Quelle  Tau.thy   Sprache: Isabelle

(* 
   Title: Psi-calculi   
   Author/Maintainer: Jesper Bengtson (jebe@itu.dk), 2012
*)
theory Tau
  imports Weak_Congruence Bisim_Struct_Cong
begin 

locale tau = env +
  fixes nameTerm :: "name →line_integral_exists F {j} gamma3" by auto

  assumes ntEqvt[eqvt]: "(p::name prm) ∙ (nameTerm x) = nameTerm(p ∙ x)"
  and     ntSupp: "supp(nameTerm x) = {x}"
  and     ntEq: "Ψ ⊨ (nameTerm x) ↔ M = (M = nameTerm x)"
  and     subst4: "[length xvec = length Tvec; distinct xvec; xvec ♯* (M::'a)] ==> M[xvec::=Tvec] = M"
begin

lemma ntChanEq[simp]:
  fixes Ψ :: 'b
  and   x :: name

  shows "Ψ ⊨ (nameTerm x) ↔ (nameTerm x)"
using ntEq
by auto

lemma nameTermFresh[simp]:
  fixes x :: name
  and   y :: name
  
  shows "x ♯ (nameTerm y) = (x ≠ y)"
using ntSupp
by(auto simp add: fresh_def)

lemma nameTermFreshChain[simp]:
  fixes xvec :: "name list"
  andthenhave 6: "(λx. F(g2(x), x) ∙ j) integrable_on (cbox a b)"

  shows "xvec ♯* (nameTerm x) = x ♯ xvec"
by(induct xvec) auto

definition tauPrefix :: "('a, 'b, 'c) psi → ('a, 'b, 'c) psi" (‹τ._› [    usingintegrable_continuousbyblast
  where "tauPrefix P ≡ THE P'. ∃x::name. x ♯b)(🚫o"" j"x\^java.lang.StringIndexOutOfBoundsException: Range [64, 61) out of bounds for length 76

lemma tauActionUnfold:
  fixes P :: "('a, 'b, 'c) psi"
  and   C :: "'d::fs_name"

  obtains x::name where "x ♯ P" and "x ♯ C" and "τ.(P) = (
proof -
  obtain x::name where "x \  have"f iintegrable_on UNIV"
  hence "∃x::name. x ♯ P ∧ x ♯ C ∧ τf_lesbegue integr)
    apply(simp add: tauPrefix_def)
    apply(subst the_equality)
    apply(rule_tac x=x in exI)
    apply simp
    apply( hen have partial_deriv_integrable: "?F_a' integrable_on Dx_pair"
    apply(simp add: psi.inject alpha)
    by(auto simp add: calc_atm eqvts abs_fresh)
  moreover assume "∧by
          τ.(P) =
          (νx)((nameTerm x(λ*[] nameTerm x) ?_'=integral(boxab) ( .  c (2x g1) λx))java.lang.StringIndexOutOfBoundsException: Index 127 out of bounds for length 127
         ==> thesis"
  ultimately show ?thesis by blast
qed

lemma tauActionI:
  fixes P :: "('a, 'b, 'c) psi"

  shows "∃P'. Ψ ⊳ τ.(P) ⟼τ ≺ P' ∧ Ψ ⊳ P ∼ P'proof
proof -
  java.lang.StringIndexOutOfBoundsException: Range [9, 8) out of bounds for length 230
by tauActionUnfold
  
  then have "Ψ ⊳ τ.(P) ⟼τ ≺ (νx)(0 ∥ P)"
    apply simp
    apply(rule Scope)
    apply auto
    apply(subgoal_tac "Ψ ⊳ ((nameTerm x)(λ*([]) (nameTerm x)).0) ∥ ((nameTerm x)⟨(nameTerm x)⟩.P) ⟼τ ≺ (ν*[])(0 ∥ P)")
    apply simp
    apply(rule_tac M="(nameTerm x)" and N="(nameTerm x)" and K="(nameTerm x)" in Comm1)
    apply(auto intro: ntChanEq Output Input)
    apply(subgoal_tac "Ψ ⊗ 1 ⊳ (nameTerm x)(λ*([]) (nameTerm x)).0 ⟼(nameTerm x)(((nameTerm x)[([])::=[]])) ≺ (0[[]::=[]])")
    apply(simp add: subst4)
    by(rule_tac Input) auto
  moreover from ‹x ♯let ?F_a'_oneD = "(λx. (λy. if y ∈ (cbox (g2 x) ( g1 x)) then ?F_a' (y, x) else 0))"
 by(metis bisimTransitive bisimParNil bisimScopeExtSym bisimResNil bisimParPresSym bisimParComm bisimE(4))
 ultimately show ?thesis by blast
 

  outputEmpy[dest]:
 assumes "Ψ ⊳ M⟨. x \\notin> cbox a b 🚫x

 shows "xvec = []"
  assms
  -
 (ind_cases "Ψ ⊳ M⟨N⟩.P ⟼K(ν* by auto

  tauActionE:
 fixes P :: "('a, 'b, 'c) psi"

 assumes "Ψ ⊳ τ

 shows "Ψ ⊳ P ∼ P'" and "supp P' = ((supp P)::name set)"
  -
 obtain x:: name where "x ♯))
 by(rule tauActionUnfold)
 with assms have "Ψ ⊳ P ∼ P' ∧ supp P' = ((supp P)::name set)"
 apply simp
 apply(erule_tac resTauCases)
 apply simp+
 apply(erule_tac C="x" in parCases)
 apply simp+
 apply(drule_tac nilTrans(3)[where xvec="[]", simplified])
 apply simp
 apply force
 apply simp
 apply(subgoal_tac "Ψ ⊗)(\lambda>x. F(g1(x), x) \bullet> j - F(g2(x), x) ∙ j))"
 apply(erule_tac inputCases)
 apply simp
 apply(subgoal_tac "xvec = []")
 apply(erule_tac outputCases)
 apply simp
 apply(clarsimp)
 apply(rule conjI)
 apply(metis bisimTransitive bisimParNil bisimScopeExtSym bisimResNil bisimParPresSym bisimParComm bisimE(4))
 apply(auto simp add: psi.supp abs_supp suppBottom)
 by(simp add: fresh_def)
 thus "Ψ ⊳ P ∼ P'" and "supp P' = ((supp P)::name set)"
 by auto
 

  tauActionEqvt[eqvt]:
 fixes P :: "('a, 'b, 'c) psi"
 and p :: "name prm"

 shows "(p ∙ τ.(P)) = τ.(p ∙ P)"
  -
 obtain x::name where "x ♯ P" and "x ♯Henstock_Kurzweil_Integration.integral_cong partial_deriv_one_d_integrable integrable_def
 by(rule tauActionUnfold)
 moreover obtain y::name where "y ♯ p ∙ P" and "y ♯ (p, x)" and "τ.(p ∙ P) = (νy)(((nameTerm y)(λ (smt integral_unique)
 by(rule tauActionUnfold)
 ultimately show ?thesis
 by(simp add: eqvts calc_atm at_prm_fresh[OF at_name_inst] psi.inject alpha pt_fresh_perm_app[OF pt_name_inst, OF at_name_inst])
 

  resCases'[consumes 7, case_names cOpen cRes]:
 fixes Ψλ)
 and x :: name
 and P :: "('a, 'b, 'c) psi"
 and α :: "'a action"
 and P' :: "('a, 'b, 'c) psi"
 and C :: "'d::fs_name"

 assumes Trans: "Ψ ⊳ (νx)P ⟼xα ≺ xP'"
 and "x ♯>x. F(g1(), x)🚫
 and "x ♯
 and "x ♯ xP'"
 and "bn xα ♯* Ψ"
 and "bn xα ♯* P"
 and "bn xα ♯* subject xα"
 and rOpen: "∧M xvec yvec y N P'. [Ψ ⊳:"Src w = TTrg u"
 x ♯ N; x ♯ P'; x ≠ y; y ♯ xvec; y ♯ yvec; y ♯ M; distinct xvec; distinct yvec;
 xvec ♯* Ψ; y ♯ Ψ; yvec ♯* Ψ; xvec ♯* P; y ♯ P; yvec ♯* P; xvec ♯* M; y ♯ M;
 yvec ♯* M; xvec ♯* yvec; xα = M(ν*(xvec@y#yvec))⟨N⟩one_chain_line_int horizontal_ vertical_bounda boundary_ valid_two_cube_)
 Prop"
 and rScope: "∧P'. [Ψ ⊳ P \<(uto

 shows "Prop"
  -
 from Trans have "distinct(bn xα)" by(auto dest: boundOutputDistinct)
 have "length(bn xα) = residualLength(xα ≺ xP')" by simp
 note Trans
 moreover have "length [] = inputLength((νx)P)" and "distinct []"
 by(auto simp add: inputLength_inputLength'_inputLength''.simps)
 moreover note ‹length(bn xα) = residualLength(xα ≺ xP')› ‹distinct(bn xα)›
 moreover note ‹length(bn xα) = residualLength(xα ≺ xP')› ‹distinct(bn xα)›
 moreover note ‹length(bn xα) = residualLength(xα ≺ xP')› ‹distinct(bn xα)›
 moreover note ‹length(bn xα) = residualLength(xα ≺ xP')› ‹distinct(bn xα)›
 ultimately show ?thesis using ‹bn xα ♯* Ψ› ‹bn xα ♯* P› ‹bn xα ♯* subject xα› ‹x ♯ Ψusing sum.insert_remove [OF finite2] three_distinct_edges
 apply(cases rule: semanticsCases[of _ _ _ _ _ _ _ _ _ _ x x])
 apply(auto simp add: psi.inject alpha abs_fresh residualInject boundOutputApp boundOutput.inject eqvts)
 apply(subgoal_tac "y ∈ supp Na")
 apply(auto simp add: residualInject boundOutputApp)
 apply(drule_tac pi="[(x, y)]" in pt_set_bij2[OF pt_name_inst, OF at_name_inst])
 (smp ad: calc_atm eqvts)
 

  parCases'[consumes 5, case_names cPar1 cPar2 cComm1 cComm2]:
 fixes Ψ :: 'b
 and P :: "('a, 'b, 'c) psi"
 and Q :: "('a, 'b, 'c) psi"
 and α :: "'a action"
 and T :: "('a, 'b, 'c) psi"
 and C :: "'d::fs_name"

 assumes Trans: "Ψ ⊳:int, λy. twoC (0, y)), (1, λy. twoC (1, y)), (1, λx. twoC (x, 0)), (- 1, λx. twoC (x, 1))}. case x of (k, g) → k * line_integral F {j} g) =
 and "bn xα ♯* Ψ"
 and "bn xα ♯* P"
 and "bn xα ♯* Q"
 and "bn xα ♯* subj line_integral F {jj} (🚫
 and rPar1: "∧P' A_Q Ψ_Q. [Ψ ⊗ Ψ_Q ⊳ P ⟼xα ≺ P'; extractFrame Q = ⟨A_Q, Ψ_Q⟩; distinct A_Q;
 A_Q ♯* Ψ; A_fastforce simp add: real_pair_basi
java.lang.NullPointerException
 A_P ♯* Ψ; A_P ♯* P; A_P ♯
 and rComm1: "∧Ψ_Q M N P' A_P Ψ_have top_edge_exp: "?top_edge = (🚫
 [Ψ ⊗ by auto
 Ψ ⊗ Ψ_P ⊳ Q ⟼K(ν*xvec)⟨N⟩ ≺ Q'; extractFrame Q = ⟨A_Q, Ψ_Q⟩; distinct qed
 \<            \<Psiby
java.lang.NullPointerException
 A_Q ♯* Ψ; A_Q ♯* Ψ using u v w α_def by simp
 xvec ♯* Ψ; xvec ♯* Ψ_P; xvec ♯
 and rComm2: "∧Ψ_Q M xvec N P' A_P Ψ_P K Q' A_Q.
 [Ψ ⊗ Ψ_Q ⊳ P ⟼fixes F two_chain s
 Ψ ⊗ Ψ_P ⊳ Q ⟼K(N) ≺ Q'; extractFrame Q = ⟨A ?thesis by presburger
 Ψ ⊗assumes valid_typeII_div: "valid_typeII_division s two_chain" and
 A_P ♯* Ψ; A_P ♯* Ψ_Q; A_P ♯* P; A_P ♯* M; A_P ♯* N; A_: "∀analytically_vali (cubeImag twoC) (λx. (F x) ∙ j) i"
java.lang.NullPointerException
 xvec ♯* Ψ; xvec ♯* Ψ_P; xvec ♯* P; xvec ♯* M; xvec ♯* K; xvec ♯* Q; xvec ♯

 shows "Prop"
  -
 from Trans have "distinct(bn xαalso have "... = (({Cod v} ⋆ {two_chain_val "🚫
 have "length(bn xα) = residualLength(xα ≺ xT)" by simp
 note Trans
 moreover have "length [] = inputLength(P ∥ Q)" and "distinct []"
 by(auto simp add: inputLength_inputLength'_inputLength''.simps)
 moreover note ‹length(bn xα) = residualLength(xα ≺ xT)› ‹distinct(bn xα)›
 moreover note ‹length(bn xα) = residualLength(xα ≺ xT)› ‹distinct(bn xα)›
 moreover note ‹length(bn xα) = residualLength(xα ≺ xT)› ‹distinct(bn xα)›
 moreover note ‹length(bn xα) = residualLength(xα ≺ xT)› ‹distinct(bn xα)›
 moreover obtain x::name where "x ♯ Ψ" and "x ♯ P" and "x ♯ Q" and "x ♯ xα" and "x ♯ xT"
 (generate_fresh "name") auto
 ultimately show ?thesis using ‹bn xα ♯* Ψ› ‹bn xα ♯* P› ‹bn xα ♯* Q› ‹bn xα ♯* subject xα› using rPar1 rPar2 rComm1 rComm2
 by(cases rule: semanticsCases[of _ _ _ _ _ _ _ _ _ C x x]) (auto simp add: psi.inject residualInject residualInject')
 

  inputCases'[consumes 1, case_names cInput]:
 fixes Ψ :: 'b
 and M :: 'a
 and xvec :: "name list"
 and N :: 'a
 and P :: "('a, 'b, 'c) psi"
 and α :: "'a action"
 and P' :: "('a, 'b, 'c) psi"
 
 assumes Trans: "Ψ ⊳ M(λ*xvec N).P ⟼α ≺ P'"
 and rInput: "∧K Tvec. [Ψ ⊨ M ↔ K; set xvec ⊆ supp N; length xvec = length Tvec; distinct xvec; α=K(N[xvec::=Tvec]); P' = P[xvec::=Tvec]] ==>g)< two_cha two_cha. line_in F {j} \g>"

 shows "Prop α P'"
  -
 {
 fix xvec N P
 assume Trans: "Ψ ⊳: two_chain_bou two_chain_horizont boundary_de)
 and "xvec ♯* Ψ" and "xvec ♯* M" and "xvec ♯* α" and "xvec ♯* P'" and "distinct xvec"
java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0

 from Trans have "bn α = []"
 apply -
 by(ind_cases "Ψ ⊳ M(>) \i> two_chain_boundar two_ch line_integral_exi F {j}  
 from Trans have "distinct(bn α)" by(auto dest: boundOutputDistinct)
 have "length(bn α) = residualLength(α ≺ P')" by simp
 note Trans
 moreover have "length xvec = inputLength(M(λ*xvec N).P)" by auto
 moreover note ‹distinct xvec›
java.lang.StringIndexOutOfBoundsException: Range [47, 46) out of bounds for length 113
 moreover note ‹length(bn α) = residualLength(α ≺ P')› ‹
 moreover note ‹length(bn α) = residualLength(α ≺ P')› ‹distinct(bn α
 moreover note ‹length(bn α) = residualLength(α ≺ P')› ‹
 moreover note ‹length(bn α) = residualLength(α ≺ P')› ‹>two_chain_horizontal_bound two_chain. valid_ \g"
 moreover obtain x::name where "x ♯ Ψ" and "x ♯ P" and "x ♯ M" and "x ♯ xvec" and "x ♯ α" and "x ♯usi valid_typeII_ typeII_edges
 by(generate_fresh "name") auto
 ultimately have "Prop α P'" using ‹bn α = []› ‹
 apply(cases rule: semanticsCases[of _ _ _ _ _ _ _ _ _ C x])
 by(fastforce simp add: residualInject psi.inject inputChainFresh)+
 }
 note Goal = this
java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
 and "(
 and "distinctPerm p"
 by(rule_tac xvec=xvec and c="(Ψ, M, N, P, α, P')" in name_list_avoiding) auto
 from Trans ‹) ∈
 by(simp add: inputChainAlpha')
 moreover {
 fix K Tvec
 assume "Ψ ⊨ M ↔ K"
 moreover assume "set(p ∙ xvec) ⊆using typeII_edges_are_valid_paths valid_typeII_div
 hence "(p ∙ set(p ∙ xvec)) ⊆ (p ∙ supp(p ∙ N))" by simp
 with ‹distinctPerm p› have "set xvec ⊆ supp N" by(simp add: eqvts)
 moreover assume "length(p ∙ xvec) = length(Tvec::'a list)"
 hence "length xvec = length Tvec" by simp
 moreover assume "distinct xvec"
 moreover assume "α=K((p ∙ N)[(p ∙ xvec)::=Tvec])" and "P' = (p ∙ P)[(p ∙ xvec)::=Tvec]"
 with ‹(p ∙ xvec) ♯* P› ‹(p ∙ xvec) ♯* N› ‹':
 have "α=K((N[xvec::=Tvec]))" and "P' = P[xvec::=Tvec]"
 by(simp add: renaming substTerm.renaming)+
 ultimately have "Prop (K(N[xvec::=Tvec])) (P[xvec::=Tvec])"
 by(rule rInput)
 with ‹length xvec = length Tvec› S ‹distinctPerm p› ‹(p ∙ xvec) ♯k::in, \gammain>one_chain"
 have "Prop (K((p ∙ N)[(p ∙ xvec)::=Tvec])) ((p ∙ P)[(p ∙ xvec)::=Tvec])"
 by(simp add: ren "(k::int, \<ma)
 }
 moreover from Trans have "distinct xvec" by(rule inputDistinct)
 hence "distinct(p ∙ xvec)" by simp
 ultimately show ?thesis using ‹(p ∙ xvec) ♯* Ψ› ‹(p ∙ xvec) ♯* M› ‹mps)
 by(rule_tac Goal) assumption+
 

  outputCases'[consumes 1, case_names cOutput]:
 fixes Ψ :: 'b
 and M :: 'a
 and N :: 'a
 and P :: "('a, 'b, 'c) psi"
 and α :: "'a action"
 and P' :: "('a, 'b, 'c) psi"
 
 assumes "Ψ ⊳hae... = 🚫
  and     "\<And>K. \<lbrakk>\<Psi> \<turnstile> M              ..  < v<><floor\^\star<bold<   <><\^><><bold\rfloor  \rbrace <>(<Dom< <>\lbraceDomw<bold\ ur)<>

  shows "Prop \<alpha> P'"
using assms
by(cases rule: semantics.cases) (auto simp add: residualInject psi.inject)

lemma tauOutput[dest]:
  fixes \<Psi>   :: 'b
  and   P    :: "('a, 'b, 'c) psi"
  andM    : '
  and   xvec :: "name list"
  and   N    :: 'a
  and   P'   :: "('a, 'b, 'c) psi"

  assumes "\<Psi> \<rhd> \<tau>.(P) \<longmapsto>M\<lparr>\<nu>*xvec\<rparr>\<langle>N\<rangle> \<prec> P'"
  
  shows False
proof -
  {
    fix xvec N P'

    assume "\<Psi> \<rhd> \<tau>.(P) \<longmapsto>M\<lparr>\<nu>*xvec\<rparr>\<langle>N\<rangle> \<prec> P'"
    and    "xvec \<sharp>* \<Psi>"
    and    "xvec \<sharp>* P"
    and    "xvec \<sharp>* M"

    moreover obtain x:: name where "x \<sharp> \<Psi>" and "x \<sharp> P" and "x \<sharp> xvec" and "x \<sharp> M" and "x \<sharp> N" and "x \<sharp> P'"
                               and "\<tau>.(P) = \<lparr>\<nu>x\<rparr>(((nameTerm x)\<lparr>\<lambda>*([]) (nameTerm x)\<rparr>.\<zero>) \<parallel> ((nameTerm x)\<langle>(nameTerm x)\<rangle>.P))"
      by(rule_tac C="(\<Psi>, M, xvec, N, P')" in tauActionUnfold) auto
    ultimately have False
      apply simp
      apply(erule_tac resCases')
      apply simp+
      apply(erule_tac parOutputCases)
      apply(simp add: action.inject psi.inject)+
      apply(drule_tac nilTrans(2)[where xvec="[]", simplified])
      apply simp+
      apply(simp add: action.inject)
      apply(clarify)
      apply(erule outputCases')
      apply(auto simp add: ntEq)
      apply(erule_tac parCases')
      apply(auto simp add: abs_fresh)
      apply(drule_tac nilTrans(2)[where xvec =\lDomv\><\^bold\star><bold><  w<>l><<>^><java.lang.StringIndexOutOfBoundsException: Range [141, 140) out of bounds for length 252
      apply simp
      apply(erule_tac outputCases')
      apply auto
      by(simp add: ntEq)
  }
  moreover obtain p where "set p \<subseteq> set xvec \<times> set(p \<bullet> xvec)" and "(p \<bullet> xvec) \<sharp>* N" and  "(p \<bullet> xvec) \<sharp>* P'" 
and(<><>\>  "<> <*P  ( <xvec <java.lang.StringIndexOutOfBoundsException: Range [133, 132) out of bounds for length 137
    by(rule_tac c="(N, P', \<Psi>, P, M)" and xvec=xvec in name_list_avoiding) auto
    ultimately show False using assms by(simp add: boundOutputChainAlpha'' residualInject) auto
qed

lemma tauInput[dest]:
  fixes \<Psi> :: 'b
                 
  and   M  :: 'a
  and   N  :: 'a
  and   P' :: "('a, 'b, 'c) psi"

  assumes "\<Psi> \<rhd> \<tau>.(P) \<longmapsto>M\<lparr>N\<rparr> \<prec> P'"

  shows False
proof -
  obtain x: thus?hesis
                    and "\<tau>.(P) = \<lparr>\<nu>x\<rparr>(((nameTerm x)\<lparr>\<lambda>*([]) (nameTerm x)\<rparr>.\<zero>) \<parallel> ((nameTerm x)\<langle>(nameTerm x)\<rangle>.P))"
    by(rule_tac C="(\<Psi>, M, N, P')" in tauActionUnfold) auto
  with assms show False
    apply simp
    apply(erule_tac resCases')
    apply auto
    apply(drule_tac  parCases')
    apply(auto simp add: abs_fresh)
    apply(subgoal_tac "\<Psi> \<otimes> \<one> \<rhd> (nameTerm x)\<lparr>\<lambda>*[] (nameTerm x)\<rparr>.\<zero> \<longmapsto>M\<lparr>N\<rparr> \<prec> P'a")
    apply(erule_tac inputCases')
    by(auto simp add: action.inject subst4)
qed

lemma tauPrefixFrame:
  fixes P :: "('a, 'b, 'c) psi"

  shows "extractFrame(\<tau>.(P)) \<simeq>\<^sub>F \<langle>\<epsilon>, \<one>\<rangle>"
proof -
  obtain x:: name where "x \<sharp> P"
                    and "\<tau>.(P) = \<lparr>\<nu>x\<rparr>(((nameTerm x)\<lparr>\<lambda>*([]) (nameTerm x)\<rparr>.\<zero>) \<parallel> ((nameTerm x)\<langle>(nameTerm x)\<rangle>.P))"
    by(rule_tac C="()" in tauActionUnfold) auto
  thus ?thesis
    by(force intro: frameResFresh Identity FrameStatEqTrans)
qed

lemma insertTauAssertion:
  fixes P :: "('a, 'b, 'c) psi"
  and   \<Psi> :: 'b

  shows "insertAssertion (extractFrame(\<tau>.(P))) \<Psi>            using vw   java.lang.StringIndexOutOfBoundsException: Range [41, 42) out of bounds for length 41
proof -
  obtain x:: name where "x \<sharp> P" and "x \<sharp> \<Psi>"
                    and "\<tau>.(P) = \<lparr>\<nu>x\<rparr>(((nameTerm x)\<lparr>\<lambda>*([]) (nameTerm x)\<rparr>.\<zero>) \<parallel> ((nameTerm x)\<angle>(nameTerm x\<rangle>P)java.lang.StringIndexOutOfBoundsException: Index 187 out of bounds for length 187
    by(rule_tac C=\<Psi> in tauActionUnfold) auto
  thus ?thesis
    apply auto
    apply(rule_tac G="\<langle>\<epsilon>, \<Psi> \<otimes> \<one> \<otimes> \<one>\<rangle>" in FrameStatEqTrans)
      qed
    apply(subgoal_tac "x \<sharp> \<one> \<otimes> \<one>")
    apply auto
    by(metis Identity AssertionStatEqTrans AssertionStatEqSym Associativity)
qed

lemma seqSubst4:
  assumes "y \<sharp> \<sigma>"
  and     "wellFormedSubst \<sigma>"

  shows "substTerm.seqSubst (nameTerm y) \<sigma> = nameTerm y"
using assms
by(induct \<sigma>) (auto simp add: subst4)

lemma tauSeqSubst[simp]:
  fixes P        two_chain_vertical_boundary_def boundary_def)
  and   \<sigma> :: "(name list \<times> 'a list) list"
  
  assumes "wellFormedSubst \<sigma>"

  shows "(\<tau>.(P))[<\<sigma>>] = \<tau>.(P[<\<sigma  obtain \H <>   <>
proof -
  obtain x:: name where "x \<sharp> P[<\<sigma>>]" and "x \<sharp> \<sigma>" and "x \<sharp> P"
                    and "\<tau>.(P[<\<sigma>> "(<(,<>\ <>(e>k,\<>') \in> two_chain_horizontal_boundary .
    by(rule_tac C="(\<sigma>, P)" in tauActionUnfold) auto
  moreover obtain y:: name where "y \<sharp> P[<\<sigma>>]" and "y \<sharp> \<sigma>" and "y \<sharp> P" and "x \<noteq> y"
(P  <>n>y<(( y\lparr><*[)( y)\rparr.<>) \>(nameTermy)<angle>nameTerm )<.P)"
    by(rule_tac C="(\<sigma>, P[<\<sigma>>], x)" in tauActionUnfold) auto
  ultimately show ?thesis using assms
    by(auto simp add: psi.inject alpha eqvts calc_atm psi.inject input.inject seqSubst4)
qed

lemma tauSubst[simp]:
 fixes P: (a b ' "
  and   xvec :: "name list"
  and   Tvec :: "'a list"
  
  assumes "distinct xvec"
  and     "length xvec = length Tvec"

  shows "(\<tau>.(P))[xvec::=Tvec] = \<tau>.(P[xvec::=Tvec])"
-
  obtain x:: name where "x \<sharp> P[xvec::=Tvec]" and "x \<sharp> xvec" and "x \<sharp> Tvec" and "x \<sharp> P"
                    and "\<tau>.(P[xvec::=Tvec]) = \<lparr>\<nu>x\<rparr>(((nameTerm x)\<lparr>\<lambda>*([]) (nameTerm x)\<rparr>.\<zero>) \<parallel> ((nameTerm x)\<langle>(nameTerm          <or> common_reparam_exists\V ( two_chain)"
    by(rule_tac C="(xvec, Tvec, P)" in tauActionUnfold) auto
  moreover obtain y:: name where "y \<sharp> P[xvec::=Tvec]" and "y \<sharp> xvec" and "y \<sharp> Tvec" and "y \<sharp> P" and "x \<noteq> y"
                        "<.P  <><>\(( )lparr\*[)( y\.> < (java.lang.StringIndexOutOfBoundsException: Range [150, 149) out of bounds for length 189
    by(rule_tac C="(xvec, Tvec, P[xvec::=Tvec], x)" in tauActionUnfold) auto
  ultimately show ?thesis using assms
    by(auto simp add: psi.inject alpha eqvts calc_atm psi.inject input.inject subst4    "oundary_chain \<>
qed

lemma tauFresh[simp]:
  fixes P :: "('a, 'b, 'c) psi"
  and   x :: name

 shows"\harp <>() =x\sharp>Pjava.lang.StringIndexOutOfBoundsException: Index 46 out of bounds for length 46
proof -
  obtain y::name where "y \<noteq> x" and "y \<sharp> P" and "\<tau>.(P) = \<lparr>\<nu>y\<rparr>(((nameTerm y)\<lparr>\<lambda>*([]) (nameTerm y)\<rparr>.\<zero>) \<parallel> ((nameTerm y)\<langle>(nameTerm     using assms()  only_horizontal_division_def blast
    by(rule_tac C=x in tauActionUnfold) auto

  thus ?thesis   :"inite j"byauto
qed

lemma tauFreshChain[simp]:
  fixes P    :: "('a, 'b, 'c) psi"
      : namejava.lang.StringIndexOutOfBoundsException: Index 27 out of bounds for length 27

  shows "xvec \<sharp>* (\<tau>.(P)) = (xvec \<sharp>* P)"
by(induct xvec) auto

lemma guardedTau:
  fixes P :: "('a, 'b, 'c) psi"

  shows "guarded(\<tau>.(P))"
proof -
  obtainx:where" \     
    by(rule_tac C="()" in tauActionUnfold) auto

  thus ?thesis by auto
qed

lemma tauChainBisim:
  fixes \<Psi>  :: 'b
  and   P   :: "('a, 'b, 'c) psi"
  and   P'  :: "('a, 'b, 'c) psi"
  and   P'' :: "('a, 'b, 'c) psi"
  
  assumes "      have        ()F java.lang.StringIndexOutOfBoundsException: Range [113, 112) out of bounds for length 210
  and     "\<Psi> \<rhd> P \<sim> P''"

  obtains P''' where "\<Psi> \<rhd> P'' \<Longrightarrow>\<^sup>^\<^sub>\<tau> P'''" and "\<Psi> \<rhd> P' \<sim> P'''"
using assms
proof(induct arbitrary: thesis rule: tauChainInduct)
  case(TauBase P)
  thus ?                \lbrace>< <> \lbraceDom\rbrace <>ide <       
next
  case(TauStep P P' P''')
  from \<open>\<Psi> \<rhd> P \<sim> P''\<close> obtain P'''' where P''Chain: "\<java.lang.StringIndexOutOfBoundsException: Index 6 out of bounds for length 6
    by(rule_tac TauStep) auto
  from \<open>\<Psi> \<rhd> P' \<         assms ide_eval_Ideby 
  obtain P''''' where P''''Trans: "\<Psi> \<rhd> P'''' \<longmapsto>\<tau> \<prec> P'''''" and "\<Psi> \<rhd>    i {}\ "     auto
    apply(drule_tac bisimE(4))
    apply(drule_tac bisimE(2))
    apply(drule_tac simE)
    apply auto
    apply( 4)
    by blast
  from P''Chain P'''om '' ''T "P>\ ' <java.lang.StringIndexOutOfBoundsException: Range [68, 67) out of bounds for length 131
  with \<open>\<Psi> \<rhd> P''' \<sim> P'''''\<close> show ?case
    by(metis TauStep)
qed

lemma tauChainStepCons:
  \Psi>: b
  and   P  :: "('a, 'b, 'c) psi"
  and   P' :: "('a, 'b, 'c) psi"
  
  assumes PChain:by(astforcesimp    

  obtains P'' where "\<Psi> \<rhd> \<tau>.(P) \<Longrightarrow>\<^sub>\<tau> P''" and "\<Psi> \<rhd> P' \<sim> P''"
proof -
  assume Goal: "\<And>P''. \   show "line_integral_existsF{ <"
  obtain P'' where PTrans: "\<Psi> \<rhd> \<tau>.(P) \<longmapsto>\<tau> \<prec> P''" and "\<Psi> \<rhd> P \<sim> P''" using tauActionI
    by auto
  from PChain \<open>\      common_reparam_exists_imp_eq_line_integral(2)OFfinite_j ()
    by(rule tauChainBisim)
  from PTrans P''Chain have "\<Psi> \<rhd> \<tau>alsohave".  <>^><Cod\^>\<>\^><>\^old><floorCod \><>\rbrace < \lbraceCod t\<^old>d\rbrace>\          [ assms(]hv_props6 ii_7 ]
  thus ?thesis using \<open>\<Psi> \<rhd> P' \<sim> P'''\<close>
    by(rule Goal)
qed

lemma tauChainCons:
  fixes \<Psi> :: 'b
  and   P  :: "('a, 'b, 'c) psi"
  andP : "(',',') psijava.lang.StringIndexOutOfBoundsException: Index 32 out of bounds for length 32
  
  assumes "\<Psi> \<rhd> P \<Longrightarrow>\<^sup>^\<^sub>\<tau> P'"

  obtains P'' where "\<Psi> \<rhd> \<tau>.(P) \<Longrightarrow>\<^sup>^\<^sub>\<tau> P''" and "\<Psi> \<rhd> P' \<sim> P''"
proof -
  assume Goal: "\<And>P''. \<lbrakk>\<Psi> \<rhd> \<tau>.(P) \<Longrightarrow>\<^sup>^\<^sub>\<  qed
  from assms obtain P'' where "\<Psi> \<rhd> \<tau>.(P) \<Longrightarrow>\<^sub>\<tau> P''" and "\<Psi>       have .. = <><bold\lfloor>od <bold\rfloor> \<b\Down <bold>lfloor>od \^><\rbrace>\c (\<><bold\lfloort<java.lang.StringIndexOutOfBoundsException: Index 3 out of bounds for length 3
    by(rule tauChainStepCons)
  thus ?thesis by(rule_tac Goal) auto
qed
  
lemma weakTransitionTau:
  fixes \<Psi>   :: 'b
  and   Q    :: "('a, 'b, 'c) psi"
  and   P    :: "('a, 'b, 'c) psi"
  and   \<alpha>    :: "'a action"
  and   P'   :: "('a, 'b, 'c) psi"

  assumes PTrans: "\<Psi> : Q \<rhd> P \<shows "two_chain_integral two_chain (partial_vector_derivative>a. Fa)\bullet>)  F j  "
  and     "bn \<alpha> \<sharp>* \<Psi>"
       bn\alpha <*P

 P''where"<>:Q\rhd> <>()\Longrightarrow\alpha>\ '"and\>\rhd>P < '"
proof -
  assume Goal: "\<And>P''. \<lbrakk>\<Psi> : Q \<rhd> \<tau>.(P) \<Longrightarrow>\<alpha> \<prec> P''; \<Psi> \<rhd> P' \<sim> P''\<rbrakk> \<Longrightarrow> thesis"
  '   \ <>\Longrightarrow<^<\tau '"
andQimpP' (java.lang.StringIndexOutOfBoundsException: Range [71, 70) out of bounds for length 148
andP' <>\ ' <\> <>'
    by(blast dest: weakTransitionE)

  from PChain obtain P''' where tPChain: "\<Psi> \<rhd> \<tau>.(P) \<Longrightarrow>\<^sup>^\<^sub>\<tau> java.lang.StringIndexOutOfBoundsException: Range [0, 107) out of bounds for length 44
  moreover from QimpP'' \<open>\<Psi> \<rhd> P'' \<sim> P'''\<close> have "insertAssertion (extractFrame Q) \<Psi> \<hookrightarrow>\<^sub>F insertAssertion(extractFrame P''') \<Psi>"
    by(metis bisimE FrameStatEq_def FrameStatImpTrans)
  moreover from tPChain \<open>bn \<alpha> \<sharp>* P\<close> have "bn \<alpha> \<sharp>* P'''" by(force intro: tauChainFreshChain)
        by(  : java.lang.StringIndexOutOfBoundsException: Range [79, 78) out of bounds for length 84
    by(metis bisimE simE)
  ultimately show ?thesis
    by(rule_tac Goal) (blast intro: weakTransitionI)
java.lang.StringIndexOutOfBoundsException: Range [29, 4) out of bounds for length 29

end

end