(* Title: The pi-calculus Author/Maintainer: Jesper Bengtson (jebe.dk), 2012 *) theory Weak_Early_Step_Sim imports Weak_Early_Sim Strong_Early_Sim begin definition weakStepSimulation :: "pi ⇒ (pi × pi) set ⇒ pi ⇒ bool" (‹_ ↝«_» _› [80, 80, 80] 80) where "P ↝«Rel» Q ≡ (∀Q' a x. Q ⟼a<νx> ≺ Q' ⟶ x ♯ P ⟶ (∃P' . P ⟹a<νx> ≺ P' ∧ (P', Q') ∈ Rel)) ∧ (∀Q' α. Q ⟼α ≺ Q' ⟶ (∃P'. P ⟹α ≺ P' ∧ (P', Q') ∈ Rel))" 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 by(simp add: weakStepSimulation_def) blast lemma simCasesCont[consumes 1, case_names Bound Free]: fixes P :: pi and Q :: pi and Rel :: "(pi × pi) set" and C :: "'a::fs_name" assumes Eqvt: "eqvt Rel" and Bound: "⋀a x Q'. ⟦x ♯ C; Q ⟼ a<νx> ≺ Q'⟧ ⟹ ∃P'. P ⟹ a<νx> ≺ P' ∧ (P', Q') ∈ Rel" and Free: "⋀α Q'. Q ⟼ α ≺ Q' ⟹ ∃P'. P ⟹ α ≺ P' ∧ (P', Q') ∈ Rel" shows "P ↝«Rel» Q" proof - from Free show ?thesis proof(auto simp add: weakStepSimulation_def) 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: alphaBoundOutput) with cFreshC have "∃P'. P ⟹ a<νc> ≺ P' ∧ (P', [(x, c)] ∙ Q') ∈ Rel" by(rule Bound) then obtain P' where PTrans: "P ⟹ a<νc> ≺ P'" and P'RelQ': "(P', [(x, c)] ∙ Q') ∈ Rel" by blast from PTrans ‹x ♯ P› ‹c ≠ x› have "P ⟹a<νx> ≺ ([(x, c)] ∙ P')" by(simp add: weakTransitionAlpha name_swap) moreover from Eqvt P'RelQ' have "([(x, c)] ∙ P', [(x, c)] ∙ [(x, c)] ∙ Q') ∈ Rel" by(rule eqvtRelI) with ‹c ≠ x› have "([(x, c)] ∙ P', Q') ∈ Rel" by simp ultimately show "∃P'. P ⟹a<νx> ≺ P' ∧ (P', Q') ∈ Rel" by blast qed qed lemma simCases[consumes 0, case_names Bound Free]: fixes P :: pi and Q :: pi and Rel :: "(pi × pi) set" and C :: "'a::fs_name" assumes "⋀a x Q'. ⟦Q ⟼ a<νx> ≺ Q'; x ♯ P⟧ ⟹ ∃P'. P ⟹ a<νx> ≺ P' ∧ (P', Q') ∈ Rel" and "⋀α Q'. Q ⟼ α ≺ Q' ⟹ ∃P'. P ⟹ α ≺ P' ∧ (P', Q') ∈ Rel" shows "P ↝«Rel» Q" using assms by(auto simp add: weakStepSimulation_def) lemma simE: fixes P :: pi and Rel :: "(pi × pi) set" and Q :: pi and a :: name and x :: name and Q' :: pi assumes "P ↝«Rel» Q" shows "Q ⟼a<νx> ≺ Q' ⟹ x ♯ P ⟹ ∃P'. P ⟹a<νx> ≺ P' ∧ (P', Q') ∈ Rel" and "Q ⟼α ≺ Q' ⟹ ∃P'. P ⟹α ≺ P' ∧ (P', Q') ∈ Rel" using assms by(simp add: weakStepSimulation_def)+ 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: "⋀R S. (R, S) ∈ Rel ⟹ R ↝<Rel> S" and Eqvt: "eqvt Rel" and PRelQ: "(P, Q) ∈ Rel" shows "Q ⟹a<νx> ≺ Q' ⟹ x ♯ P ⟹ ∃P'. P ⟹a<νx> ≺ P' ∧ (P', Q') ∈ Rel" and "Q ⟹α ≺ Q' ⟹ ∃P'. P ⟹α ≺ P' ∧ (P', Q') ∈ Rel" proof - assume QTrans: "Q ⟹a<νx> ≺ Q'" assume "x ♯ P" from QTrans obtain Q'' Q''' where QChain: "Q ⟹⇩_{τ}Q''" and Q''Trans: "Q'' ⟼a<νx> ≺ Q'''" and Q'''Chain: "Q''' ⟹⇩_{τ}Q'" by(blast dest: transitionE) from QChain PRelQ Sim have "∃P''. P ⟹⇩_{τ}P'' ∧ (P'', Q'') ∈ Rel" by(rule weakSimTauChain) then obtain P'' where PChain: "P ⟹⇩_{τ}P''" and P''RelQ'': "(P'', Q'') ∈ Rel" by blast from PChain ‹x ♯ P› have xFreshP'': "x ♯ P''" by(rule freshChain) from P''RelQ'' have "P'' ↝<Rel> Q''" by(rule Sim) with Q''Trans xFreshP'' obtain P''' where P''Trans: "P'' ⟹a<νx> ≺ P'''" and P'''RelQ''': "(P''', Q''') ∈ Rel" by(blast dest: Weak_Early_Sim.simE) have "∃P'. P''' ⟹⇩_{τ}P' ∧ (P', Q') ∈ Rel" using Q'''Chain P'''RelQ''' Sim by(rule 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 ⟹a<νx> ≺ P'" by(blast dest: Weak_Early_Step_Semantics.chainTransitionAppend) with P'RelQ' show "∃P'. P ⟹ a<νx> ≺ P' ∧ (P', Q') ∈ Rel" by blast next assume "Q ⟹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) from QChain Q''Trans Q'''Chain show "∃P'. P ⟹α ≺ P' ∧ (P', Q') ∈ Rel" proof(induct arbitrary: α Q''' Q' rule: tauChainInduct) case id from PSimQ ‹Q ⟼α ≺ Q'''› have "∃P'. P ⟹α ≺ P' ∧ (P', Q''') ∈ Rel" by(blast dest: simE) then obtain P''' where PTrans: "P ⟹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_Early_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 ⟹α ≺ P'" by(blast dest: Weak_Early_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 PChain: "P ⟹τ ≺ P''" and P''RelQ'': "(P'', Q'') ∈ Rel" by(drule_tac ih) auto from P''RelQ'' have "P'' ↝<Rel> Q''" by(rule Sim) hence "∃P'''. P'' ⟹⇧^{^}α ≺ P''' ∧ (P''', Q''') ∈ Rel" using ‹Q'' ⟼α ≺ Q'''› by(rule Weak_Early_Sim.simE) then obtain P''' where P''Trans: "P'' ⟹⇧^{^}α ≺ P'''" and P'''RelQ''': "(P''', Q''') ∈ Rel" by blast from ‹Q''' ⟹⇩_{τ}Q'› P'''RelQ''' Sim have "∃P'. P''' ⟹⇩_{τ}P' ∧ (P', Q') ∈ Rel" by(rule Weak_Early_Sim.weakSimTauChain) then obtain P' where P'''Chain: "P''' ⟹⇩_{τ}P'" and P'RelQ': "(P', Q') ∈ Rel" by blast from PChain P''Trans have "P ⟹α ≺ P'''" apply(auto simp add: freeTransition_def weakFreeTransition_def) apply(drule tauActTauChain, auto) by(rule_tac x=P'''aa in exI) auto hence "P ⟹α ≺ P'" using P'''Chain by(rule Weak_Early_Step_Semantics.chainTransitionAppend) with P'RelQ' show ?case by blast qed qed lemma eqvtI: fixes P :: pi and Q :: pi and Rel :: "(pi × pi) set" and perm :: "name prm" assumes PSimQ: "P ↝«Rel» Q" and RelRel': "Rel ⊆ Rel'" and EqvtRel': "eqvt Rel'" shows "(perm ∙ P) ↝«Rel'» (perm ∙ Q)" proof(induct rule: simCases) case(Bound a x Q') have xFreshP: "x ♯ perm ∙ P" by fact have QTrans: "(perm ∙ Q) ⟼ a<νx> ≺ Q'" by fact hence "(rev perm ∙ (perm ∙ Q)) ⟼ rev perm ∙ (a<νx> ≺ Q')" by(rule eqvt) 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 ⟹ (rev perm ∙ a)<ν(rev perm ∙ x)> ≺ P'" and P'RelQ': "(P', rev perm ∙ Q') ∈ Rel" using PSimQ by(blast dest: simE) from PTrans have "(perm ∙ P) ⟹(perm ∙ rev perm ∙ a)<ν(perm ∙ rev perm ∙ x)> ≺ perm ∙ P'" by(rule Weak_Early_Step_Semantics.eqvtI) hence L1: "(perm ∙ P) ⟹ a<νx> ≺ (perm ∙ P')" by(simp add: name_per_rev) 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) hence "(perm ∙ P', Q') ∈ Rel'" by(simp add: name_per_rev) with L1 show ?case by blast next case(Free α Q') have QTrans: "(perm ∙ Q) ⟼ α ≺ Q'" by fact hence "(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 PSimQ obtain P' where PTrans: "P ⟹ (rev perm ∙ α) ≺ P'" and PRel: "(P', (rev perm ∙ Q')) ∈ Rel" by(blast dest: simE) from PTrans have "(perm ∙ P) ⟹(perm ∙ rev perm ∙ α) ≺ perm ∙ P'" by(rule Weak_Early_Step_Semantics.eqvtI) hence L1: "(perm ∙ P) ⟹ α ≺ (perm ∙ P')" by(simp add: name_per_rev) from PRel EqvtRel' RelRel' have "((perm ∙ P'), (perm ∙ (rev perm ∙ Q'))) ∈ Rel'" by(force intro: eqvtRelI) hence "((perm ∙ P'), Q') ∈ Rel'" by(simp add: name_per_rev) with L1 show ?case by blast 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_Early_Step_Semantics.singleActionChain simp add: weakStepSimulation_def weakFreeTransition_def) 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: "⋀S T. (S, T) ∈ Rel ⟹ S ↝<Rel> T" and PRelQ: "(P, Q) ∈ Rel" shows "P ↝«Rel''» R" proof - from Eqvt'' show ?thesis proof(induct rule: simCasesCont[of _ "(P, Q)"]) case(Bound a x R') have "x ♯ (P, Q)" by fact hence xFreshP: "x ♯ P" and xFreshQ: "x ♯ Q" by(simp add: fresh_prod)+ have RTrans: "R ⟼a<νx> ≺ R'" by fact from xFreshQ QSimR RTrans obtain Q' where QTrans: "Q ⟹ a<νx> ≺ Q'" and Q'Rel'R': "(Q', R') ∈ Rel'" by(blast dest: simE) with PSimQ Sim Eqvt PRelQ QTrans xFreshP have "∃P'. P ⟹ a<νx> ≺ P' ∧ (P', Q') ∈ Rel" by(blast intro: simE2) then obtain P' where PTrans: "P ⟹ a<νx> ≺ P'" and P'RelQ': "(P', Q') ∈ Rel" by blast moreover from P'RelQ' Q'Rel'R' Trans have "(P', R') ∈ Rel''" by blast ultimately show ?case by blast next case(Free α R') have RTrans: "R ⟼ α ≺ R'" by fact with QSimR obtain Q' where QTrans: "Q ⟹ α ≺ Q'" and Q'RelR': "(Q', R') ∈ Rel'" by(blast dest: simE) from PSimQ Sim Eqvt PRelQ QTrans have "∃P'. P ⟹ α ≺ P' ∧ (P', Q') ∈ Rel" by(blast intro: simE2) then obtain P' where PTrans: "P ⟹ α ≺ P'" and P'RelQ': "(P', Q') ∈ Rel" by blast from P'RelQ' Q'RelR' Trans have "(P', R') ∈ Rel''" by blast with PTrans show ?case by blast qed qed lemma strongSimWeakSim: fixes P :: pi and Q :: pi and Rel :: "(pi × pi) set" assumes PSimQ: "P ↝[Rel] Q" shows "P ↝«Rel» Q" proof(induct rule: simCases) case(Bound a x Q') have "Q ⟼a<νx> ≺ Q'" and "x ♯ P" by fact+ with PSimQ obtain P' where PTrans: "P ⟼a<νx> ≺ P'" and P'RelQ': "(P', Q') ∈ Rel" by(blast dest: Strong_Early_Sim.elim) from PTrans have "P ⟹a<νx> ≺ P'" by(force intro: Weak_Early_Step_Semantics.singleActionChain simp add: weakFreeTransition_def) with P'RelQ' show ?case by blast next case(Free α Q') have "Q ⟼α ≺ Q'" by fact with PSimQ obtain P' where PTrans: "P ⟼α ≺ P'" and P'RelQ': "(P', Q') ∈ Rel" by(blast dest: Strong_Early_Sim.elim) from PTrans have "P ⟹α ≺ P'" by(rule Weak_Early_Step_Semantics.singleActionChain) with P'RelQ' show ?case 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: weakStepSimulation_def Weak_Early_Sim.weakSimulation_def weakFreeTransition_def) end