# Theory Relative

(*  Title:      ZF/Constructible/Relative.thy
Author:     Lawrence C Paulson, Cambridge University Computer Laboratory
With modifications by E. Gunther, M. Pagano,
and P. Sánchez Terraf
*)

section ‹Relativization and Absoluteness›

theory Relative imports ZF 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)"

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

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 ‹P› should only involve parameters
belonging to ‹M› and all its quantifiers must be relativized
to ‹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 ‹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])
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 ‹M››
lemma separation_cong [cong]:
"(⋀x. M(x) ⟹ P(x) ⟷ P'(x))
⟹ separation(M, λx. P(x)) ⟷ separation(M, λx. P'(x))"

lemma univalent_cong [cong]:
"⟦A=A'; ⋀x y. ⟦x∈A; 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 [intro,simp]:
"univalent(M, A, λx y. y = f(x))"

lemma univalent_conjI2 [intro,simp]:
"univalent(M,A,Q) ⟹ univalent(M, A, λx y. P(x,y) ∧ Q(x,y))"

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, λx y. P(x,y)) ⟷
strong_replacement(M, λx y. P'(x,y))"

text‹The extensionality axiom›
lemma "extensionality(λx. x ∈ univ(0))"
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)"

lemma "separation(λx. x ∈ univ(0), P)"
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 (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="⋃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)"
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 (cut_tac A=x in foundation)
apply (blast intro: univ0_downwards_mem)
done

lemma "replacement(λx. x ∈ univ(0), P)"
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 ∈ ⋃(⋃(r)). ∃p∈r. ∃x∈A. p=⟨x,y⟩}"
apply (rule equalityI, auto)
done

lemma vimage_iff_Collect:
"r - A = {x ∈ ⋃(⋃(r)). ∃p∈r. ∃y∈A. p=⟨x,y⟩}"
apply (rule equalityI, auto)
done

text‹These two lemmas lets us prove ‹domain_closed› and
‹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)
done

lemma range_eq_image: "range(r) = r  Union(Union(r))"
apply (rule equalityI, auto)
apply (rule imageI, assumption)
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))"

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))))"

lemma separationD:
"⟦separation(M,P); M(z)⟧ ⟹ ∃y[M]. ∀x[M]. x ∈ y ⟷ x ∈ z ∧ P(x)"

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 inhabited›
locale M_trans =
fixes M
assumes transM:   "⟦y∈x; M(x)⟧ ⟹ M(y)"
and M_inhabited: "∃x . M(x)"

lemma (in M_trans) nonempty [simp]:  "M(0)"
proof -
have "M(x) ⟶ M(0)" for x
proof (rule_tac P="λw. M(w) ⟶ M(0)" in eps_induct)
{
fix x
assume "∀y∈x. M(y) ⟶ M(0)" "M(x)"
consider (a) "∃y. y∈x" | (b) "x=0" by auto
then
have "M(x) ⟶ M(0)"
proof cases
case a
then show ?thesis using ‹∀y∈x._› ‹M(x)› transM by auto
next
case b
then show ?thesis by simp
qed
}
then show "M(x) ⟶ M(0)" if "∀y∈x. M(y) ⟶ M(0)" for x
using that by auto
qed
with M_inhabited
show "M(0)" using M_inhabited by blast
qed

text‹The class M is assumed to be transitive and to satisfy some
relativized ZF axioms›
locale M_trivial = M_trans +
assumes upair_ax:         "upair_ax(M)"
and Union_ax:         "Union_ax(M)"

lemma (in M_trans) rall_abs [simp]:
"M(A) ⟹ (∀x[M]. x∈A ⟶ P(x)) ⟷ (∀x∈A. P(x))"
by (blast intro: transM)

lemma (in M_trans) rex_abs [simp]:
"M(A) ⟹ (∃x[M]. x∈A ∧ P(x)) ⟷ (∃x∈A. P(x))"
by (blast intro: transM)

lemma (in M_trans) 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_trans) M_equalityI:
"⟦⋀x. M(x) ⟹ x∈A ⟷ x∈B; M(A); M(B)⟧ ⟹ A=B"
by (blast dest: transM)

subsubsection‹Trivial Absoluteness Proofs: Empty Set, Pairs, etc.›

lemma (in M_trans) empty_abs [simp]:
"M(z) ⟹ empty(M,z) ⟷ z=0"
apply (blast intro: transM)
done

lemma (in M_trans) subset_abs [simp]:
"M(A) ⟹ subset(M,A,B) ⟷ A ⊆ B"
apply (blast intro: transM)
done

lemma (in M_trans) upair_abs [simp]:
"M(z) ⟹ upair(M,a,b,z) ⟷ z={a,b}"
apply (blast intro: transM)
done

lemma (in M_trivial) upair_in_MI [intro!]:
" M(a) ∧ M(b) ⟹ M({a,b})"
by (insert upair_ax, simp add: upair_ax_def)

lemma (in M_trans) upair_in_MD [dest!]:
"M({a,b}) ⟹ M(a) ∧ M(b)"
by (blast intro: transM)

lemma (in M_trivial) upair_in_M_iff [simp]:
"M({a,b}) ⟷ M(a) ∧ M(b)"
by blast

lemma (in M_trivial) singleton_in_MI [intro!]:
"M(a) ⟹ M({a})"
by (insert upair_in_M_iff [of a a], simp)

lemma (in M_trans) singleton_in_MD [dest!]:
"M({a}) ⟹ M(a)"
by (insert upair_in_MD [of a a], simp)

lemma (in M_trivial) singleton_in_M_iff [simp]:
"M({a}) ⟷ M(a)"
by blast

lemma (in M_trans) pair_abs [simp]:
"M(z) ⟹ pair(M,a,b,z) ⟷ z=⟨a,b⟩"
apply (blast intro: transM)
done

lemma (in M_trans) pair_in_MD [dest!]:
"M(⟨a,b⟩) ⟹ M(a) ∧ M(b)"
by (simp add: Pair_def, blast intro: transM)

lemma (in M_trivial) pair_in_MI [intro!]:
"M(a) ∧ M(b) ⟹ M(⟨a,b⟩)"

lemma (in M_trivial) pair_in_M_iff [simp]:
"M(⟨a,b⟩) ⟷ M(a) ∧ M(b)"
by blast

lemma (in M_trans) pair_components_in_M:
"⟦⟨x,y⟩ ∈ A; M(A)⟧ ⟹ M(x) ∧ M(y)"
by (blast dest: transM)

lemma (in M_trivial) cartprod_abs [simp]:
"⟦M(A); M(B); M(z)⟧ ⟹ cartprod(M,A,B,z) ⟷ z = A*B"
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_trans) union_abs [simp]:
"⟦M(a); M(b); M(z)⟧ ⟹ union(M,a,b,z) ⟷ z = a ∪ b"
unfolding union_def
by (blast intro: transM )

lemma (in M_trans) inter_abs [simp]:
"⟦M(a); M(b); M(z)⟧ ⟹ inter(M,a,b,z) ⟷ z = a ∩ b"
unfolding inter_def
by (blast intro: transM)

lemma (in M_trans) setdiff_abs [simp]:
"⟦M(a); M(b); M(z)⟧ ⟹ setdiff(M,a,b,z) ⟷ z = a-b"
unfolding setdiff_def
by (blast intro: transM)

lemma (in M_trans) Union_abs [simp]:
"⟦M(A); M(z)⟧ ⟹ big_union(M,A,z) ⟷ z = ⋃(A)"
unfolding big_union_def
by (blast  dest: transM)

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

lemma (in M_trivial) Un_closed [intro,simp]:
"⟦M(A); M(B)⟧ ⟹ M(A ∪ 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)"

lemma (in M_trans) succ_in_MD [dest!]:
"M(succ(a)) ⟹ M(a)"
unfolding succ_def
by (blast intro: transM)

lemma (in M_trivial) succ_in_MI [intro!]:
"M(a) ⟹ M(succ(a))"
unfolding succ_def
by (blast intro: transM)

lemma (in M_trivial) succ_in_M_iff [simp]:
"M(succ(a)) ⟷ M(a)"
by blast

subsubsection‹Absoluteness for Separation and Replacement›

lemma (in M_trans) 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))"

lemma (in M_trans) Collect_abs [simp]:
"⟦M(A); M(z)⟧ ⟹ is_Collect(M,A,P,z) ⟷ z = Collect(A,P)"
unfolding is_Collect_def
by (blast dest: transM)

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, λx y. P(x,y), z) ⟷
is_Replace(M, A', λx y. P'(x,y), z')"

lemma (in M_trans) 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))"
unfolding Replace_iff univalent_def
by (blast dest: transM)

(*The last premise expresses that P takes M to M*)
lemma (in M_trans) 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 (drule_tac x=A in rspec, safe)
apply (subgoal_tac "Replace(A,P) = Y")
apply simp
apply (rule equality_iffI)
apply (blast dest: transM)
done

lemma (in M_trans) 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 (rule iffI)
apply (rule equality_iffI)
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_trans) RepFun_closed:
"⟦strong_replacement(M, λx y. y = f(x)); M(A); ∀x∈A. M(f(x))⟧
⟹ M(RepFun(A,f))"
done

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

text‹Better than ‹RepFun_closed› when having the formula \<^term>‹x∈A›
makes relativization easier.›
lemma (in M_trans) RepFun_closed2:
"⟦strong_replacement(M, λx y. x∈A ∧ y = f(x)); M(A); ∀x∈A. M(f(x))⟧
⟹ M(RepFun(A, λx. f(x)))"
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 ‹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 (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 (rule iffI)
prefer 2 apply (simp add: lam_def)
apply (rule equality_iffI)
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, λx y. is_b(x,y), z) ⟷
is_lambda(M, A', λx y. is_b'(x,y), z')"

lemma (in M_trans) image_abs [simp]:
"⟦M(r); M(A); M(z)⟧ ⟹ image(M,r,A,z) ⟷ z = rA"
apply (rule iffI)
apply (blast intro!: equalityI dest: transM, blast)
done

subsubsection‹Relativization of Powerset›

text‹What about ‹Pow_abs›?  Powerset is NOT absolute!
This result is one direction of absoluteness.›

lemma (in M_trans) powerset_Pow:
"powerset(M, x, Pow(x))"

text‹But we can't prove that the powerset in ‹M› includes the
real powerset.›

lemma (in M_trans) powerset_imp_subset_Pow:
"⟦powerset(M,x,y); M(y)⟧ ⟹ y ⊆ Pow(x)"
apply (blast dest: transM)
done

lemma (in M_trans) powerset_abs:
assumes
"M(y)"
shows
"powerset(M,x,y) ⟷ y = {a∈Pow(x) . M(a)}"
proof (intro iffI equalityI)
(* First show the converse implication by double inclusion *)
assume "powerset(M,x,y)"
with ‹M(y)›
show "y ⊆ {a∈Pow(x) . M(a)}"
using powerset_imp_subset_Pow transM by blast
from ‹powerset(M,x,y)›
show "{a∈Pow(x) . M(a)} ⊆ y"
using transM unfolding powerset_def by auto
next (* we show the direct implication *)
assume
"y = {a ∈ Pow(x) . M(a)}"
then
show "powerset(M, x, y)"
unfolding powerset_def subset_def using transM by blast
qed

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_trans) 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)
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 (elim disjE exE)
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')"

subsection‹Absoluteness for Ordinals›
text‹These results constitute Theorem IV 5.1 of Kunen (page 126).›

lemma (in M_trans) lt_closed:
"⟦j<i; M(i)⟧ ⟹ M(j)"
by (blast dest: ltD intro: transM)

lemma (in M_trans) transitive_set_abs [simp]:
"M(a) ⟹ transitive_set(M,a) ⟷ Transset(a)"

lemma (in M_trans) ordinal_abs [simp]:
"M(a) ⟹ ordinal(M,a) ⟷ Ord(a)"

lemma (in M_trivial) limit_ordinal_abs [simp]:
"M(a) ⟹ limit_ordinal(M,a) ⟷ Limit(a)"
unfolding Limit_def limit_ordinal_def
done

lemma (in M_trivial) successor_ordinal_abs [simp]:
"M(a) ⟹ successor_ordinal(M,a) ⟷ Ord(a) ∧ (∃b[M]. a = succ(b))"
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 (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 (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 (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"

lemma (in M_trivial) number2_abs [simp]:
"M(a) ⟹ number2(M,a) ⟷ a = succ(1)"

lemma (in M_trivial) number3_abs [simp]:
"M(a) ⟹ number3(M,a) ⟷ a = succ(succ(1))"

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 ‹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))"
and power_ax:         "power_ax(M)"

lemma (in M_trivial) 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 (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 ∪ 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 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 ∪ B" and P="λx. rex(M,Q(x))" for Q in rspec)
apply (blast, clarify)
apply (drule_tac x=z and P="λx. rex(M,Q(x))" for Q 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 ∪ B" and P="λx. rex(M,Q(x))" for Q in rspec)
apply (blast, clarify)
apply (drule_tac x=z and P="λx. rex(M,Q(x))" for Q 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)"

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)"

lemma (in M_trivial) Inl_abs [simp]:
"M(Z) ⟹ is_Inl(M,a,Z) ⟷ (Z = Inl(a))"

lemma (in M_trivial) Inr_in_M_iff [iff]:
"M(Inr(a)) ⟷ M(a)"

lemma (in M_trivial) Inr_abs [simp]:
"M(Z) ⟹ is_Inr(M,a,Z) ⟷ (Z = Inr(a))"

subsubsection ‹converse of a relation›

lemma (in M_basic) M_converse_iff:
"M(r) ⟹
converse(r) =
{z ∈ ⋃(⋃(r)) * ⋃(⋃(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 (blast dest: transM)
done

lemma (in M_basic) converse_closed [intro,simp]:
"M(r) ⟹ M(converse(r))"
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 (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(rA)"
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 (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)"

subsubsection‹Domain, range and field›

lemma (in M_trans) domain_abs [simp]:
"⟦M(r); M(z)⟧ ⟹ is_domain(M,r,z) ⟷ z = domain(r)"
apply (blast intro!: equalityI dest: transM)
done

lemma (in M_basic) domain_closed [intro,simp]:
"M(r) ⟹ M(domain(r))"
done

lemma (in M_trans) range_abs [simp]:
"⟦M(r); M(z)⟧ ⟹ is_range(M,r,z) ⟷ z = range(r)"
apply (blast intro!: equalityI dest: transM)
done

lemma (in M_basic) range_closed [intro,simp]:
"M(r) ⟹ M(range(r))"
done

lemma (in M_basic) field_abs [simp]:
"⟦M(r); M(z)⟧ ⟹ is_field(M,r,z) ⟷ z = field(r)"

lemma (in M_basic) field_closed [intro,simp]:
"M(r) ⟹ M(field(r))"

subsubsection‹Relations, functions and application›

lemma (in M_trans) relation_abs [simp]:
"M(r) ⟹ is_relation(M,r) ⟷ relation(r)"
apply (blast dest!: bspec dest: pair_components_in_M)+
done

lemma (in M_trivial) 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(fa)"

lemma (in M_basic) apply_abs [simp]:
"⟦M(f); M(x); M(y)⟧ ⟹ fun_apply(M,f,x,y) ⟷ fx = y"
apply (simp add: fun_apply_def apply_def, blast)
done

lemma (in M_trivial) 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 (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)"

lemma (in M_basic) bijection_abs [simp]:
"⟦M(A); M(B); M(f)⟧ ⟹ bijection(M,A,B,f) ⟷ f ∈ bij(A,B)"

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 (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 (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 (blast dest: transM)
txt‹Opposite implication›
apply (rule M_equalityI)
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 (unfold function_def, blast)
done

lemma (in M_trans) 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_trans) 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 (insert restrict_separation [of A], simp)
done

lemma (in M_trans) Inter_abs [simp]:
"⟦M(A); M(z)⟧ ⟹ big_inter(M,A,z) ⟷ z = ⋂(A)"
apply (blast intro!: equalityI dest: transM)
done

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

lemma (in M_basic) Int_closed [intro,simp]:
"⟦M(A); M(B)⟧ ⟹ M(A ∩ 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)

lemma (in M_basic) separation_conj:
"⟦separation(M,P); separation(M,Q)⟧ ⟹ separation(M, λz. P(z) ∧ Q(z))"
by (simp del: separation_closed

(*???equalities*)
lemma Collect_Un_Collect_eq:
"Collect(A,P) ∪ Collect(A,Q) = Collect(A, λx. P(x) ∨ Q(x))"
by blast

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

lemma (in M_trans) Collect_rall_eq:
"M(Y) ⟹ Collect(A, λx. ∀y[M]. y∈Y ⟶ P(x,y)) =
(if Y=0 then A else (⋂y ∈ Y. {x ∈ A. P(x,y)}))"
by (simp,blast dest: transM)

lemma (in M_basic) separation_disj:
"⟦separation(M,P); separation(M,Q)⟧ ⟹ separation(M, λz. P(z) ∨ Q(z))"
by (simp del: separation_closed

lemma (in M_basic) separation_neg:
"separation(M,P) ⟹ separation(M, λz. ¬P(z))"
by (simp del: separation_closed

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
apply (blast intro!:  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_trivial) is_funspace_abs [simp]:
"⟦M(A); M(B); M(F); M(A->B)⟧ ⟹ is_funspace(M,A,B,F) ⟷ F = A->B"
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 =
⋃{z. p ∈ (n->B)*B, ∃f[M]. ∃b[M]. p = ⟨f,b⟩ ∧ z = {cons(⟨n,b⟩, f)}}"
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
‹rtrancl_alt› in in ‹WF_absolute.thy›.›
lemma (in M_basic) finite_funspace_closed [intro,simp]:
"⟦n∈nat; M(B)⟧ ⟹ M(n->B)"
apply (induct_tac n, simp)
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)"

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))"

lemma (in M_trivial) and_closed [intro,simp]:
"⟦M(p); M(q)⟧ ⟹ M(p and q)"

lemma (in M_trivial) or_closed [intro,simp]:
"⟦M(p); M(q)⟧ ⟹ M(p or q)"

lemma (in M_trivial) not_closed [intro,simp]:
"M(p) ⟹ M(not(p))"

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)"

lemma (in M_trivial) Nil_abs [simp]: "M(Z) ⟹ is_Nil(M,Z) ⟷ (Z = Nil)"

lemma (in M_trivial) Cons_in_M_iff [iff]: "M(Cons(a,l)) ⟷ M(a) ∧ M(l)"

lemma (in M_trivial) Cons_abs [simp]:
"⟦M(a); M(l); M(Z)⟧ ⟹ is_Cons(M,a,l,Z) ⟷ (Z = Cons(a,l))"

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
"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)"

lemma [iff]: "quasilist(Cons(x,l))"

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"

lemma list_case'_Cons [simp]: "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 [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)")
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 (elim disjE exE)
done

subsubsection‹The Modified Operators \<^term>‹hd'› and \<^term>‹tl'››

lemma (in M_trivial) is_hd_Nil: "is_hd(M,[],Z) ⟷ empty(M,Z)"

lemma (in M_trivial) is_hd_Cons:
"⟦M(a); M(l)⟧ ⟹ is_hd(M,Cons(a,l),Z) ⟷ Z = a"

lemma (in M_trivial) hd_abs [simp]:
"⟦M(x); M(y)⟧ ⟹ is_hd(M,x,y) ⟷ y = hd'(x)"
apply (intro impI conjI)
prefer 2 apply (force simp add: is_hd_def)
"⟦M(x); M(y)⟧ ⟹ is_tl(M,x,y) ⟷ y = tl'(x)