13634

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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


4 
*)


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header {*Absoluteness for Order Types, Rank Functions and WellFounded


7 
Relations*}


8 

16417

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theory Rank imports WF_absolute begin

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10 


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subsection {*Order Types: A Direct Construction by Replacement*}


12 


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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))"


36 


37 


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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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51 


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text{*Kunen's Lemma I 6.1, page 14:


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there's no orderisomorphism to an initial segment of a wellordering*}


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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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78 


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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


140 


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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


160 
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)


176 
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)


219 
apply (safe elim!: predE)


220 
apply (blast intro: restrict_ord_iso ord_iso_sym ltI dest: transM)


221 
done


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223 
lemma (in M_ordertype) domain_omap:


224 
"[ omap(M,A,r,f); M(A); M(r); M(B); M(f) ]


225 
==> domain(f) = obase(M,A,r)"


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apply (simp add: domain_closed obase_def)


227 
apply (rule equality_iffI)


228 
apply (simp add: domain_iff omap_iff, blast)


229 
done


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231 
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"


234 
apply clarify


235 
apply (simp add: omap_iff obase_def)


236 
apply (force simp add: otype_iff)


237 
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"


242 
apply (simp add: domain_omap omap_subset Pi_iff function_def omap_iff)


243 
apply (blast intro: Ord_iso_implies_eq ord_iso_sym ord_iso_trans)


244 
done


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246 


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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);


249 
M(A); M(r); M(f); M(i) ] ==> f \<in> bij(obase(M,A,r),i)"


250 
apply (insert omap_funtype [of A r f i])


251 
apply (auto simp add: bij_def inj_def)


252 
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)


255 
apply (simp add: omap_iff)


256 
apply (blast intro: wellordered_iso_pred_eq ord_iso_sym ord_iso_trans)


257 
done


258 


259 


260 
text{*This is not the final result: we must show @{term "oB(A,r) = A"}*}


261 
lemma (in M_ordertype) omap_ord_iso:


262 
"[ wellordered(M,A,r); omap(M,A,r,f); otype(M,A,r,i);


263 
M(A); M(r); M(f); M(i) ] ==> f \<in> ord_iso(obase(M,A,r),r,i,Memrel(i))"


264 
apply (rule ord_isoI)


265 
apply (erule wellordered_omap_bij, assumption+)


266 
apply (insert omap_funtype [of A r f i], simp)


267 
apply (frule_tac a=x in apply_Pair, assumption)


268 
apply (frule_tac a=y in apply_Pair, assumption)


269 
apply (auto simp add: omap_iff)


270 
txt{*direction 1: assuming @{term "\<langle>x,y\<rangle> \<in> r"}*}


271 
apply (blast intro: ltD ord_iso_pred_imp_lt)


272 
txt{*direction 2: proving @{term "\<langle>x,y\<rangle> \<in> r"} using linearity of @{term r}*}


273 
apply (rename_tac x y g ga)


274 
apply (frule wellordered_is_linear, assumption,


275 
erule_tac x=x and y=y in linearE, assumption+)


276 
txt{*the case @{term "x=y"} leads to immediate contradiction*}


277 
apply (blast elim: mem_irrefl)


278 
txt{*the case @{term "\<langle>y,x\<rangle> \<in> r"}: handle like the opposite direction*}


279 
apply (blast dest: ord_iso_pred_imp_lt ltD elim: mem_asym)


280 
done


281 


282 
lemma (in M_ordertype) Ord_omap_image_pred:


283 
"[ wellordered(M,A,r); omap(M,A,r,f); otype(M,A,r,i);


284 
M(A); M(r); M(f); M(i); b \<in> A ] ==> Ord(f `` Order.pred(A,b,r))"


285 
apply (frule wellordered_is_trans_on, assumption)


286 
apply (rule OrdI)


287 
prefer 2 apply (simp add: image_iff omap_iff Ord_def, blast)


288 
txt{*Hard part is to show that the image is a transitive set.*}


289 
apply (simp add: Transset_def, clarify)


290 
apply (simp add: image_iff pred_iff apply_iff [OF omap_funtype [of A r f i]])


291 
apply (rename_tac c j, clarify)


292 
apply (frule omap_funtype [of A r f, THEN apply_funtype], assumption+)

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293 
apply (subgoal_tac "j \<in> i")

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294 
prefer 2 apply (blast intro: Ord_trans Ord_otype)

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295 
apply (subgoal_tac "converse(f) ` j \<in> obase(M,A,r)")

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296 
prefer 2


297 
apply (blast dest: wellordered_omap_bij [THEN bij_converse_bij,


298 
THEN bij_is_fun, THEN apply_funtype])


299 
apply (rule_tac x="converse(f) ` j" in bexI)


300 
apply (simp add: right_inverse_bij [OF wellordered_omap_bij])


301 
apply (intro predI conjI)


302 
apply (erule_tac b=c in trans_onD)


303 
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 orderisomorphism


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, nonOrd 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 WellFounded Relations*}


670 


671 
text{*Relativized to @{term M}: Every wellfounded 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 
wellfounded 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 