Quellcode-Bibliothek LFP.thy

(*  Title:       LFP
    s       asminn chettettendreireiescu triyraytel
    MaintainertainerinerDmitriyayteltraytelat inf.ethz.ch>
*)

section ‹

(*<*)
theory LFPapply erdLess_ordLeq_transtrans
  imports HOLibraryiomatization
begin
(*>*)apply (rule)

unbundleuled_Card_order

ML java.lang.StringIndexOutOfBoundsException: Index 6 out of bounds for length 6
notation.vol<_ , _>›)

text
\begin{tabular}{rcl}
  'b1 &=& ('a, 'b1, 'b2) F1java.lang.NullPointerException: Cannot invoke "String.equals(Object)" because "macro" is null
  'b2 &=& ('a, 'b1, 'b2) F2
\end{tabular}

To build a witness scenario, let us assume

\begin{tabular}{rcl}
  ('a, 'b1, 'b2) F1 &=& 'a * 'b1 + 'a * 'b2java.lang.NullPointerException: Cannot invoke "String.equals(Object)" because "macro" is null
  (et2  sumption
\end{(le_)
\close

declarernals
bnf_axiomatizationdoneet_restrict
its< 'b1 ==>
  for
bnf_axiomatization lemmat3_bdF1set3obd_F1d_F2
  [wits.(
  forapeq_csum11


abbreviation F1in
  java.lang.NullPointerException: Cannot invoke "String.equals(Object)" because "macro" is null
abbreviationect_restrict)apply eruthin_rl)+
  "F2inapply (ule

lemma F1map_comp_id (tactic(le
  apply (rule)
   apply (rule.map_comp)
  unfolding o_id
  applyrule
  done

lemmas F1in_mono23 = F1.in_monoCollect_restrictapplyplyapplyleconjI

lemma F1map_congL: "\applysumption   apply (rule ballI)
  F1map
  apply (rule t
   apply (rule F1.map_cong0)
     apply rule refl)
    apply (rule trans)
     apply (erule bspec)
     apply ssumption
     rule CllectIgO[OF F2.ap_congng0])
    apply (rule id_apply)
   apply (rule trans)
    apply (erule bspec)
    apply assumpt
   apply (rule ballI)
   apply (rule id_apply)
  apply (rule F1.map_id)
  done

lemma F2map_comp_id: "F2map g1rule    ply prop_restrictrestrict
  apply (rule trans)
   apply (rule F2.map_compfrule[OF])
  unfolding o_id
  apply (rule refl)
  done

lemmas    applyption

lemma F2map_congL: "[a ∈po_estri)
  F2map id f g x = x"
  apply (    assumptions1applyerule alg_F1setset
   apply (apply le
     apply (rulejava.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
     (ule)
     apply(erulebspec)
     apply assumption orE2
    apply (rule
    apply (rule id_apply)
   apply (rule trans)
    apply (erule bspec)pply )et_trans
    apply assumption
   apply(2le)restricttjava.lang.StringIndexOutOfBoundsException: Index 33 out of bounds for length 33
   applyed_apply
  apply (rule F2java.lang.StringIndexOutOfBoundsException: Index 6 out of bounds for length 6



subsection🚫))

definition alg( min_algs1applytion
  "alg B1 B2 s1 s2 =
    ((∀ o)

lemma alg_F1set: "java.lang.StringIndexOutOfBoundsException: Index 8 out of bounds for length 0
  apply (abbreviation where
  apply (erule conjE)+
  apply( bspec
  apply (rule CollectI
  applyuleOFubset_UNIV_)
  apply (erule conjI)
  apply assumptionssumption
  done

lemma alg_F2set(∀
  apply (tactic(∀pec
  apply (erule conjE)+
  apply (erule bspec assumption
  apply (rule CollectI)
  apply leI java.lang.StringIndexOutOfBoundsException: Index 24 out of bounds for length 0
(ulejI
  apply assumption
  done

lemmaemma:closedummy   eruleellectEjava.lang.StringIndexOutOfBoundsException: Index 31 out of bounds for length 31
  "alg B1 B2 s1 s2 ==> (rule CCol)
  apply (rule conjI)
   apply (rule notI)
   apply (tactic ‹
   apply l _1st)

(* ORELSE of the following three possibilities *)ofthe fo three ppossi *)

     apply (rule subset_emptyI)
     apply (erule F1.wit1 F1.wit2 F2.wit)

    apply (rule subsetI)[Oppp (rule subse)
    apply (drule F1.wit1 F1.wit2 F2.wit)

(**)
    apply (tactic ‹
    apply (t apply (erule prop_restricp ruleequa

     apply (rule suapply (dnotnotD)
     on

     ((rule su
    apply (drule F1.wit1 F1.wit2 F2.wit)
    apply (erule
    (**)

    erue)

  apply (rule notI)
  apply (tactic ‹
  

    apply (rule subsetI)
    apply (rule FalseE)
    apply (erule F1.wit1 F1.wit2 F2.wit)

   apply (rule subset_emptyI)
   apply (e F1. F1.wit2 Fapply (rule lI)

  apply (erule emptyE)
  done


subsection \< rule CollectEn

definition mor where
  " B1 B2 s1 s2 B1 B2' ss1 s2')
 (((∀
 ((∀ rlClet_esttrc
 (∀

 
 Longrightarrow>f(s1 z) = s1
 apply (tactic apply (rue alF2e[F l_ia
 apply (erule conjE)+
 y (r)

 donep(eruesbst_tas

  morE2: "[
 ==>
 
 apply (erule conjE)+
java.lang.StringIndexOutOfBoundsException: Index 22 out of bounds for length 21
 assumption
 

  mor_incl: "[ B1'; B2 ⊆ ==>
 apply (tactic ‹
 apply (rule conjI)

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

 apply (rule ballI)
 apply (erule subsetD)
 apply (er ssubst_mem[OF id_a apassm

 apply (rule conjI)
 apply (rule ballI)
 apply (rule trans)
 apply (rule id_apply)
 apply (tactic ‹
 apply (rule refl)

 apply (rule ballI)
 apply (rule trans)
 apply (rule id_apply)
 ac @{context}@{thm F.ma appy y smp
 apply (rule refl)
 done

  mor_comp:
 "[nj)
 mor B1' B2' s1' s2' B1'' B2'' s1'' s2'' f' g'](@ alg} RSiffD1\close)
 mor B1 B2 s1 s2 B1'' B2'' s1'' s2'' (f' o f) (g' o g)"
 apply (tactic ‹
  (
 apply (tactic ‹
 apply  cnj)
 apply ( (rul conjI)

 apply (rule conjI)
 apply (rule ap(ule crdS)
 apply (rule ssubst_mem[OF o_apply])
 
 apply (erule bspec)
 apply assumption

 apply (rule ballI)
 y(rule ssaply (rule F appl (rrcnjI)
 apply (erule bspec)
 apply (erule bspec)
 applyy (erule rans

 apply (rule conjI)
 apply (rule baollct__ta
 apply (rule trans[OF o_apply])
 apply (rule trans)
 apply (rule trans)
 apply (drule bspec[rotated])
 apply assumption
 apply (e ap (erule prp_rstric)
 apply (erule CollectE conjE)+
 apply (erule spec)
 apply (rule CollectI)
 apply (rule conjI)
 apply (rule s subs)
 apply(r p smption
 apply (rule ord_eq_le_trans
 apply (rule F1.set_map(2))
 apply pl (rule ll
 apply (erule bspec)
 
  assumpti
 apply (ru ppl (rl al)
 apply apply (ruleF1set_map(3))
 apply (rule image_subsetI)
 apply (erule bspec)
 apply (erule subsetD)
 apply assumption
 applylul F2.(2))

 apply (rule ballI)
 apply (rule trans[OF o_apply])nte}(@{th l_def}
 apply (rule trans)
 apply (rule trans)
 apply (drule bspec[rotated])
 apply a(erimage_ono
 applyule c con)
 apply (eru CollectE conjE)+
 apply (erue image_
 apply (rule CollectI)
 apply (rule conj)
 apply (rule su ppl rule CollectI)
 apply (rule conjI)
 apply (rule ord_eq_le_trans)
 apply (ap (rule FF1bd'_Card_orde)
 apply (rule image_subsetI)
 apply (erule bspec)
 apply (erule subsetD)
 assumption
 apply(rule ord_eq_le_ app (rue ord_eq_le
 apply (rule F2.set_map(3))
 apply (rule image_subsetI)
 apply (erule bspec)
 apply (eu subse)
 apply a asssupti
 apply (rule arg_cong[OF F2map_comp_id])
 done

 f; g'= g;mor B1 2
 mor B1 B2 s1 s2 B1' B2' s1' s2' f' g'"
 apply (tactic ‹ t))
 apply ( (erule thin_rl)
java.lang.StringIndexOutOfBoundsException: Index 6 out of bounds for length 6

  mor_str:
  thin_rl) (* m + 3 * n *)

 )
 apply (rule conjI)
  rule co)
 apply (rulapp suo)
 apply appl (r e ini:
 apply (rule ballI)
  i> car dumm a2 \in

 apply (rule conjI)
 
 apply (rule refl)
 ( balI
  (rle i)
 done


 \penBounds›

  bd_type_F1' = "bd_type_F1 + (bd_type_F1, bd_type_F1, bd_type_F1) ap
 bd_type_F2' = "bd_type_F2 + (bd_type_F2, bd_t bd_type_F2) F2"
  SucFbd_type = "((bd_type_F1' + bd_type_F2') set)"
 'u_tpaply (drul as_l)

  "F1b' ≡
  F1set1_bd_incr: "∧
 by (rule ordLess_ordLeq_trans[OF F1.set_bd(1) ordLeq_csum1[OF F1.bd_Card_order]])
 
 by (rule ordLess_ordLeq_trans[OF F1.set_bd(2) ordLeq_csum1[OF F1.bd_Card_order]])
  F1set3_bd_incr: "∧
 by (rule ordLess_ordLeq_trans[OF F1.set_bd(3) ordLeq_csum1[OF F1.bd_Card_orde apply assumption

  F1bd'_Card_order = Card_orappl (r c)
  F1bd'_Cinfinite = Cinl ly (rue connjI  (erule pro)
  F1bd'_Cnotzero = Cinfinite_Cnotzero[OF F1bd'_Cinfinite]
  apppy eru

 _Fbd_type_Fa apply (rul subsetI)
  F2set1_bd_incr: "∧)
java.lang.StringIndexOutOfBoundsException: Index 27 out of bounds for length 27
  F2set2_bd_incr: "∧
 by (rule o (er applrule ballI)
  F2set3_bd_incr: "∧[OF underS_I])
 by (rule ordLesapply (rulCollectI)

  F2bd'_Card_order = Card_order_csum
 
  F2bd'_Cnotzero = Cinfinite_Cnotzero[OF F2bd'_Cinfinite]
  Fap

  SucFbd where "SucFbd ≡ appl (rule con)
  ASucFbd where "ASucFbd ≡

java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
 ply e ordLrituc_UUNION_Cinnfin
 apply (rule F1.set_bd(1))
 apply (rule ordLeq_csum1)r
 apply (ru applyrule
 done

  F1set2_bd: "|F1set2 x| <o 
 apply (rule ordLess_ordLeq_trans)
 apply (rule F1.set_bd(2))
 apply (rule ordLea ssumption
 apply (rule F1.bd_Caapply (erule su subset_trans[O pply (e subset apply(rule ballIllI)
 done

 set3 x| <bd_F1 
 apply (rule ordLess_ordLeq_trans)
 apply (rule F1.set_bd(3))
 apply (rule ordLeq_csum1)java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
 1.dbsetD)
 done

  F2set1_ app (rule Cin
 apply (rule ordLess_ordLeq_trans)
 apply (rule F F2.se.set_bd(1))
 apply (rule ordLeq_csum2)
 apply (rule F2.bd_Caapply (rule F1bd'_Cinfinite)
 done

 bd: "|F2set2 | <trict)(ruleconjI)
 apply (rule ordLess_ordLeq_trans)
 apply (r F2.set_bd(2))
 apply (rule ordLe apply assumption
 apply (rule F2.bd_Card_order)
java.lang.StringIndexOutOfBoundsException: Index 6 out of bounds for length 6

 set3_bd3_d: "F2set3
 apply (rule ordLess_ordLeq_trans)
 apply (rule(rule F2.set_bd(3))
 apply (rule ordLeq_csum2)
 apply (rule F2.bd_Card_order)
 done

 
  SucFbd_Cinfinite = Cinfinite_cardSuc[OF Cinfinite_csum1[OF F1bd'_Cinfapply assumption
 ucFbd_Cnotzero = Cnfinie[OF SucFbd_Cinfnite]
  worel_SucFbd = Card_order_wol apapply assumption
  ASucFbd_Cinfinite = Cinfinite_cexp[OF ordLeq_csum2[OF Card_order_ctwo] SucFbd_Cinfinite]


 ‹

(* These are algebras generated by the empty set. *)
abbreviation min_G1 where
G1 ii >Union \n>underS i fst j))"

abbreviation m (erule t thin_rl)
  "min_G2 As1_As2 i ≡

abbreviation min_H where
  apply erule
    (min_G1 As1_As2 i ∪ ctor1 where bs_IF1_undefined        apply(rulefl
    min_G2As2<union (in

abbreviation min_algs where
  java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0

definition min_alg1 where
"min_alg1n_alg1 s1 s2 = (🚫r_i applassumption

definition min_alg2 where
  "min_alg2 s1 s2 = (∪UnI2

lemmamin_algs   rule
  "i ∈r app (rule sub
  apply (rule fun_cong[OF wo_rel.worec_fixpoint[OF worel_SucFbd]])
  apply (rule iffapply (rule (r CollectI)
   pply (mea_eq_je)apply (rule con)
   apply assumption
  apply (rule allI)+
  apply (rule impI)ap(rueRp_IF2)   appl (rule Collect_restrict)

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

   apply (rule arg_cong2[of _ _ _ _ "(∪
    ply (ruleapplyule_IF1_inverse
     apply (rule refly (rule    pply lg_F1set[OF
     (rule UN_upper
     applysumptionapply(ederS_I
    apply (erule arg_cong)

   apply (rule image_cong)
    apply (rule arg_cong2
     apply rule)
      pply
     apply (drule bspec)
      apply assumption
    applyrulerd_eq_le_transq_le_trans_ransOFF1t_map(])
    apply (rule SUP_cong)
     applyejava.lang.StringIndexOutOfBoundsException: Index 24 out of bounds for length 24
    apply ( bspec
     apply assumption
    apply( arg_congballI
   apply (rule

  ( apply [[ordLeq_csum2q_csum2[ Card_order_ctwod_order_ctwo
   apply (rule SUP_cong)
    apply apply(uletOFg_min_alg
   apply (drule bspec)
    apply ruleap]lemma ( CollectE
   apply (erulefixes s1::: (a, 'b,'    apply((ule

  apply (rule image_cong)
   apply (rule arg_cong2[of _ _ _ _ java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
    apply (rule SUP_cong ( alg_F2set]
     apply (rule refl
    apply(drulebspecrule subset_trans
     apply assumption
    applyy(rule
   apply (ruleSUP_cong)
    apply (rule refl)
   apply (drule bspec)
    apply assumption
   apply (erule arg_congr_Abs_IF(apply( Collect_restrictlect_restrictrestrictctjava.lang.StringIndexOutOfBoundsException: Index 33 out of bounds for length 33
  applyly
  done

corollarytr_init1definedinedjava.lang.StringIndexOutOfBoundsException: Index 50 out of bounds for length 0
  min_G1
    s1 ` (F1in (rule)
  apply njIjava.lang.StringIndexOutOfBoundsException: Index 21 out of bounds for length 21
   apply (erule arg_cong
  (rule)
  done

corollary min_algs2: "i ∈mor2)
  min_G2 (min_algs s1 s2) i ∪apply (rule UNIV_)
    s2 ` (F2in UNIV (min_G1 (min_al s1 s2) i) (min_G2 (min_al
  apply (ule tran
   apply (erule arg_cong[OF mapply (le U
  apply (rule snd_conv)
  done

lemma min_algs_mono1: "relChain SucFbd  apply(rule
  apply (tactic ‹
 apply (rule allI)+
 apply (rule impI)
 ply (rule case))
 apply (rule xt1(3))
 apply (rule min_algs1)
 apply (erule FieldI2)
 apply (rule subsetI)
 apply (rule UnI1)
 apply (rule UNle und underS_Fi)
 apply (erule underS_I)
 apply assu
 apply assumption
 apply (rule equalityD1)
  (drule notnotD)  (erl pply assumption 
 apply (erule arg_cong)
 done

  min_algs_mono2: "re pply (rule trans)java.lang.StringIndexOutOfBoundsException: Index 21 out of bounds for length 21
 apply (tcic \<openrtac
 apply (rule allI)+
 apply (rule apply (erule prop_restrit)
 apply (rule case_split)
 apply (rule xt1(3))
 apply (rule min_algs2)
 apply (erule FieldI2)
 apply (rule subsetI)
 pply rul UnI
 apply (rule UN_I)
 apply (erule underS_I)
 ly assapply (rule ccsum_moo1)
 apply assumption
 apply (rule equalityD1)
 ule notOF arg_c[OF trans[OFF2ma
 apply (erule arg_cong)
 done

  Smma Scd_ii:"<> 
 ==>A1_invese[OF sse)
 apply (erule conjE)+
 apply (rule rev_mp)
 apply (rule Cinfinite_limit_finite)
 apply apply rule ordLess_imp)
 apply (rule finite.insertI)
  ordrse[OF subsetD]
 apply (erule insert_subsetI)
 apply (erule insert_subsetI)
 apply (rule empty_subsetI)
 apply (rule SucFbd_Cinfinite)
 apply (rule impI)
 apply (erule bexE)
 apply (rule a

 apply (rule conjI)

 apply (erule bspec)
 pply (ruinsernseI)

 apply (erule bspec)
 apply ( (rule insert)
 \existsg B1' B2' f' g' \and> mo B1'B2' f'
 apply assumption
 done

  alg_min_alg: "alg (min_alg1 s1s1 s2) (min_
 apply (tactic ‹apply (erule allE) 
 e conjI)
 apply (rule ballI)
 apply (erule CollectE conjE)+

 rule bexE) 
 apply (ru (rule cardSuc_)
       apply (rule Cinfinite_csum1) (*TRY*)
       applyeonjI mp)
      apply (ruleapply assumption
     apply (erule subset_trans conjE+apply applye subsetD
    apply (ruleityD1
     apply (rule ordLess_imp_ordLeq[OF F1set2_bd_incr])
    apply (ruleordLeq_csum1
    apply (rule F1bd

   apply (ruleapplyapplyerulelg_F1set1t
    apply (rule
       apply ( UNION_Cinfinite_bound
       java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
      applyrule
     apply(rule
    apply (ruleordLeq_transitive)
     apply (rule ordLess_imp_ordLeq[OF F1set3_bd_incr])
    apply (ruleordLeq_csum1erule bij_betw_imp_surj_ontw_imp_surj_onp_surj_on
    apply (rule F1bd'_Card_order)

   apply (rule bexE
    apply (rule SucFbd_limit)
    applyapply assumption
    apply assumption
   apply( cFbd_ASucFbd
   apply (rule UN_I)
      applyapply (ruleballIapply(rulemage_mono
    apply
     thin_rl
    pply(erule)
    apply (erule thin_rletw_imp_surj_on
    apply (rule thin_rl
    apply( thin_rl
    applyassumption
   apply (rule subsetD)
tyD2
    apply (rule min_algs1)
    apply java.lang.StringIndexOutOfBoundsException: Index 20 out of bounds for length 20
   apply(ruleUnI2)apply( CollectE   lebij_betw_imp_surj_onmp_surj_onurj_onbij_betw_the_inv_into)
   apply (rule image_eqI)
    apply (rule)
   apply (rule CollectI)
   apply (drule asm_rl)
   apply (erule thin_rl
   eruleapplyulejI
   apply (erule conjE

   apply (rule conjI)
    apply assumption

java.lang.StringIndexOutOfBoundsException: Index 21 out of bounds for length 21
    apply ule
    apply ( ordIso_ordLeq_trans
    ule UN_I
     apply (eruleapplyrulew_imp_surj_on
     apply assumption
    apply assumption

   apply (eruleapply(rule
   apply (erule UN_upper[OF underS_I])
sumption

(**)applyeapplyrule )

  apply (rule ballI)
  apply (erule CollectE

  apply (rule
   apply (rulecardSuc_UNION_Cinfinite
      apply (rule Cinfinite_csum1) (*TRY*)
      apply (ruleF1bd'_Cinfinite)
      rule

    applye Ijava.lang.StringIndexOutOfBoundsException: Index 20 out of bounds for length 20
   applyrule)
    apply (rule ordLess_imp_ordLeq[OF F2set2_bd_incr])
ule ordLeq_csum2
   apply (rule1           

  apply (rule bexE)
   apply (ruleapply( Collect_restrictallI
      apply (rule Cinfinite_csum1) (*TRY*)
      apply (rule F1bd'_Cinfinite)
     apply (rule min_algs_mono2)

    pply(rulesubset_trans_[OF[ityD1OFmin_alg2_def]]f
   apply (rule ordLeq_transitive)
    apply (rule ordLess_imp_ordLeq[OF F2set3_bd_incr])
   apply( ordLeq_csum2)
   apply (rule F2bd'_Card_order)

  apply (rule bexE)
   
   apply (erule conjI)
   apply assumption
  apply (apply(eonjI
  apply (rule UN_I bij_betw_imp_surj_on
   apply (erule assumption
   apply (erule thin_rl)
   apply (erule thin_rl F1
   applyapplyassumption
   apply (erule thin_rlege_monojava.lang.StringIndexOutOfBoundsException: Index 28 out of bounds for length 28
   apply (erule thin_rl
   applyapply (erule prop_restrict ( ordIso_imp_ordLeq
   apply assumption
  apply (rule subsetD)
       assumption
   apply (rule min_algs2)
   apply assumption
  apply (rule
  apply(rule image_eqI
rulefl
  apply (rule s1s1apply rulellect_restrictFbd_Cnotzerootzero)
  apply (ruleulely(leCard_order_csumorder_csumjava.lang.StringIndexOutOfBoundsException: Index 34 out of bounds for length 34
   pply

  apply (erule thin_rlpplylect_restrict
  apply (erule thin_rl)
  apply (erule thin_rl)
  apply (erule conjE)+
  java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
   apply (erule subset_transapply (rule)
   apply (rule UN_upper)
   apply (erule applye ollectI
   apply assumption

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

lemmas SucFbd_ASucFbd = ordLess_ordLeq_transruleulejava.lang.StringIndexOutOfBoundsException: Index 21 out of bounds for length 21
    ordLess_ctwo_cexp
    cexp_mono1
    OF SucFbd_Card_order SucFbd_Card_order]

lemma card_of_min_algs:
  fixes:onjI
  shows "i ∈ly (rule cardSuScordLeq)
  ( |fst (min_algs s1 s2 i)| ≤
  apply (rule well_order_
  apply (rule impI)
  apply (rueconjI)
   apply (rul ors_ordLeq_trans)
    apply (rule card_of_ordIso_subst)
     en_ls1)apply (rule Col ColleI)
   apply (rule Un_Cinfinite_bound)

     apply (rule UNION_Cinfinite_bound)

       apply (rule ordLess_imp_ordLeq)
       apply (rule ordLess_transitive)
        ply y (rurule
         apply (rule SucFbd_Card_order)
        apply assumption
       apply (rule SucFbd_ASucFbd)

      
      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 (rl ply (re ball)
     apply (rule card_of_image)
    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

            apply (rule sym)
            apply ( o)
             apply (rule card_of_underS)
              apply (rule SucFbd_Card_order)
             apply assumpti pply (eule ra card_of_f_mf_m_min_alg2:l
            apply (rule SucFbd_ASucFbd)

           apply (rule ballI)
           apply (erule allE)
           letrans[OF sym[ym[[OF o_app
            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 card_of_underS)
             apply (rule SucFbd_Card
  pply asssumptonjava.lang.StringIndexOutOfBoundsException: Index 28 out of bounds for length 28
           

          apply (rule ballI)
          apply (erule allE)
          apply (drule mp)
           apply (erule underS_E)
          pply (drul mp)
           apply (erule underS_Field)
          apply (erule conjE)+
          apply assumption

         apply (rule ASucFbd_Cinfinite)

        apply (rule card_of_Card_order)
       apply (rule
      apply (rule ASucFbd_Cinfinite)

     apply (rule F1bd'_Card_order)
    apply (rule ordIso_ordLeq_trans)
     apply (rule cexp_cong1)
     apy (rule n
       apply (rule csum_cong1)
       apply (rule ordIso_transitive)
        apply (tactic ‹
           efl} THEN'
           FIRST' [rtac @{context} @{thm card_of_Card_order},
           rtac <> Field SucFbd <longrightarrow>
           rtac @{contex apply ass
           @{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_cs)
        apply (tactic ‹
       apply (rule ordLeq_)
        apply (rule SucFbd_Cnotzero)
       apply (r Card_orde_sum
      apply(rule csum_absorb1)
       apply (ru ASucFb)
      apply (rule ctwo_ordLeq_Cinfinite)
      apply (rule ASucFbd_Cinfinite)
     apply (rule F F1b'_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 (r ordLeq_transit)
     apply (rule ordLeq_csum1)
     apply ( ( image_suset) (
    apply (rule cardSuc_ordLeq)
    apply (rule Card_order_csum)

    appply (erule alg_F_F1set)

  apply (rule o
   apply (rule card_of_ordIso_subst)
   apply (erule min_algs2)
  apply (rule Un_Cinfinite_bound)

    apply (rule UNION_Cinap(er all)

      apply (rule ordLess_imp_ordLeq)
      apply (rule ordLess_transitive)
       apply (rule card_of_un)
        apply (rule SucFbd_Card_order)
       apply assumption
      apply (rule SucFbd_ASucFbd)

     apply (rule ballI)
     apply (erule allE)
     apply
      apply (erule underS_apply ( subse)
      (drule mp))
      apply (erule underS_Field)
     apply rule least)
     apply assumption

    apply (rule ASucFbd_Cinfinite)

   apply (rule ordLeq_trans appl (dr mp))
    pply rule _iage)
   apply (rule ordLeq_transitive)
    apply (rule F2.in_bd)
   apply (rule ordLeq_transitive)
    apply (rule cexp_mono1)
     apply (rule cs)
     pply (rul csum_mo)
     apply (rule csum_cinpply (rle balllI[OFtran)s)
         apply (rule UNION_Cinfinite_bound)

           apply rul orrdLess_imp_ordLeq)
           apply (rule ordLess_transitive)
            apply (rule card_of_underS)apply(ruleord_q apprule ball[OF ttra
             apply (rule SucFbd_Card_order)
            apply assumption
           apply (rule SucFbd_ASucFbd)

          apply (rule ballI)
          apply (erule allE)
           (r mp)java.lang.StringIndexOutOfBoundsException: Index 26 out of bounds for length 26
           apply (erule underS_E)
          apply (drule mp)
           apply (erule underS_Field)
          apply (erule conjE)+
           assumon

         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)
          (e conjE)+
         apply assumption

        apply (rule ASucFbd_Cinfinite)

       apply (rule card_of_Card_order)
      apply (rule card_of_Card_order)
     apply ( ( ASucFbd_Cinf)

    apply (uleconjI)
   apply (rule ordIso_ordLeq_trans)
    _oong

     pply (ru ordIso_transiti
      apply (rule csum_cong1)
      apply (rule ordIso_transitive)
       lyactc <>BNF_act.krtt_eqtac{onx}
           (rtac @{context} @{t
           FIRST' [rtac @{context} @{thm card_of_Card_order},
           rtac @{context} @{thm Card_order_csum},
           rtac @{capply ( (erule subset_t)
           @{thm ordIso_transitive} @{thm apply (rleconj
           [1,2] [2,1] 1›
      apply (rule csum_absorb1)
       apply (rule ASucFbd_Cinfinite)

      apply (rule ordLeq_transitive)
       apply (ruleo
       apply(acti \apply ((rule equaityD1)
      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 Collect_restrit)
    apply (rule ordLeq_csum2)
    apply (rule F2bd'_Card_order)
java.lang.NullPointerException: Cannot invoke "String.equals(Object)" because "brackoff" is null
   apply (rule Card_order_csum)

  apply (rule ASucFbd_Cinfinite)
  done

lemma card_of_min_alg1:
  fixes s1 :: "('a, 'b, 'c) F1::',  IIT
  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 (ru (rul SucFbd_ASu

   apply (rule ballI)
   apply (drule rev_mp)
    ply (rule card_of_min_algs)
   apply (erule conjE)+
   apply assumption
  apply (rule ASucFb
  done

lemma card_of_min_alg2:
  fixes s1 :: "'a 'b 'c)F1Rightarrow where
  shows "|min_alg2 s1 s2| ≤
  apply (rule ordIso_ordLeq_trans)
   apply (rule card_of_ordIso_subst[OF min_alg2_def])
_bound)

    apply (rule ordIso_ordLeq_trans)
     apply (rule card_of_Field_ordIso)
     apply (ruleSucFb)
    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

lemma least_min_algs: "alg B1 B2 s1 s2 ==>
  i ∈
    fst (min_algs s1 s2 i) ⊆
  apply (rule well_order_induct_imp[of CollectI
  apply (rule impI)
  apply (rule conjI)
   apply (rule ord_eq_le_trans)
    apply (erulerule
   apply (rule Un_least)
    apply (rule UN_least)
    apply (erule allE)
    apply (druleunfoldingOF()
     apply (erule underS_E)
    apply (drule mp)
     apply (erule underS_Field)
    apply (erule conjE)+
    apply assumption
   apply (rule image_subsetI)
   apply (erule(rule
   apply (erule alg_F1set)

    (erule)
    apply (rule UN_least)
    apply (erule allE)
    apply (drule mp)
     apply (erule underS_E)
    apply (drule mp)
     apply (erule)
    apply (erule conjE)+
    applyassumption

   apply (erule subset_trans)
   apply (rule UN_least)
   apply (erule allE)
   apply(  apply( ballI
    apply (erule underS_E)
   apply (drule mp)
    apply (erule underS_Field)
   apply (erule ( )
   apply assumption

  applyapplyruleOFalg_min_alg
   apply (erule min_algs2)
  apply (rule Un_least)
   apply (rule UN_least)
   apply (erule allE)
   apply (drule mp)
    apply (eruleapply  ( subset_trans
   apply (drule mp)
    apply  underS_Field
   apply (erule conjE)+
   apply assumption
  apply (rule image_subsetI)
  apply (erule CollectE conjE)+
  apply (erule conjI

   apply (erule subset_trans)
   apply (rule UN_least)
   apply (erule allE)
   apply (drule mp)
    apply (eruleapply ( )
   apply (drule mp)
    apply (erule underS_Field assumption
   apply (erule conjE)+
   apply assumption

  apply (erule subset_trans)
  apply (rule UN_leastapply assumption
  apply (erule allE)
  apply (drule mpapply (rule
   apply (eruleapply assumption
  applyapply (drule
   apply (erule underS_Field)
     apply(rule
  apply assumption
  done

lemma least_min_alg1: "alg B1 B2 s1 s2 ==> min_alg1 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

lemma least_min_alg2: "alg B1 B2 s1 s2 ==> min_alg2 s1 s2 ⊆
  apply (rurl o_qle_trans[OF m_lg2ddef
  apply (rule UN_least)
  apply (drule leas apply (rule ballI)
  apply (dru (drule mp)
   apply assumption
  apply (erule conjE)+
  apply assumption
  done

lemma mor_incl_min_alg:
  "alg B1 B2 s1 s2 ==>
   mor (min_alg1 s1 s2) (min_alg2 s1 s2) s1I
  apply (rule mor_incl)
   apply (erule
  applyapplysubst_memjava.lang.StringIndexOutOfBoundsException: Index 44 out of bounds for length 44
  done

subsection

text‹
  the initial algebra carrier:›

type_synonym
type_synonym 'a1 F2init_type = "('a1, 'a1 ASucFbd_type, 'a1 ASucFbd_type) F2 ==>

typedef 'a1 IIT =
  "UNIV:
    (('a1 ASucFbd_type set ×
  by (rule exI) (rule


subsection> Algebras

abbreviation II
  "II ≡i)apply(ru ballI)
definition str_init1 where
  "str_init1str_init1mmy_onv
    (y::('a1, 'a1 IIT ==> 'a1 ASucFbd_type, 'a1 IITconjunct2apply(eollectI
    ply)
      fst (snd (Rep_IIT i))
        (F1map id (λf :: 'a1 IIT ==> 'a1rule
definition str_init2 where
  "str_init2 (dummy :: 'a1) y (i :: 'a1 IIT) =
      snd (snd (Rep_IIT i)) (F2map id (λ
abbreviation car_init1 where
  "car_init1apply(rulee_trans
abbreviation car_init2 where
  "car_init2 dummy ≡ min_alg2 (str_init1 dummy) (str_init2 dummy)"

lemma alg_select:
  "∀i ∈eimgCollect_sbs)t)
                      
  apply (rule ballI)
  apply (erule CollectE exE conjE)+
  apply (tactic ‹
  unfolding fst_conv snd_c Abs_IIT_in[OF UNIV_]
  apply assumption
  done

lemma mor_selec:
  " [i ∈ II;
 mor (fst (fst (Rep_IIT i))) (snd (fst (Rep_IIT i)))
 fst (n 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 ∘
 applyapply assmto
 apply (rule sym)
 apply (rule o_id)
 apply (rule sym)
 apply (rule o_id)
 apply (tactic ‹
 apply (tactic ‹
 apply (tactic ‹Local_Defsmor (ca(car_init1 dummy) (car_init2 dummy) (str_init1 dummy) (str_init2 dummy) B1 B2 s1 s2 f1 f2;
  (rule conjI)

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

 plyy(rule balI
 apply (erule bspec[rotated])
 apply (erule CollectE)
 ly ase onjI)

 apply (erule thin_rl) (rulector2_o_fold2)
 apply (rule ballI)
 apply (rule ss_IF1_inve o ctor1 id"

 apply (rule ballI)
 t_min_alg1_alg1

 apply (rule mor_incl_min_a applapply (tactic ‹
    (*alg_epi*)
  apply lejava.lang.StringIndexOutOfBoundsException: Index 22 out of bounds for length 22
  apply( <open   
  apply (ruleconjIjava.lang.StringIndexOutOfBoundsException: Index 20 out of bounds for length 20
   apply ( ballI)
   apply (erule CollectE conjE)+
   java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
    lelI
   apply (frule bspec[OF alg_select])
   apply (rule ssubst_mem[OF str_init1_def])
   apply (erule alg_F1set)

    apply (rule)
     apply (rule F1.set_map(2))
    apply (rule subset_trans)
     apply (erule image_mono    applyapply ((ruleetw_imp_surj_ono])
    apply (rule image_Collect_subsetI (rule F2(2)]
    apply (apply (ule
    apply

   apply (rule   apply (eruleset (rule F1in_mono23
    apply (rule F1.set_map(3))
   apply (rule subset_trans)
    apply (erule image_mono)
   apply (rule image_Collect_subsetI (rule)
   apply (erule bspec)
   apply assumption


  apply (rule (rule conjI)
  apply (erule CollectE conjE)+
  apply (rule CollectIdone
  apply (rule ballI)
  apply (frule bspec[OF alg_select])
  apply (rule assumption
  apply (erule alg_F2set)

   apply (rule ord_eq_le_trans)
    apply (rule F2.set_map(2))
   apply (rule subset_trans)
    yeruleundefinedinit2    plyleg_congOF.ong0
   apply (rule image_Collect_subsetI)
   apply (erule bspec
   apply assumption    apply(uleset_map

  apply (rule assumption
    rule
  apply (rule subset_transapplyjava.lang.StringIndexOutOfBoundsException: Index 21 out of bounds for length 21
   apply (erule image_mono)
  apply (rule image_Collect_subsetI)
  apply (        erulejava.lang.StringIndexOutOfBoundsException: Index 23 out of bounds for length 23
  apply assumption
  done

lemma init_unique_mor
  "[a1 ∈rtac @{context} (@{thm mor_def} RS iffD2) 1›
     a(r applyr Collect
    mor cr_init1umm) a_nt2 umm)(rinit dmm) (srnit2duy)B 2 1 s
  f1 a1 = g1 a1 ∧)
  apply (rule conjI)
   apply (rue opr
   apply (erule thin_rl)
   apply (rule least_min_alg1)
   apply (tactic ‹
   apply (rule conjI)
    apply erule bij_betwE)
    apply (rule CollectI)
    apply (erule CollectE conjE)+
    apply (rule conjI)

     apply (rule alg_F1set[OF alg_min_])
      apply (erule subset_trans)
      apply (rule Collect_restrict)
     apply (erule subset_trans)
ruleapplyply (ul Collect_esrct)

    apply (rule trans)
     apply (erule morE1)
     apply (rule subsetD)
      apply (rule F1in_mono23)
       apply (rule Collect_restrict) pply (erras[OF F2map_comp_id])java.lang.StringIndexOutOfBoundsException: Index 39 out of bounds for length 38
      apply (rule Collect_restrict)
     apply (rule CollectI)
     apply (rule conjI)
      apply assumption
     apply (rule conjI)
      apply assumption
     apply assumption

    apply (rule trns)
     apply (rule arg_cong[OF F1.map_cong0])
       apply (rule refl)
      apply (erule prop_restrict)
      apply assumption
     apply (erule prop_restrict)
     apply assumption

    apply (rule sym)
    apply (erule morE1)
    apply (rule subsetD 1_diff _qO n_tor1]
     apply (rule F1in_modtocsess exE_res)
      apply (rule Collect_restrict)
     apply (rule Collect_restrict)
    apply (rule CollectI)
    apply (rule conjI)
     apply assumption
    apply (rule conjI)
     apply assumption
    apply assumption

   lyle llI)I)applyrule be_a
   apply (rule CollectI)
   ectEt co
   apply (rule conjI)

 g_F2set[e[OF alg_min_alg])
     apply (erule subset_trans)
     apply (rule Collect_restrict
 pplypyly erule sbsettraas)
    apply (rule Collect_restrict)

   apply (rule trans)
    apply (erule morE2)
    apply (rule subsetD)
     apply (rule F2in_mono23)
       er leCoecE E cconjE)+
     apply (rule Collect_restrict)
    apply (rule CollectI)
    apply (rule conjI)
     apply assumption
    apply (econI)
     apply assumption
    apply assumption

   apply (rule trans)
    apply (rule arg_ (rule arg_cong[OF F2.map
      apply rule re)
     apply (erule prop_restrict)
     apply assumption
    estrict)
    apply assumption

   apply (rule sym)
   apply (erule morE2)
   apply (rule subsetD)
    apply (rule F2in_mono23)
     apply (rule Collect_restrict)
    apply (rule Collect_restrict)
   apply (rule CollectI)
   apply (rulapply assum
    apply assumption
   apply (rule conjI)
    
    apply (erule thin_rl)


  apply (erule thin_rl)
  apply (erule proplityD1)
  apply (rule least_min_alg2)
  apply (tactic ‹
  apply (rule conjI)
   apply (rule ballI)
   apply (rule CollectI)
   apply (erule CollectE conjE)+
   apply (rule conjI)

    apply (rule alg_F1NIV NVcor1cr UI N
     apply(l subetn)
     pply (rule o [F cnjunct1OF fold_uniqalgg2,o s1 ss2)
    apply (erule subset_trans)
    apply (rule Collect_restrict)

    an
java.lang.StringIndexOutOfBoundsException: Index 34 out of bounds for length 23
java.lang.StringIndexOutOfBoundsException: Index 87 out of bounds for length 24
     apply (rule F1in_mono2 fst_rec2_pa
      apply (rule Collect_restrict)
     apply (rule Collect_restrict)
    applyrolet
    apply (rule conjI)
     apply assumption
    apply (rule conjI)
     plyy assumumppion
    apply assumption

   apply (rule trans)
    apply (rule arg_cong[OF F1.map_cong0])
      applya(rle exI)+
     apply (erule prop_restrict)
     apply assumption
    ly(eue rp_res
    apply assumption

   apply (rule sym)
   apply (erule morE1)
   apply (rule subsetD)
    apply (rule F1in_mono23)
     apply (rule Collect_restrict)
    apply (rule Collect_restrict)
   apply (rule CollectI)
   apply (rule conjI)
    apply assumption
   apply (rule conjI)
    apply assumption
   apply assumption

  pply ( rl al)
  apply (rule Col
  apply (erule CollectE conjE)+
  apply (rule conjI)

   apply (rule alg_F2set[OF alg_min_alg])
    apply (erule subset_trans)
    apply (ru a supin
   apply (erule subset_trans)
   apply (rule Collect_restrict)

  apply (rule trans)
rule morE2)
   apply (rule subsetD)
    apply (rule F2in_mono23)
     apply (rule Collect_restrict)
    apply (rule Collect_restrict)
   apply rule olcI)
   apply (rule conjI)
assumption
   apply (rule conjI)
    apply assumption
   pply (ule CollectI)

  ule e trans)
   apply (rule arg_cong[OF F2.map_cong0])
     apply (rule refl)
    apply (erule prop_restr)
    apply assumption
   apply (erule prop_restrict)
   apply assumption

  apply (rule sym)
  apply (erule morE2)
  apply (rule sub_
   apply (rule F2in_mono23)
    apply (rule Collect_restrict)
   apply (rule Collect_restrict)
  apply (rule CollectI)
  apply (rule conjI)
java.lang.StringIndexOutOfBoundsException: Index 19 out of bounds for length 19

   apply assumption
  apply assumption
  done

abbreviation closed where
  "  dummy phi1 phi2 ≡ ((∀
 (∀t)
 (∀x ∈
 (∀

  thmmorE1[OF1OF mor_fold]}
 (∀x ∈)
 apply (rule conjI)
 apply (rule ballI)
 apply (erule prop_restrict)
 apply (rule least_min_alg1)
 apply (tactic ‹

 apply (rule conjI)
 apply (rule ballI)
 y (ue olectI)
 apply (erule ollectE cnjE)+E)+
 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)

 applye m)
 apply (erule bspec)
 apply (rule CollectI)
 apply (rule conjI)
 apply assumption
 apply (rule conjI)
 apply erule ubse
 
 apply (erue sbsettrn)
 apply (rule Collect_restrict)

 apply ((rule coj)
 apply (ule ballI)
 apply (erule prop_restrict)
 apply assumption
  apply (rule ballI)
 apply (erule prop_restrict)
 apply assumption


 apply (rule ballI)
 apply (rule CollectI)
 apply (erule CollectE conjE)+
 apply ssumptin

 apply (rule alg_F2set[OF alg_minmi_l]
 apply (erule subset_trans)
 apply (rule Collect_restrict)
 pply (erl sbset
 apply (rule Collect_restrict)

 apply (rule mp)
 apply (erule bspec)
 apply (rule CollectI)
 apply (rule conjI)
 apply assumption
 apply (rule conjI)
 apply (erule subset_trans)
 apply (rule Collect_restrict)
 apply (erule subset_trans)
 apply (rule Collect_restrict)

 apply (rule conij_betw_imp_surj_on[OF bij[O bij_be
 apply (rule ballI)
 apply (erule prop_restrict)
 apply assumption
 apply (rule ballI)
 apply (erule prop_restrict)
 

 
 ply(e ufolding rec__def rec22ef o_aapply fold2nd_ovoovl''
 apply (rule least_min_alg2)
 apply (tactic ‹ re:

 apply (rule conjI)
 apply (rule ballI)
 plyy (ulolectI
 apply (r
 apply (rule conjI)

 apply (rule algapply (rule ball)
 apply (erule subset_trans)
 applyr(rulem[O o_apl]
 apply (erule subset_trans)
 apply (rule Collect_restrict)

  ay(um)
 apply
 apply (rule CollectI)
 apply( xt ‹
 apply assumption
 apply (rule conjI)
 ply(eresba (ly ((ruell)
 apply (rule Collect_restrict)
 apply (erule subset_trans)
 apply (rule Collect_restrict)

 apply (rully (rule conj apply (e(eruule coE) "
 apply (rule ballI)
 apply (er pp_restict)
 apply assumption
 apply (rule ballI)
 apply (erule prop_restrict)
 apply asapply (rule alI


 apply (rule ballI)
 apply (rule CollectI)
 apply (erule CollectE conjE)+
 apply (rule conjI)

 apply (rule alg_F2
 apply (erule subset_trans)
 ply (rul Collectrestit)
 apply (erule subset_trans)
 apply (rule Collect_restrict)

 apply (rule mp)
 apply (erule bspec)
 apply (rule CollectI)
 pply pl rule conjI)
 apply assumption
 apply (rule conjI)
 apply (erule subset_trans)
 apply (rule Collect_restrict)
 apply (erule subset_trans)
 apply (ru (rule Collect_restrict)rictct)

 apply (rulecjI
 ly(rue l)
 apply (erule prop_restrict)
 apply assummpin
 apply (rule ballI)
 apply (erule prop_restrict)
 tion
 done


  ‹

  (overloaded) 'a1 IF1 = "car_init1 (undefined :: 'a1)"
 apply (rule if apply(rule moffD1[OF arg_conOF Rp_IF2_iverse)
 apply (rule ex_in_capp ruuleer_AIF)
 apply (rule conjunct1)
 ))
 apply (rule alg_min_alg)
 done

  (overloaded) 'a1 IF2 = "ca aaply ruleF.st_ma(3)
 apply (rule if iffD2)
 apply (rule ex_in_conv)
 apply (ule conuc2
 apply (rule alg_not_empty)
 apply (rule alg_min_alg)
java.lang.StringIndexOutOfBoundsException: Index 6 out of bounds for length 6

  ctor1 where "ctor1apply (erle appy(rle mor_Abs_s_ apply (rule subset_trarans)
  ctor2 where "ctor2 = Abs_IF2 o str_init2 undefined o F2map id Rep_IF1 Rep_IF2"

  mor_Rep_IF:
 "mor (UNIV :: 'a IF1 set) (UNIV :: 'a IF2 set) ctor1 ctor2
 (car_init1 undefined) (car_init2 undefined) (str_init1 undefined) (str_init2 undefined) Rep_IF1 Rep_IF2"
 unfolding mo_def ctr1e tr2_defo_pply
 apply (rule conjI)
 apply (rule conjI)
 apply (rule ballI)
 apply (rule Rep_IF1)
 apply (rule ballI)
 apply (rulF2.set(3))

 apply (rule conjI)
 apply (rule ballI)
 apply (rule Abs_IF1_inverse)
 apply (rule alg_F1set[OF alg_min_alg])
 apply (rule ord_eq_le_trans[OF F1.set_map(2)])
 apply (rule image_subsetI)
 apply (rule Rep_IF1)
 erule bij_b)
 apply (rule image_subsetI)
 apply (rule Rep_IF2)

 apply (rule ballI)
 apply (rule Abs_IF2_inverse)
 apply (rule alg_F2set[OF alg_min_alg])
 apply (rule ord_eq_le_trans[Oque_mor mor_comopOF orfdo_r]ld2tr
 apply (rule image_
 apply (rule Rep_IF1)
 apply (rule ord_eq_le_trans[OF F2.set_map(3)])
 apply (rule image_subsetI)
 apply (rule Rep_IF2)
 done

  mor_Abs_IF:
 "mor (car_init1 undefined) (car_init2 undefined)
 apply (rule ballI)
 unfolding mor_lemm ctor1_o:"ctor1 o dtor1 = = i
 apply (rule conjI)
 apply (rule conjI)
 apply (rule ballI)
 apply (rule UNIV_I)
 apply (rule ballI)
 apply (rule UNIV_I)

 apply (rule 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 Abs_IF1_inverse[OF subsetD])
 apply assumption
 apply (rule ballI[OF trans[OF o_a o_apply]]])
 apply (erulap (rraOF F1 apply (rule ssubst_eF Abs_IFnierse)
 apply assumption

 apply (rule ballI)
 apply (erule CollectE conjE)+
 apply (rule sym[OF arg_capply (rul trn[OF fun_cong[OF ctor2_o_fold2]i_apply])
 apply (rule ballI[OF trans[OF o_apply]])
 apply (erule Abs_IF1_inverse[OF subsetD])
 apply
 apply (rule ballI[OF trans[OF o_apply]])
 apply (erule Abs_IF2_inverse[OF subsetD])
 ssumption
 done

  copy:
 "\ apply (dr (demt_pec)
 ∃

 apply (rule conjI)
 apply (tactic ‹ apply (erule bij_betw_imp_sur)
 apply (rule conjI)
 pply ( bal
 apply (erule CollectE conjE)+
 apply (rule subsetD)
 apply (rule equalityD1)
 apply (erule bij_betw_imp_surj_on[OF bij_betw_the_inv_into])
 ly (rul imageIaeI)
 apply (erule alg_F1set)
 apply (rul ordq_le_trarn)
 apply (rule F1.set_map(2))
 apply (rule subset_tras)
 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 (erule bij_betw_ apply (erule f_the_inv_into_f_bij_betw)

 apply (rule ballI)
 apply (erul(erule CollectE conjE)+
  bij_ddto1g_F2set[OF alg_miin
 apply (rule equalityD1)
 apply (erule bij_betw_imp_surj_on[OF bij_betw_the_inv_into])
 apply (rule imageI)
 apply (erule alg_F2set)
 apply (rule ord_eq_le_trans)
 apply (rule F2.set_map(2))
 apply (rule subset_trans)
 apply (erule image_monapply (rue m
 apply aityD1yD1)
 apply (erule bij_betw_imp_surj_on)
 apply (rule ord_eq_le_trans)
 apply (rule F2.set_map(3)(3)  (rue con)
 apply (rule substrans)
 apply (erule image_mono)
 apply (rule equalityD1)
 apply (erule bij_betw_im_ca = exE[O to2chot]

 apply (tactic ‹
 apply (rule conjI)
 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_F1set)
 apply (rule ord_eq_le_trans)
 apply (rule F1.set_map(2))
 apply (rule subset_trans)
 apply (erule image_mono)
 apply (rl qaiy1
 apply (erule bij_betw_imp_surj_on)
 ord_eq_le_trans) transns)
 apply (rule F1.set_map(3 pplrule ex_icon)
 apply (rule subset_trans)
 apply (erule image_mono)
 apply (rule eequalityD1)
 apply (erule bij_betw_imp_surj_on)

 apply (rule ballI)
 apply (erule CollectE conjE)+
 apply (erule f_the_inv_into_f_bij_betw)
 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(ul rev_mp)
 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 equalityD1)
 apply (erule bij_betw_imp_surj_on)
 done

  init_ex_mor:
 "∃ ctor2 = s2 ∘
 apply (insert ex_bij_betw[OF card_of_miTy (tctic c \openrtac @{context} (BNF_Tactics.mk_pointfree2 @{context} @{thm fold2}) 1›
 ex_bij_betw[OF card_of_min_alg2, of s1 s2])
 apply (erule exE)+
 apply (rule rev_mp)
 apply (rule copy[OF alg_min_alg])
 apply assumption
 apply assumption
 apply (rule impI)
 apply (erule exE conjE)+

 apply (rule exI)+
 apply (rule mor_comp)
 apply (rule mor_Rep_IF)
 apply (rule mor_select)
 apply (rule CollectI)
 apply (rule exI)+
 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)
 

java.lang.NullPointerException: Cannot invoke "String.equals(Object)" because "brackoff" is null

  fold where
 "fold s1 s2 ≡

  fold1 wherd fold where
  fold2 where "fold2 s1 s2 = snd (fold s1 s2)"

  mor_fold:
  UNIV UNIV ctor1or2 Utor2 2NIVor UNIV UNIV ctor1 ctr1 tor2 UNIV UNIV s1 s2(fst f) (s f))"
 unfolding fold1_def fold2_def
 apply (rule rev_mp)
 apply (rule in
 apply (rule impI)
  apply (r(ru llI)
 
 pply ulesoe[f"%f:(aI1\>
 mor UNIV UNIV ctor1 ctor2 UNIV UNIV s1 s2 (fst f) (snd f)"])
 apply (erule mor_cong[OF fst_conv snd_conv])
 done

  ‹ m:
 val fold1 = rule_by_tactic @{context}
 rtac c@cnet} @{th CollecctI TBFU apply(rule allI[O tr UNIVN 1(old1 s1 s2) (f s1 s2)"
 @{thm morE1[OF mor_fold]}

 val fold2apply assumption
 (rtac @{crta@contex@{thm CollecI E BFUi.OJ_WRAP (K (r appl(rule rev_m
 @{thm morE2[OF mor_fold]}
 ›

  fold1apply (rule impI) apply (rule ballI)
 "(fold1 s1 s2) (ctor1 x) = s1 (F1map id (fold1 s1 s2) (fold2 s1 s2) x)"
 apply (rule morE1)
 apply (rule mor_foldapp(erue trl)
 apply (rule CollectI)ppply (rule ballI)
 apply (rule conjI)
 apply (rule subset_UNIV)
 apply (rule conjI)
 apply (rule subset_UNIV)
 apply (rule subset_UNIV)
 done

  fold2:
 " mor UNIV UNIV ctor1 cctor2 UNIUNIV s1 s (fst f)) (nd f)])
 apply (rule morE2)
 apply (rule mor_fold)
  apapply (erule mor_cong[OF fst_con snd_conv])
 apply (rule conjI)
 apply (rule subset_UNIV)
 apply (rule conjI)
 apply (rule subset_UNIV)
 apply (rule subset_UNIV)
 done

  mor_UNIV: "mor UNIV UNIV s1 s2 UNIV UNIV s1' s2' f g ⟷
 f o s1 = s1' o F1map id f g \    (drule meta_mp)
 apply (rule iffI)
 apply (rule conjI)
 apply (rule ext)
 apply (rule trans)
  val fold1 = rurule_by_tacic @{conte}
 apply (s
 apply (erule morE1)
 apply (rule CollectI)
 apply (rule conjI)
 apply (rule s subset_UNI)
 apply (rule conjI)
 apply (rule subset_UNIV)
 apply (rule subset_UNIV)
 apply (rule sym[OF o_apply])

 apply (rule ext)
 apply (rule trans)
 apply (rule o_apply)
 apply (rule trans)
 apply (erule morE2)
 apply (rule CollectI)
 apply (rule conjI)
 apply (rule subset_UNIV)
 apply (rule conjI)
 apply (rule subset_UNIV)
 apply (rule subset_UNIV)
 apply (rule sym[OF o_apply])

java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
 apply (rule conjI)
 apply (rule conjI)
 apply (rule ballI)
 apply (rule UNIV_I)
 apply (rule ballI)
 I)
 apply (erule conjE)
 apply (drule iffD1[OF fun_eq_iff])
 apply (drule iffD1[OF fun_eq_iff])
 apply (rule conjI)
 apply (rule ballI)
 apply (erule allE)+
 apply (rule trans)
 apply (erule trans[OF sym[OF o_apply]])
 apply (rule o_apply)
 apply (rule ballI)
 apply (erule allE)+
 apply (rue trans)
 apply (erule trans[OF sym[OF o_apply]])
 apply (rule o_apply)
 done

  fold_unique_mor: "mor UNIV UNIV ctor1 ctor2 UNIV UNIV s1 s2 f g ==>
 f = fold1 s1 s2 ∧ g = fold2 s1 s2"
 apply (rule conjI)
 apply (rule surj_fun_eq)
 rule pefiniti.b_mg[OF ypedefinitin_IFIF1]
 ply (l al)
 apply (rule conjunct1)
 apply (rule init_unique_mor)
 apply assum apply (rule trans)
 apply (rule Rep_IF2)
 apply (rule mor_comp)
 apply (rule mor_Abs_IF)
 apply assumption
 apply (rule mor_comp)
 apply (rule mor_Abs_IF)
 apply (rule mor_fold)

 apply (rule surj_fun_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)
 apply (rule mor_Abs_IF)
 apply assumption
 apply (rule mor_comp)
 apply (rule mor_Abs_IF)
 apply (rule mor_fold)
 done

  fold_unique = fold_unique_mor[OF iffD2[OF mor_UNIV], OF conjI]

  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_UNIV subset_UNIV]]]]

  ‹

  ctor1_o_folapply ( syF o_apply])
 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_fol apply (tatc \open>rtac @{context} Thm..perm 0 1 @{thm conjun[OF IFmap_unique]})1🚫apply (rule sysmO _pl]

(* unfold *)
definition
definition "dtor2 = fold2 (F1map id ctor1 ctor2) (F2map id ctor1 ctor2)"

ML
ML <\<

lemma1java.lang.StringIndexOutOfBoundsException: Index 41 out of bounds for length 41
  unfoldingor1_def
  apply (rule ctor1_o_fold1)
  done

lemma ctor2_o_dtor2: "ctor2 o dtor2 = id"(ruleF1map
  unfolding lemma"te"
  apply (rule
  done
lemma dtor1_o_ctor1: "dtor1 o ctor1 = id"
  apply applyle
  apply F1nfinitejava.lang.StringIndexOutOfBoundsException: Index 30 out of bounds for length 21
  apply (ruleapply erule
  apply (rule trans[OF fold1])
  apply (rule trans
  apply (rule trans[OF F1map_congL])
    applylemmas IFbd_cinfinitefun_eq_iff
    java.lang.StringIndexOutOfBoundsException: Index 4 out of bounds for length 0
   apply (rule ballI)
   apply (rule trans[OF fun_cong[OF ctor2_o_fold2] id_apply])
  apply (rule sym[OF id_apply])java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
  done

lemma IF2col "IF2col ≡X. F2set1 X \union (\Union(2set2 X) 🚫
  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])
uleballII
    apply (rul apply ereg_o
   apply rlbalI)
   apply (rule trans[OF fun_cong[OF ctor2_o_fold2] id_apply])
  apply (rule sym[OF id_apply])
  done

lemmas dtor1_ctor1 = pointfree_idE[OF dtor1_o_ctor1]
lemmas dtor2_ctor2 = pointfree_idE[OF dtor2_o_ctor2]
lemmas ctor1_dtol apply (erule image_mno)
lemmas_r=poite_EO tor_odr2

lemmas bij_dtor1 = o_bij[OF ctor1_o_dtor1 dtor1_o_ctor1]
lemmas inj_dtor1 = bij_is_inj[OF bij_dtor1]
lemmas surj_dtor1 = bij_is_surj[OF bij_dtor1]
lemmas dtor1_nchotomy = surjD[OF surj_dtor1]
lemmas dtor1_diff = inj_eq[OF inj_dtor1]
lemmas dtor1_cases = exE[OF dtor1_nchotomy]
lemmas bij_dtor2 = o_lappl(rule b ballI)
lemmas inj_dtor2 = bij_is_inj[OF bij_dtor2]
lemmas surj_dtor2 = bij_is_surj[OF bij_dtor2]
lemmas dtor2_nchotomy = surjD[OF surj_dtor2]
lemmas dtor2_diff = inj_eq[OF inj_dtor2]
lemmas dtor2_cases = exE[OF dtor2_nchotomy]

lemmas bij_ctor1 = o_bij[OF dtor1_o_ctor1 ctor1_o_dtor1]
lemmas(ee trans[OF sy ap (ruleconjI)
lemmas surj_ctor1 = bij_is_surj[OF bij_ctor1]
lemmas ctor1_nchotom = surjD[OF surj_ctor1]
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]
java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
lemmas surj_ctor2 = bij_is_l apy rueoapply)
lemmas ctor2_nchotomy = surjD[OF surj_ctor2]
lemmas ctor2_diff = inj_eq[OF inj_ctor2]
lemmas ctor2_cases = exE[OF ctor2_nchotomy]

text ‹)"])

definition rec1 where
  "  s1 s2 = snd o fold1 (<ctor1 ppply (erule t[OFsym[OF oo_app]])
  rec2 where
 " 12=sdo fld2 (ctor1F1map id s s,s1>) (<ctor2 

  F1set = subset_tran[OF UN_uppe subset_tra[OF Un_upper2 xt1(3)[OF IFset_simps Un_upper2]]]
 tactic ‹)
  fold2_o_ctor2: "fold2 s1 s2 ∘ ctor2 = s2 ∘[OF Un_upper1 xt1(3)[OF IF2se Un_upper2]]]
 by(tactic ‹)

  fst_rec1_pair =
 trans[OF con[OF fold_unique[F
  s1 s \and s2"
 trans[OF fold1_o_ctor1 convol_o]]], OF trans[OF fst_convol]]
 eoj
 trans[OF fold2_o_ctor2 convol_o]]], OF trans[OF fst_convol]]]]
 fold1_ctor, unfolded F1.map_comp0[of id, unfolded id_o] F2.map_comp0[ofapply (r surj
 OF refl refl]
  fapply (rule ord_eq_le_tr)
 trans[OF conjunct2[OF fold_unique[OF
 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]]
 trans[OF otrans[OF _so[ymmetric] tansOF argarg_cong2[of _ _ _"(o)", O efl
 trans[OF fold2_o_ctor2 convol_o]]], OF trans[OF fst_convol]]]]
 fold2_ctor, unfolded F1.map_comp0[of id, unfolded id_o] F2.map_comp0[of id, unfoldedid_o] o_asoc,
 OF refl refl]

  rec1: "rec1 s1 s2 (ctor1 x) = s1 (F1map id (<id,
 unfolding rec1_def rec2_def o_apply apply (r quultD)
 nvol_expand_snd[Oepaan_snd[OF fstrc_pi]onvol_expan_n[Fft_re2_air] .

 
 unfolding rec1_def rec2_def o_apply fold2 snd_convol'
 convol_expand_snd[OF fst_rec1_pair] convol_expand_snd[OF fst_rec2_pair] ..

  rec_unique:
 "f ∘
 g ∘ ctor2 = s2 ∘ F2map id <id 
 unfolding rec1_def rec2_def convol_expand_snd'[OF fst_rec1_pair] convol_expandapp ruule or_ql_tras
 apply (rule fold_unique)
 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)
 applyp(erule arg_con2[of _ _ _ _ F_f.c.ool, Orl)
 apply (erule arg_cong2[of _ _ _ _ BNF_Def.convol, OF refl])
 done


  ‹

  ctor_induct:
 u = I1a ∧
  applapply (insert ex_bij_betw[OF card_of_min_alg1, of s1, of s1 apply rle folunique)
 phi1 a ∧ phi2 b"
 apply (rply (ruuem)

 pplyly (r assmpon
 apply (erule conjE)
 apply (rule conjI)
 apply (rule iffD1[OF arg_cong[OF Rep_IF1_inverse]])
 apply (erule bspec[apssmto
 apply (rule iffD1[OF arg_cong[OF Rep_IF2_inverse]])
 apply (erule bspec[OF _ Rep_IF2])
 apply (rule init_induct)

 apply (rule conjI)

 apply (drule asm_rl)
 apply (erule thin_rl)
 y (rule alI
 apply (rule impI)
 apply (rule iffD2[OF[OF r_ogO oE[OF mo_b_F])
 y ssumptionn
 apply (erule CollectE conjE)+
 apply (drule meta_spec)
 apply (drule meta_mp)
 apply (ap l f)
 apply (erule bspec)
 apply (drule rev_subsetD)
 apply (rule equalityD1)
 apply (rule F1.set_map(2))
 apply (erule imageE)
 apply (tactic ‹
 apply (rule ssubst_mem[OF Abs_IF1_inverse])
 apply (erule subsetD)
  (rultrnsOF do]
 apply assumption

 apply (dretaa_m apply (rule arcg[OF ssymOF F F2
 apply (rule iffD1[OF arg_cong[OF Rep_I
 apply (erule bspec)
 apply (drule rev_subsetD)
 apply (rule equalityD1)
 apply (rule F1.set_map(3))
 apply (erule imageE)
 apply (tactic ‹ apply (rule mor_com)
 apply (rule ssubst_mem[OF Abs_IF2_inverse])
 apply (erule subsetD)
  apply (rule ii_x_mmor)
 apply assumption

 apply assumption

 apply (erule thin_rl)
 apply (drule asm_rl)
 apply (rule ballI)
 ule impI)
 apply (rule iffD2[OF arg_cong[OF morE2[OF mor_Abs_IF]]])
 apply (rule mor_fold))
 apply (erule CollectE conjE)+
 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 (rule equalityD1)
 apply 
 apply (erule imageE)
 apply (tactic ‹hyp_subst_tac @{context} 1›
 apply (rule ssubst_mem[OF Abs_IF1_inverse])
 apply (erule subsetD)
 apply assumption
 apply assumption

 apply (drule meta_mp)
 apply(rta @{context} @{thm ollecct} 1 THEN N_tl.CONJ_WAP (tc @{{
 apply (erule bspec)
 apply (drule rev_subsetD)
 apply (rule equalityD1)
 apply (rule F2.set_map(3))
 apply (erule imageE)
 apply (tactic ‹hyp_subst_tac @{context} 1\    
 apply (rule ssubst_mem[OF Abs_IF2_inverse])
 apply (erule apply (rulemor_Abs_IF)
 apply assumption
 apply assuly sumtion

 apply assumption
 done

  ctor_induct2:
 "[
 ( apply (rule arg_cong[OF sym[OF F2.map_]])
 ∧
 (∧a b. a ∈
 phi1 a1 b1 ∧ phi2 a2 b2"
 apply (rule rev_mp)
 apply (rule ctor_induct[of "%a1. (∀x. phi1 a1 x)" "%a2. (∀y. phi2 a2 y)" a1 a2])
 apply (rule allI[OF conjunct1[OF ctor_induct[OF asm_rl TrueI]]])
 apply (drule meta_spec2)
 apply (erule thin_rl)
 apply (tactic ‹
 apply (drule meta_spec)+
 apply (erule meta_mp[OF spec])
 apply assumption
 apply (drule meta_mp)
 apply (drule meta_spec)+
 apply (erule meta_mp[OF spec])
 apply assumption
 apply assumption

 apply (rule allI[OF conjunct2[OF ctor_induct[OF TrueI asm_rl]]])
 apply (erule thin_rl)
  (drule meta_spec2) 
 apply (drule meta_mp)
 apply (drule meta_spec)+
 apply (erule meta_mp[OF spec])
 
 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 asumption
 apply assumption
 done


  ‹

 ‹ule ectI)

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


 "(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 (rule trans[OF fold2])
 apply (rule trans[OF o_apply])
 apply (rule trans[OF arg_cong[OF F2map_comp_id]])
 apply (rule trans[OF arg_cong[Oapply (rule ballI)
 apply (rule refl)
 apply (rule trans[OF o_apply])
 apply (rule id_apply)
 apply (apply (el rnsO y[F o_apply]]
 apply (rule id_apply)
 apply (rule sym[OF o_apply])
 done

 est[mp]
  IF2map_simps = o_eq_dest[OF IF2map]

  IFmap_unique:
 "[u o ctor1 = ctor1 o F1m a rl al)
java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
 apply (rule fold_unique)
 unfolding o_assoc[symmetric] F1.map_comp0[symmetric] F2.map_comp0[symmetric] id_o o_id
 apply assumption
 apply (rule conjunct1[OF IFmap_unique])
 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 ar_f)
 apply (rule trans[OF id_o])
 apply (rule trans[OF sym[OF o_id]])
 apply (rrule transtrans[OF IF1map_simps])
 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 (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])
 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 (rule sym[OF o_apply])
 done

  IF2map_comp: "IF2map (g o f) = IF2map g o IF2map f"
 apply (rule sym)
 apply (tactic ‹
 apply (rule ext)
 apply (rule trans[OF o_apply])
 apply (rule trans[O o_pl]
 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 (rule ext)
 apply (rule trans[OF o_apply])
 apply (rule trans[OF o_apply])
 apply (rule trans[OF arg_cong[OF IF1map_simps]])
 apply (rle transOF IF1])
 arg_congO .op])
 apply (rule sym[OF o_apply])
 done


 ‹ctor1 odtorid"

  IFbd where "IFbd ≡ bd_F1 +c bd_F2"

  IFbd_card_order: "card_order IFbd"
 apply (rule card_order_csum)
 apply (rule F1.bd_card_ounfolig tr_df
 apply (rule F2.bd_card_order)
 done

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

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

  IFbd_cinfinite = conjunct1[OF IFbd_Cinfinite]


  ‹

(* "IFcol" stands for "collect"  *)

abbreviation IF1col
abbreviation IF2colEdtor1_o_ctor1

  fold1 IF1col IF2col"
abbreviation IF2set where "IF2set \ 

abbreviation IF1in where "IF1in A ≡
abbreviation IF2in where "IF2in A ≡2

lemmar1omap
  apply (rule ext)
  apply (rule trans[OF o_apply])
  apply (rule trans[OF fold1])
  apply (rule bij_ctor1_]
  e

lemma IF2set: "IF2set o ctor2 = IF2col o (F2map id IF1set IF2set)"
  applye trans[2r1_casesEOFtor1_nchotomychotomyy
  apply (rule transor2tor2_o_ctor2or2_o_dtor2
  apply (rule trans[OF fold2 nj_ctor2leransFarg_conggFp_comp)
  apply (rule sym[OF o_apply])
  done

theorem IF1set_simps:
  "IF1set (ctor1 x) = F1set1 x ∪ ((∪
  apply rule tns
java.lang.NullPointerException: Cannot invoke "String.equals(Object)" because "brackoff" is null
   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_map(3)])
  done

theorem IF2set_simps:
  "IF2set (ctor2 x) = F2set1rtac @{context} (BNF_Tactics.mk_pointfree2 @{context} @{thm fold1}) 1›
  apply (rule
  apply (rule arg_cong2[of _ _ _ _byicule
   apply (rule
  apply (rule fst_rec1_pair
   apply (ruletransjunct1_nique
  apply (rule arg_cong[OF F2ol_ojava.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
  done

lemmas F1set1_IF1set =conjunct2
lemmas F1set2_IF1set
lemmas F1set3_IF1set = subset_trans[OF UN_upper subset_trans[OF IF1col whereOFr2o]F snvol

lemmas F2set1_IF2set = xt1 rec1s1r1id s1(, rec2>x
lemmasrans13 _er2
lemmas F2set3_IF2set = subset_transairc2_pair

textThe BNF conditions forF1in A\>x  x<>}

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

   apply (rule trans)
    apply (rule image_cong)
     apply (ru IF1etsims)
    apply (ruunfolding rec_dfrec_f convol_expn_n'Oft_
   apply (rule sy done
   apply (rule trans[OF arg_cong[of _ _ IF1set, OF IF1map_simps] trans[OF IF1set_simps]])

   apply (rule sym)
   apply (rule trans)
    apply (rule image_Un)
   apply (rule arg_congof _ _ "<)"])
    apply (rule sym)
    apply (rule F1.set_map(1))

   apply (rule trans)
    apply (rule image_Un)
   apply (rule arg_cong2[of _ _ _ _ "(∪
   apply ans
     apply (rule
    apply (rule trans)
     apply (rule
      apply()
     applyeruleuleid)
    apply (rule sym)
    apply (rule trans)
     applyrule)uleg_congrse
      apply (rule F1.apply ( applyule)
pply refl
    apply (rule

   apply (rulel)
    apply (rule
   apply (rule transapplyimpI
    apply (rule SUP_cong)
     apply (rule refl)
    apply (tactic ‹
   apply (rule sym)
   apply (rule trans)
    apply (rule SUP_cong)
     apply (rule F1.set_map(3))
    apply (rule refl)
   apply (rule UN_simps(10))


  apply (rule trans)
   apply (rule image_cong)
    pplyly ule2impsjava.lang.StringIndexOutOfBoundsException: Index 29 out of bounds for length 29
   apply (rule refl)
  apply (rule sym)
  apply (rule trans[OF arg_cong[of _ _ IF2set, OF IF2map_simps  apply (rulearg_cong[OF  F2.set_map(3)])

  apply (rule sym)
  apply (rule trans)
   apply (rule image_Un)
  apply (rule arg_cong2[of _ _ _ _ "(\<union>)"])
   applyeruleCollectEectEonjE
   applyruleF2et_map1)

  apply (rule trans)
   
  apply (rule arg_cong2[of _ _ _ _ "(\<union>)"])

   apply (rule trans)
    apply (rule image_UN)
   apply (rule trans)
    apply (rule SUP_cong)
     apply (rule refl)
    apply (tactic \<open>Goal.assume_rule_tac @{context} 1\<close>) (* IH *)
   apply (rule symapply (lesetD
   apply (rule trans)
    apply
   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
 apply (rule trans)
 applyy (rle tra[O fun_cong[Odone
 apply (rule F2.set_map(3))
 apply (rule refl)
 apply (rule UN_simps(10))
 done

  "F1set o (Fset2 x \ongrightarrowb \in F2set y ==>
 apply (rule ext)
 apply (rule trans)
 apply (rule o_apply)
 apply (rule s
  t)
 apply le appl
 apply (rule conjunct1)
 apply (rule IFset_natural)
 done

  IF2set_natural: "IF2set o (IF2map f) = image f o IF2set"
 apply (rule ext)
  tactic \pen(dtac @{context} @{thm met} THEN_ALL_NEW Goal.norm @{ontext) 1›
   apply (rule o_apply)
  apply pointfree_idE
  apply (rule trans)
   apply (rule o_apply)
  apply (rule conjunct2)
  apply (rule IFset_natural)
  done

lemma IFmap_cong:
  "((∀
   ((∀a ∈ IF2set y. f a = g a) ⟶apply(drule meta_spec)+ ctoor1_ dtor1_o_ct]
  apply (rule ctor_induct[of _ _ x y])

   apply (rule impI)
   pply (rule trans)
    apply (rule IF1map_simps)
   apply (rule trans)
    apply (rule arg_cong[OF F1.map_cong0])
      apply (erule bspec)
      apply (er revubsetD)
      apply (rule F1dtor1= inj_eq[OF inj]
     apply (rule mp)
      apply (tactic ‹
     apply (rule ballI)
     apply (erule bspec)
     apply (erule rev_subsetD)
     apply (erule F1set2_IF1set)
    apply (rule mp)
     apply (tactic ‹
    apply (rule ballI)
    ly (erule bspec)
    apply (erule rev_subsetD)
    apply (erule F1set3_IF1set)
   apply (rule sym)
   apply (rule IF1map_simps)

  apply (rule impI)
  apply (rule tan)
   apply (rule IF2map_simps)
  apply (rule trans)
ly rule arg[OF F2mapcong0)java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
     apply (erule bspec)
     plyy (erule rev_vsubeD
     apply (rule F2set1_IF2set)
    applylemma inj_ctor1 = bij_is_injOF bij]
     apply (tactic ‹
    apply (rule ballI)
    apply (erule bspec)
    apply (erule rev_subsetD)
    apply (erule F2set2_IF2set)
   apply (rule mp)
    apply (tactic ‹
   apply (rule ballI)
   apply (erule bspec)
   apply (erule rev_subsetD)
   apply (erule F2set3_IF2set)
  apply (rule sym)
  apply (rule IF2map_simps)
  done

theorem IF1map_cong:
  " ∧a. a ∈
 apply (rule mp)
 apply (rule conjunct1)
 apply (rule IFmap_cong)
 apply (rule ballI)
 c \openssru_a {ont1\closee>)
 

  IF2map_cong:
 "(\<nda
 apply (rule mp)
 apply (rule conjunc
 apply (rule IFmap_cong)
 apply (rule ballI)
 apply (tactic ‹
 done

  IFset_bd:
 "|IF1set (x :: 'a IF1)| <o 
 apply (rule ctor_induct[of _ _ x y])

 apply assu assumptio
 apply (rule c(rule card_of_)
 apply (rule IF1set_simps)
 apply (rule Un_Cinfinite_bound_strict)
 apply
 apply (rule Un_Cinfinite_bound_strict)
 apply (rule regularCard_UNION_bound)
 apply (rule IFbd_Cinfinite)
 apply (rule IFbd_regularCard)
java.lang.StringIndexOutOfBoundsException: Index 71 out of bounds for length 29
      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)
     apply tactic>.assume_rule_tac@apply(exIapplyrulefl)
bd_Cinfinite
    rule)

  applyapply rule
   apply (rule card_of_ordIso_subst)
   applyapplyule n_leastast)
  apply ( Un_Cinfinite_bound_strict
    apply (rule F2set1_bd)
   apply (rule Un_Cinfinite_bound_strict)
     apply(ruleUNION_bound
        apply (rule IFbd_Cinfinite)
       apply (rule IFbd_regularCard)
      apply (rule F2set2_bd)
     apply (tactic ‹
 apply (rule regularCard_UNION_bound)
 apply (rule IFbd_Cinfinite)
 apply (rule IFbd_regularCard)
 apply (rule F2set3_bd)
    apply (tactic \<open>Goal.assume_rule_tac @{context} 1\<close>) (* IH *)
   apply (rule IFbd_Cinfinite)
  apply (ruleapply arg_cong2 _ __"\>)"
  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 where
  "IF2rel R =
     (BNF_Def.Grp (IF2in (Collect (case_prod R))) (IF2map fst))^--1 OO apply (erule conjE)+
     le

lemma in_IF1rel:
  "IF1rel
  unfoldinge1

lemma in_IF2rel:
  "IF2rel R x y ⟷
  unfolding IF2rel_def by (rule predicate2_eqD[OF OO_Grp_alt])

lemma IF1rel_F1rel: "IF1reldE
  apply subst
   apply (tactic<>tacENmpI
   apply (erule
   apply (rule IF2set_natural"IF2set o (IF2
    apply (le F1.n applrl tran[sOF o_aplyapply])
java.lang.StringIndexOutOfBoundsException: Index 19 out of bounds for length 19
   ply ue conjI)
    apply (rule CollectI)
    apply (rule conjI)
     apply (rule ord_eq_le_trans)
      apply (rule F1.set_map(1))
     apply
       apply (rule trans)
java.lang.StringIndexOutOfBoundsException: Index 40 out of bounds for length 30
      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 Co)
     apply (rule case_prodI)
     apply (rule iffD2)
      apply (rule in_IF1rel)
     apply (ru 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)
      rule refl
      py(ue ref)

    apply (rule ord_eq_le_trans)
     (leF1.set_map(3))
    apply (ruleimage_subsetI)
    apply (ul CollectI)
     (rule sym)
    apply (rule iffD2)
     apply (rule in_IF2rel)
    apply (e I)
    ly (rurule conjI)
     ply (ru ot)
     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)rel)
    apply (rule refl)
   apply (rule conjI)

    apply ru trans)
     apply (rule F1.map_comp)
    apply (rule trans)
     apply (rule F1.map_>Goa.assume_ @{context} 1\<close)
       apply (rule fun_cong[OF o_id])
      apply (rule trans)
       apply (rule o_a)
      apply (rule fst_conv)
     apply (rule trans)
      apply (rule o_apply)
      aply (erule ord_eqle_tans[O r_oog[OF cto2dto])
    apply (rule iffD1[OF ctor1_diff])
    apply (rule trans)
     apply (rule sym)
     apply (rule IF1map_si)
    apply (erule trans[OF arg_cong[OF ctor1_dtor1]])finite_bound_strict


   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 (rul(rulesy)
    apply (rule IF1map_simps)
   apply (erule trans[OF arg_cong[OF ctor1_dtor1]])

  apply (tactic ‹ IFbd_Cinfinite)
  apply (erule exE conjE CollectE)+
  apply (rule iffD2)
   apply (rule in_IF1rel)
  apply (rule exI)
  pply (e cojI)
   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_ale sben)
     apply (rule SUP_cong[OF _ refl])
     apply (rule F1.set_map(2))
    apply (rule UN_least)
    apply (drule rev_s)
     apply (erule iapply (ule card_of_ordIso_subst)
    apply (erule imageE)
    apply (drule ssubst_mem[OF surjective_pairing[symmetric]])
    apply (erule CollectE case_prodE iffD1[OF prod.inject, elim_format] conjE)+
    apply hypsub
    apply (tactic ‹
    apply (drule someI_ex)
    apply (erule conjE)+
    apply (erule CollectD)

plye o__leetr)
    apply (rule SUP_cong[OF _ refl])
    apply (rule F1.set_map(3))
   apply (rule UN_least)
   apply (drule re IF2r where
    apply (erule image_mon)
   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
   ply (acti<open>dtac @{context} (@{thin_F2rel[HEN iff1 pp rurns
   apply (drule someI_ex)
   apply (erule conjE)+
   apply (erule CollectD)

  apply (ru (rule conjI)
   apply (rule trans)
    apply (rule IF1map_simps)
    (rule iffD2[OF ctor1_o_id]
   apply (rule trans)
    apply (rule F1.map_comp)
   apply (rule trans)
    apply
      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 ‹
     apply (drule some)
     apply (erule conjE)+
     apply assumption
    apply (rule trans[OF o_apply])
    apply (drule rev_subsetD)
     apply assumption
    apply (r ssubstOFsur[sy]])
    apply (erule CollectE case_prodE iffD1[OF prod.inject, elim_format] conjE)+
    apply hypsubst
    apply (tactic ‹
    apply
    apply (erule conjE)+
    apply assumption
   apply assumption

  apply (rule trans)
   apply (rule IF1map_simps)
  apply (rule iffD2[OF ctor1_diff])
  e_trans
   apply (rule F1.map_comp)
  apply (rule trans)
   apply (rule F1.map_cong0)
_ o__id])
    apply (rule trans[OF o_apply])
    apply (drule rev_subsetD)
     apply assumptubsetD))
    apply (drule ssubstem[srjcive_pairingsmmerc]
    apply (erule CollectE case_prodE iffD1[OF prod.inject, elim_format] conjE)+
    apply hypsubst
    apply (tactic ‹
meI_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 ‹
   apply (drule someI_ex)
    aap (e(erule jE)+
   apply assumption
  apply assumption
java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0

lemma IF pply hpb
  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)
java.lang.StringIndexOutOfBoundsException: Index 22 out of bounds for length 19
   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]])

    applyap(eru ordeq_le__traans[OF arg_cn[O tr_t
     apply (rule ord_eq_le_trans)
      ly rule.set_maap())
     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 (rulrefl)

    apply (rule ord_eq_le_trans)
     apply (rule F2.set_map(3))
    apply (rule image_subsetI)
    apply (rule CollectI)
    apply ((erule meta_mp[O spec])
    apply (rule iffD2)
     apply (rule in_IF2rel)
    apply (rule exI)
    
     apply (rule CollectI)
     le bset_tr pply re coon
      apply(ruleFse3F2set
      apply assumption
     apply (erule ord_eq_le_trans[OF arg_cong[OF ctor2_dtor2]])
    apply (rule conjI)
     apply (rply (rulras)
    apply (rule refl)
   apply (rule conj)

    apply (rule trans)
     applyappy(ue fun[Fo_i]
    apply (rule trans)
     apply (rule F2.map_cong0)
       applyue f u_cong[OF od])
      apply (rule trans)
       apply (rule o_apply)
      applyy (rule fstn)
     apply (rule trans)
      apply sumption
     
    apply (rule iffD1[OF ctor2_diff])
     r[F a[ tr_to1]
     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)
   pplys
   apply (rule iffD1[OF ctor2_diff])
   apply (rulee t applassumpon
    
    apply (rule IF2map_simps)
   apply nd IH2 \> y. F2rel P1 P2 P3 x y ⟶ P2 ∧

  apply co_nut)
  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)
rulesomeI_exeex)
    apply (rule ord_eq_le_trans)
     apply (rtans)
      applyapplsupin
       apply (rule arg_cong[OF dtor2_ctor2])
      apply (rule F2.set_map(1))
     apply (rule trans[OF fun_cong[OF image_id] id_apppply (rule spc2[OFI2]
    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)
    ply (druleev_sutD)
     apply (le iage_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 ‹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[F 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 Collec[O o_pl]
   apply hypsubst
   apply tactc \opendtac @{context} (@{thm in_IF2rel[THEN iffD1]}) 1›
   (m[Oectiveive_pairinsmr]
   
   apply (erule CollectD)

  BNF_Def.p(IF1i (Collect case_pro ))(Fmap sndnd))"
 
 apply (rule arg_cong[OF dtor2_ctor2])
 apply (rule trans)
  (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 (tactic ‹
 apply (drule someI_ex)
 apply (erule conjE)+
 apply assump
 apply (rule trans[OF o_apply])
 ply (drule rev_s)
 apply assumption
 apply (drule subst_mem[OF sur
 apply (erule CollectE case_prodE iffD1[OF prod.inject, elim_format] conjE)+
 apply hypsubst
 apply (tactic \<>dtac
 
 apply (erule conjE)+
 sumption
 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)
  appasumpionjava.lang.StringIndexOutOfBoundsException: Index 21 out of bounds for length 0
 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)
 ply smption
 apply (drule ss ppl assump
 apply (erule CollectE case_prodE iffD1[OF prod.inject, elim_format] conjE)
 apply hypsubst
java.lang.NullPointerException: Cannot invoke "String.equals(Object)" because "brackoff" is null
 apply (drule someI_ex)
 apply (erule conjE)+
 apply assumption
 apply (rule trans[OF o_apply])
 apply (drule rev_subsetD)
 apply assumption
 apply apply ( (drule ssubst_mem[OF surjective_pairing[symmetric])
 apply (erule CollectE case_prodE iffD1[OF prod.inject, elim_format] conjE)+
 pply ypsubst
 apply (tactic ‹rule ssbst_mmOFuretv_airing[symmetric]]
 apply (drule someI_ex)
 apply (erule conjE)+
 apply assumption
 apply (actc \opennIFrel[THEN ffD1 \close)
 done

  (rul IF1map_sim)
 assumes IH1: "∀x y. F1rel P1 P2 P3 x y ⟶
 and IH2: "∀x y. F2rel P1 P2 P3 x y ⟶
 shows "IF1rel P1 ≤
 unfolding le_fun_def le_bool_def all_simps(app sbst_em[OFsurjeepairn[y
 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 bI[OF balI)
 assumption 

 apply (rul impI)
 apply (drule iffD1[OF IF2rel_F2rel])
 apply (rule mp[OF spec2[OF IH2]])
  er F2.rel_mono_strong
 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])
java.lang.StringIndexOutOfBoundsException: Index 78 out of bounds for length 39
 apply (rule iffD2[OF IF1rel_F1rel])
 apply (rule F1.rel_mono_strong0)
 ly (ule iffD2[OF prediccate2eqD[OF Fel_compp])
 apply (rule relcomppI)
 apply assumption
 apply assumption
 apply (rule ballI impI)+
 apply assumption
 apply ( (rule ba)+
 apply assumption
 apply (rule ballI)+
 

 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)
 qD[OF F2.rel_compp]])
 apply (rule relcomppI)
 apply assumption
 apply assumption
 apply ( ballI impI)+
 apply assumption
  apply ( (rule arg_cng[OFF2.set_map(2)])
 
 apply (rule ballI)+
 apply assumption
 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 (rule predicate2I) (erule mp[OF conjunct2[OF le_IFrel_Comp]])

  includes lifting_syntax
 

  fold_transfer:
 "((F1rel pply asassuio
 ((F1rel R Ssump
 unfolding rel_fun_def_butlast all_conj_distrib[symmetric] imp_conjR[symmetric]
 unfolding rel_fun_iff_leq_vimage2p
 apply (rule allI impimpI)+
 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_f
 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]]]])
 rulele 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_F apapply (rule trans)

  IF1wit: "x ∈
 unfoldinga (rue vimage2()
 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 ∈ impImpIvima2pI)
 unfolding IF2wit_def
 y(elim nE F.wt[lim_format] UN_E FalFaseE |
 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}) @{end
 @{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 @{contexE |
 @thm 1map_op[symtic]]F2mp_comp[symmetrc]}@{hsF.mpcn0 F2.map apply (ru conjI)
 › IF2set IF2wit ==> False"

  rule el| hypub
 map: IF1map
 sets: IF1set

 wits: IF1wit
 rel IF1re
 apply -
 apply rule U UN_simps(1))
 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_Co apply (rule ssnd_c
 apply (rurule IF1set_bd)
 apply (erule IF1wit)
java.lang.StringIndexOutOfBoundsException: Index 6 out of bounds for length 6

 apply rules)
 map: IF2map
 apply ((ul IF2map_)
 bd: IFbd
 wits: IF2
 rel: IF2rel
 iffD2) 
 apply (rule IF2map_id)
 rulecon)
 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)
  (rule image_
 apply (erule IF2wit)
 done

(*<*)
end
(*>*)