Quelle  FrecR.thy

sectionWell-founded nnames
theory FrecR imports Names Synthetic_Definitionbegin

lemmas sep_rules'
  fun_plus_iff_sats FOL_sats_iff omega_iff_satsFOL_sats_iff 

text‹‹ is the well-founded relation on names that allows
  to d forcing for atomic formulas.🚫

(* MOVE THIS. absoluteness of higher-order composition *)
definition
  is_hcomp ::     is_f_abs:"∧ M(z) ==>(a,z) ⟷> z = f(a)" and
  "is_hcomp(M,is_f,is_g,a,w) ≡ is_f(z,w)" 

lemma (in M_trivial    g_closeda. M(a) ==>
  assumes    "M()"M(w)"
    :"<And>a z. M(a) ==> M(z) ==> is_f(a,z) ⟷ z = f(a)" and
    is_g_abs:"∧ M(z) ==> z = g(a)" and
    g_closed:"∧
    "M(a)" "M(w)"
  shows"comp_fm(f,pga,w <uv> it(ndpg(uc(a,0),pf0suc(w)))"
    is_hcomp,is_f,is_g,) ⟷
  unfolding is_hcomp_defnat ==>b<nnat z∈ 

definition
  hcomp_fm ::                 is_f,szv)\longleftrightarrow sats(M,pf(a,b),Cons(z,env))"
  "hcomp_fm(pf,pg) ≡

lemma sats_hcomp_fm:
  assumes 
    f_iff_sats:"\<Andand<And>a b z. a∈ b∈t ==> z∈Longrigh>
                 is_f(nth(a,Cons(z,env)),nth(b,Cons(z,env))) ⟷ sats(M,pf(a,b),Cons(z,env))"
    and
    g_iff_sats:"∧nat ==> b∈ z∈
                is_g(nth(a,Cons(z,env)),nth(b,Cons(z,env))) ⟷ sats(M,pg(a,b),Cons(z,env))"
    and
    "a\<in"nat" w<>natlistM)"
  shows
    "sats(M,hcomp_fmshows
proof-
  have "sats(M, pf(0, succ(w)), Cons(x, env)) ⟷ is_f(x,nth(w,env))" if "x∈nat" for x java.lang.StringIndexOutOfBoundsException: Index 120 out of bounds for length 120
    using f_iff_sats[of "succ(w)"]that
  moreover
  have "sats(M, pg(succ(a), 0), Cons(x, env)) ⟷::"\Rightarrow
    using g_iff_sats "succ(a)"0x]that bysimp
  ultimatelydefinition
    name2 :: "i==>
qed


(* Preliminary *)
definition
  ftype :: "i==>()<equiv> snd(snd(snd


definition
  name1 :: "i==>i" where
  "name1(x) ≡ fst(snd(x))"

definition
  name2 :: "i==>f,n1,n2,c⟩
  "(x) ≡

definition
  cond_ofi\Rightarrowi" where
  "cond_of(x) ≡f,n1n2,c∠def

lemma components_simp:
  "ftype(⟨
  "name1
  "ecloseN :: "i ==>ecloseN(name1\union>eclose_n(ame2,x)"
  "cond_of components_in_eclose:
  unfolding ftype_def>f,n1,n2c∠f,n1,n2,c⟩
  p_all

definition eclose_n
  "eclose_n(nam assu "\>eclosen1 x∈)"

definition
  ecloseN :: "i <>where
  

lemma_
  "n1 ∈fn1n2,c\<rangle)
  n2 \inele(<>f,n,n,\rangle)
  from <pen>u>u\<>_u \in eclose({name2(x)})"
  usingo

lemmas names_simp = components_simp(2) components_simp(3)

lemma ecloseNI1
  assumes
  shows ecloseN(⟨,n2c<>"
  unfolding ecloseN_def eclose_n_def
  using assms eclose_sing names_simp
  by auto

lemmas eclo = ecloseNI1

lemmaunfolding ecloseN_def eclose_n_def
  assumes "u ∈ ecloseN(y)" "name2(x) <>ecloseN(y)"
  shows "u ∈ ecloseN(y)"
proof -
  from ‹name2(x) ∈ _› ?thes
  consider (a) "u
    unfolding
  then
  show ?thesis
  proof cases
    case
    with                       \ot(∃z[M]. ∃w[M]. pair(M,w,z,x)) ∧ empty(M,t))"
    show ?thesis
      unfolding ecloseN_def e(x,t) \<equiv 
       eclose_singE[OF a] mem_ecloseof u "name1
  next
    case b
    with ‹ nat;env ∈
  ?thesis
 unfolding ecloseN_def eclose_n_def
 using eclose_singE[OF b] em_eclof u "name2(x)"] by auto
 qed
 


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

definition
  is_fst :: "(i==>o)==>by (simp ad: fst_m_de is_fst_def)
  "(M,x,t) ≡(∃ 
                       (¬ is_fst"

definition
  fst_fm : [i,i] \Rightarrow> i" where
  "fst_fm(x,t) ≡
                   And(Neg(Exists(Exists(pair_fm(0,1,2 #+ x)))),empty_fm(t))

lemma sats_fst_fm :
  "lbrakk nat; y ∈ ∈ 
    ==>  is_ftype_def
        is_fstnthxenv(y,env)
  by (simp add: fst_fm_def

definition 
  is_ftype
  "is_ftype \<"is_ftype)"

definition
  ftype_fm :: "[i,i] ==> i" where
  "ftype_fm ≡ fst_fm" 

lemmasats_ftype_fm 
  " (siadd:sats_ftype_fm)
    ==> :: "(i<>o==>i==>
        is_ftype#A,nth(,env(y,env
  unfolding ftype_fm_def\not(<>[M]. ∃<>mpty
  by (simp add

lemma is_ftype_iff_sats
  assumes
    "nth(a,env) = aa" "nth(b,env) = bb" "a∈nat" "b∈(Exists(pair_fm(1,0,2 #+ x)))),e(t)))"
  shows
    "is_ftype(##A,aa,bb) <eftihtrw> asA,fy_f(,) ev"
  using sats(A, snd_fmxy,) ⟷), nth))java.lang.StringIndexOutOfBoundsException: Index 44 out of bounds for length 44
  by add)

definition
  is_snd :"i<> i where
  "is_snd(M,x,t)"ame1_fmt<> hcom_m(f_fm,
                       (¬ x ∈ nat;env ∈

definition
  snd_fm :: "[i,i] ==>is_name1"
  "snd_fmequiv> Or(Exists(pair_fm(0,succ(x))),
                   And  "is_snd(##A)"] bysimp

lemma sats_snd_fmnthv b "a\in b<i>nat"env list(A)"
  "[((a,b), )"
    \using assms
        is_snd#A, th(x,env), nth(y,env))"
  by (simp

definition"s_snd_snd,)\ivhopM,i_n()i_sdM
  is_name1 :: "(isnd_snd_fm::"i,i]<>i" where
  "is_name1(M,x,t2) ≡

definition
  name1_fm :: "[i,i] sats(A,snd_snd_fmenv
  "name1_fm(x,t) ≡ snd_snd_fm_def is_nd_snd_def using sats_snd_fm

lemma sats_name1_fm :
  "java.lang.NullPointerException: Cannot invoke "String.equals(Object)" because "brackoff" is null
<sats(A, name1_fmlongleftrightarrow>
        is_name1"
  unfolding name1_fm_def is_name1_def using sats_fst_fm sats_snd_fm
    sats_hcomp_fm[oshow

lemma is_name1_iff_sat
  assumes
    "nth(a,env) = aa" "nth(b,env) = bb" "a∈nat" "by add)
  
    "is_name1(##A,aa,bb) ⟷ sats(A,name1_fm(a,b), env)"
  using assms
  by (simp add:sats_name1_fm)

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

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

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

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) \<    "formula"
  name2_fmformula"
  by simp add:s_ame2m)

definition
  is_cond_of :: "(i\Rightarrowo)==>i==>
  "is_cond_of(M,x,t4) ≡usissms

definition
  cond_of_fm
  cond_of_fm(x(x,t4) ≡)"

lemma sats_cond_of_fm
  nat<in;envin list(A) ]
    ==> = fst_fm_def snd_fm_def hcomp_fm_def
        is_cond_of name2_fm_def
  unfolding
    sats_hcomp_fm

lemmais_cond_of_iff_satsjava.lang.StringIndexOutOfBoundsException: Index 26 out of bounds for length 26
  assumes
    "nth(a,env) = aa" "nth(b,env) = bb" "a∈
  shows
    "is_cond_of(##A,aa,bb(,)equiv(And2,singleton_fm,
  using ms
  by (simp add:sats_cond_of_fm)

lemma components_type[TC]==> java.lang.StringIndexOutOfBoundsException: Index 47 out of bounds for length 47
  assumes>" "b∈
  showsis_eclose_fm,#+2)))"
    "ftype_fm(a,b)∈
    "name1_fm(a,b)∈
    "name2_fm(a,b)∈
    ula
  using assms,,en
  unfolding
    cond_of_fm_def hcomp_fm_def :: "[i,] \Rightarrow i"ere
  by simp_all

lemmas sats_components_fm[simp] = sats_ftype_fm sats_name1_fm sats_name2_fm sats_cond_of_fm

lemmas components_iff_sats = is_ftype_iff_sats is_name1_iff_sats is_name2_iff_sats
  is_cond_of_iff_sats

lemmas components_defs = fst_fm_def ftype_fm_def snd_fm_def snd_snd_fm_def hcomp_fm_def
  name1_fm_def name2_fm_def cond_of_fm_def


definition
  is_eclose_n :: "[i==>t+)uni(1,0,e#+2)))"
  "is_eclose_n(N,is_name,en,t) \< ecloseN_fm_type nat ] ecloseN_fm(en,t) ∈
        ∃n1

definition
  eclose_n1_fm :: "[i,i] ==> satsN_fm<>is_ecloseN(##A,nth(en)""
  "eclose_n1_fm is_ecloseN_def eclose_n1_fm_def eclose_n2_fm_def is_e
                                       is_eclose_fm(1,#+)))"

definition 
  eclose_n2_fm  []\> i" where
  "eclose_n2_fm(m,t) ≡ i ==>
                                       is_eclose_fm(1,m#+2))))"

definition
  is_ecloseN :: "[i==>o,i,i] ==> (ftype(x) = 0 ∧(y    🪙 domain(name2()"
  "is_ecloseN(N,en,t) ≡[N].∃
                is_eclose_n(N,is_name1 assms frecR_def
                union

definition 
  ecloseN_fm :: "[unfol frecR_def by (sp add:components_simp)
  "ecloseN_fmRI1<indomain(n1) ∪ domain(n2) ==> frecR(⟨1, s, n1<> \langle, n2rangle"
                             And(ecse_n2_f0,#+ioion_f(110,en#+2)))))"
lemmaecloseN_fm_typeTCjava.lang.StringIndexOutOfBoundsException: Index 28 out of bounds for length 28
  "[ nat ; t ∈ ==> ecloseN_fm(en,t) ∈
  unfoldinge eclose_n1_fm_ eclose_n2_fm_def by sim

lemma s [simp]:
  "<> <>  list(A) ]
    ==>
  unfolding ecloseN_fm_def is_ecloseN_def __fe_n2_fm_def
  usingonsIfm
    is_singleton_iff_sats[symmetric]
  by autofrecR_ifff

(* Relation of forces *)
definition
  frecR ::"i \> i ==>
  "frecR(x,y) ≡ (name1\>domain(name1(y)) ∪ (name2) name1 \or> name2(x) = name2(y))))
    (ftype(x) = 1 ∧
      🪙\union omainme2<and (name2(x) = name1(y) ∨)
   ∨<>  

lemma frecR_ftypeD :
  assumesy)
  shows "(ftype(x) = 0
  using assms unfolding frecR_def by auto

lemma frecRI1: "s <>() ∨ domain(n2==>frecR<<langle0, n1,n2 q'<rangle)
  unfolding frecR_def(x, ==>ftype(y) = 1 ==> ftype(x) = 0 ∧ 

lemma frecRI1': "s ∈ domain(n1) ∪ domain(n2) ==> frecR(⟨1, s, n1, q⟩, ⟨0, n1, n2, q'⟩)"
  unfolding frecR_def by (simp add:components_simp)

lemma frecRI2indomain(n1) ∨ s ∈ frecR(⟨, ⟨)"
  unfolding :

lemma frecRI2': "s ∈ domain(n2) ==> frecR<>,)"
  unfolding frecR_def by (simp add:components_simp)


ecRI3: "<><>  > ⟨)"
  unfolding frecR_def by (auto simp add:components_simp)

lemma frecRI3': "s ∈
  unfolding frecR_def domain(name1(y)) ∪ 

lemmacR_iff
  "frecR(x,y) ⟷ name1(x) = name1(y) ∧ domain(name2(y))*)
    (ftype(x) = 1 ∧[i<>o,i,i] \<Rightarrow 
      ∧ (name1(x ∈ domain(name2(y)) ∧(x) = name1(y) \<> (Mx,) \and> is_M,x,n1)\<>is_name2
   ∨ ftype(y) = 1 ∧(y) <and> name2(x)<in do(name2(y)))"
  unfolding frecR_def ..

lemma frecR_D1 :
  "frecR(x,y) ==> ftype(y) = 0 ==> is_domain(M,n1y,dn1) ∧(M,n2y,dn2) ∧
      name1(x) ∈ domain(name2(y)) ∧ name2(x) = name2(y)))"
  using frecR_iff
  by auto

lemma frecR_D2
  "frecR(x,y) ==> ftype(x) = 0 ∧
      ftype(x) = 0 ∧ name1(x) = name1(y) ∧ name2(x) ∈(y))"
  using frecR_iff
  by

lemma frecR_DI : 
  assumes "frecR(⟨a,b,c,d⟩ftype(y),name1(y),name2(y),cond_of(y)⟩
  
  using assms unfolding frecR_def by (force simp add:components_simp)

(*
name1(x) ∈ domain(name1(y)) ∪ domain(name2(y)) ∧
            2(x) = name1(y) \<oror name2(x) = name2(y))
          \or ae(x) = name1((y) \<>name2
definition
  s_frecRecR :: "i<>,i,i] ==>
  "is_frecR(M,x,y) ≡
  is_ftype(M,x,ftx) ∧
  is_ftype(M,y,fty) ∧ is_name1(M,y,n1y) ∧ frecR(x,y)"
          ∧ assms frecR_ftypeD force
          (  (number1(M,ftx
              :: "i ==>

schematic_goal sats_frecR_fm_auto:
  assumes
    "i∈] 
  shows
    "is_frecR(##,aab)\longleftrightarrow> sats(A,?fr_fm(i,j,e)"
    is_frecR_def
  by (insert assmscartprod_iff_satsiff_satsmponents_iff_sats_ff_sats
        | simp del

synthesize from_schematicsats_frecR_fm_auto

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


definition
  rank_names :: " <Rightarrowre
  "rank_names(x) ≡)

lemma rank_names_types [TC]:
  shows "Ord(rank_namesx)java.lang.StringIndexOutOfBoundsException: Index 28 out of bounds for length 28
  unfoldingrank_names_def max_def using Ord_rank by auto

definition
  mtype_form :: "i ==> i" where
  "mtype_form(x) ≡

definition
  type_form :: "i <Rightarrow" where
  "type_form(x) ≡

lemma type_form_tc[TC
  shows "type_form(x) ∈ 3"
  unfolding type_form_def type_form_defto

lemma :
  assumes  "frecR(x,y)"
  shows "rank_names(x)≤(cases)
proof -
  obtain a b c d where
    H: "a = name1(x)" "b = name2(x)"
    "c = name1(y)" "d = name2(y)"
    "(a ∈ domain(c)∪domain(d) ∧ (b=c ∨ b = d)) ∨ (a = c ∧ b ∈ domain(d))"
    using assms unfolding frecR_def by force
  then
  consider
    (m) "a ∈ (b = c ∨
    | (howf singg ong2uto
    |   ext
    byauto
  then show
    case m
    then 
    have "rank(a) < rank(c)" 
      using eclose_rank_lt  in_dom_in_eclosewith ‹ H n
 with \<>rank H m
 show ?thesis unfolding rank_names_def using Ord_rank max_cong max_cong2 leI by auto
 next
 case n
 en
  "rank(a) < rank(d)" (is "?b < ?
 ng eclose_rank_lt _dom_in_eclose by simp
 with ‹?b < ?d›
 show ow ?thesis unfolding rank_names_def
 using Ord_rank max_cong2 ma
 next
 case o
 then
 have "rank(b) < rank(d)" (is "?b < ?d") "rank(a) = rank(c)" (is "?a = _")
 using eclose_rank_lt in_dom_in_eclose by simp_all
 with H
 show ?thess unfoldiingank_names_def
 using Ord_rank max_commutes max_cong2[OF leI[OF ‹
 qed
 


 
 Γ :: "i ==> i" where
 "Γ

  Γ_type [TC]:
 shows "Ord(ΓΓ
 unfolding Γ_def by simp


  Γ_mono :
 assumes "frecR(x,y)"
 shows \Gammamo :
  -
 have F: "type_form(x) < 3" "type_form(y) < 3"
 using ltI by simp_all
  assm
 have A: "rank_names(x) ≤ ltI simp_all
 using frecR
 then
 have "Ord(?y)" unfolding rank_ using Or max_d by simp
 note leE[OF ‹?x≤?y\<closeusing
 then
 show ?thesis
 roof(cases)
 case 1
 then
 show ?thesis unfolding Γ_def using oadd_lt_mono2 ‹
 next
 case 2
 consider (a) "ftype(x) = 0 ∧ ftype(y) = 1" | (b) "ftype(x) = 1 ∧ ftype(y) = 0"
 using fre"ftype(x = 0 ∧
 then show ?thesis proof(cases)
 case b
 then
 have "type_form(y) = 1"
 using type_form_def by simp
 from b
 have H: "name2(x) = name1(y) ∨ name2(x) = name2(y) " (is "?τ = ?σ' ∨ ?τ = ?τ'")
 "name1(x) ∈ domain(name1(y)) ∪ domain(name2(y))"
 (is"?\<> 
 using assms unfolding type_form_def frecR_def by auto
 then
 have E: "rank(?τ) = rank(?σ') ∨ rank(?τ) = rank(?τ')" by auto
 from H
 consider
 using eclose_rank_lt in_dom_in_eclose by force
 then
 have "rank(?σ) < rank(?τby simp
 case a
 with ‹ = name1(y) \<>(
 show ?thesis unfolding rank_names_def mtype_form_def type_form_def using max_D2[OF E a]
 E assms Ord_rank by simp
 next
 case b
 with ‹rank_names(x) = rank_names(y) ›
 show ?thesis unfolding rank_names_def mtype_form_def type_form_def
 using max_D2[OF _ b] max_commutes E assms Ord_rank disj_commute by auto
 qed
 with b
 have "type_form(x) = 0" unfolding type_form_def mtype_form_def by simp
 with ‹
 show ?thesis
 unfolding Γ_def by auto
 next
 case a
 then
 have "name1(x) = name1(y)" (is "?σ?🚫
 "name2(x) ∈ domain(name2(y))" (is "?τ\tau) rank(?σ r(?\tau')" byauto
 "type_form(x) = 1"
 using assms unfolding type_form_def frecR_def by auto
 then
 have "rank(?σ) = rank(?σ')" "rank(?τ) < rank(?τ')"
 using eclose_rank_lt in_dom_in_eclose by simp_all
 ose>
 have "rank(?τ') ≤ rank(?σ')"
 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 assms by simp
 with ‹
 show ?thesis
 unfolding Γ_def by auto
 qed
 qed
 

 
 frecrel :: "i ==> i" where
 "frecrel(A) ≡ Rrel(frecR,A)"

  frecrelI :
 assumes "x ∈thesis unfol rank_n mtype_ t usin max_[OF E a]
 shows "⟨x,y⟩∈frecrel(A)"
 using assms unfolding frecrel_def Rrel_def by auto

  frecrelD :
 assumes "⟨x,y⟩ ∈ frecrel(A1×A2×A3×A4)"
 shows "ftype(x) ∈ A1" "ftype(x) ∈ A1"
 "name1(x) ∈
 "cond_(x) \in" ""c()\in
 "frecR(x,y)"
 using assms unfolding frecrel_def Rrel_def ftype_def by (auto simp add:components_simp)

  wf_frecrel :
 shows "wf(frecrel(A))"
  -
 have "frecrel(A) ⊆
 unfolding frecrel_def Rrel_def measure_def
 using Γ_mono by force
 then show ?hesis using wf_subset wf_measure by auto
 

  core_induction_aux:
 fixes A1 A2 :: "i"
 assumes
 "Transset(A1)"
 "∧τqed
 "∧🪙<> ,p)"
 shows "a∈2×A1×A1×A2 ==> Q(ftype(a),name1(a),name2(a),cond_of(a))"
  (induct a rule:wf_induct[OF wf_frecrel[of "2×A1×A1×A2"]])
 case (1 x)
 let ?τ = "name1(x)"
 let ?θ = "name2(x)"
 let ?D = "2×A1×A1×A2"
 assume "x ∈ ?D"
 then
 have "cond_of(x)∈A2"
 by (auto simp add:components_simp)
 from ‹x∈?D›
 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, σ, ?τ, q) ∧ Q(1, σ, ?θ, q)" if "σ ∈ domain(?τ) ∪ domain(?θ)" and "q∈A2" for q σ
 proof -
 from 1
 have A: "?τ∈A1" "?θ∈A1" "?τ∈eclose(A1)" "?θ∈eclose(A1)"
 using arg_into_eclose by (auto simp add:components_simp)
 with ‹Transset(A1)› that(1)
 have "σ∈eclose(?τ) ∪ eclose(?θ)"
 using in_dom_in_eclose by auto
 then
 have "σ∈A1"
 using mem_eclose_subset[OF ‹?τ∈A1›] mem_eclose_subset[OF ‹?θ∈A1›]
 Transset_eclose_eq_arg[OF ‹Transset(A1)›<> 
 by auto
 with ‹q∈A2› ‹?θ ∈ A1› ‹cond_of(x) show ?thesis
 have "frecR(⟨
 "fcase
 using frecRI1'[OF that(1)] frecR_DI ‹
 frecRI2'[OF that(1)]
 by (auto simp add:components_simp)
 with ‹x∈?D›t> in> doma?τ
 have "⟨?T,x⟩∈ frecrel(?D)" "⟨?U,x⟩∈ frecrel(?D)"
 using frecrelI[of ?T ?D x] frecrelI[of ?U ?D x] by (auto simp add:components_simp)
 with ‹
 have "Q(1, σ, ?τ, q)" using 1 by (force simp add:components_simp)
 moreover from ‹tau')"
 have "Q(1, σ, ?θ, q)" using 1 by (force simp add:components_simp)
 ultimately
 show ?thesis using A by simp
 qed
 then show ?thesis using assms(3) ‹ftype(x) = 0›
 next
 () = r rank_na(y) ) \<<close
 have "Q(0, ?τ, σ, "rank(\>' \<> 
 proof -
 from 1 assms
 have "?τ
 using arg_into_eclose by (auto simp add:components_simp)
 with ‹
 have \<<sigma
 using in_dom_in_eclose by auto
 then
 have "σ∈A1"
 using mem_eclose_subset[OF ‹
 by auto
 with ‹q∈A2› ‹?θ ∈ A1› ‹cond_of(x)∈ show ?thesi
 have "frecR(⟨0, ?τ qe
 using frecRI3'[OF that(1)] frecR_DI ‹
 by (auto simp add:components_simp)
 with ‹
 have "⟨?T,x⟩∈ frecrel(?D)" "?T∈?D"
 using frecrelI[of ?T ?D x] by (auto simp add:components_simp)
 with ‹q∈A2›
 show ?thesis by (force simp adassumes " \<> 
 qed
 then show ?thesis using assms(2) ‹
 qed
 

  def_frecrel : "frecrel(A) = {z∈A×A. ∃ using assms unfolding Rrel_def b auauto
 unfolding frecrel_def Rrel_def ..

  frecrel_fst_snd:
 "frecrel(A) = {z ∈ A×A .
 ftype(fst(z)) = 1 ∧
 ftype(snd(z)) = 0 ∧ name1(fst(z)) ∈ domain(name1(snd(z))) ∪ domain(name2(snd(z))) ∧
 (name2(fst(z)) = name1(snd(z)) ∨ name2(fst(z)) = name>xy\<> 
 ∨ (ftype(fst(z)) = 0 ∧
 ftype(snd(z)) = 1 ∧ name1(fst(z)) = name1(snd(z)) ∧ name2(fst(z)) ∈A1"
 unfolding def_frecrel frecR_def
 by (intro equalityI subsetI CollectI; elim CollectE; auto)