| Quellcodebibliothek Statistik Leitseite products/Sources/formale Sprachen/Isabelle/Archive-of-Formal-Proofs/thys/BNF_Operations/ (Sammlung formaler Beweise Version 2026-5©) Datei vom 31.4.2026 mit Größe 98 kB |
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Quellcode-Bibliothek LFP.thySprache: Isabelle |
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(* Title: LFP Authors: Jasmin Blanchette, Andrei Popescu, Dmitriy Traytel Maintainer: Dmitriy Traytel <traytel at inf.ethz.ch> *) section ‹Least Fixpoint (a.k.a. Datatype)› (*<*) theory LFP imports "HOL-Library.BNF_Axiomatization" begin (*>*) unbundle cardinal_syntax ML ‹open Ctr_Sugar_Util› notation BNF_Def.convol (‹<_ , _>›) text ‹ begin{tabular}{rcl} 'b1 &=& ('a, 'b1, 'b2) F1\\ 'b2 &=& ('a, 'b1, 'b2) F2 end{tabular} build a witness scenario, let us assume begin{tabular}{rcl} ('a, 'b1, 'b2) F1 &=& 'a * 'b1 + 'a * 'b2\\ ('a, 'b1, 'b2) F2 &=& unit + 'b1 * 'b2 end{tabular} › declare [[bnf_internals]] bnf_axiomatization (F1set1: 'a, F1set2: 'b1, F1set3: 'b2) F1 [wits: "'a ==> 'b1 ==> ('a, 'b1, 'b2) F1" "'a ==> 'b2 ==> ('a, 'b1, 'b2) F1"] for map: F1map rel: F1rel bnf_axiomatization (F2set1: 'a, F2set2: 'b1, F2set3: 'b2) F2 [wits: "('a, 'b1, 'b2) F2"] for map: F2map rel: F2rel abbreviation F1in :: "'a1 set ==> 'a2 set ==> 'a3 set ==> (('a1, 'a2, 'a3) F1) s "F1in A1 A2 A3 ≡ {x. F1set1 x ⊆ A1 ∧ F1set2 x ⊆ A2 ∧ F1set3 x ⊆ A3}" abbreviation F2in :: "'a1 set ==> 'a2 set ==> 'a3 set ==> (('a1, 'a2, 'a3) F2) s "F2in A1 A2 A3 ≡ {x. F2set1 x ⊆ A1 ∧ F2set2 x ⊆ A2 ∧ F2set3 x ⊆ A3}" lemma F1map_comp_id: "F1map g1 g2 g3 (F1map id f2 f3 x) = F1map g1 (g2 o f2) (g3 apply (rule trans) apply (rule F1.map_comp) unfolding o_id apply (rule refl) done lemmas F1in_mono23 = F1.in_mono[OF subset_refl] lemma F1map_congL: "[∀a ∈ F1set2 x. f a = a; ∀a ∈ F1set3 x. g a = a] ==> F1map id f g x = x" apply (rule trans) apply (rule F1.map_cong0) apply (rule refl) apply (rule trans) apply (erule bspec) apply assumption apply (rule sym) apply (rule id_apply) apply (rule trans) apply (erule bspec) apply assumption apply (rule sym) apply (rule id_apply) apply (rule F1.map_id) done lemma F2map_comp_id: "F2map g1 g2 g3 (F2map id f2 f3 x) = F2map g1 (g2 o f2) (g3 apply (rule trans) apply (rule F2.map_comp) unfolding o_id apply (rule refl) done lemmas F2in_mono23 = F2.in_mono[OF subset_refl] lemma F2map_congL: "[∀a ∈ F2set2 x. f a = a; ∀a ∈ F2set3 x. g a = a] ==> F2map id f g x = x" apply (rule trans) apply (rule F2.map_cong0) apply (rule refl) apply (rule trans) apply (erule bspec) apply assumption apply (rule sym) apply (rule id_apply) apply (rule trans) apply (erule bspec) apply assumption apply (rule sym) apply (rule id_apply) apply (rule F2.map_id) done subsection‹Algebra› definition alg where "alg B1 B2 s1 s2 = ((∀x ∈ F1in (UNIV :: 'a set) B1 B2. s1 x ∈ B1) ∧ (∀y ∈ F2in (UNIV :: 'a set) B1 lemma alg_F1set: "[alg B1 B2 s1 s2; F1set2 x ⊆ B1; F1set3 x ⊆ B2] ==> s1 x ∈ B1" apply (tactic ‹dtac @{context} @{thm iffD1[OF alg_def]} 1›) apply (erule conjE)+ apply (erule bspec) apply (rule CollectI) apply (rule conjI[OF subset_UNIV]) apply (er:) F1 applya done lemma alg_F2set: "<alg B1 B2 s1 s2; F2set2 x ⊆ B1; F2set3 x ⊆ B2] ==> s2 x ∈ B2" apply (tactic ‹F1inA1A2 A3 \equiv{x. FF1set1 x ⊆ F1set2 x ⊆ F1set3 x ⊆ apply (e conjE)+ apply (erule bspec) apply (rule CollectI) apply (r conjI[Osubset_UNIV]) apply (erule conjI) apply as done lemma a: (rule refl) apply (rule notI) apply (tactic \open>hyp_subst_tac @{co} 1› tras) apply (frule alg rletrn) (* ORELSE of the following three possibilities *)apply rl dly apply(ue use_emyII) apply (erule F1apply(ueF1mp__d apply (rule subsetI) apply (drule F1.wit1 F1.wit2 F2.wit) (**) apply (tatic‹@{context} 1›) apply (tactic ‹ apply (rule subset_emptyI) apply (erule F1.wit1 F1.wit2 F2.wit) apply (rule subsetI) apply (drule F1.wit1 F1.wit2 F2.wit) apply (erule FalseE) (**) apply (erule emptyE) apply (rule notI) apply (tactic ‹hyp_subst_tac @{context} 1›) apply (drule alg_F2set) apply (rule subsetI) apply (rule FalseE) apply (erule F1.wit1 F1.wit2 F2.wit) apply (rule subset_emptyI) apply (erule F1.wit1 F1.wit2 F2.wit) apply (erule emptyE) done subsection ‹∀ F2set2 x. f a = a; ∀ F2set3 x. g a = a] definition " e (((∀ uleans ((∀ (∀z ∈ F2in (UNIV :: 'a set) B1 B2. g (s2 z) = s2' (F2map id fg ))) morE1:((\forallx \<in : 'a set) B1B2. s2 \<in ==> f (s1 z) = s1' (F1map id f g z)" apply (tactic ‹dtac @{context} @{thm iffD1[OF mor_def]} 1›) apply (erule conjE)+ apply (erule bspec) apply assumption done morE2: "[mor B1 B2 s1 s2 B1' B2' s1' s2' f g; z ∈ F2in UNIV B1 B2] ==> g (s2 z) = s2' (F2map id f g z)" apply (tactic ‹dtac @{context} @{thm iffD1[OF mor_def]} 1›) apply (erule conjE)+ apply (erule bspec) apply assumption mor_incl: "[B1 ⊆ B1'; B2 ⊆ B2'] ==> mor B1 B2 s1 s2 B1' B2' s1 s2 id id" apply (tactic ‹ apply (erule con) apply (rule conjI) apply (rule ballI) apply (erule subsetD) pply (erul ssubst_mem[OF]) apply (apply (tac \open @{c} @{ ififf[O alg_d]}1clo>) apply (erule subsetD) apply (erule ssubst_mem[OF id_apply]) apply (rule conjI) (ererule co)+ apply (rule trans) apply (rule id_apply) apply (tactic ‹stac @{context} @{thm F1.map_id} 1›) apply (erule b) apply (rule ballI) apply (rule trans) apply (rule id_apply) apply (tactic ‹ apply (rule refl) done morcomp: "[mor B1 B2 s1 s2 B1' B2' s1' s2' f g; mor B1' B2' s1' s2' B1'' B2'' s1'' s2'' f' g'] ==> mor B1 B2 s1 s2 B1'' B2'' s1'' s2'' (f' o f) (g' o g)" apply (tactic ‹ apply (tactic ‹ apply (tactic ‹ apply (erule conjE)+ apply (ul conjI) apply (rconjI) apply (r ballI) apply (rule ssubst_mem[OF o_app apply (erule bspec) apply (erule bspec) apply assumption apply (rule ballI) apply (rule ssubst_mem[OF o_apply]) apply (erule bspec) apply erule bspec) apply assumption apply (rule conjI) drule F1.wit1 F1.wi F2.wit) apply rule ballI) apply (rule trans[OF o_apply]) apply (rule trans) apply (rule trans) apply (drule bspec[rotated]) apply assumption apply (erule arg_cong) apply (erule CollectEc)+ java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0 apply (rule CollectI) apply (rule conjI) apply (rule subset_UNIV) apply (rule conmowere apply (e or_q_leran) (rule F1.set_map(2)) apply (rule image_subsetI) apply (erule bspec) apply (erule subsetD) apply assumption apply (rule ord_eq_le_trans) apply (rule F1.set_map(3)) apply (rule image_subsetI) apply (erule bspec) apply (erule subsetD) apply assumption apply (rule arg_cong[OF F1map_comp_id]) apply (rule ballI) apply (rul(rule trans[OF o_apply]) rule trans apply apply ( bspec[rotated]) apply assumption apply (erule arg_cong) apply (erule CollectE conjE)+ apply (erule bspec) (le apply (rule conjI) apply (rrul susetI) Ft) apply(ue ordd_eqq_le_tra apply (rule.set_map() (erule emptyE) apply (eruerule subsetD) apply assumption apply (rule ord_eq_le_trans) apply (rule F2.set_map(3)) apply (rule image_subsetI) apply (erule bspec) apply (erule subsetD) apply assumption apply (rule arg_cong[OF F2map_comp_id]) done mor_cong: "[ f' = f; g' = g; mor B1 B2 s1 s2 B1' B2' s1' s2' f g] a ∈ B1') ∧a ∈ B2')) ∧ apply (tactic ‹z ∈ z)= s2' (F2map id f g z)))" java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0 done moapp (ule cE)+ "m"mpti apply (rule iffD2) apply(rule mor_def) apply (rule conjI) apply (rule conjI) apply (rule ballI) apply (rule UNIV_I) apply (rule ballI) apply (rule UNIV_I) apply aston apply (rule ballI) apply (rule refl) apply (rule ballI) apply (rule refl) done ‹ d_type_F1 ="bd_t_F1+ (b, bd_t_F1d_type1 1 ype_synonym bd_type_F2 bdype_F2(_ty_F2, bd_type_, b_tpe_2 " apply (rule ssubst_mem[m[OF id_aply]) "appy(ue id_ply)java.lang.StringIndexOutOfBoundsException: Index 25 out of boun apply (rletran) Leq_trans[OF F1.set_bd(1) ordLeq_csum1[OF F1.bd_Card_order]]) F1set2_bd_incr: "∧F1bd'" rule rdLessordLeqra[OF F1.set_bd(2)ordLLqcsm1[OF F.bd_d_Crdr] F1set3_bd_incr: "∧[moB1 B' ' s'B1' ' s' ff' g'\rbrakk> ==> by (ruatacti<dtac rtac @{context} (@{thm mor_def} RS iffD2) 1 ssst_me_apply]) F1ae bec F1bd'_Cnotzero m rlecnjj trans[O _aply]) 2 \>bd_F2 +c |UNIV :: (bd_type_F2, bd_type_F2, bd_type_F2) F2 set|" F2set1_bd_incr: "∧ (ueordLs_ordLeq_rans[OFF2..set_bd(1) oordLeq_csumm11OF F2.bd_ard_order) F2set2"\<AndxF2bd'" by (rule ordLess_ordLeq_trans[OF F2.set_bd(2) ordLeq_csum1[OF F2.bd_Card_order]] F2set3_bd_incr: "∧ e ordLessordLeq_asOFF.set(3)ordLeq_csum1[OF F2.Card_odr]) 2bdCrd_orderCarder_csum F2b (ule _eqle_trans) (rule iagesubse apply ( subs SucFbd where "SucFbd ≡ cardSuc (F1bd' +c F2bd')" ASucFbd where "ASucFbd ≡ ( |UNIV| +c ctwo ) ^c SucFbd" F1set1_bd: "|F1set1 x| <o bd_F1 +c bd_F2" apply (rule ordLess_ordLeq_trans) apply (rule F1.set_bd(1)) apply (rule ordLeq_csum1) apply (rule F1.bd_Card_order) done F1set2_bd: "|F1set2 x| <o bd_F1 +c bd_F2" apply (rule ordLess_ordLeq_trans) apply (rule F1.set_bd(2)) apply (rule ordLeq_csum1) apply (rule F1.bd_Card_order) done F1set3_bd: "|F1set3 x| <o bd_F1 +c bd_F2" apply (rule ordLess_ordLeq_trans) apply apply (rule ordLeq_c uthors: : Jasmi Blanche, And Popescu,DmitrT apply (rule F1.bd_Card_order) done F2set1_bd: "|F2set1 x| <o bd_Mainta: Traytel < atLeast Fixpoint (a.k.a. Datatype) (rul ordLess_ordLeq_) apply imports"HOL-Library.BNF_Axiom rule ordLeq_csum2) apply (r F2.bd) done F2set2_bd: "|F2set2 x| <o bd_F1 +c bd_F2" apply (rule ordLess_ordLeq_trans) apply (rule F2.set_bd(2)) apply (rule ordLeq_csum2) apply (rule F2. BNF_Def.cnvol (‹ ‹ done F2set3_bd: "|F2set3 x| <o bd_F1 +c bd_F2" apply (rule ordLess_ordLeq_trans) apply (rule F2.set_bd(3)) apply (rule ordLeq_csum2) apply (rule F2.bd_Card_order) done SucFbd_Card_order = cardSuc_Card_order[OF Card_order_csum] SucFbd_Cinfinite = Cinfinite_cardSuc[OF Cinfinite_csum1[OF F1bd'_Cinfinite]] SucFbd_Cnotzero = Cinfinite_Cnotzero[OF SucFbd_Cinfinite] worel_SucFbd = Card_order_wo_rel[OF SucFbd_Card_order] ASucFbd_Cinfinite = Cinfinite_cexp[OF ordLeq_csum2[OF Card_order_ctwo] SucFbd_C ‹Minimal Algebras› (* These are algebras generated by the empty set. *) abbreviation min_G1 where "min_G1 As1_As2 i ≡ (∪j ∈ underS SucFbd i. fst (As1_As2 j))" abbreviation min_G2 where "min_G2 As1_As2 i ≡ (∪j ∈ underS SucFbd i. snd (As1_As2 j))" abbreviation min_H where "min_H s1 s2 As1_As2 i ≡ (min_G1 As1_As2 i ∪ s1 ` (F1in (UNIV :: 'a set) (min_G1 As1_As2 i) (min_G2 As1_A min_G2 As1_As2 i ∪ s2 ` (F2in (UNIV :: 'a set) (min_G1 As1_As2 i) (min_G2 As1_As abbreviation min_algs where "min_algs s1 s2 ≡ wo_rel.worec SucFbd (min_H s1 s2)" definition min_alg1 where "min_alg1 s1 s2 = (∪i ∈ Field SucFbd. fst (min_algs s1 s2 i))" definition min_alg2 where "min_alg2 s1 s2 = (∪i ∈ Field SucFbd. snd (min_algs s1 s2 i))" lemma min_algs: "i ∈ apply (rule fun_cong[OF wo apply (rule iffD2) apply (rule meta_eq_to_obj_eq) apply (rule wo_rel.adm_wo_def[OF worel_SucFbd]) apply (rule allI pply(u) apply (rule iffD2) apply (rule prod.inject) apply (rule conjI)(rule sbet_UNI) apply (rule arg_cong2[of _ _ _ _ "(∪ apply (rule ydrule applyumption apply (erule arg_cong) apply (rule image_congsumption apply (rule arg_cong2 apply (rule SUP_cong) applymor B21 apply (drule bspec) apply assumption apply (erule arg_cong) apply (rule SUP_cong apply (rule refl) apply (drule bspec) apply assumption apply(rule apply (ruleD2 (e onjI apply (rule SUP_conglI apply (rulee ) (drulebspec apply assumption ply apply (rule image_cong apply (rule apply (rule SUP_cong) apply (rule refl) apply (drule bspec apply assumptionbd_type_F1F11d_type_F1d_type_F1 applyulerg_cong rule apply (ule apply (drule<equiv bd_F1 +c |UNIV :: (bd_type_F1, bd_type_F1 bd_type_F1et apply assumption apply (erule arg_cong) apply (rule refl) done corollaryary<>ield< fst (min_algs s1 s2 i) = min_G1 (min_algs s1 s2) i ∪ s1 ` (F1in UNIV (min_G1 (min_algs s1 s2) i) (min_G2min_algsijava.lang.StringIndexOutOfBou ( apply apply (rule fst_conv) done corollary min_algs2: "i ∈ Field SucFbd ==> snd (min_algs s1 s2 i) = min_G2_lss1 s2) i ∪ s2 ` (F2in UNIV (min_G1 (min_algs s1 s2) i) (min_G2 (min_al2set2_bd_ncr: "<>.set2F pplyeans apply F2set3_bd_incrx. |F2set3 x| <o F2bd' apply (rule 2nite lemma min_algs_mono1: "relChainoOF F2.card_order card_of_card_o_on] apply abbreviation SucFbd whr "ucFbd> cardSuc (F1bd' +c F2bd')" apply (ule allI)+ apply (rule impI) apply (rule case_split) apply (rule xt1(3)) apply (rule min_algs1) apply (erule FieldI2) apply apply (rule UnI1) apply (rule UN_I) apply (erule underS_I) apply assumption apply assumption apply (rule equalityD1) apply (drule notnotD) apply (erule arg_cong) done lemma min_algs_mono2: "relChain SucFbd (%i. snd (min_algs s1 s2 i))" apply (tactic ‹rtac @{context} @{thm iffD2[OF meta_eq_to_obj_eq[OF relChain_def] apply (ruleall)+ apply (rule impI) apply (rule case_split) apply (rule xt1(3)) apply (rule min_algs2) apply (erule FieldI2) apply (rule subsetI) apply (rule UnI1) apply (rule UN_I) apply (erule underS_I) apply assumption apply assumption apply (rule equalityD1) apply (drule notnotD) apply (erule arg_cong) done lemma SucFbd_limit: "[ ==> ∃y ∈ apply (erule conjE)+ apply (rule rev_mp) apply (rule Cinfinite_limit_finite) apply (rule finite.insertI) apply (rule finite.insertI) apply (rule finite.emptyI) apply (erule insert_subsetI) apply (erule) apply (rule empty_subsetI) apply (rule SucFbd_Cinfinite) apply (rule impI) apply (erule bexE) apply (rule bexI) apply (rule conjI) apply (erule bspec) applytacticopen@context @thm} RSiffD2› apply (erule bspec) apply (rule insertI2) apply (rule insertI1) apply assumption done lemma alg_min_alg: "alg (min_alg1 s1 s2) (min_alg2 s1 s2) s1 s2" apply (tactic ‹ apply (rule conjI) apply (rule ballI) apply (ruleCollectEconjE+ apply (rule bexE) apply (rule cardSuc_UNION_Cinfinite) apply (rule Cinfinite_csum1) (*TRY*) apply (rule F1bd'_Cinfinite) apply (rule min_algs_mono1) applyapply ( [rotated apply (rule ordLeq_transitive) apply (rule ordLess_imp_ordLeq( CollectE apply (rule ordLeq_csum1) (*or refl *) apply (rule F1bd'_Card_order) apply (rule bexE) apply (rule cardSuc_UNION_Cinfinite) apply (rule Cinfinite_csum1 apply (rule F1bd'_Cinfinite) apply ( min_algs_mono2 apply (erule subset_trans[OF _ equalityD1[OF min_alg2_def]]) apply (rule ordLeq_transitive) apply(ule[OFF1set3_bd_incr apply (rule ordLeq_csum1) (*or refl *) apply (rule F1bd'_Card_order) apply (rule bexE) apply (rule SucFbd_limit) apply (erule conjI) apply assumption apply (rule subsetD[OF equalityD2[OF apply (rule UN_I) apply (erule thin_rl) apply (erule thin_rl) apply (erule thin_rl) n_rl apply (erule thin_rl) apply (erule thin_rl) apply (erule thin_rl) (* m + 3 * n *) apply assumption apply (rule subsetD) apply (rule equalityD2) apply (rule min_algs1apply rule) apply assumption apply (rule UnI2) apply (rule image_eqI apply (rule refl) apply (rule CollectI) apply (drule asm_rl) apply (erule thin_rl) apply (erule thin_rl) apply (erule conjE)+ apply (rule conjI) apply assumption apply (apply (mor_incl_min_alg apply (erule subset_trans) apply (rule subsetI) apply (rule UN_I apply (erule underS_I) apply assumption apply assumption apply (erule subset_trans) apply (erule UN_upper[OF underS_Iapply (tactic ‹) 1› apply assumption (**) apply (rule ballI) apply erule conjE)+ apply (rule bexE) apply (erule CollectE)+ apply (rule Cinfinite_csum1) (*TRY*) apply (rule ( CollectI) apply (rule min_algs_mono1) apply (erule subset_trans (rule ballI apply (rule ordLeq_transitive) apply ordLess_imp_ordLeqOF F2set2_bd_incr apply (rule ordLeq_csum2) apply (rule F2bd'_Card_order) apply (rule bexE) apply (rule cardSuc_UNION_Cinfinite) apply (rule) (*TRY*) apply (rule F1bd'_Cinfinite) apply (rule min_algs_mono2) apply (erule subset_trans[OF _ equalityD1[OF min_alg2_def]]) apply (rule ordLeq_transitive) apply (rule ordLess_imp_ordLeq[OF F2set3_bd_incr]) apply (rule ordLeq_csum2 apply (rule F2bd'_Card_order) apply(rule) apply (rule SucFbd_limit) apply (erule conjI) apply assumption apply (rule subsetD[OF equalityD2[OF]]) apply (rule UN_I) apply (erule thin_rl) apply (erule thin_rl) apply (erule thin_rl) apply (erule thin_rl) apply (erule thin_rl) apply (erule) apply (erule thin_rl) (* m + 3 * n *) apply assumption apply (ule) apply (rule equalityD2) apply (rule min_algs2) apply assumption apply (rule UnI2) apply (rule image_eqI) apply (rule refl) apply (rule CollectI) apply (rule conjI) apply assumption apply (erule thin_rl) apply (erule thin_rl) apply (erule) apply (erule conjE)+ apply (rule conjI) apply (erule subset_trans) apply ( UN_upper) apply (erule underS_I) apply assumption apply (erule subset_trans) apply (rule UN_upper) apply (erule underS_I) apply assumption done lemmasapplyerule) ordLess_ctwo_cexp cexp_mono1 OF SucFbd_Card_order SucFbd_Card_order] lemma card_of_min_algs: fixes shows "i ∈ Field SucFbd ⟶ ballI) apply erule CollectE conjE)+ apply (rule well_order_induct_imp[of _ "%i. ( |fst (min_algs s1applyruleCollectI) ( impI apply (rule conjI) apply (rule ordIso_ordLeq_trans) apply ( card_of_ordIso_subst) apply (erule min_algs1) apply (ruleapply ( ssubst_mem[OF]) apply (rule UNION_Cinfinite_bound) apply (rule ordLess_imp_ordLeq) apply (rule ordLess_transitive) apply (rule card_of_underS) apply (rule SucFbd_Card_order) apply assumption apply (rule SucFbd_ASucFbd) apply (rule(ule.set_map(2)) apply (erule allE) apply (drule mp) apply (( underS_E) apply (drule mp) apply (erule underS_Field) apply (erule conjE)+ apply assumption apply (rule ASucFbd_Cinfinite ( image_Collect_subsetI) apply (rule ordLeq_transitive) apply (rule) apply (rule ordLeq_transitive) apply (rule F1.in_bd) apply (rule ordLeq_transitive) apply (rule cexp_mono1) apply (rule csum_mono1) apply (rule csum_mono2) (* REPEAT m *) apply (rule csum_cinfinite_bound) apply (rule UNION_Cinfinite_bound) apply (rule ordLess_imp_ordLeq) apply (rule ordLess_transitive apply (rule card_of_underS) apply (rule SucFbd_Card_order (rule subset_trans apply assumption apply (rule SucFbd_ASucFbd) apply (rule ballI) apply (erule allE) apply (drule mp apply (erule underS_E) apply (drule mp) apply (erule underS_Field) apply (erule conjE)+ apply assumption apply (rule ASucFbd_Cinfinite) apply (rule apply (rule ordLess_imp_ordLeq) apply (rule ordLess_transitive) apply "<>a1 ∈ car_init2 dummy; apply (rule SucFbd_Card_order) apply assumption apply (rule ucFb_ASFbd) apply (rule ballI) apply (erule allE) apply druleule m) apply (erule underS_E) apply (drule mp) apply(eule uderSField) apply (erule conjE)+ apply assumption apply (rule ASucFbd_Cinfinite) apply (rule card_of_Card_order) apply (rule card_of_Card_order) apply (rule ASucFbd_Cinfinite) apply (rule F1bd'_Card_order) apply (rule ordIso_ordLeq_trans) apply (rule cexp_cong1) apply (rule ordIso_transitive) apply (rule csum_cong1) apply (ruleordIso_transitive) apply (tactic ‹ (rtac @{context} @{thm ordIso_refl} THEN' FIRST' [rtac @{contex} @{thm card_of_Carorder}, rtac @{context} @{thm Card_order_csum}, rtac @{contxt} @@{tm Card_order_cexpp}]) @{thm ordIso_transitive} @{thm csum_assoc} @{thm csum_com} @{thm csum_cong} [1,2] [2,1] 1›) apply (rule csum_absorb1) apply (rule ASucFbd_Cinfinite) apply (rpy (rulrule ordLeq_trsiti) apply (rule ordLeq_csum1) apply (tactic ‹ apply (rule ordLeq_cexp1) apply (rule SucFbd_Cnotzero) apply (rule Card_order_csum) apply (rule csum_absorb1) apply (rule ASucFbd_Cinfinite) apply (rule ctwo_ordLeq_Cinfinite) apply (rule ASucFbd_Cinfinite) apply (rule F1bd'_Card_order) apply (rule ordIso_imp_ordLeq) apply (rule cexp_cprod_ordLeq) apply (rule Card_order_csum) apply (rule SucFbd_Cinfinite) apply (rule F1bd'_Cnotzero) apply (rule ordLeq_transitive) apply (rule ordLeq_csum1) apply (rule F1bd'_Card_order) apply (rule cardSuc_ordLeq) apply (rule Card_order_csum) apply (rule ASucFbd_Cinfinite) apply (rule ordIso_ordLeq_trans) apply (rule card_of_ordIso_subst) apply (erule min_gs2) apply (rule Un_Cinfinite_bound) apply (rule UNION_Cinfinite_bound) apply (rule ordLess_imp_ordLeq) apply (rule ordLess_transitive) apply (rule card_of_underS) apply (rule SucFbd_Card_order) apply assumption apply (rule SucFbd_ASucFbd) apply (rule ballI) apply (erule allE) apply (drule mp) apply (erule underS_E) apply (drule mp) pply (erle undererS_Field) apply (erule conjE)+ apply assumption apply (rule ASucFbd_Cinfinite) apply (rule ordLeq_transitive) apply (rule card_of_image) apply (rule ordLeq_transitive) apply(rle F2nbd) apply (rule ordLeq_transitive) apply (rule cexp_mono1) apply (rule csum_mono1) apply (rule csum_mono2) apply (rule csum_cinfinite_bound) apply (rule UNION_Cinfinite_bound) apply (rule ordLess_imp_ordLeq) apply rul ordLess_ransitve) apply (rule card_of_underS) apply (rule SucFbd_Card_order) apply assumption apply (rule SucFbd_ASucFbd) apply (rule ballI) apply (erule allE) apply (drule mp) apply (erule underS_E) apply (drule mp) pply (ereunrS_Field) apply (erule conjE)+ apply assumption apply (rule ASucFbd_Cinfinite) apply (rule UNION_Cinfinite_bound) apply (rule ordLess_imp_ordLeq) apply (rule ordLess_transitive) apply (rule card_of_underS) apply (rule SucFbd_Card_order) apply assumption apply (rule SucFbd_ASucFbd) apply (rule ballI) apply (erule aapply ( (eru CollectEconjE)+ apply (drule mp) apply (erule underS_E) apply (drule mp) apply (erule underS_Field) apply (erule conjE)+ apply assumption apply (rule ASucFbd_Cinfinite) apply (rule card_of_Card_order) apply (rule card_of_Card_order) apply (rule ASucFbd_Cinfinite) apply (rule F2bd'_Card_order) apply (rule ordIso_ordLeq_trans) apply (rule cexp_cong1) apply (rule ordIso_transitive) apply (rule csum_con apply (rule ordIso_transitive) applyrulmooE2 (rtac @{context} @{thm ordIso_refl} THEN' FIRST' [rtac @{context} @{thm card_of_Card_order}, rtac @{context} @{thm Card_order_csum}, rtac @{context} @{thm Card_order_cexp}]) @{thm ordIso_transitive} @{thm csum_assoc} @{thm csum_com} @{thm csum_cong} [1,2] [2,1] 1›) apply (rule csum_absorb1) apply (ruy rule ASucFbinfiite) apply (rule ordLeq_transitive) apply (rule ordLeq_csum1) apply (tactic ‹ apply (rule ordLeq_cexp1) apply (rule SucFbd_Cnotzero) apply (rule Card_order_csum) apply (rule csum_absorb1) apply (rule ASucFbd_Cinfinite) apply (rule ctwo_ordLeq_Cinfinite) apply (rule ASucFbd_Cinfinite) apply (rule F2bd'_Card_order) apply (rule ordIso_imp_ordLeq) apply (rule cexp_cprod_ordLeq) apply (rule Card_order_csum) apply (rule SucFbd_Cinfinite) apply (rule F2bd'_Cnotzero) apply (rule ordLeq_transitive) apply (rule ordLeq_csum2) apply (rule F2bd'_Card_order) apply (rule cardSuc_ordLeq) apply (rule Card_order_csum) apply (rule ASucd_Cfinite done lemma card_of_min_alg1: fixes s1 :: " shows "|min_alg1 s1 s2| ≤2 apply (rule ordIso_ordLeq_trans) apply (rule card_of_ordIso_subst[OF min_alg1_def]) apply (rule UNION_Cinfinite_bound) apply (rule ordIso_ordLeq_tapply (rle F2in_mno23)3 apply (rule card_of_Field_ordIso) apply (rule SucFbd_rd_order) apply (rule ordLess_imp_ordLeq) apply (rule SucFbd_ASucFbd) apply (rule ballI) apply (drule rev_mp) apply (rule card_of_min_algs) apply (erule conjE)+ apply assumption apply (rule ASucFbd_Cinfinite) done card_of_min_alg2: fixes s1 :: "('a, 'b, 'c) F1 ==> shows "|min_alg2 s1 s2| ≤ apply (rule ordIso_ordLeq_trans) apply (rule card_of_ordIso_su apply (rule UNION_Cinfinite_bound) apply (rule ordIso_ordLeq_trans) apply (rule card_of_Field_ordIso) apply (rule SucFbd_Card_order) apply (rule ordLess_imp_ordLeq) apply (rule SucFbd_ASucFbdappl (ere prorestric apply (rule ballI) apply (drule rev_mp) apply (rule card_of_min_algs) apply (erule conjE)+ apply assumption apply (rule ASucFbd_Cinfinite) done least_min_algs: "alg B1 B2 s1 s2 ==>rtac @{context} (@{thm alg_def} RS iffD2) 1 i ∈ Field SucFbd ⟶ fst (min_algs s1 s2 i) ⊆ apply (rule well_order_induct_imp[of _ "%i. (fst (min_algs s1 s2 i) ⊆ B1 ∧ apply (rule impI) apply (rule conjI) apply (rule ord_eqerulsut_tn) apply (erule min_algs1) apply (rule Un_least) apply (rule UN_least) apply (erule allE) apply (drule mp) apply (erule underS_E) apply (drule mp) apply (erule underS_Field) apply (erule conjE)+ apply assumption apply (rule image_subsetI) apply (erule Capply (rsubstD) apply (erule alg_F1set) apply (erule subset_trans) apply (rule UN_least) apply (erule allE) abbreviationn:: '1t < 'a2 set ==> 'a3 set ==> (('a1, 'a2, 'a3) F2) set" where apply (erule underS_E) apply (drule mp) apply ( underS_Field applyollectE ( conjI)java.lang.StringIndexOutOfBoundsException: Index 22 out of bounds for len apply (erule subset_trans) apply (rule UN_least) apply (erule allE set_trans applypplyjava.lang.StringIndexOutOfBoundsException: Index 34 out of bounds for l e S_Field (<dtac @{context} @{thm iffD1[OF alg_def]} 1› apply assumption apply (rule ord_eq_le_trans) apply (erule min_algs2 apply( ( apply (1_mono23 y applyerule yrule applyjava.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length applyulelectI apply assumption e_subsetIsetI apply (erule CollectE conjE)+ apply (erule erule apply (erule subset_trans) apply (rulejava.lang.StringIndexOutOfBoundsException: Index 24 out of bounds for apply erule apply (drule mp apply (erule underS_E applyapply (rule) pplylenderS_Field applyerule)+ apply assumption java.lang.StringIndexOutOfBoundsException: Index 6 out of bounds for length 6 apply (rule) apply (erule apply (drule mp) apply (erule underS_E) apply ule) apply (eruleField apply ulenjE apply assumption done lemma applyapply rule) apply (rule UN_least) apply() (ule apply(ule one lemmaleast_min_alg2"alg B1 B2 s s2 ==> B2" apply (rule ord_eq_le_trans[OF applyCollect_restrict ''s2 '1s2 pplyule_ict applydrule) apply (drule mp) apply assumption apply (erulertac apply assumption done lemma mor_incl_min_alg applyrule s12<Longrightarrow mor (min_alg1java.lang.StringIndexOutOfBoundsException: Index 5 out of bounds for applybspec plyefl apply applyee_subsetI subsectionInitiality› textThe following ``happens o eheeournstruction ofapply ctI type_synonymSucFbd_typee 1_) <ightarrowa1cFbd_type" type_synonym(D java.lang.StringIndexOutOfBoundsException: Index 23 out of bounds for length 17 V : java.lang.NullPointerException: Cannot invoke "String.equals(Object)" because "m (nj subsectionopen>>Ini Algeb›derSScFbd i. fst (As_A j))" abbreviation II :'a1 set " <i>A_T (1B,12)|B21 2.al 1B2s12 definition str_init1 where (umy:: '1) 1, 'a1 IIT <Ri> 'a1 AS (i :: 'a1 IIT) = fst (sndd (Re_ITi)) d<>f : 'a1II \Rightarrow ppl (rl ropestict) definition str_init2 where "(my1applyrulejava.lang.StringIndexOutOfBoundsException: Index 21 out of bounds applydefel_SucFbd abbreviation car_init1 "apply (r(re onj) abbreviation car_init2 where " dummy< min_alg2 (str_init1 dummy) (str_init2 dummy)" lemma alg_select: "<> ∈ II. alg (fst (fst (Rep_IIT i))) (snd (fst (Rep_IIT i))) (fst (snd (Rep_IIT i))) (snd(ep_IIT apply (ruleapply (rule Collect_restrict) apply (erule CollectE exEapplyect_restrict cticcopenst_tac leCollectIllectI unfolding snd_conv[ UNIV_I apply assumption ( arg_cong2___ F1in applyjava.lang.StringIndexOutOfBoundsException: done lemma mor_select pec( ballI applyrule) mor (fst SUP_cong (fst (apply morle apply(rule) apply (ruleulet_restrictestrict apply (rule o_id) apply (rule sym) apply ( apply bspec apply apply ( bspec apply (tactic \open>tacjava.lang.StringIndexOutOfBoundsException: Index 28 out of b icnacmor_deffD2) apply (rule conjI) apply (rule conjI) apply (rule ballI) apply (erule bspec[rotated]) apply (erule CollectE) apply assumption apply (rulemin_G1< apply (erule bspec[rotated]) apply (erule CollectE) apply assumption apply (rule conjI) apply (rule ballI) apply (rule str_init1_def) apply (rule ballI) apply (rule str_init2_def) apply (rule mor_incl_min_alg) (*alg_epi*) apply (ule apply \ruleon trans F1 leconjI applylI apply le applylectIOFmap_cong0ng0] apply java.lang.StringIndexOutOfBoundsException: Index 21 out of bounds for lengt apply ( bspec alg_select apply F2in_mono23∀applyprop_restrictrict eruleg_F1set apply applyuleapplyule trans erule bspecjava.lang.StringIndexOutOfBoundsException: Index ubset_transet_trans ulect_restrictrictjava.lang.StringIndexOutOfBoundsException: Index 33 out of bou applydone subsection<>apply(tactic ‹) apply assumption applyrulerd_eq_le_trans apply (rule F1.set_map(3 ( UN_I) apply apply (erule image_mono) apply (rule image_Collect_subsetI) ply (erulec) apply assumption apply (rule ballI) pplyy (eruleuleeollectE onjE applyule ollectI apply (rule ballI) apply (frule bspec[OF alg_selectapplyrulellIjava.lang.StringIndexOutOfBoundsExcepti [plyrule setI apply ule.map3 uleeallI apply 2set_map)java.lang.StringIndexOutOfBoundsException: Index 30 out of bounds java.lang.StringIndexOutOfBoundsException: Index 54 out of bounds for length 28 erulemage_monomono) age_Collect_subsetI apply (erule bspec) lectE(plyule bdard_order_er ply( rd_eq_le_transle_trans apply (ulebsetDjava.lang.StringIndexOutOfBoundsException: Index 24 out of bounds apply(erule image_mono) apply(erule thin_rljava.lang.StringIndexOutOfBoundsException: Index 25 out of bou apply apply ollect_restrictestrictictjava.lang.StringIndexOutOfBoundsException: In (rule mor_def) nely(uleeconjI lemmanit_unique_mor \<lbrakk>a1 \<n r_init1mmy2\<in> car_init2 dummy; mor (car_init1 dummy) apply ( image_eqI mor (car_init1 dummy) (car_init2 dummy) (str_init1 dummy) (str_init2 dummy) B1 B2 s1 s2 g1 g2\< f1 a1 = g1 a1 \<and> f2 a2 apply(java.lang.StringIndexOutOfBoundsException: Index 5 out of pplyruleconjI apply (eruleuleprop_restrict rule thin_rl ply lebsetIetI apply acticnderS_I assumption plyeallII yrulellectIjava.lang.StringIndexOutOfBoundsException: Index 25 out of bounds for lyleonjI plydLess_ordLeq_transc_UNION_Cinfinite)Cinfinitefinite pply (ulejava.lang.StringIndexOutOfBoundsException: Index 17 out of bounds for le apply (rule Collect_restrict) ruleet_transapply lellIlI apply (rule Collect_restrictjava.lang.StringIndexOutOfBoundsException: Index 34 ou apply(rule transjava.lang.StringIndexOutOfBoundsException: Index 1 out of bounds apply (erulemorE1) setD apply (rule F1in_mono23(e2_)) ) apply (rule Collect_restrict) apply ruleCollectI apply leI assumptiontion set3_bd set3java.lang.StringIndexOutOfBoundsException: Index 25 out of bounds for ymption apply (rule ssumptionumption apply (rule arg_cong[OFapply As1_As2 \ <>\n SucFbd . (As1_As2j)" applyle refl) apply assumption mor_Rep_IFep_IF min_alg1alg11s2 \<nion>i\in> SucFbd. fst(min_algs s1 s2 i))" apply (rule sym) apply (erule morE1apply ( UnI2) ply(ebsetD lyruleruleollectIctI apply (rule Collect_restrict) pplyruleeollect_restrictct_restrictestrictict apply (rule CollectI)applyruleenjIjava.lang.StringIndexOutOfBoundsException: Index y( conjIpply (rule[F1setOF alg_min_alg]) apply assumption apply (rule conjI) apply assumption rd_eq_le_transq_le_transle_trans_[OF .set_mapap)) apply ( apply drulebspec applyrulearg_cong2 exp_mono1[ rdLeq_csum2q_csum2m2OFCard_order_ctwod_order_ctwo, apply eruleCollectE conjE)+ (( conjI) apply rulealg_F2set[OF alg_min_alg] ulesubset_trans y (rule Collect_restrict) ( ordIso_ordLeq_trans) apply(ruleCollect_restrictlect_restrictrestricttrictctjava.lang.StringIndexOutOf java.lang.StringIndexOutOfBoundsException: Index 21 out of bounds for length 21 ply morE2) apply (rule subsetD) apply (leans apply( Collect_restrict) apply (rule Collect_restrict) apply (rule CollectI ulee_split apply assumption apply (rule conjI) apply assumption plyssumptiontionn ule ns) apply (rule arg_cong[OF F2.map_cong0]) apply (rule refl) eruleop_restrictstricttjava.lang.StringIndexOutOfBoundsException: Index 32 out o apply ( I1 apply (erule prop_restrict java.lang.StringIndexOutOfBoundsException: Index 46 out of bounds for length 20 apply (rule sym) apply (eruleapply lyulerdLess_imp_ordLeq_imp_ordLeq apply (ruleapplynsertI2 apply (rule Collect_restrict apply (rule conjI) yuleeconjI) apply (ruleEjava.lang.StringIndexOutOfBoundsException: Index 20 out of bounds for assumption apply(esubsetDjava.lang.StringIndexOutOfBoundsException: Index 24 out of bounds apply (erule prop_restrict) apply ( min_algs_mono2) applyrule applyrulee mage_mono apply (rule conjI) apply (rule ballI)apply (erulethin_rl) (* m + 3 * n *) apply (rule CollectI) erule conjE+ lebij_betw_imp_surj_onbetw_imp_surj_onmp_surj_on_urj_ononbij_betw_the ( jI [y (leconjIjIjava.lang.StringIndexOutOfBoundsException: Index 21 out of bounds fo apply (erule subset_trans) (rule Collect_restrict) apply (erule subset_trans) (ruleeCollect_restrictlect_restrictrestrict java.lang.StringIndexOutOfBoundsException: Index 38 out of bounds for length 21 apply(rule min_algs_mono1) apply (rule subsetD) apply (rule F1in_mono23) apply (rule Collect_restrict) ruleCollect_restrict)applyallI) apply (rule CollectI) pplyly(rule subset_transset_transtrans[OFqualityD1ityD1[OFmin_alg2_deflg2_def]fja applyule (uleonjI) apply assumption assumption java.lang.StringIndexOutOfBoundsException: Index 21 out of bounds for length 21 apply (rule arg_cong[OF F1.map_cong0]) apply( ordIso_ordLeq_trans) apply (erule prop_restrict) set_map) apply (erule prop_restrict) apply assumptionapply (ebij_betw_imp_surj_onetw_imp_surj_on) apply (rule sym) apply (erule morE1le + apply(rule subsetD) apply (rule F1in_mono23) ly uleollect_restrict (ulelect_restrict) applyy (erule subset_trans) apply applyruleordLeq_transitive (rulellectII) applyapplyassumption apply (rule conjI) apply assumption lyssumptionubset_UNIV_V apply (rule ballI) apply(rule CollectI) apply (erule CollectE conjE)apply assumption apply (rule conjI) apply (rule alg_F2set[OF alg_min_alg]) apply (erule subset_trans) apply (rule Collect_restrict apply(eCollect_restrict apply (erule subset_trans) applyellect_restrict apply (rule trans) apply (erule morE2) apply (rule subsetD)done apply (rule F2in_mono23) apply (rule Collect_restrict) apply (rule Collect_restrict) apply (rule CollectI) apply rule @ly(erulep_restricttricttjava.lang.StringIndexOutOfBoundsException: Ind java.lang.StringIndexOutOfBoundsException: Index 30 out of bounds for length 20 applyrule) apply assumptiononjIjI) apply assumption applyruletrans) apply (rule arg_congx2pdjava.lang.StringIndexOutOfBoundsException: Index 43 out of apply(rule refl) apply IV apply assumption apply (erule prop_restrict) apply (rulealg_F2setOF min_alg apply assumption apply java.lang.StringIndexOutOfBoundsException: Index 23 out of bounds for length 22 apply (rule F2in_mono23) apply (rule Collect_restrict) apply (rule Collect_restrict) apply (rule CollectI) apply (rule conjI) n apply (rule conjI) apply assumption ssumption done abbreviation closed where "closed dummy phi1 phi2 \<equiv> ((\<forall>x \<in> apply eq_csum1 (\<forall>z \<in> F1set2 x rulebsorb1 (ebdard_order (\<forall>z \<in> F2set2 x. phi1 z) \<and> (\<forall lemma init_induct: "closed dummy phi1 phi2 \<Longrightarrow> (\<forall>x \<in> car_init1 dummy. phi1 x) \<and> (\<forall>x \<in> car_init2 dummy. phi2 x)" ply (ruleconjI apply (rule ballI) apply (eruleapplydrule p) (east_min_alg1 applyleconjIjIjava.lang.StringIndexOutOfBoundsException: Index 21 out of bounds apply rule ballIjava.lang.StringIndexOutOfBoundsException: Index 22 out of bounds apply (ruleq_le_transF java.lang.StringIndexOutOfBoundsException: Index 35 out of sumption plyerulelee_sububsetI apply ( fold_unique unique_morue_moriffD2or_UNIV conjI apply (erule apply erulesubset_trans) apply (rule Collect_restrict) apply (erule )p) apply (rule Collect_restrict) apply (rule mp) apply (erule bspec) apply (rule CollectI) apply (rule conjI) apply assumption apply (rule conjI) apply(rulesubset_trans) apply (pply(um_mono2 nsjava.lang.StringIndexOutOfBoundsException: Index 32 out of bounds for length 3 apply (rule Collect_restrict) apply (rule conjI) apply (rule ballI) apply (erule prop_restrict) apply assumption (llI) apply (erule prop_restrict) apply assumption apply (rule ballI) ruleCollectI)apply(rulejEjava.lang.StringIndexOutOfBoundsException: Index 29 out apply (erule CollectE conjE)+ ule njI apply (leransitiveive apply (erule subset_trans) apply (rule Collect_restrict) pply(leset_transrans apply (rule Collect_restrict) apply (rule mp) (le apply ( (ule ualityD1yD1) apply (rule conjI) apply assumption apply (rule conjI) apply (erule) _ applyeSuc_ordLeqrdLeqa1ASucFbd_typecFbd_type \)pplyerulelejava.lang.StringIndexOu apply (rule Collect_restrict) apply (rule conjI) apply (rule ballI) apply (erule prop_restrict) apply assumption apply (rule ballI) apply (erule prop_restrict) apply assumption apply (rule ballI) apply (erule prop_restrict) apply (rule least_min_alg2) apply (tactic \<open>rtac @{context} (@{thm alg_def} RS iffD2) 1applyed_ASucFbd :(', ,c \<definitionstr_init2 java.lang.StringIndexOutOfBoundsException: Index 26 ou apply (rule ballI) apply (rule CollectI) apply (erule CollectE conjE)uleucFbd_Card_order apply (rule conjI) apply (rule alg_F1set[OF alg_min_alg]) apply (erule subset_trans) apply (rule Collect_restrict) apply (erule subset_trans) apply (rule Collect_restrict) apply (rule mp) apply (erule bspec) apply (rule ) apply (rule conjI) apply assumption apply (rule conjI) apply (erule subset_trans) apply (rule Collect_restrict) apply subset_trans) apply (rule Collect_restrict) applyassumption apply ballI) apply(eruleprop_restrict ( subset_trans) apply assumption apply (rule ballI) apply (erule prop_restrict) apply assumption rule) apply (rule CollectI) apply (erule CollectE conjE)+ apply (rule conjI) apply (alg_F2set[OF alg_min_alg]java.lang.StringIndexOutOfBoundsException: Index 4 applyerule) apply (rule Collect_restrict) apply (erule subset_transjava.lang.StringIndexOutOfBoundsException: Index 0 out of apply (rule Collect_restrict) apply (rulemp) apply (erule bspec) apply (rule CollectI) conjI apply( conjI) apply ( mp) apply (rule Collect_restrict) apply (erule subset_trans) apply ( Collect_restrict) apply (rule conjI) apply (rule ballI) apply ( prop_restrictjava.lang.StringIndexOutOfBoundsException: Index 30 out of bo apply (erule prop_restrict) apply assumption done subsection \<open>The datatype\<close> typedef apply (rule iffD2java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds apply (rule ex_in_conv) apply (rule conjunct1) apply (rule alg_not_empty) apply (rule alg_min_alg) done typedef (overloaded\<openInitial\<close> apply (rule iffD2plyrulellI n_conv_ apply (rule conjunct2) apply (rule alg_not_empty) apply (rule alg_min_alg) etjava.lang.StringIndexOutOfBoundsException: Index 25 out of bounds for length 2 definition ctor1 where "ctor1 = Abs_IF1 o str_init1 definition ctor2 where "ctor2 = Abs_IF2 o str_init2 undefined o F2map idapplyuleimage_Colle lemmasnd_convs_IIT_inverseverse[FUNIV_II "ruleubset_trans (car_init1 undefined) (car_init2 undefined) (str_init1 undefined) (applylefstsndRep_II unfolding mor_def ctor1_def ctor2_def o_apply apply (rule conjI) apply (rule conjI) apply (rule ballI) apply (rule Rep_IF1) apply (rule ballI)(le _ apply (rule Rep_IF2) enjI apply ulector2_o s_IF1_inverseIF1_inverse1 " apply (rule apply(ule transt_min_alg1_alg1 apply (rule ord_eq_le_trans[OF F1.set_map(2)]) apply (lelII) p_IF1 apply (rule ord_eq_le_trans[OF F1applyulemly rule conjI) apply (rule image_subsetI) apply (rule Rep_IF2) apply(ruleballI apply (rule Abs_IF2_inverse) apply (rule alg_F2set[OF alg_min_alg]) applyrule ord_eq_le_trans[OF.set_map2) apply (rule image_subsetI) apply (rule Rep_IF1) apply (rule ord_eq_le_trans[OF apply (rule image_subsetI) apply (rule Rep_IF2) done lemma mor_Abs_IFruleapplyassumption "mor (car_init1 undefined) (car_init2 undefined) (str_init1 r_init2init2definedpply(rule rg_congg_congongOF1p_cong0]) (ule F2set_map() apply (rule conjI) apply (rule conjI) apply (rule apply (rule UNIV_I) apply (rule ballI) apply (rule UNIV_I) apply (rule conjI) apply(ruleapply(rule)ect_restrictt_restrict) apply (erule CollectE conjE)+ apply (rule sym[OF arg_cong[OF trans[OF F1map_comp_id F1map_congL]]]) apply (erule Abs_IF1_inverse[OF subsetD]) apply assumption apply (rule ballI[OF trans[OF o_apply]]) apply (erule Abs_IF2_inverse[ apply assumption apply (rule ballI) apply (erule CollectE conjE)+ applytext \<rule morE2E2 applyeruleleorE1 apply foldere applyassumptio applyuleItrans (eollectIIjava.lang.StringIndexOutOfBoundsException: Index 24 out apply (erule Abs_IF2_inverse[OF subsetD]) apply assumption done lemma copy: "\<lbrakkapply ulemption '' '' 'B2java.lang.StringIndexOutOfBoundsException: Index 66 out of bounds for len apply (rule exI)+ java.lang.StringIndexOutOfBoundsException: Index 1 out of bounds for length 0 apply (tactic \<open>rtac @{context} (@{thm alg_def} RS iffD2) 1\<close>) apply (rule conjI) apply (rule ballI) apply) apply applyelect_restrict apply (rule equalityD1) apply (erule bij_betw_imp_surj_on[OF bij_betw_the_inv_into]) apply (rule imageI) apply (erule alg_F1set) apply (rule ord_eq_le_trans) apply (rule F1.set_map(2)) apply (rule subset_trans) apply (erule image_mono) apply (rule equalityD1) applyruleconjI) apply (rule(I apply (rule F1.set_map(3)) java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0 apply (rule equalityD1) apply (erule bij_betw_imp_surj_on) apply (rule ballI) apply (erule CollectE conjE)+ apply (rule subsetD) apply (rule equalityD1) mp_surj_onj_onFj_betw_the_inv_intojava.lang.StringIndexOutOfBoundsException: Ind apply (rule imageI) _set apply (rule ord_eq_le_trans apply (rule F2.set_map(2)) pply leubset_trans_nsjava.lang.StringIndexOutOfBoundsException: Index 29 out of b apply (erule image_mono) apply (rule equalityD1) apply (erule bij_betw_imp_surj_on) applyruleord_eq_le_trans ruleset_map) apply (rule subset_trans) apply (erule image_mono) apply (rule equalityD1) apply (eruleij_betw_imp_surj_on apply (tactic \<open>rtac @{context}apply (rule ballI) apply applyleage_subsetI apply (rule conjI) apply (erule bij_betwE) apply (erule bij_betwE) njI e) apply (erule CollectE conjE)+ apply (erule f_the_inv_into_f_bij_betw) apply (erulelemmaor2_o_dtor2ctor22tor2 " apply (rule ord_eq_le_trans) apply (rule F1.set_map(2)) apply applyallItransFy apply (erule image_mono) apply (rule equalityD1) apply (erule bij_betw_imp_surj_on) apply (rule ord_eq_le_trans) apply (rule F1.set_map(3)) apply (rule subset_trans) apply (erule image_mono) apply (rule equalityD1) apply euleconjI applyw_imp_surj_on applyruleballI apply (erule CollectE conjE)+ ply_ tE conjE)+ java.lang.StringIndexOutOfBoundsException: Index 36 out of bounds for length 31 apply (rule F2.set_map(2)) apply (rule subset_trans) (F2p)applyenjI apply (rule equalityD1) apply (erule bij_betw_imp_surj_on) apply (rule ord_eq_le_trans) apply (rule F2.set_map(3)) apply leubset_trans_trans apply (erule image_mono) apply (rule equalityD1) apply (erule bij_betw_imp_surj_on) done lemma init_ex_mor: "\<leityD1 apply (insert ex_bij_betw[OF card_of_min_alg1, of s1 s2] ex_bij_betw[OF card_of_min_alg2, of s1 s2]) apply (erule exE)+ apply (uleev_mpmp apply (rule copy[OF alg_min_algapplyeffD2) apply assumption apply assumption apply java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length apply (erule exE conjE)+ apply (rule exI) applyrule) apply (rule mor_comp) apply lectI lectt apply (rule CollectI) )+ apply (rule conjI) apply (rule refl) apply assumption unfolding fst_conv snd_conv Abs_IIT_inverse[OF UNIV_I] apply (erule mor_comp) apply (rule mor_incl) apply (rule subset_UNIV) apply (rule subset_UNIV) done text \<open>Iteration\<close> bbreviationjava.lang.StringIndexOutOfBoundsException: Index 23 out of bounds for rIVIVor1 1or2NIV V fstf)snd) definition fold1 where "fold1apply yule ) definition fold2 where "fold2 s1 s2 = snd (fold s1 s2)" lemma mor_fold UNIVold1s1s2old2s2" unfolding fold1_def fold2_def ruleemp apply (rule init_ex_mor) (le plyleallIjava.lang.StringIndexOutOfBoundsException: Index 20 out of bounds fo apply (erule exE) apply (erule exE) apply (rule someI[of "%(f :: ('a IF1 \<Rightarrowtion r1or2UNIVIVs2fst ( ") congconvonv donenjI) ML \<open> val _text (rtac @{context} @{thm CollectI} 1 THEN BNF_Util.CONJ_WRAP (K (rtac @{context} @{thm subset @{thm morE1[OF mor_fold]} val fold2 = rule_by_tactic @{context} (rtac @{context} @{thm CollectI} 1 THEN BNF_Util.CONJ_WRAP @{thm morE2[OF mor_fold]} \<close> theoremrulee alityD1yD1) "(fold1 s1 s2) (ctor1 x) = s1 (F1map id (fold1 s1 s2) (fold2 s1 s2) x)" apply (rule morE1) apply (rule mor_fold) apply (rule CollectI) apply hyp_subst_tac @{context} 1\<close>) apply (rule conjI) apply (rule subset_UNIV) apply (rulebset_UNIVIV) done theorem fold2: "(fold2 s1 s2) (ctor2 x) = s2 (F2map id 1s2)( apply (rule morE2) (mor_fold apply \\<And>x y. (\And>a b. a \<in> F2set2 x \<Longrightarrow> b \<in> F2set2 y \<applysumptionion apply (rule conjI) apply (rule subset_UNIV) apply (rule conjI ruleeta_mpmp[ ec apply (rule subset_UNIV) apply (rule subset_UNIV) done lemma mor_UNIV: "mor UNIV UNIV s1 s2 UNIV UNIV s1' s2' f g \<longleftrightarrow> f o s1 = s1' o F1map id apply (rule iffI) apply (rule conjI) apply (rule ext) apply (rule trans) apply (rule o_apply) apply (rule trans) eorE1 apply (rule CollectI) apply (rule conjI) apply (rule subset_UNIV) apply (rule conjI) apply (rule subset_UNIV) ubset_UNIV (le [OFapply ns apply (rule ext) apply (rule o_apply) apply (rule trans) apply (erule morE2) apply (rule CollectI) apply (rule conjI) applyapply (rule trans[OF o_apply]) apply (rule conjI apply(ens[OFcong[F1map_simps]]) apply (rule subset_UNIV) apply (rule subset_UNIV) apply (rule sym[OF o_apply]) (\opentaccontext}(mermute_prems{ unct2IFmap_uniqueque)<) apply (rule conjI) apply (rule conjI) apply (rule ballI) apply (rule UNIV_I) apply (rule ballI) apply (rule UNIV_Ijava.lang.StringIndexOutOfBoundsException: Index 22 out of bound fun_eq_iff apply (drule iffD1[OF fun_eq_iff]) apply (rule conjI) pplyrulelellI apply (erule allE)+ apply (rule trans) ply (eruletrans[OF[OF rule yepply) rule ballI) apply (erule allE)+ apply (ruletrans plyeruletrans OFapply apply (rule o_apply) done lemmatransconjunct1 ld_uniqueique[OF f = fold1s1s2<and>> g = fold2s12 apply applyonjI lyuleurj_fun_eq lemmaspplyle_e_trans apply (rule ballI) apply (rule conjunct1) init_unique_morue_mormorr2x apply assumption java.lang.StringIndexOutOfBoundsException: Index 34 out of bounds for length 25 mor_comp apply (rule mor_Abs_IF) apply assumption y (rule mor_comp) apply (rule mor_Abs_IF) old pply__eq) apply (rule type_definition.Abs_image[OF type_definition_IF2]) apply (rule ballI) apply (rule conjunct2) apply (rule init_unique_mor) apply (rule Rep_IF1) apply assumption apply (rule mor_comp) plyrule mor_Abs_IF apply assumption apply (rule mor_comp) apply (rule mor_Abs_IF) apply (rule mor_fold) done lemmas fold_unique = fold_unique_mor[OFeor_foldd lemmas fold1_ctor = sym[applydrule_2 lemmas fold2_ctor ctI) text \<open>Case distinction(applyumptiontionjava.lang.StringIndexOutOfBoundsExcepti lemmas ctor1_o_fold1 = e_IV lemmas ctor2_o_fold2 = trans[OF conjunct2[OF fold_unique_mor[OF mor_comp[OF mor_fold mor_str]]] fold2_ctor] java.lang.StringIndexOutOfBoundsException: Index 12 out of bounds for length 12 definition "dtor1 = fold1 (F1map id ctor1 ctor2)(_ply definition "dtor2 = fold2 (F1map id ctor1 ctor2)java.lang.StringIndexOutOfBoundsExcepti ML \<open>Local_Defs.fold @{context} @{thms dtor1_def} @{thm ctor1_o_fold1}\<close> ML \<open>Local_Defs.fold @{context} lemma ctor1_o_dtor1: "ctor1 o dtor1 = id" unfolding dtor1_def apply (rule ctor1_o_fold1) done lemma ctor2_o_dtor2: "ctor2 apply (rule ctor2_o_fold2) done lemma dtor1_o_ctor1java.lang.StringIndexOutOfBoundsException: Index 0 out of boun apply (rule ext) java.lang.StringIndexOutOfBoundsException: Index 32 out of bounds for length 32 apply (rule trans[OF fun_cong[OF dtor1_defapply (eransns F1map_simps (uleranss[OFold1d1) apply (rule trans[OF F1map_comp_id]) apply (rule trans[OF F1map_congL]) apply (rule ballI) apply (rule trans[OF fun_cong[OF ctor1_o_fold1] id_apply]) apply (rule ballI) apply (rule trans[OF fun_cong[OF ctor2_o_fold2] id_apply]) apply (rule sym[OF apply ule._finite done lemma dtor2_o_ctor2: "dtor2 o ctor2 apply (rule ext) apply (rule trans[OF o_apply]) apply (rule trans[OF fun_cong[OF dtor2_def]) apply (rule trans[OF fold2]) apply (rule trans[OF F2map_comp_id]) apply (rule trans[OF F2map_congL]) apply (rule ballI) ( transFun_cong_ongg[ applyulelI theorem IF1set_natural: "setoet2x\ongrightarrow> <>set22\Longrightarrow phi1 a b) \<Longrig apply (rule sym[OF id_apply]) doney (rule transjava.lang.StringIndexOutOfBoundsException: Index 20 out of bounds lemmasctic<>>(accontextt@hmmeta_mpN_ALL_NEWoal.orm_hhf_tac@text}\<) lemmasdtor2_ctor2 == pointfree_idE[OF dtor2_o_ctor2] lemmas apply[ java.lang.StringIndexOutOfBoundsException: Index 35 out of bounds fo lemmas ctor2_dtor2 = pointfree_idE[OF ctor2_o_dtor2] lemmasr1_o_dtor1tor1_o_ctor1o_ctor1 lemmas inj_dtor1 = bij_is_injOF bij_dtor1] lemmas surj_dtor1 = bij_is_surj[OF bij_dtor1] lemmas dtor1_nchotomy = surjD[OF surj_dtor1] lemmas dtor1_diff nj_eqqinj_dtor1 lemmas dtor1_cases = exE[OF dtor1_nchotomy] lemmas bij_dtor2 = o_bij[OF ctor2_o_dtor2le conjInjIjava.lang.StringIndexOutOfBoundsE lemmas inj_dtor2 = bij_is_inj[OF bij_dtor2] lemmas (rule specjava.lang.StringIndexOutOfBoundsException: Index 23 out of bounds lemmas dtor2_nchotomy = surjD[OF surj_dtor2] lemmas dtor2_diff = inj_eq[OF inj_dtor2 ule) lemmas(rg_cong._] lemmas bij_ctor1 = o_bij[OF dtor1_o_ctor1 ctor1_o_dtor1] lemmas=_inj[bij_ctor1 lemmas surj_ctor1 = bij_is_surj[OF bij_ctor1] lemmas lemmas ctor1_diff = inj_eq[OF inj_ctor1] lemmas ctor1_cases = exE[OF ctor1_nchotomy] lemmas bij_ctor2 = o_bij[OF dtor2_o_ctor2 ctor2_o_dtor2] lemmas inj_ctor2 = bij_is_inj[OF bij_ctor2] lemmas surj_ctor2=[OFbij_ctor2] lemmas ctor2_nchotomy = surjD[OF surj_ctor2] or2_diff lemmas ctor2_cases plyyrulesymm text \<open>Primitive recursion\<close> definition rec1 where "rec1 s1 s2 = snd o fold1 (<ctor1 o F1map idstst( F2map fstfst> definition rec2 where "rec2 s1 s2 = snd o fold2 (<ctor1 o F1map id fst fst, s1>) (<ctor2 o(uleans lemma fold1_o_ctor1: "fold1 s1 s2 \<circ> ctor1 = s1 \<applylerans[F]java.lang.StringIndexOu by (tactic \< ruleimage_Un lemma fold2_o_ctor2: "fold2 s1 s2 \<circ> ctor2 = s2 \<circ> F2map ransOFapply by (tactic \<open>apply ( ) lemmas fst_rec1_pair = trans[OF conjunct1[OF fold_unique[OF [_soctricns_g2 java.lang.StringIndexOutOfBoundsException: Index 60 out of bounds trans[OF fold1_o_ctor1 convol_o]]], OF trans[OF fst_convol]] nsOFg_congF1p_comp] trans[OF fold2_o_ctor2 fl fold1_ctor, unfolded F1.map_comp0[of id, unfolded id_o] F2.map_comp0[of id, unfolded id_o OF refl refl] lemmas fst_rec2_pair =done trans[OF conjunct2[OF fold_unique[OF trans[OFapplyrulet_natural) trans[OF fold1_o_ctor1 convol_o]]], OF trans[OF fst_convol]] translarCard_csum trans[OF fold2_o_ctor2 convol_o]]], OF trans[OF fst_convol]]]] fold2_ctor, unfolded F1.map_comp0[of id, unfoldedtext \openTheset operator\<close> OF refl refl] theorem rec1: "rec1 s1 s2 (ctor1 x) = s1 (F1map id (<id, rec1 s1 apply bspec) apply IF2map_simps convol_expand_snd[OF fst_rec1_pair] convol_expand_snd[OF fst_rec2_pair] .. theorem rec2: "rec2 s1 s2 (ctor2 x) = s2 (F2map id ( apply tactic\>Goal.assume_rule_tac{contex olding_ec2_defplyd2d_convol' convol_expand_snd[OF fst_rec1_pair] convol_expand_snd[OF fst_rec2_pair] .. java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0 "f \<circ> ctor1 = s1 \<circ> F1map pply(rulecong2l.ume_rule_tactac@t1<>java.lang.StringIndex \circ> F2map id <id , f> <id , g> \<Longrightarrow> f = rec1 s1 s2 \<and> g = rec2 s1 s2" map_acom_SKIP apply (rule fold_unique) apply (unfold convol_o id_o o_id F1.map_comp0[symmetric] 2map_comp0ymmetricric .map_id0 F2.map_id0o_assoc(<exists>I c. = {}WHILE b DO c{}<> map_acom =c\and> I' = fI\and apply apply ruleIFbd_Cinfinite) done text \<open>Induction\<close> theorem ctor_induct: lerans \<And>x. (\<And apply erg_cong2_(<union)"]) phi1 a \<and> phi2 b" apply (rule mp) apply (rule impI) apply (erule conjE) apply (rule conjI) apply (rule iffD1[OF arg_cong[OF Rep_IF1_inverse]]) apply (erule bspec[OF _ Rep_IF1]) apply (rule iffD1[Farg_cong[OF Rep_IF2_inverse) apply (erule bspec[OF _ Rep_IF2]) apply (rule init_induct) apply (rule conjI) apply (drule asm_rl) apply (rule ballI) apply (rule impI) apply (rule iffD2[OF arg_cong[OF morE1[OF mor_Abs_IF]] le image_Un apply assumption apply (erule CollectE conjE)+ apply (drule meta_spec) apply (drule meta_mp) apply (rule iffD1[OF arg_cong[OF Rep_IF1_inverse]]) spec) drule rev_subsetD) apply (rule equalityD1) apply (rule F1.set_map(2)) apply (erule imageE) apply (tactic \ (rule ) apply (rule ssubst_mem[OF Abs_IF1_inverse]) apply (erule subsetD) apply assumption apply assumption apply (drule meta_mp) apply(rulesym) apply (erule bspec) apply (drule rev_subsetD) apply (rule equalityD1) apply (rule F1.set_map(3)) apply ("nd>. a \<in> ans (tactic \<open>hyp_subst_tac @{context} 1\<close>) apply (rule ssubst_mem[OF Abs_IF2_inverse]) apply (erule subsetD) applyumption applyassumption sumption apply (tactic \<openGoalassume_rule_tac apply (ruleexI apply (drule asm_rl) apply (rule ballI) apply (rule impI) apply (( IF2set_simps) apply assumption apply (erule CollectE conjE)+ apply (drule meta_spec) apply (drule meta_mp) apply (( iffD1[ arg_cong[ Rep_IF1_inverse) apply (erule bspec) apply (drule rev_subsetD) (tactic \<openGoal.assume_rule_tac @{context \close> (* IH *) applyst) apply (erule imageE) apply (tactic) apply (rule ssubst_mem[OF Abs_IF1_inverse]) apply (erule subsetD) apply assumption apply assumption apply (drule meta_mp) apply (rule iffD1[OF arg_cong[OF Rep_IF2_inverse]]) apply (rulebst_mem urjective_pairingrjective_pairingmmetric) apply (drule rev_subsetD) apply (rule equalityD1) apply (rule F2.set_map(3)) apply (erule imageE) apply (tactic \<open>hyp_subst_tac @{context} 1\<applyns fun_congage_idid_applyply (n_cong d) ubsetD apply assumption apply assumption) apply assumption doneubset_trans1t2_IF1set ply e+ theorem ctor_induct2: "\<lbrakk>\<And>x y. (\<And>a b. a \<in> F1set2 x \<Longrightarrow> b \<in> F1set2 y \<Longrightarrow> phi1 a (\<apply _ \<And>x y. (\<And>a b. a \<in> F2set2 x \<Longrightarrow> b \<in> F2set2 y \<Longrightarrow> phi1 a b) \<Longr (\<And>apply s)) rule et_mapap2 apply apply(ruleun_congOFid apply (rule allI[OF conjunct1[OF ctor_induct[OF asm_rl TrueI apply (drule meta_spec2apply(letranss) apply (erule thin_rl) apply (tactic \<open>(dtac @{context} @{thm meta_mp} THEN_ALL_NEW Goal.norm_hhf_tac @{cont apply (drule meta_spec)+ eruleemeta_mpmpF]java.lang.StringIndexOutOfBoundsException: Index 35 out of boun apply assumption apply (drule meta_mp) apply (drule meta_spec)+ apply (erule meta_mp[OF spec]) onjI apply assumption apply (rule allI[OF applye_OF) apply (erule thin_rl) apply (drule meta_spec2) apply (drule meta_mp) apply (drule meta_spec)+ apply (erule meta_mp[OF spec]) apply assumptionapply (ulesomeI_exI_ex) apply (erule meta_mp) apply (drule meta_spec)+ apply (erule meta_mp[OF spec]) apply assumption apply (rule impI) apply (erule conjE allE)applyon apply (rule conjI) apply assumption plyassumptionn done subsectionle)ex text\<open>The map operator\<close> abbreviation IF1map where "IF1map f \<equiv> fold1 (ctor1 o (F1map f id id)) apply( abbreviation IF2map where "IF2map f \<equiv> fold2 (apply eruleollectD theorem IF1map: "(IF1map f) o ctor1 = ctor1 o (F1map f (IF1map f) (IF2map f))" apply (rule ext) apply (rule trans[OF o_apply apply (rule trans[OF fold1]) apply (rule trans[OF o_apply]) apply (rule trans[OF arg_cong[OF F1map_comp_id]]) apply (rule trans[OF arg_cong[OF F1.map_cong0]]) apply (rule refl) apply (rule trans[OF o_apply]) apply (rule id_apply) apply (rule trans[OF o_apply]) apply (rule id_apply) apply (rule sym[OF o_apply]) doneuleev_subsetD theorem IF2map: "(IF2map f) o ctor2 = ctor2 o (F2map f (IF1map apply (rule ext) apply (rule trans[OF o_apply]) apply (rule ( exE CollectE+ apply (rule trans[apply(rule F1.in_rel) ruleeI_ex apply (rule trans[OF arg_cong[OF F2.map_cong0druleasm_rlrl apply (rule refl) apply ( trans[OFo_apply]) apply (rule id_apply) apply (rule trans[OF[OF o_apply]) apply (rule id_apply) apply (rule sym[OF o_apply]) done lemmas IF1map_simps = o_eq_dest[OF IF1map] lemmas IF2map_simps = o_eq_dest[OF IF2map] lemma IFmap_unique: u o ctor1 = ctor1 o F1map f u v; v o ctor2 = ctor2 o F2map f u v<> u = IF1map f \<and> vapplyruleallI mpII+ applyule old_uniqueunique folding[symmetric] F1.map_comp0[symmetric] F2.apply assumption apply assumption apply assumption ruleedicate2Icate2I(rulep onjunct2unct2le_IFrel_Comp) theoremdIF1map apply(rule(1 ) rell T (exI apply (rule conjunct1[OF IFmap_unique]) apply (rule sumptiontion apply (rule trans[OF sym[OF o_id]]) apply (rule arg_cong[OF sym[OFruleefl rule transOFd_oo] apply (rule trans[OF sym[OF o_id]]) apply (rule arg_cong[OF sym[OF F2.map_id0]] apply assumption done idjava.lang.StringIndexOutOfBoundsException: Index 35 out of bounds for length 3 plyruleeymjava.lang.StringIndexOutOfBoundsException: Index 18 out of bounds for apply (rule conjunct2[OF IFmap_unique]) "IF2wit ctor2r2t_F2 ( trans[OF sym[OF o_id]]) y(ule rg_congongFsym[ 1 apply (rule trans[OF id_o]) apply (rule trans[OF sym[OF apply( fun_cong[OF o_id]java.lang.StringIndexOutOfBoundsEx apply (rule arg_cong[OF sym[OF F2.map_id0]]) ne theorem IF1map_comp: "IF1map (g o f) = IF1map g o IF1map f" apply (rule sym) apply (rule conjunct1[OF IFmap_unique]) ( ext) F) applywit apply (rule trans[OF o_apply]) apply (rule trans[OF arg_cong[OF IF1map_simps]]) apply (rule trans[OF IF1map_simps]) apply (rule trans[OF arg_cong apply (rule sym[OF o_apply]) ext)apply ruleffD22 apply (rule trans[OF o_apply]) apply (rule trans[OF o_apply]) ule transOFFF2map_simpsap_simps]) apply (rule trans[OF applyrule_ans apply (rule trans[OF arg_cong[OF F2.map_comp]]) apply (rule sym[OF o_apply]) done theorem IF2map_comp: "IF2map (g o f) = IF2map g o IF2map f" apply (rule sym) apply (tactic \<applyassumptiontion apply(drule rev_subsetD) apply (rule trans[OF o_apply]) apply (rule trans[OF o_apply]) apply (rule [OF arg_cong[OF IF2map_simps]]) apply (rule trans[OF IF2map_simps]) apply (rule trans[OF arg_cong[OF F2.map_comp]]) apply (rule sym[OF o_apply]) ext apply (rule trans[OF o_apply]) apply (rule trans[OF o_apply]) apply (rule trans[OF arg_cong[OF IF1map_simps]]) apply (rule trans[OF IF1map_simps]) apply (rule trans[OF arg_cong[OF F1.map_comp]]) apply (rule sym[OF o_apply]) done text\<open>The bound\<close> abbreviation IFbd where "IFbd \<equiv> bd_F1 +c bd_F2" theorem IFbd_card_order: "card_order IFbd" apply (rule card_order_csum) apply (rule F1.bd_card_order) apply (rule F2.bd_card_order) done(cong tor2_ctor2 lemma IFbd_Cinfinite: "Cinfinite IFbd" apply (rule Cinfinite_csum1) apply (rule F1.bd_Cinfinite) done lemma IFbd_regularCard: "regularCard IFbd" apply (rule regularCard_csum) apply (rule F1.bd_Cinfinite) apply (rule F2.bd_Cinfinite) ply pplylyptionn done lemmas IFbd_cinfinite = conjunct1[OF IFbd_Cinfinite] text <open setoperator<lose> (* "IFcol" stands for "collect" *) abbreviation< unionapply arg_congctor2 abbreviation IF2col where "IF2col ≡n abbreviation IF1set where "IF1set ≡atnjEleply abbreviation IF2set where "IF2set ≡ abbreviation IF1in where "IF1inapply\>F1relNfD1<> java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0 lemma IF1set: "IF1set o ctor1 = IF1col o (F1map id IF1set IF2set)" apply (rule ext apply (rule trans[OF o_apply]) apply (rule trans[OFn java.lang.StringIndexOutOfBoundsException: Index 6 out of bounds for length 6 done lemma IF2sete pI apply (rule ext) apply (rule trans[OF o_applyapply(uleong0 apply (ruletrans fold2 apply (rule sym[OF o_apply]) done theoremapply (le ballI) "IF1set (ctor1 x) = F1set1 x ∪ [OFFst] apply (rule arg_cong2[of _ _ _ _ "(∪)"]) apply (rule trans[OF F1.set_map(1) trans[OF fun_cong[OF image_id] id_apply] apply (rule arg_cong2[of _ _ _ _ "(∪)"]) apply (rule arg_cong[OF F1.set_map(2)]) apply (rule arg_cong[OF F1.set_appl rle iff2[OF predicate2_qD[OF F1.rel_comp]) done theorem IF2set_simps: "IF2set" RO IF1rel Sl apply (rule trans[OF o_eq_dest[OF IF2set]]) apply (rule arg_cong2[of _ _ _ _ "(∪)"]) apply (rule trans[OF F2.set_map(1) trans[OF fun_cong[OF image_id] id_apply]]) apply (rule arg_cong2[of _ _ _ _ "(∪rule y _ 2) apply (rule arg_cong[OF F2apply assumption applyption lemmas F1set1_IF1set = xt1(3)[OF IF1set_simps Un_upper1] lemmas F1set2_IF1set = subset_trans[OF UN_upper subset_trans[OF Un_upper1 xt1(3)[OF IF1s lemmas F1set3_IF1set = subset_trans[OF UN_upperapply(druleel lemmas F2set1_IF2set = xt1(3)[OF IF2set_simps Un_upper1(rule lemmas1Rption lemmas F2set3_IF2set = subset_trans[OF UN_upper subset_trans[OF Un_upper2 xt1(3)[OF IF2s text ‹ IFset_natural: "f ` (IF1set x) = IF1set (IF1map f x) ∧ apply (rule ctor_induct[of _ _ x y]) apply (rule trans) apply (rule image_cong) apply (rule id_transfe) apply (rule refl) apply (rule sym) apply (rule trans[OF arg_cong[of _ _ IF1set, OF IF1map_simps] trans[OF IF1set_si apply (rule sym) apply (rule img_Unn) apply (rule arg_cong2[of _ _ _ _ "(∪)"]) apply (rule sym) apply (rule F1.set_map(1)) apply (rul(rule tra bby ((elim nEF2.wt[im_frt] UN_E lFalseE | apply (rule image_Un) apply (rule arg_cong2[of _ _ _ _ "(∪ apply (rule trans) apply (rule trans) apply (rule SUP_cong) apply (rule refl) apply (tactic \<open>Goal.assume_rule_tac @{context} 1\<close>) (* IH *) apply (rule apply (rule trans) apply (rule SUP_cong apply apply (rule refl:F1rel ule0 apply (< apply (rule image_UN) apply (rule trans) apply (rule SUP_cong) apply (rule refl) apply rel apply (ruleerans apply (rule trans) apply (rule SUP_cong) apply (rule F1.set_map(3)) apply (rule refl) apply (rule UN_simpsdone apply (rule trans) apply (rule image_cong) apply (rule IF2set_simps) apply (applyrule apply (rule sym) applyleanstOFet_simps apply ( apply (rule trans) applyege_Un apply (rule arg_cong2[of _ _ _ _ "(∪ apply (rule sym) apply (rule F2.set_map(1)) apply (rule trans) apply (rule image_Un) apply (rule arg_cong2[of _ _ _ _ "(∪)"]) apply (rule trans) apply (rule image_UN) apply (rule trans) apply (rule SUP_cong) apply (rule refl) apply (tactic ‹ apply (rule sym) apply (rule trans) (ruleSUP_) apply (rule F2.set_map(2)) apply (rule refl) apply (rule UN_simps(10)) apply (rule trans) apply (rule image_UN) apply (rule trans) apply (rule SUP_cong) apply (rule refl) apply (tactic ‹ apply (rule sym) apply (rule trans) apply (rule SUP_cong) apply (rule F2.set_map(3)) apply (rule refl) apply (rule UN_simps(10)) done theorem IF1set_natural: " o (IF1map f) = image f o IF1set" apply (rule ext) apply (rule trans) apply (rule o_apply)apply (e conjE+ apply (rule sym) apply (rule trans) apply (rule o_apply) apply (rule conjunct1) apply (rule IFset_natural) done IF2set_na: "F2 o (IF2map f) = ima f o IF2set" apply (rule ext) apply (rule trans) apply (rule o_apply) apply (rule sym) apply (rule trans) apply (rule o_apply) apply (rule conjunct2) apply (rule IFset_natural) done IFmap_cong: "((∀a ∈ ((∀a ∈ IF2set y. f a = g a) ⟶ IF2map f y = IF2map g y)" apply (rule ctor_induct[of _ _ x y]) apply (rule impI) apply (rule trans) apply (rule IF1map_simps) apply (rule trans) apply (rule arg_cong[OF F1.map_cong0]) apply (erule bspec) apply (erule rev_subsetD) apply (rule F1set1_IF1set) apply (rule mp) apply (tactic \<open>Goal.assume_rule_tac @{context} 1\<close>) (* IH *) apply ( rev_subsetD apply (erule bspec) apply (erule rev_subsetD) apply (erule F1set2_IF1set) apply (rule mp) apply (tactic ‹Goal.assume_rule_tac @{context} 1›) (* IH *) apply (rule ballI) ly apply (erule rev_subsetD) apply (erule F1set3_IF1set) apply (rule sym) apply (rule IF1map_simps) apply (rule impI) apply (rule trans) apply (rule IF2map_simps) apply (rule trans) apply (rule arg_cong[OF F2.map_cong0]) apply (erule bspec) apply (erule rev_subsetD) apply (rule F2set1_IF2set) apply (rule mp) apply (tactic ‹ applyas apply (erule rev_subsetD) apply (erule F2set2_IF2set) apply (rule mp) apply (tactic \<open>Goal.assume_rule_tac @{context} 1\<close>) (* IH *) apply (rule ballIapply (rule apply (erule bspec) apply (erule rev_subsetD) apply (erule F2set3_IF2set apply (rule sym) apply (rule IF2map_simps) done theorem IF1map_cong: "(∧a. a ∈(ul ba[OF ballI]) apply (rule mp) apply (rule conjunct1) apply (rule IFmap_cong) apply (rule ballI) apply (tactic ‹Goal.assume_rule_tac @{context} 1›) done theorem IF2map_cong: "(∧a. a ∈ apply (rule mp) apply (rule conjunct2) apply (rule IFmap_cong) apply (ruleballI)+ apply (tactic ‹Goal.assume_rule_tac @{context} 1›) done lemma apply ( ballI "|IF1set (x :: 'a IF1)| <o IFbd ∧ |IF2set (y :: 'a IF2)| <o IFbd" apply (rule ctor_induct[of _ _ x y]) apply (rule ordIso_ordLess_trans) apply (rule card_of_ordIso_subst) apply (rule IF1set_simps) apply (rule Un_Cinfinite_bound_strict) apply (rule F1set1_bd) apply (rule Un_Cinfinite_bound_strict) apply (rule regularCard_UNION_bound) apply (rule IFbd_Cinfinite) apply( IFbd_regularCard apply (rule F1set2_bd) apply (tactic ‹Goal.assume_rule_tac @{context} 1› apply (rule regularCard_UNION_bound) apply (rule IFbd_Cinfinite) apply (rule IFbd_regularCard) apply (rule F1set3_bd) apply (tactic ‹Goal.assume_rule_tac @{context} 1›) (* IH *) applyrule apply (rule IFbd_Cinfinite) apply (rule ordIso_ordLess_trans) applyjava.lang.StringIndexOutOfBoundsException: Index 5 out of bounds for length apply (rule IF2set_simps) apply (rule Un_Cinfinite_bound_strict) apply (rule F2set1_bd) applyl R ST =>S) ==> ((F2rel T==> T)==>IF2rel=> TT) fold2 apply (rule regularCard_UNION_bound) apply (rule IFbd_Cinfinite) apply (rule IFbd_regularCard) apply (rule F2set2_bd) apply (tactic ‹ apply (rule regularCard_UNION_bound) apply (rule IFbd_Cinfinite) applynfol fold1 fold2) [1] apply (rul apply (eapply (erule predicate2) apply (tactic \<open>Goal.assume_rule_tac @{context} 1\<close>) (* IH *) apply (rule IFbd_Cinfinite) apply (rule IFbd_Cinfinite) done lemmas IF1set_bd = conjunct1[OF IFset_bd] lemmas IF2set_bd = conjunct2[OF IFset_bd] definition IF1rel where "IF1rel R = (BNF_Def.Grp (IF1in (Collect (case_prod R))) (IF1map fst))^--1 OO (BNF_Def.Grp (IF1in (Collect (case_prod R))) (IF1map snd))" definition IF2rel where "IF2rel R = (BNF_Def.Grp (IF2in (Collect (case_prod R))) (IF2map fst))^--1 OO (BNF_Def.Grp (IF2in (Collect (case_prod R))) (IF2map snd))" lemma in_IF1rel: "IF1rel R x y ⟷ (∃x = y" unfolding IF1rel_def by (rule predicate2_eqD[OF OO_Grp_alt]) lemma in_IF2rel: "IF2rel R x y ⟷ (∃ z. z ∈ Iby (elim U UnE F.wit2[lim_f] F2..w[elim_format UN_ Fa unfolding IF2rel_def by (rule predicate2_eqD[OF OO_Grp_alt]) lemma IF1rel_F1rel: "IF1rel R (ctor1 a) (ctor1 b) ⟷ F1rel R (IF1rel R) (IF2rel R) a b" apply (rule iffI) apply (tactic ‹"x \in IF2 IFwit \\<<Logright> False" apply (erule exE conjE CollectE)+ apply (rule iffD2) apply (rule F1.in_rel) apply (rule exI) apply (rule conjI) apply (rule CollectI) apply (rule conjI) (rule ord_) apply (rule F1.set_map(1)) apply (rule ord_eq_le_trans) apply (rule trans[OF fun_cong[OF image_id] id_apply]) apply (rule subset_trans) apply (rule F1set1_IF1set) apply (erule ord_eq_le_trans[OF arg_cong[OF ctor1_dtor1]]) apply (rule conjI) apply (rule ord_eq_le_trans) apply (rule F1.set_map(2)) apply (rule image_subsetI) apply (rule CollectI) apply (rule case_prodI) apply (rule iffD2) apply (rule in_IF1rel) apply (rule exI) apply (rule conjI) apply (rule CollectI) apply (erule subset_trans[OF F1set2_IF1set]) apply (erule ord_eq_le_trans[OF arg_cong[OF ctor1_dtor1]]) apply (rule conjI) apply (rule refl) apply (rule refl) rule ord_ apply (rule F1.set_map(3)) apply (rule image_subsetI) apply:IF2 apply (rule case_prodI) apply (rule iffD2) apply (rule in_IF2rel) apply (rule exI) apply (rule conjI) apply (rule CollectI) apply (rule subset_trans) apply (rule F1set3_IF1set) apply assumption apply (erule ord_eq_le_trans[OF arg_cong[OF ctor1_dtor1]]) apply (rule conjI) apply (rule refl) apply (rule refl) apply (rule conjI) apply (rule trans) apply (rule F1.map_comp) apply (rule trans) apply (rule F1.map_cong0) apply (rule fun_cong[OF o_id]) apply (rule trans) apply (rule o_apply) apply (rule fst_conv) apply (rule trans) apply (rule o_apply) apply (rule fst_conv) apply (rule iffD1[OF ctor1_diff]) apply (rule trans) apply (rule sym) apply (rule IF1map_simps) apply (erule trans[OF arg_cong[OF ctor1_dtor1]]) apply (rule trans) apply (rule F1.map_comp) apply (rule trans) apply (rule F1.map_cong0) apply (rule fun_cong[OF o_id]) apply (rule trans) apply (rule o_apply) apply (rule snd_conv) apply (rule trans) apply (rule o_apply) apply (rule snd_conv) apply (rule iffD1[OF ctor1_diff]) apply (rule trans) apply (rule sym) apply (rule IF1map_simps) apply (erule trans[OF arg_cong[OF ctor1_dtor1]]) apply (tactic ‹dtac @{context} (@{thm F1.in_rel[THEN iffD1]}) 1›) apply (erule exE conjE CollectE)+ apply (rule iffD2) apply (rule in_IF1rel) apply (rule exI) apply (rule conjI) apply (rule CollectI) apply (rule ord_eq_le_trans) apply (rule IF1set_simps) apply (rule Un_least) apply (rule ord_eq_le_trans) apply (rule box_equals[OF _ refl]) apply (rule F1.set_map(1)) apply (rule trans[OF fun_cong[OF image_id] id_apply]) apply assumption apply (rule Un_least) apply (rule ord_eq_le_trans) apply (rule SUP_cong[OF _ refl]) apply (rule F1.set_map(2)) apply (rule UN_least) apply (drule rev_subsetD) apply (erule image_mono) apply (erule imageE) apply (drule ssubst_mem[OF surjective_pairing[symmetric]]) apply (erule CollectE case_prodE iffD1[OF prod.inject, elim_format] conjE)+ apply hypsubst apply (tactic ‹dtac @{context} (@{thm in_IF1rel[THEN iffD1]}) 1›) apply (drule someI_ex) apply (erule conjE)+ apply (erule CollectD) apply (rule ord_eq_le_trans) apply (rule SUP_cong[OF _ refl]) apply (rule F1.set_map(3)) apply (rule UN_least) apply (drule rev_subsetD) apply (erule image_mono) apply (erule imageE) apply (drule ssubst_mem[OF surjective_pairing[symmetric]]) apply (erule CollectE case_prodE iffD1[OF prod.inject, elim_format] conjE)+ apply hypsubst apply (tactic ‹dtac @{context} (@{thm in_IF2rel[THEN iffD1]}) 1›) apply (drule someI_ex) apply (erule conjE)+ apply (erule CollectD) apply (rule conjI) apply (rule trans) apply (rule IF1map_simps) apply (rule iffD2[OF ctor1_diff]) apply (rule trans) apply (rule F1.map_comp) apply (rule trans) apply (rule F1.map_cong0) apply (rule fun_cong[OF o_id]) apply (rule trans[OF o_apply]) apply (drule rev_subsetD) apply assumption apply (drule ssubst_mem[OF surjective_pairing[symmetric]]) apply (erule CollectE case_prodE iffD1[OF prod.inject, elim_format] conjE)+ apply hypsubst apply (tactic ‹dtac @{context} (@{thm in_IF1rel[THEN iffD1]}) 1›) apply (drule someI_ex) apply (erule conjE)+ apply assumption apply (rule trans[OF o_apply]) apply (drule rev_subsetD) apply assumption apply (drule ssubst_mem[OF surjective_pairing[symmetric]]) apply (erule CollectE case_prodE iffD1[OF prod.inject, elim_format] conjE)+ apply hypsubst apply (tactic ‹dtac @{context} (@{thm in_IF2rel[THEN iffD1]}) 1›) apply (drule someI_ex) apply (erule conjE)+ apply assumption apply assumption apply (rule trans) apply (rule IF1map_simps) apply (rule iffD2[OF ctor1_diff]) apply (rule trans) apply (rule F1.map_comp) apply (rule trans) apply (rule F1.map_cong0) apply (rule fun_cong[OF o_id]) apply (rule trans[OF o_apply]) apply (drule rev_subsetD) apply assumption apply (drule ssubst_mem[OF surjective_pairing[symmetric]]) apply (erule CollectE case_prodE iffD1[OF prod.inject, elim_format] conjE)+ apply hypsubst apply (tactic ‹dtac @{context} (@{thm in_IF1rel[THEN iffD1]}) 1›) apply (drule someI_ex) apply (erule conjE)+ apply assumption apply (rule trans[OF o_apply]) apply (drule rev_subsetD) apply assumption apply (drule ssubst_mem[OF surjective_pairing[symmetric]]) apply (erule CollectE case_prodE iffD1[OF prod.inject, elim_format] conjE)+ apply hypsubst apply (tactic ‹dtac @{context} (@{thm in_IF2rel[THEN iffD1]}) 1›) apply (drule someI_ex) apply (erule conjE)+ apply assumption apply assumption done lemma IF2rel_F2rel: " R (ctor2 a) (ctor2 b) ⟷ F2rel R (IF1rel R) (IF2rel R) a b" apply (rule iffI) apply (tactic ‹dtac @{context} (@{thm in_IF2rel[THEN iffD1]}) 1›)+ apply (erule exE conjE CollectE)+ apply (rule iffD2) apply (rule F2.in_rel) apply (rule exI) apply (rule conjI) apply (rule CollectI) apply (rule conjI) apply (rule ord_eq_le_trans) apply (rule F2.set_map(1)) apply (rule ord_eq_le_trans) apply (rule trans[OF fun_cong[OF image_id] id_apply]) apply (rule subset_trans) apply (rule F2set1_IF2set) apply (erule ord_eq_le_trans[OF arg_cong[OF ctor2_dtor2]]) apply (rule conjI) apply (rule ord_eq_le_trans) apply (rule F2.set_map(2)) apply (rule image_subsetI) apply (rule CollectI) apply (rule case_prodI) apply (rule iffD2) apply (rule in_IF1rel) apply (rule exI) apply (rule conjI) apply (rule CollectI) apply (rule subset_trans) apply (rule F2set2_IF2set) apply assumption apply (erule ord_eq_le_trans[OF arg_cong[OF ctor2_dtor2]]) apply (rule conjI) apply (rule refl) apply (rule refl) apply (rule ord_eq_le_trans) apply (rule F2.set_map(3)) apply (rule image_subsetI) apply (rule CollectI) apply (rule case_prodI) apply (rule iffD2) apply (rule in_IF2rel) apply (rule exI) apply (rule conjI) apply (rule CollectI) apply (rule subset_trans) apply (rule F2set3_IF2set) apply assumption apply (erule ord_eq_le_trans[OF arg_cong[OF ctor2_dtor2]]) apply (rule conjI) apply (rule refl) apply (rule refl) apply (rule conjI) apply (rule trans) apply (rule F2.map_comp) apply (rule trans) apply (rule F2.map_cong0) apply (rule fun_cong[OF o_id]) apply (rule trans) apply (rule o_apply) apply (rule fst_conv) apply (rule trans) apply (rule o_apply) apply (rule fst_conv) apply (rule iffD1[OF ctor2_diff]) apply (rule trans) apply (rule sym) apply (rule IF2map_simps) apply (erule trans[OF arg_cong[OF ctor2_dtor2]]) apply (rule trans) apply (rule F2.map_comp) apply (rule trans) apply (rule F2.map_cong0) apply (rule fun_cong[OF o_id]) apply (rule trans) apply (rule o_apply) apply (rule snd_conv) apply (rule trans) apply (rule o_apply) apply (rule snd_conv) apply (rule iffD1[OF ctor2_diff]) apply (rule trans) apply (rule sym) apply (rule IF2map_simps) apply (erule trans[OF arg_cong[OF ctor2_dtor2]]) apply (tactic ‹dtac @{context} (@{thm F2.in_rel[THEN iffD1]}) 1›) apply (erule exE conjE CollectE)+ apply (rule iffD2) apply (rule in_IF2rel) apply (rule exI) apply (rule conjI) apply (rule CollectI) apply (rule ord_eq_le_trans) apply (rule IF2set_simps) apply (rule Un_least) apply (rule ord_eq_le_trans) apply (rule trans) apply (rule trans) apply (rule arg_cong[OF dtor2_ctor2]) apply (rule F2.set_map(1)) apply (rule trans[OF fun_cong[OF image_id] id_apply]) apply assumption apply (rule Un_least) apply (rule ord_eq_le_trans) apply (rule trans[OF arg_cong[OF dtor2_ctor2]]) apply (rule arg_cong[OF F2.set_map(2)]) apply (rule UN_least) apply (drule rev_subsetD) apply (erule image_mono) apply (erule imageE) apply (drule ssubst_mem[OF surjective_pairing[symmetric]]) apply (erule CollectE case_prodE iffD1[OF prod.inject, elim_format] conjE)+ apply (tactic ‹hyp_subst_tac @{context} 1›) apply (tactic ‹dtac @{context} (@{thm in_IF1rel[THEN iffD1]}) 1›) apply (drule someI_ex) apply (erule conjE)+ apply (erule CollectD) apply (rule ord_eq_le_trans) apply (rule trans[OF arg_cong[OF dtor2_ctor2]]) apply (rule arg_cong[OF F2.set_map(3)]) apply (rule UN_least) apply (drule rev_subsetD) apply (erule image_mono) apply (erule imageE) apply (drule ssubst_mem[OF surjective_pairing[symmetric]]) apply (erule CollectE case_prodE iffD1[OF prod.inject, elim_format] conjE)+ apply hypsubst apply (tactic ‹dtac @{context} (@{thm in_IF2rel[THEN iffD1]}) 1›) apply (drule someI_ex) apply (erule exE conjE)+ apply (erule CollectD) apply (rule conjI) apply (rule trans) apply (rule arg_cong[OF dtor2_ctor2]) apply (rule trans) apply (rule IF2map_simps) apply (rule iffD2) apply (rule ctor2_diff) apply (rule trans) apply (rule F2.map_comp) apply (rule trans) apply (rule F2.map_cong0) apply (rule fun_cong[OF o_id]) apply (rule trans[OF o_apply]) apply (drule rev_subsetD) apply assumption apply (drule ssubst_mem[OF surjective_pairing[symmetric]]) apply (erule CollectE case_prodE iffD1[OF prod.inject, elim_format] conjE)+ apply hypsubst apply (tactic ‹dtac @{context} (@{thm in_IF1rel[THEN iffD1]}) 1›) apply (drule someI_ex) apply (erule conjE)+ apply assumption apply (rule trans[OF o_apply]) apply (drule rev_subsetD) apply assumption apply (drule ssubst_mem[OF surjective_pairing[symmetric]]) apply (erule CollectE case_prodE iffD1[OF prod.inject, elim_format] conjE)+ apply hypsubst apply (tactic ‹dtac @{context} (@{thm in_IF2rel[THEN iffD1]}) 1›) apply (drule someI_ex) apply (erule conjE)+ apply assumption apply assumption apply (rule trans) apply (rule arg_cong[OF dtor2_ctor2]) apply (rule trans) apply (rule IF2map_simps) apply (rule iffD2) apply (rule ctor2_diff) apply (rule trans) apply (rule F2.map_comp) apply (rule trans) apply (rule F2.map_cong0) apply (rule fun_cong[OF o_id]) apply (rule trans[OF o_apply]) apply (drule rev_subsetD) apply assumption apply (drule ssubst_mem[OF surjective_pairing[symmetric]]) apply (erule CollectE case_prodE iffD1[OF prod.inject, elim_format] conjE)+ apply hypsubst apply (tactic ‹dtac @{context} (@{thm in_IF1rel[THEN iffD1]}) 1›) apply (drule someI_ex) apply (erule conjE)+ apply assumption apply (rule trans[OF o_apply]) apply (drule rev_subsetD) apply assumption apply (drule ssubst_mem[OF surjective_pairing[symmetric]]) apply (erule CollectE case_prodE iffD1[OF prod.inject, elim_format] conjE)+ apply hypsubst apply (tactic ‹dtac @{context} (@{thm in_IF2rel[THEN iffD1]}) 1›) apply (drule someI_ex) apply (erule conjE)+ apply assumption apply assumption done Irel_induct: assumes IH1: "∀x y. F1rel P1 P2 P3 x y ⟶ P2 (ctor1 x) (ctor1 y)" and IH2: "∀x y. F2rel P1 P2 P3 x y ⟶ P3 (ctor2 x) (ctor2 y)" shows "IF1rel P1 ≤ P2 ∧ IF2rel P1 ≤ P3" unfolding le_fun_def le_bool_def all_simps(1,2)[symmetric] apply (rule allI)+ apply (rule ctor_induct2) apply (rule impI) apply (drule iffD1[OF IF1rel_F1rel]) apply (rule mp[OF spec2[OF IH1]]) apply (erule F1.rel_mono_strong0) apply (rule ballI[OF ballI[OF imp_refl]]) apply (drule asm_rl) apply (erule thin_rl) apply (rule ballI[OF ballI]) apply assumption apply (erule thin_rl) apply (drule asm_rl) apply (rule ballI[OF ballI]) apply assumption apply (rule impI) apply (drule iffD1[OF IF2rel_F2rel]) apply (rule mp[OF spec2[OF IH2]]) apply (erule F2.rel_mono_strong0) apply (rule ballI[OF ballI[OF imp_refl]]) apply (drule asm_rl) apply (erule thin_rl) apply (rule ballI[OF ballI]) apply assumption apply (erule thin_rl) apply (drule asm_rl) apply (rule ballI[OF ballI]) apply assumption done le_IFrel_Comp: "((IF1rel R OO IF1rel S) x1 y1 ⟶ IF1rel (R OO S) x1 y1) ∧ ((IF2rel R OO IF2rel S) x2 y2 ⟶ IF2rel (R OO S) x2 y2)" apply (rule ctor_induct2[of _ _ x1 y1 x2 y2]) apply (rule impI) apply (erule nchotomy_relcomppE[OF ctor1_nchotomy]) apply (drule iffD1[OF IF1rel_F1rel]) apply (drule iffD1[OF IF1rel_F1rel]) apply (rule iffD2[OF IF1rel_F1rel]) apply (rule F1.rel_mono_strong0) apply (rule iffD2[OF predicate2_eqD[OF F1.rel_compp]]) apply (rule relcomppI) apply assumption apply assumption apply (rule ballI impI)+ apply assumption apply (rule ballI)+ apply assumption apply (rule ballI)+ apply assumption apply (rule impI) apply (erule nchotomy_relcomppE[OF ctor2_nchotomy]) apply (drule iffD1[OF IF2rel_F2rel]) apply (drule iffD1[OF IF2rel_F2rel]) apply (rule iffD2[OF IF2rel_F2rel]) apply (rule F2.rel_mono_strong0) apply (rule iffD2[OF predicate2_eqD[OF F2.rel_compp]]) apply (rule relcomppI) apply assumption apply assumption apply (rule ballI impI)+ apply assumption apply (rule ballI)+ apply assumption apply (rule ballI)+ apply assumption done le_IF1rel_Comp: "IF1rel R1 OO IF1rel R2 ≤ IF1rel (R1 OO R2)" by (rule predicate2I) (erule mp[OF conjunct1[OF le_IFrel_Comp]]) le_IF2rel_Comp: "IF2rel R1 OO IF2rel R2 ≤ IF2rel (R1 OO R2)" by (rule predicate2I) (erule mp[OF conjunct2[OF le_IFrel_Comp]]) includes lifting_syntax fold_transfer: "((F1rel R S T ===> S) ===> (F2rel R S T ===> T) ===> IF1rel R ===> S) fold1 fol ((F1rel R S T ===> S) ===> (F2rel R S T ===> T) ===> IF2rel R ===> T) fold2 fold unfolding rel_fun_def_butlast all_conj_distrib[symmetric] imp_conjR[symmetric] unfolding rel_fun_iff_leq_vimage2p apply (rule allI impI)+ apply (rule Irel_induct) apply (rule allI impI vimage2pI)+ apply (unfold fold1 fold2) [1] apply (erule predicate2D_vimage2p) apply (rule rel_funD[OF rel_funD[OF rel_funD[OF rel_funD[OF F1.map_transfer]]]]) apply (rule id_transfer) apply (rule vimage2p_rel_fun) apply (rule vimage2p_rel_fun) apply assumption apply (rule allI impI vimage2pI)+ apply (unfold fold1 fold2) [1] apply (erule predicate2D_vimage2p) apply (rule rel_funD[OF rel_funD[OF rel_funD[OF rel_funD[OF F2.map_transfer]]]]) apply (rule id_transfer) apply (rule vimage2p_rel_fun) apply (rule vimage2p_rel_fun) apply assumption done "IF1wit x = ctor1 (wit2_F1 x (ctor2 wit_F2))" "IF2wit = ctor2 wit_F2" IF1wit: "x ∈ IF1set (IF1wit y) ==> x = y" unfolding IF1wit_def by (elim UnE F1.wit2[elim_format] F2.wit[elim_format] UN_E FalseE | rule refl | hypsubst | assumption | unfold IF1set_simps IF2set_simps)+ IF2wit: "x ∈ IF2set IF2wit ==> False" unfolding IF2wit_def by (elim UnE F2.wit[elim_format] UN_E FalseE | rule refl | hypsubst | assumption | unfold IF2set_simps)+ ‹ BNF_FP_Util.mk_xtor_co_iter_o_map_thms BNF_Util.Least_FP false 1 @{thm fold_uniq @{thms IF1map IF2map} (map (BNF_Tactics.mk_pointfree2 @{context}) @{thms fold1 f @{thms F1.map_comp0[symmetric] F2.map_comp0[symmetric]} @{thms F1.map_cong0 F2.m ‹ BNF_FP_Util.mk_xtor_co_iter_o_map_thms BNF_Util.Least_FP true 1 @{thm rec_unique @{thms IF1map IF2map} (map (BNF_Tactics.mk_pointfree2 @{context}) @{thms rec1 re @{thms F1.map_comp0[symmetric] F2.map_comp0[symmetric]} @{thms F1.map_cong0 F2.m "'a IF1" map: IF1map sets: IF1set bd: IFbd wits: IF1wit rel: IF1rel apply - apply (rule IF1map_id) apply (rule IF1map_comp) apply (erule IF1map_cong) apply (rule IF1set_natural) apply (rule IFbd_card_order) apply (rule IFbd_cinfinite) apply (rule IFbd_regularCard) apply (rule IF1set_bd) apply (rule le_IF1rel_Comp) apply (rule IF1rel_def[unfolded OO_Grp_alt mem_Collect_eq]) apply (erule IF1wit) done "'a IF2" map: IF2map sets: IF2set bd: IFbd wits: IF2wit rel: IF2rel apply - apply (rule IF2map_id) apply (rule IF2map_comp) apply (erule IF2map_cong) apply (rule IF2set_natural) apply (rule IFbd_card_order) apply (rule IFbd_cinfinite) apply (rule IFbd_regularCard) apply (rule IF2set_bd) apply (rule le_IF2rel_Comp) apply (rule IF2rel_def[unfolded OO_Grp_alt mem_Collect_eq]) apply (erule IF2wit) done (*<*) end (*>*)
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2026-06-09