theory Weak_Equivalence_Implies_Bisimilarity imports Weak_Logical_Equivalence begin section ‹Weak Logical Equivalence Implies Weak Bisimilarity› context indexed_weak_nominal_ts begin definition is_distinguishing_formula :: "('idx, 'pred, 'act) formula ⇒ 'state ⇒ 'state ⇒ bool" ("_ distinguishes _ from _" [100,100,100] 100) where "x distinguishes P from Q ≡ P ⊨ x ∧ ¬ Q ⊨ x" lemma is_distinguishing_formula_eqvt (*[eqvt]*) [simp]: assumes "x distinguishes P from Q" shows "(p ∙ x) distinguishes (p ∙ P) from (p ∙ Q)" using assms unfolding is_distinguishing_formula_def by (metis permute_minus_cancel(2) valid_eqvt) lemma weakly_equivalent_iff_not_distinguished: "(P ≡⋅ Q) ⟷ ¬(∃x. weak_formula x ∧ x distinguishes P from Q)" by (meson is_distinguishing_formula_def weakly_logically_equivalent_def valid_Not wf_Not) text ‹There exists a distinguishing weak formula for~@{term P} and~@{term Q} whose support is contained in~@{term "supp P"}.› lemma distinguished_bounded_support: assumes "weak_formula x" and "x distinguishes P from Q" obtains y where "weak_formula y" and "supp y ⊆ supp P" and "y distinguishes P from Q" proof - let ?B = "{p ∙ x|p. supp P ♯* p}" have "supp P supports ?B" unfolding supports_def proof (clarify) fix a b assume a: "a ∉ supp P" and b: "b ∉ supp P" have "(a ⇌ b) ∙ ?B ⊆ ?B" proof fix x' assume "x' ∈ (a ⇌ b) ∙ ?B" then obtain p where 1: "x' = (a ⇌ b) ∙ p ∙ x" and 2: "supp P ♯* p" by (auto simp add: permute_set_def) let ?q = "(a ⇌ b) + p" from 1 have "x' = ?q ∙ x" by simp moreover from a and b and 2 have "supp P ♯* ?q" by (metis fresh_perm fresh_star_def fresh_star_plus swap_atom_simps(3)) ultimately show "x' ∈ ?B" by blast qed moreover have "?B ⊆ (a ⇌ b) ∙ ?B" proof fix x' assume "x' ∈ ?B" then obtain p where 1: "x' = p ∙ x" and 2: "supp P ♯* p" by auto let ?q = "(a ⇌ b) + p" from 1 have "x' = (a ⇌ b) ∙ ?q ∙ x" by simp moreover from a and b and 2 have "supp P ♯* ?q" by (metis fresh_perm fresh_star_def fresh_star_plus swap_atom_simps(3)) ultimately show "x' ∈ (a ⇌ b) ∙ ?B" using mem_permute_iff by blast qed ultimately show "(a ⇌ b) ∙ ?B = ?B" .. qed then have supp_B_subset_supp_P: "supp ?B ⊆ supp P" by (metis (erased, lifting) finite_supp supp_is_subset) then have finite_supp_B: "finite (supp ?B)" using finite_supp rev_finite_subset by blast have "?B ⊆ (λp. p ∙ x) ` UNIV" by auto then have "|?B| ≤o |UNIV :: perm set|" by (rule surj_imp_ordLeq) also have "|UNIV :: perm set| <o |UNIV :: 'idx set|" by (metis card_idx_perm) also have "|UNIV :: 'idx set| ≤o natLeq +c |UNIV :: 'idx set|" by (metis Cnotzero_UNIV ordLeq_csum2) finally have card_B: "|?B| <o natLeq +c |UNIV :: 'idx set|" . let ?y = "Conj (Abs_bset ?B) :: ('idx, 'pred, 'act) formula" have "weak_formula ?y" proof show "finite (supp (Abs_bset ?B :: _ set['idx]))" using finite_supp_B card_B by simp next fix x' assume "x' ∈ set_bset (Abs_bset ?B :: _ set['idx])" with card_B obtain p where "x' = p ∙ x" using Abs_bset_inverse mem_Collect_eq by auto then show "weak_formula x'" using ‹weak_formula x› by (metis weak_formula_eqvt) qed moreover from finite_supp_B and card_B and supp_B_subset_supp_P have "supp ?y ⊆ supp P" by simp moreover have "?y distinguishes P from Q" unfolding is_distinguishing_formula_def proof from assms show "P ⊨ ?y" by (auto simp add: card_B finite_supp_B) (metis is_distinguishing_formula_def supp_perm_eq valid_eqvt) next from assms show "¬ Q ⊨ ?y" by (auto simp add: card_B finite_supp_B) (metis is_distinguishing_formula_def permute_zero fresh_star_zero) qed ultimately show ?thesis .. qed lemma weak_equivalence_is_weak_bisimulation: "is_weak_bisimulation weakly_logically_equivalent" proof - have "symp weakly_logically_equivalent" by (metis weakly_logically_equivalent_def sympI) moreover ― ‹weak static implication› { fix P Q φ assume "P ≡⋅ Q" and "P ⊢ φ" then have "∃Q'. Q ⇒ Q' ∧ P ≡⋅ Q' ∧ Q' ⊢ φ" proof - { let ?Q' = "{Q'. Q ⇒ Q' ∧ Q' ⊢ φ}" assume "∀Q'∈?Q'. ¬ P ≡⋅ Q'" then have "∀Q'∈?Q'. ∃x :: ('idx, 'pred, 'act) formula. weak_formula x ∧ x distinguishes P from Q'" by (metis weakly_equivalent_iff_not_distinguished) then have "∀Q'∈?Q'. ∃x :: ('idx, 'pred, 'act) formula. weak_formula x ∧ supp x ⊆ supp P ∧ x distinguishes P from Q'" by (metis distinguished_bounded_support) then obtain f :: "'state ⇒ ('idx, 'pred, 'act) formula" where *: "∀Q'∈?Q'. weak_formula (f Q') ∧ supp (f Q') ⊆ supp P ∧ (f Q') distinguishes P from Q'" by metis have "supp (f ` ?Q') ⊆ supp P" by (rule set_bounded_supp, fact finite_supp, cut_tac "*", blast) then have finite_supp_image: "finite (supp (f ` ?Q'))" using finite_supp rev_finite_subset by blast have "|f ` ?Q'| ≤o |UNIV :: 'state set|" using card_of_UNIV card_of_image ordLeq_transitive by blast also have "|UNIV :: 'state set| <o |UNIV :: 'idx set|" by (metis card_idx_state) also have "|UNIV :: 'idx set| ≤o natLeq +c |UNIV :: 'idx set|" by (metis Cnotzero_UNIV ordLeq_csum2) finally have card_image: "|f ` ?Q'| <o natLeq +c |UNIV :: 'idx set|" . let ?y = "Conj (Abs_bset (f ` ?Q')) :: ('idx, 'pred, 'act) formula" have "weak_formula ?y" proof (standard+) show "finite (supp (Abs_bset (f ` ?Q') :: _ set['idx]))" using finite_supp_image card_image by simp next fix x assume "x ∈ set_bset (Abs_bset (f ` ?Q') :: _ set['idx])" with card_image obtain Q' where "Q' ∈ ?Q'" and "x = f Q'" using Abs_bset_inverse imageE set_bset set_bset_to_set_bset by auto then show "weak_formula x" using "*" by metis qed let ?z = "⟨⟨τ⟩⟩(Conj (binsert (Pred φ) (bsingleton ?y)))" have "weak_formula ?z" by standard (fact ‹weak_formula ?y›) moreover have "P ⊨ ?z" proof - have "P ⇒⟨τ⟩ P" by simp moreover { fix Q' assume "Q ⇒ Q' ∧ Q' ⊢ φ" with "*" have "P ⊨ f Q'" by (metis is_distinguishing_formula_def mem_Collect_eq) } with ‹P ⊢ φ› have "P ⊨ Conj (binsert (Pred φ) (bsingleton ?y))" by (simp add: binsert.rep_eq finite_supp_image card_image) ultimately show ?thesis using valid_weak_action_modality by blast qed moreover have "¬ Q ⊨ ?z" proof assume "Q ⊨ ?z" then obtain Q' where 1: "Q ⇒ Q'" and "Q' ⊨ Conj (binsert (Pred φ) (bsingleton ?y))" using valid_weak_action_modality by auto then have 2: "Q' ⊢ φ" and 3: "Q' ⊨ ?y" by (simp add: binsert.rep_eq finite_supp_image card_image)+ from 3 have "⋀Q''. Q ⇒ Q'' ∧ Q'' ⊢ φ ⟶ Q' ⊨ f Q''" by (simp add: finite_supp_image card_image) with 1 and 2 and "*" show False using is_distinguishing_formula_def by blast qed ultimately have False by (metis ‹P ≡⋅ Q› weakly_logically_equivalent_def) } then show ?thesis by blast qed } moreover ― ‹weak simulation› { fix P Q α P' assume "P ≡⋅ Q" and "bn α ♯* Q" and "P → ⟨α,P'⟩" then have "∃Q'. Q ⇒⟨α⟩ Q' ∧ P' ≡⋅ Q'" proof - { let ?Q' = "{Q'. Q ⇒⟨α⟩ Q'}" assume "∀Q'∈?Q'. ¬ P' ≡⋅ Q'" then have "∀Q'∈?Q'. ∃x :: ('idx, 'pred, 'act) formula. weak_formula x ∧ x distinguishes P' from Q'" by (metis weakly_equivalent_iff_not_distinguished) then have "∀Q'∈?Q'. ∃x :: ('idx, 'pred, 'act) formula. weak_formula x ∧ supp x ⊆ supp P' ∧ x distinguishes P' from Q'" by (metis distinguished_bounded_support) then obtain f :: "'state ⇒ ('idx, 'pred, 'act) formula" where *: "∀Q'∈?Q'. weak_formula (f Q') ∧ supp (f Q') ⊆ supp P' ∧ (f Q') distinguishes P' from Q'" by metis have "supp P' supports (f ` ?Q')" unfolding supports_def proof (clarify) fix a b assume a: "a ∉ supp P'" and b: "b ∉ supp P'" have "(a ⇌ b) ∙ (f ` ?Q') ⊆ f ` ?Q'" proof fix x assume "x ∈ (a ⇌ b) ∙ (f ` ?Q')" then obtain Q' where 1: "x = (a ⇌ b) ∙ f Q'" and 2: "Q ⇒⟨α⟩ Q'" by auto (metis (no_types, lifting) imageE image_eqvt mem_Collect_eq permute_set_eq_image) with "*" and a and b have "a ∉ supp (f Q')" and "b ∉ supp (f Q')" by auto with 1 have "x = f Q'" by (metis fresh_perm fresh_star_def supp_perm_eq swap_atom) with 2 show "x ∈ f ` ?Q'" by simp qed moreover have "f ` ?Q' ⊆ (a ⇌ b) ∙ (f ` ?Q')" proof fix x assume "x ∈ f ` ?Q'" then obtain Q' where 1: "x = f Q'" and 2: "Q ⇒⟨α⟩ Q'" by auto with "*" and a and b have "a ∉ supp (f Q')" and "b ∉ supp (f Q')" by auto with 1 have "x = (a ⇌ b) ∙ f Q'" by (metis fresh_perm fresh_star_def supp_perm_eq swap_atom) with 2 show "x ∈ (a ⇌ b) ∙ (f ` ?Q')" using mem_permute_iff by blast qed ultimately show "(a ⇌ b) ∙ (f ` ?Q') = f ` ?Q'" .. qed then have supp_image_subset_supp_P': "supp (f ` ?Q') ⊆ supp P'" by (metis (erased, lifting) finite_supp supp_is_subset) then have finite_supp_image: "finite (supp (f ` ?Q'))" using finite_supp rev_finite_subset by blast have "|f ` ?Q'| ≤o |UNIV :: 'state set|" by (metis card_of_UNIV card_of_image ordLeq_transitive) also have "|UNIV :: 'state set| <o |UNIV :: 'idx set|" by (metis card_idx_state) also have "|UNIV :: 'idx set| ≤o natLeq +c |UNIV :: 'idx set|" by (metis Cnotzero_UNIV ordLeq_csum2) finally have card_image: "|f ` ?Q'| <o natLeq +c |UNIV :: 'idx set|" . let ?y = "Conj (Abs_bset (f ` ?Q')) :: ('idx, 'pred, 'act) formula" have "weak_formula (⟨⟨α⟩⟩?y)" proof (standard+) show "finite (supp (Abs_bset (f ` ?Q') :: _ set['idx]))" using finite_supp_image card_image by simp next fix x assume "x ∈ set_bset (Abs_bset (f ` ?Q') :: _ set['idx])" with card_image obtain Q' where "Q' ∈ ?Q'" and "x = f Q'" using Abs_bset_inverse imageE set_bset set_bset_to_set_bset by auto then show "weak_formula x" using "*" by metis qed moreover have "P ⊨ ⟨⟨α⟩⟩?y" unfolding valid_weak_action_modality proof (standard+) from ‹P → ⟨α,P'⟩› show "P ⇒⟨α⟩ P'" by simp next { fix Q' assume "Q ⇒⟨α⟩ Q'" with "*" have "P' ⊨ f Q'" by (metis is_distinguishing_formula_def mem_Collect_eq) } then show "P' ⊨ ?y" by (simp add: finite_supp_image card_image) qed moreover have "¬ Q ⊨ ⟨⟨α⟩⟩?y" proof assume "Q ⊨ ⟨⟨α⟩⟩?y" then obtain Q' where 1: "Q ⇒⟨α⟩ Q'" and 2: "Q' ⊨ ?y" using ‹bn α ♯* Q› by (metis valid_weak_action_modality_fresh) from 2 have "⋀Q''. Q ⇒⟨α⟩ Q'' ⟶ Q' ⊨ f Q''" by (simp add: finite_supp_image card_image) with 1 and "*" show False using is_distinguishing_formula_def by blast qed ultimately have False by (metis ‹P ≡⋅ Q› weakly_logically_equivalent_def) } then show ?thesis by auto qed } ultimately show ?thesis unfolding is_weak_bisimulation_def by metis qed theorem weak_equivalence_implies_weak_bisimilarity: assumes "P ≡⋅ Q" shows "P ≈⋅ Q" using assms by (metis weakly_bisimilar_def weak_equivalence_is_weak_bisimulation) end end