Impressum FrecR.thy

section‹Well-founded relation on names›
theory ‹ relation on nam on name› begin

lemmas omega_iff_sats 
  fun_plus_iff_sats FOL_sats_iff

text‹termfrecR›
us to define forcingusefine<

(* MOVE THIS. absoluteness of higher-order composition *)
definition
  is_hcomp :: "[i==>o,i==>i==>o,i==>i==>o,i,i] ==> o" where
  "is_hcomp(M,is_f,is_g,a,w) ≡ ∃z[M]. is_g(a,z) ∧ is_f(z,w)" 

lemma (in
  assumesdefinition
    is_f_absAnda z. M(a) ==> is_flongleftrightarrow
    is_g_abs:"∧ ∃z[M]. is_g(a,z) ∧
    :"∧ M(g(a))"
    ( "w
  shows
    "is_hcomp(M,is_f,is_g,a,w) \<is_f_abs"a z. M(a) ==> is_g(a,z) ⟷
  unfolding is_hcomp_def using assms by simp

definition
  hcomp_fm :: "[
  "(g,,) equv xssAdpsc)0,(,ucw)

lemma sats_hcomp_fm:
"(Mis_f,aw longleftrightarrow w = f(g(a))"
b z. a∈ b\\i>na ==>M ==>
                 is_f(nth(a,Cons(z,env)),nth(b,Cons((z,eenv))) 🚫,a,w Exists(And(pg(succ(a),0),pf(0,succ(w))))"
    and
    g_iff_sats:"\nat ==>naM \<ngrightarrowghtarrowa b z. a∈nat ==>iM ==>
                is_g(nth(a,Cons(z,env)),nth(b,Cons(z,env))) ⟷
    and
    a∈"in" "env∈( 
  shows
    "sats(M,hcomp_fm(pf,pg,a,w),env) \<proof M" "w∈w
proof -
  have "sats(M, pf(0, succ(w)), Cons(x, env)) ⟷ is_f(x,nth(w,env))" if "x∈M" "w∈nat" for x w
    using f_iff_sats[of 0 "succ(w)" x] that by simp
  moreover
  have "sats(M, pg(succ(a), 0), Cons(x, env)) ⟷ is_g(nth(a,env),x)" if "x∈M" "a∈nat" for x a
    using g_iff_sats[of "succ(a)" 0 x] that by simp
  ultimately
  show ?thesis unfolding hcomp_fm_def is_hcomp_def using assms by simp
qed


(* Preliminary *)
definition
  ftype :: "i==>i" where
  "ftype ≡ 0 x] by simp

definition
  name1 :i\Rightarrow>i" where
  "name1 [of  x]that simp

definition
name2Rightarrow>i" where
  "name2

definition
  ftypeRightarrowi" where
  "cond_of()≡((x))))"

java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
  "ftype(⟨) = f"
  "name1name2 fst(snd(snd(x)))"
  "name2(⟨ :: "i==>
  "cond_of(⟨,n2c<angle) = c"
  unfolding ftype_def name1_def name2_decond_of_def
  by simp_all

definition eclose_n :: "[i==>
  "eclose_n(name,x) = eclose({name(x)})"

definition
  ecloseN i" where
  "(x) = eclose_n,x) \union eclose_n(ame2x)

lemma :
<langle,c<)"
  "n2 ∈ ecloseN(⟨>"
  unfolding ecloseN_def eclose_n_def
  using by simp_a

lemmas names_simp = components_simp(2) components_simp(3)

lemma ecloseNI1 :
  sumes" \in eclose() ∨eclose(n2
  shows
  unfoldingi\Rightarrow i" wh
  using assms eclose_sing names_simp
  by auto

lemmas ecloseNI = ecloseNI1

lemma ecloseN_mono :
  assumes "u ∈ ecloseN(⟨,,c<>"
  shows ""n2 \\> clsN\langle,n12c<>"
proof -
   <nu\n_<close>
  consider (a) "u∈eclose({name1(x)})" | (b) "u∈ components_simp arg_into_eclose by auto
    unfolding ecloseN_def eclose_n_def by auto
  then
  show ?thesis
  proof cases
    shows "x ∈f,n1n2,∠
    with
    showecloseNI
      unfolding
      using eclose_singE ecloseN(x)" "name1(x) ∈n
  next
    case b
    with ‹
 showthesis
 unfolding ecloseN_def eclose_n_def
 using eclose_singE[OF b] mem_eclose_trans[of u "name2(x)"] by auto
 qed
 


(* ftype(p) \<equiv> THE a. \<exists>b. p = \<langle>a, b\<rangle> *)

definition
  is_fst :: "(i==>
  "     a
                       (🚫

definition
  fst_fm :: "[i,i] ==>
  "fst_fm>Or(Exists(pair_fm(succ(t),0,succ(x))),
                   And(NegusinggEse_trans(x)" ] by auto

lemma sats_fst_fm :
  "[ x ∈ nat; y ∈ list(A) ]
    \    show 
        is_fst]eclose_transjava.lang.StringIndexOutOfBoundsException: Index 72 out of bounds for length 72
  bydd_ef

definition 
  is_ftypeis_fst<equiv z[M]. pair(M,t,z,x)) ∨
  "is_ftype ≡

definition
  ftype_fm :: " :,]<Rightarrow
  "ftype_fm ≡ Or(Exists(pair_fm(succ(t),0,succ(x))),

lemma sats_ftype_fm :
  "[)"
    ==>
         "<lbrakk> x ∈ nat;env>list(A) ]
  unfolding ftype_fm_def
  by (simp add(##A, nth(,), nthenv)"

lemma is_ftype_iff_sats:
  assumes
    "nth
  shows
    is_ftype(##A,aa,bb)  ⟷ sats(A,ftype_fm(a,b), envjava.lang.StringIndexOutOfBoundsException: Index 75 out of bounds for length 75
  using sats_ftype_fm :
  bysimp

definition
  is_snd\Rightarrow)Rightarrowi==>o" where
  "is_snd(Mis_ftype(#A nthx,), nth))"
                       (<>(∃w[M]. pair(M,z,w,x)) \and empty(M,t))"

definition
  snd_fm:
  java.lang.StringIndexOutOfBoundsException: Index 9 out of bounds for length 9
                   And(Neg(Exists(stsempty_fm

lemma sats_snd_fm :
  "[ longlefrigarow ts(,typema,,e)"
    ==>(x,y)),env
        is_snd(##A, nth(x,env(y,env"
  by (simp add: snd_fm_def is_snd_def)

definition
  is_name1 :: "(i==>o)==>i==>i==>o" where
  "is_name1(M,x,t2) ≡ (simp:sats_ftype_fm)

definition
  name1_fm: "[,i] \Rightarrowi"re
  "ame1_fm((x,) \equivpf(fst_,snd_fm,x,t)" 

lemma sats_name1_fm :
  "[ nat; y ∈ list(A) ]
    ==> sats(A, name1_fm(x,y), env) ⟷
        is_name1(##A, nth(x,env), nth(y,env))"
  unfolding name1_fm_def(x,t) ≡(t),succ
 "is_fst(##A)" _ fst_fm simp

lemma is_name1_iff_sats:
  assumes
    "nth(a,env) = aa" "nth(b,env) =bb""a<in>nat""\in""env ∈
  shows
    "is_name1(##A,aa,bb)  ⟷ sats(,name1_fm, env
  using
  by (simp add:sats_name1_fm(#,env

definition
  is_snd_snd
  "s_snd_snd(M(M,x,t <equiv>> is_cm(ssd()isn(),x,t)"

definition
  snd_snd_fm : [,Rightarrow
  "snd_snd_fm(x,t) ≡ is_hcomp(M,is_fst(M),is_snd(M),x,t2)"

lemma sats_snd2_fm :
  "[ x ∈
    ==>(x,y), env) ⟷
        is_snd_snd(##A, nth(x,env), nth(y,env))"
  unfolding _snd_def
    sats_hcomp_fm[of A "is_snd(##A)" _ snd_fm

definition
  is_name2 :: "(i==>o)==>i==>i==>o" where
  "is_name2(M,x,t3) ≡ is_hcomp(M,is_fst(M),is_snd_snd(M),x,t3)"

definition
  name2_fm :: "[i,i] ==> i" where
  "name2_fm(x,t3) ≡ hcomp_fm(fst_fm,snd_snd_fm,x,t3)"

lemma sats_name2_fm :
  "[ x ∈ nat; y ∈ nat;env ∈ list(A) ]
    ==> sats(A,name2_fm(x,y), env) ⟷
        is_name2(##A, nth(x,env), nth(y,env))"
  unfolding name2_fm_def is_name2_def using sats_fst_fm sats_snd2_fm
    sats_hcomp_fm[of A "is_fst(##A)" _ fst_fm "is_snd_snd(##A)"] by simp

lemma is_name2_iff_sats:
  assumes
    "nth(a,env) = aa" "nth(b,env) = bb" "a∈nat" "b∈nat" "env ∈ list(A)"
  shows
    "is_name2(##A,aa,bb) ⟷ sats(A,name2_fm(a,b), env)"
  using assms
  by (simp add:sats_name2_fm)

definition
  is_cond_of :: "(i==>o)==>i==>i==>o" where
  "is_cond_of(M,x,t4) ≡ is_hcomp(M,is_snd(M),is_snd_snd(M),x,t4)"

definition
  cond_of_fm :: "[i,i] ==> i" where
  "cond_of_fm(x,t4) ≡ hcomp_fm(snd_fm,snd_snd_fm,x,t4)"

lemma sats_cond_of_fm :
  "[ x ∈ nat; y ∈ nat;env ∈ list(A) ]
    ==> sats(A,cond_of_fm(x,y), env) ⟷
        is_cond_of(##A, nth(x,env), nth(y,env))"
  unfolding cond_of_fm_def is_cond_of_def using sats_snd_fm sats_snd2_fm
    sats_hcomp_fm[of A "is_snd(##A)" _ snd_fm "is_snd_snd(##A)"] by simp

lemma is_cond_of_iff_sats:
  assumes
    "nth(a,env) = aa" "nth(b,env) = bb" "a∈n \<Longrightarrow (x,y), env) ⟷(##A, nth(x,env), nth(y,env))
  ows
    "is_cond_of
java.lang.StringIndexOutOfBoundsException: Index 22 out of bounds for length 13
   (simp:sats_cond_of_fm

lemma components_type[TC] :
  assumes java.lang.NullPointerException: Cannot invoke "String.equals(Object)" because "brackoff" is null
  shows
    "ftype_fm(a,b)∈formula"
    "name1_fm(a,b)∈
    "name2_fm(a,b)∈(imp add:satssatname_f)
    "cond_of_fm(is_cond_ofi==>i==>o" where
  sing assms
  unfolding ftype_fm_def fst_fm_def snd_fm_def snd_snd_fm_def name1_fm_def name2_fm_def
    cond_of_fm_def hcomp
  by simp_all

lemmas sats"cond_of_fm>hcomp_fm(snd_fm,snd_snd_fm,x,t4

lemmas :
  is_cond_of_iff_sats; y \in> nat ∈ 

lemmas components_defs ftype_fm_def snd_snd_fm_def
  name1_fm_def cond_of_fm_def


definition
  is_eclose_n is_cond_of_iff_sats:
  "is_eclose_n(N,is_name,en,t) ≡
        ∃

definition
  eclose_n1_fm :: "[i,i] ==>shows
  "eclose_n1_fmmt) \equiv> Exists(Exists(And(And(name1_fm(t#+2,,0),si(0,1)),
                                        assms

definition
  eclose_n2_fm :: "[i,i ==>i"where
  "eclose_n2_fm(m,t) ≡ Exists "a\<innatnat"
                                       (1m+))

definition
  is_ecloseN :: "[i==>formula"
  "is_ecloseN(N,en,t) ≡formula"
                is_eclose_n(N"
                union(N,en1en2en)"

definition 
  ecloseN_fm "i \> where
  "ecloseN_fm(enjava.lang.StringIndexOutOfBoundsException: Index 13 out of bounds for length 13
                            And(eclose_n2_fm(0,#2,on_fm,n#)))
lemmaecloseN_fm_type [TC] :
  "[ en ∈ nat ; t ∈ ==> formula"
  unfolding ecloseN_fm_def eclose_n1_fm_def eclose_n2_fm_def by simp

lemma sats_ecloseN_fm [simp]:
  "[
    ==>(A, ecloseN_fm(en,t), env) ⟷,env),nth(t,env))"
  oseN_fm_defose_n1_fm_def___is_eclose_n_def
  using  nth_0(m+))
    is_singleton_iff_sats[symmetric]
  by auto

(* Relation of forces *)::"i,i \<Rightarrow 
definition
  frecR :: "i ==> o" where
  "frecR(x,y) ≡
    (ftype(x) = 1 ∧ ftype(y) = 0 
      ∧ (name1(x) ∈
   ∨ ftypey)=  1<nd name1(x) = name1(y) ∧ name2(x) ∈y))"

lemma frecR_ftypeD :
  assumes "frecR(x,y)"
  shows "(ftype(x) = 0 ∧ ∃en1<>n2[N].
  using unfolding by auto

lemma frecRI1: java.lang.NullPointerException
  ding frecR_dedef by (simpadd:coments_simp)

lemma frec': "s <> , q\rangle,<>0, n1, q'<>)
  unfolding frecR_def by (simp add:components_simp)

lemma frecRI2: "s ∈ An(ecclose2m(t#2),union_m(,0,en#+2))))))"
  unfolding ecloseN_fm_type [TC] :

lemma frecRI2': "s ∈ domain(n1) ∪ domain(n2) ==> frecR(⟨ en ∈ nat ] formula"
   ecloseN_fm_defdefse_n2_fm_defimp


lemma sats_ecloseN_fm
  unfolding frecR_def\lbrakk eninnat; t ∈ nat ; env ∈

lemma frecRI3': "s ∈ close_n1fm_def eclose_n is_eclose_n_def
  unfolding fusi nth_0 nth_ConsI sats_name1_fm sats_name2_fm

lemma frecR_iff :
  "frecR(x,y) ⟷
    (ftype(x) = 1 ∧frecR: \Rightarrow o" where
      ∧(x) \<in  domain(name2(y)) ∧(x) == name1(y)∨
   ∨ (ftype(x) = 0 ∧ ftype(y) = 0
  unfolding frecR_def ..      \> (name1(x) ∈ domain(name1(y)) \>d(name2(y)) ∧ name2(x) = name2(y))))

lemma frecR_D1 :
  "frecR(x,y) Longrightarrowftype(y) = 0 ==> ftype(x) = 1 ∧
      (name1 "frecR(x,y)"
  
  by auto

lemmaRI1\in domainn1 s ∈) <Longrightarrow (\langle>1, s, n1, q⟩, ⟨ n2,q'\>"
  "frecRy)<ongrightarrow 
      ftype(x) = 0 ∧
  using frecR_iff: "s \<>  domain(n2) ==>1, s, n2, q⟩0, n1, n2, q'⟩
  by auto

lemma_DI
  assumes "frecR(⟨ domain(n1) ∪frecR(<langle1 s, n2, q⟩, ⟨0, n1, n2, q'⟩
  shows ecRI3 "\langle>s, r\rangle ∈ n2 ==> frecR(⟨0, n1, s, q⟩1, n1, n2, q'⟩
  using assms unfolding frecR_def by (force simp add:components_simp)

(*
name1(x) ∈ domain(name2(y)) ∧
            (name2(x) = name1(y) frecR_iff :
          ∨ name2(x) ∈
definition
  is_frecR :: "[\Rightarrowi>o" where
  "is_frecR(M,x,y) ≡)in domain(name1(y)) ∪ (name2 ame1<orname2(x) = name2(y))))
  is_ftypeM,ftx <and_name1n1x<and (M,x,n2x) ∧
  is_ftype(M,y,fty) ∧ is_name1(M,y,n1y) ∧ is_name2(M,y,n2y (ftype(x) = 0 ∧ name1(x) = name1)\and2( <>main
          ∧ is_domain 
          ((name1 domain(name1(y)) ∪ (name2(x) = name1(y) ∨
           ∨

schematic_goal ftype(y) = 1 ==> 
  assumes ftype(y) =  1 ∧e2 domain(name2
    "i\< auto,⟨)"
  shows
    "is_frecR(##A,a,b) \<longleftrightarrow"
  unfolding is_frecR_def is_Collect_def
  by (insert assms ; (rule sep_rules' cartprod_iff_sats co (name2(x) = name(y) ∨
        | simp del:sats_car\>nm1x) = name1(y)\and name2(x) ∈ domain(name2(y))*)

synthesize "frecR_fm" from_schis_frecR :::: "[==> o" where

(* Third item of Kunen observations about the trcl relation in p. 257. *)
lemma eq_ftypep_not_frecrR:
  assumes "ftype(x) = ftype(y)"
  shows "¬
  usingassms by force


definition
  rank_namesRightarrow i" where
  "rank_names(x) ≡

lemma rank_names_types [TC: 
  shows "Ord(rank_names(x))"
  unfolding rank_names_def max_def using Ord_rankis_frecRA,,b <longleftrightarrow(i)nv

definitionunfolding is_Collect_def  
  mtype_form :: "i ==> i" where
  "mtype_form(x) ≡ s ; (rule sep_rules' cartprod_iff_sats componentsiff_sats

definition
  type_form :: "i ==> "frecR_fm"from_schematic sats_frecR_fm_auto
  "type_form(

lemma type_form_tc [TC]:
  shows "type_form(x) ∈ey"
  unfolding type_form_def mtype_form_def by auto

lemma frecR_le_rnk_names :
  assumes "frecR(x,y)"
  shows "rank_names(x)≤
proof -
  obtain a b c  rank_names: i \Rightarrow> i" wher
    H: "a = name1(x)" "b = name2(x)
    "c = name1(y)"showsrank_names())"
    "(a ∈ domain(c)∪domain(d) ∧ (b=c ∨ b = d  unfolding rank_names_defmax_def using Ord_Un
    using assms unfolding frecR_def by force
  then\Rightarrow> i"
  consider
    (m) "a ∈ []: 
    | (n) "a ∈ mtype_form_def by auto
    | (o) "b ∈lemma frecR_le_rnk_names:
    by auto
  then show ?thesis proofjava.lang.StringIndexOutOfBoundsException: Index 32 out of bounds for length 32
    case m
    then 
    have "rank(a) < rank(c)" 
      using eclose_rank_lt  in_dom_in_eclose  by simp
    with ‹ domain(c) ∧ b = d) "
 how ?thesis unfolding rank_names_def using Ord_rank max_con max_cong2 leI by auto
 nextnext
 by auto
 then
 have "rank(a) < rank
 using eclose_rank_lt in_dom_in_eclose by simp
 rank(a) < rank(d)›
    showopen(a) < rank(c)›
      using Ord_rank max_cong2 max_cong max_commutesnext
  th
    casehaveank(d)"
    then
    have "rank(b) < rankd") "rank(a) = rank(c)" (is "?a = _")
      using eclose_rank_lt in_dom_in_eclose using eclose_rank_lank_lt in_dom_in_e_in_eclose e by simp
    with H
    show ?thesis unfolding rank_names_def
      using Ord_rank max_commutes max_cong2[OF leI[OF ‹], of ?a] by simp
  show ?thesis unfldng rank_
java.lang.StringIndexOutOfBoundsException: Index 33 out of bounds for length 3


definitionshow isoldg rank_names_def
  Γ?b < ?d›], of ?a] by simp
  "\<Gamma>(x) = 3 ** rank_names(x) ++ type_form(x)"

lemma \<Gamma>_type [TC]: 
  shows "Ord(<Gamma(x))"
  unfolding \<Gamma>_def by simp


lemma<>_no: 
  assumes "frecR(x,y)"
  shows "\<Gamma>(x) < \<Gamma>(y)"
proof -
  have F: "type_form(x)from sms
    usingtIby p_alll
  from
  have A: "rank_names(x) \<lehave?)oldingngnames_defgOrd_rankx_def
    using frecR_le_rnk_names by simp
  then
  have "Ord(?y)" then
  note leE[OF \roofs
  then
  show ?thesis
  proof(cases)
    case 1
    then 
    show java.lang.StringIndexOutOfBoundsException: Index 6 out of bounds for length 6
  next
    case 2
     ") =0 \<>ftype(y) = 1" | (b) "ftype(x) = 1 \<and> ftype(y) = 0"
      using sigma\<in> domain(?\<sigma>') \<union> domain(?\<tau>')")
    then show ?thesis proof(cases)
      
      then 
      have "type_form(y) = 1" 
        using type_form_def imp
      from b
      have H: "name2(x)  name1me11)\or name2x) = name2(y) " (is "?\<tau> = ?\<sigma>' \<or> ?\<tau> = ?\<tau>'")
        "name1(x) \<in> domain(name1(y)) \<union> domain(name2(y))" 
        (is "?\<sigma> \<in> domain(?') \<union> domain(?\<tau>')")
        using assms unfolding type_form_def frecR_def by auto
      then 
      have E: "rank(?<tau>= nk\<') \<or> rank(?\<tau>) =ank<>)by auto
      from H
      consider (a) "rank(?\<sigma>) < rank(?\<sigma> 
        using eclose_rank_lt in_dom_in_eclose by force
      then
      have "rank(?\<sigma>) < rank(?\<tau>)" proof (cases)
        case a
        with \<open>rank_names(x) = rank_names(java.lang.StringIndexOutOfBoundsException: Index 3 out of bounds for length 3
        show ?thesis nfoldingank_names_defype_form_defform_defype_form_defsinggD2  ajava.lang.StringIndexOutOfBoundsException: Index 95 out of bounds for length 95
            E assms Ord_rank by simp
      next
        case b
        with \<open>rank_names(x) = rank_names(y) \"ond_of_of\> A4 cond_ofond_of()<in> A4"
        show ?thesis unfolding rank_names_def mtype_form_def type_form_def 
          using max_D2[OF _ b] max_commutes E assms then ?esis
      
      And\<tau> \<theta> p.  p \<in> A2 \<Longrightarrow> \<lbrakk><ndq \<sigma>. \<lbrakk> q\<in>A2 ; \<sigma>\<in>domain(\<tau>) \<union> domain(\<theta>)\<rbrakk> \<Longrightarrow> Q(1,\<sigma>,\<tau>,q) \and Q(1,\<sigma>,\<theta>,q)\<rbrakk> \<Longrightarrow> Q(0,\<tau>,\<theta>)
      have "type_form(x) = 0" unfolding type_form_def mtype_form_def by simp
      with \<open>rank_names(x) = rank_names(y) \close \<open>type_form(y) = 1\<close> \<open>type_form(x) = 0\<close>
      sis
        unfolding \<Gamma>_def by auto
    next
      a
      then 
      have "name1(x) = name1(y)" (is "?\<sigma> = ?\<sigma>'") 
        "name2(x) \<in> domain(name2(y))" (is "?\<au><in omainmain\<')")
        "type_form(x) = 1"
        using assms unfolding type_form_def frecR_def by auto
      then
      have "rank(?\<sigma>) = rank(?\<sigma>')" "rank(?\<tau>) < rank(?\<> 
        using  eclose_rank_lt in_dom_in_eclose by simp_all
      with \<open>rank_namesx =rank_namesmesy \> 
      aveanknk?<tau)\e rank(?\<sigma>')" 
        unfolding rank_names_def using Ord_rank max_D1 by simp
      with a
      have "type_form(y) = 2"
        unfolding type_form_def mtype_form_def using not_lt_iff_le       have "<sigma>\<in> eclose(?\<theta>)" 
      with \<open>rank_names(x) = rank_names(y) \<close> \<open>type_form(y) = 2\<close> \<open>type_form(x) = 1\<close>
      hesis
        unfolding \<Gamma>_def by auto
qed
  qed
qed

definition
  frecrel :: "i \<Rightarrow> i" where
  "frecrel(A) \<equiv> Rrel(frecR,A)"

lemma frecrelI : 
  umesx<n A" "y\<in>A" "frecR(x,y)"
  shows "\<langle>x,y\<rangle>\<in>frecrel(A)"
  assmsoldingfrecrel_defl_deffy tojava.lang.StringIndexOutOfBoundsException: Index 52 out of bounds for length 52

lemma frecrelD :
  assumes \<langle>,\rangle\<in> frecrel(A1\<times>A2\<times>A3\<times>A4)"
  shows "ftype(x) \<in> A1" "ftype(x) \<in> "
    "name1(x) \<in> A2" "name1(y) \<in> A2" "name2(x) \<in> A3" "name2(x) \<in> A3" 
    "cond_of(x) \<in> A4" "cond_of(y) \<in> A4" 
    "frecR(x,y)"
  using assms unfolding frecrel_def Rrel_def ftype_def by (auto simp add:components_simp)

lemma wf_frecrel : 
  shows "wf(frecrel(A))"
proof -
  have "frecrel(A) \<subseteq> measure(A,\<Gamma>)"
    unfolding frecrel_def Rrel_def measure_def
    using \<Gamma>_mono by force
  then show ?thesis using wf_subset wf_measure by auto
qed

lemma core_induction_aux:
  fixes A1 A2 :: "i"
  assumes
    "Transset(A1)"
    "\<And>\<tau> \<theta> p.  p \<in> A2 \<Longrightarrow> \<lbrakk>\<And>q \<sigma>. \<lbrakk> q\<in>A2 ; \<sigma>\<in>domain(\<theta>)\<rbrakk> \<Longrightarrow> Q(0,\<tau>,\<sigma>,q)\<rbrakk> \<Longrightarrow> Q(1,\<tau>,\<theta>,p)"
    "\<And>\<tau> \<theta> p.  p \<in> A2 \<Longrightarrow> \<lbrakk>\<And>q \<sigma>. \<lbrakk> q\<in>A2 ; \<sigma>\<in>domain(\<tau>) \<union> domain(\<theta>)\<rbrakk> \<Longrightarrow> Q(1,\<sigma>,\<tau>,q) \<and> Q(1,\<sigma>,\<theta>,q)\<rbrakk> \<Longrightarrow> Q(0,\<tau>,\<theta>,p)"
  shows "a\<in>2\<times>A1\<times>A1\<times>A2 \<Longrightarrow> Q(ftype(a),name1(a),name2(a),cond_of(a))"
proof (induct a rule:wf_induct[OF wf_frecrel[of "2\<times>A1\<times>A1\<times>A2"]])
  case (1 x)
  let ?\<tau> = "name1(x)" 
  let ?\<theta> = "name2(x)"
  let ?D = "2\<times>A1\<times>A1\<times>A2"
  assume "x \<in> ?D"
  then
  have "cond_of(x)\<in>A2" 
    by (auto simp add:components_simp)
  from \<open>x\<in>?D\<close>
  consider (eq) "ftype(x)=0" | (mem) "ftype(x)=1"
    by (auto simp add:components_simp)
  then 
  show ?case 
  proof cases
    case eq
    then 
    have "Q(1, \<sigma>, ?\<tau>, q) \<and> Q(1, \<sigma>, ?\<theta>, q)" if "\<sigma> \<in> domain(?\<tau>) \<union> domain(?\<theta>)" and "q\<in>A2" for q \<sigma>
    proof -
      from 1
      have A: "?\<tau>\<in>A1" "?\<theta>\<in>A1" "?\<tau>\<in>eclose(A1)" "?\<theta>\<in>eclose(A1)"
        using  arg_into_eclose by (auto simp add:components_simp)
      with  \<open>Transset(A1)\<close> that(1)
      have "\<sigma>\<in>eclose(?\<tau>) \<union> eclose(?\<theta>)" 
        using in_dom_in_eclose  by auto
      then
      have "\<sigma>\<in>A1"
        using mem_eclose_subset[OF \<open>?\<tau>\<in>A1\<close>] mem_eclose_subset[OF \<open>?\<theta>\<in>A1\<close>] 
          Transset_eclose_eq_arg[OF \<open>Transset(A1)\<close>] 
        by auto         
      with \<open>q\<in>A2\<close> \<open>?\<theta> \<in> A1\<close> \<open>cond_of(x)\<in>A2\<close> \<open>?\<tau>\<in>A1\<close>
      have "frecR(\<langle>1, \<sigma>, ?\<tau>, q\<rangle>, x)" (is "frecR(?T,_)")
        "frecR(\<langle>1, \<sigma>, ?\<theta>, q\<rangle>, x)" (is "frecR(?U,_)")
        using  frecRI1'[OF that(1)] frecR_DI  \<open>ftype(x) = 0\<close> 
          frecRI2'[OF that(1)] 
        by (auto simp add:components_simp)
      with \<open>x\<in>?D\<close> \<open>\<sigma>\<in>A1\<close> \<open>q\<in>A2\<close>
      have "\<langle>?T,x\<rangle>\<in> frecrel(?D)" "\<langle>?U,x\<rangle>\<in> frecrel(?D)" 
        using frecrelI[of ?T ?D x]  frecrelI[of ?U ?D x] by (auto simp add:components_simp)
      with \<open>q\<in>A2\<close> \<open>\<sigma>\<in>A1\<close> \<open>?\<tau>\<in>A1\<close> \<open>?\<theta>\<in>A1\<close>
      have "Q(1, \<sigma>, ?\<tau>, q)" using 1 by (force simp add:components_simp)
      moreover from \<open>q\<in>A2\<close> \<open>\<sigma>\<in>A1\<close> \<open>?\<tau>\<in>A1\<close> \<open>?\<theta>\<in>A1\<close> \<open>\<langle>?U,x\<rangle>\<in> frecrel(?D)\<close>
      have "Q(1, \<sigma>, ?\<theta>, q)" using 1 by (force simp add:components_simp)
      ultimately
      show ?thesis using A by simp
    qed
    then show ?thesis using assms(3) \<open>ftype(x) = 0\<close> \<open>cond_of(x)\<in>A2\<close> by auto
  next
    case mem
    have "Q(0, ?\<tau>,  \<sigma>, q)" if "\<sigma> \<in> domain(?\<theta>)" and "q\<in>A2" for q \<sigma>
    proof -
      from 1 assms
      have "?\<tau>\<in>A1" "?\<theta>\<in>A1" "cond_of(x)\<in>A2" "?\<tau>\<in>eclose(A1)" "?\<theta>\<in>eclose(A1)"
        using  arg_into_eclose by (auto simp add:components_simp)
      with  \<open>Transset(A1)\<close> that(1)
      have "\<sigma>\<in> eclose(?\<theta>)" 
        using in_dom_in_eclose  by auto
      then
      have "\<sigma>\<in>A1"
        using mem_eclose_subset[OF \<open>?\<theta>\<in>A1\<close>] Transset_eclose_eq_arg[OF \<open>Transset(A1)\<close>] 
        by auto         
      with \<open>q\<in>A2\<close> \<open>?\<theta> \<in> A1\<close> \<open>cond_of(x)\<in>A2\<close> \<open>?\<tau>\<in>A1\<close>
      have "frecR(\<langle>0, ?\<tau>, \<sigma>, q\<rangle>, x)" (is "frecR(?T,_)")
        using  frecRI3'[OF that(1)] frecR_DI  \<open>ftype(x) = 1\<close>                 
        by (auto simp add:components_simp)
      with \<open>x\<in>?D\<close> \<open>\<sigma>\<in>A1\<close> \<open>q\<in>A2\<close> \<open>?\<tau>\<in>A1\<close>
      have "\<langle>?T,x\<rangle>\<in> frecrel(?D)" "?T\<in>?D"
        using frecrelI[of ?T ?D x] by (auto simp add:components_simp)
      with \<open>q\<in>A2\<close> \<open>\<sigma>\<in>A1\<close> \<open>?\<tau>\<in>A1\<close> \<open>?\<theta>\<in>A1\<close> 1
      show ?thesis by (force simp add:components_simp)
    qed
    then show ?thesis using assms(2) \<open>ftype(x) = 1\<close> \<open>cond_of(x)\<in>A2\<close>  by auto
  qed
qed

lemma def_frecrel : "frecrel(A) = {z\<in>A\<times>A. \<exists>x y. z = \<langle>x, y\<rangle> \<and> frecR(x,y)}"
  unfolding frecrel_def Rrel_def ..

lemma frecrel_fst_snd:
  "frecrel(A) = {z \<in> A\<times>A . 
            ftype(fst(z)) = 1 \<and> 
        ftype(snd(z)) = 0 \<and> name1(fst(z)) \<in> domain(name1(snd(z))) \<union> domain(name2(snd(z))) \<and> 
            (name2(fst(z)) = name1(snd(z)) \<or> name2(fst(z)) = name2(snd(z))) 
          \<or> (ftype(fst(z)) = 0 \<and> 
        ftype(snd(z)) = 1 \<and>  name1(fst(z)) = name1(snd(z)) \<and> name2(fst(z)) \<in> domain(name2(snd(z))))}"
  unfolding def_frecrel frecR_def
  by (intro equalityI subsetI CollectI; elim CollectE; auto)

end