--- a/src/ZF/Constructible/Relative.thy Tue Sep 10 16:47:17 2002 +0200
+++ b/src/ZF/Constructible/Relative.thy Tue Sep 10 16:51:31 2002 +0200
@@ -465,7 +465,7 @@
text{*The class M is assumed to be transitive and to satisfy some
relativized ZF axioms*}
-locale M_triv_axioms =
+locale M_trivial =
fixes M
assumes transM: "[| y\<in>x; M(x) |] ==> M(y)"
and nonempty [simp]: "M(0)"
@@ -475,73 +475,73 @@
and replacement: "replacement(M,P)"
and M_nat [iff]: "M(nat)" (*i.e. the axiom of infinity*)
-lemma (in M_triv_axioms) rall_abs [simp]:
+lemma (in M_trivial) rall_abs [simp]:
"M(A) ==> (\<forall>x[M]. x\<in>A --> P(x)) <-> (\<forall>x\<in>A. P(x))"
by (blast intro: transM)
-lemma (in M_triv_axioms) rex_abs [simp]:
+lemma (in M_trivial) rex_abs [simp]:
"M(A) ==> (\<exists>x[M]. x\<in>A & P(x)) <-> (\<exists>x\<in>A. P(x))"
by (blast intro: transM)
-lemma (in M_triv_axioms) ball_iff_equiv:
+lemma (in M_trivial) ball_iff_equiv:
"M(A) ==> (\<forall>x[M]. (x\<in>A <-> P(x))) <->
(\<forall>x\<in>A. P(x)) & (\<forall>x. P(x) --> M(x) --> x\<in>A)"
by (blast intro: transM)
text{*Simplifies proofs of equalities when there's an iff-equality
available for rewriting, universally quantified over M. *}
-lemma (in M_triv_axioms) M_equalityI:
+lemma (in M_trivial) M_equalityI:
"[| !!x. M(x) ==> x\<in>A <-> x\<in>B; M(A); M(B) |] ==> A=B"
by (blast intro!: equalityI dest: transM)
subsubsection{*Trivial Absoluteness Proofs: Empty Set, Pairs, etc.*}
-lemma (in M_triv_axioms) empty_abs [simp]:
+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_triv_axioms) subset_abs [simp]:
+lemma (in M_trivial) subset_abs [simp]:
"M(A) ==> subset(M,A,B) <-> A \<subseteq> B"
apply (simp add: subset_def)
apply (blast intro: transM)
done
-lemma (in M_triv_axioms) upair_abs [simp]:
+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_triv_axioms) upair_in_M_iff [iff]:
+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_triv_axioms) singleton_in_M_iff [iff]:
+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_triv_axioms) pair_abs [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_triv_axioms) pair_in_M_iff [iff]:
+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_triv_axioms) pair_components_in_M:
+lemma (in M_trivial) pair_components_in_M:
"[| <x,y> \<in> A; M(A) |] ==> M(x) & M(y)"
apply (simp add: Pair_def)
apply (blast dest: transM)
done
-lemma (in M_triv_axioms) cartprod_abs [simp]:
+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)
@@ -551,51 +551,51 @@
subsubsection{*Absoluteness for Unions and Intersections*}
-lemma (in M_triv_axioms) union_abs [simp]:
+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_triv_axioms) inter_abs [simp]:
+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_triv_axioms) setdiff_abs [simp]:
+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_triv_axioms) Union_abs [simp]:
+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_triv_axioms) Union_closed [intro,simp]:
+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_triv_axioms) Un_closed [intro,simp]:
+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_triv_axioms) cons_closed [intro,simp]:
+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_triv_axioms) cons_abs [simp]:
+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_triv_axioms) successor_abs [simp]:
+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_triv_axioms) succ_in_M_iff [iff]:
+lemma (in M_trivial) succ_in_M_iff [iff]:
"M(succ(a)) <-> M(a)"
apply (simp add: succ_def)
apply (blast intro: transM)
@@ -603,7 +603,7 @@
subsubsection{*Absoluteness for Separation and Replacement*}
-lemma (in M_triv_axioms) separation_closed [intro,simp]:
+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)
@@ -615,14 +615,14 @@
"separation(M,P) <-> (\<forall>z[M]. \<exists>y[M]. is_Collect(M,z,P,y))"
by (simp add: separation_def is_Collect_def)
-lemma (in M_triv_axioms) Collect_abs [simp]:
+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_triv_axioms) strong_replacementI [rule_format]:
+lemma (in M_trivial) strong_replacementI [rule_format]:
"[| \<forall>A[M]. separation(M, %u. \<exists>x[M]. x\<in>A & P(x,u)) |]
==> strong_replacement(M,P)"
apply (simp add: strong_replacement_def, clarify)
@@ -645,7 +645,7 @@
is_Replace(M, A', %x y. P'(x,y), z')"
by (simp add: is_Replace_def)
-lemma (in M_triv_axioms) univalent_Replace_iff:
+lemma (in M_trivial) univalent_Replace_iff:
"[| M(A); univalent(M,A,P);
!!x y. [| x\<in>A; P(x,y) |] ==> M(y) |]
==> u \<in> Replace(A,P) <-> (\<exists>x. x\<in>A & P(x,u))"
@@ -654,7 +654,7 @@
done
(*The last premise expresses that P takes M to M*)
-lemma (in M_triv_axioms) strong_replacement_closed [intro,simp]:
+lemma (in M_trivial) strong_replacement_closed [intro,simp]:
"[| strong_replacement(M,P); M(A); univalent(M,A,P);
!!x y. [| x\<in>A; P(x,y) |] ==> M(y) |] ==> M(Replace(A,P))"
apply (simp add: strong_replacement_def)
@@ -666,7 +666,7 @@
apply (blast dest: transM)
done
-lemma (in M_triv_axioms) Replace_abs:
+lemma (in M_trivial) Replace_abs:
"[| M(A); M(z); univalent(M,A,P); strong_replacement(M, P);
!!x y. [| x\<in>A; P(x,y) |] ==> M(y) |]
==> is_Replace(M,A,P,z) <-> z = Replace(A,P)"
@@ -683,7 +683,7 @@
nonconstructible set. So we cannot assume that M(X) implies M(RepFun(X,f))
even for f : M -> M.
*)
-lemma (in M_triv_axioms) RepFun_closed:
+lemma (in M_trivial) RepFun_closed:
"[| strong_replacement(M, \<lambda>x y. y = f(x)); M(A); \<forall>x\<in>A. M(f(x)) |]
==> M(RepFun(A,f))"
apply (simp add: RepFun_def)
@@ -696,7 +696,7 @@
text{*Better than @{text RepFun_closed} when having the formula @{term "x\<in>A"}
makes relativization easier.*}
-lemma (in M_triv_axioms) RepFun_closed2:
+lemma (in M_trivial) RepFun_closed2:
"[| strong_replacement(M, \<lambda>x y. x\<in>A & y = f(x)); M(A); \<forall>x\<in>A. M(f(x)) |]
==> M(RepFun(A, %x. f(x)))"
apply (simp add: RepFun_def)
@@ -712,20 +712,20 @@
\<forall>p[M]. p \<in> z <->
(\<exists>u[M]. \<exists>v[M]. u\<in>A & pair(M,u,v,p) & is_b(u,v))"
-lemma (in M_triv_axioms) lam_closed:
+lemma (in M_trivial) lam_closed:
"[| strong_replacement(M, \<lambda>x y. y = <x,b(x)>); M(A); \<forall>x\<in>A. M(b(x)) |]
==> M(\<lambda>x\<in>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\<in>A"}*}
-lemma (in M_triv_axioms) lam_closed2:
+lemma (in M_trivial) lam_closed2:
"[|strong_replacement(M, \<lambda>x y. x\<in>A & y = \<langle>x, b(x)\<rangle>);
M(A); \<forall>m[M]. m\<in>A --> M(b(m))|] ==> M(Lambda(A,b))"
apply (simp add: lam_def)
apply (blast intro: RepFun_closed2 dest: transM)
done
-lemma (in M_triv_axioms) lambda_abs2 [simp]:
+lemma (in M_trivial) lambda_abs2 [simp]:
"[| strong_replacement(M, \<lambda>x y. x\<in>A & y = \<langle>x, b(x)\<rangle>);
Relativize1(M,A,is_b,b); M(A); \<forall>m[M]. m\<in>A --> M(b(m)); M(z) |]
==> is_lambda(M,A,is_b,z) <-> z = Lambda(A,b)"
@@ -744,7 +744,7 @@
is_lambda(M, A', %x y. is_b'(x,y), z')"
by (simp add: is_lambda_def)
-lemma (in M_triv_axioms) image_abs [simp]:
+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)
@@ -754,13 +754,13 @@
text{*What about @{text Pow_abs}? Powerset is NOT absolute!
This result is one direction of absoluteness.*}
-lemma (in M_triv_axioms) powerset_Pow:
+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_triv_axioms) powerset_imp_subset_Pow:
+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)
@@ -768,22 +768,22 @@
subsubsection{*Absoluteness for the Natural Numbers*}
-lemma (in M_triv_axioms) nat_into_M [intro]:
+lemma (in M_trivial) nat_into_M [intro]:
"n \<in> nat ==> M(n)"
by (induct n rule: nat_induct, simp_all)
-lemma (in M_triv_axioms) nat_case_closed [intro,simp]:
+lemma (in M_trivial) nat_case_closed [intro,simp]:
"[|M(k); M(a); \<forall>m[M]. M(b(m))|] ==> M(nat_case(a,b,k))"
apply (case_tac "k=0", simp)
apply (case_tac "\<exists>m. k = succ(m)", force)
apply (simp add: nat_case_def)
done
-lemma (in M_triv_axioms) quasinat_abs [simp]:
+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_triv_axioms) nat_case_abs [simp]:
+lemma (in M_trivial) nat_case_abs [simp]:
"[| relativize1(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)")
@@ -808,26 +808,26 @@
subsection{*Absoluteness for Ordinals*}
text{*These results constitute Theorem IV 5.1 of Kunen (page 126).*}
-lemma (in M_triv_axioms) lt_closed:
+lemma (in M_trivial) lt_closed:
"[| j<i; M(i) |] ==> M(j)"
by (blast dest: ltD intro: transM)
-lemma (in M_triv_axioms) transitive_set_abs [simp]:
+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_triv_axioms) ordinal_abs [simp]:
+lemma (in M_trivial) ordinal_abs [simp]:
"M(a) ==> ordinal(M,a) <-> Ord(a)"
by (simp add: ordinal_def Ord_def)
-lemma (in M_triv_axioms) limit_ordinal_abs [simp]:
+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_triv_axioms) successor_ordinal_abs [simp]:
+lemma (in M_trivial) successor_ordinal_abs [simp]:
"M(a) ==> successor_ordinal(M,a) <-> Ord(a) & (\<exists>b[M]. a = succ(b))"
apply (simp add: successor_ordinal_def, safe)
apply (drule Ord_cases_disj, auto)
@@ -840,7 +840,7 @@
lemma naturals_not_limit: "a \<in> nat ==> ~ Limit(a)"
by (induct a rule: nat_induct, auto)
-lemma (in M_triv_axioms) finite_ordinal_abs [simp]:
+lemma (in M_trivial) finite_ordinal_abs [simp]:
"M(a) ==> finite_ordinal(M,a) <-> a \<in> nat"
apply (simp add: finite_ordinal_def)
apply (blast intro: finite_Ord_is_nat intro: nat_into_Ord
@@ -856,21 +856,21 @@
apply (erule nat_le_Limit)
done
-lemma (in M_triv_axioms) omega_abs [simp]:
+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_triv_axioms) number1_abs [simp]:
+lemma (in M_trivial) number1_abs [simp]:
"M(a) ==> number1(M,a) <-> a = 1"
by (simp add: number1_def)
-lemma (in M_triv_axioms) number2_abs [simp]:
+lemma (in M_trivial) number2_abs [simp]:
"M(a) ==> number2(M,a) <-> a = succ(1)"
by (simp add: number2_def)
-lemma (in M_triv_axioms) number3_abs [simp]:
+lemma (in M_trivial) number3_abs [simp]:
"M(a) ==> number3(M,a) <-> a = succ(succ(1))"
by (simp add: number3_def)
@@ -893,13 +893,13 @@
natnumber :: "[i=>o,i,i] => o"
"natnumber(M,n,x) == natnumber_aux(M,n)`x = 1"
- lemma (in M_triv_axioms) [simp]:
+ lemma (in M_trivial) [simp]:
"natnumber(M,0,x) == x=0"
*)
subsection{*Some instances of separation and strong replacement*}
-locale M_axioms = M_triv_axioms +
+locale M_basic = M_trivial +
assumes Inter_separation:
"M(A) ==> separation(M, \<lambda>x. \<forall>y[M]. y\<in>A --> x\<in>y)"
and Diff_separation:
@@ -960,7 +960,7 @@
(\<exists>fx[M]. \<exists>gx[M]. fun_apply(M,f,x,fx) & fun_apply(M,g,x,gx) &
fx \<noteq> gx))"
-lemma (in M_axioms) cartprod_iff_lemma:
+lemma (in M_basic) cartprod_iff_lemma:
"[| M(C); \<forall>u[M]. u \<in> C <-> (\<exists>x\<in>A. \<exists>y\<in>B. u = {{x}, {x,y}});
powerset(M, A \<union> B, p1); powerset(M, p1, p2); M(p2) |]
==> C = {u \<in> p2 . \<exists>x\<in>A. \<exists>y\<in>B. u = {{x}, {x,y}}}"
@@ -973,7 +973,7 @@
apply (frule transM, assumption, force)
done
-lemma (in M_axioms) cartprod_iff:
+lemma (in M_basic) cartprod_iff:
"[| M(A); M(B); M(C) |]
==> cartprod(M,A,B,C) <->
(\<exists>p1 p2. M(p1) & M(p2) & powerset(M,A Un B,p1) & powerset(M,p1,p2) &
@@ -991,7 +991,7 @@
apply (blast intro: cartprod_iff_lemma)
done
-lemma (in M_axioms) cartprod_closed_lemma:
+lemma (in M_basic) cartprod_closed_lemma:
"[| M(A); M(B) |] ==> \<exists>C[M]. cartprod(M,A,B,C)"
apply (simp del: cartprod_abs add: cartprod_iff)
apply (insert power_ax, simp add: power_ax_def)
@@ -1008,38 +1008,38 @@
text{*All the lemmas above are necessary because Powerset is not absolute.
I should have used Replacement instead!*}
-lemma (in M_axioms) cartprod_closed [intro,simp]:
+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_axioms) sum_closed [intro,simp]:
+lemma (in M_basic) sum_closed [intro,simp]:
"[| M(A); M(B) |] ==> M(A+B)"
by (simp add: sum_def)
-lemma (in M_axioms) sum_abs [simp]:
+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_triv_axioms) Inl_in_M_iff [iff]:
+lemma (in M_trivial) Inl_in_M_iff [iff]:
"M(Inl(a)) <-> M(a)"
by (simp add: Inl_def)
-lemma (in M_triv_axioms) Inl_abs [simp]:
+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_triv_axioms) Inr_in_M_iff [iff]:
+lemma (in M_trivial) Inr_in_M_iff [iff]:
"M(Inr(a)) <-> M(a)"
by (simp add: Inr_def)
-lemma (in M_triv_axioms) Inr_abs [simp]:
+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_axioms) M_converse_iff:
+lemma (in M_basic) M_converse_iff:
"M(r) ==>
converse(r) =
{z \<in> Union(Union(r)) * Union(Union(r)).
@@ -1050,13 +1050,13 @@
apply (blast dest: transM)
done
-lemma (in M_axioms) converse_closed [intro,simp]:
+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_axioms) converse_abs [simp]:
+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)
@@ -1069,105 +1069,105 @@
subsubsection {*image, preimage, domain, range*}
-lemma (in M_axioms) image_closed [intro,simp]:
+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_axioms) vimage_abs [simp]:
+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_axioms) vimage_closed [intro,simp]:
+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_axioms) domain_abs [simp]:
+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_axioms) domain_closed [intro,simp]:
+lemma (in M_basic) domain_closed [intro,simp]:
"M(r) ==> M(domain(r))"
apply (simp add: domain_eq_vimage)
done
-lemma (in M_axioms) range_abs [simp]:
+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_axioms) range_closed [intro,simp]:
+lemma (in M_basic) range_closed [intro,simp]:
"M(r) ==> M(range(r))"
apply (simp add: range_eq_image)
done
-lemma (in M_axioms) field_abs [simp]:
+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_axioms) field_closed [intro,simp]:
+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_axioms) relation_abs [simp]:
+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_axioms) function_abs [simp]:
+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_axioms) apply_closed [intro,simp]:
+lemma (in M_basic) apply_closed [intro,simp]:
"[|M(f); M(a)|] ==> M(f`a)"
by (simp add: apply_def)
-lemma (in M_axioms) apply_abs [simp]:
+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_axioms) typed_function_abs [simp]:
+lemma (in M_basic) typed_function_abs [simp]:
"[| M(A); M(f) |] ==> typed_function(M,A,B,f) <-> f \<in> A -> B"
apply (auto simp add: typed_function_def relation_def Pi_iff)
apply (blast dest: pair_components_in_M)+
done
-lemma (in M_axioms) injection_abs [simp]:
+lemma (in M_basic) injection_abs [simp]:
"[| M(A); M(f) |] ==> injection(M,A,B,f) <-> f \<in> inj(A,B)"
apply (simp add: injection_def apply_iff inj_def apply_closed)
apply (blast dest: transM [of _ A])
done
-lemma (in M_axioms) surjection_abs [simp]:
+lemma (in M_basic) surjection_abs [simp]:
"[| M(A); M(B); M(f) |] ==> surjection(M,A,B,f) <-> f \<in> surj(A,B)"
by (simp add: surjection_def surj_def)
-lemma (in M_axioms) bijection_abs [simp]:
+lemma (in M_basic) bijection_abs [simp]:
"[| M(A); M(B); M(f) |] ==> bijection(M,A,B,f) <-> f \<in> bij(A,B)"
by (simp add: bijection_def bij_def)
subsubsection{*Composition of relations*}
-lemma (in M_axioms) M_comp_iff:
+lemma (in M_basic) M_comp_iff:
"[| M(r); M(s) |]
==> r O s =
{xz \<in> domain(s) * range(r).
@@ -1179,13 +1179,13 @@
apply (blast dest: transM)+
done
-lemma (in M_axioms) comp_closed [intro,simp]:
+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_axioms) composition_abs [simp]:
+lemma (in M_basic) composition_abs [simp]:
"[| M(r); M(s); M(t) |]
==> composition(M,r,s,t) <-> t = r O s"
apply safe
@@ -1200,7 +1200,7 @@
done
text{*no longer needed*}
-lemma (in M_axioms) restriction_is_function:
+lemma (in M_basic) restriction_is_function:
"[| restriction(M,f,A,z); function(f); M(f); M(A); M(z) |]
==> function(z)"
apply (rotate_tac 1)
@@ -1208,7 +1208,7 @@
apply (unfold function_def, blast)
done
-lemma (in M_axioms) restriction_abs [simp]:
+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)
@@ -1216,39 +1216,39 @@
done
-lemma (in M_axioms) M_restrict_iff:
+lemma (in M_basic) M_restrict_iff:
"M(r) ==> restrict(r,A) = {z \<in> r . \<exists>x\<in>A. \<exists>y[M]. z = \<langle>x, y\<rangle>}"
by (simp add: restrict_def, blast dest: transM)
-lemma (in M_axioms) restrict_closed [intro,simp]:
+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_axioms) Inter_abs [simp]:
+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_axioms) Inter_closed [intro,simp]:
+lemma (in M_basic) Inter_closed [intro,simp]:
"M(A) ==> M(Inter(A))"
by (insert Inter_separation, simp add: Inter_def)
-lemma (in M_axioms) Int_closed [intro,simp]:
+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_axioms) Diff_closed [intro,simp]:
+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_axioms) separation_conj:
+lemma (in M_basic) separation_conj:
"[|separation(M,P); separation(M,Q)|] ==> separation(M, \<lambda>z. P(z) & Q(z))"
by (simp del: separation_closed
add: separation_iff Collect_Int_Collect_eq [symmetric])
@@ -1262,24 +1262,24 @@
"A - Collect(A,P) = Collect(A, %x. ~ P(x))"
by blast
-lemma (in M_triv_axioms) Collect_rall_eq:
+lemma (in M_trivial) Collect_rall_eq:
"M(Y) ==> Collect(A, %x. \<forall>y[M]. y\<in>Y --> P(x,y)) =
(if Y=0 then A else (\<Inter>y \<in> Y. {x \<in> A. P(x,y)}))"
apply simp
apply (blast intro!: equalityI dest: transM)
done
-lemma (in M_axioms) separation_disj:
+lemma (in M_basic) separation_disj:
"[|separation(M,P); separation(M,Q)|] ==> separation(M, \<lambda>z. P(z) | Q(z))"
by (simp del: separation_closed
add: separation_iff Collect_Un_Collect_eq [symmetric])
-lemma (in M_axioms) separation_neg:
+lemma (in M_basic) separation_neg:
"separation(M,P) ==> separation(M, \<lambda>z. ~P(z))"
by (simp del: separation_closed
add: separation_iff Diff_Collect_eq [symmetric])
-lemma (in M_axioms) separation_imp:
+lemma (in M_basic) separation_imp:
"[|separation(M,P); separation(M,Q)|]
==> separation(M, \<lambda>z. P(z) --> Q(z))"
by (simp add: separation_neg separation_disj not_disj_iff_imp [symmetric])
@@ -1287,7 +1287,7 @@
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_axioms) separation_rall:
+lemma (in M_basic) separation_rall:
"[|M(Y); \<forall>y[M]. separation(M, \<lambda>x. P(x,y));
\<forall>z[M]. strong_replacement(M, \<lambda>x y. y = {u \<in> z . P(u,x)})|]
==> separation(M, \<lambda>x. \<forall>y[M]. y\<in>Y --> P(x,y))"
@@ -1300,7 +1300,7 @@
subsubsection{*Functions and function space*}
text{*M contains all finite functions*}
-lemma (in M_axioms) finite_fun_closed_lemma [rule_format]:
+lemma (in M_basic) finite_fun_closed_lemma [rule_format]:
"[| n \<in> nat; M(A) |] ==> \<forall>f \<in> n -> A. M(f)"
apply (induct_tac n, simp)
apply (rule ballI)
@@ -1312,13 +1312,13 @@
apply (blast intro: apply_funtype transM restrict_type2)
done
-lemma (in M_axioms) finite_fun_closed [rule_format]:
+lemma (in M_basic) finite_fun_closed [rule_format]:
"[| f \<in> n -> A; n \<in> nat; M(A) |] ==> M(f)"
by (blast intro: finite_fun_closed_lemma)
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_axioms) is_funspace_abs [simp]:
+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)
@@ -1327,7 +1327,7 @@
apply simp_all
done
-lemma (in M_axioms) succ_fun_eq2:
+lemma (in M_basic) succ_fun_eq2:
"[|M(B); M(n->B)|] ==>
succ(n) -> B =
\<Union>{z. p \<in> (n->B)*B, \<exists>f[M]. \<exists>b[M]. p = <f,b> & z = {cons(<n,b>, f)}}"
@@ -1335,7 +1335,7 @@
apply (blast dest: transM)
done
-lemma (in M_axioms) funspace_succ:
+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)
@@ -1343,7 +1343,7 @@
text{*@{term M} contains all finite function spaces. Needed to prove the
absoluteness of transitive closure.*}
-lemma (in M_axioms) finite_funspace_closed [intro,simp]:
+lemma (in M_basic) finite_funspace_closed [intro,simp]:
"[|n\<in>nat; M(B)|] ==> M(n->B)"
apply (induct_tac n, simp)
apply (simp add: funspace_succ nat_into_M)
@@ -1368,37 +1368,37 @@
"is_or(M,a,b,z) == (number1(M,a) & number1(M,z)) |
(~number1(M,a) & z=b)"
-lemma (in M_triv_axioms) bool_of_o_abs [simp]:
+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_triv_axioms) not_abs [simp]:
+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_triv_axioms) and_abs [simp]:
+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_triv_axioms) or_abs [simp]:
+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_triv_axioms) bool_of_o_closed [intro,simp]:
+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_triv_axioms) and_closed [intro,simp]:
+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_triv_axioms) or_closed [intro,simp]:
+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_triv_axioms) not_closed [intro,simp]:
+lemma (in M_trivial) not_closed [intro,simp]:
"M(p) ==> M(not(p))"
by (simp add: Bool.not_def cond_def)
@@ -1416,16 +1416,16 @@
"is_Cons(M,a,l,Z) == \<exists>p[M]. pair(M,a,l,p) & is_Inr(M,p,Z)"
-lemma (in M_triv_axioms) Nil_in_M [intro,simp]: "M(Nil)"
+lemma (in M_trivial) Nil_in_M [intro,simp]: "M(Nil)"
by (simp add: Nil_def)
-lemma (in M_triv_axioms) Nil_abs [simp]: "M(Z) ==> is_Nil(M,Z) <-> (Z = Nil)"
+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_triv_axioms) Cons_in_M_iff [iff]: "M(Cons(a,l)) <-> M(a) & M(l)"
+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_triv_axioms) Cons_abs [simp]:
+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)
@@ -1499,18 +1499,18 @@
"xs \<in> list(A) ==>list_case'(a,b,xs) = list_case(a,b,xs)"
by (erule list.cases, simp_all)
-lemma (in M_axioms) list_case'_closed [intro,simp]:
+lemma (in M_basic) list_case'_closed [intro,simp]:
"[|M(k); M(a); \<forall>x[M]. \<forall>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_triv_axioms) quasilist_abs [simp]:
+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_triv_axioms) list_case_abs [simp]:
+lemma (in M_trivial) list_case_abs [simp]:
"[| relativize2(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)")
@@ -1525,14 +1525,14 @@
subsubsection{*The Modified Operators @{term hd'} and @{term tl'}*}
-lemma (in M_triv_axioms) is_hd_Nil: "is_hd(M,[],Z) <-> empty(M,Z)"
+lemma (in M_trivial) is_hd_Nil: "is_hd(M,[],Z) <-> empty(M,Z)"
by (simp add: is_hd_def)
-lemma (in M_triv_axioms) is_hd_Cons:
+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_triv_axioms) hd_abs [simp]:
+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)
@@ -1541,14 +1541,14 @@
apply (elim disjE exE, auto)
done
-lemma (in M_triv_axioms) is_tl_Nil: "is_tl(M,[],Z) <-> Z = []"
+lemma (in M_trivial) is_tl_Nil: "is_tl(M,[],Z) <-> Z = []"
by (simp add: is_tl_def)
-lemma (in M_triv_axioms) is_tl_Cons:
+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_triv_axioms) tl_abs [simp]:
+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)
@@ -1557,7 +1557,7 @@
apply (elim disjE exE, auto)
done
-lemma (in M_triv_axioms) relativize1_tl: "relativize1(M, is_tl(M), tl')"
+lemma (in M_trivial) relativize1_tl: "relativize1(M, is_tl(M), tl')"
by (simp add: relativize1_def)
lemma hd'_Nil: "hd'([]) = 0"
@@ -1577,7 +1577,7 @@
apply (simp_all add: tl'_Nil)
done
-lemma (in M_axioms) tl'_closed: "M(x) ==> M(tl'(x))"
+lemma (in M_basic) tl'_closed: "M(x) ==> M(tl'(x))"
apply (simp add: tl'_def)
apply (force simp add: quasilist_def)
done