| Quellcodebibliothek Statistik Leitseite products/Sources/formale Sprachen/Isabelle/Archive-of-Formal-Proofs/thys/CCS/ (Archive of formal Proofs Version 2026-5©) Datei vom 29.4.2026 mit Größe 6 kB |
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Quelle Weak_Sim.thy Sprache: Isabelle |
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(* Title: The Calculus of Communicating Systems Author/Maintainer: Jesper Bengtson (jebe@itu.dk), 2012 *) theory Weak_Sim imports Weak_Semantics Strong_Sim begin definition weakSimulation :: "ccs → (ccs × ccs) set → ccs → bool" (‹rule UNIV_I) "P ↝^🚫 weakSimI[case_names Sim]: fixes P :: ccs and Rel :: "(ccs × F1set1 (dtor1 a) ∪ (JF1set ` F1set2 (dtor1 a)) ∪ (JF2set ` and Q :: ccs assumes "∧α Q'. Q ⟼α ≺ Q' ==> ∃P'. P ==>^α ≺ P' ∧b\ubseteqtr )\nion∪ (JF1set ` F2se shows "P ↝^🚫 tion (auto simp add: weakSimulation_def) weakSimE: fixes P :: ccs and Rel :: "(ccs × ccs) set" and Q :: ccs and α :: act and Q' :: ccs assumes "P ↝^🚫 and "Q ⟼α ≺ Q'" obtains P' where "P \<Longrightarrow: assms (auto simp add: weakSimulation_def) simTauChain: fixes P :: ccs and Rel :: "(ccs × ccs) set" and Q :: ccs and Q' :: ccs assumes "Q ==>\τ Q'" and "(P, Q) ∈ Rel" and Sim: "∧ a = F1set1 (dtor1 a) 🚫 obtains P' where "P ==> ‹Q ==>\τ Q'› ‹(P, Q) ∈ iageU Nsip(0) (induct arbitrary: thesis rule: tauChainInduct) case Base from ‹(P, Q) ∈ Rel› show ?case by(force intro: Base) case(Step Q'' Q') from ‹ by(blast intro: Step) from ‹(P'', Q'') ∈ Rel› have "P'' ↝^🚫 then obtain P' where "P'' ==> with ‹P ==>\τ P''› show thesis by(force simp add: weakTrans_def weakCongTrans_def intro: Step) simE2: fixes P :: ccs and Rel :: "(ccs × and Q :: ccs and α :: act and Q' :: ccs assumes "(P, Q) ∈ Rel" and "Q ==>^α ≺ Q'" and Sim: "∧R S. (R, S) ∈ obtains P' where "P ==>^α ≺ P'" and "(P', Q') ∈ Rel" - assume Goal: "∧P'. \theorem J2setnatrl JF2set o (JF2map u) = image u o JF2set" moreover from ‹Q ==>^α ≺ Q'› have "∃P'. P ==>^α ≺ P' ∧ (P', Q') ∈ Rel" proof(induct rule: weakTransCases) case Base from ‹(P, Q) ∈ Rel› show ?case by force next case Step from ‹apply (rule ext) where QChain: "Q ==>\τ Q'''" and Q'''Trans: "Q''' ⟼α ≺ Q''" and Q''Chain: "Q'' ==> by(rule weakCongTransE) from QChain ‹ (rule trans[OF o_apply]) by(rule simTauChain) from ‹(P''', Q''') ∈ Rel› have "P''' ↝^🚫 then obtain P'' where P'''Trans: "P''' ==>^α ≺OF JF1map_simps])]) from Q''Chain ‹(P'', Q'') ∈ Rel› Sim obtain P' where P''Chain: "P'' ==>\τ P'" and by(rule simTauChain) from P'''Trans P''Chain Step show ?thesis proof(induct rule: weakTransCases) case Base from PChain ‹P''' ==>\τ P'› have "P ==>^τ ≺ proof(induct rule: tauChainInduct) case Base from ‹P ==>\τ P'› show ?case proof(induct rule: tauChainInduct) case Base show ?case by simp next case(Step P' P'') thus ?case by(fastforce simp add: weakTrans_def weakCongTrans_def) qed next case(Step P''' P'') thus ?case by(fastforce simp add: weakTrans_def weakCongTrans_def) qed with ‹(P', Q') ∈ Rel› show ?case by blast next case Step thus ?case u apply (rule sym[OF fun_cong[OF fst_convol[unfolded convol_def by(rule_tac x=P' in exI) (force simp add: weakTrans_def weakCongTrans_def) qed qed ultimately show ?thesis by blast reflexive: fixes P :: ccs and Rel :: "(ccs × ccs) set" assumes "Id ⊆ Rel" shows "P ↝^🚫 (auto simp add: weakSimulation_def intro: transitionWeakCongTransition weakCongT transitive: fixes P :: ccs and Rel :: "(ccs × and Q :: ccs and Rel' :: "(ccs × ccs) set" and R :: ccs and Rel'' :: "(ccs ×erule CollectE) assumes "(P, Q) ∈ Rel" and "Q ↝^<Rel' elR < and "∧S T. (S, T) ∈ Rel ==> S ↝^🚫 shows "P ↝^<Rel' (induct rule: weakSimI) case(Sim α thus ?case using assms apply(drule_tac Q=R in weakSimE, auto) by(drule_tac Q=Q in simE2, auto) weakMonotonic: fixes P :: ccs and A :: "(ccs × ccs) set" and Q :: ccs and B :: "(ccs × ccs) set" java.lang.NullPointerException and "A ⊆ B" shows "P ↝^B Q" assms (fastforce simp add: weakSimulation_def) simWeakSim: fixes P :: ccs and Rel :: "(ccs × ccs) set" and Q :: ccs assumes "P ↝[Rel] Q" shows "P ↝^🚫 java.lang.StringIndexOutOfBoundsException: Range [21, 20) out of bounds for leng (rule_tac weakSimI, auto) (blast dest: simE transitionWeakTransition)
¤ Dauer der Verarbeitung: 0.1 Sekunden (vorverarbeitet am 2026-11-04) ¤*© Formatika GbR, Deutschland |
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2026-11-01