm_wrt_tm C (a_Var_f x)) (a_Type s)) ->
Typing W psi a (a_WSigma psi0 A B) ->
( forall x , x \notin L ->
forall y, y \notin L \u {{x}} ->
Typing ((x ~ ((q_join psi0 psi), A)) ++ W) psi (open_tm_wrt_tm c (a_Var_f x))
(a_Pi q_Bot (open_tm_wrt_tm B (a_Var_f x))
(close_tm_wrt_tm y (open_tm_wrt_tm C (a_WPair (a_Var_f x) psi0 (a_Var_f y)))))) ->
(Typing W psi (a_LetPair psi0 a c) (open_tm_wrt_tm C a))
| T_SSigma : forall (L:vars) (W:context) (psi psi0:grade) (A B:tm) (s3 s1 s2:sort),
Typing W psi A (a_Type s1) ->
( forall x , x \notin L -> Typing ( ( x ~( psi , A )) ++ W ) psi ( open_tm_wrt_tm B (a_Var_f x) ) (a_Type s2) ) ->
rule_sig s1 s2 s3 ->
Typing W psi (a_SSigma psi0 A B) (a_Type s3)
| T_SPair : forall (W:context) (psi:grade) (a:tm) (psi0:grade) (b A B:tm) (s:sort),
Typing (meet_ctx_l q_C W ) q_C (a_SSigma psi0 A B) (a_Type s) ->
Typing W (q_join psi0 psi ) a A ->
Typing W psi b (open_tm_wrt_tm B a ) ->
( psi0 <= q_C ) ->
Typing W psi (a_SPair a psi0 b) (a_SSigma psi0 A B)
| T_Proj1 : forall (W:context) (psi psi0:grade) (a A B:tm),
Typing W psi a (a_SSigma psi0 A B) ->
( psi0 <= psi ) ->
Typing W psi (a_Proj1 psi0 a) A
| T_Proj2 : forall (W:context) (psi psi0:grade) (a B A:tm),
Typing W psi a (a_SSigma psi0 A B) ->
( psi0 <= q_C ) ->
Typing W psi (a_Proj2 psi0 a) (open_tm_wrt_tm B (a_Proj1 psi0 a) )
| T_Sum : forall (W:context) (psi:grade) (A B:tm) (s:sort),
Typing W psi A (a_Type s) ->
Typing W psi B (a_Type s) ->
Typing W psi (a_Sum A B) (a_Type s)
| T_Inj1 : forall (W:context) (psi:grade) (a1 A1 A2:tm) (s:sort),
Typing W psi a1 A1 ->
Typing (meet_ctx_l q_C W ) q_C (a_Sum A1 A2) (a_Type s) ->
Typing W psi (a_Inj1 a1) (a_Sum A1 A2)
| T_Inj2 : forall (W:context) (psi:grade) (a2 A1 A2:tm) (s:sort),
Typing W psi a2 A2 ->
Typing (meet_ctx_l q_C W ) q_C (a_Sum A1 A2) (a_Type s) ->
Typing W psi (a_Inj2 a2) (a_Sum A1 A2)
| T_Case : forall (L:vars) (W:context) (psi psi0:grade) (a b1 b2 B:tm) (A1 A2 B1 B2:tm) s,
( forall z, z \notin L -> Typing ( ( ( z ~( q_C , (a_Sum A1 A2) )) ++ ( (meet_ctx_l q_C W ) ) ) ) q_C (open_tm_wrt_tm B (a_Var_f z)) (a_Type s)) ->
Typing W psi a (a_Sum A1 A2) ->
( forall x , x \notin L -> ( open_tm_wrt_tm B1 (a_Var_f x) ) = (open_tm_wrt_tm B (a_Inj1 (a_Var_f x)) ) ) ->
( forall y , y \notin L -> ( open_tm_wrt_tm B2 (a_Var_f y) ) = (open_tm_wrt_tm B (a_Inj2 (a_Var_f y)) ) ) ->
Typing W psi b1 (a_Pi psi0 A1 B1) ->
Typing W psi b2 (a_Pi psi0 A2 B2) ->
( psi0 <= psi ) ->
Typing W psi (a_Case psi0 a b1 b2) (open_tm_wrt_tm B a )
| T_TmUnit : forall (W:context) (psi:grade) (s:sort),
uniq W ->
( psi <= q_C ) ->
Typing W psi a_TyUnit (a_Type s)
| T_TyUnit : forall (W:context) (psi:grade),
uniq W ->
( psi <= q_C ) ->
Typing W psi a_TmUnit a_TyUnit.
(* defns JCTyping *)
Inductive CTyping : context -> grade -> tm -> tm -> Prop := (* defn CTyping *)
| CT_Leq : forall (W:context) (psi:grade) (a A:tm),
Typing W psi a A ->
( psi <= q_C ) ->
CTyping W psi a A
| CT_Top : forall (W:context) (psi:grade) (a A:tm),
Typing (meet_ctx_l q_C W ) q_C a A ->
( q_C < psi ) ->
CTyping W psi a A.
(* defns JCtx *)
Inductive Ctx : context -> Prop := (* defn Ctx *)
| Ctx_Empty :
Ctx nil
| Ctx_Cons : forall (W:context) (x:tmvar) (psi0:grade) (A:tm) (s:sort),
Ctx W ->
Typing (meet_ctx_l q_C W ) q_C A (a_Type s) ->
~ AtomSetImpl.In x (dom W ) ->
Ctx ( ( x ~( psi0 , A )) ++ W ) .
(* defns JValueType *)
Inductive ValueType : tm -> Prop := (* defn ValueType *)
| ValueType_Type : forall (s:sort),
ValueType (a_Type s)
| ValueType_Unit :
ValueType a_TyUnit
| ValueType_Pi : forall (psi:grade) (A B:tm),
lc_tm A ->
lc_tm (a_Pi psi A B) ->
ValueType (a_Pi psi A B)
| ValueType_WSigma : forall (psi:grade) (A B:tm),
lc_tm A ->
lc_tm (a_WSigma psi A B) ->
ValueType (a_WSigma psi A B)
| ValueType_SSigma : forall (psi:grade) (A B:tm),
lc_tm A ->
lc_tm (a_SSigma psi A B) ->
ValueType (a_SSigma psi A B)
| ValueType_Sum : forall (A B:tm),
lc_tm A ->
lc_tm B ->
ValueType (a_Sum A B).
(* defns JConsistent *)
Inductive Consistent : tm -> tm -> Prop := (* defn Consistent *)
| Consistent_a_Type : forall (s:sort),
Consistent (a_Type s) (a_Type s)
| Consistent_a_Unit :
Consistent a_TyUnit a_TyUnit
| Consistent_a_Pi : forall (psi:grade) (A B A' B':tm),
lc_tm A ->
lc_tm (a_Pi psi A B) ->
lc_tm A' ->
lc_tm (a_Pi psi A' B') ->
Consistent ( (a_Pi psi A B) ) ( (a_Pi psi A' B') )
| Consistent_a_WSigma : forall (psi:grade) (A B A' B':tm),
lc_tm A ->
lc_tm (a_WSigma psi A B) ->
lc_tm A' ->
lc_tm (a_WSigma psi A' B') ->
Consistent ( (a_WSigma psi A B) ) ( (a_WSigma psi A' B') )
| Consistent_a_SSigma : forall (psi:grade) (A B A' B':tm),
lc_tm A ->
lc_tm (a_SSigma psi A B) ->
lc_tm A' ->
lc_tm (a_SSigma psi A' B') ->
Consistent (a_SSigma psi A B) (a_SSigma psi A' B')
| Consistent_a_Sum : forall (A B A' B':tm),
lc_tm A ->
lc_tm B ->
lc_tm A' ->
lc_tm B' ->
Consistent ( (a_Sum A B) ) ( (a_Sum A' B') )
| Consistent_a_Step_R : forall (a b:tm),
lc_tm b ->
not ( ValueType a ) ->
Consistent a b
| Consistent_a_Step_L : forall (a b:tm),
lc_tm a ->
not ( ValueType b ) ->
Consistent a b.
(* defns JPar *)
Inductive CPar : econtext -> grade -> grade -> tm -> tm -> Prop := (* defn CPar *)
| CPar_Leq : forall (P:econtext) (psi psi0:grade) (a1 a2:tm),
( psi0 <= psi ) ->
Par P psi a1 a2 ->
CPar P psi psi0 a1 a2
| CPar_Nleq : forall (P:econtext) (psi psi0:grade) (a1 a2:tm),
lc_tm a1 ->
lc_tm a2 ->
not ( ( ( psi0 <= psi ) ) ) ->
uniq P ->
CPar P psi psi0 a1 a2
with Par : econtext -> grade -> tm -> tm -> Prop := (* defn Par *)
| Par_Refl : forall (P:econtext) (psi:grade) (a:tm),
Grade P psi a ->
Par P psi a a
| Par_Pi : forall (L:vars) (P:econtext) (psi psi1:grade) (A1 B1 A2 B2:tm),
Par P psi A1 A2 ->
( forall x , x \notin L -> Par ( ( x ~ psi ) ++ P ) psi ( open_tm_wrt_tm B1 (a_Var_f x) ) ( open_tm_wrt_tm B2 (a_Var_f x) ) ) ->
Par P psi (a_Pi psi1 A1 B1) (a_Pi psi1 A2 B2)
| Par_AppBeta : forall (P:econtext) (psi:grade) (a:tm) (psi0:grade) (b a' b' A:tm),
Par P psi a ( (a_Abs psi0 A a') ) ->
CPar P psi psi0 b b' ->
Par P psi (a_App a psi0 b) (open_tm_wrt_tm a' b' )
| Par_App : forall (P:econtext) (psi:grade) (a:tm) (psi0:grade) (b a' b':tm),
Par P psi a a' ->
CPar P psi psi0 b b' ->
Par P psi (a_App a psi0 b) (a_App a' psi0 b')
| Par_Abs : forall (L:vars) (P:econtext) (psi psi0:grade) (A1 b1 A2 b2:tm),
( forall x , x \notin L -> Par ( ( x ~ psi0 ) ++ P ) psi ( open_tm_wrt_tm b1 (a_Var_f x) ) ( open_tm_wrt_tm b2 (a_Var_f x) ) ) ->
CPar P psi q_Top A1 A2 ->
Par P psi (a_Abs psi0 A1 b1) (a_Abs psi0 A2 b2)
| Par_WSigma : forall (L:vars) (P:econtext) (psi psi1:grade) (A1 B1 A2 B2:tm),
Par P psi A1 A2 ->
( forall x , x \notin L -> Par ( ( x ~ psi ) ++ P ) psi ( open_tm_wrt_tm B1 (a_Var_f x) ) ( open_tm_wrt_tm B2 (a_Var_f x) ) ) ->
Par P psi (a_WSigma psi1 A1 B1) (a_WSigma psi1 A2 B2)
| Par_WPair : forall (P:econtext) (psi:grade) (a1:tm) (psi0:grade) (b1 a2 b2:tm),
CPar P psi psi0 a1 a2 ->
Par P psi b1 b2 ->
Par P psi (a_WPair a1 psi0 b1) (a_WPair a2 psi0 b2)
| Par_WPairBeta : forall (L:vars) (P:econtext) (psi psi0:grade) (a1 b1 b2 a1' a2':tm),
Par P psi a1 (a_WPair a1' psi0 a2') ->
( forall x , x \notin L -> Par ( ( x ~ psi0 ) ++ P ) psi ( open_tm_wrt_tm b1 (a_Var_f x) ) (open_tm_wrt_tm b2 (a_Var_f x) ) ) ->
Par P psi (a_LetPair psi0 a1 b1) (a_App (open_tm_wrt_tm b2 a1' ) q_Bot a2')
| Par_LetPair : forall (L:vars) (P:econtext) (psi psi0:grade) (a1 b1 a2 b2:tm),
Par P psi a1 a2 ->
( forall x , x \notin L -> Par ( ( x ~ psi0 ) ++ P ) psi ( open_tm_wrt_tm b1 (a_Var_f x) ) ( open_tm_wrt_tm b2 (a_Var_f x) ) ) ->
Par P psi (a_LetPair psi0 a1 b1) (a_LetPair psi0 a2 b2)
| Par_SSigma : forall (L:vars) (P:econtext) (psi psi1:grade) (A1 B1 A2 B2:tm),
Par P psi A1 A2 ->
( forall x , x \notin L -> Par ( ( x ~ psi ) ++ P ) psi ( open_tm_wrt_tm B1 (a_Var_f x) ) ( open_tm_wrt_tm B2 (a_Var_f x) ) ) ->
Par P psi (a_SSigma psi1 A1 B1) (a_SSigma psi1 A2 B2)
| Par_SPair : forall (P:econtext) (psi:grade) (a1:tm) (psi0:grade) (b1 a2 b2:tm),
CPar P psi psi0 a1 a2 ->
Par P psi b1 b2 ->
Par P psi (a_SPair a1 psi0 b1) (a_SPair a2 psi0 b2)
| Par_Proj1Beta : forall (P:econtext) (psi psi0:grade) (a1 a1' a2:tm),
Par P psi a1 (a_SPair a1' psi0 a2) ->
( psi0 <= psi ) ->
Par P psi (a_Proj1 psi0 a1) a1'
| Par_Proj1 : forall (P:econtext) (psi psi0:grade) (a1 a2:tm),
Par P psi a1 a2 ->
( psi0 <= psi ) ->
Par P psi (a_Proj1 psi0 a1) (a_Proj1 psi0 a2)
| Par_Proj2Beta : forall (P:econtext) (psi psi0:grade) (a1 a2 a1':tm),
Par P psi a1 (a_SPair a1' psi0 a2) ->
Par P psi (a_Proj2 psi0 a1) a2
| Par_Proj2 : forall (P:econtext) (psi psi0:grade) (a1 a2:tm),
Par P psi a1 a2 ->
Par P psi (a_Proj2 psi0 a1) (a_Proj2 psi0 a2)
| Par_Sum : forall (P:econtext) (psi:grade) (A1 A2 A1' A2':tm),
Par P psi A1 A1' ->
Par P psi A2 A2' ->
Par P psi (a_Sum A1 A2) (a_Sum A1' A2')
| Par_Inj1 : forall (P:econtext) (psi:grade) (a1 a1':tm),
Par P psi a1 a1' ->
Par P psi (a_Inj1 a1) (a_Inj1 a1')
| Par_Inj2 : forall (P:econtext) (psi:grade) (a2 a2':tm),
Par P psi a2 a2' ->
Par P psi (a_Inj2 a2) (a_Inj2 a2')
| Par_CaseBeta1 : forall (P:econtext) (psi psi0:grade) (a b1 b2 b1' a' b2':tm),
Par P psi a (a_Inj1 a') ->
Par P psi b1 b1' ->
Par P psi b2 b2' ->
( psi0 <= psi ) ->
Par P psi (a_Case psi0 a b1 b2) (a_App b1' psi0 a')
| Par_CaseBeta2 : forall (P:econtext) (psi psi0:grade) (a b1 b2 b2' a' b1':tm),
Par P psi a (a_Inj2 a') ->
Par P psi b1 b1' ->
Par P psi b2 b2' ->
( psi0 <= psi ) ->
Par P psi (a_Case psi0 a b1 b2) (a_App b2' psi0 a')
| Par_Case : forall (P:econtext) (psi psi0:grade) (a b1 b2 a' b1' b2':tm),
Par P psi a a' ->
Par P psi b1 b1' ->
Par P psi b2 b2' ->
( psi0 <= psi ) ->
Par P psi (a_Case psi0 a b1 b2) (a_Case psi0 a' b1' b2').
(* defns JMultiPar *)
Inductive MultiPar : econtext -> grade -> tm -> tm -> Prop := (* defn MultiPar *)
| MP_Refl : forall (P:econtext) (psi:grade) (a:tm),
Grade P psi a ->
MultiPar P psi a a
| MP_Step : forall (P:econtext) (psi:grade) (a a' b:tm),
Par P psi a b ->
MultiPar P psi b a' ->
MultiPar P psi a a'.
(* defns JJoins *)
Inductive Joins : econtext -> grade -> tm -> tm -> Prop := (* defn Joins *)
| join : forall (P:econtext) (psi:grade) (a1 a2 b1 b2:tm),
MultiPar P psi a1 b1 ->
MultiPar P psi a2 b2 ->
GEq P psi b1 b2 ->
Joins P psi a1 a2.
(* defns JValue *)
Inductive Value : tm -> Prop := (* defn Value *)
| V_ValueType : forall (a:tm),
ValueType a ->
Value a
| V_TmUnit :
Value a_TmUnit
| V_Abs : forall (psi:grade) (A a:tm),
lc_tm A ->
lc_tm (a_Abs psi A a) ->
Value (a_Abs psi A a)
| V_WPair : forall (a:tm) (psi:grade) (b:tm),
lc_tm a ->
lc_tm b ->
Value (a_WPair a psi b)
| V_SPair : forall (a:tm) (psi:grade) (b:tm),
lc_tm a ->
lc_tm b ->
Value (a_SPair a psi b)
| V_Inj1 : forall (a:tm),
lc_tm a ->
Value (a_Inj1 a)
| V_Inj2 : forall (a:tm),
lc_tm a ->
Value (a_Inj2 a).
(* defns JCMultiPar *)
Inductive CMultiPar : econtext -> grade -> grade -> tm -> tm -> Prop := (* defn CMultiPar *)
| CMP_Leq : forall (P:econtext) (psi psi0:grade) (a1 a2:tm),
( psi0 <= psi ) ->
MultiPar P psi a1 a2 ->
CMultiPar P psi psi0 a1 a2
| CMP_Nleq : forall (P:econtext) (psi psi0:grade) (a1 a2:tm),
lc_tm a1 ->
lc_tm a2 ->
not ( ( ( psi0 <= psi ) ) ) ->
uniq P ->
CMultiPar P psi psi0 a1 a2.
(* defns JCJoins *)
Inductive CJoins : econtext -> grade -> grade -> tm -> tm -> Prop := (* defn CJoins *)
| CJ_Leq : forall (P:econtext) (psi psi0:grade) (a1 a2:tm),
( psi0 <= psi ) ->
Joins P psi a1 a2 ->
CJoins P psi psi0 a1 a2
| CJ_Nleq : forall (P:econtext) (psi psi0:grade) (a1 a2:tm),
lc_tm a1 ->
lc_tm a2 ->
not ( ( ( psi0 <= psi ) ) ) ->
uniq P ->
CJoins P psi psi0 a1 a2.
(** infrastructure *)
Hint Constructors Step P_sub ctx_sub CGrade Grade CEq GEq CDefEq DefEq Typing CTyping Ctx ValueType Consistent CPar Par MultiPar Joins Value CMultiPar CJoins lc_tm : core.
|
(*
* Copyright (C) 2014, National ICT Australia Limited. All rights reserved.
*
* Redistribution and use in source and binary forms, with or without
* modification, are permitted provided that the following conditions are
* met:
*
* * Redistributions of source code must retain the above copyright
* notice, this list of conditions and the following disclaimer.
*
* * Redistributions in binary form must reproduce the above copyright
* notice, this list of conditions and the following disclaimer in the
* documentation and/or other materials provided with the distribution.
*
* * The name of National ICT Australia Limited nor the names of its
* contributors may be used to endorse or promote products derived from
* this software without specific prior written permission.
*
* THIS SOFTWARE IS PROVIDED BY THE COPYRIGHT HOLDERS AND CONTRIBUTORS "AS
* IS" AND ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED
* TO, THE IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A
* PARTICULAR PURPOSE ARE DISCLAIMED. IN NO EVENT SHALL THE COPYRIGHT
* OWNER OR CONTRIBUTORS BE LIABLE FOR ANY DIRECT, INDIRECT, INCIDENTAL,
* SPECIAL, EXEMPLARY, OR CONSEQUENTIAL DAMAGES (INCLUDING, BUT NOT
* LIMITED TO, PROCUREMENT OF SUBSTITUTE GOODS OR SERVICES; LOSS OF USE,
* DATA, OR PROFITS; OR BUSINESS INTERRUPTION) HOWEVER CAUSED AND ON ANY
* THEORY OF LIABILITY, WHETHER IN CONTRACT, STRICT LIABILITY, OR TORT
* (INCLUDING NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY OUT OF THE USE
* OF THIS SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF SUCH DAMAGE.
*)
theory WordLemmaBucket
imports
Lib
MoreDivides
Aligned
HOLLemmaBucket
DistinctPropLemmaBucket
"~~/src/HOL/Library/Sublist"
"~~/src/HOL/Library/Prefix_Order"
begin
(* Setup "quickcheck" to support words. *)
quickcheck_generator word
constructors:
"zero_class.zero :: ('a::len) word",
"numeral :: num \<Rightarrow> ('a::len) word",
"uminus :: ('a::len) word \<Rightarrow> ('a::len) word"
instantiation Enum.finite_1 :: len
begin
definition "len_of_finite_1 (x :: Enum.finite_1 itself) \<equiv> (1 :: nat)"
instance
by (default, auto simp: len_of_finite_1_def)
end
instantiation Enum.finite_2 :: len
begin
definition "len_of_finite_2 (x :: Enum.finite_2 itself) \<equiv> (2 :: nat)"
instance
by (default, auto simp: len_of_finite_2_def)
end
instantiation Enum.finite_3 :: len
begin
definition "len_of_finite_3 (x :: Enum.finite_3 itself) \<equiv> (4 :: nat)"
instance
by (default, auto simp: len_of_finite_3_def)
end
(* Provide wf and less_induct for word.
wf may be more useful in loop proofs, less_induct in recursion proofs. *)
lemma word_less_wf: "wf {(a, b). a < (b :: ('a::len) word)}"
apply (rule wf_subset)
apply (rule wf_measure)
apply safe
apply (subst in_measure)
apply (erule unat_mono)
done
lemma word_less_induct:
"\<lbrakk> \<And>x::('a::len) word. (\<And>y. y < x \<Longrightarrow> P y) \<Longrightarrow> P x \<rbrakk> \<Longrightarrow> P a"
using word_less_wf
apply induct
apply blast
done
instantiation word :: (len) wellorder
begin
instance
apply (intro_classes)
apply (metis word_less_induct)
done
end
lemma word_plus_mono_left:
fixes x :: "'a :: len word"
shows "\<lbrakk>y \<le> z; x \<le> x + z\<rbrakk> \<Longrightarrow> y + x \<le> z + x"
by unat_arith
lemma word_2p_mult_inc:
assumes x: "2 * 2 ^ n < (2::'a::len word) * 2 ^ m"
assumes suc_n: "Suc n < len_of TYPE('a::len)"
assumes suc_m: "Suc m < len_of TYPE('a::len)"
assumes 2: "unat (2::'a::len word) = 2"
shows "2^n < (2::'a::len word)^m"
proof -
from suc_n
have "(2::nat) * 2 ^ n mod 2 ^ len_of TYPE('a::len) = 2 * 2^n"
apply (subst mod_less)
apply (subst power_Suc[symmetric])
apply (rule power_strict_increasing)
apply simp
apply simp
apply simp
done
moreover
from suc_m
have "(2::nat) * 2 ^ m mod 2 ^ len_of TYPE('a::len) = 2 * 2^m"
apply (subst mod_less)
apply (subst power_Suc[symmetric])
apply (rule power_strict_increasing)
apply simp
apply simp
apply simp
done
ultimately
have "2 * 2 ^ n < (2::nat) * 2 ^ m" using x
apply (unfold word_less_nat_alt)
apply simp
apply (subst (asm) unat_word_ariths(2))+
apply (subst (asm) 2)+
apply (subst (asm) word_unat_power, subst (asm) unat_of_nat)+
apply (simp add: mod_mult_right_eq[symmetric])
done
with suc_n suc_m
show ?thesis
unfolding word_less_nat_alt
apply (subst word_unat_power, subst unat_of_nat)+
apply simp
done
qed
lemma word_shiftl_add_distrib:
fixes x :: "'a :: len word"
shows "(x + y) << n = (x << n) + (y << n)"
by (simp add: shiftl_t2n ring_distribs)
lemma upper_bits_unset_is_l2p:
"n < word_bits \<Longrightarrow> (\<forall>n' \<ge> n. n' < word_bits \<longrightarrow> \<not> p !! n') = ((p::word32) < 2 ^ n)"
apply (rule iffI)
prefer 2
apply (clarsimp simp: word_bits_def)
apply (drule bang_is_le)
apply (drule_tac y=p in order_le_less_trans, assumption)
apply (drule word_power_increasing)
apply simp
apply simp
apply simp
apply simp
apply (subst mask_eq_iff_w2p [symmetric])
apply (clarsimp simp: word_size word_bits_def)
apply (rule word_eqI)
apply (clarsimp simp: word_size word_bits_def)
apply (case_tac "na < n", auto)
done
lemma up_ucast_inj:
"\<lbrakk> ucast x = (ucast y::'b::len word); len_of TYPE('a) \<le> len_of TYPE ('b) \<rbrakk> \<Longrightarrow> x = (y::'a::len word)"
apply (subst (asm) bang_eq)
apply (fastforce simp: nth_ucast word_size intro: word_eqI)
done
lemma up_ucast_inj_eq:
"len_of TYPE('a) \<le> len_of TYPE ('b) \<Longrightarrow> (ucast x = (ucast y::'b::len word)) = (x = (y::'a::len word))"
by (fastforce dest: up_ucast_inj)
lemma ucast_up_inj:
"\<lbrakk> ucast x = (ucast y :: 'b::len word); len_of TYPE('a) \<le> len_of TYPE('b) \<rbrakk>
\<Longrightarrow> x = (y :: 'a::len word)"
apply (subst (asm) bang_eq)
apply (rule word_eqI)
apply (simp add: word_size nth_ucast)
apply (erule_tac x=n in allE)
apply simp
done
lemma ucast_8_32_inj:
"inj (ucast :: 8 word \<Rightarrow> 32 word)"
apply (rule down_ucast_inj)
apply (clarsimp simp: is_down_def target_size source_size)
done
lemma no_plus_overflow_neg:
"(x :: ('a :: len) word) < -y \<Longrightarrow> x \<le> x + y"
apply (simp add: no_plus_overflow_uint_size
word_less_alt uint_word_ariths
word_size)
apply (subst(asm) zmod_zminus1_eq_if)
apply (simp split: split_if_asm)
done
lemma ucast_ucast_eq:
fixes x :: "'a::len word"
fixes y :: "'b::len word"
shows
"\<lbrakk> ucast x = (ucast (ucast y::'a::len word)::'c::len word);
len_of TYPE('a) \<le> len_of TYPE('b);
len_of TYPE('b) \<le> len_of TYPE('c) \<rbrakk> \<Longrightarrow>
x = ucast y"
apply (rule word_eqI)
apply (subst (asm) bang_eq)
apply (erule_tac x=n in allE)
apply (simp add: nth_ucast word_size)
done
(******** GeneralLib ****************)
lemma neq_into_nprefixeq:
"\<lbrakk> x \<noteq> take (length x) y \<rbrakk> \<Longrightarrow> \<not> x \<le> y"
by (clarsimp simp: prefixeq_def less_eq_list_def)
lemma distinct_suffixeq:
assumes dx: "distinct xs"
and pf: "suffixeq ys xs"
shows "distinct ys"
using dx pf by (clarsimp elim!: suffixeqE)
lemma suffixeq_map:
assumes pf: "suffixeq ys xs"
shows "suffixeq (map f ys) (map f xs)"
using pf by (auto elim!: suffixeqE intro: suffixeqI)
lemma suffixeq_drop [simp]:
"suffixeq (drop n as) as"
unfolding suffixeq_def
apply (rule exI [where x = "take n as"])
apply simp
done
lemma suffixeq_take:
"suffixeq ys xs \<Longrightarrow> xs = take (length xs - length ys) xs @ ys"
by (clarsimp elim!: suffixeqE)
lemma suffixeq_eqI:
"\<lbrakk> suffixeq xs as; suffixeq xs bs; length as = length bs;
take (length as - length xs) as \<le> take (length bs - length xs) bs\<rbrakk> \<Longrightarrow> as = bs"
by (clarsimp elim!: prefixE suffixeqE)
lemma suffixeq_Cons_mem:
"suffixeq (x # xs) as \<Longrightarrow> x \<in> set as"
apply (drule suffixeq_set_subset)
apply simp
done
lemma list_induct_suffixeq [case_names Nil Cons]:
assumes nilr: "P []"
and consr: "\<And>x xs. \<lbrakk>P xs; suffixeq (x # xs) as \<rbrakk> \<Longrightarrow> P (x # xs)"
shows "P as"
proof -
def as' == as
have "suffixeq as as'" unfolding as'_def by simp
thus ?thesis
proof (induct as)
case Nil show ?case by fact
next
case (Cons x xs)
show ?case
proof (rule consr)
from Cons.prems show "suffixeq (x # xs) as" unfolding as'_def .
hence "suffixeq xs as'" by (auto dest: suffixeq_ConsD simp: as'_def)
thus "P xs" using Cons.hyps by simp
qed
qed
qed
text {* Parallel etc. and lemmas for list prefix *}
lemma prefix_induct [consumes 1, case_names Nil Cons]:
fixes prefix
assumes np: "prefix \<le> lst"
and base: "\<And>xs. P [] xs"
and rl: "\<And>x xs y ys. \<lbrakk> x = y; xs \<le> ys; P xs ys \<rbrakk> \<Longrightarrow> P (x#xs) (y#ys)"
shows "P prefix lst"
using np
proof (induct prefix arbitrary: lst)
case Nil show ?case by fact
next
case (Cons x xs)
have prem: "(x # xs) \<le> lst" by fact
then obtain y ys where lv: "lst = y # ys"
by (rule prefixE, auto)
have ih: "\<And>lst. xs \<le> lst \<Longrightarrow> P xs lst" by fact
show ?case using prem
by (auto simp: lv intro!: rl ih)
qed
lemma not_prefix_cases:
fixes prefix
assumes pfx: "\<not> prefix \<le> lst"
and c1: "\<lbrakk> prefix \<noteq> []; lst = [] \<rbrakk> \<Longrightarrow> R"
and c2: "\<And>a as x xs. \<lbrakk> prefix = a#as; lst = x#xs; x = a; \<not> as \<le> xs\<rbrakk> \<Longrightarrow> R"
and c3: "\<And>a as x xs. \<lbrakk> prefix = a#as; lst = x#xs; x \<noteq> a\<rbrakk> \<Longrightarrow> R"
shows "R"
proof (cases prefix)
case Nil thus ?thesis using pfx by simp
next
case (Cons a as)
have c: "prefix = a#as" by fact
show ?thesis
proof (cases lst)
case Nil thus ?thesis
by (intro c1, simp add: Cons)
next
case (Cons x xs)
show ?thesis
proof (cases "x = a")
case True
show ?thesis
proof (intro c2)
show "\<not> as \<le> xs" using pfx c Cons True
by simp
qed fact+
next
case False
show ?thesis by (rule c3) fact+
qed
qed
qed
lemma not_prefix_induct [consumes 1, case_names Nil Neq Eq]:
fixes prefix
assumes np: "\<not> prefix \<le> lst"
and base: "\<And>x xs. P (x#xs) []"
and r1: "\<And>x xs y ys. x \<noteq> y \<Longrightarrow> P (x#xs) (y#ys)"
and r2: "\<And>x xs y ys. \<lbrakk> x = y; \<not> xs \<le> ys; P xs ys \<rbrakk> \<Longrightarrow> P (x#xs) (y#ys)"
shows "P prefix lst"
using np
proof (induct lst arbitrary: prefix)
case Nil thus ?case
by (auto simp: neq_Nil_conv elim!: not_prefix_cases intro!: base)
next
case (Cons y ys)
have npfx: "\<not> prefix \<le> (y # ys)" by fact
then obtain x xs where pv: "prefix = x # xs"
by (rule not_prefix_cases) auto
have ih: "\<And>prefix. \<not> prefix \<le> ys \<Longrightarrow> P prefix ys" by fact
show ?case using npfx
by (simp only: pv) (erule not_prefix_cases, auto intro: r1 r2 ih)
qed
text {* right-padding a word to a certain length *}
lemma bl_pad_to_prefix:
"bl \<le> bl_pad_to bl sz"
by (simp add: bl_pad_to_def)
lemma same_length_is_parallel:
assumes len: "\<forall>y \<in> set as. length y = x"
shows "\<forall>x \<in> set as. \<forall>y \<in> set as - {x}. x \<parallel> y"
proof (rule, rule)
fix x y
assume xi: "x \<in> set as" and yi: "y \<in> set as - {x}"
from len obtain q where len': "\<forall>y \<in> set as. length y = q" ..
show "x \<parallel> y"
proof (rule not_equal_is_parallel)
from xi yi show "x \<noteq> y" by auto
from xi yi len' show "length x = length y" by (auto dest: bspec)
qed
qed
text {* Lemmas about words *}
lemma word_bits_len_of: "len_of TYPE (32) = word_bits"
by (simp add: word_bits_conv)
lemmas unat_power_lower32 [simp] = unat_power_lower[where 'a=32, unfolded word_bits_len_of]
lemma eq_zero_set_bl: "(w = 0) = (True \<notin> set (to_bl w))"
apply (subst word_bl.Rep_inject[symmetric])
apply (subst to_bl_0)
apply (rule iffI)
apply clarsimp
apply (drule list_of_false)
apply simp
done
lemmas and_bang = word_and_nth
lemma of_drop_to_bl:
"of_bl (drop n (to_bl x)) = (x && mask (size x - n))"
apply (clarsimp simp: bang_eq and_bang test_bit_of_bl
rev_nth
cong: rev_conj_cong)
apply (safe, simp_all add: word_size to_bl_nth)
done
lemma less_is_drop_replicate:
fixes x :: "'a :: len word"
assumes lt: "x < 2 ^ n"
shows "to_bl x = replicate (len_of TYPE('a) - n) False @ drop (len_of TYPE('a) - n) (to_bl x)"
proof -
show ?thesis
apply (subst less_mask_eq [OF lt, symmetric])
apply (subst bl_and_mask)
apply simp
done
qed
lemma word_add_offset_less:
fixes x :: "'a :: len word"
assumes yv: "y < 2 ^ n"
and xv: "x < 2 ^ m"
and mnv: "sz < len_of TYPE('a :: len)"
and xv': "x < 2 ^ (len_of TYPE('a :: len) - n)"
and mn: "sz = m + n"
shows "x * 2 ^ n + y < 2 ^ sz"
proof (subst mn)
from mnv mn have nv: "n < len_of TYPE('a)" and mv: "m < len_of TYPE('a)" by auto
have uy: "unat y < 2 ^ n"
by (rule order_less_le_trans [OF unat_mono [OF yv] order_eq_refl],
rule unat_power_lower[OF nv])
have ux: "unat x < 2 ^ m"
by (rule order_less_le_trans [OF unat_mono [OF xv] order_eq_refl],
rule unat_power_lower[OF mv])
thus "x * 2 ^ n + y < 2 ^ (m + n)" using ux uy nv mnv xv'
apply (subst word_less_nat_alt)
apply (subst unat_word_ariths word_bits_len_of)+
apply (subst mod_less)
apply (simp add: unat_power_lower)
apply (subst mult.commute)
apply (rule nat_less_power_trans [OF _ order_less_imp_le [OF nv]])
apply (rule order_less_le_trans [OF unat_mono [OF xv']])
apply (cases "n = 0")
apply (simp add: unat_power_lower)
apply (simp add: unat_power_lower)
apply (subst unat_power_lower[OF nv])
apply (subst mod_less)
apply (erule order_less_le_trans [OF nat_add_offset_less], assumption)
apply (rule mn)
apply simp
apply (simp add: mn mnv unat_power_lower)
apply (erule nat_add_offset_less)
apply simp+
done
qed
lemma word_less_power_trans:
fixes n :: "'a :: len word"
assumes nv: "n < 2 ^ (m - k)"
and kv: "k \<le> m"
and mv: "m < len_of TYPE ('a)"
shows "2 ^ k * n < 2 ^ m"
using nv kv mv
apply -
apply (subst word_less_nat_alt)
apply (subst unat_word_ariths)
apply (subst mod_less)
apply (simp add: unat_power_lower)
apply (rule nat_less_power_trans)
apply (erule order_less_trans [OF unat_mono])
apply (simp add: unat_power_lower)
apply simp
apply (simp add: unat_power_lower)
apply (rule nat_less_power_trans)
apply (subst unat_power_lower[where 'a = 'a, symmetric])
apply simp
apply (erule unat_mono)
apply simp
done
lemma word_less_sub_le[simp]:
fixes x :: "'a :: len word"
assumes nv: "n < len_of TYPE('a)"
shows "(x \<le> 2 ^ n - 1) = (x < 2 ^ n)"
proof -
have "Suc (unat ((2::'a word) ^ n - 1)) = unat ((2::'a word) ^ n)" using nv
by (metis Suc_pred' power_2_ge_iff unat_gt_0 unat_minus_one word_not_simps(1))
thus ?thesis using nv
apply -
apply (subst word_le_nat_alt)
apply (subst less_Suc_eq_le [symmetric])
apply (erule ssubst)
apply (subst word_less_nat_alt)
apply (rule refl)
done
qed
lemmas word32_less_sub_le[simp] =
word_less_sub_le[where 'a = 32, folded word_bits_def]
lemma Suc_unat_diff_1:
fixes x :: "'a :: len word"
assumes lt: "1 \<le> x"
shows "Suc (unat (x - 1)) = unat x"
proof -
have "0 < unat x"
by (rule order_less_le_trans [where y = 1], simp, subst unat_1 [symmetric], rule iffD1 [OF word_le_nat_alt lt])
thus ?thesis
by ((subst unat_sub [OF lt])+, simp only: unat_1)
qed
lemma word_div_sub:
fixes x :: "'a :: len word"
assumes yx: "y \<le> x"
and y0: "0 < y"
shows "(x - y) div y = x div y - 1"
apply (rule word_unat.Rep_eqD)
apply (subst unat_div)
apply (subst unat_sub [OF yx])
apply (subst unat_sub)
apply (subst word_le_nat_alt)
apply (subst unat_div)
apply (subst le_div_geq)
apply (rule order_le_less_trans [OF _ unat_mono [OF y0]])
apply simp
apply (subst word_le_nat_alt [symmetric], rule yx)
apply simp
apply (subst unat_div)
apply (subst le_div_geq [OF _ iffD1 [OF word_le_nat_alt yx]])
apply (rule order_le_less_trans [OF _ unat_mono [OF y0]])
apply simp
apply simp
done
lemma word_mult_less_mono1:
fixes i :: "'a :: len word"
assumes ij: "i < j"
and knz: "0 < k"
and ujk: "unat j * unat k < 2 ^ len_of TYPE ('a)"
shows "i * k < j * k"
proof -
from ij ujk knz have jk: "unat i * unat k < 2 ^ len_of TYPE ('a)"
by (auto intro: order_less_subst2 simp: word_less_nat_alt elim: mult_less_mono1)
thus ?thesis using ujk knz ij
by (auto simp: word_less_nat_alt iffD1 [OF unat_mult_lem])
qed
lemma word_mult_less_dest:
fixes i :: "'a :: len word"
assumes ij: "i * k < j * k"
and uik: "unat i * unat k < 2 ^ len_of TYPE ('a)"
and ujk: "unat j * unat k < 2 ^ len_of TYPE ('a)"
shows "i < j"
using uik ujk ij
by (auto simp: word_less_nat_alt iffD1 [OF unat_mult_lem] elim: mult_less_mono1)
lemma word_mult_less_cancel:
fixes k :: "'a :: len word"
assumes knz: "0 < k"
and uik: "unat i * unat k < 2 ^ len_of TYPE ('a)"
and ujk: "unat j * unat k < 2 ^ len_of TYPE ('a)"
shows "(i * k < j * k) = (i < j)"
by (rule iffI [OF word_mult_less_dest [OF _ uik ujk] word_mult_less_mono1 [OF _ knz ujk]])
lemma Suc_div_unat_helper:
assumes szv: "sz < len_of TYPE('a :: len)"
and usszv: "us \<le> sz"
shows "2 ^ (sz - us) = Suc (unat (((2::'a :: len word) ^ sz - 1) div 2 ^ us))"
proof -
note usv = order_le_less_trans [OF usszv szv]
from usszv obtain q where qv: "sz = us + q" by (auto simp: le_iff_add)
have "Suc (unat (((2:: 'a word) ^ sz - 1) div 2 ^ us)) =
(2 ^ us + unat ((2:: 'a word) ^ sz - 1)) div 2 ^ us"
apply (subst unat_div unat_power_lower[OF usv])+
apply (subst div_add_self1, simp+)
done
also have "\<dots> = ((2 ^ us - 1) + 2 ^ sz) div 2 ^ us" using szv
apply (subst unat_minus_one)
apply (simp add: p2_eq_0)
apply (simp add: unat_power_lower)
done
also have "\<dots> = 2 ^ q + ((2 ^ us - 1) div 2 ^ us)"
apply (subst qv)
apply (subst power_add)
apply (subst div_mult_self2)
apply simp
apply (rule refl)
done
also have "\<dots> = 2 ^ (sz - us)" using qv by simp
finally show ?thesis ..
qed
lemma upto_enum_red':
assumes lt: "1 \<le> X"
shows "[(0::'a :: len word) .e. X - 1] = map of_nat [0 ..< unat X]"
proof -
have lt': "unat X < 2 ^ len_of TYPE('a)"
by (rule unat_lt2p)
show ?thesis
apply (subst upto_enum_red)
apply (simp del: upt.simps)
apply (subst Suc_unat_diff_1 [OF lt])
apply (rule map_cong [OF refl])
apply (rule toEnum_of_nat)
apply simp
apply (erule order_less_trans [OF _ lt'])
done
qed
lemma upto_enum_red2:
assumes szv: "sz < len_of TYPE('a :: len)"
shows "[(0:: 'a :: len word) .e. 2 ^ sz - 1] =
map of_nat [0 ..< 2 ^ sz]" using szv
apply (subst unat_power_lower[OF szv, symmetric])
apply (rule upto_enum_red')
apply (subst word_le_nat_alt, simp add: unat_power_lower)
done
(* FIXME: WordEnum.upto_enum_step_def is fixed to word32. *)
lemma upto_enum_step_red:
assumes szv: "sz < word_bits"
and usszv: "us \<le> sz"
shows "[0 , 2 ^ us .e. 2 ^ sz - 1] =
map (\<lambda>x. of_nat x * 2 ^ us) [0 ..< 2 ^ (sz - us)]" using szv
unfolding upto_enum_step_def
apply (subst if_not_P)
apply (rule leD)
apply (subst word_le_nat_alt)
apply (subst unat_minus_one)
apply (simp add: p2_eq_0 word_bits_def)
apply simp
apply simp
apply (subst upto_enum_red)
apply (simp del: upt.simps)
apply (subst Suc_div_unat_helper [where 'a = 32, folded word_bits_def, OF szv usszv, symmetric])
apply clarsimp
apply (subst toEnum_of_nat)
apply (subst word_bits_len_of)
apply (erule order_less_trans)
using szv
apply simp
apply simp
done
lemma upto_enum_word:
"[x .e. y] = map of_nat [unat x ..< Suc (unat y)]"
apply (subst upto_enum_red)
apply clarsimp
apply (subst toEnum_of_nat)
prefer 2
apply (rule refl)
apply (erule disjE, simp)
apply clarsimp
apply (erule order_less_trans)
apply simp
done
text {* Lemmas about upto and upto_enum *}
lemma word_upto_Cons_eq:
"\<lbrakk>x = z; x < y; Suc (unat y) < 2 ^ len_of TYPE('a)\<rbrakk>
\<Longrightarrow> [x::'a::len word .e. y] = z # [x + 1 .e. y]"
apply (subst upto_enum_red)
apply (subst upt_conv_Cons)
apply (simp)
apply (drule unat_mono)
apply arith
apply (simp only: list.map)
apply (subst list.inject)
apply rule
apply (rule to_from_enum)
apply (subst upto_enum_red)
apply (rule map_cong [OF _ refl])
apply (rule arg_cong2 [where f = "\<lambda>x y. [x ..< y]"])
apply unat_arith
apply simp
done
lemma distinct_enum_upto:
"distinct [(0 :: 'a::len word) .e. b]"
proof -
have "\<And>(b::'a word). [0 .e. b] = sublist enum {..< Suc (fromEnum b)}"
apply (subst upto_enum_red)
apply (subst sublist_upt_eq_take)
apply (subst enum_word_def)
apply (subst take_map)
apply (subst take_upt)
apply (simp only: add_0 fromEnum_unat)
apply (rule order_trans [OF _ order_eq_refl])
apply (rule Suc_leI [OF unat_lt2p])
apply simp
apply clarsimp
apply (rule toEnum_of_nat)
apply (erule order_less_trans [OF _ unat_lt2p])
done
thus ?thesis
by (rule ssubst) (rule distinct_sublistI, simp)
qed
lemma upto_enum_set_conv [simp]:
fixes a :: "'a :: len word"
shows "set [a .e. b] = {x. a \<le> x \<and> x \<le> b}"
apply (subst upto_enum_red)
apply (subst set_map)
apply safe
apply simp
apply clarsimp
apply (erule disjE)
apply simp
apply (erule iffD2 [OF word_le_nat_alt])
apply clarsimp
apply (erule word_unat.Rep_cases [OF unat_le [OF order_less_imp_le]])
apply simp
apply (erule iffD2 [OF word_le_nat_alt])
apply simp
apply clarsimp
apply (erule disjE)
apply simp
apply clarsimp
apply (rule word_unat.Rep_cases [OF unat_le [OF order_less_imp_le]])
apply assumption
apply simp
apply (erule order_less_imp_le [OF iffD2 [OF word_less_nat_alt]])
apply clarsimp
apply (rule_tac x="fromEnum x" in image_eqI)
apply clarsimp
apply clarsimp
apply (rule conjI)
apply (subst word_le_nat_alt [symmetric])
apply simp
apply safe
apply (simp add: word_le_nat_alt [symmetric])
apply (simp add: word_less_nat_alt [symmetric])
done
lemma upto_enum_less:
assumes xin: "x \<in> set [(a::'a::len word).e.2 ^ n - 1]"
and nv: "n < len_of TYPE('a::len)"
shows "x < 2 ^ n"
proof (cases n)
case 0
thus ?thesis using xin
by (simp add: upto_enum_set_conv)
next
case (Suc m)
show ?thesis using xin nv by simp
qed
lemma upto_enum_len_less:
"\<lbrakk> n \<le> length [a, b .e. c]; n \<noteq> 0 \<rbrakk> \<Longrightarrow> a \<le> c"
unfolding upto_enum_step_def
by (simp split: split_if_asm)
lemma length_upto_enum_step:
fixes x :: word32
shows "x \<le> z \<Longrightarrow> length [x , y .e. z] = (unat ((z - x) div (y - x))) + 1"
unfolding upto_enum_step_def
by (simp add: upto_enum_red)
lemma length_upto_enum_one:
fixes x :: word32
assumes lt1: "x < y" and lt2: "z < y" and lt3: "x \<le> z"
shows "[x , y .e. z] = [x]"
unfolding upto_enum_step_def
proof (subst upto_enum_red, subst if_not_P [OF leD [OF lt3]], clarsimp, rule)
show "unat ((z - x) div (y - x)) = 0"
proof (subst unat_div, rule div_less)
have syx: "unat (y - x) = unat y - unat x"
by (rule unat_sub [OF order_less_imp_le]) fact
moreover have "unat (z - x) = unat z - unat x"
by (rule unat_sub) fact
ultimately show "unat (z - x) < unat (y - x)"
using lt3
apply simp
apply (rule diff_less_mono[OF unat_mono, OF lt2])
apply (simp add: word_le_nat_alt[symmetric])
done
qed
thus "toEnum (unat ((z - x) div (y - x))) * (y - x) = 0" by simp
qed
lemma map_length_unfold_one:
fixes x :: "'a::len word"
assumes xv: "Suc (unat x) < 2 ^ len_of TYPE('a)"
and ax: "a < x"
shows "map f [a .e. x] = f a # map f [a + 1 .e. x]"
by (subst word_upto_Cons_eq, auto, fact+)
lemma upto_enum_triv [simp]:
"[x .e. x] = [x]"
unfolding upto_enum_def by simp
lemma of_nat_unat [simp]:
"of_nat \<circ> unat = id"
by (rule ext, simp)
lemma Suc_unat_minus_one [simp]:
"x \<noteq> 0 \<Longrightarrow> Suc (unat (x - 1)) = unat x"
by (metis Suc_diff_1 unat_gt_0 unat_minus_one)
text {* Lemmas about alignment *}
lemma word_bits_size:
"size (w::word32) = word_bits"
by (simp add: word_bits_def word_size)
text {* Lemmas about defs in the specs *}
lemma and_commute:
"(X and Y) = (Y and X)"
unfolding pred_conj_def by (auto simp: fun_eq_iff)
lemma ptr_add_0 [simp]:
"ptr_add ref 0 = ref "
unfolding ptr_add_def by simp
(* Other word lemmas *)
lemma word_add_le_dest:
fixes i :: "'a :: len word"
assumes le: "i + k \<le> j + k"
and uik: "unat i + unat k < 2 ^ len_of TYPE ('a)"
and ujk: "unat j + unat k < 2 ^ len_of TYPE ('a)"
shows "i \<le> j"
using uik ujk le
by (auto simp: word_le_nat_alt iffD1 [OF unat_add_lem] elim: add_le_mono1)
lemma mask_shift:
"(x && ~~ mask y) >> y = x >> y"
apply (rule word_eqI)
apply (simp add: nth_shiftr word_size)
apply safe
apply (drule test_bit.Rep[simplified, rule_format])
apply (simp add: word_size word_ops_nth_size)
done
lemma word_add_le_mono1:
fixes i :: "'a :: len word"
assumes ij: "i \<le> j"
and ujk: "unat j + unat k < 2 ^ len_of TYPE ('a)"
shows "i + k \<le> j + k"
proof -
from ij ujk have jk: "unat i + unat k < 2 ^ len_of TYPE ('a)"
by (auto elim: order_le_less_subst2 simp: word_le_nat_alt elim: add_le_mono1)
thus ?thesis using ujk ij
by (auto simp: word_le_nat_alt iffD1 [OF unat_add_lem])
qed
lemma word_add_le_mono2:
fixes i :: "('a :: len) word"
shows "\<lbrakk>i \<le> j; unat j + unat k < 2 ^ len_of TYPE('a)\<rbrakk> \<Longrightarrow> k + i \<le> k + j"
by (subst field_simps, subst field_simps, erule (1) word_add_le_mono1)
lemma word_add_le_iff:
fixes i :: "'a :: len word"
assumes uik: "unat i + unat k < 2 ^ len_of TYPE ('a)"
and ujk: "unat j + unat k < 2 ^ len_of TYPE ('a)"
shows "(i + k \<le> j + k) = (i \<le> j)"
proof
assume "i \<le> j"
show "i + k \<le> j + k" by (rule word_add_le_mono1) fact+
next
assume "i + k \<le> j + k"
show "i \<le> j" by (rule word_add_le_dest) fact+
qed
lemma word_add_less_mono1:
fixes i :: "'a :: len word"
assumes ij: "i < j"
and ujk: "unat j + unat k < 2 ^ len_of TYPE ('a)"
shows "i + k < j + k"
proof -
from ij ujk have jk: "unat i + unat k < 2 ^ len_of TYPE ('a)"
by (auto elim: order_le_less_subst2 simp: word_less_nat_alt elim: add_less_mono1)
thus ?thesis using ujk ij
by (auto simp: word_less_nat_alt iffD1 [OF unat_add_lem])
qed
lemma word_add_less_dest:
fixes i :: "'a :: len word"
assumes le: "i + k < j + k"
and uik: "unat i + unat k < 2 ^ len_of TYPE ('a)"
and ujk: "unat j + unat k < 2 ^ len_of TYPE ('a)"
shows "i < j"
using uik ujk le
by (auto simp: word_less_nat_alt iffD1 [OF unat_add_lem] elim: add_less_mono1)
lemma word_add_less_iff:
fixes i :: "'a :: len word"
assumes uik: "unat i + unat k < 2 ^ len_of TYPE ('a)"
and ujk: "unat j + unat k < 2 ^ len_of TYPE ('a)"
shows "(i + k < j + k) = (i < j)"
proof
assume "i < j"
show "i + k < j + k" by (rule word_add_less_mono1) fact+
next
assume "i + k < j + k"
show "i < j" by (rule word_add_less_dest) fact+
qed
lemma shiftr_div_2n':
"unat (w >> n) = unat w div 2 ^ n"
apply (unfold unat_def)
apply (subst shiftr_div_2n)
apply (subst nat_div_distrib)
apply simp
apply (simp add: nat_power_eq)
done
lemma shiftl_shiftr_id:
assumes nv: "n < len_of TYPE('a :: len)"
and xv: "x < 2 ^ (len_of TYPE('a :: len) - n)"
shows "x << n >> n = (x::'a::len word)"
apply (simp add: shiftl_t2n)
apply (rule word_unat.Rep_eqD)
apply (subst shiftr_div_2n')
apply (cases n)
apply simp
apply (subst iffD1 [OF unat_mult_lem])+
apply (subst unat_power_lower[OF nv])
apply (rule nat_less_power_trans [OF _ order_less_imp_le [OF nv]])
apply (rule order_less_le_trans [OF unat_mono [OF xv] order_eq_refl])
apply (rule unat_power_lower)
apply simp
apply (subst unat_power_lower[OF nv])
apply simp
done
lemma word_mult_less_iff:
fixes i :: "'a :: len word"
assumes knz: "0 < k"
and uik: "unat i * unat k < 2 ^ len_of TYPE ('a)"
and ujk: "unat j * unat k < 2 ^ len_of TYPE ('a)"
shows "(i * k < j * k) = (i < j)"
proof
assume "i < j"
show "i * k < j * k" by (rule word_mult_less_mono1) fact+
next
assume p: "i * k < j * k"
have "0 < unat k" using knz by (simp add: word_less_nat_alt)
thus "i < j" using p
by (clarsimp simp: word_less_nat_alt iffD1 [OF unat_mult_lem uik]
iffD1 [OF unat_mult_lem ujk])
qed
lemma word_le_imp_diff_le:
fixes n :: "'a::len word"
shows "\<lbrakk>k \<le> n; n \<le> m\<rbrakk> \<Longrightarrow> n - k \<le> m"
by (clarsimp simp: unat_sub word_le_nat_alt intro!: le_imp_diff_le)
lemma word_less_imp_diff_less:
fixes n :: "'a::len word"
shows "\<lbrakk>k \<le> n; n < m\<rbrakk> \<Longrightarrow> n - k < m"
by (clarsimp simp: unat_sub word_less_nat_alt
intro!: less_imp_diff_less)
lemma word_mult_le_mono1:
fixes i :: "'a :: len word"
assumes ij: "i \<le> j"
and knz: "0 < k"
and ujk: "unat j * unat k < 2 ^ len_of TYPE ('a)"
shows "i * k \<le> j * k"
proof -
from ij ujk knz have jk: "unat i * unat k < 2 ^ len_of TYPE ('a)"
by (auto elim: order_le_less_subst2 simp: word_le_nat_alt elim: mult_le_mono1)
thus ?thesis using ujk knz ij
by (auto simp: word_le_nat_alt iffD1 [OF unat_mult_lem])
qed
lemma word_mult_le_iff:
fixes i :: "'a :: len word"
assumes knz: "0 < k"
and uik: "unat i * unat k < 2 ^ len_of TYPE ('a)"
and ujk: "unat j * unat k < 2 ^ len_of TYPE ('a)"
shows "(i * k \<le> j * k) = (i \<le> j)"
proof
assume "i \<le> j"
show "i * k \<le> j * k" by (rule word_mult_le_mono1) fact+
next
assume p: "i * k \<le> j * k"
have "0 < unat k" using knz by (simp add: word_less_nat_alt)
thus "i \<le> j" using p
by (clarsimp simp: word_le_nat_alt iffD1 [OF unat_mult_lem uik]
iffD1 [OF unat_mult_lem ujk])
qed
lemma word_diff_less:
fixes n :: "'a :: len word"
shows "\<lbrakk>0 < n; 0 < m; n \<le> m\<rbrakk> \<Longrightarrow> m - n < m"
apply (subst word_less_nat_alt)
apply (subst unat_sub)
apply assumption
apply (rule diff_less)
apply (simp_all add: word_less_nat_alt)
done
lemma MinI:
assumes fa: "finite A"
and ne: "A \<noteq> {}"
and xv: "m \<in> A"
and min: "\<forall>y \<in> A. m \<le> y"
shows "Min A = m" using fa ne xv min
proof (induct A arbitrary: m rule: finite_ne_induct)
case singleton thus ?case by simp
next
case (insert y F)
from insert.prems have yx: "m \<le> y" and fx: "\<forall>y \<in> F. m \<le> y" by auto
have "m \<in> insert y F" by fact
thus ?case
proof
assume mv: "m = y"
have mlt: "m \<le> Min F"
by (rule iffD2 [OF Min_ge_iff [OF insert.hyps(1) insert.hyps(2)] fx])
show ?case
apply (subst Min_insert [OF insert.hyps(1) insert.hyps(2)])
apply (subst mv [symmetric])
apply (rule iffD2 [OF linorder_min_same1 mlt])
done
next
assume "m \<in> F"
hence mf: "Min F = m"
by (rule insert.hyps(4) [OF _ fx])
show ?case
apply (subst Min_insert [OF insert.hyps(1) insert.hyps(2)])
apply (subst mf)
apply (rule iffD2 [OF linorder_min_same2 yx])
done
qed
qed
lemma length_upto_enum [simp]:
fixes a :: "('a :: len) word"
shows "length [a .e. b] = Suc (unat b) - unat a"
apply (simp add: word_le_nat_alt upto_enum_red)
apply (clarsimp simp: Suc_diff_le)
done
lemma length_upto_enum_less_one:
"\<lbrakk>a \<le> b; b \<noteq> 0\<rbrakk>
\<Longrightarrow> length [a .e. b - 1] = unat (b - a)"
apply clarsimp
apply (subst unat_sub[symmetric], assumption)
apply clarsimp
done
lemma drop_upto_enum:
"drop (unat n) [0 .e. m] = [n .e. m]"
apply (clarsimp simp: upto_enum_def)
apply (induct m, simp)
by (metis drop_map drop_upt plus_nat.add_0)
lemma distinct_enum_upto' [simp]:
"distinct [a::'a::len word .e. b]"
apply (subst drop_upto_enum [symmetric])
apply (rule distinct_drop)
apply (rule distinct_enum_upto)
done
lemma length_interval:
"\<lbrakk>set xs = {x. (a::'a::len word) \<le> x \<and> x \<le> b}; distinct xs\<rbrakk>
\<Longrightarrow> length xs = Suc (unat b) - unat a"
apply (frule distinct_card)
apply (subgoal_tac "set xs = set [a .e. b]")
apply (cut_tac distinct_card [where xs="[a .e. b]"])
apply (subst (asm) length_upto_enum)
apply clarsimp
apply (rule distinct_enum_upto')
apply simp
done
lemma not_empty_eq:
"(S \<noteq> {}) = (\<exists>x. x \<in> S)"
by auto
lemma range_subset_lower:
fixes c :: "'a ::linorder"
shows "\<lbrakk> {a..b} \<subseteq> {c..d}; x \<in> {a..b} \<rbrakk> \<Longrightarrow> c \<le> a"
apply (frule (1) subsetD)
apply (rule classical)
apply clarsimp
done
lemma range_subset_upper:
fixes c :: "'a ::linorder"
shows "\<lbrakk> {a..b} \<subseteq> {c..d}; x \<in> {a..b} \<rbrakk> \<Longrightarrow> b \<le> d"
apply (frule (1) subsetD)
apply (rule classical)
apply clarsimp
done
lemma range_subset_eq:
fixes a::"'a::linorder"
assumes non_empty: "a \<le> b"
shows "({a..b} \<subseteq> {c..d}) = (c \<le> a \<and> b \<le> d)"
apply (insert non_empty)
apply (rule iffI)
apply (frule range_subset_lower [where x=a], simp)
apply (drule range_subset_upper [where x=a], simp)
apply simp
apply auto
done
lemma range_eq:
fixes a::"'a::linorder"
assumes non_empty: "a \<le> b"
shows "({a..b} = {c..d}) = (a = c \<and> b = d)"
by (metis atLeastatMost_subset_iff eq_iff non_empty)
lemma range_strict_subset_eq:
fixes a::"'a::linorder"
assumes non_empty: "a \<le> b"
shows "({a..b} \<subset> {c..d}) = (c \<le> a \<and> b \<le> d \<and> (a = c \<longrightarrow> b \<noteq> d))"
apply (insert non_empty)
apply (subst psubset_eq)
apply (subst range_subset_eq, assumption+)
apply (subst range_eq, assumption+)
apply simp
done
lemma range_subsetI:
fixes x :: "'a :: order"
assumes xX: "X \<le> x"
and yY: "y \<le> Y"
shows "{x .. y} \<subseteq> {X .. Y}"
using xX yY by auto
lemma set_False [simp]:
"(set bs \<subseteq> {False}) = (True \<notin> set bs)" by auto
declare of_nat_power [simp del]
(* TODO: move to word *)
lemma unat_of_bl_length:
"unat (of_bl xs :: 'a::len word) < 2 ^ (length xs)"
proof (cases "length xs < len_of TYPE('a)")
case True
hence "(of_bl xs::'a::len word) < 2 ^ length xs"
by (simp add: of_bl_length_less)
with True
show ?thesis
by (simp add: word_less_nat_alt word_unat_power unat_of_nat)
next
case False
have "unat (of_bl xs::'a::len word) < 2 ^ len_of TYPE('a)"
by (simp split: unat_split)
also
from False
have "len_of TYPE('a) \<le> length xs" by simp
hence "2 ^ len_of TYPE('a) \<le> (2::nat) ^ length xs"
by (rule power_increasing) simp
finally
show ?thesis .
qed
lemma is_aligned_0'[simp]:
"is_aligned 0 n"
by (simp add: is_aligned_def)
lemma p_assoc_help:
fixes p :: "'a::{ring,power,numeral,one}"
shows "p + 2^sz - 1 = p + (2^sz - 1)"
by simp
lemma word_add_increasing:
fixes x :: "'a :: len word"
shows "\<lbrakk> p + w \<le> x; p \<le> p + w \<rbrakk> \<Longrightarrow> p \<le> x"
by unat_arith
lemma word_random:
fixes x :: "'a :: len word"
shows "\<lbrakk> p \<le> p + x'; x \<le> x' \<rbrakk> \<Longrightarrow> p \<le> p + x"
by unat_arith
lemma word_sub_mono:
"\<lbrakk> a \<le> c; d \<le> b; a - b \<le> a; c - d \<le> c \<rbrakk>
\<Longrightarrow> (a - b) \<le> (c - d :: ('a :: len) word)"
by unat_arith
lemma power_not_zero:
"n < len_of TYPE('a::len) \<Longrightarrow> (2 :: 'a word) ^ n \<noteq> 0"
by (metis p2_gt_0 word_neq_0_conv)
lemma word_gt_a_gt_0:
"a < n \<Longrightarrow> (0 :: 'a::len word) < n"
apply (case_tac "n = 0")
apply clarsimp
apply (clarsimp simp: word_neq_0_conv)
done
lemma word_shift_nonzero:
"\<lbrakk> (x\<Colon>'a\<Colon>len word) \<le> 2 ^ m; m + n < len_of TYPE('a\<Colon>len); x \<noteq> 0\<rbrakk>
\<Longrightarrow> x << n \<noteq> 0"
apply (simp only: word_neq_0_conv word_less_nat_alt
shiftl_t2n mod_0 unat_word_ariths
unat_power_lower word_le_nat_alt)
apply (subst mod_less)
apply (rule order_le_less_trans)
apply (erule mult_le_mono2)
apply (subst power_add[symmetric])
apply (rule power_strict_increasing)
apply simp
apply simp
apply simp
done
lemma word_power_less_1 [simp]:
"sz < len_of TYPE('a\<Colon>len) \<Longrightarrow> (2::'a word) ^ sz - 1 < 2 ^ sz"
apply (simp add: word_less_nat_alt word_bits_def)
apply (subst unat_minus_one)
apply (simp add: word_unat.Rep_inject [symmetric])
apply simp
done
lemmas word32_power_less_1[simp] =
word_power_less_1[where 'a = 32, folded word_bits_def]
lemma nasty_split_lt:
"\<lbrakk> (x :: 'a:: len word) < 2 ^ (m - n); n \<le> m; m < len_of TYPE('a\<Colon>len) \<rbrakk>
\<Longrightarrow> x * 2 ^ n + (2 ^ n - 1) \<le> 2 ^ m - 1"
apply (simp only: add_diff_eq word_bits_def)
apply (subst mult_1[symmetric], subst distrib_right[symmetric])
apply (rule word_sub_mono)
apply (rule order_trans)
apply (rule word_mult_le_mono1)
apply (rule inc_le)
apply assumption
apply (subst word_neq_0_conv[symmetric])
apply (rule power_not_zero)
apply (simp add: word_bits_def)
apply (subst unat_power_lower, simp)+
apply (subst power_add[symmetric])
apply (rule power_strict_increasing)
apply (simp add: word_bits_def)
apply simp
apply (subst power_add[symmetric])
apply simp
apply simp
apply (rule word_sub_1_le)
apply (subst mult.commute)
apply (subst shiftl_t2n[symmetric])
apply (rule word_shift_nonzero)
apply (erule inc_le)
apply (simp add: word_bits_def)
apply (unat_arith)
apply (drule word_power_less_1[unfolded word_bits_def])
apply simp
done
lemma nasty_split_less:
"\<lbrakk>m \<le> n; n \<le> nm; nm < len_of TYPE('a\<Colon>len); x < 2 ^ (nm - n)\<rbrakk>
\<Longrightarrow> (x :: 'a word) * 2 ^ n + (2 ^ m - 1) < 2 ^ nm"
apply (simp only: word_less_sub_le[symmetric])
apply (rule order_trans [OF _ nasty_split_lt])
apply (rule word_plus_mono_right)
apply (rule word_sub_mono)
apply (simp add: word_le_nat_alt)
apply simp
apply (simp add: word_sub_1_le[OF power_not_zero])
apply (simp add: word_sub_1_le[OF power_not_zero])
apply (rule is_aligned_no_wrap')
apply (rule is_aligned_mult_triv2)
apply simp
apply (erule order_le_less_trans, simp)
apply simp+
done
lemma int_not_emptyD:
"A \<inter> B \<noteq> {} \<Longrightarrow> \<exists>x. x \<in> A \<and> x \<in> B"
by (erule contrapos_np, clarsimp simp: disjoint_iff_not_equal)
lemma unat_less_power:
fixes k :: "'a::len word"
assumes szv: "sz < len_of TYPE('a)"
and kv: "k < 2 ^ sz"
shows "unat k < 2 ^ sz"
using szv unat_mono [OF kv] by simp
(* This should replace some crud \<dots> search for unat_of_nat *)
lemma unat_mult_power_lem:
assumes kv: "k < 2 ^ (len_of TYPE('a::len) - sz)"
shows "unat (2 ^ sz * of_nat k :: (('a::len) word)) = 2 ^ sz * k"
proof cases
assume szv: "sz < len_of TYPE('a::len)"
show ?thesis
proof (cases "sz = 0")
case True
thus ?thesis using kv szv
by (simp add: unat_of_nat)
next
case False
hence sne: "0 < sz" ..
have uk: "unat (of_nat k :: 'a word) = k"
apply (subst unat_of_nat)
apply (simp add: nat_mod_eq less_trans[OF kv] sne)
done
show ?thesis using szv
apply (subst iffD1 [OF unat_mult_lem])
apply (simp add: uk nat_less_power_trans[OF kv order_less_imp_le [OF szv]])+
done
qed
next
assume "\<not> sz < len_of TYPE('a)"
with kv show ?thesis by (simp add: not_less power_overflow)
qed
lemma aligned_add_offset_no_wrap:
fixes off :: "('a::len) word"
and x :: "'a word"
assumes al: "is_aligned x sz"
and offv: "off < 2 ^ sz"
shows "unat x + unat off < 2 ^ len_of TYPE('a)"
proof cases
assume szv: "sz < len_of TYPE('a)"
from al obtain k where xv: "x = 2 ^ sz * (of_nat k)"
and kl: "k < 2 ^ (len_of TYPE('a) - sz)"
by (auto elim: is_alignedE)
show ?thesis using szv
apply (subst xv)
apply (subst unat_mult_power_lem[OF kl])
apply (subst mult.commute, rule nat_add_offset_less)
apply (rule less_le_trans[OF unat_mono[OF offv, simplified]])
apply (erule eq_imp_le[OF unat_power_lower])
apply (rule kl)
apply simp
done
next
assume "\<not> sz < len_of TYPE('a)"
with offv show ?thesis by (simp add: not_less power_overflow )
qed
lemma aligned_add_offset_mod:
fixes x :: "('a::len) word"
assumes al: "is_aligned x sz"
and kv: "k < 2 ^ sz"
shows "(x + k) mod 2 ^ sz = k"
proof cases
assume szv: "sz < len_of TYPE('a)"
have ux: "unat x + unat k < 2 ^ len_of TYPE('a)"
by (rule aligned_add_offset_no_wrap) fact+
show ?thesis using al szv
apply -
apply (erule is_alignedE)
apply (subst word_unat.Rep_inject [symmetric])
apply (subst unat_mod)
apply (subst iffD1 [OF unat_add_lem], rule ux)
apply simp
apply (subst unat_mult_power_lem, assumption+)
apply (subst mod_add_left_eq)
apply (simp)
apply (rule mod_less[OF less_le_trans[OF unat_mono], OF kv])
apply (erule eq_imp_le[OF unat_power_lower])
done
next
assume "\<not> sz < len_of TYPE('a)"
with al show ?thesis
by (simp add: not_less power_overflow is_aligned_mask mask_def
word_mod_by_0)
qed
lemma word_plus_mcs_4:
"\<lbrakk>v + x \<le> w + x; x \<le> v + x\<rbrakk> \<Longrightarrow> v \<le> (w::'a::len word)"
by uint_arith
lemma word_plus_mcs_3:
"\<lbrakk>v \<le> w; x \<le> w + x\<rbrakk> \<Longrightarrow> v + x \<le> w + (x::'a::len word)"
by unat_arith
have rl: "\<And>(p::'a word) k w. \<lbrakk>uint p + uint k < 2 ^ len_of TYPE('a); w = p + k; w \<le> p + (2 ^ sz - 1) \<rbrakk>
\<Longrightarrow> k < 2 ^ sz"
apply -
apply simp
apply (subst (asm) add.commute, subst (asm) add.commute, drule word_plus_mcs_4)
apply (subst add.commute, subst no_plus_overflow_uint_size)
apply (simp add: word_size_bl)
apply (erule iffD1 [OF word_less_sub_le[OF szv]])
done
from xb obtain kx where
kx: "z = x + kx" and
kxl: "uint x + uint kx < 2 ^ len_of TYPE('a)"
by (clarsimp dest!: word_le_exists')
from yb obtain ky where
ky: "z = y + ky" and
kyl: "uint y + uint ky < 2 ^ len_of TYPE('a)"
by (clarsimp dest!: word_le_exists')
have "x = y"
proof -
have "kx = z mod 2 ^ sz"
proof (subst kx, rule sym, rule aligned_add_offset_mod)
show "kx < 2 ^ sz" by (rule rl) fact+
qed fact+
also have "\<dots> = ky"
proof (subst ky, rule aligned_add_offset_mod)
show "ky < 2 ^ sz"
using kyl ky yt by (rule rl)
qed fact+
finally have kxky: "kx = ky" .
moreover have "x + kx = y + ky" by (simp add: kx [symmetric] ky [symmetric])
ultimately show ?thesis by simp
qed
thus False using neq by simp
qed
next
assume "\<not> sz < len_of TYPE('a)"
with neq alx aly
have False by (simp add: is_aligned_mask mask_def power_overflow)
thus ?thesis ..
qed
lemma less_two_pow_divD:
"\<lbrakk> (x :: nat) < 2 ^ n div 2 ^ m \<rbrakk>
\<Longrightarrow> n \<ge> m \<and> (x < 2 ^ (n - m))"
apply (rule context_conjI)
apply (rule ccontr)
apply (simp add: div_less power_strict_increasing)
apply (simp add: power_sub)
done
lemma less_two_pow_divI:
"\<lbrakk> (x :: nat) < 2 ^ (n - m); m \<le> n \<rbrakk> \<Longrightarrow> x < 2 ^ n div 2 ^ m"
by (simp add: power_sub)
lemma word_less_two_pow_divI:
"\<lbrakk> (x :: 'a::len word) < 2 ^ (n - m); m \<le> n; n < len_of TYPE('a) \<rbrakk> \<Longrightarrow> x < 2 ^ n div 2 ^ m"
apply (simp add: word_less_nat_alt)
apply (subst unat_word_ariths)
apply (subst mod_less)
apply (rule order_le_less_trans [OF div_le_dividend])
apply (rule unat_lt2p)
apply (simp add: power_sub)
done
lemma word_less_two_pow_divD:
"\<lbrakk> (x :: 'a::len word) < 2 ^ n div 2 ^ m \<rbrakk>
\<Longrightarrow> n \<ge> m \<and> (x < 2 ^ (n - m))"
apply (cases "n < len_of TYPE('a)")
apply (cases "m < len_of TYPE('a)")
apply (simp add: word_less_nat_alt)
apply (subst(asm) unat_word_ariths)
apply (subst(asm) mod_less)
apply (rule order_le_less_trans [OF div_le_dividend])
apply (rule unat_lt2p)
apply (clarsimp dest!: less_two_pow_divD)
apply (simp add: power_overflow)
apply (simp add: word_div_def)
apply (simp add: power_overflow word_div_def)
done
lemma of_nat_less_two_pow_div_set:
"\<lbrakk> n < len_of TYPE('a) \<rbrakk> \<Longrightarrow>
{x. x < (2 ^ n div 2 ^ m :: 'a::len word)}
= of_nat ` {k. k < 2 ^ n div 2 ^ m}"
apply (simp add: image_def)
apply (safe dest!: word_less_two_pow_divD less_two_pow_divD
intro!: word_less_two_pow_divI)
apply (rule_tac x="unat x" in exI)
apply (simp add: power_sub[symmetric])
apply (subst unat_power_lower[symmetric, where 'a='a])
apply simp
apply (erule unat_mono)
apply (subst word_unat_power)
apply (rule of_nat_mono_maybe)
apply (rule power_strict_increasing)
apply simp
apply simp
apply assumption
done
(* FIXME: generalise! *)
lemma upto_2_helper:
"{0..<2 :: word32} = {0, 1}"
apply (safe, simp_all)
apply unat_arith
done
(* TODO: MOVE to word *)
lemma word_less_power_trans2:
fixes n :: "'a::len word"
shows "\<lbrakk>n < 2 ^ (m - k); k \<le> m; m < len_of TYPE('a)\<rbrakk> \<Longrightarrow> n * 2 ^ k < 2 ^ m"
by (subst field_simps, rule word_less_power_trans)
lemma ucast_less:
"len_of TYPE('b) < len_of TYPE('a) \<Longrightarrow>
(ucast (x :: ('b :: len) word) :: (('a :: len) word)) < 2 ^ len_of TYPE('b)"
apply (subst mask_eq_iff_w2p[symmetric])
apply (simp add: word_size)
apply (rule word_eqI)
apply (simp add: word_size nth_ucast)
apply safe
apply (simp add: test_bit.Rep[simplified])
done
lemma ucast_less_shiftl_helper:
"\<lbrakk> len_of TYPE('b) + 2 < word_bits;
2 ^ (len_of TYPE('b) + 2) \<le> n\<rbrakk>
\<Longrightarrow> (ucast (x :: ('b :: len) word) << 2) < (n :: word32)"
apply (erule order_less_le_trans[rotated])
apply (cut_tac ucast_less[where x=x and 'a=32])
apply (simp only: shiftl_t2n field_simps)
apply (rule word_less_power_trans2)
apply (simp_all add: word_bits_def)
done
lemma ucast_range_less:
"len_of TYPE('a :: len) < len_of TYPE('b :: len) \<Longrightarrow>
range (ucast :: 'a word \<Rightarrow> 'b word)
= {x. x < 2 ^ len_of TYPE ('a)}"
apply safe
apply (erule ucast_less)
apply (simp add: image_def)
apply (rule_tac x="ucast x" in exI)
apply (drule less_mask_eq)
apply (rule word_eqI)
apply (drule_tac x=n in word_eqD)
apply (simp add: word_size nth_ucast)
done
lemma word_power_less_diff:
"\<lbrakk>2 ^ n * q < (2::'a::len word) ^ m; q < 2 ^ (len_of TYPE('a) - n)\<rbrakk> \<Longrightarrow> q < 2 ^ (m - n)"
apply (case_tac "m \<ge> len_of TYPE('a)")
apply (simp add: power_overflow)
apply (case_tac "n \<ge> len_of TYPE('a)")
apply (simp add: power_overflow)
apply (cases "n = 0")
apply simp
apply (subst word_less_nat_alt)
apply (subst unat_power_lower)
apply simp
apply (rule nat_power_less_diff)
apply (simp add: word_less_nat_alt)
apply (subst (asm) iffD1 [OF unat_mult_lem])
apply (simp add:nat_less_power_trans)
apply simp
done
lemmas word_diff_ls' = word_diff_ls [where xa=x and x=x for x, simplified]
lemmas word_l_diffs = word_l_diffs [where xa=x and x=x for x, simplified]
lemma is_aligned_diff:
fixes m :: "'a::len word"
assumes alm: "is_aligned m s1"
and aln: "is_aligned n s2"
and s2wb: "s2 < len_of TYPE('a)"
and nm: "m \<in> {n .. n + (2 ^ s2 - 1)}"
and s1s2: "s1 \<le> s2"
and s10: "0 < s1" (* Probably can be folded into the proof \<dots> *)
shows "\<exists>q. m - n = of_nat q * 2 ^ s1 \<and> q < 2 ^ (s2 - s1)"
proof -
have rl: "\<And>m s. \<lbrakk> m < 2 ^ (len_of TYPE('a) - s); s < len_of TYPE('a) \<rbrakk> \<Longrightarrow> unat ((2::'a word) ^ s * of_nat m) = 2 ^ s * m"
proof -
fix m :: nat and s
assume m: "m < 2 ^ (len_of TYPE('a) - s)" and s: "s < len_of TYPE('a)"
hence "unat ((of_nat m) :: 'a word) = m"
apply (subst unat_of_nat)
apply (subst mod_less)
apply (erule order_less_le_trans)
apply (rule power_increasing)
apply simp_all
done
thus "?thesis m s" using s m
apply (subst iffD1 [OF unat_mult_lem])
apply (simp add: nat_less_power_trans)+
done
qed
have s1wb: "s1 < len_of TYPE('a)" using s2wb s1s2 by simp
from alm obtain mq where mmq: "m = 2 ^ s1 * of_nat mq" and mq: "mq < 2 ^ (len_of TYPE('a) - s1)"
by (auto elim: is_alignedE simp: field_simps)
from aln obtain nq where nnq: "n = 2 ^ s2 * of_nat nq" and nq: "nq < 2 ^ (len_of TYPE('a) - s2)"
by (auto elim: is_alignedE simp: field_simps)
from s1s2 obtain sq where sq: "s2 = s1 + sq" by (auto simp: le_iff_add)
note us1 = rl [OF mq s1wb]
note us2 = rl [OF nq s2wb]
from nm have "n \<le> m" by clarsimp
hence "(2::'a word) ^ s2 * of_nat nq \<le> 2 ^ s1 * of_nat mq" using nnq mmq by simp
hence "2 ^ s2 * nq \<le> 2 ^ s1 * mq" using s1wb s2wb
by (simp add: word_le_nat_alt us1 us2)
hence nqmq: "2 ^ sq * nq \<le> mq" using sq by (simp add: power_add)
have "m - n = 2 ^ s1 * of_nat mq - 2 ^ s2 * of_nat nq" using mmq nnq by simp
also have "\<dots> = 2 ^ s1 * of_nat mq - 2 ^ s1 * 2 ^ sq * of_nat nq" using sq by (simp add: power_add)
also have "\<dots> = 2 ^ s1 * (of_nat mq - 2 ^ sq * of_nat nq)" by (simp add: field_simps)
also have "\<dots> = 2 ^ s1 * of_nat (mq - 2 ^ sq * nq)" using s1wb s2wb us1 us2 nqmq
by (simp add: word_unat_power)
finally have mn: "m - n = of_nat (mq - 2 ^ sq * nq) * 2 ^ s1" by simp
moreover
from nm have "m - n \<le> 2 ^ s2 - 1"
by - (rule word_diff_ls', (simp add: field_simps)+)
hence "(2::'a word) ^ s1 * of_nat (mq - 2 ^ sq * nq) < 2 ^ s2" using mn s2wb by (simp add: field_simps word_less_sub_le)
hence "of_nat (mq - 2 ^ sq * nq) < (2::'a word) ^ (s2 - s1)"
proof (rule word_power_less_diff)
have mm: "mq - 2 ^ sq * nq < 2 ^ (len_of TYPE('a) - s1)" using mq by simp
moreover from s10 have "len_of TYPE('a) - s1 < len_of TYPE('a)"
by (rule diff_less, simp)
ultimately show "of_nat (mq - 2 ^ sq * nq) < (2::'a word) ^ (len_of TYPE('a) - s1)"
apply (simp add: word_less_nat_alt)
apply (subst unat_of_nat)
apply (subst mod_less)
apply (erule order_less_le_trans)
apply simp+
done
qed
hence "mq - 2 ^ sq * nq < 2 ^ (s2 - s1)" using mq s2wb
apply (simp add: word_less_nat_alt)
apply (subst (asm) unat_of_nat)
apply (subst (asm) mod_less)
apply (rule order_le_less_trans)
apply (rule diff_le_self)
apply (erule order_less_le_trans)
apply simp
apply assumption
done
ultimately show ?thesis by auto
qed
lemma word_less_sub_1:
"x < (y :: ('a :: len) word) \<Longrightarrow> x \<le> y - 1"
apply (erule udvd_minus_le')
apply (simp add: udvd_def)+
done
lemma word_sub_mono2:
"\<lbrakk> a + b \<le> c + d; c \<le> a; b \<le> a + b; d \<le> c + d \<rbrakk>
\<Longrightarrow> b \<le> (d :: ('a :: len) word)"
apply (drule(1) word_sub_mono)
apply simp
apply simp
apply simp
done
lemma word_subset_less:
"\<lbrakk> {x .. x + r - 1} \<subseteq> {y .. y + s - 1};
x \<le> x + r - 1; y \<le> y + (s :: ('a :: len) word) - 1;
s \<noteq> 0 \<rbrakk>
\<Longrightarrow> r \<le> s"
apply (frule subsetD[where c=x])
apply simp
apply (drule subsetD[where c="x + r - 1"])
apply simp
apply (clarsimp simp: add_diff_eq[symmetric])
apply (drule(1) word_sub_mono2)
apply (simp_all add: olen_add_eqv[symmetric])
apply (erule word_le_minus_cancel)
apply (rule ccontr)
apply (simp add: word_not_le)
done
lemma two_power_strict_part_mono:
"strict_part_mono {..31} (\<lambda>x. (2 :: word32) ^ x)"
by (simp | subst strict_part_mono_by_steps)+
lemma uint_power_lower:
"n < len_of TYPE('a) \<Longrightarrow> uint (2 ^ n :: 'a :: len word) = (2 ^ n :: int)"
by (simp add: uint_nat int_power)
lemma power_le_mono:
"\<lbrakk>2 ^ n \<le> (2::'a::len word) ^ m; n < len_of TYPE('a); m < len_of TYPE('a)\<rbrakk>
\<Longrightarrow> n \<le> m"
apply (clarsimp simp add: le_less)
apply safe
apply (simp add: word_less_nat_alt)
apply (simp only: uint_arith_simps(3))
apply (drule uint_power_lower)+
apply simp
done
lemma sublist_equal_part:
"xs \<le> ys \<Longrightarrow> take (length xs) ys = xs"
by (clarsimp simp: prefixeq_def less_eq_list_def)
lemma take_n_subset_le:
"\<lbrakk> {x. take n (to_bl x) = take n xs} \<subseteq> {y :: word32. take m (to_bl y) = take m ys};
n \<le> 32; m \<le> 32; length xs = 32; length ys = 32 \<rbrakk>
\<Longrightarrow> m \<le> n"
apply (rule ccontr, simp add: le_def)
apply (simp add: subset_iff)
apply (drule spec[where x="of_bl (take n xs @ take (32 - n) (map Not (drop n ys)))"])
apply (simp add: word_bl.Abs_inverse)
apply (subgoal_tac "\<exists>p. m = n + p")
apply clarsimp
apply (simp add: take_add take_map_Not)
apply (rule exI[where x="m - n"])
apply simp
done
lemma two_power_eq:
"\<lbrakk>n < len_of TYPE('a); m < len_of TYPE('a)\<rbrakk>
\<Longrightarrow> ((2::'a::len word) ^ n = 2 ^ m) = (n = m)"
apply safe
apply (rule order_antisym)
apply (simp add: power_le_mono[where 'a='a])+
done
lemma less_list_def': "(xs < ys) = (prefix xs ys)"
apply (metis prefix_order.eq_iff prefix_def less_list_def less_eq_list_def)
done
lemma prefix_length_less:
"xs < ys \<Longrightarrow> length xs < length ys"
apply (clarsimp simp: less_list_def' prefix_def)
apply (frule prefixeq_length_le)
apply (rule ccontr, simp)
apply (clarsimp simp: prefixeq_def)
done
lemmas strict_prefix_simps [simp, code] = prefix_simps [folded less_list_def']
lemmas take_strict_prefix = take_prefix [folded less_list_def']
lemma not_prefix_longer:
"\<lbrakk> length xs > length ys \<rbrakk> \<Longrightarrow> \<not> xs \<le> ys"
by (clarsimp dest!: prefix_length_le)
lemma of_bl_length:
"length xs < len_of TYPE('a) \<Longrightarrow> of_bl xs < (2 :: 'a::len word) ^ length xs"
by (simp add: of_bl_length_less)
(* FIXME: do we need this? *)
lemma power_overflow_simp [simp]:
"(2 ^ n = (0::'a :: len word)) = (len_of TYPE ('a) \<le> n)"
by (rule WordLib.p2_eq_0)
lemma unat_of_nat_eq:
"x < 2 ^ len_of TYPE('a) \<Longrightarrow> unat (of_nat x ::'a::len word) = x"
by (simp add: unat_of_nat)
lemmas unat_of_nat32 = unat_of_nat_eq[where 'a=32, unfolded word_bits_len_of]
lemma unat_eq_of_nat:
"n < 2 ^ len_of TYPE('a) \<Longrightarrow> (unat (x :: 'a::len word) = n) = (x = of_nat n)"
by (subst unat_of_nat_eq[where x=n, symmetric], simp+)
lemma unat_less_helper:
"x < of_nat n \<Longrightarrow> unat x < n"
apply (simp add: word_less_nat_alt)
apply (erule order_less_le_trans)
apply (simp add: unat_of_nat)
done
lemma of_nat_0:
"\<lbrakk>of_nat n = (0::('a::len) word); n < 2 ^ len_of (TYPE('a))\<rbrakk> \<Longrightarrow> n = 0"
by (drule unat_of_nat_eq, simp)
lemma of_nat32_0:
"\<lbrakk>of_nat n = (0::word32); n < 2 ^ word_bits\<rbrakk> \<Longrightarrow> n = 0"
by (erule of_nat_0, simp add: word_bits_def)
lemma unat_mask_2_less_4:
"unat (p && mask 2 :: word32) < 4"
apply (rule unat_less_helper)
apply (rule order_le_less_trans, rule word_and_le1)
apply (simp add: mask_def)
done
lemma minus_one_helper3:
"x < y \<Longrightarrow> x \<le> (y :: ('a :: len) word) - 1"
apply (simp add: word_less_nat_alt word_le_nat_alt)
apply (subst unat_minus_one)
apply clarsimp
apply arith
done
lemma minus_one_helper:
"\<lbrakk> x \<le> y; x \<noteq> 0 \<rbrakk> \<Longrightarrow> x - 1 < (y :: ('a :: len) word)"
apply (simp add: word_less_nat_alt word_le_nat_alt)
apply (subst unat_minus_one)
apply assumption
apply (cases "unat x")
apply (simp add: unat_eq_zero)
apply arith
done
lemma minus_one_helper5:
fixes x :: "'a::len word"
shows "\<lbrakk>y \<noteq> 0; x \<le> y - 1 \<rbrakk> \<Longrightarrow> x < y"
by (metis leD minus_one_helper not_leE)
lemma plus_one_helper[elim!]:
"x < n + (1 :: ('a :: len) word) \<Longrightarrow> x \<le> n"
apply (simp add: word_less_nat_alt word_le_nat_alt field_simps)
apply (case_tac "1 + n = 0")
apply simp
apply (subst(asm) unatSuc, assumption)
apply arith
done
lemma not_greatest_aligned:
"\<lbrakk> x < y; is_aligned x n; is_aligned y n \<rbrakk>
\<Longrightarrow> x + 2 ^ n \<noteq> 0"
apply (rule notI)
apply (erule is_aligned_get_word_bits[where p=y])
apply (simp add: eq_diff_eq[symmetric])
apply (frule minus_one_helper3)
apply (drule le_minus'[where a="x" and c="y - x" and b="- 1" for x y, simplified])
apply (simp add: field_simps)
apply (frule is_aligned_less_sz[where a=y])
apply clarsimp
apply (erule notE)
apply (rule minus_one_helper5)
apply simp
apply (metis is_aligned_no_overflow minus_one_helper3 order_le_less_trans)
apply simp
done
lemma of_nat_inj:
"\<lbrakk>x < 2 ^ len_of TYPE('a); y < 2 ^ len_of TYPE('a)\<rbrakk> \<Longrightarrow>
(of_nat x = (of_nat y :: 'a :: len word)) = (x = y)"
by (simp add: word_unat.norm_eq_iff [symmetric])
lemma map_prefixI:
"xs \<le> ys \<Longrightarrow> map f xs \<le> map f ys"
by (clarsimp simp: less_eq_list_def prefixeq_def)
lemma if_Some_None_eq_None:
"((if P then Some v else None) = None) = (\<not> P)"
by simp
lemma CollectPairFalse [iff]:
"{(a,b). False} = {}"
by (simp add: split_def)
lemma if_P_True1:
"Q \<Longrightarrow> (if P then True else Q)"
by simp
lemma if_P_True2:
"Q \<Longrightarrow> (if P then Q else True)"
by simp
lemma list_all2_induct [consumes 1, case_names Nil Cons]:
assumes lall: "list_all2 Q xs ys"
and nilr: "P [] []"
and consr: "\<And>x xs y ys. \<lbrakk>list_all2 Q xs ys; Q x y; P xs ys\<rbrakk> \<Longrightarrow> P (x # xs) (y # ys)"
shows "P xs ys"
using lall
proof (induct rule: list_induct2 [OF list_all2_lengthD [OF lall]])
case 1 thus ?case by auto fact+
next
case (2 x xs y ys)
show ?case
proof (rule consr)
from "2.prems" show "list_all2 Q xs ys" and "Q x y" by simp_all
thus "P xs ys" by (intro "2.hyps")
qed
qed
lemma list_all2_induct_suffixeq [consumes 1, case_names Nil Cons]:
assumes lall: "list_all2 Q as bs"
and nilr: "P [] []"
and consr: "\<And>x xs y ys.
\<lbrakk>list_all2 Q xs ys; Q x y; P xs ys; suffixeq (x # xs) as; suffixeq (y # ys) bs\<rbrakk>
\<Longrightarrow> P (x # xs) (y # ys)"
shows "P as bs"
proof -
def as' == as
def bs' == bs
have "suffixeq as as' \<and> suffixeq bs bs'" unfolding as'_def bs'_def by simp
thus ?thesis using lall
proof (induct rule: list_induct2 [OF list_all2_lengthD [OF lall]])
case 1 show ?case by fact
next
case (2 x xs y ys)
show ?case
proof (rule consr)
from "2.prems" show "list_all2 Q xs ys" and "Q x y" by simp_all
thus "P xs ys" using "2.hyps" "2.prems" by (auto dest: suffixeq_ConsD)
from "2.prems" show "suffixeq (x # xs) as" and "suffixeq (y # ys) bs"
by (auto simp: as'_def bs'_def)
qed
qed
qed
lemma distinct_prop_enum:
"\<lbrakk> \<And>x y. \<lbrakk> x \<le> stop; y \<le> stop; x \<noteq> y \<rbrakk>
\<Longrightarrow> P x y \<rbrakk>
\<Longrightarrow> distinct_prop P [(0 :: word32) .e. stop]"
apply (simp add: upto_enum_def distinct_prop_map
del: upt.simps)
apply (rule distinct_prop_distinct)
apply simp
apply (simp add: less_Suc_eq_le del: upt.simps)
apply (erule_tac x="of_nat x" in meta_allE)
apply (erule_tac x="of_nat y" in meta_allE)
apply (frule_tac y=x in unat_le)
apply (frule_tac y=y in unat_le)
apply (erule word_unat.Rep_cases)+
apply (simp add: toEnum_of_nat[OF unat_lt2p]
word_le_nat_alt)
done
lemma distinct_prop_enum_step:
"\<lbrakk> \<And>x y. \<lbrakk> x \<le> stop div step; y \<le> stop div step; x \<noteq> y \<rbrakk>
\<Longrightarrow> P (x * step) (y * step) \<rbrakk>
\<Longrightarrow> distinct_prop P [0, step .e. stop]"
apply (simp add: upto_enum_step_def distinct_prop_map)
apply (rule distinct_prop_enum)
apply simp
done
lemma if_apply_def2:
"(if P then F else G) = (\<lambda>x. (P \<longrightarrow> F x) \<and> (\<not> P \<longrightarrow> G x))"
by simp
lemma case_bool_If:
"case_bool P Q b = (if b then P else Q)"
by simp
lemma option_case_If:
"case_option P (\<lambda>x. Q) v = (if v = None then P else Q)"
by clarsimp
lemma option_case_If2:
"case_option P Q v = If (v \<noteq> None) (Q (the v)) P"
by (simp split: option.split)
lemma if3_fold:
"(if P then x else if Q then y else x)
= (if P \<or> \<not> Q then x else y)"
by simp
lemma word32_shift_by_2:
"x * 4 = (x::word32) << 2"
by (simp add: shiftl_t2n)
(* TODO: move to Aligned *)
lemma add_mask_lower_bits:
"\<lbrakk>is_aligned (x :: 'a :: len word) n;
\<forall>n' \<ge> n. n' < len_of TYPE('a) \<longrightarrow> \<not> p !! n'\<rbrakk> \<Longrightarrow> x + p && ~~mask n = x"
apply (subst word_plus_and_or_coroll)
apply (rule word_eqI)
apply (clarsimp simp: word_size is_aligned_nth)
apply (erule_tac x=na in allE)+
apply simp
apply (rule word_eqI)
apply (clarsimp simp: word_size is_aligned_nth nth_mask word_ops_nth_size)
apply (erule_tac x=na in allE)+
apply (case_tac "na < n")
apply simp
apply simp
done
lemma findSomeD:
"find P xs = Some x \<Longrightarrow> P x \<and> x \<in> set xs"
by (induct xs) (auto split: split_if_asm)
lemma findNoneD:
"find P xs = None \<Longrightarrow> \<forall>x \<in> set xs. \<not>P x"
by (induct xs) (auto split: split_if_asm)
lemma dom_upd:
"dom (\<lambda>x. if x = y then None else f x) = dom f - {y}"
by (rule set_eqI) (auto split: split_if_asm)
lemma ran_upd:
"\<lbrakk> inj_on f (dom f); f y = Some z \<rbrakk> \<Longrightarrow> ran (\<lambda>x. if x = y then None else f x) = ran f - {z}"
apply (rule set_eqI)
apply (unfold ran_def)
apply simp
apply (rule iffI)
apply clarsimp
apply (rule conjI, blast)
apply clarsimp
apply (drule_tac x=a and y=y in inj_onD, simp)
apply blast
apply blast
apply simp
apply clarsimp
apply (rule_tac x=a in exI)
apply clarsimp
done
lemma maxBound_word:
"(maxBound::'a::len word) = -1"
apply (simp add: maxBound_def enum_word_def)
apply (subst last_map)
apply clarsimp
apply simp
done
lemma minBound_word:
"(minBound::'a::len word) = 0"
apply (simp add: minBound_def enum_word_def)
apply (subst map_upt_unfold)
apply simp
apply simp
done
lemma maxBound_max_word:
"(maxBound::'a::len word) = max_word"
apply (subst maxBound_word)
apply (subst max_word_minus [symmetric])
apply (rule refl)
done
lemma is_aligned_andI1:
"is_aligned x n \<Longrightarrow> is_aligned (x && y) n"
by (simp add: is_aligned_nth)
lemma is_aligned_andI2:
"is_aligned y n \<Longrightarrow> is_aligned (x && y) n"
by (simp add: is_aligned_nth)
lemma is_aligned_shiftl:
"is_aligned w (n - m) \<Longrightarrow> is_aligned (w << m) n"
by (simp add: is_aligned_nth nth_shiftl)
lemma is_aligned_shiftr:
"is_aligned w (n + m) \<Longrightarrow> is_aligned (w >> m) n"
by (simp add: is_aligned_nth nth_shiftr)
lemma is_aligned_shiftl_self:
"is_aligned (p << n) n"
by (rule is_aligned_shiftl) simp
lemma is_aligned_neg_mask_eq:
"is_aligned p n \<Longrightarrow> p && ~~ mask n = p"
apply (simp add: is_aligned_nth)
apply (rule word_eqI)
apply (clarsimp simp: word_size word_ops_nth_size)
apply fastforce
done
lemma rtrancl_insert:
assumes x_new: "\<And>y. (x,y) \<notin> R"
shows "R^* `` insert x S = insert x (R^* `` S)"
proof -
have "R^* `` insert x S = R^* `` ({x} \<union> S)" by simp
also
have "R^* `` ({x} \<union> S) = R^* `` {x} \<union> R^* `` S"
by (subst Image_Un) simp
also
have "R^* `` {x} = {x}"
apply (clarsimp simp: Image_singleton)
apply (rule set_eqI, clarsimp)
apply (rule iffI)
apply (drule rtranclD)
apply (erule disjE, simp)
apply clarsimp
apply (drule tranclD)
apply (clarsimp simp: x_new)
apply fastforce
done
finally
show ?thesis by simp
qed
lemma ran_del_subset:
"y \<in> ran (f (x := None)) \<Longrightarrow> y \<in> ran f"
by (auto simp: ran_def split: split_if_asm)
lemma trancl_sub_lift:
assumes sub: "\<And>p p'. (p,p') \<in> r \<Longrightarrow> (p,p') \<in> r'"
shows "(p,p') \<in> r^+ \<Longrightarrow> (p,p') \<in> r'^+"
by (fastforce intro: trancl_mono sub)
lemma trancl_step_lift:
assumes x_step: "\<And>p p'. (p,p') \<in> r' \<Longrightarrow> (p,p') \<in> r \<or> (p = x \<and> p' = y)"
assumes y_new: "\<And>p'. \<not>(y,p') \<in> r"
shows "(p,p') \<in> r'^+ \<Longrightarrow> (p,p') \<in> r^+ \<or> ((p,x) \<in> r^+ \<and> p' = y) \<or> (p = x \<and> p' = y)"
apply (erule trancl_induct)
apply (drule x_step)
apply fastforce
apply (erule disjE)
apply (drule x_step)
apply (erule disjE)
apply (drule trancl_trans, drule r_into_trancl, assumption)
apply blast
apply clarsimp
apply (erule disjE)
apply clarsimp
apply (drule x_step)
apply (erule disjE)
apply (simp add: y_new)
apply simp
apply clarsimp
apply (drule x_step)
apply (simp add: y_new)
done
lemma upto_enum_step_shift:
"\<lbrakk> is_aligned p n \<rbrakk> \<Longrightarrow>
([p , p + 2 ^ m .e. p + 2 ^ n - 1])
= map (op + p) [0, 2 ^ m .e. 2 ^ n - 1]"
apply (erule is_aligned_get_word_bits)
prefer 2
apply (simp add: map_idI)
apply (clarsimp simp: upto_enum_step_def)
apply (frule is_aligned_no_overflow)
apply (simp add: linorder_not_le [symmetric])
done
lemma upto_enum_step_shift_red:
"\<lbrakk> is_aligned p sz; sz < word_bits; us \<le> sz \<rbrakk>
\<Longrightarrow> [p, p + 2 ^ us .e. p + 2 ^ sz - 1]
= map (\<lambda>x. p + of_nat x * 2 ^ us) [0 ..< 2 ^ (sz - us)]"
apply (subst upto_enum_step_shift, assumption)
apply (simp add: upto_enum_step_red)
done
lemma div_to_mult_word_lt:
"\<lbrakk> (x :: ('a :: len) word) \<le> y div z \<rbrakk> \<Longrightarrow> x * z \<le> y"
apply (cases "z = 0")
apply simp
apply (simp add: word_neq_0_conv)
apply (rule order_trans)
apply (erule(1) word_mult_le_mono1)
apply (simp add: unat_div)
apply (rule order_le_less_trans [OF div_mult_le])
apply simp
apply (rule word_div_mult_le)
done
lemma upto_enum_step_subset:
"set [x, y .e. z] \<subseteq> {x .. z}"
apply (clarsimp simp: upto_enum_step_def linorder_not_less)
apply (drule div_to_mult_word_lt)
apply (rule conjI)
apply (erule word_random[rotated])
apply simp
apply (rule order_trans)
apply (erule word_plus_mono_right)
apply simp
apply simp
done
lemma shiftr_less_t2n':
fixes x :: "('a :: len) word"
shows "\<lbrakk> x && mask (n + m) = x; m < len_of TYPE('a) \<rbrakk>
\<Longrightarrow> (x >> n) < 2 ^ m"
apply (subst mask_eq_iff_w2p[symmetric])
apply (simp add: word_size)
apply (rule word_eqI)
apply (drule_tac x="na + n" in word_eqD)
apply (simp add: nth_shiftr word_size)
apply safe
done
lemma shiftr_less_t2n:
fixes x :: "('a :: len) word"
shows "x < 2 ^ (n + m) \<Longrightarrow> (x >> n) < 2 ^ m"
apply (rule shiftr_less_t2n')
apply (erule less_mask_eq)
apply (rule ccontr)
apply (simp add: not_less)
apply (subst (asm) p2_eq_0[symmetric])
apply (simp add: power_add)
done
lemma shiftr_eq_0:
"n \<ge> len_of TYPE('a :: len) \<Longrightarrow> ((w::('a::len word)) >> n) = 0"
apply (cut_tac shiftr_less_t2n'[of w n 0], simp)
apply (simp add: mask_eq_iff)
apply (simp add: lt2p_lem)
apply simp
done
lemma shiftr_not_mask_0:
"n+m\<ge>len_of TYPE('a :: len) \<Longrightarrow> ((w::('a::len word)) >> n) && ~~ mask m = 0"
apply (simp add: and_not_mask shiftr_less_t2n shiftr_shiftr)
apply (subgoal_tac "w >> n + m = 0", simp)
apply (simp add: le_mask_iff[symmetric] mask_def le_def)
apply (subst (asm) p2_gt_0[symmetric])
apply (simp add: power_add not_less)
done
lemma shiftl_less_t2n:
fixes x :: "('a :: len) word"
shows "\<lbrakk> x < (2 ^ (m - n)); m < len_of TYPE('a) \<rbrakk> \<Longrightarrow> (x << n) < 2 ^ m"
apply (subst mask_eq_iff_w2p[symmetric])
apply (simp add: word_size)
apply (drule less_mask_eq)
apply (rule word_eqI)
apply (drule_tac x="na - n" in word_eqD)
apply (simp add: nth_shiftl word_size)
apply (cases "n \<le> m")
apply safe
apply simp
apply simp
done
lemma shiftl_less_t2n':
"(x::'a::len word) < 2 ^ m \<Longrightarrow> m+n < len_of TYPE('a) \<Longrightarrow> x << n < 2 ^ (m + n)"
by (rule shiftl_less_t2n) simp_all
lemma ucast_ucast_mask:
"(ucast :: ('a :: len) word \<Rightarrow> ('b :: len) word) (ucast x) = x && mask (len_of TYPE ('a))"
apply (rule word_eqI)
apply (simp add: nth_ucast word_size)
done
lemma ucast_ucast_len:
"\<lbrakk> x < 2 ^ len_of TYPE('b) \<rbrakk> \<Longrightarrow>
ucast (ucast x::'b::len word) = (x::'a::len word)"
apply (subst ucast_ucast_mask)
apply (erule less_mask_eq)
done
lemma unat_ucast: "unat (ucast x :: ('a :: len0) word) = unat x mod 2 ^ (len_of TYPE('a))"
apply (simp add: unat_def ucast_def)
apply (subst word_uint.eq_norm)
apply (subst nat_mod_distrib)
apply simp
apply simp
apply (subst nat_power_eq)
apply simp
apply simp
done
lemma sints_subset:
"m \<le> n \<Longrightarrow> sints m \<subseteq> sints n"
apply (simp add: sints_num)
apply clarsimp
apply (rule conjI)
apply (erule order_trans[rotated])
apply simp
apply (erule order_less_le_trans)
apply simp
done
lemma up_scast_inj:
"\<lbrakk> scast x = (scast y :: ('b :: len) word); size x \<le> len_of TYPE('b) \<rbrakk>
\<Longrightarrow> x = y"
apply (simp add: scast_def)
apply (subst(asm) word_sint.Abs_inject)
apply (erule subsetD [OF sints_subset])
apply (simp add: word_size)
apply (erule subsetD [OF sints_subset])
apply (simp add: word_size)
apply simp
done
lemma up_scast_inj_eq:
"len_of TYPE('a) \<le> len_of TYPE ('b) \<Longrightarrow> (scast x = (scast y::'b::len word)) = (x = (y::'a::len word))"
by (fastforce dest: up_scast_inj simp: word_size)
lemma nth_bounded:
"\<lbrakk>(x :: 'a :: len word) !! n; x < 2 ^ m; m \<le> len_of TYPE ('a)\<rbrakk> \<Longrightarrow> n < m"
apply (frule test_bit_size)
apply (clarsimp simp: test_bit_bl word_size)
apply (simp add: nth_rev)
apply (subst(asm) is_aligned_add_conv[OF is_aligned_0',
simplified add_0_left, rotated])
apply assumption+
apply (simp only: to_bl_0 word_bits_len_of)
apply (simp add: nth_append split: split_if_asm)
done
lemma is_aligned_add_or:
"\<lbrakk>is_aligned p n; d < 2 ^ n\<rbrakk> \<Longrightarrow> p + d = p || d"
apply (rule word_plus_and_or_coroll)
apply (erule is_aligned_get_word_bits)
apply (rule word_eqI)
apply (clarsimp simp add: is_aligned_nth)
apply (frule(1) nth_bounded)
apply simp+
done
lemma two_power_increasing:
"\<lbrakk> n \<le> m; m < len_of TYPE('a) \<rbrakk> \<Longrightarrow> (2 :: 'a :: len word) ^ n \<le> 2 ^ m"
by (simp add: word_le_nat_alt)
lemma is_aligned_add_less_t2n:
"\<lbrakk>is_aligned (p\<Colon>'a\<Colon>len word) n; d < 2^n; n \<le> m; p < 2^m\<rbrakk> \<Longrightarrow> p + d < 2^m"
apply (case_tac "m < len_of TYPE('a)")
apply (subst mask_eq_iff_w2p[symmetric])
apply (simp add: word_size)
apply (simp add: is_aligned_add_or word_ao_dist less_mask_eq)
apply (subst less_mask_eq)
apply (erule order_less_le_trans)
apply (erule(1) two_power_increasing)
apply simp
apply (simp add: power_overflow)
done
(* FIXME: generalise? *)
lemma le_2p_upper_bits:
"\<lbrakk> (p::word32) \<le> 2^n - 1; n < word_bits \<rbrakk> \<Longrightarrow> \<forall>n'\<ge>n. n' < word_bits \<longrightarrow> \<not> p !! n'"
apply (subst upper_bits_unset_is_l2p, assumption)
apply simp
done
lemma ran_upd':
"\<lbrakk>inj_on f (dom f); f y = Some z\<rbrakk>
\<Longrightarrow> ran (f (y := None)) = ran f - {z}"
apply (drule (1) ran_upd)
apply (simp add: ran_def)
done
(* FIXME: generalise? *)
lemma le2p_bits_unset:
"p \<le> 2 ^ n - 1 \<Longrightarrow> \<forall>n'\<ge>n. n' < word_bits \<longrightarrow> \<not> (p::word32) !! n'"
apply (case_tac "n < word_bits")
apply (frule upper_bits_unset_is_l2p [where p=p])
apply simp_all
done
lemma aligned_offset_non_zero:
"\<lbrakk> is_aligned x n; y < 2 ^ n; x \<noteq> 0 \<rbrakk> \<Longrightarrow> x + y \<noteq> 0"
apply (cases "y = 0")
apply simp
apply (subst word_neq_0_conv)
apply (subst gt0_iff_gem1)
apply (erule is_aligned_get_word_bits)
apply (subst field_simps[symmetric], subst plus_le_left_cancel_nowrap)
apply (rule is_aligned_no_wrap')
apply simp
apply (rule minus_one_helper)
apply simp
apply assumption
apply (erule (1) is_aligned_no_wrap')
apply (simp add: gt0_iff_gem1 [symmetric] word_neq_0_conv)
apply simp
done
lemma le_imp_power_dvd_int:
"n \<le> m \<Longrightarrow> (b ^ n :: int) dvd b ^ m"
apply (simp add: dvd_def)
apply (rule exI[where x="b ^ (m - n)"])
apply (simp add: power_add[symmetric])
done
(* FIXME: this is identical to mask_eqs(1), unnecessary? *)
lemma mask_inner_mask:
"((p && mask n) + q) && mask n
= (p + q) && mask n"
apply (rule mask_eqs(1))
done
lemma mask_add_aligned:
"is_aligned p n
\<Longrightarrow> (p + q) && mask n = q && mask n"
apply (simp add: is_aligned_mask)
apply (subst mask_inner_mask [symmetric])
apply simp
done
lemma take_prefix:
"(take (length xs) ys = xs) = (xs \<le> ys)"
proof (induct xs arbitrary: ys)
case Nil thus ?case by simp
next
case Cons thus ?case by (cases ys) auto
qed
lemma rel_comp_Image:
"(R O R') `` S = R' `` (R `` S)"
by blast
lemma trancl_power:
"x \<in> r^+ = (\<exists>n > 0. x \<in> r^^n)"
apply (cases x)
apply simp
apply (rule iffI)
apply (drule tranclD2)
apply (clarsimp simp: rtrancl_is_UN_relpow)
apply (rule_tac x="Suc n" in exI)
apply fastforce
apply clarsimp
apply (case_tac n, simp)
apply clarsimp
apply (drule relpow_imp_rtrancl)
apply fastforce
done
lemma take_is_prefix:
"take n xs \<le> xs"
apply (simp add: less_eq_list_def prefixeq_def)
apply (rule_tac x="drop n xs" in exI)
apply simp
done
lemma cart_singleton_empty:
"(S \<times> {e} = {}) = (S = {})"
by blast
lemma word_div_1:
"(n :: ('a :: len) word) div 1 = n"
by (simp add: word_div_def)
lemma word_minus_one_le:
"-1 \<le> (x :: ('a :: len) word) = (x = -1)"
apply (insert word_n1_ge[where y=x])
apply safe
apply (erule(1) order_antisym)
done
lemmas word32_minus_one_le =
word_minus_one_le[where 'a=32, simplified]
lemma mask_out_sub_mask:
"(x && ~~ mask n) = x - (x && mask n)"
by (simp add: field_simps word_plus_and_or_coroll2)
lemma is_aligned_addD1:
assumes al1: "is_aligned (x + y) n"
and al2: "is_aligned (x::'a::len word) n"
shows "is_aligned y n"
using al2
proof (rule is_aligned_get_word_bits)
assume "x = 0" thus ?thesis using al1 by simp
next
assume nv: "n < len_of TYPE('a)"
from al1 obtain q1
where xy: "x + y = 2 ^ n * of_nat q1" and "q1 < 2 ^ (len_of TYPE('a) - n)"
by (rule is_alignedE)
moreover from al2 obtain q2
where x: "x = 2 ^ n * of_nat q2" and "q2 < 2 ^ (len_of TYPE('a) - n)"
by (rule is_alignedE)
ultimately have "y = 2 ^ n * (of_nat q1 - of_nat q2)"
by (simp add: field_simps)
thus ?thesis using nv by (simp add: is_aligned_mult_triv1)
qed
lemmas is_aligned_addD2 =
is_aligned_addD1[OF subst[OF add.commute,
of "%x. is_aligned x n" for n]]
lemma is_aligned_add:
"\<lbrakk>is_aligned p n; is_aligned q n\<rbrakk> \<Longrightarrow> is_aligned (p + q) n"
by (simp add: is_aligned_mask mask_add_aligned)
lemma my_BallE: "\<lbrakk> \<forall>x \<in> A. P x; y \<in> A; P y \<Longrightarrow> Q \<rbrakk> \<Longrightarrow> Q"
by (simp add: Ball_def)
lemma word_le_add:
fixes x :: "'a :: len word"
shows "x \<le> y \<Longrightarrow> \<exists>n. y = x + of_nat n"
apply (rule exI [where x = "unat (y - x)"])
apply simp
done
lemma zipWith_nth:
"\<lbrakk> n < min (length xs) (length ys) \<rbrakk> \<Longrightarrow> zipWith f xs ys ! n = f (xs ! n) (ys ! n)"
unfolding zipWith_def by simp
lemma length_zipWith:
"length (zipWith f xs ys) = min (length xs) (length ys)"
unfolding zipWith_def by simp
lemma distinct_prop_nth:
"\<lbrakk> distinct_prop P ls; n < n'; n' < length ls \<rbrakk> \<Longrightarrow> P (ls ! n) (ls ! n')"
apply (induct ls arbitrary: n n')
apply simp
apply simp
apply (case_tac n')
apply simp
apply simp
apply (case_tac n)
apply simp
apply simp
done
lemma shiftl_mask_is_0 :
"(x << n) && mask n = 0"
apply (rule iffD1 [OF is_aligned_mask])
apply (rule is_aligned_shiftl_self)
done
lemma word_power_nonzero:
"\<lbrakk> (x :: word32) < 2 ^ (word_bits - n); n < word_bits; x \<noteq> 0 \<rbrakk> \<Longrightarrow> x * 2 ^ n \<noteq> 0"
apply (cases "n = 0")
apply simp
apply (simp only: word_neq_0_conv word_less_nat_alt
shiftl_t2n mod_0 unat_word_ariths
unat_power_lower word_le_nat_alt word_bits_def)
apply (unfold word_bits_len_of)
apply (subst mod_less)
apply (subst mult.commute, erule nat_less_power_trans)
apply simp
apply simp
done
lemmas unat_mult_simple = iffD1 [OF unat_mult_lem [where 'a = 32, unfolded word_bits_len_of]]
definition
sum_map :: "('a \<Rightarrow> 'b) \<Rightarrow> ('c \<Rightarrow> 'd) \<Rightarrow> 'a + 'c \<Rightarrow> 'b + 'd" where
"sum_map f g x \<equiv> case x of Inl v \<Rightarrow> Inl (f v) | Inr v' \<Rightarrow> Inr (g v')"
lemma sum_map_simps[simp]:
"sum_map f g (Inl v) = Inl (f v)"
"sum_map f g (Inr w) = Inr (g w)"
by (simp add: sum_map_def)+
lemma if_and_helper:
"(If x v v') && v'' = If x (v && v'') (v' && v'')"
by (simp split: split_if)
lemma unat_Suc2:
fixes n :: "('a :: len) word"
shows
"n \<noteq> -1 \<Longrightarrow> unat (n + 1) = Suc (unat n)"
apply (subst add.commute, rule unatSuc)
apply (subst eq_diff_eq[symmetric], simp add: minus_equation_iff)
done
lemmas unat_eq_1
= unat_eq_0 word_unat.Rep_inject[where y=1, simplified]
lemma cart_singleton_image:
"S \<times> {s} = (\<lambda>v. (v, s)) ` S"
by auto
lemma singleton_eq_o2s:
"({x} = set_option v) = (v = Some x)"
by (cases v, auto)
lemma ran_option_map_restrict_eq:
"\<lbrakk> x \<in> ran (option_map f o g); x \<notin> ran (option_map f o (g |` (- {y}))) \<rbrakk>
\<Longrightarrow> \<exists>v. g y = Some v \<and> f v = x"
apply (clarsimp simp: elim!: ranE)
apply (rename_tac w z)
apply (case_tac "w = y")
apply clarsimp
apply (erule notE, rule_tac a=w in ranI)
apply (simp add: restrict_map_def)
done
lemma option_set_singleton_eq:
"(set_option opt = {v}) = (opt = Some v)"
by (cases opt, simp_all)
lemmas option_set_singleton_eqs
= option_set_singleton_eq
trans[OF eq_commute option_set_singleton_eq]
lemma option_map_comp2:
"option_map (f o g) = option_map f o option_map g"
by (simp add: option.map_comp fun_eq_iff)
lemma rshift_sub_mask_eq:
"(a >> (size a - b)) && mask b = a >> (size a - b)"
using shiftl_shiftr2[where a=a and b=0 and c="size a - b"]
apply (cases "b < size a")
apply simp
apply (simp add: linorder_not_less mask_def word_size
p2_eq_0[THEN iffD2])
done
lemma shiftl_shiftr3:
"b \<le> c \<Longrightarrow> a << b >> c = (a >> c - b) && mask (size a - c)"
apply (cases "b = c")
apply (simp add: shiftl_shiftr1)
apply (simp add: shiftl_shiftr2)
done
lemma and_mask_shiftr_comm:
"m\<le>size w \<Longrightarrow> (w && mask m) >> n = (w >> n) && mask (m-n)"
by (simp add: and_mask shiftr_shiftr) (simp add: word_size shiftl_shiftr3)
lemma and_not_mask_twice:
"(w && ~~ mask n) && ~~ mask m = w && ~~ mask (max m n)"
apply (simp add: and_not_mask)
apply (case_tac "n<m")
apply (simp_all add: shiftl_shiftr2 shiftl_shiftr1 not_less max_def
shiftr_shiftr shiftl_shiftl)
apply (cut_tac and_mask_shiftr_comm
[where w=w and m="size w" and n=m, simplified,symmetric])
apply (simp add: word_size mask_def)
apply (cut_tac and_mask_shiftr_comm
[where w=w and m="size w" and n=n, simplified,symmetric])
apply (simp add: word_size mask_def)
done
(* FIXME: move *)
lemma word_less_cases:
"x < y \<Longrightarrow> x = y - 1 \<or> x < y - (1 ::'a::len word)"
apply (drule word_less_sub_1)
apply (drule order_le_imp_less_or_eq)
apply auto
done
lemma eq_eqI:
"a = b \<Longrightarrow> (a = x) = (b = x)"
by simp
lemma mask_and_mask:
"mask a && mask b = mask (min a b)"
apply (rule word_eqI)
apply (simp add: word_size)
done
lemma mask_eq_0_eq_x:
"(x && w = 0) = (x && ~~ w = x)"
using word_plus_and_or_coroll2[where x=x and w=w]
by auto
lemma mask_eq_x_eq_0:
"(x && w = x) = (x && ~~ w = 0)"
using word_plus_and_or_coroll2[where x=x and w=w]
by auto
definition
"limited_and (x :: ('a :: len) word) y = (x && y = x)"
lemma limited_and_eq_0:
"\<lbrakk> limited_and x z; y && ~~ z = y \<rbrakk> \<Longrightarrow> x && y = 0"
unfolding limited_and_def
apply (subst arg_cong2[where f="op &&"])
apply (erule sym)+
apply (simp(no_asm) add: word_bw_assocs word_bw_comms word_bw_lcs)
done
lemma limited_and_eq_id:
"\<lbrakk> limited_and x z; y && z = z \<rbrakk> \<Longrightarrow> x && y = x"
unfolding limited_and_def
by (erule subst, fastforce simp: word_bw_lcs word_bw_assocs word_bw_comms)
lemma lshift_limited_and:
"limited_and x z \<Longrightarrow> limited_and (x << n) (z << n)"
unfolding limited_and_def
by (simp add: shiftl_over_and_dist[symmetric])
lemma rshift_limited_and:
"limited_and x z \<Longrightarrow> limited_and (x >> n) (z >> n)"
unfolding limited_and_def
by (simp add: shiftr_over_and_dist[symmetric])
lemmas limited_and_simps1 = limited_and_eq_0 limited_and_eq_id
lemmas is_aligned_limited_and
= is_aligned_neg_mask_eq[unfolded mask_def, folded limited_and_def]
lemma compl_of_1: "~~ 1 = (-2 :: ('a :: len) word)"
apply (rule word_bool_alg.compl_eq_compl_iff[THEN iffD1])
apply simp
done
lemmas limited_and_simps = limited_and_simps1
limited_and_simps1[OF is_aligned_limited_and]
limited_and_simps1[OF lshift_limited_and]
limited_and_simps1[OF rshift_limited_and]
limited_and_simps1[OF rshift_limited_and, OF is_aligned_limited_and]
compl_of_1 shiftl_shiftr1[unfolded word_size mask_def]
shiftl_shiftr2[unfolded word_size mask_def]
lemma isRight_sum_case: "isRight x \<Longrightarrow> case_sum f g x = g (theRight x)"
by (clarsimp simp add: isRight_def)
lemma split_word_eq_on_mask:
"(x = y) = (x && m = y && m \<and> x && ~~ m = y && ~~ m)"
apply safe
apply (rule word_eqI)
apply (drule_tac x=n in word_eqD)+
apply (simp add: word_size word_ops_nth_size)
apply auto
done
lemma inj_case_bool:
"inj (case_bool a b) = (a \<noteq> b)"
by (auto dest: inj_onD[where x=True and y=False]
intro: inj_onI split: bool.split_asm)
lemma zip_map2:
"zip as (map f bs) = map (\<lambda>(a, b). (a, f b)) (zip as bs)"
apply (induct bs arbitrary: as)
apply simp
apply (case_tac as)
apply simp
apply simp
done
lemma zip_same: "zip xs xs = map (\<lambda>v. (v, v)) xs"
by (induct xs, simp+)
lemma foldl_fun_upd:
"foldl (\<lambda>s r. s (r := g r)) f rs
= (\<lambda>x. if x \<in> set rs then g x else f x)"
apply (induct rs arbitrary: f)
apply simp
apply (auto simp: fun_eq_iff split: split_if)
done
lemma all_rv_choice_fn_eq_pred:
"\<lbrakk> \<And>rv. P rv \<Longrightarrow> \<exists>fn. f rv = g fn \<rbrakk>
\<Longrightarrow> \<exists>fn. \<forall>rv. P rv \<longrightarrow> f rv = g (fn rv)"
apply (rule_tac x="\<lambda>rv. SOME h. f rv = g h" in exI)
apply (clarsimp split: split_if)
apply (erule meta_allE, drule(1) meta_mp, elim exE)
apply (erule someI)
done
lemma ex_const_function:
"\<exists>f. \<forall>s. f (f' s) = v"
by force
lemma sum_to_zero:
"(a :: 'a :: ring) + b = 0 \<Longrightarrow> a = (- b)"
by (drule arg_cong[where f="\<lambda> x. x - a"], simp)
lemma nat_le_Suc_less_imp:
"x < y \<Longrightarrow> x \<le> y - Suc 0"
by arith
lemma list_case_If2:
"case_list f g xs = If (xs = []) f (g (hd xs) (tl xs))"
by (simp split: list.split)
lemma length_ineq_not_Nil:
"length xs > n \<Longrightarrow> xs \<noteq> []"
"length xs \<ge> n \<Longrightarrow> n \<noteq> 0 \<longrightarrow> xs \<noteq> []"
"\<not> length xs < n \<Longrightarrow> n \<noteq> 0 \<longrightarrow> xs \<noteq> []"
"\<not> length xs \<le> n \<Longrightarrow> xs \<noteq> []"
by auto
lemma numeral_eqs:
"2 = Suc (Suc 0)"
"3 = Suc (Suc (Suc 0))"
"4 = Suc (Suc (Suc (Suc 0)))"
"5 = Suc (Suc (Suc (Suc (Suc 0))))"
"6 = Suc (Suc (Suc (Suc (Suc (Suc 0)))))"
by simp+
lemma psubset_singleton:
"(S \<subset> {x}) = (S = {})"
by blast
lemma ucast_not_helper:
fixes a::word8
assumes a: "a \<noteq> 0xFF"
shows "ucast a \<noteq> (0xFF::word32)"
proof
assume "ucast a = (0xFF::word32)"
also
have "(0xFF::word32) = ucast (0xFF::word8)" by simp
finally
show False using a
apply -
apply (drule up_ucast_inj, simp)
apply simp
done
qed
lemma length_takeWhile_ge:
"length (takeWhile f xs) = n
\<Longrightarrow> length xs = n \<or> (length xs > n \<and> \<not> f (xs ! n))"
apply (induct xs arbitrary: n)
apply simp
apply (simp split: split_if_asm)
apply (case_tac n, simp_all)
done
lemma length_takeWhile_le:
"\<not> f (xs ! n) \<Longrightarrow>
length (takeWhile f xs) \<le> n"
apply (induct xs arbitrary: n)
apply simp
apply (clarsimp split: split_if)
apply (case_tac n, simp_all)
done
lemma length_takeWhile_gt:
"n < length (takeWhile f xs)
\<Longrightarrow> (\<exists>ys zs. length ys = Suc n \<and> xs = ys @ zs \<and> takeWhile f xs = ys @ takeWhile f zs)"
apply (induct xs arbitrary: n)
apply simp
apply (simp split: split_if_asm)
apply (case_tac n, simp_all)
apply (rule_tac x="[a]" in exI)
apply simp
apply (erule meta_allE, drule(1) meta_mp)
apply clarsimp
apply (rule_tac x="a # ys" in exI)
apply simp
done
lemma hd_drop_conv_nth2:
"n < length xs \<Longrightarrow> hd (drop n xs) = xs ! n"
by (rule hd_drop_conv_nth, clarsimp+)
lemma map_upt_eq_vals_D:
"\<lbrakk> map f [0 ..< n] = ys; m < length ys \<rbrakk> \<Longrightarrow> f m = ys ! m"
by clarsimp
lemma length_le_helper:
"\<lbrakk> n \<le> length xs; n \<noteq> 0 \<rbrakk> \<Longrightarrow> xs \<noteq> [] \<and> n - 1 \<le> length (tl xs)"
by (cases xs, simp_all)
lemma all_ex_eq_helper:
"(\<forall>v. (\<exists>v'. v = f v' \<and> P v v') \<longrightarrow> Q v)
= (\<forall>v'. P (f v') v' \<longrightarrow> Q (f v'))"
by auto
lemma less_4_cases:
"(x::word32) < 4 \<Longrightarrow> x=0 \<or> x=1 \<or> x=2 \<or> x=3"
apply clarsimp
apply (drule word_less_cases, erule disjE, simp, simp)+
done
lemma if_n_0_0:
"((if P then n else 0) \<noteq> 0) = (P \<and> n \<noteq> 0)"
by (simp split: split_if)
lemma insert_dom:
assumes fx: "f x = Some y"
shows "insert x (dom f) = dom f"
unfolding dom_def using fx by auto
lemma map_comp_subset_dom:
"dom (prj \<circ>\<^sub>m f) \<subseteq> dom f"
unfolding dom_def
by (auto simp: map_comp_Some_iff)
lemmas map_comp_subset_domD = subsetD [OF map_comp_subset_dom]
lemma dom_map_comp:
"x \<in> dom (prj \<circ>\<^sub>m f) = (\<exists>y z. f x = Some y \<and> prj y = Some z)"
by (fastforce simp: dom_def map_comp_Some_iff)
lemma option_map_Some_eq2:
"(Some y = option_map f x) = (\<exists>z. x = Some z \<and> f z = y)"
by (metis map_option_eq_Some)
lemma option_map_eq_dom_eq:
assumes ome: "option_map f \<circ> g = option_map f \<circ> g'"
shows "dom g = dom g'"
proof (rule set_eqI)
fix x
{
assume "x \<in> dom g"
hence "Some (f (the (g x))) = (option_map f \<circ> g) x"
by (auto simp: map_option_case split: option.splits)
also have "\<dots> = (option_map f \<circ> g') x" by (simp add: ome)
finally have "x \<in> dom g'"
by (auto simp: map_option_case split: option.splits)
} moreover
{
assume "x \<in> dom g'"
hence "Some (f (the (g' x))) = (option_map f \<circ> g') x"
by (auto simp: map_option_case split: option.splits)
also have "\<dots> = (option_map f \<circ> g) x" by (simp add: ome)
finally have "x \<in> dom g"
by (auto simp: map_option_case split: option.splits)
} ultimately show "(x \<in> dom g) = (x \<in> dom g')" by auto
qed
lemma map_comp_eqI:
assumes dm: "dom g = dom g'"
and fg: "\<And>x. x \<in> dom g' \<Longrightarrow> f (the (g' x)) = f (the (g x))"
shows "f \<circ>\<^sub>m g = f \<circ>\<^sub>m g'"
apply (rule ext)
apply (case_tac "x \<in> dom g")
apply (frule subst [OF dm])
apply (clarsimp split: option.splits)
apply (frule domI [where m = g'])
apply (drule fg)
apply simp
apply (frule subst [OF dm])
apply clarsimp
apply (drule not_sym)
apply (clarsimp simp: map_comp_Some_iff)
done
lemma is_aligned_0:
"is_aligned 0 n"
unfolding is_aligned_def
by simp
lemma compD:
"\<lbrakk>f \<circ> g = f \<circ> g'; g x = v \<rbrakk> \<Longrightarrow> f (g' x) = f v"
apply clarsimp
apply (subgoal_tac "(f (g x)) = (f \<circ> g) x")
apply simp
apply (simp (no_asm))
done
lemma option_map_comp_eqE:
assumes om: "option_map f \<circ> mp = option_map f \<circ> mp'"
and p1: "\<lbrakk> mp x = None; mp' x = None \<rbrakk> \<Longrightarrow> P"
and p2: "\<And>v v'. \<lbrakk> mp x = Some v; mp' x = Some v'; f v = f v' \<rbrakk> \<Longrightarrow> P"
shows "P"
proof (cases "mp x")
case None
hence "x \<notin> dom mp" by (simp add: domIff)
hence "mp' x = None" by (simp add: option_map_eq_dom_eq [OF om] domIff)
with None show ?thesis by (rule p1)
next
case (Some v)
hence "x \<in> dom mp" by clarsimp
then obtain v' where Some': "mp' x = Some v'" by (clarsimp simp add: option_map_eq_dom_eq [OF om])
with Some show ?thesis
proof (rule p2)
show "f v = f v'" using Some' compD [OF om, OF Some] by simp
qed
qed
lemma Some_the:
"x \<in> dom f \<Longrightarrow> f x = Some (the (f x))"
by clarsimp
lemma map_comp_update:
"f \<circ>\<^sub>m (g(x \<mapsto> v)) = (f \<circ>\<^sub>m g)(x := f v)"
apply (rule ext)
apply clarsimp
apply (case_tac "g xa")
apply simp
apply simp
done
lemma restrict_map_eqI:
assumes req: "A |` S = B |` S"
and mem: "x \<in> S"
shows "A x = B x"
proof -
from mem have "A x = (A |` S) x" by simp
also have "\<dots> = (B |` S) x" using req by simp
also have "\<dots> = B x" using mem by simp
finally show ?thesis .
qed
lemma word_or_zero:
"(a || b = 0) = (a = 0 \<and> b = 0)"
apply (safe, simp_all)
apply (rule word_eqI, drule_tac x=n in word_eqD, simp)+
done
lemma aligned_shiftr_mask_shiftl:
"is_aligned x n \<Longrightarrow> ((x >> n) && mask v) << n = x && mask (v + n)"
apply (rule word_eqI)
apply (simp add: word_size nth_shiftl nth_shiftr)
apply (subgoal_tac "\<forall>m. x !! m \<longrightarrow> m \<ge> n")
apply auto[1]
apply (clarsimp simp: is_aligned_mask)
apply (drule_tac x=m in word_eqD)
apply (frule test_bit_size)
apply (simp add: word_size)
done
lemma word_and_1_shiftl:
fixes x :: "('a :: len) word" shows
"x && (1 << n) = (if x !! n then (1 << n) else 0)"
apply (rule word_eqI)
apply (simp add: word_size nth_shiftl word_nth_1 split: split_if
del: shiftl_t2n shiftl_1)
apply auto
done
lemmas word_and_1_shiftls
= word_and_1_shiftl[where n=0, simplified]
word_and_1_shiftl[where n=1, simplified]
word_and_1_shiftl[where n=2, simplified]
lemma word_and_mask_shiftl:
"x && (mask n << m) = ((x >> m) && mask n) << m"
apply (rule word_eqI)
apply (simp add: word_size nth_shiftl nth_shiftr)
apply auto
done
lemma toEnum_eq_to_fromEnum_eq:
fixes v :: "'a :: enum" shows
"n \<le> fromEnum (maxBound :: 'a) \<Longrightarrow> (toEnum n = v) = (n = fromEnum v)"
apply (rule iffI)
apply (drule arg_cong[where f=fromEnum])
apply simp
apply (drule arg_cong[where f="toEnum :: nat \<Rightarrow> 'a"])
apply simp
done
lemma if_Const_helper:
"If P (Con x) (Con y) = Con (If P x y)"
by (simp split: split_if)
lemmas if_Some_helper = if_Const_helper[where Con=Some]
lemma expand_restrict_map_eq:
"(m |` S = m' |` S) = (\<forall>x. x \<in> S \<longrightarrow> m x = m' x)"
by (simp add: fun_eq_iff restrict_map_def split: split_if)
lemma unat_ucast_8_32:
fixes x :: "word8"
shows "unat (ucast x :: word32) = unat x"
unfolding ucast_def unat_def
apply (subst int_word_uint)
apply (subst mod_pos_pos_trivial)
apply simp
apply (rule lt2p_lem)
apply simp
apply simp
done
lemma disj_imp_rhs:
"(P \<Longrightarrow> Q) \<Longrightarrow> (P \<or> Q) = Q"
by blast
lemma remove1_filter:
"distinct xs \<Longrightarrow> remove1 x xs = filter (\<lambda>y. x \<noteq> y) xs"
apply (induct xs)
apply simp
apply clarsimp
apply (rule sym, rule filter_True)
apply clarsimp
done
lemma if_then_1_else_0:
"((if P then 1 else 0) = (0 :: word32)) = (\<not> P)"
by simp
lemma if_then_0_else_1:
"((if P then 0 else 1) = (0 :: word32)) = (P)"
by simp
lemmas if_then_simps = if_then_0_else_1 if_then_1_else_0
lemma nat_less_cases':
"(x::nat) < y \<Longrightarrow> x = y - 1 \<or> x < y - 1"
by (fastforce intro: nat_less_cases)
lemma word32_FF_is_mask:
"0xFF = mask 8 "
by (simp add: mask_def)
lemma filter_to_shorter_upto:
"n \<le> m \<Longrightarrow> filter (\<lambda>x. x < n) [0 ..< m] = [0 ..< n]"
apply (induct m)
apply simp
apply clarsimp
apply (erule le_SucE)
apply simp
apply simp
done
lemma in_emptyE: "\<lbrakk> A = {}; x \<in> A \<rbrakk> \<Longrightarrow> P" by blast
lemma ucast_of_nat_small:
"x < 2 ^ len_of TYPE('a) \<Longrightarrow>
ucast (of_nat x :: ('a :: len) word) = (of_nat x :: ('b :: len) word)"
apply (rule sym, subst word_unat.inverse_norm)
apply (simp add: ucast_def word_of_int[symmetric]
of_nat_nat[symmetric] unat_def[symmetric])
apply (simp add: unat_of_nat)
done
lemma word_le_make_less:
fixes x :: "('a :: len) word"
shows "y \<noteq> -1 \<Longrightarrow> (x \<le> y) = (x < (y + 1))"
apply safe
apply (erule plus_one_helper2)
apply (simp add: eq_diff_eq[symmetric])
done
lemma Ball_emptyI:
"S = {} \<Longrightarrow> (\<forall>x \<in> S. P x)"
by simp
lemma allfEI:
"\<lbrakk> \<forall>x. P x; \<And>x. P (f x) \<Longrightarrow> Q x \<rbrakk> \<Longrightarrow> \<forall>x. Q x"
by fastforce
lemma arith_is_1:
"\<lbrakk> x \<le> Suc 0; x > 0 \<rbrakk> \<Longrightarrow> x = 1"
by arith
(* sjw: combining lemmas here :( *)
lemma cart_singleton_empty2:
"({x} \<times> S = {}) = (S = {})"
"({} = S \<times> {e}) = (S = {})"
by auto
lemma cases_simp_conj:
"((P \<longrightarrow> Q) \<and> (\<not> P \<longrightarrow> Q) \<and> R) = (Q \<and> R)"
by fastforce
lemma domE :
"\<lbrakk> x \<in> dom m; \<And>r. \<lbrakk>m x = Some r\<rbrakk> \<Longrightarrow> P \<rbrakk> \<Longrightarrow> P"
by clarsimp
lemma dom_eqD:
"\<lbrakk> f x = Some v; dom f = S \<rbrakk> \<Longrightarrow> x \<in> S"
by clarsimp
lemma exception_set_finite:
"finite {x. P x} \<Longrightarrow> finite {x. (x = y \<longrightarrow> Q x) \<and> P x}"
"finite {x. P x} \<Longrightarrow> finite {x. x \<noteq> y \<longrightarrow> P x}"
apply (simp add: Collect_conj_eq)
apply (subst imp_conv_disj, subst Collect_disj_eq)
apply simp
done
lemma exfEI:
"\<lbrakk> \<exists>x. P x; \<And>x. P x \<Longrightarrow> Q (f x) \<rbrakk> \<Longrightarrow> \<exists>x. Q x"
by fastforce
lemma finite_word: "finite (S :: (('a :: len) word) set)"
by (rule finite)
lemma if_f:
"(if a then f b else f c) = f (if a then b else c)"
by simp
lemma in_16_range:
"0 \<in> S \<Longrightarrow> r \<in> (\<lambda>x. r + x * (16 :: word32)) ` S"
"n - 1 \<in> S \<Longrightarrow> (r + (16 * n - 16)) \<in> (\<lambda>x :: word32. r + x * 16) ` S"
by (clarsimp simp: image_def
elim!: bexI[rotated])+
definition
"modify_map m p f \<equiv> m (p := option_map f (m p))"
lemma modify_map_id:
"modify_map m p id = m"
by (auto simp add: modify_map_def map_option_case split: option.splits)
lemma modify_map_addr_com:
assumes com: "x \<noteq> y"
shows "modify_map (modify_map m x g) y f = modify_map (modify_map m y f) x g"
by (rule ext)
(simp add: modify_map_def option_map_def com split: option.splits)
lemma modify_map_dom :
"dom (modify_map m p f) = dom m"
unfolding modify_map_def
apply (cases "m p")
apply simp
apply (simp add: dom_def)
apply simp
apply (rule insert_absorb)
apply (simp add: dom_def)
done
lemma modify_map_None:
"m x = None \<Longrightarrow> modify_map m x f = m"
by (rule ext) (simp add: modify_map_def)
lemma modify_map_ndom :
"x \<notin> dom m \<Longrightarrow> modify_map m x f = m"
by (rule modify_map_None) clarsimp
lemma modify_map_app:
"(modify_map m p f) q = (if p = q then option_map f (m p) else m q)"
unfolding modify_map_def by simp
lemma modify_map_apply:
"m p = Some x \<Longrightarrow> modify_map m p f = m (p \<mapsto> f x)"
by (simp add: modify_map_def)
lemma modify_map_com:
assumes com: "\<And>x. f (g x) = g (f x)"
shows "modify_map (modify_map m x g) y f = modify_map (modify_map m y f) x g"
using assms by (auto simp: modify_map_def map_option_case split: option.splits)
lemma modify_map_comp:
"modify_map m x (f o g) = modify_map (modify_map m x g) x f"
by (rule ext) (simp add: modify_map_def option.map_comp)
lemma modify_map_exists_eq:
"(\<exists>cte. modify_map m p' f p= Some cte) = (\<exists>cte. m p = Some cte)"
by (auto simp: modify_map_def split: if_splits)
lemma modify_map_other:
"p \<noteq> q \<Longrightarrow> (modify_map m p f) q = (m q)"
by (simp add: modify_map_app)
lemma modify_map_same:
"(modify_map m p f) p = (option_map f (m p))"
by (simp add: modify_map_app)
lemma next_update_is_modify:
"\<lbrakk> m p = Some cte'; cte = f cte' \<rbrakk> \<Longrightarrow> (m(p \<mapsto> cte)) = (modify_map m p f)"
unfolding modify_map_def by simp
lemma nat_power_minus_less:
"a < 2 ^ (x - n) \<Longrightarrow> (a :: nat) < 2 ^ x"
apply (erule order_less_le_trans)
apply simp
done
lemma neg_rtranclI:
"\<lbrakk> x \<noteq> y; (x, y) \<notin> R\<^sup>+ \<rbrakk> \<Longrightarrow> (x, y) \<notin> R\<^sup>*"
apply (erule contrapos_nn)
apply (drule rtranclD)
apply simp
done
lemma neg_rtrancl_into_trancl:
"\<not> (x, y) \<in> R\<^sup>* \<Longrightarrow> \<not> (x, y) \<in> R\<^sup>+"
by (erule contrapos_nn, erule trancl_into_rtrancl)
lemma set_neqI:
"\<lbrakk> x \<in> S; x \<notin> S' \<rbrakk> \<Longrightarrow> S \<noteq> S'"
by clarsimp
lemma set_pair_UN:
"{x. P x} = UNION {xa. \<exists>xb. P (xa, xb)} (\<lambda>xa. {xa} \<times> {xb. P (xa, xb)})"
apply safe
apply (rule_tac a=a in UN_I)
apply blast+
done
lemma singleton_elemD:
"S = {x} \<Longrightarrow> x \<in> S"
by simp
lemma word_to_1_set:
"{0 ..< (1 :: ('a :: len) word)} = {0}"
by fastforce
lemma ball_ran_eq:
"(\<forall>y \<in> ran m. P y) = (\<forall>x y. m x = Some y \<longrightarrow> P y)"
by (auto simp add: ran_def)
lemma cart_helper:
"({} = {x} \<times> S) = (S = {})"
by blast
lemmas converse_trancl_induct' = converse_trancl_induct [consumes 1, case_names base step]
lemma disjCI2: "(\<not> P \<Longrightarrow> Q) \<Longrightarrow> P \<or> Q" by blast
lemma insert_UNIV :
"insert x UNIV = UNIV"
by blast
lemma not_singletonE:
"\<lbrakk> \<forall>p. S \<noteq> {p}; S \<noteq> {}; \<And>p p'. \<lbrakk> p \<noteq> p'; p \<in> S; p' \<in> S \<rbrakk> \<Longrightarrow> R \<rbrakk> \<Longrightarrow> R"
by blast
lemma not_singleton_oneE:
"\<lbrakk> \<forall>p. S \<noteq> {p}; p \<in> S; \<And>p'. \<lbrakk> p \<noteq> p'; p' \<in> S \<rbrakk> \<Longrightarrow> R \<rbrakk> \<Longrightarrow> R"
apply (erule not_singletonE)
apply clarsimp
apply (case_tac "p = p'")
apply fastforce
apply fastforce
done
lemma interval_empty:
"({m..n} = {}) = (\<not> m \<le> (n::'a::order))"
apply (rule iffI)
apply clarsimp
apply auto
done
lemma range_subset_eq2:
"{a :: word32 .. b} \<noteq> {} \<Longrightarrow> ({a .. b} \<subseteq> {c .. d}) = (c \<le> a \<and> b \<le> d)"
by (simp add: interval_empty)
lemma singleton_eqD: "A = {x} \<Longrightarrow> x \<in> A" by blast
lemma ball_ran_fun_updI:
"\<lbrakk> \<forall>v \<in> ran m. P v; \<forall>v. y = Some v \<longrightarrow> P v \<rbrakk>
\<Longrightarrow> \<forall>v \<in> ran (m (x := y)). P v"
by (auto simp add: ran_def)
lemma ball_ran_modify_map_eq:
"\<lbrakk> \<forall>v. m x = Some v \<longrightarrow> P (f v) = P v \<rbrakk>
\<Longrightarrow> (\<forall>v \<in> ran (modify_map m x f). P v) = (\<forall>v \<in> ran m. P v)"
apply (simp add: ball_ran_eq)
apply (rule iff_allI)
apply (auto simp: modify_map_def)
done
lemma disj_imp: "(P \<or> Q) = (\<not>P \<longrightarrow> Q)" by blast
lemma eq_singleton_redux:
"\<lbrakk> S = {x} \<rbrakk> \<Longrightarrow> x \<in> S"
by simp
lemma if_eq_elem_helperE:
"\<lbrakk> x \<in> (if P then S else S');
\<lbrakk> P; x \<in> S \<rbrakk> \<Longrightarrow> a = b;
\<lbrakk> \<not> P; x \<in> S' \<rbrakk> \<Longrightarrow> a = c
\<rbrakk> \<Longrightarrow> a = (if P then b else c)"
by fastforce
lemma if_option_Some :
"((if P then None else Some x) = Some y) = (\<not>P \<and> x = y)"
by simp
lemma insert_minus_eq:
"x \<notin> A \<Longrightarrow> A - S = (A - (S - {x}))"
by auto
lemma map2_Cons_2_3:
"(map2 f xs (y # ys) = (z # zs)) = (\<exists>x xs'. xs = x # xs' \<and> f x y = z \<and> map2 f xs' ys = zs)"
by (case_tac xs, simp_all)
lemma map2_xor_replicate_False:
"map2 (\<lambda>(x\<Colon>bool) y\<Colon>bool. x = (\<not> y)) xs (replicate n False) = take n xs"
apply (induct xs arbitrary: n)
apply simp
apply (case_tac n)
apply (simp add: map2_def)
apply simp
done
lemma modify_map_K_D:
"modify_map m p (\<lambda>x. y) p' = Some v \<Longrightarrow> (m (p \<mapsto> y)) p' = Some v"
by (simp add: modify_map_def split: split_if_asm)
lemmas tranclE2' = tranclE2 [consumes 1, case_names base trancl]
lemma weak_imp_cong:
"\<lbrakk> P = R; Q = S \<rbrakk> \<Longrightarrow> (P \<longrightarrow> Q) = (R \<longrightarrow> S)"
by simp
lemma Collect_Diff_restrict_simp:
"T - {x \<in> T. Q x} = T - {x. Q x}"
by (auto intro: Collect_cong)
lemma Collect_Int_pred_eq:
"{x \<in> S. P x} \<inter> {x \<in> T. P x} = {x \<in> (S \<inter> T). P x}"
by (simp add: Collect_conj_eq [symmetric] conj_ac)
lemma Collect_restrict_predR:
"{x. P x} \<inter> T = {} \<Longrightarrow> {x. P x} \<inter> {x \<in> T. Q x} = {}"
apply (subst Collect_conj_eq [symmetric])
apply (simp add: disjoint_iff_not_equal)
apply rule
apply (drule_tac x = x in spec)
apply clarsimp
apply (drule (1) bspec)
apply simp
done
lemma Diff_Un2:
assumes emptyad: "A \<inter> D = {}"
and emptybc: "B \<inter> C = {}"
shows "(A \<union> B) - (C \<union> D) = (A - C) \<union> (B - D)"
proof -
have "(A \<union> B) - (C \<union> D) = (A \<union> B - C) \<inter> (A \<union> B - D)"
by (rule Diff_Un)
also have "\<dots> = ((A - C) \<union> B) \<inter> (A \<union> (B - D))" using emptyad emptybc
by (simp add: Un_Diff Diff_triv)
also have "\<dots> = (A - C) \<union> (B - D)"
proof -
have "(A - C) \<inter> (A \<union> (B - D)) = A - C" using emptyad emptybc
by (metis Diff_Int2 Diff_Int_distrib2 inf_sup_absorb)
moreover
have "B \<inter> (A \<union> (B - D)) = B - D" using emptyad emptybc
by (metis Int_Diff Un_Diff Un_Diff_Int Un_commute Un_empty_left inf_sup_absorb)
ultimately show ?thesis
by (simp add: Int_Un_distrib2)
qed
finally show ?thesis .
qed
lemma ballEI:
"\<lbrakk> \<forall>x \<in> S. Q x; \<And>x. \<lbrakk> x \<in> S; Q x \<rbrakk> \<Longrightarrow> P x \<rbrakk> \<Longrightarrow> \<forall>x \<in> S. P x"
by auto
lemma dom_if_None:
"dom (\<lambda>x. if P x then None else f x)
= dom f - {x. P x}"
by (simp add: dom_def, fastforce)
lemma notemptyI:
"x \<in> S \<Longrightarrow> S \<noteq> {}"
by clarsimp
lemma plus_Collect_helper:
"op + x ` {xa. P (xa :: ('a :: len) word)} = {xa. P (xa - x)}"
by (fastforce simp add: image_def)
lemma plus_Collect_helper2:
"op + (- x) ` {xa. P (xa :: ('a :: len) word)} = {xa. P (x + xa)}"
by (simp add: field_simps plus_Collect_helper)
lemma restrict_map_Some_iff:
"((m |` S) x = Some y) = (m x = Some y \<and> x \<in> S)"
by (cases "x \<in> S", simp_all)
lemma context_case_bools:
"\<lbrakk> \<And>v. P v \<Longrightarrow> R v; \<lbrakk> \<not> P v; \<And>v. P v \<Longrightarrow> R v \<rbrakk> \<Longrightarrow> R v \<rbrakk> \<Longrightarrow> R v"
by (cases "P v", simp_all)
lemma inj_on_fun_upd_strongerI:
"\<lbrakk>inj_on f A; y \<notin> f ` (A - {x})\<rbrakk> \<Longrightarrow> inj_on (f(x := y)) A"
apply (simp add: inj_on_def)
apply blast
done
lemma less_handy_casesE:
"\<lbrakk> m < n; m = 0 \<Longrightarrow> R;
\<And>m' n'. \<lbrakk> n = Suc n'; m = Suc m'; m < n \<rbrakk> \<Longrightarrow> R \<rbrakk>
\<Longrightarrow> R"
apply (case_tac n, simp_all)
apply (case_tac m, simp_all)
done
lemma subset_drop_Diff_strg:
"(A \<subseteq> C) \<longrightarrow> (A - B \<subseteq> C)"
by blast
lemma word32_count_from_top:
"n \<noteq> 0 \<Longrightarrow> {0 ..< n :: word32} = {0 ..< n - 1} \<union> {n - 1}"
apply (rule set_eqI, rule iffI)
apply simp
apply (drule minus_one_helper3)
apply (rule disjCI)
apply simp
apply simp
apply (erule minus_one_helper5)
apply fastforce
done
lemma Int_Union_empty:
"(\<And>x. x \<in> S \<Longrightarrow> A \<inter> P x = {}) \<Longrightarrow> A \<inter> (\<Union>x \<in> S. P x) = {}"
by auto
lemma UN_Int_empty:
"(\<And>x. x \<in> S \<Longrightarrow> P x \<inter> T = {}) \<Longrightarrow> (\<Union>x \<in> S. P x) \<inter> T = {}"
by auto
lemma disjointI:
"\<lbrakk>\<And>x y. \<lbrakk> x \<in> A; y \<in> B \<rbrakk> \<Longrightarrow> x \<noteq> y \<rbrakk> \<Longrightarrow> A \<inter> B = {}"
by auto
lemma UN_disjointI:
assumes rl: "\<And>x y. \<lbrakk> x \<in> A; y \<in> B \<rbrakk> \<Longrightarrow> P x \<inter> Q y = {}"
shows "(\<Union>x \<in> A. P x) \<inter> (\<Union>x \<in> B. Q x) = {}"
apply (rule disjointI)
apply clarsimp
apply (drule (1) rl)
apply auto
done
lemma UN_set_member:
assumes sub: "A \<subseteq> (\<Union>x \<in> S. P x)"
and nz: "A \<noteq> {}"
shows "\<exists>x \<in> S. P x \<inter> A \<noteq> {}"
proof -
from nz obtain z where zA: "z \<in> A" by fastforce
with sub obtain x where "x \<in> S" and "z \<in> P x" by auto
hence "P x \<inter> A \<noteq> {}" using zA by auto
thus ?thesis using sub nz by auto
qed
lemma append_Cons_cases [consumes 1, case_names pre mid post]:
"\<lbrakk>(x, y) \<in> set (as @ b # bs);
(x, y) \<in> set as \<Longrightarrow> R;
\<lbrakk>(x, y) \<notin> set as; (x, y) \<notin> set bs; (x, y) = b\<rbrakk> \<Longrightarrow> R;
(x, y) \<in> set bs \<Longrightarrow> R\<rbrakk>
\<Longrightarrow> R" by auto
lemma cart_singletons:
"{a} \<times> {b} = {(a, b)}"
by blast
lemma disjoint_subset_neg1:
"\<lbrakk> B \<inter> C = {}; A \<subseteq> B; A \<noteq> {} \<rbrakk> \<Longrightarrow> \<not> A \<subseteq> C"
by auto
lemma disjoint_subset_neg2:
"\<lbrakk> B \<inter> C = {}; A \<subseteq> C; A \<noteq> {} \<rbrakk> \<Longrightarrow> \<not> A \<subseteq> B"
by auto
lemma iffE2:
"\<lbrakk> P = Q; \<lbrakk> P; Q \<rbrakk> \<Longrightarrow> R; \<lbrakk> \<not> P; \<not> Q \<rbrakk> \<Longrightarrow> R \<rbrakk> \<Longrightarrow> R"
by blast
lemma minus_one_helper2:
"\<lbrakk> x - 1 < y \<rbrakk> \<Longrightarrow> x \<le> (y :: ('a :: len) word)"
apply (cases "x = 0")
apply simp
apply (simp add: word_less_nat_alt word_le_nat_alt)
apply (subst(asm) unat_minus_one)
apply (simp add: word_less_nat_alt)
apply (cases "unat x")
apply (simp add: unat_eq_zero)
apply arith
done
lemma mod_mod_power:
fixes k :: nat
shows "k mod 2 ^ m mod 2 ^ n = k mod 2 ^ (min m n)"
proof (cases "m \<le> n")
case True
hence "k mod 2 ^ m mod 2 ^ n = k mod 2 ^ m"
apply -
apply (subst mod_less [where n = "2 ^ n"])
apply (rule order_less_le_trans [OF mod_less_divisor])
apply simp+
done
also have "\<dots> = k mod 2 ^ (min m n)" using True by simp
finally show ?thesis .
next
case False
hence "n < m" by simp
then obtain d where md: "m = n + d"
by (auto dest: less_imp_add_positive)
hence "k mod 2 ^ m = 2 ^ n * (k div 2 ^ n mod 2 ^ d) + k mod 2 ^ n"
by (simp add: mod_mult2_eq power_add)
hence "k mod 2 ^ m mod 2 ^ n = k mod 2 ^ n"
by (simp add: mod_add_left_eq)
thus ?thesis using False
by simp
qed
lemma word_div_less:
fixes m :: "'a :: len word"
shows "m < n \<Longrightarrow> m div n = 0"
apply (rule word_unat.Rep_eqD)
apply (simp add: word_less_nat_alt unat_div)
done
lemma word_must_wrap:
"\<lbrakk> x \<le> n - 1; n \<le> x \<rbrakk> \<Longrightarrow> n = (0 :: ('a :: len) word)"
apply (rule ccontr)
apply (drule(1) order_trans)
apply (drule word_sub_1_le)
apply (drule(1) order_antisym)
apply simp
done
lemma upt_add_eq_append':
assumes a1: "i \<le> j" and a2: "j \<le> k"
shows "[i..<k] = [i..<j] @ [j..<k]"
using a1 a2
by (clarsimp simp: le_iff_add intro!: upt_add_eq_append)
lemma range_subset_card:
"\<lbrakk> {a :: ('a :: len) word .. b} \<subseteq> {c .. d}; b \<ge> a \<rbrakk>
\<Longrightarrow> d \<ge> c \<and> d - c \<ge> b - a"
apply (subgoal_tac "a \<in> {a .. b}")
apply (frule(1) range_subset_lower)
apply (frule(1) range_subset_upper)
apply (rule context_conjI, simp)
apply (rule word_sub_mono, assumption+)
apply (erule word_sub_le)
apply (erule word_sub_le)
apply simp
done
lemma less_1_simp:
"n - 1 < m = (n \<le> (m :: ('a :: len) word) \<and> n \<noteq> 0)"
by unat_arith
lemma alignUp_div_helper:
fixes a :: "'a::len word"
assumes kv: "k < 2 ^ (len_of TYPE('a) - n)"
and xk: "x = 2 ^ n * of_nat k"
and le: "a \<le> x"
and sz: "n < len_of TYPE('a)"
and anz: "a mod 2 ^ n \<noteq> 0"
shows "a div 2 ^ n < of_nat k"
proof -
have kn: "unat (of_nat k :: 'a word) * unat ((2::'a word) ^ n)
< 2 ^ len_of TYPE('a)"
using xk kv sz
apply (subst unat_of_nat_eq)
apply (erule order_less_le_trans)
apply simp
apply (subst unat_power_lower, simp add: word_bits_def)
apply (subst mult.commute)
apply (rule nat_less_power_trans)
apply simp
apply simp
done
have "unat a div 2 ^ n * 2 ^ n \<noteq> unat a"
proof -
have "unat a = unat a div 2 ^ n * 2 ^ n + unat a mod 2 ^ n"
by (simp add: mod_div_equality)
also have "\<dots> \<noteq> unat a div 2 ^ n * 2 ^ n" using sz anz
by (simp add: unat_arith_simps word_bits_def)
finally show ?thesis ..
qed
hence "a div 2 ^ n * 2 ^ n < a" using sz anz
apply (subst word_less_nat_alt)
apply (subst unat_word_ariths)
apply (subst unat_div)
apply simp
apply (rule order_le_less_trans [OF mod_le_dividend])
apply (erule order_le_neq_trans [OF div_mult_le])
done
also from xk le have "\<dots> \<le> of_nat k * 2 ^ n" by (simp add: field_simps)
finally show ?thesis using sz kv
apply -
apply (erule word_mult_less_dest [OF _ _ kn])
apply (simp add: unat_div)
apply (rule order_le_less_trans [OF div_mult_le])
apply (rule unat_lt2p)
done
qed
lemma nat_mod_power_lem:
fixes a :: nat
shows "1 < a \<Longrightarrow> a ^ n mod a ^ m = (if m \<le> n then 0 else a ^ n)"
by (clarsimp, clarsimp simp add: le_iff_add power_add)
lemma power_mod_div:
fixes x :: "nat"
shows "x mod 2 ^ n div 2 ^ m = x div 2 ^ m mod 2 ^ (n - m)" (is "?LHS = ?RHS")
proof (cases "n \<le> m")
case True
hence "?LHS = 0"
apply -
apply (rule div_less)
apply (rule order_less_le_trans [OF mod_less_divisor])
apply simp
apply simp
done
also have "\<dots> = ?RHS" using True
by simp
finally show ?thesis .
next
case False
hence lt: "m < n" by simp
then obtain q where nv: "n = m + q" and "0 < q"
by (auto dest: less_imp_Suc_add)
hence "x mod 2 ^ n = 2 ^ m * (x div 2 ^ m mod 2 ^ q) + x mod 2 ^ m"
by (simp add: power_add mod_mult2_eq)
hence "?LHS = x div 2 ^ m mod 2 ^ q"
by (simp add: div_add1_eq)
also have "\<dots> = ?RHS" using nv
by simp
finally show ?thesis .
qed
lemma word_power_mod_div:
fixes x :: "'a::len word"
shows "\<lbrakk> n < len_of TYPE('a); m < len_of TYPE('a)\<rbrakk>
\<Longrightarrow> x mod 2 ^ n div 2 ^ m = x div 2 ^ m mod 2 ^ (n - m)"
apply (simp add: word_arith_nat_div unat_mod power_mod_div)
apply (subst unat_arith_simps(3))
apply (subst unat_mod)
apply (subst unat_of_nat)+
apply (simp add: mod_mod_power min.commute)
done
(* FIXME: stronger version of GenericLib.p_assoc_help *)
lemma x_power_minus_1:
fixes x :: "'a :: {ab_group_add, power, numeral, one}"
shows "x + (2::'a) ^ n - (1::'a) = x + (2 ^ n - 1)" by simp
lemma nat_le_power_trans:
fixes n :: nat
shows "\<lbrakk>n \<le> 2 ^ (m - k); k \<le> m\<rbrakk> \<Longrightarrow> 2 ^ k * n \<le> 2 ^ m"
apply (drule order_le_imp_less_or_eq)
apply (erule disjE)
apply (drule (1) nat_less_power_trans)
apply (erule order_less_imp_le)
apply (simp add: power_add [symmetric])
done
lemma nat_diff_add:
fixes i :: nat
shows "\<lbrakk> i + j = k \<rbrakk> \<Longrightarrow> i = k - j"
by arith
lemma word_range_minus_1':
fixes a :: "'a :: len word"
shows "a \<noteq> 0 \<Longrightarrow> {a - 1<..b} = {a..b}"
by (simp add: greaterThanAtMost_def atLeastAtMost_def greaterThan_def atLeast_def less_1_simp)
lemma word_range_minus_1:
fixes a :: word32
shows "b \<noteq> 0 \<Longrightarrow> {a..b - 1} = {a..<b}"
apply (simp add: atLeastLessThan_def atLeastAtMost_def atMost_def lessThan_def)
apply (rule arg_cong [where f = "\<lambda>x. {a..} \<inter> x"])
apply rule
apply clarsimp
apply (erule contrapos_pp)
apply (simp add: linorder_not_less linorder_not_le word_must_wrap)
apply (clarsimp)
apply (drule minus_one_helper3)
apply (auto simp: word_less_sub_1)
done
lemma ucast_nat_def:
"of_nat (unat x) = (ucast :: ('a :: len) word \<Rightarrow> ('b :: len) word) x"
by (simp add: ucast_def word_of_int_nat unat_def)
lemma delete_remove1 :
"delete x xs = remove1 x xs"
by (induct xs, auto)
lemma list_case_If:
"(case xs of [] \<Rightarrow> P | _ \<Rightarrow> Q)
= (if xs = [] then P else Q)"
by (clarsimp simp: neq_Nil_conv)
lemma remove1_Nil_in_set:
"\<lbrakk> remove1 x xs = []; xs \<noteq> [] \<rbrakk> \<Longrightarrow> x \<in> set xs"
by (induct xs) (auto split: split_if_asm)
lemma remove1_empty:
"(remove1 v xs = []) = (xs = [v] \<or> xs = [])"
by (cases xs, simp_all)
lemma set_remove1:
"x \<in> set (remove1 y xs) \<Longrightarrow> x \<in> set xs"
apply (induct xs)
apply simp
apply (case_tac "y = a")
apply clarsimp+
done
lemma If_rearrage:
"(if P then if Q then x else y else z)
= (if P \<and> Q then x else if P then y else z)"
by simp
lemma cases_simp_left:
"((P \<longrightarrow> Q) \<and> (\<not> P \<longrightarrow> Q) \<and> R) = (Q \<and> R)"
by fastforce
lemma disjI2_strg:
"Q \<longrightarrow> (P \<or> Q)"
by simp
lemma eq_2_32_0:
"(2 ^ 32 :: word32) = 0"
by simp
lemma eq_imp_strg:
"P t \<longrightarrow> (t = s \<longrightarrow> P s)"
by clarsimp
lemma if_fun_split:
"(if P then \<lambda>s. Q s else (\<lambda>s. R s)) = (\<lambda>s. (P \<longrightarrow> Q s) \<and> (\<not>P \<longrightarrow> R s))"
by simp
lemma i_hate_words_helper:
"i \<le> (j - k :: nat) \<Longrightarrow> i \<le> j"
by simp
lemma i_hate_words:
"unat (a :: 'a word) \<le> unat (b :: ('a :: len) word) - Suc 0
\<Longrightarrow> a \<noteq> -1"
apply (frule i_hate_words_helper)
apply (subst(asm) word_le_nat_alt[symmetric])
apply (clarsimp simp only: word_minus_one_le)
apply (simp only: linorder_not_less[symmetric])
apply (erule notE)
apply (rule diff_Suc_less)
apply (subst neq0_conv[symmetric])
apply (subst unat_eq_0)
apply (rule notI, drule arg_cong[where f="op + 1"])
apply simp
done
lemma if_both_strengthen:
"P \<and> Q \<longrightarrow> (if G then P else Q)"
by simp
lemma if_both_strengthen2:
"P s \<and> Q s \<longrightarrow> (if G then P else Q) s"
by simp
lemma if_swap:
"(if P then Q else R) = (if \<not>P then R else Q)" by simp
lemma ignore_if:
"(y and z) s \<Longrightarrow> (if x then y else z) s"
by (clarsimp simp: pred_conj_def)
lemma imp_consequent:
"P \<longrightarrow> Q \<longrightarrow> P" by simp
lemma list_case_helper:
"xs \<noteq> [] \<Longrightarrow> case_list f g xs = g (hd xs) (tl xs)"
by (cases xs, simp_all)
lemma list_cons_rewrite:
"(\<forall>x xs. L = x # xs \<longrightarrow> P x xs) = (L \<noteq> [] \<longrightarrow> P (hd L) (tl L))"
by (auto simp: neq_Nil_conv)
lemma list_not_Nil_manip:
"\<lbrakk> xs = y # ys; case xs of [] \<Rightarrow> False | (y # ys) \<Rightarrow> P y ys \<rbrakk> \<Longrightarrow> P y ys"
by simp
lemma ran_ball_triv:
"\<And>P m S. \<lbrakk> \<forall>x \<in> (ran S). P x ; m \<in> (ran S) \<rbrakk> \<Longrightarrow> P m"
by blast
lemma singleton_tuple_cartesian:
"({(a, b)} = S \<times> T) = ({a} = S \<and> {b} = T)"
"(S \<times> T = {(a, b)}) = ({a} = S \<and> {b} = T)"
by blast+
lemma strengthen_ignore_if:
"A s \<and> B s \<longrightarrow> (if P then A else B) s"
by clarsimp
lemma sum_case_True :
"(case r of Inl a \<Rightarrow> True | Inr b \<Rightarrow> f b)
= (\<forall>b. r = Inr b \<longrightarrow> f b)"
by (cases r) auto
lemma sym_ex_elim:
"F x = y \<Longrightarrow> \<exists>x. y = F x"
by auto
lemma tl_drop_1 :
"tl xs = drop 1 xs"
by (simp add: drop_Suc)
lemma upt_lhs_sub_map:
"[x ..< y] = map (op + x) [0 ..< y - x]"
apply (induct y)
apply simp
apply (clarsimp simp: Suc_diff_le)
done
lemma upto_0_to_4:
"[0..<4] = 0 # [1..<4]"
apply (subst upt_rec)
apply simp
done
lemma disjEI:
"\<lbrakk> P \<or> Q; P \<Longrightarrow> R; Q \<Longrightarrow> S \<rbrakk>
\<Longrightarrow> R \<or> S"
by fastforce
lemma dom_fun_upd2:
"s x = Some z \<Longrightarrow> dom (s (x \<mapsto> y)) = dom s"
by (simp add: insert_absorb domI)
lemma foldl_True :
"foldl op \<or> True bs"
by (induct bs) auto
lemma image_set_comp:
"f ` {g x | x. Q x} = (f \<circ> g) ` {x. Q x}"
by fastforce
lemma mutual_exE:
"\<lbrakk> \<exists>x. P x; \<And>x. P x \<Longrightarrow> Q x \<rbrakk> \<Longrightarrow> \<exists>x. Q x"
apply clarsimp
apply blast
done
lemma nat_diff_eq:
fixes x :: nat
shows "\<lbrakk> x - y = x - z; y < x\<rbrakk> \<Longrightarrow> y = z"
by arith
lemma overflow_plus_one_self:
"(1 + p \<le> p) = (p = (-1 :: word32))"
apply (safe, simp_all)
apply (rule ccontr)
apply (drule plus_one_helper2)
apply (rule notI)
apply (drule arg_cong[where f="\<lambda>x. x - 1"])
apply simp
apply (simp add: field_simps)
done
lemma plus_1_less:
"(x + 1 \<le> (x :: ('a :: len) word)) = (x = -1)"
apply (rule iffI)
apply (rule ccontr)
apply (cut_tac plus_one_helper2[where x=x, OF order_refl])
apply simp
apply clarsimp
apply (drule arg_cong[where f="\<lambda>x. x - 1"])
apply simp
apply simp
done
lemma pos_mult_pos_ge:
"[|x > (0::int); n>=0 |] ==> n * x >= n*1"
apply (simp only: mult_left_mono)
done
lemma If_eq_obvious:
"x \<noteq> z \<Longrightarrow> ((if P then x else y) = z) = (\<not> P \<and> y = z)"
by simp
lemma Some_to_the:
"v = Some x \<Longrightarrow> x = the v"
by simp
lemma dom_if_Some:
"dom (\<lambda>x. if P x then Some (f x) else g x) = {x. P x} \<union> dom g"
by fastforce
lemma dom_insert_absorb:
"x \<in> dom f \<Longrightarrow> insert x (dom f) = dom f" by auto
lemma emptyE2:
"\<lbrakk> S = {}; x \<in> S \<rbrakk> \<Longrightarrow> P"
by simp
lemma mod_div_equality_div_eq:
"a div b * b = (a - (a mod b) :: int)"
by (simp add: field_simps)
lemma zmod_helper:
"n mod m = k \<Longrightarrow> ((n :: int) + a) mod m = (k + a) mod m"
by (clarsimp simp: pull_mods)
lemma int_div_sub_1:
"\<lbrakk> m \<ge> 1 \<rbrakk> \<Longrightarrow> (n - (1 :: int)) div m = (if m dvd n then (n div m) - 1 else n div m)"
apply (subgoal_tac "m = 0 \<or> (n - (1 :: int)) div m = (if m dvd n then (n div m) - 1 else n div m)")
apply fastforce
apply (subst mult_cancel_right[symmetric])
apply (simp only: left_diff_distrib split: split_if)
apply (simp only: mod_div_equality_div_eq)
apply (clarsimp simp: field_simps dvd_mult_div_cancel)
apply (clarsimp simp: dvd_eq_mod_eq_0)
apply (cases "m = 1")
apply simp
apply (subst mod_diff_eq, simp add: zmod_minus1 mod_pos_pos_trivial)
apply clarsimp
apply (subst diff_add_cancel[where b=1, symmetric])
apply (subst push_mods(1))
apply (simp add: field_simps mod_pos_pos_trivial)
apply (rule mod_pos_pos_trivial)
apply (subst add_0_right[where a=0, symmetric])
apply (rule add_mono)
apply simp
apply simp
apply (cases "(n - 1) mod m = m - 1")
apply (drule zmod_helper[where a=1])
apply simp
apply (subgoal_tac "1 + (n - 1) mod m \<le> m")
apply simp
apply (subst field_simps, rule zless_imp_add1_zle)
apply simp
done
lemmas nat_less_power_trans_16 =
subst [OF mult.commute, where P="\<lambda>x. x < v" for v,
OF nat_less_power_trans[where k=4, simplified]]
lemmas nat_less_power_trans_256 =
subst [OF mult.commute, where P="\<lambda>x. x < v" for v,
OF nat_less_power_trans[where k=8, simplified]]
lemmas nat_less_power_trans_4096 =
subst [OF mult.commute, where P="\<lambda>x. x < v" for v,
OF nat_less_power_trans[where k=12, simplified]]
lemma ptr_add_image_multI:
"\<lbrakk> \<And>x y. (x * val = y * val') = (x * val'' = y); x * val'' \<in> S \<rbrakk> \<Longrightarrow>
ptr_add ptr (x * val) \<in> (\<lambda>p. ptr_add ptr (p * val')) ` S"
apply (simp add: image_def)
apply (erule rev_bexI)
apply (rule arg_cong[where f="ptr_add ptr"])
apply simp
done
lemma shift_times_fold:
"(x :: word32) * (2 ^ n) << m = x << (m + n)"
by (simp add: shiftl_t2n ac_simps power_add)
lemma word_plus_strict_mono_right:
fixes x :: "'a :: len word"
shows "\<lbrakk>y < z; x \<le> x + z\<rbrakk> \<Longrightarrow> x + y < x + z"
by unat_arith
lemma comp_upd_simp:
"(f \<circ> (g (x := y))) = ((f \<circ> g) (x := f y))"
by (rule ext, simp add: o_def)
lemma dom_option_map:
"dom (option_map f o m) = dom m"
by (simp add: dom_def)
lemma drop_imp:
"P \<Longrightarrow> (A \<longrightarrow> P) \<and> (B \<longrightarrow> P)" by blast
lemma inj_on_fun_updI2:
"\<lbrakk> inj_on f A; y \<notin> f ` (A - {x}) \<rbrakk>
\<Longrightarrow> inj_on (f(x := y)) A"
apply (rule inj_onI)
apply (simp split: split_if_asm)
apply (erule notE, rule image_eqI, erule sym)
apply simp
apply (erule(3) inj_onD)
done
lemma inj_on_fun_upd_elsewhere:
"x \<notin> S \<Longrightarrow> inj_on (f (x := y)) S = inj_on f S"
apply (simp add: inj_on_def)
apply blast
done
lemma not_Some_eq_tuple:
"(\<forall>y z. x \<noteq> Some (y, z)) = (x = None)"
by (cases x, simp_all)
lemma ran_option_map:
"ran (option_map f o m) = f ` ran m"
by (auto simp add: ran_def)
lemma All_less_Ball:
"(\<forall>x < n. P x) = (\<forall>x\<in>{..< n}. P x)"
by fastforce
lemma Int_image_empty:
"\<lbrakk> \<And>x y. f x \<noteq> g y \<rbrakk>
\<Longrightarrow> f ` S \<inter> g ` T = {}"
by auto
lemma Max_prop:
"\<lbrakk> Max S \<in> S \<Longrightarrow> P (Max S); (S :: ('a :: {finite, linorder}) set) \<noteq> {} \<rbrakk> \<Longrightarrow> P (Max S)"
apply (erule meta_mp)
apply (rule Max_in)
apply simp
apply assumption
done
lemma Min_prop:
"\<lbrakk> Min S \<in> S \<Longrightarrow> P (Min S); (S :: ('a :: {finite, linorder}) set) \<noteq> {} \<rbrakk> \<Longrightarrow> P (Min S)"
apply (erule meta_mp)
apply (rule Min_in)
apply simp
apply assumption
done
definition
is_inv :: "('a \<rightharpoonup> 'b) \<Rightarrow> ('b \<rightharpoonup> 'a) \<Rightarrow> bool" where
"is_inv f g \<equiv> ran f = dom g \<and> (\<forall>x y. f x = Some y \<longrightarrow> g y = Some x)"
lemma is_inv_NoneD:
assumes "g x = None"
assumes "is_inv f g"
shows "x \<notin> ran f"
proof -
from assms
have "x \<notin> dom g" by (auto simp: ran_def)
moreover
from assms
have "ran f = dom g"
by (simp add: is_inv_def)
ultimately
show ?thesis by simp
qed
lemma is_inv_SomeD:
"\<lbrakk> f x = Some y; is_inv f g \<rbrakk> \<Longrightarrow> g y = Some x"
by (simp add: is_inv_def)
lemma is_inv_com:
"is_inv f g \<Longrightarrow> is_inv g f"
apply (unfold is_inv_def)
apply safe
apply (clarsimp simp: ran_def dom_def set_eq_iff)
apply (erule_tac x=a in allE)
apply clarsimp
apply (clarsimp simp: ran_def dom_def set_eq_iff)
apply blast
apply (clarsimp simp: ran_def dom_def set_eq_iff)
apply (erule_tac x=x in allE)
apply clarsimp
done
lemma is_inv_inj:
"is_inv f g \<Longrightarrow> inj_on f (dom f)"
apply (frule is_inv_com)
apply (clarsimp simp: inj_on_def)
apply (drule (1) is_inv_SomeD)
apply (drule_tac f=f in is_inv_SomeD, assumption)
apply simp
done
lemma is_inv_None_upd:
"\<lbrakk> is_inv f g; g x = Some y\<rbrakk> \<Longrightarrow> is_inv (f(y := None)) (g(x := None))"
apply (subst is_inv_def)
apply (clarsimp simp add: dom_upd)
apply (drule is_inv_SomeD, erule is_inv_com)
apply (frule is_inv_inj)
apply (simp add: ran_upd')
apply (rule conjI)
apply (simp add: is_inv_def)
apply (drule (1) is_inv_SomeD)
apply (clarsimp simp: is_inv_def)
done
lemma is_inv_inj2:
"is_inv f g \<Longrightarrow> inj_on g (dom g)"
apply (drule is_inv_com)
apply (erule is_inv_inj)
done
lemma range_convergence1:
"\<lbrakk> \<forall>z. x < z \<and> z \<le> y \<longrightarrow> P z; \<forall>z > y. P (z :: 'a :: linorder) \<rbrakk>
\<Longrightarrow> \<forall>z > x. P z"
apply clarsimp
apply (case_tac "z \<le> y")
apply simp
apply (simp add: linorder_not_le)
done
lemma range_convergence2:
"\<lbrakk> \<forall>z. x < z \<and> z \<le> y \<longrightarrow> P z; \<forall>z. z > y \<and> z < w \<longrightarrow> P (z :: 'a :: linorder) \<rbrakk>
\<Longrightarrow> \<forall>z. z > x \<and> z < w \<longrightarrow> P z"
apply (cut_tac range_convergence1[where P="\<lambda>z. z < w \<longrightarrow> P z" and x=x and y=y])
apply simp
apply simp
apply simp
done
lemma replicate_minus:
"k < n \<Longrightarrow> replicate n False = replicate (n - k) False @ replicate k False"
by (subst replicate_add [symmetric]) simp
lemmas map_pair_split_imageI
= map_prod_imageI[where f="split f" and g="split g"
and a="(a, b)" and b="(c, d)" for a b c d f g, simplified]
lemma word_div_mult:
fixes c :: word32
shows "\<lbrakk>0 < c; a < b * c \<rbrakk> \<Longrightarrow> a div c < b"
apply (simp add: word_less_nat_alt unat_div)
apply (subst td_gal_lt [symmetric])
apply assumption
apply (erule order_less_le_trans)
apply (subst unat_word_ariths)
apply (unfold word_bits_len_of)
apply (rule mod_le_dividend)
done
lemma word_less_power_trans_ofnat:
"\<lbrakk>n < 2 ^ (m - k); k \<le> m; m < len_of TYPE('a)\<rbrakk>
\<Longrightarrow> of_nat n * 2 ^ k < (2::'a::len word) ^ m"
apply (subst mult.commute)
apply (rule word_less_power_trans)
apply (simp add: word_less_nat_alt)
apply (subst unat_of_nat_eq)
apply (erule order_less_trans)
apply (simp add: unat_power_lower)+
done
lemma upto_enum_step_red':
"\<lbrakk> c < word_bits; b \<le> c; is_aligned a c \<rbrakk> \<Longrightarrow>
[a, a + 2 ^ b .e. a + 2 ^ c - 1] = map (op + a) [0, 2 ^ b .e. 2 ^ c - 1]"
unfolding upto_enum_step_def
by (auto simp: upto_enum_word dest:is_aligned_no_overflow)
lemma div_power_helper:
"\<lbrakk> x \<le> y; y < word_bits \<rbrakk> \<Longrightarrow> (2 ^ y - 1) div (2 ^ x :: word32) = 2 ^ (y - x) - 1"
apply (rule word_uint.Rep_eqD)
apply (simp only: uint_word_ariths uint_div uint_power_lower word_bits_len_of)
apply (subst mod_pos_pos_trivial, fastforce, fastforce)+
apply (subst mod_pos_pos_trivial)
apply (simp add: le_diff_eq uint_2p_alt[where 'a=32, unfolded word_bits_len_of])
apply (rule less_1_helper)
apply (rule power_increasing)
apply (simp add: word_bits_def)
apply simp
apply (subst mod_pos_pos_trivial)
apply (simp add: uint_2p_alt[where 'a=32, unfolded word_bits_len_of])
apply (rule less_1_helper)
apply (rule power_increasing)
apply (simp add: word_bits_def)
apply simp
apply (subst int_div_sub_1)
apply simp
apply (simp add: uint_2p_alt[where 'a=32, unfolded word_bits_len_of])
apply (subst power_0[symmetric, where a=2])
apply (simp add: uint_2p_alt[where 'a=32, unfolded word_bits_len_of]
le_imp_power_dvd_int power_sub_int)
done
lemma n_less_word_bits:
"(n < word_bits) = (n < 32)"
by (simp add: word_bits_def)
lemma of_nat_less_pow:
"\<lbrakk> x < 2 ^ n; n < word_bits \<rbrakk> \<Longrightarrow> of_nat x < (2 :: word32) ^ n"
apply (subst word_unat_power)
apply (rule of_nat_mono_maybe)
apply (rule power_strict_increasing)
apply (simp add: word_bits_def)
apply simp
apply assumption
done
lemma power_helper:
"\<lbrakk> (x :: word32) < 2 ^ (m - n); n \<le> m; m < word_bits \<rbrakk> \<Longrightarrow> x * (2 ^ n) < 2 ^ m"
apply (drule word_mult_less_mono1[where k="2 ^ n"])
apply (simp add: word_neq_0_conv[symmetric] word_bits_def)
apply (simp only: unat_power_lower[where 'a=32, unfolded word_bits_len_of]
power_add[symmetric])
apply (rule power_strict_increasing)
apply (simp add: word_bits_def)
apply simp
apply (simp add: power_add[symmetric] del: power_add)
done
lemma word_1_le_power:
"n < len_of TYPE('a) \<Longrightarrow> (1 :: 'a :: len word) \<le> 2 ^ n"
by (rule inc_le[where i=0, simplified], erule iffD2[OF p2_gt_0])
lemma enum_word_div:
fixes v :: "('a :: len) word" shows
"\<exists>xs ys. enum = xs @ [v] @ ys
\<and> (\<forall>x \<in> set xs. x < v)
\<and> (\<forall>y \<in> set ys. v < y)"
apply (simp only: enum_word_def)
apply (subst upt_add_eq_append'[where j="unat v"])
apply simp
apply (rule order_less_imp_le, simp)
apply (simp add: upt_conv_Cons)
apply (intro exI conjI)
apply fastforce
apply clarsimp
apply (drule of_nat_mono_maybe[rotated, where 'a='a])
apply simp
apply simp
apply (clarsimp simp: Suc_le_eq)
apply (drule of_nat_mono_maybe[rotated, where 'a='a])
apply simp
apply simp
done
lemma less_x_plus_1:
fixes x :: "('a :: len) word" shows
"x \<noteq> max_word \<Longrightarrow> (y < (x + 1)) = (y < x \<or> y = x)"
apply (rule iffI)
apply (rule disjCI)
apply (drule plus_one_helper)
apply simp
apply (subgoal_tac "x < x + 1")
apply (erule disjE, simp_all)
apply (rule plus_one_helper2 [OF order_refl])
apply (rule notI, drule max_word_wrap)
apply simp
done
lemma of_bool_nth:
"of_bool (x !! v) = (x >> v) && 1"
apply (rule word_eqI)
apply (simp add: nth_shiftr cong: rev_conj_cong)
done
lemma unat_1_0:
"1 \<le> (x::word32) = (0 < unat x)"
by (auto simp add: word_le_nat_alt)
lemma x_less_2_0_1:
fixes x :: word32 shows
"x < 2 \<Longrightarrow> x = 0 \<or> x = 1"
by unat_arith
lemma Collect_int_vars:
"{s. P rv s} \<inter> {s. rv = xf s} = {s. P (xf s) s} \<inter> {s. rv = xf s}"
by auto
lemma if_0_1_eq:
"((if P then 1 else 0) = (case Q of True \<Rightarrow> of_nat 1 | False \<Rightarrow> of_nat 0)) = (P = Q)"
by (simp add: case_bool_If split: split_if)
lemma modify_map_exists_cte :
"(\<exists>cte. modify_map m p f p' = Some cte) = (\<exists>cte. m p' = Some cte)"
by (simp add: modify_map_def)
lemmas word_add_le_iff2 = word_add_le_iff [folded no_olen_add_nat]
lemma mask_32_max_word :
shows "mask 32 = (max_word :: word32)"
unfolding mask_def
by (simp add: max_word_def)
lemma dom_eqI:
assumes c1: "\<And>x y. P x = Some y \<Longrightarrow> \<exists>y. Q x = Some y"
and c2: "\<And>x y. Q x = Some y \<Longrightarrow> \<exists>y. P x = Some y"
shows "dom P = dom Q"
unfolding dom_def by (auto simp: c1 c2)
lemma dvd_reduce_multiple:
fixes k :: nat
shows "(k dvd k * m + n) = (k dvd n)"
apply (induct m)
apply simp
apply simp
apply (subst add.assoc, subst add.commute)
apply (subst dvd_reduce)
apply assumption
done
lemma image_iff:
"inj f \<Longrightarrow> f x \<in> f ` S = (x \<in> S)"
apply rule
apply (erule imageE)
apply (simp add: inj_eq)
apply (erule imageI)
done
lemma of_nat_n_less_equal_power_2:
"n < len_of TYPE('a::len) \<Longrightarrow> ((of_nat n)::'a word) < 2 ^ n"
apply (induct n)
apply clarsimp
apply clarsimp
apply (metis WordLemmaBucket.of_nat_power
n_less_equal_power_2 of_nat_Suc power_Suc)
done
lemma of_nat32_n_less_equal_power_2:
"n < 32 \<Longrightarrow> ((of_nat n)::32 word) < 2 ^ n"
by (rule of_nat_n_less_equal_power_2, clarsimp simp: word_size)
lemma map_comp_restrict_map_Some_iff:
"((g \<circ>\<^sub>m (m |` S)) x = Some y) = ((g \<circ>\<^sub>m m) x = Some y \<and> x \<in> S)"
by (auto simp add: map_comp_Some_iff restrict_map_Some_iff)
lemma range_subsetD:
fixes a :: "'a :: order"
shows "\<lbrakk> {a..b} \<subseteq> {c..d}; a \<le> b \<rbrakk> \<Longrightarrow> c \<le> a \<and> b \<le> d"
apply (rule conjI)
apply (drule subsetD [where c = a])
apply simp
apply simp
apply (drule subsetD [where c = b])
apply simp
apply simp
done
lemma option_case_dom:
"(case f x of None \<Rightarrow> a | Some v \<Rightarrow> b v)
= (if x \<in> dom f then b (the (f x)) else a)"
by (auto split: split_if option.split)
lemma contrapos_imp:
"P \<longrightarrow> Q \<Longrightarrow> \<not> Q \<longrightarrow> \<not> P"
by clarsimp
lemma eq_mask_less:
fixes w :: "('a::len) word"
assumes eqm: "w = w && mask n"
and sz: "n < len_of TYPE ('a)"
shows "w < (2::'a word) ^ n"
by (subst eqm, rule and_mask_less' [OF sz])
lemma of_nat_mono_maybe':
fixes Y :: "nat"
assumes xlt: "X < 2 ^ len_of TYPE ('a :: len)"
assumes ylt: "Y < 2 ^ len_of TYPE ('a :: len)"
shows "(Y < X) = (of_nat Y < (of_nat X :: 'a :: len word))"
apply (subst word_less_nat_alt)
apply (subst unat_of_nat)+
apply (subst mod_less)
apply (rule ylt)
apply (subst mod_less)
apply (rule xlt)
apply simp
done
(* FIXME: MOVE *)
lemma shiftr_mask_eq:
fixes x :: "'a :: len word"
shows "(x >> n) && mask (size x - n) = x >> n"
apply (rule word_eqI)
apply (simp add: word_size nth_shiftr)
apply (rule iffI)
apply clarsimp
apply (clarsimp)
apply (drule test_bit_size)
apply (simp add: word_size)
done
(* FIXME: move *)
lemma shiftr_mask_eq':
fixes x :: "'a :: len word"
shows "m = (size x - n) \<Longrightarrow> (x >> n) && mask m = x >> n"
by (simp add: shiftr_mask_eq)
lemma zipWith_Nil2 :
"zipWith f xs [] = []"
unfolding zipWith_def by simp
lemma zip_upt_Cons:
"a < b \<Longrightarrow> zip [a ..< b] (x # xs)
= (a, x) # zip [Suc a ..< b] xs"
by (simp add: upt_conv_Cons)
lemma map_comp_eq:
"(f \<circ>\<^sub>m g) = (case_option None f \<circ> g)"
apply (rule ext)
apply (case_tac "g x")
apply simp
apply simp
done
lemma dom_If_Some:
"dom (\<lambda>x. if x \<in> S then Some v else f x) = (S \<union> dom f)"
by (auto split: split_if)
lemma foldl_fun_upd_const:
"foldl (\<lambda>s x. s(f x := v)) s xs
= (\<lambda>x. if x \<in> f ` set xs then v else s x)"
apply (induct xs arbitrary: s)
apply simp
apply (rule ext, simp)
done
lemma foldl_id:
"foldl (\<lambda>s x. s) s xs = s"
apply (induct xs)
apply simp
apply simp
done
lemma SucSucMinus: "2 \<le> n \<Longrightarrow> Suc (Suc (n - 2)) = n" by arith
lemma ball_to_all:
"(\<And>x. (x \<in> A) = (P x)) \<Longrightarrow> (\<forall>x \<in> A. B x) = (\<forall>x. P x \<longrightarrow> B x)"
by blast
lemma bang_big: "n \<ge> size (x::'a::len0 word) \<Longrightarrow> (x !! n) = False"
by (simp add: test_bit_bl word_size)
lemma bang_conj_lt:
fixes x :: "'a :: len word"
shows "(x !! m \<and> m < len_of TYPE('a)) = x !! m"
apply (cases "m < len_of TYPE('a)")
apply simp
apply (simp add: not_less bang_big word_size)
done
lemma dom_if:
"dom (\<lambda>a. if a \<in> addrs then Some (f a) else g a) = addrs \<union> dom g"
by (auto simp: dom_def split: split_if)
lemma less_is_non_zero_p1:
fixes a :: "'a :: len word"
shows "a < k \<Longrightarrow> a + 1 \<noteq> 0"
apply (erule contrapos_pn)
apply (drule max_word_wrap)
apply (simp add: not_less)
done
lemma lt_word_bits_lt_pow:
"sz < word_bits \<Longrightarrow> sz < 2 ^ word_bits"
by (simp add: word_bits_conv)
(* FIXME: shadows an existing thm *)
lemma of_nat_mono_maybe_le:
"\<lbrakk>X < 2 ^ len_of TYPE('a); Y < 2 ^ len_of TYPE('a)\<rbrakk> \<Longrightarrow>
(Y \<le> X) = ((of_nat Y :: 'a :: len word) \<le> of_nat X)"
apply (clarsimp simp: le_less)
apply (rule disj_cong)
apply (rule of_nat_mono_maybe', assumption+)
apply (simp add: word_unat.norm_eq_iff [symmetric])
done
lemma neg_mask_bang:
"(~~ mask n :: 'a :: len word) !! m = (n \<le> m \<and> m < len_of TYPE('a))"
apply (cases "m < len_of TYPE('a)")
apply (simp add: word_ops_nth_size word_size not_less)
apply (simp add: not_less bang_big word_size)
done
lemma mask_AND_NOT_mask:
"(w && ~~ mask n) && mask n = 0"
by (rule word_eqI) (clarsimp simp add: word_size neg_mask_bang)
lemma AND_NOT_mask_plus_AND_mask_eq:
"(w && ~~ mask n) + (w && mask n) = w"
apply (rule word_eqI)
apply (rename_tac m)
apply (simp add: word_size)
apply (cut_tac word_plus_and_or_coroll[of "w && ~~ mask n" "w && mask n"])
apply (simp add: word_ao_dist2[symmetric] word_size neg_mask_bang)
apply (rule word_eqI)
apply (rename_tac m)
apply (simp add: word_size neg_mask_bang)
done
lemma mask_eqI:
fixes x :: "'a :: len word"
assumes m1: "x && mask n = y && mask n"
and m2: "x && ~~ mask n = y && ~~ mask n"
shows "x = y"
proof (subst bang_eq, rule allI)
fix m
show "x !! m = y !! m"
proof (cases "m < n")
case True
hence "x !! m = ((x && mask n) !! m)"
by (simp add: word_size bang_conj_lt)
also have "\<dots> = ((y && mask n) !! m)" using m1 by simp
also have "\<dots> = y !! m" using True
by (simp add: word_size bang_conj_lt)
finally show ?thesis .
next
case False
hence "x !! m = ((x && ~~ mask n) !! m)"
by (simp add: neg_mask_bang bang_conj_lt)
also have "\<dots> = ((y && ~~ mask n) !! m)" using m2 by simp
also have "\<dots> = y !! m" using False
by (simp add: neg_mask_bang bang_conj_lt)
finally show ?thesis .
qed
qed
lemma nat_less_power_trans2:
fixes n :: nat
shows "\<lbrakk>n < 2 ^ (m - k); k \<le> m\<rbrakk> \<Longrightarrow> n * 2 ^ k < 2 ^ m"
by (subst mult.commute, erule (1) nat_less_power_trans)
lemma nat_move_sub_le: "(a::nat) + b \<le> c \<Longrightarrow> a \<le> c - b" by arith
lemma neq_0_no_wrap:
fixes x :: "'a :: len word"
shows "\<lbrakk> x \<le> x + y; x \<noteq> 0 \<rbrakk> \<Longrightarrow> x + y \<noteq> 0"
by clarsimp
lemma plus_minus_one_rewrite:
"v + (- 1 :: ('a :: {ring, one, uminus})) \<equiv> v - 1"
by (simp add: field_simps)
lemmas plus_minus_one_rewrite32 = plus_minus_one_rewrite[where 'a=word32, simplified]
lemma power_minus_is_div:
"b \<le> a \<Longrightarrow> (2 :: nat) ^ (a - b) = 2 ^ a div 2 ^ b"
apply (induct a arbitrary: b)
apply simp
apply (erule le_SucE)
apply (clarsimp simp:Suc_diff_le le_iff_add power_add)
apply simp
done
lemma two_pow_div_gt_le:
"v < 2 ^ n div (2 ^ m :: nat) \<Longrightarrow> m \<le> n"
by (clarsimp dest!: less_two_pow_divD)
lemma unat_less_word_bits:
fixes y :: word32
shows "x < unat y \<Longrightarrow> x < 2 ^ word_bits"
unfolding word_bits_def
by (rule order_less_trans [OF _ unat_lt2p])
lemma word_add_power_off:
fixes a :: word32
assumes ak: "a < k"
and kw: "k < 2 ^ (word_bits - m)"
and mw: "m < word_bits"
and off: "off < 2 ^ m"
shows "(a * 2 ^ m) + off < k * 2 ^ m"
proof (cases "m = 0")
case True
thus ?thesis using off ak by simp
next
case False
from ak have ak1: "a + 1 \<le> k" by (rule inc_le)
hence "(a + 1) * 2 ^ m \<noteq> 0"
apply -
apply (rule word_power_nonzero)
apply (erule order_le_less_trans [OF _ kw])
apply (rule mw)
apply (rule less_is_non_zero_p1 [OF ak])
done
hence "(a * 2 ^ m) + off < ((a + 1) * 2 ^ m)" using kw mw
apply -
apply (simp add: distrib_right)
apply (rule word_plus_strict_mono_right [OF off])
apply (rule is_aligned_no_overflow'')
apply (rule is_aligned_mult_triv2)
apply assumption
done
also have "\<dots> \<le> k * 2 ^ m" using ak1 mw kw False
apply -
apply (erule word_mult_le_mono1)
apply (simp add: p2_gt_0 word_bits_def)
apply (simp add: word_bits_len_of word_less_nat_alt word_bits_def)
apply (rule nat_less_power_trans2[where m=32, simplified])
apply (simp add: word_less_nat_alt)
apply simp
done
finally show ?thesis .
qed
lemma word_of_nat_less:
"\<lbrakk> n < unat x \<rbrakk> \<Longrightarrow> of_nat n < x"
apply (simp add: word_less_nat_alt)
apply (erule order_le_less_trans[rotated])
apply (simp add: unat_of_nat)
done
lemma word_rsplit_0:
"word_rsplit (0 :: word32) = [0, 0, 0, 0 :: word8]"
apply (simp add: word_rsplit_def bin_rsplit_def Let_def)
done
lemma word_of_nat_le:
"n \<le> unat x \<Longrightarrow> of_nat n \<le> x"
apply (simp add: word_le_nat_alt unat_of_nat)
apply (erule order_trans[rotated])
apply simp
done
lemma word_unat_less_le:
"a \<le> of_nat b \<Longrightarrow> unat a \<le> b"
by (metis eq_iff le_cases le_unat_uoi word_of_nat_le)
lemma filter_eq_If:
"distinct xs \<Longrightarrow> filter (\<lambda>v. v = x) xs = (if x \<in> set xs then [x] else [])"
apply (induct xs)
apply simp
apply (clarsimp split: split_if)
done
(*FIXME: isabelle-2012 *)
lemma (in semigroup_add) foldl_assoc:
shows "foldl op+ (x+y) zs = x + (foldl op+ y zs)"
by (induct zs arbitrary: y) (simp_all add:add.assoc)
lemma (in monoid_add) foldl_absorb0:
shows "x + (foldl op+ 0 zs) = foldl op+ x zs"
by (induct zs) (simp_all add:foldl_assoc)
lemma foldl_conv_concat:
"foldl (op @) xs xss = xs @ concat xss"
proof (induct xss arbitrary: xs)
case Nil show ?case by simp
next
interpret monoid_add "op @" "[]" proof qed simp_all
case Cons then show ?case by (simp add: foldl_absorb0)
qed
lemma foldl_concat_concat:
"foldl op @ [] (xs @ ys) = foldl op @ [] xs @ foldl op @ [] ys"
by (simp add: foldl_conv_concat)
lemma foldl_does_nothing:
"\<lbrakk> \<And>x. x \<in> set xs \<Longrightarrow> f s x = s \<rbrakk> \<Longrightarrow> foldl f s xs = s"
by (induct xs, simp_all)
lemma foldl_use_filter:
"\<lbrakk> \<And>v x. \<lbrakk> \<not> g x; x \<in> set xs \<rbrakk> \<Longrightarrow> f v x = v \<rbrakk>
\<Longrightarrow>
foldl f v xs = foldl f v (filter g xs)"
apply (induct xs arbitrary: v)
apply simp
apply (simp split: split_if)
done
lemma split_upt_on_n:
"n < m \<Longrightarrow> [0 ..< m] = [0 ..< n] @ [n] @ [Suc n ..< m]"
apply (subst upt_add_eq_append', simp, erule order_less_imp_le)
apply (simp add: upt_conv_Cons)
done
lemma unat_ucast_10_32 :
fixes x :: "10 word"
shows "unat (ucast x :: word32) = unat x"
unfolding ucast_def unat_def
apply (subst int_word_uint)
apply (subst mod_pos_pos_trivial)
apply simp
apply (rule lt2p_lem)
apply simp
apply simp
done
lemma map_comp_update_lift:
assumes fv: "f v = Some v'"
shows "(f \<circ>\<^sub>m (g(ptr \<mapsto> v))) = ((f \<circ>\<^sub>m g)(ptr \<mapsto> v'))"
unfolding map_comp_def
apply (rule ext)
apply (simp add: fv)
done
lemma restrict_map_cong:
assumes sv: "S = S'"
and rl: "\<And>p. p \<in> S' \<Longrightarrow> mp p = mp' p"
shows "mp |` S = mp' |` S'"
apply (simp add: sv)
apply (rule ext)
apply (case_tac "x \<in> S'")
apply (simp add: rl )
apply simp
done
lemma and_eq_0_is_nth:
fixes x :: "('a :: len) word"
shows "y = 1 << n \<Longrightarrow> ((x && y) = 0) = (\<not> (x !! n))"
apply safe
apply (drule_tac u="(x && (1 << n))" and x=n in word_eqD)
apply (simp add: nth_w2p)
apply (simp add: test_bit_bin)
apply (rule word_eqI)
apply (simp add: nth_w2p)
done
lemmas and_neq_0_is_nth = arg_cong [where f=Not, OF and_eq_0_is_nth, simplified]
lemma ucast_le_ucast_8_32:
"(ucast x \<le> (ucast y :: word32)) = (x \<le> (y :: word8))"
by (simp add: word_le_nat_alt unat_ucast_8_32)
lemma mask_Suc_0 : "mask (Suc 0) = 1"
by (simp add: mask_def)
lemma ucast_ucast_add:
fixes x :: "('a :: len) word"
fixes y :: "('b :: len) word"
shows
"len_of TYPE('b) \<ge> len_of TYPE('a) \<Longrightarrow>
ucast (ucast x + y) = x + ucast y"
apply (rule word_unat.Rep_eqD)
apply (simp add: unat_ucast unat_word_ariths mod_mod_power
min.absorb2 unat_of_nat)
apply (subst mod_add_left_eq)
apply (simp add: mod_mod_power min.absorb2)
apply (subst mod_add_right_eq)
apply simp
done
lemma word_shift_zero:
"\<lbrakk> x << n = 0; x \<le> 2^m; m + n < len_of TYPE('a)\<rbrakk> \<Longrightarrow> (x::'a::len word) = 0"
apply (rule ccontr)
apply (drule (2) word_shift_nonzero)
apply simp
done
lemma neg_mask_mono_le:
"(x :: 'a :: len word) \<le> y \<Longrightarrow> x && ~~ mask n \<le> y && ~~ mask n"
proof (rule ccontr, simp add: linorder_not_le, cases "n < len_of TYPE('a)")
case False
show "y && ~~ mask n < x && ~~ mask n \<Longrightarrow> False"
using False
by (simp add: mask_def linorder_not_less
power_overflow)
next
case True
assume a: "x \<le> y" and b: "y && ~~ mask n < x && ~~ mask n"
have word_bits:
"n < len_of TYPE('a)"
using True by assumption
have "y \<le> (y && ~~ mask n) + (y && mask n)"
by (simp add: word_plus_and_or_coroll2 add.commute)
also have "\<dots> \<le> (y && ~~ mask n) + 2 ^ n"
apply (rule word_plus_mono_right)
apply (rule order_less_imp_le, rule and_mask_less_size)
apply (simp add: word_size word_bits)
apply (rule is_aligned_no_overflow'',
simp_all add: is_aligned_neg_mask word_bits)
apply (rule not_greatest_aligned, rule b)
apply (simp_all add: is_aligned_neg_mask)
done
also have "\<dots> \<le> x && ~~ mask n"
using b
apply -
apply (subst add.commute, rule le_plus)
apply (rule aligned_at_least_t2n_diff,
simp_all add: is_aligned_neg_mask)
apply (rule ccontr, simp add: linorder_not_le)
apply (drule aligned_small_is_0[rotated], simp_all add: is_aligned_neg_mask)
done
also have "\<dots> \<le> x"
by (rule word_and_le2)
also have "x \<le> y" by fact
finally
show "False" using b
by simp
qed
lemma isRight_right_map:
"isRight (case_sum Inl (Inr o f) v) = isRight v"
by (simp add: isRight_def split: sum.split)
lemma bool_mask [simp]:
fixes x :: word32
shows "(0 < x && 1) = (x && 1 = 1)"
apply (rule iffI)
prefer 2
apply simp
apply (subgoal_tac "x && mask 1 < 2^1")
prefer 2
apply (rule and_mask_less_size)
apply (simp add: word_size)
apply (simp add: mask_def)
apply (drule word_less_cases [where y=2])
apply (erule disjE, simp)
apply simp
done
lemma option_case_over_if:
"case_option P Q (if G then None else Some v)
= (if G then P else Q v)"
"case_option P Q (if G then Some v else None)
= (if G then Q v else P)"
by (simp split: split_if)+
lemma scast_eq_ucast:
"\<not> msb x \<Longrightarrow> scast x = ucast x"
by (simp add: scast_def ucast_def sint_eq_uint)
(* MOVE *)
lemma lt1_neq0:
fixes x :: "'a :: len word"
shows "(1 \<le> x) = (x \<noteq> 0)" by unat_arith
lemma word_plus_one_nonzero:
fixes x :: "'a :: len word"
shows "\<lbrakk>x \<le> x + y; y \<noteq> 0\<rbrakk> \<Longrightarrow> x + 1 \<noteq> 0"
apply (subst lt1_neq0 [symmetric])
apply (subst olen_add_eqv [symmetric])
apply (erule word_random)
apply (simp add: lt1_neq0)
done
lemma word_sub_plus_one_nonzero:
fixes n :: "'a :: len word"
shows "\<lbrakk>n' \<le> n; n' \<noteq> 0\<rbrakk> \<Longrightarrow> (n - n') + 1 \<noteq> 0"
apply (subst lt1_neq0 [symmetric])
apply (subst olen_add_eqv [symmetric])
apply (rule word_random [where x' = n'])
apply simp
apply (erule word_sub_le)
apply (simp add: lt1_neq0)
done
lemma word_le_minus_mono_right:
fixes x :: "'a :: len word"
shows "\<lbrakk> z \<le> y; y \<le> x; z \<le> x \<rbrakk> \<Longrightarrow> x - y \<le> x - z"
apply (rule word_sub_mono)
apply simp
apply assumption
apply (erule word_sub_le)
apply (erule word_sub_le)
done
lemma drop_append_miracle:
"n = length xs \<Longrightarrow> drop n (xs @ ys) = ys"
by simp
lemma foldr_does_nothing_to_xf:
"\<lbrakk> \<And>x s. x \<in> set xs \<Longrightarrow> xf (f x s) = xf s \<rbrakk> \<Longrightarrow> xf (foldr f xs s) = xf s"
by (induct xs, simp_all)
lemma nat_less_mult_monoish: "\<lbrakk> a < b; c < (d :: nat) \<rbrakk> \<Longrightarrow> (a + 1) * (c + 1) <= b * d"
apply (drule Suc_leI)+
apply (drule(1) mult_le_mono)
apply simp
done
lemma word_0_sle_from_less[unfolded word_size]:
"\<lbrakk> x < 2 ^ (size x - 1) \<rbrakk> \<Longrightarrow> 0 <=s x"
apply (clarsimp simp: word_sle_msb_le)
apply (simp add: word_msb_nth)
apply (subst (asm) word_test_bit_def [symmetric])
apply (drule less_mask_eq)
apply (drule_tac x="size x - 1" in word_eqD)
apply (simp add: word_size)
done
lemma not_msb_from_less:
"(v :: 'a word) < 2 ^ (len_of TYPE('a :: len) - 1) \<Longrightarrow> \<not> msb v"
apply (clarsimp simp add: msb_nth)
apply (drule less_mask_eq)
apply (drule word_eqD, drule(1) iffD2)
apply simp
done
lemma distinct_lemma: "f x \<noteq> f y \<Longrightarrow> x \<noteq> y" by auto
lemma ucast_sub_ucast:
fixes x :: "'a::len word"
assumes "y \<le> x"
assumes T: "len_of TYPE('a) \<le> len_of TYPE('b)"
shows "ucast (x - y) = (ucast x - ucast y :: 'b::len word)"
proof -
from T
have P: "unat x < 2 ^ len_of TYPE('b)" "unat y < 2 ^ len_of TYPE('b)"
by (fastforce intro!: less_le_trans[OF unat_lt2p])+
thus ?thesis
by (simp add: unat_arith_simps unat_ucast split assms[simplified unat_arith_simps])
qed
lemma word_1_0:
"\<lbrakk>a + (1::('a::len) word) \<le> b; a < of_nat ((2::nat) ^ len_of TYPE(32) - 1)\<rbrakk> \<Longrightarrow> a < b"
by unat_arith
lemma unat_of_nat_less:"\<lbrakk> a < b; unat b = c \<rbrakk> \<Longrightarrow> a < of_nat c"
by fastforce
lemma word_le_plus_1: "\<lbrakk> (y::('a::len) word) < y + n; a < n \<rbrakk> \<Longrightarrow> y + a \<le> y + a + 1"
by unat_arith
lemma word_le_plus:"\<lbrakk>(a::('a::len) word) < a + b; c < b\<rbrakk> \<Longrightarrow> a \<le> a + c"
by (metis order_less_imp_le word_random)
(*
* Basic signed arithemetic properties.
*)
lemma sint_minus1 [simp]: "(sint x = -1) = (x = -1)"
by (metis sint_n1 word_sint.Rep_inverse')
lemma sint_0 [simp]: "(sint x = 0) = (x = 0)"
by (metis sint_0 word_sint.Rep_inverse')
(* It is not always that case that "sint 1 = 1", because of 1-bit word sizes.
* This lemma produces the different cases. *)
lemma sint_1_cases:
"\<lbrakk> \<lbrakk> len_of TYPE ('a::len) = 1; (a::'a word) = 0; sint a = 0 \<rbrakk> \<Longrightarrow> P;
\<lbrakk> len_of TYPE ('a) = 1; a = 1; sint (1 :: 'a word) = -1 \<rbrakk> \<Longrightarrow> P;
\<lbrakk> len_of TYPE ('a) > 1; sint (1 :: 'a word) = 1 \<rbrakk> \<Longrightarrow> P \<rbrakk>
\<Longrightarrow> P"
apply atomize_elim
apply (case_tac "len_of TYPE ('a) = 1")
apply clarsimp
apply (subgoal_tac "(UNIV :: 'a word set) = {0, 1}")
apply (metis UNIV_I insert_iff singletonE)
apply (subst word_unat.univ)
apply (clarsimp simp: unats_def image_def)
apply (rule set_eqI, rule iffI)
apply clarsimp
apply (metis One_nat_def less_2_cases of_nat_1 semiring_1_class.of_nat_0)
apply clarsimp
apply (metis Abs_fnat_hom_0 Suc_1 lessI of_nat_1 zero_less_Suc)
apply clarsimp
apply (metis One_nat_def arith_is_1 le_def len_gt_0 sint_eq_uint uint_1 word_msb_1)
done
lemma sint_int_min:
"sint (- (2 ^ (len_of TYPE('a) - Suc 0)) :: ('a::len) word) = - (2 ^ (len_of TYPE('a) - Suc 0))"
apply (subst word_sint.Abs_inverse' [where r="- (2 ^ (len_of TYPE('a) - Suc 0))"])
apply (clarsimp simp: sints_num)
apply (clarsimp simp: wi_hom_syms word_of_int_2p)
apply clarsimp
done
lemma sint_int_max_plus_1:
"sint (2 ^ (len_of TYPE('a) - Suc 0) :: ('a::len) word) = - (2 ^ (len_of TYPE('a) - Suc 0))"
apply (subst word_of_int_2p [symmetric])
apply (subst int_word_sint)
apply (clarsimp simp: comm_semiring_1_class.normalizing_semiring_rules(27))
done
lemma word32_bounds:
"- (2 ^ (size (x :: word32) - 1)) = (-2147483648 :: int)"
"((2 ^ (size (x :: word32) - 1)) - 1) = (2147483647 :: int)"
"- (2 ^ (size (y :: 32 signed word) - 1)) = (-2147483648 :: int)"
"((2 ^ (size (y :: 32 signed word) - 1)) - 1) = (2147483647 :: int)"
by (simp_all add: word_size)
lemma sbintrunc_If:
"- 3 * (2 ^ n) \<le> x \<and> x < 3 * (2 ^ n)
\<Longrightarrow> sbintrunc n x = (if x < - (2 ^ n) then x + 2 * (2 ^ n)
else if x \<ge> 2 ^ n then x - 2 * (2 ^ n) else x)"
apply (simp add: no_sbintr_alt2, safe)
apply (simp add: mod_pos_geq mod_pos_pos_trivial)
apply (subst mod_add_self1[symmetric], simp)
apply (simp add: mod_pos_pos_trivial)
apply (simp add: mod_pos_pos_trivial)
done
lemma signed_arith_eq_checks_to_ord:
"(sint a + sint b = sint (a + b ))
= ((a <=s a + b) = (0 <=s b))"
"(sint a - sint b = sint (a - b ))
= ((0 <=s a - b) = (b <=s a))"
"(- sint a = sint (- a)) = (0 <=s (- a) = (a <=s 0))"
using sint_range'[where x=a] sint_range'[where x=b]
apply (simp_all add: sint_word_ariths
word_sle_def word_sless_alt sbintrunc_If)
apply arith+
done
(* Basic proofs that signed word div/mod operations are
* truncations of their integer counterparts. *)
lemma signed_div_arith:
"sint ((a::('a::len) word) sdiv b) = sbintrunc (len_of TYPE('a) - 1) (sint a sdiv sint b)"
apply (subst word_sbin.norm_Rep [symmetric])
apply (subst bin_sbin_eq_iff' [symmetric])
apply simp
apply (subst uint_sint [symmetric])
apply (clarsimp simp: sdiv_int_def sdiv_word_def)
apply (metis word_ubin.eq_norm)
done
lemma signed_mod_arith:
"sint ((a::('a::len) word) smod b) = sbintrunc (len_of TYPE('a) - 1) (sint a smod sint b)"
apply (subst word_sbin.norm_Rep [symmetric])
apply (subst bin_sbin_eq_iff' [symmetric])
apply simp
apply (subst uint_sint [symmetric])
apply (clarsimp simp: smod_int_def smod_word_def)
apply (metis word_ubin.eq_norm)
done
(* Signed word arithmetic overflow constraints. *)
lemma signed_arith_ineq_checks_to_eq:
"((- (2 ^ (size a - 1)) \<le> (sint a + sint b)) \<and> (sint a + sint b \<le> (2 ^ (size a - 1) - 1)))
= (sint a + sint b = sint (a + b ))"
"((- (2 ^ (size a - 1)) \<le> (sint a - sint b)) \<and> (sint a - sint b \<le> (2 ^ (size a - 1) - 1)))
= (sint a - sint b = sint (a - b))"
"((- (2 ^ (size a - 1)) \<le> (- sint a)) \<and> (- sint a) \<le> (2 ^ (size a - 1) - 1))
= ((- sint a) = sint (- a))"
"((- (2 ^ (size a - 1)) \<le> (sint a * sint b)) \<and> (sint a * sint b \<le> (2 ^ (size a - 1) - 1)))
= (sint a * sint b = sint (a * b))"
"((- (2 ^ (size a - 1)) \<le> (sint a sdiv sint b)) \<and> (sint a sdiv sint b \<le> (2 ^ (size a - 1) - 1)))
= (sint a sdiv sint b = sint (a sdiv b))"
"((- (2 ^ (size a - 1)) \<le> (sint a smod sint b)) \<and> (sint a smod sint b \<le> (2 ^ (size a - 1) - 1)))
= (sint a smod sint b = sint (a smod b))"
by (auto simp: sint_word_ariths word_size signed_div_arith signed_mod_arith
sbintrunc_eq_in_range range_sbintrunc)
lemmas signed_arith_ineq_checks_to_eq_word32
= signed_arith_ineq_checks_to_eq[where 'a=32, unfolded word32_bounds]
signed_arith_ineq_checks_to_eq[where 'a="32 signed", unfolded word32_bounds]
lemma signed_arith_sint:
"((- (2 ^ (size a - 1)) \<le> (sint a + sint b)) \<and> (sint a + sint b \<le> (2 ^ (size a - 1) - 1)))
\<Longrightarrow> sint (a + b) = (sint a + sint b)"
"((- (2 ^ (size a - 1)) \<le> (sint a - sint b)) \<and> (sint a - sint b \<le> (2 ^ (size a - 1) - 1)))
\<Longrightarrow> sint (a - b) = (sint a - sint b)"
"((- (2 ^ (size a - 1)) \<le> (- sint a)) \<and> (- sint a) \<le> (2 ^ (size a - 1) - 1))
\<Longrightarrow> sint (- a) = (- sint a)"
"((- (2 ^ (size a - 1)) \<le> (sint a * sint b)) \<and> (sint a * sint b \<le> (2 ^ (size a - 1) - 1)))
\<Longrightarrow> sint (a * b) = (sint a * sint b)"
"((- (2 ^ (size a - 1)) \<le> (sint a sdiv sint b)) \<and> (sint a sdiv sint b \<le> (2 ^ (size a - 1) - 1)))
\<Longrightarrow> sint (a sdiv b) = (sint a sdiv sint b)"
"((- (2 ^ (size a - 1)) \<le> (sint a smod sint b)) \<and> (sint a smod sint b \<le> (2 ^ (size a - 1) - 1)))
\<Longrightarrow> sint (a smod b) = (sint a smod sint b)"
by (metis signed_arith_ineq_checks_to_eq)+
lemma signed_mult_eq_checks_double_size:
assumes mult_le: "(2 ^ (len_of TYPE ('a) - 1) + 1) ^ 2
\<le> (2 :: int) ^ (len_of TYPE ('b) - 1)"
and le: "2 ^ (len_of TYPE('a) - 1) \<le> (2 :: int) ^ (len_of TYPE ('b) - 1)"
shows
"(sint (a :: ('a :: len) word) * sint b = sint (a * b))
= (scast a * scast b = (scast (a * b) :: ('b :: len) word))"
proof -
have P: "sbintrunc (size a - 1) (sint a * sint b) \<in> range (sbintrunc (size a - 1))"
by simp
have abs: "!! x :: 'a word. abs (sint x) < 2 ^ (size a - 1) + 1"
apply (cut_tac x=x in sint_range')
apply (simp add: abs_le_iff word_size)
done
have abs_ab: "abs (sint a * sint b) < 2 ^ (len_of TYPE('b) - 1)"
using abs_mult_less[OF abs[where x=a] abs[where x=b]] mult_le
by (simp add: abs_mult power2_eq_square word_size)
show ?thesis
using P[unfolded range_sbintrunc] abs_ab le
apply (simp add: sint_word_ariths scast_def)
apply (simp add: wi_hom_mult)
apply (subst word_sint.Abs_inject, simp_all)
apply (simp add: sints_def range_sbintrunc
abs_less_iff)
apply clarsimp
apply (simp add: sints_def range_sbintrunc word_size)
apply (auto elim: order_less_le_trans order_trans[rotated])
done
qed
lemmas signed_mult_eq_checks32_to_64
= signed_mult_eq_checks_double_size[where 'a=32 and 'b=64, simplified]
signed_mult_eq_checks_double_size[where 'a="32 signed" and 'b=64, simplified]
(* Properties about signed division. *)
lemma int_sdiv_simps [simp]:
"(a :: int) sdiv 1 = a"
"(a :: int) sdiv 0 = 0"
"(a :: int) sdiv -1 = -a"
apply (auto simp: sdiv_int_def sgn_if)
done
lemma sgn_div_eq_sgn_mult:
"a div b \<noteq> 0 \<Longrightarrow> sgn ((a :: int) div b) = sgn (a * b)"
apply (clarsimp simp: sgn_if zero_le_mult_iff neg_imp_zdiv_nonneg_iff not_less)
apply (metis less_le mult_le_0_iff neg_imp_zdiv_neg_iff not_less pos_imp_zdiv_neg_iff zdiv_eq_0_iff)
done
lemma sgn_sdiv_eq_sgn_mult:
"a sdiv b \<noteq> 0 \<Longrightarrow> sgn ((a :: int) sdiv b) = sgn (a * b)"
apply (clarsimp simp: sdiv_int_def sgn_times)
apply (subst sgn_div_eq_sgn_mult)
apply simp
apply (clarsimp simp: sgn_times)
apply (metis abs_mult div_0 div_mult_self2_is_id sgn_0_0 sgn_1_pos sgn_times zero_less_abs_iff)
done
lemma int_sdiv_same_is_1 [simp]:
"a \<noteq> 0 \<Longrightarrow> ((a :: int) sdiv b = a) = (b = 1)"
apply (rule iffI)
apply (clarsimp simp: sdiv_int_def)
apply (subgoal_tac "b > 0")
apply (case_tac "a > 0")
apply (clarsimp simp: sgn_if sign_simps)
apply (clarsimp simp: sign_simps not_less)
apply (metis int_div_same_is_1 le_neq_trans minus_minus neg_0_le_iff_le neg_equal_0_iff_equal)
apply (case_tac "a > 0")
apply (case_tac "b = 0")
apply (clarsimp simp: sign_simps)
apply (rule classical)
apply (clarsimp simp: sign_simps sgn_times not_less)
apply (metis le_less neg_0_less_iff_less not_less_iff_gr_or_eq pos_imp_zdiv_neg_iff)
apply (rule classical)
apply (clarsimp simp: sign_simps sgn_times not_less sgn_if split: if_splits)
apply (metis antisym less_le neg_imp_zdiv_nonneg_iff)
apply (clarsimp simp: sdiv_int_def sgn_if)
done
lemma int_sdiv_negated_is_minus1 [simp]:
"a \<noteq> 0 \<Longrightarrow> ((a :: int) sdiv b = - a) = (b = -1)"
apply (clarsimp simp: sdiv_int_def)
apply (rule iffI)
apply (subgoal_tac "b < 0")
apply (case_tac "a > 0")
apply (clarsimp simp: sgn_if sign_simps not_less)
apply (case_tac "sgn (a * b) = -1")
apply (clarsimp simp: not_less sign_simps)
apply (clarsimp simp: sign_simps not_less)
apply (rule classical)
apply (case_tac "b = 0")
apply (clarsimp simp: sign_simps not_less sgn_times)
apply (case_tac "a > 0")
apply (clarsimp simp: sign_simps not_less sgn_times)
apply (metis less_le neg_less_0_iff_less not_less_iff_gr_or_eq pos_imp_zdiv_neg_iff)
apply (clarsimp simp: sign_simps not_less sgn_times)
apply (metis div_minus_right eq_iff neg_0_le_iff_le neg_imp_zdiv_nonneg_iff not_leE)
apply (clarsimp simp: sgn_if)
done
lemma sdiv_int_range:
"(a :: int) sdiv b \<in> { - (abs a) .. (abs a) }"
apply (unfold sdiv_int_def)
apply (subgoal_tac "(abs a) div (abs b) \<le> (abs a)")
apply (clarsimp simp: sgn_if)
apply (metis Divides.transfer_nat_int_function_closures(1) abs_ge_zero
abs_less_iff abs_of_nonneg less_asym less_minus_iff not_less)
apply (metis abs_eq_0 abs_ge_zero div_by_0 zdiv_le_dividend zero_less_abs_iff)
done
lemma sdiv_int_div_0 [simp]:
"(x :: int) sdiv 0 = 0"
by (clarsimp simp: sdiv_int_def)
lemma sdiv_int_0_div [simp]:
"0 sdiv (x :: int) = 0"
by (clarsimp simp: sdiv_int_def)
lemma word_sdiv_div0 [simp]:
"(a :: ('a::len) word) sdiv 0 = 0"
apply (auto simp: sdiv_word_def sdiv_int_def sgn_if)
done
lemma word_sdiv_div_minus1 [simp]:
"(a :: ('a::len) word) sdiv -1 = -a"
apply (auto simp: sdiv_word_def sdiv_int_def sgn_if)
apply (metis wi_hom_neg word_sint.Rep_inverse')
done
lemma word_sdiv_0 [simp]: "(x :: ('a::len) word) sdiv 0 = 0"
by (clarsimp simp: sdiv_word_def)
lemma sdiv_word_min:
"- (2 ^ (size a - 1)) \<le> sint (a :: ('a::len) word) sdiv sint (b :: ('a::len) word)"
apply (clarsimp simp: word_size)
apply (cut_tac sint_range' [where x=a])
apply (cut_tac sint_range' [where x=b])
apply clarsimp
apply (insert sdiv_int_range [where a="sint a" and b="sint b"])
apply (clarsimp simp: max_def abs_if split: split_if_asm)
done
lemma sdiv_word_max:
"(sint (a :: ('a::len) word) sdiv sint (b :: ('a::len) word) < (2 ^ (size a - 1))) =
((a \<noteq> - (2 ^ (size a - 1)) \<or> (b \<noteq> -1)))"
(is "?lhs = (\<not> ?a_int_min \<or> \<not> ?b_minus1)")
proof (rule classical)
assume not_thesis: "\<not> ?thesis"
have not_zero: "b \<noteq> 0"
using not_thesis
by (clarsimp)
have result_range: "sint a sdiv sint b \<in> (sints (size a)) \<union> {2 ^ (size a - 1)}"
apply (cut_tac sdiv_int_range [where a="sint a" and b="sint b"])
apply (erule rev_subsetD)
using sint_range' [where x=a] sint_range' [where x=b]
apply (auto simp: max_def abs_if word_size sints_num)
done
have result_range_overflow: "(sint a sdiv sint b = 2 ^ (size a - 1)) = (?a_int_min \<and> ?b_minus1)"
apply (rule iffI [rotated])
apply (clarsimp simp: sdiv_int_def sgn_if word_size sint_int_min)
apply (rule classical)
apply (case_tac "?a_int_min")
apply (clarsimp simp: word_size sint_int_min)
apply (metis diff_0_right
int_sdiv_negated_is_minus1 minus_diff_eq minus_int_code(2)
power_eq_0_iff sint_minus1 zero_neq_numeral)
apply (subgoal_tac "abs (sint a) < 2 ^ (size a - 1)")
apply (insert sdiv_int_range [where a="sint a" and b="sint b"])[1]
apply (clarsimp simp: word_size)
apply (insert sdiv_int_range [where a="sint a" and b="sint b"])[1]
apply (insert word_sint.Rep [where x="a"])[1]
apply (clarsimp simp: minus_le_iff word_size abs_if sints_num split: split_if_asm)
apply (metis minus_minus sint_int_min word_sint.Rep_inject)
done
have result_range_simple: "(sint a sdiv sint b \<in> (sints (size a))) \<Longrightarrow> ?thesis"
apply (insert sdiv_int_range [where a="sint a" and b="sint b"])
apply (clarsimp simp: word_size sints_num sint_int_min)
done
show ?thesis
apply (rule UnE [OF result_range result_range_simple])
apply simp
apply (clarsimp simp: word_size)
using result_range_overflow
apply (clarsimp simp: word_size)
done
qed
lemmas sdiv_word_min' = sdiv_word_min [simplified word_size, simplified]
lemmas sdiv_word_max' = sdiv_word_max [simplified word_size, simplified]
lemmas sdiv_word32_max = sdiv_word_max [where 'a=32, simplified word_size, simplified]
sdiv_word_max [where 'a="32 signed", simplified word_size, simplified]
lemmas sdiv_word32_min = sdiv_word_min [where 'a=32, simplified word_size, simplified]
sdiv_word_min [where 'a="32 signed", simplified word_size, simplified]
(*
* Signed modulo properties.
*)
lemma smod_int_alt_def:
"(a::int) smod b = sgn (a) * (abs a mod abs b)"
apply (clarsimp simp: smod_int_def sdiv_int_def)
apply (clarsimp simp: zmod_zdiv_equality' abs_sgn sgn_times sgn_if sign_simps)
done
lemma smod_int_range:
"b \<noteq> 0 \<Longrightarrow> (a::int) smod b \<in> { - abs b + 1 .. abs b - 1 }"
apply (case_tac "b > 0")
apply (insert pos_mod_conj [where a=a and b=b])[1]
apply (insert pos_mod_conj [where a="-a" and b=b])[1]
apply (clarsimp simp: smod_int_alt_def sign_simps sgn_if
abs_if not_less add1_zle_eq [simplified add.commute])
apply (metis add_le_cancel_left comm_monoid_add_class.add.right_neutral
int_one_le_iff_zero_less less_le_trans mod_minus_right neg_less_0_iff_less
neg_mod_conj not_less pos_mod_conj)
apply (insert neg_mod_conj [where a=a and b="b"])[1]
apply (insert neg_mod_conj [where a="-a" and b="b"])[1]
apply (clarsimp simp: smod_int_alt_def sign_simps sgn_if
abs_if not_less add1_zle_eq [simplified add.commute])
apply (metis neg_0_less_iff_less neg_mod_conj not_le not_less_iff_gr_or_eq order_trans pos_mod_conj)
done
lemma smod_int_compares:
"\<lbrakk> 0 \<le> a; 0 < b \<rbrakk> \<Longrightarrow> (a :: int) smod b < b"
"\<lbrakk> 0 \<le> a; 0 < b \<rbrakk> \<Longrightarrow> 0 \<le> (a :: int) smod b"
"\<lbrakk> a \<le> 0; 0 < b \<rbrakk> \<Longrightarrow> -b < (a :: int) smod b"
"\<lbrakk> a \<le> 0; 0 < b \<rbrakk> \<Longrightarrow> (a :: int) smod b \<le> 0"
"\<lbrakk> 0 \<le> a; b < 0 \<rbrakk> \<Longrightarrow> (a :: int) smod b < - b"
"\<lbrakk> 0 \<le> a; b < 0 \<rbrakk> \<Longrightarrow> 0 \<le> (a :: int) smod b"
"\<lbrakk> a \<le> 0; b < 0 \<rbrakk> \<Longrightarrow> (a :: int) smod b \<le> 0"
"\<lbrakk> a \<le> 0; b < 0 \<rbrakk> \<Longrightarrow> b \<le> (a :: int) smod b"
apply (insert smod_int_range [where a=a and b=b])
apply (auto simp: add1_zle_eq smod_int_alt_def sgn_if)
done
lemma smod_int_mod_0 [simp]:
"x smod (0 :: int) = x"
by (clarsimp simp: smod_int_def)
lemma smod_int_0_mod [simp]:
"0 smod (x :: int) = 0"
by (clarsimp simp: smod_int_alt_def)
lemma smod_word_mod_0 [simp]:
"x smod (0 :: ('a::len) word) = x"
by (clarsimp simp: smod_word_def)
lemma smod_word_0_mod [simp]:
"0 smod (x :: ('a::len) word) = 0"
by (clarsimp simp: smod_word_def)
lemma smod_word_max:
"sint (a::'a word) smod sint (b::'a word) < 2 ^ (len_of TYPE('a::len) - Suc 0)"
apply (case_tac "b = 0")
apply (insert word_sint.Rep [where x=a, simplified sints_num])[1]
apply (clarsimp)
apply (insert word_sint.Rep [where x="b", simplified sints_num])[1]
apply (insert smod_int_range [where a="sint a" and b="sint b"])
apply (clarsimp simp: abs_if split: split_if_asm)
done
lemma smod_word_min:
"- (2 ^ (len_of TYPE('a::len) - Suc 0)) \<le> sint (a::'a word) smod sint (b::'a word)"
apply (case_tac "b = 0")
apply (insert word_sint.Rep [where x=a, simplified sints_num])[1]
apply clarsimp
apply (insert word_sint.Rep [where x=b, simplified sints_num])[1]
apply (insert smod_int_range [where a="sint a" and b="sint b"])
apply (clarsimp simp: abs_if add1_zle_eq split: split_if_asm)
done
lemma smod_word_alt_def:
"(a :: ('a::len) word) smod b = a - (a sdiv b) * b"
apply (case_tac "a \<noteq> - (2 ^ (len_of TYPE('a) - 1)) \<or> b \<noteq> -1")
apply (clarsimp simp: smod_word_def sdiv_word_def smod_int_def
minus_word.abs_eq [symmetric] times_word.abs_eq [symmetric])
apply (clarsimp simp: smod_word_def smod_int_def)
done
lemma sint_of_int_eq:
"\<lbrakk> - (2 ^ (len_of TYPE('a) - 1)) \<le> x; x < 2 ^ (len_of TYPE('a) - 1) \<rbrakk> \<Longrightarrow> sint (of_int x :: ('a::len) word) = x"
apply (clarsimp simp: word_of_int int_word_sint)
apply (subst int_mod_eq')
apply simp
apply (subst (2) power_minus_simp)
apply clarsimp
apply clarsimp
apply clarsimp
done
lemmas sint32_of_int_eq = sint_of_int_eq [where 'a=32, simplified]
lemma of_int_sint [simp]:
"of_int (sint a) = a"
apply (insert word_sint.Rep [where x=a])
apply (clarsimp simp: word_of_int)
done
lemma ucast_of_nats [simp]:
"(ucast (of_nat x :: word32) :: sword32) = (of_nat x)"
"(ucast (of_nat x :: word32) :: sword16) = (of_nat x)"
"(ucast (of_nat x :: word32) :: sword8) = (of_nat x)"
"(ucast (of_nat x :: word16) :: sword16) = (of_nat x)"
"(ucast (of_nat x :: word16) :: sword8) = (of_nat x)"
"(ucast (of_nat x :: word8) :: sword8) = (of_nat x)"
apply (auto simp: ucast_of_nat is_down)
done
lemma nth_w2p_scast [simp]:
"((scast ((2::'a::len signed word) ^ n) :: 'a word) !! m)
\<longleftrightarrow> ((((2::'a::len word) ^ n) :: 'a word) !! m)"
apply (subst nth_w2p)
apply (case_tac "n \<ge> len_of TYPE('a)")
apply (subst power_overflow, simp)
apply clarsimp
apply (metis nth_w2p scast_def bang_conj_lt
len_signed nth_word_of_int word_sint.Rep_inverse)
done
lemma scast_2_power [simp]: "scast ((2 :: 'a::len signed word) ^ x) = ((2 :: 'a word) ^ x)"
by (clarsimp simp: word_eq_iff)
lemma scast_bit_test [simp]:
"scast ((1 :: 'a::len signed word) << n) = (1 :: 'a word) << n"
by (clarsimp simp: word_eq_iff)
lemma ucast_nat_def':
"of_nat (unat x) = (ucast :: ('a :: len) word \<Rightarrow> ('b :: len) signed word) x"
by (simp add: ucast_def word_of_int_nat unat_def)
lemma mod_mod_power_int:
fixes k :: int
shows "k mod 2 ^ m mod 2 ^ n = k mod 2 ^ (min m n)"
by (metis bintrunc_bintrunc_min bintrunc_mod2p min.commute)
(* Normalise combinations of scast and ucast. *)
lemma ucast_distrib:
fixes M :: "'a::len word \<Rightarrow> 'a::len word \<Rightarrow> 'a::len word"
fixes M' :: "'b::len word \<Rightarrow> 'b::len word \<Rightarrow> 'b::len word"
fixes L :: "int \<Rightarrow> int \<Rightarrow> int"
assumes lift_M: "\<And>x y. uint (M x y) = L (uint x) (uint y) mod 2 ^ len_of TYPE('a)"
assumes lift_M': "\<And>x y. uint (M' x y) = L (uint x) (uint y) mod 2 ^ len_of TYPE('b)"
assumes distrib: "\<And>x y. (L (x mod (2 ^ len_of TYPE('b))) (y mod (2 ^ len_of TYPE('b)))) mod (2 ^ len_of TYPE('b))
= (L x y) mod (2 ^ len_of TYPE('b))"
assumes is_down: "is_down (ucast :: 'a word \<Rightarrow> 'b word)"
shows "ucast (M a b) = M' (ucast a) (ucast b)"
apply (clarsimp simp: word_of_int ucast_def)
apply (subst lift_M)
apply (subst of_int_uint [symmetric], subst lift_M')
apply (subst (1 2) int_word_uint)
apply (subst word_of_int)
apply (subst word.abs_eq_iff)
apply (subst (1 2) bintrunc_mod2p)
apply (insert is_down)
apply (unfold is_down_def)
apply (clarsimp simp: target_size source_size)
apply (clarsimp simp: mod_mod_power_int min_def)
apply (rule distrib [symmetric])
done
lemma ucast_down_add:
"is_down (ucast:: 'a word \<Rightarrow> 'b word) \<Longrightarrow> ucast ((a :: 'a::len word) + b) = (ucast a + ucast b :: 'b::len word)"
by (rule ucast_distrib [where L="op +"], (clarsimp simp: uint_word_ariths)+, presburger, simp)
lemma ucast_down_minus:
"is_down (ucast:: 'a word \<Rightarrow> 'b word) \<Longrightarrow> ucast ((a :: 'a::len word) - b) = (ucast a - ucast b :: 'b::len word)"
apply (rule ucast_distrib [where L="op -"], (clarsimp simp: uint_word_ariths)+)
apply (metis zdiff_zmod_left zdiff_zmod_right)
apply simp
done
lemma ucast_down_mult:
"is_down (ucast:: 'a word \<Rightarrow> 'b word) \<Longrightarrow> ucast ((a :: 'a::len word) * b) = (ucast a * ucast b :: 'b::len word)"
apply (rule ucast_distrib [where L="op *"], (clarsimp simp: uint_word_ariths)+)
apply (metis mod_mult_eq)
apply simp
done
lemma scast_distrib:
fixes M :: "'a::len word \<Rightarrow> 'a::len word \<Rightarrow> 'a::len word"
fixes M' :: "'b::len word \<Rightarrow> 'b::len word \<Rightarrow> 'b::len word"
fixes L :: "int \<Rightarrow> int \<Rightarrow> int"
assumes lift_M: "\<And>x y. uint (M x y) = L (uint x) (uint y) mod 2 ^ len_of TYPE('a)"
assumes lift_M': "\<And>x y. uint (M' x y) = L (uint x) (uint y) mod 2 ^ len_of TYPE('b)"
assumes distrib: "\<And>x y. (L (x mod (2 ^ len_of TYPE('b))) (y mod (2 ^ len_of TYPE('b)))) mod (2 ^ len_of TYPE('b))
= (L x y) mod (2 ^ len_of TYPE('b))"
assumes is_down: "is_down (scast :: 'a word \<Rightarrow> 'b word)"
shows "scast (M a b) = M' (scast a) (scast b)"
apply (subst (1 2 3) down_cast_same [symmetric])
apply (insert is_down)
apply (clarsimp simp: is_down_def target_size source_size is_down)
apply (rule ucast_distrib [where L=L, OF lift_M lift_M' distrib])
apply (insert is_down)
apply (clarsimp simp: is_down_def target_size source_size is_down)
done
lemma scast_down_add:
"is_down (scast:: 'a word \<Rightarrow> 'b word) \<Longrightarrow> scast ((a :: 'a::len word) + b) = (scast a + scast b :: 'b::len word)"
by (rule scast_distrib [where L="op +"], (clarsimp simp: uint_word_ariths)+, presburger, simp)
lemma scast_down_minus:
"is_down (scast:: 'a word \<Rightarrow> 'b word) \<Longrightarrow> scast ((a :: 'a::len word) - b) = (scast a - scast b :: 'b::len word)"
apply (rule scast_distrib [where L="op -"], (clarsimp simp: uint_word_ariths)+)
apply (metis zdiff_zmod_left zdiff_zmod_right)
apply simp
done
lemma scast_down_mult:
"is_down (scast:: 'a word \<Rightarrow> 'b word) \<Longrightarrow> scast ((a :: 'a::len word) * b) = (scast a * scast b :: 'b::len word)"
apply (rule scast_distrib [where L="op *"], (clarsimp simp: uint_word_ariths)+)
apply (metis mod_mult_eq)
apply simp
done
lemma scast_ucast_3:
"\<lbrakk> is_down (ucast :: 'a word \<Rightarrow> 'c word); is_down (ucast :: 'b word \<Rightarrow> 'c word) \<rbrakk> \<Longrightarrow>
(scast (ucast (a :: 'a::len word) :: 'b::len word) :: 'c::len word) = ucast a"
by (metis down_cast_same ucast_def ucast_down_wi)
lemma scast_ucast_4:
"\<lbrakk> is_up (ucast :: 'a word \<Rightarrow> 'b word); is_down (ucast :: 'b word \<Rightarrow> 'c word) \<rbrakk> \<Longrightarrow>
(scast (ucast (a :: 'a::len word) :: 'b::len word) :: 'c::len word) = ucast a"
by (metis down_cast_same ucast_def ucast_down_wi)
lemma scast_scast_b:
"\<lbrakk> is_up (scast :: 'a word \<Rightarrow> 'b word) \<rbrakk> \<Longrightarrow>
(scast (scast (a :: 'a::len word) :: 'b::len word) :: 'c::len word) = scast a"
by (metis scast_def sint_up_scast)
lemma ucast_scast_1:
"\<lbrakk> is_down (scast :: 'a word \<Rightarrow> 'b word); is_down (ucast :: 'b word \<Rightarrow> 'c word) \<rbrakk> \<Longrightarrow>
(ucast (scast (a :: 'a::len word) :: 'b::len word) :: 'c::len word) = scast a"
by (metis scast_def ucast_down_wi)
lemma ucast_scast_4:
"\<lbrakk> is_up (scast :: 'a word \<Rightarrow> 'b word); is_down (ucast :: 'b word \<Rightarrow> 'c word) \<rbrakk> \<Longrightarrow>
(ucast (scast (a :: 'a::len word) :: 'b::len word) :: 'c::len word) = scast a"
by (metis down_cast_same scast_def sint_up_scast)
lemma ucast_ucast_a:
"\<lbrakk> is_down (ucast :: 'b word \<Rightarrow> 'c word) \<rbrakk> \<Longrightarrow>
(ucast (ucast (a :: 'a::len word) :: 'b::len word) :: 'c::len word) = ucast a"
by (metis down_cast_same ucast_def ucast_down_wi)
lemma ucast_ucast_b:
"\<lbrakk> is_up (ucast :: 'a word \<Rightarrow> 'b word) \<rbrakk> \<Longrightarrow>
(ucast (ucast (a :: 'a::len word) :: 'b::len word) :: 'c::len word) = ucast a"
by (metis ucast_up_ucast)
lemma scast_scast_a:
"\<lbrakk> is_down (scast :: 'b word \<Rightarrow> 'c word) \<rbrakk> \<Longrightarrow>
(scast (scast (a :: 'a::len word) :: 'b::len word) :: 'c::len word) = scast a"
apply (clarsimp simp: scast_def)
apply (metis down_cast_same is_up_down scast_def ucast_down_wi)
done
lemma scast_down_wi [OF refl]:
"uc = scast \<Longrightarrow> is_down uc \<Longrightarrow> uc (word_of_int x) = word_of_int x"
by (metis down_cast_same is_up_down ucast_down_wi)
lemmas cast_simps =
is_down is_up
scast_down_add scast_down_minus scast_down_mult
ucast_down_add ucast_down_minus ucast_down_mult
scast_ucast_1 scast_ucast_3 scast_ucast_4
ucast_scast_1 ucast_scast_3 ucast_scast_4
ucast_ucast_a ucast_ucast_b
scast_scast_a scast_scast_b
ucast_down_bl
ucast_down_wi scast_down_wi
ucast_of_nat scast_of_nat
uint_up_ucast sint_up_scast
up_scast_surj up_ucast_surj
lemma smod_mod_positive:
"\<lbrakk> 0 \<le> (a :: int); 0 \<le> b \<rbrakk> \<Longrightarrow> a smod b = a mod b"
by (clarsimp simp: smod_int_alt_def zsgn_def)
lemmas signed_shift_guard_simpler_32
= power_strict_increasing_iff[where b="2 :: nat" and y=31, simplified]
lemma nat_mult_power_less_eq:
"b > 0 \<Longrightarrow> (a * b ^ n < (b :: nat) ^ m) = (a < b ^ (m - n))"
using mult_less_cancel2[where m = a and k = "b ^ n" and n="b ^ (m - n)"]
mult_less_cancel2[where m="a * b ^ (n - m)" and k="b ^ m" and n=1]
apply (simp only: power_add[symmetric] nat_minus_add_max)
apply (simp only: power_add[symmetric] nat_minus_add_max ac_simps)
apply (simp add: max_def split: split_if_asm)
done
lemma signed_shift_guard_to_word:
"\<lbrakk> n < len_of TYPE ('a); n > 0 \<rbrakk>
\<Longrightarrow> (unat (x :: ('a :: len) word) * 2 ^ y < 2 ^ n)
= (x = 0 \<or> x < (1 << n >> y))"
apply (simp only: nat_mult_power_less_eq)
apply (cases "y \<le> n")
apply (simp only: shiftl_shiftr1)
apply (subst less_mask_eq)
apply (simp add: word_less_nat_alt word_size)
apply (rule order_less_le_trans[rotated], rule power_increasing[where n=1])
apply simp
apply simp
apply simp
apply (simp add: nat_mult_power_less_eq word_less_nat_alt word_size)
apply auto[1]
apply (simp only: shiftl_shiftr2, simp add: unat_eq_0)
done
lemma word32_31_less:
"31 < len_of TYPE (32 signed)" "31 > (0 :: nat)"
"31 < len_of TYPE (32)" "31 > (0 :: nat)"
by auto
lemmas signed_shift_guard_to_word_32
= signed_shift_guard_to_word[OF word32_31_less(1-2)]
signed_shift_guard_to_word[OF word32_31_less(3-4)]
lemma sint_ucast_eq_uint:
"\<lbrakk> \<not> is_down (ucast :: ('a::len word \<Rightarrow> 'b::len word)) \<rbrakk>
\<Longrightarrow> sint ((ucast :: ('a::len word \<Rightarrow> 'b::len word)) x) = uint x"
apply (subst sint_eq_uint)
apply (clarsimp simp: msb_nth nth_ucast is_down)
apply (metis Suc_leI Suc_pred bang_conj_lt len_gt_0)
apply (clarsimp simp: uint_up_ucast is_up is_down)
done
lemma word_less_nowrapI':
"(x :: 'a :: len0 word) \<le> z - k \<Longrightarrow> k \<le> z \<Longrightarrow> 0 < k \<Longrightarrow> x < x + k"
by uint_arith
lemma mask_plus_1:
"mask n + 1 = 2 ^ n"
by (clarsimp simp: mask_def)
lemma unat_inj: "inj unat"
by (metis eq_iff injI word_le_nat_alt)
lemma unat_ucast_upcast:
"is_up (ucast :: 'b word \<Rightarrow> 'a word)
\<Longrightarrow> unat (ucast x :: ('a::len) word) = unat (x :: ('b::len) word)"
unfolding ucast_def unat_def
apply (subst int_word_uint)
apply (subst mod_pos_pos_trivial)
apply simp
apply (rule lt2p_lem)
apply (clarsimp simp: is_up)
apply simp
done
lemma ucast_mono:
"\<lbrakk> (x :: 'b :: len word) < y; y < 2 ^ len_of TYPE('a) \<rbrakk>
\<Longrightarrow> ucast x < ((ucast y) :: ('a :: len) word)"
apply (simp add: ucast_nat_def [symmetric])
apply (rule of_nat_mono_maybe)
apply (rule unat_less_helper)
apply (simp add: Power.of_nat_power)
apply (simp add: word_less_nat_alt)
done
lemma ucast_mono_le:
"\<lbrakk>x \<le> y; y < 2 ^ len_of TYPE('b)\<rbrakk> \<Longrightarrow> (ucast (x :: 'a :: len word) :: 'b :: len word) \<le> ucast y"
apply (simp add: ucast_nat_def [symmetric])
apply (subst of_nat_mono_maybe_le[symmetric])
apply (rule unat_less_helper)
apply (simp add: Power.of_nat_power)
apply (rule unat_less_helper)
apply (erule le_less_trans)
apply (simp add: Power.of_nat_power)
apply (simp add: word_le_nat_alt)
done
lemma zero_sle_ucast_up:
"\<not> is_down (ucast :: 'a word \<Rightarrow> 'b signed word) \<Longrightarrow>
(0 <=s ((ucast (b::('a::len) word)) :: ('b::len) signed word))"
apply (subgoal_tac "\<not> msb (ucast b :: 'b signed word)")
apply (clarsimp simp: word_sle_msb_le)
apply (clarsimp simp: is_down not_le msb_nth nth_ucast)
apply (subst (asm) bang_conj_lt [symmetric])
apply clarsimp
apply arith
done
lemma msb_ucast_eq:
"len_of TYPE('a) = len_of TYPE('b) \<Longrightarrow>
msb (ucast x :: ('a::len) word) = msb (x :: ('b::len) word)"
apply (clarsimp simp: word_msb_alt)
apply (subst ucast_down_drop [where n=0])
apply (clarsimp simp: source_size_def target_size_def word_size)
apply clarsimp
done
lemma msb_big:
"msb (a :: ('a::len) word) = (a \<ge> 2 ^ (len_of TYPE('a) - Suc 0))"
apply (rule iffI)
apply (clarsimp simp: msb_nth)
apply (drule bang_is_le)
apply simp
apply (rule ccontr)
apply (subgoal_tac "a = a && mask (len_of TYPE('a) - Suc 0)")
apply (cut_tac and_mask_less' [where w=a and n="len_of TYPE('a) - Suc 0"])
apply (clarsimp simp: word_not_le [symmetric])
apply clarsimp
apply (rule sym, subst and_mask_eq_iff_shiftr_0)
apply (clarsimp simp: msb_shift)
done
lemma zero_sle_ucast:
"(0 <=s ((ucast (b::('a::len) word)) :: ('a::len) signed word))
= (uint b < 2 ^ (len_of (TYPE('a)) - 1))"
apply (case_tac "msb b")
apply (clarsimp simp: word_sle_msb_le not_less msb_ucast_eq del: notI)
apply (clarsimp simp: msb_big word_le_def uint_2p_alt)
apply (clarsimp simp: word_sle_msb_le not_less msb_ucast_eq del: notI)
apply (clarsimp simp: msb_big word_le_def uint_2p_alt)
done
(* to_bool / from_bool. *)
definition
from_bool :: "bool \<Rightarrow> 'a::len word" where
"from_bool b \<equiv> case b of True \<Rightarrow> of_nat 1
| False \<Rightarrow> of_nat 0"
lemma from_bool_0:
"(from_bool x = 0) = (\<not> x)"
by (simp add: from_bool_def split: bool.split)
definition
to_bool :: "'a::len word \<Rightarrow> bool" where
"to_bool \<equiv> (op \<noteq>) 0"
lemma to_bool_and_1:
"to_bool (x && 1) = (x !! 0)"
apply (simp add: to_bool_def del: word_neq_0_conv)
apply (rule iffI)
apply (rule classical, erule notE, rule word_eqI)
apply clarsimp
apply (case_tac n, simp_all)[1]
apply (rule notI, drule word_eqD[where x=0])
apply simp
done
lemma to_bool_from_bool:
"to_bool (from_bool r) = r"
unfolding from_bool_def to_bool_def
by (simp split: bool.splits)
lemma from_bool_neq_0:
"(from_bool b \<noteq> 0) = b"
by (simp add: from_bool_def split: bool.splits)
lemma from_bool_mask_simp:
"((from_bool r) :: word32) && 1 = from_bool r"
unfolding from_bool_def
apply (clarsimp split: bool.splits)
done
lemma scast_from_bool:
"scast (from_bool P::word32) = (from_bool P::word32)"
by (clarsimp simp: from_bool_def scast_id split: bool.splits)
lemma from_bool_1:
"(from_bool P = 1) = P"
by (simp add: from_bool_def split: bool.splits)
lemma ge_0_from_bool:
"(0 < from_bool P) = P"
by (simp add: from_bool_def split: bool.splits)
lemma limited_and_from_bool:
"limited_and (from_bool b) 1"
by (simp add: from_bool_def limited_and_def split: bool.split)
lemma to_bool_1 [simp]: "to_bool 1" by (simp add: to_bool_def)
lemma to_bool_0 [simp]: "\<not>to_bool 0" by (simp add: to_bool_def)
lemma from_bool_eq_if:
"(from_bool Q = (if P then 1 else 0)) = (P = Q)"
by (simp add: case_bool_If from_bool_def split: split_if)
lemma to_bool_eq_0:
"(\<not> to_bool x) = (x = 0)"
by (simp add: to_bool_def)
lemma to_bool_neq_0:
"(to_bool x) = (x \<noteq> 0)"
by (simp add: to_bool_def)
lemma from_bool_all_helper:
"(\<forall>bool. from_bool bool = val \<longrightarrow> P bool)
= ((\<exists>bool. from_bool bool = val) \<longrightarrow> P (val \<noteq> 0))"
by (auto simp: from_bool_0)
lemma word_rsplit_upt:
"\<lbrakk> size x = len_of TYPE('a :: len) * n; n \<noteq> 0 \<rbrakk>
\<Longrightarrow> word_rsplit x = map (\<lambda>i. ucast (x >> i * len_of TYPE ('a)) :: 'a word) (rev [0 ..< n])"
apply (subgoal_tac "length (word_rsplit x :: 'a word list) = n")
apply (rule nth_equalityI, simp)
apply (intro allI word_eqI impI)
apply (simp add: test_bit_rsplit_alt word_size)
apply (simp add: nth_ucast nth_shiftr nth_rev field_simps)
apply (simp add: length_word_rsplit_exp_size)
apply (metis mult.commute given_quot_alt word_size word_size_gt_0)
done
end
|
! { dg-do compile }
!
! PR 48291: [4.6/4.7 Regression] [OOP] internal compiler error, new_symbol(): Symbol name too long
!
! Contributed by Adrian Prantl <[email protected]>
module Overload_AnException_Impl
type :: Overload_AnException_impl_t
end type
contains
subroutine ctor_impl(self)
class(Overload_AnException_impl_t) :: self
end subroutine
end module
|
The view that a game of chess should end in a draw given best play prevails . Even if it cannot be proved , this assumption is considered " safe " by Rowson and " logical " by Adorján . Watson agrees that " the proper result of a perfectly played chess game ... is a draw . ... Of course , I can 't prove this , but I doubt that you can find a single strong player who would disagree . ... I remember Kasparov , after a last @-@ round draw , explaining to the waiting reporters : ' Well , chess is a draw . ' " World Champion Bobby Fischer thought that was almost definitely so .
|
(*
Copyright (C) 2017 M.A.L. Marques
This Source Code Form is subject to the terms of the Mozilla Public
License, v. 2.0. If a copy of the MPL was not distributed with this
file, You can obtain one at http://mozilla.org/MPL/2.0/.
*)
(* type: gga_exc *)
(* prefix:
gga_x_rpbe_params *params;
assert(p->params != NULL);
params = (gga_x_rpbe_params * )(p->params);
*)
$ifdef gga_x_rpbe_params
params_a_rpbe_kappa := KAPPA_PBE:
params_a_rpbe_mu := MU_PBE:
$endif
rpbe_f0 := s -> 1 + params_a_rpbe_kappa * (
1 - exp(-params_a_rpbe_mu*s^2/params_a_rpbe_kappa)
):
rpbe_f := x -> rpbe_f0(X2S*x):
f := (rs, zeta, xt, xs0, xs1) -> gga_exchange(rpbe_f, rs, zeta, xs0, xs1):
|
module RecordInWhere2
fd : Int
fd = 5 where
data X : Type where
gd : Int
gd = 5 where
data X : Type where
fr : Int
fr = 5 where
record X where
gr : Int
gr = 5 where
record X where
|
lemma real_polynomial_function_separable: fixes x :: "'a::euclidean_space" assumes "x \<noteq> y" shows "\<exists>f. real_polynomial_function f \<and> f x \<noteq> f y" |
/-
Copyright (c) 2021 Bhavik Mehta, Yaël Dillies. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Bhavik Mehta, Alena Gusakov, Yaël Dillies
-/
import algebra.big_operators.basic
import data.nat.interval
import order.antichain
/-!
# `r`-sets and slice
This file defines the `r`-th slice of a set family and provides a way to say that a set family is
made of `r`-sets.
An `r`-set is a finset of cardinality `r` (aka of *size* `r`). The `r`-th slice of a set family is
the set family made of its `r`-sets.
## Main declarations
* `set.sized`: `A.sized r` means that `A` only contains `r`-sets.
* `finset.slice`: `A.slice r` is the set of `r`-sets in `A`.
## Notation
`A # r` is notation for `A.slice r` in locale `finset_family`.
-/
open finset nat
open_locale big_operators
variables {α : Type*} {ι : Sort*} {κ : ι → Sort*}
namespace set
variables {A B : set (finset α)} {r : ℕ}
/-! ### Families of `r`-sets -/
/-- `sized r A` means that every finset in `A` has size `r`. -/
def sized (r : ℕ) (A : set (finset α)) : Prop := ∀ ⦃x⦄, x ∈ A → card x = r
lemma sized.mono (h : A ⊆ B) (hB : B.sized r) : A.sized r := λ x hx, hB $ h hx
lemma sized_union : (A ∪ B).sized r ↔ A.sized r ∧ B.sized r :=
⟨λ hA, ⟨hA.mono $ subset_union_left _ _, hA.mono $ subset_union_right _ _⟩,
λ hA x hx, hx.elim (λ h, hA.1 h) $ λ h, hA.2 h⟩
alias sized_union ↔ _ set.sized.union
--TODO: A `forall_Union` lemma would be handy here.
@[simp] lemma sized_Union {f : ι → set (finset α)} : (⋃ i, f i).sized r ↔ ∀ i, (f i).sized r :=
by { simp_rw [set.sized, set.mem_Union, forall_exists_index], exact forall_swap }
@[simp] lemma sized_Union₂ {f : Π i, κ i → set (finset α)} :
(⋃ i j, f i j).sized r ↔ ∀ i j, (f i j).sized r :=
by simp_rw sized_Union
protected lemma sized.is_antichain (hA : A.sized r) : is_antichain (⊆) A :=
λ s hs t ht h hst, h $ finset.eq_of_subset_of_card_le hst ((hA ht).trans (hA hs).symm).le
protected lemma sized.subsingleton (hA : A.sized 0) : A.subsingleton :=
subsingleton_of_forall_eq ∅ $ λ s hs, card_eq_zero.1 $ hA hs
lemma sized.subsingleton' [fintype α] (hA : A.sized (fintype.card α)) : A.subsingleton :=
subsingleton_of_forall_eq finset.univ $ λ s hs, s.card_eq_iff_eq_univ.1 $ hA hs
lemma sized.empty_mem_iff (hA : A.sized r) : ∅ ∈ A ↔ A = {∅} := hA.is_antichain.bot_mem_iff
lemma sized.univ_mem_iff [fintype α] (hA : A.sized r) : finset.univ ∈ A ↔ A = {finset.univ} :=
hA.is_antichain.top_mem_iff
lemma sized_powerset_len (s : finset α) (r : ℕ) : (powerset_len r s : set (finset α)).sized r :=
λ t ht, (mem_powerset_len.1 ht).2
end set
namespace finset
section sized
variables [fintype α] {𝒜 : finset (finset α)} {s : finset α} {r : ℕ}
lemma subset_powerset_len_univ_iff : 𝒜 ⊆ powerset_len r univ ↔ (𝒜 : set (finset α)).sized r :=
forall_congr $ λ A, by rw [mem_powerset_len_univ_iff, mem_coe]
alias subset_powerset_len_univ_iff ↔ _ set.sized.subset_powerset_len_univ
lemma _root_.set.sized.card_le (h𝒜 : (𝒜 : set (finset α)).sized r) :
card 𝒜 ≤ (fintype.card α).choose r :=
begin
rw [fintype.card, ←card_powerset_len],
exact card_le_of_subset h𝒜.subset_powerset_len_univ,
end
end sized
/-! ### Slices -/
section slice
variables {𝒜 : finset (finset α)} {A A₁ A₂ : finset α} {r r₁ r₂ : ℕ}
/-- The `r`-th slice of a set family is the subset of its elements which have cardinality `r`. -/
def slice (𝒜 : finset (finset α)) (r : ℕ) : finset (finset α) := 𝒜.filter (λ i, i.card = r)
localized "infix ` # `:90 := finset.slice" in finset_family
/-- `A` is in the `r`-th slice of `𝒜` iff it's in `𝒜` and has cardinality `r`. -/
lemma mem_slice : A ∈ 𝒜 # r ↔ A ∈ 𝒜 ∧ A.card = r := mem_filter
/-- The `r`-th slice of `𝒜` is a subset of `𝒜`. -/
lemma slice_subset : 𝒜 # r ⊆ 𝒜 := filter_subset _ _
/-- Everything in the `r`-th slice of `𝒜` has size `r`. -/
lemma sized_slice : (𝒜 # r : set (finset α)).sized r := λ _, and.right ∘ mem_slice.mp
lemma eq_of_mem_slice (h₁ : A ∈ 𝒜 # r₁) (h₂ : A ∈ 𝒜 # r₂) : r₁ = r₂ :=
(sized_slice h₁).symm.trans $ sized_slice h₂
/-- Elements in distinct slices must be distinct. -/
lemma ne_of_mem_slice (h₁ : A₁ ∈ 𝒜 # r₁) (h₂ : A₂ ∈ 𝒜 # r₂) : r₁ ≠ r₂ → A₁ ≠ A₂ :=
mt $ λ h, (sized_slice h₁).symm.trans ((congr_arg card h).trans (sized_slice h₂))
lemma pairwise_disjoint_slice [decidable_eq α] : (set.univ : set ℕ).pairwise_disjoint (slice 𝒜) :=
λ m _ n _ hmn, disjoint_filter.2 $ λ s hs hm hn, hmn $ hm.symm.trans hn
variables [fintype α] (𝒜)
@[simp] lemma bUnion_slice [decidable_eq α] : (Iic $ fintype.card α).bUnion 𝒜.slice = 𝒜 :=
subset.antisymm (bUnion_subset.2 $ λ r _, slice_subset) $ λ s hs,
mem_bUnion.2 ⟨s.card, mem_Iic.2 $ s.card_le_univ, mem_slice.2 $ ⟨hs, rfl⟩⟩
@[simp] lemma sum_card_slice : ∑ r in Iic (fintype.card α), (𝒜 # r).card = 𝒜.card :=
by { rw [←card_bUnion (finset.pairwise_disjoint_slice.subset (set.subset_univ _)), bUnion_slice],
exact classical.dec_eq _ }
end slice
end finset
|
(*
Copyright (C) 2017 M.A.L. Marques
This Source Code Form is subject to the terms of the Mozilla Public
License, v. 2.0. If a copy of the MPL was not distributed with this
file, You can obtain one at http://mozilla.org/MPL/2.0/.
*)
(* type: gga_exc *)
$include "lda_c_rc04.mpl"
msigma := 1.43:
malpha := 2.30:
Bs := s -> 1/(1 + msigma*s^malpha):
f_tcs := (rs, z, xt) -> f_rc04(rs, z)*Bs(X2S*2^(1/3)*xt):
f := (rs, z, xt, xs0, xs1) ->
f_tcs(rs, z, xt):
|
! H0 XXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXX
! H0 X
! H0 X libAtoms+QUIP: atomistic simulation library
! H0 X
! H0 X Portions of this code were written by
! H0 X Albert Bartok-Partay, Silvia Cereda, Gabor Csanyi, James Kermode,
! H0 X Ivan Solt, Wojciech Szlachta, Csilla Varnai, Steven Winfield.
! H0 X
! H0 X Copyright 2006-2010.
! H0 X
! H0 X These portions of the source code are released under the GNU General
! H0 X Public License, version 2, http://www.gnu.org/copyleft/gpl.html
! H0 X
! H0 X If you would like to license the source code under different terms,
! H0 X please contact Gabor Csanyi, [email protected]
! H0 X
! H0 X Portions of this code were written by Noam Bernstein as part of
! H0 X his employment for the U.S. Government, and are not subject
! H0 X to copyright in the USA.
! H0 X
! H0 X
! H0 X When using this software, please cite the following reference:
! H0 X
! H0 X http://www.libatoms.org
! H0 X
! H0 X Additional contributions by
! H0 X Alessio Comisso, Chiara Gattinoni, and Gianpietro Moras
! H0 X
! H0 XXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXX
!X
!X IPModel_Morse module
!X
!% Module for Morse pair potential.
!% \begin{equation}
!% \nonumber
!% V(r) = D \left( \exp(-2 \alpha (r-r_0)) - 2 \exp( -\alpha (r-r_0)) \right)
!% \end{equation}
!%
!% The IPModel_Morse object contains all the parameters read from a
!% 'Morse_params' XML stanza.
!X
!XXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXX
!XXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXX
#include "error.inc"
module IPModel_Morse_module
use error_module
use system_module, only : dp, inoutput, print, verbosity_push_decrement, verbosity_pop, split_string_simple, operator(//)
use dictionary_module
use paramreader_module
use linearalgebra_module
use atoms_types_module
use atoms_module
use mpi_context_module
use QUIP_Common_module
implicit none
private
include 'IPModel_interface.h'
public :: IPModel_Morse
type IPModel_Morse
integer :: n_types = 0 !% Number of atomic types.
integer, allocatable :: atomic_num(:), type_of_atomic_num(:) !% Atomic number dimensioned as \texttt{n_types}.
real(dp) :: cutoff = 0.0_dp !% Cutoff for computing connection.
real(dp), allocatable :: D(:,:), alpha(:,:), r0(:,:), cutoff_a(:,:) !% IP parameters.
character(len=STRING_LENGTH) label
end type IPModel_Morse
logical, private :: parse_in_ip, parse_matched_label
type(IPModel_Morse), private, pointer :: parse_ip
interface Initialise
module procedure IPModel_Morse_Initialise_str
end interface Initialise
interface Finalise
module procedure IPModel_Morse_Finalise
end interface Finalise
interface Print
module procedure IPModel_Morse_Print
end interface Print
interface Calc
module procedure IPModel_Morse_Calc
end interface Calc
contains
subroutine IPModel_Morse_Initialise_str(this, args_str, param_str)
type(IPModel_Morse), intent(inout) :: this
character(len=*), intent(in) :: args_str, param_str
type(Dictionary) :: params
call Finalise(this)
call initialise(params)
this%label = ''
call param_register(params, 'label', '', this%label, help_string="No help yet. This source file was $LastChangedBy$")
if (.not. param_read_line(params, args_str, ignore_unknown=.true.,task='IPModel_Morse_Initialise_str args_str')) then
call system_abort("IPModel_Morse_Initialise_str failed to parse label from args_str="//trim(args_str))
endif
call finalise(params)
call IPModel_Morse_read_params_xml(this, param_str)
this%cutoff = maxval(this%cutoff_a)
end subroutine IPModel_Morse_Initialise_str
subroutine IPModel_Morse_Finalise(this)
type(IPModel_Morse), intent(inout) :: this
if (allocated(this%atomic_num)) deallocate(this%atomic_num)
if (allocated(this%type_of_atomic_num)) deallocate(this%type_of_atomic_num)
if (allocated(this%r0)) deallocate(this%r0)
if (allocated(this%D)) deallocate(this%D)
if (allocated(this%alpha)) deallocate(this%alpha)
if (allocated(this%cutoff_a)) deallocate(this%cutoff_a)
this%n_types = 0
this%label = ''
end subroutine IPModel_Morse_Finalise
!XXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXX
!X
!% The potential calculator: this routine computes energy, forces and the virial.
!X
!XXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXX
subroutine IPModel_Morse_Calc(this, at, e, local_e, f, virial, local_virial, args_str, mpi, error)
type(IPModel_Morse), intent(inout) :: this
type(Atoms), intent(inout) :: at
real(dp), intent(out), optional :: e, local_e(:) !% \texttt{e} = System total energy, \texttt{local_e} = energy of each atom, vector dimensioned as \texttt{at%N}.
real(dp), intent(out), optional :: f(:,:), local_virial(:,:) !% Forces, dimensioned as \texttt{f(3,at%N)}, local virials, dimensioned as \texttt{local_virial(9,at%N)}
real(dp), intent(out), optional :: virial(3,3) !% Virial
character(len=*), intent(in), optional :: args_str
type(MPI_Context), intent(in), optional :: mpi
integer, intent(out), optional :: error
real(dp), pointer :: w_e(:)
integer i, ji, j, ti, tj
real(dp) :: dr(3), dr_mag
real(dp) :: de, de_dr
logical :: i_is_min_image
integer :: i_calc, n_extra_calcs
character(len=20) :: extra_calcs_list(10)
logical :: do_flux = .false.
real(dp), pointer :: velo(:,:)
real(dp) :: flux(3)
type(Dictionary) :: params
logical :: has_atom_mask_name
character(STRING_LENGTH) :: atom_mask_name
real(dp) :: r_scale, E_scale
logical :: do_rescale_r, do_rescale_E
INIT_ERROR(error)
if (present(e)) e = 0.0_dp
if (present(local_e)) then
call check_size('Local_E',local_e,(/at%N/),'IPModel_Morse_Calc', error)
local_e = 0.0_dp
endif
if (present(f)) then
call check_size('Force',f,(/3,at%N/),'IPModel_Morse_Calc', error)
f = 0.0_dp
end if
if (present(virial)) virial = 0.0_dp
if (present(local_virial)) then
call check_size('Local_virial',local_virial,(/9,at%N/),'IPModel_Morse_Calc', error)
local_virial = 0.0_dp
RAISE_ERROR("IPModel_Morse_Calc: local_virial calculation requested but not supported yet.", error)
endif
if (present(args_str)) then
if (len_trim(args_str) > 0) then
n_extra_calcs = parse_extra_calcs(args_str, extra_calcs_list)
if (n_extra_calcs > 0) then
do i_calc=1, n_extra_calcs
select case(trim(extra_calcs_list(i_calc)))
case("flux")
if (.not. assign_pointer(at, "velo", velo)) &
call system_abort("IPModel_Morse_Calc Flux calculation requires velo field")
do_flux = .true.
flux = 0.0_dp
case default
call system_abort("Unsupported extra_calc '"//trim(extra_calcs_list(i_calc))//"'")
end select
end do
endif ! n_extra_calcs
endif ! len_trim(args_str)
call initialise(params)
call param_register(params, 'atom_mask_name', 'NONE', atom_mask_name, has_value_target=has_atom_mask_name, help_string="No help yet. This source file was $LastChangedBy$")
call param_register(params, 'r_scale', '1.0',r_scale, has_value_target=do_rescale_r, help_string="Recaling factor for distances. Default 1.0.")
call param_register(params, 'E_scale', '1.0',E_scale, has_value_target=do_rescale_E, help_string="Recaling factor for energy. Default 1.0.")
if(.not. param_read_line(params, args_str, ignore_unknown=.true.,task='IPModel_Morse_Calc args_str')) then
RAISE_ERROR("IPModel_Morse_Calc failed to parse args_str='"//trim(args_str)//"'",error)
endif
call finalise(params)
if(has_atom_mask_name) then
RAISE_ERROR('IPModel_Morse_Calc: atom_mask_name found, but not supported', error)
endif
if (do_rescale_r .or. do_rescale_E) then
RAISE_ERROR("IPModel_Morse_Calc: rescaling of potential with r_scale and E_scale not yet implemented!", error)
end if
endif ! present(args_str)
if (.not. assign_pointer(at, "weight", w_e)) nullify(w_e)
do i = 1, at%N
i_is_min_image = at%connect%is_min_image(i)
if (present(mpi)) then
if (mpi%active) then
if (mod(i-1, mpi%n_procs) /= mpi%my_proc) cycle
endif
endif
do ji = 1, n_neighbours(at, i)
j = neighbour(at, i, ji, dr_mag, cosines = dr)
if (dr_mag .feq. 0.0_dp) cycle
if ((i < j) .and. i_is_min_image) cycle
ti = get_type(this%type_of_atomic_num, at%Z(i))
tj = get_type(this%type_of_atomic_num, at%Z(j))
if (present(e) .or. present(local_e)) then
de = IPModel_Morse_pairenergy(this, ti, tj, dr_mag)
if (present(local_e)) then
local_e(i) = local_e(i) + 0.5_dp*de
if(i_is_min_image) local_e(j) = local_e(j) + 0.5_dp*de
endif
if (present(e)) then
if (associated(w_e)) then
de = de*0.5_dp*(w_e(i)+w_e(j))
endif
if(i_is_min_image) then
e = e + de
else
e = e + 0.5_dp*de
endif
endif
endif
if (present(f) .or. present(virial) .or. do_flux) then
de_dr = IPModel_Morse_pairenergy_deriv(this, ti, tj, dr_mag)
if (associated(w_e)) then
de_dr = de_dr*0.5_dp*(w_e(i)+w_e(j))
endif
if (present(f)) then
f(:,i) = f(:,i) + de_dr*dr
if(i_is_min_image) f(:,j) = f(:,j) - de_dr*dr
endif
if (do_flux) then
! -0.5 (v_i + v_j) . F_ij * dr_ij
flux = flux - 0.5_dp*sum((velo(:,i)+velo(:,j))*(de_dr*dr))*(dr*dr_mag)
endif
if (present(virial)) then
if(i_is_min_image) then
virial = virial - de_dr*(dr .outer. dr)*dr_mag
else
virial = virial - 0.5_dp*de_dr*(dr .outer. dr)*dr_mag
endif
endif
endif
end do
end do
if (present(mpi)) then
if (present(e)) e = sum(mpi, e)
if (present(local_e)) call sum_in_place(mpi, local_e)
if (present(virial)) call sum_in_place(mpi, virial)
if (present(f)) call sum_in_place(mpi, f)
endif
if (do_flux) then
flux = flux / cell_volume(at)
if (present(mpi)) call sum_in_place(mpi, flux)
call set_value(at%params, "Flux", flux)
endif
end subroutine IPModel_Morse_Calc
!% This routine computes the two-body term for a pair of atoms separated by a distance r.
function IPModel_Morse_pairenergy(this, ti, tj, r)
type(IPModel_Morse), intent(in) :: this
integer, intent(in) :: ti, tj !% Atomic types.
real(dp), intent(in) :: r !% Distance.
real(dp) :: IPModel_Morse_pairenergy
real(dp) :: texp
if ((r .feq. 0.0_dp) .or. (r > this%cutoff_a(ti,tj))) then
IPModel_Morse_pairenergy = 0.0
return
endif
texp = exp(-this%alpha(ti,tj)*(r-this%r0(ti,tj)))
IPModel_Morse_pairenergy = this%D(ti,tj) * ( texp*texp - 2.0_dp*texp )
end function IPModel_Morse_pairenergy
!% Derivative of the two-body term.
function IPModel_Morse_pairenergy_deriv(this, ti, tj, r)
type(IPModel_Morse), intent(in) :: this
integer, intent(in) :: ti, tj !% Atomic types.
real(dp), intent(in) :: r !% Distance.
real(dp) :: IPModel_Morse_pairenergy_deriv
real(dp) :: texp, texp_d
if ((r .feq. 0.0_dp) .or. (r > this%cutoff_a(ti,tj))) then
IPModel_Morse_pairenergy_deriv = 0.0
return
endif
texp = exp(-this%alpha(ti,tj)*(r-this%r0(ti,tj)))
texp_d = -this%alpha(ti,tj) * texp
IPModel_Morse_pairenergy_deriv = this%D(ti,tj) * ( 2.0_dp*texp*texp_d - 2.0_dp * texp_d)
end function IPModel_Morse_pairenergy_deriv
!XXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXX
!X
!% XML param reader functions.
!% An example for XML stanza is given below. Please notice that
!% these are simply dummy parameters for testing purposes, with no physical meaning.
!%
!%> <Morse_params n_types="2" label="default">
!%> <pair atnum_i="6" atnum_j="6" r0="2.0" D="1.0" alpha="0.1" cutoff="4.0" />
!%> <pair atnum_i="6" atnum_j="14" r0="2.5" D="2.0" alpha="0.15" cutoff="5.0" />
!%> <pair atnum_i="14" atnum_j="14" r0="3.0" D="3.0" alpha="0.2" cutoff="6.0" />
!%> </Morse_params>
!%
!% cutoff defaults to r0 - log(1e-8) / alpha
!X
!XXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXX
subroutine IPModel_startElement_handler(URI, localname, name, attributes)
character(len=*), intent(in) :: URI
character(len=*), intent(in) :: localname
character(len=*), intent(in) :: name
type(dictionary_t), intent(in) :: attributes
integer :: status
character(len=1024) :: value
integer atnum_i, atnum_j, ti, tj, ti_a(1)
if (name == 'Morse_params') then ! new Morse stanza
if (parse_in_ip) &
call system_abort("IPModel_startElement_handler entered Morse_params with parse_in true. Probably a bug in FoX (4.0.1, e.g.)")
if (parse_matched_label) return ! we already found an exact match for this label
call QUIP_FoX_get_value(attributes, 'label', value, status)
if (status /= 0) value = ''
if (len(trim(parse_ip%label)) > 0) then ! we were passed in a label
if (value == parse_ip%label) then ! exact match
parse_matched_label = .true.
parse_in_ip = .true.
else ! no match
parse_in_ip = .false.
endif
else ! no label passed in
parse_in_ip = .true.
endif
if (parse_in_ip) then
if (parse_ip%n_types /= 0) then
call finalise(parse_ip)
endif
call QUIP_FoX_get_value(attributes, 'n_types', value, status)
if (status == 0) then
read (value, *) parse_ip%n_types
else
call system_abort("Can't find n_types in Morse_params")
endif
allocate(parse_ip%atomic_num(parse_ip%n_types))
parse_ip%atomic_num = 0
allocate(parse_ip%r0(parse_ip%n_types,parse_ip%n_types))
allocate(parse_ip%D(parse_ip%n_types,parse_ip%n_types))
allocate(parse_ip%alpha(parse_ip%n_types,parse_ip%n_types))
allocate(parse_ip%cutoff_a(parse_ip%n_types,parse_ip%n_types))
parse_ip%r0 = 0.0_dp
parse_ip%D = 0.0_dp
parse_ip%alpha = 0.0_dp
endif ! parse_in_ip
elseif (parse_in_ip .and. name == 'pair') then
call QUIP_FoX_get_value(attributes, "atnum_i", value, status)
if (status /= 0) call system_abort ("IPModel_Morse_read_params_xml cannot find atnum_i")
read (value, *) atnum_i
call QUIP_FoX_get_value(attributes, "atnum_j", value, status)
if (status /= 0) call system_abort ("IPModel_Morse_read_params_xml cannot find atnum_j")
read (value, *) atnum_j
if (all(parse_ip%atomic_num /= atnum_i)) then
ti_a = minloc(parse_ip%atomic_num)
parse_ip%atomic_num(ti_a(1)) = atnum_i
endif
if (all(parse_ip%atomic_num /= atnum_j)) then
ti_a = minloc(parse_ip%atomic_num)
parse_ip%atomic_num(ti_a(1)) = atnum_j
endif
if (allocated(parse_ip%type_of_atomic_num)) deallocate(parse_ip%type_of_atomic_num)
allocate(parse_ip%type_of_atomic_num(maxval(parse_ip%atomic_num)))
parse_ip%type_of_atomic_num = 0
do ti=1, parse_ip%n_types
if (parse_ip%atomic_num(ti) > 0) &
parse_ip%type_of_atomic_num(parse_ip%atomic_num(ti)) = ti
end do
ti = parse_ip%type_of_atomic_num(atnum_i)
tj = parse_ip%type_of_atomic_num(atnum_j)
call QUIP_FoX_get_value(attributes, "r0", value, status)
if (status /= 0) call system_abort ("IPModel_Morse_read_params_xml cannot find r0")
read (value, *) parse_ip%r0(ti,tj)
call QUIP_FoX_get_value(attributes, "D", value, status)
if (status /= 0) call system_abort ("IPModel_Morse_read_params_xml cannot find D")
read (value, *) parse_ip%D(ti,tj)
call QUIP_FoX_get_value(attributes, "alpha", value, status)
if (status /= 0) call system_abort ("IPModel_Morse_read_params_xml cannot find alpha")
read (value, *) parse_ip%alpha(ti,tj)
parse_ip%cutoff_a(ti,tj) = -1.0_dp
call QUIP_FoX_get_value(attributes, "cutoff", value, status)
if (status == 0) read (value, *) parse_ip%cutoff_a(ti,tj)
if (ti /= tj) then
parse_ip%r0(tj,ti) = parse_ip%r0(ti,tj)
parse_ip%D(tj,ti) = parse_ip%D(ti,tj)
parse_ip%alpha(tj,ti) = parse_ip%alpha(ti,tj)
parse_ip%cutoff_a(tj,ti) = parse_ip%cutoff_a(ti,tj)
endif
endif ! parse_in_ip .and. name = 'Morse'
end subroutine IPModel_startElement_handler
subroutine IPModel_endElement_handler(URI, localname, name)
character(len=*), intent(in) :: URI
character(len=*), intent(in) :: localname
character(len=*), intent(in) :: name
if (parse_in_ip) then
if (name == 'Morse_params') then
parse_in_ip = .false.
end if
endif
end subroutine IPModel_endElement_handler
subroutine IPModel_Morse_read_params_xml(this, param_str)
type(IPModel_Morse), intent(inout), target :: this
character(len=*), intent(in) :: param_str
type(xml_t) :: fxml
integer :: ti, tj
if (len(trim(param_str)) <= 0) return
parse_in_ip = .false.
parse_matched_label = .false.
parse_ip => this
call open_xml_string(fxml, param_str)
call parse(fxml, &
startElement_handler = IPModel_startElement_handler, &
endElement_handler = IPModel_endElement_handler)
call close_xml_t(fxml)
if (this%n_types == 0) then
call system_abort("IPModel_Morse_read_params_xml parsed file, but n_types = 0")
endif
! default cutoff:
! exp(-alpha (r-r0)) < 1e-8
! -alpha (r-r0) < log(1e-8)
! r - r0 < log(1e-8) / -alpha
! r < r0 + log(1e-8) / -alpha
! r < r0 + 13.8 / alpha
do ti=1, this%n_types
do tj=1, this%n_types
if (this%cutoff_a(ti,tj) < 0.0_dp) this%cutoff_a(ti,tj) = this%r0(ti,tj) - log(1.0e-8_dp) / this%alpha(ti,tj)
end do
end do
this%cutoff = maxval(this%cutoff_a)
end subroutine IPModel_Morse_read_params_xml
!XXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXX
!X
!% Printing of Morse parameters: number of different types, cutoff radius, atomic numbers, etc.
!X
!XXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXX
subroutine IPModel_Morse_Print (this, file)
type(IPModel_Morse), intent(in) :: this
type(Inoutput), intent(inout),optional :: file
integer :: ti, tj
call Print("IPModel_Morse : Morse", file=file)
call Print("IPModel_Morse : n_types = " // this%n_types // " cutoff = " // this%cutoff, file=file)
do ti=1, this%n_types
call Print ("IPModel_Morse : type " // ti // " atomic_num " // this%atomic_num(ti), file=file)
call verbosity_push_decrement()
do tj=1, this%n_types
call Print ("IPModel_Morse : interaction " // ti // " " // tj // " r0 " // this%r0(ti,tj) // " D " // &
this%D(ti,tj) // " alpha " // this%alpha(ti,tj) // " cutoff_a " // this%cutoff_a(ti,tj), file=file)
end do
call verbosity_pop()
end do
end subroutine IPModel_Morse_Print
function parse_extra_calcs(args_str, extra_calcs_list) result(n_extra_calcs)
character(len=*), intent(in) :: args_str
character(len=*), intent(out) :: extra_calcs_list(:)
integer :: n_extra_calcs
character(len=STRING_LENGTH) :: extra_calcs_str
type(Dictionary) :: params
n_extra_calcs = 0
call initialise(params)
call param_register(params, "extra_calcs", "", extra_calcs_str, help_string="No help yet. This source file was $LastChangedBy$")
if (param_read_line(params, args_str, ignore_unknown=.true.,task='parse_extra_calcs')) then
if (len_trim(extra_calcs_str) > 0) then
call split_string_simple(extra_calcs_str, extra_calcs_list, n_extra_calcs, ":")
end if
end if
call finalise(params)
end function parse_extra_calcs
end module IPModel_Morse_module
|
function Retain_Img = RemoveMinorCC(SegImg,reject_T)
if nargin < 2
reject_T = 0.2;
end
Retain_Img = zeros(size(SegImg));
% class 2
Img_C2 = (SegImg==2);
Retain_C2 = Connection_Judge_3D(Img_C2,reject_T);
Retain_C2 = imfill(Retain_C2,'hole');
Retain_Img(Retain_C2) = 2;
% class 1
Img_C1 = (SegImg==1);
Retain_C1 = Connection_Judge_3D(Img_C1,reject_T);
Retain_C1 = imfill(Retain_C1,'hole');
Retain_Img(Retain_C1) = 1;
end |
{-# OPTIONS --without-K --safe #-}
open import Algebra.Structures.Bundles.Field
module Algebra.Linear.Construct.Matrix
{k ℓ} (K : Field k ℓ)
where
open import Level using (_⊔_)
open import Data.Product hiding (map)
open import Data.Fin using (Fin; toℕ; fromℕ; _≤_)
open import Data.Fin.Properties using (¬Fin0)
open import Data.Nat hiding (_⊔_; _≤_) renaming (_+_ to _+ℕ_; _*_ to _*ℕ_)
open import Data.Nat.Properties using (1+n≢0)
open import Relation.Binary
import Data.Vec.Relation.Binary.Pointwise.Inductive as PW
open import Algebra.Linear.Core
import Data.Vec.Properties as VP
open import Algebra.Structures.Field.Utils K
import Algebra.Linear.Construct.Vector K as V
open V
using (Vec; zipWith; replicate)
renaming
( _+_ to _+v_
; _∙_ to _∙v_
; -_ to -v_
)
open import Relation.Binary.PropositionalEquality as P
using (_≡_; subst; subst-subst-sym; _≗_)
renaming
( refl to ≡-refl
; sym to ≡-sym
; trans to ≡-trans
)
import Algebra.Linear.Structures.VectorSpace as VS
open VS.VectorSpaceField K
open import Data.Nat.Properties
using
( ≤-refl
; ≤-reflexive
; ≤-antisym
; n∸n≡0
; m+[n∸m]≡n
; m≤m+n
; suc-injective
)
renaming
( +-identityˡ to +ℕ-identityˡ
; +-identityʳ to +ℕ-identityʳ
)
Matrix : ℕ -> ℕ -> Set k
Matrix n p = Vec (Vec K' p) n
private
M : ℕ -> ℕ -> Set k
M = Matrix
_≈ʰ_ : ∀ {n p n' p'} (A : M n p) (B : M n' p') → Set (k ⊔ ℓ)
_≈ʰ_ = PW.Pointwise V._≈ʰ_
module _ {n p} where
setoid : Setoid k (k ⊔ ℓ)
setoid = record
{ Carrier = M n p
; _≈_ = _≈ʰ_ {n} {p}
; isEquivalence = PW.isEquivalence (V.≈-isEquiv {p}) n
}
open Setoid setoid public
renaming
( refl to ≈-refl
; sym to ≈-sym
; trans to ≈-trans
; reflexive to ≈-reflexive
; isEquivalence to ≈-isEquiv
)
import Algebra.FunctionProperties as FP
tabulate : ∀ {n p} -> (Fin n -> Fin p -> K') -> M n p
tabulate f = V.tabulate λ i -> V.tabulate λ j -> f i j
tabulate⁺ : ∀ {n p} {f g : Fin n -> Fin p -> K'}
-> (∀ i j -> f i j ≈ᵏ g i j)
-> tabulate f ≈ tabulate g
tabulate⁺ {0} r = PW.[]
tabulate⁺ {suc n} {p} {f} {g} r =
(PW.tabulate⁺ (r Fin.zero)) PW.∷ tabulate⁺ {n} {p} λ i j → r (Fin.suc i) j
fromVec : ∀ {n p} -> V.Vec K' (n *ℕ p) -> M n p
fromVec {0} V.[] = V.[]
fromVec {suc n} {p} xs =
let (vp , vnp , _) = V.splitAt p xs
in vp V.∷ fromVec {n} vnp
toVec : ∀ {n p} -> M n p -> V.Vec K' (n *ℕ p)
toVec {n} {p} = V.concat {m = p} {n = n}
concat∘fromVec : ∀ {n p} (v : V.Vec K' (n *ℕ p)) -> toVec {n} {p} (fromVec v) V.≈ v
concat∘fromVec {0} V.[] = PW.[]
concat∘fromVec {suc n} {p} v =
let (vn , vnp , r) = V.splitAt p v
in begin
toVec {suc n} {p} (fromVec v)
≡⟨⟩
vn V.++ toVec {n} {p} (fromVec vnp)
≈⟨ V.++-cong {p} {n *ℕ p} V.≈-refl (concat∘fromVec {n} {p} vnp) ⟩
vn V.++ vnp
≈⟨ V.≈-sym (V.≈-reflexive r) ⟩
v
∎
where open import Relation.Binary.EqReasoning (V.setoid (suc n *ℕ p))
_++_ : ∀ {n p q} -> M n p -> M n q -> M n (p +ℕ q)
_++_ = zipWith V._++_
_‡_ : ∀ {n m p} -> M n p -> M m p -> M (n +ℕ m) p
_‡_ = V._++_
lookup : ∀ {n p} -> M n p -> Fin n -> Fin p -> K'
lookup A i j = V.lookup (V.lookup A i) j
_⟪_,_⟫ : ∀ {n p} -> M n p -> Fin n -> Fin p -> K'
_⟪_,_⟫ = lookup
lookup-cong : ∀ {n p} {A B : M n p} (i : Fin n) (j : Fin p) -> A ≈ B -> (A ⟪ i , j ⟫) ≈ᵏ (B ⟪ i , j ⟫)
lookup-cong i j rs = PW.lookup (PW.lookup rs i) j
tabulate∘lookup : ∀ {n p} (A : M n p) → tabulate (lookup A) ≡ A
tabulate∘lookup A =
begin
tabulate (lookup A)
≡⟨⟩
V.tabulate (λ i -> V.tabulate λ j -> V.lookup (V.lookup A i) j)
≡⟨ VP.tabulate-cong (λ i → VP.tabulate∘lookup (V.lookup A i)) ⟩
V.tabulate (λ i -> V.lookup A i)
≡⟨ VP.tabulate∘lookup A ⟩
A
∎
where open import Relation.Binary.PropositionalEquality as Eq
open Eq.≡-Reasoning
lookup∘tabulate : ∀ {n p} (f : Fin n -> Fin p -> K') (i : Fin n) (j : Fin p)
-> lookup (tabulate f) i j ≡ f i j
lookup∘tabulate {suc n} f i j =
begin
lookup (tabulate f) i j
≡⟨⟩
V.lookup (V.lookup (V.tabulate λ i′ -> V.tabulate λ j′ -> f i′ j′) i) j
≡⟨ cong (λ u → V.lookup u j) (VP.lookup∘tabulate (λ i′ -> V.tabulate λ j′ -> f i′ j′) i) ⟩
V.lookup (V.tabulate λ j′ -> f i j′) j
≡⟨ VP.lookup∘tabulate (λ j′ -> f i j′) j ⟩
f i j
∎
where open import Relation.Binary.PropositionalEquality as Eq
open Eq.≡-Reasoning
tabulate-cong-≡ : ∀ {n p} {f g : Fin n -> Fin p -> K'} -> (∀ i j -> f i j ≡ g i j) -> tabulate f ≡ tabulate g
tabulate-cong-≡ {f = f} {g = g} r = VP.tabulate-cong (λ i → VP.tabulate-cong (λ j → r i j))
tabulate-cong : ∀ {n p} {f g : Fin n -> Fin p -> K'} -> (∀ i j -> f i j ≈ᵏ g i j) -> tabulate f ≈ tabulate g
tabulate-cong {f = f} {g = g} r = PW.tabulate⁺ (λ i → V.tabulate-cong (λ j → r i j))
transpose : ∀ {n p} -> M n p -> M p n
transpose A = tabulate λ i j -> A ⟪ j , i ⟫
_ᵀ : ∀ {n p} -> M n p -> M p n
_ᵀ = transpose
transpose-involutive : ∀ {n p} (A : M n p) -> ((A ᵀ) ᵀ) ≡ A
transpose-involutive A =
begin
((A ᵀ)ᵀ)
≡⟨⟩
(tabulate λ i j -> (tabulate λ i′ j′ -> A ⟪ j′ , i′ ⟫) ⟪ j , i ⟫)
≡⟨ tabulate-cong-≡ (λ i j → lookup∘tabulate (λ i′ j′ -> A ⟪ j′ , i′ ⟫) j i) ⟩
(tabulate λ i j -> A ⟪ i , j ⟫)
≡⟨ tabulate∘lookup A ⟩
A
∎
where open import Relation.Binary.PropositionalEquality as Eq
open Eq.≡-Reasoning
map : ∀ {n p} -> (K' -> K') -> M n p -> M n p
map f = V.map (V.map f)
mapRows : ∀ {n p q} -> (V.Vec K' p -> V.Vec K' q) -> M n p -> M n q
mapRows = V.map
mapCols : ∀ {n m p} -> (V.Vec K' n -> V.Vec K' m) -> M n p -> M m p
mapCols f A = (V.map f (A ᵀ)) ᵀ
map-cong : ∀ {n p} {f g : K' -> K'} -> f ≗ g -> map {n} {p} f ≗ map g
map-cong r = VP.map-cong (VP.map-cong r)
mapRows-cong : ∀ {n p q} {f g : V.Vec K' p -> V.Vec K' q}
-> f ≗ g -> mapRows {n} f ≗ mapRows g
mapRows-cong = VP.map-cong
mapCols-cong : ∀ {n m p} {f g : V.Vec K' n -> V.Vec K' m}
-> f ≗ g -> mapCols {n} {m} {p} f ≗ mapCols g
mapCols-cong r A = P.cong transpose (VP.map-cong r (transpose A))
open import Data.Nat.DivMod
_+_ : ∀ {n p} -> FP.Op₂ (M n p)
_+_ = zipWith V._+_
_∙_ : ∀ {n p} -> ScalarMultiplication K' (M n p)
_∙_ k = V.map (k V.∙_)
-_ : ∀ {n p} -> FP.Op₁ (M n p)
-_ = V.map V.-_
0# : ∀ {n p} -> M n p
0# = replicate V.0#
+-cong : ∀ {n p} {A B C D : M n p} -> A ≈ B -> C ≈ D -> (A + C) ≈ (B + D)
+-cong PW.[] PW.[] = PW.[]
+-cong (r₁ PW.∷ rs₁) (r₂ PW.∷ rs₂) = V.+-cong r₁ r₂ PW.∷ +-cong rs₁ rs₂
+-assoc : ∀ {n p} (A B C : M n p) -> ((A + B) + C) ≈ (A + (B + C))
+-assoc V.[] V.[] V.[] = PW.[]
+-assoc (u V.∷ us) (v V.∷ vs) (w V.∷ ws) =
V.+-assoc u v w PW.∷ +-assoc us vs ws
+-identityˡ : ∀ {n p} (A : M n p) -> (0# + A) ≈ A
+-identityˡ V.[] = PW.[]
+-identityˡ (u V.∷ us) = V.+-identityˡ u PW.∷ +-identityˡ us
+-identityʳ : ∀ {n p} (A : M n p) -> (A + 0#) ≈ A
+-identityʳ V.[] = PW.[]
+-identityʳ (u V.∷ us) = V.+-identityʳ u PW.∷ +-identityʳ us
+-identity : ∀ {n p} -> ((∀ (A : M n p) -> ((0# + A) ≈ A)) × (∀ (A : M n p) -> ((A + 0#) ≈ A)))
+-identity = +-identityˡ , +-identityʳ
+-comm : ∀ {n p} (A B : M n p) -> (A + B) ≈ (B + A)
+-comm V.[] V.[] = PW.[]
+-comm (u V.∷ us) (v V.∷ vs) = (V.+-comm u v) PW.∷ (+-comm us vs)
*ᵏ-∙-compat : ∀ {n p} (a b : K') (A : M n p) -> ((a *ᵏ b) ∙ A) ≈ (a ∙ (b ∙ A))
*ᵏ-∙-compat a b V.[] = PW.[]
*ᵏ-∙-compat a b (u V.∷ us) = (V.*ᵏ-∙-compat a b u) PW.∷ (*ᵏ-∙-compat a b us)
∙-+-distrib : ∀ {n p} (a : K') (A B : M n p) -> (a ∙ (A + B)) ≈ ((a ∙ A) + (a ∙ B))
∙-+-distrib a V.[] V.[] = PW.[]
∙-+-distrib a (u V.∷ us) (v V.∷ vs) = (V.∙-+-distrib a u v) PW.∷ (∙-+-distrib a us vs)
∙-+ᵏ-distrib : ∀ {n p} (a b : K') (A : M n p) -> ((a +ᵏ b) ∙ A) ≈ ((a ∙ A) + (b ∙ A))
∙-+ᵏ-distrib a b V.[] = PW.[]
∙-+ᵏ-distrib a b (u V.∷ us) = (V.∙-+ᵏ-distrib a b u) PW.∷ (∙-+ᵏ-distrib a b us)
∙-cong : ∀ {n p} {a b : K'} {A B : M n p} → a ≈ᵏ b -> A ≈ B -> (a ∙ A) ≈ (b ∙ B)
∙-cong rᵏ PW.[] = PW.[]
∙-cong rᵏ (r PW.∷ rs) = (V.∙-cong rᵏ r) PW.∷ (∙-cong rᵏ rs)
∙-identity : ∀ {n p} (A : M n p) → (1ᵏ ∙ A) ≈ A
∙-identity V.[] = PW.[]
∙-identity (u V.∷ us) = (V.∙-identity u) PW.∷ (∙-identity us)
∙-absorbˡ : ∀ {n p} (A : M n p) → (0ᵏ ∙ A) ≈ 0#
∙-absorbˡ V.[] = PW.[]
∙-absorbˡ (u V.∷ us) = (V.∙-absorbˡ u) PW.∷ (∙-absorbˡ us)
-‿inverseˡ : ∀ {n p} (A : M n p) -> ((- A) + A) ≈ 0#
-‿inverseˡ V.[] = PW.[]
-‿inverseˡ (u V.∷ us) = (V.-‿inverseˡ u) PW.∷ (-‿inverseˡ us)
-‿inverseʳ : ∀ {n p} (A : M n p) -> (A + (- A)) ≈ 0#
-‿inverseʳ V.[] = PW.[]
-‿inverseʳ (u V.∷ us) = (V.-‿inverseʳ u) PW.∷ (-‿inverseʳ us)
-‿inverse : ∀ {n p} → (∀ (A : M n p) -> ((- A) + A) ≈ 0#) × (∀ (A : M n p) -> (A + (- A)) ≈ 0#)
-‿inverse = -‿inverseˡ , -‿inverseʳ
-‿cong : ∀ {n p} {A B : M n p} -> A ≈ B -> (- A) ≈ (- B)
-‿cong PW.[] = PW.[]
-‿cong (r PW.∷ rs) = (V.-‿cong r) PW.∷ (-‿cong rs)
concat-+ : ∀ {n p} (A B : M n p) -> V.concat (V.zipWith V._+_ A B) V.≈ (V.concat A) +v (V.concat B)
concat-+ {0} {p} V.[] V.[] =
V.≈-trans (PW.concat⁺ {m = p} {p = 0} PW.[]) (V.≈-sym (V.+-identityˡ V.[]))
concat-+ {suc n} {p} (u V.∷ us) (v V.∷ vs) =
begin
V.concat (V.zipWith V._+_ (u V.∷ us) (v V.∷ vs))
≡⟨⟩
(u V.+ v) V.++ V.concat (V.zipWith V._+_ us vs)
≈⟨ V.++-cong V.≈-refl (concat-+ {n} {p} us vs) ⟩
(u V.+ v) V.++ (V.concat us V.+ V.concat vs)
≈⟨ V.≈-sym (V.+-++-distrib u (V.concat us) v (V.concat vs)) ⟩
(V.concat (u V.∷ us)) V.+ (V.concat (v V.∷ vs))
∎
where open import Relation.Binary.EqReasoning (V.setoid (p +ℕ n *ℕ p))
concat-0# : ∀ {n p} -> V.concat (0# {n} {p}) V.≈ (V.0# {n *ℕ p})
concat-0# {0} = PW.[]
concat-0# {suc n} {p} =
begin
V.concat (0# {suc n} {p})
≡⟨⟩
V.0# {p} V.++ V.concat (0# {n} {p})
≈⟨ PW.++⁺ V.≈-refl (concat-0# {n} {p}) ⟩
V.0# {p} V.++ V.0# {n *ℕ p}
≈⟨ V.0++0≈0 {p} {n *ℕ p} ⟩
V.0# {p +ℕ n *ℕ p}
∎
where open import Relation.Binary.EqReasoning (V.setoid (p +ℕ n *ℕ p))
concat-∙ : ∀ {n p} (c : K') (A : M n p) -> V.concat (c ∙ A) V.≈ (c V.∙ V.concat A)
concat-∙ {0} c V.[] = PW.[]
concat-∙ {suc n} {p} c (u V.∷ us) =
begin
V.concat ((c V.∙ u) V.∷ (c ∙ us))
≡⟨⟩
(c V.∙ u) V.++ (V.concat (c ∙ us))
≈⟨ V.++-cong V.≈-refl (concat-∙ {n} {p} c us) ⟩
(c V.∙ u) V.++ (c V.∙ V.concat us)
≈⟨ V.≈-sym (V.∙-++-distrib c u (V.concat us)) ⟩
c V.∙ V.concat (u V.∷ us)
∎
where open import Relation.Binary.EqReasoning (V.setoid (p +ℕ n *ℕ p))
I : ∀ {n p} -> M n p
I = tabulate δ
_*_ : ∀ {n p q} -> M n p -> M p q -> M n q
A * B = tabulate λ i j -> V.sum-tab λ k -> (A ⟪ i , k ⟫) *ᵏ (B ⟪ k , j ⟫)
*-cong : ∀ {n p q} {A B : M n p} {C D : M p q}
-> A ≈ B -> C ≈ D -> (A * C) ≈ (B * D)
*-cong {n} {q = q} {A} {B} {C} {D} r₁ r₂ =
begin
A * C
≡⟨⟩
tabulate (λ i j -> V.sum-tab λ k -> (A ⟪ i , k ⟫) *ᵏ (C ⟪ k , j ⟫))
≈⟨ tabulate⁺ (λ i j -> V.sum-tab-cong λ k → *ᵏ-cong (lookup-cong i k r₁) (lookup-cong k j r₂)) ⟩
B * D
∎
where open import Relation.Binary.EqReasoning (setoid {n} {q})
*-assoc : ∀ {n p q r} (A : M n p) (B : M p q) (C : M q r)
-> ((A * B) * C) ≈ (A * (B * C))
*-assoc {n} {r = r} A B C = tabulate⁺ λ i j ->
begin
V.sum-tab (λ k′ -> ((A * B) ⟪ i , k′ ⟫) *ᵏ (C ⟪ k′ , j ⟫))
≈⟨ V.sum-tab-cong (λ k′ -> *ᵏ-cong (≈ᵏ-reflexive (lookup∘tabulate
(λ i′ j′ -> V.sum-tab λ k -> (A ⟪ i′ , k ⟫) *ᵏ (B ⟪ k , j′ ⟫)) i k′)) ≈ᵏ-refl) ⟩
V.sum-tab (λ k′ -> (V.sum-tab λ k -> (A ⟪ i , k ⟫) *ᵏ (B ⟪ k , k′ ⟫)) *ᵏ (C ⟪ k′ , j ⟫))
≈⟨ V.sum-tab-cong (λ k′ -> V.*ᵏ-sum-tab-distribʳ (C ⟪ k′ , j ⟫) λ k -> (A ⟪ i , k ⟫) *ᵏ (B ⟪ k , k′ ⟫)) ⟩
V.sum-tab (λ k′ -> V.sum-tab λ k -> ((A ⟪ i , k ⟫) *ᵏ (B ⟪ k , k′ ⟫)) *ᵏ (C ⟪ k′ , j ⟫))
≈⟨ V.sum-tab-cong (λ k′ -> V.sum-tab-cong λ k -> *ᵏ-assoc (A ⟪ i , k ⟫) (B ⟪ k , k′ ⟫) (C ⟪ k′ , j ⟫)) ⟩
V.sum-tab (λ k′ -> V.sum-tab λ k -> (A ⟪ i , k ⟫) *ᵏ ((B ⟪ k , k′ ⟫) *ᵏ (C ⟪ k′ , j ⟫)))
≈⟨ V.sum-tab-cong (λ k′ -> V.sum-tab-cong λ k -> *ᵏ-comm (A ⟪ i , k ⟫) ((B ⟪ k , k′ ⟫) *ᵏ (C ⟪ k′ , j ⟫))) ⟩
V.sum-tab (λ k′ -> V.sum-tab λ k -> ((B ⟪ k , k′ ⟫) *ᵏ (C ⟪ k′ , j ⟫)) *ᵏ (A ⟪ i , k ⟫))
≈⟨ V.sum-tab-swap (λ k k′ -> (B ⟪ k , k′ ⟫) *ᵏ (C ⟪ k′ , j ⟫)) (λ k -> A ⟪ i , k ⟫) ⟩
V.sum-tab (λ k -> V.sum-tab λ k′ -> ((B ⟪ k , k′ ⟫) *ᵏ (C ⟪ k′ , j ⟫)) *ᵏ (A ⟪ i , k ⟫))
≈⟨ V.sum-tab-cong (λ k -> ≈ᵏ-trans
(≈ᵏ-sym (V.*ᵏ-sum-tab-distribʳ (A ⟪ i , k ⟫) λ k′ -> (B ⟪ k , k′ ⟫) *ᵏ (C ⟪ k′ , j ⟫)))
(*ᵏ-comm (V.sum-tab λ k′ -> (B ⟪ k , k′ ⟫) *ᵏ (C ⟪ k′ , j ⟫)) (A ⟪ i , k ⟫))) ⟩
V.sum-tab (λ k -> (A ⟪ i , k ⟫) *ᵏ V.sum-tab (λ k′ -> (B ⟪ k , k′ ⟫) *ᵏ (C ⟪ k′ , j ⟫)))
≈⟨ V.sum-tab-cong (λ k -> *ᵏ-cong ≈ᵏ-refl
(≈ᵏ-sym (≈ᵏ-reflexive (lookup∘tabulate
(λ i′ j′ -> V.sum-tab λ k′ -> (B ⟪ i′ , k′ ⟫) *ᵏ (C ⟪ k′ , j′ ⟫)) k j)))) ⟩
V.sum-tab (λ k -> (A ⟪ i , k ⟫) *ᵏ ((B * C) ⟪ k , j ⟫))
∎
where open import Relation.Binary.EqReasoning (Field.setoid K)
*-identityˡ : ∀ {n p} (A : M n p) -> I * A ≈ A
*-identityˡ {n} {p} A =
begin
I * A
≡⟨⟩
tabulate (λ i j -> V.sum-tab λ k -> (I ⟪ i , k ⟫) *ᵏ (A ⟪ k , j ⟫))
≈⟨ tabulate-cong (λ i j -> V.sum-tab-cong {n} λ k -> *ᵏ-cong (≈ᵏ-reflexive (lookup∘tabulate δ i k)) ≈ᵏ-refl) ⟩
tabulate (λ i j -> V.sum-tab λ k -> δ i k *ᵏ (A ⟪ k , j ⟫))
≈⟨ tabulate-cong (λ i j -> V.sum-tab-δ (λ k -> A ⟪ k , j ⟫) i) ⟩
tabulate (λ i j -> A ⟪ i , j ⟫)
≡⟨ tabulate∘lookup A ⟩
A
∎
where open import Relation.Binary.EqReasoning (setoid {n} {p})
*-identityʳ : ∀ {n p} (A : M n p) -> A * I ≈ A
*-identityʳ {n} {p} A =
begin
A * I
≡⟨⟩
tabulate (λ i j -> V.sum-tab λ k -> (A ⟪ i , k ⟫) *ᵏ (I ⟪ k , j ⟫))
≈⟨ tabulate-cong (λ i j -> V.sum-tab-cong {p} λ k -> *ᵏ-cong ≈ᵏ-refl (≈ᵏ-reflexive (lookup∘tabulate δ k j))) ⟩
tabulate (λ i j -> V.sum-tab λ k -> (A ⟪ i , k ⟫) *ᵏ δ k j)
≈⟨ tabulate-cong (λ i j -> V.sum-tab-cong {p} λ k -> *ᵏ-cong ≈ᵏ-refl (δ-comm k j)) ⟩
tabulate (λ i j -> V.sum-tab λ k -> (A ⟪ i , k ⟫) *ᵏ δ j k)
≈⟨ tabulate-cong (λ i j -> ≈ᵏ-trans (V.sum-tab-cong {p} (λ k -> *ᵏ-comm ((A ⟪ i , k ⟫)) (δ j k)))
(V.sum-tab-δ (λ k -> A ⟪ i , k ⟫) j)) ⟩
tabulate (λ i j -> A ⟪ i , j ⟫)
≡⟨ tabulate∘lookup A ⟩
A
∎
where open import Relation.Binary.EqReasoning (setoid {n} {p})
module _ {n p} where
open IsEquivalence (≈-isEquiv {n} {p}) public
using ()
renaming
( refl to ≈-refl
; sym to ≈-sym
; trans to ≈-trans
)
open FP (_≈_ {n} {p})
open import Algebra.Structures (_≈_ {n} {p})
open import Algebra.Linear.Structures.Bundles
isMagma : IsMagma _+_
isMagma = record
{ isEquivalence = ≈-isEquiv
; ∙-cong = +-cong
}
isSemigroup : IsSemigroup _+_
isSemigroup = record
{ isMagma = isMagma
; assoc = +-assoc
}
isMonoid : IsMonoid _+_ 0#
isMonoid = record
{ isSemigroup = isSemigroup
; identity = +-identity
}
isGroup : IsGroup _+_ 0# -_
isGroup = record
{ isMonoid = isMonoid
; inverse = -‿inverse
; ⁻¹-cong = -‿cong
}
isAbelianGroup : IsAbelianGroup _+_ 0# -_
isAbelianGroup = record
{ isGroup = isGroup
; comm = +-comm
}
open VS K
isVectorSpace : VS.IsVectorSpace K (_≈_ {n}) _+_ _∙_ -_ 0#
isVectorSpace = record
{ isAbelianGroup = isAbelianGroup
; *ᵏ-∙-compat = *ᵏ-∙-compat
; ∙-+-distrib = ∙-+-distrib
; ∙-+ᵏ-distrib = ∙-+ᵏ-distrib
; ∙-cong = ∙-cong
; ∙-identity = ∙-identity
; ∙-absorbˡ = ∙-absorbˡ
}
vectorSpace : VectorSpace K k (k ⊔ ℓ)
vectorSpace = record { isVectorSpace = isVectorSpace }
open import Algebra.Linear.Structures.FiniteDimensional K
open import Algebra.Linear.Morphism.VectorSpace K
open import Algebra.Linear.Morphism.Bundles K
open import Algebra.Morphism.Definitions (M n p) (Vec K' (n *ℕ p)) V._≈_
open import Algebra.Linear.Morphism.Definitions K (M n p) (Vec K' (n *ℕ p)) V._≈_
import Relation.Binary.Morphism.Definitions (M n p) (Vec K' (n *ℕ p)) as R
open import Function
open import Relation.Binary.EqReasoning (V.setoid (n *ℕ p))
⟦_⟧ : M n p -> Vec K' (n *ℕ p)
⟦_⟧ = V.concat
⟦⟧-cong : R.Homomorphic₂ _≈_ (V._≈_ {n *ℕ p}) ⟦_⟧
⟦⟧-cong = PW.concat⁺
+-homo : Homomorphic₂ ⟦_⟧ _+_ _+v_
+-homo = concat-+
0#-homo : Homomorphic₀ ⟦_⟧ 0# V.0#
0#-homo = concat-0# {n} {p}
∙-homo : ScalarHomomorphism ⟦_⟧ _∙_ _∙v_
∙-homo = concat-∙
⟦⟧-injective : Injective (_≈_ {n} {p}) (V._≈_ {n *ℕ p}) ⟦_⟧
⟦⟧-injective {A} {B} r = PW.concat⁻ A B r
⟦⟧-surjective : Surjective (_≈_ {n} {p}) (V._≈_ {n *ℕ p}) ⟦_⟧
⟦⟧-surjective v = fromVec {n} {p} v , concat∘fromVec {n} {p} v
embed : LinearIsomorphism vectorSpace (V.vectorSpace {n *ℕ p})
embed = record
{ ⟦_⟧ = ⟦_⟧
; isLinearIsomorphism = record
{ isLinearMonomorphism = record
{ isLinearMap = record
{ isAbelianGroupMorphism = record
{ gp-homo = record
{ mn-homo = record
{ sm-homo = record
{ ⟦⟧-cong = ⟦⟧-cong
; ∙-homo = +-homo
}
; ε-homo = 0#-homo
}
}
}
; ∙-homo = ∙-homo
}
; injective = ⟦⟧-injective
}
; surjective = ⟦⟧-surjective
}
}
isFiniteDimensional : IsFiniteDimensional _≈_ _+_ _∙_ -_ 0# (n *ℕ p)
isFiniteDimensional = record
{ isVectorSpace = isVectorSpace
; embed = embed
}
|
lemma Inf_insert: fixes S :: "real set" shows "bounded S \<Longrightarrow> Inf (insert x S) = (if S = {} then x else min x (Inf S))" |
% [PYR, INDICES, STEERMTX, HARMONICS] = buildSFpyr(IM, HEIGHT, ORDER, TWIDTH)
%
% Construct a steerable pyramid on matrix IM, in the Fourier domain.
% This is similar to buildSpyr, except that:
%
% + Reconstruction is exact (within floating point errors)
% + It can produce any number of orientation bands.
% - Typically slower, especially for non-power-of-two sizes.
% - Boundary-handling is circular.
%
% HEIGHT (optional) specifies the number of pyramid levels to build. Default
% is maxPyrHt(size(IM),size(FILT));
%
% The squared radial functions tile the Fourier plane, with a raised-cosine
% falloff. Angular functions are cos(theta-k\pi/(K+1))^K, where K is
% the ORDER (one less than the number of orientation bands, default= 3).
%
% TWIDTH is the width of the transition region of the radial lowpass
% function, in octaves (default = 1, which gives a raised cosine for
% the bandpass filters).
%
% PYR is a vector containing the N pyramid subbands, ordered from fine
% to coarse. INDICES is an Nx2 matrix containing the sizes of
% each subband. This is compatible with the MatLab Wavelet toolbox.
% See the function STEER for a description of STEERMTX and HARMONICS.
% Eero Simoncelli, 5/97.
% See http://www.cis.upenn.edu/~eero/steerpyr.html for more
% information about the Steerable Pyramid image decomposition.
function [pyr,pind,steermtx,harmonics] = buildSFpyr(im, ht, order, twidth)
%-----------------------------------------------------------------
%% DEFAULTS:
max_ht = floor(log2(min(size(im)))+2);
if (exist('ht') ~= 1)
ht = max_ht;
else
if (ht > max_ht)
error(sprintf('Cannot build pyramid higher than %d levels.',max_ht));
end
end
if (exist('order') ~= 1)
order = 3;
elseif ((order > 15) | (order < 0))
fprintf(1,'Warning: ORDER must be an integer in the range [0,15]. Truncating.\n');
order = min(max(order,0),15);
else
order = round(order);
end
nbands = order+1;
if (exist('twidth') ~= 1)
twidth = 1;
elseif (twidth <= 0)
fprintf(1,'Warning: TWIDTH must be positive. Setting to 1.\n');
twidth = 1;
end
%-----------------------------------------------------------------
%% Steering stuff:
if (mod((nbands),2) == 0)
harmonics = [0:(nbands/2)-1]'*2 + 1;
else
harmonics = [0:(nbands-1)/2]'*2;
end
steermtx = steer2HarmMtx(harmonics, pi*[0:nbands-1]/nbands, 'even');
%-----------------------------------------------------------------
dims = size(im);
ctr = ceil((dims+0.5)/2);
[xramp,yramp] = meshgrid( ([1:dims(2)]-ctr(2))./(dims(2)/2), ...
([1:dims(1)]-ctr(1))./(dims(1)/2) );
angle = atan2(yramp,xramp);
log_rad = sqrt(xramp.^2 + yramp.^2);
log_rad(ctr(1),ctr(2)) = log_rad(ctr(1),ctr(2)-1);
log_rad = log2(log_rad);
%% Radial transition function (a raised cosine in log-frequency):
[Xrcos,Yrcos] = rcosFn(twidth,(-twidth/2),[0 1]);
Yrcos = sqrt(Yrcos);
YIrcos = sqrt(1.0 - Yrcos.^2);
lo0mask = pointOp(log_rad, YIrcos, Xrcos(1), Xrcos(2)-Xrcos(1), 0);
imdft = fftshift(fft2(im));
lo0dft = imdft .* lo0mask;
[pyr,pind] = buildSFpyrLevs(lo0dft, log_rad, Xrcos, Yrcos, angle, ht, nbands);
hi0mask = pointOp(log_rad, Yrcos, Xrcos(1), Xrcos(2)-Xrcos(1), 0);
hi0dft = imdft .* hi0mask;
hi0 = ifft2(ifftshift(hi0dft));
pyr = [real(hi0(:)) ; pyr];
pind = [size(hi0); pind];
|
(* Title: HOL/Auth/n_german_lemma_inv__9_on_rules.thy
Author: Yongjian Li and Kaiqiang Duan, State Key Lab of Computer Science, Institute of Software, Chinese Academy of Sciences
Copyright 2016 State Key Lab of Computer Science, Institute of Software, Chinese Academy of Sciences
*)
header{*The n_german Protocol Case Study*}
theory n_german_lemma_inv__9_on_rules imports n_german_lemma_on_inv__9
begin
section{*All lemmas on causal relation between inv__9*}
lemma lemma_inv__9_on_rules:
assumes b1: "r \<in> rules N" and b2: "(\<exists> p__Inv1 p__Inv2. p__Inv1\<le>N\<and>p__Inv2\<le>N\<and>p__Inv1~=p__Inv2\<and>f=inv__9 p__Inv1 p__Inv2)"
shows "invHoldForRule s f r (invariants N)"
proof -
have c1: "(\<exists> i d. i\<le>N\<and>d\<le>N\<and>r=n_Store i d)\<or>
(\<exists> i. i\<le>N\<and>r=n_SendReqS i)\<or>
(\<exists> i. i\<le>N\<and>r=n_SendReqE__part__0 i)\<or>
(\<exists> i. i\<le>N\<and>r=n_SendReqE__part__1 i)\<or>
(\<exists> i. i\<le>N\<and>r=n_RecvReqS N i)\<or>
(\<exists> i. i\<le>N\<and>r=n_RecvReqE N i)\<or>
(\<exists> i. i\<le>N\<and>r=n_SendInv__part__0 i)\<or>
(\<exists> i. i\<le>N\<and>r=n_SendInv__part__1 i)\<or>
(\<exists> i. i\<le>N\<and>r=n_SendInvAck i)\<or>
(\<exists> i. i\<le>N\<and>r=n_RecvInvAck i)\<or>
(\<exists> i. i\<le>N\<and>r=n_SendGntS i)\<or>
(\<exists> i. i\<le>N\<and>r=n_SendGntE N i)\<or>
(\<exists> i. i\<le>N\<and>r=n_RecvGntS i)\<or>
(\<exists> i. i\<le>N\<and>r=n_RecvGntE i)"
apply (cut_tac b1, auto) done
moreover {
assume d1: "(\<exists> i d. i\<le>N\<and>d\<le>N\<and>r=n_Store i d)"
have "invHoldForRule s f r (invariants N)"
apply (cut_tac b2 d1, metis n_StoreVsinv__9) done
}
moreover {
assume d1: "(\<exists> i. i\<le>N\<and>r=n_SendReqS i)"
have "invHoldForRule s f r (invariants N)"
apply (cut_tac b2 d1, metis n_SendReqSVsinv__9) done
}
moreover {
assume d1: "(\<exists> i. i\<le>N\<and>r=n_SendReqE__part__0 i)"
have "invHoldForRule s f r (invariants N)"
apply (cut_tac b2 d1, metis n_SendReqE__part__0Vsinv__9) done
}
moreover {
assume d1: "(\<exists> i. i\<le>N\<and>r=n_SendReqE__part__1 i)"
have "invHoldForRule s f r (invariants N)"
apply (cut_tac b2 d1, metis n_SendReqE__part__1Vsinv__9) done
}
moreover {
assume d1: "(\<exists> i. i\<le>N\<and>r=n_RecvReqS N i)"
have "invHoldForRule s f r (invariants N)"
apply (cut_tac b2 d1, metis n_RecvReqSVsinv__9) done
}
moreover {
assume d1: "(\<exists> i. i\<le>N\<and>r=n_RecvReqE N i)"
have "invHoldForRule s f r (invariants N)"
apply (cut_tac b2 d1, metis n_RecvReqEVsinv__9) done
}
moreover {
assume d1: "(\<exists> i. i\<le>N\<and>r=n_SendInv__part__0 i)"
have "invHoldForRule s f r (invariants N)"
apply (cut_tac b2 d1, metis n_SendInv__part__0Vsinv__9) done
}
moreover {
assume d1: "(\<exists> i. i\<le>N\<and>r=n_SendInv__part__1 i)"
have "invHoldForRule s f r (invariants N)"
apply (cut_tac b2 d1, metis n_SendInv__part__1Vsinv__9) done
}
moreover {
assume d1: "(\<exists> i. i\<le>N\<and>r=n_SendInvAck i)"
have "invHoldForRule s f r (invariants N)"
apply (cut_tac b2 d1, metis n_SendInvAckVsinv__9) done
}
moreover {
assume d1: "(\<exists> i. i\<le>N\<and>r=n_RecvInvAck i)"
have "invHoldForRule s f r (invariants N)"
apply (cut_tac b2 d1, metis n_RecvInvAckVsinv__9) done
}
moreover {
assume d1: "(\<exists> i. i\<le>N\<and>r=n_SendGntS i)"
have "invHoldForRule s f r (invariants N)"
apply (cut_tac b2 d1, metis n_SendGntSVsinv__9) done
}
moreover {
assume d1: "(\<exists> i. i\<le>N\<and>r=n_SendGntE N i)"
have "invHoldForRule s f r (invariants N)"
apply (cut_tac b2 d1, metis n_SendGntEVsinv__9) done
}
moreover {
assume d1: "(\<exists> i. i\<le>N\<and>r=n_RecvGntS i)"
have "invHoldForRule s f r (invariants N)"
apply (cut_tac b2 d1, metis n_RecvGntSVsinv__9) done
}
moreover {
assume d1: "(\<exists> i. i\<le>N\<and>r=n_RecvGntE i)"
have "invHoldForRule s f r (invariants N)"
apply (cut_tac b2 d1, metis n_RecvGntEVsinv__9) done
}
ultimately show "invHoldForRule s f r (invariants N)"
by satx
qed
end
|
/-
Copyright (c) 2023 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov
! This file was ported from Lean 3 source module data.set.list
! leanprover-community/mathlib commit 2ec920d35348cb2d13ac0e1a2ad9df0fdf1a76b4
! Please do not edit these lines, except to modify the commit id
! if you have ported upstream changes.
-/
import Mathbin.Data.Set.Image
import Mathbin.Data.List.Basic
import Mathbin.Data.Fin.Basic
/-!
# Lemmas about `list`s and `set.range`
In this file we prove lemmas about range of some operations on lists.
-/
open List
variable {α β : Type _} (l : List α)
namespace Set
#print Set.range_list_map /-
theorem range_list_map (f : α → β) : range (map f) = { l | ∀ x ∈ l, x ∈ range f } :=
by
refine'
subset.antisymm (range_subset_iff.2 fun l => forall_mem_map_iff.2 fun y _ => mem_range_self _)
fun l hl => _
induction' l with a l ihl; · exact ⟨[], rfl⟩
rcases ihl fun x hx => hl x <| subset_cons _ _ hx with ⟨l, rfl⟩
rcases hl a (mem_cons_self _ _) with ⟨a, rfl⟩
exact ⟨a :: l, map_cons _ _ _⟩
#align set.range_list_map Set.range_list_map
-/
#print Set.range_list_map_coe /-
theorem range_list_map_coe (s : Set α) : range (map (coe : s → α)) = { l | ∀ x ∈ l, x ∈ s } := by
rw [range_list_map, Subtype.range_coe]
#align set.range_list_map_coe Set.range_list_map_coe
-/
#print Set.range_list_nthLe /-
@[simp]
theorem range_list_nthLe : (range fun k : Fin l.length => l.nthLe k k.2) = { x | x ∈ l } :=
by
ext x
rw [mem_set_of_eq, mem_iff_nth_le]
exact ⟨fun ⟨⟨n, h₁⟩, h₂⟩ => ⟨n, h₁, h₂⟩, fun ⟨n, h₁, h₂⟩ => ⟨⟨n, h₁⟩, h₂⟩⟩
#align set.range_list_nth_le Set.range_list_nthLe
-/
#print Set.range_list_get? /-
theorem range_list_get? : range l.get? = insert none (some '' { x | x ∈ l }) :=
by
rw [← range_list_nth_le, ← range_comp]
refine' (range_subset_iff.2 fun n => _).antisymm (insert_subset.2 ⟨_, _⟩)
exacts[(le_or_lt l.length n).imp nth_eq_none_iff.2 fun hlt => ⟨⟨_, _⟩, (nth_le_nth hlt).symm⟩,
⟨_, nth_eq_none_iff.2 le_rfl⟩, range_subset_iff.2 fun k => ⟨_, nth_le_nth _⟩]
#align set.range_list_nth Set.range_list_get?
-/
#print Set.range_list_getD /-
@[simp]
theorem range_list_getD (d : α) : (range fun n => l.getD n d) = insert d { x | x ∈ l } :=
calc
(range fun n => l.getD n d) = (fun o : Option α => o.getD d) '' range l.get? := by
simp only [← range_comp, (· ∘ ·), nthd_eq_get_or_else_nth]
_ = insert d { x | x ∈ l } := by
simp only [range_list_nth, image_insert_eq, Option.getD, image_image, image_id']
#align set.range_list_nthd Set.range_list_getD
-/
#print Set.range_list_getI /-
@[simp]
theorem range_list_getI [Inhabited α] (l : List α) : range l.getI = insert default { x | x ∈ l } :=
range_list_getD l default
#align set.range_list_inth Set.range_list_getI
-/
end Set
#print List.canLift /-
/-- If each element of a list can be lifted to some type, then the whole list can be lifted to this
type. -/
instance List.canLift (c) (p) [CanLift α β c p] :
CanLift (List α) (List β) (List.map c) fun l => ∀ x ∈ l, p x
where prf l H := by
rw [← Set.mem_range, Set.range_list_map]
exact fun a ha => CanLift.prf a (H a ha)
#align list.can_lift List.canLift
-/
|
import NLPModels: grad, cons, jac
"""
unconstrained: return the infinite norm of the gradient of the objective function
required: state.gx (filled if nothing)
"""
function unconstrained_check(pb :: AbstractNLPModel,
state :: NLPAtX;
pnorm :: Float64 = Inf,
kwargs...)
if state.gx == nothing # should be filled if empty
update!(state, gx = grad(pb, state.x))
end
res = norm(state.gx, pnorm)
return res
end
"""
unconstrained 2nd: check the norm of the gradient and the smallest
eigenvalue of the hessian.
required: state.gx, state.Hx (filled if nothing)
"""
function unconstrained2nd_check(pb :: AbstractNLPModel,
state :: NLPAtX;
pnorm :: Float64 = Inf,
kwargs...)
if state.gx == nothing # should be filled if empty
update!(state, gx = grad(pb, state.x))
end
if state.Hx == nothing
update!(state, Hx = hess(pb, state.x))
end
res = max(norm(state.gx, pnorm),
max(- eigmin(state.Hx + state.Hx' - diagm(0 => diag(state.Hx))), 0.0))
return res
end
"""
optim_check_bounded: gradient of the objective function projected
required: state.gx (filled if void)
"""
function optim_check_bounded(pb :: AbstractNLPModel,
state :: NLPAtX;
pnorm :: Float64 = Inf,
kwargs...)
if state.gx == nothing # should be filled if void
update!(state, gx = grad(pb, state.x))
end
proj = max.(min.(state.x - state.gx, pb.meta.uvar), pb.meta.lvar)
gradproj = state.x - proj
res = norm(gradproj, pnorm)
return res
end
"""
constrained: return the violation of the KKT conditions
length(lambda) > 0
"""
function _grad_lagrangian(pb :: AbstractNLPModel,
state :: NLPAtX)
if (pb.meta.ncon == 0) & !has_bounds(pb)
return state.gx
elseif pb.meta.ncon == 0
return state.gx + state.mu
else
return state.gx + state.mu + state.Jx' * state.lambda
end
end
function _sign_multipliers_bounds(pb :: AbstractNLPModel,
state :: NLPAtX)
if has_bounds(pb)
return vcat(min.(max.( state.mu,0.0), - state.x + pb.meta.uvar),
min.(max.(-state.mu,0.0), state.x - pb.meta.lvar))
else
return zeros(0)
end
end
function _sign_multipliers_nonlin(pb :: AbstractNLPModel,
state :: NLPAtX)
if pb.meta.ncon == 0
return zeros(0)
else
return vcat(min.(max.( state.lambda,0.0), - state.cx + pb.meta.ucon),
min.(max.(-state.lambda,0.0), state.cx - pb.meta.lcon))
end
end
function _feasibility(pb :: AbstractNLPModel,
state :: NLPAtX)
if pb.meta.ncon == 0
return vcat(max.( state.x - pb.meta.uvar,0.0),
max.(- state.x + pb.meta.lvar,0.0))
else
return vcat(max.( state.cx - pb.meta.ucon,0.0),
max.(- state.cx + pb.meta.lcon,0.0),
max.( state.x - pb.meta.uvar,0.0),
max.(- state.x + pb.meta.lvar,0.0))
end
end
"""
KKT: verifies the KKT conditions
required: state.gx
+ if bounds: state.mu
+ if constraints: state.cx, state.Jx, state.lambda
"""
function KKT(pb :: AbstractNLPModel,
state :: NLPAtX;
pnorm :: Float64 = Inf,
kwargs...)
#Check the gradient of the Lagrangian
gLagx = _grad_lagrangian(pb, state)
#Check the complementarity condition for the bounds
res_bounds = _sign_multipliers_bounds(pb, state)
#Check the complementarity condition for the constraints
res_nonlin = _sign_multipliers_nonlin(pb, state)
#Check the feasibility
feas = _feasibility(pb, state)
res = vcat(gLagx, feas, res_bounds, res_nonlin)
return norm(res, pnorm)
end
|
Formal statement is: lemma continuous_on_inv_into: fixes f :: "'a::topological_space \<Rightarrow> 'b::t2_space" assumes s: "continuous_on s f" "compact s" and f: "inj_on f s" shows "continuous_on (f ` s) (the_inv_into s f)" Informal statement is: If $f$ is a continuous injective map from a compact space to a Hausdorff space, then the inverse of $f$ is continuous. |
lemma get_integrable_path: assumes "open s" "connected (s-pts)" "finite pts" "f holomorphic_on (s-pts) " "a\<in>s-pts" "b\<in>s-pts" obtains g where "valid_path g" "pathstart g = a" "pathfinish g = b" "path_image g \<subseteq> s-pts" "f contour_integrable_on g" |
check_use <- function(use)
{
match.arg(
tolower(use),
c("everything", "all.obs", "complete.obs", "pairwise.complete.obs")
)
}
is.vec <- function(x)
{
is.vector(x) && !is.list(x)
}
check.is.flag <- function(x)
{
if (!(is.logical(x) && length(x) == 1 && (!is.na(x))))
{
nm <- deparse(substitute(x))
stop(paste0("argument '", nm, "' must be TRUE or FALSE"))
}
invisible()
}
#' @useDynLib coop R_check_badvals
check_badvals <- function(x)
{
.Call(R_check_badvals, x)
}
|
[GOAL]
α : Type u_1
β : α → Type u
δ : α → Sort v
inst✝ : DecidableEq α
a : α
b : δ a
s : Finset α
hs : ¬a ∈ s
e₁ e₂ : (a : α) → a ∈ s → δ a
eq : cons s a b e₁ = cons s a b e₂
e : α
h : e ∈ a ::ₘ s.val
⊢ e ∈ insert a s
[PROOFSTEP]
simpa only [Multiset.mem_cons, mem_insert] using h
[GOAL]
α : Type u_1
β : α → Type u
δ : α → Sort v
inst✝ : DecidableEq α
a : α
b : δ a
s : Finset α
hs : ¬a ∈ s
e₁ e₂ : (a : α) → a ∈ s → δ a
eq : cons s a b e₁ = cons s a b e₂
e : α
h : e ∈ a ::ₘ s.val
⊢ e ∈ insert a s
[PROOFSTEP]
simpa only [Multiset.mem_cons, mem_insert] using h
[GOAL]
α : Type u_1
β : α → Type u
δ : α → Sort v
inst✝ : DecidableEq α
a : α
b : δ a
s : Finset α
hs : ¬a ∈ s
e₁ e₂ : (a : α) → a ∈ s → δ a
eq : cons s a b e₁ = cons s a b e₂
e : α
h : e ∈ a ::ₘ s.val
⊢ cons s a b e₁ e (_ : e ∈ insert a s) = cons s a b e₂ e (_ : e ∈ insert a s)
[PROOFSTEP]
rw [eq]
[GOAL]
α : Type u_1
β : α → Type u
δ : α → Sort v
inst✝¹ : DecidableEq α
inst✝ : (a : α) → DecidableEq (β a)
s : Finset α
t : (a : α) → Finset (β a)
a : α
ha : ¬a ∈ s
⊢ pi (insert a s) t = Finset.biUnion (t a) fun b => image (Pi.cons s a b) (pi s t)
[PROOFSTEP]
apply eq_of_veq
[GOAL]
case a
α : Type u_1
β : α → Type u
δ : α → Sort v
inst✝¹ : DecidableEq α
inst✝ : (a : α) → DecidableEq (β a)
s : Finset α
t : (a : α) → Finset (β a)
a : α
ha : ¬a ∈ s
⊢ (pi (insert a s) t).val = (Finset.biUnion (t a) fun b => image (Pi.cons s a b) (pi s t)).val
[PROOFSTEP]
rw [← (pi (insert a s) t).2.dedup]
[GOAL]
case a
α : Type u_1
β : α → Type u
δ : α → Sort v
inst✝¹ : DecidableEq α
inst✝ : (a : α) → DecidableEq (β a)
s : Finset α
t : (a : α) → Finset (β a)
a : α
ha : ¬a ∈ s
⊢ dedup (pi (insert a s) t).val = (Finset.biUnion (t a) fun b => image (Pi.cons s a b) (pi s t)).val
[PROOFSTEP]
refine'
(fun s' (h : s' = a ::ₘ s.1) =>
(_ :
dedup (Multiset.pi s' fun a => (t a).1) =
dedup
((t a).1.bind fun b =>
dedup <|
(Multiset.pi s.1 fun a : α => (t a).val).map fun f a' h' => Multiset.Pi.cons s.1 a b f a' (h ▸ h'))))
_ (insert_val_of_not_mem ha)
[GOAL]
case a
α : Type u_1
β : α → Type u
δ : α → Sort v
inst✝¹ : DecidableEq α
inst✝ : (a : α) → DecidableEq (β a)
s : Finset α
t : (a : α) → Finset (β a)
a : α
ha : ¬a ∈ s
s' : Multiset α
h : s' = a ::ₘ s.val
⊢ dedup (Multiset.pi s' fun a => (t a).val) =
dedup
(Multiset.bind (t a).val fun b =>
dedup
(Multiset.map (fun f a' h' => Multiset.Pi.cons s.val a b f a' (_ : a' ∈ a ::ₘ s.val))
(Multiset.pi s.val fun a => (t a).val)))
[PROOFSTEP]
subst s'
[GOAL]
case a
α : Type u_1
β : α → Type u
δ : α → Sort v
inst✝¹ : DecidableEq α
inst✝ : (a : α) → DecidableEq (β a)
s : Finset α
t : (a : α) → Finset (β a)
a : α
ha : ¬a ∈ s
⊢ dedup (Multiset.pi (a ::ₘ s.val) fun a => (t a).val) =
dedup
(Multiset.bind (t a).val fun b =>
dedup
(Multiset.map (fun f a' h' => Multiset.Pi.cons s.val a b f a' (_ : a' ∈ a ::ₘ s.val))
(Multiset.pi s.val fun a => (t a).val)))
[PROOFSTEP]
rw [pi_cons]
[GOAL]
case a
α : Type u_1
β : α → Type u
δ : α → Sort v
inst✝¹ : DecidableEq α
inst✝ : (a : α) → DecidableEq (β a)
s : Finset α
t : (a : α) → Finset (β a)
a : α
ha : ¬a ∈ s
⊢ dedup
(Multiset.bind (t a).val fun b =>
Multiset.map (Multiset.Pi.cons s.val a b) (Multiset.pi s.val fun a => (t a).val)) =
dedup
(Multiset.bind (t a).val fun b =>
dedup
(Multiset.map (fun f a' h' => Multiset.Pi.cons s.val a b f a' (_ : a' ∈ a ::ₘ s.val))
(Multiset.pi s.val fun a => (t a).val)))
[PROOFSTEP]
congr
[GOAL]
case a.e_s.e_f
α : Type u_1
β : α → Type u
δ : α → Sort v
inst✝¹ : DecidableEq α
inst✝ : (a : α) → DecidableEq (β a)
s : Finset α
t : (a : α) → Finset (β a)
a : α
ha : ¬a ∈ s
⊢ (fun b => Multiset.map (Multiset.Pi.cons s.val a b) (Multiset.pi s.val fun a => (t a).val)) = fun b =>
dedup
(Multiset.map (fun f a' h' => Multiset.Pi.cons s.val a b f a' (_ : a' ∈ a ::ₘ s.val))
(Multiset.pi s.val fun a => (t a).val))
[PROOFSTEP]
funext b
[GOAL]
case a.e_s.e_f.h
α : Type u_1
β : α → Type u
δ : α → Sort v
inst✝¹ : DecidableEq α
inst✝ : (a : α) → DecidableEq (β a)
s : Finset α
t : (a : α) → Finset (β a)
a : α
ha : ¬a ∈ s
b : β a
⊢ Multiset.map (Multiset.Pi.cons s.val a b) (Multiset.pi s.val fun a => (t a).val) =
dedup
(Multiset.map (fun f a' h' => Multiset.Pi.cons s.val a b f a' (_ : a' ∈ a ::ₘ s.val))
(Multiset.pi s.val fun a => (t a).val))
[PROOFSTEP]
exact ((pi s t).nodup.map <| Multiset.Pi.cons_injective ha).dedup.symm
[GOAL]
α : Type u_1
β✝ : α → Type u
δ : α → Sort v
inst✝ : DecidableEq α
β : Type u_2
s : Finset α
f : α → β
⊢ (pi s fun a => {f a}) = {fun a x => f a}
[PROOFSTEP]
rw [eq_singleton_iff_unique_mem]
[GOAL]
α : Type u_1
β✝ : α → Type u
δ : α → Sort v
inst✝ : DecidableEq α
β : Type u_2
s : Finset α
f : α → β
⊢ ((fun a x => f a) ∈ pi s fun a => {f a}) ∧ ∀ (x : (a : α) → a ∈ s → β), (x ∈ pi s fun a => {f a}) → x = fun a x => f a
[PROOFSTEP]
constructor
[GOAL]
case left
α : Type u_1
β✝ : α → Type u
δ : α → Sort v
inst✝ : DecidableEq α
β : Type u_2
s : Finset α
f : α → β
⊢ (fun a x => f a) ∈ pi s fun a => {f a}
[PROOFSTEP]
simp
[GOAL]
case right
α : Type u_1
β✝ : α → Type u
δ : α → Sort v
inst✝ : DecidableEq α
β : Type u_2
s : Finset α
f : α → β
⊢ ∀ (x : (a : α) → a ∈ s → β), (x ∈ pi s fun a => {f a}) → x = fun a x => f a
[PROOFSTEP]
intro a ha
[GOAL]
case right
α : Type u_1
β✝ : α → Type u
δ : α → Sort v
inst✝ : DecidableEq α
β : Type u_2
s : Finset α
f : α → β
a : (a : α) → a ∈ s → β
ha : a ∈ pi s fun a => {f a}
⊢ a = fun a x => f a
[PROOFSTEP]
ext i hi
[GOAL]
case right.h.h
α : Type u_1
β✝ : α → Type u
δ : α → Sort v
inst✝ : DecidableEq α
β : Type u_2
s : Finset α
f : α → β
a : (a : α) → a ∈ s → β
ha : a ∈ pi s fun a => {f a}
i : α
hi : i ∈ s
⊢ a i hi = f i
[PROOFSTEP]
rw [mem_pi] at ha
[GOAL]
case right.h.h
α : Type u_1
β✝ : α → Type u
δ : α → Sort v
inst✝ : DecidableEq α
β : Type u_2
s : Finset α
f : α → β
a : (a : α) → a ∈ s → β
ha : ∀ (a_1 : α) (h : a_1 ∈ s), a a_1 h ∈ {f a_1}
i : α
hi : i ∈ s
⊢ a i hi = f i
[PROOFSTEP]
simpa using ha i hi
|
[GOAL]
X : Type u
inst✝¹ : Lattice X
inst✝ : JordanHolderLattice X
x y : X
hxz : IsMaximal x (x ⊔ y)
hyz : IsMaximal y (x ⊔ y)
⊢ IsMaximal (x ⊓ y) y
[PROOFSTEP]
rw [inf_comm]
[GOAL]
X : Type u
inst✝¹ : Lattice X
inst✝ : JordanHolderLattice X
x y : X
hxz : IsMaximal x (x ⊔ y)
hyz : IsMaximal y (x ⊔ y)
⊢ IsMaximal (y ⊓ x) y
[PROOFSTEP]
rw [sup_comm] at hxz hyz
[GOAL]
X : Type u
inst✝¹ : Lattice X
inst✝ : JordanHolderLattice X
x y : X
hxz : IsMaximal x (y ⊔ x)
hyz : IsMaximal y (y ⊔ x)
⊢ IsMaximal (y ⊓ x) y
[PROOFSTEP]
exact isMaximal_inf_left_of_isMaximal_sup hyz hxz
[GOAL]
X : Type u
inst✝¹ : Lattice X
inst✝ : JordanHolderLattice X
x b a y : X
ha : x ⊓ y = a
hxy : x ≠ y
hxb : IsMaximal x b
hyb : IsMaximal y b
⊢ IsMaximal a y
[PROOFSTEP]
have hb : x ⊔ y = b := sup_eq_of_isMaximal hxb hyb hxy
[GOAL]
X : Type u
inst✝¹ : Lattice X
inst✝ : JordanHolderLattice X
x b a y : X
ha : x ⊓ y = a
hxy : x ≠ y
hxb : IsMaximal x b
hyb : IsMaximal y b
hb : x ⊔ y = b
⊢ IsMaximal a y
[PROOFSTEP]
substs a b
[GOAL]
X : Type u
inst✝¹ : Lattice X
inst✝ : JordanHolderLattice X
x y : X
hxy : x ≠ y
hxb : IsMaximal x (x ⊔ y)
hyb : IsMaximal y (x ⊔ y)
⊢ IsMaximal (x ⊓ y) y
[PROOFSTEP]
exact isMaximal_inf_right_of_isMaximal_sup hxb hyb
[GOAL]
X : Type u
inst✝¹ : Lattice X
inst✝ : JordanHolderLattice X
x y a b : X
hm : IsMaximal x a
ha : x ⊔ y = a
hb : x ⊓ y = b
⊢ Iso (x, a) (b, y)
[PROOFSTEP]
substs a b
[GOAL]
X : Type u
inst✝¹ : Lattice X
inst✝ : JordanHolderLattice X
x y : X
hm : IsMaximal x (x ⊔ y)
⊢ Iso (x, x ⊔ y) (x ⊓ y, y)
[PROOFSTEP]
exact second_iso hm
[GOAL]
X : Type u
inst✝¹ : Lattice X
inst✝ : JordanHolderLattice X
s : CompositionSeries X
x y : X
hx : x ∈ s
hy : y ∈ s
⊢ x ≤ y ∨ y ≤ x
[PROOFSTEP]
rcases Set.mem_range.1 hx with ⟨i, rfl⟩
[GOAL]
case intro
X : Type u
inst✝¹ : Lattice X
inst✝ : JordanHolderLattice X
s : CompositionSeries X
y : X
hy : y ∈ s
i : Fin (s.length + 1)
hx : series s i ∈ s
⊢ series s i ≤ y ∨ y ≤ series s i
[PROOFSTEP]
rcases Set.mem_range.1 hy with ⟨j, rfl⟩
[GOAL]
case intro.intro
X : Type u
inst✝¹ : Lattice X
inst✝ : JordanHolderLattice X
s : CompositionSeries X
i : Fin (s.length + 1)
hx : series s i ∈ s
j : Fin (s.length + 1)
hy : series s j ∈ s
⊢ series s i ≤ series s j ∨ series s j ≤ series s i
[PROOFSTEP]
rw [s.strictMono.le_iff_le, s.strictMono.le_iff_le]
[GOAL]
case intro.intro
X : Type u
inst✝¹ : Lattice X
inst✝ : JordanHolderLattice X
s : CompositionSeries X
i : Fin (s.length + 1)
hx : series s i ∈ s
j : Fin (s.length + 1)
hy : series s j ∈ s
⊢ i ≤ j ∨ j ≤ i
[PROOFSTEP]
exact le_total i j
[GOAL]
X : Type u
inst✝¹ : Lattice X
inst✝ : JordanHolderLattice X
s₁ s₂ : CompositionSeries X
hl : s₁.length = s₂.length
h :
∀ (i : Fin (s₁.length + 1)), series s₁ i = series s₂ (↑(Fin.castIso (_ : Nat.succ s₁.length = Nat.succ s₂.length)) i)
⊢ s₁ = s₂
[PROOFSTEP]
cases s₁
[GOAL]
case mk
X : Type u
inst✝¹ : Lattice X
inst✝ : JordanHolderLattice X
s₂ : CompositionSeries X
length✝ : ℕ
series✝ : Fin (length✝ + 1) → X
step'✝ : ∀ (i : Fin length✝), IsMaximal (series✝ (Fin.castSucc i)) (series✝ (Fin.succ i))
hl : { length := length✝, series := series✝, step' := step'✝ }.length = s₂.length
h :
∀ (i : Fin ({ length := length✝, series := series✝, step' := step'✝ }.length + 1)),
series { length := length✝, series := series✝, step' := step'✝ } i =
series s₂
(↑(Fin.castIso
(_ : Nat.succ { length := length✝, series := series✝, step' := step'✝ }.length = Nat.succ s₂.length))
i)
⊢ { length := length✝, series := series✝, step' := step'✝ } = s₂
[PROOFSTEP]
cases s₂
[GOAL]
case mk.mk
X : Type u
inst✝¹ : Lattice X
inst✝ : JordanHolderLattice X
length✝¹ : ℕ
series✝¹ : Fin (length✝¹ + 1) → X
step'✝¹ : ∀ (i : Fin length✝¹), IsMaximal (series✝¹ (Fin.castSucc i)) (series✝¹ (Fin.succ i))
length✝ : ℕ
series✝ : Fin (length✝ + 1) → X
step'✝ : ∀ (i : Fin length✝), IsMaximal (series✝ (Fin.castSucc i)) (series✝ (Fin.succ i))
hl :
{ length := length✝¹, series := series✝¹, step' := step'✝¹ }.length =
{ length := length✝, series := series✝, step' := step'✝ }.length
h :
∀ (i : Fin ({ length := length✝¹, series := series✝¹, step' := step'✝¹ }.length + 1)),
series { length := length✝¹, series := series✝¹, step' := step'✝¹ } i =
series { length := length✝, series := series✝, step' := step'✝ }
(↑(Fin.castIso
(_ :
Nat.succ { length := length✝¹, series := series✝¹, step' := step'✝¹ }.length =
Nat.succ { length := length✝, series := series✝, step' := step'✝ }.length))
i)
⊢ { length := length✝¹, series := series✝¹, step' := step'✝¹ } =
{ length := length✝, series := series✝, step' := step'✝ }
[PROOFSTEP]
dsimp at hl h
[GOAL]
case mk.mk
X : Type u
inst✝¹ : Lattice X
inst✝ : JordanHolderLattice X
length✝¹ : ℕ
series✝¹ : Fin (length✝¹ + 1) → X
step'✝¹ : ∀ (i : Fin length✝¹), IsMaximal (series✝¹ (Fin.castSucc i)) (series✝¹ (Fin.succ i))
length✝ : ℕ
series✝ : Fin (length✝ + 1) → X
step'✝ : ∀ (i : Fin length✝), IsMaximal (series✝ (Fin.castSucc i)) (series✝ (Fin.succ i))
hl : length✝¹ = length✝
h : ∀ (i : Fin (length✝¹ + 1)), series✝¹ i = series✝ (↑(Fin.castIso (_ : Nat.succ length✝¹ = Nat.succ length✝)) i)
⊢ { length := length✝¹, series := series✝¹, step' := step'✝¹ } =
{ length := length✝, series := series✝, step' := step'✝ }
[PROOFSTEP]
subst hl
[GOAL]
case mk.mk
X : Type u
inst✝¹ : Lattice X
inst✝ : JordanHolderLattice X
length✝ : ℕ
series✝¹ : Fin (length✝ + 1) → X
step'✝¹ : ∀ (i : Fin length✝), IsMaximal (series✝¹ (Fin.castSucc i)) (series✝¹ (Fin.succ i))
series✝ : Fin (length✝ + 1) → X
step'✝ : ∀ (i : Fin length✝), IsMaximal (series✝ (Fin.castSucc i)) (series✝ (Fin.succ i))
h : ∀ (i : Fin (length✝ + 1)), series✝¹ i = series✝ (↑(Fin.castIso (_ : Nat.succ length✝ = Nat.succ length✝)) i)
⊢ { length := length✝, series := series✝¹, step' := step'✝¹ } =
{ length := length✝, series := series✝, step' := step'✝ }
[PROOFSTEP]
simpa [Function.funext_iff] using h
[GOAL]
X : Type u
inst✝¹ : Lattice X
inst✝ : JordanHolderLattice X
s : CompositionSeries X
⊢ List.length (toList s) = s.length + 1
[PROOFSTEP]
rw [toList, List.length_ofFn]
[GOAL]
X : Type u
inst✝¹ : Lattice X
inst✝ : JordanHolderLattice X
s : CompositionSeries X
⊢ toList s ≠ []
[PROOFSTEP]
rw [← List.length_pos_iff_ne_nil, length_toList]
[GOAL]
X : Type u
inst✝¹ : Lattice X
inst✝ : JordanHolderLattice X
s : CompositionSeries X
⊢ 0 < s.length + 1
[PROOFSTEP]
exact Nat.succ_pos _
[GOAL]
X : Type u
inst✝¹ : Lattice X
inst✝ : JordanHolderLattice X
s₁ s₂ : CompositionSeries X
h : List.ofFn s₁.series = List.ofFn s₂.series
⊢ s₁ = s₂
[PROOFSTEP]
have h₁ : s₁.length = s₂.length :=
Nat.succ_injective ((List.length_ofFn s₁).symm.trans <| (congr_arg List.length h).trans <| List.length_ofFn s₂)
[GOAL]
X : Type u
inst✝¹ : Lattice X
inst✝ : JordanHolderLattice X
s₁ s₂ : CompositionSeries X
h : List.ofFn s₁.series = List.ofFn s₂.series
h₁ : s₁.length = s₂.length
⊢ s₁ = s₂
[PROOFSTEP]
have h₂ : ∀ i : Fin s₁.length.succ, s₁ i = s₂ (Fin.castIso (congr_arg Nat.succ h₁) i) :=
-- Porting note: `List.nthLe_ofFn` has been deprecated but `List.get_ofFn` has a
-- different type, so we do golf here.congr_fun <|
List.ofFn_injective <| h.trans <| List.ofFn_congr (congr_arg Nat.succ h₁).symm _
[GOAL]
X : Type u
inst✝¹ : Lattice X
inst✝ : JordanHolderLattice X
s₁ s₂ : CompositionSeries X
h : List.ofFn s₁.series = List.ofFn s₂.series
h₁ : s₁.length = s₂.length
h₂ :
∀ (i : Fin (Nat.succ s₁.length)),
series s₁ i = series s₂ (↑(Fin.castIso (_ : Nat.succ s₁.length = Nat.succ s₂.length)) i)
⊢ s₁ = s₂
[PROOFSTEP]
cases s₁
[GOAL]
case mk
X : Type u
inst✝¹ : Lattice X
inst✝ : JordanHolderLattice X
s₂ : CompositionSeries X
length✝ : ℕ
series✝ : Fin (length✝ + 1) → X
step'✝ : ∀ (i : Fin length✝), IsMaximal (series✝ (Fin.castSucc i)) (series✝ (Fin.succ i))
h : List.ofFn { length := length✝, series := series✝, step' := step'✝ }.series = List.ofFn s₂.series
h₁ : { length := length✝, series := series✝, step' := step'✝ }.length = s₂.length
h₂ :
∀ (i : Fin (Nat.succ { length := length✝, series := series✝, step' := step'✝ }.length)),
series { length := length✝, series := series✝, step' := step'✝ } i =
series s₂
(↑(Fin.castIso
(_ : Nat.succ { length := length✝, series := series✝, step' := step'✝ }.length = Nat.succ s₂.length))
i)
⊢ { length := length✝, series := series✝, step' := step'✝ } = s₂
[PROOFSTEP]
cases s₂
[GOAL]
case mk.mk
X : Type u
inst✝¹ : Lattice X
inst✝ : JordanHolderLattice X
length✝¹ : ℕ
series✝¹ : Fin (length✝¹ + 1) → X
step'✝¹ : ∀ (i : Fin length✝¹), IsMaximal (series✝¹ (Fin.castSucc i)) (series✝¹ (Fin.succ i))
length✝ : ℕ
series✝ : Fin (length✝ + 1) → X
step'✝ : ∀ (i : Fin length✝), IsMaximal (series✝ (Fin.castSucc i)) (series✝ (Fin.succ i))
h :
List.ofFn { length := length✝¹, series := series✝¹, step' := step'✝¹ }.series =
List.ofFn { length := length✝, series := series✝, step' := step'✝ }.series
h₁ :
{ length := length✝¹, series := series✝¹, step' := step'✝¹ }.length =
{ length := length✝, series := series✝, step' := step'✝ }.length
h₂ :
∀ (i : Fin (Nat.succ { length := length✝¹, series := series✝¹, step' := step'✝¹ }.length)),
series { length := length✝¹, series := series✝¹, step' := step'✝¹ } i =
series { length := length✝, series := series✝, step' := step'✝ }
(↑(Fin.castIso
(_ :
Nat.succ { length := length✝¹, series := series✝¹, step' := step'✝¹ }.length =
Nat.succ { length := length✝, series := series✝, step' := step'✝ }.length))
i)
⊢ { length := length✝¹, series := series✝¹, step' := step'✝¹ } =
{ length := length✝, series := series✝, step' := step'✝ }
[PROOFSTEP]
dsimp at h h₁ h₂
[GOAL]
case mk.mk
X : Type u
inst✝¹ : Lattice X
inst✝ : JordanHolderLattice X
length✝¹ : ℕ
series✝¹ : Fin (length✝¹ + 1) → X
step'✝¹ : ∀ (i : Fin length✝¹), IsMaximal (series✝¹ (Fin.castSucc i)) (series✝¹ (Fin.succ i))
length✝ : ℕ
series✝ : Fin (length✝ + 1) → X
step'✝ : ∀ (i : Fin length✝), IsMaximal (series✝ (Fin.castSucc i)) (series✝ (Fin.succ i))
h : List.ofFn series✝¹ = List.ofFn series✝
h₁ : length✝¹ = length✝
h₂ : ∀ (i : Fin (Nat.succ length✝¹)), series✝¹ i = series✝ (↑(Fin.castIso (_ : Nat.succ length✝¹ = Nat.succ length✝)) i)
⊢ { length := length✝¹, series := series✝¹, step' := step'✝¹ } =
{ length := length✝, series := series✝, step' := step'✝ }
[PROOFSTEP]
subst h₁
[GOAL]
case mk.mk
X : Type u
inst✝¹ : Lattice X
inst✝ : JordanHolderLattice X
length✝ : ℕ
series✝¹ : Fin (length✝ + 1) → X
step'✝¹ : ∀ (i : Fin length✝), IsMaximal (series✝¹ (Fin.castSucc i)) (series✝¹ (Fin.succ i))
series✝ : Fin (length✝ + 1) → X
step'✝ : ∀ (i : Fin length✝), IsMaximal (series✝ (Fin.castSucc i)) (series✝ (Fin.succ i))
h : List.ofFn series✝¹ = List.ofFn series✝
h₂ : ∀ (i : Fin (Nat.succ length✝)), series✝¹ i = series✝ (↑(Fin.castIso (_ : Nat.succ length✝ = Nat.succ length✝)) i)
⊢ { length := length✝, series := series✝¹, step' := step'✝¹ } =
{ length := length✝, series := series✝, step' := step'✝ }
[PROOFSTEP]
simp only [mk.injEq, heq_eq_eq, true_and]
[GOAL]
case mk.mk
X : Type u
inst✝¹ : Lattice X
inst✝ : JordanHolderLattice X
length✝ : ℕ
series✝¹ : Fin (length✝ + 1) → X
step'✝¹ : ∀ (i : Fin length✝), IsMaximal (series✝¹ (Fin.castSucc i)) (series✝¹ (Fin.succ i))
series✝ : Fin (length✝ + 1) → X
step'✝ : ∀ (i : Fin length✝), IsMaximal (series✝ (Fin.castSucc i)) (series✝ (Fin.succ i))
h : List.ofFn series✝¹ = List.ofFn series✝
h₂ : ∀ (i : Fin (Nat.succ length✝)), series✝¹ i = series✝ (↑(Fin.castIso (_ : Nat.succ length✝ = Nat.succ length✝)) i)
⊢ series✝¹ = series✝
[PROOFSTEP]
simp only [Fin.castIso_refl] at h₂
[GOAL]
case mk.mk
X : Type u
inst✝¹ : Lattice X
inst✝ : JordanHolderLattice X
length✝ : ℕ
series✝¹ : Fin (length✝ + 1) → X
step'✝¹ : ∀ (i : Fin length✝), IsMaximal (series✝¹ (Fin.castSucc i)) (series✝¹ (Fin.succ i))
series✝ : Fin (length✝ + 1) → X
step'✝ : ∀ (i : Fin length✝), IsMaximal (series✝ (Fin.castSucc i)) (series✝ (Fin.succ i))
h : List.ofFn series✝¹ = List.ofFn series✝
h₂ : ∀ (i : Fin (Nat.succ length✝)), series✝¹ i = series✝ (↑(OrderIso.refl (Fin (Nat.succ length✝))) i)
⊢ series✝¹ = series✝
[PROOFSTEP]
exact funext h₂
[GOAL]
X : Type u
inst✝¹ : Lattice X
inst✝ : JordanHolderLattice X
s : CompositionSeries X
⊢ ∀ (i : ℕ) (h : i < List.length (toList s) - 1),
IsMaximal (List.get (toList s) { val := i, isLt := (_ : i < List.length (toList s)) })
(List.get (toList s) { val := i + 1, isLt := (_ : Nat.succ i < List.length (toList s)) })
[PROOFSTEP]
intro i hi
[GOAL]
X : Type u
inst✝¹ : Lattice X
inst✝ : JordanHolderLattice X
s : CompositionSeries X
i : ℕ
hi : i < List.length (toList s) - 1
⊢ IsMaximal (List.get (toList s) { val := i, isLt := (_ : i < List.length (toList s)) })
(List.get (toList s) { val := i + 1, isLt := (_ : Nat.succ i < List.length (toList s)) })
[PROOFSTEP]
simp only [toList, List.get_ofFn]
[GOAL]
X : Type u
inst✝¹ : Lattice X
inst✝ : JordanHolderLattice X
s : CompositionSeries X
i : ℕ
hi : i < List.length (toList s) - 1
⊢ IsMaximal
(series s
(↑(Fin.castIso (_ : List.length (List.ofFn s.series) = s.length + 1))
{ val := i, isLt := (_ : i < List.length (toList s)) }))
(series s
(↑(Fin.castIso (_ : List.length (List.ofFn s.series) = s.length + 1))
{ val := i + 1, isLt := (_ : Nat.succ i < List.length (toList s)) }))
[PROOFSTEP]
rw [length_toList] at hi
[GOAL]
X : Type u
inst✝¹ : Lattice X
inst✝ : JordanHolderLattice X
s : CompositionSeries X
i : ℕ
hi✝ : i < List.length (toList s) - 1
hi : i < s.length + 1 - 1
⊢ IsMaximal
(series s
(↑(Fin.castIso (_ : List.length (List.ofFn s.series) = s.length + 1))
{ val := i, isLt := (_ : i < List.length (toList s)) }))
(series s
(↑(Fin.castIso (_ : List.length (List.ofFn s.series) = s.length + 1))
{ val := i + 1, isLt := (_ : Nat.succ i < List.length (toList s)) }))
[PROOFSTEP]
exact s.step ⟨i, hi⟩
[GOAL]
X : Type u
inst✝¹ : Lattice X
inst✝ : JordanHolderLattice X
s : CompositionSeries X
i j : Fin (List.length (toList s))
h : i < j
⊢ List.get (toList s) i < List.get (toList s) j
[PROOFSTEP]
dsimp [toList]
[GOAL]
X : Type u
inst✝¹ : Lattice X
inst✝ : JordanHolderLattice X
s : CompositionSeries X
i j : Fin (List.length (toList s))
h : i < j
⊢ List.get (List.ofFn s.series) i < List.get (List.ofFn s.series) j
[PROOFSTEP]
rw [List.get_ofFn, List.get_ofFn]
[GOAL]
X : Type u
inst✝¹ : Lattice X
inst✝ : JordanHolderLattice X
s : CompositionSeries X
i j : Fin (List.length (toList s))
h : i < j
⊢ series s (↑(Fin.castIso (_ : List.length (List.ofFn s.series) = s.length + 1)) i) <
series s (↑(Fin.castIso (_ : List.length (List.ofFn s.series) = s.length + 1)) j)
[PROOFSTEP]
exact s.strictMono h
[GOAL]
X : Type u
inst✝¹ : Lattice X
inst✝ : JordanHolderLattice X
s : CompositionSeries X
x : X
⊢ x ∈ toList s ↔ x ∈ s
[PROOFSTEP]
rw [toList, List.mem_ofFn, mem_def]
[GOAL]
X : Type u
inst✝¹ : Lattice X
inst✝ : JordanHolderLattice X
l : List X
hl : l ≠ []
hc : List.Chain' IsMaximal l
i : Fin (List.length l - 1 + 1)
⊢ ↑i < List.length l
[PROOFSTEP]
conv_rhs => rw [← tsub_add_cancel_of_le (Nat.succ_le_of_lt (List.length_pos_of_ne_nil hl))]
[GOAL]
X : Type u
inst✝¹ : Lattice X
inst✝ : JordanHolderLattice X
l : List X
hl : l ≠ []
hc : List.Chain' IsMaximal l
i : Fin (List.length l - 1 + 1)
| List.length l
[PROOFSTEP]
rw [← tsub_add_cancel_of_le (Nat.succ_le_of_lt (List.length_pos_of_ne_nil hl))]
[GOAL]
X : Type u
inst✝¹ : Lattice X
inst✝ : JordanHolderLattice X
l : List X
hl : l ≠ []
hc : List.Chain' IsMaximal l
i : Fin (List.length l - 1 + 1)
| List.length l
[PROOFSTEP]
rw [← tsub_add_cancel_of_le (Nat.succ_le_of_lt (List.length_pos_of_ne_nil hl))]
[GOAL]
X : Type u
inst✝¹ : Lattice X
inst✝ : JordanHolderLattice X
l : List X
hl : l ≠ []
hc : List.Chain' IsMaximal l
i : Fin (List.length l - 1 + 1)
| List.length l
[PROOFSTEP]
rw [← tsub_add_cancel_of_le (Nat.succ_le_of_lt (List.length_pos_of_ne_nil hl))]
[GOAL]
X : Type u
inst✝¹ : Lattice X
inst✝ : JordanHolderLattice X
l : List X
hl : l ≠ []
hc : List.Chain' IsMaximal l
i : Fin (List.length l - 1 + 1)
⊢ ↑i < List.length l - Nat.succ 0 + Nat.succ 0
[PROOFSTEP]
exact i.2
[GOAL]
X : Type u
inst✝¹ : Lattice X
inst✝ : JordanHolderLattice X
s : CompositionSeries X
⊢ ofList (toList s) (_ : toList s ≠ []) (_ : List.Chain' IsMaximal (toList s)) = s
[PROOFSTEP]
refine' ext_fun _ _
[GOAL]
case refine'_1
X : Type u
inst✝¹ : Lattice X
inst✝ : JordanHolderLattice X
s : CompositionSeries X
⊢ (ofList (toList s) (_ : toList s ≠ []) (_ : List.Chain' IsMaximal (toList s))).length = s.length
[PROOFSTEP]
rw [length_ofList, length_toList, Nat.succ_sub_one]
[GOAL]
case refine'_2
X : Type u
inst✝¹ : Lattice X
inst✝ : JordanHolderLattice X
s : CompositionSeries X
⊢ ∀ (i : Fin ((ofList (toList s) (_ : toList s ≠ []) (_ : List.Chain' IsMaximal (toList s))).length + 1)),
series (ofList (toList s) (_ : toList s ≠ []) (_ : List.Chain' IsMaximal (toList s))) i =
series s
(↑(Fin.castIso
(_ :
Nat.succ (ofList (toList s) (_ : toList s ≠ []) (_ : List.Chain' IsMaximal (toList s))).length =
Nat.succ s.length))
i)
[PROOFSTEP]
rintro
⟨i, hi⟩
-- Porting note: Was `dsimp [ofList, toList]; rw [List.nthLe_ofFn']`.
[GOAL]
case refine'_2.mk
X : Type u
inst✝¹ : Lattice X
inst✝ : JordanHolderLattice X
s : CompositionSeries X
i : ℕ
hi : i < (ofList (toList s) (_ : toList s ≠ []) (_ : List.Chain' IsMaximal (toList s))).length + 1
⊢ series (ofList (toList s) (_ : toList s ≠ []) (_ : List.Chain' IsMaximal (toList s))) { val := i, isLt := hi } =
series s
(↑(Fin.castIso
(_ :
Nat.succ (ofList (toList s) (_ : toList s ≠ []) (_ : List.Chain' IsMaximal (toList s))).length =
Nat.succ s.length))
{ val := i, isLt := hi })
[PROOFSTEP]
simp [ofList, toList, -List.ofFn_succ]
[GOAL]
X : Type u
inst✝¹ : Lattice X
inst✝ : JordanHolderLattice X
l : List X
hl : l ≠ []
hc : List.Chain' IsMaximal l
⊢ toList (ofList l hl hc) = l
[PROOFSTEP]
refine' List.ext_get _ _
[GOAL]
case refine'_1
X : Type u
inst✝¹ : Lattice X
inst✝ : JordanHolderLattice X
l : List X
hl : l ≠ []
hc : List.Chain' IsMaximal l
⊢ List.length (toList (ofList l hl hc)) = List.length l
[PROOFSTEP]
rw [length_toList, length_ofList, tsub_add_cancel_of_le (Nat.succ_le_of_lt <| List.length_pos_of_ne_nil hl)]
[GOAL]
case refine'_2
X : Type u
inst✝¹ : Lattice X
inst✝ : JordanHolderLattice X
l : List X
hl : l ≠ []
hc : List.Chain' IsMaximal l
⊢ ∀ (n : ℕ) (h₁ : n < List.length (toList (ofList l hl hc))) (h₂ : n < List.length l),
List.get (toList (ofList l hl hc)) { val := n, isLt := h₁ } = List.get l { val := n, isLt := h₂ }
[PROOFSTEP]
intro i hi hi'
[GOAL]
case refine'_2
X : Type u
inst✝¹ : Lattice X
inst✝ : JordanHolderLattice X
l : List X
hl : l ≠ []
hc : List.Chain' IsMaximal l
i : ℕ
hi : i < List.length (toList (ofList l hl hc))
hi' : i < List.length l
⊢ List.get (toList (ofList l hl hc)) { val := i, isLt := hi } = List.get l { val := i, isLt := hi' }
[PROOFSTEP]
dsimp [ofList, toList]
[GOAL]
case refine'_2
X : Type u
inst✝¹ : Lattice X
inst✝ : JordanHolderLattice X
l : List X
hl : l ≠ []
hc : List.Chain' IsMaximal l
i : ℕ
hi : i < List.length (toList (ofList l hl hc))
hi' : i < List.length l
⊢ List.get (List.ofFn fun i => List.nthLe l ↑i (_ : ↑i < List.length l)) { val := i, isLt := hi } =
List.get l { val := i, isLt := hi' }
[PROOFSTEP]
rw [List.get_ofFn]
[GOAL]
case refine'_2
X : Type u
inst✝¹ : Lattice X
inst✝ : JordanHolderLattice X
l : List X
hl : l ≠ []
hc : List.Chain' IsMaximal l
i : ℕ
hi : i < List.length (toList (ofList l hl hc))
hi' : i < List.length l
⊢ List.nthLe l
↑(↑(Fin.castIso
(_ : List.length (List.ofFn fun i => List.nthLe l ↑i (_ : ↑i < List.length l)) = List.length l - 1 + 1))
{ val := i, isLt := hi })
(_ :
↑(↑(Fin.castIso
(_ :
List.length (List.ofFn fun i => List.nthLe l ↑i (_ : ↑i < List.length l)) = List.length l - 1 + 1))
{ val := i, isLt := hi }) <
List.length l) =
List.get l { val := i, isLt := hi' }
[PROOFSTEP]
rfl
[GOAL]
X : Type u
inst✝¹ : Lattice X
inst✝ : JordanHolderLattice X
s₁ s₂ : CompositionSeries X
h : ∀ (x : X), x ∈ s₁ ↔ x ∈ s₂
⊢ toList s₁ ~ toList s₂
[PROOFSTEP]
classical exact List.perm_of_nodup_nodup_toFinset_eq s₁.toList_nodup s₂.toList_nodup (Finset.ext <| by simp [*])
[GOAL]
X : Type u
inst✝¹ : Lattice X
inst✝ : JordanHolderLattice X
s₁ s₂ : CompositionSeries X
h : ∀ (x : X), x ∈ s₁ ↔ x ∈ s₂
⊢ toList s₁ ~ toList s₂
[PROOFSTEP]
exact List.perm_of_nodup_nodup_toFinset_eq s₁.toList_nodup s₂.toList_nodup (Finset.ext <| by simp [*])
[GOAL]
X : Type u
inst✝¹ : Lattice X
inst✝ : JordanHolderLattice X
s₁ s₂ : CompositionSeries X
h : ∀ (x : X), x ∈ s₁ ↔ x ∈ s₂
⊢ ∀ (a : X), a ∈ List.toFinset (toList s₁) ↔ a ∈ List.toFinset (toList s₂)
[PROOFSTEP]
simp [*]
[GOAL]
X : Type u
inst✝¹ : Lattice X
inst✝ : JordanHolderLattice X
s : CompositionSeries X
i : Fin (s.length - 1)
⊢ IsMaximal ((fun i => series s { val := ↑i, isLt := (_ : ↑i < s.length + 1) }) (Fin.castSucc i))
((fun i => series s { val := ↑i, isLt := (_ : ↑i < s.length + 1) }) (Fin.succ i))
[PROOFSTEP]
have := s.step ⟨i, lt_of_lt_of_le i.2 tsub_le_self⟩
[GOAL]
X : Type u
inst✝¹ : Lattice X
inst✝ : JordanHolderLattice X
s : CompositionSeries X
i : Fin (s.length - 1)
this :
IsMaximal (series s (Fin.castSucc { val := ↑i, isLt := (_ : ↑i < s.length) }))
(series s (Fin.succ { val := ↑i, isLt := (_ : ↑i < s.length) }))
⊢ IsMaximal ((fun i => series s { val := ↑i, isLt := (_ : ↑i < s.length + 1) }) (Fin.castSucc i))
((fun i => series s { val := ↑i, isLt := (_ : ↑i < s.length + 1) }) (Fin.succ i))
[PROOFSTEP]
cases i
[GOAL]
case mk
X : Type u
inst✝¹ : Lattice X
inst✝ : JordanHolderLattice X
s : CompositionSeries X
val✝ : ℕ
isLt✝ : val✝ < s.length - 1
this :
IsMaximal
(series s
(Fin.castSucc
{ val := ↑{ val := val✝, isLt := isLt✝ }, isLt := (_ : ↑{ val := val✝, isLt := isLt✝ } < s.length) }))
(series s
(Fin.succ { val := ↑{ val := val✝, isLt := isLt✝ }, isLt := (_ : ↑{ val := val✝, isLt := isLt✝ } < s.length) }))
⊢ IsMaximal
((fun i => series s { val := ↑i, isLt := (_ : ↑i < s.length + 1) }) (Fin.castSucc { val := val✝, isLt := isLt✝ }))
((fun i => series s { val := ↑i, isLt := (_ : ↑i < s.length + 1) }) (Fin.succ { val := val✝, isLt := isLt✝ }))
[PROOFSTEP]
exact this
[GOAL]
X : Type u
inst✝¹ : Lattice X
inst✝ : JordanHolderLattice X
s : CompositionSeries X
⊢ { val := ↑(Fin.last (eraseTop s).length), isLt := (_ : ↑(Fin.last (eraseTop s).length) < s.length + 1) } =
{ val := s.length - 1, isLt := (_ : s.length - 1 < s.length + 1) }
[PROOFSTEP]
ext
[GOAL]
case h
X : Type u
inst✝¹ : Lattice X
inst✝ : JordanHolderLattice X
s : CompositionSeries X
⊢ ↑{ val := ↑(Fin.last (eraseTop s).length), isLt := (_ : ↑(Fin.last (eraseTop s).length) < s.length + 1) } =
↑{ val := s.length - 1, isLt := (_ : s.length - 1 < s.length + 1) }
[PROOFSTEP]
simp only [eraseTop_length, Fin.val_last, Fin.coe_castSucc, Fin.coe_ofNat_eq_mod, Fin.val_mk]
[GOAL]
X : Type u
inst✝¹ : Lattice X
inst✝ : JordanHolderLattice X
s : CompositionSeries X
⊢ top (eraseTop s) ≤ top s
[PROOFSTEP]
simp [eraseTop, top, s.strictMono.le_iff_le, Fin.le_iff_val_le_val, tsub_le_self]
[GOAL]
X : Type u
inst✝¹ : Lattice X
inst✝ : JordanHolderLattice X
s : CompositionSeries X
x : X
hx : x ≠ top s
hxs : x ∈ s
⊢ x ∈ eraseTop s
[PROOFSTEP]
rcases hxs with ⟨i, rfl⟩
[GOAL]
case intro
X : Type u
inst✝¹ : Lattice X
inst✝ : JordanHolderLattice X
s : CompositionSeries X
i : Fin (s.length + 1)
hx : series s i ≠ top s
⊢ series s i ∈ eraseTop s
[PROOFSTEP]
have hi : (i : ℕ) < (s.length - 1).succ :=
by
conv_rhs => rw [← Nat.succ_sub (length_pos_of_mem_ne ⟨i, rfl⟩ s.top_mem hx), Nat.succ_sub_one]
exact lt_of_le_of_ne (Nat.le_of_lt_succ i.2) (by simpa [top, s.inj, Fin.ext_iff] using hx)
[GOAL]
X : Type u
inst✝¹ : Lattice X
inst✝ : JordanHolderLattice X
s : CompositionSeries X
i : Fin (s.length + 1)
hx : series s i ≠ top s
⊢ ↑i < Nat.succ (s.length - 1)
[PROOFSTEP]
conv_rhs => rw [← Nat.succ_sub (length_pos_of_mem_ne ⟨i, rfl⟩ s.top_mem hx), Nat.succ_sub_one]
[GOAL]
X : Type u
inst✝¹ : Lattice X
inst✝ : JordanHolderLattice X
s : CompositionSeries X
i : Fin (s.length + 1)
hx : series s i ≠ top s
| Nat.succ (s.length - 1)
[PROOFSTEP]
rw [← Nat.succ_sub (length_pos_of_mem_ne ⟨i, rfl⟩ s.top_mem hx), Nat.succ_sub_one]
[GOAL]
X : Type u
inst✝¹ : Lattice X
inst✝ : JordanHolderLattice X
s : CompositionSeries X
i : Fin (s.length + 1)
hx : series s i ≠ top s
| Nat.succ (s.length - 1)
[PROOFSTEP]
rw [← Nat.succ_sub (length_pos_of_mem_ne ⟨i, rfl⟩ s.top_mem hx), Nat.succ_sub_one]
[GOAL]
X : Type u
inst✝¹ : Lattice X
inst✝ : JordanHolderLattice X
s : CompositionSeries X
i : Fin (s.length + 1)
hx : series s i ≠ top s
| Nat.succ (s.length - 1)
[PROOFSTEP]
rw [← Nat.succ_sub (length_pos_of_mem_ne ⟨i, rfl⟩ s.top_mem hx), Nat.succ_sub_one]
[GOAL]
X : Type u
inst✝¹ : Lattice X
inst✝ : JordanHolderLattice X
s : CompositionSeries X
i : Fin (s.length + 1)
hx : series s i ≠ top s
⊢ ↑i < s.length
[PROOFSTEP]
exact lt_of_le_of_ne (Nat.le_of_lt_succ i.2) (by simpa [top, s.inj, Fin.ext_iff] using hx)
[GOAL]
X : Type u
inst✝¹ : Lattice X
inst✝ : JordanHolderLattice X
s : CompositionSeries X
i : Fin (s.length + 1)
hx : series s i ≠ top s
⊢ ↑i ≠ s.length
[PROOFSTEP]
simpa [top, s.inj, Fin.ext_iff] using hx
[GOAL]
case intro
X : Type u
inst✝¹ : Lattice X
inst✝ : JordanHolderLattice X
s : CompositionSeries X
i : Fin (s.length + 1)
hx : series s i ≠ top s
hi : ↑i < Nat.succ (s.length - 1)
⊢ series s i ∈ eraseTop s
[PROOFSTEP]
refine' ⟨Fin.castSucc (n := s.length + 1) i, _⟩
[GOAL]
case intro
X : Type u
inst✝¹ : Lattice X
inst✝ : JordanHolderLattice X
s : CompositionSeries X
i : Fin (s.length + 1)
hx : series s i ≠ top s
hi : ↑i < Nat.succ (s.length - 1)
⊢ series (eraseTop s) ↑↑(Fin.castSucc i) = series s i
[PROOFSTEP]
simp [Fin.ext_iff, Nat.mod_eq_of_lt hi]
[GOAL]
X : Type u
inst✝¹ : Lattice X
inst✝ : JordanHolderLattice X
s : CompositionSeries X
x : X
h : 0 < s.length
⊢ x ∈ eraseTop s ↔ x ≠ top s ∧ x ∈ s
[PROOFSTEP]
simp only [mem_def]
[GOAL]
X : Type u
inst✝¹ : Lattice X
inst✝ : JordanHolderLattice X
s : CompositionSeries X
x : X
h : 0 < s.length
⊢ x ∈ range (eraseTop s).series ↔ x ≠ top s ∧ x ∈ range s.series
[PROOFSTEP]
dsimp only [eraseTop]
[GOAL]
X : Type u
inst✝¹ : Lattice X
inst✝ : JordanHolderLattice X
s : CompositionSeries X
x : X
h : 0 < s.length
⊢ (x ∈ range fun i => series s { val := ↑i, isLt := (_ : ↑i < s.length + 1) }) ↔ x ≠ top s ∧ x ∈ range s.series
[PROOFSTEP]
constructor
[GOAL]
case mp
X : Type u
inst✝¹ : Lattice X
inst✝ : JordanHolderLattice X
s : CompositionSeries X
x : X
h : 0 < s.length
⊢ (x ∈ range fun i => series s { val := ↑i, isLt := (_ : ↑i < s.length + 1) }) → x ≠ top s ∧ x ∈ range s.series
[PROOFSTEP]
rintro ⟨i, rfl⟩
[GOAL]
case mp.intro
X : Type u
inst✝¹ : Lattice X
inst✝ : JordanHolderLattice X
s : CompositionSeries X
h : 0 < s.length
i : Fin (s.length - 1 + 1)
⊢ (fun i => series s { val := ↑i, isLt := (_ : ↑i < s.length + 1) }) i ≠ top s ∧
(fun i => series s { val := ↑i, isLt := (_ : ↑i < s.length + 1) }) i ∈ range s.series
[PROOFSTEP]
have hi : (i : ℕ) < s.length :=
by
conv_rhs => rw [← Nat.succ_sub_one s.length, Nat.succ_sub h]
exact
i.2
-- Porting note: Was `simp [top, Fin.ext_iff, ne_of_lt hi]`.
[GOAL]
X : Type u
inst✝¹ : Lattice X
inst✝ : JordanHolderLattice X
s : CompositionSeries X
h : 0 < s.length
i : Fin (s.length - 1 + 1)
⊢ ↑i < s.length
[PROOFSTEP]
conv_rhs => rw [← Nat.succ_sub_one s.length, Nat.succ_sub h]
[GOAL]
X : Type u
inst✝¹ : Lattice X
inst✝ : JordanHolderLattice X
s : CompositionSeries X
h : 0 < s.length
i : Fin (s.length - 1 + 1)
| s.length
[PROOFSTEP]
rw [← Nat.succ_sub_one s.length, Nat.succ_sub h]
[GOAL]
X : Type u
inst✝¹ : Lattice X
inst✝ : JordanHolderLattice X
s : CompositionSeries X
h : 0 < s.length
i : Fin (s.length - 1 + 1)
| s.length
[PROOFSTEP]
rw [← Nat.succ_sub_one s.length, Nat.succ_sub h]
[GOAL]
X : Type u
inst✝¹ : Lattice X
inst✝ : JordanHolderLattice X
s : CompositionSeries X
h : 0 < s.length
i : Fin (s.length - 1 + 1)
| s.length
[PROOFSTEP]
rw [← Nat.succ_sub_one s.length, Nat.succ_sub h]
[GOAL]
X : Type u
inst✝¹ : Lattice X
inst✝ : JordanHolderLattice X
s : CompositionSeries X
h : 0 < s.length
i : Fin (s.length - 1 + 1)
⊢ ↑i < Nat.succ (s.length - Nat.succ 0)
[PROOFSTEP]
exact
i.2
-- Porting note: Was `simp [top, Fin.ext_iff, ne_of_lt hi]`.
[GOAL]
case mp.intro
X : Type u
inst✝¹ : Lattice X
inst✝ : JordanHolderLattice X
s : CompositionSeries X
h : 0 < s.length
i : Fin (s.length - 1 + 1)
hi : ↑i < s.length
⊢ (fun i => series s { val := ↑i, isLt := (_ : ↑i < s.length + 1) }) i ≠ top s ∧
(fun i => series s { val := ↑i, isLt := (_ : ↑i < s.length + 1) }) i ∈ range s.series
[PROOFSTEP]
simp [top, Fin.ext_iff, ne_of_lt hi, -Set.mem_range, Set.mem_range_self]
[GOAL]
case mpr
X : Type u
inst✝¹ : Lattice X
inst✝ : JordanHolderLattice X
s : CompositionSeries X
x : X
h : 0 < s.length
⊢ x ≠ top s ∧ x ∈ range s.series → x ∈ range fun i => series s { val := ↑i, isLt := (_ : ↑i < s.length + 1) }
[PROOFSTEP]
intro h
[GOAL]
case mpr
X : Type u
inst✝¹ : Lattice X
inst✝ : JordanHolderLattice X
s : CompositionSeries X
x : X
h✝ : 0 < s.length
h : x ≠ top s ∧ x ∈ range s.series
⊢ x ∈ range fun i => series s { val := ↑i, isLt := (_ : ↑i < s.length + 1) }
[PROOFSTEP]
exact mem_eraseTop_of_ne_of_mem h.1 h.2
[GOAL]
X : Type u
inst✝¹ : Lattice X
inst✝ : JordanHolderLattice X
s : CompositionSeries X
h : 0 < s.length
⊢ IsMaximal (top (eraseTop s)) (top s)
[PROOFSTEP]
have : s.length - 1 + 1 = s.length := by conv_rhs => rw [← Nat.succ_sub_one s.length]; rw [Nat.succ_sub h]
[GOAL]
X : Type u
inst✝¹ : Lattice X
inst✝ : JordanHolderLattice X
s : CompositionSeries X
h : 0 < s.length
⊢ s.length - 1 + 1 = s.length
[PROOFSTEP]
conv_rhs => rw [← Nat.succ_sub_one s.length]; rw [Nat.succ_sub h]
[GOAL]
X : Type u
inst✝¹ : Lattice X
inst✝ : JordanHolderLattice X
s : CompositionSeries X
h : 0 < s.length
| s.length
[PROOFSTEP]
rw [← Nat.succ_sub_one s.length]; rw [Nat.succ_sub h]
[GOAL]
X : Type u
inst✝¹ : Lattice X
inst✝ : JordanHolderLattice X
s : CompositionSeries X
h : 0 < s.length
| s.length
[PROOFSTEP]
rw [← Nat.succ_sub_one s.length]; rw [Nat.succ_sub h]
[GOAL]
X : Type u
inst✝¹ : Lattice X
inst✝ : JordanHolderLattice X
s : CompositionSeries X
h : 0 < s.length
| s.length
[PROOFSTEP]
rw [← Nat.succ_sub_one s.length]
[GOAL]
X : Type u
inst✝¹ : Lattice X
inst✝ : JordanHolderLattice X
s : CompositionSeries X
h : 0 < s.length
| Nat.succ s.length - 1
[PROOFSTEP]
rw [Nat.succ_sub h]
[GOAL]
X : Type u
inst✝¹ : Lattice X
inst✝ : JordanHolderLattice X
s : CompositionSeries X
h : 0 < s.length
this : s.length - 1 + 1 = s.length
⊢ IsMaximal (top (eraseTop s)) (top s)
[PROOFSTEP]
rw [top_eraseTop, top]
[GOAL]
X : Type u
inst✝¹ : Lattice X
inst✝ : JordanHolderLattice X
s : CompositionSeries X
h : 0 < s.length
this : s.length - 1 + 1 = s.length
⊢ IsMaximal (series s { val := s.length - 1, isLt := (_ : s.length - 1 < s.length + 1) }) (series s (Fin.last s.length))
[PROOFSTEP]
convert s.step ⟨s.length - 1, Nat.sub_lt h zero_lt_one⟩
[GOAL]
case h.e'_5.h.e'_5
X : Type u
inst✝¹ : Lattice X
inst✝ : JordanHolderLattice X
s : CompositionSeries X
h : 0 < s.length
this : s.length - 1 + 1 = s.length
⊢ Fin.last s.length = Fin.succ { val := s.length - 1, isLt := (_ : s.length - 1 < s.length) }
[PROOFSTEP]
ext
[GOAL]
case h.e'_5.h.e'_5.h
X : Type u
inst✝¹ : Lattice X
inst✝ : JordanHolderLattice X
s : CompositionSeries X
h : 0 < s.length
this : s.length - 1 + 1 = s.length
⊢ ↑(Fin.last s.length) = ↑(Fin.succ { val := s.length - 1, isLt := (_ : s.length - 1 < s.length) })
[PROOFSTEP]
simp [this]
[GOAL]
X : Type u
inst✝¹ : Lattice X
inst✝ : JordanHolderLattice X
α : Type u_1
m n : ℕ
a : Fin (Nat.succ m) → α
b : Fin (Nat.succ n) → α
i : Fin m
⊢ Matrix.vecAppend (_ : Nat.succ (m + n) = m + Nat.succ n) (a ∘ Fin.castSucc) b (Fin.castSucc (Fin.castAdd n i)) =
a (Fin.castSucc i)
[PROOFSTEP]
cases i
[GOAL]
case mk
X : Type u
inst✝¹ : Lattice X
inst✝ : JordanHolderLattice X
α : Type u_1
m n : ℕ
a : Fin (Nat.succ m) → α
b : Fin (Nat.succ n) → α
val✝ : ℕ
isLt✝ : val✝ < m
⊢ Matrix.vecAppend (_ : Nat.succ (m + n) = m + Nat.succ n) (a ∘ Fin.castSucc) b
(Fin.castSucc (Fin.castAdd n { val := val✝, isLt := isLt✝ })) =
a (Fin.castSucc { val := val✝, isLt := isLt✝ })
[PROOFSTEP]
simp [Matrix.vecAppend_eq_ite, *]
[GOAL]
X : Type u
inst✝¹ : Lattice X
inst✝ : JordanHolderLattice X
α : Type u_1
m n : ℕ
a : Fin (Nat.succ m) → α
b : Fin (Nat.succ n) → α
i : Fin m
h : a (Fin.last m) = b 0
⊢ Matrix.vecAppend (_ : Nat.succ (m + n) = m + Nat.succ n) (a ∘ Fin.castSucc) b (Fin.succ (Fin.castAdd n i)) =
a (Fin.succ i)
[PROOFSTEP]
cases' i with i hi
[GOAL]
case mk
X : Type u
inst✝¹ : Lattice X
inst✝ : JordanHolderLattice X
α : Type u_1
m n : ℕ
a : Fin (Nat.succ m) → α
b : Fin (Nat.succ n) → α
h : a (Fin.last m) = b 0
i : ℕ
hi : i < m
⊢ Matrix.vecAppend (_ : Nat.succ (m + n) = m + Nat.succ n) (a ∘ Fin.castSucc) b
(Fin.succ (Fin.castAdd n { val := i, isLt := hi })) =
a (Fin.succ { val := i, isLt := hi })
[PROOFSTEP]
simp only [Matrix.vecAppend_eq_ite, hi, Fin.succ_mk, Function.comp_apply, Fin.castSucc_mk, Fin.val_mk, Fin.castAdd_mk]
[GOAL]
case mk
X : Type u
inst✝¹ : Lattice X
inst✝ : JordanHolderLattice X
α : Type u_1
m n : ℕ
a : Fin (Nat.succ m) → α
b : Fin (Nat.succ n) → α
h : a (Fin.last m) = b 0
i : ℕ
hi : i < m
⊢ (if h : i + 1 < m then a { val := i + 1, isLt := (_ : i + 1 < Nat.succ m) }
else
b
{ val := i + 1 - m,
isLt := (_ : ↑{ val := i + 1, isLt := (_ : Nat.succ i < Nat.succ (m + n)) } - m < Nat.succ n) }) =
a { val := i + 1, isLt := (_ : Nat.succ i < Nat.succ m) }
[PROOFSTEP]
split_ifs with h_1
[GOAL]
case pos
X : Type u
inst✝¹ : Lattice X
inst✝ : JordanHolderLattice X
α : Type u_1
m n : ℕ
a : Fin (Nat.succ m) → α
b : Fin (Nat.succ n) → α
h : a (Fin.last m) = b 0
i : ℕ
hi : i < m
h_1 : i + 1 < m
⊢ a { val := i + 1, isLt := (_ : i + 1 < Nat.succ m) } = a { val := i + 1, isLt := (_ : Nat.succ i < Nat.succ m) }
[PROOFSTEP]
rfl
[GOAL]
case neg
X : Type u
inst✝¹ : Lattice X
inst✝ : JordanHolderLattice X
α : Type u_1
m n : ℕ
a : Fin (Nat.succ m) → α
b : Fin (Nat.succ n) → α
h : a (Fin.last m) = b 0
i : ℕ
hi : i < m
h_1 : ¬i + 1 < m
⊢ b
{ val := i + 1 - m,
isLt := (_ : ↑{ val := i + 1, isLt := (_ : Nat.succ i < Nat.succ (m + n)) } - m < Nat.succ n) } =
a { val := i + 1, isLt := (_ : Nat.succ i < Nat.succ m) }
[PROOFSTEP]
have : i + 1 = m := le_antisymm hi (le_of_not_gt h_1)
[GOAL]
case neg
X : Type u
inst✝¹ : Lattice X
inst✝ : JordanHolderLattice X
α : Type u_1
m n : ℕ
a : Fin (Nat.succ m) → α
b : Fin (Nat.succ n) → α
h : a (Fin.last m) = b 0
i : ℕ
hi : i < m
h_1 : ¬i + 1 < m
this : i + 1 = m
⊢ b
{ val := i + 1 - m,
isLt := (_ : ↑{ val := i + 1, isLt := (_ : Nat.succ i < Nat.succ (m + n)) } - m < Nat.succ n) } =
a { val := i + 1, isLt := (_ : Nat.succ i < Nat.succ m) }
[PROOFSTEP]
calc
b ⟨i + 1 - m, by simp [this]⟩ = b 0 := congr_arg b (by simp [Fin.ext_iff, this])
_ = a (Fin.last _) := h.symm
_ = _ := congr_arg a (by simp [Fin.ext_iff, this])
[GOAL]
X : Type u
inst✝¹ : Lattice X
inst✝ : JordanHolderLattice X
α : Type u_1
m n : ℕ
a : Fin (Nat.succ m) → α
b : Fin (Nat.succ n) → α
h : a (Fin.last m) = b 0
i : ℕ
hi : i < m
h_1 : ¬i + 1 < m
this : i + 1 = m
⊢ i + 1 - m < Nat.succ n
[PROOFSTEP]
simp [this]
[GOAL]
X : Type u
inst✝¹ : Lattice X
inst✝ : JordanHolderLattice X
α : Type u_1
m n : ℕ
a : Fin (Nat.succ m) → α
b : Fin (Nat.succ n) → α
h : a (Fin.last m) = b 0
i : ℕ
hi : i < m
h_1 : ¬i + 1 < m
this : i + 1 = m
⊢ { val := i + 1 - m, isLt := (_ : i + 1 - m < Nat.succ n) } = 0
[PROOFSTEP]
simp [Fin.ext_iff, this]
[GOAL]
X : Type u
inst✝¹ : Lattice X
inst✝ : JordanHolderLattice X
α : Type u_1
m n : ℕ
a : Fin (Nat.succ m) → α
b : Fin (Nat.succ n) → α
h : a (Fin.last m) = b 0
i : ℕ
hi : i < m
h_1 : ¬i + 1 < m
this : i + 1 = m
⊢ Fin.last m = { val := i + 1, isLt := (_ : Nat.succ i < Nat.succ m) }
[PROOFSTEP]
simp [Fin.ext_iff, this]
[GOAL]
X : Type u
inst✝¹ : Lattice X
inst✝ : JordanHolderLattice X
α : Type u_1
m n : ℕ
a : Fin (Nat.succ m) → α
b : Fin (Nat.succ n) → α
i : Fin n
⊢ Matrix.vecAppend (_ : Nat.succ (m + n) = m + Nat.succ n) (a ∘ Fin.castSucc) b (Fin.castSucc (Fin.natAdd m i)) =
b (Fin.castSucc i)
[PROOFSTEP]
cases i
[GOAL]
case mk
X : Type u
inst✝¹ : Lattice X
inst✝ : JordanHolderLattice X
α : Type u_1
m n : ℕ
a : Fin (Nat.succ m) → α
b : Fin (Nat.succ n) → α
val✝ : ℕ
isLt✝ : val✝ < n
⊢ Matrix.vecAppend (_ : Nat.succ (m + n) = m + Nat.succ n) (a ∘ Fin.castSucc) b
(Fin.castSucc (Fin.natAdd m { val := val✝, isLt := isLt✝ })) =
b (Fin.castSucc { val := val✝, isLt := isLt✝ })
[PROOFSTEP]
simp only [Matrix.vecAppend_eq_ite, Nat.not_lt_zero, Fin.natAdd_mk, add_lt_iff_neg_left, add_tsub_cancel_left, dif_neg,
Fin.castSucc_mk, not_false_iff, Fin.val_mk]
[GOAL]
X : Type u
inst✝¹ : Lattice X
inst✝ : JordanHolderLattice X
α : Type u_1
m n : ℕ
a : Fin (Nat.succ m) → α
b : Fin (Nat.succ n) → α
i : Fin n
⊢ Matrix.vecAppend (_ : Nat.succ (m + n) = m + Nat.succ n) (a ∘ Fin.castSucc) b (Fin.succ (Fin.natAdd m i)) =
b (Fin.succ i)
[PROOFSTEP]
cases' i with i hi
[GOAL]
case mk
X : Type u
inst✝¹ : Lattice X
inst✝ : JordanHolderLattice X
α : Type u_1
m n : ℕ
a : Fin (Nat.succ m) → α
b : Fin (Nat.succ n) → α
i : ℕ
hi : i < n
⊢ Matrix.vecAppend (_ : Nat.succ (m + n) = m + Nat.succ n) (a ∘ Fin.castSucc) b
(Fin.succ (Fin.natAdd m { val := i, isLt := hi })) =
b (Fin.succ { val := i, isLt := hi })
[PROOFSTEP]
simp only [Matrix.vecAppend_eq_ite, add_assoc, Nat.not_lt_zero, Fin.natAdd_mk, add_lt_iff_neg_left,
add_tsub_cancel_left, Fin.succ_mk, dif_neg, not_false_iff, Fin.val_mk]
[GOAL]
X : Type u
inst✝¹ : Lattice X
inst✝ : JordanHolderLattice X
s₁ s₂ : CompositionSeries X
h : top s₁ = bot s₂
i : Fin (s₁.length + s₂.length)
⊢ IsMaximal
(Matrix.vecAppend (_ : Nat.succ (s₁.length + s₂.length) = s₁.length + Nat.succ s₂.length) (s₁.series ∘ Fin.castSucc)
s₂.series (Fin.castSucc i))
(Matrix.vecAppend (_ : Nat.succ (s₁.length + s₂.length) = s₁.length + Nat.succ s₂.length) (s₁.series ∘ Fin.castSucc)
s₂.series (Fin.succ i))
[PROOFSTEP]
refine' Fin.addCases _ _ i
[GOAL]
case refine'_1
X : Type u
inst✝¹ : Lattice X
inst✝ : JordanHolderLattice X
s₁ s₂ : CompositionSeries X
h : top s₁ = bot s₂
i : Fin (s₁.length + s₂.length)
⊢ ∀ (i : Fin s₁.length),
IsMaximal
(Matrix.vecAppend (_ : Nat.succ (s₁.length + s₂.length) = s₁.length + Nat.succ s₂.length)
(s₁.series ∘ Fin.castSucc) s₂.series (Fin.castSucc (Fin.castAdd s₂.length i)))
(Matrix.vecAppend (_ : Nat.succ (s₁.length + s₂.length) = s₁.length + Nat.succ s₂.length)
(s₁.series ∘ Fin.castSucc) s₂.series (Fin.succ (Fin.castAdd s₂.length i)))
[PROOFSTEP]
intro i
[GOAL]
case refine'_1
X : Type u
inst✝¹ : Lattice X
inst✝ : JordanHolderLattice X
s₁ s₂ : CompositionSeries X
h : top s₁ = bot s₂
i✝ : Fin (s₁.length + s₂.length)
i : Fin s₁.length
⊢ IsMaximal
(Matrix.vecAppend (_ : Nat.succ (s₁.length + s₂.length) = s₁.length + Nat.succ s₂.length) (s₁.series ∘ Fin.castSucc)
s₂.series (Fin.castSucc (Fin.castAdd s₂.length i)))
(Matrix.vecAppend (_ : Nat.succ (s₁.length + s₂.length) = s₁.length + Nat.succ s₂.length) (s₁.series ∘ Fin.castSucc)
s₂.series (Fin.succ (Fin.castAdd s₂.length i)))
[PROOFSTEP]
rw [append_succ_castAdd_aux _ _ _ h, append_castAdd_aux]
[GOAL]
case refine'_1
X : Type u
inst✝¹ : Lattice X
inst✝ : JordanHolderLattice X
s₁ s₂ : CompositionSeries X
h : top s₁ = bot s₂
i✝ : Fin (s₁.length + s₂.length)
i : Fin s₁.length
⊢ IsMaximal (series s₁ (Fin.castSucc i)) (series s₁ (Fin.succ i))
[PROOFSTEP]
exact s₁.step i
[GOAL]
case refine'_2
X : Type u
inst✝¹ : Lattice X
inst✝ : JordanHolderLattice X
s₁ s₂ : CompositionSeries X
h : top s₁ = bot s₂
i : Fin (s₁.length + s₂.length)
⊢ ∀ (i : Fin s₂.length),
IsMaximal
(Matrix.vecAppend (_ : Nat.succ (s₁.length + s₂.length) = s₁.length + Nat.succ s₂.length)
(s₁.series ∘ Fin.castSucc) s₂.series (Fin.castSucc (Fin.natAdd s₁.length i)))
(Matrix.vecAppend (_ : Nat.succ (s₁.length + s₂.length) = s₁.length + Nat.succ s₂.length)
(s₁.series ∘ Fin.castSucc) s₂.series (Fin.succ (Fin.natAdd s₁.length i)))
[PROOFSTEP]
intro i
[GOAL]
case refine'_2
X : Type u
inst✝¹ : Lattice X
inst✝ : JordanHolderLattice X
s₁ s₂ : CompositionSeries X
h : top s₁ = bot s₂
i✝ : Fin (s₁.length + s₂.length)
i : Fin s₂.length
⊢ IsMaximal
(Matrix.vecAppend (_ : Nat.succ (s₁.length + s₂.length) = s₁.length + Nat.succ s₂.length) (s₁.series ∘ Fin.castSucc)
s₂.series (Fin.castSucc (Fin.natAdd s₁.length i)))
(Matrix.vecAppend (_ : Nat.succ (s₁.length + s₂.length) = s₁.length + Nat.succ s₂.length) (s₁.series ∘ Fin.castSucc)
s₂.series (Fin.succ (Fin.natAdd s₁.length i)))
[PROOFSTEP]
rw [append_natAdd_aux, append_succ_natAdd_aux]
[GOAL]
case refine'_2
X : Type u
inst✝¹ : Lattice X
inst✝ : JordanHolderLattice X
s₁ s₂ : CompositionSeries X
h : top s₁ = bot s₂
i✝ : Fin (s₁.length + s₂.length)
i : Fin s₂.length
⊢ IsMaximal (series s₂ (Fin.castSucc i)) (series s₂ (Fin.succ i))
[PROOFSTEP]
exact s₂.step i
[GOAL]
X : Type u
inst✝¹ : Lattice X
inst✝ : JordanHolderLattice X
s₁ s₂ : CompositionSeries X
h : top s₁ = bot s₂
i : Fin s₁.length
⊢ series (append s₁ s₂ h) (Fin.castSucc (Fin.castAdd s₂.length i)) = series s₁ (Fin.castSucc i)
[PROOFSTEP]
rw [coe_append, append_castAdd_aux _ _ i]
[GOAL]
X : Type u
inst✝¹ : Lattice X
inst✝ : JordanHolderLattice X
s₁ s₂ : CompositionSeries X
h : top s₁ = bot s₂
i : Fin s₁.length
⊢ series (append s₁ s₂ h) (Fin.succ (Fin.castAdd s₂.length i)) = series s₁ (Fin.succ i)
[PROOFSTEP]
rw [coe_append, append_succ_castAdd_aux _ _ _ h]
[GOAL]
X : Type u
inst✝¹ : Lattice X
inst✝ : JordanHolderLattice X
s₁ s₂ : CompositionSeries X
h : top s₁ = bot s₂
i : Fin s₂.length
⊢ series (append s₁ s₂ h) (Fin.castSucc (Fin.natAdd s₁.length i)) = series s₂ (Fin.castSucc i)
[PROOFSTEP]
rw [coe_append, append_natAdd_aux _ _ i]
[GOAL]
X : Type u
inst✝¹ : Lattice X
inst✝ : JordanHolderLattice X
s₁ s₂ : CompositionSeries X
h : top s₁ = bot s₂
i : Fin s₂.length
⊢ series (append s₁ s₂ h) (Fin.succ (Fin.natAdd s₁.length i)) = series s₂ (Fin.succ i)
[PROOFSTEP]
rw [coe_append, append_succ_natAdd_aux _ _ i]
[GOAL]
X : Type u
inst✝¹ : Lattice X
inst✝ : JordanHolderLattice X
s : CompositionSeries X
x : X
hsat : IsMaximal (top s) x
i : Fin (s.length + 1)
⊢ IsMaximal (Fin.snoc s.series x (Fin.castSucc i)) (Fin.snoc s.series x (Fin.succ i))
[PROOFSTEP]
refine' Fin.lastCases _ _ i
[GOAL]
case refine'_1
X : Type u
inst✝¹ : Lattice X
inst✝ : JordanHolderLattice X
s : CompositionSeries X
x : X
hsat : IsMaximal (top s) x
i : Fin (s.length + 1)
⊢ IsMaximal (Fin.snoc s.series x (Fin.castSucc (Fin.last s.length)))
(Fin.snoc s.series x (Fin.succ (Fin.last s.length)))
[PROOFSTEP]
rwa [Fin.snoc_castSucc, Fin.succ_last, Fin.snoc_last, ← top]
[GOAL]
case refine'_2
X : Type u
inst✝¹ : Lattice X
inst✝ : JordanHolderLattice X
s : CompositionSeries X
x : X
hsat : IsMaximal (top s) x
i : Fin (s.length + 1)
⊢ ∀ (i : Fin s.length),
IsMaximal (Fin.snoc s.series x (Fin.castSucc (Fin.castSucc i))) (Fin.snoc s.series x (Fin.succ (Fin.castSucc i)))
[PROOFSTEP]
intro i
[GOAL]
case refine'_2
X : Type u
inst✝¹ : Lattice X
inst✝ : JordanHolderLattice X
s : CompositionSeries X
x : X
hsat : IsMaximal (top s) x
i✝ : Fin (s.length + 1)
i : Fin s.length
⊢ IsMaximal (Fin.snoc s.series x (Fin.castSucc (Fin.castSucc i))) (Fin.snoc s.series x (Fin.succ (Fin.castSucc i)))
[PROOFSTEP]
rw [Fin.snoc_castSucc, ← Fin.castSucc_fin_succ, Fin.snoc_castSucc]
[GOAL]
case refine'_2
X : Type u
inst✝¹ : Lattice X
inst✝ : JordanHolderLattice X
s : CompositionSeries X
x : X
hsat : IsMaximal (top s) x
i✝ : Fin (s.length + 1)
i : Fin s.length
⊢ IsMaximal (series s (Fin.castSucc i)) (series s (Fin.succ i))
[PROOFSTEP]
exact s.step _
[GOAL]
X : Type u
inst✝¹ : Lattice X
inst✝ : JordanHolderLattice X
s : CompositionSeries X
x : X
hsat : IsMaximal (top s) x
⊢ bot (snoc s x hsat) = bot s
[PROOFSTEP]
rw [bot, bot, ← snoc_castSucc s x hsat 0, Fin.castSucc_zero' (n := s.length + 1)]
[GOAL]
X : Type u
inst✝¹ : Lattice X
inst✝ : JordanHolderLattice X
s : CompositionSeries X
x y : X
hsat : IsMaximal (top s) x
⊢ y ∈ snoc s x hsat ↔ y ∈ s ∨ y = x
[PROOFSTEP]
simp only [snoc, mem_def]
[GOAL]
X : Type u
inst✝¹ : Lattice X
inst✝ : JordanHolderLattice X
s : CompositionSeries X
x y : X
hsat : IsMaximal (top s) x
⊢ y ∈ range (Fin.snoc s.series x) ↔ y ∈ range s.series ∨ y = x
[PROOFSTEP]
constructor
[GOAL]
case mp
X : Type u
inst✝¹ : Lattice X
inst✝ : JordanHolderLattice X
s : CompositionSeries X
x y : X
hsat : IsMaximal (top s) x
⊢ y ∈ range (Fin.snoc s.series x) → y ∈ range s.series ∨ y = x
[PROOFSTEP]
rintro ⟨i, rfl⟩
[GOAL]
case mp.intro
X : Type u
inst✝¹ : Lattice X
inst✝ : JordanHolderLattice X
s : CompositionSeries X
x : X
hsat : IsMaximal (top s) x
i : Fin (s.length + 1 + 1)
⊢ Fin.snoc s.series x i ∈ range s.series ∨ Fin.snoc s.series x i = x
[PROOFSTEP]
refine' Fin.lastCases _ (fun i => _) i
[GOAL]
case mp.intro.refine'_1
X : Type u
inst✝¹ : Lattice X
inst✝ : JordanHolderLattice X
s : CompositionSeries X
x : X
hsat : IsMaximal (top s) x
i : Fin (s.length + 1 + 1)
⊢ Fin.snoc s.series x (Fin.last (s.length + 1)) ∈ range s.series ∨ Fin.snoc s.series x (Fin.last (s.length + 1)) = x
[PROOFSTEP]
right
[GOAL]
case mp.intro.refine'_1.h
X : Type u
inst✝¹ : Lattice X
inst✝ : JordanHolderLattice X
s : CompositionSeries X
x : X
hsat : IsMaximal (top s) x
i : Fin (s.length + 1 + 1)
⊢ Fin.snoc s.series x (Fin.last (s.length + 1)) = x
[PROOFSTEP]
simp
[GOAL]
case mp.intro.refine'_2
X : Type u
inst✝¹ : Lattice X
inst✝ : JordanHolderLattice X
s : CompositionSeries X
x : X
hsat : IsMaximal (top s) x
i✝ : Fin (s.length + 1 + 1)
i : Fin (s.length + 1)
⊢ Fin.snoc s.series x (Fin.castSucc i) ∈ range s.series ∨ Fin.snoc s.series x (Fin.castSucc i) = x
[PROOFSTEP]
left
[GOAL]
case mp.intro.refine'_2.h
X : Type u
inst✝¹ : Lattice X
inst✝ : JordanHolderLattice X
s : CompositionSeries X
x : X
hsat : IsMaximal (top s) x
i✝ : Fin (s.length + 1 + 1)
i : Fin (s.length + 1)
⊢ Fin.snoc s.series x (Fin.castSucc i) ∈ range s.series
[PROOFSTEP]
simp
[GOAL]
case mpr
X : Type u
inst✝¹ : Lattice X
inst✝ : JordanHolderLattice X
s : CompositionSeries X
x y : X
hsat : IsMaximal (top s) x
⊢ y ∈ range s.series ∨ y = x → y ∈ range (Fin.snoc s.series x)
[PROOFSTEP]
intro h
[GOAL]
case mpr
X : Type u
inst✝¹ : Lattice X
inst✝ : JordanHolderLattice X
s : CompositionSeries X
x y : X
hsat : IsMaximal (top s) x
h : y ∈ range s.series ∨ y = x
⊢ y ∈ range (Fin.snoc s.series x)
[PROOFSTEP]
rcases h with (⟨i, rfl⟩ | rfl)
[GOAL]
case mpr.inl.intro
X : Type u
inst✝¹ : Lattice X
inst✝ : JordanHolderLattice X
s : CompositionSeries X
x : X
hsat : IsMaximal (top s) x
i : Fin (s.length + 1)
⊢ series s i ∈ range (Fin.snoc s.series x)
[PROOFSTEP]
use Fin.castSucc i
[GOAL]
case h
X : Type u
inst✝¹ : Lattice X
inst✝ : JordanHolderLattice X
s : CompositionSeries X
x : X
hsat : IsMaximal (top s) x
i : Fin (s.length + 1)
⊢ Fin.snoc s.series x (Fin.castSucc i) = series s i
[PROOFSTEP]
simp
[GOAL]
case mpr.inr
X : Type u
inst✝¹ : Lattice X
inst✝ : JordanHolderLattice X
s : CompositionSeries X
y : X
hsat : IsMaximal (top s) y
⊢ y ∈ range (Fin.snoc s.series y)
[PROOFSTEP]
use Fin.last _
[GOAL]
case h
X : Type u
inst✝¹ : Lattice X
inst✝ : JordanHolderLattice X
s : CompositionSeries X
y : X
hsat : IsMaximal (top s) y
⊢ Fin.snoc s.series y (Fin.last (s.length + 1)) = y
[PROOFSTEP]
simp
[GOAL]
X : Type u
inst✝¹ : Lattice X
inst✝ : JordanHolderLattice X
s : CompositionSeries X
h : 0 < s.length
⊢ s = snoc (eraseTop s) (top s) (_ : IsMaximal (top (eraseTop s)) (top s))
[PROOFSTEP]
ext x
[GOAL]
case h
X : Type u
inst✝¹ : Lattice X
inst✝ : JordanHolderLattice X
s : CompositionSeries X
h : 0 < s.length
x : X
⊢ x ∈ s ↔ x ∈ snoc (eraseTop s) (top s) (_ : IsMaximal (top (eraseTop s)) (top s))
[PROOFSTEP]
simp [mem_snoc, mem_eraseTop h]
[GOAL]
case h
X : Type u
inst✝¹ : Lattice X
inst✝ : JordanHolderLattice X
s : CompositionSeries X
h : 0 < s.length
x : X
⊢ x ∈ s ↔ ¬x = top s ∧ x ∈ s ∨ x = top s
[PROOFSTEP]
by_cases h : x = s.top
[GOAL]
case pos
X : Type u
inst✝¹ : Lattice X
inst✝ : JordanHolderLattice X
s : CompositionSeries X
h✝ : 0 < s.length
x : X
h : x = top s
⊢ x ∈ s ↔ ¬x = top s ∧ x ∈ s ∨ x = top s
[PROOFSTEP]
simp [*, s.top_mem]
[GOAL]
case neg
X : Type u
inst✝¹ : Lattice X
inst✝ : JordanHolderLattice X
s : CompositionSeries X
h✝ : 0 < s.length
x : X
h : ¬x = top s
⊢ x ∈ s ↔ ¬x = top s ∧ x ∈ s ∨ x = top s
[PROOFSTEP]
simp [*, s.top_mem]
[GOAL]
X : Type u
inst✝¹ : Lattice X
inst✝ : JordanHolderLattice X
s : CompositionSeries X
h : IsMaximal (top (eraseTop s)) (top s)
⊢ s.length ≠ 0
[PROOFSTEP]
intro hs
[GOAL]
X : Type u
inst✝¹ : Lattice X
inst✝ : JordanHolderLattice X
s : CompositionSeries X
h : IsMaximal (top (eraseTop s)) (top s)
hs : s.length = 0
⊢ False
[PROOFSTEP]
refine' ne_of_gt (lt_of_isMaximal h) _
[GOAL]
X : Type u
inst✝¹ : Lattice X
inst✝ : JordanHolderLattice X
s : CompositionSeries X
h : IsMaximal (top (eraseTop s)) (top s)
hs : s.length = 0
⊢ top s = top (eraseTop s)
[PROOFSTEP]
simp [top, Fin.ext_iff, hs]
[GOAL]
X : Type u
inst✝¹ : Lattice X
inst✝ : JordanHolderLattice X
s₁ s₂ : CompositionSeries X
h : Equivalent s₁ s₂
i : Fin s₂.length
⊢ Iso (series s₁ (Fin.castSucc (↑(Exists.choose h).symm i)), series s₁ (Fin.succ (↑(Exists.choose h).symm i)))
(series s₂ (Fin.castSucc i), series s₂ (Fin.succ i))
[PROOFSTEP]
simpa using h.choose_spec (h.choose.symm i)
[GOAL]
X : Type u
inst✝¹ : Lattice X
inst✝ : JordanHolderLattice X
s₁ s₂ t₁ t₂ : CompositionSeries X
hs : top s₁ = bot s₂
ht : top t₁ = bot t₂
h₁ : Equivalent s₁ t₁
h₂ : Equivalent s₂ t₂
e : Fin (s₁.length + s₂.length) ≃ Fin (t₁.length + t₂.length) :=
Trans.trans (Trans.trans finSumFinEquiv.symm (Equiv.sumCongr (Exists.choose h₁) (Exists.choose h₂))) finSumFinEquiv
⊢ ∀ (i : Fin (CompositionSeries.append s₁ s₂ hs).length),
Iso
(series (CompositionSeries.append s₁ s₂ hs) (Fin.castSucc i),
series (CompositionSeries.append s₁ s₂ hs) (Fin.succ i))
(series (CompositionSeries.append t₁ t₂ ht) (Fin.castSucc (↑e i)),
series (CompositionSeries.append t₁ t₂ ht) (Fin.succ (↑e i)))
[PROOFSTEP]
intro i
[GOAL]
X : Type u
inst✝¹ : Lattice X
inst✝ : JordanHolderLattice X
s₁ s₂ t₁ t₂ : CompositionSeries X
hs : top s₁ = bot s₂
ht : top t₁ = bot t₂
h₁ : Equivalent s₁ t₁
h₂ : Equivalent s₂ t₂
e : Fin (s₁.length + s₂.length) ≃ Fin (t₁.length + t₂.length) :=
Trans.trans (Trans.trans finSumFinEquiv.symm (Equiv.sumCongr (Exists.choose h₁) (Exists.choose h₂))) finSumFinEquiv
i : Fin (CompositionSeries.append s₁ s₂ hs).length
⊢ Iso
(series (CompositionSeries.append s₁ s₂ hs) (Fin.castSucc i),
series (CompositionSeries.append s₁ s₂ hs) (Fin.succ i))
(series (CompositionSeries.append t₁ t₂ ht) (Fin.castSucc (↑e i)),
series (CompositionSeries.append t₁ t₂ ht) (Fin.succ (↑e i)))
[PROOFSTEP]
refine' Fin.addCases _ _ i
[GOAL]
case refine'_1
X : Type u
inst✝¹ : Lattice X
inst✝ : JordanHolderLattice X
s₁ s₂ t₁ t₂ : CompositionSeries X
hs : top s₁ = bot s₂
ht : top t₁ = bot t₂
h₁ : Equivalent s₁ t₁
h₂ : Equivalent s₂ t₂
e : Fin (s₁.length + s₂.length) ≃ Fin (t₁.length + t₂.length) :=
Trans.trans (Trans.trans finSumFinEquiv.symm (Equiv.sumCongr (Exists.choose h₁) (Exists.choose h₂))) finSumFinEquiv
i : Fin (CompositionSeries.append s₁ s₂ hs).length
⊢ ∀ (i : Fin s₁.length),
Iso
(series (CompositionSeries.append s₁ s₂ hs) (Fin.castSucc (Fin.castAdd s₂.length i)),
series (CompositionSeries.append s₁ s₂ hs) (Fin.succ (Fin.castAdd s₂.length i)))
(series (CompositionSeries.append t₁ t₂ ht) (Fin.castSucc (↑e (Fin.castAdd s₂.length i))),
series (CompositionSeries.append t₁ t₂ ht) (Fin.succ (↑e (Fin.castAdd s₂.length i))))
[PROOFSTEP]
intro i
[GOAL]
case refine'_1
X : Type u
inst✝¹ : Lattice X
inst✝ : JordanHolderLattice X
s₁ s₂ t₁ t₂ : CompositionSeries X
hs : top s₁ = bot s₂
ht : top t₁ = bot t₂
h₁ : Equivalent s₁ t₁
h₂ : Equivalent s₂ t₂
e : Fin (s₁.length + s₂.length) ≃ Fin (t₁.length + t₂.length) :=
Trans.trans (Trans.trans finSumFinEquiv.symm (Equiv.sumCongr (Exists.choose h₁) (Exists.choose h₂))) finSumFinEquiv
i✝ : Fin (CompositionSeries.append s₁ s₂ hs).length
i : Fin s₁.length
⊢ Iso
(series (CompositionSeries.append s₁ s₂ hs) (Fin.castSucc (Fin.castAdd s₂.length i)),
series (CompositionSeries.append s₁ s₂ hs) (Fin.succ (Fin.castAdd s₂.length i)))
(series (CompositionSeries.append t₁ t₂ ht) (Fin.castSucc (↑e (Fin.castAdd s₂.length i))),
series (CompositionSeries.append t₁ t₂ ht) (Fin.succ (↑e (Fin.castAdd s₂.length i))))
[PROOFSTEP]
simpa [top, bot] using h₁.choose_spec i
[GOAL]
case refine'_2
X : Type u
inst✝¹ : Lattice X
inst✝ : JordanHolderLattice X
s₁ s₂ t₁ t₂ : CompositionSeries X
hs : top s₁ = bot s₂
ht : top t₁ = bot t₂
h₁ : Equivalent s₁ t₁
h₂ : Equivalent s₂ t₂
e : Fin (s₁.length + s₂.length) ≃ Fin (t₁.length + t₂.length) :=
Trans.trans (Trans.trans finSumFinEquiv.symm (Equiv.sumCongr (Exists.choose h₁) (Exists.choose h₂))) finSumFinEquiv
i : Fin (CompositionSeries.append s₁ s₂ hs).length
⊢ ∀ (i : Fin s₂.length),
Iso
(series (CompositionSeries.append s₁ s₂ hs) (Fin.castSucc (Fin.natAdd s₁.length i)),
series (CompositionSeries.append s₁ s₂ hs) (Fin.succ (Fin.natAdd s₁.length i)))
(series (CompositionSeries.append t₁ t₂ ht) (Fin.castSucc (↑e (Fin.natAdd s₁.length i))),
series (CompositionSeries.append t₁ t₂ ht) (Fin.succ (↑e (Fin.natAdd s₁.length i))))
[PROOFSTEP]
intro i
[GOAL]
case refine'_2
X : Type u
inst✝¹ : Lattice X
inst✝ : JordanHolderLattice X
s₁ s₂ t₁ t₂ : CompositionSeries X
hs : top s₁ = bot s₂
ht : top t₁ = bot t₂
h₁ : Equivalent s₁ t₁
h₂ : Equivalent s₂ t₂
e : Fin (s₁.length + s₂.length) ≃ Fin (t₁.length + t₂.length) :=
Trans.trans (Trans.trans finSumFinEquiv.symm (Equiv.sumCongr (Exists.choose h₁) (Exists.choose h₂))) finSumFinEquiv
i✝ : Fin (CompositionSeries.append s₁ s₂ hs).length
i : Fin s₂.length
⊢ Iso
(series (CompositionSeries.append s₁ s₂ hs) (Fin.castSucc (Fin.natAdd s₁.length i)),
series (CompositionSeries.append s₁ s₂ hs) (Fin.succ (Fin.natAdd s₁.length i)))
(series (CompositionSeries.append t₁ t₂ ht) (Fin.castSucc (↑e (Fin.natAdd s₁.length i))),
series (CompositionSeries.append t₁ t₂ ht) (Fin.succ (↑e (Fin.natAdd s₁.length i))))
[PROOFSTEP]
simpa [top, bot] using h₂.choose_spec i
[GOAL]
X : Type u
inst✝¹ : Lattice X
inst✝ : JordanHolderLattice X
s₁ s₂ : CompositionSeries X
x₁ x₂ : X
hsat₁ : IsMaximal (top s₁) x₁
hsat₂ : IsMaximal (top s₂) x₂
hequiv : Equivalent s₁ s₂
htop : Iso (top s₁, x₁) (top s₂, x₂)
e : Fin (Nat.succ s₁.length) ≃ Fin (Nat.succ s₂.length) :=
Trans.trans (Trans.trans finSuccEquivLast (Functor.mapEquiv Option (Exists.choose hequiv))) finSuccEquivLast.symm
i : Fin (snoc s₁ x₁ hsat₁).length
⊢ Iso (series (snoc s₁ x₁ hsat₁) (Fin.castSucc i), series (snoc s₁ x₁ hsat₁) (Fin.succ i))
(series (snoc s₂ x₂ hsat₂) (Fin.castSucc (↑e i)), series (snoc s₂ x₂ hsat₂) (Fin.succ (↑e i)))
[PROOFSTEP]
refine' Fin.lastCases _ _ i
[GOAL]
case refine'_1
X : Type u
inst✝¹ : Lattice X
inst✝ : JordanHolderLattice X
s₁ s₂ : CompositionSeries X
x₁ x₂ : X
hsat₁ : IsMaximal (top s₁) x₁
hsat₂ : IsMaximal (top s₂) x₂
hequiv : Equivalent s₁ s₂
htop : Iso (top s₁, x₁) (top s₂, x₂)
e : Fin (Nat.succ s₁.length) ≃ Fin (Nat.succ s₂.length) :=
Trans.trans (Trans.trans finSuccEquivLast (Functor.mapEquiv Option (Exists.choose hequiv))) finSuccEquivLast.symm
i : Fin (snoc s₁ x₁ hsat₁).length
⊢ Iso
(series (snoc s₁ x₁ hsat₁) (Fin.castSucc (Fin.last s₁.length)),
series (snoc s₁ x₁ hsat₁) (Fin.succ (Fin.last s₁.length)))
(series (snoc s₂ x₂ hsat₂) (Fin.castSucc (↑e (Fin.last s₁.length))),
series (snoc s₂ x₂ hsat₂) (Fin.succ (↑e (Fin.last s₁.length))))
[PROOFSTEP]
simpa [top] using htop
[GOAL]
case refine'_2
X : Type u
inst✝¹ : Lattice X
inst✝ : JordanHolderLattice X
s₁ s₂ : CompositionSeries X
x₁ x₂ : X
hsat₁ : IsMaximal (top s₁) x₁
hsat₂ : IsMaximal (top s₂) x₂
hequiv : Equivalent s₁ s₂
htop : Iso (top s₁, x₁) (top s₂, x₂)
e : Fin (Nat.succ s₁.length) ≃ Fin (Nat.succ s₂.length) :=
Trans.trans (Trans.trans finSuccEquivLast (Functor.mapEquiv Option (Exists.choose hequiv))) finSuccEquivLast.symm
i : Fin (snoc s₁ x₁ hsat₁).length
⊢ ∀ (i : Fin s₁.length),
Iso
(series (snoc s₁ x₁ hsat₁) (Fin.castSucc (Fin.castSucc i)), series (snoc s₁ x₁ hsat₁) (Fin.succ (Fin.castSucc i)))
(series (snoc s₂ x₂ hsat₂) (Fin.castSucc (↑e (Fin.castSucc i))),
series (snoc s₂ x₂ hsat₂) (Fin.succ (↑e (Fin.castSucc i))))
[PROOFSTEP]
intro i
[GOAL]
case refine'_2
X : Type u
inst✝¹ : Lattice X
inst✝ : JordanHolderLattice X
s₁ s₂ : CompositionSeries X
x₁ x₂ : X
hsat₁ : IsMaximal (top s₁) x₁
hsat₂ : IsMaximal (top s₂) x₂
hequiv : Equivalent s₁ s₂
htop : Iso (top s₁, x₁) (top s₂, x₂)
e : Fin (Nat.succ s₁.length) ≃ Fin (Nat.succ s₂.length) :=
Trans.trans (Trans.trans finSuccEquivLast (Functor.mapEquiv Option (Exists.choose hequiv))) finSuccEquivLast.symm
i✝ : Fin (snoc s₁ x₁ hsat₁).length
i : Fin s₁.length
⊢ Iso (series (snoc s₁ x₁ hsat₁) (Fin.castSucc (Fin.castSucc i)), series (snoc s₁ x₁ hsat₁) (Fin.succ (Fin.castSucc i)))
(series (snoc s₂ x₂ hsat₂) (Fin.castSucc (↑e (Fin.castSucc i))),
series (snoc s₂ x₂ hsat₂) (Fin.succ (↑e (Fin.castSucc i))))
[PROOFSTEP]
simpa [Fin.succ_castSucc] using hequiv.choose_spec i
[GOAL]
X : Type u
inst✝¹ : Lattice X
inst✝ : JordanHolderLattice X
s₁ s₂ : CompositionSeries X
h : Equivalent s₁ s₂
⊢ s₁.length = s₂.length
[PROOFSTEP]
simpa using Fintype.card_congr h.choose
[GOAL]
X : Type u
inst✝¹ : Lattice X
inst✝ : JordanHolderLattice X
s : CompositionSeries X
x₁ x₂ y₁ y₂ : X
hsat₁ : IsMaximal (top s) x₁
hsat₂ : IsMaximal (top s) x₂
hsaty₁ : IsMaximal (top (snoc s x₁ hsat₁)) y₁
hsaty₂ : IsMaximal (top (snoc s x₂ hsat₂)) y₂
hr₁ : Iso (top s, x₁) (x₂, y₂)
hr₂ : Iso (x₁, y₁) (top s, x₂)
e : Fin (s.length + 1 + 1) ≃ Fin (s.length + 1 + 1) :=
Equiv.swap (Fin.last (s.length + 1)) (Fin.castSucc (Fin.last s.length))
x✝ : Fin s.length
⊢ Fin.castSucc (Fin.castSucc x✝) < Fin.castSucc (Fin.last s.length)
[PROOFSTEP]
simp [Fin.castSucc_lt_last]
[GOAL]
X : Type u
inst✝¹ : Lattice X
inst✝ : JordanHolderLattice X
s : CompositionSeries X
x₁ x₂ y₁ y₂ : X
hsat₁ : IsMaximal (top s) x₁
hsat₂ : IsMaximal (top s) x₂
hsaty₁ : IsMaximal (top (snoc s x₁ hsat₁)) y₁
hsaty₂ : IsMaximal (top (snoc s x₂ hsat₂)) y₂
hr₁ : Iso (top s, x₁) (x₂, y₂)
hr₂ : Iso (x₁, y₁) (top s, x₂)
e : Fin (s.length + 1 + 1) ≃ Fin (s.length + 1 + 1) :=
Equiv.swap (Fin.last (s.length + 1)) (Fin.castSucc (Fin.last s.length))
h1 : ∀ {i : Fin s.length}, Fin.castSucc (Fin.castSucc i) ≠ Fin.castSucc (Fin.last s.length)
x✝ : Fin s.length
⊢ Fin.castSucc (Fin.castSucc x✝) < Fin.last (s.length + 1)
[PROOFSTEP]
simp [Fin.castSucc_lt_last]
[GOAL]
X : Type u
inst✝¹ : Lattice X
inst✝ : JordanHolderLattice X
s : CompositionSeries X
x₁ x₂ y₁ y₂ : X
hsat₁ : IsMaximal (top s) x₁
hsat₂ : IsMaximal (top s) x₂
hsaty₁ : IsMaximal (top (snoc s x₁ hsat₁)) y₁
hsaty₂ : IsMaximal (top (snoc s x₂ hsat₂)) y₂
hr₁ : Iso (top s, x₁) (x₂, y₂)
hr₂ : Iso (x₁, y₁) (top s, x₂)
e : Fin (s.length + 1 + 1) ≃ Fin (s.length + 1 + 1) :=
Equiv.swap (Fin.last (s.length + 1)) (Fin.castSucc (Fin.last s.length))
h1 : ∀ {i : Fin s.length}, Fin.castSucc (Fin.castSucc i) ≠ Fin.castSucc (Fin.last s.length)
h2 : ∀ {i : Fin s.length}, Fin.castSucc (Fin.castSucc i) ≠ Fin.last (s.length + 1)
⊢ ∀ (i : Fin (snoc (snoc s x₁ hsat₁) y₁ hsaty₁).length),
Iso
(series (snoc (snoc s x₁ hsat₁) y₁ hsaty₁) (Fin.castSucc i),
series (snoc (snoc s x₁ hsat₁) y₁ hsaty₁) (Fin.succ i))
(series (snoc (snoc s x₂ hsat₂) y₂ hsaty₂) (Fin.castSucc (↑e i)),
series (snoc (snoc s x₂ hsat₂) y₂ hsaty₂) (Fin.succ (↑e i)))
[PROOFSTEP]
intro i
[GOAL]
X : Type u
inst✝¹ : Lattice X
inst✝ : JordanHolderLattice X
s : CompositionSeries X
x₁ x₂ y₁ y₂ : X
hsat₁ : IsMaximal (top s) x₁
hsat₂ : IsMaximal (top s) x₂
hsaty₁ : IsMaximal (top (snoc s x₁ hsat₁)) y₁
hsaty₂ : IsMaximal (top (snoc s x₂ hsat₂)) y₂
hr₁ : Iso (top s, x₁) (x₂, y₂)
hr₂ : Iso (x₁, y₁) (top s, x₂)
e : Fin (s.length + 1 + 1) ≃ Fin (s.length + 1 + 1) :=
Equiv.swap (Fin.last (s.length + 1)) (Fin.castSucc (Fin.last s.length))
h1 : ∀ {i : Fin s.length}, Fin.castSucc (Fin.castSucc i) ≠ Fin.castSucc (Fin.last s.length)
h2 : ∀ {i : Fin s.length}, Fin.castSucc (Fin.castSucc i) ≠ Fin.last (s.length + 1)
i : Fin (snoc (snoc s x₁ hsat₁) y₁ hsaty₁).length
⊢ Iso
(series (snoc (snoc s x₁ hsat₁) y₁ hsaty₁) (Fin.castSucc i), series (snoc (snoc s x₁ hsat₁) y₁ hsaty₁) (Fin.succ i))
(series (snoc (snoc s x₂ hsat₂) y₂ hsaty₂) (Fin.castSucc (↑e i)),
series (snoc (snoc s x₂ hsat₂) y₂ hsaty₂) (Fin.succ (↑e i)))
[PROOFSTEP]
dsimp only []
[GOAL]
X : Type u
inst✝¹ : Lattice X
inst✝ : JordanHolderLattice X
s : CompositionSeries X
x₁ x₂ y₁ y₂ : X
hsat₁ : IsMaximal (top s) x₁
hsat₂ : IsMaximal (top s) x₂
hsaty₁ : IsMaximal (top (snoc s x₁ hsat₁)) y₁
hsaty₂ : IsMaximal (top (snoc s x₂ hsat₂)) y₂
hr₁ : Iso (top s, x₁) (x₂, y₂)
hr₂ : Iso (x₁, y₁) (top s, x₂)
e : Fin (s.length + 1 + 1) ≃ Fin (s.length + 1 + 1) :=
Equiv.swap (Fin.last (s.length + 1)) (Fin.castSucc (Fin.last s.length))
h1 : ∀ {i : Fin s.length}, Fin.castSucc (Fin.castSucc i) ≠ Fin.castSucc (Fin.last s.length)
h2 : ∀ {i : Fin s.length}, Fin.castSucc (Fin.castSucc i) ≠ Fin.last (s.length + 1)
i : Fin (snoc (snoc s x₁ hsat₁) y₁ hsaty₁).length
⊢ Iso
(series (snoc (snoc s x₁ hsat₁) y₁ hsaty₁) (Fin.castSucc i), series (snoc (snoc s x₁ hsat₁) y₁ hsaty₁) (Fin.succ i))
(series (snoc (snoc s x₂ hsat₂) y₂ hsaty₂)
(Fin.castSucc (↑(Equiv.swap (Fin.last (s.length + 1)) (Fin.castSucc (Fin.last s.length))) i)),
series (snoc (snoc s x₂ hsat₂) y₂ hsaty₂)
(Fin.succ (↑(Equiv.swap (Fin.last (s.length + 1)) (Fin.castSucc (Fin.last s.length))) i)))
[PROOFSTEP]
refine' Fin.lastCases _ (fun i => _) i
[GOAL]
case refine'_1
X : Type u
inst✝¹ : Lattice X
inst✝ : JordanHolderLattice X
s : CompositionSeries X
x₁ x₂ y₁ y₂ : X
hsat₁ : IsMaximal (top s) x₁
hsat₂ : IsMaximal (top s) x₂
hsaty₁ : IsMaximal (top (snoc s x₁ hsat₁)) y₁
hsaty₂ : IsMaximal (top (snoc s x₂ hsat₂)) y₂
hr₁ : Iso (top s, x₁) (x₂, y₂)
hr₂ : Iso (x₁, y₁) (top s, x₂)
e : Fin (s.length + 1 + 1) ≃ Fin (s.length + 1 + 1) :=
Equiv.swap (Fin.last (s.length + 1)) (Fin.castSucc (Fin.last s.length))
h1 : ∀ {i : Fin s.length}, Fin.castSucc (Fin.castSucc i) ≠ Fin.castSucc (Fin.last s.length)
h2 : ∀ {i : Fin s.length}, Fin.castSucc (Fin.castSucc i) ≠ Fin.last (s.length + 1)
i : Fin (snoc (snoc s x₁ hsat₁) y₁ hsaty₁).length
⊢ Iso
(series (snoc (snoc s x₁ hsat₁) y₁ hsaty₁) (Fin.castSucc (Fin.last (snoc s x₁ hsat₁).length)),
series (snoc (snoc s x₁ hsat₁) y₁ hsaty₁) (Fin.succ (Fin.last (snoc s x₁ hsat₁).length)))
(series (snoc (snoc s x₂ hsat₂) y₂ hsaty₂)
(Fin.castSucc
(↑(Equiv.swap (Fin.last (s.length + 1)) (Fin.castSucc (Fin.last s.length)))
(Fin.last (snoc s x₁ hsat₁).length))),
series (snoc (snoc s x₂ hsat₂) y₂ hsaty₂)
(Fin.succ
(↑(Equiv.swap (Fin.last (s.length + 1)) (Fin.castSucc (Fin.last s.length)))
(Fin.last (snoc s x₁ hsat₁).length))))
[PROOFSTEP]
erw [Equiv.swap_apply_left, snoc_castSucc, snoc_last, Fin.succ_last, snoc_last, snoc_castSucc, snoc_castSucc,
Fin.succ_castSucc, snoc_castSucc, Fin.succ_last, snoc_last]
[GOAL]
case refine'_1
X : Type u
inst✝¹ : Lattice X
inst✝ : JordanHolderLattice X
s : CompositionSeries X
x₁ x₂ y₁ y₂ : X
hsat₁ : IsMaximal (top s) x₁
hsat₂ : IsMaximal (top s) x₂
hsaty₁ : IsMaximal (top (snoc s x₁ hsat₁)) y₁
hsaty₂ : IsMaximal (top (snoc s x₂ hsat₂)) y₂
hr₁ : Iso (top s, x₁) (x₂, y₂)
hr₂ : Iso (x₁, y₁) (top s, x₂)
e : Fin (s.length + 1 + 1) ≃ Fin (s.length + 1 + 1) :=
Equiv.swap (Fin.last (s.length + 1)) (Fin.castSucc (Fin.last s.length))
h1 : ∀ {i : Fin s.length}, Fin.castSucc (Fin.castSucc i) ≠ Fin.castSucc (Fin.last s.length)
h2 : ∀ {i : Fin s.length}, Fin.castSucc (Fin.castSucc i) ≠ Fin.last (s.length + 1)
i : Fin (snoc (snoc s x₁ hsat₁) y₁ hsaty₁).length
⊢ Iso (x₁, y₁) (series s (Fin.last s.length), x₂)
[PROOFSTEP]
exact hr₂
[GOAL]
case refine'_2
X : Type u
inst✝¹ : Lattice X
inst✝ : JordanHolderLattice X
s : CompositionSeries X
x₁ x₂ y₁ y₂ : X
hsat₁ : IsMaximal (top s) x₁
hsat₂ : IsMaximal (top s) x₂
hsaty₁ : IsMaximal (top (snoc s x₁ hsat₁)) y₁
hsaty₂ : IsMaximal (top (snoc s x₂ hsat₂)) y₂
hr₁ : Iso (top s, x₁) (x₂, y₂)
hr₂ : Iso (x₁, y₁) (top s, x₂)
e : Fin (s.length + 1 + 1) ≃ Fin (s.length + 1 + 1) :=
Equiv.swap (Fin.last (s.length + 1)) (Fin.castSucc (Fin.last s.length))
h1 : ∀ {i : Fin s.length}, Fin.castSucc (Fin.castSucc i) ≠ Fin.castSucc (Fin.last s.length)
h2 : ∀ {i : Fin s.length}, Fin.castSucc (Fin.castSucc i) ≠ Fin.last (s.length + 1)
i✝ : Fin (snoc (snoc s x₁ hsat₁) y₁ hsaty₁).length
i : Fin (snoc s x₁ hsat₁).length
⊢ Iso
(series (snoc (snoc s x₁ hsat₁) y₁ hsaty₁) (Fin.castSucc (Fin.castSucc i)),
series (snoc (snoc s x₁ hsat₁) y₁ hsaty₁) (Fin.succ (Fin.castSucc i)))
(series (snoc (snoc s x₂ hsat₂) y₂ hsaty₂)
(Fin.castSucc (↑(Equiv.swap (Fin.last (s.length + 1)) (Fin.castSucc (Fin.last s.length))) (Fin.castSucc i))),
series (snoc (snoc s x₂ hsat₂) y₂ hsaty₂)
(Fin.succ (↑(Equiv.swap (Fin.last (s.length + 1)) (Fin.castSucc (Fin.last s.length))) (Fin.castSucc i))))
[PROOFSTEP]
refine' Fin.lastCases _ (fun i => _) i
[GOAL]
case refine'_2.refine'_1
X : Type u
inst✝¹ : Lattice X
inst✝ : JordanHolderLattice X
s : CompositionSeries X
x₁ x₂ y₁ y₂ : X
hsat₁ : IsMaximal (top s) x₁
hsat₂ : IsMaximal (top s) x₂
hsaty₁ : IsMaximal (top (snoc s x₁ hsat₁)) y₁
hsaty₂ : IsMaximal (top (snoc s x₂ hsat₂)) y₂
hr₁ : Iso (top s, x₁) (x₂, y₂)
hr₂ : Iso (x₁, y₁) (top s, x₂)
e : Fin (s.length + 1 + 1) ≃ Fin (s.length + 1 + 1) :=
Equiv.swap (Fin.last (s.length + 1)) (Fin.castSucc (Fin.last s.length))
h1 : ∀ {i : Fin s.length}, Fin.castSucc (Fin.castSucc i) ≠ Fin.castSucc (Fin.last s.length)
h2 : ∀ {i : Fin s.length}, Fin.castSucc (Fin.castSucc i) ≠ Fin.last (s.length + 1)
i✝ : Fin (snoc (snoc s x₁ hsat₁) y₁ hsaty₁).length
i : Fin (snoc s x₁ hsat₁).length
⊢ Iso
(series (snoc (snoc s x₁ hsat₁) y₁ hsaty₁) (Fin.castSucc (Fin.castSucc (Fin.last s.length))),
series (snoc (snoc s x₁ hsat₁) y₁ hsaty₁) (Fin.succ (Fin.castSucc (Fin.last s.length))))
(series (snoc (snoc s x₂ hsat₂) y₂ hsaty₂)
(Fin.castSucc
(↑(Equiv.swap (Fin.last (s.length + 1)) (Fin.castSucc (Fin.last s.length)))
(Fin.castSucc (Fin.last s.length)))),
series (snoc (snoc s x₂ hsat₂) y₂ hsaty₂)
(Fin.succ
(↑(Equiv.swap (Fin.last (s.length + 1)) (Fin.castSucc (Fin.last s.length)))
(Fin.castSucc (Fin.last s.length)))))
[PROOFSTEP]
erw [Equiv.swap_apply_right, snoc_castSucc, snoc_castSucc, snoc_castSucc, Fin.succ_castSucc, snoc_castSucc,
Fin.succ_last, snoc_last, snoc_last, Fin.succ_last, snoc_last]
[GOAL]
case refine'_2.refine'_1
X : Type u
inst✝¹ : Lattice X
inst✝ : JordanHolderLattice X
s : CompositionSeries X
x₁ x₂ y₁ y₂ : X
hsat₁ : IsMaximal (top s) x₁
hsat₂ : IsMaximal (top s) x₂
hsaty₁ : IsMaximal (top (snoc s x₁ hsat₁)) y₁
hsaty₂ : IsMaximal (top (snoc s x₂ hsat₂)) y₂
hr₁ : Iso (top s, x₁) (x₂, y₂)
hr₂ : Iso (x₁, y₁) (top s, x₂)
e : Fin (s.length + 1 + 1) ≃ Fin (s.length + 1 + 1) :=
Equiv.swap (Fin.last (s.length + 1)) (Fin.castSucc (Fin.last s.length))
h1 : ∀ {i : Fin s.length}, Fin.castSucc (Fin.castSucc i) ≠ Fin.castSucc (Fin.last s.length)
h2 : ∀ {i : Fin s.length}, Fin.castSucc (Fin.castSucc i) ≠ Fin.last (s.length + 1)
i✝ : Fin (snoc (snoc s x₁ hsat₁) y₁ hsaty₁).length
i : Fin (snoc s x₁ hsat₁).length
⊢ Iso (series s (Fin.last s.length), x₁) (x₂, y₂)
[PROOFSTEP]
exact hr₁
[GOAL]
case refine'_2.refine'_2
X : Type u
inst✝¹ : Lattice X
inst✝ : JordanHolderLattice X
s : CompositionSeries X
x₁ x₂ y₁ y₂ : X
hsat₁ : IsMaximal (top s) x₁
hsat₂ : IsMaximal (top s) x₂
hsaty₁ : IsMaximal (top (snoc s x₁ hsat₁)) y₁
hsaty₂ : IsMaximal (top (snoc s x₂ hsat₂)) y₂
hr₁ : Iso (top s, x₁) (x₂, y₂)
hr₂ : Iso (x₁, y₁) (top s, x₂)
e : Fin (s.length + 1 + 1) ≃ Fin (s.length + 1 + 1) :=
Equiv.swap (Fin.last (s.length + 1)) (Fin.castSucc (Fin.last s.length))
h1 : ∀ {i : Fin s.length}, Fin.castSucc (Fin.castSucc i) ≠ Fin.castSucc (Fin.last s.length)
h2 : ∀ {i : Fin s.length}, Fin.castSucc (Fin.castSucc i) ≠ Fin.last (s.length + 1)
i✝¹ : Fin (snoc (snoc s x₁ hsat₁) y₁ hsaty₁).length
i✝ : Fin (snoc s x₁ hsat₁).length
i : Fin s.length
⊢ Iso
(series (snoc (snoc s x₁ hsat₁) y₁ hsaty₁) (Fin.castSucc (Fin.castSucc (Fin.castSucc i))),
series (snoc (snoc s x₁ hsat₁) y₁ hsaty₁) (Fin.succ (Fin.castSucc (Fin.castSucc i))))
(series (snoc (snoc s x₂ hsat₂) y₂ hsaty₂)
(Fin.castSucc
(↑(Equiv.swap (Fin.last (s.length + 1)) (Fin.castSucc (Fin.last s.length))) (Fin.castSucc (Fin.castSucc i)))),
series (snoc (snoc s x₂ hsat₂) y₂ hsaty₂)
(Fin.succ
(↑(Equiv.swap (Fin.last (s.length + 1)) (Fin.castSucc (Fin.last s.length))) (Fin.castSucc (Fin.castSucc i)))))
[PROOFSTEP]
erw [Equiv.swap_apply_of_ne_of_ne h2 h1, snoc_castSucc, snoc_castSucc, snoc_castSucc, snoc_castSucc, Fin.succ_castSucc,
snoc_castSucc, Fin.succ_castSucc, snoc_castSucc, snoc_castSucc, snoc_castSucc]
[GOAL]
case refine'_2.refine'_2
X : Type u
inst✝¹ : Lattice X
inst✝ : JordanHolderLattice X
s : CompositionSeries X
x₁ x₂ y₁ y₂ : X
hsat₁ : IsMaximal (top s) x₁
hsat₂ : IsMaximal (top s) x₂
hsaty₁ : IsMaximal (top (snoc s x₁ hsat₁)) y₁
hsaty₂ : IsMaximal (top (snoc s x₂ hsat₂)) y₂
hr₁ : Iso (top s, x₁) (x₂, y₂)
hr₂ : Iso (x₁, y₁) (top s, x₂)
e : Fin (s.length + 1 + 1) ≃ Fin (s.length + 1 + 1) :=
Equiv.swap (Fin.last (s.length + 1)) (Fin.castSucc (Fin.last s.length))
h1 : ∀ {i : Fin s.length}, Fin.castSucc (Fin.castSucc i) ≠ Fin.castSucc (Fin.last s.length)
h2 : ∀ {i : Fin s.length}, Fin.castSucc (Fin.castSucc i) ≠ Fin.last (s.length + 1)
i✝¹ : Fin (snoc (snoc s x₁ hsat₁) y₁ hsaty₁).length
i✝ : Fin (snoc s x₁ hsat₁).length
i : Fin s.length
⊢ Iso (series s (Fin.castSucc i), series s (Fin.succ i)) (series s (Fin.castSucc i), series s (Fin.succ i))
[PROOFSTEP]
exact (s.step i).iso_refl
[GOAL]
X : Type u
inst✝¹ : Lattice X
inst✝ : JordanHolderLattice X
s₁ s₂ : CompositionSeries X
hb : bot s₁ = bot s₂
ht : top s₁ = top s₂
hs₁ : s₁.length = 0
⊢ s₂.length = 0
[PROOFSTEP]
have : s₁.bot = s₁.top := congr_arg s₁ (Fin.ext (by simp [hs₁]))
[GOAL]
X : Type u
inst✝¹ : Lattice X
inst✝ : JordanHolderLattice X
s₁ s₂ : CompositionSeries X
hb : bot s₁ = bot s₂
ht : top s₁ = top s₂
hs₁ : s₁.length = 0
⊢ ↑0 = ↑(Fin.last s₁.length)
[PROOFSTEP]
simp [hs₁]
[GOAL]
X : Type u
inst✝¹ : Lattice X
inst✝ : JordanHolderLattice X
s₁ s₂ : CompositionSeries X
hb : bot s₁ = bot s₂
ht : top s₁ = top s₂
hs₁ : s₁.length = 0
this : bot s₁ = top s₁
⊢ s₂.length = 0
[PROOFSTEP]
have : Fin.last s₂.length = (0 : Fin s₂.length.succ) := s₂.injective (hb.symm.trans (this.trans ht)).symm
[GOAL]
X : Type u
inst✝¹ : Lattice X
inst✝ : JordanHolderLattice X
s₁ s₂ : CompositionSeries X
hb : bot s₁ = bot s₂
ht : top s₁ = top s₂
hs₁ : s₁.length = 0
this✝ : bot s₁ = top s₁
this : Fin.last s₂.length = 0
⊢ s₂.length = 0
[PROOFSTEP]
rw [Fin.ext_iff] at this
[GOAL]
X : Type u
inst✝¹ : Lattice X
inst✝ : JordanHolderLattice X
s₁ s₂ : CompositionSeries X
hb : bot s₁ = bot s₂
ht : top s₁ = top s₂
hs₁ : s₁.length = 0
this✝ : bot s₁ = top s₁
this : ↑(Fin.last s₂.length) = ↑0
⊢ s₂.length = 0
[PROOFSTEP]
simpa
[GOAL]
X : Type u
inst✝¹ : Lattice X
inst✝ : JordanHolderLattice X
s₁ s₂ : CompositionSeries X
hb : bot s₁ = bot s₂
ht : top s₁ = top s₂
⊢ ¬0 < s₂.length → ¬0 < s₁.length
[PROOFSTEP]
simp only [pos_iff_ne_zero, Ne.def, not_iff_not, Classical.not_not]
[GOAL]
X : Type u
inst✝¹ : Lattice X
inst✝ : JordanHolderLattice X
s₁ s₂ : CompositionSeries X
hb : bot s₁ = bot s₂
ht : top s₁ = top s₂
⊢ s₂.length = 0 → s₁.length = 0
[PROOFSTEP]
exact length_eq_zero_of_bot_eq_bot_of_top_eq_top_of_length_eq_zero hb.symm ht.symm
[GOAL]
X : Type u
inst✝¹ : Lattice X
inst✝ : JordanHolderLattice X
s₁ s₂ : CompositionSeries X
hb : bot s₁ = bot s₂
ht : top s₁ = top s₂
hs₁0 : s₁.length = 0
⊢ s₁ = s₂
[PROOFSTEP]
have : ∀ x, x ∈ s₁ ↔ x = s₁.top := fun x =>
⟨fun hx => forall_mem_eq_of_length_eq_zero hs₁0 hx s₁.top_mem, fun hx => hx.symm ▸ s₁.top_mem⟩
[GOAL]
X : Type u
inst✝¹ : Lattice X
inst✝ : JordanHolderLattice X
s₁ s₂ : CompositionSeries X
hb : bot s₁ = bot s₂
ht : top s₁ = top s₂
hs₁0 : s₁.length = 0
this : ∀ (x : X), x ∈ s₁ ↔ x = top s₁
⊢ s₁ = s₂
[PROOFSTEP]
have : ∀ x, x ∈ s₂ ↔ x = s₂.top := fun x =>
⟨fun hx =>
forall_mem_eq_of_length_eq_zero (length_eq_zero_of_bot_eq_bot_of_top_eq_top_of_length_eq_zero hb ht hs₁0) hx
s₂.top_mem,
fun hx => hx.symm ▸ s₂.top_mem⟩
[GOAL]
X : Type u
inst✝¹ : Lattice X
inst✝ : JordanHolderLattice X
s₁ s₂ : CompositionSeries X
hb : bot s₁ = bot s₂
ht : top s₁ = top s₂
hs₁0 : s₁.length = 0
this✝ : ∀ (x : X), x ∈ s₁ ↔ x = top s₁
this : ∀ (x : X), x ∈ s₂ ↔ x = top s₂
⊢ s₁ = s₂
[PROOFSTEP]
ext
[GOAL]
case h
X : Type u
inst✝¹ : Lattice X
inst✝ : JordanHolderLattice X
s₁ s₂ : CompositionSeries X
hb : bot s₁ = bot s₂
ht : top s₁ = top s₂
hs₁0 : s₁.length = 0
this✝ : ∀ (x : X), x ∈ s₁ ↔ x = top s₁
this : ∀ (x : X), x ∈ s₂ ↔ x = top s₂
x✝ : X
⊢ x✝ ∈ s₁ ↔ x✝ ∈ s₂
[PROOFSTEP]
simp [*]
[GOAL]
X : Type u
inst✝¹ : Lattice X
inst✝ : JordanHolderLattice X
s : CompositionSeries X
x : X
hm : IsMaximal x (top s)
hb : bot s ≤ x
⊢ ∃ t, bot t = bot s ∧ t.length + 1 = s.length ∧ ∃ htx, Equivalent s (snoc t (top s) (_ : IsMaximal (top t) (top s)))
[PROOFSTEP]
induction' hn : s.length with n ih generalizing s x
[GOAL]
case zero
X : Type u
inst✝¹ : Lattice X
inst✝ : JordanHolderLattice X
s✝ : CompositionSeries X
x✝¹ : X
hm✝ : IsMaximal x✝¹ (top s✝)
hb✝ : bot s✝ ≤ x✝¹
x✝ : ℕ
hn✝ : s✝.length = x✝
s : CompositionSeries X
x : X
hm : IsMaximal x (top s)
hb : bot s ≤ x
hn : s.length = Nat.zero
⊢ ∃ t, bot t = bot s ∧ t.length + 1 = Nat.zero ∧ ∃ htx, Equivalent s (snoc t (top s) (_ : IsMaximal (top t) (top s)))
[PROOFSTEP]
exact (ne_of_gt (lt_of_le_of_lt hb (lt_of_isMaximal hm)) (forall_mem_eq_of_length_eq_zero hn s.top_mem s.bot_mem)).elim
[GOAL]
case succ
X : Type u
inst✝¹ : Lattice X
inst✝ : JordanHolderLattice X
s✝ : CompositionSeries X
x✝¹ : X
hm✝ : IsMaximal x✝¹ (top s✝)
hb✝ : bot s✝ ≤ x✝¹
x✝ : ℕ
hn✝ : s✝.length = x✝
n : ℕ
ih :
∀ (s : CompositionSeries X) (x : X) (hm : IsMaximal x (top s)),
bot s ≤ x →
s.length = n →
∃ t, bot t = bot s ∧ t.length + 1 = n ∧ ∃ htx, Equivalent s (snoc t (top s) (_ : IsMaximal (top t) (top s)))
s : CompositionSeries X
x : X
hm : IsMaximal x (top s)
hb : bot s ≤ x
hn : s.length = Nat.succ n
⊢ ∃ t, bot t = bot s ∧ t.length + 1 = Nat.succ n ∧ ∃ htx, Equivalent s (snoc t (top s) (_ : IsMaximal (top t) (top s)))
[PROOFSTEP]
have h0s : 0 < s.length := hn.symm ▸ Nat.succ_pos _
[GOAL]
case succ
X : Type u
inst✝¹ : Lattice X
inst✝ : JordanHolderLattice X
s✝ : CompositionSeries X
x✝¹ : X
hm✝ : IsMaximal x✝¹ (top s✝)
hb✝ : bot s✝ ≤ x✝¹
x✝ : ℕ
hn✝ : s✝.length = x✝
n : ℕ
ih :
∀ (s : CompositionSeries X) (x : X) (hm : IsMaximal x (top s)),
bot s ≤ x →
s.length = n →
∃ t, bot t = bot s ∧ t.length + 1 = n ∧ ∃ htx, Equivalent s (snoc t (top s) (_ : IsMaximal (top t) (top s)))
s : CompositionSeries X
x : X
hm : IsMaximal x (top s)
hb : bot s ≤ x
hn : s.length = Nat.succ n
h0s : 0 < s.length
⊢ ∃ t, bot t = bot s ∧ t.length + 1 = Nat.succ n ∧ ∃ htx, Equivalent s (snoc t (top s) (_ : IsMaximal (top t) (top s)))
[PROOFSTEP]
by_cases hetx : s.eraseTop.top = x
[GOAL]
case pos
X : Type u
inst✝¹ : Lattice X
inst✝ : JordanHolderLattice X
s✝ : CompositionSeries X
x✝¹ : X
hm✝ : IsMaximal x✝¹ (top s✝)
hb✝ : bot s✝ ≤ x✝¹
x✝ : ℕ
hn✝ : s✝.length = x✝
n : ℕ
ih :
∀ (s : CompositionSeries X) (x : X) (hm : IsMaximal x (top s)),
bot s ≤ x →
s.length = n →
∃ t, bot t = bot s ∧ t.length + 1 = n ∧ ∃ htx, Equivalent s (snoc t (top s) (_ : IsMaximal (top t) (top s)))
s : CompositionSeries X
x : X
hm : IsMaximal x (top s)
hb : bot s ≤ x
hn : s.length = Nat.succ n
h0s : 0 < s.length
hetx : top (eraseTop s) = x
⊢ ∃ t, bot t = bot s ∧ t.length + 1 = Nat.succ n ∧ ∃ htx, Equivalent s (snoc t (top s) (_ : IsMaximal (top t) (top s)))
[PROOFSTEP]
use s.eraseTop
[GOAL]
case h
X : Type u
inst✝¹ : Lattice X
inst✝ : JordanHolderLattice X
s✝ : CompositionSeries X
x✝¹ : X
hm✝ : IsMaximal x✝¹ (top s✝)
hb✝ : bot s✝ ≤ x✝¹
x✝ : ℕ
hn✝ : s✝.length = x✝
n : ℕ
ih :
∀ (s : CompositionSeries X) (x : X) (hm : IsMaximal x (top s)),
bot s ≤ x →
s.length = n →
∃ t, bot t = bot s ∧ t.length + 1 = n ∧ ∃ htx, Equivalent s (snoc t (top s) (_ : IsMaximal (top t) (top s)))
s : CompositionSeries X
x : X
hm : IsMaximal x (top s)
hb : bot s ≤ x
hn : s.length = Nat.succ n
h0s : 0 < s.length
hetx : top (eraseTop s) = x
⊢ bot (eraseTop s) = bot s ∧
(eraseTop s).length + 1 = Nat.succ n ∧
∃ htx, Equivalent s (snoc (eraseTop s) (top s) (_ : IsMaximal (top (eraseTop s)) (top s)))
[PROOFSTEP]
simp [← hetx, hn]
-- Porting note: `rfl` is required.
[GOAL]
case h
X : Type u
inst✝¹ : Lattice X
inst✝ : JordanHolderLattice X
s✝ : CompositionSeries X
x✝¹ : X
hm✝ : IsMaximal x✝¹ (top s✝)
hb✝ : bot s✝ ≤ x✝¹
x✝ : ℕ
hn✝ : s✝.length = x✝
n : ℕ
ih :
∀ (s : CompositionSeries X) (x : X) (hm : IsMaximal x (top s)),
bot s ≤ x →
s.length = n →
∃ t, bot t = bot s ∧ t.length + 1 = n ∧ ∃ htx, Equivalent s (snoc t (top s) (_ : IsMaximal (top t) (top s)))
s : CompositionSeries X
x : X
hm : IsMaximal x (top s)
hb : bot s ≤ x
hn : s.length = Nat.succ n
h0s : 0 < s.length
hetx : top (eraseTop s) = x
⊢ Equivalent s s
[PROOFSTEP]
rfl
[GOAL]
case neg
X : Type u
inst✝¹ : Lattice X
inst✝ : JordanHolderLattice X
s✝ : CompositionSeries X
x✝¹ : X
hm✝ : IsMaximal x✝¹ (top s✝)
hb✝ : bot s✝ ≤ x✝¹
x✝ : ℕ
hn✝ : s✝.length = x✝
n : ℕ
ih :
∀ (s : CompositionSeries X) (x : X) (hm : IsMaximal x (top s)),
bot s ≤ x →
s.length = n →
∃ t, bot t = bot s ∧ t.length + 1 = n ∧ ∃ htx, Equivalent s (snoc t (top s) (_ : IsMaximal (top t) (top s)))
s : CompositionSeries X
x : X
hm : IsMaximal x (top s)
hb : bot s ≤ x
hn : s.length = Nat.succ n
h0s : 0 < s.length
hetx : ¬top (eraseTop s) = x
⊢ ∃ t, bot t = bot s ∧ t.length + 1 = Nat.succ n ∧ ∃ htx, Equivalent s (snoc t (top s) (_ : IsMaximal (top t) (top s)))
[PROOFSTEP]
have imxs : IsMaximal (x ⊓ s.eraseTop.top) s.eraseTop.top :=
isMaximal_of_eq_inf x s.top rfl (Ne.symm hetx) hm (isMaximal_eraseTop_top h0s)
[GOAL]
case neg
X : Type u
inst✝¹ : Lattice X
inst✝ : JordanHolderLattice X
s✝ : CompositionSeries X
x✝¹ : X
hm✝ : IsMaximal x✝¹ (top s✝)
hb✝ : bot s✝ ≤ x✝¹
x✝ : ℕ
hn✝ : s✝.length = x✝
n : ℕ
ih :
∀ (s : CompositionSeries X) (x : X) (hm : IsMaximal x (top s)),
bot s ≤ x →
s.length = n →
∃ t, bot t = bot s ∧ t.length + 1 = n ∧ ∃ htx, Equivalent s (snoc t (top s) (_ : IsMaximal (top t) (top s)))
s : CompositionSeries X
x : X
hm : IsMaximal x (top s)
hb : bot s ≤ x
hn : s.length = Nat.succ n
h0s : 0 < s.length
hetx : ¬top (eraseTop s) = x
imxs : IsMaximal (x ⊓ top (eraseTop s)) (top (eraseTop s))
⊢ ∃ t, bot t = bot s ∧ t.length + 1 = Nat.succ n ∧ ∃ htx, Equivalent s (snoc t (top s) (_ : IsMaximal (top t) (top s)))
[PROOFSTEP]
have := ih _ _ imxs (le_inf (by simpa) (le_top_of_mem s.eraseTop.bot_mem)) (by simp [hn])
[GOAL]
X : Type u
inst✝¹ : Lattice X
inst✝ : JordanHolderLattice X
s✝ : CompositionSeries X
x✝¹ : X
hm✝ : IsMaximal x✝¹ (top s✝)
hb✝ : bot s✝ ≤ x✝¹
x✝ : ℕ
hn✝ : s✝.length = x✝
n : ℕ
ih :
∀ (s : CompositionSeries X) (x : X) (hm : IsMaximal x (top s)),
bot s ≤ x →
s.length = n →
∃ t, bot t = bot s ∧ t.length + 1 = n ∧ ∃ htx, Equivalent s (snoc t (top s) (_ : IsMaximal (top t) (top s)))
s : CompositionSeries X
x : X
hm : IsMaximal x (top s)
hb : bot s ≤ x
hn : s.length = Nat.succ n
h0s : 0 < s.length
hetx : ¬top (eraseTop s) = x
imxs : IsMaximal (x ⊓ top (eraseTop s)) (top (eraseTop s))
⊢ bot (eraseTop s) ≤ x
[PROOFSTEP]
simpa
[GOAL]
X : Type u
inst✝¹ : Lattice X
inst✝ : JordanHolderLattice X
s✝ : CompositionSeries X
x✝¹ : X
hm✝ : IsMaximal x✝¹ (top s✝)
hb✝ : bot s✝ ≤ x✝¹
x✝ : ℕ
hn✝ : s✝.length = x✝
n : ℕ
ih :
∀ (s : CompositionSeries X) (x : X) (hm : IsMaximal x (top s)),
bot s ≤ x →
s.length = n →
∃ t, bot t = bot s ∧ t.length + 1 = n ∧ ∃ htx, Equivalent s (snoc t (top s) (_ : IsMaximal (top t) (top s)))
s : CompositionSeries X
x : X
hm : IsMaximal x (top s)
hb : bot s ≤ x
hn : s.length = Nat.succ n
h0s : 0 < s.length
hetx : ¬top (eraseTop s) = x
imxs : IsMaximal (x ⊓ top (eraseTop s)) (top (eraseTop s))
⊢ (eraseTop s).length = n
[PROOFSTEP]
simp [hn]
[GOAL]
case neg
X : Type u
inst✝¹ : Lattice X
inst✝ : JordanHolderLattice X
s✝ : CompositionSeries X
x✝¹ : X
hm✝ : IsMaximal x✝¹ (top s✝)
hb✝ : bot s✝ ≤ x✝¹
x✝ : ℕ
hn✝ : s✝.length = x✝
n : ℕ
ih :
∀ (s : CompositionSeries X) (x : X) (hm : IsMaximal x (top s)),
bot s ≤ x →
s.length = n →
∃ t, bot t = bot s ∧ t.length + 1 = n ∧ ∃ htx, Equivalent s (snoc t (top s) (_ : IsMaximal (top t) (top s)))
s : CompositionSeries X
x : X
hm : IsMaximal x (top s)
hb : bot s ≤ x
hn : s.length = Nat.succ n
h0s : 0 < s.length
hetx : ¬top (eraseTop s) = x
imxs : IsMaximal (x ⊓ top (eraseTop s)) (top (eraseTop s))
this :
∃ t,
bot t = bot (eraseTop s) ∧
t.length + 1 = n ∧
∃ htx, Equivalent (eraseTop s) (snoc t (top (eraseTop s)) (_ : IsMaximal (top t) (top (eraseTop s))))
⊢ ∃ t, bot t = bot s ∧ t.length + 1 = Nat.succ n ∧ ∃ htx, Equivalent s (snoc t (top s) (_ : IsMaximal (top t) (top s)))
[PROOFSTEP]
rcases this with ⟨t, htb, htl, htt, hteqv⟩
[GOAL]
case neg.intro.intro.intro.intro
X : Type u
inst✝¹ : Lattice X
inst✝ : JordanHolderLattice X
s✝ : CompositionSeries X
x✝¹ : X
hm✝ : IsMaximal x✝¹ (top s✝)
hb✝ : bot s✝ ≤ x✝¹
x✝ : ℕ
hn✝ : s✝.length = x✝
n : ℕ
ih :
∀ (s : CompositionSeries X) (x : X) (hm : IsMaximal x (top s)),
bot s ≤ x →
s.length = n →
∃ t, bot t = bot s ∧ t.length + 1 = n ∧ ∃ htx, Equivalent s (snoc t (top s) (_ : IsMaximal (top t) (top s)))
s : CompositionSeries X
x : X
hm : IsMaximal x (top s)
hb : bot s ≤ x
hn : s.length = Nat.succ n
h0s : 0 < s.length
hetx : ¬top (eraseTop s) = x
imxs : IsMaximal (x ⊓ top (eraseTop s)) (top (eraseTop s))
t : CompositionSeries X
htb : bot t = bot (eraseTop s)
htl : t.length + 1 = n
htt : top t = x ⊓ top (eraseTop s)
hteqv : Equivalent (eraseTop s) (snoc t (top (eraseTop s)) (_ : IsMaximal (top t) (top (eraseTop s))))
⊢ ∃ t, bot t = bot s ∧ t.length + 1 = Nat.succ n ∧ ∃ htx, Equivalent s (snoc t (top s) (_ : IsMaximal (top t) (top s)))
[PROOFSTEP]
have hmtx : IsMaximal t.top x :=
isMaximal_of_eq_inf s.eraseTop.top s.top (by rw [inf_comm, htt]) hetx (isMaximal_eraseTop_top h0s) hm
[GOAL]
X : Type u
inst✝¹ : Lattice X
inst✝ : JordanHolderLattice X
s✝ : CompositionSeries X
x✝¹ : X
hm✝ : IsMaximal x✝¹ (top s✝)
hb✝ : bot s✝ ≤ x✝¹
x✝ : ℕ
hn✝ : s✝.length = x✝
n : ℕ
ih :
∀ (s : CompositionSeries X) (x : X) (hm : IsMaximal x (top s)),
bot s ≤ x →
s.length = n →
∃ t, bot t = bot s ∧ t.length + 1 = n ∧ ∃ htx, Equivalent s (snoc t (top s) (_ : IsMaximal (top t) (top s)))
s : CompositionSeries X
x : X
hm : IsMaximal x (top s)
hb : bot s ≤ x
hn : s.length = Nat.succ n
h0s : 0 < s.length
hetx : ¬top (eraseTop s) = x
imxs : IsMaximal (x ⊓ top (eraseTop s)) (top (eraseTop s))
t : CompositionSeries X
htb : bot t = bot (eraseTop s)
htl : t.length + 1 = n
htt : top t = x ⊓ top (eraseTop s)
hteqv : Equivalent (eraseTop s) (snoc t (top (eraseTop s)) (_ : IsMaximal (top t) (top (eraseTop s))))
⊢ top (eraseTop s) ⊓ x = top t
[PROOFSTEP]
rw [inf_comm, htt]
[GOAL]
case neg.intro.intro.intro.intro
X : Type u
inst✝¹ : Lattice X
inst✝ : JordanHolderLattice X
s✝ : CompositionSeries X
x✝¹ : X
hm✝ : IsMaximal x✝¹ (top s✝)
hb✝ : bot s✝ ≤ x✝¹
x✝ : ℕ
hn✝ : s✝.length = x✝
n : ℕ
ih :
∀ (s : CompositionSeries X) (x : X) (hm : IsMaximal x (top s)),
bot s ≤ x →
s.length = n →
∃ t, bot t = bot s ∧ t.length + 1 = n ∧ ∃ htx, Equivalent s (snoc t (top s) (_ : IsMaximal (top t) (top s)))
s : CompositionSeries X
x : X
hm : IsMaximal x (top s)
hb : bot s ≤ x
hn : s.length = Nat.succ n
h0s : 0 < s.length
hetx : ¬top (eraseTop s) = x
imxs : IsMaximal (x ⊓ top (eraseTop s)) (top (eraseTop s))
t : CompositionSeries X
htb : bot t = bot (eraseTop s)
htl : t.length + 1 = n
htt : top t = x ⊓ top (eraseTop s)
hteqv : Equivalent (eraseTop s) (snoc t (top (eraseTop s)) (_ : IsMaximal (top t) (top (eraseTop s))))
hmtx : IsMaximal (top t) x
⊢ ∃ t, bot t = bot s ∧ t.length + 1 = Nat.succ n ∧ ∃ htx, Equivalent s (snoc t (top s) (_ : IsMaximal (top t) (top s)))
[PROOFSTEP]
use snoc t x hmtx
[GOAL]
case h
X : Type u
inst✝¹ : Lattice X
inst✝ : JordanHolderLattice X
s✝ : CompositionSeries X
x✝¹ : X
hm✝ : IsMaximal x✝¹ (top s✝)
hb✝ : bot s✝ ≤ x✝¹
x✝ : ℕ
hn✝ : s✝.length = x✝
n : ℕ
ih :
∀ (s : CompositionSeries X) (x : X) (hm : IsMaximal x (top s)),
bot s ≤ x →
s.length = n →
∃ t, bot t = bot s ∧ t.length + 1 = n ∧ ∃ htx, Equivalent s (snoc t (top s) (_ : IsMaximal (top t) (top s)))
s : CompositionSeries X
x : X
hm : IsMaximal x (top s)
hb : bot s ≤ x
hn : s.length = Nat.succ n
h0s : 0 < s.length
hetx : ¬top (eraseTop s) = x
imxs : IsMaximal (x ⊓ top (eraseTop s)) (top (eraseTop s))
t : CompositionSeries X
htb : bot t = bot (eraseTop s)
htl : t.length + 1 = n
htt : top t = x ⊓ top (eraseTop s)
hteqv : Equivalent (eraseTop s) (snoc t (top (eraseTop s)) (_ : IsMaximal (top t) (top (eraseTop s))))
hmtx : IsMaximal (top t) x
⊢ bot (snoc t x hmtx) = bot s ∧
(snoc t x hmtx).length + 1 = Nat.succ n ∧
∃ htx, Equivalent s (snoc (snoc t x hmtx) (top s) (_ : IsMaximal (top (snoc t x hmtx)) (top s)))
[PROOFSTEP]
refine' ⟨by simp [htb], by simp [htl], by simp, _⟩
[GOAL]
X : Type u
inst✝¹ : Lattice X
inst✝ : JordanHolderLattice X
s✝ : CompositionSeries X
x✝¹ : X
hm✝ : IsMaximal x✝¹ (top s✝)
hb✝ : bot s✝ ≤ x✝¹
x✝ : ℕ
hn✝ : s✝.length = x✝
n : ℕ
ih :
∀ (s : CompositionSeries X) (x : X) (hm : IsMaximal x (top s)),
bot s ≤ x →
s.length = n →
∃ t, bot t = bot s ∧ t.length + 1 = n ∧ ∃ htx, Equivalent s (snoc t (top s) (_ : IsMaximal (top t) (top s)))
s : CompositionSeries X
x : X
hm : IsMaximal x (top s)
hb : bot s ≤ x
hn : s.length = Nat.succ n
h0s : 0 < s.length
hetx : ¬top (eraseTop s) = x
imxs : IsMaximal (x ⊓ top (eraseTop s)) (top (eraseTop s))
t : CompositionSeries X
htb : bot t = bot (eraseTop s)
htl : t.length + 1 = n
htt : top t = x ⊓ top (eraseTop s)
hteqv : Equivalent (eraseTop s) (snoc t (top (eraseTop s)) (_ : IsMaximal (top t) (top (eraseTop s))))
hmtx : IsMaximal (top t) x
⊢ bot (snoc t x hmtx) = bot s
[PROOFSTEP]
simp [htb]
[GOAL]
X : Type u
inst✝¹ : Lattice X
inst✝ : JordanHolderLattice X
s✝ : CompositionSeries X
x✝¹ : X
hm✝ : IsMaximal x✝¹ (top s✝)
hb✝ : bot s✝ ≤ x✝¹
x✝ : ℕ
hn✝ : s✝.length = x✝
n : ℕ
ih :
∀ (s : CompositionSeries X) (x : X) (hm : IsMaximal x (top s)),
bot s ≤ x →
s.length = n →
∃ t, bot t = bot s ∧ t.length + 1 = n ∧ ∃ htx, Equivalent s (snoc t (top s) (_ : IsMaximal (top t) (top s)))
s : CompositionSeries X
x : X
hm : IsMaximal x (top s)
hb : bot s ≤ x
hn : s.length = Nat.succ n
h0s : 0 < s.length
hetx : ¬top (eraseTop s) = x
imxs : IsMaximal (x ⊓ top (eraseTop s)) (top (eraseTop s))
t : CompositionSeries X
htb : bot t = bot (eraseTop s)
htl : t.length + 1 = n
htt : top t = x ⊓ top (eraseTop s)
hteqv : Equivalent (eraseTop s) (snoc t (top (eraseTop s)) (_ : IsMaximal (top t) (top (eraseTop s))))
hmtx : IsMaximal (top t) x
⊢ (snoc t x hmtx).length + 1 = Nat.succ n
[PROOFSTEP]
simp [htl]
[GOAL]
X : Type u
inst✝¹ : Lattice X
inst✝ : JordanHolderLattice X
s✝ : CompositionSeries X
x✝¹ : X
hm✝ : IsMaximal x✝¹ (top s✝)
hb✝ : bot s✝ ≤ x✝¹
x✝ : ℕ
hn✝ : s✝.length = x✝
n : ℕ
ih :
∀ (s : CompositionSeries X) (x : X) (hm : IsMaximal x (top s)),
bot s ≤ x →
s.length = n →
∃ t, bot t = bot s ∧ t.length + 1 = n ∧ ∃ htx, Equivalent s (snoc t (top s) (_ : IsMaximal (top t) (top s)))
s : CompositionSeries X
x : X
hm : IsMaximal x (top s)
hb : bot s ≤ x
hn : s.length = Nat.succ n
h0s : 0 < s.length
hetx : ¬top (eraseTop s) = x
imxs : IsMaximal (x ⊓ top (eraseTop s)) (top (eraseTop s))
t : CompositionSeries X
htb : bot t = bot (eraseTop s)
htl : t.length + 1 = n
htt : top t = x ⊓ top (eraseTop s)
hteqv : Equivalent (eraseTop s) (snoc t (top (eraseTop s)) (_ : IsMaximal (top t) (top (eraseTop s))))
hmtx : IsMaximal (top t) x
⊢ top (snoc t x hmtx) = x
[PROOFSTEP]
simp
[GOAL]
case h
X : Type u
inst✝¹ : Lattice X
inst✝ : JordanHolderLattice X
s✝ : CompositionSeries X
x✝¹ : X
hm✝ : IsMaximal x✝¹ (top s✝)
hb✝ : bot s✝ ≤ x✝¹
x✝ : ℕ
hn✝ : s✝.length = x✝
n : ℕ
ih :
∀ (s : CompositionSeries X) (x : X) (hm : IsMaximal x (top s)),
bot s ≤ x →
s.length = n →
∃ t, bot t = bot s ∧ t.length + 1 = n ∧ ∃ htx, Equivalent s (snoc t (top s) (_ : IsMaximal (top t) (top s)))
s : CompositionSeries X
x : X
hm : IsMaximal x (top s)
hb : bot s ≤ x
hn : s.length = Nat.succ n
h0s : 0 < s.length
hetx : ¬top (eraseTop s) = x
imxs : IsMaximal (x ⊓ top (eraseTop s)) (top (eraseTop s))
t : CompositionSeries X
htb : bot t = bot (eraseTop s)
htl : t.length + 1 = n
htt : top t = x ⊓ top (eraseTop s)
hteqv : Equivalent (eraseTop s) (snoc t (top (eraseTop s)) (_ : IsMaximal (top t) (top (eraseTop s))))
hmtx : IsMaximal (top t) x
⊢ Equivalent s (snoc (snoc t x hmtx) (top s) (_ : IsMaximal (top (snoc t x hmtx)) (top s)))
[PROOFSTEP]
have :
s.Equivalent ((snoc t s.eraseTop.top (htt.symm ▸ imxs)).snoc s.top (by simpa using isMaximal_eraseTop_top h0s)) :=
by
conv_lhs => rw [eq_snoc_eraseTop h0s]
exact Equivalent.snoc hteqv (by simpa using (isMaximal_eraseTop_top h0s).iso_refl)
[GOAL]
X : Type u
inst✝¹ : Lattice X
inst✝ : JordanHolderLattice X
s✝ : CompositionSeries X
x✝¹ : X
hm✝ : IsMaximal x✝¹ (top s✝)
hb✝ : bot s✝ ≤ x✝¹
x✝ : ℕ
hn✝ : s✝.length = x✝
n : ℕ
ih :
∀ (s : CompositionSeries X) (x : X) (hm : IsMaximal x (top s)),
bot s ≤ x →
s.length = n →
∃ t, bot t = bot s ∧ t.length + 1 = n ∧ ∃ htx, Equivalent s (snoc t (top s) (_ : IsMaximal (top t) (top s)))
s : CompositionSeries X
x : X
hm : IsMaximal x (top s)
hb : bot s ≤ x
hn : s.length = Nat.succ n
h0s : 0 < s.length
hetx : ¬top (eraseTop s) = x
imxs : IsMaximal (x ⊓ top (eraseTop s)) (top (eraseTop s))
t : CompositionSeries X
htb : bot t = bot (eraseTop s)
htl : t.length + 1 = n
htt : top t = x ⊓ top (eraseTop s)
hteqv : Equivalent (eraseTop s) (snoc t (top (eraseTop s)) (_ : IsMaximal (top t) (top (eraseTop s))))
hmtx : IsMaximal (top t) x
⊢ IsMaximal (top (snoc t (top (eraseTop s)) (_ : IsMaximal (top t) (top (eraseTop s))))) (top s)
[PROOFSTEP]
simpa using isMaximal_eraseTop_top h0s
[GOAL]
X : Type u
inst✝¹ : Lattice X
inst✝ : JordanHolderLattice X
s✝ : CompositionSeries X
x✝¹ : X
hm✝ : IsMaximal x✝¹ (top s✝)
hb✝ : bot s✝ ≤ x✝¹
x✝ : ℕ
hn✝ : s✝.length = x✝
n : ℕ
ih :
∀ (s : CompositionSeries X) (x : X) (hm : IsMaximal x (top s)),
bot s ≤ x →
s.length = n →
∃ t, bot t = bot s ∧ t.length + 1 = n ∧ ∃ htx, Equivalent s (snoc t (top s) (_ : IsMaximal (top t) (top s)))
s : CompositionSeries X
x : X
hm : IsMaximal x (top s)
hb : bot s ≤ x
hn : s.length = Nat.succ n
h0s : 0 < s.length
hetx : ¬top (eraseTop s) = x
imxs : IsMaximal (x ⊓ top (eraseTop s)) (top (eraseTop s))
t : CompositionSeries X
htb : bot t = bot (eraseTop s)
htl : t.length + 1 = n
htt : top t = x ⊓ top (eraseTop s)
hteqv : Equivalent (eraseTop s) (snoc t (top (eraseTop s)) (_ : IsMaximal (top t) (top (eraseTop s))))
hmtx : IsMaximal (top t) x
⊢ Equivalent s
(snoc (snoc t (top (eraseTop s)) (_ : IsMaximal (top t) (top (eraseTop s)))) (top s)
(_ : IsMaximal (top (snoc t (top (eraseTop s)) (_ : IsMaximal (top t) (top (eraseTop s))))) (top s)))
[PROOFSTEP]
conv_lhs => rw [eq_snoc_eraseTop h0s]
[GOAL]
X : Type u
inst✝¹ : Lattice X
inst✝ : JordanHolderLattice X
s✝ : CompositionSeries X
x✝¹ : X
hm✝ : IsMaximal x✝¹ (top s✝)
hb✝ : bot s✝ ≤ x✝¹
x✝ : ℕ
hn✝ : s✝.length = x✝
n : ℕ
ih :
∀ (s : CompositionSeries X) (x : X) (hm : IsMaximal x (top s)),
bot s ≤ x →
s.length = n →
∃ t, bot t = bot s ∧ t.length + 1 = n ∧ ∃ htx, Equivalent s (snoc t (top s) (_ : IsMaximal (top t) (top s)))
s : CompositionSeries X
x : X
hm : IsMaximal x (top s)
hb : bot s ≤ x
hn : s.length = Nat.succ n
h0s : 0 < s.length
hetx : ¬top (eraseTop s) = x
imxs : IsMaximal (x ⊓ top (eraseTop s)) (top (eraseTop s))
t : CompositionSeries X
htb : bot t = bot (eraseTop s)
htl : t.length + 1 = n
htt : top t = x ⊓ top (eraseTop s)
hteqv : Equivalent (eraseTop s) (snoc t (top (eraseTop s)) (_ : IsMaximal (top t) (top (eraseTop s))))
hmtx : IsMaximal (top t) x
| s
[PROOFSTEP]
rw [eq_snoc_eraseTop h0s]
[GOAL]
X : Type u
inst✝¹ : Lattice X
inst✝ : JordanHolderLattice X
s✝ : CompositionSeries X
x✝¹ : X
hm✝ : IsMaximal x✝¹ (top s✝)
hb✝ : bot s✝ ≤ x✝¹
x✝ : ℕ
hn✝ : s✝.length = x✝
n : ℕ
ih :
∀ (s : CompositionSeries X) (x : X) (hm : IsMaximal x (top s)),
bot s ≤ x →
s.length = n →
∃ t, bot t = bot s ∧ t.length + 1 = n ∧ ∃ htx, Equivalent s (snoc t (top s) (_ : IsMaximal (top t) (top s)))
s : CompositionSeries X
x : X
hm : IsMaximal x (top s)
hb : bot s ≤ x
hn : s.length = Nat.succ n
h0s : 0 < s.length
hetx : ¬top (eraseTop s) = x
imxs : IsMaximal (x ⊓ top (eraseTop s)) (top (eraseTop s))
t : CompositionSeries X
htb : bot t = bot (eraseTop s)
htl : t.length + 1 = n
htt : top t = x ⊓ top (eraseTop s)
hteqv : Equivalent (eraseTop s) (snoc t (top (eraseTop s)) (_ : IsMaximal (top t) (top (eraseTop s))))
hmtx : IsMaximal (top t) x
| s
[PROOFSTEP]
rw [eq_snoc_eraseTop h0s]
[GOAL]
X : Type u
inst✝¹ : Lattice X
inst✝ : JordanHolderLattice X
s✝ : CompositionSeries X
x✝¹ : X
hm✝ : IsMaximal x✝¹ (top s✝)
hb✝ : bot s✝ ≤ x✝¹
x✝ : ℕ
hn✝ : s✝.length = x✝
n : ℕ
ih :
∀ (s : CompositionSeries X) (x : X) (hm : IsMaximal x (top s)),
bot s ≤ x →
s.length = n →
∃ t, bot t = bot s ∧ t.length + 1 = n ∧ ∃ htx, Equivalent s (snoc t (top s) (_ : IsMaximal (top t) (top s)))
s : CompositionSeries X
x : X
hm : IsMaximal x (top s)
hb : bot s ≤ x
hn : s.length = Nat.succ n
h0s : 0 < s.length
hetx : ¬top (eraseTop s) = x
imxs : IsMaximal (x ⊓ top (eraseTop s)) (top (eraseTop s))
t : CompositionSeries X
htb : bot t = bot (eraseTop s)
htl : t.length + 1 = n
htt : top t = x ⊓ top (eraseTop s)
hteqv : Equivalent (eraseTop s) (snoc t (top (eraseTop s)) (_ : IsMaximal (top t) (top (eraseTop s))))
hmtx : IsMaximal (top t) x
| s
[PROOFSTEP]
rw [eq_snoc_eraseTop h0s]
[GOAL]
X : Type u
inst✝¹ : Lattice X
inst✝ : JordanHolderLattice X
s✝ : CompositionSeries X
x✝¹ : X
hm✝ : IsMaximal x✝¹ (top s✝)
hb✝ : bot s✝ ≤ x✝¹
x✝ : ℕ
hn✝ : s✝.length = x✝
n : ℕ
ih :
∀ (s : CompositionSeries X) (x : X) (hm : IsMaximal x (top s)),
bot s ≤ x →
s.length = n →
∃ t, bot t = bot s ∧ t.length + 1 = n ∧ ∃ htx, Equivalent s (snoc t (top s) (_ : IsMaximal (top t) (top s)))
s : CompositionSeries X
x : X
hm : IsMaximal x (top s)
hb : bot s ≤ x
hn : s.length = Nat.succ n
h0s : 0 < s.length
hetx : ¬top (eraseTop s) = x
imxs : IsMaximal (x ⊓ top (eraseTop s)) (top (eraseTop s))
t : CompositionSeries X
htb : bot t = bot (eraseTop s)
htl : t.length + 1 = n
htt : top t = x ⊓ top (eraseTop s)
hteqv : Equivalent (eraseTop s) (snoc t (top (eraseTop s)) (_ : IsMaximal (top t) (top (eraseTop s))))
hmtx : IsMaximal (top t) x
⊢ Equivalent (snoc (eraseTop s) (top s) (_ : IsMaximal (top (eraseTop s)) (top s)))
(snoc (snoc t (top (eraseTop s)) (_ : IsMaximal (top t) (top (eraseTop s)))) (top s)
(_ : IsMaximal (top (snoc t (top (eraseTop s)) (_ : IsMaximal (top t) (top (eraseTop s))))) (top s)))
[PROOFSTEP]
exact Equivalent.snoc hteqv (by simpa using (isMaximal_eraseTop_top h0s).iso_refl)
[GOAL]
X : Type u
inst✝¹ : Lattice X
inst✝ : JordanHolderLattice X
s✝ : CompositionSeries X
x✝¹ : X
hm✝ : IsMaximal x✝¹ (top s✝)
hb✝ : bot s✝ ≤ x✝¹
x✝ : ℕ
hn✝ : s✝.length = x✝
n : ℕ
ih :
∀ (s : CompositionSeries X) (x : X) (hm : IsMaximal x (top s)),
bot s ≤ x →
s.length = n →
∃ t, bot t = bot s ∧ t.length + 1 = n ∧ ∃ htx, Equivalent s (snoc t (top s) (_ : IsMaximal (top t) (top s)))
s : CompositionSeries X
x : X
hm : IsMaximal x (top s)
hb : bot s ≤ x
hn : s.length = Nat.succ n
h0s : 0 < s.length
hetx : ¬top (eraseTop s) = x
imxs : IsMaximal (x ⊓ top (eraseTop s)) (top (eraseTop s))
t : CompositionSeries X
htb : bot t = bot (eraseTop s)
htl : t.length + 1 = n
htt : top t = x ⊓ top (eraseTop s)
hteqv : Equivalent (eraseTop s) (snoc t (top (eraseTop s)) (_ : IsMaximal (top t) (top (eraseTop s))))
hmtx : IsMaximal (top t) x
⊢ Iso (top (eraseTop s), top s) (top (snoc t (top (eraseTop s)) (_ : IsMaximal (top t) (top (eraseTop s)))), top s)
[PROOFSTEP]
simpa using (isMaximal_eraseTop_top h0s).iso_refl
[GOAL]
case h
X : Type u
inst✝¹ : Lattice X
inst✝ : JordanHolderLattice X
s✝ : CompositionSeries X
x✝¹ : X
hm✝ : IsMaximal x✝¹ (top s✝)
hb✝ : bot s✝ ≤ x✝¹
x✝ : ℕ
hn✝ : s✝.length = x✝
n : ℕ
ih :
∀ (s : CompositionSeries X) (x : X) (hm : IsMaximal x (top s)),
bot s ≤ x →
s.length = n →
∃ t, bot t = bot s ∧ t.length + 1 = n ∧ ∃ htx, Equivalent s (snoc t (top s) (_ : IsMaximal (top t) (top s)))
s : CompositionSeries X
x : X
hm : IsMaximal x (top s)
hb : bot s ≤ x
hn : s.length = Nat.succ n
h0s : 0 < s.length
hetx : ¬top (eraseTop s) = x
imxs : IsMaximal (x ⊓ top (eraseTop s)) (top (eraseTop s))
t : CompositionSeries X
htb : bot t = bot (eraseTop s)
htl : t.length + 1 = n
htt : top t = x ⊓ top (eraseTop s)
hteqv : Equivalent (eraseTop s) (snoc t (top (eraseTop s)) (_ : IsMaximal (top t) (top (eraseTop s))))
hmtx : IsMaximal (top t) x
this :
Equivalent s
(snoc (snoc t (top (eraseTop s)) (_ : IsMaximal (top t) (top (eraseTop s)))) (top s)
(_ : IsMaximal (top (snoc t (top (eraseTop s)) (_ : IsMaximal (top t) (top (eraseTop s))))) (top s)))
⊢ Equivalent s (snoc (snoc t x hmtx) (top s) (_ : IsMaximal (top (snoc t x hmtx)) (top s)))
[PROOFSTEP]
refine' this.trans _
[GOAL]
case h
X : Type u
inst✝¹ : Lattice X
inst✝ : JordanHolderLattice X
s✝ : CompositionSeries X
x✝¹ : X
hm✝ : IsMaximal x✝¹ (top s✝)
hb✝ : bot s✝ ≤ x✝¹
x✝ : ℕ
hn✝ : s✝.length = x✝
n : ℕ
ih :
∀ (s : CompositionSeries X) (x : X) (hm : IsMaximal x (top s)),
bot s ≤ x →
s.length = n →
∃ t, bot t = bot s ∧ t.length + 1 = n ∧ ∃ htx, Equivalent s (snoc t (top s) (_ : IsMaximal (top t) (top s)))
s : CompositionSeries X
x : X
hm : IsMaximal x (top s)
hb : bot s ≤ x
hn : s.length = Nat.succ n
h0s : 0 < s.length
hetx : ¬top (eraseTop s) = x
imxs : IsMaximal (x ⊓ top (eraseTop s)) (top (eraseTop s))
t : CompositionSeries X
htb : bot t = bot (eraseTop s)
htl : t.length + 1 = n
htt : top t = x ⊓ top (eraseTop s)
hteqv : Equivalent (eraseTop s) (snoc t (top (eraseTop s)) (_ : IsMaximal (top t) (top (eraseTop s))))
hmtx : IsMaximal (top t) x
this :
Equivalent s
(snoc (snoc t (top (eraseTop s)) (_ : IsMaximal (top t) (top (eraseTop s)))) (top s)
(_ : IsMaximal (top (snoc t (top (eraseTop s)) (_ : IsMaximal (top t) (top (eraseTop s))))) (top s)))
⊢ Equivalent
(snoc (snoc t (top (eraseTop s)) (_ : IsMaximal (top t) (top (eraseTop s)))) (top s)
(_ : IsMaximal (top (snoc t (top (eraseTop s)) (_ : IsMaximal (top t) (top (eraseTop s))))) (top s)))
(snoc (snoc t x hmtx) (top s) (_ : IsMaximal (top (snoc t x hmtx)) (top s)))
[PROOFSTEP]
refine' Equivalent.snoc_snoc_swap _ _
[GOAL]
case h.refine'_1
X : Type u
inst✝¹ : Lattice X
inst✝ : JordanHolderLattice X
s✝ : CompositionSeries X
x✝¹ : X
hm✝ : IsMaximal x✝¹ (top s✝)
hb✝ : bot s✝ ≤ x✝¹
x✝ : ℕ
hn✝ : s✝.length = x✝
n : ℕ
ih :
∀ (s : CompositionSeries X) (x : X) (hm : IsMaximal x (top s)),
bot s ≤ x →
s.length = n →
∃ t, bot t = bot s ∧ t.length + 1 = n ∧ ∃ htx, Equivalent s (snoc t (top s) (_ : IsMaximal (top t) (top s)))
s : CompositionSeries X
x : X
hm : IsMaximal x (top s)
hb : bot s ≤ x
hn : s.length = Nat.succ n
h0s : 0 < s.length
hetx : ¬top (eraseTop s) = x
imxs : IsMaximal (x ⊓ top (eraseTop s)) (top (eraseTop s))
t : CompositionSeries X
htb : bot t = bot (eraseTop s)
htl : t.length + 1 = n
htt : top t = x ⊓ top (eraseTop s)
hteqv : Equivalent (eraseTop s) (snoc t (top (eraseTop s)) (_ : IsMaximal (top t) (top (eraseTop s))))
hmtx : IsMaximal (top t) x
this :
Equivalent s
(snoc (snoc t (top (eraseTop s)) (_ : IsMaximal (top t) (top (eraseTop s)))) (top s)
(_ : IsMaximal (top (snoc t (top (eraseTop s)) (_ : IsMaximal (top t) (top (eraseTop s))))) (top s)))
⊢ Iso (top t, top (eraseTop s)) (x, top s)
[PROOFSTEP]
exact iso_symm (second_iso_of_eq hm (sup_eq_of_isMaximal hm (isMaximal_eraseTop_top h0s) (Ne.symm hetx)) htt.symm)
[GOAL]
case h.refine'_2
X : Type u
inst✝¹ : Lattice X
inst✝ : JordanHolderLattice X
s✝ : CompositionSeries X
x✝¹ : X
hm✝ : IsMaximal x✝¹ (top s✝)
hb✝ : bot s✝ ≤ x✝¹
x✝ : ℕ
hn✝ : s✝.length = x✝
n : ℕ
ih :
∀ (s : CompositionSeries X) (x : X) (hm : IsMaximal x (top s)),
bot s ≤ x →
s.length = n →
∃ t, bot t = bot s ∧ t.length + 1 = n ∧ ∃ htx, Equivalent s (snoc t (top s) (_ : IsMaximal (top t) (top s)))
s : CompositionSeries X
x : X
hm : IsMaximal x (top s)
hb : bot s ≤ x
hn : s.length = Nat.succ n
h0s : 0 < s.length
hetx : ¬top (eraseTop s) = x
imxs : IsMaximal (x ⊓ top (eraseTop s)) (top (eraseTop s))
t : CompositionSeries X
htb : bot t = bot (eraseTop s)
htl : t.length + 1 = n
htt : top t = x ⊓ top (eraseTop s)
hteqv : Equivalent (eraseTop s) (snoc t (top (eraseTop s)) (_ : IsMaximal (top t) (top (eraseTop s))))
hmtx : IsMaximal (top t) x
this :
Equivalent s
(snoc (snoc t (top (eraseTop s)) (_ : IsMaximal (top t) (top (eraseTop s)))) (top s)
(_ : IsMaximal (top (snoc t (top (eraseTop s)) (_ : IsMaximal (top t) (top (eraseTop s))))) (top s)))
⊢ Iso (top (eraseTop s), top s) (top t, x)
[PROOFSTEP]
exact
second_iso_of_eq (isMaximal_eraseTop_top h0s) (sup_eq_of_isMaximal (isMaximal_eraseTop_top h0s) hm hetx)
(by rw [inf_comm, htt])
[GOAL]
X : Type u
inst✝¹ : Lattice X
inst✝ : JordanHolderLattice X
s✝ : CompositionSeries X
x✝¹ : X
hm✝ : IsMaximal x✝¹ (top s✝)
hb✝ : bot s✝ ≤ x✝¹
x✝ : ℕ
hn✝ : s✝.length = x✝
n : ℕ
ih :
∀ (s : CompositionSeries X) (x : X) (hm : IsMaximal x (top s)),
bot s ≤ x →
s.length = n →
∃ t, bot t = bot s ∧ t.length + 1 = n ∧ ∃ htx, Equivalent s (snoc t (top s) (_ : IsMaximal (top t) (top s)))
s : CompositionSeries X
x : X
hm : IsMaximal x (top s)
hb : bot s ≤ x
hn : s.length = Nat.succ n
h0s : 0 < s.length
hetx : ¬top (eraseTop s) = x
imxs : IsMaximal (x ⊓ top (eraseTop s)) (top (eraseTop s))
t : CompositionSeries X
htb : bot t = bot (eraseTop s)
htl : t.length + 1 = n
htt : top t = x ⊓ top (eraseTop s)
hteqv : Equivalent (eraseTop s) (snoc t (top (eraseTop s)) (_ : IsMaximal (top t) (top (eraseTop s))))
hmtx : IsMaximal (top t) x
this :
Equivalent s
(snoc (snoc t (top (eraseTop s)) (_ : IsMaximal (top t) (top (eraseTop s)))) (top s)
(_ : IsMaximal (top (snoc t (top (eraseTop s)) (_ : IsMaximal (top t) (top (eraseTop s))))) (top s)))
⊢ top (eraseTop s) ⊓ x = top t
[PROOFSTEP]
rw [inf_comm, htt]
[GOAL]
X : Type u
inst✝¹ : Lattice X
inst✝ : JordanHolderLattice X
s₁ s₂ : CompositionSeries X
hb : bot s₁ = bot s₂
ht : top s₁ = top s₂
⊢ Equivalent s₁ s₂
[PROOFSTEP]
induction' hle : s₁.length with n ih generalizing s₁ s₂
[GOAL]
case zero
X : Type u
inst✝¹ : Lattice X
inst✝ : JordanHolderLattice X
s₁✝ s₂✝ : CompositionSeries X
hb✝ : bot s₁✝ = bot s₂✝
ht✝ : top s₁✝ = top s₂✝
x✝ : ℕ
hle✝ : s₁✝.length = x✝
s₁ s₂ : CompositionSeries X
hb : bot s₁ = bot s₂
ht : top s₁ = top s₂
hle : s₁.length = Nat.zero
⊢ Equivalent s₁ s₂
[PROOFSTEP]
rw [eq_of_bot_eq_bot_of_top_eq_top_of_length_eq_zero hb ht hle]
[GOAL]
case succ
X : Type u
inst✝¹ : Lattice X
inst✝ : JordanHolderLattice X
s₁✝ s₂✝ : CompositionSeries X
hb✝ : bot s₁✝ = bot s₂✝
ht✝ : top s₁✝ = top s₂✝
x✝ : ℕ
hle✝ : s₁✝.length = x✝
n : ℕ
ih : ∀ (s₁ s₂ : CompositionSeries X), bot s₁ = bot s₂ → top s₁ = top s₂ → s₁.length = n → Equivalent s₁ s₂
s₁ s₂ : CompositionSeries X
hb : bot s₁ = bot s₂
ht : top s₁ = top s₂
hle : s₁.length = Nat.succ n
⊢ Equivalent s₁ s₂
[PROOFSTEP]
have h0s₂ : 0 < s₂.length := length_pos_of_bot_eq_bot_of_top_eq_top_of_length_pos hb ht (hle.symm ▸ Nat.succ_pos _)
[GOAL]
case succ
X : Type u
inst✝¹ : Lattice X
inst✝ : JordanHolderLattice X
s₁✝ s₂✝ : CompositionSeries X
hb✝ : bot s₁✝ = bot s₂✝
ht✝ : top s₁✝ = top s₂✝
x✝ : ℕ
hle✝ : s₁✝.length = x✝
n : ℕ
ih : ∀ (s₁ s₂ : CompositionSeries X), bot s₁ = bot s₂ → top s₁ = top s₂ → s₁.length = n → Equivalent s₁ s₂
s₁ s₂ : CompositionSeries X
hb : bot s₁ = bot s₂
ht : top s₁ = top s₂
hle : s₁.length = Nat.succ n
h0s₂ : 0 < s₂.length
⊢ Equivalent s₁ s₂
[PROOFSTEP]
rcases exists_top_eq_snoc_equivalant s₁ s₂.eraseTop.top (ht.symm ▸ isMaximal_eraseTop_top h0s₂)
(hb.symm ▸ s₂.bot_eraseTop ▸ bot_le_of_mem (top_mem _)) with
⟨t, htb, htl, htt, hteq⟩
[GOAL]
case succ.intro.intro.intro.intro
X : Type u
inst✝¹ : Lattice X
inst✝ : JordanHolderLattice X
s₁✝ s₂✝ : CompositionSeries X
hb✝ : bot s₁✝ = bot s₂✝
ht✝ : top s₁✝ = top s₂✝
x✝ : ℕ
hle✝ : s₁✝.length = x✝
n : ℕ
ih : ∀ (s₁ s₂ : CompositionSeries X), bot s₁ = bot s₂ → top s₁ = top s₂ → s₁.length = n → Equivalent s₁ s₂
s₁ s₂ : CompositionSeries X
hb : bot s₁ = bot s₂
ht : top s₁ = top s₂
hle : s₁.length = Nat.succ n
h0s₂ : 0 < s₂.length
t : CompositionSeries X
htb : bot t = bot s₁
htl : t.length + 1 = s₁.length
htt : top t = top (eraseTop s₂)
hteq : Equivalent s₁ (snoc t (top s₁) (_ : IsMaximal (top t) (top s₁)))
⊢ Equivalent s₁ s₂
[PROOFSTEP]
have := ih t s₂.eraseTop (by simp [htb, ← hb]) htt (Nat.succ_inj'.1 (htl.trans hle))
[GOAL]
X : Type u
inst✝¹ : Lattice X
inst✝ : JordanHolderLattice X
s₁✝ s₂✝ : CompositionSeries X
hb✝ : bot s₁✝ = bot s₂✝
ht✝ : top s₁✝ = top s₂✝
x✝ : ℕ
hle✝ : s₁✝.length = x✝
n : ℕ
ih : ∀ (s₁ s₂ : CompositionSeries X), bot s₁ = bot s₂ → top s₁ = top s₂ → s₁.length = n → Equivalent s₁ s₂
s₁ s₂ : CompositionSeries X
hb : bot s₁ = bot s₂
ht : top s₁ = top s₂
hle : s₁.length = Nat.succ n
h0s₂ : 0 < s₂.length
t : CompositionSeries X
htb : bot t = bot s₁
htl : t.length + 1 = s₁.length
htt : top t = top (eraseTop s₂)
hteq : Equivalent s₁ (snoc t (top s₁) (_ : IsMaximal (top t) (top s₁)))
⊢ bot t = bot (eraseTop s₂)
[PROOFSTEP]
simp [htb, ← hb]
[GOAL]
case succ.intro.intro.intro.intro
X : Type u
inst✝¹ : Lattice X
inst✝ : JordanHolderLattice X
s₁✝ s₂✝ : CompositionSeries X
hb✝ : bot s₁✝ = bot s₂✝
ht✝ : top s₁✝ = top s₂✝
x✝ : ℕ
hle✝ : s₁✝.length = x✝
n : ℕ
ih : ∀ (s₁ s₂ : CompositionSeries X), bot s₁ = bot s₂ → top s₁ = top s₂ → s₁.length = n → Equivalent s₁ s₂
s₁ s₂ : CompositionSeries X
hb : bot s₁ = bot s₂
ht : top s₁ = top s₂
hle : s₁.length = Nat.succ n
h0s₂ : 0 < s₂.length
t : CompositionSeries X
htb : bot t = bot s₁
htl : t.length + 1 = s₁.length
htt : top t = top (eraseTop s₂)
hteq : Equivalent s₁ (snoc t (top s₁) (_ : IsMaximal (top t) (top s₁)))
this : Equivalent t (eraseTop s₂)
⊢ Equivalent s₁ s₂
[PROOFSTEP]
refine' hteq.trans _
[GOAL]
case succ.intro.intro.intro.intro
X : Type u
inst✝¹ : Lattice X
inst✝ : JordanHolderLattice X
s₁✝ s₂✝ : CompositionSeries X
hb✝ : bot s₁✝ = bot s₂✝
ht✝ : top s₁✝ = top s₂✝
x✝ : ℕ
hle✝ : s₁✝.length = x✝
n : ℕ
ih : ∀ (s₁ s₂ : CompositionSeries X), bot s₁ = bot s₂ → top s₁ = top s₂ → s₁.length = n → Equivalent s₁ s₂
s₁ s₂ : CompositionSeries X
hb : bot s₁ = bot s₂
ht : top s₁ = top s₂
hle : s₁.length = Nat.succ n
h0s₂ : 0 < s₂.length
t : CompositionSeries X
htb : bot t = bot s₁
htl : t.length + 1 = s₁.length
htt : top t = top (eraseTop s₂)
hteq : Equivalent s₁ (snoc t (top s₁) (_ : IsMaximal (top t) (top s₁)))
this : Equivalent t (eraseTop s₂)
⊢ Equivalent (snoc t (top s₁) (_ : IsMaximal (top t) (top s₁))) s₂
[PROOFSTEP]
conv_rhs => rw [eq_snoc_eraseTop h0s₂]
[GOAL]
X : Type u
inst✝¹ : Lattice X
inst✝ : JordanHolderLattice X
s₁✝ s₂✝ : CompositionSeries X
hb✝ : bot s₁✝ = bot s₂✝
ht✝ : top s₁✝ = top s₂✝
x✝ : ℕ
hle✝ : s₁✝.length = x✝
n : ℕ
ih : ∀ (s₁ s₂ : CompositionSeries X), bot s₁ = bot s₂ → top s₁ = top s₂ → s₁.length = n → Equivalent s₁ s₂
s₁ s₂ : CompositionSeries X
hb : bot s₁ = bot s₂
ht : top s₁ = top s₂
hle : s₁.length = Nat.succ n
h0s₂ : 0 < s₂.length
t : CompositionSeries X
htb : bot t = bot s₁
htl : t.length + 1 = s₁.length
htt : top t = top (eraseTop s₂)
hteq : Equivalent s₁ (snoc t (top s₁) (_ : IsMaximal (top t) (top s₁)))
this : Equivalent t (eraseTop s₂)
| s₂
[PROOFSTEP]
rw [eq_snoc_eraseTop h0s₂]
[GOAL]
X : Type u
inst✝¹ : Lattice X
inst✝ : JordanHolderLattice X
s₁✝ s₂✝ : CompositionSeries X
hb✝ : bot s₁✝ = bot s₂✝
ht✝ : top s₁✝ = top s₂✝
x✝ : ℕ
hle✝ : s₁✝.length = x✝
n : ℕ
ih : ∀ (s₁ s₂ : CompositionSeries X), bot s₁ = bot s₂ → top s₁ = top s₂ → s₁.length = n → Equivalent s₁ s₂
s₁ s₂ : CompositionSeries X
hb : bot s₁ = bot s₂
ht : top s₁ = top s₂
hle : s₁.length = Nat.succ n
h0s₂ : 0 < s₂.length
t : CompositionSeries X
htb : bot t = bot s₁
htl : t.length + 1 = s₁.length
htt : top t = top (eraseTop s₂)
hteq : Equivalent s₁ (snoc t (top s₁) (_ : IsMaximal (top t) (top s₁)))
this : Equivalent t (eraseTop s₂)
| s₂
[PROOFSTEP]
rw [eq_snoc_eraseTop h0s₂]
[GOAL]
X : Type u
inst✝¹ : Lattice X
inst✝ : JordanHolderLattice X
s₁✝ s₂✝ : CompositionSeries X
hb✝ : bot s₁✝ = bot s₂✝
ht✝ : top s₁✝ = top s₂✝
x✝ : ℕ
hle✝ : s₁✝.length = x✝
n : ℕ
ih : ∀ (s₁ s₂ : CompositionSeries X), bot s₁ = bot s₂ → top s₁ = top s₂ → s₁.length = n → Equivalent s₁ s₂
s₁ s₂ : CompositionSeries X
hb : bot s₁ = bot s₂
ht : top s₁ = top s₂
hle : s₁.length = Nat.succ n
h0s₂ : 0 < s₂.length
t : CompositionSeries X
htb : bot t = bot s₁
htl : t.length + 1 = s₁.length
htt : top t = top (eraseTop s₂)
hteq : Equivalent s₁ (snoc t (top s₁) (_ : IsMaximal (top t) (top s₁)))
this : Equivalent t (eraseTop s₂)
| s₂
[PROOFSTEP]
rw [eq_snoc_eraseTop h0s₂]
[GOAL]
case succ.intro.intro.intro.intro
X : Type u
inst✝¹ : Lattice X
inst✝ : JordanHolderLattice X
s₁✝ s₂✝ : CompositionSeries X
hb✝ : bot s₁✝ = bot s₂✝
ht✝ : top s₁✝ = top s₂✝
x✝ : ℕ
hle✝ : s₁✝.length = x✝
n : ℕ
ih : ∀ (s₁ s₂ : CompositionSeries X), bot s₁ = bot s₂ → top s₁ = top s₂ → s₁.length = n → Equivalent s₁ s₂
s₁ s₂ : CompositionSeries X
hb : bot s₁ = bot s₂
ht : top s₁ = top s₂
hle : s₁.length = Nat.succ n
h0s₂ : 0 < s₂.length
t : CompositionSeries X
htb : bot t = bot s₁
htl : t.length + 1 = s₁.length
htt : top t = top (eraseTop s₂)
hteq : Equivalent s₁ (snoc t (top s₁) (_ : IsMaximal (top t) (top s₁)))
this : Equivalent t (eraseTop s₂)
⊢ Equivalent (snoc t (top s₁) (_ : IsMaximal (top t) (top s₁)))
(snoc (eraseTop s₂) (top s₂) (_ : IsMaximal (top (eraseTop s₂)) (top s₂)))
[PROOFSTEP]
simp only [ht]
[GOAL]
case succ.intro.intro.intro.intro
X : Type u
inst✝¹ : Lattice X
inst✝ : JordanHolderLattice X
s₁✝ s₂✝ : CompositionSeries X
hb✝ : bot s₁✝ = bot s₂✝
ht✝ : top s₁✝ = top s₂✝
x✝ : ℕ
hle✝ : s₁✝.length = x✝
n : ℕ
ih : ∀ (s₁ s₂ : CompositionSeries X), bot s₁ = bot s₂ → top s₁ = top s₂ → s₁.length = n → Equivalent s₁ s₂
s₁ s₂ : CompositionSeries X
hb : bot s₁ = bot s₂
ht : top s₁ = top s₂
hle : s₁.length = Nat.succ n
h0s₂ : 0 < s₂.length
t : CompositionSeries X
htb : bot t = bot s₁
htl : t.length + 1 = s₁.length
htt : top t = top (eraseTop s₂)
hteq : Equivalent s₁ (snoc t (top s₁) (_ : IsMaximal (top t) (top s₁)))
this : Equivalent t (eraseTop s₂)
⊢ Equivalent (snoc t (top s₂) (_ : IsMaximal (top t) (top s₂)))
(snoc (eraseTop s₂) (top s₂) (_ : IsMaximal (top (eraseTop s₂)) (top s₂)))
[PROOFSTEP]
exact Equivalent.snoc this (by simp [htt, (isMaximal_eraseTop_top h0s₂).iso_refl])
[GOAL]
X : Type u
inst✝¹ : Lattice X
inst✝ : JordanHolderLattice X
s₁✝ s₂✝ : CompositionSeries X
hb✝ : bot s₁✝ = bot s₂✝
ht✝ : top s₁✝ = top s₂✝
x✝ : ℕ
hle✝ : s₁✝.length = x✝
n : ℕ
ih : ∀ (s₁ s₂ : CompositionSeries X), bot s₁ = bot s₂ → top s₁ = top s₂ → s₁.length = n → Equivalent s₁ s₂
s₁ s₂ : CompositionSeries X
hb : bot s₁ = bot s₂
ht : top s₁ = top s₂
hle : s₁.length = Nat.succ n
h0s₂ : 0 < s₂.length
t : CompositionSeries X
htb : bot t = bot s₁
htl : t.length + 1 = s₁.length
htt : top t = top (eraseTop s₂)
hteq : Equivalent s₁ (snoc t (top s₁) (_ : IsMaximal (top t) (top s₁)))
this : Equivalent t (eraseTop s₂)
⊢ Iso (top t, top s₂) (top (eraseTop s₂), top s₂)
[PROOFSTEP]
simp [htt, (isMaximal_eraseTop_top h0s₂).iso_refl]
|
module Web.Internal.IndexedDBPrim
import JS
import Web.Internal.Types
%default total
--------------------------------------------------------------------------------
-- Interfaces
--------------------------------------------------------------------------------
namespace IDBCursor
export
%foreign "browser:lambda:x=>x.direction"
prim__direction : IDBCursor -> PrimIO String
export
%foreign "browser:lambda:x=>x.key"
prim__key : IDBCursor -> PrimIO AnyPtr
export
%foreign "browser:lambda:x=>x.primaryKey"
prim__primaryKey : IDBCursor -> PrimIO AnyPtr
export
%foreign "browser:lambda:x=>x.request"
prim__request : IDBCursor -> PrimIO IDBRequest
export
%foreign "browser:lambda:x=>x.source"
prim__source : IDBCursor -> PrimIO (Union2 IDBObjectStore IDBIndex)
export
%foreign "browser:lambda:(x,a)=>x.advance(a)"
prim__advance : IDBCursor -> Bits32 -> PrimIO ()
export
%foreign "browser:lambda:(x,a)=>x.continue(a)"
prim__continue : IDBCursor -> UndefOr AnyPtr -> PrimIO ()
export
%foreign "browser:lambda:(x,a,b)=>x.continuePrimaryKey(a,b)"
prim__continuePrimaryKey : IDBCursor -> AnyPtr -> AnyPtr -> PrimIO ()
export
%foreign "browser:lambda:x=>x.delete()"
prim__delete : IDBCursor -> PrimIO IDBRequest
export
%foreign "browser:lambda:(x,a)=>x.update(a)"
prim__update : IDBCursor -> AnyPtr -> PrimIO IDBRequest
namespace IDBCursorWithValue
export
%foreign "browser:lambda:x=>x.value"
prim__value : IDBCursorWithValue -> PrimIO AnyPtr
namespace IDBDatabase
export
%foreign "browser:lambda:x=>x.name"
prim__name : IDBDatabase -> PrimIO String
export
%foreign "browser:lambda:x=>x.objectStoreNames"
prim__objectStoreNames : IDBDatabase -> PrimIO DOMStringList
export
%foreign "browser:lambda:x=>x.onabort"
prim__onabort : IDBDatabase -> PrimIO (Nullable EventHandlerNonNull)
export
%foreign "browser:lambda:(x,v)=>{x.onabort = v}"
prim__setOnabort : IDBDatabase -> Nullable EventHandlerNonNull -> PrimIO ()
export
%foreign "browser:lambda:x=>x.onclose"
prim__onclose : IDBDatabase -> PrimIO (Nullable EventHandlerNonNull)
export
%foreign "browser:lambda:(x,v)=>{x.onclose = v}"
prim__setOnclose : IDBDatabase -> Nullable EventHandlerNonNull -> PrimIO ()
export
%foreign "browser:lambda:x=>x.onerror"
prim__onerror : IDBDatabase -> PrimIO (Nullable EventHandlerNonNull)
export
%foreign "browser:lambda:(x,v)=>{x.onerror = v}"
prim__setOnerror : IDBDatabase -> Nullable EventHandlerNonNull -> PrimIO ()
export
%foreign "browser:lambda:x=>x.onversionchange"
prim__onversionchange : IDBDatabase -> PrimIO (Nullable EventHandlerNonNull)
export
%foreign "browser:lambda:(x,v)=>{x.onversionchange = v}"
prim__setOnversionchange : IDBDatabase
-> Nullable EventHandlerNonNull
-> PrimIO ()
export
%foreign "browser:lambda:x=>x.version"
prim__version : IDBDatabase -> PrimIO JSBits64
export
%foreign "browser:lambda:x=>x.close()"
prim__close : IDBDatabase -> PrimIO ()
export
%foreign "browser:lambda:(x,a,b)=>x.createObjectStore(a,b)"
prim__createObjectStore : IDBDatabase
-> String
-> UndefOr IDBObjectStoreParameters
-> PrimIO IDBObjectStore
export
%foreign "browser:lambda:(x,a)=>x.deleteObjectStore(a)"
prim__deleteObjectStore : IDBDatabase -> String -> PrimIO ()
export
%foreign "browser:lambda:(x,a,b,c)=>x.transaction(a,b,c)"
prim__transaction : IDBDatabase
-> Union2 String (Array String)
-> UndefOr String
-> UndefOr IDBTransactionOptions
-> PrimIO IDBTransaction
namespace IDBFactory
export
%foreign "browser:lambda:(x,a,b)=>x.cmp(a,b)"
prim__cmp : IDBFactory -> AnyPtr -> AnyPtr -> PrimIO Int16
export
%foreign "browser:lambda:x=>x.databases()"
prim__databases : IDBFactory -> PrimIO (Promise (Array IDBDatabaseInfo))
export
%foreign "browser:lambda:(x,a)=>x.deleteDatabase(a)"
prim__deleteDatabase : IDBFactory -> String -> PrimIO IDBOpenDBRequest
export
%foreign "browser:lambda:(x,a,b)=>x.open(a,b)"
prim__open : IDBFactory
-> String
-> UndefOr JSBits64
-> PrimIO IDBOpenDBRequest
namespace IDBIndex
export
%foreign "browser:lambda:x=>x.keyPath"
prim__keyPath : IDBIndex -> PrimIO AnyPtr
export
%foreign "browser:lambda:x=>x.multiEntry"
prim__multiEntry : IDBIndex -> PrimIO Boolean
export
%foreign "browser:lambda:x=>x.name"
prim__name : IDBIndex -> PrimIO String
export
%foreign "browser:lambda:(x,v)=>{x.name = v}"
prim__setName : IDBIndex -> String -> PrimIO ()
export
%foreign "browser:lambda:x=>x.objectStore"
prim__objectStore : IDBIndex -> PrimIO IDBObjectStore
export
%foreign "browser:lambda:x=>x.unique"
prim__unique : IDBIndex -> PrimIO Boolean
export
%foreign "browser:lambda:(x,a)=>x.count(a)"
prim__count : IDBIndex -> UndefOr AnyPtr -> PrimIO IDBRequest
export
%foreign "browser:lambda:(x,a,b)=>x.getAll(a,b)"
prim__getAll : IDBIndex
-> UndefOr AnyPtr
-> UndefOr Bits32
-> PrimIO IDBRequest
export
%foreign "browser:lambda:(x,a,b)=>x.getAllKeys(a,b)"
prim__getAllKeys : IDBIndex
-> UndefOr AnyPtr
-> UndefOr Bits32
-> PrimIO IDBRequest
export
%foreign "browser:lambda:(x,a)=>x.get(a)"
prim__get : IDBIndex -> AnyPtr -> PrimIO IDBRequest
export
%foreign "browser:lambda:(x,a)=>x.getKey(a)"
prim__getKey : IDBIndex -> AnyPtr -> PrimIO IDBRequest
export
%foreign "browser:lambda:(x,a,b)=>x.openCursor(a,b)"
prim__openCursor : IDBIndex
-> UndefOr AnyPtr
-> UndefOr String
-> PrimIO IDBRequest
export
%foreign "browser:lambda:(x,a,b)=>x.openKeyCursor(a,b)"
prim__openKeyCursor : IDBIndex
-> UndefOr AnyPtr
-> UndefOr String
-> PrimIO IDBRequest
namespace IDBKeyRange
export
%foreign "browser:lambda:(a,b,c,d)=>IDBKeyRange.bound(a,b,c,d)"
prim__bound : AnyPtr
-> AnyPtr
-> UndefOr Boolean
-> UndefOr Boolean
-> PrimIO IDBKeyRange
export
%foreign "browser:lambda:(a,b)=>IDBKeyRange.lowerBound(a,b)"
prim__lowerBound : AnyPtr -> UndefOr Boolean -> PrimIO IDBKeyRange
export
%foreign "browser:lambda:(a)=>IDBKeyRange.only(a)"
prim__only : AnyPtr -> PrimIO IDBKeyRange
export
%foreign "browser:lambda:(a,b)=>IDBKeyRange.upperBound(a,b)"
prim__upperBound : AnyPtr -> UndefOr Boolean -> PrimIO IDBKeyRange
export
%foreign "browser:lambda:x=>x.lower"
prim__lower : IDBKeyRange -> PrimIO AnyPtr
export
%foreign "browser:lambda:x=>x.lowerOpen"
prim__lowerOpen : IDBKeyRange -> PrimIO Boolean
export
%foreign "browser:lambda:x=>x.upper"
prim__upper : IDBKeyRange -> PrimIO AnyPtr
export
%foreign "browser:lambda:x=>x.upperOpen"
prim__upperOpen : IDBKeyRange -> PrimIO Boolean
export
%foreign "browser:lambda:(x,a)=>x.includes(a)"
prim__includes : IDBKeyRange -> AnyPtr -> PrimIO Boolean
namespace IDBObjectStore
export
%foreign "browser:lambda:x=>x.autoIncrement"
prim__autoIncrement : IDBObjectStore -> PrimIO Boolean
export
%foreign "browser:lambda:x=>x.indexNames"
prim__indexNames : IDBObjectStore -> PrimIO DOMStringList
export
%foreign "browser:lambda:x=>x.keyPath"
prim__keyPath : IDBObjectStore -> PrimIO AnyPtr
export
%foreign "browser:lambda:x=>x.name"
prim__name : IDBObjectStore -> PrimIO String
export
%foreign "browser:lambda:(x,v)=>{x.name = v}"
prim__setName : IDBObjectStore -> String -> PrimIO ()
export
%foreign "browser:lambda:x=>x.transaction"
prim__transaction : IDBObjectStore -> PrimIO IDBTransaction
export
%foreign "browser:lambda:(x,a,b)=>x.add(a,b)"
prim__add : IDBObjectStore -> AnyPtr -> UndefOr AnyPtr -> PrimIO IDBRequest
export
%foreign "browser:lambda:x=>x.clear()"
prim__clear : IDBObjectStore -> PrimIO IDBRequest
export
%foreign "browser:lambda:(x,a)=>x.count(a)"
prim__count : IDBObjectStore -> UndefOr AnyPtr -> PrimIO IDBRequest
export
%foreign "browser:lambda:(x,a,b,c)=>x.createIndex(a,b,c)"
prim__createIndex : IDBObjectStore
-> String
-> Union2 String (Array String)
-> UndefOr IDBIndexParameters
-> PrimIO IDBIndex
export
%foreign "browser:lambda:(x,a)=>x.delete(a)"
prim__delete : IDBObjectStore -> AnyPtr -> PrimIO IDBRequest
export
%foreign "browser:lambda:(x,a)=>x.deleteIndex(a)"
prim__deleteIndex : IDBObjectStore -> String -> PrimIO ()
export
%foreign "browser:lambda:(x,a,b)=>x.getAll(a,b)"
prim__getAll : IDBObjectStore
-> UndefOr AnyPtr
-> UndefOr Bits32
-> PrimIO IDBRequest
export
%foreign "browser:lambda:(x,a,b)=>x.getAllKeys(a,b)"
prim__getAllKeys : IDBObjectStore
-> UndefOr AnyPtr
-> UndefOr Bits32
-> PrimIO IDBRequest
export
%foreign "browser:lambda:(x,a)=>x.get(a)"
prim__get : IDBObjectStore -> AnyPtr -> PrimIO IDBRequest
export
%foreign "browser:lambda:(x,a)=>x.getKey(a)"
prim__getKey : IDBObjectStore -> AnyPtr -> PrimIO IDBRequest
export
%foreign "browser:lambda:(x,a)=>x.index(a)"
prim__index : IDBObjectStore -> String -> PrimIO IDBIndex
export
%foreign "browser:lambda:(x,a,b)=>x.openCursor(a,b)"
prim__openCursor : IDBObjectStore
-> UndefOr AnyPtr
-> UndefOr String
-> PrimIO IDBRequest
export
%foreign "browser:lambda:(x,a,b)=>x.openKeyCursor(a,b)"
prim__openKeyCursor : IDBObjectStore
-> UndefOr AnyPtr
-> UndefOr String
-> PrimIO IDBRequest
export
%foreign "browser:lambda:(x,a,b)=>x.put(a,b)"
prim__put : IDBObjectStore -> AnyPtr -> UndefOr AnyPtr -> PrimIO IDBRequest
namespace IDBOpenDBRequest
export
%foreign "browser:lambda:x=>x.onblocked"
prim__onblocked : IDBOpenDBRequest -> PrimIO (Nullable EventHandlerNonNull)
export
%foreign "browser:lambda:(x,v)=>{x.onblocked = v}"
prim__setOnblocked : IDBOpenDBRequest
-> Nullable EventHandlerNonNull
-> PrimIO ()
export
%foreign "browser:lambda:x=>x.onupgradeneeded"
prim__onupgradeneeded : IDBOpenDBRequest
-> PrimIO (Nullable EventHandlerNonNull)
export
%foreign "browser:lambda:(x,v)=>{x.onupgradeneeded = v}"
prim__setOnupgradeneeded : IDBOpenDBRequest
-> Nullable EventHandlerNonNull
-> PrimIO ()
namespace IDBRequest
export
%foreign "browser:lambda:x=>x.error"
prim__error : IDBRequest -> PrimIO (Nullable DOMException)
export
%foreign "browser:lambda:x=>x.onerror"
prim__onerror : IDBRequest -> PrimIO (Nullable EventHandlerNonNull)
export
%foreign "browser:lambda:(x,v)=>{x.onerror = v}"
prim__setOnerror : IDBRequest -> Nullable EventHandlerNonNull -> PrimIO ()
export
%foreign "browser:lambda:x=>x.onsuccess"
prim__onsuccess : IDBRequest -> PrimIO (Nullable EventHandlerNonNull)
export
%foreign "browser:lambda:(x,v)=>{x.onsuccess = v}"
prim__setOnsuccess : IDBRequest -> Nullable EventHandlerNonNull -> PrimIO ()
export
%foreign "browser:lambda:x=>x.readyState"
prim__readyState : IDBRequest -> PrimIO String
export
%foreign "browser:lambda:x=>x.result"
prim__result : IDBRequest -> PrimIO AnyPtr
export
%foreign "browser:lambda:x=>x.source"
prim__source : IDBRequest
-> PrimIO (Nullable (Union3 IDBObjectStore IDBIndex IDBCursor))
export
%foreign "browser:lambda:x=>x.transaction"
prim__transaction : IDBRequest -> PrimIO (Nullable IDBTransaction)
namespace IDBTransaction
export
%foreign "browser:lambda:x=>x.db"
prim__db : IDBTransaction -> PrimIO IDBDatabase
export
%foreign "browser:lambda:x=>x.durability"
prim__durability : IDBTransaction -> PrimIO String
export
%foreign "browser:lambda:x=>x.error"
prim__error : IDBTransaction -> PrimIO (Nullable DOMException)
export
%foreign "browser:lambda:x=>x.mode"
prim__mode : IDBTransaction -> PrimIO String
export
%foreign "browser:lambda:x=>x.objectStoreNames"
prim__objectStoreNames : IDBTransaction -> PrimIO DOMStringList
export
%foreign "browser:lambda:x=>x.onabort"
prim__onabort : IDBTransaction -> PrimIO (Nullable EventHandlerNonNull)
export
%foreign "browser:lambda:(x,v)=>{x.onabort = v}"
prim__setOnabort : IDBTransaction -> Nullable EventHandlerNonNull -> PrimIO ()
export
%foreign "browser:lambda:x=>x.oncomplete"
prim__oncomplete : IDBTransaction -> PrimIO (Nullable EventHandlerNonNull)
export
%foreign "browser:lambda:(x,v)=>{x.oncomplete = v}"
prim__setOncomplete : IDBTransaction
-> Nullable EventHandlerNonNull
-> PrimIO ()
export
%foreign "browser:lambda:x=>x.onerror"
prim__onerror : IDBTransaction -> PrimIO (Nullable EventHandlerNonNull)
export
%foreign "browser:lambda:(x,v)=>{x.onerror = v}"
prim__setOnerror : IDBTransaction -> Nullable EventHandlerNonNull -> PrimIO ()
export
%foreign "browser:lambda:x=>x.abort()"
prim__abort : IDBTransaction -> PrimIO ()
export
%foreign "browser:lambda:x=>x.commit()"
prim__commit : IDBTransaction -> PrimIO ()
export
%foreign "browser:lambda:(x,a)=>x.objectStore(a)"
prim__objectStore : IDBTransaction -> String -> PrimIO IDBObjectStore
namespace IDBVersionChangeEvent
export
%foreign "browser:lambda:(a,b)=> new IDBVersionChangeEvent(a,b)"
prim__new : String
-> UndefOr IDBVersionChangeEventInit
-> PrimIO IDBVersionChangeEvent
export
%foreign "browser:lambda:x=>x.newVersion"
prim__newVersion : IDBVersionChangeEvent -> PrimIO (Nullable JSBits64)
export
%foreign "browser:lambda:x=>x.oldVersion"
prim__oldVersion : IDBVersionChangeEvent -> PrimIO JSBits64
--------------------------------------------------------------------------------
-- Dictionaries
--------------------------------------------------------------------------------
namespace IDBDatabaseInfo
export
%foreign "browser:lambda:(a,b)=> ({name: a,version: b})"
prim__new : UndefOr String -> UndefOr JSBits64 -> PrimIO IDBDatabaseInfo
export
%foreign "browser:lambda:x=>x.name"
prim__name : IDBDatabaseInfo -> PrimIO (UndefOr String)
export
%foreign "browser:lambda:(x,v)=>{x.name = v}"
prim__setName : IDBDatabaseInfo -> UndefOr String -> PrimIO ()
export
%foreign "browser:lambda:x=>x.version"
prim__version : IDBDatabaseInfo -> PrimIO (UndefOr JSBits64)
export
%foreign "browser:lambda:(x,v)=>{x.version = v}"
prim__setVersion : IDBDatabaseInfo -> UndefOr JSBits64 -> PrimIO ()
namespace IDBIndexParameters
export
%foreign "browser:lambda:(a,b)=> ({unique: a,multiEntry: b})"
prim__new : UndefOr Boolean -> UndefOr Boolean -> PrimIO IDBIndexParameters
export
%foreign "browser:lambda:x=>x.multiEntry"
prim__multiEntry : IDBIndexParameters -> PrimIO (UndefOr Boolean)
export
%foreign "browser:lambda:(x,v)=>{x.multiEntry = v}"
prim__setMultiEntry : IDBIndexParameters -> UndefOr Boolean -> PrimIO ()
export
%foreign "browser:lambda:x=>x.unique"
prim__unique : IDBIndexParameters -> PrimIO (UndefOr Boolean)
export
%foreign "browser:lambda:(x,v)=>{x.unique = v}"
prim__setUnique : IDBIndexParameters -> UndefOr Boolean -> PrimIO ()
namespace IDBObjectStoreParameters
export
%foreign "browser:lambda:(a,b)=> ({keyPath: a,autoIncrement: b})"
prim__new : UndefOr (Nullable (Union2 String (Array String)))
-> UndefOr Boolean
-> PrimIO IDBObjectStoreParameters
export
%foreign "browser:lambda:x=>x.autoIncrement"
prim__autoIncrement : IDBObjectStoreParameters -> PrimIO (UndefOr Boolean)
export
%foreign "browser:lambda:(x,v)=>{x.autoIncrement = v}"
prim__setAutoIncrement : IDBObjectStoreParameters
-> UndefOr Boolean
-> PrimIO ()
export
%foreign "browser:lambda:x=>x.keyPath"
prim__keyPath : IDBObjectStoreParameters
-> PrimIO (UndefOr (Nullable (Union2 String (Array String))))
export
%foreign "browser:lambda:(x,v)=>{x.keyPath = v}"
prim__setKeyPath : IDBObjectStoreParameters
-> UndefOr (Nullable (Union2 String (Array String)))
-> PrimIO ()
namespace IDBTransactionOptions
export
%foreign "browser:lambda:(a)=> ({durability: a})"
prim__new : UndefOr String -> PrimIO IDBTransactionOptions
export
%foreign "browser:lambda:x=>x.durability"
prim__durability : IDBTransactionOptions -> PrimIO (UndefOr String)
export
%foreign "browser:lambda:(x,v)=>{x.durability = v}"
prim__setDurability : IDBTransactionOptions -> UndefOr String -> PrimIO ()
namespace IDBVersionChangeEventInit
export
%foreign "browser:lambda:(a,b)=> ({oldVersion: a,newVersion: b})"
prim__new : UndefOr JSBits64
-> UndefOr (Nullable JSBits64)
-> PrimIO IDBVersionChangeEventInit
export
%foreign "browser:lambda:x=>x.newVersion"
prim__newVersion : IDBVersionChangeEventInit
-> PrimIO (UndefOr (Nullable JSBits64))
export
%foreign "browser:lambda:(x,v)=>{x.newVersion = v}"
prim__setNewVersion : IDBVersionChangeEventInit
-> UndefOr (Nullable JSBits64)
-> PrimIO ()
export
%foreign "browser:lambda:x=>x.oldVersion"
prim__oldVersion : IDBVersionChangeEventInit -> PrimIO (UndefOr JSBits64)
export
%foreign "browser:lambda:(x,v)=>{x.oldVersion = v}"
prim__setOldVersion : IDBVersionChangeEventInit
-> UndefOr JSBits64
-> PrimIO ()
|
Formal statement is: lemma emeasure_countably_additive: "countably_additive (sets M) (emeasure M)" Informal statement is: The measure $\mu$ is countably additive. |
(* Title: HOL/Auth/n_germanSimp_lemma_on_inv__38.thy
Author: Yongjian Li and Kaiqiang Duan, State Key Lab of Computer Science, Institute of Software, Chinese Academy of Sciences
Copyright 2016 State Key Lab of Computer Science, Institute of Software, Chinese Academy of Sciences
*)
header{*The n_germanSimp Protocol Case Study*}
theory n_germanSimp_lemma_on_inv__38 imports n_germanSimp_base
begin
section{*All lemmas on causal relation between inv__38 and some rule r*}
lemma n_SendInv__part__0Vsinv__38:
assumes a1: "(\<exists> i. i\<le>N\<and>r=n_SendInv__part__0 i)" and
a2: "(\<exists> p__Inv4. p__Inv4\<le>N\<and>f=inv__38 p__Inv4)"
shows "invHoldForRule s f r (invariants N)" (is "?P1 s \<or> ?P2 s \<or> ?P3 s")
proof -
from a1 obtain i where a1:"i\<le>N\<and>r=n_SendInv__part__0 i" apply fastforce done
from a2 obtain p__Inv4 where a2:"p__Inv4\<le>N\<and>f=inv__38 p__Inv4" apply fastforce done
have "(i=p__Inv4)\<or>(i~=p__Inv4)" apply (cut_tac a1 a2, auto) done
moreover {
assume b1: "(i=p__Inv4)"
have "?P1 s"
proof(cut_tac a1 a2 b1, auto) qed
then have "invHoldForRule s f r (invariants N)" by auto
}
moreover {
assume b1: "(i~=p__Inv4)"
have "?P2 s"
proof(cut_tac a1 a2 b1, auto) qed
then have "invHoldForRule s f r (invariants N)" by auto
}
ultimately show "invHoldForRule s f r (invariants N)" by satx
qed
lemma n_SendInv__part__1Vsinv__38:
assumes a1: "(\<exists> i. i\<le>N\<and>r=n_SendInv__part__1 i)" and
a2: "(\<exists> p__Inv4. p__Inv4\<le>N\<and>f=inv__38 p__Inv4)"
shows "invHoldForRule s f r (invariants N)" (is "?P1 s \<or> ?P2 s \<or> ?P3 s")
proof -
from a1 obtain i where a1:"i\<le>N\<and>r=n_SendInv__part__1 i" apply fastforce done
from a2 obtain p__Inv4 where a2:"p__Inv4\<le>N\<and>f=inv__38 p__Inv4" apply fastforce done
have "(i=p__Inv4)\<or>(i~=p__Inv4)" apply (cut_tac a1 a2, auto) done
moreover {
assume b1: "(i=p__Inv4)"
have "?P1 s"
proof(cut_tac a1 a2 b1, auto) qed
then have "invHoldForRule s f r (invariants N)" by auto
}
moreover {
assume b1: "(i~=p__Inv4)"
have "?P2 s"
proof(cut_tac a1 a2 b1, auto) qed
then have "invHoldForRule s f r (invariants N)" by auto
}
ultimately show "invHoldForRule s f r (invariants N)" by satx
qed
lemma n_SendInvAckVsinv__38:
assumes a1: "(\<exists> i. i\<le>N\<and>r=n_SendInvAck i)" and
a2: "(\<exists> p__Inv4. p__Inv4\<le>N\<and>f=inv__38 p__Inv4)"
shows "invHoldForRule s f r (invariants N)" (is "?P1 s \<or> ?P2 s \<or> ?P3 s")
proof -
from a1 obtain i where a1:"i\<le>N\<and>r=n_SendInvAck i" apply fastforce done
from a2 obtain p__Inv4 where a2:"p__Inv4\<le>N\<and>f=inv__38 p__Inv4" apply fastforce done
have "(i=p__Inv4)\<or>(i~=p__Inv4)" apply (cut_tac a1 a2, auto) done
moreover {
assume b1: "(i=p__Inv4)"
have "?P1 s"
proof(cut_tac a1 a2 b1, auto) qed
then have "invHoldForRule s f r (invariants N)" by auto
}
moreover {
assume b1: "(i~=p__Inv4)"
have "?P2 s"
proof(cut_tac a1 a2 b1, auto) qed
then have "invHoldForRule s f r (invariants N)" by auto
}
ultimately show "invHoldForRule s f r (invariants N)" by satx
qed
lemma n_RecvInvAckVsinv__38:
assumes a1: "(\<exists> i. i\<le>N\<and>r=n_RecvInvAck i)" and
a2: "(\<exists> p__Inv4. p__Inv4\<le>N\<and>f=inv__38 p__Inv4)"
shows "invHoldForRule s f r (invariants N)" (is "?P1 s \<or> ?P2 s \<or> ?P3 s")
proof -
from a1 obtain i where a1:"i\<le>N\<and>r=n_RecvInvAck i" apply fastforce done
from a2 obtain p__Inv4 where a2:"p__Inv4\<le>N\<and>f=inv__38 p__Inv4" apply fastforce done
have "(i=p__Inv4)\<or>(i~=p__Inv4)" apply (cut_tac a1 a2, auto) done
moreover {
assume b1: "(i=p__Inv4)"
have "((formEval (eqn (IVar (Ident ''ExGntd'')) (Const true)) s))\<or>((formEval (neg (eqn (IVar (Ident ''ExGntd'')) (Const true))) s))" by auto
moreover {
assume c1: "((formEval (eqn (IVar (Ident ''ExGntd'')) (Const true)) s))"
have "?P1 s"
proof(cut_tac a1 a2 b1 c1, auto) qed
then have "invHoldForRule s f r (invariants N)" by auto
}
moreover {
assume c1: "((formEval (neg (eqn (IVar (Ident ''ExGntd'')) (Const true))) s))"
have "?P2 s"
proof(cut_tac a1 a2 b1 c1, auto) qed
then have "invHoldForRule s f r (invariants N)" by auto
}
ultimately have "invHoldForRule s f r (invariants N)" by satx
}
moreover {
assume b1: "(i~=p__Inv4)"
have "((formEval (eqn (IVar (Ident ''ExGntd'')) (Const true)) s))\<or>((formEval (neg (eqn (IVar (Ident ''ExGntd'')) (Const true))) s))" by auto
moreover {
assume c1: "((formEval (eqn (IVar (Ident ''ExGntd'')) (Const true)) s))"
have "?P1 s"
proof(cut_tac a1 a2 b1 c1, auto) qed
then have "invHoldForRule s f r (invariants N)" by auto
}
moreover {
assume c1: "((formEval (neg (eqn (IVar (Ident ''ExGntd'')) (Const true))) s))"
have "?P2 s"
proof(cut_tac a1 a2 b1 c1, auto) qed
then have "invHoldForRule s f r (invariants N)" by auto
}
ultimately have "invHoldForRule s f r (invariants N)" by satx
}
ultimately show "invHoldForRule s f r (invariants N)" by satx
qed
lemma n_SendGntSVsinv__38:
assumes a1: "(\<exists> i. i\<le>N\<and>r=n_SendGntS i)" and
a2: "(\<exists> p__Inv4. p__Inv4\<le>N\<and>f=inv__38 p__Inv4)"
shows "invHoldForRule s f r (invariants N)" (is "?P1 s \<or> ?P2 s \<or> ?P3 s")
proof -
from a1 obtain i where a1:"i\<le>N\<and>r=n_SendGntS i" apply fastforce done
from a2 obtain p__Inv4 where a2:"p__Inv4\<le>N\<and>f=inv__38 p__Inv4" apply fastforce done
have "(i=p__Inv4)\<or>(i~=p__Inv4)" apply (cut_tac a1 a2, auto) done
moreover {
assume b1: "(i=p__Inv4)"
have "?P1 s"
proof(cut_tac a1 a2 b1, auto) qed
then have "invHoldForRule s f r (invariants N)" by auto
}
moreover {
assume b1: "(i~=p__Inv4)"
have "?P2 s"
proof(cut_tac a1 a2 b1, auto) qed
then have "invHoldForRule s f r (invariants N)" by auto
}
ultimately show "invHoldForRule s f r (invariants N)" by satx
qed
lemma n_SendGntEVsinv__38:
assumes a1: "(\<exists> i. i\<le>N\<and>r=n_SendGntE N i)" and
a2: "(\<exists> p__Inv4. p__Inv4\<le>N\<and>f=inv__38 p__Inv4)"
shows "invHoldForRule s f r (invariants N)" (is "?P1 s \<or> ?P2 s \<or> ?P3 s")
proof -
from a1 obtain i where a1:"i\<le>N\<and>r=n_SendGntE N i" apply fastforce done
from a2 obtain p__Inv4 where a2:"p__Inv4\<le>N\<and>f=inv__38 p__Inv4" apply fastforce done
have "(i=p__Inv4)\<or>(i~=p__Inv4)" apply (cut_tac a1 a2, auto) done
moreover {
assume b1: "(i=p__Inv4)"
have "?P1 s"
proof(cut_tac a1 a2 b1, auto) qed
then have "invHoldForRule s f r (invariants N)" by auto
}
moreover {
assume b1: "(i~=p__Inv4)"
have "?P3 s"
apply (cut_tac a1 a2 b1, simp, rule_tac x="(neg (andForm (eqn (IVar (Field (Para (Ident ''Chan2'') p__Inv4) ''Cmd'')) (Const GntS)) (eqn (IVar (Para (Ident ''ShrSet'') p__Inv4)) (Const false))))" in exI, auto) done
then have "invHoldForRule s f r (invariants N)" by auto
}
ultimately show "invHoldForRule s f r (invariants N)" by satx
qed
lemma n_RecvGntSVsinv__38:
assumes a1: "(\<exists> i. i\<le>N\<and>r=n_RecvGntS i)" and
a2: "(\<exists> p__Inv4. p__Inv4\<le>N\<and>f=inv__38 p__Inv4)"
shows "invHoldForRule s f r (invariants N)" (is "?P1 s \<or> ?P2 s \<or> ?P3 s")
proof -
from a1 obtain i where a1:"i\<le>N\<and>r=n_RecvGntS i" apply fastforce done
from a2 obtain p__Inv4 where a2:"p__Inv4\<le>N\<and>f=inv__38 p__Inv4" apply fastforce done
have "(i=p__Inv4)\<or>(i~=p__Inv4)" apply (cut_tac a1 a2, auto) done
moreover {
assume b1: "(i=p__Inv4)"
have "?P1 s"
proof(cut_tac a1 a2 b1, auto) qed
then have "invHoldForRule s f r (invariants N)" by auto
}
moreover {
assume b1: "(i~=p__Inv4)"
have "?P2 s"
proof(cut_tac a1 a2 b1, auto) qed
then have "invHoldForRule s f r (invariants N)" by auto
}
ultimately show "invHoldForRule s f r (invariants N)" by satx
qed
lemma n_RecvGntEVsinv__38:
assumes a1: "(\<exists> i. i\<le>N\<and>r=n_RecvGntE i)" and
a2: "(\<exists> p__Inv4. p__Inv4\<le>N\<and>f=inv__38 p__Inv4)"
shows "invHoldForRule s f r (invariants N)" (is "?P1 s \<or> ?P2 s \<or> ?P3 s")
proof -
from a1 obtain i where a1:"i\<le>N\<and>r=n_RecvGntE i" apply fastforce done
from a2 obtain p__Inv4 where a2:"p__Inv4\<le>N\<and>f=inv__38 p__Inv4" apply fastforce done
have "(i=p__Inv4)\<or>(i~=p__Inv4)" apply (cut_tac a1 a2, auto) done
moreover {
assume b1: "(i=p__Inv4)"
have "?P1 s"
proof(cut_tac a1 a2 b1, auto) qed
then have "invHoldForRule s f r (invariants N)" by auto
}
moreover {
assume b1: "(i~=p__Inv4)"
have "?P2 s"
proof(cut_tac a1 a2 b1, auto) qed
then have "invHoldForRule s f r (invariants N)" by auto
}
ultimately show "invHoldForRule s f r (invariants N)" by satx
qed
lemma n_StoreVsinv__38:
assumes a1: "\<exists> i d. i\<le>N\<and>d\<le>N\<and>r=n_Store i d" and
a2: "(\<exists> p__Inv4. p__Inv4\<le>N\<and>f=inv__38 p__Inv4)"
shows "invHoldForRule s f r (invariants N)"
apply (rule noEffectOnRule, cut_tac a1 a2, auto) done
lemma n_RecvReqE__part__0Vsinv__38:
assumes a1: "\<exists> i. i\<le>N\<and>r=n_RecvReqE__part__0 N i" and
a2: "(\<exists> p__Inv4. p__Inv4\<le>N\<and>f=inv__38 p__Inv4)"
shows "invHoldForRule s f r (invariants N)"
apply (rule noEffectOnRule, cut_tac a1 a2, auto) done
lemma n_RecvReqE__part__1Vsinv__38:
assumes a1: "\<exists> i. i\<le>N\<and>r=n_RecvReqE__part__1 N i" and
a2: "(\<exists> p__Inv4. p__Inv4\<le>N\<and>f=inv__38 p__Inv4)"
shows "invHoldForRule s f r (invariants N)"
apply (rule noEffectOnRule, cut_tac a1 a2, auto) done
lemma n_RecvReqSVsinv__38:
assumes a1: "\<exists> i. i\<le>N\<and>r=n_RecvReqS N i" and
a2: "(\<exists> p__Inv4. p__Inv4\<le>N\<and>f=inv__38 p__Inv4)"
shows "invHoldForRule s f r (invariants N)"
apply (rule noEffectOnRule, cut_tac a1 a2, auto) done
end
|
# -*- coding: utf-8 -*-
"""
Created on Sun Dec 24 17:48:22 2017
@author: lee
"""
import numpy as np
import scipy.linalg as la
import numpy.linalg as na
import os
import aaweights
import sys
def ROPE(S, rho):
p=S.shape[0]
S=S
try:
LM=na.eigh(S)
except:
LM=la.eigh(S)
L=LM[0]
M=LM[1]
for i in range(len(L)):
if L[i]<0:
L[i]=0
lamda=2.0/(L+np.sqrt(np.power(L,2)+8*rho))
indexlamda=np.argsort(-lamda)
lamda=np.diag(-np.sort(-lamda)[:p])
hattheta=np.dot(M[:,indexlamda],lamda)
hattheta=np.dot(hattheta,M[:,indexlamda].transpose())
return hattheta
def blockshaped(arr,dim=21):
p=arr.shape[0]//dim
re=np.zeros([dim*dim,p,p])
for i in range(p):
for j in range(p):
re[:,i,j]=arr[i*dim:i*dim+dim,j*dim:j*dim+dim].flatten()
return re
def computepre(msafile,weightfile):
msa=aaweights.read_msa(msafile)
weights=np.genfromtxt(weightfile).flatten()
cov=(aaweights.cal_large_matrix1(msa,weights))
rho2=np.exp((np.arange(80)-60)/5.0)[30]
pre=ROPE(cov,rho2)
#print(pre)
return blockshaped(pre)
def computeapre(msafile,weightfile,savefile):
print(msafile)
#if not os.path.isfile(savefile+'.npy222'):
pre=computepre(msafile,weightfile)
pre=pre.astype('float32')
np.save(savefile,pre)
|
-- ∀x A(x) -> (∀x B(x) -> ∀y(A(y) ∧ B(y)))
variable U: Type
variables A B: U -> Prop
example : (∀ x, A x) -> (∀ x, B x) -> (∀ x, A x ∧ B x) :=
assume hA: ∀ x, A x,
assume hB: ∀ x, B x,
assume y,
have pAy: A y, from hA y,
have pBy: B y, from hB y,
show A y ∧ B y, from and.intro pAy pBy |
## 12장. interactive graph
#### 12-1 plotly 패키지로 인터랙티브 그래프 만들기####
## ------------- 인터랙티브 그래프 만들기 ----------------- ##
install.packages('htmlwidgets')
install.packages("plotly")
library(plotly)
library(ggplot2)
p <- ggplot(data = mpg, aes(x = displ, y = hwy, col = drv)) +
geom_point()
ggplotly(p)
# clarity : http://dsmarket.tistory.com/155
p <- ggplot(data = diamonds, aes(x = cut, fill = clarity)) +
geom_bar(position = "dodge")
ggplotly(p)
library(htmlwidgets)
saveWidget(ggplotly(p), file = "plotly.html")
#### 12-2 dygraphs 패키지로 인터랙티브 시계열 그래프 만들기 ####
## ---------- 인터랙티브 시계열 그래프 만들기 ------------- ##
install.packages("dygraphs")
library(dygraphs)
economics <- ggplot2::economics
head(economics)
# - ggplot을 이용하여 date를 x축으로 unemploy y축으로 시계열
# - 그래프를 그래보세요.
library(xts)
eco <- xts(economics$unemploy, order.by = economics$date)
head(eco)
class(eco)
str(eco)
# 그래프 생성
dygraph(eco)
# 날짜 범위 선택 기능
dygraph(eco) %>% dyRangeSelector()
# 저축률 : personal saving rate
eco_a <- xts(economics$psavert, order.by = economics$date)
# 실업자 수
eco_b <- xts(economics$unemploy/1000, order.by = economics$date)
eco2 <- cbind(eco_a, eco_b) # 데이터 결합
colnames(eco2) <- c("psavert", "unemploy") # 변수명 바꾸기
head(eco2)
dygraph(eco2) %>% dyRangeSelector()
|
module JS.Number
import Data.DPair
import Data.Bits
import JS.Inheritance
import JS.Marshall
import JS.Util
--------------------------------------------------------------------------------
-- Primitives
--------------------------------------------------------------------------------
%foreign "javascript:lambda:(a,b)=>a % b"
prim__mod : Double -> Double -> Double
%foreign "javascript:lambda:(a,b)=>Math.trunc(a / b)"
prim__div : Double -> Double -> Double
%foreign "javascript:lambda:v=>Number.isInteger(v)?v:Math.trunc(v)"
prim__toIntegral : AnyPtr -> AnyPtr
%foreign "javascript:lambda:(v,b)=>v >= b || v < (-b)?v%b:v"
prim__truncSigned : Double -> Double -> Double
%foreign "javascript:lambda:(v,b)=>v >= b || v < 0?Math.abs(v)%b:v"
prim__truncUnsigned : Double -> Double -> Double
%foreign "javascript:lambda:(a,b)=>a & b"
prim__and : Double -> Double -> Double
%foreign "javascript:lambda:(a,b)=>a | b"
prim__or : Double -> Double -> Double
%foreign "javascript:lambda:(a,b)=>a ^ b"
prim__xor : Double -> Double -> Double
%foreign "javascript:lambda:(a,b)=>a >> b"
prim__shr : Double -> Double -> Double
%foreign "javascript:lambda:(a,x,b)=>{ res = a << b; res & x ? res | (-x) : res & (x-1) }"
prim__shlSigned : Double -> Double -> Double -> Double
%foreign "javascript:lambda:(a,x,b)=> (a << b) & x"
prim__shlUnsigned : Double -> Double -> Double -> Double
%foreign "javascript:lambda:x=> Number.isInteger(x)?1:0"
prim__isInteger : AnyPtr -> Double
--------------------------------------------------------------------------------
-- JSInt64
--------------------------------------------------------------------------------
||| A 64-bit signed integer in the range [-9223372036854775808,9223372036854775807]
|||
||| This corresponds to the `Long Long` WebIDL type.
||| Internally, the number is represented by a
||| Javascript `Number`.
||| Note, that arithmetic operations on this type might result
||| in rounding errors, since values might be outside the range
||| of safe integral arithmetics (up to 2^53). Use this type only for
||| interacting with external API requiring values of this type.
export
data JSInt64 : Type where [external]
export
fromJSInt64 : JSInt64 -> Double
fromJSInt64 = believe_me
-- internal precondition: v is an integer
toJSInt64 : Double -> JSInt64
toJSInt64 = believe_me
-- internal precondition: v is an integer
truncToJSInt64 : Double -> JSInt64
truncToJSInt64 v = toJSInt64 (prim__truncSigned v 9223372036854775808.0)
export
Show JSInt64 where
show = jsShow
export
Eq JSInt64 where
(==) = (==) `on` fromJSInt64
export
Ord JSInt64 where
compare = compare `on` fromJSInt64
export
Num JSInt64 where
a + b = truncToJSInt64 $ fromJSInt64 a + fromJSInt64 b
a * b = truncToJSInt64 $ fromJSInt64 a * fromJSInt64 b
fromInteger = truncToJSInt64 . fromInteger
export
Neg JSInt64 where
negate = truncToJSInt64 . negate . fromJSInt64
a - b = truncToJSInt64 $ fromJSInt64 a - fromJSInt64 b
export
Integral JSInt64 where
a `div` b = toJSInt64 $ prim__div (fromJSInt64 a) (fromJSInt64 b)
a `mod` b = toJSInt64 $ prim__mod (fromJSInt64 a) (fromJSInt64 b)
export
ToFFI JSInt64 JSInt64 where toFFI = id
export
FromFFI JSInt64 JSInt64 where fromFFI = Just
export
SafeCast JSInt64 where
safeCast = bounded (-9223372036854775808) 9223372036854775808
--------------------------------------------------------------------------------
-- JSBits64
--------------------------------------------------------------------------------
||| A 64-bit unsigned integer in the range [0,18446744073709551615].
|||
||| This corresponds to the `Unsigned Long Long` WebIDL type.
||| Internally, the number is represented by a Javascript `Number`.
||| Note, that this type is therefore susceptible to
||| rounding errors, since values might be outside the range
||| of safe integral arithmetics (up to 2^53). Use this type only for
||| interacting with external API requiring values of this type.
export
data JSBits64 : Type where [external]
export
fromUInt64 : JSBits64 -> Double
fromUInt64 = believe_me
-- internal precondition: v is a non-negative integer
toUInt64 : Double -> JSBits64
toUInt64 = believe_me
-- internal precondition: v is an integer
truncToUInt64 : Double -> JSBits64
truncToUInt64 v = toUInt64 (prim__truncUnsigned v 18446744073709551616.0)
export
Show JSBits64 where
show = jsShow
export
Eq JSBits64 where
(==) = (==) `on` fromUInt64
export
Ord JSBits64 where
compare = compare `on` fromUInt64
export
Num JSBits64 where
a + b = truncToUInt64 $ fromUInt64 a + fromUInt64 b
a * b = truncToUInt64 $ fromUInt64 a * fromUInt64 b
fromInteger = truncToUInt64 . fromInteger
export
Integral JSBits64 where
a `div` b = toUInt64 $ prim__div (fromUInt64 a) (fromUInt64 b)
a `mod` b = toUInt64 $ prim__mod (fromUInt64 a) (fromUInt64 b)
export
ToFFI JSBits64 JSBits64 where toFFI = id
export
FromFFI JSBits64 JSBits64 where fromFFI = Just
export
SafeCast JSBits64 where
safeCast = bounded 0 18446744073709551615
|
lemma pderiv_pCons: "pderiv (pCons a p) = p + pCons 0 (pderiv p)" |
/-
Copyright (c) 2021 Anne Baanen. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johan Commelin, Anne Baanen
-/
import logic.function.iterate
import order.galois_connection
import order.hom.basic
/-!
# Lattice structure on order homomorphisms
This file defines the lattice structure on order homomorphisms, which are bundled
monotone functions.
## Main definitions
* `order_hom.complete_lattice`: if `β` is a complete lattice, so is `α →o β`
## Tags
monotone map, bundled morphism
-/
namespace order_hom
variables {α β : Type*}
section preorder
variables [preorder α]
@[simps]
instance [semilattice_sup β] : has_sup (α →o β) :=
{ sup := λ f g, ⟨λ a, f a ⊔ g a, f.mono.sup g.mono⟩ }
instance [semilattice_sup β] : semilattice_sup (α →o β) :=
{ sup := has_sup.sup,
le_sup_left := λ a b x, le_sup_left,
le_sup_right := λ a b x, le_sup_right,
sup_le := λ a b c h₀ h₁ x, sup_le (h₀ x) (h₁ x),
.. (_ : partial_order (α →o β)) }
@[simps]
instance [semilattice_inf β] : has_inf (α →o β) :=
{ inf := λ f g, ⟨λ a, f a ⊓ g a, f.mono.inf g.mono⟩ }
instance [semilattice_inf β] : semilattice_inf (α →o β) :=
{ inf := (⊓),
.. (_ : partial_order (α →o β)),
.. (dual_iso α β).symm.to_galois_insertion.lift_semilattice_inf }
instance [lattice β] : lattice (α →o β) :=
{ .. (_ : semilattice_sup (α →o β)),
.. (_ : semilattice_inf (α →o β)) }
@[simps]
instance [preorder β] [order_bot β] : has_bot (α →o β) :=
{ bot := const α ⊥ }
instance [preorder β] [order_bot β] : order_bot (α →o β) :=
{ bot := ⊥,
bot_le := λ a x, bot_le }
@[simps]
instance [preorder β] [order_top β] : has_top (α →o β) :=
{ top := const α ⊤ }
instance [preorder β] [order_top β] : order_top (α →o β) :=
{ top := ⊤,
le_top := λ a x, le_top }
instance [complete_lattice β] : has_Inf (α →o β) :=
{ Inf := λ s, ⟨λ x, ⨅ f ∈ s, (f : _) x, λ x y h, infi₂_mono $ λ f _, f.mono h⟩ }
@[simp] lemma Inf_apply [complete_lattice β] (s : set (α →o β)) (x : α) :
Inf s x = ⨅ f ∈ s, (f : _) x := rfl
lemma infi_apply {ι : Sort*} [complete_lattice β] (f : ι → α →o β) (x : α) :
(⨅ i, f i) x = ⨅ i, f i x :=
(Inf_apply _ _).trans infi_range
@[simp, norm_cast] lemma coe_infi {ι : Sort*} [complete_lattice β] (f : ι → α →o β) :
((⨅ i, f i : α →o β) : α → β) = ⨅ i, f i :=
funext $ λ x, (infi_apply f x).trans (@_root_.infi_apply _ _ _ _ (λ i, f i) _).symm
instance [complete_lattice β] : has_Sup (α →o β) :=
{ Sup := λ s, ⟨λ x, ⨆ f ∈ s, (f : _) x, λ x y h, supr₂_mono (λ f _, f.mono h)⟩ }
@[simp] lemma Sup_apply [complete_lattice β] (s : set (α →o β)) (x : α) :
Sup s x = ⨆ f ∈ s, (f : _) x := rfl
lemma supr_apply {ι : Sort*} [complete_lattice β] (f : ι → α →o β) (x : α) :
(⨆ i, f i) x = ⨆ i, f i x :=
(Sup_apply _ _).trans supr_range
@[simp, norm_cast] lemma coe_supr {ι : Sort*} [complete_lattice β] (f : ι → α →o β) :
((⨆ i, f i : α →o β) : α → β) = ⨆ i, f i :=
funext $ λ x, (supr_apply f x).trans (@_root_.supr_apply _ _ _ _ (λ i, f i) _).symm
instance [complete_lattice β] : complete_lattice (α →o β) :=
{ Sup := Sup,
le_Sup := λ s f hf x, le_supr_of_le f (le_supr _ hf),
Sup_le := λ s f hf x, supr₂_le (λ g hg, hf g hg x),
Inf := Inf,
le_Inf := λ s f hf x, le_infi₂ (λ g hg, hf g hg x),
Inf_le := λ s f hf x, infi_le_of_le f (infi_le _ hf),
.. (_ : lattice (α →o β)),
.. order_hom.order_top,
.. order_hom.order_bot }
lemma iterate_sup_le_sup_iff {α : Type*} [semilattice_sup α] (f : α →o α) :
(∀ n₁ n₂ a₁ a₂, f^[n₁ + n₂] (a₁ ⊔ a₂) ≤ (f^[n₁] a₁) ⊔ (f^[n₂] a₂)) ↔
(∀ a₁ a₂, f (a₁ ⊔ a₂) ≤ (f a₁) ⊔ a₂) :=
begin
split; intros h,
{ exact h 1 0, },
{ intros n₁ n₂ a₁ a₂, have h' : ∀ n a₁ a₂, f^[n] (a₁ ⊔ a₂) ≤ (f^[n] a₁) ⊔ a₂,
{ intros n, induction n with n ih; intros a₁ a₂,
{ refl, },
{ calc f^[n + 1] (a₁ ⊔ a₂) = (f^[n] (f (a₁ ⊔ a₂))) : function.iterate_succ_apply f n _
... ≤ (f^[n] ((f a₁) ⊔ a₂)) : f.mono.iterate n (h a₁ a₂)
... ≤ (f^[n] (f a₁)) ⊔ a₂ : ih _ _
... = (f^[n + 1] a₁) ⊔ a₂ : by rw ← function.iterate_succ_apply, }, },
calc f^[n₁ + n₂] (a₁ ⊔ a₂) = (f^[n₁] (f^[n₂] (a₁ ⊔ a₂))) : function.iterate_add_apply f n₁ n₂ _
... = (f^[n₁] (f^[n₂] (a₂ ⊔ a₁))) : by rw sup_comm
... ≤ (f^[n₁] ((f^[n₂] a₂) ⊔ a₁)) : f.mono.iterate n₁ (h' n₂ _ _)
... = (f^[n₁] (a₁ ⊔ (f^[n₂] a₂))) : by rw sup_comm
... ≤ (f^[n₁] a₁) ⊔ (f^[n₂] a₂) : h' n₁ a₁ _, },
end
end preorder
end order_hom
|
\subsection{Embeddings and immersions}
Whitney embedding theroem: all manfiolds can be embedded in \(R^n\) space for some \(n\).
|
(*
D_diamond (D<>)
T ::= Bot | Top | p.Type | { Type: S..U } | (z: T) -> T^z | T1 /\ T2 | T1 \/ T2
t ::= p | t t
p ::= x | v
v ::= { Type = T } | lambda x:T.t
*)
(* based on *)
(***************************************************************************
* Preservation and Progress for System-F with Subtyping - Definitions *
* Brian Aydemir & Arthur Charguéraud, March 2007 *
***************************************************************************)
Set Implicit Arguments.
Require Import LibLN.
Implicit Types x : var.
(* ********************************************************************** *)
(** * Description of the Language *)
(** Representation of pre-types *)
Inductive typ : Set :=
| typ_bot : typ
| typ_top : typ
| typ_and : typ -> typ -> typ
| typ_or : typ -> typ -> typ
| typ_sel : trm -> typ
| typ_mem : typ -> typ -> typ
| typ_all : typ -> typ -> typ
(** Representation of pre-terms *)
with trm : Set :=
| trm_bvar : nat -> trm
| trm_fvar : var -> trm
| trm_abs : typ -> trm -> trm
| trm_mem : typ -> trm
| trm_app : trm -> trm -> trm.
Fixpoint open_t_rec (k : nat) (f : trm) (T : typ) {struct T} : typ :=
match T with
| typ_bot => typ_bot
| typ_top => typ_top
| typ_and T1 T2 => typ_and (open_t_rec k f T1) (open_t_rec k f T2)
| typ_or T1 T2 => typ_or (open_t_rec k f T1) (open_t_rec k f T2)
| typ_sel t => typ_sel (open_e_rec k f t)
| typ_mem T1 T2 => typ_mem (open_t_rec k f T1) (open_t_rec k f T2)
| typ_all T1 T2 => typ_all (open_t_rec k f T1) (open_t_rec (S k) f T2)
end
(** Opening up a term binder occuring in a term *)
with open_e_rec (k : nat) (f : trm) (e : trm) {struct e} : trm :=
match e with
| trm_bvar i => If k = i then f else (trm_bvar i)
| trm_fvar x => trm_fvar x
| trm_abs V e1 => trm_abs (open_t_rec k f V) (open_e_rec (S k) f e1)
| trm_mem T => trm_mem (open_t_rec k f T)
| trm_app e1 e2 => trm_app (open_e_rec k f e1) (open_e_rec k f e2)
end.
Definition open_t T f := open_t_rec 0 f T.
Definition open_e t u := open_e_rec 0 u t.
(** Notation for opening up binders with variables *)
Notation "t 'open_t_var' x" := (open_t t (trm_fvar x)) (at level 67).
Notation "t 'open_e_var' x" := (open_e t (trm_fvar x)) (at level 67).
(** Types as locally closed pre-types *)
Inductive type : typ -> Prop :=
| type_bot :
type typ_bot
| type_top :
type typ_top
| type_and : forall T1 T2,
type T1 ->
type T2 ->
type (typ_and T1 T2)
| type_or : forall T1 T2,
type T1 ->
type T2 ->
type (typ_or T1 T2)
| type_sel : forall e1,
term e1 ->
type (typ_sel e1)
| type_mem : forall T1 T2,
type T1 ->
type T2 ->
type (typ_mem T1 T2)
| type_all : forall L T1 T2,
type T1 ->
(forall x, x \notin L -> type (T2 open_t_var x)) ->
type (typ_all T1 T2)
(** Terms as locally closed pre-terms *)
with term : trm -> Prop :=
| term_var : forall x,
term (trm_fvar x)
| term_abs : forall L V e1,
type V ->
(forall x, x \notin L -> term (e1 open_e_var x)) ->
term (trm_abs V e1)
| term_mem : forall T1,
type T1 ->
term (trm_mem T1)
| term_app : forall e1 e2,
term e1 ->
term e2 ->
term (trm_app e1 e2).
(** Values *)
Inductive value : trm -> Prop :=
| value_abs : forall V e1, term (trm_abs V e1) ->
value (trm_abs V e1)
| value_mem : forall V, term (trm_mem V) ->
value (trm_mem V).
(** Environment is an associative list of bindings. *)
Definition env := LibEnv.env typ.
(** Well-formedness of a pre-type T in an environment E:
all the type variables of T must be bound via a
subtyping relation in E. This predicates implies
that T is a type *)
Inductive wft : env -> typ -> Prop :=
| wft_bot : forall E,
wft E typ_bot
| wft_top : forall E,
wft E typ_top
| wft_and : forall E T1 T2,
wft E T1 ->
wft E T2 ->
wft E (typ_and T1 T2)
| wft_or : forall E T1 T2,
wft E T1 ->
wft E T2 ->
wft E (typ_or T1 T2)
| wft_sel : forall E e,
value e \/ (exists x, trm_fvar x = e) ->
wfe E e ->
wft E (typ_sel e)
| wft_mem : forall E T1 T2,
wft E T1 ->
wft E T2 ->
wft E (typ_mem T1 T2)
| wft_all : forall L E T1 T2,
wft E T1 ->
(forall x, x \notin L ->
wft (E & x ~ T1) (T2 open_t_var x)) ->
wft E (typ_all T1 T2)
with wfe : env -> trm -> Prop :=
| wfe_var : forall U E x,
binds x U E ->
wfe E (trm_fvar x)
| wfe_abs : forall L E V e,
wft E V ->
(forall x, x \notin L -> wfe (E & x ~ V) (e open_e_var x)) ->
wfe E (trm_abs V e)
| wfe_mem : forall E T,
wft E T ->
wfe E (trm_mem T)
| wfe_app : forall E e1 e2,
wfe E e1 ->
wfe E e2 ->
wfe E (trm_app e1 e2)
.
(** A environment E is well-formed if it contains no duplicate bindings
and if each type in it is well-formed with respect to the environment
it is pushed on to. *)
Inductive okt : env -> Prop :=
| okt_empty :
okt empty
| okt_push : forall E x T,
okt E -> wft E T -> x # E -> okt (E & x ~ T).
(** Subtyping relation *)
Inductive sub : env -> typ -> typ -> Prop :=
| sub_bot : forall E T,
okt E ->
wft E T ->
sub E typ_bot T
| sub_top : forall E S,
okt E ->
wft E S ->
sub E S typ_top
| sub_and11 : forall E T1 T2 T,
wft E T2 ->
sub E T1 T ->
sub E (typ_and T1 T2) T
| sub_and12 : forall E T1 T2 T,
wft E T1 ->
sub E T2 T ->
sub E (typ_and T1 T2) T
| sub_and2 : forall E T T1 T2,
sub E T T1 ->
sub E T T2 ->
sub E T (typ_and T1 T2)
| sub_or21 : forall E T T1 T2,
wft E T2 ->
sub E T T1 ->
sub E T (typ_or T1 T2)
| sub_or22 : forall E T T1 T2,
wft E T1 ->
sub E T T2 ->
sub E T (typ_or T1 T2)
| sub_or1 : forall E T1 T2 T,
sub E T1 T->
sub E T2 T ->
sub E (typ_or T1 T2) T
| sub_refl_sel : forall E t,
okt E ->
wft E (typ_sel t) ->
sub E (typ_sel t) (typ_sel t)
| sub_sel1 : forall E S U t,
has E t (typ_mem S U) ->
sub E (typ_sel t) U
| sub_sel2 : forall E S U t,
has E t (typ_mem S U) ->
sub E S (typ_sel t)
| sub_mem : forall E S1 U1 S2 U2,
sub E S2 S1 -> sub E U1 U2 ->
sub E (typ_mem S1 U1) (typ_mem S2 U2)
| sub_all : forall L E S1 S2 T1 T2,
sub E T1 S1 ->
(forall x, x \notin L ->
sub (E & x ~ T1) (S2 open_t_var x) (T2 open_t_var x)) ->
sub E (typ_all S1 S2) (typ_all T1 T2)
| sub_trans : forall E S T U,
sub E S T ->
sub E T U ->
sub E S U
with has : env -> trm -> typ -> Prop :=
| has_var : forall E x T,
okt E ->
binds x T E ->
has E (trm_fvar x) T
| has_mem : forall E T,
okt E -> wft E T ->
has E (trm_mem T) (typ_mem T T)
| has_abs : forall E V e T,(* dummy case for smooth substitution lemma, see val_typing_has *)
okt E -> wfe E (trm_abs V e) -> wft E (typ_all V T) ->
has E (trm_abs V e) (typ_all V T) (* return typ doesn't matter, as long as it's moot for sel1 and sel2 *)
| has_sub : forall E t T U,
has E t T ->
sub E T U ->
has E t U
.
(** Typing relation *)
Inductive typing : env -> trm -> typ -> Prop :=
| typing_var : forall E x T,
okt E ->
binds x T E ->
typing E (trm_fvar x) T
| typing_abs : forall L E V e1 T1,
(forall x, x \notin L ->
typing (E & x ~ V) (e1 open_e_var x) (T1 open_t_var x)) ->
typing E (trm_abs V e1) (typ_all V T1)
| typing_mem : forall E T1,
okt E ->
wft E T1 ->
typing E (trm_mem T1) (typ_mem T1 T1)
| typing_app : forall T1 E e1 e2 T2,
typing E e1 (typ_all T1 T2) ->
typing E e2 T1 ->
wft E T2 ->
typing E (trm_app e1 e2) T2
| typing_appvar : forall T1 E e1 e2 T2 T2' M,
typing E e1 (typ_all T1 T2) ->
typing E e2 T1 ->
has E e2 M ->
T2' = open_t T2 e2 ->
wft E T2' ->
typing E (trm_app e1 e2) T2'
| typing_sub : forall S E e T,
typing E e S ->
sub E S T ->
typing E e T.
(** One-step reduction *)
Inductive red : trm -> trm -> Prop :=
| red_app_1 : forall e1 e1' e2,
term e2 ->
red e1 e1' ->
red (trm_app e1 e2) (trm_app e1' e2)
| red_app_2 : forall e1 e2 e2',
value e1 ->
red e2 e2' ->
red (trm_app e1 e2) (trm_app e1 e2')
| red_abs : forall V e1 v2,
term (trm_abs V e1) ->
value v2 ->
red (trm_app (trm_abs V e1) v2) (open_e e1 v2).
(** Our goal is to prove preservation and progress *)
Definition preservation := forall e e' T,
typing empty e T ->
red e e' ->
typing empty e' T.
Definition progress := forall e T,
typing empty e T ->
value e
\/ exists e', red e e'.
(***************************************************************************
* Preservation and Progress for System-F with Subtyping - Infrastructure *
***************************************************************************)
(* ********************************************************************** *)
(** * Additional Definitions Used in the Proofs *)
(** Computing free variables in a type *)
Fixpoint fv_t (T : typ) {struct T} : vars :=
match T with
| typ_bot => \{}
| typ_top => \{}
| typ_and T1 T2 => (fv_t T1) \u (fv_t T2)
| typ_or T1 T2 => (fv_t T1) \u (fv_t T2)
| typ_sel t => fv_e t
| typ_mem T1 T2 => (fv_t T1) \u (fv_t T2)
| typ_all T1 T2 => (fv_t T1) \u (fv_t T2)
end
(** Computing free variables in a term *)
with fv_e (e : trm) {struct e} : vars :=
match e with
| trm_bvar i => \{}
| trm_fvar x => \{x}
| trm_abs V e1 => (fv_t V) \u (fv_e e1)
| trm_mem T => fv_t T
| trm_app e1 e2 => (fv_e e1) \u (fv_e e2)
end.
(** Substitution for free type variables in types. *)
Fixpoint subst_t (z : var) (u : trm) (T : typ) {struct T} : typ :=
match T with
| typ_bot => typ_bot
| typ_top => typ_top
| typ_and T1 T2 => typ_and (subst_t z u T1) (subst_t z u T2)
| typ_or T1 T2 => typ_or (subst_t z u T1) (subst_t z u T2)
| typ_sel t => typ_sel (subst_e z u t)
| typ_mem T1 T2 => typ_mem (subst_t z u T1) (subst_t z u T2)
| typ_all T1 T2 => typ_all (subst_t z u T1) (subst_t z u T2)
end
(** Substitution for free term variables in terms. *)
with subst_e (z : var) (u : trm) (e : trm) {struct e} : trm :=
match e with
| trm_bvar i => trm_bvar i
| trm_fvar x => If x = z then u else (trm_fvar x)
| trm_abs V e1 => trm_abs (subst_t z u V) (subst_e z u e1)
| trm_mem T1 => trm_mem (subst_t z u T1)
| trm_app e1 e2 => trm_app (subst_e z u e1) (subst_e z u e2)
end.
(* ********************************************************************** *)
(** * Tactics *)
(** Constructors as hints. *)
Hint Constructors type term wft wfe ok okt value red.
Hint Resolve
sub_bot sub_top sub_refl_sel
typing_var typing_app typing_sub.
(** Gathering free names already used in the proofs *)
Ltac gather_vars :=
let A := gather_vars_with (fun x : vars => x) in
let B := gather_vars_with (fun x : var => \{x}) in
let C := gather_vars_with (fun x : typ => fv_t x) in
let D := gather_vars_with (fun x : trm => fv_e x) in
let E := gather_vars_with (fun x : env => dom x) in
constr:(A \u B \u C \u D \u E).
(** "pick_fresh x" tactic create a fresh variable with name x *)
Ltac pick_fresh x :=
let L := gather_vars in (pick_fresh_gen L x).
(** "apply_fresh T as x" is used to apply inductive rule which
use an universal quantification over a cofinite set *)
Tactic Notation "apply_fresh" constr(T) "as" ident(x) :=
apply_fresh_base T gather_vars x.
Tactic Notation "apply_fresh" "*" constr(T) "as" ident(x) :=
apply_fresh T as x; auto*.
(** These tactics help applying a lemma which conclusion mentions
an environment (E & F) in the particular case when F is empty *)
Ltac get_env :=
match goal with
| |- wft ?E _ => E
| |- wfe ?E _ => E
| |- sub ?E _ _ => E
| |- has ?E _ _ => E
| |- typing ?E _ _ => E
end.
Tactic Notation "apply_empty_bis" tactic(get_env) constr(lemma) :=
let E := get_env in rewrite <- (concat_empty_r E);
eapply lemma; try rewrite concat_empty_r.
Tactic Notation "apply_empty" constr(F) :=
apply_empty_bis (get_env) F.
Tactic Notation "apply_empty" "*" constr(F) :=
apply_empty F; auto*.
Scheme typ_mut := Induction for typ Sort Prop
with trm_mut := Induction for trm Sort Prop.
Combined Scheme typ_trm_mutind from typ_mut, trm_mut.
Scheme type_mut := Induction for type Sort Prop
with term_mut := Induction for term Sort Prop.
Combined Scheme lc_mutind from type_mut, term_mut.
Scheme wft_mut := Induction for wft Sort Prop
with wfe_mut := Induction for wfe Sort Prop.
Combined Scheme wf_mutind from wft_mut, wfe_mut.
Scheme sub_mut := Induction for sub Sort Prop
with has_mut := Induction for has Sort Prop.
Combined Scheme sub_has_mutind from sub_mut, has_mut.
(* ********************************************************************** *)
(** * Properties of Substitutions *)
(** Substitution on indices is identity on well-formed terms. *)
Lemma open_rec_lc_core : (forall T j v u i, i <> j ->
(open_t_rec j v T) = open_t_rec i u (open_t_rec j v T) ->
T = open_t_rec i u T) /\ (forall e j v u i, i <> j ->
open_e_rec j v e = open_e_rec i u (open_e_rec j v e) ->
e = open_e_rec i u e).
Proof.
apply typ_trm_mutind;
try (introv IH1 IH2 Neq H);
try (introv IH Neq H);
try (introv Neq H);
simpl in *; inversion H; f_equal*.
case_nat*. case_nat*.
Qed.
Lemma open_rec_lc : (forall T,
type T -> forall u k, T = open_t_rec k u T) /\ (forall e,
term e -> forall u k, e = open_e_rec k u e).
Proof.
apply lc_mutind; intros; simpl; f_equal*.
pick_fresh x. apply* ((proj1 open_rec_lc_core) T2 0 (trm_fvar x)).
pick_fresh x. apply* ((proj2 open_rec_lc_core) e1 0 (trm_fvar x)).
Qed.
Lemma open_t_var_type : forall x T,
type T -> T open_t_var x = T.
Proof.
intros. unfold open_t. rewrite* <- (proj1 open_rec_lc).
Qed.
(** Substitution for a fresh name is identity. *)
Lemma subst_fresh : (forall T z u,
z \notin fv_t T -> subst_t z u T = T) /\ (forall e z u,
z \notin fv_e e -> subst_e z u e = e).
Proof.
apply typ_trm_mutind; simpl; intros; f_equal*.
case_var*.
Qed.
(** Substitution distributes on the open operation. *)
Lemma subst_open_rec : (forall T1 t2 x u n, term u ->
subst_t x u (open_t_rec n t2 T1) =
open_t_rec n (subst_e x u t2) (subst_t x u T1)) /\ (forall t1 t2 x u n, term u ->
subst_e x u (open_e_rec n t2 t1) =
open_e_rec n (subst_e x u t2) (subst_e x u t1)).
Proof.
apply typ_trm_mutind; intros; simpls; f_equal*.
case_nat*.
case_var*. rewrite* <- (proj2 open_rec_lc).
Qed.
Lemma subst_t_open_t : forall T1 t2 x u, term u ->
subst_t x u (open_t T1 t2) =
open_t (subst_t x u T1) (subst_e x u t2).
Proof.
unfold open_t. auto* (proj1 subst_open_rec).
Qed.
Lemma subst_e_open_e : forall t1 t2 x u, term u ->
subst_e x u (open_e t1 t2) =
open_e (subst_e x u t1) (subst_e x u t2).
Proof.
unfold open_e. auto* (proj2 subst_open_rec).
Qed.
(** Substitution and open_var for distinct names commute. *)
Lemma subst_t_open_t_var : forall x y u T, y <> x -> term u ->
(subst_t x u T) open_t_var y = subst_t x u (T open_t_var y).
Proof.
introv Neq Wu. rewrite* subst_t_open_t.
simpl. case_var*.
Qed.
Lemma subst_e_open_e_var : forall x y u e, y <> x -> term u ->
(subst_e x u e) open_e_var y = subst_e x u (e open_e_var y).
Proof.
introv Neq Wu. rewrite* subst_e_open_e.
simpl. case_var*.
Qed.
(** Opening up a body t with a type u is the same as opening
up the abstraction with a fresh name x and then substituting u for x. *)
Lemma subst_t_intro : forall x T2 u,
x \notin fv_t T2 -> term u ->
open_t T2 u = subst_t x u (T2 open_t_var x).
Proof.
introv Fr Wu. rewrite* subst_t_open_t.
rewrite* (proj1 subst_fresh). simpl. case_var*.
Qed.
Lemma subst_e_intro : forall x t2 u,
x \notin fv_e t2 -> term u ->
open_e t2 u = subst_e x u (t2 open_e_var x).
Proof.
introv Fr Wu. rewrite* subst_e_open_e.
rewrite* (proj2 subst_fresh). simpl. case_var*.
Qed.
(** Substitutions preserve local closure. *)
Lemma subst_lc :
(forall T, type T -> forall z u, term u -> type (subst_t z u T)) /\
(forall e, term e -> forall z u, term u -> term (subst_e z u e)).
Proof.
apply lc_mutind; intros; simpl; auto.
apply_fresh* type_all as X. rewrite* subst_t_open_t_var.
case_var*.
apply_fresh* term_abs as y. rewrite* subst_e_open_e_var.
Qed.
Lemma subst_t_type : forall T z u,
type T -> term u -> type (subst_t z u T).
Proof.
intros. apply* (proj1 subst_lc).
Qed.
Lemma subst_e_term : forall e1 z e2,
term e1 -> term e2 -> term (subst_e z e2 e1).
Proof.
intros. apply* (proj2 subst_lc).
Qed.
Lemma subst_e_value : forall e1 z e2,
value e1 -> term e2 -> value (subst_e z e2 e1).
Proof.
intros. inversion H; subst; simpl.
- apply value_abs.
assert (trm_abs (subst_t z e2 V) (subst_e z e2 e0) = subst_e z e2 (trm_abs V e0)) as A. {
simpl. reflexivity.
}
rewrite A. apply* subst_e_term.
- apply value_mem.
assert (trm_mem (subst_t z e2 V) = subst_e z e2 (trm_mem V)) as A. {
simpl. reflexivity.
}
rewrite A. apply* subst_e_term.
Qed.
Lemma value_is_term: forall e, value e -> term e.
Proof.
introv H. inversion H; subst; eauto.
Qed.
Hint Resolve subst_t_type subst_e_term subst_e_value value_is_term.
(* ********************************************************************** *)
(** * Properties of well-formedness of a type in an environment *)
(** If a type is well-formed in an environment then it is locally closed. *)
Lemma wf_lc : (forall E T, wft E T -> type T) /\
(forall E e, wfe E e -> term e).
Proof.
apply wf_mutind; eauto.
Qed.
Lemma wft_type : forall E T,
wft E T -> type T.
Proof.
intros. eapply (proj1 wf_lc); eauto.
Qed.
Lemma wfe_term : forall E e,
wfe E e -> term e.
Proof.
intros. eapply (proj2 wf_lc); eauto.
Qed.
(** Through weakening *)
Lemma wf_weaken :
(forall E0 T, wft E0 T ->
forall E F G, E0 = E & G ->
ok (E & F & G) ->
wft (E & F & G) T)
/\
(forall E0 e, wfe E0 e ->
forall E F G, E0 = E & G ->
ok (E & F & G) ->
wfe (E & F & G) e).
Proof.
apply wf_mutind; intros; subst; eauto.
apply_fresh* wft_all as Y. apply_ih_bind* H0.
apply (@wfe_var U). apply* binds_weaken.
apply_fresh* wfe_abs as y. apply_ih_bind* H0.
Qed.
Lemma wft_weaken : forall G T E F,
wft (E & G) T ->
ok (E & F & G) ->
wft (E & F & G) T.
Proof.
intros. eapply (proj1 wf_weaken); eauto.
Qed.
Lemma wft_weaken_empty : forall T E,
wft empty T ->
ok E ->
wft E T.
Proof.
intros.
assert (E = empty & E & empty) as A. {
rewrite concat_empty_l. rewrite concat_empty_r. reflexivity.
}
rewrite A. apply wft_weaken.
rewrite concat_empty_l. auto.
rewrite concat_empty_l. rewrite concat_empty_r. auto.
Qed.
Lemma wfe_weaken : forall G T E F,
wfe (E & G) T ->
ok (E & F & G) ->
wfe (E & F & G) T.
Proof.
intros. eapply (proj2 wf_weaken); eauto.
Qed.
Lemma wfe_weaken_empty : forall T E,
wfe empty T ->
ok E ->
wfe E T.
Proof.
intros.
assert (E = empty & E & empty) as A. {
rewrite concat_empty_l. rewrite concat_empty_r. reflexivity.
}
rewrite A. apply wfe_weaken.
rewrite concat_empty_l. auto.
rewrite concat_empty_l. rewrite concat_empty_r. auto.
Qed.
(** Through narrowing *)
Lemma wf_narrow : (forall E0 T, wft E0 T -> forall V F U E x,
E0 = (E & x ~ V & F) ->
ok (E & x ~ U & F) ->
wft (E & x ~ U & F) T)
/\
(forall E0 e, wfe E0 e -> forall V F U E x,
E0 = (E & x ~ V & F) ->
ok (E & x ~ U & F) ->
wfe (E & x ~ U & F) e).
Proof.
apply wf_mutind; intros; subst; eauto.
apply_fresh* wft_all as Y. apply_ih_bind* H0.
destruct (binds_middle_inv b) as [K|[K|K]]; try destructs K.
applys wfe_var. apply* binds_concat_right.
subst. applys wfe_var. apply~ binds_middle_eq.
applys wfe_var. apply~ binds_concat_left.
apply* binds_concat_left.
apply_fresh* wfe_abs as y. apply_ih_bind* H0.
Qed.
Lemma wft_narrow : forall V F U T E x,
wft (E & x ~ V & F) T ->
ok (E & x ~ U & F) ->
wft (E & x ~ U & F) T.
Proof.
intros. eapply (proj1 wf_narrow); eauto.
Qed.
(** Through substitution *)
Lemma wf_subst : (forall E0 T, wft E0 T -> forall F Q E Z u,
E0 = E & Z ~ Q & F ->
(value u \/ exists x, trm_fvar x = u) -> wfe E u ->
ok (E & map (subst_t Z u) F) ->
wft (E & map (subst_t Z u) F) (subst_t Z u T)) /\
(forall E0 e, wfe E0 e -> forall F Q E Z u,
E0 = E & Z ~ Q & F ->
(value u \/ exists x, trm_fvar x = u) -> wfe E u ->
ok (E & map (subst_t Z u) F) ->
wfe (E & map (subst_t Z u) F) (subst_e Z u e)).
Proof.
apply wf_mutind; intros; subst; simpl; eauto.
- destruct o as [? | [? ?]].
+ apply* wft_sel. left. apply subst_e_value. assumption. apply* wfe_term.
+ subst. simpl. case_var*.
* apply_empty* wft_weaken.
* apply* wft_sel. rewrite* <- ((proj2 subst_fresh) (trm_fvar x) Z u).
simpl. auto.
- apply_fresh* wft_all as Y.
lets: wft_type.
rewrite* subst_t_open_t_var.
apply_ih_map_bind* H0. apply* wfe_term.
- case_var*.
+ apply_empty* (proj2 wf_weaken).
+ destruct (binds_concat_inv b) as [?|[? ?]].
apply (@wfe_var (subst_t Z u U)).
apply~ binds_concat_right.
destruct (binds_push_inv H3) as [[? ?]|[? ?]].
subst. false~.
applys wfe_var. apply* binds_concat_left.
- apply_fresh* wfe_abs as y.
lets: (proj2 wf_lc).
rewrite* subst_e_open_e_var.
apply_ih_map_bind* H0.
Qed.
Lemma wft_subst : forall F Q E Z u T,
wft (E & Z ~ Q & F) T ->
(value u \/ exists x, trm_fvar x = u) -> wfe E u ->
ok (E & map (subst_t Z u) F) ->
wft (E & map (subst_t Z u) F) (subst_t Z u T).
Proof.
intros. eapply (proj1 wf_subst); eauto.
Qed.
Lemma wft_subst1 : forall F Q Z u T,
wft (Z ~ Q & F) T ->
(value u \/ exists x, trm_fvar x = u) -> wfe empty u ->
ok (map (subst_t Z u) F) ->
wft (map (subst_t Z u) F) (subst_t Z u T).
Proof.
intros.
rewrite <- (@concat_empty_l typ (map (subst_t Z u) F)).
apply* wft_subst.
rewrite concat_empty_l. eassumption.
rewrite concat_empty_l. eassumption.
Qed.
Lemma wft_subst_empty : forall Q Z u T,
wft (Z ~ Q) T ->
(value u \/ exists x, trm_fvar x = u) -> wfe empty u ->
wft empty (subst_t Z u T).
Proof.
intros.
assert (empty & map (subst_t Z u) empty = empty) as A. {
rewrite map_empty. rewrite concat_empty_l. reflexivity.
}
rewrite <- A. eapply wft_subst; eauto.
rewrite concat_empty_l. rewrite concat_empty_r. eauto.
rewrite A. eauto.
Qed.
(** Through type reduction *)
Lemma wft_open : forall E u T1 T2,
ok E ->
wft E (typ_all T1 T2) ->
(value u \/ exists x, trm_fvar x = u) -> wfe E u ->
wft E (open_t T2 u).
Proof.
introv Ok WA VU WU. inversions WA. pick_fresh X.
auto* wft_type. rewrite* (@subst_t_intro X).
lets K: (@wft_subst empty).
specializes_vars K. clean_empty K. apply* K.
apply* wfe_term.
Qed.
(* ********************************************************************** *)
(** * Relations between well-formed environment and types well-formed
in environments *)
(** If an environment is well-formed, then it does not contain duplicated keys. *)
Lemma ok_from_okt : forall E,
okt E -> ok E.
Proof.
induction 1; auto.
Qed.
Hint Extern 1 (ok _) => apply ok_from_okt.
(** Extraction from an assumption in a well-formed environments *)
Lemma wft_from_env_has : forall x U E,
okt E -> binds x U E -> wft E U.
Proof.
induction E using env_ind; intros Ok B.
false* binds_empty_inv.
inversions Ok.
false (empty_push_inv H0).
destruct (eq_push_inv H) as [? [? ?]]. subst. clear H.
destruct (binds_push_inv B) as [[? ?]|[? ?]]. subst.
apply_empty* wft_weaken.
apply_empty* wft_weaken.
Qed.
(** Extraction from a well-formed environment *)
Lemma wft_from_okt : forall x T E,
okt (E & x ~ T) -> wft E T.
Proof.
intros. inversions* H.
false (empty_push_inv H1).
destruct (eq_push_inv H0) as [? [? ?]]. subst. assumption.
Qed.
(** Automation *)
Lemma wft_weaken_right : forall T E F,
wft E T ->
ok (E & F) ->
wft (E & F) T.
Proof.
intros. apply_empty* wft_weaken.
Qed.
Hint Resolve wft_weaken_right.
Hint Resolve wft_from_okt.
Hint Immediate wft_from_env_has.
Hint Resolve wft_subst.
(* ********************************************************************** *)
(** ** Properties of well-formedness of an environment *)
(** Inversion lemma *)
Lemma okt_push_inv : forall E x T,
okt (E & x ~ T) -> okt E /\ wft E T /\ x # E.
Proof.
introv O. inverts O.
false* empty_push_inv.
lets (?&M&?): (eq_push_inv H). subst. eauto.
Qed.
Lemma okt_push_type : forall E x T,
okt (E & x ~ T) -> type T.
Proof. intros. applys wft_type. forwards*: okt_push_inv. Qed.
Hint Immediate okt_push_type.
(** Through narrowing *)
Lemma okt_narrow : forall V (E F:env) U x,
okt (E & x ~ V & F) ->
wft E U ->
okt (E & x ~ U & F).
Proof.
introv O W. induction F using env_ind.
rewrite concat_empty_r in *. lets*: (okt_push_inv O).
rewrite concat_assoc in *.
lets (?&?&?): (okt_push_inv O).
applys~ okt_push. applys* wft_narrow.
Qed.
(** Through substitution *)
Lemma okt_subst : forall Q Z u (E F:env),
okt (E & Z ~ Q & F) ->
(value u \/ exists x, trm_fvar x = u) -> wfe E u ->
okt (E & map (subst_t Z u) F).
Proof.
introv O V W. induction F using env_ind.
rewrite map_empty. rewrite concat_empty_r in *.
lets*: (okt_push_inv O).
rewrite map_push. rewrite concat_assoc in *.
lets*: (okt_push_inv O).
apply okt_push. apply* IHF. apply* wft_subst. auto*.
Qed.
Lemma okt_subst1 : forall Q Z u (F:env),
okt (Z ~ Q & F) ->
(value u \/ exists x, trm_fvar x = u) -> wfe empty u ->
okt (map (subst_t Z u) F).
Proof.
intros.
rewrite <- concat_empty_l. apply* okt_subst.
rewrite concat_empty_l. eassumption.
Qed.
(** Automation *)
Hint Resolve okt_narrow okt_subst wft_weaken.
(* ********************************************************************** *)
(** ** Environment is unchanged by substitution from a fresh name *)
Ltac destruct_notin_union :=
match goal with
| H: _ \notin _ \u _ |- _ => eapply notin_union in H; destruct H
end.
Lemma notin_fv_open_rec : (forall T k y x,
x \notin fv_t (open_t_rec k (trm_fvar y) T) ->
x \notin fv_t T) /\ (forall e k y x,
x \notin fv_e (open_e_rec k (trm_fvar y) e) ->
x \notin fv_e e).
Proof.
apply typ_trm_mutind; simpl; intros;
repeat destruct_notin_union; eauto using notin_union_l.
Qed.
Lemma notin_fv_t_open : forall y x T,
x \notin fv_t (T open_t_var y) ->
x \notin fv_t T.
Proof.
unfold open_t. intros. apply* (proj1 notin_fv_open_rec).
Qed.
Lemma notin_fv_e_open : forall y x e,
x \notin fv_e (e open_e_var y) ->
x \notin fv_e e.
Proof.
unfold open_e. intros. apply* (proj2 notin_fv_open_rec).
Qed.
Lemma notin_fv_wf_rec :
(forall E T,
wft E T -> forall x, x # E -> x \notin fv_t T) /\
(forall E e,
wfe E e -> forall x, x # E -> x \notin fv_e e).
Proof.
apply wf_mutind; intros; simpl; eauto.
notin_simpl; auto. pick_fresh Y. apply* (@notin_fv_t_open Y).
rewrite notin_singleton. intro. subst. applys binds_fresh_inv b H.
notin_simpl; auto. pick_fresh y. apply* (@notin_fv_e_open y).
Qed.
Lemma notin_fv_wf : forall E x T,
wft E T -> x # E -> x \notin fv_t T.
Proof.
intros. eapply (proj1 notin_fv_wf_rec); eauto.
Qed.
Lemma map_subst_id : forall G z u,
okt G -> z # G -> G = map (subst_t z u) G.
Proof.
induction 1; intros Fr; autorewrite with rew_env_map; simpl.
auto.
rewrite* <- IHokt. rewrite* (proj1 subst_fresh). apply* notin_fv_wf.
Qed.
(* ********************************************************************** *)
(** ** Regularity of relations *)
(** The subtyping relation is restricted to well-formed objects. *)
Lemma sub_has_regular : (forall E S T,
sub E S T -> okt E /\ wft E S /\ wft E T) /\ (forall E p T,
has E p T -> okt E /\ wft E (typ_sel p) /\ wft E T).
Proof.
apply sub_has_mutind; intros; try auto*.
splits*. destruct H as [? [? A]]. inversion A; subst. assumption.
splits*. destruct H as [? [? A]]. inversion A; subst. assumption.
split. auto*. split;
apply_fresh* wft_all as Y;
forwards~: (H0 Y); apply_empty* (@wft_narrow T1).
splits*. apply wft_sel. left. apply value_mem. apply* wfe_term. apply* wfe_mem.
splits*. apply wft_sel. left. apply value_abs. apply* wfe_term. assumption.
Qed.
Lemma sub_regular : forall E S T,
sub E S T -> okt E /\ wft E S /\ wft E T.
Proof.
intros. apply* (proj1 sub_has_regular).
Qed.
Lemma has_regular : forall E p T,
has E p T -> okt E /\ wft E (typ_sel p) /\ wft E T.
Proof.
intros. apply* (proj2 sub_has_regular).
Qed.
Lemma has_regular_e : forall E p T,
has E p T -> (value p \/ (exists x, trm_fvar x = p)) /\ wfe E p.
Proof.
intros. apply has_regular in H. destruct H as [? [A ?]].
inversion A; subst. split; assumption.
Qed.
(** The typing relation is restricted to well-formed objects. *)
Lemma typing_regular : forall E e T,
typing E e T -> okt E /\ wfe E e /\ wft E T.
Proof.
induction 1.
splits*.
splits.
pick_fresh y. specializes H0 y. destructs~ H0.
forwards*: okt_push_inv.
apply_fresh* wfe_abs as y.
pick_fresh y. forwards~ K: (H0 y). destructs K.
forwards*: okt_push_inv.
forwards~ K: (H0 y). destructs K. auto.
apply_fresh* wft_all as Y.
pick_fresh y. forwards~ K: (H0 y). destructs K.
forwards*: okt_push_inv.
forwards~ K: (H0 Y). destructs K.
forwards*: okt_push_inv.
splits*.
splits*.
splits*.
splits*. destructs~ (sub_regular H0).
Qed.
(** The value relation is restricted to well-formed objects. *)
Lemma value_regular : forall t,
value t -> term t.
Proof.
induction 1; auto*.
Qed.
(** The reduction relation is restricted to well-formed objects. *)
Lemma red_regular : forall t t',
red t t' -> term t /\ term t'.
Proof.
induction 1; split; auto* value_regular.
inversions H. pick_fresh y. rewrite* (@subst_e_intro y).
Qed.
(** Automation *)
Hint Extern 1 (okt ?E) =>
match goal with
| H: sub _ _ _ |- _ => apply (proj31 (sub_regular H))
| H: has _ _ _ |- _ => apply (proj31 (has_regular H))
| H: typing _ _ _ |- _ => apply (proj31 (typing_regular H))
end.
Hint Extern 1 (wft ?E ?T) =>
match goal with
| H: typing E _ T |- _ => apply (proj33 (typing_regular H))
| H: sub E T _ |- _ => apply (proj32 (sub_regular H))
| H: sub E _ T |- _ => apply (proj33 (sub_regular H))
| H: has E _ T |- _ => apply (proj33 (has_regular H))
end.
Hint Extern 1 (wfe ?E ?e) =>
match goal with
| H: typing E e _ |- _ => apply (proj32 (typing_regular H))
| H: has E e _ |- _ => apply (proj2 (has_regular_e H))
end.
Hint Extern 1 (type ?T) =>
let go E := apply (@wft_type E); auto in
match goal with
| H: typing ?E _ T |- _ => go E
| H: sub ?E T _ |- _ => go E
| H: sub ?E _ T |- _ => go E
end.
Hint Extern 1 (term ?e) =>
match goal with
| H: typing _ ?e _ |- _ => apply (wfe_term (proj32 (typing_regular H)))
| H: red ?e _ |- _ => apply (proj1 (red_regular H))
| H: red _ ?e |- _ => apply (proj2 (red_regular H))
end.
(***************************************************************************
* Preservation and Progress for System-F with Subtyping - Proofs *
***************************************************************************)
(** In parentheses are given the label of the corresponding
lemma in the description of the POPLMark Challenge. *)
(* ********************************************************************** *)
(** * Properties of Subtyping *)
(* ********************************************************************** *)
(** Reflexivity (1) *)
Lemma sub_reflexivity : forall E T,
okt E ->
wft E T ->
sub E T T .
Proof.
introv Ok WI. lets W: (wft_type WI). gen E.
induction W; intros; inversions WI; eauto.
apply* sub_and2. apply* sub_and11. apply* sub_and12.
apply* sub_or1. apply* sub_or21. apply* sub_or22.
apply* sub_mem.
apply_fresh* sub_all as Y.
Qed.
(* ********************************************************************** *)
(** Weakening (2) *)
Lemma sub_has_weakening : (forall E0 S T, sub E0 S T -> forall E F G,
E0 = E & G ->
okt (E & F & G) ->
sub (E & F & G) S T) /\ (forall E0 p T, has E0 p T -> forall E F G,
E0 = E & G ->
okt (E & F & G) ->
has (E & F & G) p T).
Proof.
apply sub_has_mutind; intros; subst; auto.
apply* sub_and11.
apply* sub_and12.
apply* sub_and2.
apply* sub_or21.
apply* sub_or22.
apply* sub_or1.
apply* sub_sel1.
apply* sub_sel2.
apply* sub_mem.
apply_fresh* sub_all as Y. apply_ih_bind* H0.
apply* sub_trans.
apply* has_var. apply* binds_weaken.
apply* has_mem.
apply* has_abs. apply* wfe_weaken.
apply* has_sub.
Qed.
Lemma sub_weakening : forall E F G S T,
sub (E & G) S T ->
okt (E & F & G) ->
sub (E & F & G) S T.
Proof.
intros. apply* (proj1 sub_has_weakening).
Qed.
Lemma sub_weakening1 : forall E F G S T,
sub E S T ->
okt (E & F & G) ->
sub (E & F & G) S T.
Proof.
intros.
assert (E & F & G = E & (F & G) & empty) as A. {
rewrite concat_empty_r. rewrite concat_assoc. reflexivity.
}
rewrite A.
apply* sub_weakening.
rewrite concat_empty_r. assumption.
rewrite <- A. assumption.
Qed.
Lemma sub_weakening_empty : forall E S T,
sub empty S T ->
okt E ->
sub E S T.
Proof.
intros.
assert (empty & E & empty = E) as A. {
rewrite concat_empty_r. rewrite concat_empty_l. reflexivity.
}
rewrite <- A.
apply* sub_weakening.
rewrite concat_empty_r. assumption.
rewrite A. assumption.
Qed.
Lemma has_weakening : forall E F G p T,
has (E & G) p T ->
okt (E & F & G) ->
has (E & F & G) p T.
Proof.
intros. apply* (proj2 sub_has_weakening).
Qed.
Lemma has_weakening1 : forall E F G p T,
has E p T ->
okt (E & F & G) ->
has (E & F & G) p T.
Proof.
intros.
assert (E & F & G = E & (F & G) & empty) as A. {
rewrite concat_empty_r. rewrite concat_assoc. reflexivity.
}
rewrite A.
apply* has_weakening.
rewrite concat_empty_r. assumption.
rewrite <- A. assumption.
Qed.
Lemma has_weakening_empty : forall E p T,
has empty p T ->
okt E ->
has E p T.
Proof.
intros.
assert (empty & E & empty = E) as A. {
rewrite concat_empty_r. rewrite concat_empty_l. reflexivity.
}
rewrite <- A.
apply* has_weakening.
rewrite concat_empty_r. assumption.
rewrite A. assumption.
Qed.
(* ********************************************************************** *)
(** Narrowing and transitivity (3) *)
Section NarrowTrans.
Hint Resolve wft_narrow.
Lemma sub_has_narrowing_aux :
(forall E0 S T, sub E0 S T ->
forall Q E F z P,
E0 = (E & z ~ Q & F) ->
sub E P Q ->
sub (E & z ~ P & F) S T)
/\
(forall E0 p T, has E0 p T ->
forall Q E F z P,
E0 = (E & z ~ Q & F) ->
sub E P Q ->
has (E & z ~ P & F) p T).
Proof.
Hint Constructors sub has.
apply sub_has_mutind; intros; subst; eauto 4.
apply* sub_bot.
apply* sub_top.
apply* sub_and11.
apply* sub_and12.
apply* sub_or21.
apply* sub_or22.
apply* sub_refl_sel.
apply_fresh sub_all as Y. auto*. apply_ih_bind H0; eauto 4.
tests EQ: (x = z).
lets M: (@okt_narrow Q).
apply binds_middle_eq_inv in b. subst.
eapply has_sub. eapply has_var. apply* M. apply binds_middle_eq.
eapply ok_from_okt in o. eapply ok_middle_inv in o. destruct o as [o1 o2]. apply o2.
eapply sub_weakening1; eauto.
auto*.
eapply has_var; eauto. binds_cases b; auto.
apply* has_mem.
apply* has_abs. eapply (proj2 wf_narrow); eauto.
Qed.
Lemma sub_narrowing : forall Q E F Z P S T,
sub E P Q ->
sub (E & Z ~ Q & F) S T ->
sub (E & Z ~ P & F) S T.
Proof.
intros.
apply* (proj1 sub_has_narrowing_aux).
Qed.
Lemma sub_narrowing_empty : forall Q Z P S T,
sub empty P Q ->
sub (Z ~ Q) S T ->
sub (Z ~ P) S T.
Proof.
intros.
rewrite <- (concat_empty_r (Z ~ P)).
rewrite <- (concat_empty_l (Z ~ P)).
eapply sub_narrowing; eauto 4.
rewrite concat_empty_r. rewrite concat_empty_l. auto.
Qed.
End NarrowTrans.
(* ********************************************************************** *)
(** Substitution preserves subtyping (10) *)
Lemma has_value_var : forall E u T,
has E u T ->
(value u \/ exists x, trm_fvar x = u).
Proof.
intros. apply has_regular_e in H. destruct H as [A ?].
apply A.
Qed.
Hint Resolve has_value_var.
Lemma var_typing_has: forall E x Q,
typing E (trm_fvar x) Q ->
has E (trm_fvar x) Q.
Proof.
introv H. remember (trm_fvar x) as t. gen Heqt.
induction H; intros; subst; try solve [inversion Heqt].
- inversion Heqt. subst. apply* has_var.
- eapply has_sub. eapply IHtyping; eauto. assumption.
Qed.
Lemma val_typing_has: forall E u Q,
value u ->
typing E u Q ->
has E u Q.
Proof.
introv Hv H.
lets R: (typing_regular H).
induction H; intros; subst; try solve [inversion Hv].
- apply* has_abs.
- apply* has_mem.
- apply* has_sub.
Qed.
Lemma sub_has_through_subst : (forall E0 S T, sub E0 S T -> forall Q E F Z u,
E0 = (E & Z ~ Q & F) ->
(value u \/ exists x, trm_fvar x = u) -> typing E u Q ->
sub (E & map (subst_t Z u) F) (subst_t Z u S) (subst_t Z u T)) /\
(forall E0 p T, has E0 p T -> forall Q E F Z u,
E0 = (E & Z ~ Q & F) ->
(value u \/ exists x, trm_fvar x = u) -> typing E u Q ->
has (E & map (subst_t Z u) F) (subst_e Z u p) (subst_t Z u T)).
Proof.
apply sub_has_mutind; intros; subst; simpl.
- apply* sub_bot.
- apply* sub_top.
- apply* sub_and11.
- apply* sub_and12.
- apply* sub_and2.
- apply* sub_or21.
- apply* sub_or22.
- apply* sub_or1.
- simpl. apply* sub_refl_sel.
assert (typ_sel (subst_e Z u t) = subst_t Z u (typ_sel t)) as A by auto.
rewrite A. auto*.
- apply* sub_sel1. eapply H. reflexivity. auto. auto.
- apply* sub_sel2. eapply H. reflexivity. auto. auto.
- apply* sub_mem.
- apply_fresh* sub_all as X.
rewrite* subst_t_open_t_var. rewrite* subst_t_open_t_var.
apply_ih_map_bind* H0.
- apply* sub_trans.
- case_var.
+ apply binds_middle_eq_inv in b; eauto. subst.
destruct H0 as [H0 | [x H0]].
* apply_empty* has_weakening1.
rewrite (proj1 subst_fresh). apply* val_typing_has.
apply* (@notin_fv_wf E0).
* subst. rewrite (proj1 subst_fresh).
apply_empty* has_weakening1. apply var_typing_has. assumption.
apply* (@notin_fv_wf E0).
+ destruct (binds_concat_inv b) as [?|[? ?]].
* eapply has_var. auto*.
apply binds_concat_right. apply binds_map. eassumption.
* applys has_var. apply* okt_subst.
assert (T = subst_t Z u T) as B. {
rewrite (proj1 subst_fresh). reflexivity.
apply* (@notin_fv_wf E0). apply* wft_from_env_has.
apply binds_concat_left_inv in H2. eassumption.
auto*.
}
apply binds_concat_left. rewrite <- B.
apply binds_concat_left_inv in H2. apply H2.
auto*. auto*.
- apply* has_mem.
- apply* has_abs.
assert (trm_abs (subst_t Z u V) (subst_e Z u e) =
subst_e Z u (trm_abs V e)) as A by solve [simpl; reflexivity].
rewrite A. eapply (proj2 wf_subst); eauto.
assert (typ_all (subst_t Z u V) (subst_t Z u T) =
subst_t Z u (typ_all V T)) as B by solve [simpl; reflexivity].
rewrite B. apply* wft_subst.
- apply* has_sub.
Qed.
(* ********************************************************************** *)
(** * Properties of Typing *)
(* ********************************************************************** *)
(** Weakening (5) *)
Lemma typing_weakening : forall E F G e T,
typing (E & G) e T ->
okt (E & F & G) ->
typing (E & F & G) e T.
Proof.
introv Typ. gen F. inductions Typ; introv Ok.
apply* typing_var. apply* binds_weaken.
apply_fresh* typing_abs as x. forwards~ K: (H x).
apply_ih_bind (H0 x); eauto.
apply* typing_mem.
apply* typing_app.
eapply typing_appvar; eauto. eapply (proj2 sub_has_weakening); eauto.
apply* typing_sub. apply* sub_weakening.
Qed.
(************************************************************************ *)
(** Preservation by Type Narrowing (7) *)
Lemma typing_narrowing : forall Q E F X P e T,
sub E P Q ->
typing (E & X ~ Q & F) e T ->
typing (E & X ~ P & F) e T.
Proof.
introv PsubQ Typ. gen_eq E': (E & X ~ Q & F). gen F.
inductions Typ; introv EQ; subst; simpl.
- binds_cases H0.
+ apply* typing_var.
+ subst. apply* typing_sub. apply* sub_weakening1.
+ apply* typing_var.
- apply_fresh* typing_abs as y. apply_ih_bind* H0.
- apply* typing_mem. apply* wft_narrow.
- apply* typing_app. apply* wft_narrow.
- apply* typing_appvar. apply* (proj2 sub_has_narrowing_aux). apply* wft_narrow.
- apply* typing_sub. apply* (@sub_narrowing Q).
Qed.
Lemma typing_narrowing_empty : forall Q X P e T,
sub empty P Q ->
typing (X ~ Q) e T ->
typing (X ~ P) e T.
Proof.
intros.
rewrite <- (concat_empty_r (X ~ P)).
rewrite <- (concat_empty_l (X ~ P)).
eapply typing_narrowing; eauto.
rewrite concat_empty_r. rewrite concat_empty_l. auto.
Qed.
(************************************************************************ *)
(** Preservation by Substitution (8) *)
Lemma typing_through_subst : forall U E F z T e u,
typing (E & z ~ U & F) e T ->
(value u \/ exists x, trm_fvar x = u) -> typing E u U ->
typing (E & (map (subst_t z u) F)) (subst_e z u e) (subst_t z u T).
Proof.
introv TypT Hu TypU.
inductions TypT; introv; subst; simpl.
- case_var.
+ binds_get H0.
rewrite (proj1 subst_fresh).
apply_empty typing_weakening. assumption.
apply* okt_subst.
apply* (@notin_fv_wf E).
+ binds_cases H0.
rewrite (proj1 subst_fresh). eapply typing_var; eauto.
apply (@notin_fv_wf E). eapply wft_from_env_has. auto*. eapply B0. auto*.
apply* typing_var.
- apply_fresh* typing_abs as y.
rewrite* subst_e_open_e_var. rewrite* subst_t_open_t_var.
rewrite <- concat_assoc_map_push.
eapply H0; eauto. rewrite concat_assoc. auto.
- apply* typing_mem.
- eapply typing_app. eapply IHTypT1; eauto. eapply IHTypT2; eauto.
apply* wft_subst.
- eapply typing_appvar. eapply IHTypT1; eauto. eapply IHTypT2; eauto.
apply* (proj2 sub_has_through_subst).
eapply subst_t_open_t. auto*. apply* wft_subst.
- eapply typing_sub. eapply IHTypT; eauto.
eapply (proj1 sub_has_through_subst); eauto.
Qed.
(* ********************************************************************** *)
(** * Preservation *)
(* ********************************************************************** *)
(** Inversions for Typing (13) *)
Inductive psub : typ -> typ -> Prop :=
| psub_bot : forall U,
wft empty U ->
psub typ_bot U
| psub_top : forall S,
wft empty S ->
psub S typ_top
| psub_and11 : forall T1 T2 T,
wft empty T2 ->
psub T1 T ->
psub (typ_and T1 T2) T
| psub_and12 : forall T1 T2 T,
wft empty T1 ->
psub T2 T ->
psub (typ_and T1 T2) T
| psub_and2 : forall T T1 T2,
psub T T1 ->
psub T T2 ->
psub T (typ_and T1 T2)
| psub_or21 : forall T T1 T2,
wft empty T2 ->
psub T T1 ->
psub T (typ_or T1 T2)
| psub_or22 : forall T T1 T2,
wft empty T1 ->
psub T T2 ->
psub T (typ_or T1 T2)
| psub_or1 : forall T1 T2 T,
psub T1 T->
psub T2 T ->
psub (typ_or T1 T2) T
| psub_refl_sel : forall t,
wft empty (typ_sel t) ->
psub (typ_sel t) (typ_sel t)
| psub_sel1 : forall U,
wft empty U ->
psub (typ_sel (trm_mem U)) U
| psub_sel2 : forall S,
wft empty S ->
psub S (typ_sel (trm_mem S))
| psub_mem : forall S1 U1 S2 U2,
psub S2 S1 -> psub U1 U2 ->
psub (typ_mem S1 U1) (typ_mem S2 U2)
| psub_all : forall L S1 S2 T1 T2,
psub T1 S1 ->
(forall x, x \notin L ->
sub (x ~ T1) (S2 open_t_var x) (T2 open_t_var x)) ->
psub (typ_all S1 S2) (typ_all T1 T2)
| psub_trans : forall S T U,
psub S T ->
psub T U ->
psub S U
.
Lemma has_empty_value: forall p T,
has empty p T ->
value p.
Proof.
intros. apply has_regular_e in H.
destruct H as [[HV | [x Eq]] Hwf].
- assumption.
- subst. inversion Hwf; subst. false. apply* binds_empty_inv.
Qed.
Hint Constructors psub.
Lemma psub_sub: forall S T,
psub S T -> sub empty S T.
Proof.
intros. induction H; eauto.
- apply* sub_and11.
- apply* sub_and12.
- apply* sub_and2.
- apply* sub_or21.
- apply* sub_or22.
- apply* sub_or1.
- apply* sub_sel1. apply* has_mem.
- apply* sub_sel2. apply* has_mem.
- apply* sub_mem.
- apply_fresh* sub_all as y.
rewrite concat_empty_l. auto*.
- eapply sub_trans; eauto.
Qed.
Inductive possible_types : nat -> trm -> typ -> Prop :=
| pt_top : forall n v, value v -> wfe empty v -> possible_types n v typ_top
| pt_mem : forall n T S U, psub S T -> psub T U -> possible_types n (trm_mem T) (typ_mem S U)
| pt_all : forall L n V V' e1 T1 T1',
(forall X, X \notin L -> typing (X ~ V) (e1 open_e_var X) (T1 open_t_var X)) ->
psub V' V ->
(forall X, X \notin L -> sub (X ~ V') (T1 open_t_var X) (T1' open_t_var X)) ->
possible_types (S n) (trm_abs V e1) (typ_all V' T1')
| pt_all_shallow : forall V V' e1 T1',
wfe empty (trm_abs V e1) -> wft empty (typ_all V' T1') ->
possible_types 0 (trm_abs V e1) (typ_all V' T1')
| pt_sel : forall n v S,
possible_types n v S ->
possible_types n v (typ_sel (trm_mem S))
| pt_and : forall n v T1 T2,
possible_types n v T1 ->
possible_types n v T2 ->
possible_types n v (typ_and T1 T2)
| pt_or1 : forall n v T1 T2,
possible_types n v T1 ->
wft empty T2 ->
possible_types n v (typ_or T1 T2)
| pt_or2 : forall n v T1 T2,
possible_types n v T2 ->
wft empty T1 ->
possible_types n v (typ_or T1 T2)
.
Lemma possible_types_value : forall n p T,
possible_types n p T ->
value p.
Proof.
introv Hpt. induction Hpt; eauto.
- apply psub_sub in H. auto*.
- apply value_abs. apply_fresh* term_abs as y.
apply psub_sub in H0. auto*.
assert (y \notin L) as Fr by auto.
specialize (H y Fr). apply typing_regular in H.
destruct H as [? [A ?]]. apply* wfe_term.
- apply value_abs. apply* wfe_term.
Qed.
Lemma possible_types_wfe : forall n p T,
possible_types n p T ->
wfe empty p.
Proof.
introv Hpt. induction Hpt; eauto.
- apply psub_sub in H. auto*.
- apply_fresh* wfe_abs as y.
apply psub_sub in H0. auto*.
assert (y \notin L) as FrL by auto. specialize (H y FrL).
apply typing_regular in H. destruct H as [? [A ?]].
rewrite concat_empty_l. assumption.
Qed.
Lemma possible_types_wft : forall n p T,
possible_types n p T ->
wft empty T.
Proof.
introv Hpt. induction Hpt; eauto.
- apply psub_sub in H. apply psub_sub in H0. apply* wft_mem.
- apply_fresh* wft_all as y.
apply psub_sub in H0. auto*.
assert (y \notin L) as FrL by auto. specialize (H1 y FrL).
apply sub_regular in H1. destruct H1 as [? [? A]].
rewrite concat_empty_l. assumption.
- apply wft_sel. left. apply value_mem. apply* wfe_term. apply* wfe_mem.
Grab Existential Variables.
pick_fresh y. apply y.
pick_fresh y. apply y.
pick_fresh y. apply y.
Qed.
Lemma has_empty_var_false: forall x T,
has empty (trm_fvar x) T ->
False.
Proof.
intros.
remember (trm_fvar x) as p. generalize dependent x.
remember empty as E. gen HeqE.
induction H; intros; subst; eauto.
- apply* binds_empty_inv.
- inversion Heqp.
- inversion Heqp.
Qed.
Lemma possible_types_closure_psub : forall n v T U,
possible_types n v T ->
psub T U ->
possible_types n v U.
Proof.
introv Hpt Hsub. generalize dependent v.
induction Hsub; intros; subst; eauto.
- inversion Hpt.
- apply pt_top. apply* possible_types_value. apply* possible_types_wfe.
- inversion Hpt; subst. apply IHHsub. assumption.
- inversion Hpt; subst. apply IHHsub. assumption.
- apply* pt_and.
- apply* pt_or1.
- apply* pt_or2.
- inversion Hpt; subst. apply IHHsub1. assumption. apply IHHsub2. assumption.
- inversion Hpt; subst. assumption.
- apply* pt_sel.
- inversion Hpt; subst.
apply pt_mem. eapply psub_trans; eauto. eapply psub_trans; eauto.
- inversion Hpt; subst. apply_fresh* pt_all as y.
eapply sub_trans.
eapply sub_narrowing_empty. eapply psub_sub. eassumption. auto*. auto*.
apply pt_all_shallow; eauto. apply_fresh* wft_all as y.
apply psub_sub in Hsub. auto*.
rewrite concat_empty_l.
assert (y \notin L) as Fr by auto. specialize (H y Fr). auto*.
Qed.
Lemma psub_reflexivity : forall T,
wft empty T ->
psub T T .
Proof.
introv WI. lets W: (wft_type WI). remember empty as E. gen E.
induction W; intros; inversions WI; eauto 4.
apply* psub_and2.
apply* psub_or1.
apply_fresh* psub_all as y.
assert (y \notin L0)as Fr0 by auto. specialize (H5 y Fr0).
rewrite concat_empty_l in H5.
apply* sub_reflexivity. rewrite <- concat_empty_l. apply* okt_push.
Qed.
Lemma sub_psub_aux:
(forall E S T, sub E S T -> E = empty -> psub S T) /\
(forall E p T, has E p T -> E = empty -> possible_types 0 p T).
Proof.
apply sub_has_mutind; intros; subst; eauto 4.
- specialize (H eq_refl).
inversion H; subst.
eapply psub_trans. eapply psub_sel1.
apply psub_sub in H4. auto*.
assumption.
- specialize (H eq_refl).
inversion H; subst.
eapply psub_trans. eassumption. eapply psub_sel2.
apply psub_sub in H4. auto*.
- apply_fresh* psub_all as y.
rewrite <- (@concat_empty_l typ (y ~ T1)). auto*.
- false. apply* binds_empty_inv.
- apply pt_mem; eauto.
- apply* pt_all_shallow.
- eapply possible_types_closure_psub; eauto.
Qed.
Lemma sub_psub: forall S T,
sub empty S T -> psub S T.
Proof.
intros. apply* (proj1 sub_psub_aux).
Qed.
Lemma possible_types_closure : forall n v T U,
possible_types n v T ->
sub empty T U ->
possible_types n v U.
Proof.
intros. eapply possible_types_closure_psub; eauto 4.
apply* sub_psub.
Qed.
Lemma possible_types_typing : forall v T,
typing empty v T -> value v ->
possible_types 1 v T.
Proof.
introv Ht Hv.
remember Ht as Hc. clear HeqHc.
remember empty as E. generalize HeqE. generalize Hc.
induction Ht; intros; subst; eauto; try solve [inversion Hv].
- eapply typing_regular in Hc. destruct Hc as [? [? Hc]].
inversion Hc; subst.
apply_fresh pt_all as Y.
assert (Y \notin L) as Fr by eauto.
specialize (H Y Fr). rewrite concat_empty_l in H. eapply H.
eapply psub_reflexivity; eauto. eapply sub_reflexivity; eauto.
rewrite <- concat_empty_l. eauto.
assert (Y \notin L0) as Fr by eauto.
specialize (H7 Y Fr). rewrite concat_empty_l in H7. eapply H7.
- apply pt_mem. apply* psub_reflexivity. apply* psub_reflexivity.
- eapply possible_types_closure; eauto.
Qed.
Lemma typing_inv_abs : forall S1 e1 T,
typing empty (trm_abs S1 e1) T ->
forall U1 U2, sub empty T (typ_all U1 U2) ->
sub empty U1 S1
/\ exists S2, exists L, forall x, x \notin L ->
typing (x ~ S1) (e1 open_e_var x) (S2 open_t_var x) /\ sub (x ~ U1) (S2 open_t_var x) (U2 open_t_var x).
Proof.
introv Typ Hsub.
apply possible_types_typing in Typ; eauto.
assert (possible_types 1 (trm_abs S1 e1) (typ_all U1 U2)) as Hc. {
eapply possible_types_closure; eauto.
}
inversion Hc; subst.
repeat eexists; eauto.
apply* psub_sub.
Qed.
(** Canonical Forms (14) *)
Lemma canonical_form_abs : forall t U1 U2,
value t -> typing empty t (typ_all U1 U2) ->
exists V, exists e1, t = trm_abs V e1.
Proof.
introv Val Typ.
eapply possible_types_typing in Typ; eauto.
inversion Typ; subst; eauto.
Qed.
Lemma canonical_form_mem : forall t b T,
value t -> typing empty t (typ_mem b T) ->
exists V, t = trm_mem V.
Proof.
introv Val Typ.
eapply possible_types_typing in Typ; eauto.
inversion Typ; subst; eauto.
Qed.
Lemma typing_through_subst1 : forall V y v e T,
typing (y ~ V) e T ->
value v -> typing empty v V ->
typing empty (subst_e y v e) (subst_t y v T).
Proof.
intros.
assert (empty & map (subst_t y v) empty = empty) as A. {
rewrite concat_empty_l. rewrite map_empty. reflexivity.
}
rewrite <- A. eapply typing_through_subst.
rewrite concat_empty_l. rewrite concat_empty_r. eauto.
left. assumption. assumption.
Qed.
(* ********************************************************************** *)
(** Preservation Result (20) *)
Lemma value_red_contra: forall e e',
value e -> red e e' -> False.
Proof.
introv Hv Hr. inversion Hv; subst; inversion Hr; subst; eauto.
Qed.
Lemma preservation_result : preservation.
Proof.
introv Typ. gen_eq E: (@empty typ). gen e'.
induction Typ; introv QEQ; introv Red;
try solve [inversion Typ; congruence]; try solve [ inversion Red ].
- (* case: app *)
inversions Red; try solve [ apply* typing_app ].
destruct~ (typing_inv_abs Typ1 (U1:=T1) (U2:=T2)) as [P1 [S2 [L P2]]].
apply* sub_reflexivity.
pick_fresh X. forwards~ K: (P2 X). destruct K.
rewrite* (@subst_e_intro X).
erewrite <- (proj1 subst_fresh).
eapply typing_through_subst1.
eapply typing_sub. eapply typing_narrowing_empty. eapply P1. eassumption.
rewrite <- (@open_t_var_type X).
assert (X \notin L) as FrL by auto.
specialize (P2 X FrL). destruct P2 as [P2t P2s].
eassumption. apply* wft_type. assumption. assumption. auto*.
- (* case: appvar *)
inversions Red; try solve [ apply* typing_appvar ].
lets HV2: (has_empty_value H). false. eapply value_red_contra in HV2; eauto.
destruct~ (typing_inv_abs Typ1 (U1:=T1) (U2:=T2)) as [P1 [S2 [L P2]]].
apply* sub_reflexivity.
pick_fresh X. forwards~ K: (P2 X). destruct K.
rewrite* (@subst_t_intro X).
rewrite* (@subst_e_intro X).
eapply typing_through_subst1.
eapply typing_sub. eapply typing_narrowing_empty. eapply P1. eassumption.
assert (X \notin L) as FrL by auto.
specialize (P2 X FrL). destruct P2 as [P2t P2s].
eassumption. assumption. assumption.
- (* case sub *)
apply* typing_sub.
Qed.
(* ********************************************************************** *)
(** * Progress *)
(* ********************************************************************** *)
(* ********************************************************************** *)
(** Progress Result (16) *)
Lemma progress_result : progress.
Proof.
introv Typ. gen_eq E: (@empty typ). lets Typ': Typ.
induction Typ; intros EQ; subst.
- (* case: var *)
false* binds_empty_inv.
- (* case: abs *)
left*.
- (* case: mem *)
left*.
- (* case: app *)
right. destruct* IHTyp1 as [Val1 | [e1' Rede1']].
destruct* IHTyp2 as [Val2 | [e2' Rede2']].
destruct (canonical_form_abs Val1 Typ1) as [S [e3 EQ]].
subst. exists* (open_e e3 e2).
- (* case: appvar *)
right. destruct* IHTyp1 as [Val1 | [e1' Rede1']].
destruct* IHTyp2 as [Val2 | [e2' Rede2']].
destruct (canonical_form_abs Val1 Typ1) as [S [e3 EQ]].
subst. exists* (open_e e3 e2).
- (* case: sub *)
auto*.
Qed.
|
!! these functions could be implemented via C runtime library,
!! but for speed/ease of implementation, for now we use
!! compiler-specific intrinsic functions
submodule (pathlib) pathlib_gcc
implicit none (type, external)
contains
module procedure cwd
integer :: i
character(4096) :: work
i = getcwd(work)
if(i /= 0) error stop "could not get CWD"
cwd = trim(work)
end procedure cwd
module procedure is_dir
integer :: i, statb(13)
character(:), allocatable :: wk
wk = expanduser(path)
!! must not have trailing slash on Windows
i = len_trim(wk)
if (wk(i:i) == '/') wk = wk(1:i-1)
inquire(file=wk, exist=is_dir)
if(.not.is_dir) return
i = stat(wk, statb)
if(i /= 0) then
is_dir = .false.
return
endif
i = iand(statb(3), O'0040000')
is_dir = i == 16384
! print '(O8)', statb(3)
end procedure is_dir
module procedure size_bytes
character(:), allocatable :: wk
integer :: s(13), i
size_bytes = 0
wk = expanduser(path)
i = stat(wk, s)
if(i /= 0) then
write(stderr,*) "size_bytes: could not stat file: ", wk
return
endif
if (iand(s(3), O'0040000') == 16384) then
write(stderr,*) "size_bytes: is a directory: ", wk
return
endif
size_bytes = s(8)
end procedure size_bytes
module procedure is_exe
character(:), allocatable :: wk
integer :: s(13), iu, ig, i
is_exe = .false.
wk = expanduser(path)
i = stat(wk, s)
if(i /= 0) then
write(stderr,*) "is_exe: could not stat file: ", wk
return
endif
if (iand(s(3), O'0040000') == 16384) then
write(stderr,*) "is_exe: is a directory: ", wk
return
endif
iu = iand(s(3), O'0000100')
ig = iand(s(3), O'0000010')
is_exe = (iu == 64 .or. ig == 8)
end procedure is_exe
end submodule pathlib_gcc
|
module ModSizeGitm
integer, parameter :: nLons = 10
integer, parameter :: nLats = 10
integer, parameter :: nAlts = 50
integer, parameter :: nBlocksMax = 4
integer :: nBlocks
end module ModSizeGitm
|
(** * MoreCoq: Mas Sobre Coq *)
Require Export Poly.
(** Este capitulo introduce varias tacticas que, en conjunto, nos
ayudan a demostrar muchos teoremas sobre los programas funcionales
que estuvimos escribiendo. *)
(* ###################################################### *)
(** * La Tactica [apply] *)
(** Usualmente nos encontramos en situaciones en que el objetivo a ser
demostrado es exactamente lo mismo que alguna hipotesis en el
contexto o un lema previo. *)
Theorem silly1 : forall (n m o p : nat),
n = m ->
[n;o] = [n;p] ->
[n;o] = [m;p].
Proof.
intros n m o p eq1 eq2.
rewrite <- eq1.
(* En este punto, podemos concluir con
"[rewrite -> eq2. reflexivity.]" como hemos hecho varias veces
anteriormente. Pero podemos lograr lo mismo en un solo paso
usando la tactica [apply]: *)
apply eq2. Qed.
(** La tactica [apply] tambien funciona con hipotesis _condicionales_
y lemas: si el lema siendo aplicado es una implicacion, entonces
las premisas de esta implicacion van a ser agregadas a nuestra
lista de sub-objetivos a demostrar. *)
Theorem silly2 : forall (n m o p : nat),
n = m ->
(forall (q r : nat), q = r -> [q;o] = [r;p]) ->
[n;o] = [m;p].
Proof.
intros n m o p eq1 eq2.
apply eq2. apply eq1. Qed.
(** Puede encontrar instructivo experimentar con esta prueba y ver si
existe forma de utilizar [rewrite] para resolverla, en vez de
[apply]. *)
(** Tipicamente, cuando usamos [apply H], el lema (o hipotesis) [H] va
a comenzar con un ligador [forall] ligando _variables
universales_. Cuando Coq "matchea" (unifica) el objetivo actual
contra la conclusion de [H], va a intentar encontrar valores
apropiados para estas variables. Por ejemplo, cuando hacemos
[apply eq2] en la siguente prueba, la variable universal [q] en
[eq2] es instanciada con [n] y [r] es instanciada con [m]. *)
Theorem silly2a : forall (n m : nat),
(n,n) = (m,m) ->
(forall (q r : nat), (q,q) = (r,r) -> [q] = [r]) ->
[n] = [m].
Proof.
intros n m eq1 eq2.
apply eq2. apply eq1. Qed.
(** **** Ejercicio: 2 stars, opcional (silly_ex) *)
(** Complete la siguiente prueba sin utilizar [simpl]. *)
Theorem silly_ex :
(forall n, evenb n = true -> oddb (S n) = true) ->
evenb 3 = true ->
oddb 4 = true.
Proof.
(* FILL IN HERE *) Admitted.
(** [] *)
(** Para usar la tactica [apply], la (conclusion del) lema siendo
aplicado tiene que matchear el objetivo _exactamente_ -- por
ejemplo, [apply] no va a funcionar si el lado izquierdo y el lado
derecho de la igualdad estan intercambiados. *)
Theorem silly3_firsttry : forall (n : nat),
true = beq_nat n 5 ->
beq_nat (S (S n)) 7 = true.
Proof.
intros n H.
simpl.
(* Aqui no podemos utilizar la tactica [apply] directamente *)
Abort.
(** En este caso podemos utilizar la tactica [symmetry], que
intercambia los lados izquierdo y derecho de una igualdad en el
objetivo. *)
Theorem silly3 : forall (n : nat),
true = beq_nat n 5 ->
beq_nat (S (S n)) 7 = true.
Proof.
intros n H.
symmetry.
simpl. (* De hecho, este [simpl] no es necesario, puesto
que [apply] hace un paso de [simpl] primero. *)
apply H. Qed.
(** **** Ejercicio: 3 stars (apply_exercise1) *)
(** Ayuda: usted puede usar [apply] con lemas definidos previamente,
no solo hipotesis en el contexto. Recuerde que [SearchAbout] es
su amigo! *)
Theorem rev_exercise1 : forall (l l' : list nat),
l = rev l' ->
l' = rev l.
Proof.
(* FILL IN HERE *) Admitted.
(** [] *)
(** **** Ejercicio: 1 star, optional (apply_rewrite) *)
(** Explique brevemente la diferencia entre las tacticas [apply] y
[rewrite]. Hay situaciones en las que ambas puedan ser
exitosamente aplicadas?
(* FILL IN HERE *)
*)
(** [] *)
(* ###################################################### *)
(** * La Tactica [apply ... with ...] *)
(** El siguiente ejemplo tonto utiliza dos rewrites seguidos para ir
de [[a,b]] a [[e,f]]. *)
Example trans_eq_example : forall (a b c d e f : nat),
[a;b] = [c;d] ->
[c;d] = [e;f] ->
[a;b] = [e;f].
Proof.
intros a b c d e f eq1 eq2.
rewrite -> eq1. rewrite -> eq2. reflexivity. Qed.
(** Como es comun tener este tipo de situaciones, podemos abstraer en
un lema el hecho de que la igualdad es transitiva. *)
Theorem trans_eq : forall (X:Type) (n m o : X),
n = m -> m = o -> n = o.
Proof.
intros X n m o eq1 eq2. rewrite -> eq1. rewrite -> eq2.
reflexivity. Qed.
(** Ahora, deberiamos poder utilizar [trans_eq] para provar el ejemplo
de mas arriba. Sin embargo, para hacer esto necesitamos una
pequenia variacion de la tactica [apply]. *)
Example trans_eq_example' : forall (a b c d e f : nat),
[a;b] = [c;d] ->
[c;d] = [e;f] ->
[a;b] = [e;f].
Proof.
intros a b c d e f eq1 eq2.
(* Si le decimos a Coq simplemente [apply trans_eq] en este punto,
puede deducir (mediante el matcheo del objetivo con la conclusion
del lema) que debe instanciar [X] con [[nat]], [n] con [[a,b]], y
[o] con [[e,f]]. Sin embargo, el proceso de matcheo no determina
una instanciacion para [m]: tenemos que suplir uno explicitamnete
agregando [with (m:=[c,d])] a la invocacion de [apply]. *)
apply trans_eq with (m:=[c;d]). apply eq1. apply eq2. Qed.
(** De hecho, usualmente no tenemos que incluir el nombre [m] en la
clausula [with]; Coq es muchas veces inteligente y puede darse
cuenta de que instanciacion estamos proveyendo. Podemos escribir
entonces: [apply trans_eq with [c,d]]. *)
(** **** Ejercicio: 3 stars, opcional (apply_with_exercise) *)
Example trans_eq_exercise : forall (n m o p : nat),
m = (minustwo o) ->
(n + p) = m ->
(n + p) = (minustwo o).
Proof.
(* FILL IN HERE *) Admitted.
(** [] *)
(* ###################################################### *)
(** * La Tactica [inversion] *)
(** Recuerde la definicion de numeros naturales:
Inductive nat : Type :=
| O : nat
| S : nat -> nat.
Es claro de esta definicion que cada numero tiene una de dos
formas: o es el constructor [O] o esta construido a partir de
aplicar el constructor [S] a otro numero. Pero hay mas aqui que
lo que el ojo puede ver: implicito en esta definicion (y en
nuestra forma de entender informalmente como las declaraciones de
tipo funcionan en otros lenguajes de programacion) tambien hay
otros dos hechos:
- El constructor [S] es _inyectivo_. Es decir, la unica forma de
tener [S n = S m] es si [n = m].
- Los constructores [O] y [S] son _disjuntos_. Es decir, [O] no
es igual a [S n] para ningun [n]. *)
(** Principios similares se aplican a todos los tipos definidos
inductivamente: todos los constructores son inyectivos, y los
valores construidos con distintos constructores son diferentes.
Para listas, el constructor [cons] es inyectivo y [nil] es
diferente de cualquier lista no vacia. Para booleanos, [true] and
[false] son diferentes. (Como [true] y [false] no toman ningun
argumento, su inyectividad es irrelevante). *)
(** Coq provee una tactica llamada [inversion] que nos permite
explotar estos principios en una prueba.
La tactica [inversion] es usada de la siguiente forma. Suponga
que [H] es una hipotesis en el context (o un lema ya establecido)
de la forma
c a1 a2 ... an = d b1 b2 ... bm
para dos constructores [c] y [d] y argumentos [a1 ... an] y [b1
... bm]. Entonces [inversion H] instruye a Coq a "invertir" esta
igualdad para extraer la informacion que contiene acerca de estos
terminos:
- Si [c] y [d] son el mismo constructor, entonces sabemos, por el
principio de inyectividad de este constructor, que [a1 = b1],
[a2 = b2], etc.; [inversion H] agrega estos conocimientos al
contexto, e intenta utilizarlos para reescribir el objetivo.
- Si [c] y [d] son constructores diferentes, entonces la hipotesis
[H] es contradictoria. Es decir, una premisa falsa ha aparecido
en nuestro contexto, y esto significa que cualquier objetivo es
demostrable! En este caso, [inversion H] marca el objetivo
actual como completo y lo saca del stack de objetivos por
resolver. *)
(** Posiblemente la tactica [inversion] sea mas facil de entender
viendola en accion que en descripciones generales como la de
arriba. Abajo va a encontrar ejemplos de teoremas que muestran el
uso de [inversion] y ejercicios para probar su entendimiento. *)
Theorem eq_add_S : forall (n m : nat),
S n = S m ->
n = m.
Proof.
intros n m eq. inversion eq. reflexivity. Qed.
Theorem silly4 : forall (n m : nat),
[n] = [m] ->
n = m.
Proof.
intros n o eq. inversion eq. reflexivity. Qed.
(** Como conveniencia, la tactica [inversion] tambien puede destruir
igualdades entre valores complejos, ligando multiples variables a
la vez. *)
Theorem silly5 : forall (n m o : nat),
[n;m] = [o;o] ->
[n] = [m].
Proof.
intros n m o eq. inversion eq. reflexivity. Qed.
(** **** Ejercicio: 1 star (sillyex1) *)
Example sillyex1 : forall (X : Type) (x y z : X) (l j : list X),
x :: y :: l = z :: j ->
y :: l = x :: j ->
x = y.
Proof.
(* FILL IN HERE *) Admitted.
(** [] *)
Theorem silly6 : forall (n : nat),
S n = O ->
2 + 2 = 5.
Proof.
intros n contra. inversion contra. Qed.
Theorem silly7 : forall (n m : nat),
false = true ->
[n] = [m].
Proof.
intros n m contra. inversion contra. Qed.
(** **** Ejercicio: 1 star (sillyex2) *)
Example sillyex2 : forall (X : Type) (x y z : X) (l j : list X),
x :: y :: l = [] ->
y :: l = z :: j ->
x = z.
Proof.
(* FILL IN HERE *) Admitted.
(** [] *)
(** Mientras que la inyectividad de los constructores nos permite
razonar acerca de [forall (n m : nat), S n = S m -> n = m], la
direccion inversa de la implicacion es una instancia de un caso
mas general acerca de constructores y funciones, que vamos a
encontrar util: *)
Theorem f_equal : forall (A B : Type) (f: A -> B) (x y: A),
x = y -> f x = f y.
Proof. intros A B f x y eq. rewrite eq. reflexivity. Qed.
(** Aqui hay otro ejemplo de [inversion]. Este ejemplo es una
modificacion de lo que probamos arriba. Las igualdades extras nos
forzan a hacer un poco de razonamiento ecuacional y ejercitar las
tacticas que vimos recientemente. *)
Theorem length_snoc' : forall (X : Type) (v : X)
(l : list X) (n : nat),
length l = n ->
length (snoc l v) = S n.
Proof.
intros X v l. induction l as [| v' l'].
Case "l = []". intros n eq. rewrite <- eq. reflexivity.
Case "l = v' :: l'". intros n eq. simpl. destruct n as [| n'].
SCase "n = 0". inversion eq.
SCase "n = S n'".
apply f_equal. apply IHl'. inversion eq. reflexivity. Qed.
(** **** Ejercicio: 2 stars, opcional (practice) *)
(** Un par de ejemplos no triviales pero tampoco tan complicados para
ejercitar estos conceptos. Pueden requerir lemas ya provados
anteriormente. *)
Theorem beq_nat_0_l : forall n,
beq_nat 0 n = true -> n = 0.
Proof.
(* FILL IN HERE *) Admitted.
Theorem beq_nat_0_r : forall n,
beq_nat n 0 = true -> n = 0.
Proof.
(* FILL IN HERE *) Admitted.
(** [] *)
(* ###################################################### *)
(** * Usando Tacticas en las Hipotesis *)
(** Por defecto, la mayoria de las tacticas funcionan en el objetivo y
dejan el contexto sin cambiar. Sin embargo, la mayoria de las
tacticas tambien tienen una variante que realiza una operacion
similar en una premisa del contexto.
Por ejemplo, la tactica [simpl in H] realiza una simplificaion en
la hipotesis llamada [H] en el contexto. *)
Theorem S_inj : forall (n m : nat) (b : bool),
beq_nat (S n) (S m) = b ->
beq_nat n m = b.
Proof.
intros n m b H. simpl in H. apply H. Qed.
(** De forma similar, la tactica [apply L in H] matchea algun lema o
hipotesis condicional [L] (de la forma [L1 -> L2], digamos) contra
una hipotesis [H] en el contexto. Sin embargo, a diferencia del
[apply] ordinario (que reescribe el objetivo matcheando [L2] en el
sub-objetivo [L1]), [apply L in H] matchea [H] contra [L1] y, si
es exitoso, lo reemplaza con [L2].
En otras palabras, [apply L in H] nos da una forma de
"razonamiento hacia adelante" -- de [L1 -> L2] y una hipotesis
matcheando [L1], nos da una hipotesis matcheando [L2]. En
contraste, [apply L] es "razonamiento hacia atras" -- dice que si
sabemos [L1->L2] y queremos probar [L2], es suficiente con probar
[L1].
Aqui hay una variante de una prueba de arriba, usando razonamiento
hacia adelante en toda la prueba, en vez de hacia atras. *)
Theorem silly3' : forall (n : nat),
(beq_nat n 5 = true -> beq_nat (S (S n)) 7 = true) ->
true = beq_nat n 5 ->
true = beq_nat (S (S n)) 7.
Proof.
intros n eq H.
symmetry in H. apply eq in H. symmetry in H.
apply H. Qed.
(** El razonamiento hacia adelante empieza desde lo que esta _dado_
(las premisas, teoremas provados anteriormente), e iterativamente
obtiene conclusiones desde ellos hasta que el objetivo es
encontrado. El razonamiento hacia atras empieza desde el
_objetivo_, e iterativamente razona acerca que puede implicar el
objetivo, hasta llegar a las premisas o teoremas existentes. Si
usted ha visto pruebas informales antes (por ejemplo, en
matematica o en una clase de ciencias de la computacion),
probablemente hayan utilizado razonamiento hacia adelante. En
general, Coq tiende a favorecer razonamiento hacia atras, pero en
algunas situaciones el estilo de razonamiento hacia adelante puede
ser mas facil de usar o de pensar. *)
(** **** Ejercicio: 3 stars (plus_n_n_injective) *)
(** Practique utilizando variantes de "in" en este ejercicio. *)
Theorem plus_n_n_injective : forall n m,
n + n = m + m ->
n = m.
Proof.
intros n. induction n as [| n'].
(* Ayuda: utilice el lema plus_n_Sm *)
(* FILL IN HERE *) Admitted.
(** [] *)
(* ###################################################### *)
(** * Variando la Hipotesis Inductiva *)
(** A veces es importante controlar la forma exacta de la hipotesis
inductiva. En particular, necesitamos ser cuidadosos acerca de
que premisas movemos del objetivo al contexto (usando [intros])
antes de invocar la tactica [induction]. Por ejemplo, suponga que
queremos mostrar que la funcion [double] es inyectiva -- es decir,
que siempre mapea diferentes argumentos a diferentes resultados:
Theorem double_injective: forall n m, double n = double m -> n = m.
La forma que _empezamos_ esta demostracion es un poco delicada: si
comenzamos con [intros n. induction n.] todo va bien. Pero si
comenzamos con [intros n m. induction n.] nos quedamos estancados
en el medio del caso inductivo... *)
Theorem double_injective_FAILED : forall n m,
double n = double m ->
n = m.
Proof.
intros n m. induction n as [| n'].
Case "n = O". simpl. intros eq. destruct m as [| m'].
SCase "m = O". reflexivity.
SCase "m = S m'". inversion eq.
Case "n = S n'". intros eq. destruct m as [| m'].
SCase "m = O". inversion eq.
SCase "m = S m'". apply f_equal.
(* Aqui estamos estancados. La hipotesis inductiva, [IHn'], nos
da [n' = m'] -- hay un extra [S] en el camino -- asi que este
objetivo no es demostrable. *)
Abort.
(** Que salio mal? *)
(** El problema es que, en el punto en que invocamos la hipotesis
inductiva, ya hemos introducido [m] en el contexto --
intuitivamente le dijimos a Coq "Consideremos unos [n] y [m]
particulares..." y ahora vamos a demostrar que, si [double n =
double m] para _este [n] y [m] en particular_, entonces [n = m].
La siguiente tactica, [induction n], le dice a Coq: Ahora vamos a
mostrar el objetivo por induccion en [n]. Es decir, vamos a
provar que la proposicion
- [P n] = "si [double n = double m], entonces [n = m]"
vale para todo [n] mostrando
- [P O]
(es decir, "si [double O = double m] entonces [O = m]")
- [P n -> P (S n)]
(es decir, "si [double n = double m] entonces [n = m]" implica "si
[double (S n) = double m] entonces [S n = m]").
Si miramos en detalle al segundo objetivo, esta diciendo algo un
poco extranio: dice que, para un [m] _en particular_, si sabemos
- "si [double n = double m] entonces [n = m]"
podemos probar
- "si [double (S n) = double m] entonces [S n = m]".
Para ver porque esto es extranio, pensemos en un [m] particular --
digamos, [5]. Este objetivo dice que, si sabemos
- [Q] = "si [double n = 10] entonces [n = 5]"
entonces podemos probar
- [R] = "si [double (S n) = 10] entonces [S n = 5]".
Pero sabiendo [Q] no nos ayuda a probar [R]! (Si intentaramos
probar [R] a partir de [Q], deberiamos decir algo como "Suponga
[double (S n) = 10]..." pero entonces nos quedamos estancados:
sabiendo que [double (S n)] is [10] no nos dice nada acerca de que
[double n] es [10], asi que [Q] es inutil.) *)
(** Para sumarizar: Intentar hacer esta prueba por induccion en [n]
cuando [m] esta en el contexto no funciona porque estamos tratando
de probar una relacion involucrando _todo_ [n] pero solo un
_unico_ [m]. *)
(** La prueba adecuada de [double_injective] deja [m] en el objetivo
antes de invocar [induction] en [n]: *)
Theorem double_injective : forall n m,
double n = double m ->
n = m.
Proof.
intros n. induction n as [| n'].
Case "n = O". simpl. intros m eq. destruct m as [| m'].
SCase "m = O". reflexivity.
SCase "m = S m'". inversion eq.
Case "n = S n'".
(* Note que ahora el objetivo y la hipotesis inductiva cambiaron:
el objetivo pide demostrar algo mas general (es decir, probar
la propiedad para _todo_ [m]), pero la hipotesis inductiva es
correspondientemente mas flexible, permitiendonos elegir
cualquier [m] que queramos cuando la apliquemos. *)
intros m eq.
(* Ahora elegimos al [m] en particular e introducimos la premisa
que [double n = double m]. Como estamos haciendo analisis por
caso en [n], tenemos que hacer analisis por caso en [m] para
mantener las dos "sincronizadas". *)
destruct m as [| m'].
SCase "m = O".
(* El caso 0 es trivial *)
inversion eq.
SCase "m = S m'".
apply f_equal.
(* En este punto, como estamos en la segunda rama de [destruct
m], la variable [m'] mencionada en el contexto en este punto
es de hecho el predecesor de la que veniamos hablando. Y
como ademas estamos la rama [S] de la induccion, esto es
perfecto: si instanciamos la [m] generica de la hipotesis
inductiva con el [m'] que estamos mencionando ahora
(instanciacion hecha automaticamente por [apply]), entonces
[IHn'] nos da exactamente lo que necesitamos para terminar la
prueba. *)
apply IHn'. inversion eq. reflexivity. Qed.
(** Lo que esto nos ensenia es que tenemos que ser cuidadosos cuando
usamos induccion para evitar caer en un caso muy especifico: Si
estamos provando una propiedad en [n] y [m] por induccion en [n],
tal vez sea una buena idea idea dejar [m] generica. *)
(** La demostracion de este teorema tiene que ser tratada similarmente: *)
(** **** Ejercicio: 2 stars (beq_nat_true) *)
Theorem beq_nat_true : forall n m,
beq_nat n m = true -> n = m.
Proof.
(* FILL IN HERE *) Admitted.
(** [] *)
(** **** Ejercicio: 2 stars, avanzado (beq_nat_true_informal) *)
(** De una prueba informal de [beq_nat_true], siendo tan explicita
como se pueda acerca de los cuantificadores. *)
(* FILL IN HERE *)
(** [] *)
(** La estrategia de hacer menos [intros] antes de una [induction] no
funciona siempre; a veces se necesita hacer un pequenia
_reorganizacion_ de las variabes cuantificadas. Suponga, por
ejemplo, que quisieramos demostrar [double_injective] por
induccion en [m] en vez de [n]. *)
Theorem double_injective_take2_FAILED : forall n m,
double n = double m ->
n = m.
Proof.
intros n m. induction m as [| m'].
Case "m = O". simpl. intros eq. destruct n as [| n'].
SCase "n = O". reflexivity.
SCase "n = S n'". inversion eq.
Case "m = S m'". intros eq. destruct n as [| n'].
SCase "n = O". inversion eq.
SCase "n = S n'". apply f_equal.
(* Estancados aca de nuevo, igual que antes. *)
Abort.
(** El problema es que, para acer induccion en [m], queremos pimero
introducir [n]. (Si simplemente decimos [induction m] sin
introducir nada antes, Coq va a introducir automaticamente [n] por
nosotros!) *)
(** Que podemos hacer con esto? Una posibilidad es reescribir el lema
de forma que [m] sea cuantificada antes que [n]. Esto funciona,
pero no es elegante: No queremos adaptar los lemas para satisfacer
las necesidades de la estrategia de la prueba -- queremos
especificarlo en la forma mas natural y comprensible. *)
(** Lo que podemos hacer, en vez, es introducir todas las variables
cuantificadas y luego _re-generalizar_ ona o mas variables,
tomandolas del contexto y poniendolas de vuelta en el objetivo.
La tactica [generalize dependent] hace esto. *)
Theorem double_injective_take2 : forall n m,
double n = double m ->
n = m.
Proof.
intros n m.
(* [n] y [m] estan las dos en el contexto *)
generalize dependent n.
(* Ahora [n] esta devuelta en el objetivo, y ahora podemos hacer
induccion en [m] y obtener una HI suficientemente general. *)
induction m as [| m'].
Case "m = O". simpl. intros n eq. destruct n as [| n'].
SCase "n = O". reflexivity.
SCase "n = S n'". inversion eq.
Case "m = S m'". intros n eq. destruct n as [| n'].
SCase "n = O". inversion eq.
SCase "n = S n'". apply f_equal.
apply IHm'. inversion eq. reflexivity. Qed.
(** Miremos a una prueba informal de este teorema. Note que la
proposicion que provamos por induccion deja [n] cuantificado,
correspondiendo al uso de [generalize dependent] en la prueba
formal.
_Teorema_: Para todos naturales [n] y [m], si [double n = double m],
entonces [n = m].
_Demostracion_: Sea [m] un [nat]. Provamos por induccion en [m] que,
para cualquier [n], si [double n = double m] entonces [n = m].
- Primero suponga [m = 0], y suponga que [n] es un numero
tal que [double n = double m]. Debemos mostrar que [n = 0].
Como [m = 0], por definicion de [double] tenemos [double n = 0].
Tenemos que considerar dos casos para [n]. Si [n = 0] entonces ya
esta, puesto que esto es lo que queriamos probar. En otro caso,
si [n = S n'] para algun [n'], derivamos una contradiccion: por la
definicion de [double] obtenemos que [double n = S (S (double
n'))], pero esto contradice la premisa de que [double n = 0].
- En otro caso, suponga [m = S m'] y que [n] es, de vuelta, un
number tal que [double n = double m]. Debemos mostrar que [n = S
m'], con la hipotesis inductiva que para cualquier numero [s], si
[double s = double m'] entonces [s = m'].
Dado que [m = S m'] y la definicion de [double], tenemos que
[double n = S (S (double m'))]. Hay dos casos que considerar para
[n].
Si [n = 0], entonces por definicion [double n = 0], una
contradiccion. Entonces, tenemos que asumir que [n = S n'] para
algun [n'], y de vuelta por definicion de [double] tenemos [S (S
(double n')) = S (S (double m'))], que por inversion implica
[double n' = double m'].
Instanciando la hipotesis inductiva con [n'] entonces nos permite
concluir que [n' = m'], y en consecuencia, [S n' = S m']. Como [S
n' = n] y [S m' = m], esto es justo lo que queriamos probar. [] *)
(** **** Ejercicio: 3 stars (gen_dep_practice) *)
(** Demuestre por induccion en [l]. *)
Theorem index_after_last: forall (n : nat) (X : Type) (l : list X),
length l = n ->
index n l = None.
Proof.
(* FILL IN HERE *) Admitted.
(** [] *)
(** **** Ejercicio: 3 stars, avanzado, opcional (index_after_last_informal) *)
(** Escriba una prueba informal correspondiendo a su prueba de Coq de
[index_after_last]:
_Teorema_: Para todo conjunto [X], listas [l : list X], y numero
[n], si [length l = n] entonces [index n l = None].
_Demostracion_:
(* FILL IN HERE *)
[]
*)
(** **** Ejercicio: 3 stars, opcional (gen_dep_practice_more) *)
(** Demuestre lo siguiente por induccion en [l]. *)
Theorem length_snoc''' : forall (n : nat) (X : Type)
(v : X) (l : list X),
length l = n ->
length (snoc l v) = S n.
Proof.
(* FILL IN HERE *) Admitted.
(** [] *)
(** **** Ejercicio: 3 stars, opcional (app_length_cons) *)
(** Demuestre esto por induccion en [l1], sin utilizar [app_length]. *)
Theorem app_length_cons : forall (X : Type) (l1 l2 : list X)
(x : X) (n : nat),
length (l1 ++ (x :: l2)) = n ->
S (length (l1 ++ l2)) = n.
Proof.
(* FILL IN HERE *) Admitted.
(** [] *)
(** **** Ejercicio: 4 stars, opcional (app_length_twice) *)
(** Demuestre esto por induccion en [l], sin utilizar [app_length]. *)
Theorem app_length_twice : forall (X:Type) (n:nat) (l:list X),
length l = n ->
length (l ++ l) = n + n.
Proof.
(* FILL IN HERE *) Admitted.
(** [] *)
(* ###################################################### *)
(** * Usando [destruct] en Expresiones Compuestas *)
(** Vimos muchos ejemplos donde la tactica [destruct] es utilizada
para realizar analisis por caso en el valor de una variable. Pero
a veces necesitamos razonar por casos en el resultado de alguna
_expresion_. Tambien podemos hacer esto con [destruct].
Aqui hay unos ejemplos: *)
Definition sillyfun (n : nat) : bool :=
if beq_nat n 3 then false
else if beq_nat n 5 then false
else false.
Theorem sillyfun_false : forall (n : nat),
sillyfun n = false.
Proof.
intros n. unfold sillyfun.
destruct (beq_nat n 3).
Case "beq_nat n 3 = true". reflexivity.
Case "beq_nat n 3 = false". destruct (beq_nat n 5).
SCase "beq_nat n 5 = true". reflexivity.
SCase "beq_nat n 5 = false". reflexivity. Qed.
(** Luego de expandir (unfold) [sillyfun] en la definicion de arriba,
nos encontramos con que estamos varados en [if (beq_nat n 3) then
... else ...]. Bueno, o [n] es igual a [3] o no lo es, asi que
[destruct (beq_nat n 3)] nos permite razonar acerca de los dos
casos.
En general, la tactica [destruct] puede ser utilizada para
realizar analisis por caso en los resultados de computaciones
arbitrarias. Si [e] es una expresion cuyo tipo es algun tipo
definido inductivamente [T], entonces, para cada constructor [c]
de [T], [destruct e] genera un sub-objetivo en el cual todas las
ocurrencias de [e] (en el objetivo y en el contexto) son
reemplazadas por [c].
*)
(** **** Ejercicio: 1 star (override_shadow) *)
Theorem override_shadow : forall (X:Type) x1 x2 k1 k2 (f : nat->X),
(override (override f k1 x2) k1 x1) k2 = (override f k1 x1) k2.
Proof.
(* FILL IN HERE *) Admitted.
(** [] *)
(** **** Ejercicio: 3 stars, opcional (combine_split) *)
(** Complete la demostracion de abajo *)
Theorem combine_split : forall X Y (l : list (X * Y)) l1 l2,
split l = (l1, l2) ->
combine l1 l2 = l.
Proof.
(* FILL IN HERE *) Admitted.
(** [] *)
(** A veces, haciendo un [destruct] en una expresion compuesta (una
no-variable) puede borrar informacion que necesitamos para
concluir la prueba. *)
(** Por ejemplo, suponga que definimos la funcion [sillyfun1] de la
siguiente forma: *)
Definition sillyfun1 (n : nat) : bool :=
if beq_nat n 3 then true
else if beq_nat n 5 then true
else false.
(** Y suponga que queremos convencer a Coq de la observacion bastante
obvia que [sillyfun1 n] retorna [true] solo cuando [n] es impar.
En analogia con las demostraciones que hicimos con [sillyfun] de
arriba, es natural pensar la prueba de la siguiente manera: *)
Theorem sillyfun1_odd_FAILED : forall (n : nat),
sillyfun1 n = true ->
oddb n = true.
Proof.
intros n eq. unfold sillyfun1 in eq.
destruct (beq_nat n 3).
(* stuck... *)
Abort.
(** Nos quedamos estancados en este punto porque el contexto no
contiene suficiente informacion para probar el objetivo! El
problema es que la sustitucion realizada por [destruct] es
demasiado brutal -- tira todas las ocurrencias de [beq_nat n 3],
pero a veces es necesario quedarnos con algun recuerdo de esta
expresion y como ha sido destruida, porque debemos ser capaces de
razonar que, en esta rama del analisis por caso, [beq_nat n 3 =
true], y ergo debe ser que [n = 3], de lo cual concluimos que [n]
es impar.
Lo que realmente quisieramos es sustituir todas las ocurrencias de
[beq_nat n 3], pero a la vez quedarnos con una ecuacion en el
contexto que indique en que caso estamos. el modificador [_eqn:]
(o simplemente [eqn:] en Coq 8.4) nos permite obtener esta
ecuacion (con el nombre que querramos ponerle). *)
Theorem sillyfun1_odd : forall (n : nat),
sillyfun1 n = true ->
oddb n = true.
Proof.
intros n eq. unfold sillyfun1 in eq.
destruct (beq_nat n 3) as [] _eqn:Heqe3.
(* Ahora estamos en el mismo punto en el que estabamos cuando nos
quedamos estancados arriba, excepto que ahora tenemos lo que
necesitamos para poder progresar en la demostracion. *)
Case "e3 = true". apply beq_nat_true in Heqe3.
rewrite -> Heqe3. reflexivity.
Case "e3 = false".
(* Cuando llegamos al segundo test de igualdad, podemos utilizar
[_eqn:] de vuelta para poder concluir la prueba. *)
destruct (beq_nat n 5) as [] _eqn:Heqe5.
SCase "e5 = true".
apply beq_nat_true in Heqe5.
rewrite -> Heqe5. reflexivity.
SCase "e5 = false". inversion eq. Qed.
(** **** Ejercicio: 2 stars (destruct_eqn_practice) *)
Theorem bool_fn_applied_thrice :
forall (f : bool -> bool) (b : bool),
f (f (f b)) = f b.
Proof.
(* FILL IN HERE *) Admitted.
(** [] *)
(** **** Ejercicio: 2 stars (override_same) *)
Theorem override_same : forall (X:Type) x1 k1 k2 (f : nat->X),
f k1 = x1 ->
(override f k1 x1) k2 = f k2.
Proof.
(* FILL IN HERE *) Admitted.
(** [] *)
(* ################################################################## *)
(** * Resumen *)
(** Ahora hemos visto una serie de tacticas fundamentales en Coq.
Vamos a introducir algunas mas a medida que avanzemos en los
capitulos, y mas adelante veremos algunas tacticas poderosas para
_automatizar_ mucho del trabajo. Pero basicamente tenemos todo lo
necesario para poder trabajar.
Aqui estan las que hemos vistos:
- [intros]:
mueve las hipotesis/variables desde el objetivo al contexto.
- [reflexivity]:
concluye la demostracion cuando el objetivo es de la forma [e = e].
- [apply]:
demuestra el objetivo utilizando una hipotesis, lema, o constructor.
- [apply... in H]:
aplica una hipotesis, lema, o constructor a otra hipotesis en el
contexto (razonamiento hacia adelante).
- [apply... with...]:
explicita los valores especificos para las variables que no pueden
ser determinadas por simple matcheo.
- [simpl]:
simplifica (reduce cuidadosamente) las computaciones en el objetivo...
- [simpl in H]:
... on en una hipotesis.
- [rewrite]:
utiliza una premisa (o lema) de igualdad para reescribir el objetivo...
- [rewrite ... in H]:
... o una hipotesis.
- [symmetry]:
cambia un objetivo de la forma [t=u] en [u=t].
- [symmetry in H]:
cambia una hipotesis de la forma [t=u] en [u=t]
- [unfold]:
reemplaza la definicion de una constante en el objetivo...
- [unfold... in H]:
... o en una hipotesis
- [destruct... as...]:
analiza por casos los valores de tipos definidos inductivamente.
- [destruct... _eqn:...]:
especifica el nombre de la ecuacion a ser agregada en el contexto,
guardando el resultado del analisis por caso.
- [induction... as...]:
induccion en variables de un tipo inductivo.
- [inversion]:
razonamiento por inyectividad y distincion de constructores.
- [assert (e) as H]:
introduce un "lema local" [e] y lo llama [H].
- [generalize dependent x]:
mueve la variable [x] (y todo lo que dependa de ella)
desde el contexto hacia el objetivo.
*)
(* ###################################################### *)
(** * Ejercicios Adicionales *)
(** **** Ejercicio: 3 stars (beq_nat_sym) *)
Theorem beq_nat_sym : forall (n m : nat),
beq_nat n m = beq_nat m n.
Proof.
(* FILL IN HERE *) Admitted.
(** [] *)
(** **** Ejercicio: 3 stars, avanzado, opcional (beq_nat_sym_informal) *)
(** De una prueba informal a este lema que corresponda con la
demostracion formal suya de arriba:
Teorema: Para todos [nat]s [n] [m], [beq_nat n m = beq_nat m n].
Demostracion:
(* FILL IN HERE *)
[]
*)
(** **** Ejercicio: 3 stars, opcional (beq_nat_trans) *)
Theorem beq_nat_trans : forall n m p,
beq_nat n m = true ->
beq_nat m p = true ->
beq_nat n p = true.
Proof.
(* FILL IN HERE *) Admitted.
(** [] *)
(** **** Ejercicio: 3 stars, avanzado (split_combine) *)
(** Hemos demostrado que para todas las listas de pares, [combine] es
el inverso de [split]. Como formalizaria el hecho de que
[split] es el inverso de [combine]?
Complete la definicion de [split_combine_statement] de abajo con
una propiedad que estalece que [split] es el inverso de
[combine]. Luego, pruebe que esta propiedad vale. (Asegurese de
dejar su hipotesis inductiva general evitando introducir mas
cosas que las necesarias. Ayuda: que propiedad necesita de
[l1] y [l2] para [split] que haga cierto [combine l1 l2 = (l1,l2)]?)
*)
Definition split_combine_statement : Prop :=
(* FILL IN HERE *) admit.
Theorem split_combine : split_combine_statement.
Proof.
(* FILL IN HERE *) Admitted.
(** [] *)
(** **** Ejercicio: 3 stars (override_permute) *)
Theorem override_permute : forall (X:Type) x1 x2 k1 k2 k3 (f : nat->X),
beq_nat k2 k1 = false ->
(override (override f k2 x2) k1 x1) k3 = (override (override f k1 x1) k2 x2) k3.
Proof.
(* FILL IN HERE *) Admitted.
(** [] *)
(** **** Ejercicio: 3 stars, avanzado (filter_exercise) *)
(** Este es un poco dificil. Preste atencion a la forma de su HI. *)
Theorem filter_exercise : forall (X : Type) (test : X -> bool)
(x : X) (l lf : list X),
filter test l = x :: lf ->
test x = true.
Proof.
(* FILL IN HERE *) Admitted.
(** [] *)
(** **** Ejercicio: 4 stars, avanzado (forall_exists_challenge) *)
(** Defina dos [Fixpoints], [forallb] y [existsb]. El primero
verifica que todo elemento de una lista dada satisfaga una
predicado dado:
forallb oddb [1;3;5;7;9] = true
forallb negb [false;false] = true
forallb evenb [0;2;4;5] = false
forallb (beq_nat 5) [] = true
El segundo verifica que existe al menos un elmento de la lista que
satisfaga el predicado dado:
existsb (beq_nat 5) [0;2;3;6] = false
existsb (andb true) [true;true;false] = true
existsb oddb [1;0;0;0;0;3] = true
existsb evenb [] = false
A continuacion, defina una version _no recursiva_ de [existsb] --
llamela [existsb'] -- usando [forallb] y [negb].
Demuestre que [existsb'] y [existsb] tienen el mismo
comportamiento. *)
(* FILL IN HERE *)
(** [] *)
(* $Date: 2013-07-17 16:19:11 -0400 (Wed, 17 Jul 2013) $ *)
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