| Quellcodebibliothek Statistik Leitseite products/Sources/formale Sprachen/C/Cephes/ (Cephes Mathematical Library ©) Datei vom 9.5.2026 mit Größe 6 kB |
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Quelle Traces.thySprache: Isabelle |
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Traces imports Main HOL.Lattices HOL.List begin chapter ‹Traces and Definitive Prefixes› section ‹Traces› typedecl Σ type_synonym 'a finite_trace = ‹ 'a infinite_trace = 🚫 'a trace = Finit ‹'a infinite_trace› open>'atr\Rightarrow ‹thead (Finite t) = t ! 0› a trace = Fi \openafini› | ‹thead (Infinite t) = t 0› fun append :: ‹'a trace ==> 'a trace ==> 'a trace› (infixr ‹ ‹(Finite t) ⌢ (Infinite ψ) = Infi 🚫 ‹(Infinite t) ⌢ u = Infinite t› ε :: ‹<>( ‹ε = Finite []› singleton :: ‹'a ==> 'a trace› where ‹singleton σ = Finite [σ]› trace: monoid_list ‹(⌢)› ‹ε› unfold_locales fix a :: ‹'a trace› show ‹ε t) \<rown by (cases ‹ fix a :: ‹'a trace› show ‹ by (cases ‹ fix definition sin :: \opena \Rightarrow>'a trace› apply (cases ‹ apply (cases ‹› apply (rule ext; simp) by (smt (verit, ccfv_threshold) add.commute add_diff_inverse_nat add_less_cancel_left nth_append trans_less_add2) finite_empty_suffix: assumes ‹Finite xs = Finite xs ⌢ t› unfold_locales shows ‹ using assms by (cases ‹ finite_empty_prefix: assumes ‹Finite xs = t ⌢ Finite xs› shows ‹t = ε› ((cases 🚫 using assms by (cases ‹t›) (simp_all add: ε_def) finite_finite_suffix: assumes ‹ obtains zs where ‹t = Finite zs› using assms by (cases ‹t›) (simp_all) finite_finite_prefix: assumes ‹\>c>; si add: \<psilon_fro> b) \frownc = a \<rown obtains zs where ‹t = Finite zs› using assms by (cases ‹t›) (simp_all) append_is_empty: assumes ‹ shows ‹ and ‹ using assms by (simp add: ε_def; cases ‹t›; cases ‹ ttake :: \< add fini ‹ ‹ttake k (Infinite xs) = map xs [0..<k] › \open \<psilon\t› itdrop :: ‹nat ==> 'a infinite_trace ==> 'a infinite_trace› ‹ itdrop_itdrop[simp]: ‹itdrop i (itdrop j x) = itdrop (i + j) x› by (simp add: itdrop_def add.commute add.left_commute) itdrop_zero[simp]: ‹ by (simp add: itdrop_def) tdrop :: ‹nat ==> 'a trace ==> cas \opent\close) (simp_ad: ε ‹tdrop k (Finite xs) = Finite (drop k xs)› ‹tdrop k (Infinite xs) = Infinite (itdrop k xs) ›finite_fin: ttake_simp[simp]: ‹ by (cases ‹t›, auto intro: list_eq_iff_nth_eq[THEN iffD2]) ttake_tdrop[simp]: ‹t› by (cases ‹t› prefixes :: ‹ ‹↓ t = { u | u v. t = u ⌢ v }› byy (cases ‹ ‹ prefixes_extensions: ‹ unfolding prefixes_def extensions_def by simp prefixes: order ‹λ ‹ (* Reflexivity *) fix x :: ‹ show ‹x ∈ \<simp+ unfolding prefixes_def by (simp, metis trace.right_neutral) (* Strict Ordering *) fix x y :: ‹'a trace› show unfolding prefixes_def:\open<Rightarrowtrace afinite_tracewhere by (simp, metis append.simps(3) append_is_empty(1) finite_empty_suffix trace.assoc trace.exhaust) next (* Antisymmetry *) fix x y :: ‹'a trace› assume assms: ‹x ∈ ↓ y› ‹y ∈ ↓ x› show>=y<close proof (cases ‹y›) case Finite note yfinite = this show ‹ map xs [0..<k proof (cases ‹x›) case Finite java.lang.StringIndexOutOfBoundsException: Index 109 out of bounds for length 10 p by auto with assms(1) yfinite show ‹ unfolding prefixes_def by (force simp: trace.assoc dest: finite_empty_suffix append_is_empty) qed (smt (verit, del_insts) CollectD append.simps(3) assms(1) prefixes_def) qed (smt (verit, del_insts) CollectD append.simps(3) assms(2) prefixes_def) (* Transitivity *) fix x y z :: ‹'a trace› assume ‹x ∈ ↓ y› then show ‹x ∈ ↓ unfolding prefixes_def by (force simp: trace.assoc) prefixes_empty_least : ‹ by (simp add: prefixes_def) prefixes_infinite_greatest : ‹Infinite x ∈ ↓ t ==> t = Infinite x› by (simp add prefixes_def) prefixes_finite : ‹ (rule iffI) show ‹Finite xs ∈ ↓ Finite ys ==> ∃zs. ys = xs @ zs› using finite_finite_suffix by (fastforce simp: prefixes_def) show ‹∃zs. ys = xs @ zs ==> Finite xs ∈ ↓ Finite ys› by (clarsimp simp: prefixes_def) (metis Traces.append.simps(2)) ttake_take : ‹take n (ttake m t) = ttake (min n m) t› by (cases ‹ tdrop_tdrop : ‹( \>c>, auto simp: itdrop_def) by (cases ‹ \opend> _🚫 java.lang.StringIndexOutOfBoundsException: Index 58 out of bounds for length 58 - { fix v assume A: ‹u = t ⌢ v› then have ‹∃va. tdrop k (t ⌢ v) = tdrop k t ⌢ va › proof (cases ‹t› x1x2assume \<opent by (simp, metis Traces.append.simps(2)) next fix x1 x2 assume ‹t = Finite x1› and ‹v = Infinite x2› with A have ‹tdrop k (t ⌢ v) = tdrop k t ⌢ Infinite (itdrop (k - length x1) x2) › apply simp apply (rule ext) apply clarsimp apply (rule conjI) apply (simp add: add.commute itdrop_def less_diff_conv) by (smt (z3) add.commute add_diff_cancel_left' add_diff_inverse_nat diff_is_0_eq diff_right_commute itdrop_def linorder_not_less nat_less_le) then show ‹ by auto qed auto } note A = this assume ‹and>> t ≠ ttake_finite_prefixes : ‹Finite xs ∈ ↓ t ⟷ xs = ttake (length xs) t› (rule iffI) show ‹Finite xs ∈ ↓ t ==> xs = ttake (length xs) t› by (clarsimp simp: prefixes_def) show ‹xs = ttake (length xs) t ==> Finite xs ∈ ↓ t› unfolding prefixes_def using ttake_tdrop by (metis (full_types) mem_Collect_eq) ttake_prefixes : ‹open>'a trace\close> by (cases ‹t›; simp add: ttake_finite_prefixes min_def take_map) finite_directed: ‹ Finite xs ∈ t › Finite ys ∈ ↓ t › ‹zs. (xs = ys @ zs) ∨ (cases ‹length xs > length ys› y(mpmestrce.rhnetrl) with assms show ‹ java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0 using ttake_prefixes[simplified prefixes_finite] by (metis less_le_not_le) case False from assms this[THEN leI] show ‹nd_is_empty)fi_epysfx apply (simp add: ttake_finite_prefixes) using ttake_prefixes[simplified prefixes_finite] by (metis) prefixes_directed: ‹'a trace› (cases ‹ ↓ ‹ { fix a b assume ‹x = y›y›) then have ‹ using finite_directed prefixes_finite by blast } show <pen? fix a b show ‹x› using X by auto (auto simp: prefixes_def dest: prefixes_infinite_greatest) extensions: order ‹ ↑\close ‹ t u. t ∈ u ∧ u› (auto simp: prefixes_exith s(1yteho\open>?tei› extensions_infinite[simp]: ‹ by (simp add: extensions_def) extensions_empty[simp]: ‹'a trace› by (simp add: extensions_def) prefixes_empty: ‹ apply (clarsimp simp add: set_eq_iff ε prefixes_d) apply (rule iffI) apply (metis ε_def append_is_empty(1)) by (metis ε_def trace.left_neutral) 🚫 prefix_closure :: ‹ ‹ prefix_closure_subset: ‹ε ∈ t› unfolding prefix_closure_def by auto prefix_closure_infinite: ‹ assume ‹Infinite x ∈ ↓s X› then show ‹Infinite x ∈ X› by (metis UN_E prefix_closure_def prefixes_infinite_greatest) assume ‹ by (meson in_mono prefix_closure_subset) prefix_closure_idem: ‹↓s ↓s X = ↓s X› unfolding prefix_closure_ (r iffI using prefixes.order.trans by blast \open x\ F ys\<>\ unfolding prefix_closure_def by blast java.lang.NullPointerException unfolding prefix_closure_def by simp prefix_closure_Union_distrib: ‹ java.lang.StringIndexOutOfBoundsException: Index 42 out of bounds for length 30 by simp prefix_closure_Inter: ‹ unfolding prefix_closure_def using prefixes.dual_order.trans by fastforce java.lang.NullPointerException by (rule prefix_closure_Inter[where S = ‹ prefix_closure_UNIV: ‹↓s UNIV = UNIV› prefix_closure_def by blast prefix_closure_empty: ‹ unfolding prefix_closure_def by blast prefix_closure_extensions: ‹ add.commute add.le) by (force intro: prefix_closure_subset dest: prefixes_directed simp: prefixes_extensions[THEN sym] prefix_closure_def) java.lang.NullPointerException unfolding prefix_closure_def by (force intro: prefixes.dual_order.trans) ‹> tdrop k t \<<in java.lang.NullPointerException ‹ = 🚫 dprefixes_are_prefixes : ‹↓d X ⊆ ↓s X› unfolding dprefixes_def using extensions.order.refl by blast java.lang.NullPointerException using dprefixes_are_prefixes prefix_closure_idem prefix_closure_mono by blast dprefixes_idem: ‹↓d ↓<openpen show ‹↓d ↓d X ⊆ ↓d X› using prefix_closure_dprefixes by (force simp: dprefixes_def) show ‹ .order.trans pprefix_c by (force simp: dprefixes_def) dprefixes_contains_extensions: ‹t ∈ ↓> (itdrop ((k- le x1) x2) \close unfolding dprefixes_def using extensions.dual_order.trans by auto dprefixes_infinite: ‹Infinite x ∈ ↓d X ⟷ Infinite x ∈ X› show ‹ unfolding dprefix using prefix_closure_subset by fastforce show ‹ unfolding dprefixes_def by (clarsimp simp: prefix_closure_infinite) dprefixes_UNIV: ‹ unfolding dprefixes_def using prefix_closure_UNIV by force dprefixes_empty: ‹↓d {} = {}› unfolding dprefixes_def using prefix_closure_empty by blast dprefixes_Inter_distrib: ‹<>va unfolding dprefixes_def prefix_closure_def by auto dprefixes_Inter: ‹↓d (∩ (dprefixes ` S)) = ∩ (dprefixes ` S)› show ‹ unfolding dpreqed auto } note A = this using prefixes.order.refl extensions.dual_order.trans by force show ‹↓d ∩ (dprefixes ` S) ⊆ ∩ (dprefixes ` S)› using dprefixes_idem dprefixes_Inter_distrib by blast dprefixes_mono: assumes ‹ shows ‹ using assms apply (simp add: dprefixes_def) apply (simp add: prefix_closure_def) apply (rule subsetI) using prefixes_extensions by blast dprefixes_inter: ‹↓d (↓d Xp(rule iffI) by (rule dprefixes_Inter[where S = ‹ dprefixes_inter_distrib: ‹↓d (X ∩ Y) ⊆ ↓d X ∩ using dprefixes_Inter_distrib[where S = ‹↓ ‹Definitive Sets› definitive:: ‹ ‹ definitive_image: ‹∀X ∈ S. definitive X ==> dprefixes ` S = S›: ttake_finite_pr unfolding definitive_def by auto definitive_dprefixes: ‹definitive (↓d X)› unfolding definitive_def by (rule dprefixes_idem) definitive_contains_extensions: ‹ finite_direct: unfolding definitive_def using dprefixes_contains_extensions by blast definiti: ‹ unfolding definitive_def by (rule dprefixes_UNIV) definitive_empty: ‹ unfolding definitive_def by (rule dprefixes_empty) definitive_Inter: ‹∀X ∈ S. definitive X ==> definitive (∩ S)› unfolding definitive_def using dprefixes_Inter definitive_image[simplified defin definitive_inter: ‹definitive X ==> definitive Y ==> definitive (X ∩ using definitive_Inter[where S = ‹ definitive_infinite_extension: assumes ‹ shows ‹ assms proof (cases ‹t› case (Finite xs) then show ‹ apply (intro exI[where x=‹ by (force simp: prefixes_extensions[THEN sym] prefixes_def intro!: definitive_contains_extensions[THEN subsetD, OF assms] intro: exI[where x=‹Infinite (λ_. undefined)› auto definitive_elemI: assumes ‹ shows ‹ using assms by (auto simp add: definitive_def dprefixes_def) dUnion :: ‹ ‹∪><>t dunion :: ‹'a trace set ==> 'a trace set ==> 'a trace set› (infixl ‹∪d› 65) wh ‹X ∪d Y ≡ ∪d {X,Y}› dprefixes_dUnion: ‹ by (simp add: dUnion_def dprefixes_idem) definitive_dUnion: ‹ =Finite a\Longrightarrow F b ==>› by (simp add: dprefixes_dUnion definitive_def) dUnion_contains_dprefixes: ‹t ∈ S ==> ↓d t ⊆ ∪d S› by (auto simp: dUnion_def dprefixes_def prefix_closure_def) dUnion_contains_definitive: ‹X ∈ S ==> definitive X ==> X ⊆ ∪d S› unfolding definitive_def using dUnion_contains_dprefixes by blast dUnion_empty[simp]: ‹∪d {} = {}› unfolding dUnion_def by (simp add: dpref) dUnion_least_dprefixes: ‹(∧X. X ∈ unfolding dprefixes_def prefix_closure_def subset_iff, mesonextensions.order_reflprefixesorder.tr) dUnion_least_definitive: assumes all_defn: ‹[THEN sym] dedest: prefi.leD intro:prefixe.order.tans) shows ‹(∧X. X ∈ S ==> X ⊆ Z) ==> definitive Z ==> ↓d ∪ using definitive_image[OF all_defn,THEN sym] dUnion_least_dprefixes definitive_ by metis ‹A type for definitive sets› 'a dset = ‹ using definitive_UNIV by blast type_definition_dset Inter_dset :: ‹'a dset set ==> 'a ds by (simp add extens) by (simp add: definitive_Inter) inter_dset :: ‹ ‹X ⊓ Y ≡ Union_cset :: ‹ apply (rule iffI) by (rule definitive_dUnion) union_dset :: ‹def tracleft_neut) ‹ empty_dset :: ‹ by (rule definitive_empty) univ_dset :: ‹down>\\^sub>s _› by (rule definitive_UNIV) subset_dset :: ‹'a dset ==> 'a dset ==> bool› (infix ‹⊑› 50) is ‹(⊆)› done strict_subset_cset :: ‹'a dset ==> 'a dset ==> bool› (infix ‹⊏› 50) is ‹(⊂)› done :: ‹ done notin_dset :: ‹'a trace ==> 'a dset ==> bool› is ‹(∉)› done in_dset_ε apply (transfer) using definitive_contains_extensions eby au in_dset_UNIV: ‹ by (transfer, simp) in_dset_subset: ‹A ⊑ B ==> in_dset x A ==> in_dset x B› by (transfer, auto) in_dset_inter: ‹in_dset x A ==> in_dset x B ==> in_dset x (A ⊓ B)› by (transfer, simp) dset: complete_lattice ‹ <>\ (unfold_locales;transfer) fix X Y Z :: ‹'a trace set› assume ‹definitive X› \ meson in_ prefi) then show ‹ singletonD) fix A :: ‹'a trace set set› and Z :: ‹ assume ‹ then show ‹∪d A ⊆ Z› by (simp add: dUnion_def dUnion_least_definitive) (auto simp: dUnion_contains_definitive) ‹Isomorphism of definitive sets and LTL properties› infinites :: ‹'a trace set ==> 'a infinite_trace set› where ‹infinites X = (∪x ∈ X. case x of Finite xs ==> {} | Infinite xs ==> {xs})› infinites_alt: ‹Infinite ` infinites A = A ∩ range Infinite› set_eq_iff proof fix x { assume ‹ by (clarsimp simp: infinites_def split!: trace.split_asm) } moreover { assume ‹ (x ∈ A ∩ range Infinite) › hence ‹ by (force simp: infinites_def split!: trac } ultimately show ‹ by blast infinites_append_right: ‹t ⌢ Infinite ψ ∈ range Infinite› by (cases ‹t›; auto) infinites_prefix_closure: assumes ‹ shows ‹ unfolding prefix_closure_def infinites_def using definitive_infinite_extension[OF assms] prefixes.order.trans by (force split: trace.split_asm) infinites_UNIV[simp]: ‹infinites UNIV = UNIV› by (auto simp: infinites_def split: trace.split) infinites_empty[simp]: ‹ by (auto simp: ininfinite) infinites_Inter: ‹infinites (∩ S) = ∩ (infinites ` S)› unfolding infinites_def apply (rule set_eqI; rule iffI) apply (force) apply (simp split: trace.split trace.split_asm) by (metis InterI trace.distinct(1) trace.exhaust trace.inject(2)) infinites_Union: ‹ unfolding infinites_def by auto infinites_dprefixes: ‹inter> ↓>🚫 unfolding infinites_def by (force simp: dprefixes_infinite split: trace.split trace.split_asm) infinites_dprefixes_Infinite: ‹infinites (↓ r prefi[where SS = = \open,}\<lose, show ‹infinites (↓d Infinite ` X) ⊆ X› unfolding infinites_def using prefixes_infinite_greatest by (force split: trace.split_asm simp: dprefixes_def prefix_closure_def) show ‹X ⊆ infinites (↓d Infinite ` X)› by (force simp: infinites_def dprefixes_def prefix_closure_def split: trace.spli property :: ‹'a dset ==> 'a infinite_traceby (f (for intro:: pr dest: prefixes_ done definitives :: ‹ by (rule definitive_dprefixes) property_inverse: ‹ ↓ by (transfer, simp add: infinites_dprefixes_Infinite) definitives_inverse: ‹definitives (property X) = X› (rule dset.order_antisym) show ‹ by (transfer, force simp: dprefixes_def infinites_prefix_closure intro: definitive_elemI) show ‹ apply transfer using definitive_contains_extensions definitive_infinite_extension by (force simp: dprefixes_def prefix_closure_def infinites_def) definitives_mono: ‹open>↓>↓ by (transfer, metis dprefixes_inter_distrib image_mono inf.order_iff le_infE) property_mono: ‹A ⊑ B ==> property A ⊆ property B› by (transfer, auto simp: infinites_def) definitives_reflecting: ‹ using property_inverse property_mono by metis java.lang.NullPointerException: Cannot invoke "String.equals(Object)" because "b using definitives_inverse definitives_mono by metis property_Inter: ‹property (⊓ S) = ∩ (property ` S)› by (transfer, simp add: infinites_Inter) property_Union: ‹ by (transfer, simp add: dUnion_def infinites_dprefixes infinites_Union) dset: complete_distrib_lattice ‹⊓› ‹⊔› ‹(⊓)› by (unfold_locales) (auto intro: completions_reflecting simp add: property_Inter property_Union INF_ iprepend :: ‹>d \ubseteq↓ ‹iprepend X = {t. itdrop 1 t ∈ X }› iprepend_itdrop: ‹itdrop k x ∈ iprepend B ⟷ itdrop (Suc k) x ∈ by (simp add: iprepend_def) iprepelby (force simp: dpref) prepend' :: ‹ ‹ trace_uncons_cases [case_names Cons Nil]: assumes ‹ and ‹ shows ‹P› (cases ‹> case (Finite xs) then show ‹?thesis› by (cases ‹xs›; force simp: assms(2)[simplified ε_def] intro: assms(1)[where t = ‹Finite ts›proof simplified singleton_def append.simps List.append.simps]) case (Infinite f) note A = this \\>f = (λ! n lse (f ∘))\close by (rule ext, simp) with A show ‹?thesis› using assms(1)[where σ = ‹f 0› and t = ‹Infinite (f ∘ Suc)› simplified si sing prefix_closu by fastforce by simp append_prefixes_left: ‹ by (simp add: prefixes_def) (metis trace.assoc) tdrop_singleton_append[simp]: ‹tdrop (Suc n) (singleton σ ⌢ t) = tdrop n t› by (cases ‹t› tdrop_zero[simp]: ‹ by (cases ‹ tdrop_ε[simp]: ‹ by (simp add: ε_def) prepend'_prefix_closure: ‹↓s (prepend' X) ⊆ (rule subsetI) fix x assume A: ‹ show ‹ proof (cases ‹x› rule: trace_uncons_cases) case (Cons σ t) with A show ‹ unfolding prefix_closure_def prepend'_def prefixes_def by (fastforce simp: trace.assoc) next case Nil lemma dpr: \<>< unfolding prefix_closure_def prepend'_def by (force simp: prefixes_empty_least) qed prepend'_dprefixes : ‹definitive X› ‹ show ‹↓\lemmadpr: ‹ proof (rule subsetI) fix x assume A: ‹x ∈ ↓d prepend' X› show ‹x ∈ prepend' X› proof (cases ‹x› rule: trace_uncons_cases) case (Cons σ t) with A show ‹ unfolding dprefixes_def apply (subst assms[simplified definitive_def, THEN sym]) apply (clarsimp dest!: subset_trans[OF _ prepend'_prefix_closure]) using append_prefixes_left (force simp:: dprefixes_def prepe'_def prefisubset_iff prefixes_extensions[THEN sym]) next case Nil with A show ‹?thesis› apply (subst assms[simplified definitive_def, THEN sym]) apply (clarsimp simp: prefixes_empty_least prefixes_def dprefixes_def subset_ prefixes_extensions[THEN sym]) by (metis tdrop_singleton_append tdrop_zero trace.assoc) qed qed show ‹: proof (rule subsetI) fix x assume A: ‹ proof (cases ‹x›\downsubd Y› case (Cons σ t) with A show ‹?thesis› by (clarsimp simp: dprefixes_def prefixes_def prepend'_def prefix_closure_def prefixes_extensions[THEN sym]) (metis (mono_tags, lifting) assms definitive_contains_extensions mem_Collect_eq prefixes_def prefixes_extensions subset_eq tdrop_singleton_append tdrop_zero trace.assoc) next case Nil with A show ‹?thesis› using assms definitive_contains_extensions by (force simp: d sing refixes_exteby blast qed qed prepend'_definitive : assumes ‹definitive X› shows ‹ unfolding definitive_def using assms by (rule prepe'_dpr) prepend :: ‹ by (rule prepend'_definitive) prepend_Inter: ‹⊓ apply transfer by (auto simp add: prepend'_def) in_dset_prependD: ‹ by (transfer, metis One_nat_def Traces.singleton_def mem_Collect_eq prepend'_def tdrop_singleton_append tdrop_zero) in_dset_prependI: ‹ by ((tra met One Traces.singleton_ mem_Colect pr'_def tdrop_singleton_append tdrop_zero) prepend'_mono: assumes ‹A ⊆ shows ‹ using assms unfolding prepend'_def by blast property_prepend: ‹close apply transfer by (clarsimp simp: definitive_def infinites_def prepend'_def split!: trace.split_asm trace.split intro!: set_eqI; blast) iprepend_Union: ‹ by fastforce definitives_inverse_eqI: ‹definitives (property X) = definitives (property Y) = by (simp add: definitives_inverse) java.lang.NullPointerException: Cannot invoke "String.equals(Object)" because "b apply (rule definitives_inverse_eqI) (imp add prop property_ by (metis UN_extend_simps(10) iprepend_Union) non_empty_trace: ‹ apply (cases ‹ apply (metis Traces.singleton_def ε_def append_is_empty(1) not_Cons_self2 trace.i by (metis ε_def append_is_empty(1) list.discI trace.inject(1)) thead_append: ‹ by (cases ‹x› thead_ \<>x apply (simp add: prefixes_def non_empty_trace) using thead_append [where x = ‹Finite [_]›, simplified ε_def, simplified] by (metis append_is_empty(1) thead_append) compr'_inter_thead: ‹↓d {x. x ≠ ε ∧ P (thead x)} ∩ ↓usingdp definsimplidefi java.lang.NullPointerException (rule antisym) fix x t assume ‹ and ‹∀t. x ∈ ↓ t ⟶ (∃x. x ≠ ε ∧ Q (thead x) ∧ ‹ and ‹ then have ‹ by (cases ‹ then show ‹↓d {x. x ≠ ε ∧ P (thead x)} ∩ ↓d {x. x ≠ ε ∧ Q (thead x)} ⊆ ↓ shows ‹ by (clarsimp simp: set_eq_iff subset_iff dprefixes_def prefix_closure_def prefix fix x assume ‹ then have ‹(∀t. x \< case x\in t \<ongrightarrow by fastforce } then show ‹↓d {x. x ≠ ε ∧ P (thead x)} ∩ ↓d {x. x ≠ ε ∧ Q (thead x)} 🪙 ↓d {x. x ≠ ε by (clarsimp simp: set_eq_iff subset_iff dprefixes_def prefix_closure_def prefix compr :: ‹('a trace ==> bool) ==> 'a dset› by (rule definitive_dprefixes) complement :: ‹'a dset ==> 'a dset› exI[where x=\open λ by (rule definitive_dprefixes) property_complement[simp]: ‹property (cedauto by (transfer, force simp: infinites_dprefixes[simplified infinites_def] infinite split: trace.split_asm trace.split)
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2026-06-09