Theory Rank

theory Rank
imports WF_absolute
(*  Title:      ZF/Constructible/Rank.thy
    Author:     Lawrence C Paulson, Cambridge University Computer Laboratory

section ‹Absoluteness for Order Types, Rank Functions and Well-Founded 

theory Rank imports WF_absolute begin

subsection ‹Order Types: A Direct Construction by Replacement›

locale M_ordertype = M_basic +
assumes well_ord_iso_separation:
     "[| M(A); M(f); M(r) |]
      ==> separation (M, λx. x∈A ⟶ (∃y[M]. (∃p[M].
                     fun_apply(M,f,x,y) & pair(M,y,x,p) & p ∈ r)))"
  and obase_separation:
     ‹part of the order type formalization›
     "[| M(A); M(r) |]
      ==> separation(M, λa. ∃x[M]. ∃g[M]. ∃mx[M]. ∃par[M].
             ordinal(M,x) & membership(M,x,mx) & pred_set(M,A,a,r,par) &
  and obase_equals_separation:
     "[| M(A); M(r) |]
      ==> separation (M, λx. x∈A ⟶ ~(∃y[M]. ∃g[M].
                              ordinal(M,y) & (∃my[M]. ∃pxr[M].
                              membership(M,y,my) & pred_set(M,A,x,r,pxr) &
  and omap_replacement:
     "[| M(A); M(r) |]
      ==> strong_replacement(M,
             λa z. ∃x[M]. ∃g[M]. ∃mx[M]. ∃par[M].
             ordinal(M,x) & pair(M,a,x,z) & membership(M,x,mx) &
             pred_set(M,A,a,r,par) & order_isomorphism(M,par,r,x,mx,g))"

text‹Inductive argument for Kunen's Lemma I 6.1, etc.
      Simple proof from Halmos, page 72›
lemma  (in M_ordertype) wellordered_iso_subset_lemma: 
     "[| wellordered(M,A,r);  f ∈ ord_iso(A,r, A',r);  A'<= A;  y ∈ A;  
       M(A);  M(f);  M(r) |] ==> ~ <f`y, y> ∈ r"
apply (unfold wellordered_def ord_iso_def)
apply (elim conjE CollectE) 
apply (erule wellfounded_on_induct, assumption+)
 apply (insert well_ord_iso_separation [of A f r])
 apply (simp, clarify) 
apply (drule_tac a = x in bij_is_fun [THEN apply_type], assumption, blast)

text‹Kunen's Lemma I 6.1, page 14: 
      there's no order-isomorphism to an initial segment of a well-ordering›
lemma (in M_ordertype) wellordered_iso_predD:
     "[| wellordered(M,A,r);  f ∈ ord_iso(A, r, Order.pred(A,x,r), r);  
       M(A);  M(f);  M(r) |] ==> x ∉ A"
apply (rule notI) 
apply (frule wellordered_iso_subset_lemma, assumption)
apply (auto elim: predE)  
(*Now we know  ~ (f`x < x) *)
apply (drule ord_iso_is_bij [THEN bij_is_fun, THEN apply_type], assumption)
(*Now we also know f`x  ∈ pred(A,x,r);  contradiction! *)
apply (simp add: Order.pred_def)

lemma (in M_ordertype) wellordered_iso_pred_eq_lemma:
     "[| f ∈ ⟨Order.pred(A,y,r), r⟩ ≅ ⟨Order.pred(A,x,r), r⟩;
       wellordered(M,A,r); x∈A; y∈A; M(A); M(f); M(r) |] ==> <x,y> ∉ r"
apply (frule wellordered_is_trans_on, assumption)
apply (rule notI) 
apply (drule_tac x2=y and x=x and r2=r in 
         wellordered_subset [OF _ pred_subset, THEN wellordered_iso_predD]) 
apply (simp add: trans_pred_pred_eq) 
apply (blast intro: predI dest: transM)+

text‹Simple consequence of Lemma 6.1›
lemma (in M_ordertype) wellordered_iso_pred_eq:
     "[| wellordered(M,A,r);
       f ∈ ord_iso(Order.pred(A,a,r), r, Order.pred(A,c,r), r);   
       M(A);  M(f);  M(r);  a∈A;  c∈A |] ==> a=c"
apply (frule wellordered_is_trans_on, assumption)
apply (frule wellordered_is_linear, assumption)
apply (erule_tac x=a and y=c in linearE, auto) 
apply (drule ord_iso_sym)
(*two symmetric cases*)
apply (blast dest: wellordered_iso_pred_eq_lemma)+ 

text‹Following Kunen's Theorem I 7.6, page 17.  Note that this material is
not required elsewhere.›

text‹Can't use ‹well_ord_iso_preserving› because it needs the
strong premise @{term "well_ord(A,r)"}›
lemma (in M_ordertype) ord_iso_pred_imp_lt:
     "[| f ∈ ord_iso(Order.pred(A,x,r), r, i, Memrel(i));
         g ∈ ord_iso(Order.pred(A,y,r), r, j, Memrel(j));
         wellordered(M,A,r);  x ∈ A;  y ∈ A; M(A); M(r); M(f); M(g); M(j);
         Ord(i); Ord(j); ⟨x,y⟩ ∈ r |]
      ==> i < j"
apply (frule wellordered_is_trans_on, assumption)
apply (frule_tac y=y in transM, assumption) 
apply (rule_tac i=i and j=j in Ord_linear_lt, auto)  
txt‹case @{term "i=j"} yields a contradiction›
 apply (rule_tac x1=x and A1="Order.pred(A,y,r)" in 
          wellordered_iso_predD [THEN notE]) 
   apply (blast intro: wellordered_subset [OF _ pred_subset]) 
  apply (simp add: trans_pred_pred_eq)
  apply (blast intro: Ord_iso_implies_eq ord_iso_sym ord_iso_trans) 
 apply (simp_all add: pred_iff pred_closed converse_closed comp_closed)
txt‹case @{term "j<i"} also yields a contradiction›
apply (frule restrict_ord_iso2, assumption+) 
apply (frule ord_iso_sym [THEN ord_iso_is_bij, THEN bij_is_fun]) 
apply (frule apply_type, blast intro: ltD) 
  ‹thus @{term "converse(f)`j ∈ Order.pred(A,x,r)"}›
apply (simp add: pred_iff) 
apply (subgoal_tac
       "∃h[M]. h ∈ ord_iso(Order.pred(A,y,r), r, 
                               Order.pred(A, converse(f)`j, r), r)")
 apply (clarify, frule wellordered_iso_pred_eq, assumption+)
 apply (blast dest: wellordered_asym)  
apply (intro rexI)
 apply (blast intro: Ord_iso_implies_eq ord_iso_sym ord_iso_trans)+

lemma ord_iso_converse1:
     "[| f: ord_iso(A,r,B,s);  <b, f`a>: s;  a:A;  b:B |] 
      ==> <converse(f) ` b, a> ∈ r"
apply (frule ord_iso_converse, assumption+) 
apply (blast intro: ord_iso_is_bij [THEN bij_is_fun, THEN apply_funtype]) 
apply (simp add: left_inverse_bij [OF ord_iso_is_bij])

  obase :: "[i=>o,i,i] => i" where
       ‹the domain of ‹om›, eventually shown to equal ‹A››
   "obase(M,A,r) == {a∈A. ∃x[M]. ∃g[M]. Ord(x) & 
                          g ∈ ord_iso(Order.pred(A,a,r),r,x,Memrel(x))}"

  omap :: "[i=>o,i,i,i] => o" where
    ‹the function that maps wosets to order types›
   "omap(M,A,r,f) == 
         z ∈ f ⟷ (∃a∈A. ∃x[M]. ∃g[M]. z = <a,x> & Ord(x) & 
                        g ∈ ord_iso(Order.pred(A,a,r),r,x,Memrel(x)))"

  otype :: "[i=>o,i,i,i] => o" where ‹the order types themselves›
   "otype(M,A,r,i) == ∃f[M]. omap(M,A,r,f) & is_range(M,f,i)"

text‹Can also be proved with the premise @{term "M(z)"} instead of
      @{term "M(f)"}, but that version is less useful.  This lemma
      is also more useful than the definition, ‹omap_def›.›
lemma (in M_ordertype) omap_iff:
     "[| omap(M,A,r,f); M(A); M(f) |] 
      ==> z ∈ f ⟷
          (∃a∈A. ∃x[M]. ∃g[M]. z = <a,x> & Ord(x) & 
                                g ∈ ord_iso(Order.pred(A,a,r),r,x,Memrel(x)))"
apply (simp add: omap_def Memrel_closed pred_closed) 
apply (rule iffI)
 apply (drule_tac [2] x=z in rspec)
 apply (drule_tac x=z in rspec)
 apply (blast dest: transM)+

lemma (in M_ordertype) omap_unique:
     "[| omap(M,A,r,f); omap(M,A,r,f'); M(A); M(r); M(f); M(f') |] ==> f' = f" 
apply (rule equality_iffI) 
apply (simp add: omap_iff) 

lemma (in M_ordertype) omap_yields_Ord:
     "[| omap(M,A,r,f); ⟨a,x⟩ ∈ f; M(a); M(x) |]  ==> Ord(x)"
  by (simp add: omap_def)

lemma (in M_ordertype) otype_iff:
     "[| otype(M,A,r,i); M(A); M(r); M(i) |] 
      ==> x ∈ i ⟷ 
          (M(x) & Ord(x) & 
           (∃a∈A. ∃g[M]. g ∈ ord_iso(Order.pred(A,a,r),r,x,Memrel(x))))"
apply (auto simp add: omap_iff otype_def)
 apply (blast intro: transM) 
apply (rule rangeI) 
apply (frule transM, assumption)
apply (simp add: omap_iff, blast)

lemma (in M_ordertype) otype_eq_range:
     "[| omap(M,A,r,f); otype(M,A,r,i); M(A); M(r); M(f); M(i) |] 
      ==> i = range(f)"
apply (auto simp add: otype_def omap_iff)
apply (blast dest: omap_unique) 

lemma (in M_ordertype) Ord_otype:
     "[| otype(M,A,r,i); trans[A](r); M(A); M(r); M(i) |] ==> Ord(i)"
apply (rule OrdI) 
prefer 2 
    apply (simp add: Ord_def otype_def omap_def) 
    apply clarify 
    apply (frule pair_components_in_M, assumption) 
    apply blast 
apply (auto simp add: Transset_def otype_iff) 
  apply (blast intro: transM)
 apply (blast intro: Ord_in_Ord) 
apply (rename_tac y a g)
apply (frule ord_iso_sym [THEN ord_iso_is_bij, THEN bij_is_fun, 
                          THEN apply_funtype],  assumption)  
apply (rule_tac x="converse(g)`y" in bexI)
 apply (frule_tac a="converse(g) ` y" in ord_iso_restrict_pred, assumption) 
apply (safe elim!: predE) 
apply (blast intro: restrict_ord_iso ord_iso_sym ltI dest: transM)

lemma (in M_ordertype) domain_omap:
     "[| omap(M,A,r,f);  M(A); M(r); M(B); M(f) |] 
      ==> domain(f) = obase(M,A,r)"
apply (simp add: domain_closed obase_def) 
apply (rule equality_iffI) 
apply (simp add: domain_iff omap_iff, blast) 

lemma (in M_ordertype) omap_subset: 
     "[| omap(M,A,r,f); otype(M,A,r,i); 
       M(A); M(r); M(f); M(B); M(i) |] ==> f ⊆ obase(M,A,r) * i"
apply clarify 
apply (simp add: omap_iff obase_def) 
apply (force simp add: otype_iff) 

lemma (in M_ordertype) omap_funtype: 
     "[| omap(M,A,r,f); otype(M,A,r,i); 
         M(A); M(r); M(f); M(i) |] ==> f ∈ obase(M,A,r) -> i"
apply (simp add: domain_omap omap_subset Pi_iff function_def omap_iff) 
apply (blast intro: Ord_iso_implies_eq ord_iso_sym ord_iso_trans) 

lemma (in M_ordertype) wellordered_omap_bij:
     "[| wellordered(M,A,r); omap(M,A,r,f); otype(M,A,r,i); 
       M(A); M(r); M(f); M(i) |] ==> f ∈ bij(obase(M,A,r),i)"
apply (insert omap_funtype [of A r f i]) 
apply (auto simp add: bij_def inj_def) 
prefer 2  apply (blast intro: fun_is_surj dest: otype_eq_range) 
apply (frule_tac a=w in apply_Pair, assumption) 
apply (frule_tac a=x in apply_Pair, assumption) 
apply (simp add: omap_iff) 
apply (blast intro: wellordered_iso_pred_eq ord_iso_sym ord_iso_trans) 

text‹This is not the final result: we must show @{term "oB(A,r) = A"}›
lemma (in M_ordertype) omap_ord_iso:
     "[| wellordered(M,A,r); omap(M,A,r,f); otype(M,A,r,i); 
       M(A); M(r); M(f); M(i) |] ==> f ∈ ord_iso(obase(M,A,r),r,i,Memrel(i))"
apply (rule ord_isoI)
 apply (erule wellordered_omap_bij, assumption+) 
apply (insert omap_funtype [of A r f i], simp) 
apply (frule_tac a=x in apply_Pair, assumption) 
apply (frule_tac a=y in apply_Pair, assumption) 
apply (auto simp add: omap_iff)
 txt‹direction 1: assuming @{term "⟨x,y⟩ ∈ r"}›
 apply (blast intro: ltD ord_iso_pred_imp_lt)
 txt‹direction 2: proving @{term "⟨x,y⟩ ∈ r"} using linearity of @{term r}›
apply (rename_tac x y g ga) 
apply (frule wellordered_is_linear, assumption, 
       erule_tac x=x and y=y in linearE, assumption+) 
txt‹the case @{term "x=y"} leads to immediate contradiction› 
apply (blast elim: mem_irrefl) 
txt‹the case @{term "⟨y,x⟩ ∈ r"}: handle like the opposite direction›
apply (blast dest: ord_iso_pred_imp_lt ltD elim: mem_asym) 

lemma (in M_ordertype) Ord_omap_image_pred:
     "[| wellordered(M,A,r); omap(M,A,r,f); otype(M,A,r,i); 
       M(A); M(r); M(f); M(i); b ∈ A |] ==> Ord(f `` Order.pred(A,b,r))"
apply (frule wellordered_is_trans_on, assumption)
apply (rule OrdI) 
        prefer 2 apply (simp add: image_iff omap_iff Ord_def, blast) 
txt‹Hard part is to show that the image is a transitive set.›
apply (simp add: Transset_def, clarify) 
apply (simp add: image_iff pred_iff apply_iff [OF omap_funtype [of A r f i]])
apply (rename_tac c j, clarify)
apply (frule omap_funtype [of A r f, THEN apply_funtype], assumption+)
apply (subgoal_tac "j ∈ i") 
        prefer 2 apply (blast intro: Ord_trans Ord_otype)
apply (subgoal_tac "converse(f) ` j ∈ obase(M,A,r)") 
        prefer 2 
        apply (blast dest: wellordered_omap_bij [THEN bij_converse_bij, 
                                      THEN bij_is_fun, THEN apply_funtype])
apply (rule_tac x="converse(f) ` j" in bexI) 
 apply (simp add: right_inverse_bij [OF wellordered_omap_bij]) 
apply (intro predI conjI)
 apply (erule_tac b=c in trans_onD) 
 apply (rule ord_iso_converse1 [OF omap_ord_iso [of A r f i]])
apply (auto simp add: obase_def)

lemma (in M_ordertype) restrict_omap_ord_iso:
     "[| wellordered(M,A,r); omap(M,A,r,f); otype(M,A,r,i); 
       D ⊆ obase(M,A,r); M(A); M(r); M(f); M(i) |] 
      ==> restrict(f,D) ∈ (⟨D,r⟩ ≅ ⟨f``D, Memrel(f``D)⟩)"
apply (frule ord_iso_restrict_image [OF omap_ord_iso [of A r f i]], 
apply (drule ord_iso_sym [THEN subset_ord_iso_Memrel]) 
apply (blast dest: subsetD [OF omap_subset]) 
apply (drule ord_iso_sym, simp) 

lemma (in M_ordertype) obase_equals: 
     "[| wellordered(M,A,r); omap(M,A,r,f); otype(M,A,r,i);
       M(A); M(r); M(f); M(i) |] ==> obase(M,A,r) = A"
apply (rule equalityI, force simp add: obase_def, clarify) 
apply (unfold obase_def, simp) 
apply (frule wellordered_is_wellfounded_on, assumption)
apply (erule wellfounded_on_induct, assumption+)
 apply (frule obase_equals_separation [of A r], assumption) 
 apply (simp, clarify) 
apply (rename_tac b) 
apply (subgoal_tac "Order.pred(A,b,r) ⊆ obase(M,A,r)") 
 apply (blast intro!: restrict_omap_ord_iso Ord_omap_image_pred)
apply (force simp add: pred_iff obase_def)  

text‹Main result: @{term om} gives the order-isomorphism 
      @{term "⟨A,r⟩ ≅ ⟨i, Memrel(i)⟩"}›
theorem (in M_ordertype) omap_ord_iso_otype:
     "[| wellordered(M,A,r); omap(M,A,r,f); otype(M,A,r,i);
       M(A); M(r); M(f); M(i) |] ==> f ∈ ord_iso(A, r, i, Memrel(i))"
apply (frule omap_ord_iso, assumption+)
apply (simp add: obase_equals)  

lemma (in M_ordertype) obase_exists:
     "[| M(A); M(r) |] ==> M(obase(M,A,r))"
apply (simp add: obase_def) 
apply (insert obase_separation [of A r])
apply (simp add: separation_def)  

lemma (in M_ordertype) omap_exists:
     "[| M(A); M(r) |] ==> ∃z[M]. omap(M,A,r,z)"
apply (simp add: omap_def) 
apply (insert omap_replacement [of A r])
apply (simp add: strong_replacement_def) 
apply (drule_tac x="obase(M,A,r)" in rspec) 
 apply (simp add: obase_exists) 
apply (simp add: Memrel_closed pred_closed obase_def)
apply (erule impE) 
 apply (clarsimp simp add: univalent_def)
 apply (blast intro: Ord_iso_implies_eq ord_iso_sym ord_iso_trans, clarify)  
apply (rule_tac x=Y in rexI) 
apply (simp add: Memrel_closed pred_closed obase_def, blast, assumption)

declare rall_simps [simp] rex_simps [simp]

lemma (in M_ordertype) otype_exists:
     "[| wellordered(M,A,r); M(A); M(r) |] ==> ∃i[M]. otype(M,A,r,i)"
apply (insert omap_exists [of A r])  
apply (simp add: otype_def, safe)
apply (rule_tac x="range(x)" in rexI) 
apply blast+

lemma (in M_ordertype) ordertype_exists:
     "[| wellordered(M,A,r); M(A); M(r) |]
      ==> ∃f[M]. (∃i[M]. Ord(i) & f ∈ ord_iso(A, r, i, Memrel(i)))"
apply (insert obase_exists [of A r] omap_exists [of A r] otype_exists [of A r], simp, clarify)
apply (rename_tac i) 
apply (subgoal_tac "Ord(i)", blast intro: omap_ord_iso_otype)
apply (rule Ord_otype) 
    apply (force simp add: otype_def range_closed) 
   apply (simp_all add: wellordered_is_trans_on) 

lemma (in M_ordertype) relativized_imp_well_ord: 
     "[| wellordered(M,A,r); M(A); M(r) |] ==> well_ord(A,r)" 
apply (insert ordertype_exists [of A r], simp)
apply (blast intro: well_ord_ord_iso well_ord_Memrel)  

subsection ‹Kunen's theorem 5.4, page 127›

text‹(a) The notion of Wellordering is absolute›
theorem (in M_ordertype) well_ord_abs [simp]: 
     "[| M(A); M(r) |] ==> wellordered(M,A,r) ⟷ well_ord(A,r)" 
by (blast intro: well_ord_imp_relativized relativized_imp_well_ord)  

text‹(b) Order types are absolute›
theorem (in M_ordertype) 
     "[| wellordered(M,A,r); f ∈ ord_iso(A, r, i, Memrel(i));
       M(A); M(r); M(f); M(i); Ord(i) |] ==> i = ordertype(A,r)"
by (blast intro: Ord_ordertype relativized_imp_well_ord ordertype_ord_iso
                 Ord_iso_implies_eq ord_iso_sym ord_iso_trans)

subsection‹Ordinal Arithmetic: Two Examples of Recursion›

text‹Note: the remainder of this theory is not needed elsewhere.›

subsubsection‹Ordinal Addition›

(*FIXME: update to use new techniques!!*)
 (*This expresses ordinal addition in the language of ZF.  It also 
   provides an abbreviation that can be used in the instance of strong
   replacement below.  Here j is used to define the relation, namely
   Memrel(succ(j)), while x determines the domain of f.*)
  is_oadd_fun :: "[i=>o,i,i,i,i] => o" where
    "is_oadd_fun(M,i,j,x,f) == 
       (∀sj msj. M(sj) ⟶ M(msj) ⟶ 
                 successor(M,j,sj) ⟶ membership(M,sj,msj) ⟶ 
                     %x g y. ∃gx[M]. image(M,g,x,gx) & union(M,i,gx,y),
                     msj, x, f))"

  is_oadd :: "[i=>o,i,i,i] => o" where
    "is_oadd(M,i,j,k) == 
        (~ ordinal(M,i) & ~ ordinal(M,j) & k=0) |
        (~ ordinal(M,i) & ordinal(M,j) & k=j) |
        (ordinal(M,i) & ~ ordinal(M,j) & k=i) |
        (ordinal(M,i) & ordinal(M,j) & 
         (∃f fj sj. M(f) & M(fj) & M(sj) & 
                    successor(M,j,sj) & is_oadd_fun(M,i,sj,sj,f) & 
                    fun_apply(M,f,j,fj) & fj = k))"

  omult_eqns :: "[i,i,i,i] => o" where
    "omult_eqns(i,x,g,z) ==
            Ord(x) & 
            (x=0 ⟶ z=0) &
            (∀j. x = succ(j) ⟶ z = g`j ++ i) &
            (Limit(x) ⟶ z = ⋃(g``x))"

  is_omult_fun :: "[i=>o,i,i,i] => o" where
    "is_omult_fun(M,i,j,f) == 
            (∃df. M(df) & is_function(M,f) & 
                  is_domain(M,f,df) & subset(M, j, df)) & 
            (∀x∈j. omult_eqns(i,x,f,f`x))"

  is_omult :: "[i=>o,i,i,i] => o" where
    "is_omult(M,i,j,k) == 
        ∃f fj sj. M(f) & M(fj) & M(sj) & 
                  successor(M,j,sj) & is_omult_fun(M,i,sj,f) & 
                  fun_apply(M,f,j,fj) & fj = k"

locale M_ord_arith = M_ordertype +
  assumes oadd_strong_replacement:
   "[| M(i); M(j) |] ==>
         λx z. ∃y[M]. pair(M,x,y,z) & 
                  (∃f[M]. ∃fx[M]. is_oadd_fun(M,i,j,x,f) & 
                           image(M,f,x,fx) & y = i ∪ fx))"

 and omult_strong_replacement':
   "[| M(i); M(j) |] ==>
         λx z. ∃y[M]. z = <x,y> &
             (∃g[M]. is_recfun(Memrel(succ(j)),x,%x g. THE z. omult_eqns(i,x,g,z),g) & 
             y = (THE z. omult_eqns(i, x, g, z))))" 

text‹‹is_oadd_fun›: Relating the pure "language of set theory" to Isabelle/ZF›
lemma (in M_ord_arith) is_oadd_fun_iff:
   "[| a≤j; M(i); M(j); M(a); M(f) |] 
    ==> is_oadd_fun(M,i,j,a,f) ⟷
        f ∈ a → range(f) & (∀x. M(x) ⟶ x < a ⟶ f`x = i ∪ f``x)"
apply (frule lt_Ord) 
apply (simp add: is_oadd_fun_def Memrel_closed Un_closed 
             relation2_def is_recfun_abs [of "%x g. i ∪ g``x"]
             image_closed is_recfun_iff_equation  
             Ball_def lt_trans [OF ltI, of _ a] lt_Memrel)
apply (simp add: lt_def) 
apply (blast dest: transM) 

lemma (in M_ord_arith) oadd_strong_replacement':
    "[| M(i); M(j) |] ==>
            λx z. ∃y[M]. z = <x,y> &
                  (∃g[M]. is_recfun(Memrel(succ(j)),x,%x g. i ∪ g``x,g) & 
                  y = i ∪ g``x))" 
apply (insert oadd_strong_replacement [of i j]) 
apply (simp add: is_oadd_fun_def relation2_def
                 is_recfun_abs [of "%x g. i ∪ g``x"])  

lemma (in M_ord_arith) exists_oadd:
    "[| Ord(j);  M(i);  M(j) |]
     ==> ∃f[M]. is_recfun(Memrel(succ(j)), j, %x g. i ∪ g``x, f)"
apply (rule wf_exists_is_recfun [OF wf_Memrel trans_Memrel])
    apply (simp_all add: Memrel_type oadd_strong_replacement') 

lemma (in M_ord_arith) exists_oadd_fun:
    "[| Ord(j);  M(i);  M(j) |] ==> ∃f[M]. is_oadd_fun(M,i,succ(j),succ(j),f)"
apply (rule exists_oadd [THEN rexE])
apply (erule Ord_succ, assumption, simp) 
apply (rename_tac f) 
apply (frule is_recfun_type)
apply (rule_tac x=f in rexI) 
 apply (simp add: fun_is_function domain_of_fun lt_Memrel apply_recfun lt_def
                  is_oadd_fun_iff Ord_trans [OF _ succI1], assumption)

lemma (in M_ord_arith) is_oadd_fun_apply:
    "[| x < j; M(i); M(j); M(f); is_oadd_fun(M,i,j,j,f) |] 
     ==> f`x = i ∪ (⋃k∈x. {f ` k})"
apply (simp add: is_oadd_fun_iff lt_Ord2, clarify) 
apply (frule lt_closed, simp)
apply (frule leI [THEN le_imp_subset])  
apply (simp add: image_fun, blast) 

lemma (in M_ord_arith) is_oadd_fun_iff_oadd [rule_format]:
    "[| is_oadd_fun(M,i,J,J,f); M(i); M(J); M(f); Ord(i); Ord(j) |] 
     ==> j<J ⟶ f`j = i++j"
apply (erule_tac i=j in trans_induct, clarify) 
apply (subgoal_tac "∀k∈x. k<J")
 apply (simp (no_asm_simp) add: is_oadd_def oadd_unfold is_oadd_fun_apply)
apply (blast intro: lt_trans ltI lt_Ord) 

lemma (in M_ord_arith) Ord_oadd_abs:
    "[| M(i); M(j); M(k); Ord(i); Ord(j) |] ==> is_oadd(M,i,j,k) ⟷ k = i++j"
apply (simp add: is_oadd_def is_oadd_fun_iff_oadd)
apply (frule exists_oadd_fun [of j i], blast+)

lemma (in M_ord_arith) oadd_abs:
    "[| M(i); M(j); M(k) |] ==> is_oadd(M,i,j,k) ⟷ k = i++j"
apply (case_tac "Ord(i) & Ord(j)")
 apply (simp add: Ord_oadd_abs)
apply (auto simp add: is_oadd_def oadd_eq_if_raw_oadd)

lemma (in M_ord_arith) oadd_closed [intro,simp]:
    "[| M(i); M(j) |] ==> M(i++j)"
apply (simp add: oadd_eq_if_raw_oadd, clarify) 
apply (simp add: raw_oadd_eq_oadd) 
apply (frule exists_oadd_fun [of j i], auto)
apply (simp add: apply_closed is_oadd_fun_iff_oadd [symmetric]) 

subsubsection‹Ordinal Multiplication›

lemma omult_eqns_unique:
     "[| omult_eqns(i,x,g,z); omult_eqns(i,x,g,z') |] ==> z=z'"
apply (simp add: omult_eqns_def, clarify) 
apply (erule Ord_cases, simp_all) 

lemma omult_eqns_0: "omult_eqns(i,0,g,z) ⟷ z=0"
by (simp add: omult_eqns_def)

lemma the_omult_eqns_0: "(THE z. omult_eqns(i,0,g,z)) = 0"
by (simp add: omult_eqns_0)

lemma omult_eqns_succ: "omult_eqns(i,succ(j),g,z) ⟷ Ord(j) & z = g`j ++ i"
by (simp add: omult_eqns_def)

lemma the_omult_eqns_succ:
     "Ord(j) ==> (THE z. omult_eqns(i,succ(j),g,z)) = g`j ++ i"
by (simp add: omult_eqns_succ) 

lemma omult_eqns_Limit:
     "Limit(x) ==> omult_eqns(i,x,g,z) ⟷ z = ⋃(g``x)"
apply (simp add: omult_eqns_def) 
apply (blast intro: Limit_is_Ord) 

lemma the_omult_eqns_Limit:
     "Limit(x) ==> (THE z. omult_eqns(i,x,g,z)) = ⋃(g``x)"
by (simp add: omult_eqns_Limit)

lemma omult_eqns_Not: "~ Ord(x) ==> ~ omult_eqns(i,x,g,z)"
by (simp add: omult_eqns_def)

lemma (in M_ord_arith) the_omult_eqns_closed:
    "[| M(i); M(x); M(g); function(g) |] 
     ==> M(THE z. omult_eqns(i, x, g, z))"
apply (case_tac "Ord(x)")
 prefer 2 apply (simp add: omult_eqns_Not) ‹trivial, non-Ord case›
apply (erule Ord_cases) 
  apply (simp add: omult_eqns_0)
 apply (simp add: omult_eqns_succ apply_closed oadd_closed) 
apply (simp add: omult_eqns_Limit) 

lemma (in M_ord_arith) exists_omult:
    "[| Ord(j);  M(i);  M(j) |]
     ==> ∃f[M]. is_recfun(Memrel(succ(j)), j, %x g. THE z. omult_eqns(i,x,g,z), f)"
apply (rule wf_exists_is_recfun [OF wf_Memrel trans_Memrel])
    apply (simp_all add: Memrel_type omult_strong_replacement') 
apply (blast intro: the_omult_eqns_closed) 

lemma (in M_ord_arith) exists_omult_fun:
    "[| Ord(j);  M(i);  M(j) |] ==> ∃f[M]. is_omult_fun(M,i,succ(j),f)"
apply (rule exists_omult [THEN rexE])
apply (erule Ord_succ, assumption, simp) 
apply (rename_tac f) 
apply (frule is_recfun_type)
apply (rule_tac x=f in rexI) 
apply (simp add: fun_is_function domain_of_fun lt_Memrel apply_recfun lt_def
                 is_omult_fun_def Ord_trans [OF _ succI1])
 apply (force dest: Ord_in_Ord' 
              simp add: omult_eqns_def the_omult_eqns_0 the_omult_eqns_succ
                        the_omult_eqns_Limit, assumption)

lemma (in M_ord_arith) is_omult_fun_apply_0:
    "[| 0 < j; is_omult_fun(M,i,j,f) |] ==> f`0 = 0"
by (simp add: is_omult_fun_def omult_eqns_def lt_def ball_conj_distrib)

lemma (in M_ord_arith) is_omult_fun_apply_succ:
    "[| succ(x) < j; is_omult_fun(M,i,j,f) |] ==> f`succ(x) = f`x ++ i"
by (simp add: is_omult_fun_def omult_eqns_def lt_def, blast) 

lemma (in M_ord_arith) is_omult_fun_apply_Limit:
    "[| x < j; Limit(x); M(j); M(f); is_omult_fun(M,i,j,f) |] 
     ==> f ` x = (⋃y∈x. f`y)"
apply (simp add: is_omult_fun_def omult_eqns_def domain_closed lt_def, clarify)
apply (drule subset_trans [OF OrdmemD], assumption+)  
apply (simp add: ball_conj_distrib omult_Limit image_function)

lemma (in M_ord_arith) is_omult_fun_eq_omult:
    "[| is_omult_fun(M,i,J,f); M(J); M(f); Ord(i); Ord(j) |] 
     ==> j<J ⟶ f`j = i**j"
apply (erule_tac i=j in trans_induct3)
apply (safe del: impCE)
  apply (simp add: is_omult_fun_apply_0) 
 apply (subgoal_tac "x<J") 
  apply (simp add: is_omult_fun_apply_succ omult_succ)  
 apply (blast intro: lt_trans) 
apply (subgoal_tac "∀k∈x. k<J")
 apply (simp add: is_omult_fun_apply_Limit omult_Limit) 
apply (blast intro: lt_trans ltI lt_Ord) 

lemma (in M_ord_arith) omult_abs:
    "[| M(i); M(j); M(k); Ord(i); Ord(j) |] ==> is_omult(M,i,j,k) ⟷ k = i**j"
apply (simp add: is_omult_def is_omult_fun_eq_omult)
apply (frule exists_omult_fun [of j i], blast+)

subsection ‹Absoluteness of Well-Founded Relations›

text‹Relativized to @{term M}: Every well-founded relation is a subset of some
inverse image of an ordinal.  Key step is the construction (in @{term M}) of a
rank function.›

locale M_wfrank = M_trancl +
  assumes wfrank_separation:
     "M(r) ==>
      separation (M, λx. 
         ∀rplus[M]. tran_closure(M,r,rplus) ⟶
         ~ (∃f[M]. M_is_recfun(M, %x f y. is_range(M,f,y), rplus, x, f)))"
 and wfrank_strong_replacement:
     "M(r) ==>
      strong_replacement(M, λx z. 
         ∀rplus[M]. tran_closure(M,r,rplus) ⟶
         (∃y[M]. ∃f[M]. pair(M,x,y,z)  & 
                        M_is_recfun(M, %x f y. is_range(M,f,y), rplus, x, f) &
 and Ord_wfrank_separation:
     "M(r) ==>
      separation (M, λx.
         ∀rplus[M]. tran_closure(M,r,rplus) ⟶ 
          ~ (∀f[M]. ∀rangef[M]. 
             is_range(M,f,rangef) ⟶
             M_is_recfun(M, λx f y. is_range(M,f,y), rplus, x, f) ⟶

text‹Proving that the relativized instances of Separation or Replacement
agree with the "real" ones.›

lemma (in M_wfrank) wfrank_separation':
     "M(r) ==>
           (M, λx. ~ (∃f[M]. is_recfun(r^+, x, %x f. range(f), f)))"
apply (insert wfrank_separation [of r])
apply (simp add: relation2_def is_recfun_abs [of "%x. range"])

lemma (in M_wfrank) wfrank_strong_replacement':
     "M(r) ==>
      strong_replacement(M, λx z. ∃y[M]. ∃f[M]. 
                  pair(M,x,y,z) & is_recfun(r^+, x, %x f. range(f), f) &
                  y = range(f))"
apply (insert wfrank_strong_replacement [of r])
apply (simp add: relation2_def is_recfun_abs [of "%x. range"])

lemma (in M_wfrank) Ord_wfrank_separation':
     "M(r) ==>
      separation (M, λx. 
         ~ (∀f[M]. is_recfun(r^+, x, λx. range, f) ⟶ Ord(range(f))))" 
apply (insert Ord_wfrank_separation [of r])
apply (simp add: relation2_def is_recfun_abs [of "%x. range"])

text‹This function, defined using replacement, is a rank function for
well-founded relations within the class M.›
  wellfoundedrank :: "[i=>o,i,i] => i" where
    "wellfoundedrank(M,r,A) ==
        {p. x∈A, ∃y[M]. ∃f[M]. 
                       p = <x,y> & is_recfun(r^+, x, %x f. range(f), f) &
                       y = range(f)}"

lemma (in M_wfrank) exists_wfrank:
    "[| wellfounded(M,r); M(a); M(r) |]
     ==> ∃f[M]. is_recfun(r^+, a, %x f. range(f), f)"
apply (rule wellfounded_exists_is_recfun)
      apply (blast intro: wellfounded_trancl)
     apply (rule trans_trancl)
    apply (erule wfrank_separation')
   apply (erule wfrank_strong_replacement')
apply (simp_all add: trancl_subset_times)

lemma (in M_wfrank) M_wellfoundedrank:
    "[| wellfounded(M,r); M(r); M(A) |] ==> M(wellfoundedrank(M,r,A))"
apply (insert wfrank_strong_replacement' [of r])
apply (simp add: wellfoundedrank_def)
apply (rule strong_replacement_closed)
   apply assumption+
 apply (rule univalent_is_recfun)
   apply (blast intro: wellfounded_trancl)
  apply (rule trans_trancl)
 apply (simp add: trancl_subset_times) 
apply (blast dest: transM) 

lemma (in M_wfrank) Ord_wfrank_range [rule_format]:
    "[| wellfounded(M,r); a∈A; M(r); M(A) |]
     ==> ∀f[M]. is_recfun(r^+, a, %x f. range(f), f) ⟶ Ord(range(f))"
apply (drule wellfounded_trancl, assumption)
apply (rule wellfounded_induct, assumption, erule (1) transM)
  apply simp
 apply (blast intro: Ord_wfrank_separation', clarify)
txt‹The reasoning in both cases is that we get @{term y} such that
   @{term "⟨y, x⟩ ∈ r^+"}.  We find that
   @{term "f`y = restrict(f, r^+ -`` {y})"}.›
apply (rule OrdI [OF _ Ord_is_Transset])
 txt‹An ordinal is a transitive set...›
 apply (simp add: Transset_def)
 apply clarify
 apply (frule apply_recfun2, assumption)
 apply (force simp add: restrict_iff)
txt‹...of ordinals.  This second case requires the induction hyp.›
apply clarify
apply (rename_tac i y)
apply (frule apply_recfun2, assumption)
apply (frule is_recfun_imp_in_r, assumption)
apply (frule is_recfun_restrict)
    (*simp_all won't work*)
    apply (simp add: trans_trancl trancl_subset_times)+
apply (drule spec [THEN mp], assumption)
apply (subgoal_tac "M(restrict(f, r^+ -`` {y}))")
 apply (drule_tac x="restrict(f, r^+ -`` {y})" in rspec)
apply assumption
 apply (simp add: function_apply_equality [OF _ is_recfun_imp_function])
apply (blast dest: pair_components_in_M)

lemma (in M_wfrank) Ord_range_wellfoundedrank:
    "[| wellfounded(M,r); r ⊆ A*A;  M(r); M(A) |]
     ==> Ord (range(wellfoundedrank(M,r,A)))"
apply (frule wellfounded_trancl, assumption)
apply (frule trancl_subset_times)
apply (simp add: wellfoundedrank_def)
apply (rule OrdI [OF _ Ord_is_Transset])
 prefer 2
 txt‹by our previous result the range consists of ordinals.›
 apply (blast intro: Ord_wfrank_range)
txt‹We still must show that the range is a transitive set.›
apply (simp add: Transset_def, clarify, simp)
apply (rename_tac x i f u)
apply (frule is_recfun_imp_in_r, assumption)
apply (subgoal_tac "M(u) & M(i) & M(x)")
 prefer 2 apply (blast dest: transM, clarify)
apply (rule_tac a=u in rangeI)
apply (rule_tac x=u in ReplaceI)
  apply simp 
  apply (rule_tac x="restrict(f, r^+ -`` {u})" in rexI)
   apply (blast intro: is_recfun_restrict trans_trancl dest: apply_recfun2)
  apply simp 
apply blast 
txt‹Unicity requirement of Replacement›
apply clarify
apply (frule apply_recfun2, assumption)
apply (simp add: trans_trancl is_recfun_cut)

lemma (in M_wfrank) function_wellfoundedrank:
    "[| wellfounded(M,r); M(r); M(A)|]
     ==> function(wellfoundedrank(M,r,A))"
apply (simp add: wellfoundedrank_def function_def, clarify)
txt‹Uniqueness: repeated below!›
apply (drule is_recfun_functional, assumption)
     apply (blast intro: wellfounded_trancl)
    apply (simp_all add: trancl_subset_times trans_trancl)

lemma (in M_wfrank) domain_wellfoundedrank:
    "[| wellfounded(M,r); M(r); M(A)|]
     ==> domain(wellfoundedrank(M,r,A)) = A"
apply (simp add: wellfoundedrank_def function_def)
apply (rule equalityI, auto)
apply (frule transM, assumption)
apply (frule_tac a=x in exists_wfrank, assumption+, clarify)
apply (rule_tac b="range(f)" in domainI)
apply (rule_tac x=x in ReplaceI)
  apply simp 
  apply (rule_tac x=f in rexI, blast, simp_all)
txt‹Uniqueness (for Replacement): repeated above!›
apply clarify
apply (drule is_recfun_functional, assumption)
    apply (blast intro: wellfounded_trancl)
    apply (simp_all add: trancl_subset_times trans_trancl)

lemma (in M_wfrank) wellfoundedrank_type:
    "[| wellfounded(M,r);  M(r); M(A)|]
     ==> wellfoundedrank(M,r,A) ∈ A -> range(wellfoundedrank(M,r,A))"
apply (frule function_wellfoundedrank [of r A], assumption+)
apply (frule function_imp_Pi)
 apply (simp add: wellfoundedrank_def relation_def)
 apply blast
apply (simp add: domain_wellfoundedrank)

lemma (in M_wfrank) Ord_wellfoundedrank:
    "[| wellfounded(M,r); a ∈ A; r ⊆ A*A;  M(r); M(A) |]
     ==> Ord(wellfoundedrank(M,r,A) ` a)"
by (blast intro: apply_funtype [OF wellfoundedrank_type]
                 Ord_in_Ord [OF Ord_range_wellfoundedrank])

lemma (in M_wfrank) wellfoundedrank_eq:
     "[| is_recfun(r^+, a, %x. range, f);
         wellfounded(M,r);  a ∈ A; M(f); M(r); M(A)|]
      ==> wellfoundedrank(M,r,A) ` a = range(f)"
apply (rule apply_equality)
 prefer 2 apply (blast intro: wellfoundedrank_type)
apply (simp add: wellfoundedrank_def)
apply (rule ReplaceI)
  apply (rule_tac x="range(f)" in rexI) 
  apply blast
 apply simp_all
txt‹Unicity requirement of Replacement›
apply clarify
apply (drule is_recfun_functional, assumption)
    apply (blast intro: wellfounded_trancl)
    apply (simp_all add: trancl_subset_times trans_trancl)

lemma (in M_wfrank) wellfoundedrank_lt:
     "[| <a,b> ∈ r;
         wellfounded(M,r); r ⊆ A*A;  M(r); M(A)|]
      ==> wellfoundedrank(M,r,A) ` a < wellfoundedrank(M,r,A) ` b"
apply (frule wellfounded_trancl, assumption)
apply (subgoal_tac "a∈A & b∈A")
 prefer 2 apply blast
apply (simp add: lt_def Ord_wellfoundedrank, clarify)
apply (frule exists_wfrank [of concl: _ b], erule (1) transM, assumption)
apply clarify
apply (rename_tac fb)
apply (frule is_recfun_restrict [of concl: "r^+" a])
    apply (rule trans_trancl, assumption)
   apply (simp_all add: r_into_trancl trancl_subset_times)
txt‹Still the same goal, but with new ‹is_recfun› assumptions.›
apply (simp add: wellfoundedrank_eq)
apply (frule_tac a=a in wellfoundedrank_eq, assumption+)
   apply (simp_all add: transM [of a])
txt‹We have used equations for wellfoundedrank and now must use some
    for  ‹is_recfun›.›
apply (rule_tac a=a in rangeI)
apply (simp add: is_recfun_type [THEN apply_iff] vimage_singleton_iff
                 r_into_trancl apply_recfun r_into_trancl)

lemma (in M_wfrank) wellfounded_imp_subset_rvimage:
     "[|wellfounded(M,r); r ⊆ A*A; M(r); M(A)|]
      ==> ∃i f. Ord(i) & r ⊆ rvimage(A, f, Memrel(i))"
apply (rule_tac x="range(wellfoundedrank(M,r,A))" in exI)
apply (rule_tac x="wellfoundedrank(M,r,A)" in exI)
apply (simp add: Ord_range_wellfoundedrank, clarify)
apply (frule subsetD, assumption, clarify)
apply (simp add: rvimage_iff wellfoundedrank_lt [THEN ltD])
apply (blast intro: apply_rangeI wellfoundedrank_type)

lemma (in M_wfrank) wellfounded_imp_wf:
     "[|wellfounded(M,r); relation(r); M(r)|] ==> wf(r)"
by (blast dest!: relation_field_times_field wellfounded_imp_subset_rvimage
          intro: wf_rvimage_Ord [THEN wf_subset])

lemma (in M_wfrank) wellfounded_on_imp_wf_on:
     "[|wellfounded_on(M,A,r); relation(r); M(r); M(A)|] ==> wf[A](r)"
apply (simp add: wellfounded_on_iff_wellfounded wf_on_def)
apply (rule wellfounded_imp_wf)
apply (simp_all add: relation_def)

theorem (in M_wfrank) wf_abs:
     "[|relation(r); M(r)|] ==> wellfounded(M,r) ⟷ wf(r)"
by (blast intro: wellfounded_imp_wf wf_imp_relativized)

theorem (in M_wfrank) wf_on_abs:
     "[|relation(r); M(r); M(A)|] ==> wellfounded_on(M,A,r) ⟷ wf[A](r)"
by (blast intro: wellfounded_on_imp_wf_on wf_on_imp_relativized)