--- a/src/ZF/Constructible/Datatype_absolute.thy Tue Sep 10 16:47:17 2002 +0200
+++ b/src/ZF/Constructible/Datatype_absolute.thy Tue Sep 10 16:51:31 2002 +0200
@@ -126,13 +126,13 @@
\<forall>n[M]. n\<in>nat -->
wfrec_replacement(M, iterates_MH(M,isF,v), Memrel(succ(n)))"
-lemma (in M_axioms) iterates_MH_abs:
+lemma (in M_basic) iterates_MH_abs:
"[| relativize1(M,isF,F); M(n); M(g); M(z) |]
==> iterates_MH(M,isF,v,n,g,z) <-> z = nat_case(v, \<lambda>m. F(g`m), n)"
by (simp add: nat_case_abs [of _ "\<lambda>m. F(g ` m)"]
relativize1_def iterates_MH_def)
-lemma (in M_axioms) iterates_imp_wfrec_replacement:
+lemma (in M_basic) iterates_imp_wfrec_replacement:
"[|relativize1(M,isF,F); n \<in> nat; iterates_replacement(M,isF,v)|]
==> wfrec_replacement(M, \<lambda>n f z. z = nat_case(v, \<lambda>m. F(f`m), n),
Memrel(succ(n)))"
@@ -210,7 +210,7 @@
\<exists>n1[M]. \<exists>AX[M].
number1(M,n1) & cartprod(M,A,X,AX) & is_sum(M,n1,AX,Z)"
-lemma (in M_axioms) list_functor_abs [simp]:
+lemma (in M_basic) list_functor_abs [simp]:
"[| M(A); M(X); M(Z) |] ==> is_list_functor(M,A,X,Z) <-> (Z = {0} + A*X)"
by (simp add: is_list_functor_def singleton_0 nat_into_M)
@@ -266,7 +266,7 @@
cartprod(M,X,X,XX) & is_sum(M,XX,X,X3) &
is_sum(M,natnatsum,X3,Z)"
-lemma (in M_axioms) formula_functor_abs [simp]:
+lemma (in M_basic) formula_functor_abs [simp]:
"[| M(X); M(Z) |]
==> is_formula_functor(M,X,Z) <->
Z = ((nat*nat) + (nat*nat)) + (X*X + X)"
@@ -323,7 +323,7 @@
text{*Express @{term list_rec} without using @{term rank} or @{term Vset},
neither of which is absolute.*}
-lemma (in M_triv_axioms) list_rec_eq:
+lemma (in M_trivial) list_rec_eq:
"l \<in> list(A) ==>
list_rec(a,g,l) =
transrec (succ(length(l)),
@@ -728,7 +728,7 @@
done
text{*Proof is trivial since @{term length} returns natural numbers.*}
-lemma (in M_triv_axioms) length_closed [intro,simp]:
+lemma (in M_trivial) length_closed [intro,simp]:
"l \<in> list(A) ==> M(length(l))"
by (simp add: nat_into_M)
@@ -744,7 +744,7 @@
apply (simp add: tl'_Cons iterates_commute)
done
-lemma (in M_axioms) iterates_tl'_closed:
+lemma (in M_basic) iterates_tl'_closed:
"[|n \<in> nat; M(x)|] ==> M(tl'^n (x))"
apply (induct_tac n, simp)
apply (simp add: tl'_Cons tl'_closed)
@@ -785,11 +785,11 @@
"is_Member(M,x,y,Z) ==
\<exists>p[M]. \<exists>u[M]. pair(M,x,y,p) & is_Inl(M,p,u) & is_Inl(M,u,Z)"
-lemma (in M_triv_axioms) Member_abs [simp]:
+lemma (in M_trivial) Member_abs [simp]:
"[|M(x); M(y); M(Z)|] ==> is_Member(M,x,y,Z) <-> (Z = Member(x,y))"
by (simp add: is_Member_def Member_def)
-lemma (in M_triv_axioms) Member_in_M_iff [iff]:
+lemma (in M_trivial) Member_in_M_iff [iff]:
"M(Member(x,y)) <-> M(x) & M(y)"
by (simp add: Member_def)
@@ -799,11 +799,11 @@
"is_Equal(M,x,y,Z) ==
\<exists>p[M]. \<exists>u[M]. pair(M,x,y,p) & is_Inr(M,p,u) & is_Inl(M,u,Z)"
-lemma (in M_triv_axioms) Equal_abs [simp]:
+lemma (in M_trivial) Equal_abs [simp]:
"[|M(x); M(y); M(Z)|] ==> is_Equal(M,x,y,Z) <-> (Z = Equal(x,y))"
by (simp add: is_Equal_def Equal_def)
-lemma (in M_triv_axioms) Equal_in_M_iff [iff]: "M(Equal(x,y)) <-> M(x) & M(y)"
+lemma (in M_trivial) Equal_in_M_iff [iff]: "M(Equal(x,y)) <-> M(x) & M(y)"
by (simp add: Equal_def)
constdefs
@@ -812,11 +812,11 @@
"is_Nand(M,x,y,Z) ==
\<exists>p[M]. \<exists>u[M]. pair(M,x,y,p) & is_Inl(M,p,u) & is_Inr(M,u,Z)"
-lemma (in M_triv_axioms) Nand_abs [simp]:
+lemma (in M_trivial) Nand_abs [simp]:
"[|M(x); M(y); M(Z)|] ==> is_Nand(M,x,y,Z) <-> (Z = Nand(x,y))"
by (simp add: is_Nand_def Nand_def)
-lemma (in M_triv_axioms) Nand_in_M_iff [iff]: "M(Nand(x,y)) <-> M(x) & M(y)"
+lemma (in M_trivial) Nand_in_M_iff [iff]: "M(Nand(x,y)) <-> M(x) & M(y)"
by (simp add: Nand_def)
constdefs
@@ -824,11 +824,11 @@
--{* because @{term "Forall(x) \<equiv> Inr(Inr(p))"}*}
"is_Forall(M,p,Z) == \<exists>u[M]. is_Inr(M,p,u) & is_Inr(M,u,Z)"
-lemma (in M_triv_axioms) Forall_abs [simp]:
+lemma (in M_trivial) Forall_abs [simp]:
"[|M(x); M(Z)|] ==> is_Forall(M,x,Z) <-> (Z = Forall(x))"
by (simp add: is_Forall_def Forall_def)
-lemma (in M_triv_axioms) Forall_in_M_iff [iff]: "M(Forall(x)) <-> M(x)"
+lemma (in M_trivial) Forall_in_M_iff [iff]: "M(Forall(x)) <-> M(x)"
by (simp add: Forall_def)
@@ -890,7 +890,7 @@
done
text{*Proof is trivial since @{term depth} returns natural numbers.*}
-lemma (in M_triv_axioms) depth_closed [intro,simp]:
+lemma (in M_trivial) depth_closed [intro,simp]:
"p \<in> formula ==> M(depth(p))"
by (simp add: nat_into_M)
@@ -916,7 +916,7 @@
text{*Unfold @{term formula_rec} to @{term formula_rec_case}.
Express @{term formula_rec} without using @{term rank} or @{term Vset},
neither of which is absolute.*}
-lemma (in M_triv_axioms) formula_rec_eq:
+lemma (in M_trivial) formula_rec_eq:
"p \<in> formula ==>
formula_rec(a,b,c,d,p) =
transrec (succ(depth(p)),
--- a/src/ZF/Constructible/L_axioms.thy Tue Sep 10 16:47:17 2002 +0200
+++ b/src/ZF/Constructible/L_axioms.thy Tue Sep 10 16:51:31 2002 +0200
@@ -8,7 +8,7 @@
theory L_axioms = Formula + Relative + Reflection + MetaExists:
-text {* The class L satisfies the premises of locale @{text M_triv_axioms} *}
+text {* The class L satisfies the premises of locale @{text M_trivial} *}
lemma transL: "[| y\<in>x; L(x) |] ==> L(y)"
apply (insert Transset_Lset)
@@ -80,7 +80,7 @@
apply (simp_all add: Replace_iff univalent_def, blast)
done
-subsection{*Instantiating the locale @{text M_triv_axioms}*}
+subsection{*Instantiating the locale @{text M_trivial}*}
text{*No instances of Separation yet.*}
lemma Lset_mono_le: "mono_le_subset(Lset)"
@@ -93,8 +93,8 @@
lemmas L_nat = Ord_in_L [OF Ord_nat]
-theorem M_triv_axioms_L: "PROP M_triv_axioms(L)"
- apply (rule M_triv_axioms.intro)
+theorem M_trivial_L: "PROP M_trivial(L)"
+ apply (rule M_trivial.intro)
apply (erule (1) transL)
apply (rule nonempty)
apply (rule upair_ax)
@@ -104,50 +104,50 @@
apply (rule L_nat)
done
-lemmas rall_abs = M_triv_axioms.rall_abs [OF M_triv_axioms_L]
- and rex_abs = M_triv_axioms.rex_abs [OF M_triv_axioms_L]
- and ball_iff_equiv = M_triv_axioms.ball_iff_equiv [OF M_triv_axioms_L]
- and M_equalityI = M_triv_axioms.M_equalityI [OF M_triv_axioms_L]
- and empty_abs = M_triv_axioms.empty_abs [OF M_triv_axioms_L]
- and subset_abs = M_triv_axioms.subset_abs [OF M_triv_axioms_L]
- and upair_abs = M_triv_axioms.upair_abs [OF M_triv_axioms_L]
- and upair_in_M_iff = M_triv_axioms.upair_in_M_iff [OF M_triv_axioms_L]
- and singleton_in_M_iff = M_triv_axioms.singleton_in_M_iff [OF M_triv_axioms_L]
- and pair_abs = M_triv_axioms.pair_abs [OF M_triv_axioms_L]
- and pair_in_M_iff = M_triv_axioms.pair_in_M_iff [OF M_triv_axioms_L]
- and pair_components_in_M = M_triv_axioms.pair_components_in_M [OF M_triv_axioms_L]
- and cartprod_abs = M_triv_axioms.cartprod_abs [OF M_triv_axioms_L]
- and union_abs = M_triv_axioms.union_abs [OF M_triv_axioms_L]
- and inter_abs = M_triv_axioms.inter_abs [OF M_triv_axioms_L]
- and setdiff_abs = M_triv_axioms.setdiff_abs [OF M_triv_axioms_L]
- and Union_abs = M_triv_axioms.Union_abs [OF M_triv_axioms_L]
- and Union_closed = M_triv_axioms.Union_closed [OF M_triv_axioms_L]
- and Un_closed = M_triv_axioms.Un_closed [OF M_triv_axioms_L]
- and cons_closed = M_triv_axioms.cons_closed [OF M_triv_axioms_L]
- and successor_abs = M_triv_axioms.successor_abs [OF M_triv_axioms_L]
- and succ_in_M_iff = M_triv_axioms.succ_in_M_iff [OF M_triv_axioms_L]
- and separation_closed = M_triv_axioms.separation_closed [OF M_triv_axioms_L]
- and strong_replacementI = M_triv_axioms.strong_replacementI [OF M_triv_axioms_L]
- and strong_replacement_closed = M_triv_axioms.strong_replacement_closed [OF M_triv_axioms_L]
- and RepFun_closed = M_triv_axioms.RepFun_closed [OF M_triv_axioms_L]
- and lam_closed = M_triv_axioms.lam_closed [OF M_triv_axioms_L]
- and image_abs = M_triv_axioms.image_abs [OF M_triv_axioms_L]
- and powerset_Pow = M_triv_axioms.powerset_Pow [OF M_triv_axioms_L]
- and powerset_imp_subset_Pow = M_triv_axioms.powerset_imp_subset_Pow [OF M_triv_axioms_L]
- and nat_into_M = M_triv_axioms.nat_into_M [OF M_triv_axioms_L]
- and nat_case_closed = M_triv_axioms.nat_case_closed [OF M_triv_axioms_L]
- and Inl_in_M_iff = M_triv_axioms.Inl_in_M_iff [OF M_triv_axioms_L]
- and Inr_in_M_iff = M_triv_axioms.Inr_in_M_iff [OF M_triv_axioms_L]
- and lt_closed = M_triv_axioms.lt_closed [OF M_triv_axioms_L]
- and transitive_set_abs = M_triv_axioms.transitive_set_abs [OF M_triv_axioms_L]
- and ordinal_abs = M_triv_axioms.ordinal_abs [OF M_triv_axioms_L]
- and limit_ordinal_abs = M_triv_axioms.limit_ordinal_abs [OF M_triv_axioms_L]
- and successor_ordinal_abs = M_triv_axioms.successor_ordinal_abs [OF M_triv_axioms_L]
- and finite_ordinal_abs = M_triv_axioms.finite_ordinal_abs [OF M_triv_axioms_L]
- and omega_abs = M_triv_axioms.omega_abs [OF M_triv_axioms_L]
- and number1_abs = M_triv_axioms.number1_abs [OF M_triv_axioms_L]
- and number2_abs = M_triv_axioms.number2_abs [OF M_triv_axioms_L]
- and number3_abs = M_triv_axioms.number3_abs [OF M_triv_axioms_L]
+lemmas rall_abs = M_trivial.rall_abs [OF M_trivial_L]
+ and rex_abs = M_trivial.rex_abs [OF M_trivial_L]
+ and ball_iff_equiv = M_trivial.ball_iff_equiv [OF M_trivial_L]
+ and M_equalityI = M_trivial.M_equalityI [OF M_trivial_L]
+ and empty_abs = M_trivial.empty_abs [OF M_trivial_L]
+ and subset_abs = M_trivial.subset_abs [OF M_trivial_L]
+ and upair_abs = M_trivial.upair_abs [OF M_trivial_L]
+ and upair_in_M_iff = M_trivial.upair_in_M_iff [OF M_trivial_L]
+ and singleton_in_M_iff = M_trivial.singleton_in_M_iff [OF M_trivial_L]
+ and pair_abs = M_trivial.pair_abs [OF M_trivial_L]
+ and pair_in_M_iff = M_trivial.pair_in_M_iff [OF M_trivial_L]
+ and pair_components_in_M = M_trivial.pair_components_in_M [OF M_trivial_L]
+ and cartprod_abs = M_trivial.cartprod_abs [OF M_trivial_L]
+ and union_abs = M_trivial.union_abs [OF M_trivial_L]
+ and inter_abs = M_trivial.inter_abs [OF M_trivial_L]
+ and setdiff_abs = M_trivial.setdiff_abs [OF M_trivial_L]
+ and Union_abs = M_trivial.Union_abs [OF M_trivial_L]
+ and Union_closed = M_trivial.Union_closed [OF M_trivial_L]
+ and Un_closed = M_trivial.Un_closed [OF M_trivial_L]
+ and cons_closed = M_trivial.cons_closed [OF M_trivial_L]
+ and successor_abs = M_trivial.successor_abs [OF M_trivial_L]
+ and succ_in_M_iff = M_trivial.succ_in_M_iff [OF M_trivial_L]
+ and separation_closed = M_trivial.separation_closed [OF M_trivial_L]
+ and strong_replacementI = M_trivial.strong_replacementI [OF M_trivial_L]
+ and strong_replacement_closed = M_trivial.strong_replacement_closed [OF M_trivial_L]
+ and RepFun_closed = M_trivial.RepFun_closed [OF M_trivial_L]
+ and lam_closed = M_trivial.lam_closed [OF M_trivial_L]
+ and image_abs = M_trivial.image_abs [OF M_trivial_L]
+ and powerset_Pow = M_trivial.powerset_Pow [OF M_trivial_L]
+ and powerset_imp_subset_Pow = M_trivial.powerset_imp_subset_Pow [OF M_trivial_L]
+ and nat_into_M = M_trivial.nat_into_M [OF M_trivial_L]
+ and nat_case_closed = M_trivial.nat_case_closed [OF M_trivial_L]
+ and Inl_in_M_iff = M_trivial.Inl_in_M_iff [OF M_trivial_L]
+ and Inr_in_M_iff = M_trivial.Inr_in_M_iff [OF M_trivial_L]
+ and lt_closed = M_trivial.lt_closed [OF M_trivial_L]
+ and transitive_set_abs = M_trivial.transitive_set_abs [OF M_trivial_L]
+ and ordinal_abs = M_trivial.ordinal_abs [OF M_trivial_L]
+ and limit_ordinal_abs = M_trivial.limit_ordinal_abs [OF M_trivial_L]
+ and successor_ordinal_abs = M_trivial.successor_ordinal_abs [OF M_trivial_L]
+ and finite_ordinal_abs = M_trivial.finite_ordinal_abs [OF M_trivial_L]
+ and omega_abs = M_trivial.omega_abs [OF M_trivial_L]
+ and number1_abs = M_trivial.number1_abs [OF M_trivial_L]
+ and number2_abs = M_trivial.number2_abs [OF M_trivial_L]
+ and number3_abs = M_trivial.number3_abs [OF M_trivial_L]
declare rall_abs [simp]
declare rex_abs [simp]
--- a/src/ZF/Constructible/Rec_Separation.thy Tue Sep 10 16:47:17 2002 +0200
+++ b/src/ZF/Constructible/Rec_Separation.thy Tue Sep 10 16:51:31 2002 +0200
@@ -212,7 +212,7 @@
theorem M_trancl_L: "PROP M_trancl(L)"
by (rule M_trancl.intro
- [OF M_triv_axioms_L M_axioms_axioms_L M_trancl_axioms_L])
+ [OF M_trivial_L M_basic_axioms_L M_trancl_axioms_L])
lemmas iterates_abs = M_trancl.iterates_abs [OF M_trancl_L]
and rtran_closure_rtrancl = M_trancl.rtran_closure_rtrancl [OF M_trancl_L]
--- 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
--- a/src/ZF/Constructible/Separation.thy Tue Sep 10 16:47:17 2002 +0200
+++ b/src/ZF/Constructible/Separation.thy Tue Sep 10 16:51:31 2002 +0200
@@ -8,7 +8,7 @@
theory Separation = L_axioms + WF_absolute:
-text{*This theory proves all instances needed for locale @{text "M_axioms"}*}
+text{*This theory proves all instances needed for locale @{text "M_basic"}*}
text{*Helps us solve for de Bruijn indices!*}
lemma nth_ConsI: "[|nth(n,l) = x; n \<in> nat|] ==> nth(succ(n), Cons(a,l)) = x"
@@ -470,12 +470,12 @@
done
-subsection{*Instantiating the locale @{text M_axioms}*}
+subsection{*Instantiating the locale @{text M_basic}*}
text{*Separation (and Strong Replacement) for basic set-theoretic constructions
such as intersection, Cartesian Product and image.*}
-lemma M_axioms_axioms_L: "M_axioms_axioms(L)"
- apply (rule M_axioms_axioms.intro)
+lemma M_basic_axioms_L: "M_basic_axioms(L)"
+ apply (rule M_basic_axioms.intro)
apply (assumption | rule
Inter_separation Diff_separation cartprod_separation image_separation
converse_separation restrict_separation
@@ -485,114 +485,114 @@
omap_replacement is_recfun_separation)+
done
-theorem M_axioms_L: "PROP M_axioms(L)"
-by (rule M_axioms.intro [OF M_triv_axioms_L M_axioms_axioms_L])
+theorem M_basic_L: "PROP M_basic(L)"
+by (rule M_basic.intro [OF M_trivial_L M_basic_axioms_L])
-lemmas cartprod_iff = M_axioms.cartprod_iff [OF M_axioms_L]
- and cartprod_closed = M_axioms.cartprod_closed [OF M_axioms_L]
- and sum_closed = M_axioms.sum_closed [OF M_axioms_L]
- and M_converse_iff = M_axioms.M_converse_iff [OF M_axioms_L]
- and converse_closed = M_axioms.converse_closed [OF M_axioms_L]
- and converse_abs = M_axioms.converse_abs [OF M_axioms_L]
- and image_closed = M_axioms.image_closed [OF M_axioms_L]
- and vimage_abs = M_axioms.vimage_abs [OF M_axioms_L]
- and vimage_closed = M_axioms.vimage_closed [OF M_axioms_L]
- and domain_abs = M_axioms.domain_abs [OF M_axioms_L]
- and domain_closed = M_axioms.domain_closed [OF M_axioms_L]
- and range_abs = M_axioms.range_abs [OF M_axioms_L]
- and range_closed = M_axioms.range_closed [OF M_axioms_L]
- and field_abs = M_axioms.field_abs [OF M_axioms_L]
- and field_closed = M_axioms.field_closed [OF M_axioms_L]
- and relation_abs = M_axioms.relation_abs [OF M_axioms_L]
- and function_abs = M_axioms.function_abs [OF M_axioms_L]
- and apply_closed = M_axioms.apply_closed [OF M_axioms_L]
- and apply_abs = M_axioms.apply_abs [OF M_axioms_L]
- and typed_function_abs = M_axioms.typed_function_abs [OF M_axioms_L]
- and injection_abs = M_axioms.injection_abs [OF M_axioms_L]
- and surjection_abs = M_axioms.surjection_abs [OF M_axioms_L]
- and bijection_abs = M_axioms.bijection_abs [OF M_axioms_L]
- and M_comp_iff = M_axioms.M_comp_iff [OF M_axioms_L]
- and comp_closed = M_axioms.comp_closed [OF M_axioms_L]
- and composition_abs = M_axioms.composition_abs [OF M_axioms_L]
- and restriction_is_function = M_axioms.restriction_is_function [OF M_axioms_L]
- and restriction_abs = M_axioms.restriction_abs [OF M_axioms_L]
- and M_restrict_iff = M_axioms.M_restrict_iff [OF M_axioms_L]
- and restrict_closed = M_axioms.restrict_closed [OF M_axioms_L]
- and Inter_abs = M_axioms.Inter_abs [OF M_axioms_L]
- and Inter_closed = M_axioms.Inter_closed [OF M_axioms_L]
- and Int_closed = M_axioms.Int_closed [OF M_axioms_L]
- and finite_fun_closed = M_axioms.finite_fun_closed [OF M_axioms_L]
- and is_funspace_abs = M_axioms.is_funspace_abs [OF M_axioms_L]
- and succ_fun_eq2 = M_axioms.succ_fun_eq2 [OF M_axioms_L]
- and funspace_succ = M_axioms.funspace_succ [OF M_axioms_L]
- and finite_funspace_closed = M_axioms.finite_funspace_closed [OF M_axioms_L]
+lemmas cartprod_iff = M_basic.cartprod_iff [OF M_basic_L]
+ and cartprod_closed = M_basic.cartprod_closed [OF M_basic_L]
+ and sum_closed = M_basic.sum_closed [OF M_basic_L]
+ and M_converse_iff = M_basic.M_converse_iff [OF M_basic_L]
+ and converse_closed = M_basic.converse_closed [OF M_basic_L]
+ and converse_abs = M_basic.converse_abs [OF M_basic_L]
+ and image_closed = M_basic.image_closed [OF M_basic_L]
+ and vimage_abs = M_basic.vimage_abs [OF M_basic_L]
+ and vimage_closed = M_basic.vimage_closed [OF M_basic_L]
+ and domain_abs = M_basic.domain_abs [OF M_basic_L]
+ and domain_closed = M_basic.domain_closed [OF M_basic_L]
+ and range_abs = M_basic.range_abs [OF M_basic_L]
+ and range_closed = M_basic.range_closed [OF M_basic_L]
+ and field_abs = M_basic.field_abs [OF M_basic_L]
+ and field_closed = M_basic.field_closed [OF M_basic_L]
+ and relation_abs = M_basic.relation_abs [OF M_basic_L]
+ and function_abs = M_basic.function_abs [OF M_basic_L]
+ and apply_closed = M_basic.apply_closed [OF M_basic_L]
+ and apply_abs = M_basic.apply_abs [OF M_basic_L]
+ and typed_function_abs = M_basic.typed_function_abs [OF M_basic_L]
+ and injection_abs = M_basic.injection_abs [OF M_basic_L]
+ and surjection_abs = M_basic.surjection_abs [OF M_basic_L]
+ and bijection_abs = M_basic.bijection_abs [OF M_basic_L]
+ and M_comp_iff = M_basic.M_comp_iff [OF M_basic_L]
+ and comp_closed = M_basic.comp_closed [OF M_basic_L]
+ and composition_abs = M_basic.composition_abs [OF M_basic_L]
+ and restriction_is_function = M_basic.restriction_is_function [OF M_basic_L]
+ and restriction_abs = M_basic.restriction_abs [OF M_basic_L]
+ and M_restrict_iff = M_basic.M_restrict_iff [OF M_basic_L]
+ and restrict_closed = M_basic.restrict_closed [OF M_basic_L]
+ and Inter_abs = M_basic.Inter_abs [OF M_basic_L]
+ and Inter_closed = M_basic.Inter_closed [OF M_basic_L]
+ and Int_closed = M_basic.Int_closed [OF M_basic_L]
+ and finite_fun_closed = M_basic.finite_fun_closed [OF M_basic_L]
+ and is_funspace_abs = M_basic.is_funspace_abs [OF M_basic_L]
+ and succ_fun_eq2 = M_basic.succ_fun_eq2 [OF M_basic_L]
+ and funspace_succ = M_basic.funspace_succ [OF M_basic_L]
+ and finite_funspace_closed = M_basic.finite_funspace_closed [OF M_basic_L]
-lemmas is_recfun_equal = M_axioms.is_recfun_equal [OF M_axioms_L]
- and is_recfun_cut = M_axioms.is_recfun_cut [OF M_axioms_L]
- and is_recfun_functional = M_axioms.is_recfun_functional [OF M_axioms_L]
- and is_recfun_relativize = M_axioms.is_recfun_relativize [OF M_axioms_L]
- and is_recfun_restrict = M_axioms.is_recfun_restrict [OF M_axioms_L]
- and univalent_is_recfun = M_axioms.univalent_is_recfun [OF M_axioms_L]
- and exists_is_recfun_indstep = M_axioms.exists_is_recfun_indstep [OF M_axioms_L]
- and wellfounded_exists_is_recfun = M_axioms.wellfounded_exists_is_recfun [OF M_axioms_L]
- and wf_exists_is_recfun = M_axioms.wf_exists_is_recfun [OF M_axioms_L]
- and is_recfun_abs = M_axioms.is_recfun_abs [OF M_axioms_L]
- and irreflexive_abs = M_axioms.irreflexive_abs [OF M_axioms_L]
- and transitive_rel_abs = M_axioms.transitive_rel_abs [OF M_axioms_L]
- and linear_rel_abs = M_axioms.linear_rel_abs [OF M_axioms_L]
- and wellordered_is_trans_on = M_axioms.wellordered_is_trans_on [OF M_axioms_L]
- and wellordered_is_linear = M_axioms.wellordered_is_linear [OF M_axioms_L]
- and wellordered_is_wellfounded_on = M_axioms.wellordered_is_wellfounded_on [OF M_axioms_L]
- and wellfounded_imp_wellfounded_on = M_axioms.wellfounded_imp_wellfounded_on [OF M_axioms_L]
- and wellfounded_on_subset_A = M_axioms.wellfounded_on_subset_A [OF M_axioms_L]
- and wellfounded_on_iff_wellfounded = M_axioms.wellfounded_on_iff_wellfounded [OF M_axioms_L]
- and wellfounded_on_imp_wellfounded = M_axioms.wellfounded_on_imp_wellfounded [OF M_axioms_L]
- and wellfounded_on_field_imp_wellfounded = M_axioms.wellfounded_on_field_imp_wellfounded [OF M_axioms_L]
- and wellfounded_iff_wellfounded_on_field = M_axioms.wellfounded_iff_wellfounded_on_field [OF M_axioms_L]
- and wellfounded_induct = M_axioms.wellfounded_induct [OF M_axioms_L]
- and wellfounded_on_induct = M_axioms.wellfounded_on_induct [OF M_axioms_L]
- and wellfounded_on_induct2 = M_axioms.wellfounded_on_induct2 [OF M_axioms_L]
- and linear_imp_relativized = M_axioms.linear_imp_relativized [OF M_axioms_L]
- and trans_on_imp_relativized = M_axioms.trans_on_imp_relativized [OF M_axioms_L]
- and wf_on_imp_relativized = M_axioms.wf_on_imp_relativized [OF M_axioms_L]
- and wf_imp_relativized = M_axioms.wf_imp_relativized [OF M_axioms_L]
- and well_ord_imp_relativized = M_axioms.well_ord_imp_relativized [OF M_axioms_L]
- and order_isomorphism_abs = M_axioms.order_isomorphism_abs [OF M_axioms_L]
- and pred_set_abs = M_axioms.pred_set_abs [OF M_axioms_L]
+lemmas is_recfun_equal = M_basic.is_recfun_equal [OF M_basic_L]
+ and is_recfun_cut = M_basic.is_recfun_cut [OF M_basic_L]
+ and is_recfun_functional = M_basic.is_recfun_functional [OF M_basic_L]
+ and is_recfun_relativize = M_basic.is_recfun_relativize [OF M_basic_L]
+ and is_recfun_restrict = M_basic.is_recfun_restrict [OF M_basic_L]
+ and univalent_is_recfun = M_basic.univalent_is_recfun [OF M_basic_L]
+ and exists_is_recfun_indstep = M_basic.exists_is_recfun_indstep [OF M_basic_L]
+ and wellfounded_exists_is_recfun = M_basic.wellfounded_exists_is_recfun [OF M_basic_L]
+ and wf_exists_is_recfun = M_basic.wf_exists_is_recfun [OF M_basic_L]
+ and is_recfun_abs = M_basic.is_recfun_abs [OF M_basic_L]
+ and irreflexive_abs = M_basic.irreflexive_abs [OF M_basic_L]
+ and transitive_rel_abs = M_basic.transitive_rel_abs [OF M_basic_L]
+ and linear_rel_abs = M_basic.linear_rel_abs [OF M_basic_L]
+ and wellordered_is_trans_on = M_basic.wellordered_is_trans_on [OF M_basic_L]
+ and wellordered_is_linear = M_basic.wellordered_is_linear [OF M_basic_L]
+ and wellordered_is_wellfounded_on = M_basic.wellordered_is_wellfounded_on [OF M_basic_L]
+ and wellfounded_imp_wellfounded_on = M_basic.wellfounded_imp_wellfounded_on [OF M_basic_L]
+ and wellfounded_on_subset_A = M_basic.wellfounded_on_subset_A [OF M_basic_L]
+ and wellfounded_on_iff_wellfounded = M_basic.wellfounded_on_iff_wellfounded [OF M_basic_L]
+ and wellfounded_on_imp_wellfounded = M_basic.wellfounded_on_imp_wellfounded [OF M_basic_L]
+ and wellfounded_on_field_imp_wellfounded = M_basic.wellfounded_on_field_imp_wellfounded [OF M_basic_L]
+ and wellfounded_iff_wellfounded_on_field = M_basic.wellfounded_iff_wellfounded_on_field [OF M_basic_L]
+ and wellfounded_induct = M_basic.wellfounded_induct [OF M_basic_L]
+ and wellfounded_on_induct = M_basic.wellfounded_on_induct [OF M_basic_L]
+ and wellfounded_on_induct2 = M_basic.wellfounded_on_induct2 [OF M_basic_L]
+ and linear_imp_relativized = M_basic.linear_imp_relativized [OF M_basic_L]
+ and trans_on_imp_relativized = M_basic.trans_on_imp_relativized [OF M_basic_L]
+ and wf_on_imp_relativized = M_basic.wf_on_imp_relativized [OF M_basic_L]
+ and wf_imp_relativized = M_basic.wf_imp_relativized [OF M_basic_L]
+ and well_ord_imp_relativized = M_basic.well_ord_imp_relativized [OF M_basic_L]
+ and order_isomorphism_abs = M_basic.order_isomorphism_abs [OF M_basic_L]
+ and pred_set_abs = M_basic.pred_set_abs [OF M_basic_L]
-lemmas pred_closed = M_axioms.pred_closed [OF M_axioms_L]
- and membership_abs = M_axioms.membership_abs [OF M_axioms_L]
- and M_Memrel_iff = M_axioms.M_Memrel_iff [OF M_axioms_L]
- and Memrel_closed = M_axioms.Memrel_closed [OF M_axioms_L]
- and wellordered_iso_predD = M_axioms.wellordered_iso_predD [OF M_axioms_L]
- and wellordered_iso_pred_eq = M_axioms.wellordered_iso_pred_eq [OF M_axioms_L]
- and wellfounded_on_asym = M_axioms.wellfounded_on_asym [OF M_axioms_L]
- and wellordered_asym = M_axioms.wellordered_asym [OF M_axioms_L]
- and ord_iso_pred_imp_lt = M_axioms.ord_iso_pred_imp_lt [OF M_axioms_L]
- and obase_iff = M_axioms.obase_iff [OF M_axioms_L]
- and omap_iff = M_axioms.omap_iff [OF M_axioms_L]
- and omap_unique = M_axioms.omap_unique [OF M_axioms_L]
- and omap_yields_Ord = M_axioms.omap_yields_Ord [OF M_axioms_L]
- and otype_iff = M_axioms.otype_iff [OF M_axioms_L]
- and otype_eq_range = M_axioms.otype_eq_range [OF M_axioms_L]
- and Ord_otype = M_axioms.Ord_otype [OF M_axioms_L]
- and domain_omap = M_axioms.domain_omap [OF M_axioms_L]
- and omap_subset = M_axioms.omap_subset [OF M_axioms_L]
- and omap_funtype = M_axioms.omap_funtype [OF M_axioms_L]
- and wellordered_omap_bij = M_axioms.wellordered_omap_bij [OF M_axioms_L]
- and omap_ord_iso = M_axioms.omap_ord_iso [OF M_axioms_L]
- and Ord_omap_image_pred = M_axioms.Ord_omap_image_pred [OF M_axioms_L]
- and restrict_omap_ord_iso = M_axioms.restrict_omap_ord_iso [OF M_axioms_L]
- and obase_equals = M_axioms.obase_equals [OF M_axioms_L]
- and omap_ord_iso_otype = M_axioms.omap_ord_iso_otype [OF M_axioms_L]
- and obase_exists = M_axioms.obase_exists [OF M_axioms_L]
- and omap_exists = M_axioms.omap_exists [OF M_axioms_L]
- and otype_exists = M_axioms.otype_exists [OF M_axioms_L]
- and omap_ord_iso_otype' = M_axioms.omap_ord_iso_otype' [OF M_axioms_L]
- and ordertype_exists = M_axioms.ordertype_exists [OF M_axioms_L]
- and relativized_imp_well_ord = M_axioms.relativized_imp_well_ord [OF M_axioms_L]
- and well_ord_abs = M_axioms.well_ord_abs [OF M_axioms_L]
+lemmas pred_closed = M_basic.pred_closed [OF M_basic_L]
+ and membership_abs = M_basic.membership_abs [OF M_basic_L]
+ and M_Memrel_iff = M_basic.M_Memrel_iff [OF M_basic_L]
+ and Memrel_closed = M_basic.Memrel_closed [OF M_basic_L]
+ and wellordered_iso_predD = M_basic.wellordered_iso_predD [OF M_basic_L]
+ and wellordered_iso_pred_eq = M_basic.wellordered_iso_pred_eq [OF M_basic_L]
+ and wellfounded_on_asym = M_basic.wellfounded_on_asym [OF M_basic_L]
+ and wellordered_asym = M_basic.wellordered_asym [OF M_basic_L]
+ and ord_iso_pred_imp_lt = M_basic.ord_iso_pred_imp_lt [OF M_basic_L]
+ and obase_iff = M_basic.obase_iff [OF M_basic_L]
+ and omap_iff = M_basic.omap_iff [OF M_basic_L]
+ and omap_unique = M_basic.omap_unique [OF M_basic_L]
+ and omap_yields_Ord = M_basic.omap_yields_Ord [OF M_basic_L]
+ and otype_iff = M_basic.otype_iff [OF M_basic_L]
+ and otype_eq_range = M_basic.otype_eq_range [OF M_basic_L]
+ and Ord_otype = M_basic.Ord_otype [OF M_basic_L]
+ and domain_omap = M_basic.domain_omap [OF M_basic_L]
+ and omap_subset = M_basic.omap_subset [OF M_basic_L]
+ and omap_funtype = M_basic.omap_funtype [OF M_basic_L]
+ and wellordered_omap_bij = M_basic.wellordered_omap_bij [OF M_basic_L]
+ and omap_ord_iso = M_basic.omap_ord_iso [OF M_basic_L]
+ and Ord_omap_image_pred = M_basic.Ord_omap_image_pred [OF M_basic_L]
+ and restrict_omap_ord_iso = M_basic.restrict_omap_ord_iso [OF M_basic_L]
+ and obase_equals = M_basic.obase_equals [OF M_basic_L]
+ and omap_ord_iso_otype = M_basic.omap_ord_iso_otype [OF M_basic_L]
+ and obase_exists = M_basic.obase_exists [OF M_basic_L]
+ and omap_exists = M_basic.omap_exists [OF M_basic_L]
+ and otype_exists = M_basic.otype_exists [OF M_basic_L]
+ and omap_ord_iso_otype' = M_basic.omap_ord_iso_otype' [OF M_basic_L]
+ and ordertype_exists = M_basic.ordertype_exists [OF M_basic_L]
+ and relativized_imp_well_ord = M_basic.relativized_imp_well_ord [OF M_basic_L]
+ and well_ord_abs = M_basic.well_ord_abs [OF M_basic_L]
declare cartprod_closed [intro, simp]
declare sum_closed [intro, simp]
--- a/src/ZF/Constructible/WF_absolute.thy Tue Sep 10 16:47:17 2002 +0200
+++ b/src/ZF/Constructible/WF_absolute.thy Tue Sep 10 16:51:31 2002 +0200
@@ -139,7 +139,7 @@
"tran_closure(M,r,t) ==
\<exists>s[M]. rtran_closure(M,r,s) & composition(M,r,s,t)"
-lemma (in M_axioms) rtran_closure_mem_iff:
+lemma (in M_basic) rtran_closure_mem_iff:
"[|M(A); M(r); M(p)|]
==> rtran_closure_mem(M,A,r,p) <->
(\<exists>n[M]. n\<in>nat &
@@ -149,7 +149,7 @@
by (simp add: rtran_closure_mem_def Ord_succ_mem_iff nat_0_le [THEN ltD])
-locale M_trancl = M_axioms +
+locale M_trancl = M_basic +
assumes rtrancl_separation:
"[| M(r); M(A) |] ==> separation (M, rtran_closure_mem(M,A,r))"
and wellfounded_trancl_separation:
--- a/src/ZF/Constructible/WFrec.thy Tue Sep 10 16:47:17 2002 +0200
+++ b/src/ZF/Constructible/WFrec.thy Tue Sep 10 16:51:31 2002 +0200
@@ -74,7 +74,7 @@
subsection{*Reworking of the Recursion Theory Within @{term M}*}
-lemma (in M_axioms) is_recfun_separation':
+lemma (in M_basic) is_recfun_separation':
"[| f \<in> r -`` {a} \<rightarrow> range(f); g \<in> r -`` {b} \<rightarrow> range(g);
M(r); M(f); M(g); M(a); M(b) |]
==> separation(M, \<lambda>x. \<not> (\<langle>x, a\<rangle> \<in> r \<longrightarrow> \<langle>x, b\<rangle> \<in> r \<longrightarrow> f ` x = g ` x))"
@@ -89,7 +89,7 @@
The last three M-premises are redundant because of @{term "M(r)"},
but without them we'd have to undertake
more work to set up the induction formula.*}
-lemma (in M_axioms) is_recfun_equal [rule_format]:
+lemma (in M_basic) is_recfun_equal [rule_format]:
"[|is_recfun(r,a,H,f); is_recfun(r,b,H,g);
wellfounded(M,r); trans(r);
M(f); M(g); M(r); M(x); M(a); M(b) |]
@@ -112,7 +112,7 @@
apply (blast intro: transD sym)
done
-lemma (in M_axioms) is_recfun_cut:
+lemma (in M_basic) is_recfun_cut:
"[|is_recfun(r,a,H,f); is_recfun(r,b,H,g);
wellfounded(M,r); trans(r);
M(f); M(g); M(r); <b,a> \<in> r |]
@@ -124,7 +124,7 @@
apply (blast intro: is_recfun_equal transD dest: transM)
done
-lemma (in M_axioms) is_recfun_functional:
+lemma (in M_basic) is_recfun_functional:
"[|is_recfun(r,a,H,f); is_recfun(r,a,H,g);
wellfounded(M,r); trans(r); M(f); M(g); M(r) |] ==> f=g"
apply (rule fun_extension)
@@ -133,7 +133,7 @@
done
text{*Tells us that @{text is_recfun} can (in principle) be relativized.*}
-lemma (in M_axioms) is_recfun_relativize:
+lemma (in M_basic) is_recfun_relativize:
"[| M(r); M(f); \<forall>x[M]. \<forall>g[M]. function(g) --> M(H(x,g)) |]
==> is_recfun(r,a,H,f) <->
(\<forall>z[M]. z \<in> f <->
@@ -154,7 +154,7 @@
apply (simp add: is_recfun_imp_function function_restrictI)
done
-lemma (in M_axioms) is_recfun_restrict:
+lemma (in M_basic) is_recfun_restrict:
"[| wellfounded(M,r); trans(r); is_recfun(r,x,H,f); \<langle>y,x\<rangle> \<in> r;
M(r); M(f);
\<forall>x[M]. \<forall>g[M]. function(g) --> M(H(x,g)) |]
@@ -171,7 +171,7 @@
apply (blast intro: apply_recfun dest: transD)
done
-lemma (in M_axioms) restrict_Y_lemma:
+lemma (in M_basic) restrict_Y_lemma:
"[| wellfounded(M,r); trans(r); M(r);
\<forall>x[M]. \<forall>g[M]. function(g) --> M(H(x,g)); M(Y);
\<forall>b[M].
@@ -201,7 +201,7 @@
done
text{*For typical applications of Replacement for recursive definitions*}
-lemma (in M_axioms) univalent_is_recfun:
+lemma (in M_basic) univalent_is_recfun:
"[|wellfounded(M,r); trans(r); M(r)|]
==> univalent (M, A, \<lambda>x p.
\<exists>y[M]. p = \<langle>x,y\<rangle> & (\<exists>f[M]. is_recfun(r,x,H,f) & y = H(x,f)))"
@@ -212,7 +212,7 @@
text{*Proof of the inductive step for @{text exists_is_recfun}, since
we must prove two versions.*}
-lemma (in M_axioms) exists_is_recfun_indstep:
+lemma (in M_basic) exists_is_recfun_indstep:
"[|\<forall>y. \<langle>y, a1\<rangle> \<in> r --> (\<exists>f[M]. is_recfun(r, y, H, f));
wellfounded(M,r); trans(r); M(r); M(a1);
strong_replacement(M, \<lambda>x z.
@@ -245,7 +245,7 @@
text{*Relativized version, when we have the (currently weaker) premise
@{term "wellfounded(M,r)"}*}
-lemma (in M_axioms) wellfounded_exists_is_recfun:
+lemma (in M_basic) wellfounded_exists_is_recfun:
"[|wellfounded(M,r); trans(r);
separation(M, \<lambda>x. ~ (\<exists>f[M]. is_recfun(r, x, H, f)));
strong_replacement(M, \<lambda>x z.
@@ -257,7 +257,7 @@
apply (rule exists_is_recfun_indstep, assumption+)
done
-lemma (in M_axioms) wf_exists_is_recfun [rule_format]:
+lemma (in M_basic) wf_exists_is_recfun [rule_format]:
"[|wf(r); trans(r); M(r);
strong_replacement(M, \<lambda>x z.
\<exists>y[M]. \<exists>g[M]. pair(M,x,y,z) & is_recfun(r,x,H,g) & y = H(x,g));
@@ -291,7 +291,7 @@
strong_replacement(M,
\<lambda>x z. \<exists>y[M]. pair(M,x,y,z) & is_wfrec(M,MH,r,x,y))"
-lemma (in M_axioms) is_recfun_abs:
+lemma (in M_basic) is_recfun_abs:
"[| \<forall>x[M]. \<forall>g[M]. function(g) --> M(H(x,g)); M(r); M(a); M(f);
relativize2(M,MH,H) |]
==> M_is_recfun(M,MH,r,a,f) <-> is_recfun(r,a,H,f)"
@@ -306,7 +306,7 @@
==> M_is_recfun(M,MH,r,a,f) <-> M_is_recfun(M,MH',r',a',f')"
by (simp add: M_is_recfun_def)
-lemma (in M_axioms) is_wfrec_abs:
+lemma (in M_basic) is_wfrec_abs:
"[| \<forall>x[M]. \<forall>g[M]. function(g) --> M(H(x,g));
relativize2(M,MH,H); M(r); M(a); M(z) |]
==> is_wfrec(M,MH,r,a,z) <->
@@ -314,7 +314,7 @@
by (simp add: is_wfrec_def relativize2_def is_recfun_abs)
text{*Relating @{term wfrec_replacement} to native constructs*}
-lemma (in M_axioms) wfrec_replacement':
+lemma (in M_basic) wfrec_replacement':
"[|wfrec_replacement(M,MH,r);
\<forall>x[M]. \<forall>g[M]. function(g) --> M(H(x,g));
relativize2(M,MH,H); M(r)|]
@@ -381,7 +381,7 @@
fun_apply(M,f,j,fj) & fj = k"
-locale M_ord_arith = M_axioms +
+locale M_ord_arith = M_basic +
assumes oadd_strong_replacement:
"[| M(i); M(j) |] ==>
strong_replacement(M,
--- a/src/ZF/Constructible/Wellorderings.thy Tue Sep 10 16:47:17 2002 +0200
+++ b/src/ZF/Constructible/Wellorderings.thy Tue Sep 10 16:51:31 2002 +0200
@@ -49,63 +49,63 @@
subsubsection {*Trivial absoluteness proofs*}
-lemma (in M_axioms) irreflexive_abs [simp]:
+lemma (in M_basic) irreflexive_abs [simp]:
"M(A) ==> irreflexive(M,A,r) <-> irrefl(A,r)"
by (simp add: irreflexive_def irrefl_def)
-lemma (in M_axioms) transitive_rel_abs [simp]:
+lemma (in M_basic) transitive_rel_abs [simp]:
"M(A) ==> transitive_rel(M,A,r) <-> trans[A](r)"
by (simp add: transitive_rel_def trans_on_def)
-lemma (in M_axioms) linear_rel_abs [simp]:
+lemma (in M_basic) linear_rel_abs [simp]:
"M(A) ==> linear_rel(M,A,r) <-> linear(A,r)"
by (simp add: linear_rel_def linear_def)
-lemma (in M_axioms) wellordered_is_trans_on:
+lemma (in M_basic) wellordered_is_trans_on:
"[| wellordered(M,A,r); M(A) |] ==> trans[A](r)"
by (auto simp add: wellordered_def)
-lemma (in M_axioms) wellordered_is_linear:
+lemma (in M_basic) wellordered_is_linear:
"[| wellordered(M,A,r); M(A) |] ==> linear(A,r)"
by (auto simp add: wellordered_def)
-lemma (in M_axioms) wellordered_is_wellfounded_on:
+lemma (in M_basic) wellordered_is_wellfounded_on:
"[| wellordered(M,A,r); M(A) |] ==> wellfounded_on(M,A,r)"
by (auto simp add: wellordered_def)
-lemma (in M_axioms) wellfounded_imp_wellfounded_on:
+lemma (in M_basic) wellfounded_imp_wellfounded_on:
"[| wellfounded(M,r); M(A) |] ==> wellfounded_on(M,A,r)"
by (auto simp add: wellfounded_def wellfounded_on_def)
-lemma (in M_axioms) wellfounded_on_subset_A:
+lemma (in M_basic) wellfounded_on_subset_A:
"[| wellfounded_on(M,A,r); B<=A |] ==> wellfounded_on(M,B,r)"
by (simp add: wellfounded_on_def, blast)
subsubsection {*Well-founded relations*}
-lemma (in M_axioms) wellfounded_on_iff_wellfounded:
+lemma (in M_basic) wellfounded_on_iff_wellfounded:
"wellfounded_on(M,A,r) <-> wellfounded(M, r \<inter> A*A)"
apply (simp add: wellfounded_on_def wellfounded_def, safe)
apply blast
apply (drule_tac x=x in rspec, assumption, blast)
done
-lemma (in M_axioms) wellfounded_on_imp_wellfounded:
+lemma (in M_basic) wellfounded_on_imp_wellfounded:
"[|wellfounded_on(M,A,r); r \<subseteq> A*A|] ==> wellfounded(M,r)"
by (simp add: wellfounded_on_iff_wellfounded subset_Int_iff)
-lemma (in M_axioms) wellfounded_on_field_imp_wellfounded:
+lemma (in M_basic) wellfounded_on_field_imp_wellfounded:
"wellfounded_on(M, field(r), r) ==> wellfounded(M,r)"
by (simp add: wellfounded_def wellfounded_on_iff_wellfounded, fast)
-lemma (in M_axioms) wellfounded_iff_wellfounded_on_field:
+lemma (in M_basic) wellfounded_iff_wellfounded_on_field:
"M(r) ==> wellfounded(M,r) <-> wellfounded_on(M, field(r), r)"
by (blast intro: wellfounded_imp_wellfounded_on
wellfounded_on_field_imp_wellfounded)
(*Consider the least z in domain(r) such that P(z) does not hold...*)
-lemma (in M_axioms) wellfounded_induct:
+lemma (in M_basic) wellfounded_induct:
"[| wellfounded(M,r); M(a); M(r); separation(M, \<lambda>x. ~P(x));
\<forall>x. M(x) & (\<forall>y. <y,x> \<in> r --> P(y)) --> P(x) |]
==> P(a)";
@@ -114,7 +114,7 @@
apply (blast dest: transM)+
done
-lemma (in M_axioms) wellfounded_on_induct:
+lemma (in M_basic) wellfounded_on_induct:
"[| a\<in>A; wellfounded_on(M,A,r); M(A);
separation(M, \<lambda>x. x\<in>A --> ~P(x));
\<forall>x\<in>A. M(x) & (\<forall>y\<in>A. <y,x> \<in> r --> P(y)) --> P(x) |]
@@ -126,7 +126,7 @@
text{*The assumption @{term "r \<subseteq> A*A"} justifies strengthening the induction
hypothesis by removing the restriction to @{term A}.*}
-lemma (in M_axioms) wellfounded_on_induct2:
+lemma (in M_basic) wellfounded_on_induct2:
"[| a\<in>A; wellfounded_on(M,A,r); M(A); r \<subseteq> A*A;
separation(M, \<lambda>x. x\<in>A --> ~P(x));
\<forall>x\<in>A. M(x) & (\<forall>y. <y,x> \<in> r --> P(y)) --> P(x) |]
@@ -136,27 +136,27 @@
subsubsection {*Kunen's lemma IV 3.14, page 123*}
-lemma (in M_axioms) linear_imp_relativized:
+lemma (in M_basic) linear_imp_relativized:
"linear(A,r) ==> linear_rel(M,A,r)"
by (simp add: linear_def linear_rel_def)
-lemma (in M_axioms) trans_on_imp_relativized:
+lemma (in M_basic) trans_on_imp_relativized:
"trans[A](r) ==> transitive_rel(M,A,r)"
by (unfold transitive_rel_def trans_on_def, blast)
-lemma (in M_axioms) wf_on_imp_relativized:
+lemma (in M_basic) wf_on_imp_relativized:
"wf[A](r) ==> wellfounded_on(M,A,r)"
apply (simp add: wellfounded_on_def wf_def wf_on_def, clarify)
apply (drule_tac x=x in spec, blast)
done
-lemma (in M_axioms) wf_imp_relativized:
+lemma (in M_basic) wf_imp_relativized:
"wf(r) ==> wellfounded(M,r)"
apply (simp add: wellfounded_def wf_def, clarify)
apply (drule_tac x=x in spec, blast)
done
-lemma (in M_axioms) well_ord_imp_relativized:
+lemma (in M_basic) well_ord_imp_relativized:
"well_ord(A,r) ==> wellordered(M,A,r)"
by (simp add: wellordered_def well_ord_def tot_ord_def part_ord_def
linear_imp_relativized trans_on_imp_relativized wf_on_imp_relativized)
@@ -164,24 +164,24 @@
subsection{* Relativized versions of order-isomorphisms and order types *}
-lemma (in M_axioms) order_isomorphism_abs [simp]:
+lemma (in M_basic) order_isomorphism_abs [simp]:
"[| M(A); M(B); M(f) |]
==> order_isomorphism(M,A,r,B,s,f) <-> f \<in> ord_iso(A,r,B,s)"
by (simp add: apply_closed order_isomorphism_def ord_iso_def)
-lemma (in M_axioms) pred_set_abs [simp]:
+lemma (in M_basic) pred_set_abs [simp]:
"[| M(r); M(B) |] ==> pred_set(M,A,x,r,B) <-> B = Order.pred(A,x,r)"
apply (simp add: pred_set_def Order.pred_def)
apply (blast dest: transM)
done
-lemma (in M_axioms) pred_closed [intro,simp]:
+lemma (in M_basic) pred_closed [intro,simp]:
"[| M(A); M(r); M(x) |] ==> M(Order.pred(A,x,r))"
apply (simp add: Order.pred_def)
apply (insert pred_separation [of r x], simp)
done
-lemma (in M_axioms) membership_abs [simp]:
+lemma (in M_basic) membership_abs [simp]:
"[| M(r); M(A) |] ==> membership(M,A,r) <-> r = Memrel(A)"
apply (simp add: membership_def Memrel_def, safe)
apply (rule equalityI)
@@ -194,14 +194,14 @@
apply auto
done
-lemma (in M_axioms) M_Memrel_iff:
+lemma (in M_basic) M_Memrel_iff:
"M(A) ==>
Memrel(A) = {z \<in> A*A. \<exists>x[M]. \<exists>y[M]. z = \<langle>x,y\<rangle> & x \<in> y}"
apply (simp add: Memrel_def)
apply (blast dest: transM)
done
-lemma (in M_axioms) Memrel_closed [intro,simp]:
+lemma (in M_basic) Memrel_closed [intro,simp]:
"M(A) ==> M(Memrel(A))"
apply (simp add: M_Memrel_iff)
apply (insert Memrel_separation, simp)
@@ -233,7 +233,7 @@
text{*Inductive argument for Kunen's Lemma 6.1, etc.
Simple proof from Halmos, page 72*}
-lemma (in M_axioms) wellordered_iso_subset_lemma:
+lemma (in M_basic) wellordered_iso_subset_lemma:
"[| wellordered(M,A,r); f \<in> ord_iso(A,r, A',r); A'<= A; y \<in> A;
M(A); M(f); M(r) |] ==> ~ <f`y, y> \<in> r"
apply (unfold wellordered_def ord_iso_def)
@@ -247,7 +247,7 @@
text{*Kunen's Lemma 6.1: there's no order-isomorphism to an initial segment
of a well-ordering*}
-lemma (in M_axioms) wellordered_iso_predD:
+lemma (in M_basic) wellordered_iso_predD:
"[| wellordered(M,A,r); f \<in> ord_iso(A, r, Order.pred(A,x,r), r);
M(A); M(f); M(r) |] ==> x \<notin> A"
apply (rule notI)
@@ -260,7 +260,7 @@
done
-lemma (in M_axioms) wellordered_iso_pred_eq_lemma:
+lemma (in M_basic) wellordered_iso_pred_eq_lemma:
"[| f \<in> \<langle>Order.pred(A,y,r), r\<rangle> \<cong> \<langle>Order.pred(A,x,r), r\<rangle>;
wellordered(M,A,r); x\<in>A; y\<in>A; M(A); M(f); M(r) |] ==> <x,y> \<notin> r"
apply (frule wellordered_is_trans_on, assumption)
@@ -273,7 +273,7 @@
text{*Simple consequence of Lemma 6.1*}
-lemma (in M_axioms) wellordered_iso_pred_eq:
+lemma (in M_basic) wellordered_iso_pred_eq:
"[| wellordered(M,A,r);
f \<in> ord_iso(Order.pred(A,a,r), r, Order.pred(A,c,r), r);
M(A); M(f); M(r); a\<in>A; c\<in>A |] ==> a=c"
@@ -285,21 +285,21 @@
apply (blast dest: wellordered_iso_pred_eq_lemma)+
done
-lemma (in M_axioms) wellfounded_on_asym:
+lemma (in M_basic) wellfounded_on_asym:
"[| wellfounded_on(M,A,r); <a,x>\<in>r; a\<in>A; x\<in>A; M(A) |] ==> <x,a>\<notin>r"
apply (simp add: wellfounded_on_def)
apply (drule_tac x="{x,a}" in rspec)
apply (blast dest: transM)+
done
-lemma (in M_axioms) wellordered_asym:
+lemma (in M_basic) wellordered_asym:
"[| wellordered(M,A,r); <a,x>\<in>r; a\<in>A; x\<in>A; M(A) |] ==> <x,a>\<notin>r"
by (simp add: wellordered_def, blast dest: wellfounded_on_asym)
text{*Surely a shorter proof using lemmas in @{text Order}?
Like @{text well_ord_iso_preserving}?*}
-lemma (in M_axioms) ord_iso_pred_imp_lt:
+lemma (in M_basic) ord_iso_pred_imp_lt:
"[| f \<in> ord_iso(Order.pred(A,x,r), r, i, Memrel(i));
g \<in> ord_iso(Order.pred(A,y,r), r, j, Memrel(j));
wellordered(M,A,r); x \<in> A; y \<in> A; M(A); M(r); M(f); M(g); M(j);
@@ -372,7 +372,7 @@
-lemma (in M_axioms) obase_iff:
+lemma (in M_basic) obase_iff:
"[| M(A); M(r); M(z) |]
==> obase(M,A,r,z) <->
z = {a\<in>A. \<exists>x[M]. \<exists>g[M]. Ord(x) &
@@ -387,7 +387,7 @@
text{*Can also be proved with the premise @{term "M(z)"} instead of
@{term "M(f)"}, but that version is less useful.*}
-lemma (in M_axioms) omap_iff:
+lemma (in M_basic) omap_iff:
"[| omap(M,A,r,f); M(A); M(r); M(f) |]
==> z \<in> f <->
(\<exists>a\<in>A. \<exists>x[M]. \<exists>g[M]. z = <a,x> & Ord(x) &
@@ -400,18 +400,18 @@
apply (blast dest: transM)+
done
-lemma (in M_axioms) omap_unique:
+lemma (in M_basic) omap_unique:
"[| omap(M,A,r,f); omap(M,A,r,f'); M(A); M(r); M(f); M(f') |] ==> f' = f"
apply (rule equality_iffI)
apply (simp add: omap_iff)
done
-lemma (in M_axioms) omap_yields_Ord:
+lemma (in M_basic) omap_yields_Ord:
"[| omap(M,A,r,f); \<langle>a,x\<rangle> \<in> f; M(a); M(x) |] ==> Ord(x)"
apply (simp add: omap_def, blast)
done
-lemma (in M_axioms) otype_iff:
+lemma (in M_basic) otype_iff:
"[| otype(M,A,r,i); M(A); M(r); M(i) |]
==> x \<in> i <->
(M(x) & Ord(x) &
@@ -423,7 +423,7 @@
apply (simp add: omap_iff, blast)
done
-lemma (in M_axioms) otype_eq_range:
+lemma (in M_basic) otype_eq_range:
"[| omap(M,A,r,f); otype(M,A,r,i); M(A); M(r); M(f); M(i) |]
==> i = range(f)"
apply (auto simp add: otype_def omap_iff)
@@ -431,7 +431,7 @@
done
-lemma (in M_axioms) Ord_otype:
+lemma (in M_basic) Ord_otype:
"[| otype(M,A,r,i); trans[A](r); M(A); M(r); M(i) |] ==> Ord(i)"
apply (rotate_tac 1)
apply (rule OrdI)
@@ -452,7 +452,7 @@
apply (blast intro: restrict_ord_iso ord_iso_sym ltI dest: transM)
done
-lemma (in M_axioms) domain_omap:
+lemma (in M_basic) domain_omap:
"[| omap(M,A,r,f); obase(M,A,r,B); M(A); M(r); M(B); M(f) |]
==> domain(f) = B"
apply (rotate_tac 2)
@@ -461,7 +461,7 @@
apply (simp add: domain_iff omap_iff, blast)
done
-lemma (in M_axioms) omap_subset:
+lemma (in M_basic) omap_subset:
"[| omap(M,A,r,f); obase(M,A,r,B); otype(M,A,r,i);
M(A); M(r); M(f); M(B); M(i) |] ==> f \<subseteq> B * i"
apply (rotate_tac 3, clarify)
@@ -469,7 +469,7 @@
apply (force simp add: otype_iff)
done
-lemma (in M_axioms) omap_funtype:
+lemma (in M_basic) omap_funtype:
"[| omap(M,A,r,f); obase(M,A,r,B); otype(M,A,r,i);
M(A); M(r); M(f); M(B); M(i) |] ==> f \<in> B -> i"
apply (rotate_tac 3)
@@ -478,7 +478,7 @@
done
-lemma (in M_axioms) wellordered_omap_bij:
+lemma (in M_basic) wellordered_omap_bij:
"[| wellordered(M,A,r); omap(M,A,r,f); obase(M,A,r,B); otype(M,A,r,i);
M(A); M(r); M(f); M(B); M(i) |] ==> f \<in> bij(B,i)"
apply (insert omap_funtype [of A r f B i])
@@ -492,7 +492,7 @@
text{*This is not the final result: we must show @{term "oB(A,r) = A"}*}
-lemma (in M_axioms) omap_ord_iso:
+lemma (in M_basic) omap_ord_iso:
"[| wellordered(M,A,r); omap(M,A,r,f); obase(M,A,r,B); otype(M,A,r,i);
M(A); M(r); M(f); M(B); M(i) |] ==> f \<in> ord_iso(B,r,i,Memrel(i))"
apply (rule ord_isoI)
@@ -513,7 +513,7 @@
apply (blast dest: ord_iso_pred_imp_lt ltD elim: mem_asym)
done
-lemma (in M_axioms) Ord_omap_image_pred:
+lemma (in M_basic) Ord_omap_image_pred:
"[| wellordered(M,A,r); omap(M,A,r,f); obase(M,A,r,B); otype(M,A,r,i);
M(A); M(r); M(f); M(B); M(i); b \<in> A |] ==> Ord(f `` Order.pred(A,b,r))"
apply (frule wellordered_is_trans_on, assumption)
@@ -539,7 +539,7 @@
apply (auto simp add: obase_iff)
done
-lemma (in M_axioms) restrict_omap_ord_iso:
+lemma (in M_basic) restrict_omap_ord_iso:
"[| wellordered(M,A,r); omap(M,A,r,f); obase(M,A,r,B); otype(M,A,r,i);
D \<subseteq> B; M(A); M(r); M(f); M(B); M(i) |]
==> restrict(f,D) \<in> (\<langle>D,r\<rangle> \<cong> \<langle>f``D, Memrel(f``D)\<rangle>)"
@@ -550,7 +550,7 @@
apply (drule ord_iso_sym, simp)
done
-lemma (in M_axioms) obase_equals:
+lemma (in M_basic) obase_equals:
"[| wellordered(M,A,r); omap(M,A,r,f); obase(M,A,r,B); otype(M,A,r,i);
M(A); M(r); M(f); M(B); M(i) |] ==> B = A"
apply (rotate_tac 4)
@@ -570,21 +570,21 @@
text{*Main result: @{term om} gives the order-isomorphism
@{term "\<langle>A,r\<rangle> \<cong> \<langle>i, Memrel(i)\<rangle>"} *}
-theorem (in M_axioms) omap_ord_iso_otype:
+theorem (in M_basic) omap_ord_iso_otype:
"[| wellordered(M,A,r); omap(M,A,r,f); obase(M,A,r,B); otype(M,A,r,i);
M(A); M(r); M(f); M(B); M(i) |] ==> f \<in> ord_iso(A, r, i, Memrel(i))"
apply (frule omap_ord_iso, assumption+)
apply (frule obase_equals, assumption+, blast)
done
-lemma (in M_axioms) obase_exists:
+lemma (in M_basic) obase_exists:
"[| M(A); M(r) |] ==> \<exists>z[M]. obase(M,A,r,z)"
apply (simp add: obase_def)
apply (insert obase_separation [of A r])
apply (simp add: separation_def)
done
-lemma (in M_axioms) omap_exists:
+lemma (in M_basic) omap_exists:
"[| M(A); M(r) |] ==> \<exists>z[M]. omap(M,A,r,z)"
apply (insert obase_exists [of A r])
apply (simp add: omap_def)
@@ -601,7 +601,7 @@
declare rall_simps [simp] rex_simps [simp]
-lemma (in M_axioms) otype_exists:
+lemma (in M_basic) otype_exists:
"[| wellordered(M,A,r); M(A); M(r) |] ==> \<exists>i[M]. otype(M,A,r,i)"
apply (insert omap_exists [of A r])
apply (simp add: otype_def, safe)
@@ -609,7 +609,7 @@
apply blast+
done
-theorem (in M_axioms) omap_ord_iso_otype':
+theorem (in M_basic) omap_ord_iso_otype':
"[| wellordered(M,A,r); M(A); M(r) |]
==> \<exists>f[M]. (\<exists>i[M]. Ord(i) & f \<in> ord_iso(A, r, i, Memrel(i)))"
apply (insert obase_exists [of A r] omap_exists [of A r] otype_exists [of A r], simp, clarify)
@@ -620,7 +620,7 @@
apply (simp_all add: wellordered_is_trans_on)
done
-lemma (in M_axioms) ordertype_exists:
+lemma (in M_basic) ordertype_exists:
"[| wellordered(M,A,r); M(A); M(r) |]
==> \<exists>f[M]. (\<exists>i[M]. Ord(i) & f \<in> ord_iso(A, r, i, Memrel(i)))"
apply (insert obase_exists [of A r] omap_exists [of A r] otype_exists [of A r], simp, clarify)
@@ -632,7 +632,7 @@
done
-lemma (in M_axioms) relativized_imp_well_ord:
+lemma (in M_basic) relativized_imp_well_ord:
"[| wellordered(M,A,r); M(A); M(r) |] ==> well_ord(A,r)"
apply (insert ordertype_exists [of A r], simp)
apply (blast intro: well_ord_ord_iso well_ord_Memrel)
@@ -641,13 +641,13 @@
subsection {*Kunen's theorem 5.4, poage 127*}
text{*(a) The notion of Wellordering is absolute*}
-theorem (in M_axioms) well_ord_abs [simp]:
+theorem (in M_basic) well_ord_abs [simp]:
"[| M(A); M(r) |] ==> wellordered(M,A,r) <-> well_ord(A,r)"
by (blast intro: well_ord_imp_relativized relativized_imp_well_ord)
text{*(b) Order types are absolute*}
-lemma (in M_axioms)
+lemma (in M_basic)
"[| wellordered(M,A,r); f \<in> ord_iso(A, r, i, Memrel(i));
M(A); M(r); M(f); M(i); Ord(i) |] ==> i = ordertype(A,r)"
by (blast intro: Ord_ordertype relativized_imp_well_ord ordertype_ord_iso