Quellcode-Bibliothek LFP.thy

(*  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) set" where
  "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) set" where
  "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 o f3) x"
  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 o f3) x"
  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 B2. s2 y ∈ B2))"

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 bounds for length 25
 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_Cinfinite]


 ‹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_As2 i)),
    min_G2 As1_As2 i ∪ s2 ` (F2in (UNIV :: 'a set) (min_G1 As1_As2 i) (min_G2 As1_As2 i)))"

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.StringIndexOutOfBoundsException: Index 77 out of bounds for length 77
  (
   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: "<>.set2F2bd
  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]]} 1›)
  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 length 22

   apply (erule subset_trans)
   apply (rule UN_least)
   apply (erule allE set_trans
   applypplyjava.lang.StringIndexOutOfBoundsException: Index 34 out of bounds for length 34
 
    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 0
   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 length 24
   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 length 0
  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 "macro" is null
  (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 for length 21
      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: Index 20 out of bounds for length 20
  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 bounds for length 28
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 length 21
   apply ( bspec alg_select
   apply F2in_mono23∀applyprop_restrictrict
   eruleg_F1set

    apply 
     applyuleapplyule trans      erule bspecjava.lang.StringIndexOutOfBoundsException: Index 24 out of bounds for length 24
 ubset_transet_trans
     ulect_restrictrictjava.lang.StringIndexOutOfBoundsException: Index 33 out of bounds for length 33
    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.StringIndexOutOfBoundsException: Index 21 out of bounds for length 21
[plyrule setI
        apply ule.map3

    uleeallI
    apply  2set_map)java.lang.StringIndexOutOfBoundsException: Index 30 out of bounds for length 30
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 for length 24

   apply(erule image_mono)
    apply(erule thin_rljava.lang.StringIndexOutOfBoundsException: Index 25 out of bounds for length 25
  apply apply ollect_restrictestrictictjava.lang.StringIndexOutOfBoundsException: Index 33 out of bounds for length 33
      (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\<rbrakk> \<Longrightarrow>
  f1 a1 = g1 a1 \<and> f2 a2 apply(java.lang.StringIndexOutOfBoundsException: Index 5 out of bounds for length 0
  pplyruleconjI
   apply (eruleuleprop_restrict
rule thin_rl
   ply lebsetIetI
apply acticnderS_I
    assumption
  plyeallII
   yrulellectIjava.lang.StringIndexOutOfBoundsException: Index 25 out of bounds for length 25
    
    lyleonjI

 plydLess_ordLeq_transc_UNION_Cinfinite)Cinfinitefinite
      pply (ulejava.lang.StringIndexOutOfBoundsException: Index 17 out of bounds for length 0
      apply (rule Collect_restrict)
     ruleet_transapply lellIlI
     apply (rule Collect_restrictjava.lang.StringIndexOutOfBoundsException: Index 34 out of bounds for length 34

    apply(rule transjava.lang.StringIndexOutOfBoundsException: Index 1 out of bounds for length 0
     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 length 0
      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 20 out of bounds for length 20
    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.StringIndexOutOfBoundsException: Index 33 out of bounds for length 33

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 of bounds for length 32
     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 length 20
    assumption
    apply(esubsetDjava.lang.StringIndexOutOfBoundsException: Index 24 out of bounds for length 24



  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_inv_into])
   ( jI

[y (leconjIjIjava.lang.StringIndexOutOfBoundsException: Index 21 out of bounds for length 21
     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]fjava.lang.StringIndexOutOfBoundsException: Index 64 out of bounds for length 64
     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: Index 31 out of bounds for length 31
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 bounds for length 32
     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 for length 21
    apply rule ballIjava.lang.StringIndexOutOfBoundsException: Index 22 out of bounds for length 22
    apply (ruleq_le_transF java.lang.StringIndexOutOfBoundsException: Index 35 out of bounds for length 20
    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 31
     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 of bounds for length 29
   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.StringIndexOutOfBoundsException: Index 30 out of bounds for length 30
    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 out of bounds for length 26
   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 41 out of bounds for length 41
    applyerule)
    apply (rule Collect_restrict)
   apply (erule subset_transjava.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
   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 bounds for length 30

  
  apply (erule prop_restrict)
  apply assumption
  done


subsection \<open>The datatype\<close>

typedef
  apply (rule iffD2java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
   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 25

definition ctor1 where "ctor1 = Abs_IF1 o str_init1
definition ctor2 where "ctor2 = Abs_IF2 o str_init2 undefined o F2map idapplyuleimage_Collect_subsetIllect_subsetI_bsetII

lemmasnd_convs_IIT_inverseverse[FUNIV_II
  "ruleubset_trans
     (car_init1 undefined) (car_init2 undefined) (str_init1 undefined) (applylefstsndRep_IIT) dITUNIV'2\rbrakk \<Longrightarrow>
  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 of bounds for length 24
  apply (erule Abs_IF2_inverse[OF subsetD])
  apply assumption
  done

lemma copy:
  "\<lbrakkapply ulemption
'' '' 'B2java.lang.StringIndexOutOfBoundsException: Index 66 out of bounds for length 19
  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: Index 47 out of bounds for length 32
   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 bounds for length 29
     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 0
  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 length 23
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 for length 20
  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_UNIV}       et_UNIV
    @{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 bounds for length 22
  
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.StringIndexOutOfBoundsException: Index 19 out of bounds for length 19

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.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0

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 bounds for length 0
  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) \<Longrightarrow>
  apply (rule sym[OF id_apply])
  doney (rule transjava.lang.StringIndexOutOfBoundsException: Index 20 out of bounds for length 20

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 for length 35
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.StringIndexOutOfBoundsException: Index 20 out of bounds for length 20
lemmas inj_dtor2 = bij_is_inj[OF bij_dtor2]
lemmas (rule specjava.lang.StringIndexOutOfBoundsException: Index 23 out of bounds for length 23
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.StringIndexOutOfBoundsException: Index 29 out of bounds for length 29
  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 for length 31
              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] o_assoc,
    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{context} 1\<>) (* IH *)
  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.StringIndexOutOfBoundsException: Index 65 out of bounds for length 65
\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 b) \<Longrightarrow>
      (\<apply _
    \<And>x y. (\<And>a b. a \<in> F2set2 x \<Longrightarrow> b \<in> F2set2 y \<Longrightarrow> phi1 a b) \<Longrightarrow>
      (\<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 @{context}) 1\<close>)
     apply (drule meta_spec)+
eruleemeta_mpmpF]java.lang.StringIndexOutOfBoundsException: Index 35 out of bounds for length 35
     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 35
  plyruleeymjava.lang.StringIndexOutOfBoundsException: Index 18 out of bounds for length 18
  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.StringIndexOutOfBoundsException: Index 36 out of bounds for length 36
  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 IF1set_simps (ehotomy_relcomppE[ tor2_nchotomy
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 IF2set_simps Un_upper2]]]

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_simps]])

 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 5
   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_ FalseE |
  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 fold1 ∧
 ((F1rel R S T ===> S) ===> (F2rel R S T ===> T) ===> IF2rel R ===> T) fold2 fold2"
 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_unique}
 @{thms IF1map IF2map} (map (BNF_Tactics.mk_pointfree2 @{context}) @{thms fold1 fold2})
 @{thms F1.map_comp0[symmetric] F2.map_comp0[symmetric]} @{thms F1.map_cong0 F2.map_cong0}
 ›

  ‹
 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 rec2})
 @{thms F1.map_comp0[symmetric] F2.map_comp0[symmetric]} @{thms F1.map_cong0 F2.map_cong0}
 ›

  "'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
(*>*)