| Quellcodebibliothek Statistik Leitseite products/Sources/formale Sprachen/Isabelle/Archive-of-Formal-Proofs/thys/Psi_Calculi/ (Archive of formal Proofs Version 2026-5©) Datei vom 31.4.2026 mit Größe 801 kB |
|
Quelle Semantics.thySprache: Isabelle |
|||||||||||||||
|
(* Title: Psi-calculi Author/Maintainer: Jesper Bengtson (jebe@itu.dk), 2012 *) theory Semantics imports Frame begin nominal_datatype ('a, 'b, 'c) boundOutput = BOut "'a::fs_name" "('a, 'b::fs_name, 'c::fs_name) psi" (‹_ ≺'' _› [110, 110] 110) | BStep "«name¬ ('a, 'b, 'c) boundOutput" (‹(ν_)_› [110, 110] 110) primrec BOresChain :: "name list → ('a::fs_name, 'b::fs_name, 'c::fs_name) boundOutp ('a, 'b, 'c) boundOutput" where Base: "BOresChain [] B = B" | Step: "BOresChain (x#xs) B = (νx)(BOresChain xs B)" abbreviation BOresChainJudge (‹(ν*_)_› [80, 80] 80) where "(ν*xvec)B ≡ BOresChain xvec B" lemma BOresChainEqvt[eqvt]: fixes perm :: "name prm" and lst :: "name list" and B :: "('a::fs_name, 'b::fs_name, 'c::fs_name) boundOutput" shows "perm ∙ ((ν*xvec)B) = (ν*(perm ∙ xvec))(perm ∙ B)" by(induct_tac xvec, auto) lemma BOresChainSimps[simp]: fixes xvec :: "name list" and N :: "'a::fs_name" and P :: "('a, 'b::fs_name, 'c::fs_name) psi" and N' :: 'a and P' :: "('a, 'b, 'c) psi" and B :: "('a, 'b, 'c) boundOutput" and B' :: "('a, 'b, 'c) boundOutput" shows "((ν*xvec)N ≺' P = N' ≺' P') = (xvec = [] ∧ N = N' ∧ P = P')" and "(N' ≺' P' = (ν*xvec)N ≺' P) = (xvec = [] ∧ N = N' ∧ P = P')" and "(N' ≺' P' = N ≺' P) = (N = N' ∧ P = P')" and "((ν*xvec)B = (ν*xvec)B') = (B = B')" by(induct xvec) (auto simp add: boundOutput.inject alpha) lemma outputFresh[simp]: fixes Xs :: "name set" and xvec :: "name list" and N :: "'a::fs_name" and P :: "('a, 'b::fs_name, 'c::fs_name) psi" shows "(Xs ♯* (N ≺' P)) = ((Xs ♯* N) ∧ (Xs ♯* P))" and "(xvec ♯* (N ≺' P)) = ((xvec ♯* N) ∧ (xvec ♯* P))" by(auto simp add: fresh_star_def) lemma boundOutputFresh: fixes x :: name and xvec :: "name list" and B :: "('a::fs_name, 'b::fs_name, 'c::fs_name) boundOutput" shows "(x ♯ ((ν*xvec)B)) = (x ∈ set xvec ∨ x ♯ B)" by (induct xvec) (simp_all add: abs_fresh) lemma boundOutputFreshSet: fixes Xs :: "name set" and xvec :: "name list" and B :: "('a::fs_name, 'b::fs_name, 'c::fs_name) boundOutput" and yvec :: "name list" and x :: name shows "Xs ♯* ((ν*xvec)B) = (∀x∈Xs. x ∈ set xvec ∨ x ♯ B)" and "yvec ♯* ((ν*xvec)B) = (∀x∈(set yvec). x ∈ set xvec ∨ x ♯ B)" and "Xs ♯* ((νx)B) = Xs ♯* [x].B" and "xvec ♯* ((νx)B) = xvec ♯* [x].B" by(simp add: fresh_star_def boundOutputFresh)+ lemma BOresChainSupp: fixes xvec :: "name list" and B :: "('a::fs_name, 'b::fs_name, 'c::fs_name) boundOutput" shows "(supp((ν*xvec)B)::name set) = (supp B) - (supp xvec)" by(induct xvec) (auto simp add: boundOutput.supp supp_list_nil supp_list_cons abs_supp supp_atm) lemma boundOutputFreshSimps[simp]: fixes Xs :: "name set" and xvec :: "name list" and B :: "('a::fs_name, 'b::fs_name, 'c::fs_name) boundOutput" and yvec :: "name list" and x :: name shows "Xs ♯* xvec ==> (Xs ♯* ((ν*xvec)B)) = (Xs ♯* B)" and "yvec ♯* xvec ==> yvec ♯* ((ν*xvec)B) = yvec ♯* B" and "xvec ♯* ((ν*xvec)B)" and "x ♯ xvec ==> x ♯ (ν*xvec)B = x ♯ B" apply(simp add: boundOutputFreshSet) apply(force simp add: fresh_star_def name_list_sup apply(simp add: boundOutputFreshSet) apply(force simp add: fresh_star_def name_list_sup apply(simp add: boundOutputFreshSet) by(simp add: BOresChainSupp fresh_def) lemma boundOutputChainAlpha: fixes p :: "name prm" and xvec :: "name list" and B :: "('a::fs_name, 'b::fs_name, 'c::fs_name) boundOutput" and yvec :: "name list" assumes xvecFreshB: "(p ∙ xvec) ♯* B" and S: "set p ⊆ set xvec × set (p ∙ xvec)" and "(set xvec) ⊆ (set yvec)" shows "((ν*yvec)B) = ((ν*(p ∙ yvec))(p ∙ B))" proof - note pt_name_inst at_name_inst S moreover from ‹(set xvec) ⊆ (set yvec)› have "set xvec ♯* ((ν*yvec)B)" by(force simp add: boundOutputFreshSet) moreover from xvecFreshB ‹(set xvec) ⊆ (set yvec)› have "set (p ∙ xvec) ♯* ((ν*yvec)B)" by (simp add: boundOutputFreshSet) (simp add: fresh_star_def) ultimately have "((ν*yvec)B) = p ∙ ((ν*yvec)B)" by (rule_tac pt_freshs_freshs [symmetric]) then show ?thesis by(simp add: eqvts) qed lemma boundOutputChainAlpha': fixes p :: "name prm" and xvec :: "name list" and B :: "('a::fs_name, 'b::fs_name, 'c::fs_name) boundOutput" and yvec :: "name list" and zvec :: "name list" assumes xvecFreshB: "xvec ♯* B" and S: "set p ⊆ set xvec × set yvec" and "yvec ♯* ((ν*zvec)B)" shows "((ν*zvec)B) = ((ν*(p ∙ zvec))(p ∙ B))" proof - note pt_name_inst at_name_inst S ‹yvec ♯* ((ν*zvec)B)› moreover from xvecFreshB have "set (xvec) ♯* ((ν*zvec)B)" by (simp add: boundOutputFreshSet) (simp add: fresh_star_def) ultimately have "((ν*zvec)B) = p ∙ ((ν*zvec)B)" by (rule_tac pt_freshs_freshs [symmetric]) auto then show ?thesis by(simp add: eqvts) qed lemma boundOutputChainAlpha'': fixes p :: "name prm" and xvec :: "name list" and M :: "'a::fs_name" and P :: "('a::fs_name, 'b::fs_name, 'c::fs_name) psi" and yvec :: "name list" assumes "(p ∙ xvec) ♯* M" and "(p ∙ xvec) ♯* P" and "set p ⊆ set xvec × set (p ∙ xvec)" and "(set xvec) ⊆ (set yvec)" shows "((ν*yvec)M ≺' P) = ((ν*(p ∙ yvec))(p ∙ M) ≺' (p ∙ P))" using assms by(subst boundOutputChainAlpha) auto lemma boundOutputChainSwap: fixes x :: name and y :: name and N :: "'a::fs_name" and P :: "('a, 'b::fs_name, 'c::fs_name) psi" and xvec :: "name list" assumes "y ♯ N" and "y ♯ P" and "x ∈ (set xvec)" shows "(ν*xvec)N ≺' P = (ν*([(x, y)] ∙ xvec))([(x ,y)] ∙ N) ≺' ([(x, y)] ∙ P)" proof(case_tac "x=y") assume "x=y" thus ?thesis by simp next assume "x ≠ y" with assms show ?thesis by(rule_tac xvec="[x]" in boundOutputChainAlpha'') (auto simp add: calc_atm) qed lemma alphaBoundOutput: fixes x :: name and y :: name and B :: "('a::fs_name, 'b::fs_name, 'c::fs_name) boundOutput" assumes "y ♯ B" shows "(νx)B = (νy)([(x, y)] ∙ B)" using assms by(auto simp add: boundOutput.inject alpha fresh_left calc_atm) lemma boundOutputEqFresh: fixes B :: "('a::fs_name, 'b::fs_name, 'c::fs_name) boundOutput" and C :: "('a, 'b, 'c) boundOutput" and x :: name and y :: name assumes "(νx)B = (νy)C" and "x ♯ B" shows "y ♯ C" using assms by(auto simp add: boundOutput.inject alpha fresh_left calc_atm) lemma boundOutputEqSupp: fixes B :: "('a::fs_name, 'b::fs_name, 'c::fs_name) boundOutput" and C :: "('a, 'b, 'c) boundOutput" and x :: name and y :: name assumes "(νx)B = (νy)C" and "x ∈ supp B" shows "y ∈ supp C" using assms apply(auto simp add: boundOutput.inject alpha fresh_left calc_atm) apply(drule_tac pi="[(x, y)]" in pt_set_bij2[OF pt_name_inst, OF at_name_inst]) by(simp add: eqvts calc_atm) lemma boundOutputChainEq: fixes xvec :: "name list" and B :: "('a::fs_name, 'b::fs_name, 'c::fs_name) boundOutput" and yvec :: "name list" and B' :: "('a, 'b, 'c) boundOutput" assumes "(ν*xvec)B = (ν*yvec)B'" and "xvec ♯* yvec" and "length xvec = length yvec" shows "∃p. (set p) ⊆ (set xvec) × set (yvec) ∧ distinctPerm p ∧ B = p ∙ B' ∧ (se proof - obtain n where "n = length xvec" by auto with assms show ?thesis proof(induct n arbitrary: xvec yvec B B') case(0 xvec yvec B B') have Eq: "(ν*xvec)B = (ν*yvec)B'" by fact from ‹0 = length xvec› have "xvec = []" by auto moreover with ‹length xvec = length yvec› have "yvec = []" by(case_tac yvec) auto ultimately show ?case using Eq by(simp add: boundOutput.inject) next case(Suc n xvec yvec B B') from ‹Suc n = length xvec› obtain x xvec' where "xvec = x#xvec'" and "length xvec' = n" by(case_tac xvec) auto from ‹(ν*xvec)B = (ν*yvec)B'› ‹xvec = x # xvec'› ‹length xvec = length yvec› obtain y yvec' where "(ν*(x#xvec'))B = (ν*(y#yvec'))B'" and "yvec = y#yvec'" and "length xvec' = length yvec'" by(case_tac yvec) auto hence EQ: "(νx)((ν*xvec')B) = (νy)((ν*yvec')B')" by simp from ‹xvec = x#xvec'› ‹yvec=y#yvec'› ‹xvec ♯* yvec› have "x ≠ y" and "xvec' ♯* yvec'" and "x ♯ yvec'" and "y ♯ xvec'" by auto have IH: "∧xvec yvec B B'. [(ν*xvec)(B::('a::fs_name, 'b::fs_name, 'c::fs_name) bound by fact from EQ ‹x ≠ y› have EQ': "(ν*xvec')B = ([(x, y)] ∙ ((ν*yvec')B'))" and xFreshB': "x ♯ ((ν*yvec')B')" and yFreshB: "y ♯ ((ν*xvec')B)" by(metis boundOutput.inject alpha)+ from xFreshB' ‹x ♯ yvec'› have "x ♯ B'" by(auto simp add: boundOutputFresh) (simp add: fresh_def name_list_supp)+ from yFreshB ‹y ♯ xvec'› have "y ♯ B" by(auto simp add: boundOutputFresh) (simp add: fresh_def name_list_supp)+ show ?case proof(case_tac "x ♯ (ν*xvec')B") assume xFreshB: "x ♯ (ν*xvec')B" with EQ have yFreshB': "y ♯ (ν*yvec')B'" by(rule boundOutputEqFresh) with xFreshB' EQ' have "(ν*xvec')B = (ν*yvec')B'" by(simp) with ‹xvec' ♯* yvec'› ‹length xvec' = length yvec'› ‹length xvec' = n› IH obtain p where S: "(set p) ⊆ (set xvec') × (set yvec')" and "distinctPerm p" and "B = p and "set(map fst p) ⊆ supp B" and "xvec' ♯* B'" and "yvec' ♯* B" by blast from S have "(set p) ⊆ set(x#xvec') × set(y#yvec')" by auto moreover note ‹xvec = x#xvec'› ‹yvec=y#yvec'› ‹distinctPerm p› ‹B = p ∙ B'› ‹xvec' ♯* B'› ‹x ♯ B'› ‹x ♯ B'› ‹yvec' ♯* B› ‹y ♯ B› ‹set(map fst p) ⊆ supp B› ultimately show ?case by auto next assume "¬(x ♯ (ν*xvec')B)" hence xSuppB: "x ∈ supp((ν*xvec')B)" by(simp add: fresh_def) with EQ have ySuppB': "y ∈ supp ((ν*yvec')B')" by(rule boundOutputEqSupp) hence "y ♯ yvec'" by(induct yvec') (auto simp add: boundOutput.supp abs_supp) with ‹x ♯ yvec'› EQ' have "(ν*xvec')B = (ν*yvec')([(x, y)] ∙ B')" by(simp add: eqvts) with ‹xvec' ♯* yvec'› ‹length xvec' = length yvec'› ‹length xvec' = n› IH obtain p where S: "(set p) ⊆ (set xvec') × (set yvec')" and "distinctPerm p" and "B = p and "set(map fst p) ⊆ supp B" and "xvec' ♯* ([(x, y)] ∙ B')" and "yvec' ♯* B" by blast from xSuppB have "x ♯ xvec'" by(induct xvec') (auto simp add: boundOutput.supp abs_supp) with ‹x ♯ yvec'› ‹y ♯ xvec'› ‹y ♯ yvec'› S have "x ♯ p" and "y ♯ p" apply(induct p) by(auto simp add: name_list_supp) (auto simp add: fresh_def) from S have "(set ((x, y)#p)) ⊆ (set(x#xvec')) × (set(y#yvec'))" by force moreover from ‹x ≠ y› ‹x ♯ p› ‹y ♯ p› S ‹distinctPerm p› have "distinctPerm((x,y)#p)" by simp moreover from ‹B = p ∙ [(x, y)] ∙ B'› ‹x ♯ p› ‹y ♯ p› have "B = [(x, y)] ∙ p ∙ B'" by(subst perm_compose) simp hence "B = ((x, y)#p) ∙ B'" by simp moreover from ‹xvec' ♯* ([(x, y)] ∙ B')› have "([(x, y)] ∙ xvec') ♯* ([(x, y)] ∙ [( by(simp only: pt_fresh_star_bij[OF pt_name_inst, OF at_name_inst]) with ‹x ♯ xvec'› ‹y ♯ xvec'› ‹x ♯ B'› have "(x#xvec') ♯* B'" by simp moreover from ‹y ♯ B› ‹yvec' ♯* B› have "(y#yvec') ♯* B" by simp moreover from ‹set(map fst p) ⊆ supp B› xSuppB ‹x ♯ xvec'› have "set(map fst ((x, y)#p)) ⊆ supp B" by(simp add: BOresChainSupp) ultimately show ?case using ‹xvec=x#xvec'› ‹yvec=y#yvec'› by metis qed qed qed lemma boundOutputChainEqLength: fixes xvec :: "name list" and M :: "'a::fs_name" and P :: "('a, 'b::fs_name, 'c::fs_name) psi" and yvec :: "name list" and N :: "'a::fs_name" and Q :: "('a, 'b::fs_name, 'c::fs_name) psi" assumes "(ν*xvec)M ≺' P = (ν*yvec)N ≺' Q" shows "length xvec = length yvec" proof - obtain n where "n = length xvec" by auto with assms show ?thesis proof(induct n arbitrary: xvec yvec M P N Q) case(0 xvec yvec M P N Q) from ‹0 = length xvec› have "xvec = []" by auto moreover with ‹(ν*xvec)M ≺' P = (ν*yvec)N ≺' Q› have "yvec = []" by(case_tac yvec) auto ultimately show ?case by simp next case(Suc n xvec yvec M P N Q) from ‹Suc n = length xvec› obtain x xvec' where "xvec = x#xvec'" and "length xvec' = n" by(case_tac xvec) auto from ‹(ν*xvec)M ≺' P = (ν*yvec)N ≺' Q› ‹xvec = x # xvec'› obtain y yvec' where "(ν*(x#xvec'))M ≺' P = (ν*(y#yvec'))N ≺' Q" and "yvec = y#yvec'" by(case_tac yvec) auto hence EQ: "(νx)((ν*xvec')M ≺' P) = (νy)((ν*yvec')N ≺' Q)" by simp have IH: "∧xvec yvec M P N Q. [(ν*xvec)M ≺' P = (ν*yvec)N ≺' (Q::('a, 'b, 'c) psi); n = by fact show ?case proof(case_tac "x = y") assume "x = y" with EQ have "(ν*xvec')M ≺' P = (ν*yvec')N ≺' Q" by(simp add: alpha boundOutput.inject) with IH ‹length xvec' = n› have "length xvec' = length yvec'" by blast with ‹xvec = x#xvec'› ‹yvec=y#yvec'› show ?case by simp next assume "x ≠ y" with EQ have "(ν*xvec')M ≺' P = [(x, y)] ∙ (ν*yvec')N ≺' Q" by(simp add: alpha boundOutput.inject) hence "(ν*xvec')M ≺' P = (ν*([(x, y)] ∙ yvec'))([(x, y)] ∙ N) ≺' ([(x, y)] ∙ Q)" by(simp add: eqvts) with IH ‹length xvec' = n› have "length xvec' = length ([(x, y)] ∙ yvec')" by blast hence "length xvec' = length yvec'" by simp with ‹xvec = x#xvec'› ‹yvec=y#yvec'› show ?case by simp qed qed qed lemma boundOutputChainEq': fixes xvec :: "name list" and M :: "'a::fs_name" and P :: "('a, 'b::fs_name, 'c::fs_name) psi" and yvec :: "name list" and N :: 'a and Q :: "('a::fs_name, 'b::fs_name, 'c::fs_name) psi" assumes "(ν*xvec)M ≺' P = (ν*yvec)N ≺' Q" and "xvec ♯* yvec" shows "∃p. (set p) ⊆ (set xvec) × set (yvec) ∧ distinctPerm p ∧ M = p ∙ N ∧ P = using assms apply(frule_tac boundOutputChainEqLength) apply(drule_tac boundOutputChainEq) by(auto simp add: boundOutput.inject) lemma boundOutputChainEq'': fixes xvec :: "name list" and M :: "'a::fs_name" and P :: "('a, 'b::fs_name, 'c::fs_name) psi" and yvec :: "name list" and N :: 'a and Q :: "('a::fs_name, 'b::fs_name, 'c::fs_name) psi" assumes "(ν*xvec)M ≺' P = (ν*yvec)N ≺' Q" and "xvec ♯* yvec" and "distinct xvec" and "distinct yvec" obtains p where "(set p) ⊆ (set xvec) × set (p ∙ xvec)" and "distinctPerm p" and "yvec proof - assume "∧p. [set p ⊆ set xvec × set (p ∙ xvec); distinctPerm p; yvec = p ∙ xvec; moreover obtain n where "n = length xvec" by auto with assms have "∃p. (set p) ⊆ (set xvec) × set (yvec) ∧ distinctPerm p ∧ yvec = p proof(induct n arbitrary: xvec yvec M P N Q) case(0 xvec yvec M P N Q) have Eq: "(ν*xvec)M ≺' P = (ν*yvec)N ≺' Q" by fact from ‹0 = length xvec› have "xvec = []" by auto moreover with Eq have "yvec = []" by(case_tac yvec) auto ultimately show ?case using Eq by(simp add: boundOutput.inject) next case(Suc n xvec yvec M P N Q) from ‹Suc n = length xvec› obtain x xvec' where "xvec = x#xvec'" and "length xvec' = n" by(case_tac xvec) auto from ‹(ν*xvec)M ≺' P = (ν*yvec)N ≺' Q› ‹xvec = x # xvec'› obtain y yvec' where "(ν*(x#xvec'))M ≺' P = (ν*(y#yvec'))N ≺' Q" and "yvec = y#yvec'" by(case_tac yvec) auto hence EQ: "(νx)((ν*xvec')M ≺' P) = (νy)((ν*yvec')N ≺' Q)" by simp from ‹xvec = x#xvec'› ‹yvec=y#yvec'› ‹xvec ♯* yvec› have "x ≠ y" and "xvec' ♯* yvec'" and "x ♯ yvec'" and "y ♯ xvec'" by auto from ‹distinct xvec› ‹distinct yvec› ‹xvec=x#xvec'› ‹yvec=y#yvec'› have "x ♯ xvec' by simp+ have IH: "∧xvec yvec M P N Q. [(ν*xvec)(M::'a) ≺' (P::('a, 'b, 'c) psi) = (ν*yvec)N ≺' Q by fact from EQ ‹x ≠ y› ‹x ♯ yvec'› ‹y ♯ yvec'› ‹y ♯ xvec'› ‹x ♯ xvec'› have "(ν*xvec')M ≺' P apply - apply(simp add: boundOutput.inject alpha eqvts) apply(simp add: boundOutput.inject alpha eqvts) apply(simp add: boundOutput.inject alpha eqvts) by(simp add: boundOutput.inject alpha' eqvts)+ with ‹xvec' ♯* yvec'› ‹distinct xvec'› ‹distinct yvec'› ‹length xvec' = n› IH obtain p where S: "(set p) ⊆ (set xvec') × (set yvec')" and "distinctPerm p" and "yvec' by metis from S have "set((x, y)#p) ⊆ set(x#xvec') × set(y#yvec')" by auto moreover from ‹x ♯ xvec'› ‹x ♯ yvec'› ‹y ♯ xvec'› ‹y ♯ yvec'› S have "x ♯ p" and "y ♯ apply(induct p) by(auto simp add: fresh_prod name_list_supp) (auto simp add: fresh_def) with S ‹distinctPerm p› ‹x ≠ y› have "distinctPerm((x, y)#p)" by auto moreover from ‹yvec' = p ∙ xvec'› ‹x ♯ p› ‹y ♯ p› ‹x ♯ xvec'› ‹y ♯ xvec'› have "(y# by(simp add: eqvts calc_atm perm_compose freshChainSimps) moreover from ‹([(x, y)] ∙ N) = p ∙ M› have "([(x, y)] ∙ [(x, y)] ∙ N) = [(x, y)] ∙ p ∙ M" by(simp add: pt_bij) hence "N = ((x, y)#p) ∙ M" by simp moreover from ‹([(x, y)] ∙ Q) = p ∙ P› have "([(x, y)] ∙ [(x, y)] ∙ Q) = [(x, y)] ∙ p ∙ P" by(simp add: pt_bij) hence "Q = ((x, y)#p) ∙ P" by simp moreover from ‹xvec' ♯* ([(x, y)] ∙ N)› have "([(x, y)] ∙ xvec') ♯* ([(x, y)] ∙ [(x by(subst fresh_star_bij) with ‹x ♯ xvec'› ‹y ♯ xvec'› have "xvec' ♯* N" by simp with ‹x ♯ N› have "(x#xvec') ♯* N" by simp moreover from ‹xvec' ♯* ([(x, y)] ∙ Q)› have "([(x, y)] ∙ xvec') ♯* ([(x, y)] ∙ [(x by(subst fresh_star_bij) with ‹x ♯ xvec'› ‹y ♯ xvec'› have "xvec' ♯* Q" by simp with ‹x ♯ Q› have "(x#xvec') ♯* Q" by simp moreover from ‹y ♯ M› ‹yvec' ♯* M› have "(y#yvec') ♯* M" by simp moreover from ‹y ♯ P› ‹yvec' ♯* P› have "(y#yvec') ♯* P" by simp ultimately show ?case using ‹xvec=x#xvec'› ‹yvec=y#yvec'› by metis qed ultimately show ?thesis by blast qed lemma boundOutputEqSupp': fixes x :: name and xvec :: "name list" and M :: "'a::fs_name" and P :: "('a, 'b::fs_name, 'c::fs_name) psi" and y :: name and yvec :: "name list" and N :: 'a and Q :: "('a, 'b, 'c) psi" assumes Eq: "(νx)((ν*xvec)M ≺' P) = (νy)((ν*yvec)N ≺' Q)" and "x ≠ y" and "x ♯ yvec" and "x ♯ xvec" and "y ♯ xvec" and "y ♯ yvec" and "xvec ♯* yvec" and "x ∈ supp M" shows "y ∈ supp N" proof - from Eq ‹x ≠ y› ‹x ♯ yvec› ‹y ♯ yvec› have "(ν*xvec)M ≺' P = (ν*yvec)([(x, y)] ∙ N) ≺' ([ by(simp add: boundOutput.inject alpha eqvts) then obtain p where S: "set p ⊆ set xvec × set yvec" and "M = p ∙ [(x, y)] ∙ N" and "dist by(blast dest: boundOutputChainEq') with ‹x ∈ supp M› have "x ∈ supp(p ∙ [(x, y)] ∙ N)" by simp hence "(p ∙ x) ∈ p ∙ supp(p ∙ [(x, y)] ∙ N)" by(simp add: pt_set_bij[OF pt_name_inst, OF at_name_inst]) with ‹x ♯ xvec› ‹x ♯ yvec› S ‹distinctPerm p› have "x ∈ supp([(x, y)] ∙ N)" by(simp add: eqvts) hence "([(x, y)] ∙ x) ∈ ([(x, y)] ∙ (supp([(x, y)] ∙ N)))" by(simp add: pt_set_bij[OF pt_name_inst, OF at_name_inst]) with ‹x ≠ y› show ?thesis by(simp add: calc_atm eqvts) qed lemma boundOutputChainOpenIH: fixes xvec :: "name list" and x :: name and B :: "('a::fs_name, 'b::fs_name, 'c::fs_name) boundOutput" and yvec :: "name list" and y :: name and B' :: "('a, 'b, 'c) boundOutput" assumes Eq: "(ν*xvec)((νx)B) = (ν*yvec)((νy)B')" and L: "length xvec = length yvec" and xFreshB': "x ♯ B'" and xFreshxvec: "x ♯ xvec" and xFreshyvec: "x ♯ yvec" shows "(ν*xvec)B = (ν*yvec)([(x, y)] ∙ B')" using assms proof(induct n=="length xvec" arbitrary: xvec yvec y B' rule: nat.induct) case(zero xvec yvec y B') have "0 = length xvec" and "length xvec = length yvec" by fact+ moreover have "(ν*xvec)(νx)B = (ν*yvec)(νy)B'" by fact ultimately show ?case by(auto simp add: boundOutput.inject alpha) next case(Suc n xvec yvec y B') have L: "length xvec = length yvec" and "Suc n = length xvec" by fact+ then obtain x' xvec' y' yvec' where xEq: "xvec = x'#xvec'" and yEq: "yvec = y'#yvec'" and L': "length xvec' = length yvec'" by(cases xvec, auto, cases yvec, auto) have xFreshB': "x ♯ B'" by fact have "x ♯ xvec" and "x ♯ yvec" by fact+ with xEq yEq have xineqx': "x ≠ x'" and xFreshxvec': "x ♯ xvec'" and xineqy': "x ≠ y'" and xFreshyvec': "x ♯ yvec'" by simp+ have "(ν*xvec)(νx)B = (ν*yvec)(νy)B'" by fact with xEq yEq have Eq: "(νx')((ν*xvec')(νx)B) = (νy')((ν*yvec')(νy)B')" by simp have "Suc n = length xvec" by fact with xEq have L'': "n = length xvec'" by simp have "(νx')((ν*xvec')B) = (νy')((ν*yvec')([(x, y)] ∙ B'))" proof(case_tac "x'=y'") assume x'eqy': "x' = y'" with Eq have "(ν*xvec')(νx)B = (ν*yvec')(νy)B'" by(simp add: boundOutput.inject alpha) hence "(ν*xvec')B = (ν*yvec')([(x, y)] ∙ B')" using L' xFreshB' xFreshxvec' xFreshyvec' L'' by(rule_tac Suc) with x'eqy' show ?thesis by(simp add: boundOutput.inject alpha) next assume x'ineqy': "x' ≠ y'" with Eq have Eq': "(ν*xvec')(νx)B = (ν*([(x', y')] ∙ yvec'))(ν([(x', y')] ∙ y))([(x', y')] ∙ B') and x'FreshB': "x' ♯ (ν*yvec')(νy)B'" by(simp add: boundOutput.inject alpha eqvts)+ from L' have "length xvec' = length ([(x', y')] ∙ yvec')" by simp moreover from xineqx' xineqy' xFreshB' have "x ♯ [(x', y')] ∙ B'" by(simp add: fresh_left moreover from xineqx' xineqy' xFreshyvec' have "x ♯ [(x', y')] ∙ yvec'" by(simp add: fres ultimately have "(ν*xvec')B = (ν*([(x', y')] ∙ yvec'))([(x, ([(x', y')] ∙ y))] ∙ [(x', y by(rule_tac Suc) moreover from x'FreshB' have "x' ♯ (ν*yvec')([(x, y)] ∙ B')" proof(case_tac "x' ♯ yvec'") assume "x' ♯ yvec'" with x'FreshB' have x'FreshB': "x' ♯ (νy)B'" by(simp add: fresh_def BOresChainSupp) show ?thesis proof(case_tac "x'=y") assume x'eqy: "x' = y" show ?thesis proof(case_tac "x=y") assume "x=y" with xFreshB' x'eqy show ?thesis by(simp add: BOresChainSupp fresh_def) next assume "x ≠ y" with ‹x ♯ B'› have "y ♯ [(x, y)] ∙ B'" by(simp add: fresh_left calc_atm) with x'eqy show ?thesis by(simp add: BOresChainSupp fresh_def) qed next assume x'ineqy: "x' ≠ y" with x'FreshB' have "x' ♯ B'" by(simp add: abs_fresh) with xineqx' x'ineqy have "x' ♯ ([(x, y)] ∙ B')" by(simp add: fresh_left calc_atm) thus ?thesis by(simp add: BOresChainSupp fresh_def) qed next assume "¬x' ♯ yvec'" thus ?thesis by(simp add: BOresChainSupp fresh_def) qed ultimately show ?thesis using x'ineqy' xineqx' xineqy' apply(simp add: boundOutput.inject alpha eqvts) apply(subst perm_compose[of "[(x', y')]"]) by(simp add: calc_atm) qed with xEq yEq show ?case by simp qed lemma boundOutputPar1Dest: fixes xvec :: "name list" and M :: "'a::fs_name" and P :: "('a, 'b::fs_name, 'c::fs_name) psi" and yvec :: "name list" and N :: 'a and Q :: "('a, 'b, 'c) psi" and R :: "('a, 'b, 'c) psi" assumes "(ν*xvec)M ≺' P = (ν*yvec)N ≺' (Q ∥ R)" and "xvec ♯* R" and "yvec ♯* R" obtains T where "P = T ∥ R" and "(ν*xvec)M ≺' T = (ν*yvec)N ≺' Q" proof - assume "∧T. [P = T ∥ R; (ν*xvec)M ≺' T = (ν*yvec)N ≺' Q] ==> thesis" moreover obtain n where "n = length xvec" by auto with assms have "∃T. P = T ∥ R ∧ (ν*xvec)M ≺' T = (ν*yvec)N ≺' Q" proof(induct n arbitrary: xvec yvec M N P Q R) case(0 xvec yvec M N P Q R) have Eq: "(ν*xvec)M ≺' P = (ν*yvec)N ≺' (Q ∥ R)" by fact from ‹0 = length xvec› have "xvec = []" by auto moreover with Eq have "yvec = []" by(case_tac yvec) auto ultimately show ?case using Eq by(simp add: boundOutput.inject) next case(Suc n xvec yvec M N P Q R) from ‹Suc n = length xvec› obtain x xvec' where "xvec = x#xvec'" and "length xvec' = n" by(case_tac xvec) auto from ‹(ν*xvec)M ≺' P = (ν*yvec)N ≺' (Q ∥ R)› ‹xvec = x # xvec'› obtain y yvec' where "(ν*(x#xvec'))M ≺' P = (ν*(y#yvec'))N ≺' (Q ∥ R)" and "yvec = y#yvec'" by(case_tac yvec) auto hence EQ: "(νx)((ν*xvec')M ≺' P) = (νy)((ν*yvec')N ≺' (Q ∥ R))" by simp from ‹xvec ♯* R› ‹yvec ♯* R› ‹xvec = x#xvec'› ‹yvec = y#yvec'› have "x ♯ R" and "xvec' ♯* R" and "y ♯ R" and "yvec' ♯* R" by auto show ?case proof(case_tac "x = y") assume "x = y" with EQ have "(ν*xvec')M ≺' P = (ν*yvec')N ≺' (Q ∥ R)" by(simp add: boundOutput.inject alpha) with ‹xvec' ♯* R› ‹yvec' ♯* R› ‹length xvec' = n› obtain T where "P = T ∥ R" and "(ν*xvec')M ≺' T = (ν*yvec')N ≺' Q" by(drule_tac Suc) auto with ‹xvec=x#xvec'› ‹yvec=y#yvec'› ‹x=y› show ?case by(force simp add: boundOutput.inject alpha) next assume "x ≠ y" with EQ ‹x ♯ R› ‹y ♯ R› have "(ν*xvec')M ≺' P = (ν*([(x, y)] ∙ yvec'))([(x, y)] ∙ N) ≺' (([(x, y)] ∙ Q) ∥ R)" and xFreshQR: "x ♯ (ν*yvec')N ≺' (Q ∥ R)" by(simp add: boundOutput.inject alpha eqvts)+ moreover from ‹yvec' ♯* R› have "([(x, y)] ∙ yvec') ♯* ([(x, y)] ∙ R)" by(simp add: pt_fresh_star_bij[OF pt_name_inst, OF at_name_inst]) with ‹x ♯ R› ‹y ♯ R› have "([(x, y)] ∙ yvec') ♯* R" by simp moreover note ‹xvec' ♯* R› ‹length xvec' = n› ultimately obtain T where "P = T ∥ R" and A: "(ν*xvec')M ≺' T = (ν*([(x, y)] ∙ yvec'))([(x, y by(drule_tac Suc) auto from A have "(νx)((ν*xvec')M ≺' T) = (νx)((ν*([(x, y)] ∙ yvec'))([(x, y)] ∙ N) ≺' ([(x, y)] ∙ Q by(simp add: boundOutput.inject alpha) moreover from xFreshQR have "x ♯ (ν*yvec')N ≺' Q" by(force simp add: boundOutputFresh) ultimately show ?thesis using ‹P = T ∥ R› ‹xvec=x#xvec'› ‹yvec=y#yvec'› xFreshQR by(force simp add: alphaBoundOutput name_swap eqvts) qed qed ultimately show ?thesis by blast qed lemma boundOutputPar1Dest': fixes xvec :: "name list" and M :: "'a::fs_name" and P :: "('a, 'b::fs_name, 'c::fs_name) psi" and yvec :: "name list" and N :: 'a and Q :: "('a, 'b, 'c) psi" and R :: "('a, 'b, 'c) psi" assumes "(ν*xvec)M ≺' P = (ν*yvec)N ≺' (Q ∥ R)" and "xvec ♯* yvec" obtains T p where "set p ⊆ set xvec × set yvec" and "P = T ∥ (p ∙ R)" and "(ν*xvec)M ≺' T = proof - assume "∧p T. [set p ⊆ set xvec × set yvec; P = T ∥ (p ∙ R); (ν*xvec)M ≺' T = (ν*yvec)N moreover obtain n where "n = length xvec" by auto with assms have "∃p T. set p ⊆ set xvec × set yvec ∧ P = T ∥ (p ∙ R) ∧ (ν*xvec)M ≺' T proof(induct n arbitrary: xvec yvec M N P Q R) case(0 xvec yvec M N P Q R) have Eq: "(ν*xvec)M ≺' P = (ν*yvec)N ≺' (Q ∥ R)" by fact from ‹0 = length xvec› have "xvec = []" by auto moreover with Eq have "yvec = []" by(case_tac yvec) auto ultimately show ?case using Eq by(simp add: boundOutput.inject) next case(Suc n xvec yvec M N P Q R) from ‹Suc n = length xvec› obtain x xvec' where "xvec = x#xvec'" and "length xvec' = n" by(case_tac xvec) auto from ‹(ν*xvec)M ≺' P = (ν*yvec)N ≺' (Q ∥ R)› ‹xvec = x # xvec'› obtain y yvec' where "(ν*(x#xvec'))M ≺' P = (ν*(y#yvec'))N ≺' (Q ∥ R)" and "yvec = y#yvec'" by(case_tac yvec) auto hence Eq: "(νx)((ν*xvec')M ≺' P) = (νy)((ν*yvec')N ≺' (Q ∥ R))" by simp from ‹xvec = x#xvec'› ‹yvec=y#yvec'› ‹xvec ♯* yvec› have "x ≠ y" and "x ♯ yvec'" and " by auto from Eq ‹x ≠ y› have Eq': "(ν*xvec')M ≺' P = [(x, y)] ∙ (ν*yvec')N ≺' (Q ∥ R)" and xFreshQR: "x ♯ (ν*yvec')N ≺' (Q ∥ R)" by(simp add: boundOutput.inject alpha)+ have IH: "∧xvec yvec M N P Q R. [(ν*xvec)M ≺' (P::('a, 'b, 'c) psi) = (ν*yvec)N ≺' (Q ∥ by fact show ?case proof(case_tac "x ♯ (ν*xvec')M ≺' P") assume "x ♯ (ν*xvec')M ≺' P" with Eq have yFreshQR: "y ♯ (ν*yvec')N ≺' (Q ∥ R)" by(rule boundOutputEqFresh) with Eq' xFreshQR have "(ν*xvec')M ≺' P = (ν*yvec')N ≺' (Q ∥ R)" by simp with ‹xvec' ♯* yvec'› ‹length xvec' = n› obtain p T where S: "set p ⊆ set xvec' × set yvec'" and "P = T ∥ (p ∙ R)" and A: "(ν*xvec')M by(drule_tac IH) auto from yFreshQR xFreshQR have yFreshQ: "y ♯ (ν*yvec')N ≺' Q" and xFreshQ: "x ♯ (ν*yvec')N ≺' Q" by(force simp add: BOresChainSupp fresh_def boundOutput.supp psi.supp)+ hence "(νx)((ν*yvec')N ≺' Q) = (νy)((ν*yvec')N ≺' Q)" by (subst alphaBoundOutput) simp+ with A have "(νx)((ν*xvec')M ≺' T) = (νy)((ν*yvec')N ≺' Q)" by simp with ‹xvec=x#xvec'› ‹yvec=y#yvec'› S ‹P = T ∥ (p ∙ R)› show ?case by auto next assume "¬(x ♯ (ν*xvec')M ≺' P)" hence "x ∈ supp((ν*xvec')M ≺' P)" by(simp add: fresh_def) with Eq have "y ∈ supp((ν*yvec')N ≺' (Q ∥ R))" by(rule boundOutputEqSupp) hence "y ♯ yvec'" by(simp add: BOresChainSupp fresh_def) with Eq' ‹x ♯ yvec'› have "(ν*xvec')M ≺' P = (ν*yvec')([(x, y)] ∙ N) ≺' (([(x, y)] ∙ Q) ∥ by(simp add: eqvts) moreover note ‹xvec' ♯* yvec'› ‹length xvec' = n› ultimately obtain p T where S: "set p ⊆ set xvec' × set yvec'" and "P = T ∥ (p ∙ [(x, y) by(drule_tac IH) auto from S have "set(p@[(x, y)]) ⊆ set(x#xvec') × set(y#yvec')" by auto moreover from ‹P = T ∥ (p ∙ [(x, y)] ∙ R)› have "P = T ∥ ((p @ [(x, y)]) ∙ R)" by(simp add: pt2[OF pt_name_inst]) moreover from xFreshQR have xFreshQ: "x ♯ (ν*yvec')N ≺' Q" by(force simp add: BOresChainSupp fresh_def boundOutput.supp psi.supp)+ with ‹x ♯ yvec'› ‹y ♯ yvec'› ‹x ≠ y› have "y ♯ (ν*yvec')([(x, y)] ∙ N) ≺' ([(x, y)] ∙ by(simp add: fresh_left calc_atm) with ‹x ♯ yvec'› ‹y ♯ yvec'› have "(νx)((ν*yvec')([(x, y)] ∙ N) ≺' ([(x, y)] ∙ Q)) = (νy)((ν* by(subst alphaBoundOutput) (assumption | simp add: eqvts)+ with A have "(νx)((ν*xvec')M ≺' T) = (νy)((ν*yvec')N ≺' Q)" by simp ultimately show ?thesis using ‹xvec=x#xvec'› ‹yvec=y#yvec'› by(rule_tac x="p@[(x, y)]" in exI) force qed qed ultimately show ?thesis by blast qed lemma boundOutputPar2Dest: fixes xvec :: "name list" and M :: "'a::fs_name" and P :: "('a, 'b::fs_name, 'c::fs_name) psi" and yvec :: "name list" and N :: 'a and Q :: "('a, 'b, 'c) psi" and R :: "('a, 'b, 'c) psi" assumes "(ν*xvec)M ≺' P = (ν*yvec)N ≺' (Q ∥ R)" and "xvec ♯* Q" and "yvec ♯* Q" obtains T where "P = Q ∥ T" and "(ν*xvec)M ≺' T = (ν*yvec)N ≺' R" proof - assume "∧T. [P = Q ∥ T; (ν*xvec)M ≺' T = (ν*yvec)N ≺' R] ==> thesis" moreover obtain n where "n = length xvec" by auto with assms have "∃T. P = Q ∥ T ∧ (ν*xvec)M ≺' T = (ν*yvec)N ≺' R" proof(induct n arbitrary: xvec yvec M N P Q R) case(0 xvec yvec M N P Q R) have Eq: "(ν*xvec)M ≺' P = (ν*yvec)N ≺' (Q ∥ R)" by fact from ‹0 = length xvec› have "xvec = []" by auto moreover with Eq have "yvec = []" by(case_tac yvec) auto ultimately show ?case using Eq by(simp add: boundOutput.inject) next case(Suc n xvec yvec M N P Q R) from ‹Suc n = length xvec› obtain x xvec' where "xvec = x#xvec'" and "length xvec' = n" by(case_tac xvec) auto from ‹(ν*xvec)M ≺' P = (ν*yvec)N ≺' (Q ∥ R)› ‹xvec = x # xvec'› obtain y yvec' where "(ν*(x#xvec'))M ≺' P = (ν*(y#yvec'))N ≺' (Q ∥ R)" and "yvec = y#yvec'" by(case_tac yvec) auto hence EQ: "(νx)((ν*xvec')M ≺' P) = (νy)((ν*yvec')N ≺' (Q ∥ R))" by simp from ‹xvec ♯* Q› ‹yvec ♯* Q› ‹xvec = x#xvec'› ‹yvec = y#yvec'› have "x ♯ Q" and "xvec' ♯* Q" and "y ♯ Q" and "yvec' ♯* Q" by auto have IH: "∧xvec yvec M N P Q R. [(ν*xvec)M ≺' (P::('a, 'b, 'c) psi) = (ν*yvec)N ≺' (Q ∥ by fact show ?case proof(case_tac "x = y") assume "x = y" with EQ have "(ν*xvec')M ≺' P = (ν*yvec')N ≺' (Q ∥ R)" by(simp add: boundOutput.inject alpha) with ‹xvec' ♯* Q› ‹yvec' ♯* Q› ‹length xvec' = n› obtain T where "P = Q ∥ T" and "(ν*xvec')M ≺' T = (ν*yvec')N ≺' R" by(drule_tac IH) auto with ‹xvec=x#xvec'› ‹yvec=y#yvec'› ‹x=y› show ?case by(force simp add: boundOutput.inject alpha) next assume "x ≠ y" with EQ ‹x ♯ Q› ‹y ♯ Q› have "(ν*xvec')M ≺' P = (ν*([(x, y)] ∙ yvec'))([(x, y)] ∙ N) ≺' (Q ∥ ([(x, y)] ∙ R))" and xFreshQR: "x ♯ (ν*yvec')N ≺' (Q ∥ R)" by(simp add: boundOutput.inject alpha eqvts)+ moreover from ‹yvec' ♯* Q› have "([(x, y)] ∙ yvec') ♯* ([(x, y)] ∙ Q)" by(simp add: pt_fresh_star_bij[OF pt_name_inst, OF at_name_inst]) with ‹x ♯ Q› ‹y ♯ Q› have "([(x, y)] ∙ yvec') ♯* Q" by simp moreover note ‹xvec' ♯* Q› ‹length xvec' = n› ultimately obtain T where "P = Q ∥ T" and A: "(ν*xvec')M ≺' T = (ν*([(x, y)] ∙ yvec'))([(x, y by(drule_tac IH) auto from A have "(νx)((ν*xvec')M ≺' T) = (νx)((ν*([(x, y)] ∙ yvec'))([(x, y)] ∙ N) ≺' ([(x, y)] ∙ R by(simp add: boundOutput.inject alpha) moreover from xFreshQR have "x ♯ (ν*yvec')N ≺' R" by(force simp add: boundOutputFresh) ultimately show ?thesis using ‹P = Q ∥ T› ‹xvec=x#xvec'› ‹yvec=y#yvec'› xFreshQR by(force simp add: alphaBoundOutput name_swap eqvts) qed qed ultimately show ?thesis by blast qed lemma boundOutputPar2Dest': fixes xvec :: "name list" and M :: "'a::fs_name" and P :: "('a, 'b::fs_name, 'c::fs_name) psi" and yvec :: "name list" and N :: 'a and Q :: "('a, 'b, 'c) psi" and R :: "('a, 'b, 'c) psi" assumes "(ν*xvec)M ≺' P = (ν*yvec)N ≺' (Q ∥ R)" and "xvec ♯* yvec" obtains T p where "set p ⊆ set xvec × set yvec" and "P = (p ∙ Q) ∥ T" and "(ν*xvec)M ≺' T = proof - assume "∧p T. [set p ⊆ set xvec × set yvec; P = (p ∙ Q) ∥ T; (ν*xvec)M ≺' T = (ν*yvec)N moreover obtain n where "n = length xvec" by auto with assms have "∃p T. set p ⊆ set xvec × set yvec ∧ P = (p ∙ Q) ∥ T ∧ (ν*xvec)M ≺' T proof(induct n arbitrary: xvec yvec M N P Q R) case(0 xvec yvec M N P Q R) have Eq: "(ν*xvec)M ≺' P = (ν*yvec)N ≺' (Q ∥ R)" by fact from ‹0 = length xvec› have "xvec = []" by auto moreover with Eq have "yvec = []" by(case_tac yvec) auto ultimately show ?case using Eq by(simp add: boundOutput.inject) next case(Suc n xvec yvec M N P Q R) from ‹Suc n = length xvec› obtain x xvec' where "xvec = x#xvec'" and "length xvec' = n" by(case_tac xvec) auto from ‹(ν*xvec)M ≺' P = (ν*yvec)N ≺' (Q ∥ R)› ‹xvec = x # xvec'› obtain y yvec' where "(ν*(x#xvec'))M ≺' P = (ν*(y#yvec'))N ≺' (Q ∥ R)" and "yvec = y#yvec'" by(case_tac yvec) auto hence Eq: "(νx)((ν*xvec')M ≺' P) = (νy)((ν*yvec')N ≺' (Q ∥ R))" by simp from ‹xvec = x#xvec'› ‹yvec=y#yvec'› ‹xvec ♯* yvec› have "x ≠ y" and "x ♯ yvec'" and " by auto from Eq ‹x ≠ y› have Eq': "(ν*xvec')M ≺' P = [(x, y)] ∙ (ν*yvec')N ≺' (Q ∥ R)" and xFreshQR: "x ♯ (ν*yvec')N ≺' (Q ∥ R)" by(simp add: boundOutput.inject alpha)+ have IH: "∧xvec yvec M N P Q R. [(ν*xvec)M ≺' (P::('a, 'b, 'c) psi) = (ν*yvec)N ≺' (Q ∥ by fact show ?case proof(case_tac "x ♯ (ν*xvec')M ≺' P") assume "x ♯ (ν*xvec')M ≺' P" with Eq have yFreshQR: "y ♯ (ν*yvec')N ≺' (Q ∥ R)" by(rule boundOutputEqFresh) with Eq' xFreshQR have "(ν*xvec')M ≺' P = (ν*yvec')N ≺' (Q ∥ R)" by simp with ‹xvec' ♯* yvec'› ‹length xvec' = n› obtain p T where S: "set p ⊆ set xvec' × set yvec'" and "P = (p ∙ Q) ∥ T" and A: "(ν*xvec')M by(drule_tac IH) auto from yFreshQR xFreshQR have yFreshR: "y ♯ (ν*yvec')N ≺' R" and xFreshQ: "x ♯ (ν*yvec')N ≺' R" by(force simp add: BOresChainSupp fresh_def boundOutput.supp psi.supp)+ hence "(νx)((ν*yvec')N ≺' R) = (νy)((ν*yvec')N ≺' R)" by (subst alphaBoundOutput) simp+ with A have "(νx)((ν*xvec')M ≺' T) = (νy)((ν*yvec')N ≺' R)" by simp with ‹xvec=x#xvec'› ‹yvec=y#yvec'› S ‹P = (p ∙ Q) ∥ T› show ?case by auto next assume "¬(x ♯ (ν*xvec')M ≺' P)" hence "x ∈ supp((ν*xvec')M ≺' P)" by(simp add: fresh_def) with Eq have "y ∈ supp((ν*yvec')N ≺' (Q ∥ R))" by(rule boundOutputEqSupp) hence "y ♯ yvec'" by(simp add: BOresChainSupp fresh_def) with Eq' ‹x ♯ yvec'› have "(ν*xvec')M ≺' P = (ν*yvec')([(x, y)] ∙ N) ≺' (([(x, y)] ∙ Q) ∥ by(simp add: eqvts) moreover note ‹xvec' ♯* yvec'› ‹length xvec' = n› ultimately obtain p T where S: "set p ⊆ set xvec' × set yvec'" and "P = (p ∙ [(x, y)] ∙ by(drule_tac IH) auto from S have "set(p@[(x, y)]) ⊆ set(x#xvec') × set(y#yvec')" by auto moreover from ‹P = (p ∙ [(x, y)] ∙ Q) ∥ T› have "P = ((p @ [(x, y)]) ∙ Q) ∥ T" by(simp add: pt2[OF pt_name_inst]) moreover from xFreshQR have xFreshR: "x ♯ (ν*yvec')N ≺' R" by(force simp add: BOresChainSupp fresh_def boundOutput.supp psi.supp)+ with ‹x ♯ yvec'› ‹y ♯ yvec'› ‹x ≠ y› have "y ♯ (ν*yvec')([(x, y)] ∙ N) ≺' ([(x, y)] ∙ by(simp add: fresh_left calc_atm) with ‹x ♯ yvec'› ‹y ♯ yvec'› have "(νx)((ν*yvec')([(x, y)] ∙ N) ≺' ([(x, y)] ∙ R)) = (νy)((ν* by(subst alphaBoundOutput) (assumption | simp add: eqvts)+ with A have "(νx)((ν*xvec')M ≺' T) = (νy)((ν*yvec')N ≺' R)" by simp ultimately show ?thesis using ‹xvec=x#xvec'› ‹yvec=y#yvec'› by(rule_tac x="p@[(x, y)]" in exI) force qed qed ultimately show ?thesis by blast qed lemma boundOutputApp: fixes xvec :: "name list" and yvec :: "name list" and B :: "('a::fs_name, 'b::fs_name, 'c::fs_name) boundOutput" shows "(ν*(xvec@yvec))B = (ν*xvec)((ν*yvec)B)" by(induct xvec) auto lemma openInjectAuxAuxAux: fixes x :: name and xvec :: "name list" shows "∃y yvec. x # xvec = yvec @ [y] ∧ length xvec = length yvec" apply(induct xvec arbitrary: x) apply auto apply(subgoal_tac "∃y yvec. a # xvec = yvec @ [y] ∧ length xvec = length yvec") apply(clarify) apply(rule_tac x=y in exI) by auto lemma openInjectAuxAux: fixes xvec1 :: "name list" and xvec2 :: "name list" and yvec :: "name list" assumes "length(xvec1@xvec2) = length yvec" shows "∃yvec1 yvec2. yvec = yvec1@yvec2 ∧ length xvec1 = length yvec1 ∧ length xv using assms apply(induct yvec arbitrary: xvec1) apply simp apply simp apply(case_tac xvec1) apply simp apply simp apply(subgoal_tac "∃yvec1 yvec2. yvec = yvec1 @ yvec2 ∧ length list = length yvec1 ∧ length xvec2 = length yvec2" apply(clarify) apply(rule_tac x="a#yvec1" in exI) apply(rule_tac x=yvec2 in exI) by auto lemma openInjectAux: fixes xvec1 :: "name list" and x :: name and xvec2 :: "name list" and yvec :: "name list" assumes "length(xvec1@x#xvec2) = length yvec" shows "∃yvec1 y yvec2. yvec = yvec1@y#yvec2 ∧ length xvec1 = length yvec1 ∧ lengt using assms apply(case_tac yvec) apply simp apply simp apply(subgoal_tac "∃(yvec1::name list) (yvec2::name list). yvec1@yvec2 = list ∧ l apply(clarify) apply hypsubst_thin apply simp apply(subgoal_tac "∃y (yvec::name list). a # yvec1 = yvec @ [y] ∧ length yvec1 = apply(clarify) apply(rule_tac x=yvec in exI) apply(rule_tac x=y in exI) apply simp apply(rule_tac x=yvec2 in exI) apply simp apply(rule openInjectAuxAuxAux) apply(insert openInjectAuxAux) apply simp by blast lemma boundOutputOpenDest: fixes yvec :: "name list" and M :: "'a::fs_name" and P :: "('a, 'b::fs_name, 'c::fs_name) psi" and xvec1 :: "name list" and x :: name and xvec2 :: "name list" and N :: 'a and Q :: "('a, 'b, 'c) psi" assumes Eq: "(ν*(xvec1@x#xvec2))M ≺' P = (ν*yvec)N ≺' Q" and "x ♯ xvec1" and "x ♯ yvec" and "x ♯ N" and "x ♯ Q" and "distinct yvec" obtains yvec1 y yvec2 where "yvec=yvec1@y#yvec2" and "length xvec1 = length yvec1" and "l and "(ν*(xvec1@xvec2))M ≺' P = (ν*(yvec1@yvec2))([(x, y)] ∙ N) ≺' ([(x, y)] ∙ Q)" proof - assume Ass: "∧yvec1 y yvec2. [yvec = yvec1 @ y # yvec2; length xvec1 = length yvec1; length xvec2 = length yv (ν*(xvec1 @ xvec2))M ≺' P = (ν*(yvec1 @ yvec2))([(x, y)] ∙ N) ≺' ([(x, y)] ∙ Q)] ==> thesis" from Eq have "length(xvec1@x#xvec2) = length yvec" by(rule boundOutputChainEqLength) then obtain yvec1 y yvec2 where A: "yvec = yvec1@y#yvec2" and "length xvec1 = length yvec and "length xvec2 = length yvec2" by(metis openInjectAux sym) from ‹distinct yvec› A have "y ♯ yvec2" by simp from A ‹x ♯ yvec› have "x ♯ yvec2" and "x ♯ yvec1" by simp+ with Eq ‹length xvec1 = length yvec1› ‹x ♯ N› ‹x ♯ Q› ‹y ♯ yvec2› ‹x ♯ xvec1› A have "(ν*(xvec1@xvec2))M ≺' P = (ν*(yvec1@yvec2))([(x, y)] ∙ N) ≺' ([(x, y)] ∙ Q)" by(force dest: boundOutputChainOpenIH simp add: boundOutputApp BOresChainSupp fresh_def with ‹length xvec1 = length yvec1› ‹length xvec2 = length yvec2› A Ass show ?thesis by blast qed lemma boundOutputOpenDest': fixes yvec :: "name list" and M :: "'a::fs_name" and P :: "('a, 'b::fs_name, 'c::fs_name) psi" and xvec1 :: "name list" and x :: name and xvec2 :: "name list" and N :: 'a and Q :: "('a, 'b, 'c) psi" assumes Eq: "(ν*(xvec1@x#xvec2))M ≺' P = (ν*yvec)N ≺' Q" and "x ♯ xvec1" and "x ♯ yvec" and "x ♯ N" and "x ♯ Q" obtains yvec1 y yvec2 where "yvec=yvec1@y#yvec2" and "length xvec1 = length yvec1" and "l and "(ν*(xvec1@xvec2))M ≺' P = (ν*(yvec1@[(x, y)] ∙ yvec2))([(x, y)] ∙ N) ≺' ([(x, y)] proof - assume Ass: "∧yvec1 y yvec2. [yvec = yvec1 @ y # yvec2; length xvec1 = length yvec1; length xvec2 = length yv (ν*(xvec1 @ xvec2))M ≺' P = (ν*(yvec1 @ ([(x, y)] ∙ yvec2)))([(x, y)] ∙ N) ≺' ([(x, y) ==> thesis" from Eq have "length(xvec1@x#xvec2) = length yvec" by(rule boundOutputChainEqLength) then obtain yvec1 y yvec2 where A: "yvec = yvec1@y#yvec2" and "length xvec1 = length yvec and "length xvec2 = length yvec2" by(metis openInjectAux sym) from A ‹x ♯ yvec› have "x ♯ yvec2" and "x ♯ yvec1" by simp+ with Eq ‹length xvec1 = length yvec1› ‹x ♯ N› ‹x ♯ Q› ‹x ♯ xvec1› A have "(ν*(xvec1@xvec2))M ≺' P = (ν*(yvec1@([(x, y)] ∙ yvec2)))([(x, y)] ∙ N) ≺' ([(x, y by(force dest: boundOutputChainOpenIH simp add: boundOutputApp BOresChainSupp fresh_def with ‹length xvec1 = length yvec1› ‹length xvec2 = length yvec2› A Ass show ?thesis by blast qed lemma boundOutputScopeDest: fixes xvec :: "name list" and M :: "'a::fs_name" and P :: "('a, 'b::fs_name, 'c::fs_name) psi" and yvec :: "name list" and N :: 'a and x :: name and Q :: "('a, 'b, 'c) psi" assumes "(ν*xvec)M ≺' P = (ν*yvec)N ≺' (νz)Q" and "z ♯ xvec" and "z ♯ yvec" obtains R where "P = (νz)R" and "(ν*xvec)M ≺' R = (ν*yvec)N ≺' Q" proof - assume "∧R. [P = (νz)R; (ν*xvec)M ≺' R = (ν*yvec)N ≺' Q] ==> thesis" moreover obtain n where "n = length xvec" by auto with assms have "∃R. P = (νz)R ∧ (ν*xvec)M ≺' R = (ν*yvec)N ≺' Q" proof(induct n arbitrary: xvec yvec M N P Q z) case(0 xvec yvec M N P Q z) have Eq: "(ν*xvec)M ≺' P = (ν*yvec)N ≺' (νz)Q" by fact from ‹0 = length xvec› have "xvec = []" by auto moreover with Eq have "yvec = []" by(case_tac yvec) auto ultimately show ?case using Eq by(simp add: boundOutput.inject) next case(Suc n xvec yvec M N P Q z) from ‹Suc n = length xvec› obtain x xvec' where "xvec = x#xvec'" and "length xvec' = n" by(case_tac xvec) auto from ‹(ν*xvec)M ≺' P = (ν*yvec)N ≺' ((νz)Q)› ‹xvec = x # xvec'› obtain y yvec' where "(ν*(x#xvec'))M ≺' P = (ν*(y#yvec'))N ≺' (νz)Q" and "yvec = y#yvec'" by(case_tac yvec) auto hence EQ: "(νx)((ν*xvec')M ≺' P) = (νy)((ν*yvec')N ≺' (νz)Q)" by simp from ‹z ♯ xvec› ‹z ♯ yvec› ‹xvec = x#xvec'› ‹yvec = y#yvec'› have "z ≠ x" and "z ≠ y" and "z ♯ xvec'" and "z ♯ yvec'" by simp+ have IH: "∧xvec yvec M N P Q z. [(ν*xvec)M ≺' (P::('a, 'b, 'c) psi) = (ν*yvec)N ≺' (νz)Q; z by fact show ?case proof(case_tac "x = y") assume "x = y" with EQ have "(ν*xvec')M ≺' P = (ν*yvec')N ≺' (νz)Q" by(simp add: boundOutput.inject alpha) with ‹z ♯ xvec'› ‹z ♯ yvec'› ‹length xvec' = n› obtain R where "P = (νz)R" and "(ν*xvec')M ≺' R = (ν*yvec')N ≺' Q" by(drule_tac IH) auto with ‹xvec=x#xvec'› ‹yvec=y#yvec'› ‹x=y› show ?case by(force simp add: boundOutput.inject alpha) next assume "x ≠ y" with EQ ‹z ≠ x› ‹z ≠ y› have "(ν*xvec')M ≺' P = (ν*([(x, y)] ∙ yvec'))([(x, y)] ∙ N) ≺' (νz)([(x, y)] ∙ Q)" and xFreshzQ: "x ♯ (ν*yvec')N ≺' (νz)Q" by(simp add: boundOutput.inject alpha eqvts)+ moreover from ‹z ≠ x› ‹z ≠ y› ‹z ♯ yvec'› ‹x ≠ y› have "z ♯ ([(x, y)] ∙ yvec')" by(simp add: fresh_left calc_atm) moreover note ‹z ♯ xvec'› ‹length xvec' = n› ultimately obtain R where "P = (νz)R" and A: "(ν*xvec')M ≺' R = (ν*([(x, y)] ∙ yvec'))([(x, y)] by(drule_tac IH) auto from A have "(νx)((ν*xvec')M ≺' R) = (νx)((ν*([(x, y)] ∙ yvec'))([(x, y)] ∙ N) ≺' ([(x, y)] ∙ Q by(simp add: boundOutput.inject alpha) moreover from xFreshzQ ‹z ≠ x› have "x ♯ (ν*yvec')N ≺' Q" by(simp add: boundOutputFresh abs_fresh) ultimately show ?thesis using ‹P = (νz)R› ‹xvec=x#xvec'› ‹yvec=y#yvec'› xFreshzQ by(force simp add: alphaBoundOutput name_swap eqvts) qed qed ultimately show ?thesis by blast qed nominal_datatype ('a, 'b, 'c) residual = RIn "'a::fs_name" 'a "('a, 'b::fs_name, 'c::fs_name) psi" | ROut 'a "('a, 'b, 'c) boundOutput" | RTau "('a, 'b, 'c) psi" nominal_datatype 'a action = In "'a::fs_name" 'a (‹_(_)› [90, 90] 90) | Out "'a::fs_name" "name list" 'a (‹_(ν*_)⟨_⟩› [90, 90, 90] 90) | Tau (‹τ› 90) nominal_primrec bn :: "('a::fs_name) action → name list" where "bn (M(N)) = []" | "bn (M(ν*xvec)⟨N⟩) = xvec" | "bn (τ) = []" by(rule TrueI)+ lemma bnEqvt[eqvt]: fixes p :: "name prm" and α :: "('a::fs_name) action" shows "(p ∙ bn α) = bn(p ∙ α)" by(nominal_induct α rule: action.strong_induct) auto nominal_primrec create_residual :: "('a::fs_name) action → ('a, 'b::fs_name, 'c::fs_ where "(M(N)) ≺ P = RIn M N P" | "M(ν*xvec)⟨N⟩ ≺ P = ROut M ((ν*xvec)(N ≺' P))" | "τ ≺ P = (RTau P)" by(rule TrueI)+ nominal_primrec subject :: "('a::fs_name) action → 'a option" where "subject (M(N)) = Some M" | "subject (M(ν*xvec)⟨N⟩) = Some M" | "subject (τ) = None" by(rule TrueI)+ nominal_primrec object :: "('a::fs_name) action → 'a option" where "object (M(N)) = Some N" | "object (M(ν*xvec)⟨N⟩) = Some N" | "object (τ) = None" by(rule TrueI)+ lemma optionFreshChain[simp]: fixes xvec :: "name list" and X :: "name set" shows "xvec ♯* (Some x) = xvec ♯* x" and "X ♯* (Some x) = X ♯* x" and "xvec ♯* None" and "X ♯* None" by(auto simp add: fresh_star_def fresh_some fresh_none) lemmas [simp] = fresh_some fresh_none lemma actionFresh[simp]: fixes x :: name and α :: "('a::fs_name) action" shows "(x ♯ α) = (x ♯ (subject α) ∧ x ♯ (bn α) ∧ x ♯ (object α))" by(nominal_induct α rule: action.strong_induct) auto lemma actionFreshChain[simp]: fixes X :: "name set" and α :: "('a::fs_name) action" and xvec :: "name list" shows "(X ♯* α) = (X ♯* (subject α) ∧ X ♯* (bn α) ∧ X ♯* (object α))" and "(xvec ♯* α) = (xvec ♯* (subject α) ∧ xvec ♯* (bn α) ∧ xvec ♯* (object α))" by(auto simp add: fresh_star_def) lemma subjectEqvt[eqvt]: fixes p :: "name prm" and α :: "('a::fs_name) action" shows "(p ∙ subject α) = subject(p ∙ α)" by(nominal_induct α rule: action.strong_induct) auto lemma okjectEqvt[eqvt]: fixes p :: "name prm" and α :: "('a::fs_name) action" shows "(p ∙ object α) = object(p ∙ α)" by(nominal_induct α rule: action.strong_induct) auto lemma create_residualEqvt[eqvt]: fixes p :: "name prm" and α :: "('a::fs_name) action" and P :: "('a, 'b::fs_name, 'c::fs_name) psi" shows "(p ∙ (α ≺ P)) = (p ∙ α) ≺ (p ∙ P)" by(nominal_induct α rule: action.strong_induct) (auto simp add: eqvts) lemma residualFresh: fixes x :: name and α :: "'a::fs_name action" and P :: "('a, 'b::fs_name, 'c::fs_name) psi" shows "(x ♯ (α ≺ P)) = (x ♯ (subject α) ∧ (x ∈ (set(bn(α))) ∨ (x ♯ object(α) ∧ x ♯ P) by(nominal_induct α rule: action.strong_induct) (auto simp add: fresh_some fresh_none boundOutputFresh) lemma residualFresh2[simp]: fixes x :: name and α :: "('a::fs_name) action" and P :: "('a, 'b::fs_name, 'c::fs_name) psi" assumes "x ♯ α" and "x ♯ P" shows "x ♯ α ≺ P" using assms by(nominal_induct α rule: action.strong_induct) auto lemma residualFreshChain2[simp]: fixes xvec :: "name list" and X :: "name set" and α :: "('a::fs_name) action" and P :: "('a, 'b::fs_name, 'c::fs_name) psi" shows "[xvec ♯* α; xvec ♯* P] ==> xvec ♯* (α ≺ P)" and "[X ♯* α; X ♯* P] ==> X ♯* (α ≺ P)" by(auto simp add: fresh_star_def) lemma residualFreshSimp[simp]: fixes x :: name and M :: "'a::fs_name" and N :: 'a and P :: "('a, 'b::fs_name, 'c::fs_name) psi" shows "x ♯ (M(N) ≺ P) = (x ♯ M ∧ x ♯ N ∧ x ♯ P)" and "x ♯ (M(ν*xvec)⟨N⟩ ≺ P) = (x ♯ M ∧ x ♯ ((ν*xvec)(N ≺' P)))" and "x ♯ (τ ≺ P) = (x ♯ P)" by(auto simp add: residualFresh) lemma residualInject': shows "(α ≺ P = RIn M N Q) = (P = Q ∧ α = M(N))" and "(α ≺ P = ROut M B) = (∃xvec N. α = M(ν*xvec)⟨N⟩ ∧ B = (ν*xvec)(N ≺' P))" and "(α ≺ P = RTau Q) = (α = τ ∧ P = Q)" and "(RIn M N Q = α ≺ P) = (P = Q ∧ α = M(N))" and "(ROut M B = α ≺ P) = (∃xvec N. α = M(ν*xvec)⟨N⟩ ∧ B = (ν*xvec)(N ≺' P))" and "(RTau Q = α ≺ P) = (α = τ ∧ P = Q)" proof - show "(α ≺ P = RIn M N Q) = (P = Q ∧ α = M(N))" by(nominal_induct α rule: action.strong_induct) (auto simp add: residual.inject action.inject) next show "(α ≺ P = ROut M B) = (∃xvec N. α = M(ν*xvec)⟨N⟩ ∧ B = (ν*xvec)(N ≺' P))" by(nominal_induct α rule: action.strong_induct) (auto simp add: residual.inject action.inject) next show "(α ≺ P = RTau Q) = (α = τ ∧ P = Q)" by(nominal_induct α rule: action.strong_induct) (auto simp add: residual.inject action.inject) next show "(RIn M N Q = α ≺ P) = (P = Q ∧ α = M(N))" by(nominal_induct α rule: action.strong_induct) (auto simp add: residual.inject action.inject) next show "(ROut M B = α ≺ P) = (∃xvec N. α = M(ν*xvec)⟨N⟩ ∧ B = (ν*xvec)(N ≺' P))" by(nominal_induct α rule: action.strong_induct) (auto simp add: residual.inject action.inject) next show "(RTau Q = α ≺ P) = (α = τ ∧ P = Q)" by(nominal_induct α rule: action.strong_induct) (auto simp add: residual.inject action.inject) qed lemma residualFreshChainSimp[simp]: fixes xvec :: "name list" and X :: "name set" and M :: "'a::fs_name" and N :: 'a and yvec :: "name list" and P :: "('a, 'b::fs_name, 'c::fs_name) psi" shows "xvec ♯* (M(N) ≺ P) = (xvec ♯* M ∧ xvec ♯* N ∧ xvec ♯* P)" and "xvec ♯* (M(ν*yvec)⟨N⟩ ≺ P) = (xvec ♯* M ∧ xvec ♯* ((ν*yvec)(N ≺' P)))" and "xvec ♯* (τ ≺ P) = (xvec ♯* P)" and "X ♯* (M(N) ≺ P) = (X ♯* M ∧ X ♯* N ∧ X ♯* P)" and "X ♯* (M(ν*yvec)⟨N⟩ ≺ P) = (X ♯* M ∧ X ♯* ((ν*yvec)(N ≺' P)))" and "X ♯* (τ ≺ P) = (X ♯* P)" by(auto simp add: fresh_star_def) lemma residualFreshChainSimp2[simp]: fixes xvec :: "name list" and X :: "name set" and M :: "'a::fs_name" and N :: 'a and yvec :: "name list" and P :: "('a, 'b::fs_name, 'c::fs_name) psi" shows "xvec ♯* (RIn M N P) = (xvec ♯* M ∧ xvec ♯* N ∧ xvec ♯* P)" and "xvec ♯* (ROut M B) = (xvec ♯* M ∧ xvec ♯* B)" and "xvec ♯* (RTau P) = (xvec ♯* P)" and "X ♯* (RIn M N P) = (X ♯* M ∧ X ♯* N ∧ X ♯* P)" and "X ♯* (ROut M B) = (X ♯* M ∧ X ♯* B)" and "X ♯* (RTau P) = (X ♯* P)" by(auto simp add: fresh_star_def) lemma freshResidual3[dest]: fixes x :: name and α :: "('a::fs_name) action" and P :: "('a, 'b::fs_name, 'c::fs_name) psi" assumes "x ♯ bn α" and "x ♯ α ≺ P" shows "x ♯ α" and "x ♯ P" using assms by(nominal_induct rule: action.strong_induct) auto lemma freshResidualChain3[dest]: fixes xvec :: "name list" and α :: "('a::fs_name) action" and P :: "('a, 'b::fs_name, 'c::fs_name) psi" assumes "xvec ♯* (α ≺ P)" and "xvec ♯* bn α" shows "xvec ♯* α" and "xvec ♯* P" using assms by(nominal_induct rule: action.strong_induct) auto lemma freshResidual4[dest]: fixes x :: name and α :: "('a::fs_name) action" and P :: "('a, 'b::fs_name, 'c::fs_name) psi" assumes "x ♯ α ≺ P" shows "x ♯ subject α" using assms by(nominal_induct rule: action.strong_induct) auto lemma freshResidualChain4[dest]: fixes xvec :: "name list" and α :: "('a::fs_name) action" and P :: "('a, 'b::fs_name, 'c::fs_name) psi" assumes "xvec ♯* (α ≺ P)" shows "xvec ♯* subject α" using assms by(nominal_induct rule: action.strong_induct) auto lemma alphaOutputResidual: fixes M :: "'a::fs_name" and xvec :: "name list" and N :: 'a and P :: "('a, 'b::fs_name, 'c::fs_name) psi" and p :: "name prm" assumes "(p ∙ xvec) ♯* N" and "(p ∙ xvec) ♯* P" and "set p ⊆ set xvec × set(p ∙ xvec)" and "set xvec ⊆ set yvec" shows "M(ν*yvec)⟨N⟩ ≺ P = M(ν*(p ∙ yvec))⟨(p ∙ N)⟩ ≺ (p ∙ P)" using assms by(simp add: boundOutputChainAlpha'') lemmas[simp del] = create_residual.simps lemma residualInject'': assumes "bn α = bn β" shows "(α ≺ P = β ≺ Q) = (α = β ∧ P = Q)" using assms apply(nominal_induct α rule: action.strong_induct) apply(auto simp add: residual.inject create_residual.simps residualInject' action.inje by(rule_tac x="bn β" in exI) auto lemmas residualInject = residual.inject create_residual.simps residualInject' residual lemma bnFreshResidual[simp]: fixes α :: "('a::fs_name) action" shows "(bn α) ♯* (α ≺ P) = bn α ♯* (subject α)" by(nominal_induct α rule: action.strong_induct) (auto simp add: residualFresh fresh_some fresh_star_def) lemma actionCases[case_names cInput cOutput cTau]: fixes α :: "('a::fs_name) action" assumes "∧M N. α = M(N) ==> Prop" and "∧M xvec N. α = M(ν*xvec)⟨N⟩ ==> Prop" and "α = τ ==> Prop" shows Prop using assms by(nominal_induct α rule: action.strong_induct) auto lemma actionPar1Dest: fixes α :: "('a::fs_name) action" and P :: "('a, 'b::fs_name, 'c::fs_name) psi" and β :: "('a::fs_name) action" and Q :: "('a, 'b, 'c) psi" and R :: "('a, 'b, 'c) psi" assumes "α ≺ P = β ≺ (Q ∥ R)" and "bn α ♯* bn β" obtains T p where "set p ⊆ set(bn α) × set(bn β)" and "P = T ∥ (p ∙ R)" and "α ≺ T = β ≺ Q" using assms apply(cases rule: actionCases[where α=α]) apply(auto simp add: residualInject) by(drule_tac boundOutputPar1Dest') auto lemma actionPar2Dest: fixes α :: "('a::fs_name) action" and P :: "('a, 'b::fs_name, 'c::fs_name) psi" and β :: "('a::fs_name) action" and Q :: "('a, 'b, 'c) psi" and R :: "('a, 'b, 'c) psi" assumes "α ≺ P = β ≺ (Q ∥ R)" and "bn α ♯* bn β" obtains T p where "set p ⊆ set(bn α) × set(bn β)" and "P = (p ∙ Q) ∥ T" and "α ≺ T = β ≺ R" using assms apply(cases rule: actionCases[where α=α]) apply(auto simp add: residualInject) by(drule_tac boundOutputPar2Dest') auto lemma actionScopeDest: fixes α :: "('a::fs_name) action" and P :: "('a, 'b::fs_name, 'c::fs_name) psi" fixes β :: "('a::fs_name) action" and x :: name and Q :: "('a, 'b, 'c) psi" assumes "α ≺ P = β ≺ (νx)Q" and "x ♯ bn α" and "x ♯ bn β" obtains R where "P = (νx)R" and "α ≺ R = β ≺ Q" using assms apply(cases rule: actionCases[where α=α]) apply(auto simp add: residualInject) by(drule_tac boundOutputScopeDest) auto abbreviation outputJudge (‹_⟨_⟩› [110, 110] 110) where "M⟨N⟩ ≡ M(ν*([]))⟨N⟩" declare [[unify_trace_bound=100]] locale env = substPsi substTerm substAssert substCond + assertion SCompose' SImp' SBottom' SChanEq' for substTerm :: "('a::fs_name) → name list → 'a::fs_name list → 'a" and substAssert :: "('b::fs_name) → name list → 'a::fs_name list → 'b" and substCond :: "('c::fs_name) → name list → 'a::fs_name list → 'c" and SCompose' :: "'b → 'b → 'b" and SImp' :: "'b → 'c → bool" and SBottom' :: 'b and SChanEq' :: "'a → 'a → 'c" begin notation SCompose' (infixr ‹⊗› 90) notation SImp' (‹_ ⊨ _› [85, 85] 85) notation FrameImp (‹_ ⊨F _› [85, 85] 85) abbreviation FBottomJudge (‹⊥F› 90) where "⊥F ≡ (FAssert SBottom')" notation SChanEq' (‹_ ↔ _› [90, 90] 90) notation substTerm (‹_[_::=_]› [100, 100, 100] 100) notation subs (‹_[_::=_]› [100, 100, 100] 100) notation AssertionStatEq (‹_ ≃ _› [80, 80] 80) notation FrameStatEq (‹_ ≃F _› [80, 80] 80) notation SBottom' (‹1› 190) abbreviation insertAssertion' (‹insertAssertion›) where "insertAssertion' ≡ assertio inductive semantics :: "'b → ('a, 'b, 'c) psi → ('a, 'b, 'c) residual → bool" (‹_ ⊳ _ ⟼ _› [50, 50, 50] 50) where cInput: "[Ψ ⊨ M ↔ K; distinct xvec; set xvec ⊆ supp N; xvec ♯* Tvec; length xvec = length Tvec; xvec ♯* Ψ; xvec ♯* M; xvec ♯* K] ==> Ψ ⊳ M(λ*xvec N).P ⟼ K((N[xvec::=Tvec])) ≺ P[xvec:: | Output: "[Ψ ⊨ M ↔ K] ==> Ψ ⊳ M⟨N⟩.P ⟼ K⟨N⟩ ≺ P" | Case: "[Ψ ⊳ P ⟼ Rs; (φ, P) mem Cs; Ψ ⊨ φ; guarded P] ==> Ψ ⊳ Cases Cs ⟼ Rs" | cPar1: "[(Ψ ⊗ ΨQ) ⊳ P ⟼α ≺ P'; extractFrame Q = ⟨AQ, ΨQ⟩; distinct AQ; AQ ♯* P; AQ ♯* Q; AQ ♯* Ψ; AQ ♯* α; AQ ♯* P'; distinct(bn α); bn α ♯* Ψ; bn α ♯* ΨQ; bn α ♯* Q; bn α ♯* P; bn α ♯* (subject α)] ==> Ψ ⊳ P ∥ Q ⟼α ≺ (P' ∥ Q)" | cPar2: "[(Ψ ⊗ ΨP) ⊳ Q ⟼α ≺ Q'; extractFrame P = ⟨AP, ΨP⟩; distinct AP; AP ♯* P; AP ♯* Q; AP ♯* Ψ; AP ♯* α; AP ♯* Q'; distinct(bn α); bn α ♯* Ψ; bn α ♯* ΨP; bn α ♯* P; bn α ♯* Q; bn α ♯* (subject α)] ==> Ψ ⊳ P ∥ Q ⟼α ≺ (P ∥ Q')" | cComm1: "[Ψ ⊗ ΨQ ⊳ P ⟼ M(N) ≺ P'; extractFrame P = ⟨AP, ΨP⟩; distinct AP; Ψ ⊗ ΨP ⊳ Q ⟼ K(ν*xvec)⟨N⟩ ≺ Q'; extractFrame Q = ⟨AQ, ΨQ⟩; distinct AQ; Ψ ⊗ ΨP ⊗ ΨQ ⊨ M ↔ K; AP ♯* Ψ; AP ♯* ΨQ; AP ♯* P; AP ♯* M; AP ♯* N; AP ♯* P'; AP ♯* Q; AP ♯* Q'; AP ♯* AQ; AP ♯* xvec; AQ ♯* Ψ; AQ ♯* ΨP; AQ ♯* P; AQ ♯* N; AQ ♯* P'; AQ ♯* Q; AQ ♯* K; AQ ♯* Q'; AQ ♯* xvec; distinct xvec; xvec ♯* Ψ; xvec ♯* ΨP; xvec ♯* ΨQ; xvec ♯* P; xvec ♯* M; xvec ♯* Q; xvec ♯* K] ==> Ψ ⊳ P ∥ Q ⟼ τ ≺ (ν*xvec)(P' ∥ Q')" | cComm2: "[Ψ ⊗ ΨQ ⊳ P ⟼ M(ν*xvec)⟨N⟩ ≺ P'; extractFrame P = ⟨AP, ΨP⟩; distinct AP; Ψ ⊗ ΨP ⊳ Q ⟼ K(N) ≺ Q'; extractFrame Q = ⟨AQ, ΨQ⟩; distinct AQ; Ψ ⊗ ΨP ⊗ ΨQ ⊨ M ↔ K; AP ♯* Ψ; AP ♯* ΨQ; AP ♯* P; AP ♯* M; AP ♯* N; AP ♯* P'; AP ♯* Q; AP ♯* Q'; AP ♯* AQ; AP ♯* xvec; AQ ♯* Ψ; AQ ♯* ΨP; AQ ♯* P; AQ ♯* N; AQ ♯* P'; AQ ♯* Q; AQ ♯* K; AQ ♯* Q'; AQ ♯* xvec; distinct xvec; xvec ♯* Ψ; xvec ♯* ΨP; xvec ♯* ΨQ; xvec ♯* P; xvec ♯* M; xvec ♯* Q; xvec ♯* K] ==> Ψ ⊳ P ∥ Q ⟼ τ ≺ (ν*xvec)(P' ∥ Q')" | cOpen: "[Ψ ⊳ P ⟼ M(ν*(xvec@yvec))⟨N⟩ ≺ P'; x ∈ supp N; x ♯ xvec; x ♯ yvec; x ♯ M; x ♯ Ψ distinct xvec; distinct yvec; xvec ♯* Ψ; xvec ♯* P; xvec ♯* M; xvec ♯* yvec; yvec ♯* Ψ; yvec ♯* P; yvec ♯* M] == Ψ ⊳ (νx)P ⟼ M(ν*(xvec@x#yvec))⟨N⟩ ≺ P'" | cScope: "[Ψ ⊳ P ⟼α ≺ P'; x ♯ Ψ; x ♯ α; bn α ♯* Ψ; bn α ♯* P; bn α ♯* (subject α); distinc | Bang: "[Ψ ⊳ P ∥ !P ⟼ Rs; guarded P] ==> Ψ ⊳ !P ⟼ Rs" abbreviation semanticsBottomJudge (‹_ ⟼ _› [50, 50] 50) where "P ⟼ Rs ≡ 1 ⊳ P ⟼ Rs" equivariance env.semantics nominal_inductive2 env.semantics avoids cInput: "set xvec" | cPar1: "set AQ ∪ set(bn α)" | cPar2: "set AP ∪ set(bn α)" | cComm1: "set AP ∪ set AQ ∪ set xvec" | cComm2: "set AP ∪ set AQ ∪ set xvec" | cOpen: "{x} ∪ set xvec ∪ set yvec" | cScope: "{x} ∪ set(bn α)" apply(auto intro: substTerm.subst4Chain subst4Chain simp add: abs_fresh residualFresh) apply(force simp add: fresh_star_def abs_fresh) apply(simp add: boundOutputFresh) apply(simp add: boundOutputFreshSet) apply(simp add: boundOutputFreshSet) by(simp add: fresh_star_def abs_fresh) lemma nilTrans[dest]: fixes Ψ :: 'b and Rs :: "('a, 'b, 'c) residual" and M :: 'a and xvec :: "name list" and N :: 'a and P :: "('a, 'b, 'c) psi" and K :: 'a and yvec :: "name list" and N' :: 'a and P' :: "('a, 'b, 'c) psi" and CsP :: "('c × ('a, 'b, 'c) psi) list" and Ψ' :: 'b shows "Ψ ⊳ 0 ⟼ Rs ==> False" and "Ψ ⊳ M(λ*xvec N).P ⟼K(ν*yvec)⟨N'⟩ ≺ P' ==> False" and "Ψ ⊳ M(λ*xvec N).P ⟼τ ≺ P' ==> False" and "Ψ ⊳ M⟨N⟩.P ⟼K(N') ≺ P' ==> False" and "Ψ ⊳ M⟨N⟩.P ⟼τ ≺ P' ==> False" and "Ψ ⊳ {Ψ'} ⟼ Rs ==> False" apply(cases rule: semantics.cases) apply auto apply(cases rule: semantics.cases) apply(auto simp add: residualInject) apply(cases rule: semantics.cases) apply(auto simp add: residualInject) apply(cases rule: semantics.cases) apply(auto simp add: residualInject) apply(cases rule: semantics.cases) apply(auto simp add: residualInject) by(cases rule: semantics.cases) (auto simp add: residualInject) lemma residualEq: fixes α :: "'a action" and P :: "('a, 'b, 'c) psi" and β :: "'a action" and Q :: "('a, 'b, 'c) psi" assumes "α ≺ P = β ≺ Q" and "bn α ♯* (bn β)" and "distinct(bn α)" and "distinct(bn β)" and "bn α ♯* (α ≺ P)" and "bn β ♯* (β ≺ Q)" obtains p where "set p ⊆ set(bn α) × set(bn(p ∙ α))" and "distinctPerm p" and "β = p ∙ α" an using assms proof(nominal_induct α rule: action.strong_induct) case(In M N) thus ?case by(simp add: residualInject) next case(Out M xvec N) thus ?case by(auto simp add: residualInject) (drule_tac boundOutputChainEq'', auto) next case Tau thus ?case by(simp add: residualInject) qed lemma semanticsInduct[consumes 3, case_names cAlpha cInput cOutput cCase cPar1 cPar2 cComm fixes Ψ :: 'b and P :: "('a, 'b, 'c) psi" and α :: "'a action" and P' :: "('a, 'b, 'c) psi" and Prop :: "'d::fs_name → 'b → ('a, 'b, 'c) psi → 'a action → ('a, 'b, 'c) psi → bool" and C :: "'d::fs_name" assumes "Ψ ⊳ P ⟼α ≺ P'" and "bn α ♯* (subject α)" and "distinct(bn α)" and rAlpha: "∧Ψ P α P' p C. [bn α ♯* Ψ; bn α ♯* P; bn α ♯* (subject α); bn α ♯* C; bn α ♯* (bn(p ∙ α)); set p ⊆ set(bn α) × set(bn(p ∙ α)); distinctPerm p; (bn(p ∙ α)) ♯* α; (bn(p ∙ α)) ♯* P'; Prop C Ψ P α P'] ==> Prop C Ψ P (p ∙ α) (p ∙ P')" and rInput: "∧Ψ M K xvec N Tvec P C. [Ψ ⊨ M ↔ K; distinct xvec; set xvec ⊆ supp N; length xvec = length Tvec; xvec ♯* Ψ; xvec ♯* M; xvec ♯* K; xvec ♯* C] ==> Prop C Ψ (M(λ*xvec N).P) (K((N[xvec::=Tvec]))) (P[xvec::=Tvec])" and rOutput: "∧Ψ M K N P C. [Ψ ⊨ M ↔ K] ==> Prop C Ψ (M⟨N⟩.P) (K⟨N⟩) P" and rCase: "∧Ψ P α P' φ Cs C. [Ψ ⊳ P ⟼α ≺ P'; ∧C. Prop C Ψ P α P'; (φ, P) mem Cs; Ψ ⊨ φ; gua Prop C Ψ (Cases Cs) α P'" and rPar1: "∧Ψ ΨQ P α P' AQ Q C. [Ψ ⊗ ΨQ ⊳ P ⟼α ≺ P'; extractFrame Q = ⟨AQ, ΨQ⟩; distinct AQ; ∧C. Prop C (Ψ ⊗ ΨQ) P α P'; AQ ♯* P; AQ ♯* Q; AQ ♯* Ψ; AQ ♯* α; AQ ♯* P'; AQ ♯* C; distinct(bn α); bn α ♯* Q; bn α ♯* Ψ; bn α ♯* ΨQ; bn α ♯* P; bn α ♯* subject α; bn α ♯* C] ==> Prop C Ψ (P ∥ Q) α (P' ∥ Q)" and rPar2: "∧Ψ ΨP Q α Q' AP P C. [Ψ ⊗ ΨP ⊳ Q ⟼α ≺ Q'; extractFrame P = ⟨AP, ΨP⟩; distinct AP; ∧C. Prop C (Ψ ⊗ ΨP) Q α Q'; AP ♯* P; AP ♯* Q; AP ♯* Ψ; AP ♯* α; AP ♯* Q'; AP ♯* C; distinct(bn α); bn α ♯* Q; bn α ♯* Ψ; bn α ♯* ΨP; bn α ♯* P; bn α ♯* subject α; bn α ♯* C] ==> Prop C Ψ (P ∥ Q) α (P ∥ Q')" and rComm1: "∧Ψ ΨQ P M N P' AP ΨP Q K xvec Q' AQ C. [Ψ ⊗ ΨQ ⊳ P ⟼M(N) ≺ P'; ∧C. Prop C (Ψ ⊗ ΨQ) P (M(N)) P'; extractFrame P = ⟨AP, ΨP⟩; distinct AP; Ψ ⊗ ΨP ⊳ Q ⟼K(ν*xvec)⟨N⟩ ≺ Q'; ∧C. Prop C (Ψ ⊗ ΨP) Q (K(ν*xvec)⟨N⟩) Q'; extractFrame Q = ⟨AQ, ΨQ⟩; distinct AQ; Ψ ⊗ ΨP ⊗ ΨQ ⊨ M ↔ K; AP ♯* Ψ; AP ♯* ΨQ; AP ♯* P; AP ♯* M; AP ♯* N; AP ♯* P'; AP ♯* Q; AP ♯* Q'; AP ♯* AQ; AP ♯* xvec; AQ ♯* Ψ; AQ ♯* ΨP; AQ ♯* P; AQ ♯* N; AQ ♯* P'; AQ ♯* Q; AQ ♯* K; AQ ♯* Q'; distinct xvec; AQ ♯* xvec; xvec ♯* Ψ; xvec ♯* ΨP; xvec ♯* ΨQ; xvec ♯* P; xvec ♯* M; xvec ♯* Q; xvec ♯* K; AP ♯* C; AQ ♯* C; xvec ♯* C] ==> Prop C Ψ (P ∥ Q) (τ) ((ν*xvec)(P' ∥ Q'))" and rComm2: "∧Ψ ΨQ P M xvec N P' AP ΨP Q K Q' AQ C. [Ψ ⊗ ΨQ ⊳ P ⟼M(ν*xvec)⟨N⟩ ≺ P'; ∧C. Prop C (Ψ ⊗ ΨQ) P (M(ν*xvec)⟨N⟩) P'; extractFrame P = ⟨AP, ΨP⟩; distinct AP; Ψ ⊗ ΨP ⊳ Q ⟼K(N) ≺ Q'; ∧C. Prop C (Ψ ⊗ ΨP) Q (K(N)) Q'; extractFrame Q = ⟨AQ, ΨQ⟩; distinct AQ; Ψ ⊗ ΨP ⊗ ΨQ ⊨ M ↔ K; AP ♯* Ψ; AP ♯* ΨQ; AP ♯* P; AP ♯* M; AP ♯* N; AP ♯* P'; AP ♯* Q; AP ♯* Q'; AP ♯* AQ; AP ♯* xvec; AQ ♯* Ψ; AQ ♯* ΨP; AQ ♯* P; AQ ♯* N; AQ ♯* P'; AQ ♯* Q; AQ ♯* K; AQ ♯* Q'; distinct xvec; AQ ♯* xvec; xvec ♯* Ψ; xvec ♯* ΨP; xvec ♯* ΨQ; xvec ♯* P; xvec ♯* M; xvec ♯* Q; xvec ♯* K; AP ♯* C; AQ ♯* C; xvec ♯* C] ==> Prop C Ψ (P ∥ Q) (τ) ((ν*xvec)(P' ∥ Q'))" and rOpen: "∧Ψ P M xvec yvec N P' x C. [Ψ ⊳ P ⟼M(ν*(xvec@yvec))⟨N⟩ ≺ P'; x ∈ supp N; ∧C. Prop C Ψ P (M(ν*(xvec@yvec))⟨N⟩) P'; x ♯ Ψ; x ♯ M; x ♯ xvec; x ♯ yvec; xvec ♯* Ψ; xvec ♯* P; xvec ♯* M; distinct xvec; yvec ♯* Ψ; yvec ♯* P; yvec ♯* M; yvec ♯* C; x ♯ C; xvec ♯* C] ==> Prop C Ψ ((νx)P) (M(ν*(xvec@x#yvec))⟨N⟩) P'" and rScope: "∧Ψ P α P' x C. [Ψ ⊳ P ⟼α ≺ P'; ∧C. Prop C Ψ P α P'; x ♯ Ψ; x ♯ α; bn α ♯* Ψ; bn α ♯* P; bn α ♯* (subject α); x ♯ C; bn α ♯* C; distinct(bn α)] ==> Prop C Ψ ((νx)P) α ((νx)P')" and rBang: "∧Ψ P α P' C. [Ψ ⊳ P ∥ !P ⟼α ≺ P'; guarded P; ∧C. Prop C Ψ (P ∥ !P) α P'] ==> Prop C Ψ (!P) α P'" shows "Prop C Ψ P α P'" using ‹Ψ ⊳ P ⟼α ≺ P'› ‹bn α ♯* (subject α)› ‹distinct(bn α)› proof(nominal_induct x3=="α ≺ P'" avoiding: α C arbitrary: P' rule: semantics.strong_induc case(cInput Ψ M K xvec N Tvec P α C P') thus ?case by(force intro: rInput simp add: residualInject) next case(Output Ψ M K N P α C P') thus ?case by(force intro: rOutput simp add: residualInject) next case(Case Ψ P Rs φ Cs α C) thus ?case by(auto intro: rCase) next case(cPar1 Ψ ΨQ P α P' Q AQ α' C P'') note ‹α ≺ (P' ∥ Q) = α' ≺ P''› moreover from ‹bn α ♯* α'› have "bn α ♯* (bn α')" by auto moreover note ‹distinct (bn α)› ‹distinct(bn α')› moreover from ‹bn α ♯* subject α› ‹bn α' ♯* subject α'› have "bn α ♯* (α ≺ P' ∥ Q)" and "bn α' ♯* (α' ≺ P'')" by simp+ ultimately obtain p where S: "(set p) ⊆ (set(bn α)) × (set(bn(p ∙ α)))" and "distinctPerm and αEq: "α' = p ∙ α" and P'eq: "P'' = p ∙ (P' ∥ Q)" and "(bn(p ∙ α)) ♯* α" and "(bn(p ∙ α)) ♯* (P' ∥ Q)" by(rule residualEq) note ‹Ψ ⊗ ΨQ ⊳ P ⟼α ≺ P'› ‹extractFrame Q = ⟨AQ, ΨQ⟩› ‹distinct AQ› moreover from ‹bn α ♯* subject α› ‹distinct(bn α)› have "∧C. Prop C (Ψ ⊗ ΨQ) P α P'" by(rule_tac cPar1) auto moreover note ‹AQ ♯* P› ‹AQ ♯* Q› ‹AQ ♯* Ψ› ‹AQ ♯* α› ‹AQ ♯* P'› ‹AQ ♯* C› ‹bn α ♯* Q› ‹distinct(bn α)› ‹bn α ♯* Ψ› ‹bn α ♯* ΨQ› ‹bn α ♯* P› ‹bn α ♯* subject α› ‹bn ultimately have "Prop C Ψ (P ∥ Q) α (P' ∥ Q)" by(rule_tac rPar1) with ‹bn α ♯* Ψ› ‹bn α ♯* P› ‹bn α ♯* Q› ‹bn α ♯* subject α› ‹bn α ♯* C› ‹bn α ♯* bn α'› S have "Prop C Ψ (P ∥ Q) (p ∙ α) (p ∙ (P' ∥ Q))" by(rule_tac rAlpha) auto with αEq P'eq ‹distinctPerm p› show ?case by simp next case(cPar2 Ψ ΨP Q α Q' P AP α' C Q'') note ‹α ≺ (P ∥ Q') = α' ≺ Q''› moreover from ‹bn α ♯* α'› have "bn α ♯* (bn α')" by auto moreover note ‹distinct (bn α)› ‹distinct(bn α')› moreover from ‹bn α ♯* subject α› ‹bn α' ♯* subject α'› have "bn α ♯* (α ≺ P ∥ Q')" and "bn α' ♯* (α' ≺ Q'')" by simp+ ultimately obtain p where S: "(set p) ⊆ (set(bn α)) × (set(bn(p ∙ α)))" and "distinctPerm and αEq: "α' = p ∙ α" and Q'eq: "Q'' = p ∙ (P ∥ Q')" and "(bn(p ∙ α)) ♯* α" and "(bn(p ∙ α)) ♯* (P ∥ Q')" by(rule residualEq) note ‹Ψ ⊗ ΨP ⊳ Q ⟼α ≺ Q'› ‹extractFrame P = ⟨AP, ΨP⟩› ‹distinct AP› moreover from ‹bn α ♯* subject α› ‹distinct(bn α)› have "∧C. Prop C (Ψ ⊗ ΨP) Q α Q'" by(rule_tac cPar2) auto moreover note ‹AP ♯* P› ‹AP ♯* Q› ‹AP ♯* Ψ› ‹AP ♯* α› ‹AP ♯* Q'› ‹AP ♯* C› ‹bn α ♯* Q› ‹distinct(bn α)› ‹bn α ♯* Ψ› ‹bn α ♯* ΨP› ‹bn α ♯* P› ‹bn α ♯* subject α› ‹bn ultimately have "Prop C Ψ (P ∥ Q) α (P ∥ Q')" by(rule_tac rPar2) with ‹bn α ♯* Ψ› ‹bn α ♯* P› ‹bn α ♯* Q› ‹bn α ♯* subject α› ‹bn α ♯* C› ‹bn α ♯* (bn α')› have "Prop C Ψ (P ∥ Q) (p ∙ α) (p ∙ (P ∥ Q'))" by(rule_tac rAlpha) auto with αEq Q'eq ‹distinctPerm p› show ?case by simp next case(cComm1 Ψ ΨQ P M N P' AP ΨP Q K xvec Q' AQ α C P'') hence "Prop C Ψ (P ∥ Q) (τ) ((ν*xvec)(P' ∥ Q'))" by(rule_tac rComm1) (assumption | simp)+ thus ?case using ‹τ ≺ (ν*xvec)(P' ∥ Q') = α ≺ P''› by(simp add: residualInject) next case(cComm2 Ψ ΨQ P M xvec N P' AP ΨP Q K Q' AQ α C P'') hence "Prop C Ψ (P ∥ Q) (τ) ((ν*xvec)(P' ∥ Q'))" by(rule_tac rComm2) (assumption | simp)+ thus ?case using ‹τ ≺ (ν*xvec)(P' ∥ Q') = α ≺ P''› by(simp add: residualInject) next case(cOpen Ψ P M xvec yvec N P' x α C P'') note ‹M(ν*(xvec@x#yvec))⟨N⟩ ≺ P' = α ≺ P''› moreover from ‹xvec ♯* α› ‹x ♯ α› ‹yvec ♯* α› have "(xvec@x#yvec) ♯* (bn α)" by auto moreover from ‹xvec ♯* yvec› ‹x ♯ xvec› ‹x ♯ yvec› ‹distinct xvec› ‹distinct yvec› have "distinct(xvec@x#yvec)" by(auto simp add: fresh_star_def) (simp add: fresh_def name_list_supp) moreover note ‹distinct(bn α)› moreover from ‹xvec ♯* M› ‹x ♯ M› ‹yvec ♯* M› have "(xvec@x#yvec) ♯* M" by auto hence "(xvec@x#yvec) ♯* (M(ν*(xvec@x#yvec))⟨N⟩ ≺ P')" by auto moreover from ‹bn α ♯* subject α› have "bn α ♯* (α ≺ P'')" by simp ultimately obtain p where S: "(set p) ⊆ (set(xvec@x#yvec)) × (set(p ∙ (xvec@x#yvec))) and αeq: "α = (p ∙ M)(ν*(p ∙ (xvec@x#yvec)))⟨(p ∙ N)⟩" and P'eq: "P'' = (p ∙ P')" and A: "(xvec@x#yvec) ♯* ((p ∙ M)(ν*(p ∙ (xvec@x#yvec)))⟨(p ∙ N)⟩)" and B: "(p ∙ (xvec@x#yvec)) ♯* (M(ν*(xvec@x#yvec))⟨N⟩)" and C: "(p ∙ (xvec@x#yvec)) ♯* P'" by(rule_tac residualEq) (assumption | simp)+ note ‹Ψ ⊳ P ⟼M(ν*(xvec@yvec))⟨N⟩ ≺ P'› ‹x ∈ (supp N)› moreover { fix C from ‹xvec ♯* M› ‹yvec ♯* M› have "(xvec@yvec) ♯* M" by simp moreover from ‹distinct xvec› ‹distinct yvec› ‹xvec ♯* yvec› have "distinct(xvec@yv by auto (simp add: fresh_star_def name_list_supp fresh_def) ultimately have "Prop C Ψ P (M(ν*(xvec@yvec))⟨N⟩) P'" by(rule_tac cOpen) auto } moreover note ‹x ♯ Ψ› ‹x ♯ M› ‹x ♯ xvec› ‹x ♯ yvec› ‹xvec ♯* Ψ› ‹xvec ♯* P› ‹xvec ♯* ‹yvec ♯* Ψ› ‹yvec ♯* P› ‹yvec ♯* M› ‹yvec ♯* C› ‹x ♯ C› ‹xvec ♯* C› ‹distinct xve ultimately have "Prop C Ψ ((νx)P) (M(ν*(xvec@x#yvec))⟨N⟩) P'" by(rule_tac rOpen) with ‹xvec ♯* Ψ› ‹yvec ♯* Ψ› ‹xvec ♯* P› ‹yvec ♯* P› ‹xvec ♯* M› ‹yvec ♯* M› ‹yvec ♯* C› S ‹distinctPerm p› ‹x ♯ C› ‹xvec ♯* C› ‹x ♯ Ψ› ‹x ♯ M› ‹x ♯ xvec› ‹x ♯ yvec› A B C have "Prop C Ψ ((νx)P) (p ∙ (M(ν*(xvec@x#yvec))⟨N⟩)) (p ∙ P')" apply(rule_tac α="M(ν*(xvec@x#yvec))⟨N⟩" in rAlpha) apply(assumption | simp)+ apply(fastforce simp add: fresh_star_def abs_fresh) by(assumption | simp)+ with αeq P'eq show ?case by simp next case(cScope Ψ P α P' x α' C P'') note ‹α ≺ ((νx)P') = α' ≺ P''› moreover from ‹bn α ♯* α'› have "bn α ♯* (bn α')" by auto moreover note ‹distinct (bn α)› ‹distinct(bn α')› moreover from ‹bn α ♯* subject α› ‹bn α' ♯* subject α'› have "bn α ♯* (α ≺ (νx)P')" and "bn α' ♯* (α' ≺ P'')" by simp+ ultimately obtain p where S: "(set p) ⊆ (set(bn α)) × (set(bn(p ∙ α)))" and "distinctPerm and αEq: "α' = p ∙ α" and P'eq: "P'' = p ∙ ((νx)P')" and "(bn(p ∙ α)) ♯* α" and "(bn(p ∙ α)) ♯* ((νx)P')" by(rule residualEq) note ‹Ψ ⊳ P ⟼α ≺ P'› moreover from ‹bn α ♯* subject α› ‹distinct(bn α)› have "∧C. Prop C Ψ P α P'" by(rule_tac cScope) auto moreover note ‹x ♯ Ψ› ‹x ♯ α› ‹bn α ♯* Ψ› ‹bn α ♯* P› ‹bn α ♯* subject α› ‹x ♯ C› ‹bn α ♯* C› ‹distinct(bn α)› ultimately have "Prop C Ψ ((νx)P) α ((νx)P')" by(rule rScope) with ‹bn α ♯* Ψ› ‹bn α ♯* P› ‹x ♯ α› ‹bn α ♯* subject α› ‹bn α ♯* C› ‹bn α ♯* (bn α')› S ‹d have "Prop C Ψ ((νx)P) (p ∙ α) (p ∙ ((νx)P'))" by(rule_tac rAlpha) simp+ with αEq P'eq ‹distinctPerm p› show ?case by simp next case(Bang Ψ P Rs α C) thus ?case by(rule_tac rBang) auto qed lemma outputInduct[consumes 1, case_names cOutput cCase cPar1 cPar2 cOpen cScope cBang]: fixes Ψ :: 'b and P :: "('a, 'b, 'c) psi" and M :: 'a and B :: "('a, 'b, 'c) boundOutput" and Prop :: "'d::fs_name → 'b → ('a, 'b, 'c) psi → 'a → ('a, 'b, 'c) boundOutput → bool" and C :: "'d::fs_name" assumes "Ψ ⊳ P ⟼ROut M B" and rOutput: "∧Ψ M K N P C. [Ψ ⊨ M ↔ K] ==> Prop C Ψ (M⟨N⟩.P) K (N ≺' P)" and rCase: "∧Ψ P M B φ Cs C. [Ψ ⊳ P ⟼(ROut M B); ∧C. Prop C Ψ P M B; (φ, P) mem Cs; Ψ ⊨ φ; guarded P] ==> Prop C Ψ (Cases Cs) M B" and rPar1: "∧Ψ ΨQ P M xvec N P' AQ Q C. [Ψ ⊗ ΨQ ⊳ P ⟼M(ν*xvec)⟨N⟩ ≺ P'; extractFrame Q = ⟨AQ, ΨQ⟩; distinct AQ; ∧C. Prop C (Ψ ⊗ ΨQ) P M ((ν*xvec)N ≺' P'); AQ ♯* P; AQ ♯* Q; AQ ♯* Ψ; AQ ♯* M; AQ ♯* xvec; AQ ♯* N; AQ ♯* P'; AQ ♯* C; xvec ♯* Q; xvec ♯* Ψ; xvec ♯* ΨQ; xvec ♯* P; xvec ♯* M; xvec ♯* C] ==> Prop C Ψ (P ∥ Q) M ((ν*xvec)N ≺' (P' ∥ Q))" and rPar2: "∧Ψ ΨP Q M xvec N Q' AP P C. [Ψ ⊗ ΨP ⊳ Q ⟼M(ν*xvec)⟨N⟩ ≺ Q'; extractFrame P = ⟨AP, ΨP⟩; distinct AP; ∧C. Prop C (Ψ ⊗ ΨP) Q M ((ν*xvec)N ≺' Q'); AP ♯* P; AP ♯* Q; AP ♯* Ψ; AP ♯* M; AP ♯* xvec; AP ♯* N; AP ♯* Q'; AP ♯* C; xvec ♯* P; xvec ♯* Ψ; xvec ♯* ΨP; xvec ♯* Q; xvec ♯* M; xvec ♯* C] ==> Prop C Ψ (P ∥ Q) M ((ν*xvec)N ≺' (P ∥ Q'))" and rOpen: "∧Ψ P M xvec yvec N P' x C. [Ψ ⊳ P ⟼M(ν*(xvec@yvec))⟨N⟩ ≺ P'; x ∈ supp N; ∧C. Prop C Ψ P M ((ν*(xvec@yvec))N ≺' P'); x ♯ Ψ; x ♯ M; x ♯ xvec; x ♯ yvec; xvec ♯* Ψ; xvec ♯* P; xvec ♯* M; xvec ♯* yvec; yvec ♯* Ψ; yvec ♯* P; yvec ♯* M; yvec ♯* C; x ♯ C; xvec ♯* C] ==> Prop C Ψ ((νx)P) M ((ν*(xvec@x#yvec))N ≺' P')" and rScope: "∧Ψ P M xvec N P' x C. [Ψ ⊳ P ⟼M(ν*xvec)⟨N⟩ ≺ P'; ∧C. Prop C Ψ P M ((ν*xvec)N ≺' P'); x ♯ Ψ; x ♯ M; x ♯ xvec; x ♯ N; xvec ♯* Ψ; xvec ♯* P; xvec ♯* M; x ♯ C; xvec ♯* C] ==> Prop C Ψ ((νx)P) M ((ν*xvec)N ≺' (νx)P')" and rBang: "∧Ψ P M B C. [Ψ ⊳ P ∥ !P ⟼(ROut M B); guarded P; ∧C. Prop C Ψ (P ∥ !P) M B] ==> Prop C Ψ (!P) M B" shows "Prop C Ψ P M B" using ‹Ψ ⊳ P ⟼(ROut M B)› proof(nominal_induct Ψ P Rs=="(ROut M B)" avoiding: C arbitrary: B rule: semantics.strong_ case(cInput Ψ M K xvec N Tvec P C) thus ?case by(simp add: residualInject) next case(Output Ψ M K N P C) thus ?case by(force simp add: residualInject intro: rOutput) next case(Case Ψ P Rs φ Cs C) thus ?case by(force intro: rCase) next case(cPar1 Ψ ΨQ P α P' Q AQ C) thus ?case by(force intro: rPar1 simp add: residualInject) next case(cPar2 Ψ ΨP Q α Q' P AP C) thus ?case by(force intro: rPar2 simp add: residualInject) next case cComm1 thus ?case by(simp add: residualInject) next case cComm2 thus ?case by(simp add: residualInject) next case(cOpen Ψ P M xvec yvec N P' x C B) thus ?case by(force intro: rOpen simp add: residualInject) next case(cScope Ψ P M α P' x C) thus ?case by(force intro: rScope simp add: residualInject) next case(Bang Ψ P Rs C) thus ?case by(force intro: rBang) qed lemma boundOutputBindObject: fixes Ψ :: 'b and P :: "('a, 'b, 'c) psi" and M :: 'a and yvec :: "name list" and N :: 'a and P' :: "('a, 'b, 'c) psi" and y :: name assumes "Ψ ⊳ P ⟼α ≺ P'" and "bn α ♯* subject α" and "distinct(bn α)" and "y ∈ set(bn α)" shows "y ∈ supp(object α)" using assms proof(nominal_induct avoiding: P' arbitrary: y rule: semanticsInduct) case(cAlpha Ψ P α P' p P'' y) from ‹y ∈ set(bn(p ∙ α))› have "(p ∙ y) ∈ (p ∙ set(bn(p ∙ α)))" by(rule pt_set_bij2[OF pt_name_inst, OF at_name_inst]) hence "(p ∙ y) ∈ set(bn α)" using ‹distinctPerm p› by(simp add: eqvts) hence "(p ∙ y) ∈ supp(object α)" by(rule cAlpha) hence "(p ∙ p ∙ y) ∈ (p ∙ supp(object α))" by(rule pt_set_bij2[OF pt_name_inst, OF at_name_inst]) thus ?case using ‹distinctPerm p› by(simp add: eqvts) next case cInput thus ?case by(simp add: supp_list_nil) next case cOutput thus ?case by(simp add: supp_list_nil) next case cCase thus ?case by simp next case cPar1 thus ?case by simp next case cPar2 thus ?case by simp next case cComm1 thus ?case by(simp add: supp_list_nil) next case cComm2 thus ?case by(simp add: supp_list_nil) next case cOpen thus ?case by(auto simp add: supp_list_cons supp_list_append supp_atm supp_some) next case cScope thus ?case by simp next case cBang thus ?case by simp qed lemma alphaBoundOutputChain': fixes yvec :: "name list" and xvec :: "name list" and B :: "('a, 'b, 'c) boundOutput" assumes "length xvec = length yvec" and "yvec ♯* B" and "yvec ♯* xvec" and "distinct yvec" shows "(ν*xvec)B = (ν*yvec)([xvec yvec] ∙v B)" using assms proof(induct rule: composePermInduct) case cBase show ?case by simp next case(cStep x xvec y yvec) thus ?case apply auto by(subst alphaBoundOutput[of y]) (auto simp add: eqvts) qed lemma alphaBoundOutputChain'': fixes yvec :: "name list" and xvec :: "name list" and N :: 'a and P :: "('a, 'b, 'c) psi" assumes "length xvec = length yvec" and "yvec ♯* N" and "yvec ♯* P" and "yvec ♯* xvec" and "distinct yvec" shows "(ν*xvec)(N ≺' P) = (ν*yvec)(([xvec yvec] ∙v N) ≺' ([xvec yvec] ∙v P))" proof - from assms have "(ν*xvec)(N ≺' P) = (ν*yvec)([xvec yvec] ∙v (N ≺' P))" by(simp add: alphaBoundOutputChain') thus ?thesis by simp qed lemma alphaDistinct: fixes xvec :: "name list" and N :: 'a and P :: "('a, 'b, 'c) psi" and yvec :: "name list" and M :: 'a and Q :: "('a, 'b, 'c) psi" assumes "α ≺ P = β ≺ Q" and "distinct(bn α)" and "∧x. x ∈ set(bn α) ==> x ∈ supp(object α)" and "bn α ♯* bn β" and "bn α ♯* (object β)" and "bn α ♯* Q" shows "distinct(bn β)" using assms proof(rule_tac actionCases[where α=α], auto simp add: residualInject supp_some) fix xvec M yvec N assume Eq: "(ν*xvec)N ≺' P = (ν*yvec)M ≺' Q" assume "distinct xvec" and "xvec ♯* M" and "xvec ♯* yvec" and "xvec ♯* Q" assume Mem: "∧x. x ∈ set xvec ==> x ∈ (supp N)" show "distinct yvec" proof - from Eq have "length xvec = length yvec" by(rule boundOutputChainEqLength) with Eq ‹distinct xvec› ‹xvec ♯* yvec› ‹xvec ♯* M› ‹xvec ♯* Q› Mem show ?thesis proof(induct n=="length xvec" arbitrary: xvec yvec M Q rule: nat.induct) case(zero xvec yvec M Q) thus ?case by simp next case(Suc n xvec yvec M Q) have L: "length xvec = length yvec" and "Suc n = length xvec" by fact+ then obtain x xvec' y yvec' where xEq: "xvec = x#xvec'" and yEq: "yvec = y#yvec'" and L': "length xvec' = length yvec'" by(cases xvec, auto, cases yvec, auto) have xvecFreshyvec: "xvec ♯* yvec" and xvecDist: "distinct xvec" by fact+ with xEq yEq have xineqy: "x ≠ y" and xvec'Freshyvec': "xvec' ♯* yvec'" and xvec'Dist: "distinct xvec'" and xFreshxvec': "x ♯ xvec'" and xFreshyvec': "x ♯ yvec'" and yFreshxvec': "y ♯ xvec'" by auto have Eq: "(ν*xvec)N ≺' P = (ν*yvec)M ≺' Q" by fact with xEq yEq xineqy have Eq': "(ν*xvec')N ≺' P = (ν*([(x, y)] ∙ yvec'))([(x, y)] ∙ M) ≺' ([( by(simp add: boundOutput.inject alpha eqvts) moreover have Mem:"∧x. x ∈ set xvec ==> x ∈ supp N" by fact with xEq have "∧x. x ∈ set xvec' ==> x ∈ supp N" by simp moreover have "xvec ♯* M" by fact with xEq xFreshxvec' yFreshxvec' have "xvec' ♯* ([(x, y)] ∙ M)" by simp moreover have xvecFreshQ: "xvec ♯* Q" by fact with xEq xFreshxvec' yFreshxvec' have "xvec' ♯* ([(x, y)] ∙ Q)" by simp moreover have "Suc n = length xvec" by fact with xEq have "n = length xvec'" by simp moreover from xvec'Freshyvec' xFreshxvec' yFreshxvec' have "xvec' ♯* ([(x, y)] ∙ yvec' by simp moreover from L' have "length xvec' = length([(x, y)] ∙ yvec')" by simp ultimately have "distinct([(x, y)] ∙ yvec')" using xvec'Dist by(rule_tac Suc) (assumption | simp)+ hence "distinct yvec'" by simp from Mem xEq have xSuppN: "x ∈ supp N" by simp from L ‹distinct xvec› ‹xvec ♯* yvec› ‹xvec ♯* M› ‹xvec ♯* Q› have "(ν*yvec)M ≺' Q = (ν*xvec)([yvec xvec] ∙v M) ≺' ([yvec xvec] ∙v Q)" by(simp add: alphaBoundOutputChain'') with Eq have "N = [yvec xvec] ∙v M" by simp with xEq yEq have "N = [(y, x)] ∙ [yvec' xvec'] ∙v M" by simp with xSuppN have ySuppM: "y ∈ supp([yvec' xvec'] ∙v M)" by(drule_tac pi="[(x, y)]" in pt_set_bij2[OF pt_name_inst, OF at_name_inst]) (simp add: calc_atm eqvts name_swap) have "y ♯ yvec'" proof(simp add: fresh_def, rule notI) assume "y ∈ supp yvec'" hence "y mem yvec'" by(induct yvec') (auto simp add: supp_list_nil supp_list_cons supp_atm) moreover from ‹xvec ♯* M› xEq xFreshxvec' have "xvec' ♯* M" by simp ultimately have "y ♯ [yvec' xvec'] ∙v M" using L' xvec'Freshyvec' xvec'Dist by(force intro: freshChainPerm) with ySuppM show "False" by(simp add: fresh_def) qed with ‹distinct yvec'› yEq show ?case by simp qed qed qed lemma boundOutputDistinct: fixes Ψ :: 'b and P :: "('a, 'b, 'c) psi" and α :: "'a action" and P' :: "('a, 'b, 'c) psi" assumes "Ψ ⊳ P ⟼α ≺ P'" shows "distinct(bn α)" using assms proof(nominal_induct Ψ P x3=="α ≺ P'" avoiding: α P' rule: semantics.strong_induct) case cInput thus ?case by(simp add: residualInject) next case Output thus ?case by(simp add: residualInject) next case Case thus ?case by(simp add: residualInject) next case cPar1 thus ?case by(force intro: alphaDistinct boundOutputBindObject) next case cPar2 thus ?case by(force intro: alphaDistinct boundOutputBindObject) next case cComm1 thus ?case by(simp add: residualInject) next case cComm2 thus ?case by(simp add: residualInject) next case(cOpen Ψ P M xvec yvec N P' x α P'') note ‹M(ν*(xvec@x#yvec))⟨N⟩ ≺ P' = α ≺ P''› moreover from ‹xvec ♯* yvec› ‹x ♯ xvec› ‹x ♯ yvec› ‹distinct xvec› ‹distinct yvec› have "distinct(bn(M(ν*(xvec@x#yvec))⟨N⟩))" by auto (simp add: fresh_star_def fresh_def name_list_supp) moreover { fix y from ‹Ψ ⊳ P ⟼M(ν*(xvec@yvec))⟨N⟩ ≺ P'› ‹x ∈ supp N› ‹x ♯ xvec› ‹x ♯ yvec› ‹x ♯ M› ‹x ♯ Ψ have "Ψ ⊳ (νx)P ⟼M(ν*(xvec@x#yvec))⟨N⟩ ≺ P'" by(rule semantics.cOpen) moreover moreover from ‹xvec ♯* M› ‹x ♯ M› ‹yvec ♯* M› have "bn(M(ν*(xvec@x#yvec))⟨N⟩) ♯* (subject(M(ν*(xvec@x#yvec))⟨N⟩))" by simp moreover note ‹distinct(bn(M(ν*(xvec@x#yvec))⟨N⟩))› moreover assume "y ∈ set(bn(M(ν*(xvec@x#yvec))⟨N⟩))" ultimately have "y ∈ supp(object(M(ν*(xvec@x#yvec))⟨N⟩))" by(rule_tac boundOutputBindObject) } moreover from ‹xvec ♯* α› ‹x ♯ α› ‹yvec ♯* α› have "bn(M(ν*(xvec@x#yvec))⟨N⟩) ♯* bn α" and "bn(M(ν*(xvec@x#yvec))⟨N⟩) ♯* object α" by simp+ moreover from ‹xvec ♯* P''› ‹x ♯ P''› ‹yvec ♯* P''› have "bn(M(ν*(xvec@x#yvec))⟨N⟩) ♯* P''" by simp ultimately show ?case by(rule alphaDistinct) next case cScope thus ?case by(rule_tac alphaDistinct, auto) (rule_tac boundOutputBindObject, auto) next case Bang thus ?case by simp qed lemma inputDistinct: fixes Ψ :: 'b and M :: 'a and xvec :: "name list" and N :: 'a and P :: "('a, 'b, 'c) psi" and Rs :: "('a, 'b, 'c) residual" assumes "Ψ ⊳ M(λ*xvec N).P ⟼ Rs" shows "distinct xvec" using assms by(nominal_induct Ψ P=="M(λ*xvec N).P" Rs avoiding: xvec N P rule: semantics.strong_induct) (auto simp add: psi.inject intro: alphaInputDistinct) lemma outputInduct'[consumes 2, case_names cAlpha cOutput cCase cPar1 cPar2 cOpen cScope cB fixes Ψ :: 'b and P :: "('a, 'b, 'c) psi" and M :: 'a and yvec :: "name list" and N :: 'a and P' :: "('a, 'b, 'c) psi" and Prop :: "'d::fs_name → 'b → ('a, 'b, 'c) psi → 'a → name list → 'a → ('a, 'b, 'c) psi → bool" and C :: "'d::fs_name" assumes "Ψ ⊳ P ⟼M(ν*xvec)⟨N⟩ ≺ P'" and "xvec ♯* M" and rAlpha: "∧Ψ P M xvec N P' p C. [xvec ♯* Ψ; xvec ♯* P; xvec ♯* M; xvec ♯* C; xve set p ⊆ set xvec × set(p ∙ xvec); distinctPerm p; (p ∙ xvec) ♯* N; (p ∙ xvec) ♯* P'; Prop C Ψ P M xvec N P'] ==> Prop C Ψ P M (p ∙ xvec) (p ∙ N) (p ∙ P')" and rOutput: "∧Ψ M K N P C. [Ψ ⊨ M ↔ K] ==> Prop C Ψ (M⟨N⟩.P) K ([]) N P" and rCase: "∧Ψ P M xvec N P' φ Cs C. [Ψ ⊳ P ⟼M(ν*xvec)⟨N⟩ ≺ P'; ∧C. Prop C Ψ P M xvec N P'; Prop C Ψ (Cases Cs) M xvec N P'" and rPar1: "∧Ψ ΨQ P M xvec N P' AQ Q C. [Ψ ⊗ ΨQ ⊳ P ⟼M(ν*xvec)⟨N⟩ ≺ P'; extractFrame Q = ⟨AQ, ΨQ⟩; distinct AQ; ∧C. Prop C (Ψ ⊗ ΨQ) P M xvec N P'; AQ ♯* P; AQ ♯* Q; AQ ♯* Ψ; AQ ♯* M; AQ ♯* xvec; AQ ♯* N; AQ ♯* P'; AQ ♯* C; xvec ♯* Q; xvec ♯* Ψ; xvec ♯* ΨQ; xvec ♯* P; xvec ♯* M; xvec ♯* C] ==> Prop C Ψ (P ∥ Q) M xvec N (P' ∥ Q)" and rPar2: "∧Ψ ΨP Q M xvec N Q' AP P C. [Ψ ⊗ ΨP ⊳ Q ⟼M(ν*xvec)⟨N⟩ ≺ Q'; extractFrame P = ⟨AP, ΨP⟩; distinct AP; ∧C. Prop C (Ψ ⊗ ΨP) Q M xvec N Q'; AP ♯* P; AP ♯* Q; AP ♯* Ψ; AP ♯* M; AP ♯* xvec; AP ♯* N; AP ♯* Q'; AP ♯* C; xvec ♯* Q; xvec ♯* Ψ; xvec ♯* ΨP; xvec ♯* P; xvec ♯* M; xvec ♯* C] ==> Prop C Ψ (P ∥ Q) M xvec N (P ∥ Q')" and rOpen: "∧Ψ P M xvec yvec N P' x C. [Ψ ⊳ P ⟼M(ν*(xvec@yvec))⟨N⟩ ≺ P'; x ∈ supp N; ∧C. Prop C Ψ P M (xvec@yvec) N P'; x ♯ Ψ; x ♯ M; x ♯ xvec; x ♯ yvec; xvec ♯* Ψ; xvec ♯* P; xvec ♯* M; yvec ♯* Ψ; yvec ♯* P; yvec ♯* M; yvec ♯* C; x ♯ C; xvec ♯* C] ==> Prop C Ψ ((νx)P) M (xvec@x#yvec) N P'" and rScope: "∧Ψ P M xvec N P' x C. [Ψ ⊳ P ⟼M(ν*xvec)⟨N⟩ ≺ P'; ∧C. Prop C Ψ P M xvec N P'; x ♯ Ψ; x ♯ M; x ♯ xvec; x ♯ N; xvec ♯* Ψ; xvec ♯* P; xvec ♯* M; x ♯ C; xvec ♯* C] ==> Prop C Ψ ((νx)P) M xvec N ((νx)P')" and rBang: "∧Ψ P M xvec N P' C. [Ψ ⊳ P ∥ !P ⟼M(ν*xvec)⟨N⟩ ≺ P'; guarded P; ∧C. Prop C Ψ (P ∥ !P) M xvec N P'] ==> Prop C Ψ (!P) M xvec N P'" shows "Prop C Ψ P M xvec N P'" proof - note ‹Ψ ⊳ P ⟼M(ν*xvec)⟨N⟩ ≺ P'› moreover from ‹xvec ♯* M› have "bn(M(ν*xvec)⟨N⟩) ♯* subject(M(ν*xvec)⟨N⟩)" by simp moreover from ‹Ψ ⊳ P ⟼M(ν*xvec)⟨N⟩ ≺ P'› have "distinct(bn(M(ν*xvec)⟨N⟩))" by(rule boundOutputDistinct) ultimately show ?thesis proof(nominal_induct Ψ P α=="M(ν*xvec)⟨N⟩" P' avoiding: C arbitrary: M xvec N rule: semanticsInduct case(cAlpha Ψ P α P' p C M xvec N) from ‹(p ∙ α) = M(ν*xvec)⟨N⟩› have "(p ∙ p ∙ α) = p ∙ (M(ν*xvec)⟨N⟩)" by(simp add: fresh_bij) with ‹distinctPerm p› have A: "α = (p ∙ M)(ν*(p ∙ xvec))⟨(p ∙ N)⟩" by(simp add: eqvts) with ‹bn α ♯* Ψ› ‹bn α ♯* P› ‹bn α ♯* subject α › ‹bn α ♯* C› ‹bn α ♯* bn(p ∙ α)› ‹distin have "(p ∙ xvec) ♯* Ψ" and "(p ∙ xvec) ♯* P" and "(p ∙ xvec) ♯* (p ∙ M)" and "(p ∙ xvec) by auto moreover from A ‹set p ⊆ set(bn α) × set(bn(p ∙ α))› ‹distinctPerm p› have S: "set p ⊆ set(p ∙ xvec) × set(p ∙ p ∙ xvec)" by simp moreover note ‹distinctPerm p› moreover from A ‹bn(p ∙ α) ♯* α› ‹bn(p ∙ α) ♯* P'› have "(p ∙ p ∙ xvec) ♯* (p ∙ N)" and "(p ∙ p ∙ xvec) ♯* P'" by simp+ moreover from A have "Prop C Ψ P (p ∙ M) (p ∙ xvec) (p ∙ N) P'" by(rule cAlpha) ultimately have "Prop C Ψ P (p ∙ M) (p ∙ p ∙ xvec) (p ∙ p ∙ N) (p ∙ P')" by(rule rAlpha) moreover from A ‹bn α ♯* subject α› have "(p ∙ xvec) ♯* (p ∙ M)" by simp hence "xvec ♯* M" by(simp add: fresh_star_bij) from A ‹bn(p ∙ α) ♯* α› ‹distinctPerm p› have "xvec ♯* (p ∙ M)" by simp hence "(p ∙ xvec) ♯* (p ∙ p ∙ M)" by(simp add: fresh_star_bij) with ‹distinctPerm p› have "(p ∙ xvec) ♯* M" by simp with ‹xvec ♯* M› S ‹distinctPerm p› have "(p ∙ M) = M" by simp ultimately show ?case using S ‹distinctPerm p› by simp next case cInput thus ?case by(simp add: residualInject) next case cOutput thus ?case by(force dest: rOutput simp add: action.inject) next case cCase thus ?case by(force intro: rCase) next case cPar1 thus ?case by(force intro: rPar1) next case cPar2 thus ?case by(force intro: rPar2) next case cComm1 thus ?case by(simp add: action.inject) next case cComm2 thus ?case by(simp add: action.inject) next case cOpen thus ?case by(fastforce intro: rOpen simp add: action.inject) next case cScope thus ?case by(fastforce intro: rScope) next case cBang thus ?case by(fastforce intro: rBang) qed qed lemma inputInduct[consumes 1, case_names cInput cCase cPar1 cPar2 cScope cBang]: fixes Ψ :: 'b and P :: "('a, 'b, 'c) psi" and M :: 'a and N :: 'a and P' :: "('a, 'b, 'c) psi" and Prop :: "'d::fs_name → 'b → ('a, 'b, 'c) psi → 'a → 'a → ('a, 'b, 'c) psi → bool" and C :: "'d::fs_name" assumes Trans: "Ψ ⊳ P ⟼M(N) ≺ P'" and rInput: "∧Ψ M K xvec N Tvec P C. [Ψ ⊨ M ↔ K; distinct xvec; set xvec ⊆ supp N; length xvec = length Tvec; xvec ♯* Ψ; xvec ♯* M; xvec ♯* K; xvec ♯* C] ==> Prop C Ψ (M(λ*xvec N).P) K (N[xvec::=Tvec]) (P[xvec::=Tvec])" and rCase: "∧Ψ P M N P' φ Cs C. [Ψ ⊳ P ⟼M(N) ≺ P'; ∧C. Prop C Ψ P M N P'; (φ, P) mem Cs; Ψ Prop C Ψ (Cases Cs) M N P'" and rPar1: "∧Ψ ΨQ P M N P' AQ Q C. [Ψ ⊗ ΨQ ⊳ P ⟼M(N) ≺ P'; extractFrame Q = ⟨AQ, ΨQ⟩; distinct AQ; ∧C. Prop C (Ψ ⊗ ΨQ) P M N P'; distinct AQ; AQ ♯* P; AQ ♯* Q; AQ ♯* Ψ; AQ ♯* M; AQ ♯* N; AQ ♯* P'; AQ ♯* C] ==> Prop C Ψ (P ∥ Q) M N (P' ∥ Q)" and rPar2: "∧Ψ ΨP Q M N Q' AP P C. [Ψ ⊗ ΨP ⊳ Q ⟼M(N) ≺ Q'; extractFrame P = ⟨AP, ΨP⟩; distinct AP; ∧C. Prop C (Ψ ⊗ ΨP) Q M N Q'; distinct AP; AP ♯* P; AP ♯* Q; AP ♯* Ψ; AP ♯* M; AP ♯* N; AP ♯* Q'; AP ♯* C] ==> Prop C Ψ (P ∥ Q) M N (P ∥ Q')" and rScope: "∧Ψ P M N P' x C. [Ψ ⊳ P ⟼M(N) ≺ P'; ∧C. Prop C Ψ P M N P'; x ♯ Ψ; x ♯ M; x ♯ N; x ♯ C] ==> Prop C Ψ ((νx)P) M N ((νx)P')" and rBang: "∧Ψ P M N P' C. [Ψ ⊳ P ∥ !P ⟼M(N) ≺ P'; guarded P; ∧C. Prop C Ψ (P ∥ !P) M N P'] ==> Prop C Ψ (!P) M shows "Prop C Ψ P M N P'" using Trans proof(nominal_induct Ψ P Rs=="M(N) ≺ P'" avoiding: C arbitrary: P' rule: semantics.strong_in case(cInput Ψ M K xvec N Tvec P C) thus ?case by(force intro: rInput simp add: residualInject action.inject) next case(Output Ψ M K N P C) thus ?case by(simp add: residualInject) next case(Case Ψ P Rs φ CS C) thus ?case by(force intro: rCase) next case(cPar1 Ψ ΨQ P α P' Q AQ C P'') thus ?case by(force intro: rPar1 simp add: residualInject) next case(cPar2 Ψ ΨP Q α Q' xvec P C Q'') thus ?case by(force intro: rPar2 simp add: residualInject) next case(cComm1 Ψ ΨQ P M N P' xvec ΨP Q K zvec Q' yvec C PQ) thus ?case by(simp add: residualInject) next case(cComm2 Ψ ΨQ P M zvec N P' xvec ΨP Q K yvec Q' C PQ) thus ?case by(simp add: residualInject) next case(cOpen Ψ P M xvec N P' x yvec C P'') thus ?case by(simp add: residualInject) next case(cScope Ψ P α P' x C P'') thus ?case by(force intro: rScope simp add: residualInject) next case(Bang Ψ P Rs C) thus ?case by(force intro: rBang) qed lemma tauInduct[consumes 1, case_names cCase cPar1 cPar2 cComm1 cComm2 cScope cBang]: fixes Ψ :: 'b and P :: "('a, 'b, 'c) psi" and Rs :: "('a, 'b, 'c) residual" and Prop :: "'d::fs_name → 'b → ('a, 'b, 'c) psi → ('a, 'b, 'c) psi → bool" and C :: "'d::fs_name" assumes Trans: "Ψ ⊳ P ⟼τ ≺ P'" and rCase: "∧Ψ P P' φ Cs C. [Ψ ⊳ P ⟼τ ≺ P'; ∧C. Prop C Ψ P P'; (φ, P) mem Cs; Ψ ⊨ φ; guard Prop C Ψ (Cases Cs) P'" and rPar1: "∧Ψ ΨQ P P' AQ Q C. [Ψ ⊗ ΨQ ⊳ P ⟼τ ≺ P'; extractFrame Q = ⟨AQ, ΨQ⟩; distinct AQ; ∧C. Prop C (Ψ ⊗ ΨQ) P P'; AQ ♯* P; AQ ♯* Q; AQ ♯* Ψ; AQ ♯* P'; AQ ♯* C] ==> Prop C Ψ (P ∥ Q) (P' ∥ Q)" and rPar2: "∧Ψ ΨP Q Q' AP P C. [Ψ ⊗ ΨP ⊳ Q ⟼τ ≺ Q'; extractFrame P = ⟨AP, ΨP⟩; distinct AP; ∧C. Prop C (Ψ ⊗ ΨP) Q Q'; AP ♯* P; AP ♯* Q; AP ♯* Ψ; AP ♯* Q'; AP ♯* C] ==> Prop C Ψ (P ∥ Q) (P ∥ Q')" and rComm1: "∧Ψ ΨQ P M N P' AP ΨP Q K xvec Q' AQ C. [Ψ ⊗ ΨQ ⊳ P ⟼M(N) ≺ P'; extractFrame P = ⟨AP, ΨP⟩; distinct AP; Ψ ⊗ ΨP ⊳ Q ⟼K(ν*xvec)⟨N⟩ ≺ Q'; extractFrame Q = ⟨AQ, ΨQ⟩; distinct AQ; Ψ ⊗ ΨP ⊗ ΨQ ⊨ M ↔ K; AP ♯* Ψ; AP ♯* ΨQ; AP ♯* P; AP ♯* M; AP ♯* N; AP ♯* P'; AP ♯* Q; AP ♯* Q'; AP ♯* AQ; AP ♯* xvec; AQ ♯* Ψ; AQ ♯* ΨP; AQ ♯* P; AQ ♯* N; AQ ♯* P'; AQ ♯* Q; AQ ♯* K; AQ ♯* Q'; AQ ♯* xvec; xvec ♯* Ψ; xvec ♯* ΨP; xvec ♯* ΨQ; xvec ♯* P; xvec ♯* M; xvec ♯* Q; xvec ♯* K; AP ♯* C; AQ ♯* C; xvec ♯* C] ==> Prop C Ψ (P ∥ Q) ((ν*xvec)(P' ∥ Q'))" and rComm2: "∧Ψ ΨQ P M xvec N P' AP ΨP Q K Q' AQ C. [Ψ ⊗ ΨQ ⊳ P ⟼M(ν*xvec)⟨N⟩ ≺ P'; extractFrame P = ⟨AP, ΨP⟩; distinct AP; Ψ ⊗ ΨP ⊳ Q ⟼K(N) ≺ Q'; extractFrame Q = ⟨AQ, ΨQ⟩; distinct AQ; Ψ ⊗ ΨP ⊗ ΨQ ⊨ M ↔ K; AP ♯* Ψ; AP ♯* ΨQ; AP ♯* P; AP ♯* M; AP ♯* N; AP ♯* P'; AP ♯* Q; AP ♯* Q'; AP ♯* AQ; AP ♯* xvec; AQ ♯* Ψ; AQ ♯* ΨP; AQ ♯* P; AQ ♯* N; AQ ♯* P'; AQ ♯* Q; AQ ♯* K; AQ ♯* Q'; AQ ♯* xvec; xvec ♯* Ψ; xvec ♯* ΨP; xvec ♯* ΨQ; xvec ♯* P; xvec ♯* M; xvec ♯* Q; xvec ♯* K; AP ♯* C; AQ ♯* C; xvec ♯* C] ==> Prop C Ψ (P ∥ Q) ((ν*xvec)(P' ∥ Q'))" and rScope: "∧Ψ P P' x C. [Ψ ⊳ P ⟼τ ≺ P'; ∧C. Prop C Ψ P P'; x ♯ Ψ; x ♯ C] ==> Prop C Ψ ((νx)P) ((νx)P')" and rBang: "∧Ψ P P' C. [Ψ ⊳ P ∥ !P ⟼τ ≺ P'; guarded P; ∧C. Prop C Ψ (P ∥ !P) P'] ==> Prop C Ψ (!P) P'" shows "Prop C Ψ P P'" using Trans proof(nominal_induct Ψ P Rs=="τ ≺ P'" avoiding: C arbitrary: P' rule: semantics.strong_indu case(cInput M K xvec N Tvec P C) thus ?case by(simp add: residualInject) next case(Output Ψ M K N P C) thus ?case by(simp add: residualInject) next case(Case Ψ P Rs φ Cs C) thus ?case by(force intro: rCase simp add: residualInject) next case(cPar1 Ψ ΨQ P α P' AQ Q C P'') thus ?case by(force intro: rPar1 simp add: residualInject) next case(cPar2 Ψ ΨP Q α Q' AP P C Q'') thus ?case by(force intro: rPar2 simp add: residualInject) next case(cComm1 Ψ ΨQ P M N P' AP ΨP Q K xvec Q' AQ C PQ) thus ?case by(force intro: rComm1 simp add: residualInject) next case(cComm2 Ψ ΨQ P M xvec N P' AP ΨP Q' AQ C PQ) thus ?case by(force intro: rComm2 simp add: residualInject) next case(cOpen Ψ P M xvec N P' x yvec C P'') thus ?case by(simp add: residualInject) next case(cScope Ψ P α P' x C P'') thus ?case by(force intro: rScope simp add: residualInject) next case(Bang Ψ P Rs C ) thus ?case by(force intro: rBang simp add: residualInject) qed lemma semanticsFrameInduct[consumes 3, case_names cAlpha cInput cOutput cCase cPar1 cPar2 fixes Ψ :: 'b and P :: "('a, 'b, 'c) psi" and Rs :: "('a, 'b, 'c) residual" and AP :: "name list" and ΨP :: 'b and Prop :: "'d::fs_name → 'b → ('a, 'b, 'c) psi → ('a, 'b, 'c) residual → name list → 'b → bool" and C :: "'d::fs_name" assumes Trans: "Ψ ⊳ P ⟼ Rs" and FrP: "extractFrame P = ⟨AP, ΨP⟩" and "distinct AP" and rAlpha: "∧Ψ P AP ΨP p Rs C. [AP ♯* Ψ; AP ♯* P; AP ♯* (p ∙ AP); AP ♯* Rs; AP ♯* C; set p ⊆ set AP × set(p ∙ AP); distinctPerm p; Prop C Ψ P Rs AP ΨP] ==> Prop C Ψ P Rs (p ∙ AP) (p ∙ ΨP)" and rInput: "∧Ψ M K xvec N Tvec P C. [Ψ ⊨ M ↔ K; distinct xvec; set xvec ⊆ supp N; length xvec = length Tvec; xvec ♯* Ψ; xvec ♯* M; xvec ♯* K; xvec ♯* C] ==> Prop C Ψ (M(λ*xvec N).P) (K((N[xvec::=Tvec])) ≺ (P[xvec::=Tvec])) ([]) (1)" and rOutput: "∧Ψ M K N P C. Ψ ⊨ M ↔ K ==> Prop C Ψ (M⟨N⟩.P) (K⟨N⟩ ≺ P) ([]) (1)" and rCase: "∧Ψ P Rs φ Cs AP ΨP C. [Ψ ⊳ P ⟼ Rs; extractFrame P = ⟨AP, ΨP⟩; distinct AP; ∧C (φ, P) mem Cs; Ψ ⊨ φ; guarded P; ΨP ≃ 1; (supp ΨP) = ({}::name set); AP ♯* Ψ; AP ♯* P; AP ♯* Rs; AP ♯* C] ==> Prop C Ψ (Cases Cs) Rs ([]) (1)" and rPar1: "∧Ψ ΨQ P α P' AQ Q AP ΨP C. [Ψ ⊗ ΨQ ⊳ P ⟼α ≺ P'; extractFrame P = ⟨AP, ΨP⟩; distinct AP; extractFrame Q = ⟨AQ, ΨQ⟩; distinct AQ; ∧C. Prop C (Ψ ⊗ ΨQ) P (α ≺ P') AP ΨP; distinct(bn α); AP ♯* P; AP ♯* Q; AP ♯* Ψ; AP ♯* α; AP ♯* P'; AP ♯* AQ; AP ♯* ΨQ; AQ ♯* P; AQ ♯* Q; AQ ♯* Ψ; AQ ♯* α; AQ ♯* P'; AQ ♯* ΨP; bn α ♯* Ψ; bn α ♯* P; bn α ♯* Q; bn α ♯* subject α; bn α ♯* ΨP; bn α ♯* ΨQ; AP ♯* C; AQ ♯* C; bn α ♯* C] ==> Prop C Ψ (P ∥ Q) (α ≺ (P' ∥ Q)) (AP@AQ) (ΨP ⊗ ΨQ)" and rPar2: "∧Ψ ΨP Q α Q' AP P AQ ΨQ C. [Ψ ⊗ ΨP ⊳ Q ⟼α ≺ Q'; extractFrame P = ⟨AP, ΨP⟩; distinct AP; extractFrame Q = ⟨AQ, ΨQ⟩; distinct AQ; ∧C. Prop C (Ψ ⊗ ΨP) Q (α ≺ Q') AQ ΨQ; distinct(bn α); AP ♯* P; AP ♯* Q; AP ♯* Ψ; AP ♯* α; AP ♯* Q'; AP ♯* AQ; AP ♯* ΨQ; AQ ♯* P; AQ ♯* Q; AQ ♯* Ψ; AQ ♯* α; AQ ♯* Q'; AQ ♯* ΨP; bn α ♯* Ψ; bn α ♯* P; bn α ♯* Q; bn α ♯* subject α; bn α ♯* ΨP; bn α ♯* ΨQ; AP ♯* C; AQ ♯* C; bn α ♯* C] ==> Prop C Ψ (P ∥ Q) (α ≺ (P ∥ Q')) (AP@AQ) (ΨP ⊗ ΨQ)" and rComm1: "∧Ψ ΨQ P M N P' AP ΨP Q K xvec Q' AQ C. [Ψ ⊗ ΨQ ⊳ P ⟼M(N) ≺ P'; extractFrame P = ⟨AP, ΨP⟩; distinct AP; ∧C. Prop C (Ψ ⊗ ΨQ) P ((M(N)) ≺ P') AP ΨP; Ψ ⊗ ΨP ⊳ Q ⟼K(ν*xvec)⟨N⟩ ≺ Q'; extractFrame Q = ⟨AQ, ΨQ⟩; distinct AQ; Ψ ⊗ ΨP ⊗ ΨQ ⊨ M ↔ K; ∧C. Prop C (Ψ ⊗ ΨP) Q (K(ν*xvec)⟨N⟩ ≺ Q') AQ ΨQ; distinct xvec; AP ♯* Ψ; AP ♯* ΨQ; AP ♯* P; AP ♯* M; AP ♯* N; AP ♯* P'; AP ♯* Q; AP ♯* Q'; AP ♯* AQ; AP ♯* xvec; AQ ♯* Ψ; AQ ♯* ΨP; AQ ♯* P; AQ ♯* N; AQ ♯* P'; AQ ♯* Q; AQ ♯* K; AQ ♯* Q'; AQ ♯* xvec; xvec ♯* Ψ; xvec ♯* ΨP; xvec ♯* ΨQ; xvec ♯* P; xvec ♯* M; xvec ♯* Q; xvec ♯* K; AP ♯* C; AQ ♯* C; xvec ♯* C] ==> Prop C Ψ (P ∥ Q) (τ ≺ (ν*xvec)(P' ∥ Q')) (AP@AQ) (ΨP ⊗ ΨQ)" and rComm2: "∧Ψ ΨQ P M xvec N P' AP ΨP Q K Q' AQ C. [Ψ ⊗ ΨQ ⊳ P ⟼M(ν*xvec)⟨N⟩ ≺ P'; extractFrame P = ⟨AP, ΨP⟩; distinct AP; ∧C. Prop C (Ψ ⊗ ΨQ) P (M(ν*xvec)⟨N⟩ ≺ P') AP ΨP; Ψ ⊗ ΨP ⊳ Q ⟼K(N) ≺ Q'; extractFrame Q = ⟨AQ, ΨQ⟩; distinct AQ; ∧C. Prop C (Ψ ⊗ ΨP) Q (K(N) ≺ Q') AQ ΨQ; Ψ ⊗ ΨP ⊗ ΨQ ⊨ M ↔ K; distinct xvec; AP ♯* Ψ; AP ♯* ΨQ; AP ♯* P; AP ♯* M; AP ♯* N; AP ♯* P'; AP ♯* Q; AP ♯* Q'; AP ♯* AQ; AP ♯* xvec; AQ ♯* Ψ; AQ ♯* ΨP; AQ ♯* P; AQ ♯* N; AQ ♯* P'; AQ ♯* Q; AQ ♯* K; AQ ♯* Q'; AQ ♯* xvec; xvec ♯* Ψ; xvec ♯* ΨP; xvec ♯* ΨQ; xvec ♯* P; xvec ♯* M; xvec ♯* Q; xvec ♯* K; AP ♯* C; AQ ♯* C; xvec ♯* C] ==> Prop C Ψ (P ∥ Q) (τ ≺ (ν*xvec)(P' ∥ Q')) (AP@AQ) (ΨP ⊗ ΨQ)" and rOpen: "∧Ψ P M xvec yvec N P' x AP ΨP C. [Ψ ⊳ P ⟼M(ν*(xvec@yvec))⟨N⟩ ≺ P'; extractFrame P = ⟨AP, ΨP⟩; distinct AP; ∧C. Prop C Ψ P (M(ν*(xvec@yvec))⟨N⟩ ≺ P') AP ΨP; x ∈ supp N; x ♯ Ψ; x ♯ M; x ♯ AP; x ♯ xvec; x ♯ yvec; AP ♯* Ψ; AP ♯* P; AP ♯* M; AP ♯* N; AP ♯* P'; AP ♯* xvec; AP ♯* yvec; xvec ♯* yvec; distinct xvec; distinct yvec; xvec ♯* Ψ; xvec ♯* P; xvec ♯* M; xvec ♯* ΨP; yvec ♯* ΨP; yvec ♯* Ψ; yvec ♯* P; yvec ♯* M; AP ♯* C; x ♯ C; xvec ♯* C; yvec ♯* C] ==> Prop C Ψ ((νx)P) (M(ν*(xvec@x#yvec))⟨N⟩ ≺ P') (x#AP) ΨP" and rScope: "∧Ψ P α P' x AP ΨP C. [Ψ ⊳ P ⟼α ≺ P'; extractFrame P = ⟨AP, ΨP⟩; distinct AP; ∧C. Prop C Ψ P (α ≺ P') AP ΨP; x ♯ Ψ; x ♯ α; x ♯ AP; AP ♯* Ψ; AP ♯* P; AP ♯* α; AP ♯* P'; distinct(bn α); bn α ♯* Ψ; bn α ♯* P; bn α ♯* subject α; bn α ♯* ΨP; AP ♯* C; x ♯ C; bn α ♯* C] ==> Prop C Ψ ((νx)P) (α ≺ ((νx)P')) (x#AP) ΨP" and rBang: "∧Ψ P Rs AP ΨP C. [Ψ ⊳ P ∥ !P ⟼ Rs; guarded P; extractFrame P = ⟨AP, ΨP⟩; distinct AP; ∧C. Prop C Ψ (P ∥ !P) Rs AP (ΨP ⊗ 1); ΨP ≃ 1; supp ΨP = ({}::name set); AP ♯* Ψ; AP ♯* P; AP ♯* Rs; AP ♯* C] ==> Prop C Ψ (!P) Rs ([]) (1)" shows "Prop C Ψ P Rs AP ΨP" using Trans FrP ‹distinct AP› proof(nominal_induct avoiding: AP ΨP C rule: semantics.strong_induct) case(cInput Ψ M K xvec N Tvec P AP ΨP C) from ‹extractFrame (M(λ*xvec N).P) = ⟨AP, ΨP⟩› have "AP = []" and "ΨP = 1" by auto with ‹Ψ ⊨ M ↔ K› ‹distinct xvec› ‹set xvec ⊆ supp N› ‹length xvec = length Tvec› ‹xvec ♯* Ψ› ‹xvec ♯* M› ‹xvec ♯* K› ‹xvec ♯* C› show ?case by(blast intro: rInput) next case(Output Ψ M K N P AP ΨP) from ‹extractFrame (M⟨N⟩.P) = ⟨AP, ΨP⟩› have "AP = []" and "ΨP = 1" by auto with ‹Ψ ⊨ M ↔ K› show ?case by(blast intro: rOutput) next case(Case Ψ P Rs φ Cs AcP ΨcP C) obtain AP ΨP where FrP: "extractFrame P = ⟨AP, ΨP⟩" and "distinct AP" and "AP ♯* (Ψ, P, Rs, C)" by(rule freshFrame) hence "AP ♯* Ψ" and "AP ♯* P" and "AP ♯* Rs" and "AP ♯* C" by simp+ note ‹Ψ ⊳ P ⟼ Rs› FrP ‹distinct AP› moreover from FrP ‹distinct AP› ‹∧AP ΨP C. [extractFrame P = ⟨AP, ΨP⟩; distinct AP] ==> have "∧C. Prop C Ψ P Rs AP ΨP" by simp moreover note ‹(φ, P) mem Cs› ‹Ψ ⊨ φ› ‹guarded P› moreover from ‹guarded P› FrP have "ΨP ≃ 1" and "supp ΨP = ({}::name set)" by(metis guarde moreover note ‹AP ♯* Ψ› ‹AP ♯* P› ‹AP ♯* Rs› ‹AP ♯* C› ultimately have "Prop C Ψ (Cases Cs) Rs ([]) (1)" by(rule rCase) thus ?case using ‹extractFrame(Cases Cs) = ⟨AcP, ΨcP⟩› by simp next case(cPar1 Ψ ΨQ P α P' Q AQ APQ ΨPQ C) obtain AP ΨP where FrP: "extractFrame P = ⟨AP, ΨP⟩" and "distinct AP" "AP ♯* (P, Q, Ψ, α, P', AQ, APQ, C, ΨQ)" by(rule freshFrame) hence "AP ♯* P" and "AP ♯* Q" and "AP ♯* Ψ" and "AP ♯* α" and "AP ♯* P'" and "AP ♯* AQ" and "AP ♯* APQ" and "AP ♯* C" and "AP ♯* ΨQ" by simp+ have FrQ: "extractFrame Q = ⟨AQ, ΨQ⟩" by fact from ‹AQ ♯* P› ‹AP ♯* AQ› FrP have "AQ ♯* ΨP" by(force dest: extractFrameFreshChain) from ‹bn α ♯* P› ‹AP ♯* α› FrP have "bn α ♯* ΨP" by(force dest: extractFrameFreshChain) from ‹extractFrame(P ∥ Q) = ⟨APQ, ΨPQ⟩› FrP FrQ ‹AP ♯* AQ› ‹AP ♯* ΨQ› ‹AQ ♯* ΨP› have "⟨(AP@AQ), ΨP ⊗ ΨQ⟩ = ⟨APQ, ΨPQ⟩" by simp moreover from ‹distinct AP› ‹distinct AQ› ‹AP ♯* AQ› have "distinct(AP@AQ)" by(auto simp add: fresh_star_def fresh_def name_list_supp) ultimately obtain p where S: "set p ⊆ set(AP@AQ) × set((p ∙ AP)@(p ∙ AQ))" and "distinc and Ψeq: "ΨPQ = p ∙ (ΨP ⊗ ΨQ)" and Aeq: "APQ = (p ∙ AP)@(p ∙ AQ)" using ‹AP ♯* APQ› ‹AQ ♯* APQ› ‹distinct APQ› by(rule_tac frameChainEq') (assumption | simp add: eqvts)+ note ‹Ψ ⊗ ΨQ ⊳ P ⟼α ≺ P'› FrP ‹distinct AP› FrQ ‹distinct AQ› moreover from FrP ‹distinct AP› ‹∧AP ΨP C. [extractFrame P = ⟨AP, ΨP⟩; distinct AP] ==> have "∧C. Prop C (Ψ ⊗ ΨQ) P (α ≺ P') AP ΨP" by simp moreover note ‹AP ♯* P› ‹AP ♯* Q› ‹AP ♯* Ψ› ‹AP ♯* α› ‹AP ♯* P'› ‹AP ♯* AQ› ‹AP ♯* ΨQ ‹AQ ♯* P› ‹AQ ♯* Q› ‹AQ ♯* Ψ› ‹AQ ♯* α› ‹AQ ♯* P'› ‹AQ ♯* ΨP› ‹distinct(bn α)› ‹bn α ♯* Ψ› ‹bn α ♯* P› ‹bn α ♯* Q› ‹bn α ♯* subject α› ‹bn α ♯* ΨP› ‹bn α ♯* ΨQ› ‹AP ♯* C› ‹AQ ♯* C› ‹bn α ♯* C› ultimately have "Prop C Ψ (P ∥ Q) (α ≺ (P' ∥ Q)) (AP@AQ) (ΨP ⊗ ΨQ)" by(rule_tac rPar1) with ‹AP ♯* Ψ› ‹AP ♯* P› ‹AP ♯* Q› ‹AP ♯* α› ‹AP ♯* P'› ‹AP ♯* APQ› ‹AP ♯* C› ‹AQ ♯* Ψ› ‹AQ ♯* P› ‹AQ ♯* Q› ‹AQ ♯* α› ‹AQ ♯* P'› ‹AQ ♯* APQ› ‹AQ ♯* C› S ‹distinctPerm p› Aeq have "Prop C Ψ (P ∥ Q) (α ≺ (P' ∥ Q)) (p ∙ (AP@AQ)) (p ∙ (ΨP ⊗ ΨQ))" by(rule_tac rAlpha) (assumption | simp add: eqvts)+ with Ψeq Aeq show ?case by(simp add: eqvts) next case(cPar2 Ψ ΨP Q α Q' P AP APQ ΨPQ C) obtain AQ ΨQ where FrQ: "extractFrame Q = ⟨AQ, ΨQ⟩" and "distinct AQ" "AQ ♯* (P, Q, Ψ, α, Q', AP, APQ, C, ΨP)" by(rule freshFrame) hence "AQ ♯* P" and "AQ ♯* Q" and "AQ ♯* Ψ" and "AQ ♯* α" and "AQ ♯* Q'" and "AQ ♯* AP" and "AQ ♯* APQ" and "AQ ♯* C" and "AQ ♯* ΨP" by simp+ from ‹AQ ♯* AP› have "AP ♯* AQ" by simp have FrP: "extractFrame P = ⟨AP, ΨP⟩" by fact from ‹AP ♯* Q› ‹AQ ♯* AP› FrQ have "AP ♯* ΨQ" by(force dest: extractFrameFreshChain) from ‹bn α ♯* Q› ‹AQ ♯* α› FrQ have "bn α ♯* ΨQ" by(force dest: extractFrameFreshChain) from ‹extractFrame(P ∥ Q) = ⟨APQ, ΨPQ⟩› FrP FrQ ‹AP ♯* AQ› ‹AP ♯* ΨQ› ‹AQ ♯* ΨP› have "⟨(AP@AQ), ΨP ⊗ ΨQ⟩ = ⟨APQ, ΨPQ⟩" by simp moreover from ‹distinct AP› ‹distinct AQ› ‹AP ♯* AQ› have "distinct(AP@AQ)" by(auto simp add: fresh_star_def fresh_def name_list_supp) ultimately obtain p where S: "(set p ⊆ (set(AP@AQ)) × (set APQ))" and "distinctPerm p" and Ψeq: "ΨPQ = p ∙ (ΨP ⊗ ΨQ)" and Aeq: "APQ = ((p ∙ AP)@(p ∙ AQ))" using ‹AP ♯* APQ› ‹AQ ♯* APQ› ‹distinct APQ› by(rule_tac frameChainEq') (assumption | simp add: eqvts)+ note ‹Ψ ⊗ ΨP ⊳ Q ⟼α ≺ Q'› FrP ‹distinct AP› FrQ ‹distinct AQ› moreover from FrQ ‹distinct AQ› ‹∧AQ ΨQ C. [extractFrame Q = ⟨AQ, ΨQ⟩; distinct AQ] ==> have "∧C. Prop C (Ψ ⊗ ΨP) Q (α ≺ Q') AQ ΨQ" by simp moreover note ‹AP ♯* P› ‹AP ♯* Q› ‹AP ♯* Ψ› ‹AP ♯* α› ‹AP ♯* Q'› ‹AP ♯* AQ› ‹AP ♯* ΨQ ‹AQ ♯* P› ‹AQ ♯* Q› ‹AQ ♯* Ψ› ‹AQ ♯* α› ‹AQ ♯* Q'› ‹AQ ♯* ΨP› ‹distinct(bn α)› ‹bn α ♯* Ψ› ‹bn α ♯* P› ‹bn α ♯* Q› ‹bn α ♯* subject α› ‹bn α ♯* ΨP› ‹bn α ♯* ΨQ› ‹AP ♯* C› ‹AQ ♯* C› ‹bn α ♯* C› ultimately have "Prop C Ψ (P ∥ Q) (α ≺ (P ∥ Q')) (AP@AQ) (ΨP ⊗ ΨQ)" by(rule_tac rPar2) with ‹AP ♯* Ψ› ‹AP ♯* P› ‹AP ♯* Q› ‹AP ♯* α› ‹AP ♯* Q'› ‹AP ♯* APQ› ‹AP ♯* C› ‹AQ ♯* Ψ› ‹AQ ♯* P› ‹AQ ♯* Q› ‹AQ ♯* α› ‹AQ ♯* Q'› ‹AQ ♯* APQ› ‹AQ ♯* C› S ‹distinctPerm p› Aeq have "Prop C Ψ (P ∥ Q) (α ≺ (P ∥ Q')) (p ∙ (AP@AQ)) (p ∙ (ΨP ⊗ ΨQ))" by(rule_tac rAlpha) (assumption | simp add: eqvts)+ with Ψeq Aeq show ?case by(simp add: eqvts) next case(cComm1 Ψ ΨQ P M N P' AP ΨP Q K xvec Q' AQ APQ ΨPQ C) from ‹distinct AP› ‹distinct AQ› ‹AP ♯* AQ› have "distinct(AP@AQ)" by(auto simp add: fresh_star_def fresh_def name_list_supp) from cComm1 have "Prop C Ψ (P ∥ Q) (τ ≺ (ν*xvec)(P' ∥ Q')) (AP@AQ) (ΨP ⊗ ΨQ)" by(rule_tac rComm1) moreover from ‹extractFrame(P ∥ Q) = ⟨APQ, ΨPQ⟩› ‹extractFrame P = ⟨AP, ΨP⟩› ‹extractFra ‹AP ♯* AQ› ‹AP ♯* ΨQ› ‹AQ ♯* ΨP› have "⟨(AP@AQ), (ΨP ⊗ ΨQ)⟩ = ⟨APQ, ΨPQ⟩" by simp with ‹AP ♯* APQ› ‹AQ ♯* APQ› ‹distinct(AP@AQ)› ‹distinct APQ› obtain p where S: "(set p ⊆ (set(AP@AQ)) × (set APQ))" and "distinctPerm p" and Ψeq: "ΨPQ = p ∙ (ΨP ⊗ ΨQ)" and Aeq: "APQ = p ∙ (AP@AQ)" by(rule_tac frameChainEq') (assumption | simp)+ moreover note ‹AP ♯* Ψ› ‹AQ ♯* Ψ› ‹AP ♯* P› ‹AQ ♯* P› ‹AP ♯* Q› ‹AQ ♯* Q› ‹AP ♯* xve ‹AQ ♯* xvec› ‹AP ♯* P'› ‹AQ ♯* P'› ‹AP ♯* Q'› ‹AQ ♯* Q'› ‹AP ♯* APQ› ‹AQ ♯* APQ› ‹AP ♯* C› ‹AQ ♯* C› ultimately have "Prop C Ψ (P ∥ Q) (τ ≺ (ν*xvec)(P' ∥ Q')) (p ∙ (AP@AQ)) (p ∙ (ΨP ⊗ ΨQ))" by(rule_tac rAlpha) auto with Ψeq Aeq show ?case by simp next case(cComm2 Ψ ΨQ P M xvec N P' AP ΨP Q K Q' AQ APQ ΨPQ C) from ‹distinct AP› ‹distinct AQ› ‹AP ♯* AQ› have "distinct(AP@AQ)" by(auto simp add: fresh_star_def fresh_def name_list_supp) from cComm2 have "Prop C Ψ (P ∥ Q) (τ ≺ (ν*xvec)(P' ∥ Q')) (AP@AQ) (ΨP ⊗ ΨQ)" by(rule_tac rComm2) moreover from ‹extractFrame(P ∥ Q) = ⟨APQ, ΨPQ⟩› ‹extractFrame P = ⟨AP, ΨP⟩› ‹extractFra ‹AP ♯* AQ› ‹AP ♯* ΨQ› ‹AQ ♯* ΨP› have "⟨(AP@AQ), (ΨP ⊗ ΨQ)⟩ = ⟨APQ, ΨPQ⟩" by simp with ‹AP ♯* APQ› ‹AQ ♯* APQ› ‹distinct(AP@AQ)› ‹distinct APQ› obtain p where S: "(set p ⊆ (set(AP@AQ)) × (set APQ))" and "distinctPerm p" and Ψeq: "ΨPQ = p ∙ (ΨP ⊗ ΨQ)" and Aeq: "APQ = p ∙ (AP@AQ)" by(rule_tac frameChainEq') (assumption | simp)+ moreover note ‹AP ♯* Ψ› ‹AQ ♯* Ψ› ‹AP ♯* P› ‹AQ ♯* P› ‹AP ♯* Q› ‹AQ ♯* Q› ‹AP ♯* xve ‹AQ ♯* xvec› ‹AP ♯* P'› ‹AQ ♯* P'› ‹AP ♯* Q'› ‹AQ ♯* Q'› ‹AP ♯* APQ› ‹AQ ♯* APQ› ‹AP ♯* C› ‹AQ ♯* C› ultimately have "Prop C Ψ (P ∥ Q) (τ ≺ (ν*xvec)(P' ∥ Q')) (p ∙ (AP@AQ)) (p ∙ (ΨP ⊗ ΨQ))" by(rule_tac rAlpha) auto with Ψeq Aeq show ?case by simp next case(cOpen Ψ P M xvec yvec N P' x AxP ΨxP C) obtain AP ΨP where FrP: "extractFrame P = ⟨AP, ΨP⟩" and "distinct AP" and "AP ♯* (Ψ, P, M, xvec, yvec, N, P', AxP, ΨxP, C, x)" by(rule freshFrame) hence "AP ♯* Ψ" and "AP ♯* P" and "AP ♯* M" and "AP ♯* xvec"and "AP ♯* yvec" and "AP ♯* N" a and "AP ♯* AxP" and "AP ♯* ΨxP" and "AP ♯* C" and "x ♯ AP" by simp+ from ‹xvec ♯* P› ‹AP ♯* xvec› FrP have "xvec ♯* ΨP" by(force dest: extractFrameFreshChain) from ‹yvec ♯* P› ‹AP ♯* yvec› FrP have "yvec ♯* ΨP" by(force dest: extractFrameFreshChain) from ‹extractFrame((νx)P) = ⟨AxP, ΨxP⟩› FrP have "⟨(x#AP), ΨP⟩ = ⟨AxP, ΨxP⟩" by simp moreover from ‹x ♯ AP› ‹distinct AP› have "distinct(x#AP)" by simp ultimately obtain p where S: "set p ⊆ set (x#AP) × set (p ∙ (x#AP))" and "distinctPerm and Ψeq: "ΨxP = p ∙ ΨP" and Aeq: "AxP = (p ∙ x)#(p ∙ AP)" using ‹AP ♯* AxP›\‹x ♯ AxP› ‹distinct AxP› by(rule_tac frameChainEq') (assumption | simp add: eqvts)+ note ‹Ψ ⊳ P ⟼M(ν*(xvec@yvec))⟨N⟩ ≺ P'› FrP ‹distinct AP› moreover from FrP ‹distinct AP› ‹∧AP ΨP C. [extractFrame P = ⟨AP, ΨP⟩; distinct AP] ==> have "∧C. Prop C Ψ P (M(ν*(xvec@yvec))⟨N⟩ ≺ P') AP ΨP" by simp moreover note ‹x ♯ Ψ› ‹x ♯ M› ‹x ♯ xvec› ‹x ♯ yvec› ‹x ∈ supp N› ‹x ♯ AP› ‹AP ♯* Ψ› ‹xvec ♯* Ψ› ‹xvec ♯* P› ‹xvec ♯* M› ‹xvec ♯* ΨP› ‹yvec ♯* Ψ› ‹yvec ♯* P› ‹yvec ♯* M ‹AP ♯* C› ‹x ♯ C› ‹xvec ♯* C› ‹yvec ♯* C› ‹xvec ♯* yvec› ‹distinct xvec› ‹distin ultimately have "Prop C Ψ ((νx)P) (M(ν*(xvec@x#yvec))⟨N⟩ ≺ P') (x#AP) ΨP" by(rule_tac rOpen) with ‹AP ♯* Ψ› ‹AP ♯* P› ‹AP ♯* M› ‹AP ♯* xvec› ‹AP ♯* yvec› ‹AP ♯* N› ‹AP ♯* P'› ‹x ♯ Ψ› ‹x ♯ M› ‹x ♯ C› ‹x ♯ xvec› ‹x ♯ yvec› Aeq S ‹distinctPerm p› have "Prop C Ψ ((νx)P) (M(ν*(xvec@x#yvec))⟨N⟩ ≺ P') (p ∙ (x#AP)) (p ∙ ΨP)" by(rule_tac AP="x#AP" in rAlpha) (assumption | simp add: abs_fresh fresh_star_def boundOut with Ψeq Aeq show ?case by(simp add: eqvts) next case(cScope Ψ P α P' x AxP ΨxP C) obtain AP ΨP where FrP: "extractFrame P = ⟨AP, ΨP⟩" and "distinct AP" and "AP ♯* (Ψ, P, α, P', AxP, ΨxP, C, x)" by(rule freshFrame) hence "AP ♯* Ψ" and "AP ♯* P" and "AP ♯* α" and "AP ♯* P'" and "AP ♯* AxP" and "AP ♯* ΨxP" and "AP ♯* C" and "x ♯ AP" by simp+ from ‹bn α ♯* P› ‹AP ♯* α› FrP have "bn α ♯* ΨP" by(force dest: extractFrameFreshChain) from ‹extractFrame((νx)P) = ⟨AxP, ΨxP⟩› FrP have "⟨(x#AP), ΨP⟩ = ⟨AxP, ΨxP⟩" by simp moreover from ‹x ♯ AP› ‹distinct AP› have "distinct(x#AP)" by simp ultimately obtain p where S: "set p ⊆ set (x#AP) × set (p ∙ (x#AP))" and "distinctPerm and Ψeq: "ΨxP = p ∙ ΨP" and Aeq: "AxP = (p ∙ x)#(p ∙ AP)" using ‹AP ♯* AxP›\‹x ♯ AxP› ‹distinct AxP› by(rule_tac frameChainEq') (assumption | simp add: eqvts)+ note ‹Ψ ⊳ P ⟼α ≺ P'› FrP ‹distinct AP› moreover from FrP ‹distinct AP› ‹∧AP ΨP C. [extractFrame P = ⟨AP, ΨP⟩; distinct AP] ==> have "∧C. Prop C Ψ P (α ≺ P') AP ΨP" by simp moreover note ‹x ♯ Ψ› ‹x ♯ α› ‹x ♯ AP› ‹AP ♯* Ψ› ‹AP ♯* Ψ› ‹AP ♯* P› ‹AP ♯* α› ‹AP ♯* P ‹bn α ♯* Ψ› ‹bn α ♯* P› ‹bn α ♯* subject α› ‹bn α ♯* ΨP› ‹AP ♯* C› ‹x ♯ C› ‹bn α ♯* C› ultimately have "Prop C Ψ ((νx)P) (α ≺ ((νx)P')) (x#AP) ΨP" by(rule_tac rScope) with ‹AP ♯* Ψ› ‹AP ♯* P› ‹AP ♯* α› ‹AP ♯* P'› ‹AP ♯* AxP› ‹AP ♯* C› ‹x ♯ AxP› ‹AP ♯ ‹x ♯ Ψ› ‹x ♯ α› ‹x ♯ C› Aeq S ‹distinctPerm p› have "Prop C Ψ ((νx)P) (α ≺ ((νx)P')) (p ∙ (x#AP)) (p ∙ ΨP)" by(rule_tac AP="x#AP" in rAlpha) (assumption | simp add: abs_fresh fresh_star_def)+ with Ψeq Aeq show ?case by(simp add: eqvts) next case(Bang Ψ P Rs AbP ΨbP C) obtain AP ΨP where FrP: "extractFrame P = ⟨AP, ΨP⟩" and "distinct AP" and "AP ♯* (Ψ, P, Rs, C)" by(rule freshFrame) hence "AP ♯* Ψ" and "AP ♯* P" and "AP ♯* Rs" and "AP ♯* C" by simp+ note ‹Ψ ⊳ P ∥ !P ⟼ Rs› ‹guarded P› FrP ‹distinct AP› moreover from FrP have "extractFrame (P ∥ !P) = ⟨AP, ΨP ⊗ 1⟩" by simp with ‹distinct AP› ‹∧AP ΨP C. [extractFrame (P ∥ !P) = ⟨AP, ΨP⟩; distinct AP] ==> Pro have "∧C. Prop C Ψ (P ∥ !P) Rs AP (ΨP ⊗ 1)" by simp moreover from ‹guarded P› FrP have "ΨP ≃ 1" and "supp ΨP = ({}::name set)" by(metis guarde moreover note ‹AP ♯* Ψ› ‹AP ♯* P› ‹AP ♯* Rs› ‹AP ♯* C› ultimately have "Prop C Ψ (!P) Rs ([]) (1)" by(rule rBang) thus ?case using ‹extractFrame(!P) = ⟨AbP, ΨbP⟩› by simp qed lemma semanticsFrameInduct'[consumes 5, case_names cAlpha cFrameAlpha cInput cOutput cCa fixes Ψ :: 'b and P :: "('a, 'b, 'c) psi" and Rs :: "('a, 'b, 'c) residual" and AP :: "name list" and ΨP :: 'b and Prop :: "'d::fs_name → 'b → ('a, 'b, 'c) psi → 'a action → ('a, 'b, 'c) psi → name list → 'b → bool" and C :: "'d::fs_name" assumes Trans: "Ψ ⊳ P ⟼α ≺ P'" and FrP: "extractFrame P = ⟨AP, ΨP⟩" and "distinct AP" and "bn α ♯* subject α" and "distinct(bn α)" and rAlpha: "∧Ψ P α P' p AP ΨP C. [bn α ♯* Ψ; bn α ♯* P; bn α ♯* subject α; bn α ♯* ΨP; bn α ♯* C; bn α ♯* (p ∙ α); AP ♯* Ψ; AP ♯* P; AP ♯* α; AP ♯* P'; AP ♯* C; set p ⊆ set(bn α) × set(bn(p ∙ α)); distinctPerm p; bn(p ∙ α) ♯* α; (bn(p ∙ α)) ♯* P'; Prop C Ψ P α P' AP ΨP] ==> Prop C Ψ P (p ∙ α) (p ∙ P') AP ΨP" and rFrameAlpha: "∧Ψ P AP ΨP p α P' C. [AP ♯* Ψ; AP ♯* P; AP ♯* (p ∙ AP); AP ♯* α; AP ♯ set p ⊆ set AP × set(p ∙ AP); distinctPerm p; AP ♯* subject α; Prop C Ψ P α P' AP ΨP] ==> Prop C Ψ P α P' (p ∙ AP) (p ∙ ΨP)" and rInput: "∧Ψ M K xvec N Tvec P C. [Ψ ⊨ M ↔ K; distinct xvec; set xvec ⊆ supp N; length xvec = length Tvec; xvec ♯* Ψ; xvec ♯* M; xvec ♯* K; xvec ♯* C] ==> Prop C Ψ (M(λ*xvec N).P) (K((N[xvec::=Tvec]))) (P[xvec::=Tvec]) ([]) (1)" and rOutput: "∧Ψ M K N P C. Ψ ⊨ M ↔ K ==> Prop C Ψ (M⟨N⟩.P) (K⟨N⟩) P ([]) (1)" and rCase: "∧Ψ P α P' φ Cs AP ΨP C. [Ψ ⊳ P ⟼α ≺ P'; extractFrame P = ⟨AP, ΨP⟩; distinct AP; (φ, P) mem Cs; Ψ ⊨ φ; guarded P; ΨP ≃ 1; (supp ΨP) = ({}::name set); AP ♯* Ψ; AP ♯* P; AP ♯* α; AP ♯* P'; AP ♯* C] ==> Prop C Ψ (Cases Cs) α P' ([]) (1)" and rPar1: "∧Ψ ΨQ P α P' AQ Q AP ΨP C. [Ψ ⊗ ΨQ ⊳ P ⟼α ≺ P'; extractFrame P = ⟨AP, ΨP⟩; distinct AP; extractFrame Q = ⟨AQ, ΨQ⟩; distinct AQ; ∧C. Prop C (Ψ ⊗ ΨQ) P α P' AP ΨP; AP ♯* P; AP ♯* Q; AP ♯* Ψ; AP ♯* α; AP ♯* P'; AP ♯* AQ; AP ♯* ΨQ; AQ ♯* P; AQ ♯* Q; AQ ♯* Ψ; AQ ♯* α; AQ ♯* P'; AQ ♯* ΨP; bn α ♯* Ψ; bn α ♯* P; bn α ♯* Q; bn α ♯* subject α; bn α ♯* ΨP; bn α ♯* ΨQ; AP ♯* C; AQ ♯* C; bn α ♯* C] ==> Prop C Ψ (P ∥ Q) α (P' ∥ Q) (AP@AQ) (ΨP ⊗ ΨQ)" and rPar2: "∧Ψ ΨP Q α Q' AP P AQ ΨQ C. [Ψ ⊗ ΨP ⊳ Q ⟼α ≺ Q'; extractFrame P = ⟨AP, ΨP⟩; distinct AP; extractFrame Q = ⟨AQ, ΨQ⟩; distinct AQ; ∧C. Prop C (Ψ ⊗ ΨP) Q α Q' AQ ΨQ; AP ♯* P; AP ♯* Q; AP ♯* Ψ; AP ♯* α; AP ♯* Q'; AP ♯* AQ; AP ♯* ΨQ; AQ ♯* P; AQ ♯* Q; AQ ♯* Ψ; AQ ♯* α; AQ ♯* Q'; AQ ♯* ΨP; bn α ♯* Ψ; bn α ♯* P; bn α ♯* Q; bn α ♯* subject α; bn α ♯* ΨP; bn α ♯* ΨQ; AP ♯* C; AQ ♯* C; bn α ♯* C] ==> Prop C Ψ (P ∥ Q) α (P ∥ Q') (AP@AQ) (ΨP ⊗ ΨQ)" and rComm1: "∧Ψ ΨQ P M N P' AP ΨP Q K xvec Q' AQ C. [Ψ ⊗ ΨQ ⊳ P ⟼M(N) ≺ P'; extractFrame P = ⟨AP, ΨP⟩; distinct AP; ∧C. Prop C (Ψ ⊗ ΨQ) P (M(N)) P' AP ΨP; Ψ ⊗ ΨP ⊳ Q ⟼K(ν*xvec)⟨N⟩ ≺ Q'; extractFrame Q = ⟨AQ, ΨQ⟩; distinct AQ; Ψ ⊗ ΨP ⊗ ΨQ ⊨ M ↔ K; distinct xvec; ∧C. Prop C (Ψ ⊗ ΨP) Q (K(ν*xvec)⟨N⟩) Q' AQ ΨQ; AP ♯* Ψ; AP ♯* ΨQ; AP ♯* P; AP ♯* M; AP ♯* N; AP ♯* P'; AP ♯* Q; AP ♯* Q'; AP ♯* AQ; AP ♯* xvec; AQ ♯* Ψ; AQ ♯* ΨP; AQ ♯* P; AQ ♯* N; AQ ♯* P'; AQ ♯* Q; AQ ♯* K; AQ ♯* Q'; AQ ♯* xvec; xvec ♯* Ψ; xvec ♯* ΨP; xvec ♯* ΨQ; xvec ♯* P; xvec ♯* M; xvec ♯* Q; xvec ♯* K; AP ♯* C; AQ ♯* C; xvec ♯* C] ==> Prop C Ψ (P ∥ Q) (τ) ((ν*xvec)(P' ∥ Q')) (AP@AQ) (ΨP ⊗ ΨQ)" and rComm2: "∧Ψ ΨQ P M xvec N P' AP ΨP Q K Q' AQ C. [Ψ ⊗ ΨQ ⊳ P ⟼M(ν*xvec)⟨N⟩ ≺ P'; extractFrame P = ⟨AP, ΨP⟩; distinct AP; ∧C. Prop C (Ψ ⊗ ΨQ) P (M(ν*xvec)⟨N⟩) P' AP ΨP; Ψ ⊗ ΨP ⊳ Q ⟼K(N) ≺ Q'; extractFrame Q = ⟨AQ, ΨQ⟩; distinct AQ; ∧C. Prop C (Ψ ⊗ ΨP) Q (K(N)) Q' AQ ΨQ; Ψ ⊗ ΨP ⊗ ΨQ ⊨ M ↔ K; distinct xvec; AP ♯* Ψ; AP ♯* ΨQ; AP ♯* P; AP ♯* M; AP ♯* N; AP ♯* P'; AP ♯* Q; AP ♯* Q'; AP ♯* AQ; AP ♯* xvec; AQ ♯* Ψ; AQ ♯* ΨP; AQ ♯* P; AQ ♯* N; AQ ♯* P'; AQ ♯* Q; AQ ♯* K; AQ ♯* Q'; AQ ♯* xvec; xvec ♯* Ψ; xvec ♯* ΨP; xvec ♯* ΨQ; xvec ♯* P; xvec ♯* M; xvec ♯* Q; xvec ♯* K; AP ♯* C; AQ ♯* C; xvec ♯* C] ==> Prop C Ψ (P ∥ Q) (τ) ((ν*xvec)(P' ∥ Q')) (AP@AQ) (ΨP ⊗ ΨQ)" and rOpen: "∧Ψ P M xvec yvec N P' x AP ΨP y C. [Ψ ⊳ P ⟼M(ν*(xvec@yvec))⟨N⟩ ≺ P'; extractFrame P = ⟨AP, ΨP⟩; distinct AP; ∧C. Prop C Ψ P (M(ν*(xvec@yvec))⟨N⟩) P' AP ΨP; x ∈ supp N; x ♯ Ψ; x ♯ M; x ♯ AP; x ♯ xvec; x ♯ yvec; AP ♯* Ψ; AP ♯* P; AP ♯* M; AP ♯* N; AP ♯* P'; AP ♯* xvec; AP ♯* yvec; xvec ♯* yvec; distinct xvec; distinct yvec; xvec ♯* Ψ; xvec ♯* P; xvec ♯* M; xvec ♯* ΨP; yvec ♯* Ψ; yvec ♯* P; yvec ♯* M; AP ♯* C; x ♯ C; xvec ♯* C; yvec ♯* C; y ≠ x; y ♯ Ψ; y ♯ P; y ♯ M; y ♯ xvec; y ♯ yvec; y ♯ N; y ♯ P'; y ♯ AP; y ♯ ΨP; y ♯ Prop C Ψ ((νx)P) (M(ν*(xvec@y#yvec))⟨([(x, y)] ∙ N)⟩) ([(x, y)] ∙ P') (x#AP) ΨP" and rScope: "∧Ψ P α P' x AP ΨP C. [Ψ ⊳ P ⟼α ≺ P'; extractFrame P = ⟨AP, ΨP⟩; distinct AP; ∧C. Prop C Ψ P α P' AP ΨP; x ♯ Ψ; x ♯ α; x ♯ AP; AP ♯* Ψ; AP ♯* P; AP ♯* α; AP ♯* P'; bn α ♯* Ψ; bn α ♯* P; bn α ♯* subject α; bn α ♯* ΨP; AP ♯* C; x ♯ C; bn α ♯* C] ==> Prop C Ψ ((νx)P) α ((νx)P') (x#AP) ΨP" and rBang: "∧Ψ P α P' AP ΨP C. [Ψ ⊳ P ∥ !P ⟼α ≺ P'; guarded P; extractFrame P = ⟨AP, ΨP⟩; distinct AP; ∧C. Prop C Ψ (P ∥ !P) α P' AP (ΨP ⊗ 1); ΨP ≃ 1; supp ΨP = ({}::name set); AP ♯* Ψ; AP ♯* P; AP ♯* α; AP ♯* P'; AP ♯* C] ==> Prop C Ψ (!P) α P' ([]) (1)" shows "Prop C Ψ P α P' AP ΨP" using Trans FrP ‹distinct AP› ‹bn α ♯* subject α› ‹distinct(bn α)› proof(nominal_induct Ψ P Rs=="α ≺ P'" AP ΨP avoiding: C α P' rule: semanticsFrameInduct) case cAlpha thus ?case using rFrameAlpha by auto next case cInput thus ?case using rInput by(auto simp add: residualInject) next case cOutput thus ?case using rOutput by(auto simp add: residualInject) next case cCase thus ?case using rCase by(auto simp add: residualInject) next case(cPar1 Ψ ΨQ P α P' AQ Q AP ΨP C α' P'') note ‹α ≺ (P' ∥ Q) = α' ≺ P''› moreover from ‹bn α ♯* α'› have "bn α ♯* (bn α')" by auto moreover note ‹distinct (bn α)› ‹distinct(bn α')› moreover from ‹bn α ♯* subject α› ‹bn α' ♯* subject α'› have "bn α ♯* (α ≺ P' ∥ Q)" and "bn α' ♯* (α' ≺ P'')" by simp+ ultimately obtain p where S: "(set p) ⊆ (set(bn α)) × (set(bn(p ∙ α)))" and "distinctPerm and αEq: "α' = p ∙ α" and P'eq: "P'' = p ∙ (P' ∥ Q)" and "(bn(p ∙ α)) ♯* α" and "(bn(p ∙ α)) ♯* (P' ∥ Q)" by(rule residualEq) note ‹Ψ ⊗ ΨQ ⊳ P ⟼α ≺ P'› ‹extractFrame Q = ⟨AQ, ΨQ⟩› ‹distinct AQ› moreover from ‹bn α ♯* subject α› ‹distinct(bn α)› ‹AP ♯* α› have "∧C. Prop C (Ψ ⊗ ΨQ) P α P' AP ΨP" by(rule_tac cPar1) auto moreover note ‹AQ ♯* P› ‹AQ ♯* Q› ‹AQ ♯* Ψ› ‹AQ ♯* α› ‹AQ ♯* P'› ‹AQ ♯* C› ‹AP ♯* P› ‹bn α ♯* Q› ‹distinct(bn α)› ‹bn α ♯* Ψ› ‹bn α ♯* ΨQ› ‹bn α ♯* P› ‹bn α ♯* subject α› ‹bn ‹extractFrame P = ⟨AP, ΨP⟩› ‹distinct AP› ‹AP ♯* AQ› ‹AP ♯* ΨQ› ‹AQ ♯* ΨP› ‹bn α ♯* ΨP› ultimately have "Prop C Ψ (P ∥ Q) α (P' ∥ Q) (AP@AQ) (ΨP ⊗ ΨQ)" by(rule_tac rPar1) with ‹bn α ♯* Ψ› ‹bn α ♯* P› ‹bn α ♯* Q› ‹bn α ♯* subject α› ‹bn α ♯* C› ‹bn α ♯* (bn α')› have "Prop C Ψ (P ∥ Q) (p ∙ α) (p ∙ (P' ∥ Q)) (AP@AQ) (ΨP ⊗ ΨQ)" by(rule_tac rAlpha) auto with αEq P'eq ‹distinctPerm p› show ?case by simp next case(cPar2 Ψ ΨP Q α Q' AP P AQ ΨQ C α' Q'') note ‹α ≺ (P ∥ Q') = α' ≺ Q''› moreover from ‹bn α ♯* α'› have "bn α ♯* (bn α')" by auto moreover note ‹distinct (bn α)› ‹distinct(bn α')› moreover from ‹bn α ♯* subject α› ‹bn α' ♯* subject α'› have "bn α ♯* (α ≺ P ∥ Q')" and "bn α' ♯* (α' ≺ Q'')" by simp+ ultimately obtain p where S: "(set p) ⊆ (set(bn α)) × (set(bn(p ∙ α)))" and "distinctPerm and αEq: "α' = p ∙ α" and Q'eq: "Q'' = p ∙ (P ∥ Q')" and "(bn(p ∙ α)) ♯* α" and "(bn(p ∙ α)) ♯* (P ∥ Q')" by(rule residualEq) note ‹Ψ ⊗ ΨP ⊳ Q ⟼α ≺ Q'› ‹extractFrame P = ⟨AP, ΨP⟩› ‹distinct AP› moreover from ‹bn α ♯* subject α› ‹distinct(bn α)› ‹AQ ♯* α› have "∧C. Prop C (Ψ ⊗ ΨP) Q α Q' AQ ΨQ" by(rule_tac cPar2) auto moreover note ‹AQ ♯* P› ‹AQ ♯* Q› ‹AQ ♯* Ψ› ‹AQ ♯* α› ‹AQ ♯* Q'› ‹AQ ♯* C› ‹AP ♯* P› ‹bn α ♯* Q› ‹distinct(bn α)› ‹bn α ♯* Ψ› ‹bn α ♯* ΨQ› ‹bn α ♯* P› ‹bn α ♯* subject α› ‹bn ‹extractFrame Q = ⟨AQ, ΨQ⟩› ‹distinct AQ› ‹AP ♯* AQ› ‹AP ♯* ΨQ› ‹AQ ♯* ΨP› ‹bn α ♯* ΨP› ultimately have "Prop C Ψ (P ∥ Q) α (P ∥ Q') (AP@AQ) (ΨP ⊗ ΨQ)" by(rule_tac rPar2) auto with ‹bn α ♯* Ψ› ‹bn α ♯* P› ‹bn α ♯* Q› ‹bn α ♯* subject α› ‹bn α ♯* C› ‹bn α ♯* (bn α')› have "Prop C Ψ (P ∥ Q) (p ∙ α) (p ∙ (P ∥ Q')) (AP@AQ) (ΨP ⊗ ΨQ)" by(rule_tac rAlpha) auto with αEq Q'eq ‹distinctPerm p› show ?case by simp next case(cComm1 Ψ ΨQ P M N P' AP ΨP Q K xvec Q' AQ C α P'') thus ?case using rComm1 apply(auto) apply(drule_tac x="M(N)" in meta_spec) apply(drule_tac x="K(ν*xvec)⟨N⟩" in meta_spec) apply(drule_tac x=P' in meta_spec) apply(drule_tac x=Q' in meta_spec) apply auto apply(drule_tac x=Ψ in meta_spec) apply(drule_tac x=ΨQ in meta_spec) apply(drule_tac x=P in meta_spec) apply(drule_tac x=M in meta_spec) apply(drule_tac x=N in meta_spec) apply(drule_tac x=P' in meta_spec) apply(drule_tac x=AP in meta_spec) apply(drule_tac x=ΨP in meta_spec) apply(drule_tac x=Q in meta_spec) apply(drule_tac x=K in meta_spec) apply(drule_tac x=xvec in meta_spec) apply(drule_tac x=Q' in meta_spec) apply(drule_tac x=AQ in meta_spec) apply auto apply(subgoal_tac "∧C. Prop C (Ψ ⊗ ΨP) Q (K(ν*xvec)⟨N⟩) Q' AQ ΨQ") apply(subgoal_tac "∧C. Prop C (Ψ ⊗ ΨQ) P (M(N)) P' AP ΨP") by(auto simp add: residualInject) next case(cComm2 Ψ ΨQ P M xvec N P' AP ΨP Q K Q' AQ C α Q'') thus ?case using rComm2 apply(drule_tac x="M(ν*xvec)⟨N⟩" in meta_spec) apply(drule_tac x="K(N)" in meta_spec) apply(drule_tac x=P' in meta_spec) apply(drule_tac x=Q' in meta_spec) apply auto apply(drule_tac x=Ψ in meta_spec) apply(drule_tac x=ΨQ in meta_spec) apply(drule_tac x=P in meta_spec) apply(drule_tac x=M in meta_spec) apply(drule_tac x=xvec in meta_spec) apply(drule_tac x=N in meta_spec) apply(drule_tac x=P' in meta_spec) apply(drule_tac x=AP in meta_spec) apply(drule_tac x=ΨP in meta_spec) apply(drule_tac x=Q in meta_spec) apply(drule_tac x=K in meta_spec) apply(drule_tac x=Q' in meta_spec) apply(drule_tac x=AQ in meta_spec) apply auto apply(subgoal_tac "∧C. Prop C (Ψ ⊗ ΨQ) P (M(ν*xvec)⟨N⟩) P' AP ΨP") apply(subgoal_tac "∧C. Prop C (Ψ ⊗ ΨP) Q (K(N)) Q' AQ ΨQ") by(auto simp add: residualInject) next case(cOpen Ψ P M xvec yvec N P' x AP ΨP C α P'') note ‹M(ν*(xvec@x#yvec))⟨N⟩ ≺ P' = α ≺ P''› moreover from ‹xvec ♯* α› ‹x ♯ α› ‹yvec ♯* α› have "(xvec@x#yvec) ♯* (bn α)" by auto moreover from ‹xvec ♯* yvec› ‹x ♯ xvec› ‹x ♯ yvec› ‹distinct xvec› ‹distinct yvec› have "distinct(xvec@x#yvec)" by(auto simp add: fresh_star_def) (simp add: fresh_def name_list_supp) moreover note ‹distinct(bn α)› moreover from ‹xvec ♯* M› ‹x ♯ M› ‹yvec ♯* M› have "(xvec@x#yvec) ♯* M" by auto hence "(xvec@x#yvec) ♯* (M(ν*(xvec@x#yvec))⟨N⟩ ≺ P')" by auto moreover from ‹bn α ♯* subject α› have "bn α ♯* (α ≺ P'')" by simp ultimately obtain p where S: "(set p) ⊆ (set(xvec@x#yvec)) × (set(p ∙ (xvec@x#yvec))) and αeq: "α = (p ∙ M)(ν*(p ∙ (xvec@x#yvec)))⟨(p ∙ N)⟩" and P'eq: "P'' = (p ∙ P')" and A: "(xvec@x#yvec) ♯* ((p ∙ M)(ν*(p ∙ (xvec@x#yvec)))⟨(p ∙ N)⟩)" and B: "(p ∙ (xvec@x#yvec)) ♯* (M(ν*(xvec@x#yvec))⟨N⟩)" and C: "(p ∙ (xvec@x#yvec)) ♯* P'" by(rule_tac residualEq) (assumption | simp)+ note ‹Ψ ⊳ P ⟼M(ν*(xvec@yvec))⟨N⟩ ≺ P'› ‹x ∈ (supp N)› moreover { fix C from ‹xvec ♯* M› ‹yvec ♯* M› have "(xvec@yvec) ♯* M" by simp moreover from ‹distinct xvec› ‹distinct yvec› ‹xvec ♯* yvec› have "distinct(xvec@yv by auto (simp add: fresh_star_def name_list_supp fresh_def) ultimately have "Prop C Ψ P (M(ν*(xvec@yvec))⟨N⟩) P' AP ΨP" using ‹AP ♯* xvec› ‹AP ♯* yvec› by(rule_tac cOpen) auto } moreover obtain y::name where "y ♯ Ψ" and "y ≠ x" and "y ♯ P" and "y ♯ xvec" and "y ♯ yvec" an by(generate_fresh "name") auto moreover note ‹x ♯ Ψ› ‹x ♯ M› ‹x ♯ xvec› ‹x ♯ yvec› ‹xvec ♯* Ψ› ‹xvec ♯* P› ‹xvec ♯* ‹yvec ♯* Ψ› ‹yvec ♯* P› ‹yvec ♯* M› ‹yvec ♯* C› ‹x ♯ C› ‹xvec ♯* C› ‹distinct xve ‹extractFrame P = ⟨AP, ΨP⟩› ‹distinct AP› ‹x ♯ AP› ‹xvec ♯* yvec› ‹xvec ♯* ΨP› ‹AP ♯* Ψ› ‹AP ♯* P› ‹AP ♯* M› ‹AP ♯* xvec› ‹AP ♯* yvec› ‹AP ♯* N› ‹AP ♯* P'› ‹AP ultimately have "Prop C Ψ ((νx)P) (M(ν*(xvec@y#yvec))⟨([(x, y)] ∙ N)⟩) ([(x, y)] ∙ P') (x#A by(rule_tac rOpen) moreover have "(([(x, y)] ∙ p) ∙ [(x, y)] ∙ M) = [(x, y)] ∙ p ∙ M" by(subst perm_compose[symmetric]) simp with ‹y ♯ M› ‹x ♯ α› αeq ‹y ♯ p› ‹x ♯ M› have D: "(([(x, y)] ∙ p) ∙ M) = p ∙ M" by(auto simp add: eqvts freshChainSimps) moreover have "(([(x, y)] ∙ p) ∙ [(x, y)] ∙ xvec) = [(x, y)] ∙ p ∙ xvec" by(subst perm_compose[symmetric]) simp with ‹y ♯ xvec› ‹x ♯ α› αeq ‹y ♯ p› ‹x ♯ xvec› have E: "(([(x, y)] ∙ p) ∙ xvec) = p ∙ by(auto simp add: eqvts freshChainSimps) moreover have "(([(x, y)] ∙ p) ∙ [(x, y)] ∙ yvec) = [(x, y)] ∙ p ∙ yvec" by(subst perm_compose[symmetric]) simp with ‹y ♯ yvec› ‹x ♯ α› αeq ‹y ♯ p› ‹x ♯ yvec› have F: "(([(x, y)] ∙ p) ∙ yvec) = p ∙ by(auto simp add: eqvts freshChainSimps) moreover have "(([(x, y)] ∙ p) ∙ [(x, y)] ∙ x) = [(x, y)] ∙ p ∙ x" by(subst perm_compose[symmetric]) simp with ‹y ≠ x› ‹y ♯ p› have G: "(([(x, y)] ∙ p) ∙ y) = p ∙ x" apply(simp add: freshChainSimps calc_atm) apply(subgoal_tac "y ≠ p ∙ x") apply(clarsimp) using A αeq apply(simp add: eqvts) apply(subst fresh_atm[symmetric]) apply(simp only: freshChainSimps) by simp moreover have "(([(x, y)] ∙ p) ∙ [(x, y)] ∙ N) = [(x, y)] ∙ p ∙ N" by(subst perm_compose[symmetric]) simp with ‹y ♯ N› ‹x ♯ α› ‹y ♯ p› αeq have H: "(([(x, y)] ∙ p) ∙ [(x, y)] ∙ N) = p ∙ N" by(auto simp add: eqvts freshChainSimps) moreover have "(([(x, y)] ∙ p) ∙ [(x, y)] ∙ P') = [(x, y)] ∙ p ∙ P'" by(subst perm_compose[symmetric]) simp with ‹y ♯ P'› ‹x ♯ P''› ‹y ♯ p› P'eq have I: "(([(x, y)] ∙ p) ∙ [(x, y)] ∙ P') = p ∙ by(auto simp add: eqvts freshChainSimps) from ‹y ♯ p› ‹y ≠ x› have "y ≠ p ∙ x" apply(subst fresh_atm[symmetric]) apply(simp only: freshChainSimps) by simp moreover from S have "([(x, y)] ∙ set p) ⊆ [(x, y)] ∙ (set(xvec@x#yvec) × set(p ∙ (x by(simp) with ‹y ≠ p ∙ x› ‹(([(x, y)] ∙ p) ∙ y) = p ∙ x› ‹x ♯ xvec› ‹y ♯ xvec› ‹x ♯ yvec› "set([(x, y)] ∙ p) ⊆ set(xvec@y#yvec) × set(([(x, y)] ∙ p) ∙ (xvec@y#yvec))" by(simp add: eqvts calc_atm perm_compose) moreover note ‹xvec ♯* Ψ› ‹yvec ♯* Ψ› ‹xvec ♯* P› ‹yvec ♯* P› ‹xvec ♯* M› ‹yvec ♯* M ‹yvec ♯* C› S ‹distinctPerm p› ‹x ♯ C› ‹xvec ♯* C› ‹xvec ♯* ΨP› ‹yvec ♯* ΨP› ‹x ♯ Ψ› ‹AP ♯* xvec› ‹x ♯ AP› ‹AP ♯* yvec› ‹AP ♯* M› ‹x ♯ xvec› ‹x ♯ yvec› ‹x ♯ M› ‹x ♯ A B C αeq ‹AP ♯* α› ‹y ♯ Ψ› ‹y ≠ x› ‹y ♯ P› ‹y ♯ M› ‹y ♯ ΨP› ‹y ♯ C› ‹xvec ♯* α› ‹x ♯ α› ‹ ultimately have "Prop C Ψ ((νx)P) (([(x, y)] ∙ p) ∙ (M(ν*(xvec@y#yvec))⟨([(x, y)] ∙ N)⟩)) ( apply(rule_tac α="M(ν*(xvec@y#yvec))⟨([(x, y)] ∙ N)⟩" in rAlpha) apply(assumption | simp)+ apply(simp add: eqvts) apply(assumption | simp add: abs_fresh)+ apply(simp add: fresh_left calc_atm) apply(assumption | simp)+ apply(simp add: fresh_left calc_atm) apply(assumption | simp)+ by(simp add: eqvts fresh_left)+ with αeq P'eq D E F G H I show ?case by(simp add: eqvts) next case(cScope Ψ P α P' x AP ΨP C α' P'') note ‹α ≺ ((νx)P') = α' ≺ P''› moreover from ‹bn α ♯* α'› have "bn α ♯* (bn α')" by auto moreover note ‹distinct (bn α)› ‹distinct(bn α')› moreover from ‹bn α ♯* subject α› ‹bn α' ♯* subject α'› have "bn α ♯* (α ≺ (νx)P')" and "bn α' ♯* (α' ≺ P'')" by simp+ ultimately obtain p where S: "(set p) ⊆ (set(bn α)) × (set(bn(p ∙ α)))" and "distinctPerm and αEq: "α' = p ∙ α" and P'eq: "P'' = p ∙ ((νx)P')" and "(bn(p ∙ α)) ♯* α" and "(bn(p ∙ α)) ♯* ((νx)P')" by(rule residualEq) note ‹Ψ ⊳ P ⟼α ≺ P'› moreover from ‹bn α ♯* subject α› ‹distinct(bn α)› have "∧C. Prop C Ψ P α P' AP ΨP" by(rule_tac cScope) auto moreover note ‹x ♯ Ψ› ‹x ♯ α› ‹bn α ♯* Ψ› ‹bn α ♯* P› ‹bn α ♯* subject α› ‹bn α ♯* ΨP› ‹x ♯ C› ‹bn α ♯* C› ‹distinct(bn α)› ‹extractFrame P = ⟨AP, ΨP⟩› ‹distinct AP› ‹x ♯ AP› ‹AP ♯* Ψ› ‹AP ♯* P› ‹AP ♯* α› ‹AP ♯* P'› ‹AP ♯* C› ultimately have "Prop C Ψ ((νx)P) α ((νx)P') (x#AP) ΨP" by(rule_tac rScope) with ‹bn α ♯* Ψ› ‹bn α ♯* P› ‹x ♯ α› ‹bn α ♯* subject α› ‹bn α ♯* C› ‹bn α ♯* (bn α')› S ‹d have "Prop C Ψ ((νx)P) (p ∙ α) (p ∙ ((νx)P')) (x#AP) ΨP" by(rule_tac rAlpha) (simp add: abs_fresh)+ with αEq P'eq ‹distinctPerm p› show ?case by simp next case(cBang Ψ P Rs AP ΨP C α) thus ?case by(rule_tac rBang) auto qed lemma inputFrameInduct[consumes 3, case_names cAlpha cInput cCase cPar1 cPar2 cScope cBang fixes Ψ :: 'b and P :: "('a, 'b, 'c) psi" and M :: 'a and N :: 'a and P' :: "('a, 'b, 'c) psi" and Prop :: "'d::fs_name → 'b → ('a, 'b, 'c) psi → 'a → 'a → ('a, 'b, 'c) psi → name list → 'b → bool" and C :: "'d::fs_name" assumes Trans: "Ψ ⊳ P ⟼M(N) ≺ P'" and FrP: "extractFrame P = ⟨AP, ΨP⟩" and "distinct AP" and rAlpha: "∧Ψ P M N P' AP ΨP p C. [AP ♯* Ψ; AP ♯* P; AP ♯* M; AP ♯* N; AP ♯* P'; AP set p ⊆ set AP × set(p ∙ AP); distinctPerm p; Prop C Ψ P M N P' AP ΨP] ==> Prop C Ψ P M N P' (p ∙ AP) (p ∙ ΨP)" and rInput: "∧Ψ M K xvec N Tvec P C. [Ψ ⊨ M ↔ K; distinct xvec; set xvec ⊆ supp N; length xvec = length Tvec; xvec ♯* Ψ; xvec ♯* M; xvec ♯* K; xvec ♯* C] ==> Prop C Ψ (M(λ*xvec N).P) K (N[xvec::=Tvec]) (P[xvec::=Tvec]) ([]) (1)" and rCase: "∧Ψ P M N P' φ Cs AP ΨP C. [Ψ ⊳ P ⟼M(N) ≺ P'; extractFrame P = ⟨AP, ΨP⟩; distinc (φ, P) mem Cs; Ψ ⊨ φ; guarded P; ΨP ≃ 1; (supp ΨP) = ({}::name set); AP ♯* Ψ; AP ♯* P; AP ♯* M; AP ♯* N; AP ♯* P'; AP ♯* C] ==> Prop C Ψ (Cases Cs) M N and rPar1: "∧Ψ ΨQ P M N P' AQ Q AP ΨP C. [Ψ ⊗ ΨQ ⊳ P ⟼M(N) ≺ P'; extractFrame P = ⟨AP, ΨP⟩; distinct AP; extractFrame Q = ⟨AQ, ΨQ⟩; distinct AQ; ∧C. Prop C (Ψ ⊗ ΨQ) P M N P' AP ΨP; AP ♯* P; AP ♯* Q; AP ♯* Ψ; AP ♯* M; AP ♯* N; AP ♯* P'; AP ♯* AQ; AP ♯* ΨQ; AQ ♯* P; AQ ♯* Q; AQ ♯* Ψ; AQ ♯* M; AQ ♯* N; AQ ♯* P'; AQ ♯* ΨP; AP ♯* C; AQ ♯* C] ==> Prop C Ψ (P ∥ Q) M N (P' ∥ Q) (AP@AQ) (ΨP ⊗ ΨQ)" and rPar2: "∧Ψ ΨP Q M N Q' AP P AQ ΨQ C. [Ψ ⊗ ΨP ⊳ Q ⟼M(N) ≺ Q'; extractFrame P = ⟨AP, ΨP⟩; distinct AP; extractFrame Q = ⟨AQ, ΨQ⟩; distinct AQ; ∧C. Prop C (Ψ ⊗ ΨP) Q M N Q' AQ ΨQ; AP ♯* P; AP ♯* Q; AP ♯* Ψ; AP ♯* M; AP ♯* N; AP ♯* Q'; AP ♯* AQ; AP ♯* ΨQ; AQ ♯* P; AQ ♯* Q; AQ ♯* Ψ; AQ ♯* M; AQ ♯* N; AQ ♯* Q'; AQ ♯* ΨP; AP ♯* C; AQ ♯* C] ==> Prop C Ψ (P ∥ Q) M N (P ∥ Q') (AP@AQ) (ΨP ⊗ ΨQ)" and rScope: "∧Ψ P M N P' x AP ΨP C. [Ψ ⊳ P ⟼M(N) ≺ P'; extractFrame P = ⟨AP, ΨP⟩; distinct AP; ∧C. Prop C Ψ P M N P' AP ΨP; x ♯ Ψ; x ♯ M; x ♯ N; x ♯ AP; AP ♯* Ψ; AP ♯* P; AP ♯* M; AP ♯* N; AP ♯* P'; AP ♯* C; x ♯ C] ==> Prop C Ψ ((νx)P) M N ((νx)P') (x#AP) ΨP" and rBang: "∧Ψ P M N P' AP ΨP C. [Ψ ⊳ P ∥ !P ⟼M(N) ≺ P'; guarded P; extractFrame P = ⟨AP, ΨP⟩; distinct AP; ∧C. Prop C Ψ (P ∥ !P) M N P' AP (ΨP ⊗ 1); ΨP ≃ 1; (supp ΨP) = ({}::name set); AP ♯* Ψ; AP ♯* P; AP ♯* M; AP ♯* N; AP ♯* P'; AP ♯* C] ==> Prop C Ψ (!P) M N P' ([ shows "Prop C Ψ P M N P' AP ΨP" using assms by(nominal_induct Ψ P Rs=="M(N) ≺ P'" AP ΨP avoiding: C arbitrary: P' rule: semanticsFrameInduc (auto simp add: residualInject) lemma outputFrameInduct[consumes 3, case_names cAlpha cOutput cCase cPar1 cPar2 cOpen cSco fixes Ψ :: 'b and P :: "('a, 'b, 'c) psi" and M :: 'a and B :: "('a, 'b, 'c) boundOutput" and AP :: "name list" and ΨP :: 'b and Prop :: "'d::fs_name → 'b → ('a, 'b, 'c) psi → 'a → ('a, 'b, 'c) boundOutput → name list → 'b → bool" and C :: "'d::fs_name" assumes Trans: "Ψ ⊳ P ⟼ROut M B" and FrP: "extractFrame P = ⟨AP, ΨP⟩" and "distinct AP" and rAlpha: "∧Ψ P M AP ΨP p B C. [AP ♯* Ψ; AP ♯* P; AP ♯* M; AP ♯* (p ∙ AP); AP ♯* B; set p ⊆ set AP × set(p ∙ AP); distinctPerm p; Prop C Ψ P M B AP ΨP] ==> Prop C Ψ P M B (p ∙ AP) (p ∙ ΨP)" and rOutput: "∧Ψ M K N P C. Ψ ⊨ M ↔ K ==> Prop C Ψ (M⟨N⟩.P) K (N ≺' P) ([]) (1)" and rCase: "∧Ψ P M B φ Cs AP ΨP C. [Ψ ⊳ P ⟼(ROut M B); extractFrame P = ⟨AP, ΨP⟩; distinc (φ, P) mem Cs; Ψ ⊨ φ; guarded P; ΨP ≃ 1; (supp ΨP) = ({}::name set); AP ♯* Ψ; AP ♯* P; AP ♯* M; AP ♯* B; AP ♯* C] ==> Prop C Ψ (Cases Cs) M B ([]) (1)" and rPar1: "∧Ψ ΨQ P M xvec N P' AQ Q AP ΨP C. [Ψ ⊗ ΨQ ⊳ P ⟼M(ν*xvec)⟨N⟩ ≺ P'; extractFrame P = ⟨AP, ΨP⟩; distinct AP; extractFrame Q = ⟨AQ, ΨQ⟩; distinct AQ; ∧C. Prop C (Ψ ⊗ ΨQ) P M ((ν*xvec)N ≺' P') AP ΨP; AP ♯* P; AP ♯* Q; AP ♯* Ψ; AP ♯* M; AP ♯* xvec; AP ♯* N; AP ♯* P'; AP ♯* AQ; AP AQ ♯* P; AQ ♯* Q; AQ ♯* Ψ; AQ ♯* M; AQ ♯* xvec; AQ ♯* N; AQ ♯* P'; AQ ♯* ΨP; xvec ♯* Ψ; xvec ♯* P; xvec ♯* Q; xvec ♯* M; xvec ♯* ΨP; xvec ♯* ΨQ; AP ♯* C; AQ ♯* C; xvec ♯* C] ==> Prop C Ψ (P ∥ Q) M ((ν*xvec)N ≺' (P' ∥ Q)) (AP@AQ) (ΨP ⊗ ΨQ)" and rPar2: "∧Ψ ΨP Q M xvec N Q' AP P AQ ΨQ C. [Ψ ⊗ ΨP ⊳ Q ⟼M(ν*xvec)⟨N⟩ ≺ Q'; extractFrame P = ⟨AP, ΨP⟩; distinct AP; extractFrame Q = ⟨AQ, ΨQ⟩; distinct AQ; ∧C. Prop C (Ψ ⊗ ΨP) Q M ((ν*xvec)N ≺' Q') AQ ΨQ; AP ♯* P; AP ♯* Q; AP ♯* Ψ; AP ♯* M; AP ♯* xvec; AP ♯* N; AP ♯* Q'; AP ♯* AQ; AP ♯ AQ ♯* P; AQ ♯* Q; AQ ♯* Ψ; AQ ♯* M; AQ ♯* xvec; AQ ♯* N; AQ ♯* Q'; AQ ♯* ΨP; xvec ♯* Ψ; xvec ♯* P; xvec ♯* Q; xvec ♯* M; xvec ♯* ΨP; xvec ♯* ΨQ; AP ♯* C; AQ ♯* C; xvec ♯* C] ==> Prop C Ψ (P ∥ Q) M ((ν*xvec)N ≺' (P ∥ Q')) (AP@AQ) (ΨP ⊗ ΨQ)" and rOpen: "∧Ψ P M xvec yvec N P' x AP ΨP C. [Ψ ⊳ P ⟼M(ν*(xvec@yvec))⟨N⟩ ≺ P'; extractFrame P = ⟨AP, ΨP⟩; distinct AP; ∧C. Prop C Ψ P M ((ν*(xvec@yvec))N ≺' P') AP ΨP; x ∈ supp N; x ♯ Ψ; x ♯ M; x ♯ AP; x ♯ xvec; x ♯ yvec; AP ♯* Ψ; AP ♯* P; AP ♯* M; AP ♯* N; AP ♯* P'; AP ♯* xvec; AP ♯* yvec; xvec ♯* Ψ; xvec ♯* P; xvec ♯* M; xvec ♯* ΨP; yvec ♯* Ψ; yvec ♯* P; yvec ♯* M; AP ♯* C; x ♯ C; xvec ♯* C; yvec ♯* C] ==> Prop C Ψ ((νx)P) M ((ν*(xvec@x#yvec))N ≺' P') (x#AP) ΨP" and rScope: "∧Ψ P M xvec N P' x AP ΨP C. [Ψ ⊳ P ⟼M(ν*xvec)⟨N⟩ ≺ P'; extractFrame P = ⟨AP, ΨP⟩; distinct AP; ∧C. Prop C Ψ P M ((ν*xvec)N ≺' P') AP ΨP; x ♯ Ψ; x ♯ M; x ♯ xvec; x ♯ N; x ♯ AP; AP ♯* Ψ; AP ♯* P; AP ♯* M; AP ♯* N; AP ♯* P'; AP ♯* xvec; xvec ♯* Ψ; xvec ♯* P; xvec ♯* M; xvec ♯* ΨP; AP ♯* C; x ♯ C; xvec ♯* C] ==> Prop C Ψ ((νx)P) M ((ν*xvec)N ≺' ((νx)P')) (x#AP) ΨP" and rBang: "∧Ψ P M B AP ΨP C. [Ψ ⊳ P ∥ !P ⟼ROut M B; guarded P; extractFrame P = ⟨AP, ΨP⟩; distinct AP; ∧C. Prop C Ψ (P ∥ !P) M B AP (ΨP ⊗ 1); ΨP ≃ 1; supp ΨP = ({}::name set); AP ♯* Ψ; AP ♯* P; AP ♯* M; AP ♯* C] ==> Prop C Ψ (!P) M B ([]) (1)" shows "Prop C Ψ P M B AP ΨP" proof - { fix B assume "Ψ ⊳ P ⟼ROut M B" hence "Prop C Ψ P M B AP ΨP" using FrP ‹distinct AP› proof(nominal_induct Ψ P Rs=="ROut M B" AP ΨP avoiding: C arbitrary: B rule: semanticsFrameIn case cAlpha thus ?case by(fastforce intro: rAlpha) next case cInput thus ?case by(simp add: residualInject) next case cOutput thus ?case by(force intro: rOutput simp add: residualInject) next case cCase thus ?case by(force intro: rCase simp add: residualInject) next case cPar1 thus ?case by(fastforce intro: rPar1 simp add: residualInject) next case cPar2 thus ?case by(fastforce intro: rPar2 simp add: residualInject) next case cComm1 thus ?case by(simp add: residualInject) next case cComm2 thus ?case by(simp add: residualInject) next case cOpen thus ?case by(fastforce intro: rOpen simp add: residualInject) next case cScope thus ?case by(force intro: rScope simp add: residualInject) next case cBang thus ?case by(force intro: rBang simp add: residualInject) qed } with Trans show ?thesis by(simp add: residualInject) qed lemma tauFrameInduct[consumes 3, case_names cAlpha cCase cPar1 cPar2 cComm1 cComm2 cScope c fixes Ψ :: 'b and P :: "('a, 'b, 'c) psi" and P' :: "('a, 'b, 'c) psi" and Prop :: "'d::fs_name → 'b → ('a, 'b, 'c) psi → ('a, 'b, 'c) psi → name list → 'b → bool" and C :: "'d::fs_name" assumes Trans: "Ψ ⊳ P ⟼τ ≺ P'" and FrP: "extractFrame P = ⟨AP, ΨP⟩" and "distinct AP" and rAlpha: "∧Ψ P P' AP ΨP p C. [AP ♯* Ψ; AP ♯* P; AP ♯* P'; AP ♯* (p ∙ AP); AP ♯* C; set p ⊆ set AP × set (p ∙ AP); distinctPerm p; Prop C Ψ P P' AP ΨP] ==> Prop C Ψ P P' (p ∙ AP) (p ∙ ΨP)" and rCase: "∧Ψ P P' φ Cs AP ΨP C. [Ψ ⊳ P ⟼τ ≺ P'; extractFrame P = ⟨AP, ΨP⟩; distinct AP; (φ, P) mem Cs; Ψ ⊨ φ; guarded P; ΨP ≃ 1; (supp ΨP) = ({}::name set); AP ♯* Ψ; AP ♯* P; AP ♯* P'; AP ♯* C] ==> Prop C Ψ (Cases Cs) P' ([]) (1)" and rPar1: "∧Ψ ΨQ P P' AQ Q AP ΨP C. [Ψ ⊗ ΨQ ⊳ P ⟼τ ≺ P'; extractFrame P = ⟨AP, ΨP⟩; distinct AP; extractFrame Q = ⟨AQ, ΨQ⟩; distinct AQ; ∧C. Prop C (Ψ ⊗ ΨQ) P P' AP ΨP; AP ♯* P; AP ♯* Q; AP ♯* Ψ; AP ♯* P'; AP ♯* AQ; AP ♯* ΨQ; AQ ♯* P; AQ ♯* Q; AQ ♯* Ψ; AQ ♯* P'; AQ ♯* ΨP; AP ♯* C; AQ ♯* C] ==> Prop C Ψ (P ∥ Q) (P' ∥ Q) (AP@AQ) (ΨP ⊗ ΨQ)" and rPar2: "∧Ψ ΨP Q Q' AP P AQ ΨQ C. [Ψ ⊗ ΨP ⊳ Q ⟼τ ≺ Q'; extractFrame P = ⟨AP, ΨP⟩; distinct AP; extractFrame Q = ⟨AQ, ΨQ⟩; distinct AQ; ∧C. Prop C (Ψ ⊗ ΨP) Q Q' AQ ΨQ; AP ♯* P; AP ♯* Q; AP ♯* Ψ; AP ♯* Q'; AP ♯* AQ; AP ♯* ΨQ; AQ ♯* P; AQ ♯* Q; AQ ♯* Ψ; AQ ♯* Q'; AQ ♯* ΨP; AP ♯* C; AQ ♯* C] ==> Prop C Ψ (P ∥ Q) (P ∥ Q') (AP@AQ) (ΨP ⊗ ΨQ)" and rComm1: "∧Ψ ΨQ P M N P' AP ΨP Q K xvec Q' AQ C. [Ψ ⊗ ΨQ ⊳ P ⟼M(N) ≺ P'; extractFrame P = ⟨AP, ΨP⟩; distinct AP; Ψ ⊗ ΨP ⊳ Q ⟼K(ν*xvec)⟨N⟩ ≺ Q'; extractFrame Q = ⟨AQ, ΨQ⟩; distinct AQ; Ψ ⊗ ΨP ⊗ ΨQ ⊨ M ↔ K; distinct xvec; AP ♯* Ψ; AP ♯* ΨQ; AP ♯* P; AP ♯* M; AP ♯* N; AP ♯* P'; AP ♯* Q; AP ♯* Q'; AP ♯* AQ; AP ♯* xvec; AQ ♯* Ψ; AQ ♯* ΨP; AQ ♯* P; AQ ♯* N; AQ ♯* P'; AQ ♯* Q; AQ ♯* K; AQ ♯* Q'; AQ ♯* xvec; xvec ♯* Ψ; xvec ♯* ΨP; xvec ♯* ΨQ; xvec ♯* P; xvec ♯* M; xvec ♯* Q; xvec ♯* K; AP ♯* C; AQ ♯* C; xvec ♯* C] ==> Prop C Ψ (P ∥ Q) ((ν*xvec)(P' ∥ Q')) (AP@AQ) (ΨP ⊗ ΨQ)" and rComm2: "∧Ψ ΨQ P M xvec N P' AP ΨP Q K Q' AQ C. [Ψ ⊗ ΨQ ⊳ P ⟼M(ν*xvec)⟨N⟩ ≺ P'; extractFrame P = ⟨AP, ΨP⟩; distinct AP; Ψ ⊗ ΨP ⊳ Q ⟼K(N) ≺ Q'; extractFrame Q = ⟨AQ, ΨQ⟩; distinct AQ; Ψ ⊗ ΨP ⊗ ΨQ ⊨ M ↔ K; distinct xvec; AP ♯* Ψ; AP ♯* ΨQ; AP ♯* P; AP ♯* M; AP ♯* N; AP ♯* P'; AP ♯* Q; AP ♯* Q'; AP ♯* AQ; AP ♯* xvec; AQ ♯* Ψ; AQ ♯* ΨP; AQ ♯* P; AQ ♯* N; AQ ♯* P'; AQ ♯* Q; AQ ♯* K; AQ ♯* Q'; AQ ♯* xvec; xvec ♯* Ψ; xvec ♯* ΨP; xvec ♯* ΨQ; xvec ♯* P; xvec ♯* M; xvec ♯* Q; xvec ♯* K; AP ♯* C; AQ ♯* C; xvec ♯* C] ==> Prop C Ψ (P ∥ Q) ((ν*xvec)(P' ∥ Q')) (AP@AQ) (ΨP ⊗ ΨQ)" and rScope: "∧Ψ P P' x AP ΨP C. [Ψ ⊳ P ⟼τ ≺ P'; extractFrame P = ⟨AP, ΨP⟩; distinct AP; ∧C. Prop C Ψ P P' AP ΨP; x ♯ Ψ; x ♯ AP; AP ♯* Ψ; AP ♯* P; AP ♯* P'; AP ♯* C; x ♯ C] ==> Prop C Ψ ((νx)P) ((νx)P') (x#AP) ΨP" and rBang: "∧Ψ P P' AP ΨP C. [Ψ ⊳ P ∥ !P ⟼τ ≺ P'; guarded P; extractFrame P = ⟨AP, ΨP⟩; distinct AP; ∧C. Prop C Ψ (P ∥ !P) P' AP (ΨP ⊗ 1); ΨP ≃ 1; supp ΨP = ({}::name set); AP ♯* Ψ; AP ♯* P; AP ♯* P'; AP ♯* C] ==> Prop C Ψ (!P) P' ([]) (1)" shows "Prop C Ψ P P' AP ΨP" using Trans FrP ‹distinct AP› proof(nominal_induct Ψ P Rs=="τ ≺ P'" AP ΨP avoiding: C arbitrary: P' rule: semanticsFrameIndu case cAlpha thus ?case by(force intro: rAlpha simp add: residualInject) next case cInput thus ?case by(simp add: residualInject) next case cOutput thus ?case by(simp add: residualInject) next case cCase thus ?case by(force intro: rCase simp add: residualInject) next case cPar1 thus ?case by(force intro: rPar1 simp add: residualInject) next case cPar2 thus ?case by(force intro: rPar2 simp add: residualInject) next case cComm1 thus ?case by(force intro: rComm1 simp add: residualInject) next case cComm2 thus ?case by(force intro: rComm2 simp add: residualInject) next case cOpen thus ?case by(simp add: residualInject) next case cScope thus ?case by(force intro: rScope simp add: residualInject) next case cBang thus ?case by(force intro: rBang simp add: residualInject) qed lemma inputFreshDerivative: fixes Ψ :: 'b and P :: "('a, 'b, 'c) psi" and M :: 'a and N :: 'a and P' :: "('a, 'b, 'c) psi" and x :: name assumes "Ψ ⊳ P ⟼M(N) ≺ P'" and "x ♯ P" and "x ♯ N" shows "x ♯ P'" proof - have "bn(M(N)) ♯* subject(M(N))" and "distinct(bn(M(N)))" by simp+ with ‹Ψ ⊳ P ⟼M(N) ≺ P'› show ?thesis using ‹x ♯ P› ‹x ♯ N› proof(nominal_induct Ψ P α=="M(N)" P' avoiding: x rule: semanticsInduct) case(cAlpha Ψ P α P' p x) thus ?case by simp next case(cInput Ψ M' K xvec N' Tvec P x) from ‹K((N'[xvec::=Tvec])) = M(N)› have "M = K" and NeqN': "N = N'[xvec::=Tvec]" by(simp ad note ‹length xvec = length Tvec› ‹distinct xvec› then moreover have "x ♯ Tvec" using ‹set xvec ⊆ supp N'› ‹x ♯ N› NeqN' by(blast intro: substTerm.subst3) moreover from ‹xvec ♯* x› ‹x ♯ M'(λ*xvec N').P› have "x ♯ P" by(simp add: inputChainFresh) (simp add: name_list_supp fresh_def) ultimately show ?case using ‹xvec ♯* x› by auto next case(cOutput Ψ M K N P x) thus ?case by simp next case(cCase Ψ P P' φ Cs x) thus ?case by(induct Cs, auto) next case(cPar1 Ψ ΨQ P P' xvec Q x) thus ?case by simp next case(cPar2 Ψ ΨP Q Q' xvec P x) thus ?case by simp next case(cComm1 Ψ ΨQ P M N P' AP ΨP Q K xvec Q' AQ x) thus ?case by simp next case(cComm2 Ψ ΨQ P M xwec N P' AP ΨP Q K Q' AQ x) thus ?case by simp next case(cOpen Ψ P M xvec yvec N P' x y) thus ?case by simp next case(cScope Ψ P P' x y) thus ?case by(simp add: abs_fresh) next case(cBang Ψ P P' x) thus ?case by simp qed qed lemma inputFreshChainDerivative: fixes Ψ :: 'b and P :: "('a, 'b, 'c) psi" and M :: 'a and N :: 'a and P' :: "('a, 'b, 'c) psi" and xvec :: "name list" assumes "Ψ ⊳ P ⟼M(N) ≺ P'" and "xvec ♯* P" and "xvec ♯* N" shows "xvec ♯* P'" using assms by(induct xvec) (auto intro: inputFreshDerivative) lemma outputFreshDerivative: fixes Ψ :: 'b and P :: "('a, 'b, 'c) psi" and M :: 'a and xvec :: "name list" and N :: 'a and P' :: "('a, 'b, 'c) psi" and x :: name assumes "Ψ ⊳ P ⟼M(ν*xvec)⟨N⟩ ≺ P'" and "xvec ♯* M" and "distinct xvec" and "x ♯ P" and "x ♯ xvec" shows "x ♯ N" and "x ♯ P'" proof - note ‹Ψ ⊳ P ⟼M(ν*xvec)⟨N⟩ ≺ P'› moreover from ‹xvec ♯* M› have "bn(M(ν*xvec)⟨N⟩) ♯* subject(M(ν*xvec)⟨N⟩)" by simp moreover from ‹distinct xvec› have "distinct(bn(M(ν*xvec)⟨N⟩))" by simp ultimately show "x ♯ N" using ‹x ♯ P› ‹x ♯ xvec› proof(nominal_induct Ψ P α=="M(ν*xvec)⟨N⟩" P' avoiding: x arbitrary: M xvec N rule: semanticsInduct case(cAlpha Ψ P α P' p x M xvec N) have S: "set p ⊆ set(bn α) × set(bn(p ∙ α))" by fact from ‹(p ∙ α) = M(ν*xvec)⟨N⟩› have "(p ∙ p ∙ α) = p ∙ (M(ν*xvec)⟨N⟩)" by(simp add: fresh_star_bij with ‹distinctPerm p› have "α = (p ∙ M)(ν*(p ∙ xvec))⟨(p ∙ N)⟩" by simp moreover from ‹(p ∙ α) = M(ν*xvec)⟨N⟩› ‹x ♯ xvec› have "x ♯ (bn(p ∙ α))" by simp with ‹(bn α) ♯* x› ‹x ♯ xvec› S have "x ♯ (p ∙ xvec)" by(drule_tac pt_fresh_bij1[OF pt_name_inst, OF at_name_inst, where pi=p and x=xvec]) simp ultimately have "x ♯ (p ∙ N)" using ‹x ♯ P› by(rule_tac cAlpha) hence "(p ∙ x) ♯ (p ∙ p ∙ N)" by(simp add: pt_fresh_bij1[OF pt_name_inst, OF at_name_in with ‹distinctPerm p› ‹bn(α) ♯* x› ‹x ♯ (bn(p ∙ α))›S show ?case by simp next case cInput thus ?case by simp next case cOutput thus ?case by(simp add: action.inject) next case cCase thus ?case by(rule_tac cCase) (auto dest: memFresh) next case cPar1 thus ?case by simp next case cPar2 thus ?case by simp next case cComm1 thus ?case by simp next case cComm2 thus ?case by simp next case(cOpen Ψ P M xvec yvec N P' x y M' zvec N') from ‹M(ν*(xvec@x#yvec))⟨N⟩ = M'(ν*zvec)⟨N'⟩› have "zvec = xvec@x#yvec" and "N = N'" by(simp add: action.inject)+ from ‹y ♯ (νx)P› ‹x ♯ y› have "y ♯ P" by(simp add: abs_fresh) moreover from ‹y ♯ zvec› ‹zvec = xvec@x#yvec›have "y ♯ (xvec@yvec)" by simp ultimately have "y ♯ N" by(rule_tac cOpen) auto with ‹N = N'› show ?case by simp next case cScope thus ?case by(auto simp add: abs_fresh) next case cBang thus ?case by simp qed next note ‹Ψ ⊳ P ⟼M(ν*xvec)⟨N⟩ ≺ P'› moreover from ‹xvec ♯* M› have "bn(M(ν*xvec)⟨N⟩) ♯* subject(M(ν*xvec)⟨N⟩)" by simp moreover from ‹distinct xvec› have "distinct(bn(M(ν*xvec)⟨N⟩))" by simp ultimately show "x ♯ P'" using ‹x ♯ P› ‹x ♯ xvec› proof(nominal_induct Ψ P α=="M(ν*xvec)⟨N⟩" P' avoiding: x arbitrary: M xvec N rule: semanticsInduct case(cAlpha Ψ P α P' p x M xvec N) have S: "set p ⊆ set(bn α) × set(bn(p ∙ α))" by fact from ‹(p ∙ α) = M(ν*xvec)⟨N⟩› have "(p ∙ p ∙ α) = p ∙ (M(ν*xvec)⟨N⟩)" by(simp add: fresh_star_bij with ‹distinctPerm p› have "α = (p ∙ M)(ν*(p ∙ xvec))⟨(p ∙ N)⟩" by simp moreover from ‹(p ∙ α) = M(ν*xvec)⟨N⟩› ‹x ♯ xvec› have "x ♯ (bn(p ∙ α))" by simp with ‹(bn α) ♯* x› ‹x ♯ xvec› S have "x ♯ (p ∙ xvec)" by(drule_tac pt_fresh_bij1[OF pt_name_inst, OF at_name_inst, where pi=p and x=xvec]) simp ultimately have "x ♯ P'" using ‹x ♯ P› by(rule_tac cAlpha) hence "(p ∙ x) ♯ (p ∙ P')" by(simp add: pt_fresh_bij1[OF pt_name_inst, OF at_name_inst] with ‹distinctPerm p› ‹bn(α) ♯* x› ‹x ♯ (bn(p ∙ α))›S show ?case by simp next case cInput thus ?case by simp next case cOutput thus ?case by(simp add: action.inject) next case cCase thus ?case by(fastforce simp add: action.inject dest: memFresh) next case cPar1 thus ?case by simp next case cPar2 thus ?case by simp next case cComm1 thus ?case by simp next case cComm2 thus ?case by simp next case(cOpen Ψ P M xvec yvec N P' x y M' zvec N') from ‹M(ν*(xvec@x#yvec))⟨N⟩ = M'(ν*zvec)⟨N'⟩› have "zvec = xvec@x#yvec" by(simp add: action.inject) from ‹y ♯ (νx)P› ‹x ♯ y› have "y ♯ P" by(simp add: abs_fresh) moreover from ‹y ♯ zvec› ‹zvec = xvec@x#yvec›have "y ♯ (xvec@yvec)" by simp ultimately show "y ♯ P'" by(rule_tac cOpen) auto next case cScope thus ?case by(auto simp add: abs_fresh) next case cBang thus ?case by simp qed qed lemma outputFreshChainDerivative: fixes Ψ :: 'b and P :: "('a, 'b, 'c) psi" and M :: 'a and xvec :: "name list" and N :: 'a and P' :: "('a, 'b, 'c) psi" and yvec :: "name list" assumes "Ψ ⊳ P ⟼M(ν*xvec)⟨N⟩ ≺ P'" and "xvec ♯* M" and "distinct xvec" and "yvec ♯* P" and "yvec ♯* xvec" shows "yvec ♯* N" and "yvec ♯* P'" using assms by(induct yvec) (auto intro: outputFreshDerivative) lemma tauFreshDerivative: fixes Ψ :: 'b and P :: "('a, 'b, 'c) psi" and P' :: "('a, 'b, 'c) psi" and x :: name assumes "Ψ ⊳ P ⟼τ ≺ P'" and "x ♯ P" shows "x ♯ P'" proof - have "bn(τ) ♯* subject(τ)" and "distinct(bn(τ))" by simp+ with ‹Ψ ⊳ P ⟼τ ≺ P'› show ?thesis using ‹x ♯ P› proof(nominal_induct Ψ P α=="(τ::('a action))" P' avoiding: x rule: semanticsInduct) case cAlpha thus ?case by simp next case cInput thus ?case by simp next case cOutput thus ?case by simp next case cCase thus ?case by(auto dest: memFresh) next case cPar1 thus ?case by simp next case cPar2 thus ?case by simp next case cComm1 thus ?case by(fastforce dest: inputFreshDerivative outputFreshDerivative simp add: resChainFresh) next case cComm2 thus ?case by(fastforce dest: inputFreshDerivative outputFreshDerivative simp add: resChainFresh) next case cOpen thus ?case by simp next case cScope thus ?case by(simp add: abs_fresh) next case cBang thus ?case by simp qed qed lemma tauFreshChainDerivative: fixes Ψ :: 'b and P :: "('a, 'b, 'c) psi" and M :: 'a and N :: 'a and P' :: "('a, 'b, 'c) psi" and xvec :: "name list" assumes "Ψ ⊳ P ⟼τ ≺ P'" and "xvec ♯* P" shows "xvec ♯* P'" using assms by(induct xvec) (auto intro: tauFreshDerivative) lemma freeFreshDerivative: fixes Ψ :: 'b and P :: "('a, 'b, 'c) psi" and α :: "'a action" and P' :: "('a, 'b, 'c) psi" and x :: name assumes "Ψ ⊳ P ⟼α ≺ P'" and "bn α ♯* subject α" and "distinct(bn α)" and "x ♯ α" and "x ♯ P" shows "x ♯ P'" using assms by(rule_tac actionCases[where α=α]) (auto intro: inputFreshDerivative tauFreshDerivative outputFreshDerivative) lemma freeFreshChainDerivative: fixes Ψ :: 'b and P :: "('a, 'b, 'c) psi" and α :: "'a action" and P' :: "('a, 'b, 'c) psi" and xvec :: "name list" assumes "Ψ ⊳ P ⟼α ≺ P'" and "bn α ♯* subject α" and "distinct(bn α)" and "xvec ♯* P" and "xvec ♯* α" shows "xvec ♯* P'" using assms by(auto intro: freeFreshDerivative simp add: fresh_star_def) lemma Input: fixes Ψ :: 'b and M :: 'a and K :: 'a and xvec :: "name list" and N :: 'a and Tvec :: "'a list" assumes "Ψ ⊨ M ↔ K" and "distinct xvec" and "set xvec ⊆ supp N" and "length xvec = length Tvec" shows "Ψ ⊳ M(λ*xvec N).P ⟼K(N[xvec::=Tvec]) ≺ P[xvec::=Tvec]" proof - obtain p where xvecFreshPsi: "((p::name prm) ∙ (xvec::name list)) ♯* Ψ" and xvecFreshM: "(p ∙ xvec) ♯* M" and xvecFreshN: "(p ∙ xvec) ♯* N" and xvecFreshK: "(p ∙ xvec) ♯* K" and xvecFreshTvec: "(p ∙ xvec) ♯* Tvec" and xvecFreshP: "(p ∙ xvec) ♯* P" and S: "(set p) ⊆ (set xvec) × (set(p ∙ xvec))" and dp: "distinctPerm p" by(rule_tac xvec=xvec and c="(Ψ, M, K, N, P, Tvec)" in name_list_avoiding) (auto simp add: eqvts fresh_star_prod) note ‹Ψ ⊨ M ↔ K› moreover from ‹distinct xvec› have "distinct(p ∙ xvec)" by simp moreover from ‹(set xvec) ⊆ (supp N)› have "(p ∙ (set xvec)) ⊆ (p ∙ (supp N))" by simp hence "set(p ∙ xvec) ⊆ supp(p ∙ N)" by(simp add: eqvts) moreover from ‹length xvec = length Tvec› have "length(p ∙ xvec) = length Tvec" by simp ultimately have "Ψ ⊳ M(λ*(p ∙ xvec) (p ∙ N)).(p ∙ P) ⟼K((p ∙ N)[(p ∙ xvec)::=Tvec]) ≺ (p using xvecFreshPsi xvecFreshM xvecFreshK xvecFreshTvec by(rule_tac cInput) thus ?thesis using xvecFreshN xvecFreshP S ‹length xvec = length Tvec› dp by(auto simp add: inputChainAlpha' substTerm.renaming renaming) qed lemma residualAlpha: fixes p :: "name prm" and α :: "'a action" and P :: "('a, 'b, 'c) psi" assumes "bn(p ∙ α) ♯* object α" and "bn(p ∙ α) ♯* P" and "bn α ♯* subject α" and "bn(p ∙ α) ♯* subject α" and "set p ⊆ set(bn α) × set(bn(p ∙ α))" shows "α ≺ P = (p ∙ α) ≺ (p ∙ P)" using assms apply(rule_tac α=α in actionCases) apply(simp only: eqvts bn.simps) apply simp apply(simp add: boundOutputChainAlpha'' residualInject) by simp lemma Par1: fixes Ψ :: 'b and ΨQ :: 'b and P :: "('a, 'b, 'c) psi" and α :: "'a action" and P' :: "('a, 'b, 'c) psi" and AQ :: "name list" and Q :: "('a, 'b, 'c) psi" assumes Trans: "Ψ ⊗ ΨQ ⊳ P ⟼α ≺ P'" and "extractFrame Q = ⟨AQ, ΨQ⟩" and "bn α ♯* Q" and "AQ ♯* Ψ" and "AQ ♯* P" and "AQ ♯* α" shows "Ψ ⊳ P ∥ Q ⟼α ≺ (P' ∥ Q)" proof - { fix Ψ :: 'b and ΨQ :: 'b and P :: "('a, 'b, 'c) psi" and α :: "'a action" and P' :: "('a, 'b, 'c) psi" and AQ :: "name list" and Q :: "('a, 'b, 'c) psi" assume "Ψ ⊗ ΨQ ⊳ P ⟼α ≺ P'" and "extractFrame Q = ⟨AQ, ΨQ⟩" and "bn α ♯* Q" and "bn α ♯* subject α" and "AQ ♯* Ψ" and "AQ ♯* P" and "AQ ♯* α" and "distinct AQ" have "Ψ ⊳ P ∥ Q ⟼α ≺ (P' ∥ Q)" proof - from ‹Ψ ⊗ ΨQ ⊳ P ⟼α ≺ P'› have "distinct(bn α)" by(rule boundOutputDistinct) obtain q::"name prm" where "bn(q ∙ α) ♯* Ψ" and "bn(q ∙ α) ♯* P" and "bn(q ∙ α) ♯* Q" and "bn and "bn(q ∙ α) ♯* AQ" and "bn(q ∙ α) ♯* P'" and "bn(q ∙ α) ♯* ΨQ" and Sq: "(set q) ⊆ (set (bn α)) × (set(bn(q ∙ α)))" by(rule_tac xvec="bn α" and c="(Ψ, P, Q, α, AQ, ΨQ, P')" in name_list_avoiding) (auto simp a obtain p::"name prm" where "(p ∙ AQ) ♯* Ψ" and "(p ∙ AQ) ♯* P" and "(p ∙ AQ) ♯* Q" and "(p and "(p ∙ AQ) ♯* α" and "(p ∙ AQ) ♯* (q ∙ α)" and "(p ∙ AQ) ♯* P'" and "(p ∙ AQ) ♯* (q ∙ P')" and "(p ∙ AQ) ♯* ΨQ" and Sp: "(set p) ⊆ (set AQ) × (set(p ∙ by(rule_tac xvec=AQ and c="(Ψ, P, Q, α, bn α, q ∙ α, P', (q ∙ P'), ΨQ)" in name_list_avoid from ‹distinct(bn α)› have "distinct(bn(q ∙ α))" by(rule_tac α=α in actionCases) (auto simp add: eqvts) from ‹AQ ♯* α› ‹bn(q ∙ α) ♯* AQ› Sq have "AQ ♯* (q ∙ α)" apply(rule_tac α=α in actionCases) apply(simp only: bn.simps eqvts, simp) apply(simp add: freshChainSimps) by simp from ‹bn α ♯* subject α› have "(q ∙ (bn α)) ♯* (q ∙ (subject α))" by(simp add: fresh_star_bij) hence "bn(q ∙ α) ♯* subject(q ∙ α)" by(simp add: eqvts) from ‹Ψ ⊗ ΨQ ⊳ P ⟼α ≺ P'› ‹bn(q ∙ α) ♯* α› ‹bn(q ∙ α) ♯* P'› ‹bn α ♯* (subject α)› Sq have Trans: "Ψ ⊗ ΨQ ⊳ P ⟼(q ∙ α) ≺ (q ∙ P')" by(force simp add: residualAlpha) hence "AQ ♯* (q ∙ P')" using ‹bn(q ∙ α) ♯* subject(q ∙ α)› ‹distinct(bn(q ∙ α))› ‹AQ ♯ by(auto intro: freeFreshChainDerivative) from Trans have "(p ∙ (Ψ ⊗ ΨQ)) ⊳ (p ∙ P) ⟼p ∙ ((q ∙ α) ≺ (q ∙ P'))" by(rule semantics.eqvt) with ‹AQ ♯* Ψ› ‹AQ ♯* P› ‹AQ ♯* (q ∙ α)› ‹(p ∙ AQ) ♯* (q ∙ α)› ‹AQ ♯* (q ∙ P')› ‹(p ∙ AQ) ♯* Ψ› ‹(p ∙ AQ) ♯* P› ‹(p ∙ AQ) ♯* (q ∙ P')› Sp have "Ψ ⊗ (p ∙ ΨQ) ⊳ P ⟼(q ∙ α) ≺ (q ∙ P')" by(simp add: eqvts) moreover from ‹extractFrame Q = ⟨AQ, ΨQ⟩› ‹(p ∙ AQ) ♯* ΨQ› Sp have "extractFrame Q = ⟨(p by(simp add: frameChainAlpha' eqvts) moreover from ‹(bn(q ∙ α)) ♯* ΨQ› ‹(bn(q ∙ α)) ♯* AQ› ‹(p ∙ AQ) ♯* (q ∙ α)› Sp have "(bn(q ∙ α)) ♯* (p ∙ ΨQ)" by(simp add: freshAlphaPerm) moreover from ‹distinct AQ› have "distinct(p ∙ AQ)" by simp ultimately have "Ψ ⊳ P ∥ Q ⟼(q ∙ α) ≺ ((q ∙ P') ∥ Q)" using ‹(p ∙ AQ) ♯* P› ‹(p ∙ AQ) ♯* Q› ‹(p ∙ AQ) ♯* Ψ› ‹(p ∙ AQ) ♯* (q ∙ α)› ‹(p ∙ AQ) ♯* (q ∙ P')› ‹(bn(q ∙ α)) ♯* Ψ› ‹(bn(q ∙ α)) ♯* Q› ‹(bn(q ∙ α)) ♯* P› ‹(bn(q ∙ α)) ♯* (subject (q ∙ α))› ‹distinct(bn(q ∙ α))› by(rule_tac cPar1) thus ?thesis using ‹bn(q ∙ α) ♯* α› ‹bn(q ∙ α) ♯* P'› ‹bn α ♯* subject α› ‹bn(q ∙ α) ♯* Q by(force simp add: residualAlpha) qed } note Goal = this from ‹extractFrame Q = ⟨AQ, ΨQ⟩› ‹AQ ♯* Ψ› ‹AQ ♯* P› ‹AQ ♯* α› obtain AQ' where FrQ: "extractFrame Q = ⟨AQ', ΨQ⟩" and "distinct AQ'" and "AQ' ♯* Ψ" and "AQ' by(rule_tac C="(Ψ, P, α)" in distinctFrame) auto show ?thesis proof(induct rule: actionCases[where α=α]) case(cInput M N) from Trans FrQ ‹AQ' ♯* Ψ› ‹AQ' ♯* P› ‹AQ' ♯* α› ‹distinct AQ'› ‹bn α ♯* Q› show ?case using ‹α = M(N)› by(force intro: Goal) next case cTau from Trans FrQ ‹AQ' ♯* Ψ› ‹AQ' ♯* P› ‹AQ' ♯* α› ‹distinct AQ'› ‹bn α ♯* Q› show ?case using ‹α = τ› by(force intro: Goal) next case(cOutput M xvec N) from ‹α = M(ν*xvec)⟨N⟩› ‹AQ' ♯* α› ‹bn α ♯* Q› have "xvec ♯* AQ'" and "xvec ♯* Q" by simp+ obtain p where "(p ∙ xvec) ♯* N" and "(p ∙ xvec) ♯* P'" and "(p ∙ xvec) ♯* Q" and "(p ∙ xvec) ♯* M" and "(p ∙ xvec) ♯* AQ'" and S: "set p ⊆ set xvec × set(p ∙ xvec)" by(rule_tac xvec=xvec and c="(N, P', Q, M, AQ')" in name_list_avoiding) auto from Trans ‹α=M(ν*xvec)⟨N⟩› have "Ψ ⊗ ΨQ ⊳ P ⟼M(ν*xvec)⟨N⟩ ≺ P'" by simp with ‹(p ∙ xvec) ♯* N› ‹(p ∙ xvec) ♯* P'› S have "Ψ ⊗ ΨQ ⊳ P ⟼M(ν*(p ∙ xvec))⟨(p ∙ N)⟩ ≺ (p ∙ P')" by(simp add: boundOutputChainAlpha'' create_residual.simps) moreover from ‹xvec ♯* AQ'› ‹(p ∙ xvec) ♯* AQ'› ‹AQ' ♯* α› S have "AQ' ♯* (p ∙ α)" by(simp add: freshChainSimps del: actionFreshChain) ultimately have "Ψ ⊳ P ∥ Q ⟼M(ν*(p ∙ xvec))⟨(p ∙ N)⟩ ≺ (p ∙ P') ∥ Q" using FrQ ‹AQ' ♯* Ψ› ‹AQ' ♯* P› ‹distinct AQ'› ‹(p ∙ xvec) ♯* Q› ‹AQ' ♯* α› ‹(p ∙ xvec) ♯* M› ‹α = M(ν*xvec)⟨N⟩› by(force intro: Goal) with ‹(p ∙ xvec) ♯* N› ‹(p ∙ xvec) ♯* P'› ‹(p ∙ xvec) ♯* Q› ‹xvec ♯* Q› S ‹α = M(ν*xv show ?case by(simp add: boundOutputChainAlpha'' eqvts create_residual.simps) qed qed lemma Par2: fixes Ψ :: 'b and ΨP :: 'b and Q :: "('a, 'b, 'c) psi" and α :: "'a action" and Q' :: "('a, 'b, 'c) psi" and AP :: "name list" and P :: "('a, 'b, 'c) psi" assumes Trans: "Ψ ⊗ ΨP ⊳ Q ⟼α ≺ Q'" and "extractFrame P = ⟨AP, ΨP⟩" and "bn α ♯* P" and "AP ♯* Ψ" and "AP ♯* Q" and "AP ♯* α" shows "Ψ ⊳ P ∥ Q ⟼α ≺ (P ∥ Q')" proof - { fix Ψ :: 'b and ΨP :: 'b and Q :: "('a, 'b, 'c) psi" and α :: "'a action" and Q' :: "('a, 'b, 'c) psi" and AP :: "name list" and P :: "('a, 'b, 'c) psi" assume "Ψ ⊗ ΨP ⊳ Q ⟼α ≺ Q'" and "extractFrame P = ⟨AP, ΨP⟩" and "bn α ♯* P" and "bn α ♯* subject α" and "AP ♯* Ψ" and "AP ♯* Q" and "AP ♯* α" and "distinct AP" have "Ψ ⊳ P ∥ Q ⟼α ≺ (P ∥ Q')" proof - from ‹Ψ ⊗ ΨP ⊳ Q ⟼α ≺ Q'› have "distinct(bn α)" by(rule boundOutputDistinct) obtain q::"name prm" where "bn(q ∙ α) ♯* Ψ" and "bn(q ∙ α) ♯* P" and "bn(q ∙ α) ♯* Q" and "bn and "bn(q ∙ α) ♯* AP" and "bn(q ∙ α) ♯* Q'" and "bn(q ∙ α) ♯* ΨP" and Sq: "(set q) ⊆ (set (bn α)) × (set(bn(q ∙ α)))" by(rule_tac xvec="bn α" and c="(Ψ, P, Q, α, AP, ΨP, Q')" in name_list_avoiding) (auto simp a obtain p::"name prm" where "(p ∙ AP) ♯* Ψ" and "(p ∙ AP) ♯* P" and "(p ∙ AP) ♯* Q" and "(p and "(p ∙ AP) ♯* α" and "(p ∙ AP) ♯* (q ∙ α)" and "(p ∙ AP) ♯* Q'" and "(p ∙ AP) ♯* (q ∙ Q')" and "(p ∙ AP) ♯* ΨP" and Sp: "(set p) ⊆ (set AP) × (set(p ∙ AP))" by(rule_tac xvec=AP and c="(Ψ, P, Q, α, q ∙ α, Q', (q ∙ Q'), ΨP)" in name_list_avoiding) a from ‹distinct(bn α)› have "distinct(bn(q ∙ α))" by(rule_tac α=α in actionCases) (auto simp add: eqvts) from ‹AP ♯* α› ‹bn(q ∙ α) ♯* AP› Sq have "AP ♯* (q ∙ α)" apply(rule_tac α=α in actionCases) apply(simp only: bn.simps eqvts, simp) apply(simp add: freshChainSimps) by simp from ‹bn α ♯* subject α› have "(q ∙ (bn α)) ♯* (q ∙ (subject α))" by(simp add: fresh_star_bij) hence "bn(q ∙ α) ♯* subject(q ∙ α)" by(simp add: eqvts) from ‹Ψ ⊗ ΨP ⊳ Q ⟼α ≺ Q'› ‹bn(q ∙ α) ♯* α› ‹bn(q ∙ α) ♯* Q'› ‹bn α ♯* (subject α)› Sq have Trans: "Ψ ⊗ ΨP ⊳ Q ⟼(q ∙ α) ≺ (q ∙ Q')" by(force simp add: residualAlpha) hence "AP ♯* (q ∙ Q')" using ‹bn(q ∙ α) ♯* subject(q ∙ α)› ‹distinct(bn(q ∙ α))› ‹AP ♯ by(auto intro: freeFreshChainDerivative) from Trans have "(p ∙ (Ψ ⊗ ΨP)) ⊳ (p ∙ Q) ⟼p ∙ ((q ∙ α) ≺ (q ∙ Q'))" by(rule semantics.eqvt) with ‹AP ♯* Ψ› ‹AP ♯* Q› ‹AP ♯* (q ∙ α)› ‹(p ∙ AP) ♯* (q ∙ α)› ‹AP ♯* (q ∙ Q')› ‹(p ∙ AP) ♯* Ψ› ‹(p ∙ AP) ♯* Q› ‹(p ∙ AP) ♯* (q ∙ Q')› Sp have "Ψ ⊗ (p ∙ ΨP) ⊳ Q ⟼(q ∙ α) ≺ (q ∙ Q')" by(simp add: eqvts) moreover from ‹extractFrame P = ⟨AP, ΨP⟩› ‹(p ∙ AP) ♯* ΨP› Sp have "extractFrame P = ⟨(p by(simp add: frameChainAlpha' eqvts) moreover from ‹(bn(q ∙ α)) ♯* ΨP› ‹(bn(q ∙ α)) ♯* AP› ‹(p ∙ AP) ♯* (q ∙ α)› Sp have "(bn(q ∙ α)) ♯* (p ∙ ΨP)" by(simp add: freshAlphaPerm) moreover from ‹distinct AP› have "distinct(p ∙ AP)" by simp ultimately have "Ψ ⊳ P ∥ Q ⟼(q ∙ α) ≺ (P ∥ (q ∙ Q'))" using ‹(p ∙ AP) ♯* P› ‹(p ∙ AP) ♯* Q› ‹(p ∙ AP) ♯* Ψ› ‹(p ∙ AP) ♯* (q ∙ α)› ‹(p ∙ AP) ♯* (q ∙ Q')› ‹(bn(q ∙ α)) ♯* Ψ› ‹(bn(q ∙ α)) ♯* Q› ‹(bn(q ∙ α)) ♯* P› ‹(bn(q ∙ α)) ♯* (subject (q ∙ α))› ‹distinct(bn(q ∙ α))› by(rule_tac cPar2) thus ?thesis using ‹bn(q ∙ α) ♯* α› ‹bn(q ∙ α) ♯* Q'› ‹bn α ♯* subject α› ‹bn(q ∙ α) ♯* P by(force simp add: residualAlpha) qed } note Goal = this from ‹extractFrame P = ⟨AP, ΨP⟩› ‹AP ♯* Ψ› ‹AP ♯* Q› ‹AP ♯* α› obtain AP' where FrP: "extractFrame P = ⟨AP', ΨP⟩" and "distinct AP'" and "AP' ♯* Ψ" and "AP' by(rule_tac C="(Ψ, Q, α)" in distinctFrame) auto show ?thesis proof(induct rule: actionCases[where α=α]) case(cInput M N) from Trans FrP ‹AP' ♯* Ψ› ‹AP' ♯* Q› ‹AP' ♯* α› ‹distinct AP'› ‹bn α ♯* P› show ?case using ‹α = M(N)› by(force intro: Goal) next case cTau from Trans FrP ‹AP' ♯* Ψ› ‹AP' ♯* Q› ‹AP' ♯* α› ‹distinct AP'› ‹bn α ♯* P› show ?case using ‹α = τ› by(force intro: Goal) next case(cOutput M xvec N) from ‹α = M(ν*xvec)⟨N⟩› ‹AP' ♯* α› ‹bn α ♯* P› have "xvec ♯* AP'" and "xvec ♯* P" by simp+ obtain p where "(p ∙ xvec) ♯* N" and "(p ∙ xvec) ♯* Q'" and "(p ∙ xvec) ♯* P" and "(p ∙ xvec) ♯* M" and "(p ∙ xvec) ♯* AP'" and S: "set p ⊆ set xvec × set(p ∙ xvec)" by(rule_tac xvec=xvec and c="(N, Q', P, M, AP')" in name_list_avoiding) auto from Trans ‹α=M(ν*xvec)⟨N⟩› have "Ψ ⊗ ΨP ⊳ Q ⟼M(ν*xvec)⟨N⟩ ≺ Q'" by simp with ‹(p ∙ xvec) ♯* N› ‹(p ∙ xvec) ♯* Q'› S have "Ψ ⊗ ΨP ⊳ Q ⟼M(ν*(p ∙ xvec))⟨(p ∙ N)⟩ ≺ (p ∙ Q')" by(simp add: boundOutputChainAlpha'' create_residual.simps) moreover from ‹xvec ♯* AP'› ‹(p ∙ xvec) ♯* AP'› ‹AP' ♯* α› S have "AP' ♯* (p ∙ α)" by(simp add: freshChainSimps del: actionFreshChain) ultimately have "Ψ ⊳ P ∥ Q ⟼M(ν*(p ∙ xvec))⟨(p ∙ N)⟩ ≺ P ∥ (p ∙ Q')" using FrP ‹AP' ♯* Ψ› ‹AP' ♯* Q› ‹distinct AP'› ‹(p ∙ xvec) ♯* P› ‹AP' ♯* α› ‹(p ∙ xvec) ♯* M› ‹α = M(ν*xvec)⟨N⟩› by(force intro: Goal) with ‹(p ∙ xvec) ♯* N› ‹(p ∙ xvec) ♯* Q'› ‹(p ∙ xvec) ♯* P› ‹xvec ♯* P› S ‹α = M(ν*xv show ?case by(simp add: boundOutputChainAlpha'' eqvts create_residual.simps) qed qed lemma Open: fixes Ψ :: 'b and P :: "('a, 'b, 'c) psi" and M :: 'a and xvec :: "name list" and yvec :: "name list" and N :: 'a and P' :: "('a, 'b, 'c) psi" and x :: name assumes Trans: "Ψ ⊳ P ⟼M(ν*(xvec@yvec))⟨N⟩ ≺ P'" and "x ∈ supp N" and "x ♯ Ψ" and "x ♯ M" and "x ♯ xvec" and "x ♯ yvec" shows "Ψ ⊳ (νx)P ⟼M(ν*(xvec@x#yvec))⟨N⟩ ≺ P'" proof - from Trans have "distinct(xvec@yvec)" by(force dest: boundOutputDistinct) hence "xvec ♯* yvec" by(induct xvec) auto obtain p where "(p ∙ yvec) ♯* Ψ" and "(p ∙ yvec) ♯* P" and "(p ∙ yvec) ♯* M" and "(p ∙ yvec) ♯* yvec" and "(p ∙ yvec) ♯* N" and "(p ∙ yvec) ♯* P'" and "x ♯ (p ∙ yvec)" and "(p ∙ yvec) ♯* xvec" and Sp: "(set p) ⊆ (set yvec) × (set(p ∙ yvec))" by(rule_tac xvec=yvec and c="(Ψ, P, M, xvec, yvec, N, P', x)" in name_list_avoiding) (auto simp add: eqvts fresh_star_prod) obtain q where "(q ∙ xvec) ♯* Ψ" and "(q ∙ xvec) ♯* P" and "(q ∙ xvec) ♯* M" and "(q ∙ xvec) ♯* xvec" and "(q ∙ xvec) ♯* N" and "(q ∙ xvec) ♯* P'" and "x ♯ (q ∙ xvec)" and "(q ∙ xvec) ♯* yvec" and "(q ∙ xvec) ♯* p" and "(q ∙ xvec) ♯* (p ∙ yvec)" and Sq: "(set q) ⊆ (set xvec) × (set(q ∙ xvec))" by(rule_tac xvec=xvec and c="(Ψ, P, M, xvec, yvec, p ∙ yvec, N, P', x, p)" in name_lis (auto simp add: eqvts fresh_star_prod) note ‹Ψ ⊳ P ⟼M(ν*(xvec@yvec))⟨N⟩ ≺ P'› moreover from ‹(p ∙ yvec) ♯* N› ‹(q ∙ xvec) ♯* N› ‹xvec ♯* yvec› ‹(q ∙ xvec) ♯* yv have "((p@q) ∙ (xvec @ yvec)) ♯* N" apply(simp only: eqvts) apply(simp only: pt2[OF pt_n by simp moreover from ‹(p ∙ yvec) ♯* P'› ‹(q ∙ xvec) ♯* P'› ‹xvec ♯* yvec› ‹(q ∙ xvec) ♯* have "((p@q) ∙ (xvec @ yvec)) ♯* P'" by(simp del: freshAlphaPerm add: eqvts pt2[OF pt_na moreover from Sp Sq ‹xvec ♯* yvec› ‹(q ∙ xvec) ♯* yvec› ‹(q ∙ xvec) ♯* (p ∙ yvec)› ‹ have Spq: "set(p@q) ⊆ set(xvec@yvec) × set((p@q) ∙ (xvec@yvec))" by(simp add: pt2[OF pt_name_inst] eqvts) blast ultimately have "Ψ ⊳ P ⟼M(ν*((p@q) ∙ (xvec@yvec)))⟨((p@q) ∙ N)⟩ ≺ ((p@q) ∙ P')" apply(simp add: create_residual.simps) by(erule_tac rev_mp) (subst boundOutputChainAlpha, auto) with Sp Sq ‹xvec ♯* yvec› ‹(q ∙ xvec) ♯* yvec› ‹(q ∙ xvec) ♯* (p ∙ yvec)› ‹(p ∙ yve have "Ψ ⊳ P ⟼M(ν*((q ∙ xvec)@(p ∙ yvec)))⟨((p@q) ∙ N)⟩ ≺ ((p@q) ∙ P')" by(simp add: eqvts pt2[OF pt_name_inst] del: freshAlphaPerm) moreover from ‹x ∈ supp N› have "((p@q) ∙ x) ∈ (p@q) ∙ (supp N)" by(simp add: pt_set_bij[OF pt_name_inst, OF at_name_inst]) with ‹x ♯ xvec› ‹x ♯ yvec› ‹x ♯ (q ∙ xvec)› ‹x ♯ (p ∙ yvec)› Sp Sq have "x ∈ supp((p@q)∙ N)" by(simp add: eqvts pt2[OF pt_name_inst]) moreover from ‹distinct(xvec@yvec)› have "distinct(q ∙ xvec)" and "distinct(p ∙ yvec) by auto moreover note ‹x ♯ (q ∙ xvec)› ‹x ♯ (p ∙ yvec)› ‹x ♯ M› ‹x ♯ Ψ› ‹(q ∙ xvec) ♯* Ψ› ‹(q ∙ xvec) ♯* P› ‹(q ∙ xvec) ♯* M› ‹(q ∙ xvec) ♯* (p ∙ yvec)› ‹(p ∙ yvec) ♯* Ψ› ‹(p ∙ yvec) ♯* P› ‹(p ∙ yvec) ♯* M› ‹distinct(q ∙ xvec)› ultimately have "Ψ ⊳ (νx)P ⟼M(ν*((q ∙ xvec)@x#(p ∙ yvec)))⟨((p@q) ∙ N)⟩ ≺ ((p@q) ∙ P')" by(rule_tac cOpen) with ‹x ♯ xvec› ‹x ♯ yvec› ‹x ♯ (q ∙ xvec)› ‹x ♯ (p ∙ yvec)› ‹xvec ♯* yvec› ‹(q ∙ xvec) ♯* yvec› ‹(q ∙ xvec) ♯* (p ∙ yvec)› ‹(p ∙ yvec) ♯* xv have "Ψ ⊳ (νx)P ⟼M(ν*((p@q) ∙ (xvec@x#yvec)))⟨((p@q) ∙ N)⟩ ≺ ((p@q) ∙ P')" by(simp add: eqvts pt2[OF pt_name_inst] del: freshAlphaPerm) thus ?thesis using ‹((p@q) ∙ (xvec @ yvec)) ♯* N› ‹((p@q) ∙ (xvec @ yvec)) ♯* P'› S apply(simp add: create_residual.simps) by(erule_tac rev_mp) (subst boundOutputChainAlpha, auto) qed lemma Scope: fixes Ψ :: 'b and P :: "('a, 'b, 'c) psi" and α :: "'a action" and P' :: "('a, 'b, 'c) psi" and x :: name assumes "Ψ ⊳ P ⟼α ≺ P'" and "x ♯ Ψ" and "x ♯ α" shows "Ψ ⊳ (νx)P ⟼α ≺ (νx)P'" proof - { fix Ψ P M xvec N P' x assume "Ψ ⊳ P ⟼M(ν*xvec)⟨N⟩ ≺ P'" and "(x::name) ♯ Ψ" and "x ♯ M" and "x ♯ xvec" and "x ♯ N" obtain p::"name prm" where "(p ∙ xvec) ♯* Ψ" and "(p ∙ xvec) ♯* P" and "(p ∙ xvec) ♯* M" and "(p ∙ xvec) ♯* N" and "(p ∙ xvec) ♯* P'" and "x ♯ (p ∙ xvec)" and S: "(set p) ⊆ (set xvec) × (set(p ∙ xvec))" by(rule_tac xvec=xvec and c="(Ψ, P, M, xvec, N, P', x)" in name_list_avoiding) (auto simp add: eqvts fresh_star_prod) from ‹Ψ ⊳ P ⟼M(ν*xvec)⟨N⟩ ≺ P'› ‹(p ∙ xvec) ♯* N› ‹(p ∙ xvec) ♯* P'› S have "Ψ ⊳ P ⟼M(ν*(p ∙ xvec))⟨(p ∙ N)⟩ ≺ (p ∙ P')" by(simp add: boundOutputChainAlpha'' create_residual.simps) moreover hence "distinct(p ∙ xvec)" by(force dest: boundOutputDistinct) moreover note ‹x ♯ Ψ› ‹x ♯ M› ‹x ♯ (p ∙ xvec)› moreover from ‹x ♯ xvec› ‹x ♯ p ∙ xvec› ‹x ♯ N› S have "x ♯ (p ∙ N)" by(simp add: fresh_left del: freshAlphaSwap) ultimately have "Ψ ⊳ (νx)P ⟼M(ν*(p ∙ xvec))⟨(p ∙ N)⟩ ≺ (νx)(p ∙ P')" using ‹(p ∙ xvec) ♯* Ψ› ‹(p by(rule_tac cScope) auto moreover from ‹x ♯ xvec› ‹x ♯ p ∙ xvec› S have "p ∙ x = x" by simp ultimately have "Ψ ⊳ (νx)P ⟼M(ν*(p ∙ xvec))⟨(p ∙ N)⟩ ≺ (p ∙ ((νx)P'))" by simp moreover from ‹(p ∙ xvec) ♯* P'› ‹x ♯ xvec› ‹x ♯ (p ∙ xvec)› have "(p ∙ xvec) ♯* (νx)P by(simp add: abs_fresh_star) ultimately have"Ψ ⊳ (νx)P ⟼M(ν*xvec)⟨N⟩ ≺ (νx)P'" using ‹(p ∙ xvec) ♯* N› S by(simp add: boundOutputChainAlpha'' create_residual.simps) } note Goal = this show ?thesis proof(induct rule: actionCases[where α=α]) case(cInput M N) with assms show ?case by(force intro: cScope) next case(cOutput M xvec N) with assms show ?case by(force intro: Goal) next case cTau with assms show ?case by(force intro: cScope) qed qed lemma inputSwapFrameSubject: fixes Ψ :: 'b and P :: "('a, 'b, 'c) psi" and M :: 'a and N :: 'a and P' :: "('a, 'b, 'c) psi" and x :: name and y :: name assumes "Ψ ⊳ P ⟼M(N) ≺ P'" and "x ♯ P" and "y ♯ P" shows "([(x, y)] ∙ Ψ) ⊳ P ⟼ ([(x, y)] ∙ M)(N) ≺ P'" using assms proof(nominal_induct avoiding: x y rule: inputInduct) case(cInput Ψ M K xvec N Tvec P x y) from ‹x ♯ M(λ*xvec N).P› have "x ♯ M" by simp from ‹y ♯ M(λ*xvec N).P› have "y ♯ M" by simp from ‹Ψ ⊨ M ↔ K› have "([(x, y)] ∙ Ψ) ⊨ ([(x, y)] ∙ M) ↔ ([(x, y)] ∙ K)" by(rule chanEqClosed) with ‹x ♯ M› ‹y ♯ M› have "([(x, y)] ∙ Ψ) ⊨ M ↔ ([(x, y)] ∙ K)" by(simp) thus ?case using ‹distinct xvec› ‹set xvec ⊆ supp N› ‹length xvec = length Tvec› by(rule Input) next case(cCase Ψ P M N P' φ Cs x y) from ‹x ♯ Cases Cs› ‹y ♯ Cases Cs› ‹(φ, P) mem Cs› have "x ♯ φ" and "x ♯ P" and "y ♯ φ" an by(auto dest: memFresh) from ‹x ♯ P› ‹y ♯ P› have "([(x ,y)] ∙ Ψ) ⊳ P ⟼ ([(x, y)] ∙ M)(N) ≺ P'" by(rule cCase) moreover note ‹(φ, P) mem Cs› moreover from ‹Ψ ⊨ φ› have "([(x, y)] ∙ Ψ) ⊨ ([(x, y)] ∙ φ)" by(rule statClosed) with ‹x ♯ φ› ‹y ♯ φ› have "([(x, y)] ∙ Ψ) ⊨ φ" by simp ultimately show ?case using ‹guarded P› by(rule Case) next case(cPar1 Ψ ΨQ P M N P' AQ Q x y) from ‹x ♯ P ∥ Q› have "x ♯ P" and "x ♯ Q" by simp+ from ‹y ♯ P ∥ Q› have "y ♯ P" and "y ♯ Q" by simp+ from ‹x ♯ P› ‹y ♯ P› ‹∧x y. [x ♯ P; y ♯ P] ==> ([(x, y)] ∙ (Ψ ⊗ ΨQ)) ⊳ P ⟼([(x, y)] have "([(x, y)] ∙ Ψ) ⊗ ([(x, y)] ∙ ΨQ) ⊳ P ⟼([(x, y)] ∙ M)(N) ≺ P'" by(simp add: eqvts) moreover from ‹extractFrame Q = ⟨AQ, ΨQ⟩› have "([(x, y)] ∙ (extractFrame Q)) = ([(x, by simp with ‹AQ ♯* x› ‹x ♯ Q› ‹AQ ♯* y› ‹y ♯ Q› have "⟨AQ, ([(x, y)] ∙ ΨQ)⟩ = extractFrame Q by(simp add: eqvts) moreover from ‹AQ ♯* Ψ› have "([(x, y)] ∙ AQ) ♯* ([(x, y)] ∙ Ψ)" by(simp add: pt_fresh_star_bij[OF pt_name_inst, OF at_name_inst]) with ‹AQ ♯* x› ‹AQ ♯* y› have "AQ ♯* ([(x, y)] ∙ Ψ)" by simp moreover from ‹AQ ♯* M› have "([(x, y)] ∙ AQ) ♯* ([(x, y)] ∙ M)" by(simp add: pt_fresh_star_bij[OF pt_name_inst, OF at_name_inst]) with ‹AQ ♯* x› ‹AQ ♯* y› have "AQ ♯* ([(x, y)] ∙ M)" by simp ultimately show ?case using ‹AQ ♯* P› ‹AQ ♯* N› by(rule_tac Par1) auto next case(cPar2 Ψ ΨP Q M N Q' AP P x y) from ‹x ♯ P ∥ Q› have "x ♯ P" and "x ♯ Q" by simp+ from ‹y ♯ P ∥ Q› have "y ♯ P" and "y ♯ Q" by simp+ from ‹x ♯ Q› ‹y ♯ Q› ‹∧x y. [x ♯ Q; y ♯ Q] ==> ([(x, y)] ∙ (Ψ ⊗ ΨP)) ⊳ Q ⟼([(x, y)] have "([(x, y)] ∙ Ψ) ⊗ ([(x, y)] ∙ ΨP) ⊳ Q ⟼([(x, y)] ∙ M)(N) ≺ Q'" by(simp add: eqvts) moreover from ‹extractFrame P = ⟨AP, ΨP⟩› have "([(x, y)] ∙ (extractFrame P)) = ([(x, by simp with ‹AP ♯* x› ‹x ♯ P› ‹AP ♯* y› ‹y ♯ P› have "⟨AP, ([(x, y)] ∙ ΨP)⟩ = extractFrame P by(simp add: eqvts) moreover from ‹AP ♯* Ψ› have "([(x, y)] ∙ AP) ♯* ([(x, y)] ∙ Ψ)" by(simp add: pt_fresh_star_bij[OF pt_name_inst, OF at_name_inst]) with ‹AP ♯* x› ‹AP ♯* y› have "AP ♯* ([(x, y)] ∙ Ψ)" by simp moreover from ‹AP ♯* M› have "([(x, y)] ∙ AP) ♯* ([(x, y)] ∙ M)" by(simp add: pt_fresh_star_bij[OF pt_name_inst, OF at_name_inst]) with ‹AP ♯* x› ‹AP ♯* y› have "AP ♯* ([(x, y)] ∙ M)" by simp ultimately show ?case using ‹AP ♯* Q› ‹AP ♯* N› by(rule_tac Par2) auto next case(cScope Ψ P M N P' z x y) from ‹x ♯ (νz)P› ‹z ♯ x› have "x ♯ P" by(simp add: abs_fresh) from ‹y ♯ (νz)P› ‹z ♯ y› have "y ♯ P" by(simp add: abs_fresh) from ‹x ♯ P› ‹y ♯ P› ‹∧x y. [x ♯ P; y ♯ P] ==> ([(x, y)] ∙ Ψ) ⊳ P ⟼([(x, y)] ∙ M)(N) have "([(x, y)] ∙ Ψ) ⊳ P ⟼([(x, y)] ∙ M)(N) ≺ P'" by simp moreover with ‹z ♯ Ψ› have "([(x, y)] ∙ z) ♯ [(x, y)] ∙ Ψ" by(simp add: pt_fresh_bij[OF pt_name_inst, OF at_name_inst]) with ‹z ♯ x› ‹z ♯ y› have "z ♯ [(x, y)] ∙ Ψ" by simp moreover with ‹z ♯ M› have "([(x, y)] ∙ z) ♯ [(x, y)] ∙ M" by(simp add: pt_fresh_bij[OF pt_name_inst, OF at_name_inst]) with ‹z ♯ x› ‹z ♯ y› have "z ♯ [(x, y)] ∙ M" by simp ultimately show ?case using ‹z ♯ N› by(rule_tac Scope) (assumption | simp)+ next case(cBang Ψ P M N P' x y) thus ?case by(force intro: Bang) qed lemma inputPermFrameSubject: fixes Ψ :: 'b and P :: "('a, 'b, 'c) psi" and M :: 'a and N :: 'a and P' :: "('a, 'b, 'c) psi" and p :: "name prm" and Xs :: "name set" and Ys :: "name set" assumes "Ψ ⊳ P ⟼M(N) ≺ P'" and S: "set p ⊆ Xs × Ys" and "Xs ♯* P" and "Ys ♯* P" shows "(p ∙ Ψ) ⊳ P ⟼ (p ∙ M)(N) ≺ P'" using S proof(induct p) case Nil from ‹Ψ ⊳ P ⟼M(N) ≺ P'› show ?case by simp next case(Cons a p) from ‹set(a#p) ⊆ Xs × Ys› have "set p ⊆ Xs × Ys" by auto with ‹set p ⊆ Xs × Ys ==> (p ∙ Ψ) ⊳ P ⟼ (p ∙ M)(N) ≺ P'› have Trans: "(p ∙ Ψ) ⊳ P ⟼ (p ∙ M)(N) ≺ P'" by simp from ‹set(a#p) ⊆ Xs × Ys› show ?case proof(cases a, clarsimp) fix a b assume "a ∈ Xs" and "b ∈ Ys" with ‹Xs ♯* P› ‹Ys ♯* P› have "a ♯ P" and "b ♯ P" by(auto simp add: fresh_star_def) with Trans show "([(a, b)] ∙ p ∙ Ψ) ⊳ P ⟼ ([(a, b)] ∙ p ∙ M)(N) ≺ P'" by(rule inputSwapFrameSubject) qed qed lemma inputSwapSubject: fixes Ψ :: 'b and P :: "('a, 'b, 'c) psi" and M :: 'a and N :: 'a and P' :: "('a, 'b, 'c) psi" and x :: name and y :: name assumes "Ψ ⊳ P ⟼M(N) ≺ P'" and "x ♯ P" and "y ♯ P" and "x ♯ Ψ" and "y ♯ Ψ" shows "Ψ ⊳ P ⟼ ([(x, y)] ∙ M)(N) ≺ P'" proof - from ‹Ψ ⊳ P ⟼M(N) ≺ P'› ‹x ♯ P› ‹y ♯ P› have "([(x, y)] ∙ Ψ) ⊳ P ⟼([(x, y)] ∙ M)(N) ≺ P'" by(rule inputSwapFrameSubject) with ‹x ♯ Ψ› ‹y ♯ Ψ› show ?thesis by simp qed lemma inputPermSubject: fixes Ψ :: 'b and P :: "('a, 'b, 'c) psi" and M :: 'a and N :: 'a and P' :: "('a, 'b, 'c) psi" and p :: "name prm" and Xs :: "name set" and Ys :: "name set" assumes "Ψ ⊳ P ⟼M(N) ≺ P'" and S: "set p ⊆ Xs × Ys" and "Xs ♯* P" and "Ys ♯* P" and "Xs ♯* Ψ" and "Ys ♯* Ψ" shows "Ψ ⊳ P ⟼ (p ∙ M)(N) ≺ P'" proof - from ‹Ψ ⊳ P ⟼M(N) ≺ P'› S ‹Xs ♯* P› ‹Ys ♯* P› have "(p ∙ Ψ) ⊳ P ⟼(p ∙ M)(N) ≺ P'" by(rule inputPermFrameSubject) with ‹Xs ♯* Ψ› ‹Ys ♯* Ψ› S show ?thesis by simp qed lemma inputSwapFrame: fixes Ψ :: 'b and P :: "('a, 'b, 'c) psi" and M :: 'a and N :: 'a and P' :: "('a, 'b, 'c) psi" and x :: name and y :: name assumes "Ψ ⊳ P ⟼M(N) ≺ P'" and "x ♯ P" and "y ♯ P" and "x ♯ M" and "y ♯ M" shows "([(x, y)] ∙ Ψ) ⊳ P ⟼ M(N) ≺ P'" proof - from ‹Ψ ⊳ P ⟼M(N) ≺ P'› ‹x ♯ P› ‹y ♯ P› have "([(x, y)] ∙ Ψ) ⊳ P ⟼([(x, y)] ∙ M)(N) ≺ P'" by(rule inputSwapFrameSubject) with ‹x ♯ M› ‹y ♯ M› show ?thesis by simp qed lemma inputPermFrame: fixes Ψ :: 'b and P :: "('a, 'b, 'c) psi" and M :: 'a and N :: 'a and P' :: "('a, 'b, 'c) psi" and p :: "name prm" and Xs :: "name set" and Ys :: "name set" assumes "Ψ ⊳ P ⟼M(N) ≺ P'" and S: "set p ⊆ Xs × Ys" and "Xs ♯* P" and "Ys ♯* P" and "Xs ♯* M" and "Ys ♯* M" shows "(p ∙ Ψ) ⊳ P ⟼ M(N) ≺ P'" proof - from ‹Ψ ⊳ P ⟼M(N) ≺ P'› S ‹Xs ♯* P› ‹Ys ♯* P› have "(p ∙ Ψ) ⊳ P ⟼(p ∙ M)(N) ≺ P'" by(rule inputPermFrameSubject) with ‹Xs ♯* M› ‹Ys ♯* M› S show ?thesis by simp qed lemma inputAlpha: fixes Ψ :: 'b and P :: "('a, 'b, 'c) psi" and M :: 'a and N :: 'a and P' :: "('a, 'b, 'c) psi" and p :: "name prm" and xvec :: "name list" assumes "Ψ ⊳ P ⟼M(N) ≺ P'" and "set p ⊆ (set xvec) × (set (p ∙ xvec))" and "distinctPerm p" and "xvec ♯* P" and "(p ∙ xvec) ♯* P" shows "Ψ ⊳ P ⟼M((p ∙ N)) ≺ (p ∙ P')" proof - from ‹Ψ ⊳ P ⟼M(N) ≺ P'› ‹set p ⊆ (set xvec) × (set (p ∙ xvec))› ‹xvec ♯* P› ‹(p ∙ xv have "(p ∙ Ψ) ⊳ P ⟼(p ∙ M)(N) ≺ P'" by(rule_tac inputPermFrameSubject) auto hence "(p ∙ p ∙ Ψ) ⊳ (p ∙ P) ⟼(p ∙ ((p ∙ M)(N) ≺ P'))" by(rule eqvts) with ‹distinctPerm p› ‹xvec ♯* P› ‹(p ∙ xvec) ♯* P› ‹set p ⊆ (set xvec) × (set (p show ?thesis by(simp add: eqvts) qed lemma frameFresh[dest]: fixes x :: name and AF :: "name list" and ΨF :: 'b assumes "x ♯ AF" and "x ♯ ⟨AF, ΨF⟩" shows "x ♯ ΨF" using assms by(simp add: frameResChainFresh) (simp add: fresh_def name_list_supp) lemma outputSwapFrameSubject: fixes Ψ :: 'b and P :: "('a, 'b, 'c) psi" and M :: 'a and xvec :: "name list" and N :: 'a and x :: name and y :: name assumes "Ψ ⊳ P ⟼M(ν*xvec)⟨N⟩ ≺ P'" and "xvec ♯* M" and "x ♯ P" and "y ♯ P" shows "([(x, y)] ∙ Ψ) ⊳ P ⟼([(x, y)] ∙ M)(ν*xvec)⟨N⟩ ≺ P'" using assms proof(nominal_induct avoiding: x y rule: outputInduct') case cAlpha thus ?case by(simp add: create_residual.simps boundOutputChainAlpha'') next case(cOutput Ψ M K N P x y) from ‹x ♯ M⟨N⟩.P› have "x ♯ M" by simp from ‹y ♯ M⟨N⟩.P› have "y ♯ M" by simp from ‹Ψ ⊨ M ↔ K› have "([(x, y)] ∙ Ψ) ⊨ ([(x, y)] ∙ M) ↔ ([(x, y)] ∙ K)" by(rule chanEqClosed) with ‹x ♯ M› ‹y ♯ M› have "([(x, y)] ∙ Ψ) ⊨ M ↔ ([(x, y)] ∙ K)" by(simp) thus ?case by(rule Output) next case(cCase Ψ P M xvec N P' φ Cs x y) from ‹x ♯ Cases Cs› ‹y ♯ Cases Cs› ‹(φ, P) mem Cs› have "x ♯ φ" and "x ♯ P" and "y ♯ φ" an by(auto dest: memFresh) from ‹x ♯ P› ‹y ♯ P› have "([(x ,y)] ∙ Ψ) ⊳ P ⟼([(x, y)] ∙ M)(ν*xvec)⟨N⟩ ≺ P'" by(rule cCa moreover note ‹(φ, P) mem Cs› moreover from ‹Ψ ⊨ φ› have "([(x, y)] ∙ Ψ) ⊨ ([(x, y)] ∙ φ)" by(rule statClosed) with ‹x ♯ φ› ‹y ♯ φ› have "([(x, y)] ∙ Ψ) ⊨ φ" by simp ultimately show ?case using ‹guarded P› by(rule Case) next case(cPar1 Ψ ΨQ P M xvec N P' AQ Q x y) from ‹x ♯ P ∥ Q› have "x ♯ P" and "x ♯ Q" by simp+ from ‹y ♯ P ∥ Q› have "y ♯ P" and "y ♯ Q" by simp+ from ‹x ♯ P› ‹y ♯ P› ‹∧x y. [x ♯ P; y ♯ P] ==> ([(x, y)] ∙ (Ψ ⊗ ΨQ)) ⊳ P ⟼([(x, y)] have "([(x, y)] ∙ Ψ) ⊗ ([(x, y)] ∙ ΨQ) ⊳ P ⟼([(x, y)] ∙ M)(ν*xvec)⟨N⟩ ≺ P'" by(simp add: eqvts) moreover from ‹extractFrame Q = ⟨AQ, ΨQ⟩› have "([(x, y)] ∙ ⟨AQ, ΨQ⟩) = ([(x, y)] ∙ (extr by simp with ‹AQ ♯* x› ‹x ♯ Q› ‹AQ ♯* y› ‹y ♯ Q› have "⟨AQ, ([(x, y)] ∙ ΨQ)⟩ = extractFrame Q by(simp add: eqvts) moreover from ‹AQ ♯* Ψ› have "([(x, y)] ∙ AQ) ♯* ([(x, y)] ∙ Ψ)" by(simp add: pt_fresh_star_bij[OF pt_name_inst, OF at_name_inst]) with ‹AQ ♯* x› ‹AQ ♯* y› have "AQ ♯* ([(x, y)] ∙ Ψ)" by simp moreover from ‹AQ ♯* M› have "([(x, y)] ∙ AQ) ♯* ([(x, y)] ∙ M)" by(simp add: pt_fresh_star_bij[OF pt_name_inst, OF at_name_inst]) with ‹AQ ♯* x› ‹AQ ♯* y› have "AQ ♯* ([(x, y)] ∙ M)" by simp ultimately show ?case using ‹AQ ♯* P› ‹AQ ♯* N› ‹xvec ♯* Q› ‹AQ ♯* xvec› by(rule_tac Par1) auto next case(cPar2 Ψ ΨP Q M xvec N Q' AP P x y) from ‹x ♯ P ∥ Q› have "x ♯ P" and "x ♯ Q" by simp+ from ‹y ♯ P ∥ Q› have "y ♯ P" and "y ♯ Q" by simp+ from ‹x ♯ Q› ‹y ♯ Q› ‹∧x y. [x ♯ Q; y ♯ Q] ==> ([(x, y)] ∙ (Ψ ⊗ ΨP)) ⊳ Q ⟼([(x, y)] have "([(x, y)] ∙ Ψ) ⊗ ([(x, y)] ∙ ΨP) ⊳ Q ⟼([(x, y)] ∙ M)(ν*xvec)⟨N⟩ ≺ Q'" by(simp add: eqvts) moreover from ‹extractFrame P = ⟨AP, ΨP⟩› have "([(x, y)] ∙ ⟨AP, ΨP⟩) = ([(x, y)] ∙ (extr by simp with ‹AP ♯* x› ‹x ♯ P› ‹AP ♯* y› ‹y ♯ P› have "⟨AP, ([(x, y)] ∙ ΨP)⟩ = extractFrame P by(simp add: eqvts) moreover from ‹AP ♯* Ψ› have "([(x, y)] ∙ AP) ♯* ([(x, y)] ∙ Ψ)" by(simp add: pt_fresh_star_bij[OF pt_name_inst, OF at_name_inst]) with ‹AP ♯* x› ‹AP ♯* y› have "AP ♯* ([(x, y)] ∙ Ψ)" by simp moreover from ‹AP ♯* M› have "([(x, y)] ∙ AP) ♯* ([(x, y)] ∙ M)" by(simp add: pt_fresh_star_bij[OF pt_name_inst, OF at_name_inst]) with ‹AP ♯* x› ‹AP ♯* y› have "AP ♯* ([(x, y)] ∙ M)" by simp ultimately show ?case using ‹AP ♯* Q› ‹AP ♯* N› ‹xvec ♯* P› ‹AP ♯* xvec› by(rule_tac Par2) auto next case(cOpen Ψ P M xvec yvec N P' z x y) from ‹x ♯ (νz)P› ‹z ♯ x› have "x ♯ P" by(simp add: abs_fresh) from ‹y ♯ (νz)P› ‹z ♯ y› have "y ♯ P" by(simp add: abs_fresh) from ‹x ♯ P› ‹y ♯ P› ‹∧x y. [x ♯ P; y ♯ P] ==> ([(x, y)] ∙ Ψ) ⊳ P ⟼([(x, y)] ∙ M)(ν* have "([(x, y)] ∙ Ψ) ⊳ P ⟼([(x, y)] ∙ M)(ν*(xvec@yvec))⟨N⟩ ≺ P'" by simp moreover with ‹z ♯ Ψ› have "([(x, y)] ∙ z) ♯ [(x, y)] ∙ Ψ" by(simp add: pt_fresh_bij[OF pt_name_inst, OF at_name_inst]) with ‹z ♯ x› ‹z ♯ y› have "z ♯ [(x, y)] ∙ Ψ" by simp moreover with ‹z ♯ M› have "([(x, y)] ∙ z) ♯ [(x, y)] ∙ M" by(simp add: pt_fresh_bij[OF pt_name_inst, OF at_name_inst]) with ‹z ♯ x› ‹z ♯ y› have "z ♯ [(x, y)] ∙ M" by simp ultimately show ?case using ‹z ∈ supp N› ‹z ♯ xvec› ‹z ♯ yvec› by(rule_tac Open) (assumption | simp)+ next case(cScope Ψ P M xvec N P' z x y) from ‹x ♯ (νz)P› ‹z ♯ x› have "x ♯ P" by(simp add: abs_fresh) from ‹y ♯ (νz)P› ‹z ♯ y› have "y ♯ P" by(simp add: abs_fresh) from ‹x ♯ P› ‹y ♯ P› ‹∧x y. [x ♯ P; y ♯ P] ==> ([(x, y)] ∙ Ψ) ⊳ P ⟼([(x, y)] ∙ M)(ν* have "([(x, y)] ∙ Ψ) ⊳ P ⟼([(x, y)] ∙ M)(ν*xvec)⟨N⟩ ≺ P'" by simp moreover with ‹z ♯ Ψ› have "([(x, y)] ∙ z) ♯ [(x, y)] ∙ Ψ" by(simp add: pt_fresh_bij[OF pt_name_inst, OF at_name_inst]) with ‹z ♯ x› ‹z ♯ y› have "z ♯ [(x, y)] ∙ Ψ" by simp moreover with ‹z ♯ M› have "([(x, y)] ∙ z) ♯ [(x, y)] ∙ M" by(simp add: pt_fresh_bij[OF pt_name_inst, OF at_name_inst]) with ‹z ♯ x› ‹z ♯ y› have "z ♯ [(x, y)] ∙ M" by simp ultimately show ?case using ‹z ♯ N› ‹z ♯ xvec› by(rule_tac Scope) (assumption | simp)+ next case(cBang Ψ P M B x y) thus ?case by(force intro: Bang) qed lemma outputPermFrameSubject: fixes Ψ :: 'b and P :: "('a, 'b, 'c) psi" and M :: 'a and xvec :: "name list" and N :: 'a and P' :: "('a, 'b, 'c) psi" and p :: "name prm" and yvec :: "name list" and zvec :: "name list" assumes "Ψ ⊳ P ⟼M(ν*xvec)⟨N⟩ ≺ P'" and S: "set p ⊆ set yvec × set zvec" and "yvec ♯* P" and "zvec ♯* P" shows "(p ∙ Ψ) ⊳ P ⟼(p ∙ M)(ν*xvec)⟨N⟩ ≺ P'" proof - { fix xvec N P' Xs YS assume "Ψ ⊳ P ⟼M(ν*xvec)⟨N⟩ ≺ P'" and "xvec ♯* M" and "xvec ♯* yvec" and "xvec ♯* zvec" have "(p ∙ Ψ) ⊳ P ⟼(p ∙ M)(ν*xvec)⟨N⟩ ≺ P'" using S proof(induct p) case Nil from ‹Ψ ⊳ P ⟼M(ν*xvec)⟨N⟩ ≺ P'› show ?case by simp next case(Cons a p) from ‹set(a#p) ⊆ set yvec × set zvec› have "set p ⊆ set yvec × set zvec" by auto then have Trans: "(p ∙ Ψ) ⊳ P ⟼(p ∙ M)(ν*xvec)⟨N⟩ ≺ P'" by(rule Cons) from ‹set(a#p) ⊆ set yvec × set zvec› show ?case proof(cases a, clarsimp) fix x y note Trans moreover from ‹xvec ♯* yvec› ‹xvec ♯* zvec› ‹set p ⊆ set yvec × set zvec› ‹xvec ♯* by(simp add: freshChainSimps) moreover assume "x ∈ set yvec" and "y ∈ set zvec" with ‹yvec ♯* P› ‹zvec ♯* P› have "x ♯ P" and "y ♯ P" by(auto simp add: fresh_star_def) ultimately show "([(x, y)] ∙ p ∙ Ψ) ⊳ P ⟼([(x, y)] ∙ p ∙ M)(ν*xvec)⟨N⟩ ≺ P'" by(rule outputSwapFrameSubject) qed qed } note Goal = this obtain q::"name prm" where "(q ∙ xvec) ♯* yvec" and "(q ∙ xvec) ♯* zvec" and "(q ∙ xvec and "(q ∙ xvec) ♯* N" and "(q ∙ xvec) ♯* P'" and "(q ∙ xvec) ♯* M" and Sq: "(set q) ⊆ (set xvec) × (set(q ∙ xvec))" by(rule_tac xvec=xvec and c="(P, xvec, yvec, zvec, N, M, P')" in name_list_avoiding) a with ‹Ψ ⊳ P ⟼M(ν*xvec)⟨N⟩ ≺ P'› have "Ψ ⊳ P ⟼M(ν*(q ∙ xvec))⟨(q ∙ N)⟩ ≺ (q ∙ P')" by(simp add: boundOutputChainAlpha'' residualInject) hence "(p ∙ Ψ) ⊳ P ⟼(p ∙ M)(ν*(q ∙ xvec))⟨(q ∙ N)⟩ ≺ (q ∙ P')" using ‹(q ∙ xvec) ♯* M› ‹(q ∙ xvec) ♯* yvec› ‹(q ∙ xvec) ♯* zvec› by(rule Goal) with ‹(q ∙ xvec) ♯* N› ‹(q ∙ xvec) ♯* P'› Sq show ?thesis by(simp add: boundOutputChainAlpha'' residualInject) qed lemma outputSwapSubject: fixes Ψ :: 'b and P :: "('a, 'b, 'c) psi" and M :: 'a and B :: "('a, 'b, 'c) boundOutput" and x :: name and y :: name assumes "Ψ ⊳ P ⟼M(ν*xvec)⟨N⟩ ≺ P'" and "xvec ♯* M" and "x ♯ P" and "y ♯ P" and "x ♯ Ψ" and "y ♯ Ψ" shows "Ψ ⊳ P ⟼([(x, y)] ∙ M)(ν*xvec)⟨N⟩ ≺ P'" proof - from ‹Ψ ⊳ P ⟼M(ν*xvec)⟨N⟩ ≺ P'› ‹xvec ♯* M› ‹x ♯ P› ‹y ♯ P› have "([(x, y)] ∙ Ψ) ⊳ P ⟼([(x, y)] ∙ M)(ν*xvec)⟨N⟩ ≺ P'" by(rule outputSwapFrameSubject) with ‹x ♯ Ψ› ‹y ♯ Ψ› show ?thesis by simp qed lemma outputPermSubject: fixes Ψ :: 'b and P :: "('a, 'b, 'c) psi" and M :: 'a and B :: "('a, 'b, 'c) boundOutput" and p :: "name prm" and yvec :: "name list" and zvec :: "name list" assumes "Ψ ⊳ P ⟼M(ν*xvec)⟨N⟩ ≺ P'" and S: "set p ⊆ set yvec × set zvec" and "yvec ♯* P" and "zvec ♯* P" and "yvec ♯* Ψ" and "zvec ♯* Ψ" shows "Ψ ⊳ P ⟼(p ∙ M)(ν*xvec)⟨N⟩ ≺ P'" proof - from assms have "(p ∙ Ψ) ⊳ P ⟼(p ∙ M)(ν*xvec)⟨N⟩ ≺ P'" by(rule_tac outputPermFrameSubject) with S ‹yvec ♯* Ψ› ‹zvec ♯* Ψ› show ?thesis by simp qed lemma outputSwapFrame: fixes Ψ :: 'b and P :: "('a, 'b, 'c) psi" and M :: 'a and B :: "('a, 'b, 'c) boundOutput" and x :: name and y :: name assumes "Ψ ⊳ P ⟼M(ν*xvec)⟨N⟩ ≺ P'" and "xvec ♯* M" and "x ♯ P" and "y ♯ P" and "x ♯ M" and "y ♯ M" shows "([(x, y)] ∙ Ψ) ⊳ P ⟼M(ν*xvec)⟨N⟩ ≺ P'" proof - from ‹Ψ ⊳ P ⟼M(ν*xvec)⟨N⟩ ≺ P'› ‹xvec ♯* M› ‹x ♯ P› ‹y ♯ P› have "([(x, y)] ∙ Ψ) ⊳ P ⟼([(x, y)] ∙ M)(ν*xvec)⟨N⟩ ≺ P'" by(rule outputSwapFrameSubject) with ‹x ♯ M› ‹y ♯ M› show ?thesis by simp qed lemma outputPermFrame: fixes Ψ :: 'b and P :: "('a, 'b, 'c) psi" and M :: 'a and B :: "('a, 'b, 'c) boundOutput" and p :: "name prm" and yvec :: "name list" and zvec :: "name list" assumes "Ψ ⊳ P ⟼M(ν*xvec)⟨N⟩ ≺ P'" and S: "set p ⊆ set yvec × set zvec" and "yvec ♯* P" and "zvec ♯* P" and "yvec ♯* M" and "zvec ♯* M" shows "(p ∙ Ψ) ⊳ P ⟼M(ν*xvec)⟨N⟩ ≺ P'" proof - from assms have "(p ∙ Ψ) ⊳ P ⟼(p ∙ M)(ν*xvec)⟨N⟩ ≺ P'" by(rule_tac outputPermFrameSubject) with S ‹yvec ♯* M› ‹zvec ♯* M› show ?thesis by simp qed lemma Comm1: fixes Ψ :: 'b and ΨQ :: 'b and P :: "('a, 'b, 'c) psi" and M :: 'a and N :: 'a and P' :: "('a, 'b, 'c) psi" and AP :: "name list" and ΨP :: 'b and Q :: "('a, 'b, 'c) psi" and K :: 'a and xvec :: "name list" and Q' :: "('a, 'b, 'c) psi" and AQ :: "name list" assumes "Ψ ⊗ ΨQ ⊳ P ⟼M(N) ≺ P'" and "extractFrame P = ⟨AP, ΨP⟩" and "Ψ ⊗ ΨP ⊳ Q ⟼K(ν*xvec)⟨N⟩ ≺ Q'" and "extractFrame Q = ⟨AQ, ΨQ⟩" and "Ψ ⊗ ΨP ⊗ ΨQ ⊨ M ↔ K" and "AP ♯* Ψ" and "AP ♯* P" and "AP ♯* Q" and "AP ♯* M" and "AP ♯* AQ" and "AQ ♯* Ψ" and "AQ ♯* P" and "AQ ♯* Q" and "AQ ♯* K" and "xvec ♯* P" shows "Ψ ⊳ P ∥ Q ⟼τ ≺ (ν*xvec)(P' ∥ Q')" proof - { fix Ψ :: 'b and ΨQ :: 'b and P :: "('a, 'b, 'c) psi" and M :: 'a and N :: 'a and P' :: "('a, 'b, 'c) psi" and AP :: "name list" and ΨP :: 'b and Q :: "('a, 'b, 'c) psi" and K :: 'a and xvec :: "name list" and Q' :: "('a, 'b, 'c) psi" and AQ :: "name list" assume "Ψ ⊗ ΨQ ⊳ P ⟼M(N) ≺ P'" and "extractFrame P = ⟨AP, ΨP⟩" and "distinct AP" and "Ψ ⊗ ΨP ⊳ Q ⟼K(ν*xvec)⟨N⟩ ≺ Q'" and "extractFrame Q = ⟨AQ, ΨQ⟩" and "distinct AQ" and "Ψ ⊗ ΨP ⊗ ΨQ ⊨ M ↔ K" and "AP ♯* Ψ" and "AP ♯* P" and "AP ♯* Q" and "AP ♯* M" and "AP ♯* AQ" and "AQ ♯* Ψ" and "AQ ♯* P" and "AQ ♯* Q" and "AQ ♯* K" and "xvec ♯* P" have "Ψ ⊳ P ∥ Q ⟼τ ≺ (ν*xvec)(P' ∥ Q')" proof - obtain r::"name prm" where "(r ∙ xvec) ♯* Ψ" and "(r ∙ xvec) ♯* P" and "(r ∙ xvec) ♯* Q" and "(r ∙ xvec) ♯* K" and "(r ∙ xvec) ♯* N" and "(r ∙ xvec) ♯* AP" and "(r ∙ xvec) ♯* A and "(r ∙ xvec) ♯* P'" and "(r ∙ xvec) ♯* Q'" and "(r ∙ xvec) ♯* ΨP" and "(r ∙ xvec) ♯* Ψ and Sr: "(set r) ⊆ (set xvec) × (set(r ∙ xvec))" and "distinctPerm r" by(rule_tac xvec=xvec and c="(Ψ, P, Q, M, K, N, AP, AQ, ΨP, ΨQ, P', Q')" in name_list_av (auto simp add: eqvts fresh_star_prod) obtain q::"name prm" where "(q ∙ AQ) ♯* Ψ" and "(q ∙ AQ) ♯* P" and "(q ∙ AQ) ♯* Q" and "(q and "(q ∙ AQ) ♯* (r ∙ N)" and "(q ∙ AQ) ♯* (r ∙ xvec)" and "(q ∙ AQ) ♯* (r ∙ Q')" and "(q ∙ AQ) ♯* (r ∙ P')" and "(q ∙ AQ) ♯* ΨP" and "(q ∙ AQ) ♯* AP" and "(q ∙ AQ) ♯* ΨQ" and Sq: "set q ⊆ set AQ × set(q ∙ AQ)" by(rule_tac xvec=AQ and c="(Ψ, P, Q, K, r ∙ N, r ∙ xvec, ΨQ, AP, ΨP, r ∙ Q', r ∙ P')" i (auto simp add: eqvts fresh_star_prod) obtain p::"name prm" where "(p ∙ AP) ♯* Ψ" and "(p ∙ AP) ♯* P" and "(p ∙ AP) ♯* Q" and "(p and "(p ∙ AP) ♯* (r ∙ N)" and "(p ∙ AP) ♯* (r ∙ xvec)" and "(p ∙ AP) ♯* (r ∙ Q')" and "(p ∙ AP) ♯* (r ∙ P')" and "(p ∙ AP) ♯* ΨP" and "(p ∙ AP) ♯* ΨQ" and "(p ∙ AP) ♯* AQ" and "(p ∙ AP) ♯* (q ∙ AQ)" and Sp: "(set p) ⊆ (set AP) × (set(p ∙ AP))" by(rule_tac xvec=AP and c="(Ψ, P, Q, M, r ∙ N, r ∙ xvec, AQ, q ∙ AQ, ΨQ, ΨP, r ∙ Q', r (auto simp add: eqvts fresh_star_prod) have FrP: "extractFrame P = ⟨AP, ΨP⟩" by fact have FrQ: "extractFrame Q = ⟨AQ, ΨQ⟩" by fact from ‹AP ♯* Q› FrQ ‹AP ♯* AQ› have "AP ♯* ΨQ" by(drule_tac extractFrameFreshChain) auto from ‹AQ ♯* P› FrP ‹AP ♯* AQ› have "AQ ♯* ΨP" by(drule_tac extractFrameFreshChain) auto from ‹(r ∙ xvec) ♯* AP› ‹(p ∙ AP) ♯* (r ∙ xvec)› ‹(r ∙ xvec) ♯* AP› Sp have "(r ∙ x by(simp add: freshChainSimps) from ‹Ψ ⊗ ΨQ ⊳ P ⟼M(N) ≺ P'› Sr ‹distinctPerm r› ‹xvec ♯* P› ‹(r ∙ xvec) ♯* P› have "Ψ ⊗ ΨQ ⊳ P ⟼M((r ∙ N)) ≺ (r ∙ P')" by(rule inputAlpha) hence "(q ∙ (Ψ ⊗ ΨQ)) ⊳ P ⟼(q ∙ M)((r ∙ N)) ≺ (r ∙ P')" using Sq ‹AQ ♯* P› ‹(q ∙ AQ) ♯* by(rule_tac inputPermFrameSubject) (assumption | simp)+ hence PTrans: "Ψ ⊗ (q ∙ ΨQ) ⊳ P ⟼(q ∙ M)((r ∙ N)) ≺ (r ∙ P')" using Sq ‹AQ ♯* Ψ› ‹(q ∙ AQ) by(simp add: eqvts) moreover from ‹extractFrame P = ⟨AP, ΨP⟩› Sp ‹(p ∙ AP) ♯* ΨP› have FrP: "extractFrame P = ⟨(p ∙ AP), (p ∙ ΨP)⟩" by(simp add: frameChainAlpha) moreover from ‹distinct AP› have "distinct(p ∙ AP)" by simp moreover from ‹Ψ ⊗ ΨP ⊳ Q ⟼K(ν*xvec)⟨N⟩ ≺ Q'› Sr ‹(r ∙ xvec) ♯* N› ‹(r ∙ xvec) ♯* Q'› have "Ψ ⊗ ΨP ⊳ Q ⟼K(ν*(r ∙ xvec))⟨(r ∙ N)⟩ ≺ (r ∙ Q')" by(simp add: boundOutputChainAlpha'' create_residual.simps) hence "(p ∙ (Ψ ⊗ ΨP)) ⊳ Q ⟼(p ∙ K)(ν*(r ∙ xvec))⟨(r ∙ N)⟩ ≺ (r ∙ Q')" using Sp ‹AP ♯* Q› ‹( by(rule_tac outputPermFrameSubject) (assumption | auto) hence QTrans: "Ψ ⊗ (p ∙ ΨP) ⊳ Q ⟼(p ∙ K)(ν*(r ∙ xvec))⟨(r ∙ N)⟩ ≺ (r ∙ Q')" using Sp ‹AP ♯* Ψ by(simp add: eqvts) moreover hence "distinct(r ∙ xvec)" by(force dest: boundOutputDistinct) moreover from ‹extractFrame Q = ⟨AQ, ΨQ⟩› Sq ‹(q ∙ AQ) ♯* ΨQ› have FrQ: "extractFrame Q = ⟨(q ∙ AQ), (q ∙ ΨQ)⟩" by(simp add: frameChainAlpha) moreover from ‹distinct AQ› have "distinct(q ∙ AQ)" by simp moreover from ‹Ψ ⊗ ΨP ⊗ ΨQ ⊨ M ↔ K› have "(p ∙ q ∙ (Ψ ⊗ ΨP ⊗ ΨQ)) ⊨ (p ∙ q ∙ M) ↔ (p ∙ q ∙ by(rule_tac chanEqClosed)+ with ‹AP ♯* Ψ› ‹(p ∙ AP) ♯* Ψ› ‹AQ ♯* Ψ› ‹(q ∙ AQ) ♯* Ψ› ‹AQ ♯* ΨP› ‹(q ∙ AQ) ♯* ΨP› ‹AP ♯* ΨQ› ‹(p ∙ AP) ♯* ΨQ› ‹AP ♯* M› ‹(p ∙ AP) ♯* M› ‹(q ∙ AQ) ♯* AP› ‹(p ∙ AP) ♯ ‹AQ ♯* K› ‹(q ∙ AQ) ♯* K› ‹AP ♯* AQ› ‹(p ∙ AP) ♯* AQ› Sp Sq have "Ψ ⊗ (p ∙ ΨP) ⊗ (q ∙ ΨQ) ⊨ (q ∙ M) ↔ (p ∙ K)" by(simp add: eqvts freshChainSimps) moreover note ‹(p ∙ AP) ♯* Ψ› moreover from ‹(p ∙ AP) ♯* AQ› ‹(p ∙ AP) ♯* (q ∙ AQ)› ‹(p ∙ AP) ♯* ΨQ› Sq have "(p ∙ by(simp add: freshChainSimps) moreover note ‹(p ∙ AP) ♯* P› moreover from ‹(p ∙ AP) ♯* AQ› ‹(p ∙ AP) ♯* (q ∙ AQ)› ‹(p ∙ AP) ♯* M› Sq have "(p ∙ by(simp add: freshChainSimps) moreover note ‹(p ∙ AP) ♯* (r ∙ N)› ‹(p ∙ AP) ♯* (r ∙ P')› ‹(p ∙ AP) ♯* Q› ‹(p ∙ A ‹(p ∙ AP) ♯* (r ∙ xvec)› ‹(q ∙ AQ) ♯* Ψ› moreover from ‹(q ∙ AQ) ♯* AP› ‹(p ∙ AP) ♯* AQ› ‹(q ∙ AQ) ♯* ΨP› ‹(p ∙ AP) ♯* (q ∙ by(simp add: freshChainSimps) moreover note ‹(q ∙ AQ) ♯* P› ‹(q ∙ AQ) ♯* (r ∙ N)›\‹(q ∙ AQ) ♯* (r ∙ P')› ‹(q ∙ A moreover from ‹(q ∙ AQ) ♯* AP› ‹(p ∙ AP) ♯* AQ› ‹(q ∙ AQ) ♯* K› ‹(p ∙ AP) ♯* (q ∙ by(simp add: freshChainSimps) moreover note ‹(q ∙ AQ) ♯* (r ∙ Q')› ‹(q ∙ AQ) ♯* (r ∙ xvec)› ‹(r ∙ xvec) ♯* Ψ› moreover from ‹(r ∙ xvec) ♯* AP› ‹(p ∙ AP) ♯* (r ∙ xvec)› ‹(r ∙ xvec) ♯* ΨP› Sp have by(simp add: freshChainSimps) moreover from ‹(r ∙ xvec) ♯* AQ› ‹(q ∙ AQ) ♯* (r ∙ xvec)› ‹(r ∙ xvec) ♯* ΨQ› Sq have by(simp add: freshChainSimps) moreover note ‹(r ∙ xvec) ♯* P› moreover from ‹(r ∙ xvec) ♯* AQ› ‹(q ∙ AQ) ♯* (r ∙ xvec)› ‹(r ∙ xvec) ♯* M› Sq have by(simp add: freshChainSimps) moreover note ‹(r ∙ xvec) ♯* Q› moreover from ‹(r ∙ xvec) ♯* AP› ‹(p ∙ AP) ♯* (r ∙ xvec)› ‹(r ∙ xvec) ♯* K› Sp have by(simp add: freshChainSimps) ultimately have "Ψ ⊳ P ∥ Q ⟼τ ≺ (ν*(r ∙ xvec))((r ∙ P') ∥ (r ∙ Q'))" by(rule_tac cComm1) with ‹(r ∙ xvec) ♯* P'› ‹(r ∙ xvec) ♯* Q'› Sr show ?thesis by(subst resChainAlpha) auto qed } note Goal = this note ‹Ψ ⊗ ΨQ ⊳ P ⟼M(N) ≺ P'› ‹Ψ ⊗ ΨP ⊳ Q ⟼K(ν*xvec)⟨N⟩ ≺ Q'› ‹Ψ ⊗ ΨP ⊗ ΨQ ⊨ M ↔ K› moreover from ‹extractFrame P = ⟨AP, ΨP⟩› ‹AP ♯* Ψ› ‹AP ♯* P› ‹AP ♯* Q› ‹AP ♯* M› ‹AP obtain AP' where "extractFrame P = ⟨AP', ΨP⟩" and "distinct AP'" and "AP' ♯* Ψ" and "AP' ♯* P by(rule_tac C="(Ψ, P, Q, M, AQ)" in distinctFrame) auto moreover from ‹extractFrame Q = ⟨AQ, ΨQ⟩› ‹AQ ♯* Ψ› ‹AQ ♯* P› ‹AQ ♯* Q› ‹AQ ♯* K› ‹AP' obtain AQ' where "extractFrame Q = ⟨AQ', ΨQ⟩" and "distinct AQ'" and "AQ' ♯* Ψ" and "AQ' ♯* P apply(rule_tac C="(Ψ, P, Q, K, AP')" in distinctFrame) by auto ultimately show ?thesis using ‹xvec ♯* P› by(rule_tac Goal) qed lemma Comm2: fixes Ψ :: 'b and ΨQ :: 'b and P :: "('a, 'b, 'c) psi" and M :: 'a and xvec :: "name list" and N :: 'a and P' :: "('a, 'b, 'c) psi" and AP :: "name list" and ΨP :: 'b and Q :: "('a, 'b, 'c) psi" and K :: 'a and Q' :: "('a, 'b, 'c) psi" and AQ :: "name list" assumes "Ψ ⊗ ΨQ ⊳ P ⟼M(ν*xvec)⟨N⟩ ≺ P'" and "extractFrame P = ⟨AP, ΨP⟩" and "Ψ ⊗ ΨP ⊳ Q ⟼K(N) ≺ Q'" and "extractFrame Q = ⟨AQ, ΨQ⟩" and "Ψ ⊗ ΨP ⊗ ΨQ ⊨ M ↔ K" and "AP ♯* Ψ" and "AP ♯* P" and "AP ♯* Q" and "AP ♯* M" and "AP ♯* AQ" and "AQ ♯* Ψ" and "AQ ♯* P" and "AQ ♯* Q" and "AQ ♯* K" and "xvec ♯* Q" shows "Ψ ⊳ P ∥ Q ⟼τ ≺ (ν*xvec)(P' ∥ Q')" proof - { fix Ψ :: 'b and ΨQ :: 'b and P :: "('a, 'b, 'c) psi" and M :: 'a and xvec :: "name list" and N :: 'a and P' :: "('a, 'b, 'c) psi" and AP :: "name list" and ΨP :: 'b and Q :: "('a, 'b, 'c) psi" and K :: 'a and Q' :: "('a, 'b, 'c) psi" and AQ :: "name list" assume "Ψ ⊗ ΨQ ⊳ P ⟼M(ν*xvec)⟨N⟩ ≺ P'" and "extractFrame P = ⟨AP, ΨP⟩" and "distinct AP" and "Ψ ⊗ ΨP ⊳ Q ⟼K(N) ≺ Q'" and "extractFrame Q = ⟨AQ, ΨQ⟩" and "distinct AQ" and "Ψ ⊗ ΨP ⊗ ΨQ ⊨ M ↔ K" and "AP ♯* Ψ" and "AP ♯* P" and "AP ♯* Q" and "AP ♯* M" and "AP ♯* AQ" and "AQ ♯* Ψ" and "AQ ♯* P" and "AQ ♯* Q" and "AQ ♯* K" and "xvec ♯* Q" have "Ψ ⊳ P ∥ Q ⟼τ ≺ (ν*xvec)(P' ∥ Q')" proof - obtain r::"name prm" where "(r ∙ xvec) ♯* Ψ" and "(r ∙ xvec) ♯* P" and "(r ∙ xvec) ♯* Q" and "(r ∙ xvec) ♯* K" and "(r ∙ xvec) ♯* N" and "(r ∙ xvec) ♯* AP" and "(r ∙ xvec) ♯* A and "(r ∙ xvec) ♯* P'" and "(r ∙ xvec) ♯* Q'" and "(r ∙ xvec) ♯* ΨP" and "(r ∙ xvec) ♯* Ψ and Sr: "(set r) ⊆ (set xvec) × (set(r ∙ xvec))" and "distinctPerm r" by(rule_tac xvec=xvec and c="(Ψ, P, Q, M, K, N, AP, AQ, ΨP, ΨQ, P', Q')" in name_list_av (auto simp add: eqvts fresh_star_prod) obtain q::"name prm" where "(q ∙ AQ) ♯* Ψ" and "(q ∙ AQ) ♯* P" and "(q ∙ AQ) ♯* Q" and "(q and "(q ∙ AQ) ♯* (r ∙ N)" and "(q ∙ AQ) ♯* (r ∙ xvec)" and "(q ∙ AQ) ♯* (r ∙ Q')" and "(q ∙ AQ) ♯* (r ∙ P')" and "(q ∙ AQ) ♯* ΨP" and "(q ∙ AQ) ♯* AP" and "(q ∙ AQ) ♯* ΨQ" and Sq: "set q ⊆ set AQ × set(q ∙ AQ)" by(rule_tac xvec=AQ and c="(Ψ, P, Q, K, r ∙ N, r ∙ xvec, ΨQ, AP, ΨP, r ∙ Q', r ∙ P')" i (auto simp add: eqvts fresh_star_prod) obtain p::"name prm" where "(p ∙ AP) ♯* Ψ" and "(p ∙ AP) ♯* P" and "(p ∙ AP) ♯* Q" and "(p and "(p ∙ AP) ♯* (r ∙ N)" and "(p ∙ AP) ♯* (r ∙ xvec)" and "(p ∙ AP) ♯* (r ∙ Q')" and "(p ∙ AP) ♯* (r ∙ P')" and "(p ∙ AP) ♯* ΨP" and "(p ∙ AP) ♯* ΨQ" and "(p ∙ AP) ♯* AQ" and "(p ∙ AP) ♯* (q ∙ AQ)" and Sp: "(set p) ⊆ (set AP) × (set(p ∙ AP))" by(rule_tac xvec=AP and c="(Ψ, P, Q, M, r ∙ N, r ∙ xvec, AQ, q ∙ AQ, ΨQ, ΨP, r ∙ Q', r (auto simp add: eqvts fresh_star_prod) have FrP: "extractFrame P = ⟨AP, ΨP⟩" by fact have FrQ: "extractFrame Q = ⟨AQ, ΨQ⟩" by fact from ‹AP ♯* Q› FrQ ‹AP ♯* AQ› have "AP ♯* ΨQ" by(drule_tac extractFrameFreshChain) auto from ‹AQ ♯* P› FrP ‹AP ♯* AQ› have "AQ ♯* ΨP" by(drule_tac extractFrameFreshChain) auto from ‹(r ∙ xvec) ♯* AQ› ‹(q ∙ AQ) ♯* (r ∙ xvec)› ‹(r ∙ xvec) ♯* AQ› Sq have "(r ∙ x by(simp add: freshChainSimps) from ‹Ψ ⊗ ΨQ ⊳ P ⟼M(ν*xvec)⟨N⟩ ≺ P'› Sr ‹(r ∙ xvec) ♯* N› ‹(r ∙ xvec) ♯* P'› have "Ψ ⊗ ΨQ ⊳ P ⟼M(ν*(r ∙ xvec))⟨(r ∙ N)⟩ ≺ (r ∙ P')" by(simp add: boundOutputChainAlpha'' create_residual.simps) hence "(q ∙ (Ψ ⊗ ΨQ)) ⊳ P ⟼(q ∙ M)(ν*(r ∙ xvec))⟨(r ∙ N)⟩ ≺ (r ∙ P')" using Sq ‹AQ ♯* P› ‹( by(rule_tac outputPermFrameSubject) (assumption | auto) hence PTrans: "Ψ ⊗ (q ∙ ΨQ) ⊳ P ⟼(q ∙ M)(ν*(r ∙ xvec))⟨(r ∙ N)⟩ ≺ (r ∙ P')" using Sq ‹AQ ♯* Ψ by(simp add: eqvts) moreover hence "distinct(r ∙ xvec)" by(force dest: boundOutputDistinct) moreover from ‹extractFrame P = ⟨AP, ΨP⟩› Sp ‹(p ∙ AP) ♯* ΨP› have FrP: "extractFrame P = ⟨(p ∙ AP), (p ∙ ΨP)⟩" by(simp add: frameChainAlpha) moreover from ‹distinct AP› have "distinct(p ∙ AP)" by simp moreover from ‹Ψ ⊗ ΨP ⊳ Q ⟼K(N) ≺ Q'› Sr ‹distinctPerm r› ‹xvec ♯* Q› ‹(r ∙ xvec) ♯* Q› have "Ψ ⊗ ΨP ⊳ Q ⟼K((r ∙ N)) ≺ (r ∙ Q')" by(rule inputAlpha) hence "(p ∙ (Ψ ⊗ ΨP)) ⊳ Q ⟼(p ∙ K)((r ∙ N)) ≺ (r ∙ Q')" using Sp ‹AP ♯* Q› ‹(p ∙ AP) ♯* by(rule_tac inputPermFrameSubject) (assumption | simp)+ hence QTrans: "Ψ ⊗ (p ∙ ΨP) ⊳ Q ⟼(p ∙ K)((r ∙ N)) ≺ (r ∙ Q')" using Sp ‹AP ♯* Ψ› ‹(p ∙ AP) by(simp add: eqvts) moreover from ‹extractFrame Q = ⟨AQ, ΨQ⟩› Sq ‹(q ∙ AQ) ♯* ΨQ› have FrQ: "extractFrame Q = ⟨(q ∙ AQ), (q ∙ ΨQ)⟩" by(simp add: frameChainAlpha) moreover from ‹distinct AQ› have "distinct(q ∙ AQ)" by simp moreover from ‹Ψ ⊗ ΨP ⊗ ΨQ ⊨ M ↔ K› have "(p ∙ q ∙ (Ψ ⊗ ΨP ⊗ ΨQ)) ⊨ (p ∙ q ∙ M) ↔ (p ∙ q ∙ by(rule_tac chanEqClosed)+ with ‹AP ♯* Ψ› ‹(p ∙ AP) ♯* Ψ› ‹AQ ♯* Ψ› ‹(q ∙ AQ) ♯* Ψ› ‹AQ ♯* ΨP› ‹(q ∙ AQ) ♯* ΨP› ‹AP ♯* ΨQ› ‹(p ∙ AP) ♯* ΨQ› ‹AP ♯* M› ‹(p ∙ AP) ♯* M› ‹(q ∙ AQ) ♯* AP› ‹(p ∙ AP) ♯ ‹AQ ♯* K› ‹(q ∙ AQ) ♯* K› ‹AP ♯* AQ› ‹(p ∙ AP) ♯* AQ› Sp Sq have "Ψ ⊗ (p ∙ ΨP) ⊗ (q ∙ ΨQ) ⊨ (q ∙ M) ↔ (p ∙ K)" by(simp add: eqvts freshChainSimps) moreover note ‹(p ∙ AP) ♯* Ψ› moreover from ‹(p ∙ AP) ♯* AQ› ‹(p ∙ AP) ♯* (q ∙ AQ)› ‹(p ∙ AP) ♯* ΨQ› Sq have "(p ∙ by(simp add: freshChainSimps) moreover note ‹(p ∙ AP) ♯* P› moreover from ‹(p ∙ AP) ♯* AQ› ‹(p ∙ AP) ♯* (q ∙ AQ)› ‹(p ∙ AP) ♯* M› Sq have "(p ∙ by(simp add: freshChainSimps) moreover note ‹(p ∙ AP) ♯* (r ∙ N)› ‹(p ∙ AP) ♯* (r ∙ P')› ‹(p ∙ AP) ♯* Q› ‹(p ∙ A ‹(p ∙ AP) ♯* (r ∙ xvec)› ‹(q ∙ AQ) ♯* Ψ› moreover from ‹(q ∙ AQ) ♯* AP› ‹(p ∙ AP) ♯* AQ› ‹(q ∙ AQ) ♯* ΨP› ‹(p ∙ AP) ♯* (q ∙ by(simp add: freshChainSimps) moreover note ‹(q ∙ AQ) ♯* P› ‹(q ∙ AQ) ♯* (r ∙ N)›\‹(q ∙ AQ) ♯* (r ∙ P')› ‹(q ∙ A moreover from ‹(q ∙ AQ) ♯* AP› ‹(p ∙ AP) ♯* AQ› ‹(q ∙ AQ) ♯* K› ‹(p ∙ AP) ♯* (q ∙ by(simp add: freshChainSimps) moreover note ‹(q ∙ AQ) ♯* (r ∙ Q')› ‹(q ∙ AQ) ♯* (r ∙ xvec)› ‹(r ∙ xvec) ♯* Ψ› moreover from ‹(r ∙ xvec) ♯* AP› ‹(p ∙ AP) ♯* (r ∙ xvec)› ‹(r ∙ xvec) ♯* ΨP› Sp have by(simp add: freshChainSimps) moreover from ‹(r ∙ xvec) ♯* AQ› ‹(q ∙ AQ) ♯* (r ∙ xvec)› ‹(r ∙ xvec) ♯* ΨQ› Sq have by(simp add: freshChainSimps) moreover note ‹(r ∙ xvec) ♯* P› moreover from ‹(r ∙ xvec) ♯* AQ› ‹(q ∙ AQ) ♯* (r ∙ xvec)› ‹(r ∙ xvec) ♯* M› Sq have by(simp add: freshChainSimps) moreover note ‹(r ∙ xvec) ♯* Q› moreover from ‹(r ∙ xvec) ♯* AP› ‹(p ∙ AP) ♯* (r ∙ xvec)› ‹(r ∙ xvec) ♯* K› Sp have by(simp add: freshChainSimps) ultimately have "Ψ ⊳ P ∥ Q ⟼τ ≺ (ν*(r ∙ xvec))((r ∙ P') ∥ (r ∙ Q'))" by(rule_tac cComm2) with ‹(r ∙ xvec) ♯* P'› ‹(r ∙ xvec) ♯* Q'› Sr show ?thesis by(subst resChainAlpha) auto qed } note Goal = this note ‹Ψ ⊗ ΨQ ⊳ P ⟼M(ν*xvec)⟨N⟩ ≺ P'› ‹Ψ ⊗ ΨP ⊳ Q ⟼K(N) ≺ Q'› ‹Ψ ⊗ ΨP ⊗ ΨQ ⊨ M ↔ K› moreover from ‹extractFrame P = ⟨AP, ΨP⟩› ‹AP ♯* Ψ› ‹AP ♯* P› ‹AP ♯* Q› ‹AP ♯* M› ‹AP obtain AP' where "extractFrame P = ⟨AP', ΨP⟩" and "distinct AP'" and "AP' ♯* Ψ" and "AP' ♯* P by(rule_tac C="(Ψ, P, Q, M, AQ)" in distinctFrame) auto moreover from ‹extractFrame Q = ⟨AQ, ΨQ⟩› ‹AQ ♯* Ψ› ‹AQ ♯* P› ‹AQ ♯* Q› ‹AQ ♯* K› ‹AP' obtain AQ' where "extractFrame Q = ⟨AQ', ΨQ⟩" and "distinct AQ'" and "AQ' ♯* Ψ" and "AQ' ♯* P by(rule_tac C="(Ψ, P, Q, K, AP')" in distinctFrame) auto ultimately show ?thesis using ‹xvec ♯* Q› by(rule_tac Goal) qed lemma semanticsCasesAux[consumes 1, case_names cInput cOutput cCase cPar1 cPar2 cComm1 cCo fixes Ψ :: 'b and cP :: "('a, 'b, 'c) psi" and cRs :: "('a, 'b, 'c) residual" and C :: "'d::fs_name" and x :: name assumes "Ψ ⊳ cP ⟼ cRs" and rInput: "∧M K xvec N Tvec P. [cP = M(λ*xvec N).P; cRs = K((N[xvec::=Tvec])) ≺ P[xv Ψ ⊨ M ↔ K; distinct xvec; set xvec ⊆ supp N; length xvec=length Tvec; xvec ♯* Tvec; xvec ♯* Ψ; xvec ♯* M; xvec ♯* K; xvec ♯* C] ==> Prop" and rOutput: "∧M K N P. [cP = M⟨N⟩.P; cRs = K⟨N⟩ ≺ P; Ψ ⊨ M ↔ K] ==> Prop" and rCase: "∧Cs P φ. [cP = Cases Cs; Ψ ⊳ P ⟼ cRs; (φ, P) mem Cs; Ψ ⊨ φ; guarded P] ==> and rPar1: "∧ΨQ P α P' Q AQ. [cP = P ∥ Q; cRs = α ≺ (P' ∥ Q); (Ψ ⊗ ΨQ) ⊳ P ⟼ (α ≺ P'); extractFrame Q = ⟨AQ, ΨQ⟩; distinct AQ; AQ ♯* P; AQ ♯* Q; AQ ♯* Ψ; AQ ♯* α; AQ ♯* C; AQ ♯* P'; bn α ♯* Ψ; bn α ♯* ΨQ; bn α ♯* Q; bn α ♯* P; bn α ♯* subject α; bn α ♯* C; distinct(bn α)] ==> Prop" and rPar2: "∧ΨP Q α Q' P AP. [cP = P ∥ Q; cRs = α ≺ (P ∥ Q'); (Ψ ⊗ ΨP) ⊳ Q ⟼α ≺ Q'; extractFrame P = ⟨AP, ΨP⟩; distinct AP; AP ♯* P; AP ♯* Q; AP ♯* Ψ; AP ♯* α; AP ♯* C; AP ♯* Q'; bn α ♯* Ψ; bn α ♯* ΨP; bn α ♯* P; bn α ♯* Q; bn α ♯* subject α; bn α ♯* C; dist and rComm1: "∧ΨQ P M N P' AP ΨP Q K xvec Q' AQ. [cP = P ∥ Q; cRs = τ ≺ (ν*xvec)P' ∥ Q'; Ψ ⊗ ΨQ ⊳ P ⟼M(N) ≺ P'; extractFrame P = ⟨AP, ΨP⟩; distinct AP; Ψ ⊗ ΨP ⊳ Q ⟼ K(ν*xvec)⟨N⟩ ≺ Q'; extractFrame Q = ⟨AQ, ΨQ⟩; distinct AQ; Ψ ⊗ ΨP ⊗ ΨQ ⊨ M ↔ K; AP ♯* Ψ; AP ♯* ΨQ; AP ♯* P; AP ♯* M; AP ♯* N; AP ♯* P'; AP ♯* Q; AP ♯* Q'; AP ♯* AQ; AP ♯* xvec; AQ ♯* Ψ; AQ ♯* ΨP; AQ ♯* P; AQ ♯* K; AQ ♯* N; AQ ♯* P'; AQ ♯* Q; AQ ♯* Q'; AQ ♯* xvec; xvec ♯* Ψ; xvec ♯* ΨP; xvec ♯* ΨQ; xvec ♯* P; xvec ♯* M; xvec ♯* Q; xvec ♯* K; AP ♯* C; AQ ♯* C; xvec ♯* C; distinct xvec] ==> Prop" and rComm2: "∧ΨQ P M xvec N P' AP ΨP Q K Q' AQ. [cP = P ∥ Q; cRs = τ ≺ (ν*xvec)P' ∥ Q'; Ψ ⊗ ΨQ ⊳ P ⟼M(ν*xvec)⟨N⟩ ≺ P'; extractFrame P = ⟨AP, ΨP⟩; distinct AP; Ψ ⊗ ΨP ⊳ Q ⟼ K(N) ≺ Q'; extractFrame Q = ⟨AQ, ΨQ⟩; distinct AQ; Ψ ⊗ ΨP ⊗ ΨQ ⊨ M ↔ K; AP ♯* Ψ; AP ♯* ΨQ; AP ♯* P; AP ♯* M; AP ♯* N; AP ♯* P'; AP ♯* Q; AP ♯* Q'; AP ♯* AQ; AP ♯* xvec; AQ ♯* Ψ; AQ ♯* ΨP; AQ ♯* P; AQ ♯* K; AQ ♯* N; AQ ♯* P'; AQ ♯* Q; AQ ♯* Q'; AQ ♯* xvec; xvec ♯* Ψ; xvec ♯* ΨP; xvec ♯* ΨQ; xvec ♯* P; xvec ♯* M; xvec ♯* Q; xvec ♯* K; AP ♯* C; AQ ♯* C; xvec ♯* C; distinct xvec] ==> Prop" and rOpen: "∧P M xvec yvec N P' x. [cP = (νx)P; cRs = M(ν*(xvec@x#yvec))⟨N⟩ ≺ P'; Ψ ⊳ P ⟼ M(ν*(xvec@yvec))⟨N⟩ ≺ P'; x ∈ supp N; x ♯ xvec; x ♯ yvec; x ♯ M; x ♯ Ψ; distinc xvec ♯* Ψ; xvec ♯* P; xvec ♯* M; xvec ♯* yvec; yvec ♯* Ψ; yvec ♯* P; yvec ♯* M; xv Prop" and rScope: "∧P α P' x. [cP = (νx)P; cRs = α ≺ (νx)P'; Ψ ⊳ P ⟼α ≺ P'; x ♯ Ψ; x ♯ α; x ♯ C; bn α ♯* Ψ; bn α ♯* P; bn α ♯* subject α; bn α ♯* C; di and rBang: "∧P. [cP = !P; Ψ ⊳ P ∥ !P ⟼ cRs; guarded P] ==> Prop" shows Prop using ‹Ψ ⊳ cP ⟼ cRs› proof(cases rule: semantics.cases) case(cInput M K xvec N Tvec P) obtain p::"name prm" where "(p ∙ xvec) ♯* Ψ" and "(p ∙ xvec) ♯* M" and "(p ∙ xvec) ♯* N" and "(p ∙ xvec) ♯* Tvec" and "(p ∙ xvec) ♯* P" and "(p ∙ xvec) ♯* C" and S: "(set p) ⊆ (set xvec) × (set(p ∙ xvec))" and "distinctPerm p" by(rule_tac xvec=xvec and c="(Ψ, M, K, N, P, C, Tvec)" in name_list_avoiding) (auto simp add: eqvts fresh_star_prod) from ‹cP = M(λ*xvec N).P› ‹(p ∙ xvec) ♯* N› ‹(p ∙ xvec) ♯* P› S have "cP = M(λ*(p ∙ xvec) (p ∙ N)).(p ∙ P)" by(simp add: inputChainAlpha') moreover from ‹cRs = K((N[xvec::=Tvec])) ≺ P[xvec::=Tvec]› ‹(p ∙ xvec) ♯* N› ‹(p ∙ x have "cRs = K(((p ∙ N)[(p ∙ xvec)::=Tvec])) ≺ (p ∙ P)[(p ∙ xvec)::=Tvec]" by(auto simp add: substTerm.renaming renaming residualInject) moreover note ‹Ψ ⊨ M ↔ K› moreover from ‹distinct xvec› have "distinct(p ∙ xvec)" by simp moreover from ‹(set xvec) ⊆ (supp N)› have "(p ∙ (set xvec)) ⊆ (p ∙ (supp N))" by simp hence "set(p ∙ xvec) ⊆ supp(p ∙ N)" by(simp add: eqvts) moreover from ‹length xvec = length Tvec› have "length(p ∙ xvec) = length Tvec" by simp ultimately show ?thesis using ‹(p ∙ xvec) ♯* Tvec› ‹(p ∙ xvec) ♯* Ψ› ‹(p ∙ xvec) ♯* M ‹(p ∙ xvec) ♯* C› by(rule rInput) next case(Output M K N P) thus ?thesis by(rule rOutput) next case(Case P φ Cs) thus ?thesis by(rule rCase) next case(cPar1 ΨQ P α P' Q AQ) obtain q::"name prm" where "(bn(q ∙ α)) ♯* Ψ" and "(bn(q ∙ α)) ♯* P" and "(bn(q ∙ α)) ♯* Q" and "(bn(q ∙ α)) ♯* α" and "(bn(q ∙ α)) ♯* AQ" and "(bn(q ∙ α)) ♯* P'" and "(bn(q ∙ α)) ♯* ΨQ and "distinctPerm q" and "(bn(q ∙ α)) ♯* C" and Sq: "(set q) ⊆ set(bn α) × (set(bn(q ∙ α)))" by(rule_tac xvec="bn α" and c="(Ψ, P, Q, α, AQ, ΨQ, P', C)" in name_list_avoiding) (auto si obtain p::"name prm" where "(p ∙ AQ) ♯* Ψ" and "(p ∙ AQ) ♯* P" and "(p ∙ AQ) ♯* Q" and "(p ∙ AQ) ♯* α" and "(p ∙ AQ) ♯* (q ∙ α)" and "(p ∙ AQ) ♯* P'" and "(p ∙ AQ) ♯* (q ∙ P')" and "(p ∙ AQ) ♯* ΨQ" and "(p ∙ AQ) ♯* C" and Sp: "(set p) ⊆ (set AQ) × (set(p ∙ AQ))" and "distinctPerm p" by(rule_tac xvec=AQ and c="(Ψ, P, Q, α, q ∙ α, P', (q ∙ P'), ΨQ, C)" in name_list_avoidin from ‹AQ ♯* α› ‹bn(q ∙ α) ♯* AQ› Sq ‹distinctPerm q› have "AQ ♯* (q ∙ α)" by(subst fresh_star_bij[symmetric, of _ _ q]) (simp add: eqvts) from ‹bn α ♯* subject α› ‹distinctPerm q› have "bn(q ∙ α) ♯* subject(q ∙ α)" by(subst fresh_star_bij[symmetric, of _ _ q]) (simp add: eqvts) from ‹distinct(bn α)› ‹distinctPerm q› have "distinct(bn(q ∙ α))" by(subst distinctClosed[symmetric, of _ q]) (simp add: eqvts) note ‹cP = P ∥ Q› moreover from ‹cRs = α ≺ (P' ∥ Q)› ‹bn α ♯* subject α› ‹(bn(q ∙ α)) ♯* α› ‹(bn(q ∙ α)) ♯ have "cRs = (q ∙ α) ≺ (q ∙ P') ∥ Q" by(force simp add: residualAlpha) moreover from ‹Ψ ⊗ ΨQ ⊳ P ⟼α ≺ P'› ‹bn α ♯* subject α› ‹(bn(q ∙ α)) ♯* α› ‹(bn(q ∙ α)) ♯* have Trans: "Ψ ⊗ ΨQ ⊳ P ⟼(q ∙ α) ≺ (q ∙ P')" by(force simp add: residualAlpha) hence "AQ ♯* (q ∙ P')" using ‹bn(q ∙ α) ♯* subject(q ∙ α)› ‹distinct(bn(q ∙ α))› ‹AQ ♯ by(drule_tac freeFreshChainDerivative) auto from Trans have "(p ∙ (Ψ ⊗ ΨQ)) ⊳ (p ∙ P) ⟼p ∙ ((q ∙ α) ≺ (q ∙ P'))" by(rule semantics.eqvt) with ‹AQ ♯* Ψ› ‹AQ ♯* P› ‹AQ ♯* (q ∙ α)› ‹AQ ♯* (q ∙ P')› ‹(bn(q ∙ α)) ♯* AQ› ‹(p ∙ ‹(p ∙ AQ) ♯* Ψ› ‹(p ∙ AQ) ♯* P› ‹(p ∙ AQ) ♯* (q ∙ P')› Sp have "Ψ ⊗ (p ∙ ΨQ) ⊳ P ⟼(q ∙ α) ≺ (q ∙ P')" by(simp add: eqvts) moreover from ‹extractFrame Q = ⟨AQ, ΨQ⟩› ‹(p ∙ AQ) ♯* ΨQ› Sp have "extractFrame Q = ⟨(p by(simp add: frameChainAlpha' eqvts) moreover from ‹(bn(q ∙ α)) ♯* ΨQ› ‹(bn(q ∙ α)) ♯* AQ› ‹(p ∙ AQ) ♯* (q ∙ α)› Sp have "(bn by(simp add: freshAlphaPerm) moreover from ‹distinct AQ› have "distinct(p ∙ AQ)" by simp ultimately show ?thesis using ‹(p ∙ AQ) ♯* P› ‹(p ∙ AQ) ♯* Q› ‹(p ∙ AQ) ♯* Ψ› ‹(p ∙ AQ) ♯* (q ∙ α)› ‹(p ∙ AQ) ♯* (q ∙ P')› ‹(bn(q ∙ α)) ♯* Ψ› ‹(bn(q ∙ α)) ♯* Q› ‹(bn(q ∙ α)) ♯* P› ‹(bn(q ∙ α)) ♯* C› ‹(p ∙ AQ) ♯* C› ‹bn (q ∙ α) ♯* subject (q ∙ α)› ‹distinct(bn(q ∙ by(rule_tac rPar1) next case(cPar2 ΨP Q α Q' P AP) obtain q::"name prm" where "(bn(q ∙ α)) ♯* Ψ" and "(bn(q ∙ α)) ♯* P" and "(bn(q ∙ α)) ♯* Q" and "(bn(q ∙ α)) ♯* α" and "(bn(q ∙ α)) ♯* AP" and "(bn(q ∙ α)) ♯* Q'" and "(bn(q ∙ α)) ♯* ΨP and "distinctPerm q" and "(bn(q ∙ α)) ♯* C" and Sq: "(set q) ⊆ set(bn α) × (set(bn(q ∙ α)))" by(rule_tac xvec="bn α" and c="(Ψ, P, Q, α, AP, ΨP, Q', C)" in name_list_avoiding) (auto si obtain p::"name prm" where "(p ∙ AP) ♯* Ψ" and "(p ∙ AP) ♯* P" and "(p ∙ AP) ♯* Q" and "(p ∙ AP) ♯* α" and "(p ∙ AP) ♯* (q ∙ α)" and "(p ∙ AP) ♯* Q'" and "(p ∙ AP) ♯* (q ∙ Q')" and "(p ∙ AP) ♯* ΨP" and "(p ∙ AP) ♯* C" and Sp: "(set p) ⊆ (set AP) × (set(p ∙ AP))" and "distinctPerm p" by(rule_tac xvec=AP and c="(Ψ, P, Q, α, q ∙ α, Q', (q ∙ Q'), ΨP, C)" in name_list_avoidin from ‹AP ♯* α› ‹bn(q ∙ α) ♯* AP› Sq ‹distinctPerm q› have "AP ♯* (q ∙ α)" by(subst fresh_star_bij[symmetric, of _ _ q]) (simp add: eqvts) from ‹bn α ♯* subject α› ‹distinctPerm q› have "bn(q ∙ α) ♯* subject(q ∙ α)" by(subst fresh_star_bij[symmetric, of _ _ q]) (simp add: eqvts) from ‹distinct(bn α)› ‹distinctPerm q› have "distinct(bn(q ∙ α))" by(subst distinctClosed[symmetric, of _ q]) (simp add: eqvts) note ‹cP = P ∥ Q› moreover from ‹cRs = α ≺ (P ∥ Q')› ‹bn α ♯* subject α› ‹(bn(q ∙ α)) ♯* α› ‹(bn(q ∙ α)) ♯ have "cRs = (q ∙ α) ≺ P ∥ (q ∙ Q')" by(force simp add: residualAlpha) moreover from ‹Ψ ⊗ ΨP ⊳ Q ⟼α ≺ Q'› ‹bn α ♯* subject α› ‹(bn(q ∙ α)) ♯* α› ‹(bn(q ∙ α)) ♯* have Trans: "Ψ ⊗ ΨP ⊳ Q ⟼(q ∙ α) ≺ (q ∙ Q')" by(force simp add: residualAlpha) hence "AP ♯* (q ∙ Q')" using ‹bn(q ∙ α) ♯* subject(q ∙ α)› ‹distinct(bn(q ∙ α))› ‹AP ♯ by(drule_tac freeFreshChainDerivative) auto from Trans have "(p ∙ (Ψ ⊗ ΨP)) ⊳ (p ∙ Q) ⟼p ∙ ((q ∙ α) ≺ (q ∙ Q'))" by(rule semantics.eqvt) with ‹AP ♯* Ψ› ‹AP ♯* Q› ‹AP ♯* (q ∙ α)› ‹AP ♯* (q ∙ Q')› ‹(bn(q ∙ α)) ♯* AP› ‹(p ∙ ‹(p ∙ AP) ♯* Ψ› ‹(p ∙ AP) ♯* Q› ‹(p ∙ AP) ♯* (q ∙ Q')› Sp have "Ψ ⊗ (p ∙ ΨP) ⊳ Q ⟼(q ∙ α) ≺ (q ∙ Q')" by(simp add: eqvts) moreover from ‹extractFrame P = ⟨AP, ΨP⟩› ‹(p ∙ AP) ♯* ΨP› Sp have "extractFrame P = ⟨(p by(simp add: frameChainAlpha' eqvts) moreover from ‹(bn(q ∙ α)) ♯* ΨP› ‹(bn(q ∙ α)) ♯* AP› ‹(p ∙ AP) ♯* (q ∙ α)› Sp have "(bn by(simp add: freshAlphaPerm) moreover from ‹distinct AP› have "distinct(p ∙ AP)" by simp ultimately show ?thesis using ‹(p ∙ AP) ♯* P› ‹(p ∙ AP) ♯* Q› ‹(p ∙ AP) ♯* Ψ› ‹(p ∙ AP) ♯* (q ∙ α)› ‹(p ∙ AP) ♯* (q ∙ Q')› ‹(bn(q ∙ α)) ♯* Ψ› ‹(bn(q ∙ α)) ♯* Q› ‹(bn(q ∙ α)) ♯* P› ‹(bn(q ∙ α)) ♯* C› ‹(p ∙ AP) ♯* C› ‹bn (q ∙ α) ♯* subject (q ∙ α)› ‹distinct(bn(q ∙ by(rule_tac rPar2) next case(cComm1 ΨQ P M N P' AP ΨP Q K xvec Q' AQ) obtain r::"name prm" where "(r ∙ xvec) ♯* Ψ" and "(r ∙ xvec) ♯* P" and "(r ∙ xvec) ♯* Q" and "(r ∙ xvec) ♯* K" and "(r ∙ xvec) ♯* N" and "(r ∙ xvec) ♯* AP" and "(r ∙ xvec) ♯* A and "(r ∙ xvec) ♯* P'" and "(r ∙ xvec) ♯* Q'" and "(r ∙ xvec) ♯* ΨP" and "(r ∙ xvec) ♯* Ψ and "(r ∙ xvec) ♯* C" and Sr: "(set r) ⊆ (set xvec) × (set(r ∙ xvec))" and "distinctPe by(rule_tac xvec=xvec and c="(Ψ, P, Q, M, K, N, AP, AQ, ΨP, ΨQ, P', Q', C)" in name_list (auto simp add: eqvts) obtain q::"name prm" where "(q ∙ AQ) ♯* Ψ" and "(q ∙ AQ) ♯* P" and "(q ∙ AQ) ♯* Q" and "(q and "(q ∙ AQ) ♯* N" and "(q ∙ AQ) ♯* xvec" and "(q ∙ AQ) ♯* Q'" and "(q ∙ AQ) ♯* P'" and "(q ∙ AQ) ♯* ΨP" and "(q ∙ AQ) ♯* AP" and "(q ∙ AQ) ♯* ΨQ" and "(q ∙ AQ) ♯* (r ∙ xvec and "(q ∙ AQ) ♯* C" and Sq: "(set q) ⊆ (set AQ) × (set(q ∙ AQ))" by(rule_tac xvec=AQ and c="(Ψ, P, Q, K, N, xvec, r ∙ xvec, ΨQ, AP, ΨP, Q', P', C)" in na obtain p::"name prm" where "(p ∙ AP) ♯* Ψ" and "(p ∙ AP) ♯* P" and "(p ∙ AP) ♯* Q" and "(p and "(p ∙ AP) ♯* N" and "(p ∙ AP) ♯* xvec" and "(p ∙ AP) ♯* Q'" and "(p ∙ AP) ♯* AQ" and "(p ∙ AP) ♯* P'" and "(p ∙ AP) ♯* ΨP" and "(p ∙ AP) ♯* ΨQ" and "(p ∙ AP) ♯* (q ∙ AQ)" and "(p ∙ AP) ♯* C" and "(p ∙ AP) ♯* (r ∙ xvec)" and Sp: "(set p) ⊆ (set AP) × (set(p by(rule_tac xvec=AP and c="(Ψ, P, Q, M, N, xvec, r ∙ xvec, AQ, q ∙ AQ, ΨQ, ΨP, Q', P', (auto simp add: eqvts fresh_star_prod) have FrP: "extractFrame P = ⟨AP, ΨP⟩" by fact have FrQ: "extractFrame Q = ⟨AQ, ΨQ⟩" by fact from ‹AP ♯* Q› FrQ ‹AP ♯* AQ› have "AP ♯* ΨQ" by(drule_tac extractFrameFreshChain) auto from ‹AQ ♯* P› FrP ‹AP ♯* AQ› have "AQ ♯* ΨP" by(drule_tac extractFrameFreshChain) auto note ‹cP = P ∥ Q› moreover from ‹(r ∙ xvec) ♯* P'› ‹(r ∙ xvec) ♯* Q'› have "(r ∙ xvec) ♯* (P' ∥ Q')" by simp with ‹cRs = τ ≺ (ν*xvec)(P' ∥ Q')› ‹(r ∙ xvec) ♯* N› Sr have "cRs = τ ≺ (ν*(r ∙ xvec))(r ∙ (P' ∥ Q'))" by(simp add: resChainAlpha residualInject) hence "cRs = τ ≺ (ν*(r ∙ xvec))((r ∙ P') ∥ (r ∙ Q'))" by simp moreover from ‹Ψ ⊗ ΨQ ⊳ P ⟼M(N) ≺ P'› Sr ‹distinctPerm r› ‹xvec ♯* P› ‹(r ∙ xvec) ♯* P› have "Ψ ⊗ ΨQ ⊳ P ⟼M((r ∙ N)) ≺ (r ∙ P')" by(rule inputAlpha) hence "(q ∙ (Ψ ⊗ ΨQ)) ⊳ P ⟼(q ∙ M)((r ∙ N)) ≺ (r ∙ P')" using Sq ‹AQ ♯* P› ‹(q ∙ AQ) ♯* by(rule_tac inputPermFrameSubject) (assumption | simp)+ hence PTrans: "Ψ ⊗ (q ∙ ΨQ) ⊳ P ⟼(q ∙ M)((r ∙ N)) ≺ (r ∙ P')" using Sq ‹AQ ♯* Ψ› ‹(q ∙ AQ) by(simp add: eqvts) moreover from ‹extractFrame P = ⟨AP, ΨP⟩› Sp ‹(p ∙ AP) ♯* ΨP› have FrP: "extractFrame P = ⟨(p ∙ AP), (p ∙ ΨP)⟩" by(simp add: frameChainAlpha) moreover from ‹distinct AP› have "distinct(p ∙ AP)" by simp moreover from ‹Ψ ⊗ ΨP ⊳ Q ⟼K(ν*xvec)⟨N⟩ ≺ Q'› Sr ‹(r ∙ xvec) ♯* N› ‹(r ∙ xvec) ♯* Q'› have "Ψ ⊗ ΨP ⊳ Q ⟼K(ν*(r ∙ xvec))⟨(r ∙ N)⟩ ≺ (r ∙ Q')" by(simp add: boundOutputChainAlpha'' residualInject) hence "(p ∙ (Ψ ⊗ ΨP)) ⊳ Q ⟼(p ∙ K)(ν*(r ∙ xvec))⟨(r ∙ N)⟩ ≺ (r ∙ Q')" using Sp ‹AP ♯* Q› ‹( by(rule_tac outputPermFrameSubject) (assumption | auto) hence QTrans: "Ψ ⊗ (p ∙ ΨP) ⊳ Q ⟼(p ∙ K)(ν*(r ∙ xvec))⟨(r ∙ N)⟩ ≺ (r ∙ Q')" using Sp ‹AP ♯* Ψ by(simp add: eqvts) moreover from ‹extractFrame Q = ⟨AQ, ΨQ⟩› Sq ‹(q ∙ AQ) ♯* ΨQ› have FrQ: "extractFrame Q = ⟨(q ∙ AQ), (q ∙ ΨQ)⟩" by(simp add: frameChainAlpha) moreover from ‹distinct AQ› have "distinct(q ∙ AQ)" by simp moreover from ‹Ψ ⊗ ΨP ⊗ ΨQ ⊨ M ↔ K› have "(p ∙ q ∙ (Ψ ⊗ ΨP ⊗ ΨQ)) ⊨ (p ∙ q ∙ M) ↔ (p ∙ q ∙ by(rule_tac chanEqClosed)+ with ‹AP ♯* Ψ› ‹(p ∙ AP) ♯* Ψ› ‹AQ ♯* Ψ› ‹(q ∙ AQ) ♯* Ψ› ‹AQ ♯* ΨP› ‹(q ∙ AQ) ♯* ΨP› ‹AP ♯* ΨQ› ‹(p ∙ AP) ♯* ΨQ› ‹AP ♯* M› ‹(p ∙ AP) ♯* M› ‹(q ∙ AQ) ♯* AP› ‹(p ∙ AP) ♯ ‹AQ ♯* K› ‹(q ∙ AQ) ♯* K› ‹AP ♯* AQ› ‹(p ∙ AP) ♯* AQ› Sp Sq have "Ψ ⊗ (p ∙ ΨP) ⊗ (q ∙ ΨQ) ⊨ (q ∙ M) ↔ (p ∙ K)" by(simp add: eqvts freshChainSimps) moreover note ‹(p ∙ AP) ♯* Ψ› moreover from ‹(p ∙ AP) ♯* AQ› ‹(p ∙ AP) ♯* (q ∙ AQ)› ‹(p ∙ AP) ♯* ΨQ› Sq have "(p ∙ by(simp add: freshChainSimps) moreover note ‹(p ∙ AP) ♯* P› moreover from ‹(p ∙ AP) ♯* AQ› ‹(p ∙ AP) ♯* (q ∙ AQ)› ‹(p ∙ AP) ♯* M› Sq have "(p ∙ by(simp add: freshChainSimps) moreover from ‹(p ∙ AP) ♯* xvec› ‹(p ∙ AP) ♯* (r ∙ xvec)› ‹(p ∙ AP) ♯* N› Sr have "( by(simp add: freshChainSimps) moreover from ‹(p ∙ AP) ♯* xvec› ‹(p ∙ AP) ♯* (r ∙ xvec)› ‹(p ∙ AP) ♯* P'› Sr have " by(simp add: freshChainSimps) moreover note ‹(p ∙ AP) ♯* Q› moreover from ‹(p ∙ AP) ♯* xvec› ‹(p ∙ AP) ♯* (r ∙ xvec)› ‹(p ∙ AP) ♯* Q'› Sr have " by(simp add: freshChainSimps) moreover note ‹(p ∙ AP) ♯* (q ∙ AQ)› ‹(p ∙ AP) ♯* (r ∙ xvec)› ‹(q ∙ AQ) ♯* Ψ› moreover from ‹(q ∙ AQ) ♯* AP› ‹(p ∙ AP) ♯* AQ› ‹(q ∙ AQ) ♯* ΨP› ‹(p ∙ AP) ♯* (q ∙ by(simp add: freshChainSimps) moreover note ‹(q ∙ AQ) ♯* P› moreover from ‹(q ∙ AQ) ♯* AP› ‹(p ∙ AP) ♯* AQ› ‹(q ∙ AQ) ♯* K› ‹(p ∙ AP) ♯* (q ∙ by(simp add: freshChainSimps) moreover from ‹(q ∙ AQ) ♯* xvec› ‹(q ∙ AQ) ♯* (r ∙ xvec)› ‹(q ∙ AQ) ♯* N› Sr have "( by(simp add: freshChainSimps) moreover from ‹(q ∙ AQ) ♯* xvec› ‹(q ∙ AQ) ♯* (r ∙ xvec)› ‹(q ∙ AQ) ♯* P'› Sr have " by(simp add: freshChainSimps) moreover note ‹(q ∙ AQ) ♯* Q› moreover from ‹(q ∙ AQ) ♯* xvec› ‹(q ∙ AQ) ♯* (r ∙ xvec)› ‹(q ∙ AQ) ♯* Q'› Sr have " by(simp add: freshChainSimps) moreover note ‹(q ∙ AQ) ♯* (r ∙ xvec)› ‹(r ∙ xvec) ♯* Ψ› moreover from ‹(r ∙ xvec) ♯* AP› ‹(p ∙ AP) ♯* (r ∙ xvec)› ‹(r ∙ xvec) ♯* ΨP› Sp have by(simp add: freshChainSimps) moreover from ‹(r ∙ xvec) ♯* AQ› ‹(q ∙ AQ) ♯* (r ∙ xvec)› ‹(r ∙ xvec) ♯* ΨQ› Sq have by(simp add: freshChainSimps) moreover note ‹(r ∙ xvec) ♯* P› moreover from ‹(r ∙ xvec) ♯* AQ› ‹(q ∙ AQ) ♯* (r ∙ xvec)› ‹(r ∙ xvec) ♯* M› Sq have by(simp add: freshChainSimps) moreover note ‹(r ∙ xvec) ♯* Q› moreover from ‹(r ∙ xvec) ♯* AP› ‹(p ∙ AP) ♯* (r ∙ xvec)› ‹(r ∙ xvec) ♯* K› Sp have by(simp add: freshChainSimps) moreover note ‹(p ∙ AP) ♯* C› ‹(q ∙ AQ) ♯* C› ‹(r ∙ xvec) ♯* C› moreover from ‹distinct xvec› have "distinct(r ∙ xvec)" by simp ultimately show ?thesis by(rule rComm1) next case(cComm2 ΨQ P M xvec N P' AP ΨP Q K Q' AQ) obtain r::"name prm" where "(r ∙ xvec) ♯* Ψ" and "(r ∙ xvec) ♯* P" and "(r ∙ xvec) ♯* Q" and "(r ∙ xvec) ♯* K" and "(r ∙ xvec) ♯* N" and "(r ∙ xvec) ♯* AP" and "(r ∙ xvec) ♯* A and "(r ∙ xvec) ♯* P'" and "(r ∙ xvec) ♯* Q'" and "(r ∙ xvec) ♯* ΨP" and "(r ∙ xvec) ♯* Ψ and "(r ∙ xvec) ♯* C" and Sr: "(set r) ⊆ (set xvec) × (set(r ∙ xvec))" and "distinctPe by(rule_tac xvec=xvec and c="(Ψ, P, Q, M, K, N, AP, AQ, ΨP, ΨQ, P', Q', C)" in name_list (auto simp add: eqvts) obtain q::"name prm" where "(q ∙ AQ) ♯* Ψ" and "(q ∙ AQ) ♯* P" and "(q ∙ AQ) ♯* Q" and "(q and "(q ∙ AQ) ♯* N" and "(q ∙ AQ) ♯* xvec" and "(q ∙ AQ) ♯* Q'" and "(q ∙ AQ) ♯* P'" and "(q ∙ AQ) ♯* ΨP" and "(q ∙ AQ) ♯* AP" and "(q ∙ AQ) ♯* ΨQ" and "(q ∙ AQ) ♯* (r ∙ xvec and "(q ∙ AQ) ♯* C" and Sq: "(set q) ⊆ (set AQ) × (set(q ∙ AQ))" by(rule_tac xvec=AQ and c="(Ψ, P, Q, K, N, xvec, r ∙ xvec, ΨQ, AP, ΨP, Q', P', C)" in na obtain p::"name prm" where "(p ∙ AP) ♯* Ψ" and "(p ∙ AP) ♯* P" and "(p ∙ AP) ♯* Q" and "(p and "(p ∙ AP) ♯* N" and "(p ∙ AP) ♯* xvec" and "(p ∙ AP) ♯* Q'" and "(p ∙ AP) ♯* AQ" and "(p ∙ AP) ♯* P'" and "(p ∙ AP) ♯* ΨP" and "(p ∙ AP) ♯* ΨQ" and "(p ∙ AP) ♯* (q ∙ AQ)" and "(p ∙ AP) ♯* C" and "(p ∙ AP) ♯* (r ∙ xvec)" and Sp: "(set p) ⊆ (set AP) × (set(p by(rule_tac xvec=AP and c="(Ψ, P, Q, M, N, xvec, r ∙ xvec, AQ, q ∙ AQ, ΨQ, ΨP, Q', P', (auto simp add: eqvts fresh_star_prod) have FrP: "extractFrame P = ⟨AP, ΨP⟩" by fact have FrQ: "extractFrame Q = ⟨AQ, ΨQ⟩" by fact from ‹AP ♯* Q› FrQ ‹AP ♯* AQ› have "AP ♯* ΨQ" by(drule_tac extractFrameFreshChain) auto from ‹AQ ♯* P› FrP ‹AP ♯* AQ› have "AQ ♯* ΨP" by(drule_tac extractFrameFreshChain) auto note ‹cP = P ∥ Q› moreover from ‹(r ∙ xvec) ♯* P'› ‹(r ∙ xvec) ♯* Q'› have "(r ∙ xvec) ♯* (P' ∥ Q')" by simp with ‹cRs = τ ≺ (ν*xvec)(P' ∥ Q')› ‹(r ∙ xvec) ♯* N› Sr have "cRs = τ ≺ (ν*(r ∙ xvec))(r ∙ (P' ∥ Q'))" by(simp add: resChainAlpha residualInject) hence "cRs = τ ≺ (ν*(r ∙ xvec))((r ∙ P') ∥ (r ∙ Q'))" by simp moreover from ‹Ψ ⊗ ΨQ ⊳ P ⟼M(ν*xvec)⟨N⟩ ≺ P'› Sr ‹(r ∙ xvec) ♯* N› ‹(r ∙ xvec) ♯* P'› have "Ψ ⊗ ΨQ ⊳ P ⟼M(ν*(r ∙ xvec))⟨(r ∙ N)⟩ ≺ (r ∙ P')" by(simp add: boundOutputChainAlpha'' hence "(q ∙ (Ψ ⊗ ΨQ)) ⊳ P ⟼(q ∙ M)(ν*(r ∙ xvec))⟨(r ∙ N)⟩ ≺ (r ∙ P')" using Sq ‹AQ ♯* P› ‹( by(rule_tac outputPermFrameSubject) (assumption | auto) hence PTrans: "Ψ ⊗ (q ∙ ΨQ) ⊳ P ⟼(q ∙ M)(ν*(r ∙ xvec))⟨(r ∙ N)⟩ ≺ (r ∙ P')" using Sq ‹AQ ♯* Ψ by(simp add: eqvts) moreover from ‹extractFrame P = ⟨AP, ΨP⟩› Sp ‹(p ∙ AP) ♯* ΨP› have FrP: "extractFrame P = ⟨(p ∙ AP), (p ∙ ΨP)⟩" by(simp add: frameChainAlpha) moreover from ‹distinct AP› have "distinct(p ∙ AP)" by simp moreover from ‹Ψ ⊗ ΨP ⊳ Q ⟼K(N) ≺ Q'› Sr ‹distinctPerm r› ‹xvec ♯* Q› ‹(r ∙ xvec) ♯* Q› have "Ψ ⊗ ΨP ⊳ Q ⟼K((r ∙ N)) ≺ (r ∙ Q')" by(rule inputAlpha) hence "(p ∙ (Ψ ⊗ ΨP)) ⊳ Q ⟼(p ∙ K)((r ∙ N)) ≺ (r ∙ Q')" using Sp ‹AP ♯* Q› ‹(p ∙ AP) ♯* by(rule_tac inputPermFrameSubject) (assumption | simp)+ hence QTrans: "Ψ ⊗ (p ∙ ΨP) ⊳ Q ⟼(p ∙ K)((r ∙ N)) ≺ (r ∙ Q')" using Sp ‹AP ♯* Ψ› ‹(p ∙ AP) by(simp add: eqvts) moreover from ‹extractFrame Q = ⟨AQ, ΨQ⟩› Sq ‹(q ∙ AQ) ♯* ΨQ› have FrQ: "extractFrame Q = ⟨(q ∙ AQ), (q ∙ ΨQ)⟩" by(simp add: frameChainAlpha) moreover from ‹distinct AQ› have "distinct(q ∙ AQ)" by simp moreover from ‹Ψ ⊗ ΨP ⊗ ΨQ ⊨ M ↔ K› have "(p ∙ q ∙ (Ψ ⊗ ΨP ⊗ ΨQ)) ⊨ (p ∙ q ∙ M) ↔ (p ∙ q ∙ by(rule_tac chanEqClosed)+ with ‹AP ♯* Ψ› ‹(p ∙ AP) ♯* Ψ› ‹AQ ♯* Ψ› ‹(q ∙ AQ) ♯* Ψ› ‹AQ ♯* ΨP› ‹(q ∙ AQ) ♯* ΨP› ‹AP ♯* ΨQ› ‹(p ∙ AP) ♯* ΨQ› ‹AP ♯* M› ‹(p ∙ AP) ♯* M› ‹(q ∙ AQ) ♯* AP› ‹(p ∙ AP) ♯ ‹AQ ♯* K› ‹(q ∙ AQ) ♯* K› ‹AP ♯* AQ› ‹(p ∙ AP) ♯* AQ› Sp Sq have "Ψ ⊗ (p ∙ ΨP) ⊗ (q ∙ ΨQ) ⊨ (q ∙ M) ↔ (p ∙ K)" by(simp add: eqvts freshChainSimps) moreover note ‹(p ∙ AP) ♯* Ψ› moreover from ‹(p ∙ AP) ♯* AQ› ‹(p ∙ AP) ♯* (q ∙ AQ)› ‹(p ∙ AP) ♯* ΨQ› Sq have "(p ∙ by(simp add: freshChainSimps) moreover note ‹(p ∙ AP) ♯* P› moreover from ‹(p ∙ AP) ♯* AQ› ‹(p ∙ AP) ♯* (q ∙ AQ)› ‹(p ∙ AP) ♯* M› Sq have "(p ∙ by(simp add: freshChainSimps) moreover from ‹(p ∙ AP) ♯* xvec› ‹(p ∙ AP) ♯* (r ∙ xvec)› ‹(p ∙ AP) ♯* N› Sr have "( by(simp add: freshChainSimps) moreover from ‹(p ∙ AP) ♯* xvec› ‹(p ∙ AP) ♯* (r ∙ xvec)› ‹(p ∙ AP) ♯* P'› Sr have " by(simp add: freshChainSimps) moreover note ‹(p ∙ AP) ♯* Q› moreover from ‹(p ∙ AP) ♯* xvec› ‹(p ∙ AP) ♯* (r ∙ xvec)› ‹(p ∙ AP) ♯* Q'› Sr have " by(simp add: freshChainSimps) moreover note ‹(p ∙ AP) ♯* (q ∙ AQ)› ‹(p ∙ AP) ♯* (r ∙ xvec)› ‹(q ∙ AQ) ♯* Ψ› moreover from ‹(q ∙ AQ) ♯* AP› ‹(p ∙ AP) ♯* AQ› ‹(q ∙ AQ) ♯* ΨP› ‹(p ∙ AP) ♯* (q ∙ by(simp add: freshChainSimps) moreover note ‹(q ∙ AQ) ♯* P› moreover from ‹(q ∙ AQ) ♯* AP› ‹(p ∙ AP) ♯* AQ› ‹(q ∙ AQ) ♯* K› ‹(p ∙ AP) ♯* (q ∙ by(simp add: freshChainSimps) moreover from ‹(q ∙ AQ) ♯* xvec› ‹(q ∙ AQ) ♯* (r ∙ xvec)› ‹(q ∙ AQ) ♯* N› Sr have "( by(simp add: freshChainSimps) moreover from ‹(q ∙ AQ) ♯* xvec› ‹(q ∙ AQ) ♯* (r ∙ xvec)› ‹(q ∙ AQ) ♯* P'› Sr have " by(simp add: freshChainSimps) moreover note ‹(q ∙ AQ) ♯* Q› moreover from ‹(q ∙ AQ) ♯* xvec› ‹(q ∙ AQ) ♯* (r ∙ xvec)› ‹(q ∙ AQ) ♯* Q'› Sr have " by(simp add: freshChainSimps) moreover note ‹(q ∙ AQ) ♯* (r ∙ xvec)› ‹(r ∙ xvec) ♯* Ψ› moreover from ‹(r ∙ xvec) ♯* AP› ‹(p ∙ AP) ♯* (r ∙ xvec)› ‹(r ∙ xvec) ♯* ΨP› Sp have by(simp add: freshChainSimps) moreover from ‹(r ∙ xvec) ♯* AQ› ‹(q ∙ AQ) ♯* (r ∙ xvec)› ‹(r ∙ xvec) ♯* ΨQ› Sq have by(simp add: freshChainSimps) moreover note ‹(r ∙ xvec) ♯* P› moreover from ‹(r ∙ xvec) ♯* AQ› ‹(q ∙ AQ) ♯* (r ∙ xvec)› ‹(r ∙ xvec) ♯* M› Sq have by(simp add: freshChainSimps) moreover note ‹(r ∙ xvec) ♯* Q› moreover from ‹(r ∙ xvec) ♯* AP› ‹(p ∙ AP) ♯* (r ∙ xvec)› ‹(r ∙ xvec) ♯* K› Sp have by(simp add: freshChainSimps) moreover note ‹(p ∙ AP) ♯* C› ‹(q ∙ AQ) ♯* C› ‹(r ∙ xvec) ♯* C› moreover from ‹distinct xvec› have "distinct(r ∙ xvec)" by simp ultimately show ?thesis by(rule rComm2) next case(cOpen P M xvec yvec N P' x) from ‹Ψ ⊳ P ⟼ M(ν*(xvec@yvec))⟨N⟩ ≺ P'› have "distinct(xvec@yvec)" by(force dest: boundOutp hence "xvec ♯* yvec" by(induct xvec) auto obtain p where "(p ∙ yvec) ♯* Ψ" and "(p ∙ yvec) ♯* P" and "(p ∙ yvec) ♯* M" and "(p ∙ yvec) ♯* yvec" and "(p ∙ yvec) ♯* N" and "(p ∙ yvec) ♯* P'" and "x ♯ (p ∙ yvec)" and "(p ∙ yvec) ♯* xvec" and "(p ∙ yvec) ♯* C" and Sp: "(set p) ⊆ (set yvec) × (set(p ∙ yvec))" by(rule_tac xvec=yvec and c="(Ψ, P, M, xvec, yvec, N, P', x, C)" in name_list_avoiding (auto simp add: eqvts fresh_star_prod) obtain q where "(q ∙ xvec) ♯* Ψ" and "(q ∙ xvec) ♯* P" and "(q ∙ xvec) ♯* M" and "(q ∙ xvec) ♯* xvec" and "(q ∙ xvec) ♯* N" and "(q ∙ xvec) ♯* P'" and "x ♯ (q ∙ xvec)" and "(q ∙ xvec) ♯* yvec" and "(q ∙ xvec) ♯* p" and "(q ∙ xvec) ♯* (p ∙ yvec)" and "(q ∙ xvec) ♯* C" and Sq: "(set q) ⊆ (set xvec) × (set(q ∙ xvec))" by(rule_tac xvec=xvec and c="(Ψ, P, M, xvec, yvec, p ∙ yvec, N, P', x, p, C)" in name_ (auto simp add: eqvts fresh_star_prod) obtain y::name where "y ♯ P" and "y ♯ C" and "y ♯ xvec" and "y ♯ yvec" and "y ≠ x" and "y ♯ N and "y ♯ (q ∙ xvec)" and "y ♯ (p ∙ yvec)" and "y ♯ M" and "y ♯ Ψ" and "y ♯ P'" by(generate_fresh "name") (auto simp add: freshChainSimps) from ‹cP = (νx)P› ‹y ♯ P› have "cP = (νy)([(x, y)] ∙ P)" by(simp add: alphaRes) moreover have "cRs = M(ν*((q ∙ xvec)@y#(p ∙ yvec)))⟨((q@(x, y)#p) ∙ N)⟩ ≺ ((q@(x, y)#p) proof - note ‹cRs = M(ν*(xvec@x#yvec))⟨N⟩ ≺ P'› moreover have "(ν*(xvec@x#yvec))N ≺' P' = (ν*xvec)((νx)((ν*yvec)N ≺' P'))" by(simp add: boundOutp moreover from ‹(p ∙ yvec) ♯* N› ‹(p ∙ yvec) ♯* P'› Sp have "… = (ν*xvec)((νx)((ν*(p ∙ yvec))( by(simp add: boundOutputChainAlpha'') moreover with ‹y ♯ N› ‹y ♯ P'› ‹y ♯ (p ∙ yvec)› ‹y ♯ yvec› ‹x ♯ yvec› ‹x ♯ (p ∙ yv have "… = (ν*xvec)((νy)((ν*(p ∙ yvec))(([(x, y)] ∙ p ∙ N) ≺' ([(x, y)] ∙ p ∙ P'))))" by(subst alphaBoundOutput[where y=y]) (simp add: freshChainSimps eqvts)+ moreover hence "… = (ν*xvec)((νy)((ν*(p ∙ yvec))((((x, y)#p) ∙ N) ≺' (((x, y)#p) ∙ P'))))" by simp moreover from ‹(q ∙ xvec) ♯* N› ‹(q ∙ xvec) ♯* P'› ‹xvec ♯* yvec› ‹(p ∙ yvec) ♯* x ‹y ♯ xvec› ‹y ♯ (q ∙ xvec)› ‹x ♯ xvec› ‹x ♯ (q ∙ xvec)› Sp Sq have "… = (ν*(q ∙ xvec))((νy)((ν*(p ∙ yvec))((q ∙ ((x, y)#p) ∙ N) ≺' (q ∙ ((x, y)#p) ∙ P') apply(subst boundOutputChainAlpha[where p=q and xvec=xvec and yvec="xvec"]) defer apply assumption apply simp apply(simp add: eqvts) apply(simp add: eqvts) apply(simp add: boundOutputFreshSet(4)) apply(rule conjI) apply(simp add: freshChainSimps) apply(simp add: freshChainSimps) done moreover hence "… = (ν*(q ∙ xvec@y#(p ∙ yvec)))((q@(x, y)#p) ∙ N) ≺' ((q@(x, y)#p) ∙ by(simp only: pt2[OF pt_name_inst] boundOutputApp BOresChain.simps) ultimately show ?thesis by(simp only: residualInject) qed moreover have "Ψ ⊳ ([(x, y)] ∙ P) ⟼M(ν*((q ∙ xvec)@(p ∙ yvec)))⟨((q@(x, y)#p) ∙ N)⟩ ≺ (( proof - note‹Ψ ⊳ P ⟼M(ν*(xvec@yvec))⟨N⟩ ≺ P'› moreover from ‹(p ∙ yvec) ♯* N› ‹(q ∙ xvec) ♯* N› ‹xvec ♯* yvec› ‹(q ∙ xvec) ♯* yv have "((q@p) ∙ (xvec @ yvec)) ♯* N" apply(simp only: eqvts) apply(simp only: pt2[OF pt_n by simp moreover from ‹(p ∙ yvec) ♯* P'› ‹(q ∙ xvec) ♯* P'› ‹xvec ♯* yvec› ‹(q ∙ xvec) ♯* have "((q@p) ∙ (xvec @ yvec)) ♯* P'" by(simp del: freshAlphaPerm add: eqvts pt2[OF pt_na moreover from Sp Sq ‹xvec ♯* yvec› ‹(q ∙ xvec) ♯* yvec› ‹(q ∙ xvec) ♯* (p ∙ yvec)› ‹ have Spq: "set(q@p) ⊆ set(xvec@yvec) × set((q@p) ∙ (xvec@yvec))" by(simp add: pt2[OF pt_name_inst] eqvts) blast ultimately have "Ψ ⊳ P ⟼M(ν*((q@p) ∙ (xvec@yvec)))⟨((q@p) ∙ N)⟩ ≺ ((q@p) ∙ P')" apply(simp only: residualInject) by(erule_tac rev_mp) (subst boundOutputChainAlpha, auto) with Sp Sq ‹xvec ♯* yvec› ‹(q ∙ xvec) ♯* yvec› ‹(q ∙ xvec) ♯* (p ∙ yvec)› ‹(p ∙ yve have "Ψ ⊳ P ⟼M(ν*((q ∙ xvec)@(p ∙ yvec)))⟨((q@p) ∙ N)⟩ ≺ ((q@p) ∙ P')" by(simp add: eqvts pt2[OF pt_name_inst] del: freshAlphaPerm) hence "([(x, y)] ∙ Ψ) ⊳ ([(x, y)] ∙ P) ⟼ [(x, y)] ∙ (M(ν*((q ∙ xvec)@(p ∙ yvec)))⟨((q@ by(rule semantics.eqvt) with ‹x ♯ Ψ› ‹y ♯ Ψ› ‹x ♯ M› ‹y ♯ M› ‹x ♯ xvec› ‹y ♯ xvec› ‹x ♯ (q ∙ xvec)› ‹y ♯ (q show ?thesis apply(simp add: eqvts pt2[OF pt_name_inst]) by(subst perm_compose[of q], simp)+ qed moreover from ‹x ∈ supp N› have "((q@(x, y)#p) ∙ x) ∈ ((q@(x, y)#p) ∙ (supp N))" by(simp add: pt_set_bij[OF pt_name_inst, OF at_name_inst]) with ‹x ♯ xvec› ‹x ♯ yvec› ‹x ♯ (q ∙ xvec)› ‹x ♯ (p ∙ yvec)› ‹y ♯ xvec› ‹y ♯ (q ∙ have "y ∈ supp((q@(x, y)#p)∙ N)" by(simp add: pt2[OF pt_name_inst] calc_atm eqvts) moreover from ‹distinct xvec› have "distinct(q ∙ xvec)" by simp moreover from ‹distinct yvec› have "distinct(p ∙ yvec)" by simp moreover note ‹x ♯ (q ∙ xvec)› ‹x ♯ (p ∙ yvec)› ‹x ♯ M› ‹x ♯ Ψ› ‹(q ∙ xvec) ♯* Ψ› ‹(q ∙ xvec) ♯* P› ‹(q ∙ xvec) ♯* M› ‹(q ∙ xvec) ♯* (p ∙ yvec)› ‹(p ∙ yvec) ♯* Ψ› ‹(p ∙ yvec) ♯* P› ‹(p ∙ yvec) ♯* M› ‹y ♯ (q ∙ xvec)› ‹y ♯ (p ∙ ‹(p ∙ yvec) ♯* C› ‹(q ∙ xvec) ♯* C› ultimately show Prop by(rule_tac rOpen) (assumption | simp)+ next case(cScope P α P' x) obtain p::"name prm" where "(bn(p ∙ α)) ♯* Ψ" and "(bn(p ∙ α)) ♯* P" and "(bn(p ∙ α)) ♯* α" and "(bn(p ∙ α)) ♯* P'" and "x ♯ bn(p ∙ α)" and "distinctPerm p" and "(bn(p ∙ α)) ♯* C" and Sp: "(set p) ⊆ set(bn α) × (set(bn(p ∙ α)))" by(rule_tac xvec="bn α" and c="(Ψ, P, α, x, P', C)" in name_list_avoiding) (auto simp add: e obtain y::name where "y ♯ Ψ" and "y ♯ P" and "y ♯ (p ∙ P')" and "y ♯ (p ∙ α)" and "y ♯ C" by(generate_fresh "name") (auto simp add: freshChainSimps simp del: actionFresh) from ‹bn α ♯* subject α› ‹distinctPerm p› have "bn(p ∙ α) ♯* subject(p ∙ α)" by(subst fresh_star_bij[symmetric, of _ _ p]) (simp add: eqvts) from ‹distinct(bn α)› ‹distinctPerm p› have "distinct(bn(p ∙ α))" by(subst distinctClosed[symmetric, of _ p]) (simp add: eqvts) from ‹x ♯ α› ‹x ♯ (bn(p ∙ α))› ‹distinctPerm p› Sp have "x ♯ (p ∙ α)" by(subst fresh_bij[symmetric, of _ _ p]) (simp add: eqvts freshChainSimps) moreover from ‹cP = (νx)P› ‹y ♯ P› have "cP = (νy)([(x, y)] ∙ P)" by(simp add: alphaRes) moreover from ‹cRs = α ≺ (νx)P'› ‹bn α ♯* subject α› ‹(bn(p ∙ α)) ♯* α› ‹x ♯ bn(p ∙ α)› ‹(b have "cRs = (p ∙ α) ≺ (νx)(p ∙ P')" by(force simp add: residualAlpha) with ‹y ♯ (p ∙ P')› have "cRs = (p ∙ α) ≺ (νy)([(x, y)] ∙ p ∙ P')" by(simp add: alphaRes) moreover from ‹Ψ ⊳ P ⟼α ≺ P'› ‹bn α ♯* subject α› ‹(bn(p ∙ α)) ♯* α› ‹(bn(p ∙ α)) ♯* P'› have "Ψ ⊳ P ⟼(p ∙ α) ≺ (p ∙ P')" by(force simp add: residualAlpha) hence"([(x, y)] ∙ Ψ) ⊳ ([(x, y)] ∙ P) ⟼[(x, y)] ∙ ((p ∙ α) ≺ (p ∙ P'))" by(rule eqvts) with ‹x ♯ Ψ› ‹y ♯ Ψ› ‹y ♯ (p ∙ α)› ‹x ♯ (p ∙ α)› Sp ‹distinctPerm p› have "Ψ ⊳ ([(x, y)] ∙ P) ⟼(p ∙ α) ≺ ([(x, y)] ∙ p ∙ P')" by(simp add: eqvts) moreover from ‹bn(p ∙ α) ♯* P› ‹y ♯ (p ∙ α)› ‹y ♯ P› have "bn(p ∙ α) ♯* ([(x, y)] ∙ P) by(auto simp add: fresh_star_def fresh_left calc_atm) (simp add: fresh_def name_list_supp moreover from ‹distinct(bn α)› have "distinct(p ∙ bn α)" by simp hence "distinct(bn(p ∙ α))" by(simp add: eqvts) ultimately show ?thesis using ‹y ♯ Ψ› ‹y ♯ (p ∙ α)› ‹y ♯ C› ‹bn(p ∙ α) ♯* Ψ› ‹bn(p ∙ α) ♯* subject(p ∙ α)› ‹bn( by(rule_tac rScope) next case(Bang P) thus ?thesis by(rule_tac rBang) qed nominal_primrec inputLength :: "('a::fs_name, 'b::fs_name, 'c::fs_name) psi → nat" and inputLength' :: "('a::fs_name, 'b::fs_name, 'c::fs_name) input → nat" and inputLength'' :: "('a::fs_name, 'b::fs_name, 'c::fs_name) psiCase → nat" where "inputLength (0) = 0" | "inputLength (M⟨N⟩.P) = 0" | "inputLength (M(I) = inputLength' I" | "inputLength (Case C) = 0" | "inputLength (P ∥ Q) = 0" | "inputLength ((νx)P) = 0" | "inputLength ({Ψ}) = 0" | "inputLength (!P) = 0" | "inputLength' (Trm M P) = 0" | "inputLength' (ν y I) = 1 + (inputLength' I)" | "inputLength'' (⊥c) = 0" | "inputLength'' (◻Φ → P C) = 0" apply(finite_guess)+ apply(rule TrueI)+ by(fresh_guess add: fresh_nat)+ nominal_primrec boundOutputLength :: "('a, 'b, 'c) boundOutput → nat" where "boundOutputLength (BOut M P) = 0" | "boundOutputLength (BStep x B) = (boundOutputLength B) + 1" apply(finite_guess)+ apply(rule TrueI)+ by(fresh_guess add: fresh_nat)+ nominal_primrec residualLength :: "('a, 'b, 'c) residual → nat" where "residualLength (RIn M N P) = 0" | "residualLength (ROut M B) = boundOutputLength B" | "residualLength (RTau P) = 0" by(rule TrueI)+ lemma inputLengthProc[simp]: shows "inputLength(M(λ*xvec N).P) = length xvec" by(induct xvec) auto lemma boundOutputLengthSimp[simp]: shows "residualLength(M(ν*xvec)⟨N⟩ ≺ P) = length xvec" by(induct xvec) (auto simp add: residualInject) lemma boundOuputLengthSimp2[simp]: shows "residualLength(α ≺ P) = length(bn α)" by(nominal_induct α rule: action.strong_induct, auto) (auto simp add: residualInject) lemmas [simp del] = inputLength_inputLength'_inputLength''.simps residualLength.simps lemma constructPerm: fixes xvec :: "name list" and yvec :: "name list" assumes "length xvec = length yvec" and "xvec ♯* yvec" and "distinct xvec" and "distinct yvec" obtains p where "set p ⊆ set xvec × set(p ∙ xvec)" and "distinctPerm p" and "yvec = p ∙ proof - assume "∧p. [set p ⊆ set xvec × set (p ∙ xvec); distinctPerm p; yvec = p ∙ xvec] moreover obtain n where "n = length xvec" by auto with assms have "∃p. (set p) ⊆ (set xvec) × set (yvec) ∧ distinctPerm p ∧ yvec = p proof(induct n arbitrary: xvec yvec) case(0 xvec yvec) thus ?case by simp next case(Suc n xvec yvec) from ‹Suc n = length xvec› obtain x xvec' where "xvec = x#xvec'" and "length xvec' = n" by(case_tac xvec) auto from ‹length xvec = length yvec› ‹xvec = x # xvec'› obtain y yvec' where "length xvec' = length yvec'" and "yvec = y#yvec'" by(case_tac yvec) auto from ‹xvec = x#xvec'› ‹yvec=y#yvec'› ‹xvec ♯* yvec› have "x ≠ y" and "xvec' ♯* yvec'" and "x ♯ yvec'" and "y ♯ xvec'" by(auto simp add: fresh_list_cons) from ‹distinct xvec› ‹distinct yvec› ‹xvec=x#xvec'› ‹yvec=y#yvec'› have "x ♯ xvec' by simp+ from ‹Suc n = length xvec› ‹xvec=x#xvec'› have "n = length xvec'" by simp with ‹length xvec' = length yvec'› ‹xvec' ♯* yvec'› ‹distinct xvec'› ‹distinct yv obtain p where S: "set p ⊆ set xvec' × set yvec'" and "distinctPerm p" and "yvec' = p ∙ by(drule_tac Suc) auto from S have "set((x, y)#p) ⊆ set(x#xvec') × set(y#yvec')" by auto moreover from ‹x ♯ xvec'› ‹x ♯ yvec'› ‹y ♯ xvec'› ‹y ♯ yvec'› S have "x ♯ p" and "y ♯ apply(induct p) by(auto simp add: fresh_list_nil fresh_list_cons fresh_prod name_list_supp) (auto simp ad with S ‹distinctPerm p› ‹x ≠ y› have "distinctPerm((x, y)#p)" by auto moreover from ‹yvec' = p ∙ xvec'› ‹x ♯ p› ‹y ♯ p› ‹x ♯ xvec'› ‹y ♯ xvec'› have "(y# by(simp add: calc_atm freshChainSimps) ultimately show ?case using ‹xvec=x#xvec'› ‹yvec=y#yvec'› by blast qed ultimately show ?thesis by blast qed lemma distinctApend[simp]: fixes xvec :: "name list" and yvec :: "name list" shows "(set xvec ∩ set yvec = {}) = xvec ♯* yvec" by(auto simp add: fresh_star_def name_list_supp fresh_def) lemma lengthAux: fixes xvec :: "name list" and y :: name and yvec :: "name list" assumes "length xvec = length(y#yvec)" obtains z zvec where "xvec = z#zvec" and "length zvec = length yvec" using assms by(induct xvec arbitrary: yvec y) auto lemma lengthAux2: fixes xvec :: "name list" and yvec :: "name list" and zvec :: "name list" assumes "length xvec = length(yvec@y#zvec)" obtains xvec1 x xvec2 where "xvec=xvec1@x#xvec2" and "length xvec1 = length yvec" and "le proof - assume "∧xvec1 x xvec2. [xvec = xvec1 @ x # xvec2; length xvec1 = length yvec; length xvec2 = length zvec] ==> thesis" moreover from assms have "∃xvec1 x xvec2. xvec=xvec1@x#xvec2 ∧ length xvec1 = length apply(rule_tac x="take (length yvec) xvec" in exI) apply(rule_tac x="hd(drop (length yvec) xvec)" in exI) apply(rule_tac x="tl(drop (length yvec) xvec)" in exI) by auto ultimately show ?thesis by blast qed lemma semanticsCases[consumes 11, case_names cInput cCase cPar1 cPar2 cComm1 cComm2 cScope fixes Ψ :: 'b and cP :: "('a, 'b, 'c) psi" and cRs :: "('a, 'b, 'c) residual" and C :: "'d::fs_name" and x1 :: name and x2 :: name and xvec1 :: "name list" and xvec2 :: "name list" and xvec3 :: "name list" and xvec4 :: "name list" and xvec5 :: "name list" assumes "Ψ ⊳ cP ⟼cRs" and "length xvec1 = inputLength cP" and "distinct xvec1" and "length xvec2 = residualLength cRs" and "distinct xvec2" and "length xvec3 = residualLength cRs" and "distinct xvec3" and "length xvec4 = residualLength cRs" and "distinct xvec4" and "length xvec5 = residualLength cRs" and "distinct xvec5" and rInput: "∧M K N Tvec P. ([xvec1 ♯* Ψ; xvec1 ♯* cP; xvec1 ♯* cRs] ==> cP = M(λ*xve Ψ ⊨ M ↔ K ∧ distinct xvec1 ∧ set xvec1 ⊆ supp N ∧ length xvec1=length Tvec ∧ xvec1 ♯* Tvec ∧ xvec1 ♯* Ψ ∧ xvec1 ♯* M ∧ xvec1 ♯* K) ==> Prop" and rOutput: "∧M K N P. [cP = M⟨N⟩.P; cRs = K⟨N⟩ ≺ P; Ψ ⊨ M ↔ K] ==> Prop" and rCase: "∧Cs P φ. [cP = Cases Cs; Ψ ⊳ P ⟼ cRs; (φ, P) mem Cs; Ψ ⊨ φ; guarded P] ==> and rPar1: "∧ΨQ P α P' Q AQ. ([xvec2 ♯* Ψ; xvec2 ♯* cP; xvec2 ♯* cRs] ==> cP = P ∥ Q ∧ cRs = α ≺ (P' ∥ Q) ∧ xvec2 = bn α ∧ Ψ ⊗ ΨQ ⊳ P ⟼α ≺ P' ∧ extractFrame Q = ⟨AQ, ΨQ⟩ ∧ distinct AQ ∧ AQ ♯* P ∧ AQ ♯* Q ∧ AQ ♯* Ψ ∧ AQ ♯* α ∧ AQ ♯* P' ∧ AQ ♯* C) ==> Prop" and rPar2: "∧ΨP Q α Q' P AP. ([xvec3 ♯* Ψ; xvec3 ♯* cP; xvec3 ♯* cRs] ==> cP = P ∥ Q ∧ cRs = α ≺ (P ∥ Q') ∧ xvec3 = bn α ∧ Ψ ⊗ ΨP ⊳ Q ⟼α ≺ Q' ∧ extractFrame P = ⟨AP, ΨP⟩ ∧ distinct AP ∧ AP ♯* P ∧ AP ♯* Q ∧ AP ♯* Ψ ∧ AP ♯* α ∧ AP ♯* Q' ∧ AP ♯* C) ==> Prop" and rComm1: "∧ΨQ P M N P' AP ΨP Q K xvec Q' AQ. [cP = P ∥ Q; cRs = τ ≺ (ν*xvec)P' ∥ Q'; Ψ ⊗ ΨQ ⊳ P ⟼M(N) ≺ P'; extractFrame P = ⟨AP, ΨP⟩; distinct AP; Ψ ⊗ ΨP ⊳ Q ⟼ K(ν*xvec)⟨N⟩ ≺ Q'; extractFrame Q = ⟨AQ, ΨQ⟩; distinct AQ; Ψ ⊗ ΨP ⊗ ΨQ ⊨ M ↔ K; AP ♯* Ψ; AP ♯* ΨQ; AP ♯* P; AP ♯* M; AP ♯* N; AP ♯* P'; AP ♯* Q; AP ♯* Q'; AP ♯* AQ; AP ♯* xvec; AQ ♯* Ψ; AQ ♯* ΨP; AQ ♯* P; AQ ♯* K; AQ ♯* N; AQ ♯* P'; AQ ♯* Q; AQ ♯* Q'; AQ ♯* xvec; xvec ♯* Ψ; xvec ♯* ΨP; xvec ♯* ΨQ; xvec ♯* P; xvec ♯* M; xvec ♯* Q; xvec ♯* K; AP ♯* C; AQ ♯* C; xvec ♯* C; distinct xvec] ==> Prop" and rComm2: "∧ΨQ P M xvec N P' AP ΨP Q K Q' AQ. [cP = P ∥ Q; cRs = τ ≺ (ν*xvec)P' ∥ Q'; Ψ ⊗ ΨQ ⊳ P ⟼M(ν*xvec)⟨N⟩ ≺ P'; extractFrame P = ⟨AP, ΨP⟩; distinct AP; Ψ ⊗ ΨP ⊳ Q ⟼ K(N) ≺ Q'; extractFrame Q = ⟨AQ, ΨQ⟩; distinct AQ; Ψ ⊗ ΨP ⊗ ΨQ ⊨ M ↔ K; AP ♯* Ψ; AP ♯* ΨQ; AP ♯* P; AP ♯* M; AP ♯* N; AP ♯* P'; AP ♯* Q; AP ♯* Q'; AP ♯* AQ; AP ♯* xvec; AQ ♯* Ψ; AQ ♯* ΨP; AQ ♯* P; AQ ♯* K; AQ ♯* N; AQ ♯* P'; AQ ♯* Q; AQ ♯* Q'; AQ ♯* xvec; xvec ♯* Ψ; xvec ♯* ΨP; xvec ♯* ΨQ; xvec ♯* P; xvec ♯* M; xvec ♯* Q; xvec ♯* K; AP ♯* C; AQ ♯* C; xvec ♯* C; distinct xvec] ==> Prop" and rOpen: "∧P M xvec y yvec N P'. ([xvec4 ♯* Ψ; xvec4 ♯* cP; xvec4 ♯* cRs; x1 ♯ Ψ; x1 ♯ cP; x1 ♯ cRs; x1 ♯ xvec4] == cP = (νx1)P ∧ cRs = M(ν*(xvec@x1#yvec))⟨N⟩ ≺ P' ∧ xvec4=xvec@y#yvec ∧ Ψ ⊳ P ⟼M(ν*(xvec@yvec))⟨N⟩ ≺ P' ∧ x1 ∈ supp N ∧ x1 ♯ xvec ∧ x1 ♯ yvec ∧ distinct xvec ∧ distinct yvec ∧ xvec ♯* Ψ ∧ xvec ♯* P ∧ xvec ♯* M ∧ xvec ♯* yvec yvec ♯* Ψ ∧ yvec ♯* P ∧ yvec ♯* M) ==> Prop" and rScope: "∧P α P'. ([xvec5 ♯* Ψ; xvec5 ♯* cP; xvec5 ♯* cRs; x2 ♯ Ψ; x2 ♯ cP; x2 ♯ cP = (νx2)P ∧ cRs = α ≺ (νx2)P' ∧ xvec5 = bn α ∧ Ψ ⊳ P ⟼α ≺ P' ∧ x2 ♯ Ψ ∧ x2 ♯ α ∧ bn α ♯* subject α ∧ distinct(bn α)) ==> Prop" and rBang: "∧P. [cP = !P; Ψ ⊳ P ∥ !P ⟼ cRs; guarded P] ==> Prop" shows Prop using ‹Ψ ⊳ cP ⟼cRs› proof(cases rule: semanticsCasesAux[where C="(xvec1, xvec2, xvec3, xvec4, xvec5, x1 case(cInput M K xvec N Tvec P) have B: "cP = M(λ*xvec N).P" and C: "cRs = K((N[xvec::=Tvec])) ≺ (P[xvec::=Tvec])" by fact+ from ‹xvec ♯* (xvec1, xvec2, xvec3, xvec4, xvec5, x1, x2, cP, cRs, C)› have "xvec from ‹length xvec1 = inputLength cP› B have "length xvec1 = length xvec" by simp then obtain p where S: "set p ⊆ set xvec × set(p ∙ xvec)" and "distinctPerm p" and "xvec1 using ‹xvec ♯* xvec1› ‹distinct xvec› ‹distinct xvec1› by(rule_tac constructPerm[where xvec=xvec and yvec=xvec1]) auto show ?thesis proof(rule rInput[where M=M and K=K and N = "p ∙ N" and Tvec=Tvec and P="p ∙ P"], goal_cases) case 1 from B ‹xvec ♯* xvec1› ‹xvec1 ♯* cP› have "xvec1 ♯* N" and "xvec1 ♯* P" by(auto simp add: fresh_star_def inputChainFresh name_list_supp) (auto simp add: fresh_de from ‹cP = M(λ*xvec N).P› S ‹xvec1 ♯* N› ‹xvec1 ♯* P› ‹xvec1 = p ∙ xvec› have "cP = M(λ*xvec1 (p ∙ N)).(p ∙ P)" apply simp by(subst inputChainAlpha) auto moreover from ‹cRs = K((N[xvec::=Tvec])) ≺ P[xvec::=Tvec]› S ‹xvec1 ♯* N› ‹xvec1 ♯* P have "cRs = K(((p ∙ N)[xvec1::=Tvec])) ≺ (p ∙ P)[xvec1::=Tvec]" by(simp add: renaming substTerm.renaming) moreover note ‹Ψ ⊨ M ↔ K› moreover from ‹distinct xvec› ‹xvec1 = p ∙ xvec› have "distinct xvec1" by simp moreover from ‹set xvec ⊆ supp N› have "(p ∙ set xvec) ⊆ (p ∙ (supp N))" by(simp add: eqvts) with ‹xvec1 = p ∙ xvec› have "set xvec1 ⊆ supp(p ∙ N)" by(simp add: eqvts) moreover from ‹length xvec = length Tvec› ‹xvec1 = p ∙ xvec› have "length xvec1 = l by simp moreover from ‹xvec1 ♯* cRs› C ‹length xvec = length Tvec› ‹distinct xvec› ‹set xve have "(set xvec1) ♯* Tvec" by(rule_tac substTerm.subst3Chain[where T=N]) auto hence "xvec1 ♯* Tvec" by simp moreover from ‹xvec ♯* Tvec› have "(p ∙ xvec) ♯* (p ∙ Tvec)" by(simp add: fresh_star_b with S ‹xvec ♯* Tvec› ‹xvec1 ♯* Tvec› ‹xvec1 = p ∙ xvec› have "xvec1 ♯* Tvec" by simp moreover from ‹xvec ♯* M› have "(p ∙ xvec) ♯* (p ∙ M)" by(simp add: fresh_star_bij) with S ‹xvec ♯* M› ‹xvec1 ♯* cP› B ‹xvec1 = p ∙ xvec› have "xvec1 ♯* M" by simp moreover from ‹xvec ♯* K› have "(p ∙ xvec) ♯* (p ∙ K)" by(simp add: fresh_star_bij) with S ‹xvec ♯* K› ‹xvec1 ♯* cRs› C ‹xvec1 = p ∙ xvec› have "xvec1 ♯* K" by simp ultimately show ?case using ‹xvec1 ♯* Ψ› by blast qed next case(cOutput M K N P) thus ?thesis by(rule rOutput) next case(cCase Cs P φ) thus ?thesis by(rule rCase) next case(cPar1 ΨQ P α P' Q AQ) have B: "cP = P ∥ Q" and C: "cRs = α ≺ P' ∥ Q" by fact+ from ‹bn α ♯* (xvec1, xvec2, xvec3, xvec4, xvec5, x1, x2, cP, cRs, C)› have "bn α ♯* from ‹AQ ♯* (xvec1, xvec2, xvec3, xvec4, xvec5, x1, x2, cP, cRs, C)› have "AQ ♯* x from ‹length xvec2 = residualLength cRs› C have "length xvec2 = length(bn α)" by simp then obtain p where S: "set p ⊆ set(bn α) × set(bn(p ∙ α))" and "distinctPerm p" and "xvec2 using ‹bn α ♯* xvec2› ‹distinct(bn α)› ‹distinct xvec2› by(rule_tac constructPerm[where xvec="bn α" and yvec=xvec2]) (auto simp add: eqvts) show ?thesis proof(rule rPar1[where P=P and Q=Q and α="p ∙ α" and P'="p ∙ P'" and AQ=AQ and ΨQ=ΨQ], goal_cases case 1 note ‹cP = P ∥ Q› moreover from B C S ‹bn α ♯* xvec2› ‹xvec2 ♯* cRs› ‹xvec2 = bn(p ∙ α)› ‹bn α ♯* subject α have "cRs = (p ∙ α) ≺ (p ∙ P') ∥ Q" apply auto by(subst residualAlpha[where p=p]) auto moreover note ‹xvec2 = bn(p ∙ α)› moreover from ‹Ψ ⊗ ΨQ ⊳ P ⟼α ≺ P'› S B C S ‹bn α ♯* xvec2› ‹xvec2 ♯* cRs› ‹xvec2 = bn(p ∙ α have "Ψ ⊗ ΨQ ⊳ P ⟼(p ∙ α) ≺ (p ∙ P')" by(subst residualAlpha[symmetric]) auto moreover note ‹extractFrame Q = ⟨AQ, ΨQ⟩› ‹distinct AQ› ‹AQ ♯* P› ‹AQ ♯* Q› ‹AQ ♯* Ψ› moreover from ‹AQ ♯* α› ‹AQ ♯* xvec2› S ‹xvec2 = bn(p ∙ α)› ‹distinctPerm p› have "AQ by(subst fresh_star_bij[symmetric, where pi=p]) simp moreover from ‹AQ ♯* P'› ‹AQ ♯* α› ‹AQ ♯* xvec2› S ‹xvec2 = bn(p ∙ α)› ‹distinctPerm by(subst fresh_star_bij[symmetric, where pi=p]) simp moreover note ‹AQ ♯* C› ultimately show ?case by blast qed next case(cPar2 ΨP Q α Q' P AP) have B: "cP = P ∥ Q" and C: "cRs = α ≺ P ∥ Q'" by fact+ from ‹bn α ♯* (xvec1, xvec2, xvec3, xvec4, xvec5, x1, x2, cP, cRs, C)› have "bn α ♯* from ‹AP ♯* (xvec1, xvec2, xvec3, xvec4, xvec5, x1, x2, cP, cRs, C)› have "AP ♯* x from ‹length xvec3 = residualLength cRs› C have "length xvec3 = length(bn α)" by simp then obtain p where S: "set p ⊆ set(bn α) × set(bn(p ∙ α))" and "distinctPerm p" and "xvec3 using ‹bn α ♯* xvec3› ‹distinct(bn α)› ‹distinct xvec3› by(rule_tac constructPerm[where xvec="bn α" and yvec=xvec3]) (auto simp add: eqvts) show ?thesis proof(rule rPar2[where P=P and Q=Q and α="p ∙ α" and Q'="p ∙ Q'" and AP=AP and ΨP=ΨP], goal_cases case 1 note ‹cP = P ∥ Q› moreover from B C S ‹bn α ♯* xvec3› ‹xvec3 ♯* cRs› ‹xvec3 = bn(p ∙ α)› ‹bn α ♯* subject α have "cRs = (p ∙ α) ≺ P ∥ (p ∙ Q')" apply auto by(subst residualAlpha[where p=p]) auto moreover note ‹xvec3 = bn(p ∙ α)› moreover from ‹Ψ ⊗ ΨP ⊳ Q ⟼α ≺ Q'› S B C S ‹bn α ♯* xvec3› ‹xvec3 ♯* cRs› ‹xvec3 = bn(p ∙ α have "Ψ ⊗ ΨP ⊳ Q ⟼(p ∙ α) ≺ (p ∙ Q')" by(subst residualAlpha[symmetric]) auto moreover note ‹extractFrame P = ⟨AP, ΨP⟩› ‹distinct AP› ‹AP ♯* P› ‹AP ♯* Q› ‹AP ♯* Ψ› moreover from ‹AP ♯* α› ‹AP ♯* xvec3› S ‹xvec3 = bn(p ∙ α)› ‹distinctPerm p› have "AP by(subst fresh_star_bij[symmetric, where pi=p]) simp moreover from ‹AP ♯* Q'› ‹AP ♯* α› ‹AP ♯* xvec3› S ‹xvec3 = bn(p ∙ α)› ‹distinctPerm by(subst fresh_star_bij[symmetric, where pi=p]) simp moreover note ‹AP ♯* C› ultimately show ?case by blast qed next case(cComm1 ΨQ P M N P' AP ΨP Q K xvec Q' AQ) thus ?thesis by(rule_tac rComm1[where P=P and Q=Q]) (assumption | simp)+ next case(cComm2 ΨQ P M xvec N P' AP ΨP Q K Q' AQ) thus ?thesis by(rule_tac rComm2[where P=P and Q=Q]) (assumption | simp)+ next case(cOpen P M xvec yvec N P' x) have B: "cP = (νx)P" and C: "cRs = M(ν*(xvec@x#yvec))⟨N⟩ ≺ P'" by fact+ from ‹xvec ♯* (xvec1, xvec2, xvec3, xvec4, xvec5, x1, x2, cP, cRs, C)› have "xvec from ‹x ♯ (xvec1, xvec2, xvec3, xvec4, xvec5, x1, x2, cP, cRs, C)› have "x ♯ xvec4 from ‹yvec ♯* (xvec1, xvec2, xvec3, xvec4, xvec5, x1, x2, cP, cRs, C)› have "yvec from ‹xvec ♯* cRs› ‹x ♯ cRs› ‹yvec ♯* cRs› C have "(xvec@x#yvec) ♯* M" by simp from ‹xvec ♯* Ψ› ‹x ♯ Ψ› ‹yvec ♯* Ψ› have "(xvec@x#yvec) ♯* Ψ" by simp from ‹length xvec4 = residualLength cRs› C obtain xvec' y yvec' where D: "xvec4 = xvec'@ by(rule_tac lengthAux2) auto with ‹distinct xvec› ‹distinct yvec› ‹x ♯ xvec› ‹x ♯ yvec› ‹xvec ♯* yvec› ‹xvec ♯ have "distinct xvec'" and "distinct yvec'" and "xvec' ♯* yvec'" and "x ≠ y" and "y ♯ xvec and "x ♯ xvec'" and "x ♯ yvec'" and "y ♯ xvec" and "y ♯ yvec" and "xvec ♯* xvec'" and "yvec by auto from ‹length xvec' = length xvec› ‹xvec ♯* xvec'› ‹distinct xvec› ‹distinct xvec' obtain p where Sp: "set p ⊆ set xvec × set(p ∙ xvec)" and "distinctPerm p" and E: "xvec' by(metis constructPerm) from ‹length yvec' = length yvec› ‹yvec ♯* yvec'› ‹distinct yvec› ‹distinct yvec' obtain q where Sq: "set q ⊆ set yvec × set(q ∙ yvec)" and "distinctPerm q" and F: "yvec' by(metis constructPerm) show ?thesis proof(rule rOpen[where P="([(x, x1)] ∙ P)" and xvec="p ∙ xvec" and y="y" and yvec="q ∙ y case 1 from ‹xvec ♯* xvec4› ‹x ♯ xvec4› ‹x1 ♯ xvec4› ‹yvec ♯* xvec4› D E F have "x ≠ y" and "x1 ≠ y" and "x1 ♯ p ∙ xvec" and "x1 ♯ q ∙ yvec" by simp+ from ‹xvec4 ♯* cRs› ‹x1 ♯ cRs› C have "xvec4 ♯* M" and "x1 ♯ M" by simp+ from ‹cP = (νx)P› ‹x ♯ cP› ‹x ≠ x1› have "([(x, x1)] ∙ cP) = [(x, x1)] ∙ (νx)P" by simp with ‹x ♯ cP› ‹x1 ♯ cP› have "cP = (νx1)([(x, x1)] ∙ P)" by(simp add: eqvts calc_atm) moreover from C have "((p@(x1, x)#q) ∙ cRs) = (p@(x1, x)#q) ∙ (M(ν*(xvec@x#yvec))⟨N⟩ ≺ P' with Sp Sq ‹xvec4 ♯* cRs› D E F ‹xvec ♯* cRs› ‹x ♯ cRs› ‹yvec ♯* cRs› ‹xvec4 ♯* M› ‹(x have "cRs = M(ν*((p ∙ xvec)@x1#(q ∙ yvec)))⟨((p@(x1, x)#q) ∙ N)⟩ ≺ ((p@(x1, x)#q) ∙ P' by(simp add: eqvts pt2[OF pt_name_inst] calc_atm) moreover from D E F have "xvec4 = (p ∙ xvec)@y#(q ∙ yvec)" by simp moreover from ‹Ψ ⊳ P ⟼M(ν*(xvec@yvec))⟨N⟩ ≺ P'› have "((p@(x1, x)#q) ∙ Ψ) ⊳ ((p@(x1, x)#q) by(intro eqvts) with Sp Sq B C D E F ‹xvec4 ♯* Ψ› ‹(xvec@x#yvec) ♯* Ψ› ‹xvec4 ♯* cRs› ‹x ♯ xvec4› C D ‹x ♯ c have "Ψ ⊳ ([(x, x1)] ∙ P) ⟼M(ν*((p ∙ xvec)@(q ∙ yvec)))⟨((p@(x1, x)#q) ∙ N)⟩ ≺ ((p@(x1, by(simp add: eqvts pt_fresh_bij[OF pt_name_inst, OF at_name_inst] pt2[OF pt_name_inst] na moreover from ‹x ∈ supp N› have "((p@(x1, x)#q) ∙ x) ∈ ((p@(x1, x)#q) ∙ supp N)" by(simp add: pt_set_bij[OF pt_name_inst, OF at_name_inst]) hence "x1 ∈ supp((p@(x1, x)#q) ∙ N)" using ‹x ♯ xvec› ‹x ♯ yvec› ‹x1 ♯ xvec› ‹x1 ♯ yvec› ‹x ♯ xvec4› ‹x1 ♯ xvec4› ‹xve by(simp add: eqvts pt2[OF pt_name_inst] calc_atm) moreover from ‹x1 ♯ xvec4› D E F have "x1 ♯ (p ∙ xvec)" and "x1 ♯ (q ∙ yvec)" by simp+ moreover from ‹distinct xvec'› ‹distinct yvec'› E F have "distinct(p ∙ xvec)" and "dist moreover from ‹xvec' ♯* yvec'› E F have "(p ∙ xvec) ♯* (q ∙ yvec)" by auto moreover from ‹xvec ♯* Ψ› have "(p ∙ xvec) ♯* (p ∙ Ψ)" by(simp add: pt_fresh_star_bij[OF with Sp D E ‹xvec4 ♯* Ψ› ‹xvec ♯* Ψ› have "(p ∙ xvec) ♯* Ψ" by(simp add: eqvts) moreover from ‹yvec ♯* Ψ› have "(p ∙ yvec) ♯* (p ∙ Ψ)" by(simp add: pt_fresh_star_bij[OF with Sq D F ‹xvec4 ♯* Ψ› ‹yvec ♯* Ψ› have "(q ∙ yvec) ♯* Ψ" by(simp add: eqvts) moreover from ‹xvec4 ♯* cP› ‹x ♯ xvec4› ‹x1 ♯ xvec4› B D E F have "(p ∙ xvec) ♯* ([(x, by simp+ moreover from ‹xvec4 ♯* M› C D E F have "(p ∙ xvec) ♯* M" and "(q ∙ yvec) ♯* M" by simp+ ultimately show ?case by blast qed next case(cScope P α P' x) have B: "cP = (νx)P" and C: "cRs = α ≺ (νx)P'" by fact+ from ‹bn α ♯* (xvec1, xvec2, xvec3, xvec4, xvec5, x1, x2, cP, cRs, C)› have "bn α ♯* from ‹x ♯ (xvec1, xvec2, xvec3, xvec4, xvec5, x1, x2, cP, cRs, C)› have "x ♯ xvec5 from ‹length xvec5 = residualLength cRs› C have "length xvec5 = length(bn α)" by simp then obtain p where S: "set p ⊆ set(bn α) × set(bn(p ∙ α))" and "distinctPerm p" and "xvec5 using ‹bn α ♯* xvec5› ‹distinct(bn α)› ‹distinct xvec5› by(rule_tac constructPerm[where xvec="bn α" and yvec=xvec5]) (auto simp add: eqvts) show ?thesis proof(rule rScope[where P="[(x, x2)] ∙ P" and α="[(x, x2)] ∙ p ∙ α" and P'="[(x, x2)] ∙ case 1 from ‹x2 ♯ cRs› C ‹x2 ♯ bn α› ‹x ≠ x2› have "x2 ♯ α" and "x2 ♯ P'" by(auto simp add: abs_fr from ‹cP = (νx)P› ‹x2 ♯ cP› ‹x ≠ x2› have "cP = (νx2)([(x, x2)] ∙ P)" by(simp add: alphaRes abs_fresh) moreover from B C S ‹bn α ♯* xvec5› ‹xvec5 ♯* cRs› ‹xvec5 = bn(p ∙ α)› ‹bn α ♯* subject α have "cRs = (p ∙ α) ≺ (νx)(p ∙ P')" apply auto by(subst residualAlpha[where p=p] alphaRes) (auto simp del: actionFresh) hence "([(x, x2)] ∙ cRs) = [(x, x2)] ∙ ((p ∙ α) ≺ (νx)(p ∙ P'))" by simp with ‹x2 ♯ cRs› ‹x ♯ cRs› have "cRs = ([(x, x2)] ∙ p ∙ α) ≺ (νx2)([(x, x2)] ∙ p ∙ P')" by(simp add: eqvts calc_atm) moreover from ‹xvec5= bn(p ∙ α)› have "([(x, x2)] ∙ xvec5) = ([(x, x2)] ∙ bn(p ∙ α))" by simp with ‹x ♯ xvec5› ‹x2 ♯ xvec5› have "xvec5 = bn([(x, x2)] ∙ p ∙ α)" by(simp add: eqvts) moreover from ‹Ψ ⊳ P ⟼α ≺ P'› S B C S ‹bn α ♯* xvec5› ‹xvec5 ♯* cRs› ‹xvec5 = bn(p ∙ α)› ‹ have "Ψ ⊳ P ⟼(p ∙ α) ≺ (p ∙ P')" by(subst residualAlpha[symmetric]) auto hence "([(x, x2)] ∙ Ψ) ⊳ ([(x, x2)] ∙ P) ⟼([(x, x2)] ∙ ((p ∙ α) ≺ (p ∙ P')))" by(rule eqvt) with ‹x ♯ Ψ› ‹x2 ♯ Ψ› have "Ψ ⊳ ([(x, x2)] ∙ P) ⟼([(x, x2)] ∙ p ∙ α) ≺ ([(x, x2)] ∙ p by(simp add: eqvts) moreover note ‹x2 ♯ Ψ› moreover from ‹x ♯ α› ‹x2 ♯ α› ‹x ♯ xvec5› ‹x2 ♯ xvec5› S ‹x ≠ x2› ‹xvec5 = bn(p ∙ α)› apply(subgoal_tac "x ♯ p ∧ x2 ♯ p") apply(simp add: perm_compose freshChainSimps del: actionFresh) by(auto dest: freshAlphaSwap) moreover from ‹bn α ♯* subject α› have "([(x, x2)] ∙ p ∙ (bn α)) ♯* ([(x, x2)] ∙ p ∙ ( by(simp add: pt_fresh_star_bij[OF pt_name_inst, OF at_name_inst]) hence "bn([(x, x2)] ∙ p ∙ α) ♯* subject([(x, x2)] ∙ p ∙ α)" by(simp add: eqvts) moreover from ‹distinct(bn α)› have "distinct([(x, x2)] ∙ p ∙ (bn α))" by simp hence "distinct(bn([(x, x2)] ∙ p ∙ α))" by(simp add: eqvts) ultimately show ?case by blast qed next case(cBang P) thus ?thesis by(rule_tac rBang) auto qed lemma 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 ⊳ P ∥ Q ⟼α ≺ T" and "bn α ♯* Ψ" and "bn α ♯* P" and "bn α ♯* Q" and "bn α ♯* subject α" and rPar1: "∧P' AQ ΨQ. [Ψ ⊗ ΨQ ⊳ P ⟼α ≺ P'; extractFrame Q = ⟨AQ, ΨQ⟩; distinct AQ; AQ ♯* Ψ; AQ ♯* P; AQ ♯* Q; AQ ♯* α; AQ ♯* P'; AQ ♯* C] ==> Prop α (P' ∥ Q)" and rPar2: "∧Q' AP ΨP. [Ψ ⊗ ΨP ⊳ Q ⟼α ≺ Q'; extractFrame P = ⟨AP, ΨP⟩; distinct AP; AP ♯* Ψ; AP ♯* P; AP ♯* Q; AP ♯* α; AP ♯* Q'; AP ♯* C] ==> Prop α (P ∥ Q')" and rComm1: "∧ΨQ M N P' AP ΨP K xvec Q' AQ. [Ψ ⊗ ΨQ ⊳ P ⟼M(N) ≺ P'; extractFrame P = ⟨AP, ΨP⟩; distinct AP; Ψ ⊗ ΨP ⊳ Q ⟼K(ν*xvec)⟨N⟩ ≺ Q'; extractFrame Q = ⟨AQ, ΨQ⟩; distinct AQ; Ψ ⊗ ΨP ⊗ ΨQ ⊨ M ↔ K; distinct xvec; AP ♯* Ψ; AP ♯* ΨQ; AP ♯* P; AP ♯* M; AP ♯* N; AP ♯* P'; AP ♯* Q; AP ♯* xvec AQ ♯* Ψ; AQ ♯* ΨP; AQ ♯* P; AQ ♯* K; AQ ♯* N; AQ ♯* P'; AQ ♯* Q; AQ ♯* xvec; xvec ♯* Ψ; xvec ♯* ΨP; xvec ♯* P; xvec ♯* M; xvec ♯* K; xvec ♯* Q; xvec ♯* ΨQ; Prop (τ) ((ν*xvec)(P' ∥ Q'))" and rComm2: "∧ΨQ M xvec N P' AP ΨP K Q' AQ. [Ψ ⊗ ΨQ ⊳ P ⟼M(ν*xvec)⟨N⟩ ≺ P'; extractFrame P = ⟨AP, ΨP⟩; distinct AP; Ψ ⊗ ΨP ⊳ Q ⟼K(N) ≺ Q'; extractFrame Q = ⟨AQ, ΨQ⟩; distinct AQ; Ψ ⊗ ΨP ⊗ ΨQ ⊨ M ↔ K; distinct xvec; AP ♯* Ψ; AP ♯* ΨQ; AP ♯* P; AP ♯* M; AP ♯* N; AP ♯* P'; AP ♯* Q; AP ♯* xvec AQ ♯* Ψ; AQ ♯* ΨP; AQ ♯* P; AQ ♯* K; AQ ♯* N; AQ ♯* P'; AQ ♯* Q; AQ ♯* xvec; xvec ♯* Ψ; xvec ♯* ΨP; xvec ♯* P; xvec ♯* M; xvec ♯* K; xvec ♯* Q; xvec ♯* ΨQ; Prop (τ) ((ν*xvec)(P' ∥ Q'))" shows "Prop α T" proof - from Trans have "distinct(bn α)" by(auto dest: boundOutputDistinct) have "length(bn α) = residualLength(α ≺ T)" by simp note Trans moreover have "length [] = inputLength(P ∥ Q)" and "distinct []" by(auto simp add: inputLength_inputLength'_inputLength''.simps) moreover note ‹length(bn α) = residualLength(α ≺ T)› ‹distinct(bn α)› moreover note ‹length(bn α) = residualLength(α ≺ T)› ‹distinct(bn α)› moreover note ‹length(bn α) = residualLength(α ≺ T)› ‹distinct(bn α)› moreover note ‹length(bn α) = residualLength(α ≺ T)› ‹distinct(bn α)› moreover obtain x::name where "x ♯ Ψ" and "x ♯ P" and "x ♯ Q" and "x ♯ α" and "x ♯ T" by(generate_fresh "name") auto ultimately show ?thesis using ‹bn α ♯* Ψ› ‹bn α ♯* P› ‹bn α ♯* Q› ‹bn α ♯* subject α› apply(cases rule: semanticsCases[of _ _ _ _ _ _ _ _ _ C x x]) apply(auto simp add: psi.inject) apply(force simp add: residualInject residualInject' intro: rPar1) apply(force simp add: residualInject residualInject' intro: rPar2) apply(fastforce simp add: residualInject residualInject' intro: rComm1) by(fastforce simp add: residualInject residualInject' intro: rComm2) qed lemma parInputCases[consumes 1, case_names cPar1 cPar2]: fixes Ψ :: 'b and P :: "('a, 'b, 'c) psi" and Q :: "('a, 'b, 'c) psi" and M :: 'a and N :: 'a and R :: "('a, 'b, 'c) psi" and C :: "'d::fs_name" assumes Trans: "Ψ ⊳ P ∥ Q ⟼M(N) ≺ R" and rPar1: "∧P' AQ ΨQ. [Ψ ⊗ ΨQ ⊳ P ⟼M(N) ≺ P'; extractFrame Q = ⟨AQ, ΨQ⟩; distinct AQ; AQ ♯* Ψ; AQ ♯* P; AQ ♯* Q; AQ ♯* M; AQ ♯* N; AQ ♯* C] ==> Prop (P' ∥ Q)" and rPar2: "∧Q' AP ΨP. [Ψ ⊗ ΨP ⊳ Q ⟼M(N) ≺ Q'; extractFrame P = ⟨AP, ΨP⟩; distinct AP; AP ♯* Ψ; AP ♯* P; AP ♯* Q; AP ♯* M; AP ♯* N; AP ♯* C] ==> Prop (P ∥ Q')" shows "Prop R" proof - from Trans obtain α where "Ψ ⊳ P ∥ Q ⟼α ≺ R" and "bn α ♯* Ψ" and "bn α ♯* P" and "bn α ♯* Q" and "bn thus ?thesis using rPar1 rPar2 by(induct rule: parCases) (auto simp add: residualInject) qed lemma parOutputCases[consumes 5, case_names cPar1 cPar2]: fixes Ψ :: 'b and P :: "('a, 'b, 'c) psi" and Q :: "('a, 'b, 'c) psi" and M :: 'a and xvec :: "name list" and N :: 'a and R :: "('a, 'b, 'c) psi" and C :: "'d::fs_name" assumes Trans: "Ψ ⊳ P ∥ Q ⟼M(ν*xvec)⟨N⟩ ≺ R" and "xvec ♯* Ψ" and "xvec ♯* P" and "xvec ♯* Q" and "xvec ♯* M" and rPar1: "∧P' AQ ΨQ. [Ψ ⊗ ΨQ ⊳ P ⟼M(ν*xvec)⟨N⟩ ≺ P'; extractFrame Q = ⟨AQ, ΨQ⟩; distinct A AQ ♯* Ψ; AQ ♯* P; AQ ♯* Q; AQ ♯* M; AQ ♯* xvec; AQ ♯* N; AQ ♯* C; AQ ♯* xvec; dis and rPar2: "∧Q' AP ΨP. [Ψ ⊗ ΨP ⊳ Q ⟼M(ν*xvec)⟨N⟩ ≺ Q'; extractFrame P = ⟨AP, ΨP⟩; distinct AP AP ♯* Ψ; AP ♯* P; AP ♯* Q; AP ♯* M; AP ♯* xvec; AP ♯* N; AP ♯* C; AP ♯* xvec; di shows "Prop R" proof - from Trans have "distinct xvec" by(auto dest: boundOutputDistinct) obtain α where "α=M(ν*xvec)⟨N⟩" by simp with Trans ‹xvec ♯* Ψ› ‹xvec ♯* P› ‹xvec ♯* Q› ‹xvec ♯* M› have "Ψ ⊳ P ∥ Q ⟼α ≺ R" and "bn α ♯* Ψ" and "bn α ♯* P" and "bn α ♯* Q" "bn α ♯* subject α" by simp+ thus ?thesis using ‹α=M(ν*xvec)⟨N⟩› rPar1 rPar2 ‹distinct xvec› by(induct rule: parCases[where C="(xvec, C)"]) (auto simp add: residualInject) qed lemma theEqvt[eqvt_force]: fixes p :: "name prm" and α :: "'a action" assumes "α ≠ τ" shows "(p ∙ the(subject α)) = the(p ∙ (subject α))" using assms by(induct rule: actionCases[where α=α]) auto lemma theSubjectFresh[simp]: fixes α :: "'a action" and x :: name assumes "α ≠ τ" shows "x ♯ the(subject α) = x ♯ subject α" using assms by(cases rule: actionCases) auto lemma theSubjectFreshChain[simp]: fixes α :: "'a action" and xvec :: "name list" assumes "α ≠ τ" shows "xvec ♯* the(subject α) = xvec ♯* subject α" using assms by(cases rule: actionCases) auto lemma obtainPrefix: fixes Ψ :: 'b and P :: "('a, 'b, 'c) psi" and α :: "'a action" and P' :: "('a, 'b, 'c) psi" and AP :: "name list" and ΨP :: 'b and B :: "name list" assumes "Ψ ⊳ P ⟼α ≺ P'" and "extractFrame P = ⟨AP, ΨP⟩" and "distinct AP" and "bn α ♯* subject α" and "distinct(bn α)" and "α ≠ τ" and "B ♯* P" and "AP ♯* Ψ" and "AP ♯* B" and "AP ♯* P" and "AP ♯* subject α" obtains M where "Ψ ⊗ ΨP ⊨ the(subject α) ↔ M" and "B ♯* M" using assms proof(nominal_induct avoiding: B arbitrary: thesis rule: semanticsFrameInduct') case(cAlpha Ψ P α P' p AP ΨP B) then obtain M where subjEq: "Ψ ⊗ ΨP ⊨ the(subject α) ↔ M" and "B ♯* M" by(rule_tac cAlpha) auto from ‹set p ⊆ set(bn α) × set(bn(p ∙ α))› ‹bn α ♯* subject α› ‹bn(p ∙ α) ♯* α› subjEq have "Ψ ⊗ ΨP ⊨ the(subject(p ∙ α)) ↔ M" by(simp add: subjectEqvt[symmetric]) thus ?case using cAlpha ‹B ♯* M› by auto next case(cFrameAlpha Ψ P AP ΨP p α P' B) then obtain M where subjEq: "Ψ ⊗ ΨP ⊨ the(subject α) ↔ M" and "B ♯* M" by(rule_tac cFrameAlpha) auto have S: "set p ⊆ set AP × set (p ∙ AP)" by fact from subjEq have "(p ∙ (Ψ ⊗ ΨP)) ⊨ (p ∙ the(subject α)) ↔ (p ∙ M)" by(rule chanEqClosed) with ‹AP ♯* Ψ› ‹(p ∙ AP) ♯* Ψ› ‹(p ∙ AP) ♯* subject α› S ‹α ≠ τ› ‹AP ♯* α› have "Ψ ⊗ (p ∙ ΨP) ⊨ the(subject α) ↔ (p ∙ M)" by(simp add: eqvts del: subjectEqvt) moreover from ‹B ♯* M› have "(p ∙ B) ♯* (p ∙ M)" by(simp add: pt_fresh_star_bij[OF pt_name_inst, OF at_name_inst]) with ‹AP ♯* B› ‹(p ∙ AP) ♯* B› S have "B ♯* (p ∙ M)" by(simp add: eqvts) ultimately show ?case by(rule cFrameAlpha) next case(cInput Ψ M K xvec N Tvec P B) from ‹Ψ ⊨ M ↔ K› have "Ψ ⊗ 1 ⊨ M ↔ K" by(blast intro: statEqEnt AssertionStatEqSym[OF Identity]) hence "Ψ ⊗ 1 ⊨ K ↔ M" by(rule chanEqSym) moreover from ‹B ♯* (M(λ*xvec N).P)› have "B ♯* M" by simp ultimately show ?case by(rule_tac cInput) auto next case(cOutput Ψ M K N P B) from ‹Ψ ⊨ M ↔ K› have "Ψ ⊗ 1 ⊨ M ↔ K" by(blast intro: statEqEnt AssertionStatEqSym[OF Identity]) hence "Ψ ⊗ 1 ⊨ K ↔ M" by(rule chanEqSym) moreover from ‹B ♯* (M⟨N⟩.P)› have "B ♯* M" by simp ultimately show ?case by(rule_tac cOutput) auto next case(cCase Ψ P α P' φ Cs AP ΨP B) then obtain M where "Ψ ⊗ ΨP ⊨ the(subject α) ↔ M" and "B ♯* M" by(rule_tac cCase) (auto dest: memFreshChain) with ‹ΨP ≃ 1› show ?case by(blast intro: cCase statEqEnt compositionSym Identity) next case(cPar1 Ψ ΨQ P α P' AQ Q AP ΨP B) then obtain M where "(Ψ ⊗ ΨQ) ⊗ ΨP ⊨ the(subject α) ↔ M" and "B ♯* M" apply(rule_tac cPar1) by assumption auto thus ?case by(metis cPar1 statEqEnt Associativity Commutativity AssertionStatEqTrans Composition) next case(cPar2 Ψ ΨP Q α Q' AP P AQ ΨQ B) then obtain M where "(Ψ ⊗ ΨP) ⊗ ΨQ ⊨ the(subject α) ↔ M" and "B ♯* M" by(rule_tac cPar2) auto thus ?case by(metis cPar2 statEqEnt Associativity) next case cComm1 thus ?case by simp next case cComm2 thus ?case by simp next case(cOpen Ψ P M xvec yvec N P' x AP ΨP B) then obtain K where "Ψ ⊗ ΨP ⊨ M ↔ K" and "B ♯* K" apply(rule_tac cOpen) by force auto thus ?case by(fastforce intro: cOpen) next case(cScope Ψ P α P' x AP ΨP B) then obtain M where "Ψ ⊗ ΨP ⊨ the(subject α) ↔ M" and "B ♯* M" by(rule_tac cScope) auto thus ?case by(fastforce intro: cScope) next case(cBang Ψ P α P' AP ΨP B) then obtain K where "Ψ ⊗ ΨP ⊗ 1 ⊨ the(subject α) ↔ K" and "B ♯* K" by(rule_tac cBang) auto with ‹ΨP ≃ 1› show ?case by(metis cBang statEqEnt compositionSym Identity) qed lemma inputRenameSubject: fixes Ψ :: 'b and P :: "('a, 'b, 'c) psi" and M :: 'a and N :: 'a and P' :: "('a, 'b, 'c) psi" and AP :: "name list" and ΨP :: 'b assumes "Ψ ⊳ P ⟼M(N) ≺ P'" and "extractFrame P = ⟨AP, ΨP⟩" and "distinct AP" and "Ψ ⊗ ΨP ⊨ M ↔ K" and "AP ♯* Ψ" and "AP ♯* P" and "AP ♯* M" and "AP ♯* K" shows "Ψ ⊳ P ⟼K(N) ≺ P'" using assms proof(nominal_induct avoiding: K rule: inputFrameInduct) case(cAlpha Ψ P M N P' AP ΨP p K) have S: "set p ⊆ set AP × set (p ∙ AP)" by fact from ‹Ψ ⊗ (p ∙ ΨP) ⊨ M ↔ K› have "(p ∙ (Ψ ⊗ (p ∙ ΨP))) ⊨ (p ∙ M) ↔ (p ∙ K)" by(rule chanEqClosed) with S ‹distinctPerm p› ‹AP ♯* Ψ› ‹AP ♯* M› ‹AP ♯* K› ‹(p ∙ AP) ♯* Ψ› ‹(p ∙ AP) ♯* M have "Ψ ⊗ ΨP ⊨ M ↔ K" by(simp add: eqvts) with ‹AP ♯* Ψ› ‹AP ♯* P› ‹AP ♯* M› ‹AP ♯* K› ‹[Ψ ⊗ ΨP ⊨ M ↔ K; AP ♯* Ψ; AP ♯* P; AP ♯* M; AP ♯* K] ==> Ψ ⊳ P ⟼K(N) ≺ P'› show ?case by blast next case(cInput Ψ M K xvec N Tvec P K') from ‹Ψ ⊗ 1 ⊨ K ↔ K'› have "Ψ ⊨ K ↔ K'" by(blast intro: statEqEnt Identity) with ‹Ψ ⊨ M ↔ K› have "Ψ ⊨ M ↔ K'" by(rule chanEqTrans) thus ?case using ‹distinct xvec› ‹set xvec ⊆ supp N› ‹length xvec = length Tvec› by(rule Input) next case(cCase Ψ P M N P' φ Cs AP ΨP K) from ‹Ψ ⊗ 1 ⊨ M ↔ K› ‹ΨP ≃ 1› have "Ψ ⊗ ΨP ⊨ M ↔ K" by(blast intro: statEqEnt Identity compositionSym AssertionStatEqSym) with ‹AP ♯* Ψ› ‹AP ♯* P› ‹AP ♯* M› ‹AP ♯* K› ‹∧K. [Ψ ⊗ ΨP ⊨ M ↔ K; AP ♯* Ψ; AP ♯* P; AP ♯* M; AP ♯* K] ==> Ψ ⊳ P ⟼K(N) ≺ P'› have "Ψ ⊳ P ⟼K(N) ≺ P'" by force thus ?case using ‹(φ, P) mem Cs› ‹Ψ ⊨ φ› ‹guarded P› by(rule Case) next case(cPar1 Ψ ΨQ P M N P' AQ Q AP ΨP K) from ‹Ψ ⊗ ΨP ⊗ ΨQ ⊨ M ↔ K› have "(Ψ ⊗ ΨQ) ⊗ ΨP ⊨ M ↔ K" by(metis statEqEnt Associativity Composition AssertionStatEqTrans Commutativity) with ‹AP ♯* Ψ› ‹AP ♯* ΨQ› ‹AP ♯* P› ‹AP ♯* M› ‹AP ♯* K› ‹∧K. [(Ψ ⊗ ΨQ) ⊗ ΨP ⊨ M ↔ K; AP ♯* (Ψ ⊗ ΨQ); AP ♯* P; AP ♯* M; AP ♯* K] ==> Ψ ⊗ ΨQ ⊳ P ⟼ have "Ψ ⊗ ΨQ ⊳ P ⟼K(N) ≺ P'" by force thus ?case using ‹extractFrame Q = ⟨AQ, ΨQ⟩› ‹AQ ♯* Ψ› ‹AQ ♯* P› ‹AQ ♯* K› ‹AQ ♯* N› by(rule_tac Par1) auto next case(cPar2 Ψ ΨP Q M N Q' AP P AQ ΨQ K) from ‹Ψ ⊗ ΨP ⊗ ΨQ ⊨ M ↔ K› have "(Ψ ⊗ ΨP) ⊗ ΨQ ⊨ M ↔ K" by(rule statEqEnt[OF AssertionStatEqSym[OF Associativity]]) with ‹AQ ♯* Ψ› ‹AQ ♯* ΨP› ‹AQ ♯* Q› ‹AQ ♯* M› ‹AQ ♯* K› ‹∧K. [(Ψ ⊗ ΨP) ⊗ ΨQ ⊨ M ↔ K; AQ ♯* (Ψ ⊗ ΨP); AQ ♯* Q; AQ ♯* M; AQ ♯* K] ==> Ψ ⊗ ΨP ⊳ Q ⟼ have "Ψ ⊗ ΨP ⊳ Q ⟼K(N) ≺ Q'" by force thus ?case using ‹extractFrame P = ⟨AP, ΨP⟩› ‹AP ♯* Ψ› ‹AP ♯* Q› ‹AP ♯* K› ‹AP ♯* N› by(rule_tac Par2) auto next case(cScope Ψ P M N P' x AP ΨP) hence "Ψ ⊳ P ⟼K(N) ≺ P'" by force with ‹x ♯ Ψ› ‹x ♯ K› ‹x ♯ N› show ?case by(rule_tac Scope) auto next case(cBang Ψ P M N P' AP ΨP K) from ‹Ψ ⊗ 1 ⊨ M ↔ K› ‹ΨP ≃ 1› have "Ψ ⊗ ΨP ⊗ 1 ⊨ M ↔ K" by(blast intro: statEqEnt Identity compositionSym AssertionStatEqSym) with ‹AP ♯* Ψ› ‹AP ♯* P› ‹AP ♯* M› ‹AP ♯* K› ‹∧K. [Ψ ⊗ ΨP ⊗ 1 ⊨ M ↔ K; AP ♯* Ψ; AP ♯* (P ∥ !P); AP ♯* M; AP ♯* K] ==> Ψ ⊳ P ∥ !P ⟼ have "Ψ ⊳ P ∥ !P ⟼K(N) ≺ P'" by force thus ?case using ‹guarded P› by(rule Bang) qed lemma outputRenameSubject: fixes Ψ :: 'b and P :: "('a, 'b, 'c) psi" and M :: 'a and xvec :: "name list" and N :: 'a and P' :: "('a, 'b, 'c) psi" and AP :: "name list" and ΨP :: 'b assumes "Ψ ⊳ P ⟼M(ν*xvec)⟨N⟩ ≺ P'" and "extractFrame P = ⟨AP, ΨP⟩" and "distinct AP" and "Ψ ⊗ ΨP ⊨ M ↔ K" and "AP ♯* Ψ" and "AP ♯* P" and "AP ♯* M" and "AP ♯* K" shows "Ψ ⊳ P ⟼K(ν*xvec)⟨N⟩ ≺ P'" using assms apply(simp add: residualInject) proof(nominal_induct avoiding: K rule: outputFrameInduct) case(cAlpha Ψ P M AP ΨP p B K) have S: "set p ⊆ set AP × set(p ∙ AP)" by fact from ‹Ψ ⊗ (p ∙ ΨP) ⊨ M ↔ K› have "(p ∙ (Ψ ⊗ (p ∙ ΨP))) ⊨ (p ∙ M) ↔ (p ∙ K)" by(rule chanEqClosed) with S ‹distinctPerm p› ‹AP ♯* Ψ› ‹AP ♯* M› ‹AP ♯* K› ‹(p ∙ AP) ♯* Ψ› ‹(p ∙ AP) ♯* M have "Ψ ⊗ ΨP ⊨ M ↔ K" by(simp add: eqvts) with ‹AP ♯* Ψ› ‹AP ♯* P› ‹AP ♯* M› ‹AP ♯* K› show ?case by(blast intro: cAlpha) next case(cOutput Ψ M K N P K') from ‹Ψ ⊗ 1 ⊨ K ↔ K'› have "Ψ ⊨ K ↔ K'" by(blast intro: statEqEnt Identity) with ‹Ψ ⊨ M ↔ K› have "Ψ ⊨ M ↔ K'" by(rule chanEqTrans) thus ?case using Output by(force simp add: residualInject) next case(cCase Ψ P M B φ Cs AP ΨP K) from ‹Ψ ⊗ 1 ⊨ M ↔ K› ‹ΨP ≃ 1› have "Ψ ⊗ ΨP ⊨ M ↔ K" by(blast intro: statEqEnt Identity compositionSym AssertionStatEqSym) with ‹AP ♯* Ψ› ‹AP ♯* P› ‹AP ♯* M› ‹AP ♯* K› ‹∧K. [Ψ ⊗ ΨP ⊨ M ↔ K; AP ♯* Ψ; AP ♯* P; AP ♯* M; AP ♯* K] ==> Ψ ⊳ P ⟼(ROut K B)› have "Ψ ⊳ P ⟼ROut K B" by force thus ?case using ‹(φ, P) mem Cs› ‹Ψ ⊨ φ› ‹guarded P› by(rule Case) next case(cPar1 Ψ ΨQ P M xvec N P' AQ Q AP ΨP K) from ‹Ψ ⊗ ΨP ⊗ ΨQ ⊨ M ↔ K› have "(Ψ ⊗ ΨQ) ⊗ ΨP ⊨ M ↔ K" by(metis statEqEnt Associativity Composition AssertionStatEqTrans Commutativity) with ‹AP ♯* Ψ› ‹AP ♯* ΨQ› ‹AP ♯* P› ‹AP ♯* M› ‹AP ♯* K› ‹∧K. [(Ψ ⊗ ΨQ) ⊗ ΨP ⊨ M ↔ K; AP ♯* (Ψ ⊗ ΨQ); AP ♯* P; AP ♯* M; AP ♯* K] ==> Ψ ⊗ ΨQ ⊳ P ⟼ have "Ψ ⊗ ΨQ ⊳ P ⟼K(ν*xvec)⟨N⟩ ≺ P'" by(force simp add: residualInject) thus ?case using ‹extractFrame Q = ⟨AQ, ΨQ⟩› ‹xvec ♯* Q› ‹AQ ♯* Ψ› ‹AQ ♯* P› ‹AQ ♯* K› by(auto simp add: residualInject) next case(cPar2 Ψ ΨP Q M xvec N Q' AP P AQ ΨQ K) from ‹Ψ ⊗ ΨP ⊗ ΨQ ⊨ M ↔ K› have "(Ψ ⊗ ΨP) ⊗ ΨQ ⊨ M ↔ K" by(rule statEqEnt[OF AssertionStatEqSym[OF Associativity]]) with ‹AQ ♯* Ψ› ‹AQ ♯* ΨP› ‹AQ ♯* Q› ‹AQ ♯* M› ‹AQ ♯* K› ‹∧K. [(Ψ ⊗ ΨP) ⊗ ΨQ ⊨ M ↔ K; AQ ♯* (Ψ ⊗ ΨP); AQ ♯* Q; AQ ♯* M; AQ ♯* K] ==> Ψ ⊗ ΨP ⊳ Q ⟼ have "Ψ ⊗ ΨP ⊳ Q ⟼ROut K ((ν*xvec)N ≺' Q')" by force thus ?case using ‹extractFrame P = ⟨AP, ΨP⟩› ‹xvec ♯* P› ‹AP ♯* Ψ› ‹AP ♯* Q› ‹AP ♯* K› by(auto simp add: residualInject) next case(cOpen Ψ P M xvec yvec N P' x AP ΨP) hence "Ψ ⊳ P ⟼K(ν*(xvec@yvec))⟨N⟩ ≺ P'" by(force simp add: residualInject) with ‹x ∈ supp N› ‹x ♯ Ψ› ‹x ♯ K› ‹x ♯ xvec› ‹x ♯ yvec› Open show ?case by(auto simp add: residualInject) next case(cScope Ψ P M xvec N P' x AP ΨP) hence "Ψ ⊳ P ⟼K(ν*xvec)⟨N⟩ ≺ P'" by(force simp add: residualInject) with ‹x ♯ Ψ› ‹x ♯ K› ‹x ♯ xvec› ‹x ♯ N› Scope[where α="K(ν*xvec)⟨N⟩"] show ?case by(auto simp add: residualInject) next case(cBang Ψ P M B AP ΨP K) from ‹Ψ ⊗ 1 ⊨ M ↔ K› ‹ΨP ≃ 1› have "Ψ ⊗ ΨP ⊗ 1 ⊨ M ↔ K" by(blast intro: statEqEnt Identity compositionSym AssertionStatEqSym) with ‹AP ♯* Ψ› ‹AP ♯* P› ‹AP ♯* M› ‹AP ♯* K› ‹∧K. [Ψ ⊗ ΨP ⊗ 1 ⊨ M ↔ K; AP ♯* Ψ; AP ♯* (P ∥ !P); AP ♯* M; AP ♯* K] ==> Ψ ⊳ P ∥ !P ⟼ have "Ψ ⊳ P ∥ !P ⟼ROut K B" by force thus ?case using ‹guarded P› by(rule Bang) qed lemma parCasesSubject[consumes 7, 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 R :: "('a, 'b, 'c) psi" and C :: "'d::fs_name" and yvec :: "name list" assumes Trans: "Ψ ⊳ P ∥ Q ⟼α ≺ R" and "bn α ♯* Ψ" and "bn α ♯* P" and "bn α ♯* Q" and "bn α ♯* subject α" and "yvec ♯* P" and "yvec ♯* Q" and rPar1: "∧P' AQ ΨQ. [Ψ ⊗ ΨQ ⊳ P ⟼α ≺ P'; extractFrame Q = ⟨AQ, ΨQ⟩; distinct AQ; AQ ♯* Ψ; AQ ♯* P; AQ ♯* α; AQ ♯* C] ==> Prop α (P' ∥ Q)" and rPar2: "∧Q' AP ΨP. [Ψ ⊗ ΨP ⊳ Q ⟼α ≺ Q'; extractFrame P = ⟨AP, ΨP⟩; distinct AP; AP ♯* Ψ; AP ♯* Q; AP ♯* α; AP ♯* C] ==> Prop α (P ∥ Q')" and rComm1: "∧ΨQ M N P' AP ΨP K xvec Q' AQ. [Ψ ⊗ ΨQ ⊳ P ⟼M(N) ≺ P'; extractFrame P = ⟨AP, ΨP⟩; distinct AP; Ψ ⊗ ΨP ⊳ Q ⟼K(ν*xvec)⟨N⟩ ≺ Q'; extractFrame Q = ⟨AQ, ΨQ⟩; distinct AQ; Ψ ⊗ ΨP ⊗ ΨQ ⊨ M ↔ K; yvec ♯* M; yvec ♯* K; distinct xvec; AP ♯* Ψ; AP ♯* ΨQ; AP ♯* P; AP ♯* M; AP ♯* N; AP ♯* P'; AP ♯* Q; AP ♯* xvec AQ ♯* Ψ; AQ ♯* ΨP; AQ ♯* P; AQ ♯* K; AQ ♯* N; AQ ♯* P'; AQ ♯* Q; AQ ♯* xvec; xvec ♯* Ψ; xvec ♯* ΨP; xvec ♯* P; xvec ♯* M; xvec ♯* K; xvec ♯* Q; xvec ♯* ΨQ; Prop (τ) ((ν*xvec)(P' ∥ Q'))" and rComm2: "∧ΨQ M xvec N P' AP ΨP K Q' AQ. [Ψ ⊗ ΨQ ⊳ P ⟼M(ν*xvec)⟨N⟩ ≺ P'; extractFrame P = ⟨AP, ΨP⟩; distinct AP; Ψ ⊗ ΨP ⊳ Q ⟼K(N) ≺ Q'; extractFrame Q = ⟨AQ, ΨQ⟩; distinct AQ; Ψ ⊗ ΨP ⊗ ΨQ ⊨ M ↔ K; yvec ♯* M; yvec ♯* K; distinct xvec; AP ♯* Ψ; AP ♯* ΨQ; AP ♯* P; AP ♯* M; AP ♯* N; AP ♯* P'; AP ♯* Q; AP ♯* xvec AQ ♯* Ψ; AQ ♯* ΨP; AQ ♯* P; AQ ♯* K; AQ ♯* N; AQ ♯* P'; AQ ♯* Q; AQ ♯* xvec; xvec ♯* Ψ; xvec ♯* ΨP; xvec ♯* P; xvec ♯* M; xvec ♯* K; xvec ♯* Q; xvec ♯* ΨQ; Prop (τ) ((ν*xvec)(P' ∥ Q'))" shows "Prop α R" using Trans ‹bn α ♯* Ψ› ‹bn α ♯* P› ‹bn α ♯* Q› ‹bn α ♯* subject α› proof(induct rule: parCases[where C="(C, yvec)"]) case(cPar1 P' AQ ΨQ) thus ?case by(rule_tac rPar1) auto next case(cPar2 Q' AP ΨP) thus ?case by(rule_tac rPar2) auto next case(cComm1 ΨQ M N P' AP ΨP K xvec Q' AQ) from ‹AP ♯* (C, yvec)› ‹AQ ♯* (C, yvec)› ‹xvec ♯* (C, yvec)› have "AP ♯* C" and "AQ ♯* C" and "xvec ♯* C" and "AP ♯* yvec" and "AQ ♯* yvec" and "xvec ♯* by simp+ have FrP: "extractFrame P = ⟨AP, ΨP⟩" and FrQ: "extractFrame Q = ⟨AQ, ΨQ⟩" and MeqK: "Ψ ⊗ ΨP ⊗ ΨQ ⊨ M ↔ K" by fact+ from ‹Ψ ⊗ ΨQ ⊳ P ⟼M(N) ≺ P'› FrP ‹distinct AP› ‹AP ♯* P› ‹AQ ♯* P› ‹yvec ♯* P› ‹AP ♯* Ψ ‹AP ♯* AQ› ‹AP ♯* yvec› ‹AP ♯* xvec› ‹AP ♯* P› ‹AP ♯* M› ‹xvec ♯* P› ‹AP ♯* ΨQ› obtain M' where MeqM': "(Ψ ⊗ ΨQ) ⊗ ΨP ⊨ M ↔ M'" and "xvec ♯* M'" and "yvec ♯* M'" and "AQ ♯* M by(rule_tac B="xvec@yvec@AQ" in obtainPrefix) (assumption | force)+ from ‹Ψ ⊗ ΨP ⊳ Q ⟼K(ν*xvec)⟨N⟩ ≺ Q'› FrQ ‹distinct AQ› ‹AP ♯* Q› ‹AQ ♯* Q› ‹yvec ♯* Q› ‹A ‹AP ♯* AQ› ‹AQ ♯* yvec› ‹AQ ♯* xvec› ‹AQ ♯* Q› ‹AQ ♯* K› ‹xvec ♯* Q› ‹AQ ♯* ΨP› ‹ obtain K' where KeqK': "(Ψ ⊗ ΨP) ⊗ ΨQ ⊨ K ↔ K'" and "xvec ♯* K'" and "yvec ♯* K'" and "AP ♯* K by(rule_tac B="xvec@yvec@AP" in obtainPrefix) (assumption | force | metis freshChainSym)+ from MeqK KeqK' have "(Ψ ⊗ ΨQ) ⊗ ΨP ⊨ M ↔ K'" by(metis statEqEnt Associativity Commutativity Composition chanEqTrans) with ‹Ψ ⊗ ΨQ ⊳ P ⟼M(N) ≺ P'› FrP ‹distinct AP› have "Ψ ⊗ ΨQ ⊳ P ⟼K'(N) ≺ P'" using ‹AP ♯* Ψ› ‹AP ♯* ΨQ› ‹AP ♯* P› ‹AP ♯* M› ‹AP ♯* K'› by(rule_tac inputRenameSubject) (assumption | force)+ moreover note FrP ‹distinct AP› moreover from MeqK MeqM' have "(Ψ ⊗ ΨP) ⊗ ΨQ ⊨ K ↔ M'" by(metis statEqEnt Associativity Commutativity Composition chanEqTrans chanEqSym) with ‹Ψ ⊗ ΨP ⊳ Q ⟼K(ν*xvec)⟨N⟩ ≺ Q'› FrQ ‹distinct AQ› have "Ψ ⊗ ΨP ⊳ Q ⟼M'(ν*xvec)⟨N⟩ ≺ Q'" using ‹AQ ♯* Ψ› ‹AQ ♯* ΨP› ‹AQ ♯* Q› ‹AQ ♯* K› ‹AQ ♯* by(rule_tac outputRenameSubject) (assumption | force)+ moreover note FrQ ‹distinct AQ› moreover from MeqM' KeqK' MeqK have "Ψ ⊗ ΨP ⊗ ΨQ ⊨ K' ↔ M'" by(metis statEqEnt Associativity Commutativity Composition chanEqTrans chanEqSym) moreover note ‹AP ♯* Ψ› ‹AP ♯* ΨQ› ‹AP ♯* P› ‹AP ♯* K'› ‹AP ♯* N› ‹AP ♯* P'› ‹AP ♯* ‹AQ ♯* Ψ› ‹AQ ♯* ΨP› ‹AQ ♯* Q› ‹AQ ♯* M'› ‹AQ ♯* N› ‹AQ ♯* Q'› ‹AQ ♯* P› ‹AQ ♯* xv ‹xvec ♯* Ψ› ‹xvec ♯* ΨP› ‹xvec ♯* P› ‹xvec ♯* M'› ‹xvec ♯* K'› ‹xvec ♯* Q› ‹xvec ♯ ultimately show ?case by(rule_tac rComm1) next case(cComm2 ΨQ M xvec N P' AP ΨP K Q' AQ) from ‹AP ♯* (C, yvec)› ‹AQ ♯* (C, yvec)› ‹xvec ♯* (C, yvec)› have "AP ♯* C" and "AQ ♯* C" and "xvec ♯* C" and "AP ♯* yvec" and "AQ ♯* yvec" and "xvec ♯* by simp+ have FrP: "extractFrame P = ⟨AP, ΨP⟩" and FrQ: "extractFrame Q = ⟨AQ, ΨQ⟩" and MeqK: "Ψ ⊗ ΨP ⊗ ΨQ ⊨ M ↔ K" by fact+ from ‹Ψ ⊗ ΨQ ⊳ P ⟼M(ν*xvec)⟨N⟩ ≺ P'› FrP ‹distinct AP› ‹AP ♯* P› ‹AQ ♯* P› ‹yvec ♯* P› ‹A ‹AP ♯* AQ› ‹AP ♯* yvec› ‹AP ♯* xvec› ‹AP ♯* P› ‹AP ♯* M› ‹xvec ♯* P› ‹AP ♯* ΨQ› ‹ obtain M' where MeqM': "(Ψ ⊗ ΨQ) ⊗ ΨP ⊨ M ↔ M'" and "xvec ♯* M'" and "yvec ♯* M'" and "AQ ♯* M by(rule_tac B="xvec@yvec@AQ" in obtainPrefix) (assumption | force)+ from ‹Ψ ⊗ ΨP ⊳ Q ⟼K(N) ≺ Q'› FrQ ‹distinct AQ› ‹AP ♯* Q› ‹AQ ♯* Q› ‹yvec ♯* Q› ‹AQ ♯* Ψ ‹AP ♯* AQ› ‹AQ ♯* yvec› ‹AQ ♯* xvec› ‹AQ ♯* Q› ‹AQ ♯* K› ‹xvec ♯* Q› ‹AQ ♯* ΨP› obtain K' where KeqK': "(Ψ ⊗ ΨP) ⊗ ΨQ ⊨ K ↔ K'" and "xvec ♯* K'" and "yvec ♯* K'" and "AP ♯* K by(rule_tac B="xvec@yvec@AP" in obtainPrefix) (assumption | force | metis freshChainSym)+ from MeqK KeqK' have "(Ψ ⊗ ΨQ) ⊗ ΨP ⊨ M ↔ K'" by(metis statEqEnt Associativity Commutativity Composition chanEqTrans) with ‹Ψ ⊗ ΨQ ⊳ P ⟼M(ν*xvec)⟨N⟩ ≺ P'› FrP ‹distinct AP› have "Ψ ⊗ ΨQ ⊳ P ⟼K'(ν*xvec)⟨N⟩ ≺ P'" using ‹AP ♯* Ψ› ‹AP ♯* ΨQ› ‹AP ♯* P› ‹AP ♯* M› ‹AP ♯* by(rule_tac outputRenameSubject) (assumption | force)+ moreover note FrP ‹distinct AP› moreover from MeqK MeqM' have "(Ψ ⊗ ΨP) ⊗ ΨQ ⊨ K ↔ M'" by(metis statEqEnt Associativity Commutativity Composition chanEqTrans chanEqSym) with ‹Ψ ⊗ ΨP ⊳ Q ⟼K(N) ≺ Q'› FrQ ‹distinct AQ› have "Ψ ⊗ ΨP ⊳ Q ⟼M'(N) ≺ Q'" using ‹AQ ♯* Ψ› ‹AQ ♯* ΨP› ‹AQ ♯* Q› ‹AQ ♯* K› ‹AQ ♯* M'› by(rule_tac inputRenameSubject) (assumption | force)+ moreover note FrQ ‹distinct AQ› moreover from MeqM' KeqK' MeqK have "Ψ ⊗ ΨP ⊗ ΨQ ⊨ K' ↔ M'" by(metis statEqEnt Associativity Commutativity Composition chanEqTrans chanEqSym) moreover note ‹AP ♯* Ψ› ‹AP ♯* ΨQ› ‹AP ♯* P› ‹AP ♯* K'› ‹AP ♯* N› ‹AP ♯* P'› ‹AP ♯* ‹AQ ♯* Ψ› ‹AQ ♯* ΨP› ‹AQ ♯* Q› ‹AQ ♯* M'› ‹AQ ♯* N› ‹AQ ♯* Q'› ‹AQ ♯* P› ‹AQ ♯* xv ‹xvec ♯* Ψ› ‹xvec ♯* ΨP› ‹xvec ♯* P› ‹xvec ♯* M'› ‹xvec ♯* K'› ‹xvec ♯* Q› ‹xvec ♯ ultimately show ?case by(rule_tac rComm2) qed lemma 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; dist shows "Prop α P'" proof - { fix xvec N P assume Trans: "Ψ ⊳ M(λ*xvec N).P ⟼α ≺ P'" and "xvec ♯* Ψ" and "xvec ♯* M" and "xvec ♯* α" and "xvec ♯* P'" and "distinct xvec" and rInput: "∧K Tvec. [Ψ ⊨ M ↔ K; set xvec ⊆ supp N; length xvec = length Tvec; dist from Trans have "bn α = []" apply - by(ind_cases "Ψ ⊳ M(λ*xvec N).P ⟼α ≺ P'") (auto simp add: residualInject) 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› moreover note ‹length(bn α) = residualLength(α ≺ P')› ‹distinct(bn α)› moreover note ‹length(bn α) = residualLength(α ≺ P')› ‹distinct(bn α)› moreover note ‹length(bn α) = residualLength(α ≺ P')› ‹distinct(bn α)› moreover note ‹length(bn α) = residualLength(α ≺ P')› ‹distinct(bn α)› moreover note ‹length(bn α) = residualLength(α ≺ P')› ‹distinct(bn α)› moreover obtain x::name where "x ♯ Ψ" and "x ♯ P" and "x ♯ M" and "x ♯ xvec" and "x ♯ α" and "x by(generate_fresh "name") auto ultimately have "Prop α P'" using ‹bn α = []› ‹xvec ♯* Ψ›\‹xvec ♯* M› ‹xvec ♯* α› ‹xvec apply(cases rule: semanticsCases[of _ _ _ _ _ _ _ _ _ C x]) apply(force simp add: residualInject psi.inject rInput) by(fastforce simp add: residualInject psi.inject inputChainFresh)+ } note Goal = this moreover obtain p :: "name prm" where "(p ∙ xvec) ♯* Ψ" and "(p ∙ xvec) ♯* M" and "(p ∙ xve and "(p ∙ xvec) ♯* α" and "(p ∙ xvec) ♯* P'" and S: "set p ⊆ set xvec × set(p ∙ xvec)" and "distinctPerm p" by(rule_tac xvec=xvec and c="(Ψ, M, N, P, α, P')" in name_list_avoiding) auto from Trans ‹(p ∙ xvec) ♯* N› ‹(p ∙ xvec) ♯* P› S have "Ψ ⊳ M(λ*(p ∙ xvec) (p ∙ N)).(p ∙ by(simp add: inputChainAlpha') moreover { fix K Tvec assume "Ψ ⊨ M ↔ K" moreover assume "set(p ∙ xvec) ⊆ supp(p ∙ N)" 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)" --> -------------------- --> maximum size reached --> --------------------
¤ Dauer der Verarbeitung: 1.19 Sekunden (vorverarbeitet am 2026-09-03) ¤*© Formatika GbR, Deutschland |
|
||||||||||||||
2026-08-25