--- a/src/ZF/Constructible/Datatype_absolute.thy Thu Jul 04 10:53:52 2002 +0200
+++ b/src/ZF/Constructible/Datatype_absolute.thy Thu Jul 04 10:54:04 2002 +0200
@@ -138,10 +138,10 @@
lemma (in M_datatypes) list_replacement1':
"[|M(A); n \<in> nat|]
==> strong_replacement
- (M, \<lambda>x y. \<exists>z[M]. \<exists>g[M]. y = \<langle>x,z\<rangle> &
- is_recfun (Memrel(succ(n)), x,
+ (M, \<lambda>x y. \<exists>z[M]. y = \<langle>x,z\<rangle> &
+ (\<exists>g[M]. is_recfun (Memrel(succ(n)), x,
\<lambda>n f. nat_case(0, \<lambda>m. {0} + A \<times> f`m, n), g) &
- z = nat_case(0, \<lambda>m. {0} + A \<times> g ` m, x))"
+ z = nat_case(0, \<lambda>m. {0} + A \<times> g ` m, x)))"
by (insert list_replacement1, simp add: nat_into_M)
@@ -149,4 +149,5 @@
"M(A) ==> M(list(A))"
by (simp add: list_eq_Union list_replacement1' list_replacement2')
+
end
--- a/src/ZF/Constructible/WF_absolute.thy Thu Jul 04 10:53:52 2002 +0200
+++ b/src/ZF/Constructible/WF_absolute.thy Thu Jul 04 10:54:04 2002 +0200
@@ -335,20 +335,21 @@
apply (blast intro: Ord_wfrank_range)
txt{*We still must show that the range is a transitive set.*}
apply (simp add: Transset_def, clarify, simp)
-apply (rename_tac x f i u)
+apply (rename_tac x i f u)
apply (frule is_recfun_imp_in_r, assumption)
apply (subgoal_tac "M(u) & M(i) & M(x)")
prefer 2 apply (blast dest: transM, clarify)
apply (rule_tac a=u in rangeI)
-apply (rule ReplaceI)
- apply (rule_tac x=i in rexI, simp)
- apply (rule_tac x="restrict(f, r^+ -`` {u})" in rexI)
- apply (blast intro: is_recfun_restrict trans_trancl dest: apply_recfun2)
- apply (simp, simp, blast)
+apply (rule_tac x=u in ReplaceI)
+ apply simp
+ apply (rule_tac x="restrict(f, r^+ -`` {u})" in rexI)
+ apply (blast intro: is_recfun_restrict trans_trancl dest: apply_recfun2)
+ apply simp
+apply blast
txt{*Unicity requirement of Replacement*}
apply clarify
apply (frule apply_recfun2, assumption)
-apply (simp add: trans_trancl is_recfun_cut)+
+apply (simp add: trans_trancl is_recfun_cut)
done
lemma (in M_wfrank) function_wellfoundedrank:
@@ -368,10 +369,9 @@
apply (rule equalityI, auto)
apply (frule transM, assumption)
apply (frule_tac a=x in exists_wfrank, assumption+, clarify)
-apply (rule domainI)
-apply (rule ReplaceI)
- apply (rule_tac x="range(f)" in rexI)
- apply simp
+apply (rule_tac b="range(f)" in domainI)
+apply (rule_tac x=x in ReplaceI)
+ apply simp
apply (rule_tac x=f in rexI, blast, simp_all)
txt{*Uniqueness (for Replacement): repeated above!*}
apply clarify
@@ -502,8 +502,8 @@
before we can replace @{term "r^+"} by @{term r}. *}
theorem (in M_trancl) trans_wfrec_relativize:
"[|wf(r); trans(r); relation(r); M(r); M(a);
- 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));
+ strong_replacement(M, \<lambda>x z. \<exists>y[M].
+ pair(M,x,y,z) & (\<exists>g[M]. is_recfun(r,x,H,g) & y = H(x,g)));
\<forall>x[M]. \<forall>g[M]. function(g) --> M(H(x,g))|]
==> wfrec(r,a,H) = z <-> (\<exists>f[M]. is_recfun(r,a,H,f) & z = H(a,f))"
by (simp cong: is_recfun_cong
@@ -513,13 +513,13 @@
lemma (in M_trancl) trans_eq_pair_wfrec_iff:
"[|wf(r); trans(r); relation(r); M(r); M(y);
- 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));
+ strong_replacement(M, \<lambda>x z. \<exists>y[M].
+ pair(M,x,y,z) & (\<exists>g[M]. is_recfun(r,x,H,g) & y = H(x,g)));
\<forall>x[M]. \<forall>g[M]. function(g) --> M(H(x,g))|]
==> y = <x, wfrec(r, x, H)> <->
(\<exists>f[M]. is_recfun(r,x,H,f) & y = <x, H(x,f)>)"
-apply safe
- apply (simp add: trans_wfrec_relativize [THEN iff_sym])
+apply safe
+ apply (simp add: trans_wfrec_relativize [THEN iff_sym, of concl: _ x])
txt{*converse direction*}
apply (rule sym)
apply (simp add: trans_wfrec_relativize, blast)
@@ -577,7 +577,7 @@
(\<exists>f[M]. is_recfun(r^+, x, \<lambda>x f. H(x, restrict(f, r -`` {x})), f) &
y = <x, H(x,restrict(f,r-``{x}))>)"
apply safe
- apply (simp add: wfrec_relativize [THEN iff_sym])
+ apply (simp add: wfrec_relativize [THEN iff_sym, of concl: _ x])
txt{*converse direction*}
apply (rule sym)
apply (simp add: wfrec_relativize, blast)
--- a/src/ZF/Constructible/WFrec.thy Thu Jul 04 10:53:52 2002 +0200
+++ b/src/ZF/Constructible/WFrec.thy Thu Jul 04 10:54:04 2002 +0200
@@ -187,10 +187,10 @@
apply (clarsimp simp add: vimage_singleton_iff is_recfun_type [THEN apply_iff]
apply_recfun is_recfun_cut)
txt{*Opposite inclusion: something in f, show in Y*}
-apply (frule_tac y="<y,z>" in transM, assumption, simp)
-apply (rule_tac x=y in bexI)
-prefer 2 apply (blast dest: transD)
-apply (rule_tac x=z in rexI)
+apply (frule_tac y="<y,z>" in transM, assumption)
+apply (simp add: vimage_singleton_iff)
+apply (rule conjI)
+ apply (blast dest: transD)
apply (rule_tac x="restrict(f, r -`` {y})" in rexI)
apply (simp_all add: is_recfun_restrict
apply_recfun is_recfun_type [THEN apply_iff])
@@ -200,7 +200,7 @@
lemma (in M_axioms) univalent_is_recfun:
"[|wellfounded(M,r); trans(r); M(r)|]
==> univalent (M, A, \<lambda>x p.
- \<exists>y[M]. \<exists>f[M]. p = \<langle>x, y\<rangle> & is_recfun(r,x,H,f) & y = H(x,f))"
+ \<exists>y[M]. p = \<langle>x,y\<rangle> & (\<exists>f[M]. is_recfun(r,x,H,f) & y = H(x,f)))"
apply (simp add: univalent_def)
apply (blast dest: is_recfun_functional)
done
@@ -228,13 +228,10 @@
apply (simp add: vimage_singleton_iff restrict_Y_lemma [of r H])
txt{*one more case*}
apply (simp (no_asm_simp) add: Bex_def vimage_singleton_iff)
-apply (rename_tac x)
-apply (rule_tac x=x in exI, simp)
-apply (rule_tac x="H(x, restrict(Y, r -`` {x}))" in rexI)
apply (drule_tac x1=x in spec [THEN mp], assumption, clarify)
apply (rename_tac f)
apply (rule_tac x=f in rexI)
-apply (simp add: restrict_Y_lemma [of r H], simp_all)
+apply (simp_all add: restrict_Y_lemma [of r H])
done
text{*Relativized version, when we have the (currently weaker) premise
@@ -337,14 +334,16 @@
assumes oadd_strong_replacement:
"[| M(i); M(j) |] ==>
strong_replacement(M,
- \<lambda>x z. \<exists>y[M]. \<exists>f[M]. \<exists>fx[M]. pair(M,x,y,z) & is_oadd_fun(M,i,j,x,f) &
- image(M,f,x,fx) & y = i Un fx)"
+ \<lambda>x z. \<exists>y[M]. pair(M,x,y,z) &
+ (\<exists>f[M]. \<exists>fx[M]. is_oadd_fun(M,i,j,x,f) &
+ image(M,f,x,fx) & y = i Un fx))"
+
and omult_strong_replacement':
"[| M(i); M(j) |] ==>
- strong_replacement(M, \<lambda>x z. \<exists>y[M]. \<exists>g[M].
- z = <x,y> &
- is_recfun(Memrel(succ(j)),x,%x g. THE z. omult_eqns(i,x,g,z),g) &
- y = (THE z. omult_eqns(i, x, g, z)))"
+ strong_replacement(M,
+ \<lambda>x z. \<exists>y[M]. z = <x,y> &
+ (\<exists>g[M]. is_recfun(Memrel(succ(j)),x,%x g. THE z. omult_eqns(i,x,g,z),g) &
+ y = (THE z. omult_eqns(i, x, g, z))))"
@@ -365,10 +364,10 @@
lemma (in M_ord_arith) oadd_strong_replacement':
"[| M(i); M(j) |] ==>
- strong_replacement(M, \<lambda>x z. \<exists>y[M]. \<exists>g[M].
- z = <x,y> &
- is_recfun(Memrel(succ(j)),x,%x g. i Un g``x,g) &
- y = i Un g``x)"
+ strong_replacement(M,
+ \<lambda>x z. \<exists>y[M]. z = <x,y> &
+ (\<exists>g[M]. is_recfun(Memrel(succ(j)),x,%x g. i Un g``x,g) &
+ y = i Un g``x))"
apply (insert oadd_strong_replacement [of i j])
apply (simp add: Memrel_closed Un_closed image_closed is_oadd_fun_def
is_recfun_iff_M)
--- a/src/ZF/Constructible/Wellorderings.thy Thu Jul 04 10:53:52 2002 +0200
+++ b/src/ZF/Constructible/Wellorderings.thy Thu Jul 04 10:54:04 2002 +0200
@@ -346,23 +346,23 @@
obase :: "[i=>o,i,i,i] => o"
--{*the domain of @{text om}, eventually shown to equal @{text A}*}
"obase(M,A,r,z) ==
- \<forall>a. M(a) -->
- (a \<in> z <->
+ \<forall>a[M].
+ a \<in> z <->
(a\<in>A & (\<exists>x g mx par. M(x) & M(g) & M(mx) & M(par) & ordinal(M,x) &
membership(M,x,mx) & pred_set(M,A,a,r,par) &
- order_isomorphism(M,par,r,x,mx,g))))"
+ order_isomorphism(M,par,r,x,mx,g)))"
omap :: "[i=>o,i,i,i] => o"
--{*the function that maps wosets to order types*}
"omap(M,A,r,f) ==
- \<forall>z. M(z) -->
- (z \<in> f <->
+ \<forall>z[M].
+ z \<in> f <->
(\<exists>a\<in>A. M(a) &
(\<exists>x g mx par. M(x) & M(g) & M(mx) & M(par) & ordinal(M,x) &
pair(M,a,x,z) & membership(M,x,mx) &
pred_set(M,A,a,r,par) &
- order_isomorphism(M,par,r,x,mx,g))))"
+ order_isomorphism(M,par,r,x,mx,g)))"
otype :: "[i=>o,i,i,i] => o" --{*the order types themselves*}
@@ -392,8 +392,10 @@
g \<in> ord_iso(Order.pred(A,a,r),r,x,Memrel(x)))"
apply (rotate_tac 1)
apply (simp add: omap_def Memrel_closed pred_closed)
-apply (rule iffI)
-apply (drule_tac x=z in spec, blast dest: transM)+
+apply (rule iffI)
+ apply (drule_tac [2] x=z in rspec)
+ apply (drule_tac x=z in rspec)
+ apply (blast dest: transM)+
done
lemma (in M_axioms) omap_unique:
@@ -576,35 +578,37 @@
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
+done
lemma (in M_axioms) obase_exists:
- "[| M(A); M(r) |] ==> \<exists>z. M(z) & obase(M,A,r,z)"
+ "[| 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:
- "[| M(A); M(r) |] ==> \<exists>z. M(z) & omap(M,A,r,z)"
+ "[| M(A); M(r) |] ==> \<exists>z[M]. omap(M,A,r,z)"
apply (insert obase_exists [of A r])
apply (simp add: omap_def)
apply (insert omap_replacement [of A r])
apply (simp add: strong_replacement_def, clarify)
-apply (drule_tac x=z in spec, clarify)
+apply (drule_tac x=x in spec, clarify)
apply (simp add: Memrel_closed pred_closed obase_iff)
apply (erule impE)
apply (clarsimp simp add: univalent_def)
apply (blast intro: Ord_iso_implies_eq ord_iso_sym ord_iso_trans, clarify)
-apply (rule_tac x=Y in exI)
-apply (simp add: Memrel_closed pred_closed obase_iff, blast)
+apply (rule_tac x=Y in rexI)
+apply (simp add: Memrel_closed pred_closed obase_iff, blast, assumption)
done
+declare rall_simps [simp] rex_simps [simp]
+
lemma (in M_axioms) otype_exists:
"[| wellordered(M,A,r); M(A); M(r) |] ==> \<exists>i. M(i) & otype(M,A,r,i)"
-apply (insert omap_exists [of A r])
-apply (simp add: otype_def, clarify)
-apply (rule_tac x="range(z)" in exI)
+apply (insert omap_exists [of A r])
+apply (simp add: otype_def, safe)
+apply (rule_tac x="range(x)" in exI)
apply blast
done