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(* Title: ZF/Constructible/Rank.thy
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ID: $Id$
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Author: Lawrence C Paulson, Cambridge University Computer Laboratory
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*)
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header {*Absoluteness for Order Types, Rank Functions and Well-Founded
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Relations*}
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16417
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theory Rank imports WF_absolute begin
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subsection {*Order Types: A Direct Construction by Replacement*}
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locale M_ordertype = M_basic +
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assumes well_ord_iso_separation:
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"[| M(A); M(f); M(r) |]
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==> separation (M, \<lambda>x. x\<in>A --> (\<exists>y[M]. (\<exists>p[M].
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fun_apply(M,f,x,y) & pair(M,y,x,p) & p \<in> r)))"
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and obase_separation:
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--{*part of the order type formalization*}
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"[| M(A); M(r) |]
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==> separation(M, \<lambda>a. \<exists>x[M]. \<exists>g[M]. \<exists>mx[M]. \<exists>par[M].
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ordinal(M,x) & membership(M,x,mx) & pred_set(M,A,a,r,par) &
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order_isomorphism(M,par,r,x,mx,g))"
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and obase_equals_separation:
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"[| M(A); M(r) |]
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==> separation (M, \<lambda>x. x\<in>A --> ~(\<exists>y[M]. \<exists>g[M].
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ordinal(M,y) & (\<exists>my[M]. \<exists>pxr[M].
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membership(M,y,my) & pred_set(M,A,x,r,pxr) &
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order_isomorphism(M,pxr,r,y,my,g))))"
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and omap_replacement:
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"[| M(A); M(r) |]
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==> strong_replacement(M,
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\<lambda>a z. \<exists>x[M]. \<exists>g[M]. \<exists>mx[M]. \<exists>par[M].
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ordinal(M,x) & pair(M,a,x,z) & membership(M,x,mx) &
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pred_set(M,A,a,r,par) & order_isomorphism(M,par,r,x,mx,g))"
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text{*Inductive argument for Kunen's Lemma I 6.1, etc.
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Simple proof from Halmos, page 72*}
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lemma (in M_ordertype) wellordered_iso_subset_lemma:
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"[| wellordered(M,A,r); f \<in> ord_iso(A,r, A',r); A'<= A; y \<in> A;
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M(A); M(f); M(r) |] ==> ~ <f`y, y> \<in> r"
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apply (unfold wellordered_def ord_iso_def)
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apply (elim conjE CollectE)
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apply (erule wellfounded_on_induct, assumption+)
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apply (insert well_ord_iso_separation [of A f r])
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apply (simp, clarify)
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apply (drule_tac a = x in bij_is_fun [THEN apply_type], assumption, blast)
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done
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text{*Kunen's Lemma I 6.1, page 14:
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there's no order-isomorphism to an initial segment of a well-ordering*}
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lemma (in M_ordertype) wellordered_iso_predD:
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"[| wellordered(M,A,r); f \<in> ord_iso(A, r, Order.pred(A,x,r), r);
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M(A); M(f); M(r) |] ==> x \<notin> A"
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apply (rule notI)
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apply (frule wellordered_iso_subset_lemma, assumption)
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apply (auto elim: predE)
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(*Now we know ~ (f`x < x) *)
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apply (drule ord_iso_is_bij [THEN bij_is_fun, THEN apply_type], assumption)
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(*Now we also know f`x \<in> pred(A,x,r); contradiction! *)
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apply (simp add: Order.pred_def)
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done
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lemma (in M_ordertype) wellordered_iso_pred_eq_lemma:
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"[| f \<in> \<langle>Order.pred(A,y,r), r\<rangle> \<cong> \<langle>Order.pred(A,x,r), r\<rangle>;
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wellordered(M,A,r); x\<in>A; y\<in>A; M(A); M(f); M(r) |] ==> <x,y> \<notin> r"
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apply (frule wellordered_is_trans_on, assumption)
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apply (rule notI)
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apply (drule_tac x2=y and x=x and r2=r in
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wellordered_subset [OF _ pred_subset, THEN wellordered_iso_predD])
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apply (simp add: trans_pred_pred_eq)
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apply (blast intro: predI dest: transM)+
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done
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text{*Simple consequence of Lemma 6.1*}
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lemma (in M_ordertype) wellordered_iso_pred_eq:
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"[| wellordered(M,A,r);
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f \<in> ord_iso(Order.pred(A,a,r), r, Order.pred(A,c,r), r);
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M(A); M(f); M(r); a\<in>A; c\<in>A |] ==> a=c"
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apply (frule wellordered_is_trans_on, assumption)
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apply (frule wellordered_is_linear, assumption)
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apply (erule_tac x=a and y=c in linearE, auto)
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apply (drule ord_iso_sym)
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(*two symmetric cases*)
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apply (blast dest: wellordered_iso_pred_eq_lemma)+
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done
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text{*Following Kunen's Theorem I 7.6, page 17. Note that this material is
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not required elsewhere.*}
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text{*Can't use @{text well_ord_iso_preserving} because it needs the
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strong premise @{term "well_ord(A,r)"}*}
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lemma (in M_ordertype) ord_iso_pred_imp_lt:
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"[| f \<in> ord_iso(Order.pred(A,x,r), r, i, Memrel(i));
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g \<in> ord_iso(Order.pred(A,y,r), r, j, Memrel(j));
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wellordered(M,A,r); x \<in> A; y \<in> A; M(A); M(r); M(f); M(g); M(j);
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Ord(i); Ord(j); \<langle>x,y\<rangle> \<in> r |]
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==> i < j"
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apply (frule wellordered_is_trans_on, assumption)
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apply (frule_tac y=y in transM, assumption)
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apply (rule_tac i=i and j=j in Ord_linear_lt, auto)
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txt{*case @{term "i=j"} yields a contradiction*}
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apply (rule_tac x1=x and A1="Order.pred(A,y,r)" in
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wellordered_iso_predD [THEN notE])
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apply (blast intro: wellordered_subset [OF _ pred_subset])
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apply (simp add: trans_pred_pred_eq)
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apply (blast intro: Ord_iso_implies_eq ord_iso_sym ord_iso_trans)
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apply (simp_all add: pred_iff pred_closed converse_closed comp_closed)
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txt{*case @{term "j<i"} also yields a contradiction*}
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apply (frule restrict_ord_iso2, assumption+)
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apply (frule ord_iso_sym [THEN ord_iso_is_bij, THEN bij_is_fun])
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apply (frule apply_type, blast intro: ltD)
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--{*thus @{term "converse(f)`j \<in> Order.pred(A,x,r)"}*}
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apply (simp add: pred_iff)
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apply (subgoal_tac
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"\<exists>h[M]. h \<in> ord_iso(Order.pred(A,y,r), r,
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Order.pred(A, converse(f)`j, r), r)")
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apply (clarify, frule wellordered_iso_pred_eq, assumption+)
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apply (blast dest: wellordered_asym)
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apply (intro rexI)
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apply (blast intro: Ord_iso_implies_eq ord_iso_sym ord_iso_trans)+
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done
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lemma ord_iso_converse1:
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"[| f: ord_iso(A,r,B,s); <b, f`a>: s; a:A; b:B |]
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==> <converse(f) ` b, a> \<in> r"
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apply (frule ord_iso_converse, assumption+)
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apply (blast intro: ord_iso_is_bij [THEN bij_is_fun, THEN apply_funtype])
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apply (simp add: left_inverse_bij [OF ord_iso_is_bij])
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done
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constdefs
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obase :: "[i=>o,i,i] => i"
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--{*the domain of @{text om}, eventually shown to equal @{text A}*}
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"obase(M,A,r) == {a\<in>A. \<exists>x[M]. \<exists>g[M]. Ord(x) &
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g \<in> ord_iso(Order.pred(A,a,r),r,x,Memrel(x))}"
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omap :: "[i=>o,i,i,i] => o"
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--{*the function that maps wosets to order types*}
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"omap(M,A,r,f) ==
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\<forall>z[M].
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z \<in> f <-> (\<exists>a\<in>A. \<exists>x[M]. \<exists>g[M]. z = <a,x> & Ord(x) &
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g \<in> ord_iso(Order.pred(A,a,r),r,x,Memrel(x)))"
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otype :: "[i=>o,i,i,i] => o" --{*the order types themselves*}
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"otype(M,A,r,i) == \<exists>f[M]. omap(M,A,r,f) & is_range(M,f,i)"
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text{*Can also be proved with the premise @{term "M(z)"} instead of
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@{term "M(f)"}, but that version is less useful. This lemma
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is also more useful than the definition, @{text omap_def}.*}
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lemma (in M_ordertype) omap_iff:
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"[| omap(M,A,r,f); M(A); M(f) |]
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==> z \<in> f <->
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(\<exists>a\<in>A. \<exists>x[M]. \<exists>g[M]. z = <a,x> & Ord(x) &
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g \<in> ord_iso(Order.pred(A,a,r),r,x,Memrel(x)))"
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apply (simp add: omap_def Memrel_closed pred_closed)
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apply (rule iffI)
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apply (drule_tac [2] x=z in rspec)
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apply (drule_tac x=z in rspec)
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apply (blast dest: transM)+
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done
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lemma (in M_ordertype) omap_unique:
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"[| omap(M,A,r,f); omap(M,A,r,f'); M(A); M(r); M(f); M(f') |] ==> f' = f"
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apply (rule equality_iffI)
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apply (simp add: omap_iff)
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done
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lemma (in M_ordertype) omap_yields_Ord:
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"[| omap(M,A,r,f); \<langle>a,x\<rangle> \<in> f; M(a); M(x) |] ==> Ord(x)"
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by (simp add: omap_def)
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lemma (in M_ordertype) otype_iff:
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"[| otype(M,A,r,i); M(A); M(r); M(i) |]
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==> x \<in> i <->
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(M(x) & Ord(x) &
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(\<exists>a\<in>A. \<exists>g[M]. g \<in> ord_iso(Order.pred(A,a,r),r,x,Memrel(x))))"
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apply (auto simp add: omap_iff otype_def)
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apply (blast intro: transM)
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apply (rule rangeI)
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apply (frule transM, assumption)
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apply (simp add: omap_iff, blast)
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done
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lemma (in M_ordertype) otype_eq_range:
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"[| omap(M,A,r,f); otype(M,A,r,i); M(A); M(r); M(f); M(i) |]
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==> i = range(f)"
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apply (auto simp add: otype_def omap_iff)
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apply (blast dest: omap_unique)
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done
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lemma (in M_ordertype) Ord_otype:
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"[| otype(M,A,r,i); trans[A](r); M(A); M(r); M(i) |] ==> Ord(i)"
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apply (rule OrdI)
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prefer 2
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apply (simp add: Ord_def otype_def omap_def)
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apply clarify
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apply (frule pair_components_in_M, assumption)
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apply blast
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apply (auto simp add: Transset_def otype_iff)
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apply (blast intro: transM)
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apply (blast intro: Ord_in_Ord)
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apply (rename_tac y a g)
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apply (frule ord_iso_sym [THEN ord_iso_is_bij, THEN bij_is_fun,
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THEN apply_funtype], assumption)
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apply (rule_tac x="converse(g)`y" in bexI)
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apply (frule_tac a="converse(g) ` y" in ord_iso_restrict_pred, assumption)
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apply (safe elim!: predE)
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apply (blast intro: restrict_ord_iso ord_iso_sym ltI dest: transM)
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done
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lemma (in M_ordertype) domain_omap:
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"[| omap(M,A,r,f); M(A); M(r); M(B); M(f) |]
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==> domain(f) = obase(M,A,r)"
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apply (simp add: domain_closed obase_def)
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apply (rule equality_iffI)
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apply (simp add: domain_iff omap_iff, blast)
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done
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lemma (in M_ordertype) omap_subset:
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"[| omap(M,A,r,f); otype(M,A,r,i);
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M(A); M(r); M(f); M(B); M(i) |] ==> f \<subseteq> obase(M,A,r) * i"
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apply clarify
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apply (simp add: omap_iff obase_def)
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apply (force simp add: otype_iff)
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done
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lemma (in M_ordertype) omap_funtype:
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"[| omap(M,A,r,f); otype(M,A,r,i);
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M(A); M(r); M(f); M(i) |] ==> f \<in> obase(M,A,r) -> i"
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apply (simp add: domain_omap omap_subset Pi_iff function_def omap_iff)
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apply (blast intro: Ord_iso_implies_eq ord_iso_sym ord_iso_trans)
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done
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lemma (in M_ordertype) wellordered_omap_bij:
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"[| wellordered(M,A,r); omap(M,A,r,f); otype(M,A,r,i);
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M(A); M(r); M(f); M(i) |] ==> f \<in> bij(obase(M,A,r),i)"
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apply (insert omap_funtype [of A r f i])
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apply (auto simp add: bij_def inj_def)
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prefer 2 apply (blast intro: fun_is_surj dest: otype_eq_range)
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apply (frule_tac a=w in apply_Pair, assumption)
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apply (frule_tac a=x in apply_Pair, assumption)
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apply (simp add: omap_iff)
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apply (blast intro: wellordered_iso_pred_eq ord_iso_sym ord_iso_trans)
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done
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text{*This is not the final result: we must show @{term "oB(A,r) = A"}*}
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lemma (in M_ordertype) omap_ord_iso:
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"[| wellordered(M,A,r); omap(M,A,r,f); otype(M,A,r,i);
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M(A); M(r); M(f); M(i) |] ==> f \<in> ord_iso(obase(M,A,r),r,i,Memrel(i))"
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apply (rule ord_isoI)
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apply (erule wellordered_omap_bij, assumption+)
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apply (insert omap_funtype [of A r f i], simp)
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apply (frule_tac a=x in apply_Pair, assumption)
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apply (frule_tac a=y in apply_Pair, assumption)
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apply (auto simp add: omap_iff)
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txt{*direction 1: assuming @{term "\<langle>x,y\<rangle> \<in> r"}*}
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apply (blast intro: ltD ord_iso_pred_imp_lt)
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txt{*direction 2: proving @{term "\<langle>x,y\<rangle> \<in> r"} using linearity of @{term r}*}
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apply (rename_tac x y g ga)
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apply (frule wellordered_is_linear, assumption,
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erule_tac x=x and y=y in linearE, assumption+)
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txt{*the case @{term "x=y"} leads to immediate contradiction*}
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apply (blast elim: mem_irrefl)
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txt{*the case @{term "\<langle>y,x\<rangle> \<in> r"}: handle like the opposite direction*}
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apply (blast dest: ord_iso_pred_imp_lt ltD elim: mem_asym)
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done
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lemma (in M_ordertype) Ord_omap_image_pred:
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"[| wellordered(M,A,r); omap(M,A,r,f); otype(M,A,r,i);
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M(A); M(r); M(f); M(i); b \<in> A |] ==> Ord(f `` Order.pred(A,b,r))"
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apply (frule wellordered_is_trans_on, assumption)
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apply (rule OrdI)
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prefer 2 apply (simp add: image_iff omap_iff Ord_def, blast)
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txt{*Hard part is to show that the image is a transitive set.*}
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apply (simp add: Transset_def, clarify)
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apply (simp add: image_iff pred_iff apply_iff [OF omap_funtype [of A r f i]])
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apply (rename_tac c j, clarify)
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apply (frule omap_funtype [of A r f, THEN apply_funtype], assumption+)
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apply (subgoal_tac "j \<in> i")
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prefer 2 apply (blast intro: Ord_trans Ord_otype)
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apply (subgoal_tac "converse(f) ` j \<in> obase(M,A,r)")
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prefer 2
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apply (blast dest: wellordered_omap_bij [THEN bij_converse_bij,
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THEN bij_is_fun, THEN apply_funtype])
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apply (rule_tac x="converse(f) ` j" in bexI)
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apply (simp add: right_inverse_bij [OF wellordered_omap_bij])
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apply (intro predI conjI)
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apply (erule_tac b=c in trans_onD)
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apply (rule ord_iso_converse1 [OF omap_ord_iso [of A r f i]])
|
|
304 |
apply (auto simp add: obase_def)
|
|
305 |
done
|
|
306 |
|
|
307 |
lemma (in M_ordertype) restrict_omap_ord_iso:
|
|
308 |
"[| wellordered(M,A,r); omap(M,A,r,f); otype(M,A,r,i);
|
|
309 |
D \<subseteq> obase(M,A,r); M(A); M(r); M(f); M(i) |]
|
|
310 |
==> restrict(f,D) \<in> (\<langle>D,r\<rangle> \<cong> \<langle>f``D, Memrel(f``D)\<rangle>)"
|
|
311 |
apply (frule ord_iso_restrict_image [OF omap_ord_iso [of A r f i]],
|
|
312 |
assumption+)
|
|
313 |
apply (drule ord_iso_sym [THEN subset_ord_iso_Memrel])
|
|
314 |
apply (blast dest: subsetD [OF omap_subset])
|
|
315 |
apply (drule ord_iso_sym, simp)
|
|
316 |
done
|
|
317 |
|
|
318 |
lemma (in M_ordertype) obase_equals:
|
|
319 |
"[| wellordered(M,A,r); omap(M,A,r,f); otype(M,A,r,i);
|
|
320 |
M(A); M(r); M(f); M(i) |] ==> obase(M,A,r) = A"
|
|
321 |
apply (rule equalityI, force simp add: obase_def, clarify)
|
|
322 |
apply (unfold obase_def, simp)
|
|
323 |
apply (frule wellordered_is_wellfounded_on, assumption)
|
|
324 |
apply (erule wellfounded_on_induct, assumption+)
|
|
325 |
apply (frule obase_equals_separation [of A r], assumption)
|
|
326 |
apply (simp, clarify)
|
|
327 |
apply (rename_tac b)
|
|
328 |
apply (subgoal_tac "Order.pred(A,b,r) <= obase(M,A,r)")
|
|
329 |
apply (blast intro!: restrict_omap_ord_iso Ord_omap_image_pred)
|
|
330 |
apply (force simp add: pred_iff obase_def)
|
|
331 |
done
|
|
332 |
|
|
333 |
|
|
334 |
|
|
335 |
text{*Main result: @{term om} gives the order-isomorphism
|
|
336 |
@{term "\<langle>A,r\<rangle> \<cong> \<langle>i, Memrel(i)\<rangle>"} *}
|
|
337 |
theorem (in M_ordertype) omap_ord_iso_otype:
|
|
338 |
"[| wellordered(M,A,r); omap(M,A,r,f); otype(M,A,r,i);
|
|
339 |
M(A); M(r); M(f); M(i) |] ==> f \<in> ord_iso(A, r, i, Memrel(i))"
|
|
340 |
apply (frule omap_ord_iso, assumption+)
|
|
341 |
apply (simp add: obase_equals)
|
|
342 |
done
|
|
343 |
|
|
344 |
lemma (in M_ordertype) obase_exists:
|
|
345 |
"[| M(A); M(r) |] ==> M(obase(M,A,r))"
|
|
346 |
apply (simp add: obase_def)
|
|
347 |
apply (insert obase_separation [of A r])
|
|
348 |
apply (simp add: separation_def)
|
|
349 |
done
|
|
350 |
|
|
351 |
lemma (in M_ordertype) omap_exists:
|
|
352 |
"[| M(A); M(r) |] ==> \<exists>z[M]. omap(M,A,r,z)"
|
|
353 |
apply (simp add: omap_def)
|
|
354 |
apply (insert omap_replacement [of A r])
|
|
355 |
apply (simp add: strong_replacement_def)
|
|
356 |
apply (drule_tac x="obase(M,A,r)" in rspec)
|
|
357 |
apply (simp add: obase_exists)
|
|
358 |
apply (simp add: Memrel_closed pred_closed obase_def)
|
|
359 |
apply (erule impE)
|
|
360 |
apply (clarsimp simp add: univalent_def)
|
|
361 |
apply (blast intro: Ord_iso_implies_eq ord_iso_sym ord_iso_trans, clarify)
|
|
362 |
apply (rule_tac x=Y in rexI)
|
|
363 |
apply (simp add: Memrel_closed pred_closed obase_def, blast, assumption)
|
|
364 |
done
|
|
365 |
|
|
366 |
declare rall_simps [simp] rex_simps [simp]
|
|
367 |
|
|
368 |
lemma (in M_ordertype) otype_exists:
|
|
369 |
"[| wellordered(M,A,r); M(A); M(r) |] ==> \<exists>i[M]. otype(M,A,r,i)"
|
|
370 |
apply (insert omap_exists [of A r])
|
|
371 |
apply (simp add: otype_def, safe)
|
|
372 |
apply (rule_tac x="range(x)" in rexI)
|
|
373 |
apply blast+
|
|
374 |
done
|
|
375 |
|
|
376 |
lemma (in M_ordertype) ordertype_exists:
|
|
377 |
"[| wellordered(M,A,r); M(A); M(r) |]
|
|
378 |
==> \<exists>f[M]. (\<exists>i[M]. Ord(i) & f \<in> ord_iso(A, r, i, Memrel(i)))"
|
|
379 |
apply (insert obase_exists [of A r] omap_exists [of A r] otype_exists [of A r], simp, clarify)
|
|
380 |
apply (rename_tac i)
|
|
381 |
apply (subgoal_tac "Ord(i)", blast intro: omap_ord_iso_otype)
|
|
382 |
apply (rule Ord_otype)
|
|
383 |
apply (force simp add: otype_def range_closed)
|
|
384 |
apply (simp_all add: wellordered_is_trans_on)
|
|
385 |
done
|
|
386 |
|
|
387 |
|
|
388 |
lemma (in M_ordertype) relativized_imp_well_ord:
|
|
389 |
"[| wellordered(M,A,r); M(A); M(r) |] ==> well_ord(A,r)"
|
|
390 |
apply (insert ordertype_exists [of A r], simp)
|
|
391 |
apply (blast intro: well_ord_ord_iso well_ord_Memrel)
|
|
392 |
done
|
|
393 |
|
|
394 |
subsection {*Kunen's theorem 5.4, page 127*}
|
|
395 |
|
|
396 |
text{*(a) The notion of Wellordering is absolute*}
|
|
397 |
theorem (in M_ordertype) well_ord_abs [simp]:
|
|
398 |
"[| M(A); M(r) |] ==> wellordered(M,A,r) <-> well_ord(A,r)"
|
|
399 |
by (blast intro: well_ord_imp_relativized relativized_imp_well_ord)
|
|
400 |
|
|
401 |
|
|
402 |
text{*(b) Order types are absolute*}
|
|
403 |
theorem (in M_ordertype)
|
|
404 |
"[| wellordered(M,A,r); f \<in> ord_iso(A, r, i, Memrel(i));
|
|
405 |
M(A); M(r); M(f); M(i); Ord(i) |] ==> i = ordertype(A,r)"
|
|
406 |
by (blast intro: Ord_ordertype relativized_imp_well_ord ordertype_ord_iso
|
|
407 |
Ord_iso_implies_eq ord_iso_sym ord_iso_trans)
|
|
408 |
|
|
409 |
|
|
410 |
subsection{*Ordinal Arithmetic: Two Examples of Recursion*}
|
|
411 |
|
|
412 |
text{*Note: the remainder of this theory is not needed elsewhere.*}
|
|
413 |
|
|
414 |
subsubsection{*Ordinal Addition*}
|
|
415 |
|
|
416 |
(*FIXME: update to use new techniques!!*)
|
|
417 |
constdefs
|
|
418 |
(*This expresses ordinal addition in the language of ZF. It also
|
|
419 |
provides an abbreviation that can be used in the instance of strong
|
|
420 |
replacement below. Here j is used to define the relation, namely
|
|
421 |
Memrel(succ(j)), while x determines the domain of f.*)
|
|
422 |
is_oadd_fun :: "[i=>o,i,i,i,i] => o"
|
|
423 |
"is_oadd_fun(M,i,j,x,f) ==
|
|
424 |
(\<forall>sj msj. M(sj) --> M(msj) -->
|
|
425 |
successor(M,j,sj) --> membership(M,sj,msj) -->
|
|
426 |
M_is_recfun(M,
|
|
427 |
%x g y. \<exists>gx[M]. image(M,g,x,gx) & union(M,i,gx,y),
|
|
428 |
msj, x, f))"
|
|
429 |
|
|
430 |
is_oadd :: "[i=>o,i,i,i] => o"
|
|
431 |
"is_oadd(M,i,j,k) ==
|
|
432 |
(~ ordinal(M,i) & ~ ordinal(M,j) & k=0) |
|
|
433 |
(~ ordinal(M,i) & ordinal(M,j) & k=j) |
|
|
434 |
(ordinal(M,i) & ~ ordinal(M,j) & k=i) |
|
|
435 |
(ordinal(M,i) & ordinal(M,j) &
|
|
436 |
(\<exists>f fj sj. M(f) & M(fj) & M(sj) &
|
|
437 |
successor(M,j,sj) & is_oadd_fun(M,i,sj,sj,f) &
|
|
438 |
fun_apply(M,f,j,fj) & fj = k))"
|
|
439 |
|
|
440 |
(*NEEDS RELATIVIZATION*)
|
|
441 |
omult_eqns :: "[i,i,i,i] => o"
|
|
442 |
"omult_eqns(i,x,g,z) ==
|
|
443 |
Ord(x) &
|
|
444 |
(x=0 --> z=0) &
|
|
445 |
(\<forall>j. x = succ(j) --> z = g`j ++ i) &
|
|
446 |
(Limit(x) --> z = \<Union>(g``x))"
|
|
447 |
|
|
448 |
is_omult_fun :: "[i=>o,i,i,i] => o"
|
|
449 |
"is_omult_fun(M,i,j,f) ==
|
|
450 |
(\<exists>df. M(df) & is_function(M,f) &
|
|
451 |
is_domain(M,f,df) & subset(M, j, df)) &
|
|
452 |
(\<forall>x\<in>j. omult_eqns(i,x,f,f`x))"
|
|
453 |
|
|
454 |
is_omult :: "[i=>o,i,i,i] => o"
|
|
455 |
"is_omult(M,i,j,k) ==
|
|
456 |
\<exists>f fj sj. M(f) & M(fj) & M(sj) &
|
|
457 |
successor(M,j,sj) & is_omult_fun(M,i,sj,f) &
|
|
458 |
fun_apply(M,f,j,fj) & fj = k"
|
|
459 |
|
|
460 |
|
|
461 |
locale M_ord_arith = M_ordertype +
|
|
462 |
assumes oadd_strong_replacement:
|
|
463 |
"[| M(i); M(j) |] ==>
|
|
464 |
strong_replacement(M,
|
|
465 |
\<lambda>x z. \<exists>y[M]. pair(M,x,y,z) &
|
|
466 |
(\<exists>f[M]. \<exists>fx[M]. is_oadd_fun(M,i,j,x,f) &
|
|
467 |
image(M,f,x,fx) & y = i Un fx))"
|
|
468 |
|
|
469 |
and omult_strong_replacement':
|
|
470 |
"[| M(i); M(j) |] ==>
|
|
471 |
strong_replacement(M,
|
|
472 |
\<lambda>x z. \<exists>y[M]. z = <x,y> &
|
|
473 |
(\<exists>g[M]. is_recfun(Memrel(succ(j)),x,%x g. THE z. omult_eqns(i,x,g,z),g) &
|
|
474 |
y = (THE z. omult_eqns(i, x, g, z))))"
|
|
475 |
|
|
476 |
|
|
477 |
|
|
478 |
text{*@{text is_oadd_fun}: Relating the pure "language of set theory" to Isabelle/ZF*}
|
|
479 |
lemma (in M_ord_arith) is_oadd_fun_iff:
|
|
480 |
"[| a\<le>j; M(i); M(j); M(a); M(f) |]
|
|
481 |
==> is_oadd_fun(M,i,j,a,f) <->
|
|
482 |
f \<in> a \<rightarrow> range(f) & (\<forall>x. M(x) --> x < a --> f`x = i Un f``x)"
|
|
483 |
apply (frule lt_Ord)
|
|
484 |
apply (simp add: is_oadd_fun_def Memrel_closed Un_closed
|
|
485 |
relation2_def is_recfun_abs [of "%x g. i Un g``x"]
|
|
486 |
image_closed is_recfun_iff_equation
|
|
487 |
Ball_def lt_trans [OF ltI, of _ a] lt_Memrel)
|
|
488 |
apply (simp add: lt_def)
|
|
489 |
apply (blast dest: transM)
|
|
490 |
done
|
|
491 |
|
|
492 |
|
|
493 |
lemma (in M_ord_arith) oadd_strong_replacement':
|
|
494 |
"[| M(i); M(j) |] ==>
|
|
495 |
strong_replacement(M,
|
|
496 |
\<lambda>x z. \<exists>y[M]. z = <x,y> &
|
|
497 |
(\<exists>g[M]. is_recfun(Memrel(succ(j)),x,%x g. i Un g``x,g) &
|
|
498 |
y = i Un g``x))"
|
|
499 |
apply (insert oadd_strong_replacement [of i j])
|
|
500 |
apply (simp add: is_oadd_fun_def relation2_def
|
|
501 |
is_recfun_abs [of "%x g. i Un g``x"])
|
|
502 |
done
|
|
503 |
|
|
504 |
|
|
505 |
lemma (in M_ord_arith) exists_oadd:
|
|
506 |
"[| Ord(j); M(i); M(j) |]
|
|
507 |
==> \<exists>f[M]. is_recfun(Memrel(succ(j)), j, %x g. i Un g``x, f)"
|
|
508 |
apply (rule wf_exists_is_recfun [OF wf_Memrel trans_Memrel])
|
|
509 |
apply (simp_all add: Memrel_type oadd_strong_replacement')
|
|
510 |
done
|
|
511 |
|
|
512 |
lemma (in M_ord_arith) exists_oadd_fun:
|
|
513 |
"[| Ord(j); M(i); M(j) |] ==> \<exists>f[M]. is_oadd_fun(M,i,succ(j),succ(j),f)"
|
|
514 |
apply (rule exists_oadd [THEN rexE])
|
|
515 |
apply (erule Ord_succ, assumption, simp)
|
|
516 |
apply (rename_tac f)
|
|
517 |
apply (frule is_recfun_type)
|
|
518 |
apply (rule_tac x=f in rexI)
|
|
519 |
apply (simp add: fun_is_function domain_of_fun lt_Memrel apply_recfun lt_def
|
|
520 |
is_oadd_fun_iff Ord_trans [OF _ succI1], assumption)
|
|
521 |
done
|
|
522 |
|
|
523 |
lemma (in M_ord_arith) is_oadd_fun_apply:
|
|
524 |
"[| x < j; M(i); M(j); M(f); is_oadd_fun(M,i,j,j,f) |]
|
|
525 |
==> f`x = i Un (\<Union>k\<in>x. {f ` k})"
|
|
526 |
apply (simp add: is_oadd_fun_iff lt_Ord2, clarify)
|
|
527 |
apply (frule lt_closed, simp)
|
|
528 |
apply (frule leI [THEN le_imp_subset])
|
|
529 |
apply (simp add: image_fun, blast)
|
|
530 |
done
|
|
531 |
|
|
532 |
lemma (in M_ord_arith) is_oadd_fun_iff_oadd [rule_format]:
|
|
533 |
"[| is_oadd_fun(M,i,J,J,f); M(i); M(J); M(f); Ord(i); Ord(j) |]
|
|
534 |
==> j<J --> f`j = i++j"
|
|
535 |
apply (erule_tac i=j in trans_induct, clarify)
|
|
536 |
apply (subgoal_tac "\<forall>k\<in>x. k<J")
|
|
537 |
apply (simp (no_asm_simp) add: is_oadd_def oadd_unfold is_oadd_fun_apply)
|
|
538 |
apply (blast intro: lt_trans ltI lt_Ord)
|
|
539 |
done
|
|
540 |
|
|
541 |
lemma (in M_ord_arith) Ord_oadd_abs:
|
|
542 |
"[| M(i); M(j); M(k); Ord(i); Ord(j) |] ==> is_oadd(M,i,j,k) <-> k = i++j"
|
|
543 |
apply (simp add: is_oadd_def is_oadd_fun_iff_oadd)
|
|
544 |
apply (frule exists_oadd_fun [of j i], blast+)
|
|
545 |
done
|
|
546 |
|
|
547 |
lemma (in M_ord_arith) oadd_abs:
|
|
548 |
"[| M(i); M(j); M(k) |] ==> is_oadd(M,i,j,k) <-> k = i++j"
|
|
549 |
apply (case_tac "Ord(i) & Ord(j)")
|
|
550 |
apply (simp add: Ord_oadd_abs)
|
|
551 |
apply (auto simp add: is_oadd_def oadd_eq_if_raw_oadd)
|
|
552 |
done
|
|
553 |
|
|
554 |
lemma (in M_ord_arith) oadd_closed [intro,simp]:
|
|
555 |
"[| M(i); M(j) |] ==> M(i++j)"
|
|
556 |
apply (simp add: oadd_eq_if_raw_oadd, clarify)
|
|
557 |
apply (simp add: raw_oadd_eq_oadd)
|
|
558 |
apply (frule exists_oadd_fun [of j i], auto)
|
|
559 |
apply (simp add: apply_closed is_oadd_fun_iff_oadd [symmetric])
|
|
560 |
done
|
|
561 |
|
|
562 |
|
|
563 |
subsubsection{*Ordinal Multiplication*}
|
|
564 |
|
|
565 |
lemma omult_eqns_unique:
|
|
566 |
"[| omult_eqns(i,x,g,z); omult_eqns(i,x,g,z') |] ==> z=z'";
|
|
567 |
apply (simp add: omult_eqns_def, clarify)
|
|
568 |
apply (erule Ord_cases, simp_all)
|
|
569 |
done
|
|
570 |
|
|
571 |
lemma omult_eqns_0: "omult_eqns(i,0,g,z) <-> z=0"
|
|
572 |
by (simp add: omult_eqns_def)
|
|
573 |
|
|
574 |
lemma the_omult_eqns_0: "(THE z. omult_eqns(i,0,g,z)) = 0"
|
|
575 |
by (simp add: omult_eqns_0)
|
|
576 |
|
|
577 |
lemma omult_eqns_succ: "omult_eqns(i,succ(j),g,z) <-> Ord(j) & z = g`j ++ i"
|
|
578 |
by (simp add: omult_eqns_def)
|
|
579 |
|
|
580 |
lemma the_omult_eqns_succ:
|
|
581 |
"Ord(j) ==> (THE z. omult_eqns(i,succ(j),g,z)) = g`j ++ i"
|
|
582 |
by (simp add: omult_eqns_succ)
|
|
583 |
|
|
584 |
lemma omult_eqns_Limit:
|
|
585 |
"Limit(x) ==> omult_eqns(i,x,g,z) <-> z = \<Union>(g``x)"
|
|
586 |
apply (simp add: omult_eqns_def)
|
|
587 |
apply (blast intro: Limit_is_Ord)
|
|
588 |
done
|
|
589 |
|
|
590 |
lemma the_omult_eqns_Limit:
|
|
591 |
"Limit(x) ==> (THE z. omult_eqns(i,x,g,z)) = \<Union>(g``x)"
|
|
592 |
by (simp add: omult_eqns_Limit)
|
|
593 |
|
|
594 |
lemma omult_eqns_Not: "~ Ord(x) ==> ~ omult_eqns(i,x,g,z)"
|
|
595 |
by (simp add: omult_eqns_def)
|
|
596 |
|
|
597 |
|
|
598 |
lemma (in M_ord_arith) the_omult_eqns_closed:
|
|
599 |
"[| M(i); M(x); M(g); function(g) |]
|
|
600 |
==> M(THE z. omult_eqns(i, x, g, z))"
|
|
601 |
apply (case_tac "Ord(x)")
|
|
602 |
prefer 2 apply (simp add: omult_eqns_Not) --{*trivial, non-Ord case*}
|
|
603 |
apply (erule Ord_cases)
|
|
604 |
apply (simp add: omult_eqns_0)
|
|
605 |
apply (simp add: omult_eqns_succ apply_closed oadd_closed)
|
|
606 |
apply (simp add: omult_eqns_Limit)
|
|
607 |
done
|
|
608 |
|
|
609 |
lemma (in M_ord_arith) exists_omult:
|
|
610 |
"[| Ord(j); M(i); M(j) |]
|
|
611 |
==> \<exists>f[M]. is_recfun(Memrel(succ(j)), j, %x g. THE z. omult_eqns(i,x,g,z), f)"
|
|
612 |
apply (rule wf_exists_is_recfun [OF wf_Memrel trans_Memrel])
|
|
613 |
apply (simp_all add: Memrel_type omult_strong_replacement')
|
|
614 |
apply (blast intro: the_omult_eqns_closed)
|
|
615 |
done
|
|
616 |
|
|
617 |
lemma (in M_ord_arith) exists_omult_fun:
|
|
618 |
"[| Ord(j); M(i); M(j) |] ==> \<exists>f[M]. is_omult_fun(M,i,succ(j),f)"
|
|
619 |
apply (rule exists_omult [THEN rexE])
|
|
620 |
apply (erule Ord_succ, assumption, simp)
|
|
621 |
apply (rename_tac f)
|
|
622 |
apply (frule is_recfun_type)
|
|
623 |
apply (rule_tac x=f in rexI)
|
|
624 |
apply (simp add: fun_is_function domain_of_fun lt_Memrel apply_recfun lt_def
|
|
625 |
is_omult_fun_def Ord_trans [OF _ succI1])
|
|
626 |
apply (force dest: Ord_in_Ord'
|
|
627 |
simp add: omult_eqns_def the_omult_eqns_0 the_omult_eqns_succ
|
|
628 |
the_omult_eqns_Limit, assumption)
|
|
629 |
done
|
|
630 |
|
|
631 |
lemma (in M_ord_arith) is_omult_fun_apply_0:
|
|
632 |
"[| 0 < j; is_omult_fun(M,i,j,f) |] ==> f`0 = 0"
|
|
633 |
by (simp add: is_omult_fun_def omult_eqns_def lt_def ball_conj_distrib)
|
|
634 |
|
|
635 |
lemma (in M_ord_arith) is_omult_fun_apply_succ:
|
|
636 |
"[| succ(x) < j; is_omult_fun(M,i,j,f) |] ==> f`succ(x) = f`x ++ i"
|
|
637 |
by (simp add: is_omult_fun_def omult_eqns_def lt_def, blast)
|
|
638 |
|
|
639 |
lemma (in M_ord_arith) is_omult_fun_apply_Limit:
|
|
640 |
"[| x < j; Limit(x); M(j); M(f); is_omult_fun(M,i,j,f) |]
|
|
641 |
==> f ` x = (\<Union>y\<in>x. f`y)"
|
|
642 |
apply (simp add: is_omult_fun_def omult_eqns_def domain_closed lt_def, clarify)
|
|
643 |
apply (drule subset_trans [OF OrdmemD], assumption+)
|
|
644 |
apply (simp add: ball_conj_distrib omult_Limit image_function)
|
|
645 |
done
|
|
646 |
|
|
647 |
lemma (in M_ord_arith) is_omult_fun_eq_omult:
|
|
648 |
"[| is_omult_fun(M,i,J,f); M(J); M(f); Ord(i); Ord(j) |]
|
|
649 |
==> j<J --> f`j = i**j"
|
|
650 |
apply (erule_tac i=j in trans_induct3)
|
|
651 |
apply (safe del: impCE)
|
|
652 |
apply (simp add: is_omult_fun_apply_0)
|
|
653 |
apply (subgoal_tac "x<J")
|
|
654 |
apply (simp add: is_omult_fun_apply_succ omult_succ)
|
|
655 |
apply (blast intro: lt_trans)
|
|
656 |
apply (subgoal_tac "\<forall>k\<in>x. k<J")
|
|
657 |
apply (simp add: is_omult_fun_apply_Limit omult_Limit)
|
|
658 |
apply (blast intro: lt_trans ltI lt_Ord)
|
|
659 |
done
|
|
660 |
|
|
661 |
lemma (in M_ord_arith) omult_abs:
|
|
662 |
"[| M(i); M(j); M(k); Ord(i); Ord(j) |] ==> is_omult(M,i,j,k) <-> k = i**j"
|
|
663 |
apply (simp add: is_omult_def is_omult_fun_eq_omult)
|
|
664 |
apply (frule exists_omult_fun [of j i], blast+)
|
|
665 |
done
|
|
666 |
|
|
667 |
|
|
668 |
|
13647
|
669 |
subsection {*Absoluteness of Well-Founded Relations*}
|
|
670 |
|
|
671 |
text{*Relativized to @{term M}: Every well-founded relation is a subset of some
|
|
672 |
inverse image of an ordinal. Key step is the construction (in @{term M}) of a
|
|
673 |
rank function.*}
|
|
674 |
|
13634
|
675 |
locale M_wfrank = M_trancl +
|
|
676 |
assumes wfrank_separation:
|
|
677 |
"M(r) ==>
|
|
678 |
separation (M, \<lambda>x.
|
|
679 |
\<forall>rplus[M]. tran_closure(M,r,rplus) -->
|
|
680 |
~ (\<exists>f[M]. M_is_recfun(M, %x f y. is_range(M,f,y), rplus, x, f)))"
|
|
681 |
and wfrank_strong_replacement:
|
|
682 |
"M(r) ==>
|
|
683 |
strong_replacement(M, \<lambda>x z.
|
|
684 |
\<forall>rplus[M]. tran_closure(M,r,rplus) -->
|
|
685 |
(\<exists>y[M]. \<exists>f[M]. pair(M,x,y,z) &
|
|
686 |
M_is_recfun(M, %x f y. is_range(M,f,y), rplus, x, f) &
|
|
687 |
is_range(M,f,y)))"
|
|
688 |
and Ord_wfrank_separation:
|
|
689 |
"M(r) ==>
|
|
690 |
separation (M, \<lambda>x.
|
|
691 |
\<forall>rplus[M]. tran_closure(M,r,rplus) -->
|
|
692 |
~ (\<forall>f[M]. \<forall>rangef[M].
|
|
693 |
is_range(M,f,rangef) -->
|
|
694 |
M_is_recfun(M, \<lambda>x f y. is_range(M,f,y), rplus, x, f) -->
|
|
695 |
ordinal(M,rangef)))"
|
|
696 |
|
|
697 |
|
|
698 |
text{*Proving that the relativized instances of Separation or Replacement
|
|
699 |
agree with the "real" ones.*}
|
|
700 |
|
|
701 |
lemma (in M_wfrank) wfrank_separation':
|
|
702 |
"M(r) ==>
|
|
703 |
separation
|
|
704 |
(M, \<lambda>x. ~ (\<exists>f[M]. is_recfun(r^+, x, %x f. range(f), f)))"
|
|
705 |
apply (insert wfrank_separation [of r])
|
|
706 |
apply (simp add: relation2_def is_recfun_abs [of "%x. range"])
|
|
707 |
done
|
|
708 |
|
|
709 |
lemma (in M_wfrank) wfrank_strong_replacement':
|
|
710 |
"M(r) ==>
|
|
711 |
strong_replacement(M, \<lambda>x z. \<exists>y[M]. \<exists>f[M].
|
|
712 |
pair(M,x,y,z) & is_recfun(r^+, x, %x f. range(f), f) &
|
|
713 |
y = range(f))"
|
|
714 |
apply (insert wfrank_strong_replacement [of r])
|
|
715 |
apply (simp add: relation2_def is_recfun_abs [of "%x. range"])
|
|
716 |
done
|
|
717 |
|
|
718 |
lemma (in M_wfrank) Ord_wfrank_separation':
|
|
719 |
"M(r) ==>
|
|
720 |
separation (M, \<lambda>x.
|
|
721 |
~ (\<forall>f[M]. is_recfun(r^+, x, \<lambda>x. range, f) --> Ord(range(f))))"
|
|
722 |
apply (insert Ord_wfrank_separation [of r])
|
|
723 |
apply (simp add: relation2_def is_recfun_abs [of "%x. range"])
|
|
724 |
done
|
|
725 |
|
|
726 |
text{*This function, defined using replacement, is a rank function for
|
|
727 |
well-founded relations within the class M.*}
|
|
728 |
constdefs
|
|
729 |
wellfoundedrank :: "[i=>o,i,i] => i"
|
|
730 |
"wellfoundedrank(M,r,A) ==
|
|
731 |
{p. x\<in>A, \<exists>y[M]. \<exists>f[M].
|
|
732 |
p = <x,y> & is_recfun(r^+, x, %x f. range(f), f) &
|
|
733 |
y = range(f)}"
|
|
734 |
|
|
735 |
lemma (in M_wfrank) exists_wfrank:
|
|
736 |
"[| wellfounded(M,r); M(a); M(r) |]
|
|
737 |
==> \<exists>f[M]. is_recfun(r^+, a, %x f. range(f), f)"
|
|
738 |
apply (rule wellfounded_exists_is_recfun)
|
|
739 |
apply (blast intro: wellfounded_trancl)
|
|
740 |
apply (rule trans_trancl)
|
|
741 |
apply (erule wfrank_separation')
|
|
742 |
apply (erule wfrank_strong_replacement')
|
|
743 |
apply (simp_all add: trancl_subset_times)
|
|
744 |
done
|
|
745 |
|
|
746 |
lemma (in M_wfrank) M_wellfoundedrank:
|
|
747 |
"[| wellfounded(M,r); M(r); M(A) |] ==> M(wellfoundedrank(M,r,A))"
|
|
748 |
apply (insert wfrank_strong_replacement' [of r])
|
|
749 |
apply (simp add: wellfoundedrank_def)
|
|
750 |
apply (rule strong_replacement_closed)
|
|
751 |
apply assumption+
|
|
752 |
apply (rule univalent_is_recfun)
|
|
753 |
apply (blast intro: wellfounded_trancl)
|
|
754 |
apply (rule trans_trancl)
|
|
755 |
apply (simp add: trancl_subset_times)
|
|
756 |
apply (blast dest: transM)
|
|
757 |
done
|
|
758 |
|
|
759 |
lemma (in M_wfrank) Ord_wfrank_range [rule_format]:
|
|
760 |
"[| wellfounded(M,r); a\<in>A; M(r); M(A) |]
|
|
761 |
==> \<forall>f[M]. is_recfun(r^+, a, %x f. range(f), f) --> Ord(range(f))"
|
|
762 |
apply (drule wellfounded_trancl, assumption)
|
|
763 |
apply (rule wellfounded_induct, assumption, erule (1) transM)
|
|
764 |
apply simp
|
|
765 |
apply (blast intro: Ord_wfrank_separation', clarify)
|
|
766 |
txt{*The reasoning in both cases is that we get @{term y} such that
|
|
767 |
@{term "\<langle>y, x\<rangle> \<in> r^+"}. We find that
|
|
768 |
@{term "f`y = restrict(f, r^+ -`` {y})"}. *}
|
|
769 |
apply (rule OrdI [OF _ Ord_is_Transset])
|
|
770 |
txt{*An ordinal is a transitive set...*}
|
|
771 |
apply (simp add: Transset_def)
|
|
772 |
apply clarify
|
|
773 |
apply (frule apply_recfun2, assumption)
|
|
774 |
apply (force simp add: restrict_iff)
|
|
775 |
txt{*...of ordinals. This second case requires the induction hyp.*}
|
|
776 |
apply clarify
|
|
777 |
apply (rename_tac i y)
|
|
778 |
apply (frule apply_recfun2, assumption)
|
|
779 |
apply (frule is_recfun_imp_in_r, assumption)
|
|
780 |
apply (frule is_recfun_restrict)
|
|
781 |
(*simp_all won't work*)
|
|
782 |
apply (simp add: trans_trancl trancl_subset_times)+
|
|
783 |
apply (drule spec [THEN mp], assumption)
|
|
784 |
apply (subgoal_tac "M(restrict(f, r^+ -`` {y}))")
|
|
785 |
apply (drule_tac x="restrict(f, r^+ -`` {y})" in rspec)
|
|
786 |
apply assumption
|
|
787 |
apply (simp add: function_apply_equality [OF _ is_recfun_imp_function])
|
|
788 |
apply (blast dest: pair_components_in_M)
|
|
789 |
done
|
|
790 |
|
|
791 |
lemma (in M_wfrank) Ord_range_wellfoundedrank:
|
|
792 |
"[| wellfounded(M,r); r \<subseteq> A*A; M(r); M(A) |]
|
|
793 |
==> Ord (range(wellfoundedrank(M,r,A)))"
|
|
794 |
apply (frule wellfounded_trancl, assumption)
|
|
795 |
apply (frule trancl_subset_times)
|
|
796 |
apply (simp add: wellfoundedrank_def)
|
|
797 |
apply (rule OrdI [OF _ Ord_is_Transset])
|
|
798 |
prefer 2
|
|
799 |
txt{*by our previous result the range consists of ordinals.*}
|
|
800 |
apply (blast intro: Ord_wfrank_range)
|
|
801 |
txt{*We still must show that the range is a transitive set.*}
|
|
802 |
apply (simp add: Transset_def, clarify, simp)
|
|
803 |
apply (rename_tac x i f u)
|
|
804 |
apply (frule is_recfun_imp_in_r, assumption)
|
|
805 |
apply (subgoal_tac "M(u) & M(i) & M(x)")
|
|
806 |
prefer 2 apply (blast dest: transM, clarify)
|
|
807 |
apply (rule_tac a=u in rangeI)
|
|
808 |
apply (rule_tac x=u in ReplaceI)
|
|
809 |
apply simp
|
|
810 |
apply (rule_tac x="restrict(f, r^+ -`` {u})" in rexI)
|
|
811 |
apply (blast intro: is_recfun_restrict trans_trancl dest: apply_recfun2)
|
|
812 |
apply simp
|
|
813 |
apply blast
|
|
814 |
txt{*Unicity requirement of Replacement*}
|
|
815 |
apply clarify
|
|
816 |
apply (frule apply_recfun2, assumption)
|
|
817 |
apply (simp add: trans_trancl is_recfun_cut)
|
|
818 |
done
|
|
819 |
|
|
820 |
lemma (in M_wfrank) function_wellfoundedrank:
|
|
821 |
"[| wellfounded(M,r); M(r); M(A)|]
|
|
822 |
==> function(wellfoundedrank(M,r,A))"
|
|
823 |
apply (simp add: wellfoundedrank_def function_def, clarify)
|
|
824 |
txt{*Uniqueness: repeated below!*}
|
|
825 |
apply (drule is_recfun_functional, assumption)
|
|
826 |
apply (blast intro: wellfounded_trancl)
|
|
827 |
apply (simp_all add: trancl_subset_times trans_trancl)
|
|
828 |
done
|
|
829 |
|
|
830 |
lemma (in M_wfrank) domain_wellfoundedrank:
|
|
831 |
"[| wellfounded(M,r); M(r); M(A)|]
|
|
832 |
==> domain(wellfoundedrank(M,r,A)) = A"
|
|
833 |
apply (simp add: wellfoundedrank_def function_def)
|
|
834 |
apply (rule equalityI, auto)
|
|
835 |
apply (frule transM, assumption)
|
|
836 |
apply (frule_tac a=x in exists_wfrank, assumption+, clarify)
|
|
837 |
apply (rule_tac b="range(f)" in domainI)
|
|
838 |
apply (rule_tac x=x in ReplaceI)
|
|
839 |
apply simp
|
|
840 |
apply (rule_tac x=f in rexI, blast, simp_all)
|
|
841 |
txt{*Uniqueness (for Replacement): repeated above!*}
|
|
842 |
apply clarify
|
|
843 |
apply (drule is_recfun_functional, assumption)
|
|
844 |
apply (blast intro: wellfounded_trancl)
|
|
845 |
apply (simp_all add: trancl_subset_times trans_trancl)
|
|
846 |
done
|
|
847 |
|
|
848 |
lemma (in M_wfrank) wellfoundedrank_type:
|
|
849 |
"[| wellfounded(M,r); M(r); M(A)|]
|
|
850 |
==> wellfoundedrank(M,r,A) \<in> A -> range(wellfoundedrank(M,r,A))"
|
|
851 |
apply (frule function_wellfoundedrank [of r A], assumption+)
|
|
852 |
apply (frule function_imp_Pi)
|
|
853 |
apply (simp add: wellfoundedrank_def relation_def)
|
|
854 |
apply blast
|
|
855 |
apply (simp add: domain_wellfoundedrank)
|
|
856 |
done
|
|
857 |
|
|
858 |
lemma (in M_wfrank) Ord_wellfoundedrank:
|
|
859 |
"[| wellfounded(M,r); a \<in> A; r \<subseteq> A*A; M(r); M(A) |]
|
|
860 |
==> Ord(wellfoundedrank(M,r,A) ` a)"
|
|
861 |
by (blast intro: apply_funtype [OF wellfoundedrank_type]
|
|
862 |
Ord_in_Ord [OF Ord_range_wellfoundedrank])
|
|
863 |
|
|
864 |
lemma (in M_wfrank) wellfoundedrank_eq:
|
|
865 |
"[| is_recfun(r^+, a, %x. range, f);
|
|
866 |
wellfounded(M,r); a \<in> A; M(f); M(r); M(A)|]
|
|
867 |
==> wellfoundedrank(M,r,A) ` a = range(f)"
|
|
868 |
apply (rule apply_equality)
|
|
869 |
prefer 2 apply (blast intro: wellfoundedrank_type)
|
|
870 |
apply (simp add: wellfoundedrank_def)
|
|
871 |
apply (rule ReplaceI)
|
|
872 |
apply (rule_tac x="range(f)" in rexI)
|
|
873 |
apply blast
|
|
874 |
apply simp_all
|
|
875 |
txt{*Unicity requirement of Replacement*}
|
|
876 |
apply clarify
|
|
877 |
apply (drule is_recfun_functional, assumption)
|
|
878 |
apply (blast intro: wellfounded_trancl)
|
|
879 |
apply (simp_all add: trancl_subset_times trans_trancl)
|
|
880 |
done
|
|
881 |
|
|
882 |
|
|
883 |
lemma (in M_wfrank) wellfoundedrank_lt:
|
|
884 |
"[| <a,b> \<in> r;
|
|
885 |
wellfounded(M,r); r \<subseteq> A*A; M(r); M(A)|]
|
|
886 |
==> wellfoundedrank(M,r,A) ` a < wellfoundedrank(M,r,A) ` b"
|
|
887 |
apply (frule wellfounded_trancl, assumption)
|
|
888 |
apply (subgoal_tac "a\<in>A & b\<in>A")
|
|
889 |
prefer 2 apply blast
|
|
890 |
apply (simp add: lt_def Ord_wellfoundedrank, clarify)
|
|
891 |
apply (frule exists_wfrank [of concl: _ b], erule (1) transM, assumption)
|
|
892 |
apply clarify
|
|
893 |
apply (rename_tac fb)
|
|
894 |
apply (frule is_recfun_restrict [of concl: "r^+" a])
|
|
895 |
apply (rule trans_trancl, assumption)
|
|
896 |
apply (simp_all add: r_into_trancl trancl_subset_times)
|
|
897 |
txt{*Still the same goal, but with new @{text is_recfun} assumptions.*}
|
|
898 |
apply (simp add: wellfoundedrank_eq)
|
|
899 |
apply (frule_tac a=a in wellfoundedrank_eq, assumption+)
|
|
900 |
apply (simp_all add: transM [of a])
|
|
901 |
txt{*We have used equations for wellfoundedrank and now must use some
|
|
902 |
for @{text is_recfun}. *}
|
|
903 |
apply (rule_tac a=a in rangeI)
|
|
904 |
apply (simp add: is_recfun_type [THEN apply_iff] vimage_singleton_iff
|
|
905 |
r_into_trancl apply_recfun r_into_trancl)
|
|
906 |
done
|
|
907 |
|
|
908 |
|
|
909 |
lemma (in M_wfrank) wellfounded_imp_subset_rvimage:
|
|
910 |
"[|wellfounded(M,r); r \<subseteq> A*A; M(r); M(A)|]
|
|
911 |
==> \<exists>i f. Ord(i) & r <= rvimage(A, f, Memrel(i))"
|
|
912 |
apply (rule_tac x="range(wellfoundedrank(M,r,A))" in exI)
|
|
913 |
apply (rule_tac x="wellfoundedrank(M,r,A)" in exI)
|
|
914 |
apply (simp add: Ord_range_wellfoundedrank, clarify)
|
|
915 |
apply (frule subsetD, assumption, clarify)
|
|
916 |
apply (simp add: rvimage_iff wellfoundedrank_lt [THEN ltD])
|
|
917 |
apply (blast intro: apply_rangeI wellfoundedrank_type)
|
|
918 |
done
|
|
919 |
|
|
920 |
lemma (in M_wfrank) wellfounded_imp_wf:
|
|
921 |
"[|wellfounded(M,r); relation(r); M(r)|] ==> wf(r)"
|
|
922 |
by (blast dest!: relation_field_times_field wellfounded_imp_subset_rvimage
|
|
923 |
intro: wf_rvimage_Ord [THEN wf_subset])
|
|
924 |
|
|
925 |
lemma (in M_wfrank) wellfounded_on_imp_wf_on:
|
|
926 |
"[|wellfounded_on(M,A,r); relation(r); M(r); M(A)|] ==> wf[A](r)"
|
|
927 |
apply (simp add: wellfounded_on_iff_wellfounded wf_on_def)
|
|
928 |
apply (rule wellfounded_imp_wf)
|
|
929 |
apply (simp_all add: relation_def)
|
|
930 |
done
|
|
931 |
|
|
932 |
|
|
933 |
theorem (in M_wfrank) wf_abs:
|
|
934 |
"[|relation(r); M(r)|] ==> wellfounded(M,r) <-> wf(r)"
|
|
935 |
by (blast intro: wellfounded_imp_wf wf_imp_relativized)
|
|
936 |
|
|
937 |
theorem (in M_wfrank) wf_on_abs:
|
|
938 |
"[|relation(r); M(r); M(A)|] ==> wellfounded_on(M,A,r) <-> wf[A](r)"
|
|
939 |
by (blast intro: wellfounded_on_imp_wf_on wf_on_imp_relativized)
|
|
940 |
|
|
941 |
end |