Theory Relative

Up to index of Isabelle/ZF/Constructible

theory Relative
imports Main
begin

(*  Title:      ZF/Constructible/Relative.thy
    ID:         $Id: Relative.thy,v 1.43 2007/04/15 21:25:53 wenzelm Exp $
    Author:     Lawrence C Paulson, Cambridge University Computer Laboratory
*)

header {*Relativization and Absoluteness*}

theory Relative imports Main begin

subsection{* Relativized versions of standard set-theoretic concepts *}

definition
  empty :: "[i=>o,i] => o" where
    "empty(M,z) == ∀x[M]. x ∉ z"

definition
  subset :: "[i=>o,i,i] => o" where
    "subset(M,A,B) == ∀x[M]. x∈A --> x ∈ B"

definition
  upair :: "[i=>o,i,i,i] => o" where
    "upair(M,a,b,z) == a ∈ z & b ∈ z & (∀x[M]. x∈z --> x = a | x = b)"

definition
  pair :: "[i=>o,i,i,i] => o" where
    "pair(M,a,b,z) == ∃x[M]. upair(M,a,a,x) &
                     (∃y[M]. upair(M,a,b,y) & upair(M,x,y,z))"


definition
  union :: "[i=>o,i,i,i] => o" where
    "union(M,a,b,z) == ∀x[M]. x ∈ z <-> x ∈ a | x ∈ b"

definition
  is_cons :: "[i=>o,i,i,i] => o" where
    "is_cons(M,a,b,z) == ∃x[M]. upair(M,a,a,x) & union(M,x,b,z)"

definition
  successor :: "[i=>o,i,i] => o" where
    "successor(M,a,z) == is_cons(M,a,a,z)"

definition
  number1 :: "[i=>o,i] => o" where
    "number1(M,a) == ∃x[M]. empty(M,x) & successor(M,x,a)"

definition
  number2 :: "[i=>o,i] => o" where
    "number2(M,a) == ∃x[M]. number1(M,x) & successor(M,x,a)"

definition
  number3 :: "[i=>o,i] => o" where
    "number3(M,a) == ∃x[M]. number2(M,x) & successor(M,x,a)"

definition
  powerset :: "[i=>o,i,i] => o" where
    "powerset(M,A,z) == ∀x[M]. x ∈ z <-> subset(M,x,A)"

definition
  is_Collect :: "[i=>o,i,i=>o,i] => o" where
    "is_Collect(M,A,P,z) == ∀x[M]. x ∈ z <-> x ∈ A & P(x)"

definition
  is_Replace :: "[i=>o,i,[i,i]=>o,i] => o" where
    "is_Replace(M,A,P,z) == ∀u[M]. u ∈ z <-> (∃x[M]. x∈A & P(x,u))"

definition
  inter :: "[i=>o,i,i,i] => o" where
    "inter(M,a,b,z) == ∀x[M]. x ∈ z <-> x ∈ a & x ∈ b"

definition
  setdiff :: "[i=>o,i,i,i] => o" where
    "setdiff(M,a,b,z) == ∀x[M]. x ∈ z <-> x ∈ a & x ∉ b"

definition
  big_union :: "[i=>o,i,i] => o" where
    "big_union(M,A,z) == ∀x[M]. x ∈ z <-> (∃y[M]. y∈A & x ∈ y)"

definition
  big_inter :: "[i=>o,i,i] => o" where
    "big_inter(M,A,z) ==
             (A=0 --> z=0) &
             (A≠0 --> (∀x[M]. x ∈ z <-> (∀y[M]. y∈A --> x ∈ y)))"

definition
  cartprod :: "[i=>o,i,i,i] => o" where
    "cartprod(M,A,B,z) ==
        ∀u[M]. u ∈ z <-> (∃x[M]. x∈A & (∃y[M]. y∈B & pair(M,x,y,u)))"

definition
  is_sum :: "[i=>o,i,i,i] => o" where
    "is_sum(M,A,B,Z) ==
       ∃A0[M]. ∃n1[M]. ∃s1[M]. ∃B1[M].
       number1(M,n1) & cartprod(M,n1,A,A0) & upair(M,n1,n1,s1) &
       cartprod(M,s1,B,B1) & union(M,A0,B1,Z)"

definition
  is_Inl :: "[i=>o,i,i] => o" where
    "is_Inl(M,a,z) == ∃zero[M]. empty(M,zero) & pair(M,zero,a,z)"

definition
  is_Inr :: "[i=>o,i,i] => o" where
    "is_Inr(M,a,z) == ∃n1[M]. number1(M,n1) & pair(M,n1,a,z)"

definition
  is_converse :: "[i=>o,i,i] => o" where
    "is_converse(M,r,z) ==
        ∀x[M]. x ∈ z <->
             (∃w[M]. w∈r & (∃u[M]. ∃v[M]. pair(M,u,v,w) & pair(M,v,u,x)))"

definition
  pre_image :: "[i=>o,i,i,i] => o" where
    "pre_image(M,r,A,z) ==
        ∀x[M]. x ∈ z <-> (∃w[M]. w∈r & (∃y[M]. y∈A & pair(M,x,y,w)))"

definition
  is_domain :: "[i=>o,i,i] => o" where
    "is_domain(M,r,z) ==
        ∀x[M]. x ∈ z <-> (∃w[M]. w∈r & (∃y[M]. pair(M,x,y,w)))"

definition
  image :: "[i=>o,i,i,i] => o" where
    "image(M,r,A,z) ==
        ∀y[M]. y ∈ z <-> (∃w[M]. w∈r & (∃x[M]. x∈A & pair(M,x,y,w)))"

definition
  is_range :: "[i=>o,i,i] => o" where
    --{*the cleaner
      @{term "∃r'[M]. is_converse(M,r,r') & is_domain(M,r',z)"}
      unfortunately needs an instance of separation in order to prove
        @{term "M(converse(r))"}.*}
    "is_range(M,r,z) ==
        ∀y[M]. y ∈ z <-> (∃w[M]. w∈r & (∃x[M]. pair(M,x,y,w)))"

definition
  is_field :: "[i=>o,i,i] => o" where
    "is_field(M,r,z) ==
        ∃dr[M]. ∃rr[M]. is_domain(M,r,dr) & is_range(M,r,rr) &
                        union(M,dr,rr,z)"

definition
  is_relation :: "[i=>o,i] => o" where
    "is_relation(M,r) ==
        (∀z[M]. z∈r --> (∃x[M]. ∃y[M]. pair(M,x,y,z)))"

definition
  is_function :: "[i=>o,i] => o" where
    "is_function(M,r) ==
        ∀x[M]. ∀y[M]. ∀y'[M]. ∀p[M]. ∀p'[M].
           pair(M,x,y,p) --> pair(M,x,y',p') --> p∈r --> p'∈r --> y=y'"

definition
  fun_apply :: "[i=>o,i,i,i] => o" where
    "fun_apply(M,f,x,y) ==
        (∃xs[M]. ∃fxs[M].
         upair(M,x,x,xs) & image(M,f,xs,fxs) & big_union(M,fxs,y))"

definition
  typed_function :: "[i=>o,i,i,i] => o" where
    "typed_function(M,A,B,r) ==
        is_function(M,r) & is_relation(M,r) & is_domain(M,r,A) &
        (∀u[M]. u∈r --> (∀x[M]. ∀y[M]. pair(M,x,y,u) --> y∈B))"

definition
  is_funspace :: "[i=>o,i,i,i] => o" where
    "is_funspace(M,A,B,F) ==
        ∀f[M]. f ∈ F <-> typed_function(M,A,B,f)"

definition
  composition :: "[i=>o,i,i,i] => o" where
    "composition(M,r,s,t) ==
        ∀p[M]. p ∈ t <->
               (∃x[M]. ∃y[M]. ∃z[M]. ∃xy[M]. ∃yz[M].
                pair(M,x,z,p) & pair(M,x,y,xy) & pair(M,y,z,yz) &
                xy ∈ s & yz ∈ r)"

definition
  injection :: "[i=>o,i,i,i] => o" where
    "injection(M,A,B,f) ==
        typed_function(M,A,B,f) &
        (∀x[M]. ∀x'[M]. ∀y[M]. ∀p[M]. ∀p'[M].
          pair(M,x,y,p) --> pair(M,x',y,p') --> p∈f --> p'∈f --> x=x')"

definition
  surjection :: "[i=>o,i,i,i] => o" where
    "surjection(M,A,B,f) ==
        typed_function(M,A,B,f) &
        (∀y[M]. y∈B --> (∃x[M]. x∈A & fun_apply(M,f,x,y)))"

definition
  bijection :: "[i=>o,i,i,i] => o" where
    "bijection(M,A,B,f) == injection(M,A,B,f) & surjection(M,A,B,f)"

definition
  restriction :: "[i=>o,i,i,i] => o" where
    "restriction(M,r,A,z) ==
        ∀x[M]. x ∈ z <-> (x ∈ r & (∃u[M]. u∈A & (∃v[M]. pair(M,u,v,x))))"

definition
  transitive_set :: "[i=>o,i] => o" where
    "transitive_set(M,a) == ∀x[M]. x∈a --> subset(M,x,a)"

definition
  ordinal :: "[i=>o,i] => o" where
     --{*an ordinal is a transitive set of transitive sets*}
    "ordinal(M,a) == transitive_set(M,a) & (∀x[M]. x∈a --> transitive_set(M,x))"

definition
  limit_ordinal :: "[i=>o,i] => o" where
    --{*a limit ordinal is a non-empty, successor-closed ordinal*}
    "limit_ordinal(M,a) ==
        ordinal(M,a) & ~ empty(M,a) &
        (∀x[M]. x∈a --> (∃y[M]. y∈a & successor(M,x,y)))"

definition
  successor_ordinal :: "[i=>o,i] => o" where
    --{*a successor ordinal is any ordinal that is neither empty nor limit*}
    "successor_ordinal(M,a) ==
        ordinal(M,a) & ~ empty(M,a) & ~ limit_ordinal(M,a)"

definition
  finite_ordinal :: "[i=>o,i] => o" where
    --{*an ordinal is finite if neither it nor any of its elements are limit*}
    "finite_ordinal(M,a) ==
        ordinal(M,a) & ~ limit_ordinal(M,a) &
        (∀x[M]. x∈a --> ~ limit_ordinal(M,x))"

definition
  omega :: "[i=>o,i] => o" where
    --{*omega is a limit ordinal none of whose elements are limit*}
    "omega(M,a) == limit_ordinal(M,a) & (∀x[M]. x∈a --> ~ limit_ordinal(M,x))"

definition
  is_quasinat :: "[i=>o,i] => o" where
    "is_quasinat(M,z) == empty(M,z) | (∃m[M]. successor(M,m,z))"

definition
  is_nat_case :: "[i=>o, i, [i,i]=>o, i, i] => o" where
    "is_nat_case(M, a, is_b, k, z) ==
       (empty(M,k) --> z=a) &
       (∀m[M]. successor(M,m,k) --> is_b(m,z)) &
       (is_quasinat(M,k) | empty(M,z))"

definition
  relation1 :: "[i=>o, [i,i]=>o, i=>i] => o" where
    "relation1(M,is_f,f) == ∀x[M]. ∀y[M]. is_f(x,y) <-> y = f(x)"

definition
  Relation1 :: "[i=>o, i, [i,i]=>o, i=>i] => o" where
    --{*as above, but typed*}
    "Relation1(M,A,is_f,f) ==
        ∀x[M]. ∀y[M]. x∈A --> is_f(x,y) <-> y = f(x)"

definition
  relation2 :: "[i=>o, [i,i,i]=>o, [i,i]=>i] => o" where
    "relation2(M,is_f,f) == ∀x[M]. ∀y[M]. ∀z[M]. is_f(x,y,z) <-> z = f(x,y)"

definition
  Relation2 :: "[i=>o, i, i, [i,i,i]=>o, [i,i]=>i] => o" where
    "Relation2(M,A,B,is_f,f) ==
        ∀x[M]. ∀y[M]. ∀z[M]. x∈A --> y∈B --> is_f(x,y,z) <-> z = f(x,y)"

definition
  relation3 :: "[i=>o, [i,i,i,i]=>o, [i,i,i]=>i] => o" where
    "relation3(M,is_f,f) ==
       ∀x[M]. ∀y[M]. ∀z[M]. ∀u[M]. is_f(x,y,z,u) <-> u = f(x,y,z)"

definition
  Relation3 :: "[i=>o, i, i, i, [i,i,i,i]=>o, [i,i,i]=>i] => o" where
    "Relation3(M,A,B,C,is_f,f) ==
       ∀x[M]. ∀y[M]. ∀z[M]. ∀u[M].
         x∈A --> y∈B --> z∈C --> is_f(x,y,z,u) <-> u = f(x,y,z)"

definition
  relation4 :: "[i=>o, [i,i,i,i,i]=>o, [i,i,i,i]=>i] => o" where
    "relation4(M,is_f,f) ==
       ∀u[M]. ∀x[M]. ∀y[M]. ∀z[M]. ∀a[M]. is_f(u,x,y,z,a) <-> a = f(u,x,y,z)"


text{*Useful when absoluteness reasoning has replaced the predicates by terms*}
lemma triv_Relation1:
     "Relation1(M, A, λx y. y = f(x), f)"
by (simp add: Relation1_def)

lemma triv_Relation2:
     "Relation2(M, A, B, λx y a. a = f(x,y), f)"
by (simp add: Relation2_def)


subsection {*The relativized ZF axioms*}

definition
  extensionality :: "(i=>o) => o" where
    "extensionality(M) ==
        ∀x[M]. ∀y[M]. (∀z[M]. z ∈ x <-> z ∈ y) --> x=y"

definition
  separation :: "[i=>o, i=>o] => o" where
    --{*The formula @{text P} should only involve parameters
        belonging to @{text M} and all its quantifiers must be relativized
        to @{text M}.  We do not have separation as a scheme; every instance
        that we need must be assumed (and later proved) separately.*}
    "separation(M,P) ==
        ∀z[M]. ∃y[M]. ∀x[M]. x ∈ y <-> x ∈ z & P(x)"

definition
  upair_ax :: "(i=>o) => o" where
    "upair_ax(M) == ∀x[M]. ∀y[M]. ∃z[M]. upair(M,x,y,z)"

definition
  Union_ax :: "(i=>o) => o" where
    "Union_ax(M) == ∀x[M]. ∃z[M]. big_union(M,x,z)"

definition
  power_ax :: "(i=>o) => o" where
    "power_ax(M) == ∀x[M]. ∃z[M]. powerset(M,x,z)"

definition
  univalent :: "[i=>o, i, [i,i]=>o] => o" where
    "univalent(M,A,P) ==
        ∀x[M]. x∈A --> (∀y[M]. ∀z[M]. P(x,y) & P(x,z) --> y=z)"

definition
  replacement :: "[i=>o, [i,i]=>o] => o" where
    "replacement(M,P) ==
      ∀A[M]. univalent(M,A,P) -->
      (∃Y[M]. ∀b[M]. (∃x[M]. x∈A & P(x,b)) --> b ∈ Y)"

definition
  strong_replacement :: "[i=>o, [i,i]=>o] => o" where
    "strong_replacement(M,P) ==
      ∀A[M]. univalent(M,A,P) -->
      (∃Y[M]. ∀b[M]. b ∈ Y <-> (∃x[M]. x∈A & P(x,b)))"

definition
  foundation_ax :: "(i=>o) => o" where
    "foundation_ax(M) ==
        ∀x[M]. (∃y[M]. y∈x) --> (∃y[M]. y∈x & ~(∃z[M]. z∈x & z ∈ y))"


subsection{*A trivial consistency proof for $V_\omega$ *}

text{*We prove that $V_\omega$
      (or @{text univ} in Isabelle) satisfies some ZF axioms.
     Kunen, Theorem IV 3.13, page 123.*}

lemma univ0_downwards_mem: "[| y ∈ x; x ∈ univ(0) |] ==> y ∈ univ(0)"
apply (insert Transset_univ [OF Transset_0])
apply (simp add: Transset_def, blast)
done

lemma univ0_Ball_abs [simp]:
     "A ∈ univ(0) ==> (∀x∈A. x ∈ univ(0) --> P(x)) <-> (∀x∈A. P(x))"
by (blast intro: univ0_downwards_mem)

lemma univ0_Bex_abs [simp]:
     "A ∈ univ(0) ==> (∃x∈A. x ∈ univ(0) & P(x)) <-> (∃x∈A. P(x))"
by (blast intro: univ0_downwards_mem)

text{*Congruence rule for separation: can assume the variable is in @{text M}*}
lemma separation_cong [cong]:
     "(!!x. M(x) ==> P(x) <-> P'(x))
      ==> separation(M, %x. P(x)) <-> separation(M, %x. P'(x))"
by (simp add: separation_def)

lemma univalent_cong [cong]:
     "[| A=A'; !!x y. [| x∈A; M(x); M(y) |] ==> P(x,y) <-> P'(x,y) |]
      ==> univalent(M, A, %x y. P(x,y)) <-> univalent(M, A', %x y. P'(x,y))"
by (simp add: univalent_def)

lemma univalent_triv [intro,simp]:
     "univalent(M, A, λx y. y = f(x))"
by (simp add: univalent_def)

lemma univalent_conjI2 [intro,simp]:
     "univalent(M,A,Q) ==> univalent(M, A, λx y. P(x,y) & Q(x,y))"
by (simp add: univalent_def, blast)

text{*Congruence rule for replacement*}
lemma strong_replacement_cong [cong]:
     "[| !!x y. [| M(x); M(y) |] ==> P(x,y) <-> P'(x,y) |]
      ==> strong_replacement(M, %x y. P(x,y)) <->
          strong_replacement(M, %x y. P'(x,y))"
by (simp add: strong_replacement_def)

text{*The extensionality axiom*}
lemma "extensionality(λx. x ∈ univ(0))"
apply (simp add: extensionality_def)
apply (blast intro: univ0_downwards_mem)
done

text{*The separation axiom requires some lemmas*}
lemma Collect_in_Vfrom:
     "[| X ∈ Vfrom(A,j);  Transset(A) |] ==> Collect(X,P) ∈ Vfrom(A, succ(j))"
apply (drule Transset_Vfrom)
apply (rule subset_mem_Vfrom)
apply (unfold Transset_def, blast)
done

lemma Collect_in_VLimit:
     "[| X ∈ Vfrom(A,i);  Limit(i);  Transset(A) |]
      ==> Collect(X,P) ∈ Vfrom(A,i)"
apply (rule Limit_VfromE, assumption+)
apply (blast intro: Limit_has_succ VfromI Collect_in_Vfrom)
done

lemma Collect_in_univ:
     "[| X ∈ univ(A);  Transset(A) |] ==> Collect(X,P) ∈ univ(A)"
by (simp add: univ_def Collect_in_VLimit Limit_nat)

lemma "separation(λx. x ∈ univ(0), P)"
apply (simp add: separation_def, clarify)
apply (rule_tac x = "Collect(z,P)" in bexI)
apply (blast intro: Collect_in_univ Transset_0)+
done

text{*Unordered pairing axiom*}
lemma "upair_ax(λx. x ∈ univ(0))"
apply (simp add: upair_ax_def upair_def)
apply (blast intro: doubleton_in_univ)
done

text{*Union axiom*}
lemma "Union_ax(λx. x ∈ univ(0))"
apply (simp add: Union_ax_def big_union_def, clarify)
apply (rule_tac x="\<Union>x" in bexI)
 apply (blast intro: univ0_downwards_mem)
apply (blast intro: Union_in_univ Transset_0)
done

text{*Powerset axiom*}

lemma Pow_in_univ:
     "[| X ∈ univ(A);  Transset(A) |] ==> Pow(X) ∈ univ(A)"
apply (simp add: univ_def Pow_in_VLimit Limit_nat)
done

lemma "power_ax(λx. x ∈ univ(0))"
apply (simp add: power_ax_def powerset_def subset_def, clarify)
apply (rule_tac x="Pow(x)" in bexI)
 apply (blast intro: univ0_downwards_mem)
apply (blast intro: Pow_in_univ Transset_0)
done

text{*Foundation axiom*}
lemma "foundation_ax(λx. x ∈ univ(0))"
apply (simp add: foundation_ax_def, clarify)
apply (cut_tac A=x in foundation)
apply (blast intro: univ0_downwards_mem)
done

lemma "replacement(λx. x ∈ univ(0), P)"
apply (simp add: replacement_def, clarify)
oops
text{*no idea: maybe prove by induction on the rank of A?*}

text{*Still missing: Replacement, Choice*}

subsection{*Lemmas Needed to Reduce Some Set Constructions to Instances
      of Separation*}

lemma image_iff_Collect: "r `` A = {y ∈ Union(Union(r)). ∃p∈r. ∃x∈A. p=<x,y>}"
apply (rule equalityI, auto)
apply (simp add: Pair_def, blast)
done

lemma vimage_iff_Collect:
     "r -`` A = {x ∈ Union(Union(r)). ∃p∈r. ∃y∈A. p=<x,y>}"
apply (rule equalityI, auto)
apply (simp add: Pair_def, blast)
done

text{*These two lemmas lets us prove @{text domain_closed} and
      @{text range_closed} without new instances of separation*}

lemma domain_eq_vimage: "domain(r) = r -`` Union(Union(r))"
apply (rule equalityI, auto)
apply (rule vimageI, assumption)
apply (simp add: Pair_def, blast)
done

lemma range_eq_image: "range(r) = r `` Union(Union(r))"
apply (rule equalityI, auto)
apply (rule imageI, assumption)
apply (simp add: Pair_def, blast)
done

lemma replacementD:
    "[| replacement(M,P); M(A);  univalent(M,A,P) |]
     ==> ∃Y[M]. (∀b[M]. ((∃x[M]. x∈A & P(x,b)) --> b ∈ Y))"
by (simp add: replacement_def)

lemma strong_replacementD:
    "[| strong_replacement(M,P); M(A);  univalent(M,A,P) |]
     ==> ∃Y[M]. (∀b[M]. (b ∈ Y <-> (∃x[M]. x∈A & P(x,b))))"
by (simp add: strong_replacement_def)

lemma separationD:
    "[| separation(M,P); M(z) |] ==> ∃y[M]. ∀x[M]. x ∈ y <-> x ∈ z & P(x)"
by (simp add: separation_def)


text{*More constants, for order types*}

definition
  order_isomorphism :: "[i=>o,i,i,i,i,i] => o" where
    "order_isomorphism(M,A,r,B,s,f) ==
        bijection(M,A,B,f) &
        (∀x[M]. x∈A --> (∀y[M]. y∈A -->
          (∀p[M]. ∀fx[M]. ∀fy[M]. ∀q[M].
            pair(M,x,y,p) --> fun_apply(M,f,x,fx) --> fun_apply(M,f,y,fy) -->
            pair(M,fx,fy,q) --> (p∈r <-> q∈s))))"

definition
  pred_set :: "[i=>o,i,i,i,i] => o" where
    "pred_set(M,A,x,r,B) ==
        ∀y[M]. y ∈ B <-> (∃p[M]. p∈r & y ∈ A & pair(M,y,x,p))"

definition
  membership :: "[i=>o,i,i] => o" where --{*membership relation*}
    "membership(M,A,r) ==
        ∀p[M]. p ∈ r <-> (∃x[M]. x∈A & (∃y[M]. y∈A & x∈y & pair(M,x,y,p)))"


subsection{*Introducing a Transitive Class Model*}

text{*The class M is assumed to be transitive and to satisfy some
      relativized ZF axioms*}
locale M_trivial =
  fixes M
  assumes transM:           "[| y∈x; M(x) |] ==> M(y)"
      and upair_ax:         "upair_ax(M)"
      and Union_ax:         "Union_ax(M)"
      and power_ax:         "power_ax(M)"
      and replacement:      "replacement(M,P)"
      and M_nat [iff]:      "M(nat)"           (*i.e. the axiom of infinity*)


text{*Automatically discovers the proof using @{text transM}, @{text nat_0I}
and @{text M_nat}.*}
lemma (in M_trivial) nonempty [simp]: "M(0)"
by (blast intro: transM)

lemma (in M_trivial) rall_abs [simp]:
     "M(A) ==> (∀x[M]. x∈A --> P(x)) <-> (∀x∈A. P(x))"
by (blast intro: transM)

lemma (in M_trivial) rex_abs [simp]:
     "M(A) ==> (∃x[M]. x∈A & P(x)) <-> (∃x∈A. P(x))"
by (blast intro: transM)

lemma (in M_trivial) ball_iff_equiv:
     "M(A) ==> (∀x[M]. (x∈A <-> P(x))) <->
               (∀x∈A. P(x)) & (∀x. P(x) --> M(x) --> x∈A)"
by (blast intro: transM)

text{*Simplifies proofs of equalities when there's an iff-equality
      available for rewriting, universally quantified over M.  
      But it's not the only way to prove such equalities: its
      premises @{term "M(A)"} and  @{term "M(B)"} can be too strong.*}
lemma (in M_trivial) M_equalityI:
     "[| !!x. M(x) ==> x∈A <-> x∈B; M(A); M(B) |] ==> A=B"
by (blast intro!: equalityI dest: transM)


subsubsection{*Trivial Absoluteness Proofs: Empty Set, Pairs, etc.*}

lemma (in M_trivial) empty_abs [simp]:
     "M(z) ==> empty(M,z) <-> z=0"
apply (simp add: empty_def)
apply (blast intro: transM)
done

lemma (in M_trivial) subset_abs [simp]:
     "M(A) ==> subset(M,A,B) <-> A ⊆ B"
apply (simp add: subset_def)
apply (blast intro: transM)
done

lemma (in M_trivial) upair_abs [simp]:
     "M(z) ==> upair(M,a,b,z) <-> z={a,b}"
apply (simp add: upair_def)
apply (blast intro: transM)
done

lemma (in M_trivial) upair_in_M_iff [iff]:
     "M({a,b}) <-> M(a) & M(b)"
apply (insert upair_ax, simp add: upair_ax_def)
apply (blast intro: transM)
done

lemma (in M_trivial) singleton_in_M_iff [iff]:
     "M({a}) <-> M(a)"
by (insert upair_in_M_iff [of a a], simp)

lemma (in M_trivial) pair_abs [simp]:
     "M(z) ==> pair(M,a,b,z) <-> z=<a,b>"
apply (simp add: pair_def ZF.Pair_def)
apply (blast intro: transM)
done

lemma (in M_trivial) pair_in_M_iff [iff]:
     "M(<a,b>) <-> M(a) & M(b)"
by (simp add: ZF.Pair_def)

lemma (in M_trivial) pair_components_in_M:
     "[| <x,y> ∈ A; M(A) |] ==> M(x) & M(y)"
apply (simp add: Pair_def)
apply (blast dest: transM)
done

lemma (in M_trivial) cartprod_abs [simp]:
     "[| M(A); M(B); M(z) |] ==> cartprod(M,A,B,z) <-> z = A*B"
apply (simp add: cartprod_def)
apply (rule iffI)
 apply (blast intro!: equalityI intro: transM dest!: rspec)
apply (blast dest: transM)
done

subsubsection{*Absoluteness for Unions and Intersections*}

lemma (in M_trivial) union_abs [simp]:
     "[| M(a); M(b); M(z) |] ==> union(M,a,b,z) <-> z = a Un b"
apply (simp add: union_def)
apply (blast intro: transM)
done

lemma (in M_trivial) inter_abs [simp]:
     "[| M(a); M(b); M(z) |] ==> inter(M,a,b,z) <-> z = a Int b"
apply (simp add: inter_def)
apply (blast intro: transM)
done

lemma (in M_trivial) setdiff_abs [simp]:
     "[| M(a); M(b); M(z) |] ==> setdiff(M,a,b,z) <-> z = a-b"
apply (simp add: setdiff_def)
apply (blast intro: transM)
done

lemma (in M_trivial) Union_abs [simp]:
     "[| M(A); M(z) |] ==> big_union(M,A,z) <-> z = Union(A)"
apply (simp add: big_union_def)
apply (blast intro!: equalityI dest: transM)
done

lemma (in M_trivial) Union_closed [intro,simp]:
     "M(A) ==> M(Union(A))"
by (insert Union_ax, simp add: Union_ax_def)

lemma (in M_trivial) Un_closed [intro,simp]:
     "[| M(A); M(B) |] ==> M(A Un B)"
by (simp only: Un_eq_Union, blast)

lemma (in M_trivial) cons_closed [intro,simp]:
     "[| M(a); M(A) |] ==> M(cons(a,A))"
by (subst cons_eq [symmetric], blast)

lemma (in M_trivial) cons_abs [simp]:
     "[| M(b); M(z) |] ==> is_cons(M,a,b,z) <-> z = cons(a,b)"
by (simp add: is_cons_def, blast intro: transM)

lemma (in M_trivial) successor_abs [simp]:
     "[| M(a); M(z) |] ==> successor(M,a,z) <-> z = succ(a)"
by (simp add: successor_def, blast)

lemma (in M_trivial) succ_in_M_iff [iff]:
     "M(succ(a)) <-> M(a)"
apply (simp add: succ_def)
apply (blast intro: transM)
done

subsubsection{*Absoluteness for Separation and Replacement*}

lemma (in M_trivial) separation_closed [intro,simp]:
     "[| separation(M,P); M(A) |] ==> M(Collect(A,P))"
apply (insert separation, simp add: separation_def)
apply (drule rspec, assumption, clarify)
apply (subgoal_tac "y = Collect(A,P)", blast)
apply (blast dest: transM)
done

lemma separation_iff:
     "separation(M,P) <-> (∀z[M]. ∃y[M]. is_Collect(M,z,P,y))"
by (simp add: separation_def is_Collect_def)

lemma (in M_trivial) Collect_abs [simp]:
     "[| M(A); M(z) |] ==> is_Collect(M,A,P,z) <-> z = Collect(A,P)"
apply (simp add: is_Collect_def)
apply (blast intro!: equalityI dest: transM)
done

text{*Probably the premise and conclusion are equivalent*}
lemma (in M_trivial) strong_replacementI [rule_format]:
    "[| ∀B[M]. separation(M, %u. ∃x[M]. x∈B & P(x,u)) |]
     ==> strong_replacement(M,P)"
apply (simp add: strong_replacement_def, clarify)
apply (frule replacementD [OF replacement], assumption, clarify)
apply (drule_tac x=A in rspec, clarify)
apply (drule_tac z=Y in separationD, assumption, clarify)
apply (rule_tac x=y in rexI, force, assumption)
done

subsubsection{*The Operator @{term is_Replace}*}


lemma is_Replace_cong [cong]:
     "[| A=A';
         !!x y. [| M(x); M(y) |] ==> P(x,y) <-> P'(x,y);
         z=z' |]
      ==> is_Replace(M, A, %x y. P(x,y), z) <->
          is_Replace(M, A', %x y. P'(x,y), z')"
by (simp add: is_Replace_def)

lemma (in M_trivial) univalent_Replace_iff:
     "[| M(A); univalent(M,A,P);
         !!x y. [| x∈A; P(x,y) |] ==> M(y) |]
      ==> u ∈ Replace(A,P) <-> (∃x. x∈A & P(x,u))"
apply (simp add: Replace_iff univalent_def)
apply (blast dest: transM)
done

(*The last premise expresses that P takes M to M*)
lemma (in M_trivial) strong_replacement_closed [intro,simp]:
     "[| strong_replacement(M,P); M(A); univalent(M,A,P);
         !!x y. [| x∈A; P(x,y) |] ==> M(y) |] ==> M(Replace(A,P))"
apply (simp add: strong_replacement_def)
apply (drule_tac x=A in rspec, safe)
apply (subgoal_tac "Replace(A,P) = Y")
 apply simp
apply (rule equality_iffI)
apply (simp add: univalent_Replace_iff)
apply (blast dest: transM)
done

lemma (in M_trivial) Replace_abs:
     "[| M(A); M(z); univalent(M,A,P); 
         !!x y. [| x∈A; P(x,y) |] ==> M(y)  |]
      ==> is_Replace(M,A,P,z) <-> z = Replace(A,P)"
apply (simp add: is_Replace_def)
apply (rule iffI)
 apply (rule equality_iffI)
 apply (simp_all add: univalent_Replace_iff) 
 apply (blast dest: transM)+
done


(*The first premise can't simply be assumed as a schema.
  It is essential to take care when asserting instances of Replacement.
  Let K be a nonconstructible subset of nat and define
  f(x) = x if x:K and f(x)=0 otherwise.  Then RepFun(nat,f) = cons(0,K), a
  nonconstructible set.  So we cannot assume that M(X) implies M(RepFun(X,f))
  even for f : M -> M.
*)
lemma (in M_trivial) RepFun_closed:
     "[| strong_replacement(M, λx y. y = f(x)); M(A); ∀x∈A. M(f(x)) |]
      ==> M(RepFun(A,f))"
apply (simp add: RepFun_def)
apply (rule strong_replacement_closed)
apply (auto dest: transM  simp add: univalent_def)
done

lemma Replace_conj_eq: "{y . x ∈ A, x∈A & y=f(x)} = {y . x∈A, y=f(x)}"
by simp

text{*Better than @{text RepFun_closed} when having the formula @{term "x∈A"}
      makes relativization easier.*}
lemma (in M_trivial) RepFun_closed2:
     "[| strong_replacement(M, λx y. x∈A & y = f(x)); M(A); ∀x∈A. M(f(x)) |]
      ==> M(RepFun(A, %x. f(x)))"
apply (simp add: RepFun_def)
apply (frule strong_replacement_closed, assumption)
apply (auto dest: transM  simp add: Replace_conj_eq univalent_def)
done

subsubsection {*Absoluteness for @{term Lambda}*}

definition
 is_lambda :: "[i=>o, i, [i,i]=>o, i] => o" where
    "is_lambda(M, A, is_b, z) ==
       ∀p[M]. p ∈ z <->
        (∃u[M]. ∃v[M]. u∈A & pair(M,u,v,p) & is_b(u,v))"

lemma (in M_trivial) lam_closed:
     "[| strong_replacement(M, λx y. y = <x,b(x)>); M(A); ∀x∈A. M(b(x)) |]
      ==> M(λx∈A. b(x))"
by (simp add: lam_def, blast intro: RepFun_closed dest: transM)

text{*Better than @{text lam_closed}: has the formula @{term "x∈A"}*}
lemma (in M_trivial) lam_closed2:
  "[|strong_replacement(M, λx y. x∈A & y = ⟨x, b(x)⟩);
     M(A); ∀m[M]. m∈A --> M(b(m))|] ==> M(Lambda(A,b))"
apply (simp add: lam_def)
apply (blast intro: RepFun_closed2 dest: transM)
done

lemma (in M_trivial) lambda_abs2:
     "[| Relation1(M,A,is_b,b); M(A); ∀m[M]. m∈A --> M(b(m)); M(z) |]
      ==> is_lambda(M,A,is_b,z) <-> z = Lambda(A,b)"
apply (simp add: Relation1_def is_lambda_def)
apply (rule iffI)
 prefer 2 apply (simp add: lam_def)
apply (rule equality_iffI)
apply (simp add: lam_def) 
apply (rule iffI) 
 apply (blast dest: transM) 
apply (auto simp add: transM [of _ A]) 
done

lemma is_lambda_cong [cong]:
     "[| A=A';  z=z';
         !!x y. [| x∈A; M(x); M(y) |] ==> is_b(x,y) <-> is_b'(x,y) |]
      ==> is_lambda(M, A, %x y. is_b(x,y), z) <->
          is_lambda(M, A', %x y. is_b'(x,y), z')"
by (simp add: is_lambda_def)

lemma (in M_trivial) image_abs [simp]:
     "[| M(r); M(A); M(z) |] ==> image(M,r,A,z) <-> z = r``A"
apply (simp add: image_def)
apply (rule iffI)
 apply (blast intro!: equalityI dest: transM, blast)
done

text{*What about @{text Pow_abs}?  Powerset is NOT absolute!
      This result is one direction of absoluteness.*}

lemma (in M_trivial) powerset_Pow:
     "powerset(M, x, Pow(x))"
by (simp add: powerset_def)

text{*But we can't prove that the powerset in @{text M} includes the
      real powerset.*}
lemma (in M_trivial) powerset_imp_subset_Pow:
     "[| powerset(M,x,y); M(y) |] ==> y <= Pow(x)"
apply (simp add: powerset_def)
apply (blast dest: transM)
done

subsubsection{*Absoluteness for the Natural Numbers*}

lemma (in M_trivial) nat_into_M [intro]:
     "n ∈ nat ==> M(n)"
by (induct n rule: nat_induct, simp_all)

lemma (in M_trivial) nat_case_closed [intro,simp]:
  "[|M(k); M(a); ∀m[M]. M(b(m))|] ==> M(nat_case(a,b,k))"
apply (case_tac "k=0", simp)
apply (case_tac "∃m. k = succ(m)", force)
apply (simp add: nat_case_def)
done

lemma (in M_trivial) quasinat_abs [simp]:
     "M(z) ==> is_quasinat(M,z) <-> quasinat(z)"
by (auto simp add: is_quasinat_def quasinat_def)

lemma (in M_trivial) nat_case_abs [simp]:
     "[| relation1(M,is_b,b); M(k); M(z) |]
      ==> is_nat_case(M,a,is_b,k,z) <-> z = nat_case(a,b,k)"
apply (case_tac "quasinat(k)")
 prefer 2
 apply (simp add: is_nat_case_def non_nat_case)
 apply (force simp add: quasinat_def)
apply (simp add: quasinat_def is_nat_case_def)
apply (elim disjE exE)
 apply (simp_all add: relation1_def)
done

(*NOT for the simplifier.  The assumption M(z') is apparently necessary, but
  causes the error "Failed congruence proof!"  It may be better to replace
  is_nat_case by nat_case before attempting congruence reasoning.*)
lemma is_nat_case_cong:
     "[| a = a'; k = k';  z = z';  M(z');
       !!x y. [| M(x); M(y) |] ==> is_b(x,y) <-> is_b'(x,y) |]
      ==> is_nat_case(M, a, is_b, k, z) <-> is_nat_case(M, a', is_b', k', z')"
by (simp add: is_nat_case_def)


subsection{*Absoluteness for Ordinals*}
text{*These results constitute Theorem IV 5.1 of Kunen (page 126).*}

lemma (in M_trivial) lt_closed:
     "[| j<i; M(i) |] ==> M(j)"
by (blast dest: ltD intro: transM)

lemma (in M_trivial) transitive_set_abs [simp]:
     "M(a) ==> transitive_set(M,a) <-> Transset(a)"
by (simp add: transitive_set_def Transset_def)

lemma (in M_trivial) ordinal_abs [simp]:
     "M(a) ==> ordinal(M,a) <-> Ord(a)"
by (simp add: ordinal_def Ord_def)

lemma (in M_trivial) limit_ordinal_abs [simp]:
     "M(a) ==> limit_ordinal(M,a) <-> Limit(a)"
apply (unfold Limit_def limit_ordinal_def)
apply (simp add: Ord_0_lt_iff)
apply (simp add: lt_def, blast)
done

lemma (in M_trivial) successor_ordinal_abs [simp]:
     "M(a) ==> successor_ordinal(M,a) <-> Ord(a) & (∃b[M]. a = succ(b))"
apply (simp add: successor_ordinal_def, safe)
apply (drule Ord_cases_disj, auto)
done

lemma finite_Ord_is_nat:
      "[| Ord(a); ~ Limit(a); ∀x∈a. ~ Limit(x) |] ==> a ∈ nat"
by (induct a rule: trans_induct3, simp_all)

lemma (in M_trivial) finite_ordinal_abs [simp]:
     "M(a) ==> finite_ordinal(M,a) <-> a ∈ nat"
apply (simp add: finite_ordinal_def)
apply (blast intro: finite_Ord_is_nat intro: nat_into_Ord
             dest: Ord_trans naturals_not_limit)
done

lemma Limit_non_Limit_implies_nat:
     "[| Limit(a); ∀x∈a. ~ Limit(x) |] ==> a = nat"
apply (rule le_anti_sym)
apply (rule all_lt_imp_le, blast, blast intro: Limit_is_Ord)
 apply (simp add: lt_def)
 apply (blast intro: Ord_in_Ord Ord_trans finite_Ord_is_nat)
apply (erule nat_le_Limit)
done

lemma (in M_trivial) omega_abs [simp]:
     "M(a) ==> omega(M,a) <-> a = nat"
apply (simp add: omega_def)
apply (blast intro: Limit_non_Limit_implies_nat dest: naturals_not_limit)
done

lemma (in M_trivial) number1_abs [simp]:
     "M(a) ==> number1(M,a) <-> a = 1"
by (simp add: number1_def)

lemma (in M_trivial) number2_abs [simp]:
     "M(a) ==> number2(M,a) <-> a = succ(1)"
by (simp add: number2_def)

lemma (in M_trivial) number3_abs [simp]:
     "M(a) ==> number3(M,a) <-> a = succ(succ(1))"
by (simp add: number3_def)

text{*Kunen continued to 20...*}

(*Could not get this to work.  The λx∈nat is essential because everything
  but the recursion variable must stay unchanged.  But then the recursion
  equations only hold for x∈nat (or in some other set) and not for the
  whole of the class M.
  consts
    natnumber_aux :: "[i=>o,i] => i"

  primrec
      "natnumber_aux(M,0) = (λx∈nat. if empty(M,x) then 1 else 0)"
      "natnumber_aux(M,succ(n)) =
           (λx∈nat. if (∃y[M]. natnumber_aux(M,n)`y=1 & successor(M,y,x))
                     then 1 else 0)"

  definition
    natnumber :: "[i=>o,i,i] => o"
      "natnumber(M,n,x) == natnumber_aux(M,n)`x = 1"

  lemma (in M_trivial) [simp]:
       "natnumber(M,0,x) == x=0"
*)

subsection{*Some instances of separation and strong replacement*}

locale M_basic = M_trivial +
assumes Inter_separation:
     "M(A) ==> separation(M, λx. ∀y[M]. y∈A --> x∈y)"
  and Diff_separation:
     "M(B) ==> separation(M, λx. x ∉ B)"
  and cartprod_separation:
     "[| M(A); M(B) |]
      ==> separation(M, λz. ∃x[M]. x∈A & (∃y[M]. y∈B & pair(M,x,y,z)))"
  and image_separation:
     "[| M(A); M(r) |]
      ==> separation(M, λy. ∃p[M]. p∈r & (∃x[M]. x∈A & pair(M,x,y,p)))"
  and converse_separation:
     "M(r) ==> separation(M,
         λz. ∃p[M]. p∈r & (∃x[M]. ∃y[M]. pair(M,x,y,p) & pair(M,y,x,z)))"
  and restrict_separation:
     "M(A) ==> separation(M, λz. ∃x[M]. x∈A & (∃y[M]. pair(M,x,y,z)))"
  and comp_separation:
     "[| M(r); M(s) |]
      ==> separation(M, λxz. ∃x[M]. ∃y[M]. ∃z[M]. ∃xy[M]. ∃yz[M].
                  pair(M,x,z,xz) & pair(M,x,y,xy) & pair(M,y,z,yz) &
                  xy∈s & yz∈r)"
  and pred_separation:
     "[| M(r); M(x) |] ==> separation(M, λy. ∃p[M]. p∈r & pair(M,y,x,p))"
  and Memrel_separation:
     "separation(M, λz. ∃x[M]. ∃y[M]. pair(M,x,y,z) & x ∈ y)"
  and funspace_succ_replacement:
     "M(n) ==>
      strong_replacement(M, λp z. ∃f[M]. ∃b[M]. ∃nb[M]. ∃cnbf[M].
                pair(M,f,b,p) & pair(M,n,b,nb) & is_cons(M,nb,f,cnbf) &
                upair(M,cnbf,cnbf,z))"
  and is_recfun_separation:
     --{*for well-founded recursion: used to prove @{text is_recfun_equal}*}
     "[| M(r); M(f); M(g); M(a); M(b) |]
     ==> separation(M,
            λx. ∃xa[M]. ∃xb[M].
                pair(M,x,a,xa) & xa ∈ r & pair(M,x,b,xb) & xb ∈ r &
                (∃fx[M]. ∃gx[M]. fun_apply(M,f,x,fx) & fun_apply(M,g,x,gx) &
                                   fx ≠ gx))"

lemma (in M_basic) cartprod_iff_lemma:
     "[| M(C);  ∀u[M]. u ∈ C <-> (∃x∈A. ∃y∈B. u = {{x}, {x,y}});
         powerset(M, A ∪ B, p1); powerset(M, p1, p2);  M(p2) |]
       ==> C = {u ∈ p2 . ∃x∈A. ∃y∈B. u = {{x}, {x,y}}}"
apply (simp add: powerset_def)
apply (rule equalityI, clarify, simp)
 apply (frule transM, assumption)
 apply (frule transM, assumption, simp (no_asm_simp))
 apply blast
apply clarify
apply (frule transM, assumption, force)
done

lemma (in M_basic) cartprod_iff:
     "[| M(A); M(B); M(C) |]
      ==> cartprod(M,A,B,C) <->
          (∃p1[M]. ∃p2[M]. powerset(M,A Un B,p1) & powerset(M,p1,p2) &
                   C = {z ∈ p2. ∃x∈A. ∃y∈B. z = <x,y>})"
apply (simp add: Pair_def cartprod_def, safe)
defer 1
  apply (simp add: powerset_def)
 apply blast
txt{*Final, difficult case: the left-to-right direction of the theorem.*}
apply (insert power_ax, simp add: power_ax_def)
apply (frule_tac x="A Un B" and P="λx. rex(M,?Q(x))" in rspec)
apply (blast, clarify)
apply (drule_tac x=z and P="λx. rex(M,?Q(x))" in rspec)
apply assumption
apply (blast intro: cartprod_iff_lemma)
done

lemma (in M_basic) cartprod_closed_lemma:
     "[| M(A); M(B) |] ==> ∃C[M]. cartprod(M,A,B,C)"
apply (simp del: cartprod_abs add: cartprod_iff)
apply (insert power_ax, simp add: power_ax_def)
apply (frule_tac x="A Un B" and P="λx. rex(M,?Q(x))" in rspec)
apply (blast, clarify)
apply (drule_tac x=z and P="λx. rex(M,?Q(x))" in rspec, auto)
apply (intro rexI conjI, simp+)
apply (insert cartprod_separation [of A B], simp)
done

text{*All the lemmas above are necessary because Powerset is not absolute.
      I should have used Replacement instead!*}
lemma (in M_basic) cartprod_closed [intro,simp]:
     "[| M(A); M(B) |] ==> M(A*B)"
by (frule cartprod_closed_lemma, assumption, force)

lemma (in M_basic) sum_closed [intro,simp]:
     "[| M(A); M(B) |] ==> M(A+B)"
by (simp add: sum_def)

lemma (in M_basic) sum_abs [simp]:
     "[| M(A); M(B); M(Z) |] ==> is_sum(M,A,B,Z) <-> (Z = A+B)"
by (simp add: is_sum_def sum_def singleton_0 nat_into_M)

lemma (in M_trivial) Inl_in_M_iff [iff]:
     "M(Inl(a)) <-> M(a)"
by (simp add: Inl_def)

lemma (in M_trivial) Inl_abs [simp]:
     "M(Z) ==> is_Inl(M,a,Z) <-> (Z = Inl(a))"
by (simp add: is_Inl_def Inl_def)

lemma (in M_trivial) Inr_in_M_iff [iff]:
     "M(Inr(a)) <-> M(a)"
by (simp add: Inr_def)

lemma (in M_trivial) Inr_abs [simp]:
     "M(Z) ==> is_Inr(M,a,Z) <-> (Z = Inr(a))"
by (simp add: is_Inr_def Inr_def)


subsubsection {*converse of a relation*}

lemma (in M_basic) M_converse_iff:
     "M(r) ==>
      converse(r) =
      {z ∈ Union(Union(r)) * Union(Union(r)).
       ∃p∈r. ∃x[M]. ∃y[M]. p = ⟨x,y⟩ & z = ⟨y,x⟩}"
apply (rule equalityI)
 prefer 2 apply (blast dest: transM, clarify, simp)
apply (simp add: Pair_def)
apply (blast dest: transM)
done

lemma (in M_basic) converse_closed [intro,simp]:
     "M(r) ==> M(converse(r))"
apply (simp add: M_converse_iff)
apply (insert converse_separation [of r], simp)
done

lemma (in M_basic) converse_abs [simp]:
     "[| M(r); M(z) |] ==> is_converse(M,r,z) <-> z = converse(r)"
apply (simp add: is_converse_def)
apply (rule iffI)
 prefer 2 apply blast
apply (rule M_equalityI)
  apply simp
  apply (blast dest: transM)+
done


subsubsection {*image, preimage, domain, range*}

lemma (in M_basic) image_closed [intro,simp]:
     "[| M(A); M(r) |] ==> M(r``A)"
apply (simp add: image_iff_Collect)
apply (insert image_separation [of A r], simp)
done

lemma (in M_basic) vimage_abs [simp]:
     "[| M(r); M(A); M(z) |] ==> pre_image(M,r,A,z) <-> z = r-``A"
apply (simp add: pre_image_def)
apply (rule iffI)
 apply (blast intro!: equalityI dest: transM, blast)
done

lemma (in M_basic) vimage_closed [intro,simp]:
     "[| M(A); M(r) |] ==> M(r-``A)"
by (simp add: vimage_def)


subsubsection{*Domain, range and field*}

lemma (in M_basic) domain_abs [simp]:
     "[| M(r); M(z) |] ==> is_domain(M,r,z) <-> z = domain(r)"
apply (simp add: is_domain_def)
apply (blast intro!: equalityI dest: transM)
done

lemma (in M_basic) domain_closed [intro,simp]:
     "M(r) ==> M(domain(r))"
apply (simp add: domain_eq_vimage)
done

lemma (in M_basic) range_abs [simp]:
     "[| M(r); M(z) |] ==> is_range(M,r,z) <-> z = range(r)"
apply (simp add: is_range_def)
apply (blast intro!: equalityI dest: transM)
done

lemma (in M_basic) range_closed [intro,simp]:
     "M(r) ==> M(range(r))"
apply (simp add: range_eq_image)
done

lemma (in M_basic) field_abs [simp]:
     "[| M(r); M(z) |] ==> is_field(M,r,z) <-> z = field(r)"
by (simp add: domain_closed range_closed is_field_def field_def)

lemma (in M_basic) field_closed [intro,simp]:
     "M(r) ==> M(field(r))"
by (simp add: domain_closed range_closed Un_closed field_def)


subsubsection{*Relations, functions and application*}

lemma (in M_basic) relation_abs [simp]:
     "M(r) ==> is_relation(M,r) <-> relation(r)"
apply (simp add: is_relation_def relation_def)
apply (blast dest!: bspec dest: pair_components_in_M)+
done

lemma (in M_basic) function_abs [simp]:
     "M(r) ==> is_function(M,r) <-> function(r)"
apply (simp add: is_function_def function_def, safe)
   apply (frule transM, assumption)
  apply (blast dest: pair_components_in_M)+
done

lemma (in M_basic) apply_closed [intro,simp]:
     "[|M(f); M(a)|] ==> M(f`a)"
by (simp add: apply_def)

lemma (in M_basic) apply_abs [simp]:
     "[| M(f); M(x); M(y) |] ==> fun_apply(M,f,x,y) <-> f`x = y"
apply (simp add: fun_apply_def apply_def, blast)
done

lemma (in M_basic) typed_function_abs [simp]:
     "[| M(A); M(f) |] ==> typed_function(M,A,B,f) <-> f ∈ A -> B"
apply (auto simp add: typed_function_def relation_def Pi_iff)
apply (blast dest: pair_components_in_M)+
done

lemma (in M_basic) injection_abs [simp]:
     "[| M(A); M(f) |] ==> injection(M,A,B,f) <-> f ∈ inj(A,B)"
apply (simp add: injection_def apply_iff inj_def apply_closed)
apply (blast dest: transM [of _ A])
done

lemma (in M_basic) surjection_abs [simp]:
     "[| M(A); M(B); M(f) |] ==> surjection(M,A,B,f) <-> f ∈ surj(A,B)"
by (simp add: surjection_def surj_def)

lemma (in M_basic) bijection_abs [simp]:
     "[| M(A); M(B); M(f) |] ==> bijection(M,A,B,f) <-> f ∈ bij(A,B)"
by (simp add: bijection_def bij_def)


subsubsection{*Composition of relations*}

lemma (in M_basic) M_comp_iff:
     "[| M(r); M(s) |]
      ==> r O s =
          {xz ∈ domain(s) * range(r).
            ∃x[M]. ∃y[M]. ∃z[M]. xz = ⟨x,z⟩ & ⟨x,y⟩ ∈ s & ⟨y,z⟩ ∈ r}"
apply (simp add: comp_def)
apply (rule equalityI)
 apply clarify
 apply simp
 apply  (blast dest:  transM)+
done

lemma (in M_basic) comp_closed [intro,simp]:
     "[| M(r); M(s) |] ==> M(r O s)"
apply (simp add: M_comp_iff)
apply (insert comp_separation [of r s], simp)
done

lemma (in M_basic) composition_abs [simp]:
     "[| M(r); M(s); M(t) |] ==> composition(M,r,s,t) <-> t = r O s"
apply safe
 txt{*Proving @{term "composition(M, r, s, r O s)"}*}
 prefer 2
 apply (simp add: composition_def comp_def)
 apply (blast dest: transM)
txt{*Opposite implication*}
apply (rule M_equalityI)
  apply (simp add: composition_def comp_def)
  apply (blast del: allE dest: transM)+
done

text{*no longer needed*}
lemma (in M_basic) restriction_is_function:
     "[| restriction(M,f,A,z); function(f); M(f); M(A); M(z) |]
      ==> function(z)"
apply (simp add: restriction_def ball_iff_equiv)
apply (unfold function_def, blast)
done

lemma (in M_basic) restriction_abs [simp]:
     "[| M(f); M(A); M(z) |]
      ==> restriction(M,f,A,z) <-> z = restrict(f,A)"
apply (simp add: ball_iff_equiv restriction_def restrict_def)
apply (blast intro!: equalityI dest: transM)
done


lemma (in M_basic) M_restrict_iff:
     "M(r) ==> restrict(r,A) = {z ∈ r . ∃x∈A. ∃y[M]. z = ⟨x, y⟩}"
by (simp add: restrict_def, blast dest: transM)

lemma (in M_basic) restrict_closed [intro,simp]:
     "[| M(A); M(r) |] ==> M(restrict(r,A))"
apply (simp add: M_restrict_iff)
apply (insert restrict_separation [of A], simp)
done

lemma (in M_basic) Inter_abs [simp]:
     "[| M(A); M(z) |] ==> big_inter(M,A,z) <-> z = Inter(A)"
apply (simp add: big_inter_def Inter_def)
apply (blast intro!: equalityI dest: transM)
done

lemma (in M_basic) Inter_closed [intro,simp]:
     "M(A) ==> M(Inter(A))"
by (insert Inter_separation, simp add: Inter_def)

lemma (in M_basic) Int_closed [intro,simp]:
     "[| M(A); M(B) |] ==> M(A Int B)"
apply (subgoal_tac "M({A,B})")
apply (frule Inter_closed, force+)
done

lemma (in M_basic) Diff_closed [intro,simp]:
     "[|M(A); M(B)|] ==> M(A-B)"
by (insert Diff_separation, simp add: Diff_def)

subsubsection{*Some Facts About Separation Axioms*}

lemma (in M_basic) separation_conj:
     "[|separation(M,P); separation(M,Q)|] ==> separation(M, λz. P(z) & Q(z))"
by (simp del: separation_closed
         add: separation_iff Collect_Int_Collect_eq [symmetric])

(*???equalities*)
lemma Collect_Un_Collect_eq:
     "Collect(A,P) Un Collect(A,Q) = Collect(A, %x. P(x) | Q(x))"
by blast

lemma Diff_Collect_eq:
     "A - Collect(A,P) = Collect(A, %x. ~ P(x))"
by blast

lemma (in M_trivial) Collect_rall_eq:
     "M(Y) ==> Collect(A, %x. ∀y[M]. y∈Y --> P(x,y)) =
               (if Y=0 then A else (\<Inter>y ∈ Y. {x ∈ A. P(x,y)}))"
apply simp
apply (blast intro!: equalityI dest: transM)
done

lemma (in M_basic) separation_disj:
     "[|separation(M,P); separation(M,Q)|] ==> separation(M, λz. P(z) | Q(z))"
by (simp del: separation_closed
         add: separation_iff Collect_Un_Collect_eq [symmetric])

lemma (in M_basic) separation_neg:
     "separation(M,P) ==> separation(M, λz. ~P(z))"
by (simp del: separation_closed
         add: separation_iff Diff_Collect_eq [symmetric])

lemma (in M_basic) separation_imp:
     "[|separation(M,P); separation(M,Q)|]
      ==> separation(M, λz. P(z) --> Q(z))"
by (simp add: separation_neg separation_disj not_disj_iff_imp [symmetric])

text{*This result is a hint of how little can be done without the Reflection
  Theorem.  The quantifier has to be bounded by a set.  We also need another
  instance of Separation!*}
lemma (in M_basic) separation_rall:
     "[|M(Y); ∀y[M]. separation(M, λx. P(x,y));
        ∀z[M]. strong_replacement(M, λx y. y = {u ∈ z . P(u,x)})|]
      ==> separation(M, λx. ∀y[M]. y∈Y --> P(x,y))"
apply (simp del: separation_closed rall_abs
         add: separation_iff Collect_rall_eq)
apply (blast intro!: Inter_closed RepFun_closed dest: transM)
done


subsubsection{*Functions and function space*}

text{*The assumption @{term "M(A->B)"} is unusual, but essential: in
all but trivial cases, A->B cannot be expected to belong to @{term M}.*}
lemma (in M_basic) is_funspace_abs [simp]:
     "[|M(A); M(B); M(F); M(A->B)|] ==> is_funspace(M,A,B,F) <-> F = A->B";
apply (simp add: is_funspace_def)
apply (rule iffI)
 prefer 2 apply blast
apply (rule M_equalityI)
  apply simp_all
done

lemma (in M_basic) succ_fun_eq2:
     "[|M(B); M(n->B)|] ==>
      succ(n) -> B =
      \<Union>{z. p ∈ (n->B)*B, ∃f[M]. ∃b[M]. p = <f,b> & z = {cons(<n,b>, f)}}"
apply (simp add: succ_fun_eq)
apply (blast dest: transM)
done

lemma (in M_basic) funspace_succ:
     "[|M(n); M(B); M(n->B) |] ==> M(succ(n) -> B)"
apply (insert funspace_succ_replacement [of n], simp)
apply (force simp add: succ_fun_eq2 univalent_def)
done

text{*@{term M} contains all finite function spaces.  Needed to prove the
absoluteness of transitive closure.  See the definition of
@{text rtrancl_alt} in in @{text WF_absolute.thy}.*}
lemma (in M_basic) finite_funspace_closed [intro,simp]:
     "[|n∈nat; M(B)|] ==> M(n->B)"
apply (induct_tac n, simp)
apply (simp add: funspace_succ nat_into_M)
done


subsection{*Relativization and Absoluteness for Boolean Operators*}

definition
  is_bool_of_o :: "[i=>o, o, i] => o" where
   "is_bool_of_o(M,P,z) == (P & number1(M,z)) | (~P & empty(M,z))"

definition
  is_not :: "[i=>o, i, i] => o" where
   "is_not(M,a,z) == (number1(M,a)  & empty(M,z)) |
                     (~number1(M,a) & number1(M,z))"

definition
  is_and :: "[i=>o, i, i, i] => o" where
   "is_and(M,a,b,z) == (number1(M,a)  & z=b) |
                       (~number1(M,a) & empty(M,z))"

definition
  is_or :: "[i=>o, i, i, i] => o" where
   "is_or(M,a,b,z) == (number1(M,a)  & number1(M,z)) |
                      (~number1(M,a) & z=b)"

lemma (in M_trivial) bool_of_o_abs [simp]:
     "M(z) ==> is_bool_of_o(M,P,z) <-> z = bool_of_o(P)"
by (simp add: is_bool_of_o_def bool_of_o_def)


lemma (in M_trivial) not_abs [simp]:
     "[| M(a); M(z)|] ==> is_not(M,a,z) <-> z = not(a)"
by (simp add: Bool.not_def cond_def is_not_def)

lemma (in M_trivial) and_abs [simp]:
     "[| M(a); M(b); M(z)|] ==> is_and(M,a,b,z) <-> z = a and b"
by (simp add: Bool.and_def cond_def is_and_def)

lemma (in M_trivial) or_abs [simp]:
     "[| M(a); M(b); M(z)|] ==> is_or(M,a,b,z) <-> z = a or b"
by (simp add: Bool.or_def cond_def is_or_def)


lemma (in M_trivial) bool_of_o_closed [intro,simp]:
     "M(bool_of_o(P))"
by (simp add: bool_of_o_def)

lemma (in M_trivial) and_closed [intro,simp]:
     "[| M(p); M(q) |] ==> M(p and q)"
by (simp add: and_def cond_def)

lemma (in M_trivial) or_closed [intro,simp]:
     "[| M(p); M(q) |] ==> M(p or q)"
by (simp add: or_def cond_def)

lemma (in M_trivial) not_closed [intro,simp]:
     "M(p) ==> M(not(p))"
by (simp add: Bool.not_def cond_def)


subsection{*Relativization and Absoluteness for List Operators*}

definition
  is_Nil :: "[i=>o, i] => o" where
     --{* because @{prop "[] ≡ Inl(0)"}*}
    "is_Nil(M,xs) == ∃zero[M]. empty(M,zero) & is_Inl(M,zero,xs)"

definition
  is_Cons :: "[i=>o,i,i,i] => o" where
     --{* because @{prop "Cons(a, l) ≡ Inr(⟨a,l⟩)"}*}
    "is_Cons(M,a,l,Z) == ∃p[M]. pair(M,a,l,p) & is_Inr(M,p,Z)"


lemma (in M_trivial) Nil_in_M [intro,simp]: "M(Nil)"
by (simp add: Nil_def)

lemma (in M_trivial) Nil_abs [simp]: "M(Z) ==> is_Nil(M,Z) <-> (Z = Nil)"
by (simp add: is_Nil_def Nil_def)

lemma (in M_trivial) Cons_in_M_iff [iff]: "M(Cons(a,l)) <-> M(a) & M(l)"
by (simp add: Cons_def)

lemma (in M_trivial) Cons_abs [simp]:
     "[|M(a); M(l); M(Z)|] ==> is_Cons(M,a,l,Z) <-> (Z = Cons(a,l))"
by (simp add: is_Cons_def Cons_def)


definition
  quasilist :: "i => o" where
    "quasilist(xs) == xs=Nil | (∃x l. xs = Cons(x,l))"

definition
  is_quasilist :: "[i=>o,i] => o" where
    "is_quasilist(M,z) == is_Nil(M,z) | (∃x[M]. ∃l[M]. is_Cons(M,x,l,z))"

definition
  list_case' :: "[i, [i,i]=>i, i] => i" where
    --{*A version of @{term list_case} that's always defined.*}
    "list_case'(a,b,xs) ==
       if quasilist(xs) then list_case(a,b,xs) else 0"

definition
  is_list_case :: "[i=>o, i, [i,i,i]=>o, i, i] => o" where
    --{*Returns 0 for non-lists*}
    "is_list_case(M, a, is_b, xs, z) ==
       (is_Nil(M,xs) --> z=a) &
       (∀x[M]. ∀l[M]. is_Cons(M,x,l,xs) --> is_b(x,l,z)) &
       (is_quasilist(M,xs) | empty(M,z))"

definition
  hd' :: "i => i" where
    --{*A version of @{term hd} that's always defined.*}
    "hd'(xs) == if quasilist(xs) then hd(xs) else 0"

definition
  tl' :: "i => i" where
    --{*A version of @{term tl} that's always defined.*}
    "tl'(xs) == if quasilist(xs) then tl(xs) else 0"

definition
  is_hd :: "[i=>o,i,i] => o" where
     --{* @{term "hd([]) = 0"} no constraints if not a list.
          Avoiding implication prevents the simplifier's looping.*}
    "is_hd(M,xs,H) ==
       (is_Nil(M,xs) --> empty(M,H)) &
       (∀x[M]. ∀l[M]. ~ is_Cons(M,x,l,xs) | H=x) &
       (is_quasilist(M,xs) | empty(M,H))"

definition
  is_tl :: "[i=>o,i,i] => o" where
     --{* @{term "tl([]) = []"}; see comments about @{term is_hd}*}
    "is_tl(M,xs,T) ==
       (is_Nil(M,xs) --> T=xs) &
       (∀x[M]. ∀l[M]. ~ is_Cons(M,x,l,xs) | T=l) &
       (is_quasilist(M,xs) | empty(M,T))"

subsubsection{*@{term quasilist}: For Case-Splitting with @{term list_case'}*}

lemma [iff]: "quasilist(Nil)"
by (simp add: quasilist_def)

lemma [iff]: "quasilist(Cons(x,l))"
by (simp add: quasilist_def)

lemma list_imp_quasilist: "l ∈ list(A) ==> quasilist(l)"
by (erule list.cases, simp_all)

subsubsection{*@{term list_case'}, the Modified Version of @{term list_case}*}

lemma list_case'_Nil [simp]: "list_case'(a,b,Nil) = a"
by (simp add: list_case'_def quasilist_def)

lemma list_case'_Cons [simp]: "list_case'(a,b,Cons(x,l)) = b(x,l)"
by (simp add: list_case'_def quasilist_def)

lemma non_list_case: "~ quasilist(x) ==> list_case'(a,b,x) = 0"
by (simp add: quasilist_def list_case'_def)

lemma list_case'_eq_list_case [simp]:
     "xs ∈ list(A) ==>list_case'(a,b,xs) = list_case(a,b,xs)"
by (erule list.cases, simp_all)

lemma (in M_basic) list_case'_closed [intro,simp]:
  "[|M(k); M(a); ∀x[M]. ∀y[M]. M(b(x,y))|] ==> M(list_case'(a,b,k))"
apply (case_tac "quasilist(k)")
 apply (simp add: quasilist_def, force)
apply (simp add: non_list_case)
done

lemma (in M_trivial) quasilist_abs [simp]:
     "M(z) ==> is_quasilist(M,z) <-> quasilist(z)"
by (auto simp add: is_quasilist_def quasilist_def)

lemma (in M_trivial) list_case_abs [simp]:
     "[| relation2(M,is_b,b); M(k); M(z) |]
      ==> is_list_case(M,a,is_b,k,z) <-> z = list_case'(a,b,k)"
apply (case_tac "quasilist(k)")
 prefer 2
 apply (simp add: is_list_case_def non_list_case)
 apply (force simp add: quasilist_def)
apply (simp add: quasilist_def is_list_case_def)
apply (elim disjE exE)
 apply (simp_all add: relation2_def)
done


subsubsection{*The Modified Operators @{term hd'} and @{term tl'}*}

lemma (in M_trivial) is_hd_Nil: "is_hd(M,[],Z) <-> empty(M,Z)"
by (simp add: is_hd_def)

lemma (in M_trivial) is_hd_Cons:
     "[|M(a); M(l)|] ==> is_hd(M,Cons(a,l),Z) <-> Z = a"
by (force simp add: is_hd_def)

lemma (in M_trivial) hd_abs [simp]:
     "[|M(x); M(y)|] ==> is_hd(M,x,y) <-> y = hd'(x)"
apply (simp add: hd'_def)
apply (intro impI conjI)
 prefer 2 apply (force simp add: is_hd_def)
apply (simp add: quasilist_def is_hd_def)
apply (elim disjE exE, auto)
done

lemma (in M_trivial) is_tl_Nil: "is_tl(M,[],Z) <-> Z = []"
by (simp add: is_tl_def)

lemma (in M_trivial) is_tl_Cons:
     "[|M(a); M(l)|] ==> is_tl(M,Cons(a,l),Z) <-> Z = l"
by (force simp add: is_tl_def)

lemma (in M_trivial) tl_abs [simp]:
     "[|M(x); M(y)|] ==> is_tl(M,x,y) <-> y = tl'(x)"
apply (simp add: tl'_def)
apply (intro impI conjI)
 prefer 2 apply (force simp add: is_tl_def)
apply (simp add: quasilist_def is_tl_def)
apply (elim disjE exE, auto)
done

lemma (in M_trivial) relation1_tl: "relation1(M, is_tl(M), tl')"
by (simp add: relation1_def)

lemma hd'_Nil: "hd'([]) = 0"
by (simp add: hd'_def)

lemma hd'_Cons: "hd'(Cons(a,l)) = a"
by (simp add: hd'_def)

lemma tl'_Nil: "tl'([]) = []"
by (simp add: tl'_def)

lemma tl'_Cons: "tl'(Cons(a,l)) = l"
by (simp add: tl'_def)

lemma iterates_tl_Nil: "n ∈ nat ==> tl'^n ([]) = []"
apply (induct_tac n)
apply (simp_all add: tl'_Nil)
done

lemma (in M_basic) tl'_closed: "M(x) ==> M(tl'(x))"
apply (simp add: tl'_def)
apply (force simp add: quasilist_def)
done


end

Relativized versions of standard set-theoretic concepts

lemma triv_Relation1:

  Relation1(M, A, λx y. y = f(x), f)

lemma triv_Relation2:

  Relation2(M, A, B, λx y a. a = f(x, y), f)

The relativized ZF axioms

A trivial consistency proof for $V_\omega$

lemma univ0_downwards_mem:

  [| yx; xuniv(0) |] ==> yuniv(0)

lemma univ0_Ball_abs:

  Auniv(0) ==> (∀xA. xuniv(0) --> P(x)) <-> (∀xA. P(x))

lemma univ0_Bex_abs:

  Auniv(0) ==> (∃xA. xuniv(0) ∧ P(x)) <-> (∃xA. P(x))

lemma separation_cong:

  (!!x. M(x) ==> P(x) <-> P'(x))
  ==> separation(M, λx. P(x)) <-> separation(M, λx. P'(x))

lemma univalent_cong:

  [| A = A'; !!x y. [| xA; M(x); M(y) |] ==> P(x, y) <-> P'(x, y) |]
  ==> univalent(M, A, λx y. P(x, y)) <-> univalent(M, A', λx y. P'(x, y))

lemma univalent_triv:

  univalent(M, A, λx y. y = f(x))

lemma univalent_conjI2:

  univalent(M, A, Q) ==> univalent(M, A, λx y. P(x, y) ∧ Q(x, y))

lemma strong_replacement_cong:

  (!!x y. [| M(x); M(y) |] ==> P(x, y) <-> P'(x, y))
  ==> strong_replacement(M, λx y. P(x, y)) <->
      strong_replacement(M, λx y. P'(x, y))

lemma

  extensionalityx. xuniv(0))

lemma Collect_in_Vfrom:

  [| XVfrom(A, j); Transset(A) |] ==> Collect(X, P) ∈ Vfrom(A, succ(j))

lemma Collect_in_VLimit:

  [| XVfrom(A, i); Limit(i); Transset(A) |] ==> Collect(X, P) ∈ Vfrom(A, i)

lemma Collect_in_univ:

  [| Xuniv(A); Transset(A) |] ==> Collect(X, P) ∈ univ(A)

lemma

  separationx. xuniv(0), P)

lemma

  upair_axx. xuniv(0))

lemma

  Union_axx. xuniv(0))

lemma Pow_in_univ:

  [| Xuniv(A); Transset(A) |] ==> Pow(X) ∈ univ(A)

lemma

  power_axx. xuniv(0))

lemma

  foundation_axx. xuniv(0))

Lemmas Needed to Reduce Some Set Constructions to Instances of Separation

lemma image_iff_Collect:

  r `` A = {y ∈ \<Union>\<Union>r . ∃pr. ∃xA. p = ⟨x, y⟩}

lemma vimage_iff_Collect:

  r -`` A = {x ∈ \<Union>\<Union>r . ∃pr. ∃yA. p = ⟨x, y⟩}

lemma domain_eq_vimage:

  domain(r) = r -`` (\<Union>\<Union>r)

lemma range_eq_image:

  range(r) = r `` (\<Union>\<Union>r)

lemma replacementD:

  [| replacement(M, P); M(A); univalent(M, A, P) |]
  ==> ∃Y[M]. ∀b[M]. (∃x[M]. xAP(x, b)) --> bY

lemma strong_replacementD:

  [| strong_replacement(M, P); M(A); univalent(M, A, P) |]
  ==> ∃Y[M]. ∀b[M]. bY <-> (∃x[M]. xAP(x, b))

lemma separationD:

  [| separation(M, P); M(z) |] ==> ∃y[M]. ∀x[M]. xy <-> xzP(x)

Introducing a Transitive Class Model

lemma nonempty:

  M(0)

lemma rall_abs:

  M(A) ==> (∀x[M]. xA --> P(x)) <-> (∀xA. P(x))

lemma rex_abs:

  M(A) ==> (∃x[M]. xAP(x)) <-> (∃xA. P(x))

lemma ball_iff_equiv:

  M(A)
  ==> (∀x[M]. xA <-> P(x)) <-> (∀xA. P(x)) ∧ (∀x. P(x) --> M(x) --> xA)

lemma M_equalityI:

  [| !!x. M(x) ==> xA <-> xB; M(A); M(B) |] ==> A = B

Trivial Absoluteness Proofs: Empty Set, Pairs, etc.

lemma empty_abs:

  M(z) ==> empty(M, z) <-> z = 0

lemma subset_abs:

  M(A) ==> subset(M, A, B) <-> AB

lemma upair_abs:

  M(z) ==> upair(M, a, b, z) <-> z = {a, b}

lemma upair_in_M_iff:

  M({a, b}) <-> M(a) ∧ M(b)

lemma singleton_in_M_iff:

  M({a}) <-> M(a)

lemma pair_abs:

  M(z) ==> pair(M, a, b, z) <-> z = ⟨a, b

lemma pair_in_M_iff:

  M(⟨a, b⟩) <-> M(a) ∧ M(b)

lemma pair_components_in_M:

  [| ⟨x, y⟩ ∈ A; M(A) |] ==> M(x) ∧ M(y)

lemma cartprod_abs:

  [| M(A); M(B); M(z) |] ==> cartprod(M, A, B, z) <-> z = A × B

Absoluteness for Unions and Intersections

lemma union_abs:

  [| M(a); M(b); M(z) |] ==> union(M, a, b, z) <-> z = ab

lemma inter_abs:

  [| M(a); M(b); M(z) |] ==> inter(M, a, b, z) <-> z = ab

lemma setdiff_abs:

  [| M(a); M(b); M(z) |] ==> setdiff(M, a, b, z) <-> z = a - b

lemma Union_abs:

  [| M(A); M(z) |] ==> big_union(M, A, z) <-> z = \<Union>A

lemma Union_closed:

  M(A) ==> M(\<Union>A)

lemma Un_closed:

  [| M(A); M(B) |] ==> M(AB)

lemma cons_closed:

  [| M(a); M(A) |] ==> M(cons(a, A))

lemma cons_abs:

  [| M(b); M(z) |] ==> is_cons(M, a, b, z) <-> z = cons(a, b)

lemma successor_abs:

  [| M(a); M(z) |] ==> successor(M, a, z) <-> z = succ(a)

lemma succ_in_M_iff:

  M(succ(a)) <-> M(a)

Absoluteness for Separation and Replacement

lemma separation_closed:

  [| separation(M, P); M(A) |] ==> M(Collect(A, P))

lemma separation_iff:

  separation(M, P) <-> (∀z[M]. ∃y[M]. is_Collect(M, z, P, y))

lemma Collect_abs:

  [| M(A); M(z) |] ==> is_Collect(M, A, P, z) <-> z = Collect(A, P)

lemma strong_replacementI:

  (!!B. M(B) ==> separation(M, λu. ∃x[M]. xBP(x, u)))
  ==> strong_replacement(M, P)

The Operator @{term is_Replace}

lemma is_Replace_cong:

  [| A = A'; !!x y. [| M(x); M(y) |] ==> P(x, y) <-> P'(x, y); z = z' |]
  ==> is_Replace(M, A, λx y. P(x, y), z) <-> is_Replace(M, A', λx y. P'(x, y), z')

lemma univalent_Replace_iff:

  [| M(A); univalent(M, A, P); !!x y. [| xA; P(x, y) |] ==> M(y) |]
  ==> u ∈ Replace(A, P) <-> (∃x. xAP(x, u))

lemma strong_replacement_closed:

  [| strong_replacement(M, P); M(A); univalent(M, A, P);
     !!x y. [| xA; P(x, y) |] ==> M(y) |]
  ==> M(Replace(A, P))

lemma Replace_abs:

  [| M(A); M(z); univalent(M, A, P); !!x y. [| xA; P(x, y) |] ==> M(y) |]
  ==> is_Replace(M, A, P, z) <-> z = Replace(A, P)

lemma RepFun_closed:

  [| strong_replacement(M, λx y. y = f(x)); M(A); ∀xA. M(f(x)) |]
  ==> M(RepFun(A, f))

lemma Replace_conj_eq:

  {y . xA, xAy = f(x)} = {y . xA, y = f(x)}

lemma RepFun_closed2:

  [| strong_replacement(M, λx y. xAy = f(x)); M(A); ∀xA. M(f(x)) |]
  ==> M({f(x) . xA})

Absoluteness for @{term Lambda}

lemma lam_closed:

  [| strong_replacement(M, λx y. y = ⟨x, b(x)⟩); M(A); ∀xA. M(b(x)) |]
  ==> MxA. b(x))

lemma lam_closed2:

  [| strong_replacement(M, λx y. xAy = ⟨x, b(x)⟩); M(A);
     ∀m[M]. mA --> M(b(m)) |]
  ==> M(Lambda(A, b))

lemma lambda_abs2:

  [| Relation1(M, A, is_b, b); M(A); ∀m[M]. mA --> M(b(m)); M(z) |]
  ==> is_lambda(M, A, is_b, z) <-> z = Lambda(A, b)

lemma is_lambda_cong:

  [| A = A'; z = z';
     !!x y. [| xA; M(x); M(y) |] ==> is_b(x, y) <-> is_b'(x, y) |]
  ==> is_lambda(M, A, λx y. is_b(x, y), z) <->
      is_lambda(M, A', λx y. is_b'(x, y), z')

lemma image_abs:

  [| M(r); M(A); M(z) |] ==> image(M, r, A, z) <-> z = r `` A

lemma powerset_Pow:

  powerset(M, x, Pow(x))

lemma powerset_imp_subset_Pow:

  [| powerset(M, x, y); M(y) |] ==> y ⊆ Pow(x)

Absoluteness for the Natural Numbers

lemma nat_into_M:

  nnat ==> M(n)

lemma nat_case_closed:

  [| M(k); M(a); ∀m[M]. M(b(m)) |] ==> M(nat_case(a, b, k))

lemma quasinat_abs:

  M(z) ==> is_quasinat(M, z) <-> quasinat(z)

lemma nat_case_abs:

  [| relation1(M, is_b, b); M(k); M(z) |]
  ==> is_nat_case(M, a, is_b, k, z) <-> z = nat_case(a, b, k)

lemma is_nat_case_cong:

  [| a = a'; k = k'; z = z'; M(z');
     !!x y. [| M(x); M(y) |] ==> is_b(x, y) <-> is_b'(x, y) |]
  ==> is_nat_case(M, a, is_b, k, z) <-> is_nat_case(M, a', is_b', k', z')

Absoluteness for Ordinals

lemma lt_closed:

  [| j < i; M(i) |] ==> M(j)

lemma transitive_set_abs:

  M(a) ==> transitive_set(M, a) <-> Transset(a)

lemma ordinal_abs:

  M(a) ==> ordinal(M, a) <-> Ord(a)

lemma limit_ordinal_abs:

  M(a) ==> limit_ordinal(M, a) <-> Limit(a)

lemma successor_ordinal_abs:

  M(a) ==> successor_ordinal(M, a) <-> Ord(a) ∧ (∃b[M]. a = succ(b))

lemma finite_Ord_is_nat:

  [| Ord(a); ¬ Limit(a); ∀xa. ¬ Limit(x) |] ==> anat

lemma finite_ordinal_abs:

  M(a) ==> finite_ordinal(M, a) <-> anat

lemma Limit_non_Limit_implies_nat:

  [| Limit(a); ∀xa. ¬ Limit(x) |] ==> a = nat

lemma omega_abs:

  M(a) ==> omega(M, a) <-> a = nat

lemma number1_abs:

  M(a) ==> number1(M, a) <-> a = 1

lemma number2_abs:

  M(a) ==> number2(M, a) <-> a = 2

lemma number3_abs:

  M(a) ==> number3(M, a) <-> a = succ(2)

Some instances of separation and strong replacement

lemma cartprod_iff_lemma:

  [| M(C); ∀u[M]. uC <-> (∃xA. ∃yB. u = {{x}, {x, y}});
     powerset(M, AB, p1.0); powerset(M, p1.0, p2.0); M(p2.0) |]
  ==> C = {up2.0 . ∃xA. ∃yB. u = {{x}, {x, y}}}

lemma cartprod_iff:

  [| M(A); M(B); M(C) |]
  ==> cartprod(M, A, B, C) <->
      (∃p1[M].
          ∃p2[M].
             powerset(M, AB, p1) ∧
             powerset(M, p1, p2) ∧ C = {zp2 . ∃xA. ∃yB. z = ⟨x, y⟩})

lemma cartprod_closed_lemma:

  [| M(A); M(B) |] ==> ∃C[M]. cartprod(M, A, B, C)

lemma cartprod_closed:

  [| M(A); M(B) |] ==> M(A × B)

lemma sum_closed:

  [| M(A); M(B) |] ==> M(A + B)

lemma sum_abs:

  [| M(A); M(B); M(Z) |] ==> is_sum(M, A, B, Z) <-> Z = A + B

lemma Inl_in_M_iff:

  M(Inl(a)) <-> M(a)

lemma Inl_abs:

  M(Z) ==> is_Inl(M, a, Z) <-> Z = Inl(a)

lemma Inr_in_M_iff:

  M(Inr(a)) <-> M(a)

lemma Inr_abs:

  M(Z) ==> is_Inr(M, a, Z) <-> Z = Inr(a)

converse of a relation

lemma M_converse_iff:

  M(r)
  ==> converse(r) =
      {z ∈ \<Union>\<Union>r × \<Union>\<Union>r .
       ∃pr. ∃x[M]. ∃y[M]. p = ⟨x, y⟩ ∧ z = ⟨y, x⟩}

lemma converse_closed:

  M(r) ==> M(converse(r))

lemma converse_abs:

  [| M(r); M(z) |] ==> is_converse(M, r, z) <-> z = converse(r)

image, preimage, domain, range

lemma image_closed:

  [| M(A); M(r) |] ==> M(r `` A)

lemma vimage_abs:

  [| M(r); M(A); M(z) |] ==> pre_image(M, r, A, z) <-> z = r -`` A

lemma vimage_closed:

  [| M(A); M(r) |] ==> M(r -`` A)

Domain, range and field

lemma domain_abs:

  [| M(r); M(z) |] ==> is_domain(M, r, z) <-> z = domain(r)

lemma domain_closed:

  M(r) ==> M(domain(r))

lemma range_abs:

  [| M(r); M(z) |] ==> is_range(M, r, z) <-> z = range(r)

lemma range_closed:

  M(r) ==> M(range(r))

lemma field_abs:

  [| M(r); M(z) |] ==> is_field(M, r, z) <-> z = field(r)

lemma field_closed:

  M(r) ==> M(field(r))

Relations, functions and application

lemma relation_abs:

  M(r) ==> is_relation(M, r) <-> relation(r)

lemma function_abs:

  M(r) ==> is_function(M, r) <-> function(r)

lemma apply_closed:

  [| M(f); M(a) |] ==> M(f ` a)

lemma apply_abs:

  [| M(f); M(x); M(y) |] ==> fun_apply(M, f, x, y) <-> f ` x = y

lemma typed_function_abs:

  [| M(A); M(f) |] ==> typed_function(M, A, B, f) <-> fA -> B

lemma injection_abs:

  [| M(A); M(f) |] ==> injection(M, A, B, f) <-> finj(A, B)

lemma surjection_abs:

  [| M(A); M(B); M(f) |] ==> surjection(M, A, B, f) <-> fsurj(A, B)

lemma bijection_abs:

  [| M(A); M(B); M(f) |] ==> bijection(M, A, B, f) <-> fbij(A, B)

Composition of relations

lemma M_comp_iff:

  [| M(r); M(s) |]
  ==> r O s =
      {xz ∈ domain(s) × range(r) .
       ∃x[M]. ∃y[M]. ∃z[M]. xz = ⟨x, z⟩ ∧ ⟨x, y⟩ ∈ s ∧ ⟨y, z⟩ ∈ r}

lemma comp_closed:

  [| M(r); M(s) |] ==> M(r O s)

lemma composition_abs:

  [| M(r); M(s); M(t) |] ==> composition(M, r, s, t) <-> t = r O s

lemma restriction_is_function:

  [| restriction(M, f, A, z); function(f); M(f); M(A); M(z) |] ==> function(z)

lemma restriction_abs:

  [| M(f); M(A); M(z) |] ==> restriction(M, f, A, z) <-> z = restrict(f, A)

lemma M_restrict_iff:

  M(r) ==> restrict(r, A) = {zr . ∃xA. ∃y[M]. z = ⟨x, y⟩}

lemma restrict_closed:

  [| M(A); M(r) |] ==> M(restrict(r, A))

lemma Inter_abs:

  [| M(A); M(z) |] ==> big_inter(M, A, z) <-> z = \<Inter>A

lemma Inter_closed:

  M(A) ==> M(\<Inter>A)

lemma Int_closed:

  [| M(A); M(B) |] ==> M(AB)

lemma Diff_closed:

  [| M(A); M(B) |] ==> M(A - B)

Some Facts About Separation Axioms

lemma separation_conj:

  [| separation(M, P); separation(M, Q) |] ==> separation(M, λz. P(z) ∧ Q(z))

lemma Collect_Un_Collect_eq:

  Collect(A, P) ∪ Collect(A, Q) = {xA . P(x) ∨ Q(x)}

lemma Diff_Collect_eq:

  A - Collect(A, P) = {xA . ¬ P(x)}

lemma Collect_rall_eq:

  M(Y)
  ==> {xA . ∀y[M]. yY --> P(x, y)} =
      (if Y = 0 then A else \<Inter>yY. {xA . P(x, y)})

lemma separation_disj:

  [| separation(M, P); separation(M, Q) |] ==> separation(M, λz. P(z) ∨ Q(z))

lemma separation_neg:

  separation(M, P) ==> separation(M, λz. ¬ P(z))

lemma separation_imp:

  [| separation(M, P); separation(M, Q) |] ==> separation(M, λz. P(z) --> Q(z))

lemma separation_rall:

  [| M(Y); ∀y[M]. separation(M, λx. P(x, y));
     ∀z[M]. strong_replacement(M, λx y. y = {uz . P(u, x)}) |]
  ==> separation(M, λx. ∀y[M]. yY --> P(x, y))

Functions and function space

lemma is_funspace_abs:

  [| M(A); M(B); M(F); M(A -> B) |] ==> is_funspace(M, A, B, F) <-> F = A -> B

lemma succ_fun_eq2:

  [| M(B); M(n -> B) |]
  ==> succ(n) -> B =
      \<Union>{z . p ∈ (n -> B) ×
                       B, ∃f[M]. ∃b[M]. p = ⟨f, b⟩ ∧ z = {cons(⟨n, b⟩, f)}}

lemma funspace_succ:

  [| M(n); M(B); M(n -> B) |] ==> M(succ(n) -> B)

lemma finite_funspace_closed:

  [| nnat; M(B) |] ==> M(n -> B)

Relativization and Absoluteness for Boolean Operators

lemma bool_of_o_abs:

  M(z) ==> is_bool_of_o(M, P, z) <-> z = bool_of_o(P)

lemma not_abs:

  [| M(a); M(z) |] ==> is_not(M, a, z) <-> z = not(a)

lemma and_abs:

  [| M(a); M(b); M(z) |] ==> is_and(M, a, b, z) <-> z = a and b

lemma or_abs:

  [| M(a); M(b); M(z) |] ==> is_or(M, a, b, z) <-> z = a or b

lemma bool_of_o_closed:

  M(bool_of_o(P))

lemma and_closed:

  [| M(p); M(q) |] ==> M(p and q)

lemma or_closed:

  [| M(p); M(q) |] ==> M(p or q)

lemma not_closed:

  M(p) ==> M(not(p))

Relativization and Absoluteness for List Operators

lemma Nil_in_M:

  M([])

lemma Nil_abs:

  M(Z) ==> is_Nil(M, Z) <-> Z = []

lemma Cons_in_M_iff:

  M(Cons(a, l)) <-> M(a) ∧ M(l)

lemma Cons_abs:

  [| M(a); M(l); M(Z) |] ==> is_Cons(M, a, l, Z) <-> Z = Cons(a, l)

@{term quasilist}: For Case-Splitting with @{term list_case'}

lemma

  quasilist([])

lemma

  quasilist(Cons(x, l))

lemma list_imp_quasilist:

  l ∈ list(A) ==> quasilist(l)

@{term list_case'}, the Modified Version of @{term list_case}

lemma list_case'_Nil:

  list_case'(a, b, []) = a

lemma list_case'_Cons:

  list_case'(a, b, Cons(x, l)) = b(x, l)

lemma non_list_case:

  ¬ quasilist(x) ==> list_case'(a, b, x) = 0

lemma list_case'_eq_list_case:

  xs ∈ list(A) ==> list_case'(a, b, xs) = list_case(a, b, xs)

lemma list_case'_closed:

  [| M(k); M(a); ∀x[M]. ∀y[M]. M(b(x, y)) |] ==> M(list_case'(a, b, k))

lemma quasilist_abs:

  M(z) ==> is_quasilist(M, z) <-> quasilist(z)

lemma list_case_abs:

  [| relation2(M, is_b, b); M(k); M(z) |]
  ==> is_list_case(M, a, is_b, k, z) <-> z = list_case'(a, b, k)

The Modified Operators @{term hd'} and @{term tl'}

lemma is_hd_Nil:

  is_hd(M, [], Z) <-> empty(M, Z)

lemma is_hd_Cons:

  [| M(a); M(l) |] ==> is_hd(M, Cons(a, l), Z) <-> Z = a

lemma hd_abs:

  [| M(x); M(y) |] ==> is_hd(M, x, y) <-> y = hd'(x)

lemma is_tl_Nil:

  is_tl(M, [], Z) <-> Z = []

lemma is_tl_Cons:

  [| M(a); M(l) |] ==> is_tl(M, Cons(a, l), Z) <-> Z = l

lemma tl_abs:

  [| M(x); M(y) |] ==> is_tl(M, x, y) <-> y = tl'(x)

lemma relation1_tl:

  relation1(M, is_tl(M), tl')

lemma hd'_Nil:

  hd'([]) = 0

lemma hd'_Cons:

  hd'(Cons(a, l)) = a

lemma tl'_Nil:

  tl'([]) = []

lemma tl'_Cons:

  tl'(Cons(a, l)) = l

lemma iterates_tl_Nil:

  nnat ==> tl'^n ([]) = []

lemma tl'_closed:

  M(x) ==> M(tl'(x))