src/ZF/Constructible/Wellorderings.thy
 changeset 13634 99a593b49b04 parent 13628 87482b5e3f2e child 13780 af7b79271364
```--- a/src/ZF/Constructible/Wellorderings.thy	Tue Oct 08 14:09:18 2002 +0200
+++ b/src/ZF/Constructible/Wellorderings.thy	Wed Oct 09 11:07:13 2002 +0200
@@ -1,7 +1,6 @@
(*  Title:      ZF/Constructible/Wellorderings.thy
ID:         \$Id\$
Author:     Lawrence C Paulson, Cambridge University Computer Laboratory
-    Copyright   2002  University of Cambridge
*)

@@ -220,60 +219,6 @@
wellfounded_on_subset)
done

-text{*Inductive argument for Kunen's Lemma 6.1, etc.
-      Simple proof from Halmos, page 72*}
-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)
-apply (elim conjE CollectE)
-apply (erule wellfounded_on_induct, assumption+)
- apply (insert well_ord_iso_separation [of A f r])
- apply (simp, clarify)
-apply (drule_tac a = x in bij_is_fun [THEN apply_type], assumption, blast)
-done
-
-
-text{*Kunen's Lemma 6.1: there's no order-isomorphism to an initial segment
-      of a well-ordering*}
-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)
-apply (frule wellordered_iso_subset_lemma, assumption)
-apply (auto elim: predE)
-(*Now we know  ~ (f`x < x) *)
-apply (drule ord_iso_is_bij [THEN bij_is_fun, THEN apply_type], assumption)
-(*Now we also know f`x  \<in> pred(A,x,r);  contradiction! *)
-done
-
-
-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)
-apply (rule notI)
-apply (drule_tac x2=y and x=x and r2=r in
-         wellordered_subset [OF _ pred_subset, THEN wellordered_iso_predD])
-apply (blast intro: predI dest: transM)+
-done
-
-
-text{*Simple consequence of Lemma 6.1*}
-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"
-apply (frule wellordered_is_trans_on, assumption)
-apply (frule wellordered_is_linear, assumption)
-apply (erule_tac x=a and y=c in linearE, auto)
-apply (drule ord_iso_sym)
-(*two symmetric cases*)
-apply (blast dest: wellordered_iso_pred_eq_lemma)+
-done
-
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"
@@ -285,353 +230,4 @@
"[| 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{*Can't use @{text well_ord_iso_preserving} because it needs the
-strong premise @{term "well_ord(A,r)"}*}
-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);
-         Ord(i); Ord(j); \<langle>x,y\<rangle> \<in> r |]
-      ==> i < j"
-apply (frule wellordered_is_trans_on, assumption)
-apply (frule_tac y=y in transM, assumption)
-apply (rule_tac i=i and j=j in Ord_linear_lt, auto)
-txt{*case @{term "i=j"} yields a contradiction*}
- apply (rule_tac x1=x and A1="Order.pred(A,y,r)" in
-          wellordered_iso_predD [THEN notE])
-   apply (blast intro: wellordered_subset [OF _ pred_subset])
-  apply (blast intro: Ord_iso_implies_eq ord_iso_sym ord_iso_trans)
- apply (simp_all add: pred_iff pred_closed converse_closed comp_closed)
-txt{*case @{term "j<i"} also yields a contradiction*}
-apply (frule restrict_ord_iso2, assumption+)
-apply (frule ord_iso_sym [THEN ord_iso_is_bij, THEN bij_is_fun])
-apply (frule apply_type, blast intro: ltD)
-  --{*thus @{term "converse(f)`j \<in> Order.pred(A,x,r)"}*}
-apply (subgoal_tac
-       "\<exists>h[M]. h \<in> ord_iso(Order.pred(A,y,r), r,
-                               Order.pred(A, converse(f)`j, r), r)")
- apply (clarify, frule wellordered_iso_pred_eq, assumption+)
- apply (blast dest: wellordered_asym)
-apply (intro rexI)
- apply (blast intro: Ord_iso_implies_eq ord_iso_sym ord_iso_trans)+
-done
-
-
-lemma ord_iso_converse1:
-     "[| f: ord_iso(A,r,B,s);  <b, f`a>: s;  a:A;  b:B |]
-      ==> <converse(f) ` b, a> : r"
-apply (frule ord_iso_converse, assumption+)
-apply (blast intro: ord_iso_is_bij [THEN bij_is_fun, THEN apply_funtype])
-apply (simp add: left_inverse_bij [OF ord_iso_is_bij])
-done
-
-
-subsection {* Order Types: A Direct Construction by Replacement*}
-
-text{*This follows Kunen's Theorem I 7.6, page 17.*}
-
-constdefs
-
-  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 \<in> z <->
-          (a\<in>A & (\<exists>x[M]. \<exists>g[M]. Ord(x) &
-                   order_isomorphism(M,Order.pred(A,a,r),r,x,Memrel(x),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 \<in> f <->
-          (\<exists>a[M]. a\<in>A &
-           (\<exists>x[M]. \<exists>g[M]. \<exists>mx[M]. \<exists>par[M].
-                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)))"
-
-
-  otype :: "[i=>o,i,i,i] => o"  --{*the order types themselves*}
-   "otype(M,A,r,i) == \<exists>f[M]. omap(M,A,r,f) & is_range(M,f,i)"
-
-
-
-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) &
-                          g \<in> ord_iso(Order.pred(A,a,r),r,x,Memrel(x))}"
-apply (simp add: obase_def Memrel_closed pred_closed)
-apply (rule iffI)
- prefer 2 apply blast
-apply (rule equalityI)
- apply (clarify, frule transM, assumption, simp)
-apply (clarify, frule transM, assumption, force)
-done
-
-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_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) &
-                        g \<in> ord_iso(Order.pred(A,a,r),r,x,Memrel(x)))"
-apply (simp add: omap_def Memrel_closed pred_closed)
-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_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)
-done
-
-lemma (in M_basic) omap_yields_Ord:
-     "[| omap(M,A,r,f); \<langle>a,x\<rangle> \<in> f; M(a); M(x) |]  ==> Ord(x)"
-
-lemma (in M_basic) otype_iff:
-     "[| otype(M,A,r,i); M(A); M(r); M(i) |]
-      ==> x \<in> i <->
-          (M(x) & Ord(x) &
-           (\<exists>a\<in>A. \<exists>g[M]. g \<in> ord_iso(Order.pred(A,a,r),r,x,Memrel(x))))"
-apply (auto simp add: omap_iff otype_def)
- apply (blast intro: transM)
-apply (rule rangeI)
-apply (frule transM, assumption)
-done
-
-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)
-apply (blast dest: omap_unique)
-done
-
-
-lemma (in M_basic) Ord_otype:
-     "[| otype(M,A,r,i); trans[A](r); M(A); M(r); M(i) |] ==> Ord(i)"
-apply (rule OrdI)
-prefer 2
-    apply (simp add: Ord_def otype_def omap_def)
-    apply clarify
-    apply (frule pair_components_in_M, assumption)
-    apply blast
-apply (auto simp add: Transset_def otype_iff)
-  apply (blast intro: transM)
- apply (blast intro: Ord_in_Ord)
-apply (rename_tac y a g)
-apply (frule ord_iso_sym [THEN ord_iso_is_bij, THEN bij_is_fun,
-			  THEN apply_funtype],  assumption)
-apply (rule_tac x="converse(g)`y" in bexI)
- apply (frule_tac a="converse(g) ` y" in ord_iso_restrict_pred, assumption)
-apply (safe elim!: predE)
-apply (blast intro: restrict_ord_iso ord_iso_sym ltI dest: transM)
-done
-
-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 (rule equality_iffI)
-apply (simp add: domain_iff omap_iff, blast)
-done
-
-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 clarify
-done
-
-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 (simp add: domain_omap omap_subset Pi_iff function_def omap_iff)
-apply (blast intro: Ord_iso_implies_eq ord_iso_sym ord_iso_trans)
-done
-
-
-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])
-apply (auto simp add: bij_def inj_def)
-prefer 2  apply (blast intro: fun_is_surj dest: otype_eq_range)
-apply (frule_tac a=w in apply_Pair, assumption)
-apply (frule_tac a=x in apply_Pair, assumption)
-apply (blast intro: wellordered_iso_pred_eq ord_iso_sym ord_iso_trans)
-done
-
-
-text{*This is not the final result: we must show @{term "oB(A,r) = A"}*}
-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)
- apply (erule wellordered_omap_bij, assumption+)
-apply (insert omap_funtype [of A r f B i], simp)
-apply (frule_tac a=x in apply_Pair, assumption)
-apply (frule_tac a=y in apply_Pair, assumption)
- txt{*direction 1: assuming @{term "\<langle>x,y\<rangle> \<in> r"}*}
- apply (blast intro: ltD ord_iso_pred_imp_lt)
- txt{*direction 2: proving @{term "\<langle>x,y\<rangle> \<in> r"} using linearity of @{term r}*}
-apply (rename_tac x y g ga)
-apply (frule wellordered_is_linear, assumption,
-       erule_tac x=x and y=y in linearE, assumption+)
-apply (blast elim: mem_irrefl)
-txt{*the case @{term "\<langle>y,x\<rangle> \<in> r"}: handle like the opposite direction*}
-apply (blast dest: ord_iso_pred_imp_lt ltD elim: mem_asym)
-done
-
-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)
-apply (rule OrdI)
-	prefer 2 apply (simp add: image_iff omap_iff Ord_def, blast)
-txt{*Hard part is to show that the image is a transitive set.*}
-apply (simp add: image_iff pred_iff apply_iff [OF omap_funtype [of A r f B i]])
-apply (rename_tac c j, clarify)
-apply (frule omap_funtype [of A r f B, THEN apply_funtype], assumption+)
-apply (subgoal_tac "j : i")
-	prefer 2 apply (blast intro: Ord_trans Ord_otype)
-apply (subgoal_tac "converse(f) ` j : B")
-	prefer 2
-	apply (blast dest: wellordered_omap_bij [THEN bij_converse_bij,
-                                      THEN bij_is_fun, THEN apply_funtype])
-apply (rule_tac x="converse(f) ` j" in bexI)
- apply (simp add: right_inverse_bij [OF wellordered_omap_bij])
-apply (intro predI conjI)
- apply (erule_tac b=c in trans_onD)
- apply (rule ord_iso_converse1 [OF omap_ord_iso [of A r f B i]])
-done
-
-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>)"
-apply (frule ord_iso_restrict_image [OF omap_ord_iso [of A r f B i]],
-       assumption+)
-apply (drule ord_iso_sym [THEN subset_ord_iso_Memrel])
-apply (blast dest: subsetD [OF omap_subset])
-apply (drule ord_iso_sym, simp)
-done
-
-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 (rule equalityI, force simp add: obase_iff, clarify)
-apply (subst obase_iff [of A r B, THEN iffD1], assumption+, simp)
-apply (frule wellordered_is_wellfounded_on, assumption)
-apply (erule wellfounded_on_induct, assumption+)
- apply (frule obase_equals_separation [of A r], assumption)
- apply (simp, clarify)
-apply (rename_tac b)
-apply (subgoal_tac "Order.pred(A,b,r) <= B")
- apply (blast intro!: restrict_omap_ord_iso Ord_omap_image_pred)
-apply (force simp add: pred_iff obase_iff)
-done
-
-
-
-text{*Main result: @{term om} gives the order-isomorphism
-      @{term "\<langle>A,r\<rangle> \<cong> \<langle>i, Memrel(i)\<rangle>"} *}
-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_basic) obase_exists:
-     "[| M(A); M(r) |] ==> \<exists>z[M]. obase(M,A,r,z)"
-apply (insert obase_separation [of A r])
-done
-
-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 (insert omap_replacement [of A r])
-apply (drule_tac x=x in rspec, 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 rexI)
-apply (simp add: Memrel_closed pred_closed obase_iff, blast, assumption)
-done
-
-declare rall_simps [simp] rex_simps [simp]
-
-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 (rule_tac x="range(x)" in rexI)
-apply blast+
-done
-
-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)
-apply (rename_tac i)
-apply (subgoal_tac "Ord(i)", blast intro: omap_ord_iso_otype)
-apply (rule Ord_otype)
-    apply (force simp add: otype_def range_closed)
-done
-
-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)
-apply (rename_tac i)
-apply (subgoal_tac "Ord(i)", blast intro: omap_ord_iso_otype')
-apply (rule Ord_otype)
-    apply (force simp add: otype_def range_closed)
-done
-
-
-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)
-done
-
-subsection {*Kunen's theorem 5.4, poage 127*}
-
-text{*(a) The notion of Wellordering is absolute*}
-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_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
-                 Ord_iso_implies_eq ord_iso_sym ord_iso_trans)
-
end```