(* Title: The pi-calculus Author/Maintainer: Jesper Bengtson (jebe.dk), 2012 *) theory Weak_Late_Step_Sim imports Weak_Late_Step_Semantics Weak_Late_Sim Strong_Late_Sim begin definition weakStepSimAct :: "pi ⇒ residual ⇒ ('a::fs_name) ⇒ (pi × pi) set ⇒ bool" where "weakStepSimAct P Rs C Rel ≡ (∀Q' a x. Rs = a<νx> ≺ Q' ⟶ x ♯ C ⟶ (∃P' . P ⟹⇩_{l}a<νx> ≺ P' ∧ (P', Q') ∈ Rel)) ∧ (∀Q' a x. Rs = a<x> ≺ Q' ⟶ x ♯ C ⟶ (∃P''. ∀u. ∃P'. P ⟹⇩_{l}u in P''→a<x> ≺ P' ∧ (P', Q'[x::=u]) ∈ Rel)) ∧ (∀Q' α. Rs = α ≺ Q' ⟶ (∃P'. P ⟹⇩_{l}α ≺ P' ∧ (P', Q') ∈ Rel))" definition weakStepSimAux :: "pi ⇒ (pi × pi) set ⇒ pi ⇒ bool" where "weakStepSimAux P Rel Q ≡ (∀Q' a x. (Q ⟼a<νx> ≺ Q' ∧ x ♯ P) ⟶ (∃P' . P ⟹⇩_{l}a<νx> ≺ P' ∧ (P', Q') ∈ Rel)) ∧ (∀Q' a x. (Q ⟼a<x> ≺ Q' ∧ x ♯ P) ⟶ (∃P''. ∀u. ∃P'. P ⟹⇩_{l}u in P''→a<x> ≺ P' ∧ (P', Q'[x::=u]) ∈ Rel)) ∧ (∀Q' α. Q ⟼α ≺ Q' ⟶ (∃P'. P ⟹⇩_{l}α ≺ P' ∧ (P', Q') ∈ Rel))" definition weakStepSim :: "pi ⇒ (pi × pi) set ⇒ pi ⇒ bool" (‹_ ↝<_> _› [80, 80, 80] 80) where "P ↝<Rel> Q ≡ (∀Rs. Q ⟼ Rs ⟶ weakStepSimAct P Rs P Rel)" lemmas weakStepSimDef = weakStepSimAct_def weakStepSim_def lemma "weakStepSimAux P Rel Q = weakStepSim P Rel Q" by(auto simp add: weakStepSimDef weakStepSimAux_def) lemma monotonic: fixes A :: "(pi × pi) set" and B :: "(pi × pi) set" and P :: pi and P' :: pi assumes "P ↝<A> P'" and "A ⊆ B" shows "P ↝<B> P'" using assms apply(auto simp add: weakStepSimDef) apply blast apply(erule_tac x="a<x> ≺ Q'" in allE) apply(clarsimp) apply(rotate_tac 4) apply(erule_tac x=Q' in allE) apply(erule_tac x=a in allE) apply(erule_tac x=x in allE) by blast+ lemma simCasesCont[consumes 1, case_names Bound Input Free]: fixes P :: pi and Q :: pi and Rel :: "(pi × pi) set" and C :: "'a::fs_name" assumes Eqvt: "eqvt Rel" and Bound: "⋀Q' a x. ⟦x ♯ C; Q ⟼a<νx> ≺ Q'⟧ ⟹ ∃P'. P ⟹⇩_{l}a<νx> ≺ P' ∧ (P', Q') ∈ Rel" and Input: "⋀Q' a x. ⟦x ♯ C; Q ⟼a<x> ≺ Q'⟧ ⟹ ∃P''. ∀u. ∃P'. P ⟹⇩_{l}u in P''→a<x> ≺ P' ∧ (P', Q'[x::=u]) ∈ Rel" and Free: "⋀Q' α. Q ⟼ α ≺ Q' ⟹ (∃P'. P ⟹⇩_{l}α ≺ P' ∧ (P', Q') ∈ Rel)" shows "P ↝<Rel> Q" using Free proof(auto simp add: weakStepSimDef) fix Q' a x assume xFreshP: "(x::name) ♯ P" assume Trans: "Q ⟼ a<νx> ≺ Q'" have "∃c::name. c ♯ (P, Q', x, C)" by(blast intro: name_exists_fresh) then obtain c::name where cFreshP: "c ♯ P" and cFreshQ': "c ♯ Q'" and cFreshC: "c ♯ C" and cineqx: "c ≠ x" by(force simp add: fresh_prod) from Trans cFreshQ' have "Q ⟼ a<νc> ≺ ([(x, c)] ∙ Q')" by(simp add: alphaBoundResidual) with cFreshC have "∃P'. P ⟹⇩_{l}a<νc> ≺ P' ∧ (P', [(x, c)] ∙ Q') ∈ Rel" by(rule Bound) then obtain P' where PTrans: "P ⟹⇩_{l}a<νc> ≺ P'" and P'RelQ': "(P', [(x, c)] ∙ Q') ∈ Rel" by blast from PTrans xFreshP cineqx have xFreshP': "x ♯ P'" by(force dest: Weak_Late_Step_Semantics.freshTransition) with PTrans have "P ⟹⇩_{l}a<νx> ≺ ([(x, c)] ∙ P')" by(simp add: alphaBoundResidual name_swap) moreover have "([(x, c)] ∙ P', Q') ∈ Rel" (is "?goal") proof - from Eqvt P'RelQ' have "([(x, c)] ∙ P', [(x, c)] ∙ [(x, c)] ∙ Q') ∈ Rel" by(rule eqvtRelI) with cineqx show ?goal by(simp add: name_calc) qed ultimately show "∃P'. P ⟹⇩_{l}a<νx> ≺ P' ∧ (P', Q') ∈ Rel" by blast next fix Q' a x u assume QTrans: "Q ⟼a<x> ≺ (Q'::pi)" and xFreshP: "x ♯ P" have "∃c::name. c ♯ (P, Q', C, x)" by(blast intro: name_exists_fresh) then obtain c::name where cFreshP: "c ♯ P" and cFreshQ': "c ♯ Q'" and cFreshC: "c ♯ C" and cineqx: "c ≠ x" by(force simp add: fresh_prod) from QTrans cFreshQ' have "Q ⟼a<c> ≺ ([(x, c)] ∙ Q')" by(simp add: alphaBoundResidual) with cFreshC have "∃P''. ∀u. ∃P'. P ⟹⇩_{l}u in P''→a<c> ≺ P' ∧ (P', ([(x, c)] ∙ Q')[c::=u]) ∈ Rel" by(rule Input) then obtain P'' where L1: "∀u. ∃P'. P ⟹⇩_{l}u in P''→a<c> ≺ P' ∧ (P', ([(x, c)] ∙ Q')[c::=u]) ∈ Rel" by blast have "∀u. ∃P'. P ⟹⇩_{l}u in ([(c, x)] ∙ P'')→a<x> ≺ P' ∧ (P', Q'[x::=u]) ∈ Rel" proof(auto) fix u from L1 obtain P' where PTrans: "P ⟹⇩_{l}u in P''→a<c> ≺ P'" and P'RelQ': "(P', ([(x, c)] ∙ Q')[c::=u]) ∈ Rel" by blast from PTrans xFreshP have "P ⟹⇩_{l}u in ([(c, x)] ∙ P'')→a<x> ≺ P'" by(rule alphaInput) moreover from P'RelQ' cFreshQ' have "(P', Q'[x::=u]) ∈ Rel" by(simp add: renaming[THEN sym] name_swap) ultimately show "∃P'. P ⟹⇩_{l}u in ([(c, x)] ∙ P'')→a<x> ≺ P' ∧ (P', Q'[x::=u]) ∈ Rel" by blast qed thus "∃P''. ∀u. ∃P'. P ⟹⇩_{l}u in P''→a<x> ≺ P' ∧ (P', Q'[x::=u]) ∈ Rel" by blast qed lemma simCases[consumes 0, case_names Bound Input Free]: fixes P :: pi and Q :: pi and Rel :: "(pi × pi) set" and C :: "'a::fs_name" assumes Bound: "⋀Q' a x. ⟦Q ⟼a<νx> ≺ Q'; x ♯ P⟧ ⟹ ∃P'. P ⟹⇩_{l}a<νx> ≺ P' ∧ (P', Q') ∈ Rel" and Input: "⋀Q' a x. ⟦Q ⟼a<x> ≺ Q'; x ♯ P⟧ ⟹ ∃P''. ∀u. ∃P'. P ⟹⇩_{l}u in P''→a<x> ≺ P' ∧ (P', Q'[x::=u]) ∈ Rel" and Free: "⋀Q' α. Q ⟼ α ≺ Q' ⟹ (∃P'. P ⟹⇩_{l}α ≺ P' ∧ (P', Q') ∈ Rel)" shows "P ↝<Rel> Q" using assms by(auto simp add: weakStepSimDef) lemma simActBoundCases[consumes 1, case_names Input BoundOutput]: fixes P :: pi and a :: subject and x :: name and Q' :: pi and C :: "'a::fs_name" and Rel :: "(pi × pi) set" assumes EqvtRel: "eqvt Rel" and DerInput: "⋀b. a = InputS b ⟹ (∃P''. ∀u. ∃P'. (P ⟹⇩_{l}u in P''→b<x> ≺ P') ∧ (P', Q'[x::=u]) ∈ Rel)" and DerBoundOutput: "⋀b. a = BoundOutputS b ⟹ (∃P'. (P ⟹⇩_{l}b<νx> ≺ P') ∧ (P', Q') ∈ Rel)" shows "weakStepSimAct P (a«x» ≺ Q') P Rel" proof(simp add: weakStepSimAct_def fresh_prod, auto) fix Q'' b y assume Eq: "a«x» ≺ Q' = b<νy> ≺ Q''" assume yFreshP: "y ♯ P" from Eq have "a = BoundOutputS b" by(simp add: residual.inject) from yFreshP DerBoundOutput[OF this] Eq show "∃P'. P ⟹⇩_{l}b<νy> ≺ P' ∧ (P', Q'') ∈ Rel" proof(cases "x=y", auto simp add: residual.inject name_abs_eq) fix P' assume PTrans: "P ⟹⇩_{l}b<νx> ≺ P'" assume P'RelQ': "(P', ([(x, y)] ∙ Q'')) ∈ Rel" assume xineqy: "x ≠ y" with PTrans yFreshP have yFreshP': "y ♯ P'" by(force intro: Weak_Late_Step_Semantics.freshTransition) hence "b<νx> ≺ P' = b<νy> ≺ [(x, y)] ∙ P'" by(rule alphaBoundResidual) moreover have "([(x, y)] ∙ P', Q'') ∈ Rel" proof - from EqvtRel P'RelQ' have "([(x, y)] ∙ P', [(x, y)] ∙ ([(x, y)] ∙ Q''))∈ Rel" by(rule eqvtRelI) thus ?thesis by(simp add: name_calc) qed ultimately show "∃P'. P ⟹⇩_{l}b<νy> ≺ P' ∧ (P', Q'') ∈ Rel" using PTrans by auto qed next fix Q'' b y u assume Eq: "a«x» ≺ Q' = b<y> ≺ Q''" assume yFreshP: "y ♯ P" from Eq have "a = InputS b" by(simp add: residual.inject) from DerInput[OF this] obtain P'' where L1: "∀u. ∃P'. P ⟹⇩_{l}u in P''→b<x> ≺ P' ∧ (P', Q'[x::=u]) ∈ Rel" by blast have "∀u. ∃P'. P ⟹⇩_{l}u in ([(x, y)] ∙ P'')→b<y> ≺ P' ∧ (P', Q''[y::=u]) ∈ Rel" proof(rule allI) fix u from L1 Eq show "∃P'. P ⟹⇩_{l}u in ([(x, y)] ∙ P'')→b<y> ≺ P' ∧ (P', Q''[y::=u]) ∈ Rel" proof(cases "x=y", auto simp add: residual.inject name_abs_eq) assume Der: "∀u. ∃P'. P ⟹⇩_{l}u in P''→b<x> ≺ P' ∧ (P', ([(x, y)] ∙ Q'')[x::=u]) ∈ Rel" assume xFreshQ'': "x ♯ Q''" from Der obtain P' where PTrans: "P ⟹⇩_{l}u in P''→b<x> ≺ P'" and P'RelQ': "(P', ([(x, y)] ∙ Q'')[x::=u]) ∈ Rel" by force from PTrans yFreshP have "P ⟹⇩_{l}u in ([(x, y)] ∙ P'')→b<y> ≺ P'" by(rule alphaInput) moreover from xFreshQ'' P'RelQ' have "(P', Q''[y::=u]) ∈ Rel" by(simp add: renaming) ultimately show ?thesis by force qed qed thus "∃P''. ∀u. ∃P'. P ⟹⇩_{l}u in P''→b<y> ≺ P' ∧ (P', Q''[y::=u]) ∈ Rel" by blast qed lemma simActFreeCases[consumes 0, case_names Free]: fixes P :: pi and α :: freeRes and C :: "'a::fs_name" and Rel :: "(pi × pi) set" assumes Der: "∃P'. (P ⟹⇩_{l}α ≺ P') ∧ (P', Q') ∈ Rel" shows "weakStepSimAct P (α ≺ Q') P Rel" using assms by(simp add: weakStepSimAct_def residual.inject) lemma simE: fixes P :: pi and Rel :: "(pi × pi) set" and Q :: pi and a :: name and x :: name and u :: name and Q' :: pi assumes "P ↝<Rel> Q" shows "Q ⟼a<νx> ≺ Q' ⟹ x ♯ P ⟹ ∃P'. P ⟹⇩_{l}a<νx> ≺ P' ∧ (P', Q') ∈ Rel" and "Q ⟼a<x> ≺ Q' ⟹ x ♯ P ⟹ ∃P''. ∀u. ∃P'. P ⟹⇩_{l}u in P''→a<x> ≺ P' ∧ (P', Q'[x::=u]) ∈ Rel" and "Q ⟼α ≺ Q' ⟹ (∃P'. P ⟹⇩_{l}α ≺ P' ∧ (P', Q') ∈ Rel)" using assms by(simp add: weakStepSimDef)+ lemma weakSimTauChain: fixes P :: pi and Rel :: "(pi × pi) set" and Q :: pi and Q' :: pi assumes QChain: "Q ⟹⇩_{τ}Q'" and PRelQ: "(P, Q) ∈ Rel" and Sim: "⋀P Q. (P, Q) ∈ Rel ⟹ P ↝<Rel> Q" shows "∃P'. P ⟹⇩_{τ}P' ∧ (P', Q') ∈ Rel" proof - from QChain show ?thesis proof(induct rule: tauChainInduct) case id have "P ⟹⇩_{τ}P" by simp with PRelQ show ?case by blast next case(ih Q' Q'') have IH: "∃P'. P ⟹⇩_{τ}P' ∧ (P', Q') ∈ Rel" by fact then obtain P' where PChain: "P ⟹⇩_{τ}P'" and P'RelQ': "(P', Q') ∈ Rel" by blast from P'RelQ' have "P' ↝<Rel> Q'" by(rule Sim) moreover have Q'Trans: "Q' ⟼τ ≺ Q''" by fact ultimately have "∃P''. P' ⟹⇩_{l}τ ≺ P'' ∧ (P'', Q'') ∈ Rel" by(rule simE) then obtain P'' where P'Trans: "P' ⟹⇩_{l}τ ≺ P''" and P''RelQ'': "(P'', Q'') ∈ Rel" by blast from P'Trans have "P' ⟹⇩_{τ}P''" by(rule Weak_Late_Step_Semantics.tauTransitionChain) with PChain have "P ⟹⇩_{τ}P''" by auto with P''RelQ'' show ?case by blast qed qed lemma strongSimWeakEqSim: fixes P :: pi and Q :: pi and Rel :: "(pi × pi) set" assumes PSimQ: "P ↝[Rel] Q" shows "P ↝<Rel> Q" proof(auto simp add: weakStepSimDef) fix Q' a x assume "Q ⟼a<νx> ≺ Q'" and "x ♯ P" with PSimQ have "∃P'. P ⟼a<νx> ≺ P' ∧ derivative P' Q' (BoundOutputS a) x Rel" by(rule Strong_Late_Sim.simE) then obtain P' where PTrans: "P ⟼a<νx> ≺ P'" and P'RelQ': "(P', Q') ∈ Rel" by(force simp add: derivative_def) from PTrans have "P ⟹⇩_{l}a<νx> ≺ P'" by(rule Weak_Late_Step_Semantics.singleActionChain) thus "∃P'. P ⟹⇩_{l}a<νx> ≺ P' ∧ (P', Q') ∈ Rel" using P'RelQ' by blast next fix Q' a x u assume "Q ⟼a<x> ≺ Q'" and "x ♯ P" with PSimQ have L1: "∃P'. P ⟼a<x> ≺ P' ∧ derivative P' Q' (InputS a) x Rel" by(blast intro: Strong_Late_Sim.simE) then obtain P' where PTrans: "P ⟼a<x> ≺ P'" and PDer: "derivative P' Q' (InputS a) x Rel" by blast have "∀u. ∃P''. P ⟹⇩_{l}u in P'→a<x> ≺ P'' ∧ (P'', Q'[x::=u]) ∈ Rel" proof(rule allI) fix u from PTrans have "P ⟹⇩_{l}u in P'→a<x> ≺ P'[x::=u]" by(blast intro: Weak_Late_Step_Semantics.singleActionChain) moreover from PDer have "(P'[x::=u], Q'[x::=u]) ∈ Rel" by(force simp add: derivative_def) ultimately show "∃P''. P ⟹⇩_{l}u in P'→a<x> ≺ P'' ∧ (P'', Q'[x::=u]) ∈ Rel" by auto qed thus "∃P''. ∀u. ∃P'. P ⟹⇩_{l}u in P''→a<x> ≺ P' ∧ (P', Q'[x::=u]) ∈ Rel" by blast next fix Q' α assume "Q ⟼α ≺ Q'" with PSimQ have "∃P'. P ⟼α ≺ P' ∧ (P', Q') ∈ Rel" by(rule Strong_Late_Sim.simE) then obtain P' where PTrans: "P ⟼α ≺ P'" and P'RelQ': "(P', Q') ∈ Rel" by blast from PTrans have "P ⟹⇩_{l}α ≺ P'" by(rule Weak_Late_Step_Semantics.singleActionChain) thus "∃P'. P ⟹⇩_{l}α ≺ P' ∧ (P', Q') ∈ Rel" using P'RelQ' by blast qed lemma weakSimWeakEqSim: fixes P :: pi and Q :: pi and Rel :: "(pi × pi) set" assumes "P ↝<Rel> Q" shows "P ↝⇧^{^}<Rel> Q" using assms by(force simp add: weakStepSimDef simDef weakTransition_def) lemma eqvtI: fixes P :: pi and Q :: pi and Rel :: "(pi × pi) set" and perm :: "name prm" assumes Sim: "P ↝<Rel> Q" and RelRel': "Rel ⊆ Rel'" and EqvtRel': "eqvt Rel'" shows "(perm ∙ P) ↝<Rel'> (perm ∙ Q)" using EqvtRel' proof(induct rule: simCasesCont[of _ "perm ∙ P"]) case(Bound Q' a x) have QTrans: "(perm ∙ Q) ⟼ a<νx> ≺ Q'" by fact have xFreshP: "x ♯ perm ∙ P" by fact from QTrans have "(rev perm ∙ (perm ∙ Q)) ⟼ rev perm ∙ (a<νx> ≺ Q')" by(rule eqvts) hence "Q ⟼ (rev perm ∙ a)<ν(rev perm ∙ x)> ≺ (rev perm ∙ Q')" by(simp add: name_rev_per) moreover from xFreshP have "(rev perm ∙ x) ♯ P" by(simp add: name_fresh_left) ultimately obtain P' where PTrans: "P ⟹⇩_{l}(rev perm ∙ a)<ν(rev perm ∙ x)> ≺ P'" and P'RelQ': "(P', rev perm ∙ Q') ∈ Rel" using Sim by(blast dest: simE) from PTrans have "(perm ∙ P) ⟹⇩_{l}perm ∙ ((rev perm ∙ a)<ν(rev perm ∙ x)> ≺ P')" by(rule Weak_Late_Step_Semantics.eqvtI) hence "(perm ∙ P) ⟹⇩_{l}a<νx> ≺ (perm ∙ P')" by(simp add: name_per_rev) moreover have "(perm ∙ P', Q') ∈ Rel'" proof - from P'RelQ' RelRel' have "(P', rev perm ∙ Q') ∈ Rel'" by blast with EqvtRel' have "(perm ∙ P', perm ∙ (rev perm ∙ Q')) ∈ Rel'" by(rule eqvtRelI) thus ?thesis by(simp add: name_per_rev) qed ultimately show ?case by blast next case(Input Q' a x) have QTrans: "(perm ∙ Q) ⟼a<x> ≺ Q'" by fact have xFreshP: "x ♯ perm ∙ P" by fact from QTrans have "(rev perm ∙ (perm ∙ Q)) ⟼ rev perm ∙ (a<x> ≺ Q')" by(rule eqvts) hence "Q ⟼ (rev perm ∙ a)<(rev perm ∙ x)> ≺ (rev perm ∙ Q')" by(simp add: name_rev_per) moreover from xFreshP have xFreshP: "(rev perm ∙ x) ♯ P" by(simp add: name_fresh_left) ultimately obtain P'' where L1: "∀u. ∃P'. P ⟹⇩_{l}u in P''→(rev perm ∙ a)<(rev perm ∙ x)> ≺ P' ∧ (P', (rev perm ∙ Q')[(rev perm ∙ x)::=u]) ∈ Rel" using Sim by(blast dest: simE) have "∀u. ∃P'. (perm ∙ P) ⟹⇩_{l}u in (perm ∙ P'')→a<x> ≺ P' ∧ (P', Q'[x::=u]) ∈ Rel'" proof(rule allI) fix u from L1 obtain P' where PTrans: "P ⟹⇩_{l}(rev perm ∙ u) in P''→(rev perm ∙ a)<(rev perm ∙ x)> ≺ P'" and P'RelQ': "(P', (rev perm ∙ Q')[(rev perm ∙ x)::=(rev perm ∙ u)]) ∈ Rel" by blast from PTrans have "(perm ∙ P) ⟹⇩_{l}(perm ∙ (rev perm ∙ u)) in (perm ∙ P'')→(perm ∙ rev perm ∙ a)<(perm ∙ rev perm ∙ x)> ≺ (perm ∙ P')" by(rule_tac Weak_Late_Step_Semantics.eqvtI, auto) hence "(perm ∙ P) ⟹⇩_{l}u in (perm ∙ P'')→a<x> ≺ (perm ∙ P')" by(simp add: name_per_rev) moreover have "(perm ∙ P', Q'[x::=u]) ∈ Rel'" proof - from P'RelQ' RelRel' have "(P', (rev perm ∙ Q')[(rev perm ∙ x)::=(rev perm ∙ u)]) ∈ Rel'" by blast with EqvtRel' have "(perm ∙ P', perm ∙ ((rev perm ∙ Q')[(rev perm ∙ x)::=(rev perm ∙ u)])) ∈ Rel'" by(rule eqvtRelI) thus ?thesis by(simp add: name_per_rev eqvt_subs[THEN sym] name_calc) qed ultimately show "∃P'. (perm ∙ P) ⟹⇩_{l}u in (perm ∙ P'')→a<x> ≺ P' ∧ (P', Q'[x::=u]) ∈ Rel'" by blast qed thus ?case by blast next case(Free Q' α) have QTrans: "(perm ∙ Q) ⟼ α ≺ Q'" by fact from QTrans have "(rev perm ∙ (perm ∙ Q)) ⟼ rev perm ∙ (α ≺ Q')" by(rule eqvts) hence "Q ⟼ (rev perm ∙ α) ≺ (rev perm ∙ Q')" by(simp add: name_rev_per) with Sim obtain P' where PTrans: "P ⟹⇩_{l}(rev perm ∙ α) ≺ P'" and PRel: "(P', (rev perm ∙ Q')) ∈ Rel" by(blast dest: simE) from PTrans have "(perm ∙ P) ⟹⇩_{l}perm ∙ ((rev perm ∙ α)≺ P')" by(rule Weak_Late_Step_Semantics.eqvtI) hence "(perm ∙ P) ⟹⇩_{l}α ≺ (perm ∙ P')" by(simp add: name_per_rev) moreover have "((perm ∙ P'), Q') ∈ Rel'" proof - from PRel EqvtRel' RelRel' have "((perm ∙ P'), (perm ∙ (rev perm ∙ Q'))) ∈ Rel'" by(force intro: eqvtRelI) thus ?thesis by(simp add: name_per_rev) qed ultimately show ?case by blast qed lemma simE2: fixes P :: pi and Rel :: "(pi × pi) set" and Q :: pi and a :: name and x :: name and Q' :: pi assumes PSimQ: "P ↝<Rel> Q" and Sim: "⋀P Q. (P, Q) ∈ Rel ⟹ P ↝⇧^{^}<Rel> Q" and Eqvt: "eqvt Rel" and PRelQ: "(P, Q) ∈ Rel" shows "Q ⟹⇩_{l}a<νx> ≺ Q' ⟹ x ♯ P ⟹ ∃P'. P ⟹⇩_{l}a<νx> ≺ P' ∧ (P', Q') ∈ Rel" and "Q ⟹⇩_{l}α ≺ Q' ⟹ ∃P'. P ⟹⇩_{l}α ≺ P' ∧ (P', Q') ∈ Rel" proof - assume QTrans: "Q ⟹⇩_{l}a<νx> ≺ Q'" assume xFreshP: "x ♯ P" have Goal: "⋀P Q a x Q'. ⟦P ↝<Rel> Q; Q ⟹⇩_{l}a<νx> ≺ Q'; x ♯ P; x ♯ Q; (P, Q) ∈ Rel⟧ ⟹ ∃P'. P ⟹⇩_{l}a<νx> ≺ P' ∧ (P', Q') ∈ Rel" proof - fix P Q a x Q' assume PSimQ: "P ↝<Rel> Q" assume QTrans: "Q ⟹⇩_{l}a<νx> ≺ Q'" assume xFreshP: "x ♯ P" assume xFreshQ: "x ♯ Q" assume PRelQ: "(P, Q) ∈ Rel" from QTrans xFreshQ obtain Q'' Q''' where QChain: "Q ⟹⇩_{τ}Q''" and Q''Trans: "Q'' ⟼a<νx> ≺ Q'''" and Q'''Chain: "Q''' ⟹⇩_{τ}Q'" by(force dest: transitionE simp add: weakTransition_def) from QChain PRelQ Sim have "∃P''. P ⟹⇩_{τ}P'' ∧ (P'', Q'') ∈ Rel" by(rule Weak_Late_Sim.weakSimTauChain) then obtain P'' where PChain: "P ⟹⇩_{τ}P''" and P''RelQ'': "(P'', Q'') ∈ Rel" by blast from PChain xFreshP have xFreshP'': "x ♯ P''" by(rule freshChain) from P''RelQ'' have "P'' ↝⇧^{^}<Rel> Q''" by(rule Sim) hence "∃P'''. P'' ⟹⇩_{l}⇧^{^}a<νx> ≺ P''' ∧ (P''', Q''') ∈ Rel" using Q''Trans xFreshP'' by(rule Weak_Late_Sim.simE) then obtain P''' where P''Trans: "P'' ⟹⇩_{l}a<νx> ≺ P'''" and P'''RelQ''': "(P''', Q''') ∈ Rel" by(force simp add: weakTransition_def) have "∃P'. P''' ⟹⇩_{τ}P' ∧ (P', Q') ∈ Rel" using Q'''Chain P'''RelQ''' Sim by(rule Weak_Late_Sim.weakSimTauChain) then obtain P' where P'''Chain: "P''' ⟹⇩_{τ}P'" and P'RelQ': "(P', Q') ∈ Rel" by blast from PChain P''Trans P'''Chain xFreshP'' have "P ⟹⇩_{l}a<νx> ≺ P'" by(blast dest: Weak_Late_Step_Semantics.chainTransitionAppend) with P'RelQ' show "∃P'. P ⟹⇩_{l}a<νx> ≺ P' ∧ (P', Q') ∈ Rel" by blast qed have "∃c::name. c ♯ (Q, Q', P, x)" by(blast intro: name_exists_fresh) then obtain c::name where cFreshQ: "c ♯ Q" and cFreshQ': "c ♯ Q'" and cFreshP: "c ♯ P" and xineqc: "x ≠ c" by(force simp add: fresh_prod) from QTrans cFreshQ' have "Q ⟹⇩_{l}a<νc> ≺ ([(x, c)] ∙ Q')" by(simp add: alphaBoundResidual) with PSimQ have "∃P'. P ⟹⇩_{l}a<νc> ≺ P' ∧ (P', [(x, c)] ∙ Q') ∈ Rel" using cFreshP cFreshQ PRelQ by(rule Goal) then obtain P' where PTrans: "P ⟹⇩_{l}a<νc> ≺ P'" and P'RelQ': "(P', [(x, c)] ∙ Q') ∈ Rel" by force have "P ⟹⇩_{l}a<νx> ≺ ([(x, c)] ∙ P')" proof - from PTrans xFreshP xineqc have "x ♯ P'" by(rule Weak_Late_Step_Semantics.freshTransition) with PTrans show ?thesis by(simp add: alphaBoundResidual name_swap) qed moreover have "([(x, c)] ∙ P', Q') ∈ Rel" proof - from Eqvt P'RelQ' have "([(x, c)] ∙ P', [(x, c)] ∙ [(x, c)] ∙ Q') ∈ Rel" by(rule eqvtRelI) thus ?thesis by simp qed ultimately show "∃P'. P ⟹⇩_{l}a<νx> ≺ P' ∧ (P', Q') ∈ Rel" by blast next assume QTrans: "Q ⟹⇩_{l}α ≺ Q'" then obtain Q'' Q''' where QChain: "Q ⟹⇩_{τ}Q''" and Q''Trans: "Q'' ⟼α ≺ Q'''" and Q'''Chain: "Q''' ⟹⇩_{τ}Q'" by(blast dest: transitionE) thus "∃P'. P ⟹⇩_{l}α ≺ P' ∧ (P', Q') ∈ Rel" proof(induct arbitrary: α Q''' Q' rule: tauChainInduct) case(id α Q''') from PSimQ ‹Q ⟼α ≺ Q'''› have "∃P'. P ⟹⇩_{l}α ≺ P' ∧ (P', Q''') ∈ Rel" by(blast dest: simE) then obtain P''' where PTrans: "P ⟹⇩_{l}α ≺ P'''" and P'RelQ''': "(P''', Q''') ∈ Rel" by blast have "∃P'. P''' ⟹⇩_{τ}P' ∧ (P', Q') ∈ Rel" using ‹Q''' ⟹⇩_{τ}Q'› P'RelQ''' Sim by(rule Weak_Late_Sim.weakSimTauChain) then obtain P' where P'''Chain: "P''' ⟹⇩_{τ}P'" and P'RelQ': "(P', Q') ∈ Rel" by blast from P'''Chain PTrans have "P ⟹⇩_{l}α ≺ P'" by(blast dest: Weak_Late_Step_Semantics.chainTransitionAppend) with P'RelQ' show ?case by blast next case(ih Q'''' Q''' α Q'' Q') have "Q''' ⟹⇩_{τ}Q'''" by simp with ‹Q'''' ⟼τ ≺ Q'''› obtain P''' where PTrans: "P ⟹⇩_{l}τ ≺ P'''" and P'''RelQ''': "(P''', Q''') ∈ Rel" by(drule_tac ih) auto from P'''RelQ''' ‹Q''' ⟼α ≺ Q''› obtain P'' where P'''Trans: "P''' ⟹⇩_{l}⇧^{^}α ≺ P''" and P''RelQ'': "(P'', Q'') ∈ Rel" by(blast dest: Weak_Late_Sim.simE Sim) from P''RelQ'' ‹Q'' ⟹⇩_{τ}Q'› Sim obtain P' where P''Chain: "P'' ⟹⇩_{τ}P'" and P'RelQ': "(P', Q')∈ Rel" by(drule_tac Weak_Late_Sim.weakSimTauChain) auto from PTrans P'''Trans P''Chain have "P ⟹⇩_{l}α ≺ P'" apply(auto simp add: weakTransition_def residual.inject) apply(drule_tac Weak_Late_Step_Semantics.tauTransitionChain, auto) apply(drule_tac Weak_Late_Step_Semantics.chainTransitionAppend, simp) apply(rule Weak_Late_Step_Semantics.chainTransitionAppend, auto) by(drule_tac Weak_Late_Step_Semantics.chainTransitionAppend, auto) with ‹(P', Q') ∈ Rel› show ?case by blast qed qed (*****************Reflexivity and transitivity*********************) lemma reflexive: fixes P :: pi and Rel :: "(pi × pi) set" assumes "Id ⊆ Rel" shows "P ↝<Rel> P" using assms by(auto intro: Weak_Late_Step_Semantics.singleActionChain simp add: weakStepSimDef) lemma transitive: fixes P :: pi and Q :: pi and R :: pi and Rel :: "(pi × pi) set" and Rel' :: "(pi × pi) set" and Rel'' :: "(pi × pi) set" assumes PSimQ: "P ↝<Rel> Q" and QSimR: "Q ↝<Rel'> R" and Eqvt: "eqvt Rel" and Eqvt': "eqvt Rel''" and Trans: "Rel O Rel' ⊆ Rel''" and Sim: "⋀P Q. (P, Q) ∈ Rel ⟹ P ↝⇧^{^}<Rel> Q" and PRelQ: "(P, Q) ∈ Rel" shows "P ↝<Rel''> R" using Eqvt' proof(induct rule: simCasesCont[of _ "(P, Q)"]) case(Bound R' a x) have RTrans: "R ⟼ a<νx> ≺ R'" by fact have "x ♯ (P, Q)" by fact hence xFreshP: "x ♯ P" and xFreshQ: "x ♯ Q" by(simp add: fresh_prod)+ from QSimR RTrans xFreshQ obtain Q' where QTrans: "Q ⟹⇩_{l}a<νx> ≺ Q'" and Q'RelR': "(Q', R') ∈ Rel'" by(blast dest: simE) from PSimQ Sim Eqvt PRelQ QTrans xFreshP obtain P' where PTrans: "P ⟹⇩_{l}a<νx> ≺ P'" and P'RelQ': "(P', Q') ∈ Rel" by(blast dest: simE2) moreover from P'RelQ' Q'RelR' Trans have "(P', R') ∈ Rel''" by blast ultimately show ?case by blast next case(Input R' a x) have RTrans: "R ⟼ a<x> ≺ R'" by fact have "x ♯ (P, Q)" by fact hence xFreshP: "x ♯ P" and xFreshQ: "x ♯ Q" by(simp add: fresh_prod)+ from QSimR RTrans xFreshQ obtain Q'' where "∀u. ∃Q'. Q ⟹⇩_{l}u in Q''→a<x> ≺ Q' ∧ (Q', R'[x::=u]) ∈ Rel'" by(blast dest: simE) hence "∃Q'''. Q ⟹⇩_{τ}Q''' ∧ Q'''⟼a<x> ≺ Q'' ∧ (∀u. ∃Q'. Q''[x::=u]⟹⇩_{τ}Q' ∧ (Q', R'[x::=u]) ∈ Rel')" by(simp add: inputTransition_def, blast) then obtain Q''' where QChain: "Q ⟹⇩_{τ}Q'''" and Q'''Trans: "Q''' ⟼a<x> ≺ Q''" and L1: "∀u. ∃Q'. Q''[x::=u]⟹⇩_{τ}Q' ∧ (Q', R'[x::=u]) ∈ Rel'" by blast from QChain PRelQ Sim obtain P''' where PChain: "P ⟹⇩_{τ}P'''" and P'''RelQ''': "(P''', Q''') ∈ Rel" by(drule_tac Weak_Late_Sim.weakSimTauChain) auto from PChain xFreshP have xFreshP''': "x ♯ P'''" by(rule freshChain) from P'''RelQ''' have "P''' ↝⇧^{^}<Rel> Q'''" by(rule Sim) with xFreshP''' Q'''Trans obtain P'''' where L2: "∀u. ∃P''. P''' ⟹⇩_{l}u in P''''→a<x> ≺ P'' ∧ (P'', Q''[x::=u]) ∈ Rel" by(blast dest: Weak_Late_Sim.simE) have "∀u. ∃P' Q'. P ⟹⇩_{l}u in P''''→a<x> ≺ P' ∧ (P', R'[x::=u]) ∈ Rel''" proof(rule allI) fix u from L1 obtain Q' where Q''Chain: "Q''[x::=u] ⟹⇩_{τ}Q'" and Q'RelR': "(Q', R'[x::=u]) ∈ Rel'" by blast from L2 obtain P'' where P'''Trans: "P''' ⟹⇩_{l}u in P''''→a<x> ≺ P''" and P''RelQ'': "(P'', Q''[x::=u]) ∈ Rel" by blast from P''RelQ'' have "P'' ↝⇧^{^}<Rel> Q''[x::=u]" by(rule Sim) have "∃P'. P'' ⟹⇩_{τ}P' ∧ (P', Q') ∈ Rel" using Q''Chain P''RelQ'' Sim by(rule Weak_Late_Sim.weakSimTauChain) then obtain P' where P''Chain: "P'' ⟹⇩_{τ}P'" and P'RelQ': "(P', Q') ∈ Rel" by blast from PChain P'''Trans P''Chain have "P ⟹⇩_{l}u in P''''→a<x> ≺ P'" by(blast dest: Weak_Late_Step_Semantics.chainTransitionAppend) moreover from P'RelQ' Q'RelR' have "(P', R'[x::=u]) ∈ Rel''" by(insert Trans, auto) ultimately show "∃P' Q'. P ⟹⇩_{l}u in P''''→a<x> ≺ P' ∧ (P', R'[x::=u]) ∈ Rel''" by blast qed thus ?case by force next case(Free R' α) have RTrans: "R ⟼ α ≺ R'" by fact with QSimR obtain Q' where QTrans: "Q ⟹⇩_{l}α ≺ Q'" and Q'RelR': "(Q', R') ∈ Rel'" by(blast dest: simE) from PSimQ Sim Eqvt PRelQ QTrans obtain P' where PTrans: "P ⟹⇩_{l}α ≺ P'" and P'RelQ': "(P', Q') ∈ Rel" by(blast dest: simE2) from P'RelQ' Q'RelR' Trans have "(P', R') ∈ Rel''" by blast with PTrans show ?case by blast qed end