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
  [witsapply ssumption
  for
bnf_axiomatization
  [witsalg_F2set🚫
  for map: F2map rel: F2rel


abbreviation F1in :: "'a1 set ==>dtac @{context} @{thm iffD1[OF alg_def]} 1›)
  "  2<> x F1set1> A1 ∧ A2 ∧ A3}"
abbreviation F2in :: "'a1 set ==> 'a2 set(rule
  ruleOF)

lemma(leonjI
  applyapplyassumption
   apply (rule
  unfolding alg_not_empty
  applyle
  done

lemmas F1in_mono23

lemma F1map_congL <open@ntext)
  F1map id f g x = x"
  apply rule trns)
   apply (rule F1.map_cong0)
     apply (rule refl)
    apply(ule as
     apply (erule bspec)
     apply assumption
    apply (rule sym)
    (uleid__apply)
   apply (rule trans)
    apply (erule bspec)
    
   apply (rule sy rlesustemptyI
   apply (rule id_apply)
  pply rl 1.api)
  done

lemma F2ma
  apply (actic \<hyp_subst_tac FIRST' (map (fn thm => rtac @{context} thm THEN' assume_tac @{context}) @{thms alg_F1set alg_F2set}) 1›)
java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
  unfolding o_id
  apply (rule refl)
  done

lemmas F2in_mono23 = F2.in_mono[OF subset_refl]

lemma F2map_congL: "[a ∈a ∈ ==>
  F2map id f g x = x"
  apply (rule trans) mor where
   apply (rule F2.map_cong0)
     apply (rule refl)
    apply (rule trans)
     apply (erule bspec)
     apply assumption
    apply (rule sym)
    apply (rule id_apply)
   pply (re tans)
    apply (erule bspec)
    apply assumption
   apply (rule sym)
   apply (rule id_apply)
  apply (rule F2.map_id)
  done


subsection‹ g z)z)))"

definition alg where
  "alg
    <forall>x<> F1in (UNIV :: 'a set) B1 B2. s1 x \<in> B1) \<and> (\<forall>y \<in> F2in (UNIV:at B12y<> B2))"

lemmadone
  apply (tactic \<open>dtac @{context} @{thm iffD1[OF alg_def]} 1\<close>)
  apply (erule conjE)+
  apply (erule bspec)
  apply (rule CollectI)
  apply (rule conjI[OF subset_UNIV])
  applyrule onjI)
  apply assumption
  done

lemmaeruleesubst_memt_memOF id_apply
  plyactic<open>dtac@{ontext@thm ffD1D1Flg_defef \<se
  pply (eruleconjEnjEjava.lang.StringIndexOutOfBoundsException: Index 22 out of bounds for length 22
  (rulespec
  apply (rule CollectI)
  apply (rule conjI[OF subset_UNIV])
  apply (erule conjI)
  apply assumption
  done

lemma alg_not_empty:
  "alg B1 B2 s1 s2 \<Longrightarrowmor_comp_ompjava.lang.StringIndexOutOfBoundsException: Index 15 out of bounds for length 15
  apply (rule conjI)
   apply (rule notI)
   apply (tactic \<open>hyp_subst_tac @{context} 1\<close>)
   apply (frule alg_F1set)

(* ORELSE of the following three possibilities *)

     apply ((ule
     apply

    apply (rulerule
    apply (drulerule

(**)
    (erulepec
    apply (tacticssumption

     apply (rulerulemem)
     apply (erule F1.(rule

    apply (rule
    apply ( F1t2
    apply (erule( Ijava.lang.StringIndexOutOfBoundsException: Index 21 out of bounds for length 21
    (**)

   apply (eruleleEonjE

  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 ‹

  r whre
 "mor B1 B2 s1 s2 rul ord_d_eqle_tanss)
 (((\<apply 
 ((∀z ∈ F1in (UNIV :: 'a set) B1 B2. f (s1 z) = s1' (F1map id f g z)) ∧
 (∀z ∈ F2in (UNIV :: 'a set) B1 B2. g (s2 z) = s2' (F2map id f g z))))"

  morE1: "[mor B1 B2 s1 s2 B1' B2' s1' s2' f g; z ∈ F1in UNIV B1 B2]
 ==> f (s1 z) = s1' (F1map id f g"
 apply (tact >
 apply (erule conjE)+
 apply (erule bspebegin{tabula}rcl}
 apply a'b2&& ('1, 'b22
 ne

 mor B1 B2 s1 s2 B1' B2' s1s' g; z <>F2in 'b1 \<ghtarrowarrowx subseteq> A3}"
 ==> g (s2 z) = s2' (F2map id f g z)"
 apply (tactic ‹D1[OF mor_def]} 1\<close)
 apply (erule conjE)+
 apply (erule bspec)
 apply as
 done

  mor_incl: "[B1 ⊆ B1'; B2 ⊆ B2'] ==>
 apply app(rul 1ap_comp
 applyly (uecnjI)

 apply (rule conjI)
 apply (rule ballI)
 F1map i f g x =
 apply (y(eessubst_me[OF id_appyy])

 apply (rule ballI)
 apply (erule subsetD)
 apply (erule ssubst_mem[OF id_apply])

 apply (rule rule trras)
 y blI)
 apply (rule tans
 (rule 1.map__i
 apply (tactic ‹a ∈ ==>
 apply (rule refl)

 pply y(rlballI)
 apply (asumptionjava.lang.StringIndexOutOfBoundsException: Index 20 out of bounds for length 20
 apply (rule id_apply)
 apply (tactic ‹stac @{context} @{thm F2.map_id} 1›)
 apply (rule refl)
 done

  mor_comp:
 "[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 (tactc\open @{context} (@{thm mor_def} RS iffD1) 1›
  cnj[ substI])
 done
 apply rulle conjI)

 apply (rule conjI)
 apply (rule ballI)
 apply (rule ssubst_mem[OF o_apply])
 apply (erule bspec)
 apply (erule bspec)
 apply assumption

 plyy (rule balI)
 (rule [OF subet_UNIV])
 erule
 alg_not_empty:
 apply assumption

 apply rule conjI)
 apply apply (tactc \openhyp_subst_tac @{context} 1›)
 apply (rule trans[OF o_apply])
 apply (rule trans)
 apply (
 apply (drule bspec[rotated])
 apply assumpti
 apply erule arg_cong)
 apply (e (e CollectE conjE)+
 apply (e(erule bspec)
 apply (rule CollectI)
 apply (rua
 apply (rule su)
 apply (rule conjI)
 apply (rule ord_eq_le_trans)
 apply (rule F1.set_map(2))
 apply (rule image_subsetI)
 apply (erule bspec)
 apply (erule subsetD)
 apply assumption
 apply pply ( (rule subset_emptyI
 F1.set_map(3))
 apply (rapply rule subse)
 apply (erule bspec)
 apply (erule subsetD)
 apply assumption
 apply (rule arg_cong[OF F1map_comp_id])

 apply (rule ballI)
 apply (rule trans[OF o_apply])
 apply (rule trans)
 apply (rule trans)
 apply (drule bspec[rotated])
 apply)+
 apply ( rle impI
 apply (erule CollectE conjE)+
 apply (erule bspec)
 apply (rule CollectI)
 apply (rule conjI)
 apply (ruleubset_UNUNIV)
 apply (rule conjI)
 apply (rule ord_eq_le_trans)
 apply (rule F2.set_map(2))
 apply (rule image_subsetI)
 apply (erule bspec)
 apply (erule subsetD)
 apply assumption
 applyy ( bspec)
 apply (rule F2.set_m assumption
 apply (rule image_subsetI)
 apply (erule bspec)
 apply (erule subsetD)
 apply assumption
 apply (rule arg_cong[OF F2map_comp_id])
 done

  mor_cong: "[
 mor B1 B2 s1 s2 B1' 2' s1' ' s2' f' g'"
 apply (tactic ‹SUP_cong))
 apply assumption
 done

  mor_str:
 "mor UNIV UNIV (F1map id s1 s2) (F2map id s1 s2) UNIV UNIV s1 s2 s erule arg_cong)
 apply (rule iffD2)
 apply (rule mor_def)
 apply ruleconjI)
 apply (rule conjI)
 apply (rule bal)
 apply (rule UNIV_I)
 apply (rul ballI
 apply (drule b

 apply (rule conjI)
 apply (rule ballI)
 apply (ruleapply (erule arg_cong)
 apply (rule ball
 apply (rule refl)
 done


 ‹)

  bd_type_F1' = "bd_type_F1 + + (bd_type_F1, bd_type_F1, bd_ty) F1"
  (er arg_cong)
  SucFbd_typetapply(rul SUP_cong)
  'rule refl)

  "F1bd' ≡,b) F1 set|"
  F1set1_bd_incr: "∧as
 by (rule ordLess_ordLeq_trans[OF F1.set_
  F1set2_bd_inc
 by (rule ordLess_ordLecorollary min_algs1: "i \in Field SucFbd 🚫 (m s1 s2) i))"
  F1set3_bd_incr: "∧x. |F1set3 x| <o F1bd'"
 by (rule ordLess_ordLeq_trans[OF F1.set_bd(3) ordLeq_csum1[OF F1.bd_Card_order]])

  F1bd'_Card_order = Card_order_csum
  F1bd'_Cinfinite = Cinfinite_csum1[OF F1.bd_Cinfinite]
  F1bd'_Cnotzero = Cinfinite_Cnotzero[OF F1bd'_Cinfinite]
  F1bd'_card_order = card_order_csum[OF F1.bd_card_order card_of_apply (ruletrans)

  "F2bd' ≡
java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
 by (rule ordLess_ordLeq_trans[OF F2.set_bd(1) ordLeq_csum1[OF F2.bd_Card min(minagss1 s2) 2) i ∪
  Fet2__incr: ""\Andx. |F2s x| <o '"
 by (rule ordLess_ordLapply (rule t trans)
 F2set3_bd_incr: "∧"
 by (rule ordLess_ordLeq_trans[OF F2.set_bd(3) ordLeq_csum1[OF F2.bd_Card_order]])

  F2bd'_Card_order = Card_order_csum
  F2bd'_C2bd'_Cibd'_Cinfinite = Cinfinite_csum1[OF F2.bd_Cinfinite]
  F2bd'_Cnotzero = Cinfinite_Cnotzero[OF F2bd'_Cinfinite]
 F2.bd_card_order card_of_card_ordern]

 Fbdhere "SucFbd ≡
  (rule al)+

  F1set1_bd: "|F1set1 x| <ole
 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 
 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 (rule F1.set_bd(3))
 apply (rule ordLeq_csum1)
 apply (rule F1.bd_Card_order)
 done

  F2set1_bd: "|F2set1 x| <o bd_F1 +c bd_F2"
  s_ooLe_s
 pplyy (rrule Fet_bd(1)
 apply _or Card_ordesum
 apply (rulotzeroo = = nfte_Cnnotzero[[O Fd'_infinite]
 donene

 et2_bd:"Fset2 xx| <bd_F1 
 apply (ule ordLLess_ordLeq_trans)
 apply (rule F2.set_bd(2))
java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
 done
  ScFd_adodr=crSc_Cr_rdr[F Card_ordercm

  F2set3_bd: "|F2 worl_Sud =rorder_welucFbd_ar_order]
 apply(rl odss_rdeq_trans)
 
 apply(rule e ordeq_csum2)
 apply (rule F2.bd_Card_order)
 

  SucFbd_Card_order = cardSuc_Card_order[OF Card_order_c]
  SucFd_Cnfiite Cinfite_cardu[FCnie_csum1OF Fbd'_Cie]]
  SucFbd_Cnotzero = Cinfinite_Cnotzero[OF SucFbd_Ci
  worel_SucminG2 As1A2 i \equiv> (∪j ∈ underS SucFbd i. snd (As_s22 ))"
  ASucFbd_Cinfinite = Cinfinite_cexp[OF ordLeq_csum2[OF Card_order_ctwo] SucFbd_"min_H s1 s2 As1_As2 i 🚫),,


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

(* These are algebras generated by the empty set. *)
abbreviation min_G1 min_alg1
  "min_G1 As1_As2 i ≡j ∈

abbreviation m
  "min_alg2s2i ∈  (min_algs i))

abbreviation min_H Field SucFbd ==>
  "min_Hs1 As1_A i ≡ iffD2)
    (min_G1 A i ∪ :: 'aset) (minG1 A i) min_G2As1_As2i))
    min_G2 As1_ i ∪ i) ( As1_As2 i))))

abbreviation mia
  "min_algs

definition min_alg1 where
  "min_alg1rrefl)

definition min_alg2 where
  "min_alg2

lemma:
  "<> Field SucFbd ==>
  apply (rule fun_cong[OF wo_rapp(drule bspec)
  apply (ruleiffD)
   apply (rul SUP_cong)
   apply rule wo_re.adm_[OF worel_SucFbd])
  apply (rul allI)+
  apply ((rule )

  apply (rule if
   apply (rule prod.inject)
  ule conjI)

   apply (rule arg_cong2[of _ _ _ _ "applyuleefl
    apply (rule
     ly
    ply
     apply assumption
    apply (erule

   apply (rule image_cong)
    apply (rule arg_cong2[of _ _ _ _ "F1in UNIV"])
     apply (rule
      applye l)
     apply (druleinField SucFbd ==> fst (min_algs  i)=
      apply assumption
     apply (erule( UNIV min_algs ) ) min_G2 s1 ))java.lang.StringIndexOutOfBoundsException: Index 77 out of bounds for length 77
    applyrule)
     apply (rule refl)
    apply (drule bspec)
     pply
    apply (rule)
   apply( refl

  apply (
    ule
    ply
   apply (drule
     nI1) le
   apply (erulen

  apply (rule image_cong)
   apply (rule arg_cong2[of _ _ _ _ "F2in UNIV"])
    apply (rule SUP_cong
     apply (rule refl)
    apply (druletacticrtac @{context} @{thm iffD2[OF meta_eq_to_obj_eq[OF relChain_def]]} 1›apply
     apply assumption
    apply (erule arg_cong
   apply (rule
    pply (rule refl)
   \Longrightarrow∃y ∈ y ∧ SucFbd) ∧ y ∧ SucFbd)"
    apply assumption
   apply (erule arg_cong)
  apply (e ir_ubet)
  done

corollary min_algs1: pplrleI)
  min_G1 (min_algs s1 sappr)
    s1 ` (F1in UNIV (min_G1 (a rle cnj
  apply (rule trans)
   apply (erule arg_cong[OF min_algsapp(rulesec)
  apply (rule fst_conv)
  done

corollary min_algs2: "i ∈ Field SucFbd ==>2
  min_G2EE+
    s2
  apply (rule transapply
   apply( arg_cong])
  apply        ( min_algs_mono1
  

lemma: "relChain SucFbd (%i. fst (min_algs1 s2 i))"
   meta_eq_to_obj_eq]]}1\close)
  apply (rule)+
  apply (rule'_Card_order
  apply (rule case_split )
   apply (rule xt1apply rule)(TRY
    applyrule)
    apply (erule)
    rule)
   apply ( (rule)
   apply rule)
    apply  ordLeq_csum1
     
   apply assumption)
  apply (rule applyrule)
  apply (drule)
  apply (erule)
     erule

lemma: "relChain SucFbd (%i. snd (min_algs s1 s2i))"
  apply (tactic ‹ (ruUnI2)
  apply rule )+
  apply( impI)
  apply (rule case_split)
   apply (rule xt1(3))
     rule)
    apply (erule FieldI2)
   apply (rulejava.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
   apply (rule UnI1)
   apply (rule UN_I)
    apply ( underS_I)
    apply assumption
   apply assumption
  
  apply drulenotnotD
  apply (erule arg_cong)
  done

lemma SucFbd_limit
 \<LongrightarrowrulecardSuc_UNION_Cinfinite
  apply
  apply rule rev_mp)
   apply( Cinfinite_limit_finite
     apply (rule F2bd_Card_order)
     apply (rule finite.insertI)
     applyrule finite.emptyI)
     erule insert_subsetI
    apply (erule insert_subsetI)
    applyruleempty_subsetI)
   apply ruleSucFbd_Cinfinite)
  apply( impI)
  apply (erule bexE)
  apply (rule bexI)

   apply (rule conjI)

    apply ( bspec)
    ly(rule insertI1)

   apply (erule bspec)
   apply (rule insertI2
   apply (rule insertI1)
   assumption
  done

lemma: "algmin_alg1 s1 s2) ( s1 s2) s1 "
  apply(tactic \<open>rtac @context ({thm alg_def}RS iffD2 \<lose)
  apply ulenjI
   apply (rule ballI)


   applyrule ordIso_ordLeq_trans
    apply (rule cardSuc_UNION_Cinfinite)
       apply (rule Cinfinite_csum1) (*TRY*)
       applyUNION_Cinfinite_bound
      apply min_algs_mono1
     erule[ _ [OF]]
    apply (rule rule)
      ( ordLess_imp_ordLeq F1set2_bd_incr
    apply (rule ordLeq_csum1SucFbd_ASucFbd
    applyapply( ballI

   apply(bexE
    apply (cardSuc_UNION_Cinfinite
       apply (rule erule)
       apply (apply assumption
      apply ( min_algs_mono2
     apply(erule[OF[OF]java.lang.StringIndexOutOfBoundsException: Index 65 out of bounds for length 65
    apply(rule)
     apply rule])
    apply ( csum_mono2
    apply ( F1bdjava.lang.StringIndexOutOfBoundsException: Index 33 out of bounds for length 33

   apply (rule bexE)
    apply (rule SucFbd_limit)
    apply (erule conjI)
    applyjava.lang.StringIndexOutOfBoundsException: Index 20 out of bounds for length 20
    rule [OF]]
   apply (rule
    applyerule)
    apply( thin_rl
    apply ( thin_rl
    apply (erule thin_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 (rulerule)
    applyassumption
   apply (rule UnI2)
   apply (rule image_eqI)
    apply (rule refl)
   apply (rule CollectI)
   apply (drule asm_rl <>FIRST
   apply (erule thin_rl)
   apply (erule thin_rl)
   apply (erule)+

   apply rule)
    apply assumptionruleCard_order_csum

   apply (applyrule)
    apply erule)
    apply (rule subsetI(ruleASucFbd_Cinfinite)
    apply (rule UN_I F1bdCard_order
     apply erule underS_I
     apply assumption
    apply assumption ( Card_order_csum

    rule'_Cnotzero
   apply ( UN_upperOFunderS_I]java.lang.StringIndexOutOfBoundsException: Index 38 out of bounds for length 38
   apply assumption

(**) ( ordIso_ordLeq_trans)

  apply erule)
  apply (erule conjE

  apply rule bexE
   apply (rule cardSuc_UNION_Cinfinite)
      apply (rule Cinfinite_csum1) (*TRY*)
       rule'Cinfinite
     apply (rule min_algs_mono1)

    applyerule[OFequalityD1OF min_alg1_def]])
   apply (ruleassumption
    apply (rule)
   apply (rule ordLeq_csum2)
   apply (rule F2bd allE

  apply (ruleerule)
   apply (rule)
      apply       (erule underS_Field
      apply (ruleF1bd_Cinfinite)
     apply (rule min_algs_mono2)

    apply (erule subset_trans[OF
   apply rule)
    apply (rule
   apply (rule)
   apply (rule F2bd'_Card_order)

  apply (rule bexE ordLeq_transitive
   apply rule)
   apply (erule (rule ordLeq_transitive)
   apply assumption
  apply (rule[OF[OF]])
  apply (rule UN_I )
   apply (erule ( csum_cinfinite_bound
   apply (erule thin_rl)
   apply (erule thin_rl)
   apply(erule thin_rl
   apply (erule thin_rl)
   apply( thin_rl)
   apply (erule thin_rl) (* m + 3 * n *)
   apply assumption
  apply (rule)
   apply (rule equalityD2)
   apply (rule min_algs2)
   apply assumption
  apply (ruleapply(rule ballI
  apply( image_eqI
   apply (applydrule)
  apply (rule)
  apply (rule( mp
   apply assumption

  apply (erule thin_rl)
  apply (erule thin_rl)
  apply (erule thin_rl)
  apply (eruleapply ( UNION_Cinfinite_bound
  apply (rule conjI)
   apply (erule subset_trans)
   apply ( UN_upper
   apply (erule underS_I (rule)
   apply assumption

  apply (erule subset_trans)
  apply (rule UN_upper)
  apply (erule underS_I)
  apply assumption
  done

lemmas SucFbd_ASucFbd)
    ordLess_ctwo_cexp(drulemp
    cexp_mono1apply( underS_Field
    OF SucFbd_Card_order]

lemmajava.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
  fixes s1 :: "('a, 'b, 'c) F1 ==>
  shows "i java.lang.NullPointerException: Cannot invoke "String.equals(Object)" because "brackoff" is null
  ( fst  i))| <> (SucFbda ASucFbd_type ∧rel
  apply (rule well_order_induct_imp[of _ "%i. ( |fst (min_algs s1 s2 i)| ≤o ASucFbd ∧ |snd (min_algs s1 s2 i)| ≤o ASucFbd )", OF worel_SucFbd])
  apply (rule impI)
  apply (rule conjI)
   apply (rule ordIso_ordLeq_trans)
    apply (rule card_of_ordIso_subst)
    apply (erule min_algs1)
   apply (rule Un_Cinfinite_bound)

     apply (rule UNION_Cinfinite_bound)

       apply (rule ordLess_imp_ordLeq)
       apply ruleordLess_transitive
        apply (rule card_of_underS)
          (rule)
        apply assumption
       apply (rule SucFbd_ASucFbd)

      apply (rule ballI)
      apply (erule allE)
      apply drule)
       apply (erule underS_E)
      apply (drule mp)
       apply (eruleunderS_Field
      apply (erule ASucFbd_Cinfinite
      apply assumption

     apply (rule ASucFbd_Cinfinite)

    apply (rule ordLeq_transitive)
     
    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 F2bd
             apply assumption
            apply (rule SucFbd_ASucFbd)

           apply (rule ballI)apply (ule
           apply (erule rule)
           apply (drule mp)
            apply (erule underS_E)
           apply (drule (ule
            apply (erule underS_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 card_of_min_alg1
             ()
            apply assumption
           apply (rule SucFbd_ASucFbd)

          apply (rule ballI ordIso_ordLeq_trans
          apply (erule rule[ min_alg1_def
          apply (drule mp)
           apply (erule underS_E)
          apply (drule
           apply (erule underS_Field)
          apply (erule f_Field_ordIso
          apply assumption

         apply (rule ASucFbd_Cinfinite)

        apply (
       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
       apply (rule ordIso_transitive)
        apply (tactic ‹BNF_Tactics.mk_rotate_eq_tac @{context}
 (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 (rule ASucFbd_Cinfinite)

pply
        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_algs2)
 apply (rule Un_Cinfinite_bound)

 apply (rule UNION_Cinfinite_bound)

  (ruleordLess_imp_ordLeq)
 apply (rule ordLess_transitive)
 apply (rule card_of_underS)
 apply (rule SucFbd_Card_order)
 apply assumption
 apply (rule SucFbd_ASucFbd)

 ply (r 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 ordLeq_transitive)
 apply (rule card_of_image)
 apply (rule ordLeq_transitive)
 apply (rule F2.in_bd)
 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 (rule ordLess_transitive)
 apply (rule card_of_nderS)
 apply (rule SucFbd_Card_order)
 apply assumption
 apply (rule SucFbd_ASucFbd)

 applyapply (erule u (erulunderS_E)
 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 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)
 apply (erule underS_Field)
 apply (drule mp)
 apply assumption

 apply (rule ASucFbd_Cinfinite)

 apply eruleunde
 apply (rule card_of_Card_order)
 apply ( apply (erule conjE+

 apply (rule F2bd'_Card_order)
 apply (rule ordIso_ordLeq_trans)
 apply (rule cexp_cong1)

 apply (rule ordIso_transitive)
 apply (rule csum_cong1)
 apply (rule ordIso_transitive)
 apply (tactic ‹
 (rtac @{context} @{thm ordIso_refl} THEN'
 FIRST' [rtac @{context} @{thm card_of_Card_order},
 rtac @{context} @{thm Card_order_csum},
 rtac @{c (erule al)
 @{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 (rule ordLeq_transitive)
 apply (rule ordLeq_csum1)
 apply (tactic ‹FIRST' [rtac @{context} @{thm Card_order_csum}, rtaapply (erule con)+
 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 ASucFbd_Cinfinite)
 done

  card_of_min_alg1:
 fixes s1 :: "('a, 'b, 'c) F1 ==> 'b" and s2 :: "('a, 'b, 'c) F2 ==> 'c"
 shows "|min_alg1 s1 s2| ≤
 apply (rule ordIso_ordLeq_trans)
 apply (rule card_of_ordIso_subst[OF min_alg1_def])
 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_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 ==> 'b" and s2 :: "('a, 'b, 'c) F2 ==>
 shows "|min_alg2 s1 s2| ≤o (ASucFbd :: 'a ASucFbd_type rel)"
 apply (rule ordIso_ordLeq_trans)
 apply (rule card_of_ordIso_subst[OF min_alg2_def])
 apply (rule UNION_Cinfinite_bound)

 apply(rule ordIso_ordLeq_tr)
 apply (rule card_of_Field_ordIso)
 apply (rule SucFbd_Card_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

  least_min_algs: "alg B1 B2 s1 s2 ==>
 i ∈ Field SucFbd ⟶
 fst (min_a s1s2 i) ⊆ B2"
 apply (rule well_order_induct_imp[of _ "%i. (fst (min_algs s1 s2 i) ⊆ B1 ∧ snd (min_algs s1 s2 i) ⊆ B2)", OF worel_SucFbd])
 apply (rule impI)
 apply (rule conjI)
 apply (rule ord_eq_le_trans)
 apply (erule min_algs1)
 apply (rule Un_least)
 apply (rule UN_least)
 apply (erule allE)
 apply (drule mp)
 apply (erul apply assumpt
 apply (drule mp)
 apply (erule underS_Field)
 apply (erule conjE)+
 apply assumption
 apply (rule image_subsetI)
 apply (erule CollectE conjE)+
 apply (erule alg_F1set)

 apply (erule subset_trans)
 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 (erule subset_trans)
 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 ord_eq_le_trans)
 apply (erule min_algs2)
 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 (eru apply (pply (rule ord_eq_le_tras[OF min_alg1_de])
 apply assumption
 apply (rule image_subsetI)
 apply (erule CollectE conjE)+
 apply (erule alg_F2set)

 apply (erule subset_trans)
 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 (erule subset_trans)
 
 apply (erule allE)
 apply (drule mp)
 apply (erule underS_E)
 apply (drule mp)
 apply (erule underS_Field)
 apply (erule conjE)+
 apply assumption
 done

  least_min_alg1: "alg B1 B2 s1 s2 ==>
 apply (rule ord_eq_le_trans[OF min_alg1_def])
 apply (rule UN_least)
 apply (drule least_min_algs)
 apply (drule mp)
 apply assumption
 apply (erule conjE)+
 apply assumption
 done

  least_min_alg2: "alg B1 B2 s1 s2 ==>
 apply (rule ord_eq_le_trans[OF min_alg2_def])
 apply (rule UN_least)
 apply (drule least_min_algs)
 apply (drule mpmp)
 apply assumption
 apply (erule conjE)+
 apply assumption
 

  mor_incl_min_alg:
 "alg B1 B2 s1 s2 ==>
 mor (min_alg1 s1 s2) (min_alg2 s1 s2) s1 s2 B1 B2 s1 s2 id iappl (e least)
 apply (rule mor_incl)
 apply (erule least_min_alg1)
 apply (erule least_min_alg2)
 done

  ‹

 ‹
  the initial algebra carrier:›

  'a1 F1init_type = "('a1, 'a1 ASucFbd_type, 'a1 ASucFbd_type) F1 ==> 'a1 ASucFbd_type"
  'a1 F2init_type = "('a1, 'a1 ASucFbd_type, 'a1 ASucFbd_type) F2 ==> 'a1 ASucFbd_type"

  'a1 IIT =
 "UNIV ::
 
 by (rule exI) (rule UNIV_I)


 ‹Initial Algebras›

  II :: "'a1 IIT set" where
 "II ≡ s1 s2"
  str_init1 where
 "str_init1 (dummy :: 'a1)
 (y::('a1, 'a1 IIT ==> 'a1 ASucFbd_type, 'a1 IIT ==> 'a1 ASucFbd_type) F1)
 (i :: 'a1 IIT) =
 fst (snd (Rep_IIT i))
 (F1map id (λ
  str_init2 where
 "str_init2 (dummy :: 'a1) y (i :: 'a1 IIT) =
 snd (snd (Rep_IIT i)) (F2map id (λf. f i) (λf. f i) y)"
 breviation car_init1 where
 "car_init1 dummy ≡×
  car_init2 where
 "car_init2 dummy ≡ min_alg2 (str_init1 dummy) (str_init2 dummy)"

  alg_select:
 "∀i ∈ II. alg (fst (fst (Rep_IIT i))) (snd (fst (Rep_IIT i)))
 (fst (snd (Rep_IIT i))) (snd (snd (Rep_IIT i)))"
 apply (rule ballI)
 apply (erule CollectE exE conjE)+
 apply (tactic ‹
 unfolding fst_conv snd_conv Abs_IIT_inverse[OF UNIV_I]
 apply assumption
 done

  mor_select:
 "[i ∈
 mor (fst (fst (Rep_IIT i))) (snd (fst (Rep_IIT i)))
 (fst (snd (Rep_IIT i))) (snd (snd (Rep_IIT i))) UNIV UNIV s1' s2' f g] ==>
 mor (car_init1 dummy) (car_init2 dummy) (str_init1 dummy) (str_init2 dummy) UNIV UNIV s1' s2' (f ∘
 apply (rule mor_cong)
 apply (rule sym)
 apply (rule o_id)
  apply ( sym)
 apply (rule o_id)
 apply (tactic ‹"str_init1 (dummy:: 'a1)
 apply (tactic ‹ ASucFbd_typ, 'a1 I \Rightarrowa1 ASuc) F1)
 apply (tactic ‹rtac @{context} (@{thm mor_def} RS iffD2) 1›)
 apply (rule conjI)

 apply (rule conjI)
  rule bal)
 apply (erule bspec[rotated])
 apply (erule CollectE)
 apply assumption

 apply (rule ballI)
 apply (erule bspec[rotated])
 apply (eruleColle)
 apply assumption

 apply (rule conjI)
 apply (rule ballI)
 apply (rule str_init1_def)

  (rule ballI)
 apply (rule str_init2_def)

 apply (rule mor_incl_min_alg)
    (*alg_epi*)
  apply (erule thin_rl)+
  apply (tactic ‹rtac @{context} (@{thm alg_def} RS iffD2) 1›f i) (\lambdafi yjava.lang.StringIndexOutOfBoundsException: Index 77 out of bounds for length 77
  apply (rule conjI)
   apply (rule ballI)
   apply (erule 
   apply (rule CollectI  \equiv min_alg1 dummystr_init2
   apply (rule ballI)
   apply (frule bspec[OF alg_select])
   apply (rule ssubst_mem[OF str_init1_def])
   apply (erule alg_F1set)

    apply (rule ord_eq_le_trans)
     apply (rule"car_init2 dummy\equiv min_alg2 str_int1dumm) (strinit2 dummy)"
    apply (rule subset_trans)
     apply (erule image_mono)
    apply (rule image_Collect_subsetI)
    apply (erule bspec)
    apply assumption

   apply (rule ord_eq_le_trans)
    apply (rule F1.set_map(3))
   apply (rule subset_trans)
    apply (erule image_mono)
   apply (rule image_Collect_subsetI)
   apply (erule bspec)
   apply assumption


  apply (rule ballI)
  apply (erule CollectE conjE)+
  apply (rule CollectI)
  apply (rule ballI)
  apply (frule bspec[OF alg_select])
  apply ( rule[ str_init2_def)
  apply (erule alg_F2set)

   apply (rule ord_eq_le_trans)
    apply (rule F2.set_map(2))
   apply (rule subset_trans)
    apply (erule image_mono)
   apply (rule image_Collect_subsetI)
   apply (erule bspec)
   apply assumption

  apply (rule ord_eq_le_trans)
   apply (rule F2.set_map(3))
  apply (rule subset_trans)
   apply (erule image_mono)
  apply (rule image_Collect_subsetI)
  apply (erule bspec)
  apply assumption
  done

lemma init_unique_mor:
  "[a1 ∈ car_init1 dummy; a2 ∈ car_init2 dummy;
    mor (car_init1 dummy) (car_init2 dummy) (str_init1 dummy) (str_init2 dummy) B1 B2 s1 s2 f1 f2;
java.lang.NullPointerException: Cannot invoke "String.equals(Object)" because "brackoff" is null
  f1>f2 a2 = g2 a2"
   t3]
   apply (erule prop_restrict
   
   apply (rule least_min_alg1)
   apply (tactic <  
   apply (rule conjIerule)
    apply (rule ballI)
    apply (rule CollectI)
CollectE
    apply rule     (uleconjI

     applyulelg_F1setmin_alg
      ulef =
      pplyjava.lang.StringIndexOutOfBoundsException: Index 35 out of bounds for length 35
     apply (rulerans
     pply ollect_restrictapply ules)

    apply (rule
     apply
     apply (rule
      apply  e njE
       apply (rulejava.lang.StringIndexOutOfBoundsException: Index 6 out of bounds for length 6
      uleeotI
     apply (rule CollectI)
     
      apply assumption
     apply ( conjI
      apply assumption
     ssumption

    applye wit1ule
     apply ( notI
       apply (rule refl)
      apply (erule prop_restrict)
      apply assumption
     apply (conjI_subsetIetI
     apply assumption

    apply (rule
    apply (erule morE1)
    apply (rule subsetD)
     apply (rule ( allE
      apply (rule(forall F2in)
   (p
    applyectIrulenderS_Field
    apply (    ( conjE
     apply assumption
    apply (rule conjI)
     apply assumption
    apply assumptionptionjava.lang.StringIndexOutOfBoundsException: Index 6 out of bounds for length 6

   apply (rule ballI)
   apply(ule
   apply (erule CollectEption
   Ijava.lang.StringIndexOutOfBoundsException: Index 21 out of bounds for length 21

    java.lang.StringIndexOutOfBoundsException: Index 6 out of bounds for length 6
     apply (erulelemma
     apply( Collect_restrict
    ly( )
    apply (ruleleD)

   apply (rule_ply
    apply (erule)
    apply (ruleleply
     apply (rule least_min_alg2: alg1<Longrightarrow min_alg2 s1 s2 ⊆lbrakkmor B1 B2 s1 s2 B1''''f;
   pply( lect_restrictestrictict
     t_restrict
    apply (ruleapply ( least_min_algs
    apply (rule conjI)
     apply assumption
    apply (rule conjI)
     plyjava.lang.StringIndexOutOfBoundsException: Index 1 out of bounds for length 0
    y    <

  pply
       drule
      apply rule
     apply      applyly
     applylyion ‹" o b the type (for our particular const)
 ()
 apply aply (e bspnit_ype = "('a1, 'a1 ASucFbd_type, 'a1 ASucbddype) F1 \<ightarrow'

 apply (rule sym)
 apply (erule morE2)
 apply (rule subsetD)
 y (r u 2n_mmnoo23)
 apply (rule Collect_restrict)
 plyy (rueCol_mo2))
 apply (rule CollectI)
 
 apply astin
 apply (rule conjI)
 apply supon
 apply assumption


 apply (erule thin_rl)
 apply (rule Col
 apply (rule least_min_alg2)
 applyby (rue ordLsorLeq
 apply (rule c)
 apply (rule ballI)
 apply (rule nfinit = Cinfnite_ce__csumO 2bd_ifnt]
 apply (erule CollectE con SucFbd wheree"uFd\equivcardSuc (F1bd' +c F2bd')"
 apply (rul coI)

 done
 apply (erule subset_trans)
 le Collect_erestrrit)
 apply (erule subset_trans)
 apply (r Fset3_bd: "|F1set3| <o  map:ule ordLeq_csu1

 apply (rule trans)
 apply (erule morE1)
 apply (rle ssuetD
 pply (r (rl Fi> Fi UNIV (car_ini1dummy) (c (ccnit dumy).
 pply (rule Coletrsrc)
 apply (rrelFbd rd_ore_o_rel[SucFbb_Card_ore]
 by (rule exI) (rule UNIV_I)
 apply (rul conjI)
 apply assumption
 apply (r algebras generated by the empty set. *)
 \<openInitial undeS uFbd i. fst (As2 j)"
java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0

java.lang.StringIndexOutOfBoundsException: Index 22 out of bounds for length 21
 apply (rule arg_cong[OF F1.map_cong0])
 apply dummy: 'a)
 apply (er init_1, 'a1 IIT \Rightarrow> ''a1 ASu
 fst ((snRep_Ii))
 rop_restrict)trc)
  aassumption"stst (dummy :: '1)ap (ruleballI)

 apply ule sym))
  apapply (ru allI)+
 apply" (rule onI
 apply(tact ‹ (snd (Rep_ i)))"
 
 apply (rule Collectg)
 apply (rule Collec appl(tactic \openhyp_subst_tac p (ule CollectI)
 apply (rule conjI)
 
 apply (rule co aapply (rule conjI)
 apply assumption
  apass

 apply (rulball
 apply (rule CollectI)
 apply (erule Col CollectE conjE)+
 

 apply (rule alg_F2set[OF alg_min_alg])
 apply (erule subset_trans)
 apply (rule Collect_restr)
 apply (erule subset_trans)
 apply (rule Collect_rest

 
 pply (er (erule morE2)
 apply (rule subsetD)
 apply (rule )
 apply (rule Collect_r min_algs1: "i \inield uFd\Longrightarrow(mialgs1 2c apply ac 🚫
 ollect_restrict
 apply (rule CollectI)
 (ul conjI)
 
  ( refl)
  apply assupt (rule conjI)
 apply asumption

 apply (rule tranasumpion
 g[OF F2.map_cong0]
 ion
 apply (ele prop_tric)
  aassumptio
 apply (erule prprstri)
 applyass

 apply (rule sym)
 apply (erule morE)
 apply (rule subsetD)
  ue su)
 ct_restrict) rictt)
  (rule id_apply)
 lemma min_algs_mo
 apply (rule conjI)
 apply assumpti
  (rule conjI)
 apply assumption
 apply assumpt
 done

  close where
  (rucnj[OF subsetIV])
 \> ∈)
 (\forall> ∈) 
 (∀(rulconjI[I[[OF su

 it_induct: clos dummyl pply ((erule Col cco)+
 (∀ B1 ≠
 apply (rule conjI)
 apply (rule (r ballI)
 apply (erule prop_ (fre algF1se
 apply (rule least_min_alg1)
java.lang.NullPointerException: Cannot invoke "String.equals(Object)" because "brackoff" is null

 apply (rle balI)
 apply (rule ballI)
 apply (rule C(rule CollectI)
 apply (erule CollectE conjE)+
 apply (rule conjI)

 
 apply (erule subset_trans)
 apply(ue olct_restric)
 apply (erule subset_trans)
 apply (rule Collect_restrict)

 apply (rule mp)
  apply erule spec)  (rule finiteins)
 apply (ru
 apply (rule conjI)
 apply asse Collect_restrict)
 apply (rule conjI)
 apply (erule (r emptempty
 apply (rule Collect_restrict)
 apply (erule subset_trans)
 apply rule Cubset_t)  assumption   ⊆ B2'] mor B1 B2 s1 s2 B1' B2' s1 s2 id id"

 apply (rule conjI)
 apply (rule ballI)
 apply (erule prop_restrict)
 apply assumpo
 apply (rule ballI)
 apply (erule prop_restrict)
  aumption


  (rule ins)
 apply (rule CollectI)
 apply (erule CollectE conjE)+
 applyply y assupio

 njI)
 apply (erule subset_trans)
 apply (rule Collect_restrict)
 apply (erule subset_trans)
 apply (rule Collect_restrict)

 
 apply (erule bspec)
 apply rule Col)
 apply (rule con cononjI
 apply assumption
 ply (rle carduc
 apply (rule Cinfinite_csum1) (*TRY
 apply (ruleapply rule ssapplypply(rue I) rule conjI)
 y (eeru subset_rans)
 ollect_restrict)let_rtra

 apply (e aa(erule prop_resstrit)
 apply (rule ballI)
  e passuption
 apply assumptimaapply (rule l)
 pply (ruly (reblI
 apply (erule prop_restrict)
 apply assumption

 apply (rule ballI)
 apply (eruaapply (rule apply (rle F2.set(2)
 apply (rule least_min_alg2)
 ntext} (@thm ag_def} R

 apply (rule coonjIjI
 apply (rule ballI)
 pply ly (rule CollectI
 apply (erule ollapp p(rrle 1bbd'_Card_oder)
 apply assumption

 apply applyassumption
 apply (erule subset_trans) '= ; mor B1B s1
 apply (rule Collec (erule thin_rl)
 apply (erule subset_tdone
 apply (ru (ruleCollect_restr)

 apply (rule mp)
 apply (erule bspe
 apply (rule CollectI)
 
 nit_unique_mor:
 apply (rule conjI)
 apply rule m min_algs1)
 apply (ru (rule Col)
 apply (rule UnI2 apply (rule lI)
 restrict)

 apply (rule conjI)
 apply (rule ballI)
 apply (erule prop_restrict)
 ply assumption
 apply (rule ballI)
 apply (erule prop_re 'a1ASucFbdtypee =apply (drule asdrul asmrl)
 apply assumption


 apply (rule ballI)
 apply (rule CollectI)
 apply
 apply (rule conjI)

  app (rule conjI) 
 apply (erule subset_Fl apy rule )
 apply (rule Collect_restrictF2 +c |UNIV :: (bd_type_F2,2, d_t_type_2, b pply (rule subsetI)
 by rule Ls_ ly tactic \apply)
 apply (rule Collect_restrict)

 apply (rule ballI)
 apply (erule bspec)
 apply (rule CollectI)
 apply (rule conjI)
 apply assumption
 apply (rule conjI)
 apply (erule s subset_trans)
 apply (rule Collect_restrict)
 apply applap (rul ordLdSuc_UNION_CinCinfin
 

 apply (rule conjI)
 apply (rule ballI)
  (rule min_
 pplyassmption
 apply y (rule ballI)
 apply (erule p "|F1set3 o bd_F1+c bd_F2"
 apply assumption
 done


  \\

 y (rue F1.bdbD)

 apply (rule ex_in_conv)
java.lang.StringIndexOutOfBoundsException: Index 37 out of bounds for length 36
 apply(ruldone
 lemma F2set3b: "|Fset33
 done

  (overloaded) 'a1 IF2 SucFbdinfinit[OF SucFbd_Cinniite]
 apply (rule iffD2)
  ass
 apply (rule conjunct2)
  ii \equiv (\j n
 apply (rule alg)
 done

  ctor1 where "cto = bs_IF1 o str_init1 und a (rule rrefl)
  As1_A i \union s2 ` (F2in

  mor_R:
 "mor (UNIV :: 'a IF1 se "min_alg1 s11 s2 = (\nion < Field
 (car_nt a a assumptionjava.lang.StringIndexOutOfBoundsException: Index 19 out of bounds for length 19
 unfolding mor_def ctor1_def ctor2_def o_apply
 apply (rule conjI)

 apply (rule ballI)
 rule ta_eq_t_eq_to_ob_q
 apply (rule ballI)
 apply (u ep_IF2) apply (ay(rulCollct_resetrict)

 ))
 apply (rule ballI)
  (rule Abs_IF1_in)
 rule algalg assumpapply (eule und)
  assum
 setI)
  apply (rule refl)
  ord_eq_le_trans[OF F1.set_map(3)]))
 apply (apply (ru underS_I)
 apply (rule Rep_IF2)

 apply (rule ballI
 apply ((cexp_mono1[OF ordLeq_csum2[OF Card_or]],
 apply (rule alg_F2set[OF [OF alg])
 rule ord ord_eq_le_trans[OF F2.set_map(2)])
 apply (rule image_subsetI)
java.lang.StringIndexOutOfBoundsException: Index 31 out of bounds for length 23
 apply (rule ord_eq_le_trans[OF F2.set_map(3)])
 apply (rule image_subsetI) (drule bs a (rule s
 apply (rule Rep_IF2)
 done

  mor_Aapply (e apply (rCoCollect_restrict)))
 "mor (car_init1 undefined) (car_iapp (rule refl)
 (str_iundefined) (str_init2 undefin) UN
 unfolding mor_def ctor1_def ctor2_def o_apply
 apply (ruleconjIs)
 apply (rule conjI)
  apply (eru morE)
 UNIV_I)
 
 ply rul UN

 apply (u conjI)
 apply (rule ballI)
 apply (erule CollectE conjE)+
 apply (rule sym[OF arg_cong[OF trans[OF F1map_comp_id F1map_congL]]])
 apply (rule ballI[OF trans[OF o_apply]])
 apply (erule apply asssumption
 apply assumption
 apply (rule ballI[OF trans[OF o_apply]])
 pply (erule Abs_IF2_inverse[OF subsetD])
 apply aapply (tatc ‹restrict)tri)

 apply pp(ruleUnI1)
 apply (erule CollectE conjE)apply a aspply (rule csum_monno1))
 _cong[OF trrans[O F2ma
 apply (rule ballI[OF trans[OFlemma SuScFbd_lmt \lbrakkx1 ∈ Fiapply (rule csum_cinfini
 apply (eru(erule Abs_IF1_invese[OF subsetD]
 apply assumption
 apply apply (rul orr)
 rse[[OF subsetD])
 apply assumption
 done

  copy:
 "le inset1
 exists>f>f' g'. algB12' f' g' 🪙"alg (_alg1 s1 s) (minmin
 apply (rule exI)+
 apply (rule conj apply ale con)
 apply (tactic ‹
 apply (rule conjI)
 apply (rule ballI)
 apply (erule CollectE conjE)+
 apply ( (r rule subse)
 apply (rule equalityD1)
 apply (erule bij_betw_imp_surj_on[OF bij_betw_the_inv_into])
 apply (rule imageI)
 apply (eru 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)
 apply (erule bbij_betw_imp)
 apply (rule ord_eq_le_trans)
 apply (rule F1.set_map(3))
 apply( appruleSucFb)
  (erule im)
 apply(erule thin_rl)
 apply (erule bij_be)

  assumption
 apply (erule CollectyD2)
 apply (rule subsetD)
 apply (rule equalityD1)
 apply (erule bij_betw_imp_surj_on[OF b])java.lang.StringIndexOutOfBoundsException: Index 65 out of bounds for length 64
 apply (erule (rule conjI)
 apply (erule alg_F2set)
 apply (rule ord_eq_le_trans)
 apply (rule F2.set_t y (rle con)
 apply (rule subset_trans)
 apply (erule image_mono)
 apply (rule equalityD1)
 apply (erule bij_betw_imp_surj_on)
 apply (rule ord_eq_le_trans)
 apply (rule F2.set_map(3))
 apply (rule subset_trans)
 apply (erule image_mono)
 apply (rule equalityDapply (rule (rule CCollect_)
 apply (erule bij_betw_imp_surj_on)

 apply (tactic ‹
 apply (rule onjI))
 apply (rule conjI)
 apply (erule bij_betwE)
 apply (erule bij_betwE)

 apply (rule conjI)
 apply (rule ballI
 apply (erule CollectE conjE)+
 apply (erule f_the_inv_into_f_bij_betw)
 apply (erule alg_F1  (ru(rue ord apply (errule subset_trans[OF _ equalityD1[OF min_alg2ef]])
 apply (rule ord_eq_)
 apply (rule F1apply ( (tactic ‹ apply (rule ordLeq_cexp1)
 apply (ru subs
 apply (erule image_mono)
 apply (rule conjI)
 apply (erule bij_)
 apply (rule ord_eq_le_trans)
 apply (ruleF1.set_map(3))
 apply (rule subset_trans)
 apply (erul image_mono)  (rule ASucFbd_Cinfinite)
 apply (rule equalityD1)
 apply (erule bij_betw_imp_surj_on)

 apply (rule ballI)
 apply (erule refl)
 apply (erlpply ( (rule F1bd'_Card_order)
 2
 ule q_le_trans)
 
 apply (rule s subset_trans)
 apply (erule image_mono)
 apply (rule equalityD1)
 apply (erule bij_betw_imp_surj_on)
 apply (rule oa (erule min_algs2)
 apply (ruleF2.set_map(3)
 apply (rule subset_trans)
 
 apply (rule equalityD1)
 apply (erule bij_betI)
 done

 
 "∃
 apply (erule allE)
 ex_bij_betw[OFapply (d(drule mp))
 apply (erule exE)+
 rulerv_mp)
 apply (rule copy[OF alg_min_alg])
 apply assumption
 apply assumption
 apply (rule impI)
 apply (erule exE conjE)+

 apply (rurule Collect_)
 apply (rule mor_comp)
 apply (rule mor_Repapply (rule ordL)
 apply (rule mor_select)
 apply(rule CollectI)
 apply (rule exI)+
 apply rule ordLeq_transitive)
 apply (rule refl)
 apply assumption
 unfolding fst_conv snd_conv Abs_IIT_inverse[O apply (rule csum_mono1)
 apply (erule mor_comp)
 apply (rule mapply (rule csum_cinfinite_bound)
 e ubset_UNIV)
 apply (rule subset_UNIV)
 done

  ‹

  where
 "fold s1 s2 ≡ SucFbd_ASucFbd)

  fold1 where "fold1 s1 s2 = fst (fold s1 s2)"
 definition ruleul conjI)jI)

  mor_fold:
 "mor UNIV UNIV ctor1 c (drubset_trn)
 unfolding fold1_def fold2_def
 apply (rule rev_mp)
 apply   Co
 apply
 apply (erule exE)
 apply (erule exE)
 apply (rule someI[of "%(f :: ('a IF1 ==> (rul Co)
java.lang.StringIndexOutOfBoundsException: Index 2 out of bounds for length 0
 apply (erule mor_cong[OF fst_conv snd_conv])
 done

  \<openML
 val fold1 = rule_by_tactic @{context}
 (rtac @{context}
 @{thm morE1[OF mor_fold]}

 val fold2 = rule_by_tactic @{context}
 (rtac @{context} @{thm CollectI} 1 THEN BNF_Util.CONJ_Warule card_of_Car)
 @@tl (eru(erule prop_estrictt)
 ›

  fold1:
 "(fold1 s1 s2) (ctor1 x) = s
 apply (rule morE1)
 apply (rule mor_fold)
 apply (rule CollectI)
 apply (rule conjI)
 bset_UNIV)
 apply (rule conjI)
 apply (rule subset_UNIV)
 apply (rule susubset)
 done

  fold2:
  x) = s) = s (F2map id (f
 apply (rule morE2)
 apply (rule mor_fold)
 apply (rule CollectI)
 apply (rule conjI)
 t_UNIVUNIVIV)
 apply (rule conjI)
 rule subset_UNIV)
 apply (rule subset_UNIV)
 done

  mor_UNIV: "mor UNIV UNIV s1 s2 UNIV UNIV s1' s2' f g \longleftrightarrow
 f o s1 = ' pply (rul CollucFbd_Cnotzero)
 apply (rule iapply (ru Card_order_csum)
 apply (rule conjI)
 apply (rule pply a assumption
 apply (rulepply (rule Collect_restrict)
 apply (rule o_apply)
 apply (rule trans)
java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
 apply (rule CollectI)
 apply (rule conjI)
 apply (rule subset_UNIV)
 apply (rule conjI)
 apply (rule subset_UNIV)
 apply (ruleonjI)
 apply (rule sym[OF o_apply])

 apply (rule ext)
  ::e con)
 apply (rule apply (ule cardu_rdd) 
 apply (rule trans)
 (rue orIo_ordLeq_trans)
 eotI)
 apply (rule conjI)
 apply (rule subset_UNIV)
 apply (rule conjI)
 apply ply (rule c
 apply (rule subset_UNIV)
 apply (rule sym[OF o_apply])

 apply (tactic ‹
 apply
 apply (rule conjI)
 apply (rule ballI)
 ly ue apl (r (rue ballI)
 apply (rule ballI)
 apply (rule UNIV_I)
 apply (erule conjE)
 apply (drule iffD1[OF fun_eq_iff])
 
 onjI)
 apply (rule ballI)
 apply (erule allE)+
 apply (rule trans)
 plyy (errule tn card_of_min_a_lg:
 apply (rule o_apply)
 apply (rule ballI)
 apply (erule allE)+
 apply (rule trans)
 apply (erul trans[OF sym[OF o_app
 apply (rule o_apply)
 done

  fold_unique_mor: "mor UNIV UNIV ctor1 ctor2 UNIV UNIV s1 s2
 f = fold1 s1 sapply (
 apply (rule conjI)
 ule surj_fun_eq)
 apply (rule type_definition.Abs_image[OF type_definition_IF1])
 apply (rule ords apply assuumption
 apply (rule conjunct1)
 apply (rule init_unique_mor)
 apply assumption
 apply (rule Rep_IF2)
 e mor_comp
 apply (ru pply e mp)
 apply assumption
 apply (rule mor_comp)
 apply (rule mor_Abs_IF)
 apply (rule mor_fold)

 apply (rule conj)
 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 (rule mor_comp)
 apply (rule mor_Abs_IF)
 apply assumptio
 apply (rule mor_comp)
 apply (rule mor_A apply (erule con_sub)
 apply (rule mor_fold)
 done

 fold_unique = f= fold_unique_[OF iffD2[OF mor_UNIV], OF con]

  fold1_ctor = sym[OF conjunct1[OF fold_unique_mor[OF mor_incl[OF subset_UNIV subset_UNIV]]]]
  fold2_ctor = sym[OF conjunct2[OF fold_unique_mor[OF mor_incl[OF subset_UN

  ‹

  ctor1_o_fold1 =
 trans[OF conjunct1[OF fold_unique_mor[OF mor_comp[OF mor_fold mor_str]]] fold1_ctor]
  ctor2_o_fold2 =
 trans[OF conjunct2[OF fold_unique_mor[OF mor_comp[OF mor_fold mor_str]]] fold2_ctor]

(* unfold *)
definition
definition  erule

ML ‹>tac @{context} (Thm.permute_prems 0 1 @{thm mor_comp}) 1›
ML ‹

 
 unfolding dtor1_def
 fold1)
 (apply (rule ballI)

  ctor2_o_dtor2: "ctor2 o dtor2 = id" pply ase conjI
 unfolding dtor2_def
 r2_o_fold2)
 done
 1 d"
 apply (rule ext)
  (ule tstr_t_min_alg1
 apply (rule trans[OF fun_cong[OF dtor1_def]])
 apply (rule trans[OF fold1])
 apply (rule trans[OF F1map_comp_id])
 apply (rrul trans[OFF1map_congL
 y (rul ballI)
 apply (rule trans[OF fun_concong[OF ctor1] id_apply])
 apply (rulee Rep_IF1)
 apply (rule trans[OF fun_conOF ctor2_o_fold2] id_apply])
 apply (rule (rule sym[OF id_apply])
 done

  dtor2_o_ctor2: "dtor2 o ctor2 = id"
 apply (rule ext)
 apply (rule trans[OF o_apply (rule ballI))
 
 apply (rule trans[OF fold2])
 sOF F2maap_comp_id])
 apply (rule trans[OF F2map_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 id_apply])
 done

 
  dtor2_app (rule t)
  ctor1_dtor1 = pointfree_idE[OF ctor1_o_dtor1]
  ctor2_dtor2 = pointfree_idE[OF ctor2_o_dtor2]

  bij_dtor1 = o_bij[OF ctor1_o_dtor1 dtor1_o_ctor1]
  inj_dtor1 = bij_is_inj[OF bij_dtor1]
  surj_dtor1 = bij_is_surj[OF bij_dtor1]
  dtor1_nchotomy = surjD[OF surj_dtor1]
  dtoriff = inje[Fijdor1]
 dtor1_cases =1_cses = exE_restri
  bij_dtor2 = o_bij[OF ctor2_o_dtor2 dtor2_o_ctor2]
  inj_dtor2 = bij_is_inj[OF bij_dtor2]
  surj_dtor2 = bij_is_surj[OF bij_dtor2]
  dtor2_nchotomy = surjD[OF surj_dtor2]
 
  dtor2_cases = exE[OF dtor2_nchotomy]

  bij_ctor1 = o_bij[OF dtor1_o_ctor1 ctopp (ru ballI
  inj_ctor1 = bij_is_inj[OF bij_ctor1]
  surj_ctor1 =lectE conjE
  ctor1_nchotomy = surjD[OF surj_ctor1]
  ctor1_diff = inj_eq[OF inj_ctor1aappy (ruleealg_F2_F2t[Oalg_])
  ctor1_cases = exE[OF ctor1_nchotomy]
 _r= objO tr2oco2cto apply (erule ubtrns)
  inj_ctor2 = bij_is_inj[OF bij_ctor2]
  surj_ctor2 = bij_is_surj[OF bij_ctor2]
  ctor2_nchotomy = surjD[OF surj_ctor2]
  ctor2_diff = inj_eq[OF inj_ctor2]
  ctor2_cases = exE[OF ctor2_nchotomy]  CollectE nj)+
 
  ‹

  rec1 where
 "rec1 s1 s2 = snd o fold1 (<ctor1 
 
 "rec2 s1 s2 = snd o fold2 (<ctor1 

  fold1_o_ctor1: "fold1 s1 s2 ∘
 by (tactic ‹rtac @{context} (@{thm alg_def} RS iffD2) 1›
  fold2_o_ctor2: "fold2 s1 s2 ∘
 by (tactic ‹

 apply (erle ubset_tran)
 F onjunt1[OF fold_uniqalg22, of s1 ss2])
 trans[OF o_assoc[symmetric] trans[OF arg_cong2[of _ _ _ _ "(o)", OF refl
 trans[OF fold1_o_ctor1 convol_o]]], OF trans[OF fst_convol]]
  apply (rule ra)
 trans[OF fold2_o_ctor2 convol_o]]], OF trans[OF fst_convol]]]]
 _unfolded F1ma_comp0[of id, unfolded id_o] F2mpcomp0[f unfolddee
 OF refl refl]
 st_rec2_pair=
 trans[OF conjunct2[OF fold_unique[OF
 apply (rule CllcI)
 trans[OF fold1_o_ctor1 convol_o]]], OF trans[OF fst_convol]]
 trans[OF o_assoc[symmetric] trans[OF arg_cong2[of _ _ _ _ "(o)", OF refl
  rans[[OFfol2_ctor_o]]] Ftranpp pply assumptption
 fold2_ctor, unfolded F1.map_comp0[of id, un apply (rule F11_mono23))
 OF refl refl]

  rec1: "rec1 s1 s2 (ctor1 x) = s1 (F1map id (<id,prest
 unfolding rec1_def rec2_def o_apply fold1 snd_convol'
 convol_expand_snd[OF fst_rec1_pair] convol_exand_snd[dO s_e2pair] ..

 (rule ballI)
 oldingling rec1_deffec2_def o_aply fold2 snd_convovol'
 convol_expand_snd[OF fst_rec1_pair] convol_expand_snd[OF fst_rec2_pair] ..

 rec_unique:
 
 g ∘
 unfolding rec1_def rec2 applyrul ClcI
java.lang.StringIndexOutOfBoundsException: Index 10 out of bounds for length 0
 apply (unfold convol_o id_o o_id F1.map_comp0[symmetric] F2.map_comp0[symmetric]
 F1.map_id0 F2.map_id0 o_assoc[symmetric] fst_convol)
 apply (erule arg_cong2[of _ _ _ _ BNF_Def.convol, OF refl])
 apply (erule arg_cong2[of _ _ _ _ BNF_Def.convol, OF refl])
  aply rule p)


 \Induction›

  capply erulesbsapply rueballI)
 "[∧
 ∧x. (∧
 
 apply (rule mp)

 apply (rule impI)
 apply (erule conjE)
 apply (rule conjI)
 (ue iffD1D[O r_o
 apply (erule bspec[OF _ Rep_IF1])
 apply (rule iffD1[OF arg_cong[OF Rep_IF2_inverse]])
 apply (erule bspec[OF _ Rep_IF2])
 apply euesbe_trans)

 apply (rule conjI)

 apply (dr pp rl ol_sc
 apply (erule thin_rl)
 apply (rule ballI)
 apply (rule impI)
 apply (rule iffD2[OF arg_cong[OF morE1[OF mor_Abs_IF]]])
 apply assumapp apply rule onjnI)
 applyapply assumpto
 apply (drule meta_spec)
 apply (drule meta_mp)
 apply (rule iffD1[OF arg_cong[OF Rep_IF1_inverse]])
 apply (erule bspec)
 apply (drule rev_subsetD)
 apply ule ply (rruebl)
 apply rule F.e_ap2)) assumptionuumptin
 apply (erule imageE)
 lyssmption
 apply (rule ssubst_mem[OF Abs_IF1_inverse])
 apply
 apply assumption
 apply assumption

 apply (drule meta_mp)
 [F arg_cggO Rep_IF2_ivers])
 
  apapply(ulep)no
 apply (rule equalityD1)
  pyul1ep3)
 apply (erule imageE)
java.lang.StringIndexOutOfBoundsException: Index 47 out of bounds for length 47
  aappyly (rule ssubsmem Abs_IF2_invrrse])
 apply (erule subsetD)
 apply assumption
 apply assumption

 apply assumption

 apply (erule thin_rl)
 apply (drule asm_rl)
 apply (rule ballI)
 applylemma d dtassuassumption
 apply (rule iffD2[OF arg_cong[OF morE2[OF mor_Abs_IF]]])
 apply assumption
 apply (erule CollectE conjE)+
 apply ul easec)
 apply (drule meta_mp)
 apply (rule iffD1[OF arg_cong[OF Rep_IF1_inverse]])
 apply (erule bspec)
 apply (drule rev
 apply (ruleequalityD1)
 apply (rule F2.set_map(2))
 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 (erule bspec)
 apply (drule rev_subsetD)
 apply (rule equalityD1)
 apply (rule F2.set_map(3))
 apply (erule imageE)
 apply (tactic ‹
 m[OF_F22_nverse])
 apply (erule subsetD)
 apply assumption
 apply assumption

 apply assumption
 done

  ctor_induct2:
 "[∧r subset_UNIV)
 (∧a b. a ∈
 ∧
 (∧a b. a ∈
 phi1 a1 b1 ∧done
 apply (rule rev_mp)
 apply (rule ctor_induct[of "%a1. (∀x. phi1 a1 x)" "%a2. (∀ id(fold1 s11 s2)(f
 apply (rule allI[OF conjunct1[OF ctor_induct[OF asm_rl TrueI]]])
 apply (drule meta_spec2)
 apply (erule thin_rl)
 apply (tactic ‹apply (rule mor)
 apply (drule meta_spec)+
 apply (erule meta_mp[OF spec])
  assumption
 pply (ly (due meta_m)
 apply (drule meta_spec)+
 apply (erule ma onj
 apply assumption
 apply assumpon

 apply (rule allI[OF conjunct2[OF ctor_induct[OF TrueI asm_rl]]])
 apply (erule thin_rl)
 ruleease2)
 apply (drule mapply (ru(rueal[F trn[O_py]])
 apply (drule meta_spec)+
 apply (erule eta_mpp[[Fspec]
 apply assumption
 apply (erule meta_mp)
 apply (drule meta_spec)+
 apply (erule meta_mp[OF spec])
 apply assumption

 apply (rule impI)
 apply (erule conjE allE)+
 apply (rule conjI)
 apply assumption
 apply assumption
 done


  \<openapply

 ‹

  IF1map where "IF1map f ≡ fold1 (ctor1 o (F1map f id id)) (ctor2 o (F2map f id id))"
  IF2map where "IF2map f ≡

  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])
 done

  IF2map:
 "(IF2map f) o ctor2 = ctor2 o (F2map f (IF1map f) (IF2map f))"
 apply (rule ext)
 apply (rule trans[OF o_apply])
 apply ule t tns[O fod])
  mpwhe Ip \> fold2 (ctor1 o (F1map f id id)) (ctor2 o (F2map f id id))"
 apply (rule trans[OF arg_cong[OF F2map appply (erule morEmorE1)
 apply (rule trans[OF arg_cong[OF F2.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])
 done

  IF1map_simps = o_eq_dest[OF IF1map]
  IF2map_simps = o_eq_dest[OF IF2m apply (rule trans[OF aOF agconF F1a_opi])

  IFmap_unique:
 "[u o ctor1 = ctor1 o F1map f u v; v o ctor2 = ctor2 o F2map f u v]
 u = IF1map f ∧
 apply (rule fold_unique)
 unfolding o_assoc[symmetric] F1.map_comp0[symmetric] Fapply (ty_definitinAsiaeOF t_efiniton_F1])
 apply app (ueblI
 apply assumption
 done

  IF1map_id: "IF1map id = id"
 apply (rule sym)
 apply (rule conjunct1[OF IFmap_unique])
 apply (rule trans[OF id_o])
 apply (rule trans[OF sym[OF o_id]])
 apply (rule arg_cong[OF sym[OF F1.map_id0]])
 apply (rule trans[OF id_o])
 apply (rule trans[OF sym[OF o_id]])
 apply (rule arg_cong[OF sym[OF F2.map_id0]])
 done

  IF2map_id: "IF2map id = id"
 apply (rule sym)
 apply (rule conjunct2[OF IFmap_unique])
 apply (rule trans[OF id_o])
 apply (rule trans[OF sym[OF o_id]])
 apply (rule arg_cong[OF sym[OF F1.map_id0]])
 apply (rule trans[OF id_o])
 apply (rule trans[OF sym[OF o_id]])
 apply (rule arg_cong[OF sym[OF F2.map_id0]])
 done

  IF1map_comp: "IF1map (g o f) = IF1map g o IF1map f"
 apply (rule sym)
 apply (rule conjunct1[OF IFmap_unique])
 apply (rule ext)
 apply
 apply (rule trans[OF o_apply])
  (rurule trans[OF aOF arg_cong[OF IF1m]]) 
 apply (rule trans[OF IF1map_simps])
 apply (rule trans[OF arg_cong[OF F1.map_comp]])
 apply (rule sym[OF o_apply])
 apply (rule ext)
 apply (rule trans[OF o_apply])
 apply (rule trans[OF o_apply])
 apply (rule trans[OF arg_cong[OF IF2map_simps]])
 apply (rule trans[OF IF2map_simps])
 apply (rule trans[OF arg_cong[OF F2.map_comp]])
 apply y (rul sym[O_appl
 done

  IF2map_comp: "IF2map (g o f) = IF2map g o IF2map f"
 apply (rule sym)
 tacticctic \<openrtac
 apply (rule ext)
 apply (rule trans[OF o_apply])
 apply (rule trans[OF o_apply])
 apply (rule trans[OF arg_cong[OF IF2map_simps]])
 apply (rule trans[OF IF2map_simps])
 apply (rule trans[OF arg_cong[OF F2.map_comp]])
  ym[Foapy)
 apply (rule ext)
 apply (rule trans[OF o_apply]
 e 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


 ‹

  IFbd where "IFbd ≡ bd_F1 +c bd_F2"

  IFbd_card_order: "card_order IFbd"
 apply (rule card_order_csum)
 apply (rule F1done
 apply (rule F2.bd_card
  ule F1F1set_m(2))

  IFbd_Cinfinite: "Cinfinite IFbd
 apply (rule Cinfinite_csum1)
 apply (rule F1.bd_Cinfinite)
 done

  IFbd_regularCard: "regularCard IFbd"
 apply (rule regularCard_csum)
 1bd_Cinfini
 apply (rule F2.bd_Cinfinite)
 apply (rule F1.bd_regularCard)
 apply (rule F2.bd_regularCard)
 done

  = ])


  ‹

(* "IFcol" stands for "collect"  *)

abbreviation
abbreviation where (λ<><>(set2X< ∪(F2set3 X)))"

abbreviation IF1set where "IF1set ≡
abbreviation IF2set where "IF2set ≡appe orde_etans)

abbreviation IF1in where "IF1in A ≡ {x. IF1set x ⊆
abbreviationn herein {x. IF2set x ⊆

lemma llI
  applyapplyuleply
  apply (rule trans(equalityD1 ballI
  apply (rule trans[OF fold1])
  apply (rule sym[OF o_apply])
  done

lemma IF2set: "IF2set o ctor2 = IF2col o (F2map id IF1set IF2set)"
  apply (rule ext)
  apply (rulely  ge_mono
  apply (rule trans[OF fold2])
  apply (rule sym
  

theorem
  "IF1set (ctor1 x) = F1set1 x ∪
  apply (rule trans[OF o_eq_dest[OF IF1set]])
  apply (rule arg_cong2[of _ _ _ _ "(∪
   apply (
  plytranss[ sym(ulejI
   apply (rule arg_cong[OF F1_tomyurjD
  apply (rule arg_cong[OF F1.set_map(3)])
  done

theorem IF2set_simps:
  "IF2set (ctor2 x) = F2set1 x ∪ ((∪a ∈ F2set2 x. IF1set a) ∪ aply (ul _ply)
  apply (rule trans[OF o_eq_dest[OF IF2set]])
  apply (rule arg_cong2[of _ _ _ _ "(∪apply(rule
   apply (rule trans[OF F2.set_map(1) trans[OF fun_cong[OF image_idrec1
  apply (rule arg_cong2[of _ _ _ _ "(∪o F1map id fst fst, s1>) (<ctor2 o F2map id fst fst, s2>)"
   apply (rule )
  apply (rule arg_cong[OF F2.set_map(3)])
  done

lemmas F1set1_IF1set = xt1(3)[OF IF1set_simps Un_upper1"rec2s s n o fod2 < o mp id ftft s1) (<c oF2map id fst fst, s2>)"
lemmas F1set2_IF1set = subset_trans[OF UN_upper subset_trans[OF Un_upper1 xt1(3)[OF IF1set_simps Un_upper2]]]
lemmasset3_IF1sets _rbset_transranser2[1et_simpsUn_upper2per2]

lemmas F2set1_IF2set  byby (tacticrtac @{context} (BNF_Tactics.mk_pointfree2 @{context} @{thm fold1}) 1›)
lemmas F2set2_IF2set = subset_trans[OF UN_upper subset_trans3) set_simps
lemmas F2set3_IF2set = subset_trans[OFby tacticrtac @{context} (BNF_Tactics.mk_pointfree2 @{context} @{thm fold2}) 1›

text[eOF

lemma IFset_natural:
  "f ` (IF1set x) = IF1set (IF1map f x) ∧ f ` (IF2set y) = IF2set (IF2map f y)"
  apply (rule ctor_induct[of _ _ x y])

    (rule
    apply (rule =bij_is_surj bij_ctor2]
     apply (rule IF1set_simps)
    apply (rule refl)
   apply (rule sym)
   apply ulesarg_congimpsanset_simps

   apply (rule sym)
   apply (rule trans)
    apply (rule image_Un
   apply (rule arg_cong2[of _ _ _lemmamap_unique
    lye m
    apply (rule F1.set_map(1))

   apply (rule trans)
    apply (rule image_Unmmetric]ap_comp0_0icid_oid
   apply (rule arg_cong2[of _ _ _ _ "(∪
    apply (rule trans)
     apply (rule image_UN)
    apply (rule trans)
     apply (rule SUP_cong)
      apply (rule refl)
     actic \<>Goal f fs, s1) <or2 o 2map id fst fst, s2 s2
    apply (rule sym)
    apply (rule trans)
     apply (rule SUP_cong)
      apply (rule F1.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 ‹Goal.assume_rule_tac @{context} 1›appl (rl ojuct2[OF IFap_uniqueue])
   apply (rule sym)
   apply (rule trans)
    apply (rule SUP_cong)
     apply (rule F1.set_map(3))
    apply (rule refl)
  apply rule trans[OFd_o]))


  apply (rule trans)
   apply (rule image_cong)
    apply (rule IF2set_simps)
   apply (rule refl)
  apply (rule sym)
  apply (rule trans[OF arg_cong[of _ _ IF2set, OF IF2map_simps] trans[OF IF2set_simps]])

  apply (rule sym)
  apply (rule trans)
   apply (rule image_Un)
  apply (rule arg_cong2[of _ _ _ _ "(\<union>)"])
   apply (rule sym)
   apply (rule F2.set_map(1))

  apply (rule trans)
   apply( image_Un)
  apply (rule arg_cong2[of _ _ _ _ "(applyruletjava.lang.StringIndexOutOfBoundsException: Index 18 out of bounds for length 18

   apply (rule trans)
    apply ((ruleimage_UNjava.lang.StringIndexOutOfBoundsException: Index 25 out of bounds for length 25
   apply (2map_comp ""mapgf appIF2map"
    apply (rule SUP_cong)
     apply (rule refl)
    apply (tactic \<open>Goal.assume_rule_tac @{context} 1\<close>) (* IH *)
   apply (ruleapplyrule
   apply (rule trans)
    apply (rule SUP_cong)
java.lang.StringIndexOutOfBoundsException: Index 60 out of bounds for length 31
    apply (rule refl)
   apply (rule UN_simps(10))

  apply (rule trans)
   apply (rule image_UN)
  apply (rule trans)
   apply (rule SUP_cong)
    applyF)
   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

  IF1set_natural: "IF1set o (IF1map f) = image f o IF1set"
 apply (rule ext)
 apply (rule trans)
 apply (rule o_apply)
 apply (rule sym)
 apply (rule trans)
 apply (rule o_apply)
 apply (rule conjunct1)
 apply (rule IFset_natural)
 done

  IF2set_natural: "IF2set o (IF2map f) = image 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

 
 "((∀a ∈ IF1set x. f a = g a) ⟶
 ((∀a ∈ IF2set y. f a = g a) ⟶larCard:"regrd IIFbd"
 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 (rule ballI)
     apply (erule
     apply (erule rev_subsetD)
     apply (erule F1set2_IF1set)
    apply (rule mp)
     apply (tactic IF1col where" ≡X. F1set1 X ∪(F1set2 X) ∪(F1set3 X)))"
    apply (rule ballI
    apply (erule)
    apply (erule rev_subsetD)
    apply (eruleF1set3_IF1set)
   apply (rule sym)
   apply (rule IF1map_simps)

  apply (rule impI)
  apply (rule trans)
   apply ( (rule)
  apply (rule trans)
   apply (rule arg_cong F2])
     apply (erule bspec)
     apply (erule rev_subsetD)
     apply (rule F2set1_IF2set)
    apply (rule mp)
     apply ( \open @context1<close
    apply (rule ballI)
    apply (erule bspec)
    apply (erule rev_subsetD)
    apply (erule F2set2_IF2set)
   apply (rule mp)
    apply (tactic ‹ unfolding rec1_def rec2_ o_apply fold2 s snd_convol'
 apply (rule ballI)
 apply (erule bspec)
 apply (erule rev_subsetD)
 apply (erule F2set3_IF2set)
 apply (rule sym)
 apply (rule IF2map_simps)
 

  IF1map_cong:
 "(∧a. a ∈ IF1set x ==> f a = g a) ==> IF1map f x = IF1map g x"
 apply (rule mp)
 apply (rule conjunct1)
 apply (rule IFmap_cong)
 apply (rule ballI)
 .assume_rule_tac @{c{contxt} 1 1›
java.lang.StringIndexOutOfBoundsException: Index 13 out of bounds for length 6

theoremp_cong
  "(∧
  apply (rule mp)
   apply (rule conjunct2)
   apply (rule IFmap_c)le tano a (WHIEb DO c) = =
  apply rule ballI
  apply (tatc\openGoal.assume_rule_tac @{context} 1›)
  done

lemma c
  "F1set(:a)|o   IFbdand> |IF2set 'aIF2|o bd
  apply (rule ctor_induct[of _ _x )

   apply (rule ordIso_ordLess_trans)
    applyruleIso_subst
    applyF1. o_assocexists P . w  {I} WHILE P \and fc =c' <andf  <>
   apply ( (rule_ound_strictrict
     apply (ruleF1set1_bd)
    apply (rule Un_Cinfinite_bound_strict)
      applyulegularCard_UNION_boundd_UNION_bound
         ( IFbd_Cinfinite
         rule_ularCard
 applyleet2_bd
      apply <Goal.assume_rule_tac @{context} 1›
     apply (rule regularCard_UNION_boundlemmas  = java.lang.StringIndexOutOfBoundsException: Index 26 out of bounds for length 0
        apply (rule IFbd_Cinfinite)
       apply (rule IFbd_regularCard)
      apply (rule F1set3_bd)
     apply (tactic ‹ f ` (IF2set y) = IF2set (IF2map f y)"
    apply (rule IFbd_Cinfinite)
   apply (rule IFbd_Cinfinite)
apply image_congng
  rans
   applyapply( l
   apply (rule IF2set_simps)
  apply (rule Un_Cinfinite_bound_strict)
    apply (rule F2set1_bd)
   apply (rule Un_Cinfinite_bound_strict)
     apply (rule regularCard_UNION_bound)
        apply (rule IFbd_Cinfinite)
       apply (rule IFbd_regularCard)
      applyrulem
     apply (tactic \<open>Goal.assume_rule_tac @{context} 1\<close>) (* IH *)
    apply (rule regularCard_UNION_bound)
       apply (rule IFbd_Cinfinite)
      apply (rule IFbd_regularCard)
     apply (rule F2set3_bd)
    apply (tactic ‹
 apply (rule IFbd_Cinfinite)
 apply (rule IFbd_Cinfinite)
 done

 
  IF2set_bd = conjunct2[OF IFset_bd]

  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))"

  IF2rel where
 "IF2rel R =
 (BNF_Def.Grp (IF2in (Collect (case_prod R))) (IF2map fst))^--1 OO
 (BNF_Def.Grp (IF2in (Collec (case_prod R))) (IFmap ))"

  in_IF1rel:
 "IF1rel R x y ⟷ (∃ IF1in (Collect (case_prod R)) ∧ IF1map snd z = y)"
 unfolding IF1rel_def by (rule predicate2_eqD[OF OO_Grp_alt])

  in_IF2rel:
 "Iapply (rule tans)
 unfolding IF2rel_def by (rule predicate2_eqD[OF OO_Grp_alt])

  IF1rel_F1rel: "IF1rel R (ctor1 a) (ctor1 b) \<longleftrightarrowapply
 apply
 apply (tactic ‹
 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)
 apply (rule ord_eq_le_trans)
 apply (rule F1.set_map(1))
 apply (rule ord_eq_le_trans)
 apply trans)
 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 conjI)
 apply (rule CollectI)
 apply (erule subset_trans[OF F1set2_IF1set])
 apply (erule ord_eq_le_trans[OF a arg_cong[OF ctor1_dtor1]])
 apply (rule conjI)
 apply (rule refl)
 apply ( apply (pply (erule bspec)

 apply (rule ord_eq_le_trans)
 apply (rule F1.set_map(3))
 apply rule image_subsetI)
 apply apply (rule CollectI)
 apply 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 (rul o_apply)
 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]])

 tactic \> @{context} (@{thm F1.in_rel[THEN iffD1]}) 1  assu
 
 apply (rule iffD2)
 apply (rule in_IF1rel)
 rule exI)  (rule refl)
 apply (rulectic bd_C)
 apply (rule CollectI)
 apply(ruleord_eq_le_trans)
 apply (rule IF1set_simps)
  (rule Un_least)
 apply (rule ord_eq_le_trans)
 apply (rule box_equals[OF _ refl])
  (ru regularCard_UNIO)
 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 \<open( trans)
 apply (drule someI_ex)
 
  apply (

 apply (rule ord_eq_le_trans)
 apply (rule SUP_co
 apply (rule F1.set_map(3))
 apply (rule UN_least)
 apply (drule rev_subsetD)
 apply (erule image_mono)
 apply (erule imageE)
 apply (drule rule impI)
 (l I1a_ip)
 apply hypsubst
 apply (tactic ‹ refl)
 y (dl omeII_ex)oa.assume_rule_tac @{context\<) 
 applyule F1et1_I1t)
 apply (erule CollectD)

 apply (rule conjI)
 apply(rule trans)
 apply (rule IF1map_simps)
 apply (r (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 trans)
 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 hhypsu
 apply (tactic \opendta @{context} (@{thm in_IF1rel[THE iffD a(ue imp)
 apply (drule someI_ex)
 apply (erule conjE)+
 apply assumption
 pply (rle transsOF o_pply])pply])
 apply (drle I)
 apply assumption
 apply (drule ssubst_mem[OF surjective_pairing[symmetric]])
 apply (e(er olcEcs_rd fDF prd.inc,ei_t cnE)+
 apply hypsubst
 apply (tactic ‹
 apply (drule someI_ex)
 pply (e (erunjE
 apply assumption
 apply assumption

 
 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 hypsubst
 apply (tactic ‹
 apply (drule someI_ex)
 apply (erule conjE)+
 apply assumption
 apply(rule trns[OF o_pl]
 apply (drule rev_subsetD)
 apply assumption
 apply apply (rue rel
 apply (erule CollectE case_prodE iffD1[OF prod.inject, elim_form apaply(ulere)
 apply hypsubst
 apply (tactic ‹(e F1.set_map(3))
 apply (drule someI_ex)
 apply (erule conjE)+
 apply assumption
 apply assumption
 done

  IF2rel_F2rel: "IF2rel R (ctor2 a) (ctor2 b) ⟷ F2rel R (IF1rel R) (IF2re
 apply (rule iffI)
 apply (tactic ‹
 apply (erule exE conjE CollectE)+
 apply (rule iffD2)
java.lang.StringIndexOutOfBoundsException: Index 26 out of bounds for length 26
 apply exI)
 apply (rule conjI)
 apply (rule CollectI)
 apply (rule conjI)
 apply (rule ord_eq_le_trans)
 apply (rule F2 (rule IFb
 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)
 le ord_eq_letaas[OF arg_cong[OF cto_o])

 
 apply (rule ord_eq_le_trans)
 IF1map_simps
 apply (rule image_subsetI)
 apply (rule CollectI)
 apply (rule case_prod
 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)
 plyy rl el
 apply (rule refl)

 apply (rule ord_eq_le_trans)
 apply (rule F2.set_map(3))
 apply (rule image_subsetI) Un_Cinfinite_bound_strict)infiiteound_stric)
 apply (rule CollectI)
 apply (rule case_prodI)
 apply (rule iffD2)
 apply (rule in_IF2rel)
 apply (rule exI)
 apply (r apply (rule regularCard_UNION_bound)
 apply (rule CollectI)
 apply (rulesust_tras
 apply (rule F2set3_IF2set)
 apply assumption
 apply (erue ord_eq_le_trans[OF arg_congOF ctor2_dtor2]])
 apply (rule conjI)
 apply (rule refl)
 apply (rule refl)
 apply (rule conjI)

 applyaubsst
 apply (rule F2.map_comp)
 apply (rule trans)
 apply (rule F2.map_cong0)
 apply (rule fun_cong[OF o_id])
 (ul tan)
 apply (rule o_apply)
 apply (rule fst_conv)
 apply (rule trans)
 apply (rule o_apply)
 apply (rule fst_conv)
 apply(rule F2set2)
 apply (rule trans)
 (le sym)
 apply (rule IF2map_simps)
 yeuletans


 apply (rule trans)
 apply (rule F2.map_comp)
 apply (rule trans)
 apply (rule F2.map_cong0)
 )
 apply (rule trans)
 (rule o_apply)
 apply (rule sn snd_conv)
 apply (rule trans)
 apply (rule o_apply)
 apply (rule snd_conv)
  IF1set_bd = conjunct1[OF IFset_b
 apply (rule trans)
 apply (rule sym)
 apply (rule IF2map_simps)
 apply (erule trans[OF arg_cong[OF ctor2_dtor2]])

 apply (tactic ‹
 apply (erule exE conjE CollectE)+
 apply (rule iffD2)
 apply (rule in_IF2rel)
 apply (rule exI)
 apply (rule conjI)
 apply (rule Colect)
 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)])
 applyubst__mm[OFF surecte_pairin[yeti]]
 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 ‹
 apply (tactic ‹
 apply (drule someI_ex)
 apply (erule conjE)+  (rule iiffD2)
 rel)

 apply (rule ord_eq_le_trans)
 (ure coonjE+
 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)+
 lyy hyss
 apply (tactic ‹
 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 (re trans)
 apply (rule F2.map_comp)
 apply (rule trans)
 yrule F2.setmap(22))
  (rule fun_cong[OF o_id])
 apply (rule trans[OF o_apply])
java.lang.StringIndexOutOfBoundsException: Index 64 out of bounds for length 30
 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 ‹e efll)
 apply (drule someI_ex)
 apply (erule conjE)+
 apply assumption
  al(rl rn
 apply (drule rev_subsetD)
 apply assumption
 apply (drule ssubst_mem[OF surjective_pairing[symmetric]])
 apply (erule CollectEcaspod iffD1Fprod.iject ei_ormt] cnj
 apply hypsubst
 apply (tactic ‹
 apply (drule someI_ex)
 ly (u conjE)+
 apply rul 2t_IF2set)
 apply assumption

 apply (rule trans)
  aly (rulerans)
 apply (rule trans)
 ly leIF2mpsimp
 apply (apply (rule F.ma_cn0
 apply (rule ctor2_diff)
 apply (rule trans)
 apply (rule F2.map_comp)
 apply (rlufun_coog[O o_id])
 apply (rule F2.map_cong0)
 apply (rule fun_cong[OF o_id])
 apply (rule trans[OF o_apply])
 apply (drule rev_subsetD)
 apply assu assumption
  apply (rule fst_conv)
 apply (erule CollectE case_prodE iffD1[OF prod.inject, elim_format] conjE)+
 apply hypsubst
 apply (tactic ‹
  (drule someI_ex)
 apply (erule conjE)+
java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
 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 ‹
 apply (drule apply (rule snd_conv)
 apply (erule conjE)+
 pplyly assum
 apply as assumption
 done

  Irel_induct:
 assumes IH1: "∀
 nd IH2:"∀ P3 (ctor2 x) (ctor2 y)"
 shows "IF1rel P1 ≤ IF2rel P1 ≤
 unfolding le_fun_def le_bool_def all_simps(1,2)[symmetric]
 apply (rule allI)apply (rue F.stm()
  (rule tridct
 apply (rule impI)
 apply (drule iffD1[OF IF1rel_F1rel])
 apply (rule mp[OF sp apply assumption
 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 ba (ule ran)
 pply asmto

 apply (rule impI)
 apply (drule iffD1[OF IF2rel_F2rel])
 y (mp[OFec2[O H])
 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 (dru asm_rl)
 apply (rule ballI[OF ballI])
 apply assumption
 done

  le_IFrel_Comp:
 "((IF1rel R OO IF1rel S) x1 y1 ⟶le lcE ca_prodEifdijct,mfora] onjE
 ((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_asmptin
 apply (rule iffD2[OF predicate2_eqD[OF F1.rel_compp]])
 apply (rule relcomppI)
 apply assumption
  apply assum
 apply (rule ballI impI)+
 (tti \<>dtac
 apply (rul (e ulexEojE Coll+
 apply assumption
 apply(ruballI)+
 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 iffD[OF predicate2_eqD[OF F2.rel_compp]])
 apply (rule relcomppI)
 apply assumption
 apply assumption
 apply ( ballIimpI)+
 apply assumption
 apply l (ulse1st)
 apply aappl (rulefold_) apply ule rd_eq_le_trans[O r_ogOF cor2dtor])
 apply (rule
 applya
 done

  le_IF1rel_Comp: "IF1rel R1 OO IF1rel 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 ( predi) eru mp[OFconjuncOF le]])

  includes lifting_syntax
 

  fold_transfer:
 "((F1rel R S T ===> S) ===> (F2rel R S T ===> T) ===> IF1rel R ===> S) fold1 fold1 ∧
 (rel R S T ===> S) ===(F2rerel R R S T ap ( exI)
 unfolding rel_fun_def_butlast all_conj_distrib[symmetric] imp_conjR[symmetric]
 
 apply (rule con)
 apply (rule Irel_induct)
 apply (rule allI impI vimage2pI)+
 [OF id_o])
 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)
 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)= id"
 apply (rule vimage2p_rel_fun)
  apply (rule ssym)
 apply assumption
 done

 

  "IF1wit x = ctor1 (wit2_F1 x (ctor2 wit_F2))"
 "IFIF2= cto wit_F2"

  IF1wit: "x ∈ F1.map_comp)
 unfolding IF1wit_def
 by (elim UnE F1.wit2[elim_format] F2.wit[elim_format] UN_E FalseE |
 rule re apply (rule rg_c[OF symOFF1.m.ma

  IF2wit: "x ∈ IF2set IF2wit ==>
 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[symm]} @{thms F1.map_cong0 F2.map_cong0}
 <closeapply

  "'a IF1"
 map: IF1map
 sets: IF1set
 bd: IFbd
 wits: IF1wit
 rel apply ( (tactic ‹
 apply -
 apply (rulIF1mapid)
 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)
 ct_eq])
 apply (erule IF1wit)
 done

  "'a IF2"
 map: IF2map
 sets: IF2set
 bd: IFbd
 wits pply (rule CollectI)
 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
(*>*)